Physics Reports vol.357

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Physics Reports 357 (2002) 1–111

Semiconductor superlattices: a model system for nonlinear transport Andreas Wacker ∗ Institut fur Theoretische Physik, Technische Universitat Berlin, Hardenbergstrae 36, 10623 Berlin, Germany Received March 2001; editor: C:W:J: Beenakker

Contents 0. Notation and list of symbols 1. Introduction 1.1. Experimental summary 1.2. Outline of this work 2. States in superlattices 2.1. Minibands 2.2. Bloch states of the three-dimensional superlattice 2.3. Wannier functions 2.4. Wannier–Stark ladder 3. The standard approaches for superlattice transport 3.1. Miniband transport 3.2. Wannier–Stark hopping 3.3. Sequential tunneling 3.4. Comparison of the approaches 4. Quantum transport 4.1. Nonequilibrium Green functions applied to stationery transport 4.2. Application to the superlattice structure

3 4 6 7 8 9 11 12 15 16 18 25 28 36 37 38 47

4.3. Solution for constant self-energy 4.4. Results 5. Formation of ;eld domains 5.1. The model 5.2. Numerical results 5.3. Traveling fronts 5.4. The injecting contact 5.5. Global behavior 5.6. Summary 6. Transport under irradiation 6.1. Low-frequency limit 6.2. Results for miniband transport 6.3. Sequential tunneling 6.4. Discussion 7. Summary 8. Outlook Acknowledgements Appendix A. Sequential tunneling with density matrices A.1. The model A.2. Density matrix theory

∗

Tel: +49-30-314-23002; fax: +49-30-314-21130. E-mail address: [email protected] (A. Wacker). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter  PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 2 9 - 1

50 51 56 57 59 60 65 65 69 71 72 73 75 81 82 83 84 84 85 86

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Appendix B. Derivation of the standard approaches B.1. Sequential tunneling B.2. Miniband conduction B.3. Wannier–Stark hopping Appendix C. Quantum transport under irradiation

89 89 91 93

C.1. General formulation C.2. Sequential tunneling C.3. Tunneling between diDerent levels References

96 98 99 101

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Abstract Electric transport in semiconductor superlattices is dominated by pronounced negative diDerential conductivity. In this report, the standard transport theories for superlattices, i.e. miniband conduction, Wannier–Stark hopping, and sequential tunneling, are reviewed in detail. Their relation to each other is clari;ed by a comparison with a quantum transport model based on nonequilibrium Green functions. It is demonstrated that the occurrence of negative diDerential conductivity causes inhomogeneous electric ;eld distributions, yielding either a characteristic sawtooth shape of the current–voltage characteristic or self-sustained current oscillations. An additional ac-voltage in the THz range is included in the theory as well. The results display absolute negative conductance, photon-assisted tunneling, the possibility of c 2002 Elsevier Science B.V. All rights reserved. gain, and a negative tunneling capacitance.
PACS: 72.20.Ht; 72.10.−d; 73.40.Gk; 73.21.Cd Keywords: Superlattice transport; Nonequilibrium Green functions; THz irradiation; Formation of ;eld domains

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0. Notation and list of symbols Throughout this work, we consider a superlattice, which is grown in the z direction. Vectors within the (x; y) plane parallel to the interfaces are denoted by bold face letters k; r, while vectors in 3-dimensional space are ˜r; ˜k; : : : : All sums and integrals extend from −∞ to ∞ if not stated otherwise. The following relations are frequently used in this work and are given here for easy reference: J−n () = (−1)n Jn () ;


Jn ()Jn+h () = h; 0 ;

n

ei sin(x) =




Jn ()einx ;

n

2n Jn+1 () + Jn−1 () = Jn () ;    1 1 ∓ i (x − x0 ) : =P x − x0 ± i0+ x − x0 A cross section A(k; E) spectral function a; a† electron annihilation and creation operators † b; b phonon annihilation and creation operators d period of the superlattice structure d integration and diDerentiation symbol E energy E center of energy for miniband  Ek =˝2 k 2 =2mc kinetic energy in the direction parallel to the layers e =2:718 : : : base of natural logarithm e charge of the electron (e ¡ 0) F electric ;eld in the superlattice direction f(k) semiclassical distribution function Hˆ Hamilton operator i imaginary unit I =AJ electric current. In Section 5 there is an additional prefactor sgn(e) so that the direction is identical with the electron Jow J current density in the superlattice direction Jl (x) Bessel function of ;rst kind and order l k wavevector in (x; y)-plane [i.e., plane  to superlattice interfaces] kB Boltzmann constant L length in superlattice direction

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m; n mc m0 N ND nB (E) nF (E) nm q Th T U  $ %0 %ˆ %$ &;  &m # ' ’q (z) )m (z) *j (z) , -(x) Im{} Re{} P{} O(xn ) [a; b] {a; b}

well indices eDective mass of conduction band electron mass 9:11 × 10−31 kg number of wells doping density per period and area (unit [cm−2 ]) =(eE=kB T − 1)−1 Bose distribution function =(eE=kB T + 1)−1 Fermi distribution function electron density per period and area (unit [cm−2 ]) in well m Bloch vector in superlattice direction coupling between Wannier-states of miniband  separated by h barriers temperature bias applied to the superlattice =eFac d= ˝# argument of Bessel function for irradiation =2T1 =eFd argument of Bessel functions for Wannier–Stark states =mc =˝2 free-particle density of states for the 2D electron gas density operator one-particle density matrix indices of energy bands=levels chemical potential in well m, measured with respect to the bottom of the well frequency of the radiation ;eld electrical potential Bloch function of band  Wannier function of band  localized in well m Wannier–Stark function of band  centered around well j scattering time Heavyside function -(x) = 0 for x ¡ 0 and -(x) = 1 for x ¿ 0 imaginary part real part principal value order of xn =ab − ba commutator =ab + ba anticommutator

1. Introduction In this review, the transport properties of semiconductor superlattices are studied. These nanostructures consist of two diDerent semiconductor materials (exhibiting similar lattice constants, e.g., GaAs and AlAs), which are deposited alternately on each other to form a periodic structure in the growth direction. The technical development of growth techniques allows one to control the thicknesses of these layers with a high precision, so that the interfaces are well de;ned within one atomic monolayer. In this way, it is possible to tailor arti;cial periodic structures which show features similar to conventional crystals.

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Crystal structures exhibit a periodic arrangement of the atoms with a lattice period a. This has strong implications for the energy spectrum of the electronic states: energy bands [1] appear instead of discrete levels, which are characteristic for atoms and molecules. The corresponding extended states are called Bloch states and are characterized by the band index  and the Bloch ˜ is applied, vector ˜k. Their energy is given by the dispersion relation E  (˜k). If an electric ;eld F the Bloch states are no longer eigenstates of the Hamiltonian, but the Bloch vector ˜k becomes time dependent according to the acceleration theorem ˝

d˜k ˜ ; = eF dt

(1)

where e ¡ 0 is the charge of the electron. Since the Bloch vectors are restricted to the Brillouin zone, which has a size ∼ 2=a, a special feature arises when the acceleration of the state lasts for a time of ,Bloch ≈ 2˝=(eFa): if interband transitions are neglected, the initial state ˜k is then reached again, and the electron performs a periodic motion both in the Brillouin zone and in real space [2], which is conventionally referred to as a Bloch oscillation. For typical materials and electric ;elds, ,Bloch is much larger than the scattering time, and therefore this surprising eDect has not been observed yet in standard crystals. In 1970, Esaki and Tsu suggested that superlattice structures with an arti;cial period d can be realized by the periodically repeated deposition of alternate layers from diDerent materials [3]. This leads to spatial variations in the conduction and valence band of the material with period d implying the formation of energy bands as sketched in Fig. 1. Both the energy width / of these bands, as well as the extension 2=d of the Brillouin zone, are much smaller than the corresponding values for conventional conduction bands. Thus, the energy bands originating from the superlattice structure are called minibands. As d can be signi;cantly larger than the period a of the crystal, ,Bloch can become smaller than the scattering time for available structures and applicable electric ;elds. It is crucial to note that the picture of Bloch oscillations is not the only possibility to understand the behavior of semiconductor superlattices in an electric ;eld. The combination of a constant electric ;eld and a periodic structure causes the formation of a Wannier–Stark ladder [4], a periodic sequence of energy levels separated by eFd in energy space. This concept is complementary to the Bloch oscillation picture, where the frequency !Bloch =eFd= ˝ corresponds to the energy diDerence between the Wannier–Stark levels.

Fig. 1. Sketch of the spatial variation for the conduction band edge Ec (z), together with minibands  = a; b (shaded areas) for a semiconductor superlattice.

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Both the occurrence of Bloch oscillations and the nature of the Wannier–Stark states predict an increasing localization of the electrons with increasing electric ;eld. This causes a signi;cant drop in the conductivity at moderate ;elds, associated with the occurrence of negative di6erential conductivity [3]. Similar to the Gunn diode, this eDect is likely to cause the formation of inhomogeneous 7eld distributions. These provide various kinds of interesting nonlinear behavior, but make it diMcult to observe the Bloch oscillations. The presence of a strong alternating electric ;eld (with frequency # in the THz range) along the superlattice structure provides further interesting features. Both photon-assisted resonances (shifted by ˝# from the original resonance) and negative dynamical conductance have been predicted on the basis of a simple analysis [5]. For speci;c ratios between the ;eld strength and its frequency, dynamical localization [6] occurs, i.e. the dc-conductance becomes zero, which can be attributed to the collapse of the miniband [7]. Some fundamental aspects of superlattice physics have already been reviewed in Ref. [8]. Refs. [9,10] consider the electronic structure in detail and the review article [11] focuses on infrared spectroscopy. Much information regarding the growth processes as well as transport measurements can be found in Ref. [12]. The relation between Bloch oscillations and Wannier– Stark states has been analyzed in [13]. Ref. [14] provides an early review on high-frequency phenomena. In addition to superlattices consisting of diDerent semiconductor materials, it is possible to achieve similar properties by a periodic sequence of n- and p-type doped layers [15]. 1.1. Experimental summary A large variety of superlattice structures has been studied since the original proposal of Esaki and Tsu in 1970. These investigations can be divided into four diDerent areas: the nonlinear current-;eld relation and its implications, the Wannier–Stark ladder, the search for Bloch oscillations, and the interaction with THz-;elds. The simple model by Esaki and Tsu [3] predicts a nonlinear current-7eld relation exhibiting a maximum for ;eld strengths of eFd= ˝=,, where , denotes the scattering time. For higher ;elds, the current drops with increasing ;eld, yielding a region of negative di6erential conductivity. Such behavior was ;rst observed in the experiment by Esaki and Chang [16] in 1974, where the conductance exhibited a sequence of dips, reaching negative values, as a function of bias voltage. This complicated behavior was attributed to the formation of domains with two diDerent ;eld values in the superlattice. With improving sample quality the sawtooth structure of the current– voltage characteristic due to domain formation could be resolved [17] more than a decade later. Traveling ;eld domains (already proposed in 1977 [18]) were observed as well [19]. They cause self-generated current oscillations with frequencies up to 150 GHz [20]. Domain formation eDects typically hinder the direct observation of negative diDerential conductance, as there is no simple proportionality between the measured bias and local electric ;eld in the sample. In Ref. [21], the local relation between current and ;eld could be extracted from an analysis of the global current–voltage characteristic. A direct observation of the Esaki–Tsu shape was possible from time-of-Jight-measurements [22], and the analysis of the frequency response [23].

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The concept of the Wannier–Stark ladder could be corroborated by the observation of the typical spacing eFd in the optical excitation spectrum of superlattice structures [24,25]. More recent studies refer to the transition between the Franz–Keldysh oscillations and the Wannier– Stark ladder [26], and the inJuence of higher valleys in the band structure [27]. The dynamical nature of Bloch oscillations with period ,Bloch was observed by transient four-wave mixing [28] and by a direct observation of the THz emission at !Bloch [29]. Under stationary conditions, the phases for the oscillation cycles of individual electrons are randomized by scattering processes, and the global signal averages out. Therefore, decaying signals have been observed in these experiments after a short pulse excitation, which synchronizes the dynamics in the very beginning. More recently, the spatial extension of the Bloch oscillation was resolved by measuring its dipole ;eld [30]. With the development of strong THz sources, the interaction of THz 7elds with transport through superlattices have been studied in the last few years. In particular, dynamical localization, photon-assisted tunneling, and absolute negative conductance were observed under irradiation by a free-electron laser [31]. Recent work aims at applying these eDects to the detection of THz signals [32]. Further experiments will be discussed in the subsequent sections in direct comparison with the theory. 1.2. Outline of this work In this work, the theory of electrical transport in semiconductor superlattices is reviewed with a strong emphasis on nonlinear electric transport. Here two diDerent issues arise, which will be treated thoroughly: How can the electric transport in semiconductor structures be described quantitatively? This is not a straightforward issue as diDerent energy scales like the miniband width, the scattering induced broadening, and the potential drop per period are typically of the same order of magnitude in semiconductor superlattices. For this reason, standard concepts from bulk transport (like the semiclassical Boltzmann equation), which rely on the large band width, become questionable. Therefore, diDerent approaches, such as miniband transport [3], Wannier–Stark hopping [33], or sequential tunneling [34,35], have been suggested to study the transport properties of semiconductor superlattices. These standard approaches imply diDerent approximations, and their relation to each other was only recently resolved within a quantum transport theory [36]. As a result, one ;nds that all standard approaches are likely to fail if the miniband width, the scattering induced broadening, and the potential drop per period take similar values. In this case, one has to apply a full quantum transport calculation. For the linear response, the quantum aspects of the problem can be treated within the Kubo formula [37], which is evaluated in thermal equilibrium. For the nonlinear transport discussed here, this is not suMcient and a more involved treatment of nonequilibrium quantum transport is necessary. An overview regarding diDerent aspects of quantum transport in mesoscopic systems can be found in recent textbooks [38– 42]. What is the implication of a strongly nonlinear relation between the current-density and the local 7eld? As long as the ;eld and current distribution remain (approximately) homogeneous, the ratio between this local relation and the global current–voltage characteristic is given by geometrical factors. In the region of negative diDerential conductivity, the stationary

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homogeneous ;eld distribution becomes unstable, which may lead to complex spatio-temporal behavior. Typically, one observes complex scenarios, where current ;laments or electric ;eld domains are formed, which may yield both stationary or oscillating behavior. In some cases, chaotic behavior is observed as well. Such eDects can be treated within standard concepts of nonlinear dynamics for a variety of diDerent semiconductor systems [43– 47]. The nature of quantum transport as well as pronounced nonlinearities are characteristic problems of high-;eld transport in semiconductor nanostructures. In such structures, the electric transport is determined by various quantum phenomena such as resonant tunneling (e.g. the resonant tunneling diode [48]), or transmission through funnel injectors (e.g. in the quantum cascade laser [49]). In such cases, neither standard semiclassical bulk transport models nor linear-response theories apply, and more advanced simulation techniques are required. The excellent possibilities for tailoring diDerent structures with speci;c superlattice periods, miniband widths, or doping densities make semiconductor superlattices an ideal testing ground for nonlinear quantum transport. This review is organized as follows. Section 2 introduces the basis state functions, such as Bloch, Wannier, or Wannier–Stark states, which will be used in the subsequent sections. In particular, it is shown how the miniband widths and coupling parameters can be calculated from the material parameters on the basis of the envelope function theory. Section 3 reviews the three standard approaches of superlattice transport: miniband transport, Wannier–Stark hopping, and sequential tunneling. Each of these approaches is valid in a certain parameter range and allows for a quantitative determination of the current density. It is a common feature of these approaches that they display negative diDerential conductivity in qualitative agreement with the simple Esaki–Tsu result. These approaches can be viewed as limiting cases of a quantum transport theory, which is derived in Section 4 on the basis of nonequilibrium Green functions. The occurrence of stationary and traveling ;eld domains in long superlattices structures is discussed in Section 5. Here, speci;c criteria are presented, which allow the prediction of the global behavior on the basis of the current-;eld relation and the contact conditions. Finally, transport under irradiation by a THz ;eld is addressed in Section 6. Some technical matters are presented in the appendices. 2. States in superlattices In order to perform any quantum calculation one has to de;ne a basis set of states to be used. While, in principle, the exact result of any calculation must not depend on the choice of basis states, this does not hold if approximations are made, which is necessary for almost any realistic problem. Now diDerent sets of basis states suggest diDerent kinds of approximations and therefore a good choice of basis states is a crucial question. For practical purposes, the basis set is usually chosen as the set of eigenstates of a soluble part H0 of the total Hamiltonian. If the remaining part H − H0 is small, it can be treated in the lowest order of the perturbation theory (e.g., Fermi’s golden rule for transition rates) which allows for a signi;cant simpli;cation. This provides an indication for the practicability of a set of states. In the following discussion, we restrict ourselves to the states arising from the conduction band of the superlattice, which is assumed to be a single band with spin degeneracy.

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All wave functions employed in the following have to be considered as envelope functions fc (˜r) with respect to this conduction band, which are determined by the SchrQodinger-like equation   ˝2 Ec (˜r) − ∇ ∇ + e'(˜r) fc (˜r) = Efc (˜r) (2) 2mc where mc denotes the eDective mass in the conduction band. Assuming ideal interfaces, the structure is translational invariant within the x and y directions perpendicular to the growth direction. Therefore, the (x; y) dependence can be taken in the form of plane waves eik·r where k and r are vectors within the two-dimensional (x; y) plane. The crucial point in this section is the choice of the z dependence of the basis states. This reJects the current direction considered here and has, therefore, strong implications on the description of transport. Semiconductor superlattices are designed as periodic structures with period d in the growth direction. Thus, their eigenstates can be chosen as Bloch states ’q (z) (where q ∈ [ − =d; =d] denotes the Bloch vector and  is the band index) which extend over the whole structure. The corresponding eigenvalues E  (q) of the Hamiltonian form the miniband (Section 2.1). This provides the exact solution for a perfect superlattice without applied electric ;eld. An alternative set of basis functions can be constructed by employing localized wave functions which resemble eigenstates of single quantum wells labeled by the index n. Here we use the Wannier-states ) (z − nd), which can be constructed separately for each miniband  (Section 2.3). Thirdly, we may consider the Hamiltonian of the superlattice in the presence of a ;nite electric ;eld F. Then the energy levels take the form En = E0 − neFd and one obtains the Wannier–Stark states * (z − nd), where we neglect the ;eld-dependent coupling between the subbands (Section 2.4). The spatial extension of these states is inversely proportional to the electric ;eld. In the subsequent subsections the diDerent basis sets will be derived and their properties will be studied in detail. 2.1. Minibands The periodicity of the superlattice structure within the z direction implies that the eigenstates of the Hamiltonian can be written as Bloch states ’q (z), where q ∈ [ − =d; =d] denotes the Bloch vector. The construction of these eigenstates can be performed straightforwardly within the transfer matrix formulation, see e.g. [50,51]. Within a region zj ¡ z ¡ zj+1 of constant potential and constant material composition the envelope function can be written as f(z) = Aj eikj (E)(z−zj ) + Bj e−ikj (E)(z−zj ) . Then the connection rules [52] (see also Refs. [10,51,53] for a detailed discussion) fc (˜r)z→zj+1 +0− = fc (˜r)z→zj+1 +0+ ;

(3)

1 9fc (˜r) 1 9fc (˜r) = − mc; j 9z z→zj+1 +0 mc; j+1 9z z→zj+1 +0+

(4)

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apply 1 where mc; j is the eDective mass in region zj ¡ z ¡ zj+1 .     Aj+1 Aj = Mj (E) Bj+1 Bj with

(5)



    kj mc; j+1 ikj (zj+1 −zj ) kj mc; j+1 −ikj (zj+1 −zj ) 1 + e 1 − e  kj+1 mc; j kj+1 mc; j 1   Mj (E) =    :    2 kj mc; j+1 −ikj (zj+1 −zj )  kj mc; j+1 ikj (zj+1 −zj ) 1− e 1+ e kj+1 mc; j kj+1 mc; j

(6)

If a single period of the superlattice consists of M regions with constant material composition, the Bloch condition ’q (z + d) = eiqd ’q (z) implies        M AM +1 A1 ! iqd A1 = Mj (E) =e : (7) BM +1 B1 B1 j=1

For standard superlattices (M = 2) the solutions resemble those of the Kronig–Penney model [61], except for the use of eDective masses associated with connection rule (4). Within the transfer formalism the extension to superlattices with a basis [62] (larger M ) is straightforward. For given q, Eq. (7) is only solvable for selected values of E which de;ne the miniband structure E  (q). An example is shown in Fig. 2. Next to the envelope function approximation discussed here, diDerent approaches can be used to calculate the superlattice band structure [9,63]. For a given miniband  the following quantities may be de;ned:  =d center of miniband: E  = d=(2) dq E  (q) (8) −=d

miniband width: / = Maxq {E  (q)} − Minq {E  (q)}

(9)

which characterize the miniband structure. For the miniband structure shown in Fig. 2, the values E a = 54:5 meV, /a = 23:6 meV, E b = 220 meV, /b = 98 meV, E c = 491 meV, and /c = 233 meV are found, where the band indices are labeled by  = a; b; c : : : : The increase of / with  can be easily understood in terms of the increasing transparency of the barrier with the electron

1 Throughout this work, we use the energy dispersion E(˜k)=Ec +˝2 k 2 =2mc with the band edge Ec =0:8x meV and the eDective mass mc = (0:067 + 0:083x)m0 for the conduction band of Alx Ga1−x As with x ¡ 0:45 [54]. For GaAs=AlAs structures, nonparabolicity eDects are included using the energy-dependent eDective mass mc (E)=mc (E −Ev )=(Ec −Ev ) with the parameters mc =0:067m0 , Ec =0, Ev =−1:52 meV for GaAs and mc =0:152m0 , Ec =1:06 meV, Ev =−2:07 meV for AlAs [55,56], where Ev denotes the edge of the valence band. X,L-related eDects are neglected for simplicity. They become relevant in some transport studies [57]. Some approaches to the theoretical study of tunneling via these minima can be found in [58– 60].

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Fig. 2. Calculated miniband structure for the GaAs–Al0:3 Ga0:7 As superlattice used in [64] with well width 6:5 nm and barrier width 2:5 nm.

energy. As E  (q) is restricted to −=d ¡ q ¡ =d and E  (q) = E  (−q) by Kramers degeneracy, the function E  (q) can be expanded as follows: 



E (q) = E +

∞


2Th cos(hdq) :

(10)

h=1

Typically, the terms Th for h ¿ 2 are much smaller than the T1 term. E.g., for the bandstructure of Fig. 2 one obtains T1a = −5:84 meV, T2a = 0:48 meV for the lowest band, T1b = 23:7 meV, T2b = 2:1 meV for the second band, and T1c = −53:3 meV, T2c = 5:3 meV for the third band. This demonstrates that the band structure is essentially of cosine shape and thus / ≈ 4|T1 |. The dispersion E  (q) = E  + 2T1 cos(dq) can be viewed as the result of a standard tight-binding calculation with next-neighbor coupling T1 . In order to perform many-particle calculations the formalism of second quantization, see, e.g.,  [65], is appropriate. Let a† q and aq be the creation and annihilation operator for electrons in the Bloch state of band  with Bloch vector q. Then the Hamiltonian reads as
 =d  ˆ H SL = dq E  (q)a† (11) q aq 

−=d

which is diagonal in the Bloch states, as the Bloch states are eigenstates of the unperturbed superlattice. 2.2. Bloch states of the three-dimensional superlattice In the preceding subsection only the z direction of the superlattice was taken into account. For an ideal superlattice Eq. (2) does not exhibit an (x; y)-dependence and thus a complete set of eigenstates states can be constructed by products of plane waves eik·r =(2) and a z-dependent function fk (z) which satis;es the eigenvalue equation   ˝2 k 2 9 ˝2 9 Ec (z) − fk (z) = Efk (z) : (12) + 9z 2mc (z) 9z 2mc (z)

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As Ec (z) and mc (z) are periodic functions with the superlattice period d, the eigenstates are Bloch state fk (z) = ’q; k (z) with energy E  (q; k). 2 Within ;rst-order perturbation theory in k2 , one obtains the energy  2 2          ˝ k   E (q; k) ≈ E (q; 0) + ’q; 0  ’ : (13) 2mc (z)  q; 0 Now, ’q; 0 (z) will exhibit a larger probability in the well, so that it seems reasonable to replace the second term by Ek =

˝2 k 2

(14)

2mw

where mw is the eDective mass of the quantum well. In analogy to Eq. (11), the full Hamiltonian reads as 
 =d  Hˆ SL = dq d 2 k[E  (q) + Ek ]a† (15) q (k)aq (k) 

−=d

 where a† q (k) and aq (k) are the creation and annihilation operators for electrons in the Bloch state of band  with Bloch vector q and wave vector k in (x; y) plane. In order to evaluate matrix elements for scattering processes the zeroth-order envelope wave-functions ’q; 0 (z)eik·r =(2) are applied in subsequent sections. The treatment is completely analogous for Wannier and Wannier– Stark states discussed in the subsequent subsections.

2.3. Wannier functions By de;nition the Bloch functions are delocalized over the whole superlattice structure. This may provide diMculties if electric ;elds are applied or eDects due to the ;nite length of the superlattice are considered. Therefore, it is often helpful to use diDerent sets of basis states which are better localized. A tempting choice would be the use of eigenstates of single quantum wells, see, e.g., [66,67]. Nevertheless such a choice has a severe shortcoming: the corresponding states are solutions of two diDerent Hamiltonians, each neglecting the presence of the other well. Thus these states are not orthogonal which provides complications. Typically, the coupling is estimated by the transfer Hamiltonian [68] within this approach. For these reasons, it is more convenient to use the set of Wannier functions [69]   d =d  ) (z − nd) = dq e−inqd ’q (z) (16) 2 −=d   which are constructed from the Bloch functions with the normalization d z[’q (z)]∗ ’q (z) =

 (q − q ). Here some care has to be taken: The Bloch functions are only de;ned up to a complex phase which can be chosen arbitrarily for each value of q. The functions )(z) depend 2 Eq. (12) shows that the eDective Hamiltonian is not exactly separable in a z- and r-dependent part, as the z-dependent eDective mass aDects the k-dependence, describing the behavior in the (x; y) plane. Nevertheless, this subtlety is not taken into account here, as discussed below.

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13

Fig. 3. Wannier functions for the superlattice from Fig. 2. Full line: )a (z), dashed line )b (z). The thin line indicates the conduction band edge pro;le.

strongly on the choice of these phases. In [70] it has been shown that the Wannier functions are maximally localized if the phase is chosen in the following way at a symmetry point zsym of the superlattice: If ’0 (zsym ) = 0 choose ’q (zsym ) ∈ R (i). Otherwise choose i’q (zsym ) ∈ R (ii). In both cases, the phase is chosen such that ’q (z) is an analytic function in q. (The latter requirement de;nes the phase when ’q (zsym ) becomes zero and prevents arbitrary sign changes of ’q (zsym ).) For such a choice the Wannier functions ) (z) are real and symmetric (i) or antisymmetric (ii) around zsym . Now there are two symmetry points, one in the center of well ) and one in the barrier (z barrier ), for a typical superlattice. If the energy of the the well (zsym sym miniband is below the conduction band of the barrier, the Wannier functions seem to be strongly well , while z barrier may be suited as well for larger energies, where the minibands localized for zsym sym are above the barrier. This point has also been addressed in [71]. In Fig. 3, the Wannier functions well ) are displayed. One ;nds that both functions are essentially for the ;rst two bands (using zsym localized to the central quantum well where they resemble the bound states. Outside the well they exhibit a decaying oscillatory behavior which ensures the orthonormality relation 3  d z ) (z − nd))& (z − md) = n; m ; & : (17)  Within second quantization, the creation a† n and annihilation an operators of the states asso ciated with the Wannier functions ) (z − nd) are de;ned via  d
−iqnd   e an : (18) aq = 2 n

3

Note that the orthonormality is not strictly ful;lled for diDerent bands ; & if an energy-dependent eDective mass is used. In this case, energy-dependent Hamiltonians (2) are used for the envelope functions, and therefore the orthonormality of eigenfunctions belonging to diDerent energies is not guaranteed. In principle, this problem could be cured by reconstructing the full wave functions from the envelope functions under consideration of the admixtures from diDerent bands.

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Inserting into Eq. (11) and using Eq. (10), one obtains the Hamiltonian within the Wannier basis   ∞

†    Hˆ SL = E  a† Th (a† (19) n an + n+h an + an−h an ) : n;

h=1

As the Wannier functions are linear combinations of Bloch functions with diDerent energies, they do not represent stationary states. Neglecting terms with h ¿ 1 the time evolution of the annihilation operators in the Heisenberg representation is given by d i˝ an (t) = E  an (t) + T1 [an+1 (t) + an−1 (t)] : (20) dt For the initial condition, an (t = 0) = n; 0 a0 this set of equations has the solution    2T1  an (t) = i−n Jn t e−iE t=˝ a0 ; (21) ˝ where Jn (x) is the Bessel function of the ;rst kind [72]. This shows that the initially occupied Wannier state decays on a time scale of ˝ ,Wannier ∼ : (22) 2T1 At this time a0 † (t)a0 (t) = J0 (1)2 ≈ 0:586, thus ,Wannier may be viewed as a kind of half-life period, although there is no exponential decay. If an electric ;eld F is applied to the superlattice, the additional potential '(z) = −Fz has to be taken into account. Within the Wannier basis the corresponding terms of the Hamiltonian  can be evaluated directly by the corresponding matrix elements d z )& (z − md)eFz) (z − nd). Including the parallel degrees of freedom k, the total Hamiltonian Hˆ = Hˆ 0 + Hˆ 1 + Hˆ 2 reads as
 †  d 2 k(E  + Ek − eFR (23) Hˆ 0 = 0 − eFdn)an (k)an (k) ; n;

Hˆ 1 =

∞ 



 †  d 2 k {Th [a† n+h (k)an (k) + an (k)an+h (k)] &; 

n;;& h=1 &†  −eFR& h [an+h (k)an (k)

Hˆ 2 =


 n;;& =&

& + a† n (k)an+h (k)]} ;

&†  d 2 k(−eFR& 0 )an (k)an (k)

(24) (25)

 &  with the couplings R& h = d z ) (z − hd)z) (z). If the superlattice exhibits inversion symmetry ˆ the coeMcients R h vanish for h ¿ 0. Finally, note that the expression of H is still exact except for the separation of z and (x; y) direction. The term Hˆ 0 describes the energy of the states in the superlattice neglecting any couplings to diDerent bands or diDerent wells. Hˆ 1 gives the coupling between diDerent wells. Finally, Hˆ 2 describes the ;eld-dependent mixing of the levels inside a given well. In particular, it is responsible for the Stark shift.

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The term Hˆ 0 + Hˆ 2 for n = 0 can be diagonalized [73] by constructing the new basis
|&˜  = U &˜ |

15

(26)

 &˜ satisfying (Hˆ 0 + Hˆ 2 )|&˜  = E˜ |&˜  where the columns of U &˜ are the components of |&˜  with respect to the basis |. This shows that the level separation becomes ;eld dependent, which has been recently observed in superlattice transport [74] under irradiation. In the new basis, the ren ren Hamiltonian is given by Hˆ = Hˆ 0 + Hˆ 1 with
 ren   Hˆ 0 = d 2 k(E˜ + Ek − eFdn)a† (27) n (k)an (k) n;

ren Hˆ 1 =

∞ 

n;;& h=1

& &  † d 2 k H˜ h [a&† n+h (k)an (k) + an (k)an+h (k)]

and the matrix elements
&˜˜ ˜ H˜ h = (U &&˜ )∗ (Th ; & − eFR& : h )U

(28)

(29)

&;

It will turn out later that in the limit of sequential tunneling it is more appropriate to use ren Hˆ 1 as a perturbation instead of Hˆ 1 . 2.4. Wannier–Stark ladder If an electric ;eld F is applied to the superlattice structure the Hamiltonian exhibits an additional scalar potential e'(z) = −eFz which destroys the translational invariance. In this case, we can easily see that if there exists an eigenstate with wavefunction *0 (z) and energy E0 , then the set of states corresponding to wavefunctions *j (z) = *0 (z − jd) are eigenstates of the Hamiltonian with energies Ej = E0 − jeFd as well. These states are equally spaced both in energy and real space and form the so-called Wannier–Stark ladder [4]. This feature has to be considered with some care, as the potential e'(z) is not bounded for the in;nite crystal, which implies a continuous energy spectrum [75]. Nevertheless, the characteristic energy spectrum of these Wannier–Stark ladders could be resolved experimentally [24,25] in semiconductor superlattices. For a more detailed discussion of this subject see [13,76,77]. If one restricts the Hamiltonian Hˆ SL in Eq. (11) to a given miniband , an analytical solution for the eigenstates of Hˆ SL − eFz exists [78]:      q d =d i      |*j  = dq exp dq [Ej − E (q )] |’q  ; (30) 2 −=d eF 0 where Ej = E  − jeFd is the ladder of energies corresponding to the th miniband with average energy E  similar to the discussion of the Wannier states. 4 The ;eld-induced coupling to 4

This representation depends crucially on the relative choice of phases in the Bloch functions ’q (z). The situation resembles the construction of Wannier states (16) and suggests the use of the same choice of phase although I am not aware of a proof. For consistency, the origin of z has to be chosen such that zsym = 0 holds for the symmetry point of the superlattice.

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Fig. 4. Wannier–Stark states calculated from Eq. (30) for the superlattice from Fig. 2. The thin line indicates the conduction band edge pro;le. (a) Full line: *a (z), dashed line *b (z), (b) *a (z) for diDerent ;elds.

diDerent bands induces a ;nite lifetime of these single band Wannier–Stark states due to Zener tunneling [78]. Thus these Wannier–Stark states can be viewed as resonant states (an explicit calculation of these resonances has been performed in [79]). For a cosine-shaped band E  (q) = E  + 2T1 cos(qd) the Wannier–Stark states from Eq. (30) can be expanded in Wannier states |)n  [80]   ∞
2T1 |)n  for next-neighbor coupling ; |*j  = Jn−j (31) eFd n=−∞ where de;nition (16) has been used. This relation can be obtained directly by diagonalizing Eqs. (23) and (24) within the restriction to a single band, nearest-neighbor coupling T1 , and R 0 = 0, i.e. zsym = 0. In Fig. 4 examples for the Wannier–Stark states are shown. It can be clearly seen that the localization of these states increases with the electric ;eld. They exhibit an oscillatory structure within a region of approximately / =eFd periods decay ∞and a 2strong 2 outside this region. This magnitude can be estimated via Eq. (31). As n=−∞ n Jn (x) = x2 =2 [Eq. (8:536(2) of [81]] we can conclude that Jn (2T1 =eFd) deviates from zero essentially in the range −2|T1 |=eFd . n . 2|T1 |=eFd which, together with / ≈ 4|T1 |, provides the result given above. 3. The standard approaches for superlattice transport If an external bias is applied to a conductor, such as a metal or a semiconductor, typically an electrical current is generated. The magnitude of this current is determined by the band structure of the material, scattering processes, the applied ;eld strength, as well as the equilibrium carrier distribution of the conductor. In this section, the following question is addressed: How does the special design of a semiconductor superlattice, which allows the variation of the band structure in a wide range, inJuences the transport behavior?. Throughout this section, we assume that a homogeneous electrical ;eld F is applied in the direction of the superlattice (the z direction) and consider the current parallel to this ;eld. Due to symmetry reasons, the transverse current parallel to the layers should vanish. A very elementary solution to the problem has been provided by Esaki and Tsu in their pioneering paper [3]. Consider the lowest miniband of the superlattice labeled by the superscript

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17

Fig. 5. (a) Dispersion E a (q) and velocity v a (q) for the lowest miniband. (b) Velocity–;eld relation according to Esaki–Tsu [3].

a. The eigenstates are the Bloch states ’aq (z) with the Bloch vector q and the dispersion is approximately given by E a (q) ≈ E a − 2|T1a | cos(qd) (see Section 2.1 for details) as depicted in Fig. 5a. At low temperatures, the states close to the minimum at q ≈ 0 are occupied in thermal equilibrium. If an electric ;eld is applied (in z direction) the Bloch states are no longer eigenstates of the full Hamiltonian but change in time. According to the acceleration theorem [1] dq eF ; (32) = dt ˝ the states remain Bloch states in time, but the Bloch vector becomes time dependent and we ;nd q(t) = eFt= ˝ if the electron starts in the minimum of the band at t = 0. For t = ˝=(eFd) the boundary of the Brillouin zone (q = =d) is reached. This point is equivalent with the point at q = −=d, so that the trajectory continues there which is often called Bragg-reJection. Finally, at t = ,Bloch = 2˝=(eFd) the origin is reached again. Neglecting transitions to diDerent bands (Zener transitions, whose probability is extremely small for low ;elds) the state remains in the given band and thus the same state is reached after ,Bloch resulting in a periodical motion of the state through the Brillouin zone [2]. This oscillation is called Bloch oscillation and is quite general for arbitrary bandstructures. It could be observed in superlattices [28,29]. The Bloch states q travel with the velocity 1 9E a (q) 2d|T1a | v a (q) = ≈ sin(qd) : (33) ˝ 9q ˝ Thus, we ;nd v(t) = vm sin(eF dt= ˝) and the position of a wave packet z(t) = z0 + zm {1 − cos(eF dt= ˝)} with vm = 2d|T1a |= ˝ and zm = 2|T1a |=eF. In [82] this behavior has been nicely demonstrated by an explicit solution of the SchrQodinger equation. The spatial amplitude of this oscillation has been resolved recently [30]. Scattering processes will interrupt this oscillatory behavior. As scattering processes are likely to restore thermal equilibrium, it makes sense to assume that the scattered electron will be found close to q = 0, the initial point used before. As long as the average scattering time , is much smaller than ˝=(2eFd), the electrons will remain in the range 0 . q ¡ =(2d) where the velocity increases with q and thus an increase of F will generate larger average drift velocities. Thus for eFd˝=(2,) a linear increase of vdrift (F) is expected. In contrast, if , & ˝=(eFd) the electrons reach the region −=d ¡ q ¡ 0 with negative velocities and thus

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the average drift velocity can be expected to drop with the ;eld for eFd & ˝=,. For high ;elds, eFd2˝=,, the electrons perform many periods of the Bloch oscillation before they are scattered and thus the average drift velocity tends to zero for F → ∞. A detailed analysis for a constant (momentum-independent) scattering time , gives the Esaki–Tsu relation [3]: eFd˝=, vdrift (F) = vET (F) = vm : (34) (eFd)2 + (˝=,)2 This result will be derived in Section 3.1.1 as well. The drift velocity exhibits a linear increase with F for low ;elds, a maximum at eFd = ˝=, and negative diDerential conductivity for eFd ¿ ˝=,, see Fig. 5b. This general behavior could be observed experimentally [83,22]. The rather simple argument given above neglects the plane-wave states in (x; y)-direction, the thermal distribution of carriers, and treats scattering processes in an extremely simpli;ed manner. In Section 3.1, a more realistic treatment is given within the miniband transport model where the electrons occupy Bloch states and the dynamical evolution of the single states is described by the acceleration theorem (32). A complementary approach to miniband transport is the use of Wannier–Stark states, which are the ‘real’ eigenstates of the superlattice in an electric ;eld [see Section 2.4 for a discussion of the problems involved with these states]. Scattering processes cause transitions between these states yielding a net current in the direction of the electric ;eld [33]. This approach is called Wannier–Stark hopping and will be described in detail in Section 3.2. For superlattices with thick barriers (i.e. narrow minibands) it seems more appropriate to view the structure as a series of weakly coupled quantum wells with localized eigenstates. Due to the residual coupling between the wells, tunneling processes through the barriers are possible and the electrical transport results from sequential tunneling from well to well, which will be discussed in Section 3.3. Generally, the lowest states in adjacent wells are energetically aligned for zero potential drop. Thus, the energy conservation in Fermi’s golden rule would forbid tunneling transitions for ;nite ;elds. Here it is essential to include the scattering induced broadening of the states which allows for such transitions. These three complementary approaches are schematically depicted in Fig. 6. They treat the basic ingredients to transport, band structure (coupling T1 ), ;eld strength (eFd), and scattering (7 = ˝=,), in completely diDerent ways. In Section 4 and Appendix B these approaches will be compared with a quantum transport model, which will determine their respective range of validity as sketched in Fig. 7. It is an intriguing feature that all three approaches provide a velocity–;eld relation in qualitative agreement with Fig. 5b, except that the linear increase is missing in the Wannier–Stark hopping model. Therefore, the qualitative features from the Esaki–Tsu model persist but details as well as the magnitude of the current may be strongly altered. These points will be discussed in detail in the subsequent subsections. 3.1. Miniband transport Conventionally, the electrical transport in semiconductors or metals is described within a semiclassical approach. Due to the periodicity of the crystal, a basis of eigenstates from Blochfunctions can be constructed. For the superlattice structure considered here it is convenient

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19

Fig. 6. Overview of the diDerent standard approaches for superlattice transport.

Fig. 7. Ranges of validity for the diDerent standard approaches for superlattice transport. Miniband transport holds for eFd2|T1 | and 72|T1 |; Wannier–Stark hopping holds for 7eFd; Sequential tunneling holds for 2|T1 |eFd or 2|T1 |7, from [36].

to treat the Bloch vector q in superlattice direction and the Bloch vector k in the direction parallel to the layers separately. The eigenfunctions are ’q (z)eik·r =2 and the corresponding energies are E(q; k; ) = E  (q) + Ek with the superlattice dispersion E  (q) and the in-plane energy Ek = ˝2 k2 =2mc , for details see Section 2.1. The occupation of these states is given by the distribution function f(q; k; ; t) describing the probability that the state (q; k; ) is occupied and

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f(q; k; ; t) dq d 2 k=(2)3 is the particle density within the volume element dq d 2 k in momentum space. An electric ;eld breaks the translational invariance of the system and the Bloch states are no longer eigenstates. Within the semiclassical theory the temporal evolution of the distribution function is given by the Boltzmann equation 5   9f 9f(q; k; ; t) eF 9f (35) + = 9t ˝ 9q 9t scatt which is derived in most textbooks of solid state physics. (The inclusion of magnetic ;elds is straightforward and results for superlattice structures are given in [84,85]; see also [86] for corresponding experimental results. Recently, a mechanism for spontaneous current generation due to the presence of hot electron in a magnetic ;eld perpendicular to the superlattice direction was proposed [87].) The right-hand side describes the change of the distribution function due to scattering. For impurity or phonon scattering, the scattering term reads as   9f(q; k; ; t) 9t scatt  
=d  = dq d 2 k  {P(q ; k ;  → q; k; )f(q ; k ;  ; t) 

−=d

×[1 − f(q; k; ; t)] − P(q; k;  → q ; k ;  )f(q; k; ; t)[1 − f(q ; k ;  ; t)]} :

(36)

P(q; k;  → q ; k ;  ) denotes the scattering probability from state (q; k; ) to state (q ; k ;  ) which can be calculated by Fermi’s golden rule using appropriate matrix elements for the diDerent scattering processes. Details regarding these scattering processes (for bulk systems) can be found in textbooks such as [88,89]. Speci;c calculations for superlattice structures can be found in [90 –92]. Once the distribution function is known, the current density JMBT for miniband transport in the z direction can be evaluated directly by   2(for Spin)e
=d JMBT = dq d 2 k f(q; k; ; t)v (q) (37) (2)3 −=d  and the electron density per superlattice period (in units [1=cm2 ]) is given by   2(for Spin)d
=d n= dq d 2 k f(q; k; ; t) : (2)3 −=d 

(38)

This approach to the electric transport is called miniband transport. An earlier review has been given in [93] where several experimental details are provided. One has to be aware that Boltzmann’s equation holds for classical particles under the assumption of independent scattering events. The only quantum mechanical ingredient is the use of the dispersion E q (q; k), thus, the term semiclassical approach is often used. Therefore, deviations may result from various quantum features, such as scattering induced broadening of the 5

If spatially inhomogeneous distributions f(r; q; k; ; t) are considered, the convection term v (q)9f= 9z + ˝k=mc · 9f= 9r has to be added on the left-hand side.

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states, the intracollisional ;eld eDect, or correlations between scattering eDects leading, e.g., to weak localization. While these features are notoriously diMcult to describe, operational solution methods like Monte-Carlo methods [94,95] exist for the Boltzmann equation explaining the popularity of the semiclassical approach. 3.1.1. Relaxation time approximation Boltzmann’s equation can be solved easily if the scattering term is approximated by   nF (E(q; k) − &) − f(q; k; t) 9f(q; k; t) = 9t ,scatt (q; k) scatt

(39)

with the relaxation time ,scatt (q; k). The relaxation time approximation is correct in the linear response regime for a variety of scattering processes. Here it is applied to nonlinear transport, in order to obtain some insight into the general features. The underlying assumption is that any scattering process restores the thermal equilibrium described by the Fermi function nF (E) = [exp(E=kB T ) + 1]−1 and the chemical potential &. (The discussion is restricted to the lowest miniband here and thus the miniband index  is neglected.) Then the stationary Boltzmann equation reads as eF,scatt (q; k) 9f(q; k) (40) + f(q; k) = nF (E(q; k) − &) : ˝ 9q This is an inhomogeneous linear partial diDerential equation which, together with the boundary condition f(−=d; k; t) = f(=d; k; t), can be integrated directly and one ;nds   q   =d ˝nF (E(q0 ; k) − &) ˝  f(q; k) = dq0 dq exp − eF,scatt (q0 ; k) eF,scatt (q ; k) −=d q0     1 × -(q − q0 ) + : (41)   exp[ =d dq ˝=eF,scatt (q ; k)] − 1  −=d

Assuming a constant scattering rate ,scatt (q; k) = ,, the q integrals become trivial. For the simpli;ed miniband structure E(q) = E a − 2|T1a | cos(qd) one obtains the electron density from Eq. (38)  =d  2d n = neq (&; T ) = dq0 d 2 k nF (E(q0 ; k) − &) (42) (2)3 −=d and the current density from Eq. (37) 2|T a | eFd˝=, JMBT = e 1 ceq (&; T ) ˝ (˝=,)2 + (eFd)2 with 2d ceq (&; T ) = (2)3



=d

−=d



dq0

d 2 k cos(q0 d)nF (E(q0 ; k) − &) :

(43)

(44)

Note that the ;eld dependence of the current density is identical with the simple Esaki–Tsu result (34), but the prefactor has a complicated form, which will be analyzed in the following.

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The k-integration can be performed analytically in neq (&; T ) and ceq (&; T ).   ∞ 2 1 2 d k nF (E(q; k) − &) = %0 dEk (E(q)+Ek −&)=kB T 2 (2) e +1 0 = %0 kB T log[e(&−E(q))=kB T + 1] :

(45)

Here %0 = mc =˝2 is the density of states of the two-dimensional electron gas parallel to the layers including spin degeneracy. At ;rst, consider the degenerate case T = 0 which holds for low temperatures kB T 2|T1a |. If & ¿ E a + 2|T1a | one obtains [96] n = %0 (& − E a ) and  2%0 |T1a |2 eFd˝=, kB T 2|T1a | ; (46) for JMBT = e n ¿ 2|T1a |%0 : ˝ (˝=,)2 + (eFd)2 This expression is independent from the carrier density. (For superlattices with very thin barriers a diDerent behavior has been reported [97] which was explained by the one-dimensional character of the transport due to inhomogeneities.) A second instructive result is obtained for very low densities 0 ¡ (& − E a )=2|T1a | + 11 when    Ea − & 3 a 2 arccos (47) neq ≈ ceq ≈ %0 |T1 | 3 2|T1a | holds and thus 2|T a | eFd˝=, JMBT = en 1 ˝ (˝=,)2 + (eFd)2



for

kB T 2|T1a | ; n2|T1a |%0 ;

(48)

which gives J = e(n=d)vET (F) providing the Esaki–Tsu result (34) mentioned above. In the non-degenerate case &E a − 2|T1a | one obtains [8] n = %0 kB T e(&−E

a

)=kB T

I0 (2|T1a |=kB T ) ;   2|T1a | 2|T1a | eFd˝=, (&−E a )=kB T J = e%0 kB T e I1 kB T ˝ (˝=,)2 + (eFd)2

(49) (50)

with the modi;ed Bessel functions I0 ; I1 , see Eq. (9:6:19) of [72]. For low temperatures kB T 2|T1a | the argument x of the Bessel functions becomes large and I0 (x) ∼ I1 (x). Then Eq. (48) is recovered again. For high temperatures kB T 2|T1a | the Bessel functions behave as I0 (x) ∼ 1 and I1 (x) ∼ x=2, and  2|T1a |2 eFd˝=, kB T 2|T1a | ; (51) for JMBT = en n%0 kB T : kB T ˝ (˝=,)2 + (eFd)2 Such a 1=T dependence of the current density has been observed experimentally in [98,99] albeit the superlattices considered there exhibit a rather small miniband width and the justi;cation of the miniband transport approach is not straightforward. 3.1.2. Two scattering times A severe problem of the relaxation-time model is the fact that all scattering processes restore thermal equilibrium. While this may be correct for phonon scattering, where energy can be

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transferred to the phonon systems, this assumption is clearly wrong for impurity scattering, which does not change the energy of the particle. The signi;cance of this distinction can be studied by applying the following scattering term [5,100]:   9f(q; k; t) nF (E(q; k) − &) − f(q; k; t) f(−q; k; t) − f(q; k; t) = + : (52) 9t ,e 2,elast scatt Here scattering processes which change both momentum and energy are contained in the energy scattering time ,e and elastic scattering events, changing only the momentum, are taken into account by ,elast . While the Boltzmann equation was solved explicitly in Section 3.1.1, dynamical equations for the physical quantities of interest are derived here by taking the appropriate averages with the distribution function. At ;rst, consider the electron density (38). Performing  the integral (2d=(2)3 ) dq d 2 k of Eq. (35) one obtains dn(t) neq − n(t) ; (53) = dt ,e where the periodicity of f(q) has been used to eliminate the term ˙ F. Therefore, the electron density is again given by neq (&; T ), see Eq. (42). In the same manner, one obtains (using integration by parts in the term ˙ F) dJ (t) e2 Fd J (t) − 2 2|T1a |c(t) = − (54) dt ˝ ,m with the momentum relaxation time 1=,m = 1=,e + 1=,elast and the average  =d  2d c(t) = dq d 2 k cos(qd)f(q; k; t) : (55) (2)3 −=d Finally, the dynamical evolution of c(t) is given by ceq (&; T ) − c(t) dc(t) Fd J (t) = : (56) + a dt 2|T1 | ,e This equation can be considered as a balance equation for the kinetic energy in the superlattice  3 2 direction as (2d=(2) ) dq d k E(q)f(q; k; t) = E a n(t) − 2|T1a |c(t). The stationary solution of Eqs. (54) and (56) gives 2|T a | eFd˝=,eD JMBT = e 1 ceq (&; T ) (57) ˝ (˝=,eD )2 + (eFd)2  √ with the eDective scattering time ,eD = ,e ,m and = ,m =,e . This is just result (43) with the additional factor reducing the magnitude of the current, as ,m 6 ,e . The prefactors neq (&; T ) and ceq (&; T ) are identical with those introduced in the last subsection. The relaxation time approximation has proven to be useful for the analysis of experimental data by ;tting the phenomenological scattering times ,e ; ,elast . In [101] the times 1=,e =9 × 1012 =s and 1=,elast = 2 × 1013 =s has been obtained for a variety of highly doped and strongly coupled have superlattices at T = 300 K. It should be noted that there is an instructive interpretation [102] of Eq. (57) which may be rewritten as   1 1 1 d (58) = + J eceq (&; T ) vlf (F) vhf (F)

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with the low-;eld velocity e,m vlf (F) = F; mminiband where 1

mminiband

 2|T1a |d2 1 92 E a (q)  = 2 = : ˝ 9q2 q=0 ˝2

(59)

(60)

This is just the standard expression for the linear conductivity which is dominated by momentum relaxation. The high-;eld velocity P vhf (F) = loss (61) eF is determined by the maximal energy loss per particle given by Ploss = 2|T1a |=,e for the particular scattering term (52) where k is conserved. This relation results from the energy balance providing negative diDerential conductivity as already pointed out in [103]. While both expressions for the low-;eld velocity and the high-;eld velocity are quite general, it is not clear if the interpolation (58) in the form of a generalized Matthiessen’s rule [102] holds beyond the relaxation time model. 3.1.3. Results for real scattering processes The relaxation time approximations discussed above contain several problems: • The scattering processes conserve k, which is arti;cial. An adequate improvement to this point has been suggested in [104]. • The magnitude of the scattering times is not directly related to physical scattering processes. • Energy relaxation is treated in a very crude way by assuming that in-scattering occurs from a thermal distribution. In [105] balance equations have been derived for the condition of stationary drift velocity and stationary mean energy. Here the distribution function was parameterized by a drifted Fermi-function similar to the concepts of the hydrodynamic model for semiconductor transport (see [106] and references cited therein). This approach allows one to take into account the microscopic scattering matrix elements for impurity and electron–phonon scattering and good results were obtained for the peak position and peak velocity observed in [83]. Self-consistent solutions of the Boltzmann equation have been performed by various groups. In [91] results for optical phonon and interface roughness have been presented where Boltzmann’s equation was solved using a conjugate gradient algorithm. Using Monte-Carlo methods [94], the Boltzmann equation can be solved to a desired degree of numerical accuracy in a rather straightforward way (at least in the non-degenerate case and without electron– electron scattering). Results have been given in [107] for acoustic phonon scattering and in [108] for optical phonon and impurity scattering (using constant matrix elements). Modi;ed scattering rates due to collisional broadening have been applied in [109] without signi;cant changes in the result. Recently, extensive Monte-Carlo simulations [110,111,92] have been performed where both optical and acoustic phonon scattering as well as impurity scattering has been considered using the microscopic matrix elements. Results of these calculations are presented in Fig. 8a. The general shape of the velocity–;eld relations resembles the Esaki–Tsu result shown

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Fig. 8. (a) Drift-velocity versus electric ;eld for a superlattice with miniband width 20:3 meV and period d=5:1 nm with homogeneous doping N3D =1016 =cm3 . The calculation has been performed within the miniband transport model employing the Monte-Carlo method (Fig. 3:8 from [92]). (b) Fit by the two-time model (57) using scattering times ,elast = 4, 1.6, 1.1, 0:12 ps and ,e = 10, 2, 0.9, 0:4 ps for T = 4, 77, 150, 300 K, respectively.

in Fig. 5b both here and in all other calculations mentioned above. This is demonstrated by a comparison with the two-time model (57), where the scattering times have been chosen to give good agreement with the Monte-Carlo simulations, see Fig. 8b. The increase of the scattering rate with lattice temperature can be attributed to the enhanced phonon occupation. In contrast, the high-;eld behavior does not strongly depend on lattice temperature. Here the drift velocity is limited by energy relaxation (61) which is dominated by spontaneous emission of phonons and thus does not depend on the thermal occupation of phonon modes. 3.2. Wannier–Stark hopping If a ;nite electric ;eld is applied to a semiconductor superlattice, the Bloch states are no longer eigenstates. Within the restriction to a given miniband , Wannier–Stark states *j; k (r; z)= √  − jeFd + E diagonalize the Hamiltonian as discussed in *j (z)eik·r = A with energy Ej; k = EWS k Section 2.4. (We apply a normalization area A in the (x; y) direction here yielding discrete values   of k and normalizable states. For practical calculations the continuum limit k → A=(2)2 d 2 k is applied.) These states are approximately centered around well n = j. In the following, we restrict ourselves to the lowest band  = a and omit the index . In a semiclassical approach, the occupation of the states is given by the distribution function fj (k). Scattering causes hopping between these states [33,112]. Thus, this approach is called Wannier–Stark hopping. Within

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Fermi’s golden rule the hopping rate is given by Ri; k→j; k =

2 a ˆ scatt a 2 |*i; k | (Ek  − jeFd − Ek + ieFd[ ± ˝!phonon ])[1 − fj (k )] ; |*j; k |H ˝

(62)

where the term [ ± ˝!phonon ] has to be included if emission or absorption of phonons is considered. For details regarding the evaluation of scattering matrix elements see [33,92,113]. The current through the barrier between the wells m = 0 and 1 is then obtained by the sum of all transitions between states centered around wells m 6 0 and those centered around m ¿ 1, i.e. JWSH =

2(for Spin)e


[Ri; k→j; k fi (k) − Rj; k →i; k fj (k )] : A 

(63)

i60 j¿1 k;k

If the occupation fi (k) = f(k) is independent of the index i, i.e., the electron distribution is homogeneous in superlattice direction, one ;nds 2e

JWSH = h[R0; k→h; k f(k) − Rh; k →0; k f(k )] ; (64) A  h¿1 k;k

where Ri; k→j; k = R0; k→( j−i); k has been used. Typically, in the evaluation of Eq. (64) thermal a + E − & ) are employed. The underlying idea is the distribution functions fi (k) = nF (EWS i k assumption that the scattering rates Ri; k→i; k inside each Wannier–Stark state are suMciently fast to restore thermal equilibrium. In this case, Eq. (64) can be further simpli;ed to    2e

heFd a JWSH = hR0; k→h; k nF (EWS + Ek − &) 1 − exp − : (65) A kB T  h¿1 k;k

Evaluating this expression for various types of scattering processes, one obtains a drop of the current density with electrical ;eld as shown in Fig. 9. This is caused by the increasing localization of the Wannier–Stark functions (see Fig. 4) which reduces the matrix elements scatt *j; k |Hˆ |*i; k  with increasing ;eld. This behavior can be analyzed by expanding the scattering matrix elements in terms of Wannier states |)n;a k  from Eq. (31):  a  a
2T1 2T1 scatt a ˆ scatt |)n;a k  : )m; *i; k |Hˆ |*j; k  = Jm−i Jn−j (66) k |H eFd eFd n; m As the Wannier states are essentially localized to single quantum wells the diagonal parts m = n a |H a  for diDerent ˆ scatt |)m; dominate. Neglecting correlations between the matrix elements )m; k k wells m one obtains   a  a 2 2T1 2T1 2
a a 2 ˆ scatt |)m; R0; k→h; k = Jm Jm−h |)m; k | k |H ˝ m eFd eFd × (Ek  − heFd − Ek [ ± ˝!phonon ])[1 − f(k )] :

(67)

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Fig. 9. Drift-velocity versus electric ;eld for a superlattice with miniband width 20:3 meV and period d = 5:1 nm with homogeneous doping N3D = 1016 =cm3 . The transport model of Wannier–Stark hopping has been applied. Both self-consistent (thick lines) and thermal (thin lines) distributions of electrons in the k direction have been used (Fig. 4:20 from [92]).

For eFd2T1a the Bessel functions behave as Jn (2x) ∼ xn =n! giving a ;eld dependence ˙ (T1a =eFd)2|h| . Therefore, the transitions 0 → 1 dominate and   2 T1a 2 scatt a 2 a ˆ scatt |)1;a k |2 ) (|)0;a k |Hˆ |)0; R0; k→1; k ∼ k | + |)1; k |H ˝ eFd × (Ek  − Ek − eFd[ ± ˝!phonon ])[1 − f(k )] :

(68)

For high electric ;elds the wave vector k must be large in order to satisfy energy conservation and thus the scattering process transfers a large momentum. If the scattering matrix element does not strongly depend on momentum (such as deformation potential scattering at acoustic phonons) JWSH ˙ 1=F 2 is found, while diDerent power laws occur for momentum-dependent matrix elements (such as JWSH ˙ 1=F 3:5 for impurity scattering [113]). For optical phonon scattering, resonances can be found at heFd = ˝!opt when hopping to states in distance h becomes possible under the emission of one optical phonon [110], see also Fig. 9. If the sum in Eq. (64) is restricted to h 6 hmax , one obtains a linear increase of the current– ;eld relation for low ;elds [33,112] and a maximum at intermediate ;elds before the current drops with higher ;elds as discussed above. In [113] it has been shown that this is an artifact

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Fig. 10. Sketch of the levels in neighboring wells n = m + 1, where & = a is the lowest and  = b is the ;rst excited level in the respective quantum wells. The chemical potentials &m as well as the electrical potentials e'm are marked. For a constant electric ;eld F one ;nds 'm+1 = 'm − Fd.

and the correct J (F) relation is proportional to 1=F for low ;elds. Thus, the linear response region for low ;elds cannot be recovered by the Wannier–Stark hopping approach. While most calculations have been performed assuming thermal distribution functions f(k), recently, self-consistent calculations of the distribution functions have been obtained by solving the semiclassical Boltzmann equation for the Wannier–Stark states: 9fi (k; t)
Rj; k →i; k fj (k ; t) − Ri; k→j; k fi (k; t) : (69) = 9t  j;k

The self-consistent stationary solution of this equation can be used for the evaluation of Eq. (64). As can be seen in Fig. 9 signi;cant deviations between both approaches occur for low lattice temperatures, when electron heating eDects become important. 3.3. Sequential tunneling If the barrier width of a superlattice is large, the structure essentially consists of several decoupled √quantum wells. In each quantum well n, we have a basis set of wave functions )n (z)eik·r = A, where )n (z) is the th eigenfunction of the quantum well potential. The states have the energy E  + Ek + e'n , where the potential energy due to an electrical potential 'n has been considered separately. The notation is clari;ed in Fig. 10. If the wells are not coupled to each other (in;nite barrier width or height) no current Jows in the superlattice direction. For ;nite barrier width the states from diDerent wells become coupled to each other which can be described by a tunnel matrix element Hn;;m& in the spirit of [68] inducing transitions between the wells. In lowest-order perturbation theory the transition rate is given by Fermi’s golden rule: 2 &;   2  & R(m; &; k → n; ; k ) = |Hm; (70) n (k; k )| (E + Ek  + e'n − E − Ek − e'm ) : ˝ As the superlattice is assumed to be translational invariant in the (x; y)-plane, the matrix element is diagonal in k. Thus, transitions are only possible if E  − E & = e'm − e'n , suggesting sharp resonances when the potential drop between diDerent wells 'm − 'n equals the energy spacing

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of the bound states. This could be nicely demonstrated experimentally in Ref. [114] for simple superlattices and in Ref. [62] for superlattices with a basis where even tunneling over 5 barriers was observed. In the presence of a strong magnetic ;eld along the superlattice direction further peaks due to transitions between diDerent Landau levels are observed [115]. The resonance condition from Eq. (70) implies vanishing electric ;eld for transitions between equivalent levels (in particular the lowest level). As for zero ;eld the current vanishes (provided the electron density is equal in both wells), it was concluded that only phonon-assisted tunneling processes are possible. Thus neither a linear increase of the current for low ;elds nor a peak at low ;elds was expected for weakly coupled superlattices [8]. This conclusion is in contradiction with experimental ;ndings [22], where a drift-velocity in qualitative agreement with the Esaki– Tsu result (Fig. 5) has been obtained for weakly coupled superlattices as well. This discrepancy is due to the neglect of broadening in the argument given above. 6 3.3.1. General theory √ In a real quantum well the states )n (z)eik·r = A with energy E  + Ek + e'n are not exact eigenstates of the full Hamiltonian due to the presence of phonons and nonperiodic impurity potentials. The respective scattering processes lead to an energy shift XE and a ;nite lifetime , of the states. These features can be treated within the theory of Green functions (see, e.g., Ref. [65]). While a general treatment is postponed to Section 4, a motivation of the concept and a heuristic derivation of the current formula (79) is given in the following. For a stationary Juctuating potential V (x; y) due to impurities or interface Juctuations one ;nds
|)k |V |)k |2 XE(k) ≈ ; (71) Ek − Ek  k 2
1 ≈ |)k |V |)k |2 (Ek − Ek  ) ; (72) ,(k) ˝  k

where the second order of stationary perturbation theory as well as Fermi’s golden rule was applied for a stationary Juctuating potential V (x; y) due to impurities or interface Juctuations. For a particle which is injected at t = 0, the time dependence of the wave function is then given by 

)n; k (t) = Gn ret (k; t; 0) = −i-(t)e−i(E +Ek +e'n +<

ret

)t=˝

(73)

with
Broadening had been included in earlier theories [73] but there a term was missing which is essential for the transition between equivalent levels. This point is discussed in Appendix A.

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and the spectral function is de;ned by An (k; E) = −2 Im{Gn ret (k; E)} :

(75)

For in;nite lifetime , → ∞, one ;nds An (k; E) → 2 (E − [E  + Ek + e'n + XE(k)]) which is (except for the factor 2) just the contribution of the state (n; ; k) to the total density of states. This relation is more general and An (k; E)=(2) can be viewed as the contribution of the state (n; ; k) if the system is probed with an energy E. Here expressions (71) and (72) approximately correspond to the Born approximation for
with the density of states  2(for Spin)
 %0 ∞  %n (E) = An (k; E) = dEk An (k; E) 2A 2 0

(78)

k

in well n belonging to the level . Regarding the transitions from level & in well m to level  in well n, Eq. (70) is modi;ed as follows: • The energy E is conserved instead of the free particle energy E  + Ek + e'n . • The energy conserving -function is replaced by An (k ; E)=2. • At each energy the net particle Jow from well m to well n is proportional to (nF (E − e'm − &m )[1 − nF (E − e'n − &n )]) while the net particle Jow from n to m is proportional to (nF (E − e'n − &n )[1 − nF (E − e'm − &m )]). Therefore, the total current is proportional to the diDerence [nF (E − e'm − &m ) − nF (E − e'n − &n )] in occupation between both wells. De;ning

the electrochemical potential EF (n) = e'n + &n it becomes clear that the current is driven by the diDerence in the electrochemical potential.

Then the current density from level & in well m to level  in well n is given by  2(for spin)e
&;  &→ Jm→n = |Hm; k;n; k |2 dE A&m (k; E)An (k ; E) 2˝A  k;k

×[nF (E − e'm − &m ) − nF (E − e'n − &n )] :

(79)

While the derivation given above is heuristic, a microscopic derivation is given in Section 9.3 of [65]. In Appendix B.1, it will be shown that Eq. (79) is the limiting case of the full

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quantum transport theory based on nonequilibrium Green functions in the limit 2T1 7. The same approach has been used for tunneling between neighboring two-dimensional electron gases [117,118]. It should be noted that Eq. (79) only holds if the correlations between scattering events in diDerent wells are not signi;cant; otherwise, disorder vertex corrections must be taken into account [119]. &;  An important task is the determination of the matrix elements Hm; k;n; k . One possibility is to start with eigenfunctions of single quantum wells and consider the overlap between those functions obtained for diDerent wells m; n. This is the procedure suggested by Bardeen [68] which essentially has been applied in [66,67] for superlattice transport. Another possibility is to start from the Wannier states (see Section 2.3). If a constant electric ;eld is applied to the superlattice structure, the electric potential reads 'n = −eFd and the matrix elements can ren be obtained directly from Hˆ 1 in Eq. (24) or from Hˆ 1 in Eq. (28). In Section 3.3.3, it will be shown, by a comparison with experimental data, that the latter Hamiltonian is more appropriate (as already suggested in [73]). These matrix elements are calculated for a perfect superlattice structure. Thus they conserve the parallel momentum k. In addition, scattering processes at impurities, phonons, or interface Juctuations may cause transitions between diDerent wells. The respective matrix elements can be obtained from the respective scattering potential and the Wannier functions as well. Examples will be given in Section 3.3.3, where it is shown that these processes give a background current while the peaks in the current–;eld relation are ren typically dominated by the k-conserving terms of Hˆ 1 . An important feature of Eq. (79) is the fact that the current is driven by the diDerence of the electrochemical potential in both wells. This is in contrast to the ;ndings of [73] (based on density matrix theory) where the current is driven by the diDerence fm& (k) − fn (k ) where fn (k) = nF (E  + Ek − &n ) is the occupation of the state (n; ; k). In the latter case, the &;  current vanishes for equivalent levels (& = ) if the matrix element Hm; k;n; k is diagonal in k. In Appendix A, it will be shown how the factor [nF (E − e'm − &m ) − nF (E − e'n − &n )] is recovered from density matrix theory. 3.3.2. Evaluation for constant broadening Here we want to derive simple expressions for the current under the assumption of a constant broadening for the electronic states. We assume that only the lowest level & = a is occupied (i.e., &m ; kB T E b ). First, we consider nearest-neighbor tunneling with k-conserving matrix elements. Then Eq. (79) can be rewritten as follows:  e ; a 2 a→ Jm→m+1 = |Hm+1; | dE %am (E)Am+1 (E; Fm ) m ˝ ×[nF (E − e'm − &m ) − nF (E − e'm+1 − &m+1 )] (80) with Am+1 (E; F)

∞

=

0

dEk Aam (k; E)Am+1 (k; E) ∞ : a 0 dEk Am (k; E)

(81)

Here the eDective ;eld F = ('m − 'm+1 )=d has been introduced and the density of states %am (E) from Eq. (78) has been applied. Let us assume a constant self-energy < ret (k; E) = −i7 =2 in

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Eq. (81) for the sake of simplicity. Then the spectral functions become Lorentzians Am (k; E) = 7 =[(E − Ek − E  − e'm )2 + (7 =2)2 ]. As %am (E) is essentially zero for energies below E a we may restrict ourselves to E & E a in the evaluation of Eq. (81). For these energies the function Am+1 (k; E) takes its maximum at Ek ¿ eFd + E a − E  which is larger than zero provided we restrict ourselves to eFd ¿ E  − E a (this is always the case for the a → a resonance and eFd ¿ 0). In this case, the integrand of Eq. (81) does not take large values for Ek ¡ 0 and it is justi;ed to extend the lower limit of integration to −∞. A straightforward evaluation using the calculus of residues yields Am+1  =

7; eD (E  − E a − eFd)2 + (7; eD =2)2

with 7; eD = 7a + 7

(82)

which depends only on F. Numerical evaluations indicate that these simpli;cations are quite good for ;elds above the resonance, i.e. eFd ¿ E  − E a . In the case of  = a and eFd ¡ E  − E a the choice 7; eD = 7 may be better. Note that this simple model with a constant self-energy cannot be used in the calculation of the density of states (78) as the integral for the electron density (77) diverges. Therefore, we use the free-electron density of states %m (E) = %0 -(E − E a − e'm ) in the following. With these simpli;cations Eq. (80) becomes ; a 2 |Hm+1; 7; eD m| a→ n (eFd; nm ; nm+1 ) (83) Jm→m+1 = e ˝ (E  − E a − eFd)2 + (7; eD =2)2 eD with the eDective electron density (here E˜ = E − e'm )  ∞ d E˜ %0 [nF (E˜ − &m ) − nF (E˜ − &m+1 + eFd)] ; (84) neD (F; nm ; nm+1 ) = Ea

which describes the diDerence of occupation in both wells. The electron density is related to the chemical potential via     ∞ &m − E a d E˜ %0 nF (E˜ − &m ) = %0 kB T log 1 + exp : (85) nm = kB T Ea Here we assumed that only the lowest level is occupied. The inclusion of higher levels is straightforward, but leads to more complicated expressions. Inserting into Eq. (84) yields neD (F; nm ; nm+1 ) = nm − %0 kB T log[(enm+1 =%0 kB T − 1)e−eFd=kB T + 1] : In the nondegenerate limit we have nm+1 %0 kB T and Eq. (86) is further simpli;ed as   eFd : neD (F; nm ; nm+1 ) = nm − nm+1 exp − kB T

(86) (87)

It is interesting to note that the total current can be written as a discrete version of the drift-diDusion model in this case: nm nm+1 − nm Jm→m+1 = e v(F) − eD(F) (88) d d2 with the velocity   
d ; a 7; eD eFd 2 |H | 1 − exp − (89) v(F) = ˝ m+1; m (E  − E a − eFd)2 + (7; eD =2)2 kB T 

A. Wacker / Physics Reports 357 (2002) 1–111

and the diDusion coeMcient  
d2 ; a 7; eD eFd 2 |H D(F) = | exp − ˝ m+1; m (E  − E a − eFd)2 + (7; eD =2)2 kB T 

33

(90)

which satis;es the Einstein relation D(F) = v (F)kB T=e for F = 0. Remember that these expressions hold only for eFd ¿ 0. For eFd ¡ 0, the expressions for the opposite direction can be applied. In the last part of this subsection some explicit results for the current a → a are given. a; a a Here we assume equal densities nm = nm+1 = n in the wells and set Hm+1; m = T1 according to Eq. (24). In the degenerate limit (n%0 kB T ) we ;nd neD =n−(n−%0 eFd)-(n−%0 eFd). For eFd ¡ n=%0 we obtain  |T1a |2 27a eFd n%0 kB T; (91) for JST (F) = e%0 n ¿ %0 eFd : ˝ (eFd)2 + (7a )2 In the nondegenerate limit, we obtain for eFdkB T  en |T1a |2 27a eFd n%0 kB T; (92) for JST (F) = 2 a 2 k kB T ˝ (eFd) + (7 ) B T eFd : In both cases, the ;eld dependence is given by the Esaki–Tsu result (34) with 7a = ˝=,. 3.3.3. Results A variety of calculations for diDerent superlattice structures have been performed within this model. These calculations consist of the following steps: (i) Calculation of the miniband structure E  (q) and the associated wave functions ’q (z) according to Section 2.1.  (ii) Evaluation of the Wannier-functions and respective couplings Th ; R&; h , see Section 2.3. scatt (iii) Evaluation of (intrawell) scattering matrix elements )n; k |Hˆ |)n; k  for the dominant scattering processes. For doped superlattices ionized impurity scattering dominates and the respective calculations including screening are presented in [120,35]. Scattering processes at interface roughness [35] and phonons have been considered as well. (iv) Calculation of the self-energies
34

A. Wacker / Physics Reports 357 (2002) 1–111

Fig. 11. Current–;eld relation for the superlattice studied in [123] for constant electron density nm = nm+1 = ND . The dashed line shows the current from resonant transitions with momentum independent matrix elements, while both resonant and non-resonant currents contribute to the full line. An electron temperature Te = 20 K was used in the calculation. Fig. 12. Current–;eld relation for the superlattice studied in [31] for constant electron density nm = nm+1 = ND . The bias was taken to be NFd for the theoretical results, where N = 10 is the number of quantum wells (from [120]).

(The respective matrix elements for impurity scattering are negligible.) These scattering events represent an additional current channel in Eq. (79) yielding a background current which dominates between the resonances, but is negligible compared to the resonant currents. The height of the a → b peak (Imax = 1:75 mA) is in good agreement with the experimental data exhibiting Imax = 1:45 mA (Fig. 6 of [124]). The low-;eld peak is not resolved experimentally due to domain formation yielding a current of about 0:076 mA. Results for the superlattice studied in [31] (15 nm wide GaAs wells, 5 nm Al0:3 Ga0:7 As barriers, doping density ND = 6 × 109 =cm2 , cross section A = 8 m2 ) are shown in Fig. 12. The general shape is like that of the results from the sample mentioned before. While the latter (highly doped) sample did not exhibit a strong temperature dependence, the situation is different for the low-doped sample considered here. For low electron temperatures Te 6 15 K the electrons are located in the impurity band (about 10 meV below the free electron states). Therefore, a current exhibits a peak at eFhigh d ≈ 10 meV when these electrons can tunnel into the free electron states of the next well. For higher temperatures the electrons occupy the free electron states and the maximum occurs at eFlow d ≈ 7a ≈ 2 meV as suggested by Eq. (92). Due to the same eDect the peak at eFd ≈ 50 meV due to a → b tunneling shifts with temperature. The experimental data (taken at a lattice temperature of 4 K) are shown for comparison. While the low-;eld conductance is in good agreement with the Te = 4 K calculations, close to the ;rst maximum the agreement becomes better for the Te = 35 K curve, which can be caused by electron heating. The heights of both maxima show excellent agreement between theory and experiment. The diDerence in the position of the second maxima may be caused by an additional voltage drop in the contact, which is not taken into account in the theory, where the ;eld was just multiplied by the length of the sample. Finally, the saw-tooth shape of the experimental current–voltage characteristic is due to the formation of electric ;eld domains as discussed in Section 5.

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35

Fig. 13. Current–;eld relation for the superlattice studied in [125]. The bias was taken to be NFd for the theoretical results, where N = 30 is the number of quantum wells. Full line: result from sequential tunneling with renormalized matrix elements. Dashed line: result from sequential tunneling with bare matrix elements. Dots: experimental data (courtesy of Yu.A. Mityagin). Fig. 14. Current–;eld relation for the superlattice studied in [126] for constant electron density nm = ND . Transitions between nearest and next nearest wells have been taken into account and the number of levels was max = 2 (full line), max = 3 (dashed line), and max = 4 (dotted line). T = 20 K was used in the calculation.

In Fig. 13 results are shown for the superlattice structure from Ref. [125] (25 nm wide GaAs wells, 10 nm Al0:3 Ga0:7 As barriers, doping density ND = 1:75 × 1010 =cm2 ). Due to the large well-width the level separation is small and several resonances E a → E  can be observed with increasing ;eld. The calculations (for T = 4 K and within the approximation (83) applying phenomenological broadenings 7 =4 meV for all levels) have been performed with and without taking into account the renormalization of the matrix elements according to Eq. (29) including the 6 lowest levels. Fig. 13 shows that the result with renormalized matrix elements (full line), see Eq. (28), is in good agreement with the experimental result but exhibits higher peak currents. This may be due to an overestimation of the couplings in the calculation. E.g. assuming barriers of 10:6 nm, the current drops by a factor of 2. An increase of the Al-content in the barriers would give a similar trend. The results with the bare matrix elements (dashed line), see Eq. (24), deviates strongly from the experimental result. In particular, the peak currents do not increase for resonances at higher ;elds. This shows that the renormalization procedure is essential if higher resonances are considered. 3.3.4. Tunneling over several barriers Up to now the discussion was restricted to next-neighbor coupling, which is described by & matrix elements H˜ 1 in Eq. (28). The extension to tunneling over h barriers can be treated & analogously taking into account the matrix element H˜ h . The discussion of Section 3.3.2 can be performed in the same way applying the bias drop heFd. Thus, we expect resonances at ;eld  a strengths eFd = (E˜ − E˜ )=h. The total current is then given by JST =

hmax
max
h=1 =1

a→ hJ0→h ;

(93)

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A. Wacker / Physics Reports 357 (2002) 1–111

a→ are evaluated according to Eq. (79). Here it is where the individual current densities J0→h assumed that only the lowest level a is occupied. For the samples discussed in the last section, as well as for most other samples considered, the respective currents for h ¿ 1 are negligible. This is diDerent for the sample discussed in [126] (5 nm wide GaAs wells, 8 nm Al0:29 Ga0:71 As barriers, doping density ND =2:25 × 1011 =cm2 ), where the second miniband is located around the conduction band of the barrier and the subsequent minibands  ¿ 3 resemble free particle states. Results of the calculation with hmax =2, i.e. taking into account tunneling to next-nearest-neighbor wells, are shown in Fig. 14. If the calculation is restricted to the two lowest levels (max = 2) the current–;eld relation resembles the ;ndings of Fig. 11. There is a peak at low ;elds due to a → a tunneling and a peak at eFd ≈ E˜ b − E˜ a = 0:177 meV due to tunneling a → b(1) into the nearest-neighbor well. The matrix element H˜ ab 2 is small, thus no transitions a → b(2) to the next-nearest-neighbor well can be seen. This changes completely if the third level (max = 3) is taken into account for the renormalization of the energy levels and couplings. First, the strong coupling to the third level diminishes the value of E˜ b by 15 meV close to the a → b(1) resonance. Thus, the position of this resonance is shifted to eFd = 162 meV where the new resonance condition is ful;lled. Secondly, a new peak arises at eFd = 85 meV ≈ (E˜ b − E˜ a )=2 due to next-nearest-well tunneling a → b(2) (remember that the renormalized level energies E˜ b are ;eld dependent and thus the local ;eld must be taken into account at each comparison). The reason is the strong admixture of Rbc 2 (which is quite large for the superlattice structure considered) in the renormalization of the matrix element H˜ ab 2 due to Eq. (29). For the same reason, a third peak appears at eFd = 136 meV ≈ (E˜ c − E˜ a )=2 due to a → c(2) tunneling. If the fourth level (max = 4) is taken into account as well, the result hardly changes, thus providing con;dence in the results. These ;ndings are in agreement with the experiments [126] where a strong increase of the current was observed at ;eld strengths of eFd ≈ 80 meV and the current density becomes larger than 0:15 kA=cm2 . Nevertheless, no current peak has been observed so far in this ;eld region. Tunneling over more than one barrier has also been observed experimentally in [62,127]. Current peaks at eFd = (E b − E a )=h corresponding to resonances between hth next neighbors have also been found in the calculation by Zhao and Hone [128]. The height of these peaks was quite small, probably due to the neglect of interband couplings Rab h in their calculation. These ;ndings show that next-nearest-neighbor tunneling is possible in superlattice structures. Nevertheless, the quantitative description is still an open issue. The inclusion of results from Zener tunneling [129] may be helpful in future research here.

3.4. Comparison of the approaches Let us now compare the results from the diDerent approaches, namely, miniband transport (MBT), sequential tunneling (ST), and Wannier–Stark hopping (WSH). MBT-ST: Comparing Figs. 8 and 11 one notices that the global behavior with linear increase of the current for low ;elds and a maximum at moderate ;elds is in qualitative agreement for the MBT and ST approach. While the current scales with the square of the coupling for ST, the Esaki–Tsu drift velocity is proportional to T1 . This discrepancy is resolved if either the temperature or the electron density is high and a |T1 |2 dependence of the current density is

A. Wacker / Physics Reports 357 (2002) 1–111

37

recovered for MBT as well, see Eqs. (46) and (51). Comparing these results with Eqs. (91) and (92), we ;nd that the simpli;ed expression become identical for MBT and ST if either kB T or n=%0 are large with respect to both 2T1a and eFd. This explains the 1=T dependence of the current density observed experimentally in [98,99] for superlattices exhibiting a rather small coupling T1a ≈ 1 meV. As the experiments are performed at T ¿ 77 K, estimations (51) and (92) hold simultaneously and the ;ndings cannot be taken as a manifestation of miniband transport. ST-WSH: Both approaches exhibit negative diDerential conductivity for high electric ;elds. Let us restrict ourselves to the a → a resonance and consider a superlattice with nearest-neighbor coupling T1a . For large electric ;elds eFd − 2 Im {
−2 Im{<1a ret (k; Ek + E a )}

(eFd)2

−2 Im{<0a ret (k; Ek + E a − eFd)}

(eFd)2

2 (E − Ek − E a + eFd) :

Within the Born approximation for the scattering:
 a ret a a 2 a ˆ scatt |)m; −Im{<m (k; E)} = |)m; k | (E − E − Ek  + meFd) : k |H ˝ 

(94)

(95)

k

Applying approximation (68) for eFd2T1a expression (79) for the current in the ST model becomes, after several lines of algebra,    eFd 2e
a→a J0→1 = R0; k→1; k nF (Ek + Ea − &) 1 − exp − : (96) ˝A  kB T k;k

This is the dominating term of the current for Wannier–Stark hopping (65). Thus, the expressions of ST and WSH become identical within the limit of eFd7 and eFd2|T1 |. This is just the overlapping region between the ranges of validity of both approaches as depicted in Fig. 7. The transition between WSH and MBT is even more diMcult. MBT typically exhibits a 1=F behavior for eFd7 as predicted for the Esaki–Tsu relation, while 1=F r with various exponents r ¿ 2 is found for eFd2|T1a | from the WSH model, see Section 3.2. As mentioned there, a 1=F behavior can be recovered from the WSH approach by summing all contributions h in Eq. (64) for eFd2|T1a |. In [110,113] it is shown that in the ;eld region 7eFd2|T1a | the results of both approaches agree fairly well. Again, this agrees with the joint range of validity depicted in Fig. 7. 4. Quantum transport In semiconductor superlattices the miniband width /, the potential drop per period eFd, and the scattering induced broadening 7 are often of comparable magnitudes. This requires the application of a consistent quantum transport theory combining scattering and the quantum

38

A. Wacker / Physics Reports 357 (2002) 1–111

mechanical temporal evolution. DiDerent formulations applying nonequilibrium Green functions [130], density matrix theory [131], the master-equation approach [132], or Wigner functions [133] have recently been used to tackle this general problem for a variety of diDerent model structures. Here, the formalism of nonequilibrium Green functions is applied to study electrical transport in superlattices. This approach allows for a systematic study of both quantum eDects and scattering to arbitrary order of perturbation theory. Although the calculations involved are quite tedious (as well as the acquaintance with this method) such calculations are of importance for two purposes: On the one hand, it is possible to derive simpler expressions like those studied in the preceding section from a general theory, thus, shedding light into the question of applicability. On the other hand, there are situations where no simple theory exists and thus one has to pay the price to work with a more elaborate formalism. A variety of diDerent quantum transport calculations for semiconductor superlattices have been reported in the literature: In [134] an analysis within the density matrix theory has been presented, which was simpli;ed to diDerent approaches for low, medium, and high ;eld. The same method was applied to study transport in a perpendicular magnetic ;eld [135]. A similar approach was performed in [136], where the quantum kinetic approach was solved in the limit of Wannier–Stark hopping and the nature of phonon resonances were analyzed. The formation of Landau levels in a longitudinal magnetic ;eld causes additional resonances [137]. A transport model based on nonequilibrium Greens function [138] has been proposed as well, although explicit calculations could only be performed in the high-temperature limit and within hopping between next-neighbor Wannier–Stark states there. This section is organized as follows. At ;rst, the general formalism of nonequilibrium Green functions for stationary transport is brieJy reviewed in a form which can be applied to a variety of devices. The special notation to consider transport in homogeneous semiconductor superlattices as well as the approximations used are described in the second subsection. In the third subsection, the standard approaches (miniband transport, Wannier–Stark hopping, and sequential tunneling as discussed in Section 3) will be explicitly derived as limiting cases of the quantum transport model. This proves the regions of validity given in Fig. 7. Finally, in the fourth subsection results are presented for diDerent samples. The results from the self-consistent quantum transport model will be compared with simpler calculations within the standard approaches applying identical sample parameters. This will demonstrate that the standard approaches work quantitatively well in their respective range of applicability. The reader who is less interested in the theoretical concept and underlying equations may skip Sections 4.1– 4.3 and continue with the results in Section 4.4. 4.1. Nonequilibrium Green functions applied to stationary transport In this subsection the underlying theory of nonequilibrium Green functions is brieJy reviewed. The notation of [39,65] is followed here and the reader is referred to these textbooks for a detailed study as well as for proofs of several properties addressed here. We consider a set of one-particle basis states | and a (t); a† (t) are the corresponding annihilation and creation operators. The time dependence stems from the Heisenberg picture. Most

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39

physical one-particle observables can be expressed by the one-particle density matrix %; $ (t) = a† (t)a$ (t) = Tr {%a ˆ † (t)a$ (t)}

(97)

which is the corresponding quantum mechanical expectation value with the density operator %. ˆ In particular, the occupation of the state | is given by %;  (t). The task of any many-particle quantum theory is the evaluation of %$ (t) in the presence of a Hamiltonian Hˆ = Hˆ 0 + Uˆ + Hˆ scatt ; (98) where Hˆ 0 =




E a† (t)a (t)

is diagonal in the basis |,
Uˆ = U; $ (t)a† (t)a$ (t)

(99)

(100)

;$

describes an additional potential term, and Hˆ scatt refers to interactions with phonons, random impurity potentials (which are treated within impurity averaging), or interactions between the particles. Within density matrix theory the temporal evolution of %; $ (t) is studied directly by applying Heisenberg’s equation of motion for the product a† (t)a$ (t). E.g., the time dependence of the occupation %;  (t) is given by d † i a (t)a (t) = [Hˆ ; a† (t)a (t)] dt ˝
i = [U$;  a†$ (t)a (t) − U; $ a† (t)a$ (t) + [Hˆ scatt ; a† (t)a (t)]] ; (101) ˝ $

where [a; b] = ab − ba denotes the commutator. In order to satisfy the equation of continuity, the particle currents j(t) between the basic states have to be identi;ed by 2 $→ j $→ (t) = Re{iU$;  a†$ (t)a (t)} + jscatt (t) ; (102) ˝ which satis;es j $→ (t) = −j →$ (t) as each part of Hˆ is a Hermitian operator. The scattering $→ induced current jscatt (t) can be determined once Hˆ scatt is speci;ed. This term typically contains higher-order density matrices like a†$ (t)a†$ (t)a (t)a (t) in the case of electron–electron scattering. Thus, there is no closed set of dynamical equations for %; $ (t) and the dynamical evolution generates a hierarchy of many-particle density matrices, which has to be closed by approximations, see, e.g., [139,140] as well as references cited therein. A conceptually diDerent approach to many-particle physics constitutes the theory of nonequilibrium Green functions which has been developed by KadanoD and Baym [141] and independently by Keldysh [142]. In this theory, the time dependence of a† (t1 ) and a$ (t2 ) is considered separately; thus, two diDerent times appear in the calculation. The corresponding generalization of the density matrix is the correlation function (or ‘lesser’ Green function) G¡1 ;2 (t1 ; t2 ) = ia†2 (t2 )a1 (t1 )

(103)

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A. Wacker / Physics Reports 357 (2002) 1–111

which describes the occupation of the states together with the respective correlations both in time and state index. Note the unusual order of indices, which will be helpful in later stages of the theory. The notation follows [39,65] here. Sometimes (e.g., [141,143]) the factor i is dropped so that G ¡ agrees directly with the density matrix for equal times. Next to this correlation function the retarded and advanced Green functions are de;ned by Gret1 ;2 (t1 ; t2 ) = −i-(t1 − t2 ){a1 (t1 ); a†2 (t2 )} ;

(104)

Gadv (t1 ; t2 ) = i-(t2 − t1 ){a1 (t1 ); a†2 (t2 )} = [Gret2 ;1 (t2 ; t1 )]∗ ; 1 ;2

(105)

respectively, where {a; b} =ab+ba denotes the anticommutator which is appropriate for fermion operators a considered here. These functions describe the response of the system at time t1 in state 1 which is excited at time t2 in state 2 . 4.1.1. Temporal evolution The time dependence of these Green functions is given by the following set equations:  
9 ¡ i˝ − E1 G¡1 ;2 (t1 ; t2 ) − U1 ;$ (t1 )G$; 2 (t1 ; t2 ) 9t1 $
 dt ¡ ¡ adv = [
which are derived in Section 5 of [39]. The same result is obtained from the matrix equations (3:7:5) and (3:7:6) of [65] if the de;nitions of the retarded and advanced Green functions are inserted. While the equations for G ret and G adv exhibit a (t1 − t2 ) inhomogeneity typical for Green functions, this is not the case for G ¡ which is, strictly speaking, not a Green function (although this term is often used). The self-energies < describe the inJuence of scattering (compare the simpli;ed description in Section 3.3.1). They can be expressed by functionals of the Green

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41

functions which depend on the approximation used, such as the self-consistent Born approximation. Here it is crucial to pay attention to the fact that
¡ (t; t  ) can be considered as an in-scattering term, which which has a nice interpretation: <$; $ creates a correlated one-particle excitation at times (t; t  ) as a result of a scattering event. The retarded and advanced Green functions provide the action of this excitation at the later times t1 and t2 , at which the correlation function G ¡ is observed. The relation Eq. (110) is a particular solution of the diDerential equations (106) and (107). The general solution contains a further term (proportional to the free evolution of G ¡ without scattering) to satisfy initial conditions, see, e.g., Eq. (5:11) of [39]. Typically, the contribution of these terms decays in time if scattering is present, so that Eq. (110) holds in the long time limit. If we consider a stationary state without any time dependence of the external potential U , all functions depend only on the time diDerence t1 − t2 and it is convenient to work in Fourier space de;ned by  1 F1 ;2 (E) = dt eiEt=˝ F1 ;2 (t + t2 ; t2 ) ; (111) ˝  1 F1 ;2 (t1 ; t2 ) = dE e−iE(t1 −t2 )=˝ F1 ;2 (E) (112) 2

both for self-energies and Green functions. 7 Then the following relations hold: adv {G;ret$ (E)}∗ = G$;  (E)

Eqs. (106) – (109) yield (E − E1 )G¡1 ;2 (E) −

and


$

(E − E2 )G¡1 ;2 (E) −

¡ U1 ;$ G$; 2 (E) =


$

¡ ∗ G;¡$ (E) = −{G$;  (E)} :

G¡1 ;$ (E)U$; 2 =


$

¡ ¡ adv [<ret1 ;$ (E)G$; 2 (E) + <1 ;$ (E)G$; 2 (E)] ; (114)


$

(113)

¡ ¡ adv [Gret1 ;$ (E)<$; 2 (E) + G1 ;$ (E)<$; 2 (E)] ; (115)

7 DiDerent de;nitions have been suggested which produce gauge invariant equations when U is due to a combination of an electric and magnetic ;eld [144,39]. This is important for various approximations to treat slowly varying ;elds.

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A. Wacker / Physics Reports 357 (2002) 1–111

(E − E1 )Gret=adv (E) − 1 ;2


$

(E − E2 )Gret=adv (E) − 1 ;2


$

ret=adv U1 ;$ G$; 2 (E) = 1 ;2 +

Gret=adv (E)U$; 2 = 1 ;2 + 1 ;$


$

ret=adv <ret=adv (E)G$; 2 (E) ; 1 ;$


$

ret=adv Gret=adv (E)<$; 2 (E) : 1 ;$

(116) (117)

ret=adv If U$;  and <$; (E) are symmetric matrices, then Gret=adv (E) = Gret=adv (E) holds as well. This 1 ;2 2 ;1  can be shown by subtracting Eq. (117) from Eq. (116), where 1 and 2 are exchanged. In the same way, the Keldysh relation becomes
¡ adv G¡1 ;2 (E) = Gret1 ;$ (E)<$; (118) $ (E)G$ ; 2 (E) : $;$

A quite elementary derivation of Eqs. (116) and (118) is given in Section 8 of [38]. From G ¡ (E) the density matrix can be evaluated directly via  dE ¡ ¡ (119) %; $ = −iG$;  (t; t) = −i G (E) ; 2 $;  which provides us with the one-particle expectation values for most quantities of interest. 4.1.2. Self-energies The self-energies can be obtained from the usual diagrammatic rules which are derived in most textbooks on many-particle theory such as [65]. In the following, the results are given within the self-consistent Born-approximation. For impurity scattering, one considers an impurity potential V (˜r; {˜ri }) which depends on the locations ˜ri of the impurities. The respective Hamiltonian is given by
V$1 ;$2 ({˜ri })a†$1 a$2 (120) Hˆ imp = $1 ;$2

with V$1 ;$2 ({˜ri })= $1 |V (˜r; {˜ri })|$2 . For large systems the average V : : : V imp over all possible impurity con;gurations {˜ri } has to be taken (interface roughness can be treated in a similar way). Within the self-consistent Born-approximation only correlations between two scattering matrix elements are taken into account. Thus, one ;nds
<¡=ret=adv (E) = V1 ;$1 ({˜ri })V$2 ;2 ({˜ri })imp G$¡=ret=adv (E) : (121) ; 1 2 1 ;$2 $1 ;$2

For phonon scattering the respective Hamiltonian reads as  

˝!l (˜ p)b†p˜ ; l bp˜ ; l + M$1 ;$2 (˜ p; l)(bp˜ ; l + b†−˜p; l )a†$1 a$2  ; Hˆ phonon = p ˜ ;l

(122)

$1 ;$2

where bp˜ ; l ; b†p˜ ; l are the (bosonic) annihilation and creation operators of the phonon mode l (such as acoustic=optical or longitudinal=transverse) with wave vector p ˜ . Using Langreth rules (Section 4.3 of [39]), one obtains the retarded self-energy within the self-consistent

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43

Born approximation: 

dE1 ret ret <1 ;2 (E) = i M1 ;$1 (˜ p; l)M$2 ;2 (˜ p; l) (E − E1 )Dlret (˜ p; E1 ) [G 2 $1 ;$2 p ˜ ;l $1 ;$2 + G$ret1 ;$2 (E

− E1 )Dl¡ (˜ p; E1 ) + G$¡1 ;$2 (E − E1 )Dlret (˜ p; E1 )] ;

(123)

where Dl (˜ p; E1 ) refers to the phonon Green function. Now, we replace the phonon function by its unperturbed equilibrium values (Section 4.3 of [39] and Eq. (2:9:9) 8 of 1 1 Dlret 0 (˜ p; E1 ) = − ; + E1 − ˝!l (˜ p) + i0 E1 + ˝!l (˜ p) + i0+ 1 1 Dladv 0 (˜ p; E1 ) = − ; + E1 − ˝!l (˜ E1 + ˝!l (˜ p) − i0 p) − i0+ Dl¡ 0 (˜ p; E1 ) = −2i{nB (˝!l (˜ p)) (E1 − ˝!l (˜ p))

Green [65]).

p)) + 1] (E1 + ˝!l (˜ p))} ; +[nB (˝!l (˜ ¡ 0 ret 0 adv 0 i.e., Dl (˜ p; E1 ) = nB (E1 )[Dl (˜ p; E1 ) − Dl (˜ p; E1 )] where nB (E) = [exp(E=kB T ) − 1]−1 Bose distribution. As G$ret1 ;$2 (E − E1 ) has only poles for Im{E1 } ¿ 0, we ;nd  ∞ dE1 ret p; E1 ) = −i[G$ret1 ;$2 (E − ˝!l (˜ p)) − G$ret1 ;$2 (E + ˝!l (˜ p))] G$1 ;$2 (E − E1 )Dlret 0 (˜ 2 −∞

(126)

(124) (125)

is the

from the residua of the contour over the complex plane with Im{E1 } ¡ 0. Putting things together we obtain 

ret <1 ;2 (E) = M1 ;$1 (˜ p; l)M$2 ;2 (˜ p; l) [nB (˝!l (˜ p)) + 1]G$ret1 ;$2 (E − ˝!l (˜ p)) p ˜ ;l $1 ;$2

1 + nB (˝!l (˜ p))G$ret1 ;$2 (E + ˝!l (˜ p)) + G$¡1 ;$2 (E − ˝!l (˜ p)) 2  1 dE1 ¡ − G$¡1 ;$2 (E + ˝!l (˜ p)) + i (E − E1 ) G 2 2 $1 ;$2       1 1 −P × P : E1 − ˝!l (˜ E1 + ˝!l (˜ p) p)

(127)

The lesser self-energy reads as 

dE1 ¡ <¡1 ;2 (E) = i M1 ;$1 (˜ p; l)M$2 ;2 (˜ p; l) (E − E1 )Dl¡ (˜ p; E1 ) G 2 $1 ;$2 p ˜ ;l $1 ;$2

=





p ˜ ;l $1 ;$2

M1 ;$1 (˜ p; l)M$2 ;2 (˜ p; l)[nB (˝!l (˜ p))G$¡1 ;$2 (E − ˝!l (˜ p))

+ [nB (˝!l (˜ p)) + 1]G$¡1 ;$2 (E + ˝!l (˜ p))] ; 8

Note the sign error for D0adv in Eq. (2:9:9) of [65].

(128)

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A. Wacker / Physics Reports 357 (2002) 1–111

which describes the in-scattering from correlated states $1 ; $2 by phonon absorption as well as stimulated and spontaneous emission of phonons. The Coulomb interaction can be easily included the Hartree–Fock approximation, which  within ¡ provides an additional potential (depending on dE G (E)), see, e.g., Chapter 8 of Ref. [38]. Higher-order approximations (describing electron–electron scattering) are diMcult to implement. All these eDects have been neglected in this work. The combination of these functionals for the self-energies with Eqs. (116) and (118) for the Green functions provides a coupled set of equations which has to be solved self-consistently to obtain the functions G ¡ (E) for the stationary state. Afterwards, the physical quantities of interest can be evaluated by Eq. (119). 4.1.3. Thermal equilibrium In thermal equilibrium, the electron distribution is governed by the Fermi distribution. In the language of Green functions this can be written as G¡1 ;2 (E) = inF (E)A1 ;2 (E)

(129)

with the spectral function A1 ;2 (E) = i[Gret1 ;2 (E) − Gadv (E)] ; 1 ;2

(130)

which is derived, e.g., in Section 3.7 of [65]. As discussed before, the occupation of the state  is given by   1 1 ¡ %;  = dE G;  (E) = dE A;  (E)nF (E) : (131) 2i 2 Thus, A;  (E)=2 represents the energy resolved density of state . Here it is crucial to note the diDerence to the classical value of the occupation nF (E ), which is only recovered in the free-particle case U = < = 0 when A;  (E) = 2 (E − E ) holds. In contrast %;  and nF (E ) will in general diDer, if scattering induced broadening leads to a ;nite width of the spectral function. These eDects can be estimated assuming U =0 and <ret1 ;2 (E) ≈ −i-(E) 1 ;2 7=2 which mimics the fact that there are no scattering states below a band edge of E = 0. Then one ;nds from Eqs. (116) and (130) that 7 A1 ;2 (E) ≈ 1 ;2 -(E) (132) (E − E )2 + 72 =4 and in the limit E kB T one obtains CkB T 7 %;  ∼ (133) 2 (E )2 + 72 =4 for a nondegenerate distribution nF (E) ≈ Ce−E=kB T . Thus, the occupation of the high-energy states is larger than one would estimate from a semiclassical distribution nF (E ) ≈ Ce−E =kB T . Finally, it should be pointed out that the diDerent Green functions G ¡ ; G ret , and G adv are related to each other in thermal equilibrium which allows for a description in terms of a single Green function. Thus, the theory of equilibrium Green functions is signi;cantly simpler than its nonequilibrium counterpart discussed here.

A. Wacker / Physics Reports 357 (2002) 1–111

45

4.1.4. Spatial boundary conditions and contacts Although we are concerned with homogeneous in;nite systems in this section, some remarks concerning boundary conditions in real structures are appropriate. They are needed in the discussion of transmission through superlattices [145]. In order to solve the system of equations discussed above in a ;nite system, boundary conditions have to be speci;ed. Here two types can be distinguished. On the one hand, there are regions where the device is terminated by an insulating layer. Here it is appropriate to neglect states in these regions, as their energy is signi;cantly larger than the relevant energies in the device. A far more interesting point is the treatment of contacts, which act as a source or drain for the electric current. We separate the system into a central region with index C and a lead region with index L. The matrix G; $ can then be divided in submatrices of the type GCL , where the index  belongs to the central region and $ to one of the leads. Then the matrix equation (116) can be written in the form    ret    ret E − EC + i0+ 0 GCC (E) GCL (E) 1 0 · = ret ret 0 1 0 E − EL + i0+ GLC (E) GLL (E)    ret  ret UCC UCL GCC (E) GCL (E) · + ret ret ULC ULL GLC (E) GLL (E)  ret   ret  ret
CC (E)
ret (E) (E) GCC (E) GCL CL · + : (134) ret ret ret GLC
ret (E) GLL (E) LC (E)
LL (E) Now, let us assume that
CL =
LC = 0, i.e., there is no scattering between the lead regions and the central region. Then we can write: ret ret ret ret (E − EL + i0+ )GLC (E) = ULC · GCC (E) + ULL · GLC (E) +
ret LL (E) · GLC (E) : This equation is solved by

(135)

ret ret ret GLC (E) = GL0 (E) · ULC · GCC (E) ; ret (E) is the Green function of the lead satisfying the equation where GL0

(136)

ret ret ret (E − EL + i0+ ) · GL0 (E) = 1 + ULL · GL0 (E) +
ret (137) LL (E) · GL0 (E) : ret It is important to note that GL0 (E) is not exactly the Green function of the pure lead. In fact, ret ret
ret LL (E) has to be evaluated from the full Green function G (E) and not only from GL0 (E). This diDerence vanishes under the usual assumption that scattering is negligible in the leads. ret (E) can be written as Now, the part of Eq. (134) for GCC ret ret (E − EC + i0+ ) · GCC (E) = 1 + UCC · GCC (E)

ret ret (138) + [
ret CC (E) + UCL · GL0 (E) · ULC ] · GCC (E) : ret This is a closed equation for the matrix GCC (E) concerning the states inside the structure. ret (E) · U The term UCL · GL0 LC can be viewed as an additional self-energy, due to the transitions between the central region and the lead. In a similar way, Eq. (134) yields ret ret ret GLL (E) = GL0 (E) · [1 + ULC · GCL (E)] :

(139)

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A. Wacker / Physics Reports 357 (2002) 1–111

From Eq. (117), a similar structure as Eq. (134) can be obtained which provides: ret ret ret (E) = GCC (E) · UCL · GL0 (E) : GCL

(140)

Furthermore, all relations hold for the advanced Green functions in the same way. With these ingredients, Keldysh relation (118) can be rewritten as ¡ ret adv ret ¡ adv GCC (E) = GCC (E) ·
¡ CC (E) · GCC (E) + GCL (E) ·
LL (E) · GLC (E) ret ¡ adv (E) · [
¡ = GCC CC (E) + UCL · GL0 (E) · ULC ] · GCC (E) ;

(141)

¡ (E) de;ned by where GL0 ¡ ret adv (E) = GL0 (E) ·
¡ GL0 LL (E) · GL0 (E)

(142)

is not exactly the lesser Green function of the pure lead as
¡ LL (E) depends on the full Green ret function G (E). E.g., this reJects the fact, that the presence of a current through the central region will, in principle, aDect the electron distribution in the lead. Nevertheless, this reaction is typically negligible. Again, the complications vanish under the assumption, that scattering is neglected in the leads. In the same way, one obtains ¡ ret adv ret ¡ adv GCL (E) = GCC (E) ·
¡ CC (E) · GCL (E) + GCL (E) ·
LL (E) · GLL (E) ret adv adv (E) ·
¡ = GCC CC (E) · GCC (E) · UCL · GL0 (E) ret ret adv adv (E) · UCL · GL0 (E) ·
¡ + GCC LL (E) · GL0 (E)[1 + ULC · GCL (E)] ¡ adv ret ¡ (E) · UCL · GL0 (E) + GCC (E) · UCL · GL0 (E) ; = GCC

(143)

where Eqs. (139) – (141) have been subsequently applied (partially in the form for advanced functions). Let us consider a typical structure where the central region is connected to several independent leads ‘, which are translational invariant in their current direction. It is assumed that each lead ‘ is disorder-free so that the eigenstates can be separated into transverse and longitudinal parts, '‘Aq (˜r) = B‘A (r)’‘q (z), where z is the spatial coordinate in the direction of the central structure and r is a two-dimensional vector perpendicular to z. (A diDerent coordinate system is applied for each lead.) The index A numbers the transverse modes within a given lead and q denotes the behavior far away from the central region where ’‘q (z) ∼ eiqz is assumed. The corresponding matrices GL0 are diagonal to matrix elements G‘Aq (E) in this basis. The electric current from lead ‘ and mode A into the central region can be obtained from Eqs. (102) and (119).  2 dE

I‘A = 2(for spin)e Re{U‘Aq;  G;¡‘Aq (E)} ; (144) ˝ 2 q 

A. Wacker / Physics Reports 357 (2002) 1–111

where sum

47



runs over all states belonging to the central region. Eq. (143) provides:    
4e dE

 ¡ adv ret ¡ Re U‘Aq;  [G$ (E)U$; ‘Aq G‘Aq (E) + G$ (E)U$; ‘Aq G‘Aq (E)] I‘A =   ˝ 2 q  =



2e ˝



$

dE

¡ adv ¡ ret U$; ‘Aq U‘Aq;  [G$ (E)G‘Aq (E) − G$ (E)G‘Aq (E) 2 q $

ret ¡ adv ¡ + G$ (E)G‘Aq (E) − G$ (E)G‘Aq (E)]  2e dE ¡ ret adv = (E) + f‘A (E)i(GCC (E) − GCC (E))]} : Tr {CC (‘A; E) · [iGCC ˝ 2

(145)

In the second line, the real part was taken by adding the complex conjugated expression (with  and $ exchanged). In the last line, the de;nition
ret adv 7$ (‘A; E) = i U$; ‘Aq U‘Aq;  [G‘Aq (E) − G‘Aq (E)] (146) q

is introduced and it is assumed that the occupation of the modes A in the lead ‘ can be treated by a distribution function f‘A (E) with ¡ adv ret (E) = f‘A (E)[G‘Aq (q; E) − G‘Aq (q; E)] : G‘Aq

(147)

Eq. (145) has been derived in [146,38,39]. Together with Eqs. (138) and (141), one obtains a closed set of equations for the Green functions in the central regions which is a convenient starting point for the simulation of quantum devices, see. Eq. (145) can also be used as a starting point to derive the Landauer–BQuttiker formalism [147] which can be easily applied if < = 0 inside the central region [148]. Otherwise some complications arise as discussed in [145]. A generalization to take into account time-dependent phenomena is straightforward [149]. 4.2. Application to the superlattice structure While the discussion of nonequilibrium Green functions was quite general in the preceding subsection, the general formalism will now be applied to a superlattice structure, which is assumed to be in;nitely long. Therefore, we restrict ourselves to the central region here. We use the basis given by the products √ of Wannier states multiplied by plane waves in the direction parallel to the layers, )n (z)eik·r = A. Then the general states are given by | = |n; ; k. The Hamiltonian Hˆ 0 is given by Eq. (23) and Uˆ is given by Eqs. (24) and (25). For simplicity, we restrict ourselves to the lowest level &=a (and omit the respective indices) and nearest-neighbor coupling T1 in the following. Furthermore we set E a = 0. (The inclusion of higher levels is straightforward but tedious.) We assume that the superlattice is (after impurity averaging) spatially homogeneous in the x; y plane. Then the expectation values [am (k1 )]† an (k2 ) must vanish for k1 = k2 . Thus, the Green functions are diagonal in the wave vector k and can be written as Gn; m (k; t1 ; t2 ) in the following.

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A. Wacker / Physics Reports 357 (2002) 1–111

4.2.1. Basic equations The Wannier functions are essentially localized within a single quantum well. Therefore, the scattering matrix elements connecting states of diDerent quantum wells are small compared to those describing intrawell scattering. Thus, we restrict ourselves to scattering matrix elements Vmk; nk ({˜ri }); Mmk; nk (˜ p; l) which are diagonal in the well indices m; n in the following (no interwell scattering). In this case, the scattering-induced currents between diDerent wells vanish in Eq. (102) and the total electric current density well m to well m + 1 is given by  dE 2(for Spin)e
2 ¡ Jm→m+1 = (148) Re{T1 Gm+1; m (k; E)} ; A ˝ 2 k

where Eq. (119) has been applied. Similarly, the electron density in well m is given by  2(for Spin)
dE ¡ nm = (149) Im{Gm; m (k; E)} : A 2 k

We consider impurity and phonon scattering within the self-consistent Born approximation. Furthermore, we neglect correlations between the scattering matrix elements in diDerent wells. This means that Vmk; mk ({˜ri })Vnk ; nk ({˜ri })imp vanishes in Eq. (121) for n = m. This assumption is realistic for short-range potentials of random impurities but becomes problematic if interface roughness scattering is considered where signi;cant correlations between neighboring wells may occur [150,151]. For phonon scattering, this approximation makes sense if localized phonons such as interface phonons are considered which are diDerent in each well. Under this assumption, the self-energies become diagonal in the well index and can be written as

and in the same way for phonon scattering with phonon modes l; p ˜ we ;nd from Eqs. (127) and (128): 
ret; phonon 2
+ ˝!l (˜ p)) 1 1 + Gn;¡n (k ; E − ˝!l (˜ p)) − Gn;¡n (k ; E + ˝!l (˜ p)) 2 2        dE1 ¡  1 1 +i −P G (k ; E − E1 ) P 2 n; n E1 − ˝!l (˜ p) E1 + ˝!l (˜ p)

and



(151)

|Mnk; nk (˜ p; l)|2 [nB (˝!l (˜ p))Gn;¡n (k ; E − ˝!l (˜ p))

p ˜ ;l;k

p)) + 1]Gn;¡n (k ; E + ˝!l (˜ p))] : +[nB (˝!l (˜

(152)

A. Wacker / Physics Reports 357 (2002) 1–111

49

Then Eq. (116) becomes (E − Ek + eFdm1 − <mret1 (k; E))Gmret1 ;m2 (k; E) − T1 Gmret1 −1;m2 (k; E) − T1 Gmret1 +1;m2 (k; E) = m1 ;m2 and the Keldysh relation becomes
Gmret1 ;n (k; E)
(153) (154)

n

Together with the functionals for the self-energies, Eqs. (153) and (154) form a closed set of equations which will be solved in Section 4.4, where explicit results are presented. In the same way, the diDerence between Eqs. (114) and (115) gives ¡ ¡ ¡ ¡ ¡ (m − n)eFdGm; n (k; E) = T1 [Gm−1; n (k; E) + Gm+1; n (k; E) − Gm; n−1 (k; E) − Gm; n+1 (k; E)] ¡ ¡ adv + <mret (k; E)Gm; n (k; E) + <m (k; E)Gm; n (k; E) ret ¡ ¡ adv −Gm; (155) n (k; E)
which will be the starting point for the derivation of the miniband conduction and Wannier–Stark hopping model in Appendix B. 4.2.2. Constant scattering matrix elements For numerical calculations, the k-dependence of the self-energy is diMcult to handle, as a two-dimensional array of <(k; E) has to be evaluated and stored in each calculation cycle. Thus, the problem becomes much simpler if constant (i.e. momentum independent) scattering matrix elements are assumed. Then the self-energy for impurity scattering can be written as

%0 = AVnn ({˜rj })Vnn ({˜rj })imp 2

 0

∞

 dEk  Gn;¡=ret n (k ; E) ;

(156)

√ which does not depend on k. (Note that Gn;¡=ret only depends on |k| = 2Ek m= ˝ due to the n rotational symmetry in the (x; y)-plane.) As Gn;retn (k ; E) ∼ −1=Ek  for large Ek  , the respective integral exhibits a logarithmic divergence. This can be cured either by applying a ;nite cut-oD Ekmax or by adding 1=(Ek  + 7) in the integrand with an arbitrary but constant value 7 ¿ 0. Throughout this work, the total scattering rate of free-particle states is used here. This procedure adds a constant real term to the retarded and advanced self-energy, which eDectively renormalizes the energy scale to E + 12 AVnn ({˜rj })Vnn ({˜rj })imp %0 log(Ekmax =7 + 1) but does not change the physics, which only depends on energy diDerences. For free-particle Green functions Gn;retn (k ; E) = 1=(E − Ek  + neFd + i0+ ) the integral yields       %0 7 ret  
implying a scattering rate (for E ¿ − neFd) %0 AVnn ({˜rj })Vnn ({˜rj })imp 1 2 = − {
(158)

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A. Wacker / Physics Reports 357 (2002) 1–111

which will be used as the parameter for the impurity scattering strength in the subsequent calculations. 9 Phonon scattering is treated in the same way with a constant matrix element Mn (˜ p; l) = Ml . Here the scattering rate is de;ned via %0 AMl2 1 = ; (159) ,phonon; l ˝ which is the free-particle spontaneous phonon emission rate if the ;nal state is energetically available and Pauli blocking is negligible at low densities. 4.3. Solution for constant self-energy The nature of the retarded Green function determined by Eq. (153) can be analyzed by an analytical evaluation for the constant self-energy
Now, only the times 0 6 (t1 − t2 ) . 2˝=7 contribute to G ret due to the exponential decay. If eFd ¿ 7 the sine may oscillate over many periods with the average absolute value of 2= ≈ 12 . Then Eq. (162) gives 2|T1 | & eFd. If eFd ¡ 7, we may replace sin(x) ≈ x and Eq. (162) gives 2|T1 | & 7 at t1 − t2 = ˝=7. Thus, we conclude that the states are essentially delocalized if 2|T1 |7

and

2|T1 |eFd :

(163)

On the other hand, for 2|T1 |7 ret Gm; n

or

2|T1 |eFd ;

(164)

becomes small for m = n and the states are essentially localized. Furthermore, for eFd7 ;

(165)

the poles at E = −i7=2+jeFd are clearly resolved in the energy dependence of Eq. (160) which indicates the persistence of the Wannier–Stark ladder under scattering. The ranges (163) – (165) 9

Note that the product AVnn ({˜rj })Vnn ({˜rj }) imp typically does not depend on the sample area A.

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51

correspond to the regimes of validity for miniband transport, sequential tunneling and Wannier– Stark hopping, respectively, as given in Fig. 7. In Appendix B, it will be shown explicitly that the respective transport equations can be recovered from the model of nonequilibrium Green functions in these ranges. 4.4. Results Now, the results from explicit calculations of the full quantum transport model are presented. Let us ;rst summarize the underlying assumptions as applied successively in Section 4.2. • Stationary transport in a homogeneous electric ;eld is considered where only the lowest subband of the superlattice is taken into account. • The scattering matrix elements are assumed to be diagonal in the well index (no interwell scattering) and no correlations between scattering matrix elements belonging to diDerent wells exist. • The scattering is treated within the self-consistent Born approximation (thus neglecting eDects due to weak localization [152,153]) using momentum-independent matrix elements. (The latter assumption is equivalent to a localized scattering potential, which may be problematic for phonon scattering.) Within these assumptions, it is possible to solve the system of equations, Eqs. (150) – (154) self-consistently. Note that in all these calculations temperature only enters the occupation of the phonon modes, which are assumed to be in thermal equilibrium. The electronic distribution is calculated self-consistently without any assumption of (heated) equilibrium. Throughout this subsection we use the scattering time ,opt = 0:125 ps and the phonon energy ˝!opt =36 meV for optical phonon scattering, which is the dominating energy relaxation process in III–V materials. In order to guarantee energy relaxation for particle energies below 36 meV, acoustic phonon scattering has to be taken into √ account. We mimic these phonons by a second phonon with constant energy ˝!ac = ˝!opt ( 5 − 1)=10 ≈ 4:4498 meV. Here ˝!ac should be less than kB T (≈ 6:4 meV at 77 K), so that this mechanism can be eMcient close to thermal equilibrium. Furthermore, the ratio !ac =!opt was chosen to be irrational in order to avoid spurious resonances. The respective scattering time is chosen as ,ac = 5 ps. At ;rst, consider the strongly coupled superlattice studied experimentally in [154]. It consists of 100 periods with 3:45 nm GaAs wells and 0:96 nm AlAs barriers, yielding a coupling T1 = −20:5 meV. The doping density provides ND = 3:6 × 1010 =cm2 and an impurity scattering time ,imp =0:12 ps is applied, which can be estimated from the impurity scattering rate for this doping range [120]. Results for the lattice temperature T = 300 K (which only enters the phonon occupation number nB ) are shown in Fig. 15 (full line). One encounters the typical shape for superlattice transport. While the current peak is in good quantitative agreement with the experimental observation, the peak position is shifted signi;cantly. The agreement becomes excellent if a serial resistance of the order of 10 Z (for the experimental sample area) is included which may result from contacts, leads, or the substrate. Several simulations have been performed for a model superlattice with T1 = −5:075 meV; d = 5:1 nm, and ND = 5:1 × 109 =cm2 , which has been extensively studied by Rott [92] (see also Figs. 8 and 9 in Section 3). The rather low doping is taken into account by an impurity scattering time ,imp = 0:333 ps in agreement with semiclassical Monte-Carlo simulations. The

52

A. Wacker / Physics Reports 357 (2002) 1–111

Fig. 15. Current–voltage characteristic for the superlattice studied in [154]. Full line: result from quantum transport for N = 100 wells. Dots: experimental data (courtesy of E. Schomburg). Dashed line: theoretical result with a serial resistance of 7:2 × 10−6 Zcm2 =A (T = 300 K).

Fig. 16. Drift velocity versus ;eld for a superlattice with / = 20:3 meV; d = 5:1 nm; ND = 5:1 × 109 =cm2 , and ,imp = 0:333 ps. Full line: quantum transport. Dashed line: miniband transport (from [155]).

results for the drift velocity vdrift = J=(eND =d) for T = 77 K and T = 300 K are shown in Fig. 16 (full line). For comparison, Monte-Carlo simulations within the miniband transport model (Section 3.1) have been performed by S. Rott and A. Markus (Institut fQur Technische Physik der UniversitQat Erlangen) applying the same scattering matrix elements. For low and moderate ;eld strengths up to eFd ≈ 20 meV the relation obtained from miniband transport agrees extremely well with the full quantum transport result. At larger ;elds, eFd exceeds the miniband width and miniband transport is no longer valid as discussed in Section 4.3. In Fig. 17, the electron distribution functions (for T = 77 K) are depicted for various electric ;eld strengths. Comparison of the dashed and full lines for miniband conduction and quantum transport shows that the distribution functions agree very well for eFd . 20 meV in the energy range Ek ¡ 30 meV. For higher values of Ek the quantum mechanical distribution function is larger than its semiclassical counterpart. Here the occupation is quite small, so that the tail from the broadened spectral function ∼ 7=Ek2 becomes visible [156,157], in good agreement with the estimate of Eq. (133). In compensation, the quantum result is slightly smaller than its semiclassical counterpart for low values of Ek , as the total density has to be the same. Nevertheless, these eDects are small compared to the typical occupation numbers and one can conclude that the Boltzmann equation for miniband transport gives reliable results both for

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the current density and the electron distribution. Actually, the quality of agreement is stunning regarding the crude assumptions necessary for the derivation of the Boltzmann equation in Appendix B.2. The velocity–;eld relations from Fig. 16 can be directly compared with those in Fig. 8, where the correct matrix elements have been applied within the semiclassical miniband transport model. There are no qualitative diDerences, so that one may conclude that the assumption of constant matrix elements as well as the arti;cial acoustic phonons do not cause unphysical results. Let us now study the details in the velocity–;eld relation through a glance at the respective distribution functions. At very low ;elds, one encounters a linear increase of the drift velocity with the electric ;eld. Here the distribution function resembles a thermal distribution with the lattice temperature (dotted lines for eFd = 0:03 meV). The slight shift can be easily treated within linear response, yielding a ;eld-independent mobility. For higher ;elds (eFd = 0:3 meV), one encounters a sublinear increase of the drift velocity. Here signi;cant heating occurs, which can be seen from the respective distribution function in Fig. 17. The distribution function resemble a shifted Fermi distribution with an electron temperature Te ≈ 140 K for Ek . ˝!opt . For higher energies optical phonon scattering is still eMcient and one encounters a steeper decrease of f(Ek ) with energy. This shows that the concept of an electron temperature makes sense in this range both for quantum and semiclassical transport even without electron–electron scattering. For electric ;elds close to the current maximum (eFd = 2 meV), the distribution functions strongly deviates form any thermal or heated distribution function. In the semiclassical picture this can be viewed as a result of frequent Bragg-scattering where particles reach the states with q = =d. If the electric ;eld becomes even stronger, the electrons may traverse the q-Brillouin zone several times without scattering for eFd˝=, in the semiclassical miniband picture. This leads to a Jat electron distribution in q as can be observed for eFd=50 meV in Fig. 17. The same holds for the quantum distribution function although the reason is diDerent. In this ;eld range (2|T1 |eFd), the Wannier–Stark states are essentially localized to a single well. Thus, a semiclassical occupation of the Wannier–State creates a Jat distribution in q-space as well. While both approaches explain the Jat distribution in q-space, signi;cant diDerences in scattering arise. The electron running through the q states exhibits diDerent energies E(q) during passage. This provides diDerent selection rules for scattering compared to the situation of a ;xed energy for Wannier–Stark states. Therefore, the f(Ek )-distribution calculated by the quantum transport model exhibits pronounced features on the energy scales eFd and ˝!opt as well as the diDerence eFd − ˝!opt because at these energies new scattering channels appear. If these scales match, the phonon resonance at eFd = ˝!opt appears in the velocity–;eld characteristics, see Fig. 16. In contrast, the semiclassical result exhibits a rather Jat f(Ek ) distribution and no phonon resonance can be observed. The situation changes for small coupling and strong scattering, when 7 ¿ 2|T1 | holds. Fig. 18 shows the velocity ;eld relations for T1 = −1 meV and ,imp = 0:0666 ps. Here the semiclassical miniband transport calculation strongly deviates from the full quantum result both in the low-;eld and in the high-;eld region. Nevertheless, the agreement gets better for higher temperatures. In Fig. 19, the quantum transport calculations are compared with results from the models of Wannier–Stark hopping and sequential tunneling. In all calculations identical scattering matrix

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Fig. 17. Distribution functions f(q; k) and f(Ek ) = d=(2) dq f(q; k) of the carriers for diDerent ;eld strengths for the superlattice studied in Fig. 16 at T = 77 K. Full line: diagonal elements of the density matrix %(q; k) as calculated from the quantum transport model. Dashed line: semiclassical distribution function as calculated from the Boltzmann equation for miniband transport. Dotted line: thermal distribution nF (Eq + Ek − &) applying an electron temperature Te = 77 K for eFd = 0:03 meV and Te = 140 K for eFd = 0:3 meV.

elements are used. The calculations for Wannier–Stark hopping have been performed by S. Rott and A. Markus (Institut fQur Technische Physik der UniversitQat Erlangen) using self-consistent distribution functions, see Section 3.2 for details. One ;nds that the Wannier–Stark hopping model provides good results (including the phonon resonance) in the high-;eld region where eFd7 (7 ≈ 7 and 15 meV for the left and right superlattice, respectively, for energies at which phonon emission is possible). The agreement deteriorates signi;cantly if the simple version of Wannier–Stark hopping without self-consistency is applied.

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Fig. 18. Drift velocity versus ;eld for a superlattice with T1 = 1 meV; d = 5:1 nm; ND = 5:1 × 109 =cm2 , and ,imp = 0:0666 ps. Full line: quantum transport. Dashed line: miniband transport.

Fig. 19. Drift velocity versus ;eld for a superlattice with d=5:1 nm; ND =5:1×109 =cm2 ; T1 =5:075 meV; ,imp =0:33 ps (left) as well as T1 = 1 meV, ,imp = 0:0666 ps (right). Full line: Quantum transport. Dashed line: sequential tunneling. Dotted line: self-consistent Wannier–Stark hopping.

The sequential tunneling model clearly fails at low and moderate ;eld strengths for the strongly coupled superlattice (left part of Fig. 19) as 2|T1 | ¿ 7. In the high-;eld region eFd2|T1 | it provides valid results and the phonon resonance can be observed within the sequential tunneling model. For a weakly coupled superlattice (right part of Fig. 19), the sequential tunneling model should be applicable and the low-;eld conductance is in good agreement with the full quantum transport result. Signi;cant deviations occur at intermediate ;elds because electron heating is not included in the sequential tunneling model. As the current density drops with electron temperature (see Section 3.3.2), the drift velocities are too high, if heating is neglected. These heating eDects have been recently taken into account within the sequential tunneling model [158,159]. In Fig. 20, a comparison between quantum transport and sequential tunneling is shown for the weakly coupled superlattice studied in [123,124], see also Section 3.3.3. Excellent agreement is found between both approaches and the results agree well with those presented in Fig. 11 when realistic impurity scattering matrix elements are applied but phonon scattering is neglected. The observations presented here correspond to the boundaries of validity given in Fig. 7, which can be considered as a reliable guide. Furthermore, the results indicate that the models of miniband transport and sequential tunneling give qualitatively reasonable results even outside

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Fig. 20. Current–voltage characteristic for the superlattice studied in [123,124] with parameters T1 = −0:02 meV; ND = 1:5 × 1011 =cm2 ; d = 13 nm; ,imp = 0:0666 ps, and T = 4 K. Full line: quantum transport. Dashed line: sequential tunneling.

their range of applicability. Thus, they can be used as a rough guide to obtain a correct order of magnitude of the current. 5. Formation of &eld domains As shown in Sections 3 and 4, semiconductor superlattices typically exhibit ranges of negative diDerential conductivity. This occurs after the ;rst current peak for miniband transport (or sequential tunneling) as well as after the subsequent resonances at higher ;elds when different levels align. The shape of the current–;eld relation is thus typically of N-type and domain-formation eDects are likely to occur (see [45] for a general overview). The prototype of an extended device with N-shaped current–;eld relation is the Gunn diode which exhibits self-sustained current oscillation due to traveling ;eld domains [160 –166]. A similar behavior has been suggested for semiconductor superlattices [18,167], and oscillatory behavior has indeed been found experimentally in previous years [19,168] with frequencies over 100 GHz [20,169]. In contrast to the Gunn diode, semiconductor superlattices frequently exhibit the formation of stable stationary domains which lead to a characteristic saw-tooth pattern in the current–voltage characteristic (see Fig. 21a) as observed by many diDerent groups [16,17,123,124,170 –174]. (See also [175,176] for domain formation in a parallel magnetic ;eld.) The measurements are typically performed by applying a continuous sweep of the bias. As the branches overlap, one observes diDerent parts of the branches for sweep-up and sweep-down of the bias 10 Sometimes a quite complicated behavior is observed as well. Particularly for the superlattice structure of Fig. 21 stationary ;eld domains [179], self-sustained oscillations [19], as well as bistability between stationary and oscillatory behavior [180] has been observed under ;xed bias conditions within the ;rst plateau of the current–voltage characteristic. In this section, such complex behavior will be analyzed by a comparison between numerical results based on the sequential tunneling model from Section 3.3 with analytical studies. It will be shown that most of the observed eDects can be understood as the result of a competition 10

Of course the variation of the bias must be slow, so that the ;eld domains can follow adiabatically. Otherwise, the sawtooth structure disappears [177,178].

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Fig. 21. Experimental results for a semiconductor superlattice with 40 periods consisting of 4 nm AlAs barriers and 9 nm GaAs wells. The doping density is ND = 1:5 × 1011 =cm2 and the sample area is A = 1:13 × 10−8 m2 . (a) Current–voltage characteristic for negative bias applied to the top contact, where a stationary current is observed (from [179]). (b) Current-time signal for a constant positive bias exhibiting self-sustained current oscillations (from [181]).

between two mechanisms: (i) The motion of fronts connecting low- and high-;eld domains and (ii) the dynamical evolution of the ;eld close to the injecting contact, i.e., the cathode. This section is organized as follows: First, a model is described which is able to reproduce most of the experimental ;ndings for weakly coupled superlattices. In Section 5.2 numerical results are presented reproducing the behavior shown in Fig. 21. In the subsequent subsections, the key elements, namely the traveling fronts (Section 5.3) and the cathode behavior (Section 5.4) are discussed separately. In Section 5.5 it will be demonstrated that their combination explains the behavior observed both in experiment and simulation. Finally, the ;ndings are summarized and an instruction for the analysis of nonlinear superlattice transport behavior is presented. 5.1. The model In weakly coupled multiple quantum wells the electronic states are essentially localized in single wells forming energy levels E  . Transport then occurs by sequential tunneling between neighboring wells. The current from well m to well m + 1 is modeled by a function Jm→(m+1) = J (Fm ; nm ; nm+1 ), where Fm is the average ;eld drop between the respective wells and nm denotes the electron density (per unit area) in well m as depicted in Fig. 22. Considering a superlattice with N wells embedded between N + 1 barriers, the dynamics is determined by the continuity equation e

dnm = J(m−1)→m − Jm→(m+1) dt

for m = 1; : : : ; N :

(166)

The electric ;eld satis;es Poisson’s equation Cr C0 (Fm − Fm−1 ) = e(nm − ND )

for m = 1; : : : ; N ;

(167)

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Fig. 22. Sketch of a superlattice structure with an inhomogeneous ;eld distribution.

where ND is the doping density per period (per unit area), and Cr and C0 are the relative and absolute permittivities. Finally, the total voltage U is determined by U (t) =

N


Fm d ;

(168)

m=0

where we have neglected a possible voltage drop at the contacts for simplicity. Eqs. (166) – (168) can be transformed into an equivalent set of equations: dFm Cr C0 (169) = (J (t) − Jm→(m+1) ) for m = 0; : : : ; N ; dt Cr C0 nm = ND + (170) (Fm − Fm−1 ) for m = 1; : : : ; N ; e N
Cr C0 dU (t) (N + 1)J (t) = Jm→(m+1) + : (171) d dt j=0

This shows that the dynamical evolution of the local ;elds is driven by the total current density J (t), which itself is determined by the global behavior of the sample. Thus, J (t) represents a global coupling. In order to obtain a closed set of equations, the currents across the ;rst (cathode) and the last (anode) barrier, J0→1 and JN →(N +1) , respectively, have to be speci;ed. Here the following approaches were followed previously. In [66,182–184] these contact currents were calculated within the assumption of two ;ctitious additional wells, one before the ;rst and one after the last barrier. Then the current J0→1 is given by J (F0 ; n0 ; n1 ) for tunneling between two wells, where the ;ctitious density n0 has to be speci;ed, usually assuming n0 = (1 + c)ND , with c ¿ − 1. The current across the last barrier is treated analogously by introducing a ;ctitious density nN +1 . This approach will be referred to as constant density boundary condition in the following. Alternatively, one may assume that the current is proportional to the local ;eld, i.e., AJ0→1 = DF0 d with an Ohmic conductance D. For the anode condition JN →(N +1) one has to take into account the condition, that the current must vanish if nN tends to zero, as otherwise the

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Fig. 23. Current–;eld relation I (eFd; nm = ND ; nm+1 = ND ) calculated from the microscopic model for the superlattice of Fig. 21. The relation for the cathode current I0→1 = sgn(e)DFd is also shown for two diDerent values of D dom dom ¡ I ¡ Imax where stable domains are found. (broken lines). The hatched area indicates the current range Imin

density nN can become negative. This can be ensured by an additional factor nN =ND and AJN →(N +1) = DFN d nN =ND is a reasonable choice. This will be referred to as Ohmic boundary condition. The actual potential distribution at the boundary could be taken into account within a transmission-type formalism [67]. Here I will restrict myself to the two approaches sketched above, which are simpler and provide an understanding of most of the underlying physics. They contain the main ingredients to understand more complicated and physically better motivated versions. 5.2. Numerical results Let us consider the GaAs=AlAs superlattice structure from Fig. 21 (which exhibits a rather small miniband width of 0:08 meV) as a model system for subsequent calculations. In the following, the function J (Fm ; nm ; nm+1 ) is calculated from the nominal sample parameters by the model described in Section 3.3 (see also Fig. 11). The result for a homogeneous electron density nm =nm+1 =ND is shown in Fig. 23. Here, the current I =sgn(e)AJ is shown applying the experimental sample area A = 1:13 × 10−8 m2 in order to facilitate comparison with experimental data. In this section, the factor sgn(e) is included, so that the sign of the current equals the particle current in transport direction which simpli;es the following discussion. Fig. 23 shows that the current–;eld characteristics exhibits a linear conductivity for low ;elds, a maximum at eFd = eFmax d, a range with negative diDerential conductivity for eFd ¿ eFmax d, and a second sharp rise of the current for higher ;elds eFd ¿ eFmin d due to resonant tunneling from the lowest level to the second level of adjacent wells. DiDerent approaches to obtain the local current density J (Fm ; nm ; nm+1 ) can be applied as well and the overall results do not depend on this choice, provided the general shape resembles Fig. 23. E.g., in Refs. [19,66,67,182,174,185], one can ;nd results similar to those presented in this section, where various types of local current density functions are applied. By simulating Eqs. (166) – (168) for a ;xed bias U until a stationary state is reached, the stationary current–voltage characteristics shown in Fig. 24 are obtained. Here the initial condition

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Fig. 24. Current–voltage characteristics exhibiting stationary ;eld domains as shown in the insets for diDerent boundary conditions: (a) I0→1 =I (F0 ; 1:5ND ; n1 ) and IN →N +1 =I (F0 ; nN ; 1:5ND ), (b) I0→1 =DF0 d and IN →N +1 =DFN d(nN =ND ) with D = 5 mA=V (parameters as in Fig. 21).

for the calculation is taken from the result of the previous voltage point, simulating a sweep-up of the bias U . In Fig. 24a, we use the constant density boundary condition with c = 0:5, while the Ohmic boundary condition with D = 5 mA=V was applied in Fig. 24b. Both characteristics are almost identical and in reasonable quantitative agreement with the experimental data presented in Fig. 21a. As shown in the insets, the branches are due to the formation of electric ;eld domains inside the sample, where a low-;eld domain is located at the cathode and a high-;eld domain is located at the anode. Close to the contacts, we ;nd a small transition layer, which depends on the contact boundary conditions. The domain branches span a ;xed current dom ¡ I ¡ I dom with I dom = 24:7 A and I dom = 85:5 A, indicated by the hatched area range Imin max max min in Fig. 23. For diDerent boundary conditions oscillatory behavior can be obtained for the same sample parameters. This is shown in Fig. 25 for the Ohmic boundary condition with D = 0:5 mA=V. The current signal resembles the measured signal displayed in Fig. 21b. Nevertheless, diDerent current signals have been observed as well, both experimentally and in numerical simulations (see, e.g., Fig. 10:12 in [35] for results obtained from the same model with diDerent boundary conditions). The oscillations reported here are due to traveling high-;eld domains which are diDerent from the calculations presented in [181,186,187] where traveling monopoles have been reported. This is due to the diDerence in the boundary conditions and in the transport model used in both calculations, see also Refs. [185,188], where this point is analyzed. 5.3. Traveling fronts In this subsection traveling fronts will be examined which will form one of the building blocks in understanding the global dynamics. Eq. (169) shows that the dynamical evolution of the electric ;eld is determined by the total current I (t). The corresponding dynamics can be studied most easily for a constant total current corresponding to current controlled conditions. For Imin = 7:1 A ¡ I ¡ Imax = 326 A we have three intersections of I with the homogeneous current–;eld relation shown in Fig. 23. They correspond to three stationary homogeneous ;eld distributions eF I ¡ eF II ¡ eF III . Linearization of Eq. (169) shows that the ;eld F II is unstable

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Fig. 25. Self-sustained current oscillations for ;xed bias conditions sgn(e)U = 1:2 V and an Ohmic boundary condition D =0:5 mA=V. (a) Current I versus time t. (b) Density plot of the electric ;eld distribution. The high-;eld region is black. (c) Electric ;eld pro;le at diDerent times (parameters as Fig. 21).

under current-controlled conditions as dJ=dF ¡ 0. Thus a homogeneous initial ;eld distribution will either tend to F I or to F III in its temporal evolution. For an appropriate inhomogeneous initial ;eld distribution a part develops to F I while another part reaches F III . Thus, a front between these two spatial regimes appears. Calculations on the basis of Eqs. (169) and (170) show that the front develops on a typical time scale of less than 1 s and afterwards travels through the sample with unaltered shape, as shown in Fig. 26. Here two types of fronts appear. Accumulation fronts connect a low-;eld region on the left-hand side to a high-;eld region on the right-hand side as shown in Figs. 26a and c. Eq. (170) shows that the carriers accumulate (nm ¿ ND ) in the transition region. If, on the other hand, the high-;eld region is located on the left side, see Figs. 26b and d, a depletion front is present. As can be seen from Fig. 26, the front velocities depend on the external current and diDer for accumulation fronts with velocity cacc and depletion fronts with velocity cdep . These velocities have been determined from a series of simulations as a function of I and are given in Fig. 27. Note that cacc (I ) becomes zero in a ;nite range of currents 24:7 ¡ I ¡ 85:5 A corresponding dom ¡ I ¡ I dom of the stationary domains discussed above. (Such a stationary to the range Imin max front is shown in the inset of Fig. 24.) For higher currents cacc (I ) is negative, i.e., the front travels upstream against the direction of the average drift velocity of the electrons as shown in Fig. 26c. In contrast cdep (I ) is positive for all currents for the sample parameters used. These functions cacc (I ); cdep (I ) have been shown to be very helpful to understand and analyze Gunn oscillations [166,189]. Here they are applied in the context of semiconductor superlattices where some peculiarities can be found. In the following two subsections the special shape of cacc (I ) and cdep (I ) will be explained.

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Fig. 26. Temporal evolution of diDerent initial ;eld pro;les under constant current conditions (parameters as in Fig. 21).

Fig. 27. Front velocities for accumulation fronts (cacc , full line) and depletion fronts (cdep , dashed lines). The velocities are approximately linear functions of the current outside the range shown here. I1 and I2 denote the current where cacc = cdep and 2cacc = cdep , respectively (parameters as in Fig. 21).

5.3.1. Continuum limit and fronts traveling backwards In order to understand the simulations, let us ;rst discuss the continuum limit of vanishing superlattice period. We can approximate nm nm+1 − nm Jm→(m+1) ≈ e v(Fm ) − eD(Fm ) ; (172) d d2 with the average drift velocity v(F) and an eDective diDusion constant D(F) given by v(F) =

d J (F; ND ; ND ) eND

and

D(F) = −

d2 9J (F; ND ; n2 ) : e 9n2

(173)

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This approximation becomes exact in the nondegenerate limit, see Eq. (88), which is not applicable for the sample discussed here. Replacing ;nite diDerences by derivatives, Eqs. (169) and (170) are transformed to J 9F(z; t) eND 9F(z; t) 92 F(x; t) = + D(F) − v(F) − v(F) : (174) 9t Cr C0 dCr C0 9z 9z 2 This is the standard equation for the Gunn eDect, for which dcacc (I )=dI ¡ 0 and dcdep (I )=dI ¿ 0 hold [163]. Neglecting diDusion, the fronts can only travel in the direction of the drift velocity, i.e. c(I ) ¿ 0, which can be shown by the method of characteristics [190]. We conclude without diDusion term :

cdep ¿ 0 and cacc ¿ 0

for Imin ¡ I ¡ Imax :

In contrast, if the drift term v(F) 9F= 9z is neglected, one obtains a nonlinear reaction–diDusion system. Here solutions of traveling accumulation fronts F(z; t) = f(z − cacc t) exist with a monotonic function f(z) and f(−∞) = F I and f(∞) = F III . Furthermore, one ;nds cacc ¿ 0 if Aarea (J ) ¡ 0, where    F III 1 J eND Aarea (J ) = dF − v(F) ; (175) D(F) Cr C0 dCr C0 FI see Section 2 of [191]. The depletion fronts can be obtained by applying the symmetry operation z → −z, thus cdep (J ) = −cacc (J ). Typically, the integral Aarea is positive for Ico ¡ I ¡ Imax and negative for Imin ¡ I ¡ Ico , where Ico is the current, where both ;eld domains coexist. (This value satis;es the equal area rule [192] Aarea (Ico ) = 0.) Then one ;nds the generic scenario: cdep ¡ 0 and cacc ¿ 0 for Imin ¡ I ¡ Ico ; without drift term : cdep = 0 and cacc = 0 for I = Ico ; cdep ¿ 0 and cacc ¡ 0 for Ico ¡ I ¡ Imax : Including the drift term, it is obvious that both velocities will increase and the intersection point cdep = cacc is shifted to a positive velocity in agreement with Fig. 27. If the diDusion constant is not too small, a range of negative velocity is likely to remain. This can be estimated in the following way: let F0 satisfy J = eND v(F0 ). Linearization of Eq. (174) for F(z; t) = F0 + F(k) exp[i(kz − !t)] gives the dispersion ! = −iE + v0 k − iD0 k 2

(176) E = eND v (F0 )=(dCr C0 ).

with v0 = v(F0 ), D0 = D(F0 ), and the dielectric relaxation rate Thus, an initial condition F(z; 0) = F0 + C (z − z0 ) has the solution   C (z − z0 − v0 t)2 √ ; (177) exp −Et − F(z; t) = 4D0 t 4D0 t which essentially travels in the direction of the drift velocity. Nevertheless, F(z; t) also grows exponentially in the range z ¡ z0 provided −E ¿ v02 =4D0 . This can be interpreted as the occurrence of an absolute instability [193] for v(F)2 Cr C0 d dv(F) −e (178) ND ¿ 4D(F) dF when a local perturbation from a homogeneous ;eld spreads in both directions. If D is larger than the bound given by (178), spatial variations do not necessarily travel through the structure

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and either stationary domain structures or backward traveling ;eld domains are likely to occur. This condition has been applied to superlattice transport in [194] and seems to be in good agreement with data from a variety of strongly coupled superlattices [195]. Nevertheless, the condition for a stationary front (c = 0) is only valid for a speci;c current, as the functions c(I ) are strictly monotonic for the continuous drift diDusion model (174). This is in contrast to the results shown in Fig. 27 where cacc (I ) is zero in a ;nite interval. Thus, we can draw the following conclusions from the continuum limit: • Backward traveling fronts can occur for large values of DND for both accumulation and depletion fronts. • The front velocities c(I ) are decreasing with I for accumulation fronts and increasing for depletion fronts. Finally, note that the simple model for the current used in [181,182,184] (where the nm+1 dependence of J (F; nm ; nm+1 ) is neglected) implies D = 0 in Eq. (172). These models are appropriate for the second plateau but diDerences occur for the ;rst plateau discussed here. In particular, these models cannot reveal fronts traveling backwards. 5.3.2. Discreteness of the superlattice and stationary fronts While the considerations of the previous subsection apply to a continuous system, weakly coupled superlattices form a system where the discretization due to the ;nite superlattice period is essential. As explained in [35,184,196] this leads to stationary domain states, provided the transition region between the two domains becomes of the order of the superlattice period. In this case, the accumulation or depletion front gets trapped within one well. As this pinning can dom ¡ I ¡ I dom , one observes extended branches in the occur within a certain range of currents Imin max current–voltage characteristic (Figs. 21 and 24). As shown in [35,184], the suMcient condition for a stationary accumulation front reads as vmin Cr C0 ND & NDacc ≡ (179) (Fmin − Fmax ) : vmax − vmin e (Similar results have been given in [182,183,196] as well.) For the superlattice structure considered, one obtains NDacc = 1:2 × 1010 =cm2 , which is smaller than the actual doping density ND = 1:5 × 1011 =cm2 . Therefore, stationary accumulation fronts can exist in a certain range of currents for this sample. Let us remark that condition (179) strongly resembles (178) if the diDusion constant D(F) = v(F)d=2 for shot noise [194] is used. Again the discreteness of the structure is responsible for the occurrence of stable domains. For more strongly doped superlattices the velocity of the depletion fronts may become zero as well. The corresponding condition is given by [35] vmax Cr C0 ND & NDdep ≡ (180) (Fmin − Fmax ) : vmax − vmin e For the superlattice structure considered, this relation gives NDdep =4:4 × 1011 =cm2 , which is larger than the actual doping. Therefore, no stationary depletion fronts occur in this sample. Together with the monotony arguments depicted in the previous subsection these estimations explain the shape of the c(I ) functions shown in Fig. 27. Rigorous proofs of some of the features discussed here are given in [197].

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In general, one can distinguish three types of superlattice: For ND . NDacc there are no stationary fronts. For NDacc . ND . NDdep accumulation fronts are stationary within a certain current range. Finally, for highly doped superlattices with ND & NDdep both accumulation fronts and depletion fronts can be stationary, i.e., pinned at a certain well. 5.4. The injecting contact Now, the inJuence of the contact boundary condition at the cathode J0→1 (F0 ) shall be investigated which is essential for the dynamical behavior. For a given current J , the evolution of the ;eld at the cathode, F0 , is given by Cr C0

dF0 = J − J0→1 (F0 ) : dt

(181)

Let Fc (J ) be the solution of J0→1 (F) = J , which forms an attracting point of Eq. (181), i.e., dJ0→1 =dF ¿ 0. Then F0 (t) will tend towards Fc (J ). Provided the relaxation time at the contact is much smaller than the corresponding time scale on which J changes, one may use the boundary condition F0 = Fc (J (t)) to describe superlattice dynamics. If Fc (J ) is close to the value of F I (J ) or F III (J ), a low-;eld domain or a high-;eld domain will be injected into the sample, respectively [190]. For a pure drift system the condition for the injection of a low-;eld domain is given by eFc (J ) ¡ eF II (J ) [45]. This gives a qualitative bound but this does not hold strictly in the superlattice system due to discreteness and diDusion. For the Ohmic boundary condition depicted in Fig. 23, we thus ;nd that a low-;eld domain forms close to the injecting contact for I . Ic and a high-;eld domain forms there for I & Ic , where Ic is the current at the intersection of I0→1 (F0 ) with the homogeneous current–;eld characteristic. In contrast to the cathode where electrons are injected into the sample, the anode contact conditions do not play a major role. A boundary layer exists there, which is typically stable (see e.g. [190] for a discussion within a drift model). This can be understood from the fact that the perturbations mainly travel through the sample in the direction of the current Jow, see Eq. (177), even if there might be some response in the opposite direction as well. 5.5. Global behavior The behavior found numerically in Section 5.2 will now be explained within the interplay between the dynamics of fronts and contacts. Let us restrict ourselves to rather long superlattices where the N3D L criterion ND N ¿ NDnL ≡ 2:09

v(F)Cr C0

−e dv=dF

(182)

is satis;ed so that the homogeneous ;eld distribution is unstable [161,162]. In this case either stationary domain states, self-sustained periodic current oscillations, or aperiodic behavior occur. Calculations yield NDnL ≈ 1011 =cm2 for the superlattice structure under consideration, which is much smaller than 40ND . This criterion has been successfully applied to a variety of superlattice structures [195].

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Fig. 28. Results for a superlattice with 5 nm Al0:3 Ga0:7 As barriers and 8 nm GaAs quantum wells with a high doping density ND = 5 × 1011 =cm2 . (a) Current–;eld relation (full line) evaluated by Eq. (83) with 7a = 7b = 8 meV. The dashed lines depict the cathode current densities used in parts (b), (c), and (d), respectively. (b) Current–bias relation for an Ohmic boundary condition with large D. (c) Current–bias relation for an Ohmic boundary condition with small D. (d) Current–bias relation for a constant-density boundary condition with c = −0:2. The insets display examples of the respective ;eld distributions.

5.5.1. Formation of stable stationary 7eld domains For the sample considered, accumulation fronts become stationary in the current range dom ¡ I ¡ I dom . Obviously, these domain states are only possible if the low-;eld domain is Imin max maintained at the cathode, i.e. eFc (I ) . eF II (I ), as discussed in the previous subsection. For dom . Thus, these domain states are in agreement with the D = 5 mA/V, we ;nd Ic = 110 A ¿ Imax numerical ;ndings from Fig. 24. The same behavior can be found for the constant density boundary condition with c ¿ 0. In this case, the ;eld F0 remains in the low-;eld region as long as I . (1 + c)Imax , which is hardly exceeded unless the superlattice is operated in the second tunneling resonance. Therefore similar behavior is observed for both boundary conditions. For the superlattice structure considered here, stationary depletion fronts do not occur. Therefore, one expects that the high-;eld domain is always located at the anode for stationary behavior. This agrees with the experimental ;ndings for this superlattice [198]. For superlattices with a higher doping, the situation is diDerent and ;eld distributions have been observed where the high-;eld domain is located at the cathode [172]. These experimental ;ndings are in good agreement with criterion (180). Simulations [122] give such a ;eld distribution for appropriate contact conditions. A further example is shown in Figs. 28c and d. dom ¡ I ¡ I dom . If the maximum These stationary fronts appear within a ;nite current range Imin max accumulation occurs in well mdom , the total bias for accumulation fronts is given by U (I; mdom ) = mdom F I (I )d + (N + 1 − mdom )F III (I )d + Uc (I ) + Ufront (I ) :

(183)

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Here Uc and Ufront are correction terms due to the inhomogeneous ;eld pro;les at the contacts and in the front region, which do not depend on the front position mdom , provided this position is not too close to either of the contacts. Due to the periodicity of the superlattice, a shift of the front by one period does not change the current. Thus, one obtains up to N branches U (I; mdom ) of Eq. (183) with mdom = 1; : : : ; N . This results in the characteristic saw-tooth pattern in the current–voltage characteristics for Figs. 21a and 24. In addition, two ;eld distributions exist, which exhibit a homogeneous (except for boundary eDects) ;eld distribution Fm = F I (I ) and Fm = F III (I ) for all m. The actual number of branches can be smaller due to the inhomogeneous ;eld distributions at the boundaries as shown in the insets in Fig. 24. The voltages corresponding to neighboring branches diDer by [F III (I ) − F I (I )]d. As this diDerence depends on the current, these branches are not exactly reproduced for diDerent mdom , but change their shape. Neglecting boundary eDects, their slopes are given by dF I (I ) dF III (I ) dU = mdom d + (N − mdom + 1)d ; (184) dI dI dI which varies between LdF I (I )=dI for small biases (accumulation front close to the right contact) and L dF III (I )=dI for large bias (accumulation front close to the left contact). The behavior for depletion fronts is identical except that they are located close to the left=right contact for small=large bias and the ;elds F III and F I have to be exchanged in Eq. (183). 5.5.2. Self-sustained current oscillations dom . Therefore no stationary For D = 0:5 mA=V we ;nd Ic = 22 A, which is smaller than Imin domain states are possible. In contrast, self-sustained oscillations occur, as shown in Fig. 25. The mechanism of these oscillations can be understood by using ideas reminiscent of the Gunn eDect asymptotic [166]: for t = 0:3 s the total bias is distributed between a high-;eld domain and a low-;eld domain. As the total bias is constant, there is a certain part of each oscillation period during which both boundaries must travel at the same velocity [166,189]. Then cacc (I1 )=cdep (I1 ) giving I1 = 0:0114 mA (see Fig. 26). As I1 ¡ Ic the cathode remains in the low-;eld domain. We observe a current signal which is constant in average and exhibits fast oscillations with the period c=d due to well-to-well hopping of the accumulation front. This feature has been discussed in the analysis of switching behavior [199 –201] and is explained in detail in [185]. After the leading edge of the high-;eld domain has reached the anode, the size of the high-;eld domain shrinks as the trailing edge travels further with unaltered velocity. In order to maintain the total bias, the ;elds increase in both domains, leading to an increase of I (t) via the global condition (171), as can be seen at t = 2:8 s. When I (t) becomes larger than Ic the ;eld at the cathode injects a high-;eld domain into the superlattice (t = 4:1 s). As cacc is quite large in this range of currents the newly formed domain expands relatively fast and the ;elds drop in order to maintain the bias. Therefore, the current shrinks again below Ic and the cathode injects a low-;eld domain into the superlattice (t = 5:6 s). In this situation three boundaries are present in the superlattice. The old accumulation front (around well 35) and the depletion and accumulation front limiting the newly formed high-;eld domain (from well 5 –10). In this situation, the sum of the extensions of both high-;eld regions is kept constant if the depletion front travels with twice the velocity of the accumulation fronts, i.e. 2cacc (I2 ) = cdep (I2 ) yielding I2 = 0:015 mA (see Fig. 26). This is just the average current

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Fig. 29. Same as Fig. 28 for a medium doping density ND = 1011 =cm2 . (c) and (d) show current oscillations at a ;xed bias.

observed in the simulation observed in the plateau around t = 5:6 s. The oscillatory part of the current exhibits a lower fundamental frequency compared to the frequency observed around t = 0:3 s as cacc (I2 ) ¡ cacc (I1 ). In addition, one observes two peaks per period resulting from the presence of two accumulation fronts. This behavior is maintained upto t = 7:1 s when the old accumulation front reaches the anode and only the newly formed high-;eld domain remains. Afterwards the cycle is repeated. DiDerent scenarios for oscillatory behavior are also possible. For the constant density boundary condition with c ¡ 0 oscillations due to traveling depletion fronts have been found for the same model, see Refs. [181,35] and Fig. 29d for a further example. In this case, the ;eld at the cathode remains above eFmin , once this range of ;elds has been reached. For eU ¡ NeFmin the high-;eld domain cannot extend over the whole superlattice and thus a range of ;elds with eFd ¡ eFmin d must be present. This requires the existence of a depletion front, which cannot be stationary for the superlattice considered here as ND ¡ NDdep . Thus the front travels towards the anode increasing the width of the high-;eld domain. In order to keep the bias constant, the ;eld inside the domain shrinks during this process. If F becomes lower than Fmin the domain becomes unstable and the ;eld tends to the low-;eld value F I . As the ;eld F0 remains high due to the boundary condition a new depletion front is formed and the cycle repeats itself. An equivalent oscillation mechanism is discussed in detail in [181,190] for accumulation fronts. (This mechanism is also active in the oscillation shown in Fig. 30b.) It becomes relevant for lower doped samples, i.e., for ND ¡ NDacc , when these fronts cannot become stationary.

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Fig. 30. Same as Fig. 28 for a low doping density ND = 1010 =cm2 . (b) and (c) show current oscillations at a ;xed bias.

5.6. Summary Let us summarize the ;ndings from the previous subsections in order to obtain a general outline for the analysis of semiconductor superlattices. To visualize the general trends, various types of behavior are shown for a superlattice test structure in Figs. 28–30 for diDerent doping densities. (i) Determine the local current–;eld relation J (Fm ; nm ; nm+1 ) from some transport model, see, e.g., Section 3. The corresponding results for nm = nm+1 = ND are displayed in Figs. 28a, 29a, 30a. (ii) Determine NDacc and NDdep from Eqs. (179) and (180). According to their de;nition NDacc ¡ NDdep holds. From Fig. 29a we ;nd NDacc ≈ 1:7 × 1010 =cm2 and NDdep ≈ 3 × 1011 =cm2 . (iii) Compare these quantities with the actual doping density ND NDdep ¡ ND : Stable domains are formed for both good and bad contacts at the cathode. The domain boundaries can be formed by accumulation layers, see Fig. 28b, or depletion layers, dom ; I dom ] overlap. In see Figs. 28c and d. Both may even coexist if the respective ranges [Imin max all cases, the current–voltage characteristic exhibits the typical saw-tooth behavior, where the number of jumps roughly equals the number of periods. NDacc ¡ ND ¡ Ndep D : The behavior depends crucially on the boundary condition at the injecting contact: If the current can be injected at a fairly low electric ;eld in the cathode (good contact), stable domains are found which exhibit accumulation layers, i.e., the high-;eld region is located at the anode, see Fig. 29b. Otherwise, for bad contacts, one observes self-sustained current oscillations, see Figs. 29c and d. The shape of these oscillations as well as their frequency depends on the type of contacts.

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ND ¡ Nacc D : Stable domains associated with a saw-tooth current–voltage characteristic do not occur. The behavior of the sample is either dominated by current oscillations, see Figs. 30b and c or a rather smooth stationary current–voltage relation, see Fig. 30d. According to the N3D L-criterion (182), the latter case dominates for short samples and lower doping. Here it seems to be crucial that the inhomogeneous ;eld distribution essentially takes values from a range where dv=dF is small, suggesting large values of NDnL . (Similar behavior is found for D = 105 A=Vcm2 if the number of periods is reduced.) Experimentally, this scenario can be veri;ed by varying the electron density by optical irradiation [202,203]. Transitions between oscillating and stationary domains can also be provoked by changing the lattice temperature [204]. This eDectively alters the shape of the j(F) relation and thereby via Eqs. (179) and (180) the critical doping densities. There are several other aspects complicating the picture sketched above, which are not addressed in the discussion given here. Real superlattices are not perfect structures, where the properties of each well are repeated exactly. In contrast there will be Juctuations in the doping density, the barrier and well width as well as the material composition from period to period. Some information can be obtained by X-ray analysis and indeed it has been possible to relate some global properties to the extent of disorder [195,205] in the respective samples. By extensive simulations it has been shown that the presence of such disorder inside the superlattice aDects the behavior signi;cantly [187,206]. The nature of the current oscillations, the actual shape of the domain branches, as well as critical doping densities can be aDected by the amount of disorder. Frequently, one observes a direct correlation between some global current signals and the actual realization in a particular sample [159,207]. Recently, an additional S-type current–voltage characteristic has been found in strongly coupled superlattices due to electron heating [208]. The combination of N-type and S-type negative diDerential conductivity may provide additional interesting eDects. In all calculations performed here the transport model for sequential tunneling has been applied, which provides rate equations between the quantum wells. For strongly coupled superlattices, it is questionable if the electrons can be con;ned to accumulation layers extending over a few wells. Therefore it is not clear as to how far these stationary domain structures persist. The alternative is to start from the miniband model. Such calculations have been performed in [209,210] using the drift velocity from the relaxation time model. A more microscopic approach can be performed within the hydrodynamic model [167,211]. Nevertheless, one has to be aware that the miniband transport model becomes questionable for large ;eld strengths (cf. Fig. 7), which are typically reached within a high-;eld domain. Thus, a quantum transport calculation would be desirable to clarify the situation. Except for the stability analysis of the homogeneous state with respect to spatial Juctuations in [138], I am not aware of any quantum transport simulations concerning inhomogeneous ;eld distributions in superlattices. 11 Thus, it remains an open question as to how far quantum eDects modify the behavior discussed in this section.

11 Very recently stationary inhomogeneous ;eld distributions could be modeled for short superlattices within the quantum transport model discussed in Section 4. First results are in qualitative agreement with those from the sequential tunneling model [212].

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6. Transport under irradiation In this section, we consider superlattice transport under irradiation by an external microwave ;eld with frequency #=2. In this case, a further energy scale, the photon energy ˝# of the radiation ;eld, comes into play. For frequencies in the THz range (1 THz , 4:14 meV) this energy is of the same order of magnitude as typical miniband widths, scattering induced broadening, and the potential drop per period. This provides an interesting ;eld to study various types of quantum eDects. Transport under irradiation has ;rst been studied theoretically within the simple Esaki–Tsu model. In this context, it was shown that negative diDerential dynamical conductance [5,213] occurs. In the recti;ed response replica of the current peaks appear at ;eld strengths obeying eFd=eFpeak d+‘˝# (with ‘ ∈ Z) indicating the quantum nature of the radiation ;eld [100,214]. Furthermore, absolute negative conductance (i.e. a negative current for positive bias) is possible under certain conditions [215,216]. With the development of the free-electron laser as a high power THz source it became possible to study these eDects experimentally and indeed the photon-assisted replica of current peaks [217,218] as well as absolute negative conductance [31] were observed. The superlattices used in these experiments exhibited rather small miniband widths, so that the application of miniband transport (as done in the theories mentioned above) is questionable as discussed in Sections 3 and 4. Nevertheless, these ;ndings could be explained qualitatively [31,219 –221] within the standard theory of photon-assisted tunneling [222–224] as well, which is applicable for sequential tunneling. A quantitative description of these experiments [120] was possible within the sequential tunneling approach described in Section 3.3. Photon-assisted peaks in the current–voltage characteristic could also be observed in strongly coupled superlattices [225] although the results are less clear in this case. Furthermore, the reduction of current due to the irradiation could be nicely demonstrated in experiments for superlattices with large miniband widths [226], which gave an excellent agreement with the simple miniband models mentioned above. In this section, the basic ingredients of the transport theory under irradiation are reviewed. The ;rst subsection deals with the simple quasi-static response, which holds for low frequencies. This will be the basis for the discussion of quantum eDects in the subsequent subsections. The main results within the miniband transport will be reviewed in the second subsection. Here a form will be chosen which simpli;es the comparison with sequential tunneling which is discussed in the third subsection. Throughout this section we consider a homogeneous electric ;eld F(t) along the superlattice structure which can be separated into a dc-part Fdc and a cosine-shaped time dependence with amplitude Fac ¿ 0, i.e., F(t) = Fdc + Fac cos(#t) :

(185)

The transport problem has been considered for Fac = 0 in the preceding sections where the relation Idc (Fdc ) was obtained. Now, we are looking for periodic solutions with period 2=#, neglecting transient eDects as well as the possibility of aperiodic behavior. Then the general

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current response can be written as I (t) = I0 +

∞


[Ihcos cos(h#t) + Ihsin sin(h#t)] :

(186)

h=1

In general, four diDerent aspects of transport under irradiation can be identi;ed: (i) The recti7ed response I0 is considered in many diDerent experiments. It is easily accessible and can be used for the detection of high-frequency signals. (ii) The active current I1cos provides the direct interaction with the irradiation ;eld. If dI1cos =dFac ¡ 0 we observe gain at the given frequency #. (iii) The reactive current I1sin describes the response out of phase, which can be described by an inductance (I1sin = Uac =L#) or a capacitance (I1sin = −C#Uac ) in standard circuit theory. There are two possibilities to de;ne inductive and capacitive eDects: (i) One assumes that C and L are always positive. Then positive=negative I1sin is referred to as an inductive=capacitive eDect, respectively [227]. (ii) The other considers the low-frequency limit. It will be shown later that typically I1sin ˙ # for low frequencies. This resembles the behavior of a capacitor which can either be positive or negative [228]. (iv) Harmonic generation Ih for h ¿ 2: These terms describe the occurrence of higher harmonics and can be used to generate higher frequencies [229 –231]. If Fdc = 0 only the odd multiplies h are present for symmetric structures with Idc (F) = −Idc (−F). In this section we restrict ourselves to homogeneous ;eld distributions with the time dependence (185). In this case, the current is homogeneous over the superlattice direction and no charge accumulation inside the structure occurs. Such complications can be treated within the general formalism discussed in [232,233]. Domain-formation eDects in superlattices under irradiation have been studied in [234]. If higher harmonics are present in the time dependence of the ;eld, the superlattice may act as a recti;er [235]. Furthermore, the response of an external circuit is neglected here. A detailed discussion of the latter issue can be found in [236].

6.1. Low-frequency limit In the range of radio frequencies (say #1 THz) the frequency # is slow with respect to the internal degrees of freedom, such as carrier heating or XE= ˝ (where XE describes typical energy scales of the transport problem). Then one can assume that the current follows the ;eld instantaneously: Irf (t) = Idc (F(t))  n ∞
1 d n Idc (Fdc ) ei#t + e−i#t = Fac n! dF n 2 n=0       j  ∞

1 Fac 2j d 2j Idc (Fdc ) 2j 2j = + (ei2k#t + e−i2k#t ) j j+k (2j)! 2 dF 2j j=0

k=1

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+

∞
j=0

1 (2j + 1)!



Fac 2

2j+1

j

d 2j+1 Idc (Fdc )
dF 2j+1 k=0



2j + 1 j+1+k



73

(ei(2k+1)#t + e−i(2k+1)#t ) : (187)

This shows that all terms Ihsin vanish in the radio-frequency limit. Furthermore, we obtain the following expressions in lowest order of the irradiation ;eld: Fac2 d 2 Idc (Fdc ) ; 4 dF 2 dI (F ) = Fac dc dc ; dF  h h 2 Fac d Idc (Fdc ) = : h! 2 dF h

I0; rf = Idc +

(188)

I1;cosrf

(189)

Ih;cosrf

(190)

They will be compared with the results discussed in the next subsections. 6.2. Results for miniband transport In the miniband transport model the time dependence of the electric ;eld enters the Boltzmann equation (35) which complicates the problem tremendously. The easiest way to deal with this situation is the relaxation-time approximation. Assuming ,m = ,e = ,, Eqs. (54) and (56) can be solved for an arbitrary time dependence of the electric ;eld. One obtains [215,237]   t  2e|T1 |ceq (&; T ) t eF(t2 )d −(t−t1 )=, J (t) = dt1 e sin dt2 ; (191) ˝, ˝ −∞ t1   t  ceq (&; T ) t eF(t2 )d −(t−t1 )=, dt1 e cos dt2 : (192) c(t) = , ˝ −∞ t1 Let us now consider the ;eld dependence (185). The crucial parameter in the following will be the ratio between the ac-;eld strength and the photon energy eFac d = ; (193) ˝# which will appear in most of the following results as the argument of the integer Bessel functions Jn . In order to avoid confusion with the symbol J for the current density, the results are given in terms of the current I = AJ in the following. Then one ;nds     2eA|T1 |ceq (&; T ) t eFdc d −(t−t1 )=, I (t) = dt1 e Im exp i (t − t1 ) + i(sin(#t) − sin(#t1 )) ˝, ˝ −∞ # $ t

2eA|T1 |ceq (&; T ) −(t−t1 )=, i(eFdc d=˝)(t−t1 ) i‘ #t −i‘#t1 = dt1 e e J‘ () e J‘ ()e Im ˝, −∞  ‘ ‘ # $

2eA|T1 |ceq (&; T ) 1 i(‘ −‘)#t = J‘ ()J‘ ()Im 1 e ˝,  , − i eFdc d= ˝ − i‘# ‘ ‘

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Fig. 31. Recti;ed current–;eld relation from Eq. (197) within the Esaki–Tsu model for  = 1:2 (a) and  = 2:4 (b). The thick line gives the total current I0 (Fdc ). The thin lines show the contributions (J0 ())2 Idc (eFdc d) (full line), (J±1 ())2 Idc (eFdc d ± ˝#) (dashed line), and (J±2 ())2 Idc (eFdc d ± 2˝#) (dotted line). Parameters 7 = 4 meV, ˝# = 6 meV.

 # $ 2eA|T1 |ceq (&; T )

1 = Jh+‘ ()J‘ () Im 1 cos(h#t) ˝, , − i eFdc d= ˝ − i‘# h ‘ # $  1 + Re 1 sin(h#t) : , − i eFdc d= ˝ − i‘#

From Eq. (57) one can identify 2eA|T1 |ceq (&; T ) Idc (eFd) = Im ˝,

# 1 ,

1 − i eFd= ˝

$

=

2eA|T1 |ceq (&; T ) 7eFd ˝ (eFd)2 + 72

with 7 = ˝=,. Furthermore, we de;ne $ # 2eA|T1 |ceq (&; T ) 2eA|T1 |ceq (&; T ) 1 72 = K(eFd) = Re 1 ˝, ˝ (eFd)2 + 72 , − i eFd= ˝    dE Idc (E) = P ;  E − eFd

(194)

(195)

(196)

where the Kramers–Kronig relation, connecting the imaginary and real part of ˝=(7 − iE), has been applied. Then we can evaluate the components of Eq. (186) in the form
I0 = (J‘ ())2 Idc (eFdc d + ‘˝#) ; (197) ‘

Ihcos

=




J‘ ()(J‘+h () + J‘−h ())Idc (eFdc d + ‘˝#) ;

(198)

J‘ ()(J‘+h () − J‘−h ())K(eFdc d + ‘˝#) :

(199)

‘

Ihsin

=


‘

Let us ;rst consider the recti;ed response I0 , which is a sum of several dc-curves (as shown in Fig. 5b) shifted by integer multiples of the photon energy. This explains the occurrence of photon-assisted peaks at biases eFdc d ≈ 7 + ‘˝# as shown in Fig. 31a. If the coeMcient for ‘ = 0 becomes small, i.e., close to a zero of J0 () (the ;rst zero occurs at  = 2:4048 : : :) the

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terms with ‘ = ±1 dominate, which can provide absolute negative conductance for ˝# & 7 as shown in Fig. 31b [216]. For small ac-;elds eFac d˝# we may set J0 () ≈ 1; J±1 () ≈ ±eFac d=(2˝#), while the higher-order Bessel functions are approximately zero. Thus, Ih ≈ 0 for h ¿ 2 and I1cos ≈

Idc (eFdc d + ˝#) − Idc (eFdc d − ˝#) eFac d= : gdyn (#)Fac d ; 2˝#

I1sin ≈ −

˝# K(eFdc d + ˝#) − 2K(eFdc d) + K(eFdc d − ˝#) eFac d= : −#ct (#)Fac d ; 2 (˝#)2

(200) (201)

which can be viewed as a resistor 1=gdyn and a capacitor ct in parallel yielding a complex admittance z −1 = gdyn + i#ct (small letters indicate quantities per period and the engineering convention I (t) ˙ ei#t is used here). Similar expressions were derived in Ref. [5] for the model with two scattering times. Both gdyn and ct may take positive or negative values. Furthermore, note that ct is not the sample capacitance which has to be added in parallel as well, but originates from a quantum eDect. (For small frequencies ct ≈ e˝=2 d 2 K(E)=d E2 vanishes in the limit ˝ → 0, if the functions Idc (E) and K(E) are kept constant.) Eq. (200) shows that the derivative in the low-frequency response (189) is replaced by a ;nite diDerence on the quantum scale. A straightforward calculation for the Esaki–Tsu model gives gdyn (#) ¡ 0 for |eFdc d| ¿ 72 + (˝#)2 . In this range, the superlattice structure can provide gain. Nevertheless, one has to note that this occurs in the range of negative diDerential conductivity, where the homogeneous ;eld distribution is typically unstable as discussed in Section 5. In the limit of small scattering 7 → 0, the functions Idc (eFd) and K(eFd) vanish unless F ≈ 0. From Eqs. (197) – (199), one ;nds for Fdc ≈ 0 all components I0 ; Ihcos ; Ihsin of the current that vanish if J0 () = 0. This can be interpreted as a dynamical localization [6] or the collapse of the miniband [7]: for a certain strength of the irradiation ;eld the periodic structure does not conduct any current. In addition, a ;nite conductivity appears at eFdc d ≈ ‘˝# opening up new transport channels, which are not present for  = 0 in the limit of 7 → 0. All these results have been obtained within the simple Esaki–Tsu model. Calculations within the energy balance model [238] provide similar results. If the dielectric relaxation is included, chaotic behavior [239] as well as a spontaneous generation of dc current [240] can be found. Recently, Monte-Carlo simulations of the Boltzmann equation under THz-irradiation have been performed, too [241]. 6.3. Sequential tunneling For sequential tunneling between two neighboring wells (m and m + 1) Eqs. (197) – (199) hold again with the dc-expression from Eq. (79):
; &  ∞ dE &  Idc (eFd) = 2e |H1 |2 A˜ m (k; E)A˜ m+1 (k; E + eFd) −∞ 2˝ k;;&

×[nF (E − &m ) − nF (E + eFd − &m+1 )]

(202)

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and the quantity K(eFd) = −4e


k;;&

|H1; & |2



∞

−∞

dE &  adv [nF (E − &m )A˜ m (k; E)Re{G˜ m+1 (k; E + eFd)} 2˝

& ret  + Re{G˜ m (k; E)}nF (E + eFd − &j+1 )A˜ m+1 (k; E + eFd)]    dE Idc (E) = P ;  E − eFd

(203)

where G˜ m (k; E) equals Gm (k; E − meFd) in the limit of decoupled wells. (The de;nition used here diDers by an m-dependent shift of the energy scale from the one used in Sections 3.3 and (B.1).) These expressions have been derived in [223] for a constant matrix element H1; & . A similar derivation is provided in Appendix C. Furthermore, it is shown there that Eq. (197) also applies to the case of a ;eld-dependent matrix element H1; & = eFdR;1 & , which is relevant for tunneling between nonequivalent levels. Note that the THz ;eld also couples the diDerent subbands via the term R& 0 , which may cause further eDects [242]. For sequential tunneling Eq. (197) has a simple interpretation. In the evaluation of the dc-current, eFd gives the energy mismatch between the levels in well m and m + 1. Under irradiation photons of energy ˝# can be absorbed or emitted during the tunneling process, which provides the energy mismatch eFdc d ± ‘˝# for the absorption=emission of ‘ photons during the tunneling process. The Bessel functions represent the probability that an ‘-photon process occurs assuming a classical radiation ;eld (i.e. neglecting spontaneous emission). Eq. (197) then consists of the weighted sum of all possible photon-assisted tunneling processes. Similar results have been obtained in [243], where a strictly one-dimensional tight-binding lattice coupled to a heat bath has been considered (the approximation of incoherent tunneling dynamics applied there corresponds to sequential tunneling). An extended discussion of the methods used there can be found in [244]. As discussed in Section 3 the structure of the ;rst peak at low electric ;elds eFd ≈ 7 is similar for miniband transport and sequential tunneling. Therefore, the discussion for miniband transport given in the preceding subsection holds for sequential tunneling as well. In addition, Eqs. (202) and (203) also describe the current peaks at resonances between diDerent levels (a,b) in neighboring quantum wells. There one typically ;nds, see Eq. (83), Idc (eFd) = eND A

|H ab |2 7b; eD : ˝ (eFd + E a − E b )2 + (7b; eD =2)2

(204)

With the Kramers–Kronig relation one obtains K(eFd) = −eND A

|H ab |2 2(eFd + E a − E b ) ; ˝ (eFd + E a − E b )2 + (7b; eD =2)2

(205)

where the ;eld dependence of H ab has been neglected. In the following, some results are presented for the superlattice structure studied experimentally in [31,120,129] (15 nm wide GaAs wells, 5 nm Al0:3 Ga0:7 As barriers, doping density ND = 6 × 109 =cm2 , cross section A = 8 m2 ). Results for this structure have been presented already in Fig. 12, where the complicated temperature dependence was discussed. Here we will

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Fig. 32. Recti;ed current response for the superlattice of [31] displaying absolute negative conductance. (a) Theoretical results for ˝# = 6:3 meV and diDerent ;eld strength eFac d = ˝# of the irradiation. (b) Experimental results for ˝# = 6:3 meV and diDerent laser intensities increasing from the top to the bottom. The actual values Fac inside the sample are not accessible. (c) Theoretical results for  = 2:4 and diDerent photon energies. The thin line depicts ˝# = 5:3 meV and  = 2:1. (d) Experimental results for diDerent photon energies. The laser intensity was tuned to give maximum negative conductance (from [120]).

focus on the behavior under irradiation. For these calculations, a constant electron temperature Te = 35 K is assumed, which provides the best agreement around the ;rst maximum. Due to the presence of the irradiation ;eld, it seems realistic that the electron gas does not even reach thermal equilibrium for vanishing dc-bias. 6.3.1. Recti7ed THz response In Fig. 32 results are shown for the recti;ed current response I0 (eFdc d) under diDerent strengths and frequencies of the irradiation ;eld. Both from experiment and theory one observes a range of absolute negative conductance (i.e., I0 ¡ 0 for Fdc ¿ 0) for low biases. Furthermore, photon-assisted peaks are visible at ;eld strengths eFdc d ≈ eFmax d + ‘˝#. These ;ndings are in qualitative agreement with the discussion of Fig. 31. Quantitative agreement between

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theory (Fig. 32a) and experiment (Fig. 32b) is found for ˝# = 6:3 meV (1:5 THz) for diDerent strengths of the laser ;eld. The low-;eld peak occurs at Udir = NFmax d ≈ 20 mV corresponding to direct tunneling. Photon replicas can be observed at U ≈ Udir + N ˝#=e = 83 mV and U ≈ Udir + 2N ˝#=e = 146 mV. For low bias and high intensities ( = 2:0) there is a region of absolute negative conductance [31], which will be discussed in the following. In Fig. 32d, the laser intensity has been tuned such that maximal absolute negative conductance occurred for each of the diDerent laser frequencies. Then one observes a minimum in the current at U ≈ −Udir + N ˝#=e which is just the ;rst photon replica of the direct tunneling peak on the negative bias side. This replica dominates the current if the direct tunneling channel is suppressed close to the zero of J0 () in Eq. (197), i.e.,  ≈ 2:4, as used in the calculation of Fig. 32c. Both the theoretical and experimental results show that absolute negative conductance persists in a wide range of frequencies but becomes less pronounced with decreasing photon energy. In the calculation absolute negative conductance vanishes for ˝# ¡ 1:8 meV which is approximately equal to ˝# . eFmax d. (The latter relation has been veri;ed by calculations for diDerent samples as well.) For ˝# = 5:3 meV a smaller value of  = 2:1 (thin line) agrees better with the experimental data (in the same sense the value  = 2:0 agrees better for ˝# = 6:3 meV, cf. Fig. 32a). This may be explained as follows: If strong NDC is present in doped superlattices, the homogeneous ;eld distribution becomes unstable and either self-sustained oscillations or stable ;eld domains form as discussed in Section 5. Then the current–voltage characteristic deviates from the relation for homogeneous ;eld distribution, where U = NFd, and typically shows less pronounced NDC. Therefore maximal negative conductance is observed at a laser ;eld corresponding to a value of  ¡ 2:4, where the NDC is weaker and the ;eld distribution is still homogeneous. The presence of an inhomogeneous ;eld distribution could also explain the deviations between theory and experiment for U ¿ 150 mV. Quantitative agreement between theory and experiment regarding the recti;ed response was also obtained for a diDerent superlattice structure, where up to seven photon replica of the ;rst current peak could be observed [245]. 6.3.2. Negative dynamical conductance A semiconductor element is able to give gain at the given frequency if dI1cos =dFac ¡ 0. In the low-frequency limit (Section 6.1), one obtains I1; rf =dIdc (F)=dF × Fac if Fac is not too large. This means that gain is related to negative diDerential conductance in the static current–;eld relation Idc (F). As this situation is typically unstable with respect to the formation of inhomogeneous ;eld distributions, it is diMcult to apply this concept for a real device. Thus, it would be of interest to have a system exhibiting dI1cos =dFac ¡ 0 in the THz frequency range considered but a positive diDerential conductance at # → 0. Inspection of Eq. (200) for the standard expressions (195) and (204) shows that I1cos is always positive as long as dIdc =dF ¿ 0 holds. Thus, it is a nontrivial task to ;nd the opposite case in a semiconductor superlattice. Fig. 33a shows the calculated current–;eld relation Idc (eFd) for the superlattice structure discussed here. It exhibits a kind of plateau in the range 2 meV ¡ eFd ¡ 9 meV, due to the presence of impurity bands which is observed experimentally as well (see Fig. 12). Experiments 12 show that this plateau is almost unchanged for lattice temperatures between 4 and 12

Private communication from Stefan Zeuner.

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79

Fig. 33. Dynamical response for the superlattice of [31]. (a) Current–;eld relation without irradiation. (b) Current response, full line I1cos , dashed line I1sin for ;xed eFdc d = 8 meV and ˝# = 5 meV. (c) ditto for ;xed eFac d = 1 meV and ˝# = 5 meV. (d) ditto for ;xed eFac d = 1 meV and eFdc d = 8 meV.

35 K. At eFd = 8 meV we have positive diDerential conductance, but the ;nite distance derivative for ˝# = 5 meV is clearly negative as indicated in the ;gure (see also [246]). Thus I1cos will be negative as long as eFac d is not too large and the terms with ‘ ¿ 1 become important in Eq. (198). From Fig. 33b, one obtains a negative dynamical conductance gdyn = −33 A=V at ˝# = 5 meV. Due to the higher-order terms, the dynamical conductance becomes positive for ac-;eld strengths larger than 8:5 mV per period. Fig. 33c and d show that negative dynamical conductance persists over a wide range of dc-bias and frequency. Nevertheless, one must note that for eFdc d ¿ 9 meV the static conductance dIdc =d(Fd) becomes negative yielding domain formation as observed experimentally (see Fig. 12). Unfortunately, the negative dynamical conductance of the device considered here is compensated by the contact resistance. Measurements of the temperature-dependent conductance of the sample yield values up to 20 S (around T = 30 K) at zero bias [120]. As a part of the resistance is from the superlattice itself one may conclude that the contact resistance Rc is definitely smaller than 50 kZ in the sample. In comparison, the dynamical resistance for 10 wells is Rdyn = 10=gdyn = −300 kZ, which seems to dominate. However, capacitive eDects have to be taken into account: the sample capacitance per period is given by cs = ACC0 =d ≈ 46 fF. This gives a total impedance Z = Rc + N

1 − i#(cs + ct )=gdyn 1 : = Rc + Rdyn gdyn + i#(cs + ct ) 1 + (#(cs + ct )=gdyn )2

(206)

As cs =gdyn = −1:4 ns, the negative dynamical resistance will be compensated even by a small contact resistance at THz frequencies. Thus, signi;cantly larger values of gdyn (i.e. higher current

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Fig. 34. Dynamical response for the superlattice of [31]. (a) Current response (full line I1cos , dashed line I1sin ) for ;xed ˝# = 1 meV, (b) for ;xed eFdc d = 50 meV. In both cases eFac d = 0:1 meV.

densities) are necessary for the observation of gain. Nevertheless, the eDect discussed here is quite general and therefore gain should be observable in superlattice structures with special shapes for the dc-characteristic. 6.3.3. Tunneling capacitance Now, we want to investigate the reactive current I1sin from Eq. (199). As can be seen in Fig. 33d as well as in Fig. 34b, I1sin ˙ # for low frequencies in the linear response region (i.e., small irradiation ;elds). This can be interpreted as a tunneling capacitance I1sin ≈ −ct #Fac d

(207)

as shown in Eq. (201). In Fig. 34a, the reactive current I1sin is displayed as a function of Fdc using a small irradiation ;eld and a small frequency, so that Eq. (207) holds. The quantity I1sin shows a very characteristic behavior around the a → b resonance with a minimum at eFdc d = 50 meV, where the diDerential conductance is positive. This behavior can be understood within the approximation (205), yielding ct =

a b a b 2 b; eD =2)2 ] e˝ d 2 K(eFd) 2 ab 2 2(eFd + E − E )[(eFd + E − E ) − 3(7 = e N A | H | ; D 2 d(eFd)2 [(eFd + E a − E b )2 + (7b; eD =2)2 ]3 (208)

which provides just the structure observed around eFdc e ≈ 50 meV. From Fig. 34b, one obtains ct = 0:67 fF at eFdc d = 50 meV, which is about 1.7% of the sample capacitance cs . It would be interesting if this tunneling capacitance could be measured. Here it may be useful to study superlattice structures with higher doping and larger coupling, which enhances the ratio between ct =cs , as can be seen from Eq. (208). 6.3.4. Harmonic generation % In order to investigate harmonic generation, the quantities |Ih | = (Ihcos )2 + (Ihsin )2 have been plotted for Fdc = 0 in Fig. 35. The basic frequency |I1 | dominates for low irradiation ;elds, while the other quantities vanish like |I3 | ∼ (eFac d)3 and |I5 | ∼ (eFac d)5 as expected from the low-frequency limit (190). The current of the third harmonic is in the range of the dc-current at the ;rst peak indicating strong harmonic generation. In the range eFac d ¿ 50 meV; the

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81

Fig. 35. Generation of harmonics |I1 | (full line), |I3 | (dashed line), and |I5 | (dotted line) for two diDerent frequencies for the superlattice of [31]. As Fdc = 0, the even harmonics vanish.

a → b resonance becomes of importance which can be understood from the classical behavior Idc (eFac d cos(#t)). This provides a possibility to probe the second resonance even if it is not directly accessible due to sample heating. A similar idea to probe a peak by its response to an applied frequency has been performed in [247]. 6.4. Discussion In this section, it has been shown that transport under irradiation is essentially governed by Eqs. (197) – (199). This scheme holds both for miniband transport within the relaxation time approximation and sequential tunneling for constant coupling matrix elements, albeit with diDerent functions Idc (eFd) and K(eFd). Therefore the question arises, whether this structure might be general. Although this issue has not been settled ;nally, I believe that this is not the case. If one considers next-nearest-neighbor tunneling processes within the model of sequential tunneling, it becomes obvious that photon-assisted peaks occur at ;eld strengths 2eFdc d = 2eFpeak d + ‘˝# in contrast to the structure of Eq. (197). (This might provide an interesting tool to investigate tunneling processes where tunneling occurs between wells separated by more than one barrier [62,127,126].) The same behavior is found in a miniband model containing higher Fourier components in the band structure [248]. Therefore, it seems that the structure of Eqs. (197) – (199) is limited to next-neighbor tunneling, or a cosine-shaped band structure E(q), respectively. Further deviations may occur if the THz-;eld causes additional heating of the electron gas, 13 which had been neglected in the models applied here. The generic structure of the functions I (eFd) and K(eFd) is given by Eqs. (195), (196), (204) and (205). These imply the following typical eDects under irradiation: • Photon-assisted peaks in the recti;ed current response at characteristic ;eld strengths eFd ≈ eFpeak d + ‘˝# with integer values ‘ for next-neighbor coupling. • Absolute negative conductance if the normalized ac-;eld strength  is close to zeros of J0 () and ˝# & 7. • Gain in most of the region of negative diDerential conductivity dIdc =dF ¡ 0. 13

This was pointed out by A.A. Ignatov (private communication, 1998).

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• A quantum capacitance with the characteristic dependence (208) on the dc-;eld close to the

resonances.

• Generation of higher harmonics.

These eDects rely on the structure of Eqs. (197) – (199) and should therefore (at least approximately) hold for both wide and narrow minibands. In addition, it has been shown that gain in the THz range is also possible in the region of positive diDerential conductance for appropriate shapes of the Idc (eFd)-relation. 7. Summary In this review, the transport properties of semiconductor superlattices have been analyzed. Strong emphasis has been given to the microscopic modeling of the stationary transport for a homogeneous electrical ;eld as well as the formation of inhomogeneous ;eld distributions leading to stationary ;eld domains and self-sustained current oscillations. The three diDerent standard approaches, miniband transport, Wannier–Stark hopping, and sequential tunneling, have been reviewed in detail. Although the concepts applied are quite diDerent, each approach provides negative diDerential conductivity for suMciently high electric ;elds. While miniband transport and sequential tunneling provide an Ohmic behavior for low electric ;elds together with a maximum current around eFd ≈ 7, Wannier–Stark hopping fails in the low-;eld region. In particular, good quantitative agreement with various transport measurements in weakly coupled superlattice structures has been obtained within the sequential tunneling model, both with and without irradiation. The relation between the standard transport approaches could be identi;ed by considering quantum transport based on nonequilibrium Green functions. It has been explicitly shown that the equations used for these simpli;ed models can be obtained from the quantum transport model by applying various approximations. This justi;es each of these approaches and sheds light on the respective ranges of applicability sketched in Fig. 7. Good quantitative agreement was found between self-consistent solutions of the quantum transport model with each of the simpli;ed approaches in their respective ranges of applicability. We have shown how diDerent aspects of nonlinear pattern formation in semiconductor superlattices, such as the formation of stationary ;eld domains as well as self-sustained oscillations, can be understood by the properties of traveling fronts. Fig. 27 shows that depletion and accumulation fronts travel with diDerent velocities, which depend on the total current acting as a global coupling. This situation is similar to the Gunn diode albeit the discreteness of the superlattice structure allows for stationary fronts in a ;nite interval of current, when the front becomes trapped. As this trapping may occur in each of the wells, one obtains several branches (of the order of superlattice periods) in the global current–voltage characteristics, which exhibits a typical saw-tooth shape. The calculated current–voltage characteristics as well as the self-sustained oscillations are in reasonable quantitative agreement with experimental ;ndings. Furthermore, the conditions for the occurrence of self-sustained oscillations have been discussed. Under strong irradiation by a THz ;eld, photon-assisted peaks appear, and absolute negative conductivity is possible. Both eDects are predicted in a similar way by the sequential tunneling model and miniband transport. It has been demonstrated that there is a possibility for gain even

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83

in the region where the low-frequency conductivity is positive (i.e. no ;eld-domain formation eDects are expected). In addition, the tunneling processes are connected with characteristic variations in the capacitance of the structure, which have not been observed so far.

8. Outlook Even 30 years after the proposal by Esaki and Tsu, semiconductor superlattices continue to be a hot topic of ongoing research. Presently the following directions stand out: Superlattices with lower dimension: In this work, superlattice structures have been considered, where a free electron behavior is present in the two directions parallel to the layers. Presently, the ;rst experiments have been performed where these lateral directions are con;ned. In Ref. [249] the measurement of the conduction through a stack of 50 InAs quantum dots was reported (see also [250]). Such structures can be regarded as a superlattice structure consisting of zero-dimensional boxes. Negative diDerential resistance was observed recently in a superlattice formed by quantum wires, fabricated by the method of cleaved edge overgrowth [251]. Due to the restricted phase space perpendicular to the transport direction, scattering should be strongly reduced in these structures, and strong eDects related to the miniband structure are likely to be observed. Theoretical approaches to transport in such structures can be found in Refs. [252,253] for quantum box superlattices and in Ref. [254] for a superlattice formed by quantum wires. Self-sustained current oscillations: As discussed in Section 5, traveling ;eld domains cause self-generated current oscillations in superlattices. While the ;rst experiments reported frequencies in the MHz range [19] at 5 K for a weakly coupled superlattice, frequencies up to 150 GHz could be observed even at room temperature in specially designed superlattices with a large miniband width [20]. While it may be diMcult to increase the fundamental frequency, the use of higher harmonics can be helpful for possible devices in the THz-range, although the prospects of this approach are under debate [255]. Field domains in strongly coupled superlattices: The formation of ;eld domains is usually considered within the model of rate equation presented in Section 5.1. This model is justi;ed for weakly coupled superlattices, where the electrons can be considered to be localized in single quantum wells. This localization becomes questionable for strongly coupled superlattices and it would be desirable to develop a quantum transport model, which takes into account both the inhomogeneity and the time dependence of the ;eld distributions. (See also the discussion at the end of Section 5.) Chaos and nonlinear dynamics: If the superlattice is driven by an ac-bias, chaotic behavior is likely to occur due to the presence of incommensurable frequencies, which is well-known for the Gunn diode [256]. Simulations for semiconductor superlattices [257,258] yield similar results. These chaotic oscillations could be observed experimentally [259,260] and provide a good example, where many aspects of chaotic systems [261] can be observed. There are also reports indicating undriven chaos under dc-bias conditions in superlattices [259], which is not understood yet. In addition, frequency locking associated with the occurrence of a Devil’s staircase has been observed [262]. These examples provide an interesting ;eld for the study of general phenomena in nonlinear physics.

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Generation of THz signals: Since the original proposal by Esaki and Tsu, it has been tempting to use the Bloch oscillator as a source for THz radiation. Unfortunately, only transient signals have been observed so far. Another possibility lies in the occurrence of a negative dynamical conductance. Here the problem arises that domain formation is a competing process in this region of parameter space. Two diDerent scenarios have been proposed as a solution to this problem: In Ref. [246] it was proposed that for special shapes of the local current–;eld characteristic gain is possible even in the region where the low-frequency conductivity is positive, i.e., no ;eld-domain formation eDects are expected (see also Section 6.3.2). A diDerent possibility could be related to the fact that the low-frequency conductivity can become positive for large amplitudes of the ac ;elds [255], thus stabilizing the oscillation. It is not clear by now, if one of these eDects may be useful to establish THz devices on the basis of semiconductor superlattices. Acknowledgements This work resulted from a long collaboration with many colleagues from several diDerent groups. First, I want to thank E. SchQoll for initiating the project as well as for many stimulating discussions. In particular, several of the concepts to combine semiconductor transport and nonlinear physics have their origin in his suggestions. During my stay at the Mikroelektronik Centret in Lyngby (DK) as well as in the time afterwards I pro;ted substantially from A.-P. Jauho, who introduced me to the application of nonequilibrium Green functions in semiconductor transport. From my visits at the groups of L.L. Bonilla (Madrid) and S.J. Allen (Santa Barbara, USA) I gained much insight into the mathematical modeling of instabilities and the problems related to transport under strong THz irradiation, respectively. I am grateful for having been able to work together with A. Amann, S. Bose, J. Damzog, G. DQohler, F. Elsholz, E. Gornik, H.T. Grahn, J. Grenzer, M. Helm, B. Yu-Kuang Hu, A. Ignatov, K. Johnson, J. Kastrup, G. Kie\lich, A. Kristensen, A. Markus, Y.A. Mityagin, M. Moscoso, P.E. Lindelof, M. Patra, G. Platero, F. Prengel, C. Rauch, K.F. Renk, S. Rott, E. Schomburg, G. Schwarz, J.S. Scott, H. Steuer, M.C. Wanke, S. Winnerl, and S. Zeuner, who contributed substantially to the development of the basic ideas reported here. Last but not the least I would like to thank the Deutsche Forschungsgemeinschaft for providing me with a grant for my stay in Lyngby as well as for ;nancial support within the Sfb296. Appendix A. Sequential tunneling with density matrices As discussed in Section 3.3, the electrical transport in weakly coupled superlattices can be described by sequential tunneling. The idea of this concept is that the states are essentially localized in single wells and the residual coupling causes transitions between neighboring wells. As this coupling (T1 for equivalent levels) is small, one can restrict the theory to the lowest order T12 , which provides, essentially, Fermi’s golden rule. Already in Section 3.3 it was mentioned that scattering induced broadening is essential to recover the correct behavior for ;nite ;elds. This complication was treated within the theory of nonequilibrium Green functions which provided Eq. (79) as derived in Appendix B.1.

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An alternative way to treat quantum transport is the density matrix theory which also gives an exact treatment of both quantum eDects and scattering. A recent overview can be found in [140]. Density matrix theory has been applied to superlattice transport in [73] for the evaluation of currents between nonequivalent levels. To my knowledge no such calculations exist regarding transport between equivalent levels with identical particle densities. In this case, the calculation of [73] provides zero current independent of the electric ;eld as shown below. A possible resolution of this problem will be presented in this appendix and a form similar to Eq. (83) will be derived. This demonstrates the equivalence of both approaches and highlights the diDerences in performance of both approaches. A.1. The model We use the Hamiltonian (23) and (24) in the basis of Wannier states and restrict ourselves to the lowest band for simplicity. Furthermore, impurity and phonon scattering is taken into account within the restriction of intrawell scattering. Then the Hamiltonin reads as Hˆ = Hˆ 0 + Uˆ + Vˆimp + Vˆphon with
(E a + Ek − eFdn)a†n (k)an (k) ; (A.1) Hˆ 0 = n;k

Uˆ =


n;k

T1 [a†n+1 (k)an (k) + a†n (k)an+1 (k)] ;




Vˆimp =

n;k;k

Vˆphon =

Vkn k a†n (k )an (k) ;


n;p

˝!p b†n (p)bn (p) +

(A.2) (A.3)




Mpn a†n (k + p)[bn (p) + b†n (−p)]an (k) :

(A.4)

n;k;p

In the following, we assume that phonon scattering is strong enough to establish thermal equilibrium. Thus, a thermal distribution function with chemical distribution &n can be assumed for each well. Furthermore, let us assume that impurity scattering is stronger than phonon scattering and dominates the broadening of the states, which essentially simpli;es the following calculations. Correlations between the scattering matrix elements V n in diDerent wells are neglected. (These drastic approximations make sense as we are mainly interested in the structure of the theory, not in quantitative results. The inclusion of diDerent scattering mechanisms should be possible in an analogous way.) In this case, the self-energy within the Born approximation for impurity averaging is given by

k

|Vkkn  |2

≈ −i


k

1 E − Ek  + neFd + i0+

|Vkkn  |2 (E − Ek  + neFd) ≈ −i7=2 ;

(A.5)

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where we neglected the real part and assumed that the scattering rate is energy independent for simplicity. Using Eq. (102) one obtains the current density from well n to well n + 1 2(for spin)e
2 Jn→n+1 = Im{T1 a†n+1 (k)an (k)} : (A.6) A ˝ k

The task is the evaluation of the current in lowest order T12 with respect to the coupling (sequential tunneling). Under these conditions the theory of nonequilibrium Green functions provides Eq. (79) which can be simpli;ed to Eq. (83) for the self-energy (A.5). A.2. Density matrix theory The key point is the temporal evaluation of a†n+1 (k; t)an (k; t). Similar to Eq. (101) the dynamics is given by ˝d

i dt

a†n+1 (k; t)an (k; t) = [Hˆ ; a†n+1 (k)an (k)

= −eFda†n+1 (k)an (k) + T1 (a†n (k)an (k) − a†n+1 (k)an+1 (k))
† †  n  + [Vkn+1 (A.7)  k an+1 (k )an (k) − Vkk  an+1 (k)an (k )] : k

Here (and in the following) terms containing density matrices a†n+2 (k )an (k) will be neglected as they provide terms of order T14 in the current. As stationary states will be considered, the time dependence is dropped in most of the density matrices. On the right-hand side new expressions of the type a†n+1 (k )an (k) appear. Their temporal evolution is given by ˝d

†  n+1 †  Vkn+1  k an+1 (k ; t)an (k; t) = (Ek  − Ek − eFd)Vk  k an+1 (k )an (k) i dt

† n+1 †    + Vkn+1 Vkkn  Vkn+1  k  Vk  k an+1 (k )an (k) −  k an+1 (k )an (k ) k

†  +T1 [Vkn+1  k an (k )an (k)

−

k n+1 † Vk  k an+1 (k )an+1 (k)]

(A.8)

and ˝d

Vkkn  a†n+1 (k; t)an (k ; t) = (Ek − Ek  − eFd)Vkkn  a†n+1 (k)an (k ) i dt

† n   + Vkn+1 Vkn k  Vkkn  a†n+1 (k)an (k )  k Vkk  an+1 (k )an (k ) − k

+T1 [Vkkn  a†n (k)an (k )

−

k † n Vkk  an+1 (k)an+1 (k )]

:

(A.9)

These equations have again to be solved in the stationary state. In the following, the key quantities of interest are the occupation fn (k) of the mode k in well n and the corresponding polarizations Pn (k) which provide the current. They are given by fn (k) = a†n (k)an (k) and

Pn (k) = a†n+1 (k)an (k) :

(A.10)

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In the following, the more complicated density matrices appearing in Eqs. (A.8) and (A.9) have to be related to these quantities. A.2.1. Lowest-order calculation In [73] only the lowest-order terms within the impurity averaging process and the coupling have been considered for the evaluation of Eq. (A.7). As the current is already of order T1 a†n+1 (k)an (k) only terms up to T1 or V 2 will be taken into account. Due to impurity averagn+1 2 remains then in Eq. (A.8) and we ;nd in the stationary = |Vkn+1 ing only the term Vkkn+1  Vk  k k | state: †  Vkn+1  k an+1 (k )an (k) =

−1 |V n+1 |2 Pn (k) ;  Ek  − Ek − eFd + i0+ k k

(A.11)

where the term i0+ ensures that the correlations vanish for t → −∞. (An alternative way to obtain the factor i0+ is the application of the Markov limit, see [140].) Using the same argument we obtain from Eq. (A.9) Vkkn  a†n+1 (k)an (k ) =

1 |V n |2 Pn (k) : Ek − Ek  − eFd + i0+ k k

(A.12)

Inserting into Eq. (A.7) gives ˝d

i dt

Pn (k; t) = −eFdPn (k) + T1 [fn (k) − fn+1 (k)] + i7Pn (k) ;

(A.13)

where approximation (A.5) has been used. This equation has also been applied in [263] to study time-dependent phenomena. The stationary solution yields Pn (k) =

−1

−eFd + i7

T1 [fn (k) − fn+1 (k)]

(A.14)

and we obtain the current density via Eq. (A.6) Jn→n+1 = 2e

T12
27 [fn (k) − fn+1 (k)] : ˝A (eFd)2 + 72 k

(A.15)

 Taking into account the fact that %(E) = 2(forspin)=A k (E − Ek ) is the density of states, this expression is almost identical with Eq. (83). Nevertheless, there is a signi;cant diDerence: while in Eq. (83) the transport is driven by the diDerence of the occupation at the same energy, now the transport is driven by the diDerence of the occupation of the state k in both wells. The latter diDerence becomes zero, if both wells have the same electron density. This has led to the conclusion that no resonant tunneling peak occurs in weakly coupled superlattices for tunneling between equivalent levels [8], in contrast to the ;ndings of Section 3.3. Regarding tunneling between the ground state and excited states (which are typically empty) the respective formula provides a ;nite current which essentially agrees with the corresponding result of Eq. (83). The corresponding calculations have been presented in [73], where interwell correlations between scattering matrix elements were also taken into account.

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A.2.2. Improved treatment This problem can be circumvented by taking into account the last term in Eqs. (A.8) and (A.9) as well. Neglecting terms containing the coupling T1 (which provide terms ∼ T14 in the  current) the dynamics of Vkn k a†n (k )an (k) is given by ˝d

i dt





Vkn k a†n (k ; t)an (k; t) = (Ek  − Ek )Vkn k a†n (k )an (k) + −


k

n kk


k



Vkn k  Vkn k a†n (k )an (k)

Vkkn  V a†n (k )an (k ) :

(A.16)

By impurity averaging all terms V 2 vanish unless n = n and k  = k or k  = k  . Thus, we ;nd in the stationary state: 

Vkn k a†n (k )an (k) =

−1

Ek  − Ek + i0+

|Vkn k |2 n; n [fn (k) − fn (k )] :

(A.17)

Then the evaluation of Eq. (A.8) yields †  Vkn+1  k an+1 (k )an (k)   −1 T1 n+1 2 n+1 2  |Vk  k | Pn (k) + = |V  | [fn+1 (k) − fn+1 (k )] Ek  − Ek − eFd + i0+ Ek  − Ek + i0+ k k −1 = |V n+1 |2 Pn (k)  Ek  − Ek − eFd+ i0+ k k  1 1 T1 n+1 2 |V  | − [fn+1 (k) − fn+1 (k )] + eFd k k Ek  − Ek + i0+ Ek  − Ek − eFd + i0+ (A.18)

and Eq. (A.9) gives 1 Vkkn  a†n+1 (k)an (k ) = |V n |2 Pn (k) + kk  E − E − eFd + i0 k k   T1 1 1 n 2 − |V  | − [fn (k ) − fn (k)] : eFd k k Ek − Ek  + i0+ Ek − Ek  − eFd + i0+

(A.19)

This provides further terms for Eq. (A.7), which becomes ˝d

Pn (k; t) = −eFdPn (k) + T1 [fn (k) − fn+1 (k)] + i7Pn (k)   T1
1 1 2  |Vkn+1 − −  k | [fn+1 (k ) − fn+1 (k)] eFd  Ek  − Ek + i0+ Ek  − Ek − eFd + i0+ k   1 1 T1
+ |Vkn k |2 [fn (k ) − fn (k)] : (A.20) − eFd  Ek − Ek  + i0+ Ek − Ek  − eFd + i0+

i dt

k

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89

Now, we apply the relation 1=(x +i0+ )= P{1=x} − i (x) and restrict ourselves to the imaginary parts. The isotropy in k-space gives fn (k) = fn (Ek ) and we ;nd with Eq. (A.5), the stationary solution Pn (k) =

1 {T1 [fn (Ek ) − fn+1 (Ek )] eFd − i7 T1 i7 + [fn+1 (Ek ) − fn+1 (Ek + eFd) − fn (Ek ) + fn (Ek − eFd)]} eFd 2

(A.21)

and the current (A.6) becomes T12
27 ˝A (eFd)2 + 72 k   fn (Ek ) + fn (Ek − eFd) fn+1 (Ek + eFd) + fn+1 (Ek ) × − : 2 2

Jn→n+1 = 2e

(A.22)

In contrast to Eq. (A.15), the current is now driven by the occupation diDerence taken at diDerent values of Ek so that the total energy Ek − neFd is equal in both wells. This structure agrees with Eq. (83) and a ;nite current is found for identical occupation function fn (Ek ) = fn+1 (Ek ).  This shows that the terms Vkn k a†n (k )an (k), which had been neglected in the derivation of Eq. (A.15), are of crucial importance. They describe the internal correlations in the single quantum wells. These correlations correspond to the broadening of states in the Green function formalism, where they are taken into account by treating the probe energy E and the energy of the bare state Ek separately. As the broadening of the states is of crucial importance for the tunneling current, it becomes clear that the density matrix theory gives a wrong result if the corresponding matrices are neglected. Appendix B. Derivation of the standard approaches In this appendix the relations between the quantum transport equations of Section 4 and the standard approaches (miniband transport, Wannier–Stark hopping, and sequential tunneling as discussed in Section 3) are examined. It will be shown, that the transport equations for the diDerent standard approaches can be derived explicitly from the quantum transport model using various types of approximations. In each case, the respective approximations can be justi;ed within the range of validity of the given standard approach sketched in Fig. 7 and motivated in Section 4.3. B.1. Sequential tunneling In the parameter ranges 2T1 7 or 2T1 eFd the diagonal elements of Gn; m dominate (see Section 4.3) and an expansion of Eqs. (153) and (154) in T1 is appropriate. In this way, the formula for sequential tunneling (79) will be recovered as the leading order in T1 of the general Eq. (148).

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A. Wacker / Physics Reports 357 (2002) 1–111 ¡=ret

˜ For T1 = 0 we obtain Gm¡=ret 1 ;m2 (k; E) = m1 ;m2 G m1

(k; E) which are determined by

ret ret (E − Ek + eFdm − <˜ m (k; E))G˜ m (k; E) = 1

(B.1)

¡ ret ¡ adv G˜ m (k; E) = G˜ m (k; E)<˜ m (k; E)G˜ m (k; E) ;

(B.2)

and where the self-energies <˜ m are evaluated applying the Green-functions G˜ m . These equations decouple in the well index. As no current Jows in this case, one obtains the equilibrium solution ¡ G˜ m (k; E) = iA˜ m (k; E)nF (E − &m + meFd) ;

(B.3)

¡ adv ret <˜ m (k; E) = [<˜ m (k; E) − <˜ m (k; E)]nF (E − &m + meFd)

(B.4)

with the spectral function ret adv ret adv A˜ m (k; E) = i[G˜ m (k; E) − G˜ m (k; E)] = −2 Im{G˜ m (k; E)} = 2 Im{G˜ m (k; E)} :

(B.5)

To ;rst order in T1 , Eq. (153) together with Eq. (B.1) gives ret 2 ˜ ret ˜ ret Gm±1; m (k; E) = G m±1 (k; E)T1 G m (k; E) + O(T1 ) ;

(B.6)

ret 2 ˜ ret Gm; m (k; E) = G m (k; E) + O(T1 ) ;

(B.7)

Gn;retm (k; E) = O(T12 )

(B.8)

for |m − n| ¿ 2

and Eq. (154) gives ¡ ˜ ret ˜¡ ˜ adv ˜ ret Gm+1; m (k; E) = G m+1 (k; E)T1 G m (k; E)<m (k; E)G m (k; E)

ret ¡ adv adv + G˜ m+1 (k; E)<˜ m+1 (k; E)G˜ m+1 (k; E)T1 G˜ m (k; E) + O(T12 ) ret ¡ ¡ adv = T1 [G˜ m+1 (k; E)G˜ m (k; E) + G˜ m+1 (k; E)G˜ m (k; E)] + O(T12 ) ret

= iT1 [G˜ m+1 (k; E)A˜ m (k; E)nF (E − &m + meFd) adv + A˜ m+1 (k; E)nF (E − &m+1 + (m + 1)eFd)G˜ m (k; E)] + O(T12 ) :

(B.9)

Then the current is evaluated by Eq. (148)  2e
2 dE ¡ a→a Jm→m+1 = Re{T1 Gm+1; m (k; E)} A ˝ 2 k  2e
|T1 |2 dE ˜ = Am+1 (k; E)A˜ m (k; E) A ˝ 2 k

×[nF (E − &m + meFd) − nF (E − &m+1 + (m + 1)eFd)] + O(T13 ) ;

which is just the expression (79) used for sequential tunneling.

(B.10)

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91

B.2. Miniband conduction If 2|T1 |7; eFd, the states in a semiconductor superlattice are essentially delocalized, as shown in Section 4.3. In this case it makes sense to work in an extended basis like the Bloch states q. The key point is the idea, that the occupation of these states can be treated as a semiclassical distribution function f(q; k) neglecting quantum mechanical correlations. We will show that the Boltzmann equation for f(q; k) as well as the formula for the current density (see Section 3.1) can be derived from the full quantum transport model under this assumption. We consider a superlattice with a homogeneous electric ;eld F and a homogeneous carrier distribution. In this case, it makes sense to use a local energy scale E = E − e'n which refers to the bottom of the respective quantum well (here e'n = −neFd holds). We de;ne   n+m ] G n; m (k; E) = Gn; m k; E − eFd (B.11) 2 and <] n; m (k; E) in the same way. Now, G] m; n (k; E) only depends on the diDerence m − n and ] E) due to the homogeneity of the system under stationary transport. Therefore, <] m (k; E) = <(k; it is helpful to de;ne the spatial Fourier transform as
] k; E) = G(q; e−iqhd G] n+h; n (k; E) : (B.12) h

This corresponds to the Fourier representation
1  ] k; E) = G(q; dt ei(E+e ('n +'m )=2)t=˝ e−iqhd Gn+h; n (k; t + t2 ; t2 ) ; ˝

(B.13)

h

which is the special case for a homogeneous electric ;eld of the general gauge invariant version used in Section 7 of [39] to obtain gauge invariant quantities [144]. The respective diagonal element of the density matrix is de;ned by  
1 1 ¡ ¡ ] f(q; k) = d E G (q; k; E) = dE e−iqhd G] n+h; n (k; E) : (B.14) 2i 2i h

Applying Eq. (155) we obtain with these de;nitions      eFd 9 ]¡ eFd ¡ ¡ ] ] ieF G (q; k; E) = 2i sin(qd)T1 G −G q; k; E + q; k; E − 9q 2 2     
heFd heFd ¡ adv ret ¡ −iqhd ] ] ] ] + e k; E + k; E − < G h; 0 (k; E) − G h; 0 (k; E)< 2 2 h       heFd heFd adv ret ¡ − <] k; E − k; E + G] h; 0 (k; E) ; − <] 2 2

(B.15)

which is still exact. In the same way, Eq. (153) yields       ieF 9 ] ret eFd eFd ret ret iqd −iqd −e E − Ek + q; k; E + T1 G] q; k; E − G (q; k; E) − e T1 G] 2 9q 2 2  
heFd ] ret ret =1 + e−iqhd <] k; E + (B.16) G h; 0 (k; E) : 2 h

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Assuming that the self-energy does not depend strongly on E 14 within the energy scale eFd, ret ret i.e. <] (k; E + heFd=2) ≈ <] (k; E), this equation is solved by $ # ret 2G ] 9 (q; k; E ) 1 ret : (B.17) + O (eFd)2 G] (q; k; E) = ret 9E2 ] E − Ek − 2T1 cos(qd) − < (k; E) ret On the energy scale 2|T1 |; the Green function G] (q; k; E) essentially resembles the free particle ret Green function 1=(E − Ek − 2T1 cos(qd) + i0+ ) if <] ∼ 72|T1 | and eFd2|T1 | (so that the last term is negligible), which is just the condition (163). Now, we integrate both sides of Eq. (B.15) over energy E, so that the T1 terms on the right-hand side cancel each other. In the terms containing the self-energies, the following approximations are performed: • The heFd=2 terms in energy dependence of the self-energies are neglected. adv ret • The expression [G] (q; k; E) − G] (q; k; E)] is approximated by 2i (E − 2T1 cos(qd) − Ek ) which holds exactly for the free particle Green functions. ¡ • We use G] (q; k; E) ≈ f(q; k)2i (E− 2T1 cos(qd) − Ek ). This means that the energetical width of the respective states is neglected. As information about quantum mechanical correlations is stored in the energy dependence (the Fourier transform of the time diDerence t1 − t2 ), this approximation provides quasiclassical particles with speci;c momenta q; k. Then one ;nds 9 ¡ adv ret ieF f(q; k) = <] (k; Eq + Ek ) − (<] (k; Eq + Ek ) − <] (k; Eq + Ek ))f(q; k) : (B.18) 9q Now, the same approximations are used in the evaluations of the self-energy for impurity scattering (150): 
d =d  ¡   ¡ < (k; Eq + Ek ) = Vnk; nk ({˜ri })Vnk ; nk ({˜ri })imp dq G (q ; k ; Eq + Ek ) 2 −=d k
d  =d dq 2Vnk; nk ({˜ri })Vnk ; nk ({˜ri })imp = 2 −=d  k

×f(q ; k )i (Eq + Ek − Eq − Ek  ) ;

(B.19) which is just (up to the factor i) the in-scattering term from Fermi’s golden rule. In the same way the term
A. Wacker / Physics Reports 357 (2002) 1–111

Finally, the current density is evaluated via Eq. (148) which takes the form # $   2e
2 dE d =d Jn→n+1 = dq eiqd G ¡ (q; k; E) Re T1 A ˝ 2 2 −=d k 
2 d =d 2e = dq Re{T1 eiqd if(q; k)} A ˝ 2 −=d k 
2e 1 =d −2T1 d sin(qd) = dq f(q; k) ; A 2 −=d ˝

93

(B.20)

k

which is just Eq. (37). B.3. Wannier–Stark hopping In Section 4.3 it was shown that the Wannier–Stark states become resolved for eFd7. Here it will be shown that Eqs. (64) and (69) for the self-consistent Wannier–Stark hopping model can be derived from that of the quantum transport in this limit. Similar to the derivation of miniband transport, the key point is the idea that the occupation of the Wannier–Stark states j can be treated as a semiclassical distribution function fj (k) neglecting quantum mechanical correlations. Similar derivations have been presented in [112,252], where essentially the same approximations have been applied as those like in the quantum transport model discussed in this work.  According to Eq. (31) the Wannier–Stark states are |*j  = n Jn−j ($)|)n  with $ = 2T1 =eFd. The respective Green functions are given by
GWSj1 ;j2 (k; E) = Jm1 −j1 ($)Jm2 −j2 ($)Gm1 ;m2 (k; E) : (B.21) m1 ; m2

Within this new basis Eq. (153) becomes ret (k; E) = j1 ;j2 + (E − Ek + j1 eFd)GWSj 1 ;j2


j

ret ret <WSj (k; E)GWSj; j2 (k; E) 1 ;j

(B.22)

with the de;nition <WSj1 ;j2 (k; E) =




Jm−j1 ($)Jm−j2 ($)<m (k; E)

m

=





Jm−j1 ($)Jm−j2 ($)|Vk k |2 Jm−j3 ($)Jm−j4 ($)GWSj3 ;j4 (k ; E) ;

(B.23)

m; j3 ; j4 k

where the Born approximation was applied under the assumption that scattering is diagonal in the Wannier basis, independent on the well number m, and that no correlations exist between diDerent wells m. Furthermore, we restrict ourselves to impurity scattering with matrix element

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)m; k |Hˆ

scatt

|)m; k  = Vk k for simplicity. In the same way, Eq. (155) gives


¡ (j1 − j2 )eFdGWSj (k; E) = 1 ;j2

ret ¡ ¡ adv [<WSj (k; E)GWSj; j2 (k; E) + <WSj1 ;j (k; E)GWSj; j2 (k; E) 1 ;j

j ret ¡ −GWSj1 ;j (k; E)<WSj; j2 (k; E) −

¡ adv GWSj (k; E)<WSj; j2 (k; E)] : 1 ;j

(B.24)

The typical energy scale is given by eFd. If the self-energies (of the order of 7) are small in comparison to eFd, Eq. (B.22) gives 1 ret (k; E) ≈ j1 ;j2 (B.25) GWSj 1 ;j2 E − Ek + j1 eFd + i0+ describing free-particle Wannier–Stark states. The occupation of these states is governed by a semiclassical distribution fj1 (k) yielding ¡ GWSj (k; E) ≈ 2i j1 ;j2 fj1 (k) (E − Ek + j1 eFd) : 1 ;j2

(B.26)

Let us ;rst consider the case j1 =j2 , when the left-hand side of Eq. (B.24) vanishes. Applying approximations (B.25)and (B.26) in the scattering term on the right-hand side and performing the integration 1=(2) dE we obtain ret adv ¡ 0 = i[<WSj (k; Ek − j1 eFd) − <WSj (k; Ek − j1 eFd)]fj1 (k) + i<WSj (k; Ek − j1 eFd) : 1 ;j1 1 ;j1 1 ;j1 (B.27)

Inserting Eq. (B.23) one obtains together with the approximations (B.25) and (B.26):
0= [Jm−j1 ($)]2 |Vk k |2 [Jm−j2 ($)]2 2 (Ek − j1 eFd − (Ek  − j2 eFd)) [fj1 (k) − fj2 (k )] j2 ; m;k

=˝


k j

[Rj1 ;k→j2 ;k fj1 (k) − Rj2 ;k →j1 ;k fj2 (k )] ;

(B.28)

2

where Eq. (67) has been inserted. This is just the condition for self-consistency (69) in the stationary case for the Wannier–Stark hopping approach. The current is determined from Eq. (148) which can be rewritten as  2e
dE J0→1 = T1 Re{G1;¡0 (k; E) − G0;¡1 (k; E)} A˝ 2 k  dE 2e

¡ = mJm ($)Jm−j1 ($) (k; E)} : (B.29) Re{j1 eFdGWSj 1 ;0 A˝ 2 j1 ;k m  ¡ (k; E) = this identity insert all de;nitions into the second line and use dE Gm; n (To verify ¡ dE Gm−n; 0 (k; E) for the homogeneous system.) Now, Eq. (B.24) can be inserted, where the approximations (B.25) and (B.26) are applied to the scattering terms. This yields  2e

1 ret ¡ J= mJm ($)Jm−j1 ($) −f0 (k)Im{<WSj (k; Ek )} − Im{<WSj (k; Ek )} 1 ;0 1 ;0 A˝ 2 j1 ;k m  1 ¡ adv (B.30) − Im{<WSj1 ;0 (k; Ek − j1 eFd)} + fj1 (k)Im{<WSj1 ;0 (k; Ek − j1 eFd)} ; 2

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95

where it has been used that <¡ is purely imaginary. Now, we de;ne the auxiliary function
|Vk k |2  (Ek − jeFd − Ek  + j  eFd) [fj (k) − fj (k )] (B.31) faux (j  − j) = k;k

which is an odd function of the diDerence j  − j for a homogeneous situation, when the occupation functions are independent from the index j. Inserting Eq. (B.23) the 7rst two summands of Eq. (B.30) yield 2e
mJm ($)Jm−j1 ($)Jn−j1 ($)Jn ($) [Jn−j2 ($)]2 faux (j2 ) A˝ j ; m; n; j 1 2 2e
= n[Jn ($)]2 [Jn−j2 ($)]2 faux (j2 ) A˝ n; j 2 2e
= h[Jn ($)]2 [Jn−h ($)]2 faux (h) ; (B.32) A˝ n;h¿1   where, in the last line, the relation n n[Jn ($)]2 [Jn−j2 ($)]2 = j22 n [Jn ($)]2 [Jn−j2 ($)]2 and the symmetry for j2 = ±h was applied. Comparing with the relation (67) one obtains e
[R0; k→h; k f0 (k) − Rh; k →0; k fh (k )] ; (B.33) A  h¿1;k;k

which is just half the current density for Wannier–Stark hopping (64). Similarly, the last two summands of Eq. (B.30) yield 2e
mJm ($)Jm−j1 ($)Jn−j1 ($)Jn ($) [Jn−j2 ($)]2 faux (j2 − j1 ) A˝ j ; m; n; j 1 2 2e
= (m + j1 )Jm +j1 ($)Jm ($)Jn ($)Jn +j1 ($) [Jn −h ($)]2 faux (h) A˝   j1 ; m ; n ;h 2e
 = n [Jn ($)]2 [Jn −h ($)]2 faux (h) ; (B.34) A˝  n ;h

where in the second line n = n − j1 , m = m − j1 , and h = j2 − j1 have been introduced. In the derivation of the last line, the term with prefactor j1 vanishes by performing the m sum. The ;nal expression is identical to the second line of Eq. (B.32) and thus the full current density JWSH for Wannier–Stark hopping (64) is recovered from Eq. (B.30). Appendix C. Quantum transport under irradiation In this appendix, the quantum transport in superlattices under irradiation with a THz ;eld Fac is investigated. Together with a static ;eld Fdc one obtains a potential with diagonal elements Un (t) = −eFdc dn − ˝# cos(#t)n

(C.1)

in the Wannier basis, where  = eFac d= ˝# is the ratio between the radiation ;eld strength and its energy quantum. Similar to Section 4 the theory of nonequilibrium Green functions is

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applied. The respective equations are derived in Section C.1 for transport in the lowest miniband. In Section C.2 it is shown that Eqs. (197) – (199) together with Eqs. (202) and (203) hold for sequential tunneling between equivalent levels a → a, where the tunneling matrix element T1a does not depend on the ;eld. In contrast, the matrix element H ba = eF(t)dRba 1 = [eFdc d + ˝# cos(#t1 )]Rba for the transition between diDerent levels [see Eq. (24)] depends on time 1 which provides further complications. This will be analyzed in Section C.3, where it will be shown that Eq. (197) for the recti;ed response still holds in this case. C.1. General formulation In this subsection essentially the same approximations as in Section 4.2 are applied. In particular, the self-energies <m¡=ret (t3 ; t4 ) are assumed to be diagonal in the well index. For simplicity, we set E a = 0 here. Neglecting scattering and coupling between the wells, the bare Green function reads as gnret (t1 ; t2 ) = −i-(t1 − t2 )e−i(Ek −eFdc dn)(t1 −t2 )=˝ ein[sin(#t1 )−sin(#t2 )]

(C.2)

for potential (C.1). Similar to [268], we de;ne the on-site evolution Sm (t) = e−im sin(#t)

(C.3)

of the states and apply the following Fourier expansion for retarded functions: 
1 ret † ret Cm; (t ; t ) = S (t ) dE e−ir#t1 e−iE(t1 −t2 )=˝ Cm; n 1 2 m 1 n;r (E)Sn (t2 ) : 2 r In particular, this de;nition gives 1 ret gn;r (E) =

r; 0 : E − Ek + neFdc d + i0+ Now, investigate  1 ret ret dt3 gm (t1 ; t3 )T1 Gm±1; n (t3 ; t2 ) ˝   1 1 1 dt3 Sm† (t1 ) dE1 e−iE1 (t1 −t3 )=˝ Sm (t3 ) = ˝ 2 E1 − Ek + meFdc d + i0+ 
1 † ret ×T1 Sm±1 (t3 ) dE2 e−ir2 #t3 e−iE2 (t3 −t2 )=˝ Gm±1; n;r2 (E2 )Sn (t2 ) 2 r2   1 1 1 † = Sm (t1 ) dE1 dt3 e−iE1 (t1 −t3 )=˝ 2 ˝ E1 − Ek + meFdc d + i0+ 

1 ret ×T1 Js ()e±is#t3 dE2 e−ir2 #t3 e−iE2 (t3 −t2 )=˝ Gm±1; n;r2 (E2 )Sn (t2 ) 2 s r2 
1 1 † = Sm (t1 ) dE2 e−i(E2 =˝+r2 #∓s#)t1 2 E2 + r2 ˝# ∓ s˝# − Ek + meFd + i0+ r ;s 2

iE2 t2 =˝

×T1 Js ()e

ret Gm±1; n;r2 (E2 )Sn (t2 )

(C.4)

(C.5)

A. Wacker / Physics Reports 357 (2002) 1–111

= Sm† (t1 )

1 2



dE




e−ir#t1 e−iE(t1 −t2 )=˝ Sn (t2 )

r


1 ret T J ()Gm±1; 1 n;r2 (E) : E + r ˝# − Ek + meFdc d + i0+ r ±(r2 −r)

×

97

(C.6)

2

Similarly,   1 ret ret dt3 dt4 gm (t1 ; t3 )<mret (t3 ; t4 )Gm; n (t4 ; t2 ) ˝2 
1 dE e−ir#t1 e−iE(t1 −t2 )=˝ Sn (t2 ) = Sm† (t1 ) 2 r
1 ret ×
(C.7)

3

With these relations Eq. (108) gives the recursion for the retarded Green function: ret (E + r ˝# − Ek + meFdc d + i0+ )Gm; n;r (E)
ret ret = m; n r; 0 + T1 [J(r2 −r) ()Gm+1; n;r2 (E) + J(r−r2 ) ()Gm−1; n;r2 (E)]




+

r3

r2

ret <m;r−r (E 3

ret + r3 ˝#)Gm; n;r3 (E) :

For the advanced functions we de;ne 
1 adv † adv dE eir#t2 e−iE(t1 −t2 )=˝ Cm; Cm; n (t1 ; t2 ) = Sm (t1 ) n;r (E)Sn (t2 ) ; 2 r

(C.8)

(C.9)

adv (E) = {G ret (E)}∗ holds. The Fourier expansion of the lesser functions is de;ned so that Gm; n;r n; m;r by 
1 ¡ † ¡ Cm; n (t1 ; t2 ) = Sm (t1 ) dE e−ir#(t1 +t2 )=2 e−iE(t1 −t2 )=˝ Cm; (C.10) n;r (E)Sn (t2 ) : 2 r

Then we ;nd the product   1 ret ¡ adv dt3 dt4 Gm; m1 (t1 ; t3 )<m1 (t3 ; t4 )Gm1 ;n (t4 ; t2 ) ˝2 & &r ' '
1 
† ret dE e−ir#(t1 +t2 )=2 e−iE(t1 −t2 )=˝ Sn (t2 ) Gm; − r1 ˝# = Sm (t1 ) m1 ;r1 E + 2 2 r r1 ;r3   & ' ' &r r1 + r3 ¡ adv ×<m ˝# ; E − ˝ # G E − (C.11) + r 3 m1 ;n;r3 1 ;r−r1 +r3 2 2 so that the Keldysh relation becomes & &r ' '
Gn;retm1 ;r1 E + − r1 ˝ # Gn;¡m;r (E) = 2 r1 ;r3   & ' ' &r r1 + r3 ¡ ×<m1 ;r−r1 +r3 E − ˝# Gmadv E− + r3 ˝ # : 1 ;m;r3 2 2

(C.12)

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Eqs. (C.8) and (C.12) allow for a self-consistent solution provided the functionals for the self-energy are known. Within the self-consistent Born approximation they are given by the same functionals as in Section 4.1.2 where the same index r is added in the self-energies as well as the Green functions. This diagonal structure in r is due to the fact that the scattering matrix element is either not time dependent (for impurity scattering) or only depends on the time diDerence t1 − t2 (for phonon scattering). Finally, the current is given by 4e
¡ In→n+1 = Re{T1 Gn+1; n (t; t; k)} ˝ k # $ 
4e

dE ¡ ir#t = Js+r ()T1 Gn+1; ; (C.13) Re n;s (E; k)e ˝ 2 r s k

† Sn (t) = where Sn+1



ir  #t r  Jr  ()e

has been used. Note that for homogeneous systems

Gm; n;r (E) = Gm−n; 0;r (E + neFdc d)

and


(C.14)

holds which can simplify the calculation signi;cantly. C.2. Sequential tunneling Now, we want to derive the expression for sequential tunneling used in Section 6.3. Like in Appendix B.1 the lowest order in the coupling yields a current ∼ T12 . For vanishing coupling ˜ ¡=ret (k; E + m1 eFdc d) and <m¡=ret (k; E) = Eqs. (C.8) and (C.12) give Gm¡=ret 1 ;m2 ;r (k; E) = m1 ;m2 r; 0 G m1 ¡=ret <˜ m;r (k; E + meFdc d) r; 0 which are [up to the shift in the energy argument which eliminates the Fdc -dependence of G˜ m (k; E)] the same functions as applied in Appendix B.1 without irradiation. Thus, ¡ ret ¡ adv ˜ E)nF (E − &m ) : G˜ m (k; E) = G˜ m (k; E)<˜ m (k; E)G˜ m (k; E) = iA(k;

(C.15)

In lowest order of the coupling one ;nds from Eq. (C.8): ret ˜ ret ˜ ret Gm±1; m;r (k; E) = G m±1 (k; E + (m ± 1)eFdc d + r ˝#)T1 J±r ()G m (k; E + meFdc d) + O(T12 ) ; ret

(C.16)

ret 2 ˜ Gm; m;r (k; E) = r; 0 G m (k; E + meFdc d) + O(T1 ) ;

(C.17)

Gn;retm;r (k; E) = O(T12 )

(C.18)

for |m − n| ¿ 2

and in a similar way: 2 adv ˜ adv ˜ adv Gm±1; m;r (k; E) = G m±1 (k; E + (m ± 1)eFdc d)T1 J∓r ()G m (k; E + r ˝# + meFdc d) + O(T1 ) (C.19)

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99

With these relations, Eq. (C.12) gives ¡ Gm+1; m;s (k; E) ret

ret

= G˜ m+1 (k; E + (m + 1)eFdc d + s=2˝#)T1 Js ()G˜ m (k; E + meFdc d − s=2˝#) ¡

adv

×<˜ m (k; E + meFdc d − s=2˝#)G˜ m (k; E + me Fdc d − s=2˝#) ret

¡

+ G˜ m+1 (k; E + (m + 1)eFdc d + s=2˝#)<˜ m+1 (k; E + (m + 1)eFdc d + s=2˝#) adv

adv

×G˜ m+1 (k; E + (m + 1)eFdc d + s=2˝#)T1 Js G˜ m (k; E + meFdc d − s=2˝#) ret = iT1 Js [G˜ m+1 (k; E˜ + eFdc d + s˝#)A˜ m (k; E)nF (E˜ − &m ) adv ˜ + A˜ m+1 (k; E˜ + eFdc d + s˝#)nF (E˜ + eFdc d + s˝# − &m+1 )G˜ m (k; E)]

+ O(T12 ) ;

(C.20)

where E˜ =E +meFdc d − s=2˝# was inserted in the last line. Inserting into Eq. (C.13) and sorting with respect to cos(r#t) and sin(r#t) we obtain Eqs. (197) – (199) with  2e
dE ˜ ret Idc (eFd) = T12 A (k; E˜ + eFd)A˜ m (k; E) ˝ 2 m+1 k

×[nF (E˜ − &m ) − nF (E˜ + eFd − &m+1 )]

and 4e
K(eFd) = − T12 ˝ k



(C.21)

dE ret [Re{G˜ m+1 (k; E˜ + eFd)}A˜ m (k; E)nF (E˜ − &m ) 2

adv ˜ }] : + A˜ m+1 (k; E˜ + eFd)nF (E˜ + eFd − &m+1 )Re{G˜ m (k; E)

(C.22)

This proves the formulas applied in Section 6.3 for a → a tunneling. C.3. Tunneling between di6erent levels Now, we want to consider tunneling between diDerent levels. We will focus on the transitions from level a in well m=0 to the level b in well m=1 and omit the well indices in the following. According to Eq. (24), the matrix element for this transition [eFdc d+ ˝# cos(#t1 )]Rba 1 becomes time dependent. Neglecting the coupling one obtains again the functions G˜ a=b (k; E) as in the preceding subsection. In the lowest order Eq. (108) yields   9 b i˝ − Ek − E + eFdc d Gb;ret=adv (k; t1 ; t2 ) a 9t1  dt ˜ ret=adv ret=adv ˜ = [eFdc d + ˝# cos(#t1 )]Rba (k; t ; t ) + (k; t1 ; t)Gb;ret=adv (k; t; t2 ) : G < 1 2 a; a 1 a ˝ b (C.23)

100

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In the same way, as Eq. (C.6) we obtain  1 ˜ ret dt3 gbret (t1 ; t3 )[eFdc d + ˝# cos(#t3 )]Rba 1 G a (t3 ; t2 ) ˝   1 1 1 † dt3 S1 (t1 ) dE1 e−iE1 (t1 −t3 )=˝ S1 (t3 ) = b ˝ 2 E1− E − Ek + eFdc d + i0+ 1 ret † ×[eFdc d + ˝# cos(#t3 )]Rba dE2 e−iE2 (t3 −t2 )=˝ G˜ a (E2 )S0 (t2 ) 1 S0 (t3 ) 2   1 1 1
† dt3 S1 (t1 ) dE1 e−iE1 (t1 −t3 )=˝ Js () = b ˝ s 2 E1 − E − Ek + eFdc d + i0+    ei#t3 + e−i#t3 ba 1 ret ×e−is#t3 eFdc d + ˝# R1 dE2 e−iE2 (t3 −t2 )=˝ G˜ a (E2 )S0 (t2 ) 2 2 
1 † 1 = S1 (t1 )S0 (t2 ) dE1 e−iE1 t1 =˝ Js () b 2 E1 − E − Ek + eFdc d + i0+ s  ret ba ×R1 eFdc dei(E1 =˝−s#)t2 =˝ G˜ a (E1 − s˝#) +

˝#

ret

ei(E1 =˝−(s+1)#)t2 =˝ G˜ a (E1 − (s + 1)˝#)

2  ˝# i(E1 =˝−(s−1)#)t2 =˝ ˜ ret + G a (E1 − (s − 1)˝#) e 2 
1 † 1 = S1 (t1 ) dE e−ir#t1 e−iE(t1 −t2 )=˝ S0 (t2 ) b − E + eF d + i0+ 2 E + r ˝ # − E k dc r ret

˜ ×Rba 1 [Jr ()eFdc d + Jr+1 ()˝#=2 + Jr−1 ()˝#=2]G a (E) :

(C.24)

Therefore, we ;nd ret

ret

2 ˜ Gb;reta;r (k; E) = G˜ b (k; E + eFdc d + r ˝#)Jr ()Rba 1 [eFdc d + r ˝#]G a (k; E) + O(T1 )

(C.25)

and the Keldysh relation becomes, in analogy to the preceding subsection ˜ ret ˜ ˜ ˜ Gb;¡a;s (k; E) = iJs ()Rba 1 [eFdc d + s˝#][G b (k; E + eFdc d + s˝#)Aa (k; E)nF (E − &0 )

adv ˜ + O(T12 ) : + A˜ b (k; E˜ + eFdc d + s˝#)nF (E˜ + eFdc d + s˝# − &1 )G˜ a (k; E)] (C.26)

Finally, the current density can be obtained by inserting the time-dependent matrix element into Eq. (C.13) # $ 
4e

dE i#t −i#t ba ¡ ir  #t Ia→b = Re Js+r  ()[eFdc d + ˝#(e + e )=2]R1 Gb; a;s (E; k)e ˝ 2  k r # s $ 


4e dE ba ¡ ir#t = Js+r ()[eFdc d + (s + r)˝#]R1 Gb; a;s (E; k)e : (C.27) Re ˝ 2 r s k

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101

For r = 0, Eq. (197) can be recovered even in the case of a linear ;eld dependence of the matrix element. In contrast, for r ¿ 1 Eqs. (198) and (199) only hold if the ;eld dependence of the matrix element is negligible, which is appropriate if eFac deFdc d and ˝#eFdc d hold. References Q [1] F. Bloch, Uber die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52 (1928) 555. [2] C. Zener, A theory of the electrical breakdown of solid dielectrics, Proc. Roy. Soc. A 145 (1934) 523. [3] L. Esaki, R. Tsu, Superlattice and negative diDerential conductivity in semiconductors, IBM J. Res. Develop. 14 (1970) 61. [4] G.H. Wannier, Wave functions and eDective Hamiltonian for Bloch electrons in an electric ;eld, Phys. Rev. 117 (1960) 432. [5] S.A. Ktitorov, G.S. Simin, V.Y. Sindalovskii, Bragg reJections and the high-frequency conductivity of an electronic solid-state plasma, Sov. Phys.-Sol. State 13 (1972) 1872 [Fizika Tverdogo Tela 13 (1971) 2230]. [6] D.H. Dunlap, V.M. Kenkre, Dynamic localization of a charged particle under the inJuence of an electric ;eld, Phys. Rev. B 34 (1986) 3625. [7] M. Holthaus, Collapse of minibands in far-infrared irradiated superlattices, Phys. Rev. Lett. 69 (1992) 351. [8] A.Y. Shik, Superlattices-periodic semiconductor structures, Sov. Phys. Semicond. 8 (1975) 1195 [Fiz. Tekh. Poluprov. 8 (1974) 1841]. [9] D.L. Smith, C. Mailhiot, Theory of semiconductor superlattice electronic structure, Rev. Mod. Phys. 62 (1990) 173. [10] E.L. Ivchenko, G. Pikus, Superlattices and other Heterostructures, Springer, Berlin, 1995. [11] M. Helm, Infrared spectroscopy and transport of electrons in semiconductor superlattices, Semicond. Sci. Technol. 10 (1995) 557. [12] H.T. Grahn (Ed.), Semiconductor Superlattices, Growth and Electronic Properties, World Scienti;c, Singapore, 1995. [13] F. Rossi, Bloch oscillations and Wannier–Stark localization in semiconductor superlattices, in: E. SchQoll (Ed.), Theory of Transport Properties of Semiconductor Nanostructures, Chapman & Hall, London, 1998 (Chapter 9). [14] F.G. Bass, A.P. Tetervov, High-frequency phenomena in semiconductor superlattices, Phys. Rep. 140 (1986) 237. [15] G.H. DQohler, n–i–p–i doping superlattice-semiconductors with tunable electronic properties, in: H.L. Grubin, K. Hess, G.J. Iafrate, D.K. Ferry (Eds.), The Physics of Submicron Structures, Plenum Press, New York, 1984. [16] L. Esaki, L.L. Chang, New transport phenomenon in a semiconductor superlattice, Phys. Rev. Lett. 33 (8) (1974) 495. [17] Y. Kawamura, K. Wakita, H. Asahi, K. Kurumada, Observation of room temperature current oscillation in InGaAs=InAlAs MQW pin diodes, Jpn. J. Appl. Phys. 25 (1986) L928. [18] M. BQuttiker, H. Thomas, Current instability and domain propagation due to Bragg scattering, Phys. Rev. Lett. 38 (1977) 78. [19] J. Kastrup, R. Klann, H.T. Grahn, K. Ploog, L.L. Bonilla, J. Gal_an, M. Kindelan, M. Moscoso, R. Merlin, Self-oscillations of domains in doped GaAs–AlAs superlattices, Phys. Rev. B 52 (1995) 13761. [20] E. Schomburg, R. Scheurer, S. Brandl, K.F. Renk, D.G. Pavel’ev, Y. Koschurinov, V. Ustinov, A. Zhukov, A. Kovsh, P.S. Kop’ev, InGaAs=InAlAs superlattice oscillator at 147 GHz, Electron. Lett. 35 (1999) 1491. [21] A. Sibille, J.F. Palmier, F. Mollot, H. Wang, J.C. Esnault, Negative diDerential conductance in GaAs=AlAs superlattices, Phys. Rev. B 39 (1989) 6272. [22] H.T. Grahn, K. von Klitzing, K. Ploog, G.H. DQohler, Electrical transport in narrow-miniband semiconductor superlattices, Phys. Rev. B 43 (1991) 12094. [23] E. Schomburg, A.A. Ignatov, S. Winnerl, J. Grenzer, K.F. Renk, D.G. Pavel’ev, Y. Koschurinov, B.Y. Melzer, V. Ustinov, S. Ivanov, S.S. ov, A. Zhukov, P.S. Kop’ev, Determination of the velocity-;eld characteristic of

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Physics Reports 357 (2002) 113–213

Physical aspects of the space–time torsion I.L. Shapiro ∗; 1 Departamento de F sica, Universidad de Federal de Juiz de Fora, CEP: 36036-330, MG, Brazil Received March 2001; editor : A: Schwimmer

Contents 1. Introduction 2. Classical torsion 2.1. De*nitions, notations and basic concepts 2.2. Einstein–Cartan theory and non-dynamical torsion 2.3. Interaction of torsion with matter *elds 2.4. Conformal properties of torsion 2.5. Gauge approach to gravity: higher derivative gravity theories with torsion 2.6. An example of the possible e4ect of classical torsion 3. Renormalization and anomalies in curved space–time with torsion 3.1. General description of renormalizable theory 3.2. One-loop calculations in the vacuum sector 3.3. One-loop calculations in the matter *elds sector 3.4. Renormalization group and universality in the non-minimal sector 3.5. E4ective potential of scalar *eld in the space–time with torsion: spontaneous symmetry breaking and phase transitions induced by curvature and torsion

115 116 116 119 120 125 129 131 133 134 138 142 146

151

3.6. Conformal anomaly in the spaces with torsion: trace anomaly and modi*ed trace anomaly 3.7. Integration of conformal anomaly and anomaly-induced e4ective actions of vacuum: application to in9ationary cosmology 3.8. Chiral anomaly in the spaces with torsion: cancellation of anomalies 4. Spinning and spinless particles and the possible e4ects on the classical background of torsion 4.1. Generalized Pauli equation with torsion 4.2. Foldy–Wouthuysen transformation with torsion 4.3. Non-relativistic particle in the external torsion *eld 4.4. Path-integral approach for the relativistic particle with torsion 4.5. Space–time trajectories for the spinning and spinless particles in an external torsion *eld 4.6. Experimental constraints for the constant background torsion

∗

155

158 160 162 163 164 166 167 175 177

Corresponding author. Departamento de F?@sica Teorica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain. Tel.: +34-976-76-12-62; fax: +34-976-76-12-64. E-mail address: [email protected] (I.L. Shapiro). 1 On leave from Department of Mathematical Analysis, Tomsk State Pedagogical University, Russia. c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter  PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 3 0 - 8

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5. The e4ective quantum *eld theory approach for the dynamical torsion 5.1. Early works on the quantum gravity with torsion 5.2. General note about the e4ective approach to torsion 5.3. Torsion–fermion interaction again: softly broken symmetry associated with torsion and the unique possibility for the low-energy torsion action 5.4. Brief review of the possible torsion e4ects in high-energy physics 5.5. First test of consistency: loops in the fermion–scalar systems break unitarity

180 180 183

183 185

5.6. Second test: problems with the quantized fermion–torsion systems 5.7. Interpretation of the results: do we have a chance to meet propagating torsion? 5.8. What is the di4erence with metric? 6. Alternative approaches: induced torsion 6.1. Is that torsion induced in string theory? 6.2. Gravity with torsion induced by quantum e4ects of matter 7. Conclusions Acknowledgements References

191 197 199 199 200 203 205 206 207

188

Abstract We review many quantum aspects of torsion theory and discuss the possibility of the space–time torsion to exist and to be detected. The paper starts, in Section 2, with a pedagogical introduction to the classical gravity with torsion, that includes also interaction of torsion with matter *elds. Special attention is paid to the conformal properties of the theory. In Section 3, the renormalization of quantum theory of matter *elds and related topics, like renormalization group, e4ective potential and anomalies, are considered. Section 4 is devoted to the action of spinning and spinless particles in a space–time with torsion, and to the discussion of possible physical e4ects generated by the background torsion. In particular, we review the upper bounds for the magnitude of the background torsion which are known from the literature. In Section 5, the comprehensive study of the possibility of a theory for the propagating completely antisymmetric torsion *eld is presented. It is supposed that the propagating *eld should be quantized, and that its quantum e4ects must be described by, at least, some e4ective low-energy quantum *eld theory. We show, that the propagating torsion may be consistent with the principles of quantum theory only in the case when the torsion mass is much greater than the mass of the heaviest fermion coupled to torsion. Then, universality of the fermion–torsion interaction implies that torsion itself has a huge mass, and cannot be observed in realistic experiments. Thus, the theory of quantum matter *elds on the classical torsion background can be formulated in a consistent way, while the theory of dynamical torsion meets serious obstacles. In Section 6, we brie9y discuss the string-induced torsion and the possibility to c 2002 Elsevier induce torsion action and torsion itself through the quantum e4ects of matter *elds.
Science B.V. All rights reserved. PACS: 04.50.+h; 04.62+v; 11.10.Gh; 11:10: − z Keywords: Torsion; Renormalization in curved space–time; Limits on new interactions; Unitarity and renormalizability

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1. Introduction The development of physics, until recent times, went from experiment to theory. New theories were created when the previous ones did not *t with some existing phenomena, or when the theories describing di4erent classes of phenomena have shown some mutual contradictions. At some point this process almost stopped. However, recent data on the neutrino oscillations should be, perhaps, interpreted in such a way that the Minimal Standard Model of particle physics does not describe the full spectrum of existing particles. On the other hand, one can mention the supernova observational evidence for a positive cosmological constant and the lack of the natural explanation for the in9ation. This probably indicates that the extension of the Standard Model must also include gravity. The desired fundamental theory is expected to provide the solution to the quantum gravity problem, hopefully explain the observable value of the cosmological constant and maybe even predict the low-energy observable particle spectrum. The construction of such a fundamental theory meets obvious diLculties: besides purely theoretical ones, there is an extremely small link with the experiments or observations. Nowadays, the number of theoretical models or ideas has overwhelming majority over the number of their possible veri*cations. In this sense, today the theory is very far ahead of experiment. In such a situation, when a fundamental theory is unknown or it cannot be veri*ed, one might apply some e4ective approach and ask what could be the traces of such a theory at low energies. In principle, there can be two kinds of evidences: new *elds or new low-energy symmetries. The Standard Model is composed by three types of *elds: spinors, vectors and scalars. On the other hand, General Relativity yields one more *eld—metric, which describes the properties of the space–time. Now, if there is some low-energy manifestation of the fundamental theory, it could be some additional characteristics of the space–time, di4erent from the *elds included into the Standard Model. One of the candidates for this role could be the space–time torsion, which we are going to discuss in this paper. Torsion is some independent characteristic of the space–time, which has a very long history of study (see [99] for the extensive review and references, mainly on various aspects of classical gravity theory with torsion). In this paper we shall concentrate on the quantum aspects of the theory, and will look at the problem, mainly, from quantum point of view. Our purpose will be to apply the approach which is standard in the high-energy physics when things concern the search for some new particle or interaction. One has to formulate the corresponding theory in a consistent way, *rst at the level of lower complexity, and then investigate the possibility of experimental manifestations. After that, it is possible to study more complicated models. Indeed, for the case of torsion, which has not been ever observed, the study of experimental manifestations reduces to the upper bounds on the torsion parameters from various experiments. Besides this principal line, the extensive introduction to the gravity with torsion will be given in Section 2. For us, the simplest level of the torsion theory will be the classical background for quantum matter *elds. As we shall see, such a theory can be formulated in a consistent way. The next level is, naturally, the theory of dynamical (propagating) torsion, which should be considered in the same manner as metric or as the constituents of the Standard Model. We shall present the general review of the original publications [18,22] discussing the restrictions in implementing torsion into a gauge theory such as the Standard Model. Towards the end of the paper, we

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discuss the possibility of torsion induced in string theory and through the quantum e4ects of matter *elds. 2. Classical torsion This section mainly contains introductory material, which is necessary for the next sections, where the quantum aspects of torsion will be discussed. 2.1. De5nitions, notations and basic concepts Let us start with the basic notions of gravity with torsion. In general, our notations correspond  are independent characteristics of the space– to those in [165,34]. The metric g and torsion T· time. In order to understand better, how the introduction of torsion becomes possible, let us brie9y review the construction of covariant derivative in General Relativity. We shall mainly consider the algebraic aspect of the covariant derivative. For the geometric aspects, related to the notion of parallel transport, the reader is referred, for example, to [99]. The partial derivative of a scalar *eld is a covariant vector (one-form). However, the partial derivative of any other tensor *eld does not form a tensor. But, one can add to the partial derivative some additional term such that the sum is a tensor. The sum of partial derivative and this additional term is called covariant derivative. For instance, in the case of the (contravariant) vector A the covariant derivative looks like  ∇ A = 9 A +  A ;

(2.1)

where the last term is a necessary addition. The covariant derivative (2.1) is a tensor if and only  transforms in a special non-tensor way. The rule for constructing if the aLne connection  the covariant derivatives of other tensors immediately follows from the following facts: (i) The product of co- and contravariant vectors A and B should be a scalar. Then ∇ (A B ) = 9 (A B )

and therefore ∇ B = 9 B −  B :

(2.2)

(ii) In the same manner, one can note that the contraction of any tensor with some appropriate set of vectors is a scalar, and arrive at the standard expression for the covariant derivative of an arbitrary tensor 1 :::  ∇ T 11:::::: = 9 T 11:::::: +  T 1 ::: + · · · −  T 1 ::: − · · · : 1 :::

(2.3)

Now, (2.1) and (2.2) become particular cases of (2.3). At this point, it becomes clear that the  contains, from the very beginning, some ambiguity. Indeed, (2.3) remains a de*nition of   any tensor C  : tensor if one adds to      →  + C : 

(2.4)

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 , which is used in General Relativity, appears as a consequence of A very special choice of  two requirements:  =  and (i) symmetry   (ii) metricity of the covariant derivative ∇ g = 0.  : If these conditions are satis*ed, one can apply (2.3) and obtain the unique solution for 
    = = 12 g (9 g + 9 g  − 9 g ) : (2.5) 

Expression (2.5) is called Christo4el symbol, it is a particular case of the aLne connection. Indeed, (2.5) is a very important object, because it depends on the metric only. Eq. (2.5) is the simplest one among all possible aLne connections. It is very useful to consider (2.5) as some “reference point” for all the connections. Other connections can be considered as (2.5) plus some additional tensor as in (2.4). It is easy to prove that the di4erence between any two connections is a tensor. When the space–time is 9at, the metric and expression (2.5) depend just on the choice of  the coordinates, and one can choose them in such a way that {  } vanishes everywhere. On the contrary, if we consider, as in (2.4),
   ˆ  = + C· ; (2.6)   (and, consequently, the whole connection  ˆ  ) cannot be eliminated by a then the tensor C·  choice of the coordinates. Even if one takes the 9at metric, the covariant derivative based on ˆ  does not reduce to the coordinate transform of partial derivative. Thus, the introduction of an aLne connection di4erent from Christo4el symbol means that the geometry is not completely  . The described by the metric, but has another, absolutely independent characteristic-tensor C  ambiguity in the de*nition of  is very important, for it enables one to introduce gauge *elds di4erent from gravity, and thus describe various interactions.  . Namely, we suppose In this paper we shall consider the particular choice of the tensor C that the aLne connection ˜  is not symmetric:  = 0 : ˜  − ˜   = T·

(2.7)

At the same time, we postulate that the corresponding covariant derivative satis*es the metricity  is called torsion. condition ∇˜  g = 0. 2 The tensor T· Below, we use notation (2.5) for the Christo4el symbol, and the notation with tilde for the connection with torsion and for the corresponding covariant derivative. The metricity condition enables one to express the connection through the metric and torsion in a unique way as   ˜  =  + K· ;

(2.8)

2 The breaking of this condition means that one adds one more tensor to the aLne connection. This term is called non-metricity, and it may be important, for example, in the consideration of the *rst order formalism for General Relativity. However, we will not consider the theories with non-metricity here.

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where     K· = 12 (T· − T· − T · )

(2.9)

is called the contorsion tensor. The indices are raised and lowered by means of the metric. It is worthwhile noting that the contorsion is antisymmetric in the *rst two indices: K = −K ,  itself is antisymmetric in the last two indices. while torsion T· The commutator of covariant derivatives in the space with torsion depends on the torsion and on the curvature tensor. First of all, consider the commutator acting on the scalar *eld ’. We obtain

[∇˜  ; ∇˜  ]’ = T· 9 ’ ;

(2.10)

that indicates a di4erence with respect to the commutator of the covariant derivatives ∇ based on the Christo4el symbol (2.5). In the case of a vector, after some simple algebra we arrive at the expression  ˜ [∇˜  ; ∇˜  ]P = T· ∇ P + R˜ · P  ;

(2.11)

where R˜ · is the curvature tensor in the space with torsion: R˜ · = 9 ˜ · − 9 ˜ · + ˜ ·  ˜ · − ˜ ·  ˜ · :

(2.12)

Using (2.10), (2.11) and that the product P B is a scalar, one can easily derive the commutator of covariant derivatives acting on a one-form B and then calculate such a commutator acting on any tensor. In all cases the commutator is the linear combination of curvature (2.12) and torsion. Curvature (2.12) can be easily expressed through the Riemann tensor (curvature tensor depending only on the metric), covariant derivative ∇ (torsionless) and contorsion as



R˜ · = R · + ∇ K· − ∇ K· + K·  K· − K·  K· :

(2.13)

Similar formulas can be written for the Ricci tensor and for the scalar curvature with torsion:



R˜  = R˜ · = R + ∇ K· − ∇ K· + K· K· − K· K· 

(2.14)

(note that it is not symmetric) and 

 R˜ = g R˜  = R + 2∇ K·  − K · K·· + K K  :

It proves useful to divide torsion into three irreducible components:  ; (i) the trace vector T = T· (ii) the (sometimes, it is called pseudotrace) axial vector S  = j T and  , which satis*es two conditions q = 0 and j q (iii) the tensor q·  = 0. ·

(2.15)

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Then, the torsion *eld can be expressed through these new *elds as 3 T = 13 (T g − T g ) − 16  S  + q :

(2.16)

Using the above formulas, it is not diLcult to express curvatures (2.13) – (2.15) through these irreducible components. We shall write only the expression for scalar curvature, which will be useful in what follows: R˜ = R − 2∇ T  − 43 T T  + 12 q q +

1  24 S S

:

(2.17)

2.2. Einstein–Cartan theory and non-dynamical torsion In order to start the discussion of gravity with torsion, we *rst consider a direct generalization of General Relativity, which is usually called Einstein–Cartan theory. Indeed, our consideration will be very brief. For further information one is recommended to look at the review [99]. Our *rst aim is to generalize the Einstein–Hilbert action  √ 1 SEH = − 2 d 4 x −gR  for the space with torsion. It is natural to substitute scalar curvature R by (2.17), despite the change of the coeLcients for the torsion terms in (2.17) cannot be viewed as something wrong. As we shall see later, in quantum theory the action of gravity with torsion is induced with the coeLcients which are, generally, di4erent from the ones in (2.18). The choice of the volume element d V4 should be done in such a manner that it transforms like a scalar and also reduces to the usual d 4 x for the case of a 9at space–time and global orthonormal coordinates. Since we have two independent tensors: metric and torsion, the correct transformation property√could be, in principle, satis*ed in in*nitely many ways. For example, one can take d V4 = d 4 x −g as in 4 General Relativity, or d V4 = d x det(S T − S T ), or choose some other form. However, if we request that the determinant becomes d 4 x in a 9at space–time limit with zero torsion, all the expressions similar to the last one are excluded. In this paper we postulate, as usual, that the volume element √ in the space with torsion depends only on the metric and hence it has the form d V4 = d 4 x −g. Then, according to (2.17), the most natural expression for the action of gravity with torsion will be     1 2 1 1 4 2 1 2 4 √ 4 √  ˜ d x −g R = − 2 d x −g R − 2∇ T − T + q + S : SEC = − 2   3 2 24 (2.18) The second term in the last integrand is a total derivative, so it does not a4ect the equations of  = 0. Therefore, on the mass shell the theory motion, which have the non-dynamical form T· (2.18) is completely equivalent to General Relativity. The di4erence appears when we add the 3 In the most of this paper, we consider the four-dimensional space–time. More general, n-dimensional formulas concerning classical gravity with torsion can be found in Ref. [102]. More detailed classi*cation of the torsion components can be found in Ref. [44].

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external source for torsion. Imagine that torsion is coupled to some matter *elds, and that the action of these *elds depends on torsion in such a way that it contains the term  √  Sm = d 4 x −g K· · ; (2.19) where the tensor · is constructed from the matter *elds (it is similar to the dynamically de*ned energy–momentum tensor) but may also depend on metric and torsion. One can check that, for the Dirac fermion minimally coupled to torsion, the · is nothing but the expression for the spin tensor of this *eld. One can use this as a hint and choose 1 Sm · = √ (2.20)  −g K· as the dynamical de*nition of the spin tensor for the theory with the classical action Sm . Unfortunately, in some theories this formula gives the result di4erent from the one coming from the Noether theorem, and only for the minimally coupled Dirac spinor (see the next section) the result is the same. Next, since there is no experimental evidence for torsion, we can safely suppose it to be very weak. Then, as an approximation, · can be considered independent of torsion. In this case, the equations following from the action SEC + Sm have the structure 1 K ∼ 2  ∼ 2 ·  ; (2.21) MP where MP = 1= is the Planck mass. Then, torsion leads to the contact spin–spin interaction with the classical potential 1 V () ∼ 2 · 2 : (2.22) MP Some discussion of this contact interaction can be found in [99]. In Section 2.6 we shall provide an example, illustrating the possible importance of this interaction in the early Universe. Since the last expression (2.22) contains a 1=MP2 ≈ 10−38 GeV−2 factor, it can only lead to some extremely weak e4ects at low energies. Therefore, the e4ects of torsion, in the Einstein–Cartan theory, are suppressed by the torsion mass which is of the Planck order MP . Even if one introduces kinetic terms for the torsion components, the situation would remain essentially the same, as far as we consider low-energy e4ects. An alternative possibility is to suppose that torsion is light or even massless. In this case torsion can propagate, and there would be a chance to meet really independent torsion *eld. The review of theoretical limitations on this kind of theory [18,22] is one of the main subject of the present review (see Section 4). These limitations come from the consistency requirement for the e4ective quantum *eld theory for such a “light” torsion. 2.3. Interaction of torsion with matter 5elds In order to construct the actions of matter *elds in an external gravitational *eld with torsion we impose the principles of locality and general covariance. Furthermore, in order to preserve

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the fundamental features of the original 9at-space theory, one has to require the symmetries of a given theory (gauge invariance) in 9at space–time to hold for the theory in curved space–time with torsion. It is also natural to forbid the introduction of new parameters with the dimension of inverse mass. This set of conditions enables one to construct the consistent quantum theory of matter *elds on the classical gravitational background with torsion. The form of the action of a matter *eld is *xed except the values of some new parameters (non-minimal and vacuum ones) which remain arbitrary. This procedure which we have described above, leads to the so-called non-minimal actions. Along with the non-minimal scheme, there is a (more traditional) minimal one. According to it the partial derivatives 9 are substituted by the covariant √ ones ∇˜  , the 9at metric " by g and the volume element d 4 x by the covariant expression d 4 x −g. We remark, that the minimal scheme gives, for the case of the Einstein–Cartan theory, action (2.18), while the non-minimal scheme would make the coeLcients at the torsion terms arbitrary. We de*ne the minimal generalization of the action for the scalar *eld as    1  4 √ S0 = d x −g (2.23) g 9 ’9 ’ − V (’) : 2 Obviously, the last action does not contain torsion. One has to note some peculiar property of the last statement. If one starts from the equivalent 9at-space expression    1 2 4 S0 = d x − ’9 ’ − V (’) ; 2 then the generalized action does contain torsion. This can be easily seen from the following simple calculation: ˜ 2 ’ = g ∇˜  ∇˜  ’ = ’ + T  9 ’ : ∇ (2.24) Thus, at this point, the minimal scheme contains a small ambiguity which can be cured through the introduction of the non-minimal interaction. For one real scalar, one meets *ve possible non-minimal structures (cf. to expression (2.18)) ’2 Pi , where [39] P1 = R; P2 = ∇ T  ; P3 = T T  ; P4 = S S  ; P5 = q q : (2.25) Correspondingly, there are *ve non-minimal parameters $1 : : : $5 . The general non-minimal free *eld action has the form
  5 √ 1 1 1 S0 = d 4 −g $i Pi ’2 : (2.26) g ∇ ’∇ ’ + m2 ’2 + 2 2 2 i=1

A more complicated scalar content gives rise to more non-minimal terms [36]. In particular, for the complex scalar *eld & one can introduce, into the covariant Lagrangian, the following additional term: UL(&; &† ) = i$0 T  (&† · 9 & − 9 &† · &) : In the case of a scalar ’ coupled to a pseudoscalar (, there are other possible non-minimal terms: 10 1   UL(’; () = $0 S (’9 ( − (9 ’) + ’( $j Dj ; 2 j=6

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where D6 = ∇  S  ;

D7 = T S  ;

D9 = j q  q 

D8 = j S  q ;

   D10 = j q· · q :

More general scalar models can be treated in a similar way. For the Dirac spinor, 4 the minimal procedure leads to the expression for the hermitian action  √ i ˜  − ∇˜  V  − 2im V ) : d 4 x g( V  ∇ (2.27) S1=2; min = 2 Here  =ea a , where a is usual (9at-space) -matrix, and ea is tetrad (vierbein) de*ned through the standard relations ea · eb = "ab ;

ea · ea = g ;

ea · ea = g ;

ea · eb = "ab :

The covariant derivative of a Dirac spinor ∇˜  should be de*ned to be consistent with the covariant derivative of tensors. We suppose that ˜  = 9 + i w˜ ab ∇  /ab 2

;

(2.28)

where w˜ ab  is a new object which is usually called spinor connection, and i /ab = ( a b − b a ) : 2 The conjugated expression is

˜  V = 9 V − i V w˜ ab ∇  /ab : 2

(2.29)

Now, we consider how the covariant derivative acts on the vector V  . As we already learned in Section 2.1, if the connection provides the proper transformation law for the vector, it does so with any tensor. Therefore, the only one equation for the spinor connection w˜ ab  is  ˜  ( V  ) = 9 ( V  ) + ˜ ·  ( V  ) = ∇ ( V  ) + K·  ∇ ( V  ) :

(2.30)

Substituting (2.28) and (2.29) into (2.30), after some algebra, we arrive at the formula for the spinor connection  w˜ ab = wab + 14 K·  (ea eb − eb ea ) ;

(2.31)

 wab = 14 (eb 9 ea − ea 9 eb ) + 14   (eb ea − ea eb )

(2.32)

where is spinor connection in the space–time without torsion. 4

Various aspects of the minimal fermion–torsion interaction have been considered in many papers, e.g. [57,12,99].

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Substituting (2.31) into (2.37), and performing integration by parts, we arrive at two equivalent forms for the spinor action    √ 1 S1=2; min = i d 4 x g V  ∇˜  −  T − im 2    i 5  4 √ V  ∇ − S − im ; (2.33) =i d x g 8 where we use standard representation for the Dirac matrices, such that 5 = −i 0 1 2 3 and ( 5 )2 = 1. Also, 5 = 5 . The *rst integral is written in terms of the covariant derivative ∇˜  with torsion, while the last is expressed through the torsionless covariant derivative ∇ . Indeed the last form is more informative, for it tells us that only the axial vector S couples to fermion, and  completely decouple. It is important to note, that in the the other two components: T and q· *rst of integrals (2.33) the  T -term has an extra factor of i as compared to the electromagnetic (or any vector) *eld. This indicates, that if being taken separately from the *rst term  ∇˜  , the  T -term would introduce an imaginary part into the spinor action, and therefore it has no sense. Of course, the same concerns the term i V  ∇˜  . The imaginary terms, coming from the two parts, cancel, and give rise to a hermitian action for the spinor, equivalent to (2.27) or to the second integral in (2.33). The non-minimal interaction is a bit more complicated. Using covariance, locality, dimension, and requesting that the action does not break parity one can construct only two non-minimal (real, of course) terms with the structures already known from (2.33):    √ S1=2; non-min = i d 4 x g V   ∇ + "j Qj − im (2.34) j=1;2

with Q1 = i 5  S  ;

Q2 = i  T 

and two arbitrary non-minimal parameters "1 ; "2 . The minimal theory corresponds to "1 = − 18 ; "2 = 0. Let us again note that the T -dependent term in (2.34) is di4erent from the last term in the *rst representation in (2.33). In (2.34) all the terms are real. We observe, that the interaction of the torsion trace T with fermion is identical to the one of the electromagnetic *eld. Therefore, in some situations when torsion is considered simultaneously with the external electromagnetic *eld A , one can simply rede*ne A such that the torsion trace T disappears. Consider the symmetries of the Dirac spinor action non-minimally coupled to the vector and torsion *elds  S1=2 = i d 4 x V [  (9 + ieA + i" 5 S ) − im] : (2.35) As compared to (2.34), here we have changed the notation for the non-minimal parameter of interaction between spinor *elds and the axial part S of torsion: "1 → ". Furthermore, we used the possibility to rede*ne the external electromagnetic potential A in such a way that it absorbs the torsion trace T .

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The new interaction with torsion does not spoil the invariance of the action under usual gauge transformation: 

= e(x) ;

V  = V e−(x) ;

A = A − e−1 9 (x) :

(2.36)

Furthermore, the massless part of action (2.35) is invariant under the transformation in which the axial vector S plays the role of the gauge *eld 

= e 5 (x) ;

V  = V e 5 (x) ;

S = S − "−1 9 (x) :

(2.37)

Thus, in the massless sector of the theory one faces generalized gauge symmetry depending on the scalar, (x), and pseudoscalar, (x), parameters of the transformations, while the massive term is not invariant under the last transformation. Consider whether the massless vector *eld might couple to torsion. Here, one has to use the principle of preserving the symmetry. The minimal interaction with torsion breaks the gauge invariance for the vector *eld, since

F˜  = ∇˜  A − ∇˜  A = F + 2A K·[]

is not invariant. The possibility to modify the gauge transformation in the theory with torsion has been studied in [139,104]. In this paper we opt to keep the form of the gauge transformation unaltered, and postulate that the gauge vector does not couple to torsion. The reasons for this choice is the following. First of all, when one is investigating the quantum *eld theory in external torsion *eld, it is natural to separate the e4ects of external *eld from the purely matter sector. Thus, the modi*cation of the gauge transformation does not *t with our approach. Furthermore, the most important part S of the torsion tensor does not admit the *ne-tuning of the gauge transformation. In other words, for the most interesting case of purely antisymmetric torsion it is not possible to save gauge invariance for the vector coupled to torsion in a minimal way. Let us consider the non-minimal interaction for the special case of an abelian gauge vector.

4 One √ can introduce several non-minimal terms which do not break the gauge invariance: d x −gF K  . The most general form of K  is

K  = 21 j T S + 22 j 9 S + 23 j q· S

+ 24 q· T + 25 (9 T − 9 T ) + 26 9 q· :

(2.38)

For the non-abelian vector, the non-minimal structures like (2.38) are algebraically impossible. In the Standard Model, where all vectors are non-abelian, the non-minimal terms (2.38) do not exist. And so, we have constructed the actions of free scalar, spinor and vector *elds coupled to torsion. In general, there are two types of actions: minimal and non-minimal. As we shall see in the next sections, the non-minimal interactions with spinors and scalars provide certain advantages at the quantum level, for they give the possibility to construct renormalizable theory [36].

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2.4. Conformal properties of torsion Conformal symmetry with torsion has been studied in many papers (see, for example, [141,36, 102] and references therein). Here, we shall summarize the results obtained in [36,102]. For the torsionless theory the conformal transformation of the metric, scalar, spinor and vector *elds take the form  g → g = g e2/ ; ’ → ’ = ’e−/ ; →  = e−3=2 / ; A → A = A ; (2.39) where / = /(x). In the absence of torsion, the actions of free *elds are invariant if they are massless, besides in the scalar sector one has to put $ = 16 . Indeed, the interaction terms of the gauge theory (gauge, Yukawa and 4-scalar terms) are conformally invariant. The problem is to de*ne the conformal transformation for torsion, such that the free actions formulated in the previous section would be invariant for these or that values of the non-minimal parameters. It turns out that there are three di4erent ways to choose the conformal transformation for torsion:

→ T  = T . The (i) Weak conformal symmetry [36]: Torsion does not transform at all, T· · · conditions of conformal symmetry are absolutely the same as in the torsionless theory. (ii) Strong conformal symmetry [36]: In this version, torsion transforms as



T· → T· = T· + !(  9 −   9 )/(x) :

(2.40)

This transformation includes an arbitrary parameter, 5 !. Indeed the above transformation means that only the torsion trace transforms T → T = T + 3!9 /(x) : Other components of torsion remain inert under (2.40). For any value of !, the free actions

, but (2.26) and (2.34) are invariant if they depend only on the axial vector S and tensor q· not on the trace T . Of course, this is quite natural, because only T transforms. The restrictions imposed by the symmetry are $2 = $3 = "2 = 0. The immediate result of the strong conformal symmetry is the modi*ed Noether identity, which now reads S S S  g + =0 ; (2.41) 4 +  T· g 4 T· where 4 is the full set of matter *elds. √ Using the equations of motion, the de*nition of the energy–momentum tensor T = −(2= −g)S=g , and the de*nition of the spin tensor given in (2.20), we get 

T  g + ( · − 12 ·· )T· =0 ; that gives

2T − 3!∇ · = 0 :

(2.42) T

Along with the standard relation for the trace of the energy–momentum tensor = 0, here we meet an additional identity ∇ · = 0. It is interesting that this identity is exact, for it is 5 Instead of introducing an arbitrary numerical parameter !, one could replace, in the transformation rule for torsion (2.40), the parameter / for some other, independent parameter. This observation has been done in 1985 by A.O. Barvinsky and V.N. Ponomaryev in the report on my Ph.D. Thesis [165].

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not violated by the anomaly on quantum level. In order to understand this, we remember that the T -dependence is purely non-minimal, and if it does not exist in classical theory, it cannot appear at the quantum level. (iii) Compensating conformal symmetry [141,102]: This is the most interesting and complicated version of the conformal transformation. Let us consider the massless scalar, non-minimally coupled to metric and torsion (2.26). One can add to it the ’4 -term, without great changes in the results. Then
  5 √ 1 1

S = d 4 x −g $i Pi ’2 − ’4 ; (2.43) g 9 ’9 ’ + 2 2 4! i=1

where Pi were de*ned at (2.25). The equations of motion for the torsion tensor can be split into three independent equations written for the components T ; S ; q ; they yield $2 ∇  ’ T = · (2.44) ; S = q = 0 : $3 ’ Replacing these expressions back into action (2.43), we obtain the on-shell action      $22 1 1

4 4 √  2 1− g 9 ’9 ’ + $1 ’ R − ’ ; (2.45) S = d x −g 2 $3 2 4! that can be immediately reduced to the torsionless conformal action    1 

 4 1 2 4 √ S = d x −g ; (2.46) g 9 ’9 ’ + ’ R − ’ 2 12 4! by an obvious change of variables, whenever the non-minimal parameters satisfy the condition   $22 1 1− (2.47) $1 = 6 $3 and some relation between and  . Some important observation is in order. It is well known, that the conformal action (2.46) is classically equivalent to the Einstein–Hilbert action of General Relativity, but with the opposite sign [59,168]. In order to check this, we take such a wrong-sign action:    1 4 SEH [g ] = d x −gˆ Rˆ + 5 · (2.48) 2 This action depends on the metric gˆ . Performing conformal transformation gˆ = g · e2/(x) , we use the standard relations between geometric quantities of the original and transformed metrics:  √ −gˆ = −ge4/ ; Rˆ = e−2/ [R − 6 / − 6(∇/)2 ] : (2.49) Substituting (2.49) into (2.48), after integration by parts, we arrive at    6 2/ e2/ 4 √ 2 4/ SEH [g ] = d x −g e (∇/) + 2 R + 5e ; 2  where (∇/)2 = g 9 /9 /. If one denotes  12 / ’=e · ; 2

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action (2.48) becomes
  2 2  1 1  4 √ 2 2 4 S = d x −g ·’ ; (∇’) + R’ + 5 2 12 12

127

(2.50)

that is nothing but (2.46). And so, the metric–scalar theory described by the action of Eq. (2.50) and the metric–torsion–scalar theory (2.43) with constraint (2.47) are equivalent to the General Relativity with cosmological constant. One has to notice that the *rst two theories exhibit an extra local conformal symmetry, which compensates an extra (with respect to (2.48)) scalar degree of freedom. Moreover, (2.50) is a particular case of a family of similar actions, linked to each other by the reparametrization of the scalar or (and) the conformal transformation of the metric [168]. The symmetry transformation which leaves action (2.50) stable is  g = g · e26(x) ;

’ = ’ · e−6(x) :

(2.51)

The version of the Brans-Dicke theory with torsion (2.43) is conformally equivalent to General Relativity (2.48) provided that the new condition (2.47) is satis*ed and there are only external  and for the transverse component of T . Such a sources conformally covariant sources for S ; q·  do not spoil the conformal symmetry. Now, we can see that the introduction of torsion provides some theoretical advantage. If we start from the positively de*ned gravitational action (2.48), the sign of the scalar action (2.50) should be negative, indicating to the well-known instability of the conformal mode of General Relativity (see, for example, [97] and also [189,130] for the recent account of this problem and further references). It is easy to see that the metric–torsion–scalar theory may be free of this problem, if we choose $22 =$3 − 1 ¿ 0. In this case one meets the equivalence of the positively de*ned scalar action (2.43) to action (2.48) with the negative sign. The negative sign in (2.48) signi*es, in turn, the positively de*ned gravitational action. Without torsion one can achieve positivity in the gravitational action only by the expense of taking the negative kinetic energy for the scalar action in (2.50). The equation of motion (2.44) for T may be regarded as a constraint that *xes the conformal transformation for this vector to be consistent with the one for the metric and scalar. Then, instead of (2.51), one has  g = g · e26(x) ;

’ = ’ · e−6(x) ;

T = T −

$2 · 9 6(x) : $3

(2.52)

The on-shell equivalence can be also veri*ed using the equations of motion [102]. It is easy to check, by direct inspection, that even o4-shell, the theory with torsion (2.43), satisfying relation (2.47), may be conformally invariant whenever we de*ne the transformation law for the torsion  , do trace according to (2.52): and also postulate that the other pieces of torsion: S and q· √ not transform. The quantities −g and R transform as in (2.49). One may introduce into the action other conformal invariant terms depending on torsion. For instance,  √ 1 S =− d 4 x −gT T  ; 4

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where T = 9 T − 9 T . In the spinor sector one has to request "2 = 0 in (2.34), as it was for the strong conformal symmetry. It is important to notice, for the future, that on the quantum level this condition does not break renormalizability, even if $2; 3 are non-zero. One can better understand the equivalence between General Relativity and conformal metric– scalar–torsion theory (2.43), (2.47) after presenting an alternative form for the symmetric action. All torsion-dependent terms in (2.43) may be uni*ed in the expression 1 $2 2 P = − 2 R + $2 (∇ T  ) + $3 T T  + $4 S2 + $5 q : (2.53) 6 $3 It is not diLcult to check that the transformation law for this new quantity is especially simple: P = e−26 P. Using new quantity, the conformal invariance of the action becomes obvious:    1  1 1 4 √ 2 2 Sinv = d x −g : (2.54) g 9 ’9 ’ + R’ + P’ 2 12 2 In order to clarify the role of torsion in our conformal model, we construct one more representation for the metric–scalar–torsion action with local conformal symmetry. Let us start, once again, from action (2.43), (2.47) and perform only part of transformations (2.52): $2 ’ → ’ = ’ · e−6(x) ; T → T = T − · 9 6(x) : (2.55) $3 Of course, if we supplement (2.55) by the transformation of the metric, we arrive at (2.52) and the action does not change. On the other hand, (2.55) alone might lead to an alternative conformally equivalent description of the theory. Taking 6 such that   $22 12 −6(x) ’·e = 2 1− = const: ;  $3 we obtain, after some algebra, the following action:
  2   3 1 $2 4 √ 2 2 $4 S + $5 q + $3 T − ∇ ln ’ : S = 2 d x −g R + 2  $3  (1 − $22 =$3 ) (2.56) This form of the action does not contain interaction between curvature and the scalar *eld. At the same time, the latter is present until we use the equations of motion (2.44) for torsion. Torsion trace looks here like a Lagrange multiplier, and only using torsion equations of motion, one can obtain the action of GR. It is clear that one can arrive at the same action (2.56), making the transformation of the metric as in (2.52) instead of (2.55). To complete this part of our consideration, we mention that the direct generalization of the Einstein–Cartan theory including an extra scalar may be conformally equivalent to General Relativity, provided that the non-minimal parameter takes an appropriate value. To see this, one uses relation (2.17) and replace it into the “minimal” action    1 ˜ 2 1  4 √ SECBD = d x −g : (2.57) g 9 ’9 ’ + $R’ 2 2 It is easy to see that condition (2.47) is satis*ed for the special value $ = 13 , contrary to the famous $ = 16 in the torsionless case. The e4ect of changing conformal value of $ due to the

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non-trivial transformation of torsion has been discussed in [148,102] (see also further references there). 2.5. Gauge approach to gravity: higher derivative gravity theories with torsion There are many good reviews on the gauge approach to gravity (see, for example, [99]), and since the aim of the present paper is to treat torsion from the *eld-theoretical point of view, we shall restrict ourselves to a brief account of the results and some observations. In Section 2.1 we have introduced covariant derivative and found, that this can be done in  (see Eq. (2.6)). di4erent ways, because one can add to the aLne connection any tensor C· Any such extension of the aLne connection is related to some additional physical *eld, ex is a tensor and cannot be removed by a coordinate transformation. It was actly because C· already mentioned in Section 2.1, that the introduction of covariant derivative is related to the general coordinate transformations. The aim of the gauge approach to gravity is to show that the same construction, including the metric and the covariant derivative based on an arbitrary connection, can be achieved through the local version of the Lorentz–Poincare symmetry. If one requests the theory to be invariant with respect to the Poincare group with the in*nitesimal parameters depending on the space–time point, one has to introduce two compensating *elds: vierbein ea and some independent spinor connection Wbc [179,115,99,100] (see also [112,34] and [176,136,110,151,89] for alternative considerations). In this way, one naturally arrives at the gauge approach to gravity. One of the important applications of this approach is the natural and compact formulation of simple supergravity [63], where the supersymmetric generalization of the Einstein–Cartan theory emerges. The gauge approach, exactly as the one of Section 2.1, does not provide any reasonable restrictions on Wbc , and one has to introduce (or not) these restrictions additionally. In this paper we suppose that the covariant derivative possess metricity ∇ ea = 0. Then Wbc becomes w˜ bc  -spinor connection with torsion. The interesting question to answer is whether the description of the gravity with torsion in terms of the variables (ea ; w˜ bc  ) is equivalent to the description in  ). terms of the variables (g ; T· One can make the following observation. The *rst set (ea ; w˜ bc  ) corresponds to the *rst order  ) to the second order formalism. The origin of this is formalism, while the second set (g ; T· that in the last case the non-torsional part of the aLne connection is a function of the metric, while, within the gauge approach, the variables (ea ; w˜ bc  ) are mutually independent completely. One has to note that, in gravity, the equivalence between the *rst and second order formalism is a subtle matter. The simplest situation is the following. One takes the action  √ S1 = d 4 x −gg R () ; (2.58)

− 9  +   −   depends on the connection  which is where R () = 9         independent on the metric g . Then the equations for these two *elds

S1 = 0; g

S1 =0



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lead to the conventional Einstein equations and also to the standard expression for the aLne connection (2.5). In this case the *rst order formalism is equivalent to the usual second order formalism. However, this is not true if one chooses some other action for gravity. For instance, introducing higher derivative terms or adding to (2.58) additional terms depending on the non-metricity, one can indeed lose classical equivalence between two formalisms. 6 For the general action, the only possibility to link the connection with the metric is to impose the metricity condition. Let us come back to our case of gravity with torsion. From the consideration above it is clear  that the descriptions in terms of the variables (ea ; w˜ bc  ) and (g ; T· ) cannot be equivalent unlike we work with the Einstein–Cartan action (2.58). But, the non-equivalence comes only from the usual di4erence between *rst and second order formalisms, and has nothing to do with torsion. In order to have comparable situations, we have to replace the *rst set by (ea ; Uwbc ), where bc bc Uwbc = w˜ bc  − w and w depends on the vierbein through (2.32). The equivalence between  a the sets (g ; T· ) and (e ; Uwbc ) really takes place and this can be checked explicitly. First of all, one has to establish the equivalence (invertible relation) between metric and vierbein. This can be achieved by deriving eb g 1  = 2 ( e)a and = eb( ) (2.59)  : ea g 2 In a similar fashion, one can calculate the derivatives Uwab 1 [b a][6 /] 1 a[6 /]b T 6/ = e e e +   e e and = 4 e [a eb][/ 6] : T 6/ 2 4 Uwab

(2.60)

 ) to another one (ea ; Uw bc ) is Thus, the transformation from one set of variables (g ; T·   non-degenerate and two (second order in the torsion-independent part) descriptions are equiva ) variables, it is also true for the lent. If some statement about torsion is true for the (g ; T· (ea ; Uwbc ) variables, and v.v. Now, since we established the equivalence between di4erent variables, we can try to formulate some torsion theories more general than the one based on the Einstein–Cartan action. On the classical level the consideration can be based only on the general covariance and other symmetries. Let us restrict ourselves to the local actions only. As we have already learned, there are three possible descriptions of gravity with torsion:  and Riemann curvature R . When useful, torsion (a) In terms of the metric g , torsion T· ·

. tensor can be replaced by its irreducible components T ; S ; q·   and curvature (2.12) with torsion R ˜ . The (b) In terms of the metric g , torsion T· · relation between two curvature tensors, with and without torsion, is given by (2.13). (c) In terms of the variables (ec ; w˜ ab  ) and the corresponding curvature

Rab = 9 w˜ ab ˜ ab ˜ ac ˜ bc − w˜ ac ˜ bc :  − 9 w  +w  w  w We consider possibility (a) as the most useful one and will follow it in this paper, in particular for the construction of the new actions. 6

We remark that on the quantum level there is no equivalence even for the (2.58) action [35].

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The invariant local action can be always expanded into the power series in derivatives of the metric and torsion. It is natural to consider torsion to be of the same order as the aLne connection, that is T ∼ 9g. Then, in the second order (in the metric derivatives) we *nd just those terms which were already included into the Einstein–Cartan action (2.18), but possibly with other coeLcients. In the next order one meets numerous possible structures of the mass dimension 4, which were analyzed in Ref. [48]. Indeed, this action, which includes more than 100 dynamical terms, and many surface terms, does not look attractive for deriving physical predictions of the theory. It is important that this general action, and many its particular cases, describe torsion dynamics. In fact, all interesting and physically important cases, like the action of vacuum for the quantized matter *elds, second order (in  ) string e4ective action, possible candidates for the torsion action [18] are nothing but the particular cases of the bulky action of [48]. In what follows, we consider some of the mentioned particular cases of [48]. Some other torsion actions which will not be presented here: the general fourth derivative actions with absolutely antisymmetric torsion, with and without local conformal symmetry, were described in [34]. 2.6. An example of the possible e=ect of classical torsion There is an extensive bibliography on di4erent aspects of classical gravity with torsion. We are not going to review these publications here, just because our main target is the quantum theory. However, it is worthwhile to present a short general remark and consider a simple but interesting example. The expression “classical action of torsion” can be used only in some special sense. In the Einstein–Cartan theory, with or without matter, torsion does not have dynamics and therefore can only lead to the contact interaction between spins. On the other hand, the spin of the particle is essentially quantum characteristic. Therefore, the classical torsion can be understood only as the result of a semi-classical approximation in some quantum theory. Now, without going into the details, let us suppose that such an approximation can be done and consider its possible e4ects. The most natural possibility is the application to early cosmology, which has been studied long ago (see, for example, discussion in [99]). Here, we are going to consider this issue in a very simple manner. One can suppose that in the early Universe, due to quantum e4ects of matter, the average spin (axial) current is non-zero. Let us demonstrate that this might lead to a non-singular cosmological solution. For simplicity, we suppose that torsion is completely antisymmetric and that there is a conformally constant spinor current J  = V 5  : The Einstein–Cartan action (2.18), with this additional current is [38] 7    1 4 √   (R + 2 S S ) + S J : S = d x −g − 16=G 7

(2.61)

(2.62)

The one-loop quantum calculations in this model, and the construction of the on-shell renormalization group, has been performed in [38].

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We have included an arbitrary coeLcient 2 into the Einstein–Cartan action, but it could be equally well included into the de*nition of the global current (2.61). It is worth mentioning that at the quantum level the introduction of such coeLcient is justi*ed. Since torsion does not have its own dynamics, on shell it is simply expressed through the current 8=G  S = (2.63) J : 2 Replacing (2.63) back into action (2.62) we arrive at the expression    4=G 1 4 √  S = d x −g − ; (2.64) R+ J J 16=G 2 which resembles the Einstein–Hilbert action with the cosmological constant. However, the analogy is incomplete, because the square of the current J  has conformal properties di4erent from the ones of the cosmological constant. Consider, for the sake of simplicity, the conformally 9at metric g = " · a2 (") ; where " is the conformal time. According to (2.61), the current J  has to be replaced by   J  = a−4 (") · JV , where JV is constant. Denoting 32 =G 2  " JV JV  = K = const: ; 3 2 we arrive at the action and the corresponding equation of motion for a("):     3 K d2 a K 3 2 d " d x (∇a) − 2 ; = 3 : (2.65) S =− 2 8=G a d" a This equation can be rewritten in terms of physical time t (where, as usual, a(") d " = d t) as a2 aY + aa˙2 = Ka−3 : After the standard reduction of order, the integral solving this equation is written in the form  a2 d a √ (2.66) = t − t0 ; Ca2 − K where C is the integration constant. The last integral has di4erent solutions depending on the signs of K and C. Consider all the possibilities: (1) The global spinor current is time-like and  K ¿ 0. Then, Eq. (2.66) shows that: (i) C is positive, and (ii) a(t) has minimal value a0 = K=C ¿ 0. Thus, the presence of the global time-like spinor current, in the Einstein–Cartan theory, prevents the singularity. Indeed, since such a global spinor current can appear only as a result of some quantum e4ects, one can consider this as an example of quantum elimination of the Big Bang singularity. The singularity is prevented, in this example, at the scale comparable to the Planck one. This is indeed natural, since the dimensional unity in the theory is the Newton constant. The dimensional considerations [99] show that in the Einstein–Cartan theory the e4ects of torsion become relevant only at the Planck scale. Finally, the explicit solution of Eq. (2.66) has the form    C K 2C 3=2 arccosh (2.67) a + a a2 − = (t − t0 ) ; K C K

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where C is an integration constant. The value of C can be easily related to the minimal possible value of a. In the long-time limit we meet the asymptotic behavior a ∼ t 2=3 . The importance of torsion, in the Einstein–Cartantheory, is seen only at small distances and times and for the scale factor comparable to a0 = K=C. At this scale torsion prevents singularity and provides the cosmological solution with bounce. (2) The spinor current is space-like and K ¡ 0. Then, for any value of C, there are singularities. In the case of positive C the solution is     a C C 2 |K |1=2 C 2 1+ a − 3=2 ln a+ 1+ a = 2 (t − t0 ) ; (2.68) C |K | |K | |K | C while in case of negative C the solution is      1=2   K  C   C 1=2 a − 1 −   a2 +  3  arcsin   a = 2 (t − t0 ) |C | K C K

(2.69)

and for C = 0 it is the simplest one a(t) = [3 |K | (t − t0 )]1=3 ∼ t 1=3 : (3) The last case is when the spinor current is light-like and K = 0. Then, C ¿ 0 and the solution is √ a(t) = [2 C (t − t0 )]1=2 ∼ t 1=2 : (2.70) This is, of course, exactly the same solution as one meets in the theory without torsion. Light-like spin vector decouples from the conformal factor of the metric. The above solutions are, up to our knowledge, new (see, however, Refs. [160,190,8] where other, similar, non-singular solutions were obtained) and may have some interest for cosmology. We notice that the second-derivative in9ationary models with torsion have attracted some interest recently, in particular they were used for the analysis of the cosmic perturbations [147,81]. 3. Renormalization and anomalies in curved space--time with torsion The classical theory of torsion, which has been reviewed in the previous section, is not really consistent, unless quantum corrections are taken into account. The consistency of a quantum theory usually includes such requirements as unitarity, renormalizability and the conservation of fundamental symmetries on the quantum level. In many cases, these requirements help to restrict the form of the classical theories, and thus improve their predictive power. The condition of unitarity is relevant for the propagating torsion. But, for the study of the quantum theory of matter on classical curved background with torsion, it is useless. Therefore, we have to start by formulating the renormalizable quantum *eld theory of the matter *elds on curved background with torsion and related issues like anomalies. There is an extensive literature devoted to the quantum *eld theory in curved space–time (see, for example, books [23,90,80,34] and references therein). In the book [34] the quantum theory

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on curved background with torsion has been also considered. We shall rely on the formalism developed in [36,165,37,4,33,32,34,102], and consider some additional applications. 8 3.1. General description of renormalizable theory Let us start out with some gauge theory (some version of SM or GUT) which is renormalizable in 9at space–time, and describe its generalization for the curved background with torsion. The theory includes spinor, vector and scalar *elds linked by gauge, Yukawa and 4-scalar interactions, and is characterized by gauge invariance and may be by some other symmetries. It is useful to introduce, from the very beginning, the non-minimal interactions between matter *elds and torsion. One can note, that the terms describing the matter self-interaction have dimensionless couplings and hence they cannot (according to our intention not to introduce the inverse-mass dimension parameters), be a4ected by torsion. Thus, the general action can be presented in the form [36]    1  1 a 2 1  4 √ S = d x g − (G ) + g D &D & + $i Pi + M 2 &2 − Vint (&) 4 2 2    "j Qj − im + h& + Svac ; (3.1) + i V  D + where D denotes derivatives which are covariant with respect to both gravitational and gauge *elds but do not contain torsion. $i Pi and "j Qj are non-minimal terms described in Section 2.3. The last term in (3.1) represents the vacuum action, which is a necessary element of the renormalizable theory. We shall discuss it below, especially in the next section. The action of a renormalizable theory must include all the terms that can show up as counterterms. So, let us investigate which kind of counterterms one can meet in the matter *elds sector of the theory with torsion. We shall consider both general non-minimal theory (3.1) and its particular minimal version. Some remark is in order. The general consideration of renormalization in curved space–time, based on the BRST symmetry, has been performed in [30,34]. The generalization to the theory with torsion is straightforward and it is not worth to present it here. Instead, we are going to discuss the renormalization in a more simple form, using the language of Feynman diagrams, and also will refer to the general statements about the renormalization of the gauge theories in presence of the background *elds [64,7,113]. The generating functional of Green’s functions, in the curved space–time with torsion, can be postulated in the form   Z[J; g ; T·  ] = N d 4 exp{i S[4; g; T ] + i&J } ; (3.2) where 4 denotes all the matter (non-gravitational) *elds & (with spins 0; 1=2; 1) and the Faddeev–Popov ghosts c; c. V J are the external sources for the matter *elds &. In the last term, in the exponential, we are using condensed (DeWitt) notations. N =Z −1 [J =0] is the normalization factor. 8

In part, we repeat here the content of Section 4 of [34], but some essential portion of information was not known at the time when [34] was written, or has not been included into that edition.

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Fig. 1. The straight lines correspond to the matter (in this case scalar with ’3 interaction) *eld, and wavy lines to the external *eld (in this case metric). A single diagram in 9at space–time generates an in*nite set of families of diagrams in curved space–time. The *rst of these generated diagrams is exactly the one in the 9at space–time, and the rest have external gravity lines.

Besides the source term, (3.2) depends on the external *elds g and T· . One has to de*ne how to modify the perturbation theory in 9at space–time so that it incorporates the external *elds. The corresponding procedure is similar to that for the purely metric background. One has to consider the metric as a sum of " and of the perturbation h g = " + h : Then, we expand the action S[4; g; T ] such that the propagators and vertices of all the *elds (quantum and background) are the usual ones in the 9at space–time. The internal lines of all the diagrams are only those of the matter *elds, while external lines are both of matter and background gravitational *elds (metric h and torsion). As a result, any 9at-space diagram gives rise to the in*nite set of diagrams, with increasing number of the background *elds tails. An example of such set is depicted at Fig. 1. Let us now remind three relevant facts. First, when the number of vertices increases, the super*cial degree of divergence for the given diagram may only decrease. Therefore, the insertion of new vertices of interaction with the background *elds g ; T· cannot increase the degree of divergence. In other words, for any 9at-space diagram, all generated diagrams with gravitational external tails have the same or smaller index of divergence than the original diagram. Second, since we are working with the renormalizable theory, the number of the divergent n-loop diagrams, in 9at space–time, is *nite. As a result, after generating the diagrams with external gravity (metric and torsion) tails, we meet a *nite number of families of divergent diagrams at any loop order. Furthermore, including an extra vertex of interaction with external *eld one can convert the quadratically divergent diagram into a logarithmically divergent one. For example, the quadratically divergent diagram of Fig. 2a generates the logarithmically divergent ones

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Fig. 2. (a) Quadratically divergent graph for the ’4 -theory. (b) The example of logarithmically divergent graph generated by the graph at Fig. 2(a) and the procedure presented at Fig. 1. This diagram contributes to the R’2 -type counterterm. (c) Dashed lines represent spinor, continuous lines represent scalar and double line—external torsion S . This diagram gives rise to the V 5  S -type counterterm. (d) This diagram produces ’2 S 2 -type counterterm.

of Fig. 2b. The diagrams from Fig. 2b give rise to the R’2 -type counterterm. Similarly, the diagrams of Figs. 2c and d produce V 5  S and ’2 S 2 -type counterterms. Third, there are general proofs [182] that the divergences of a gauge invariant theory can be removed, at any loop, by the gauge invariant and local counterterms. 9 Indeed these theorems apply only in the situation when there is no anomaly. In the present case we have regularizations (say, dimensional [106,123], or properly used higher derivative [171,10]) which preserve, on the quantum level, both general covariance and gauge invariance of the model. Thus, we are in a position to use general covariance and gauge invariance for the analysis of the counterterms. Anomalies do not threaten these symmetries, for in the four-dimensional space–time there are no gravitational anomalies. Taking all three points into account, we arrive at the following conclusion. The counterterms of the theory in an external gravitational background with torsion have the same dimension as the counterterms for the corresponding theory in 9at space–time. These counterterms possess general covariance and gauge invariance, which are the most important symmetries of the classical action. 9

In case of the di4eomorphism invariance, this can be also proved using the existence of the explicitly covariant perturbation technique based on the local momentum representation and Riemann normal coordinates. For example, in [149] this technique has been described in details and applied to the extensive one-loop calculations. In the case of gravity with torsion, the local momentum representation have been used in Ref. [53].

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At this stage one can explain why the introduction of the non-minimal interaction between torsion and matter (spin-1=2 and spin-0 *elds) is so important. The reason is that the appearance of the non-minimal counterterms is possible, for they have proper symmetries and proper dimensions. Let us imagine that we have started from the minimal theory, that is take "1 = 18 ; "2 = 0 and $1; 2; 3; 4; 5 = 0. Then, the classical action depends on the metric g and on the axial vector component S of torsion. Thus, the vertices of interaction with these two *elds will modify the diagrams and one can expect that the counterterms depending on g and S will appear. According to our analysis these counterterms should be of three possible forms (see Figs. 2b–d):    √ √ √ d 4 x −g S S  ’2 ; d 4 x −g R’2 ; d 4 x −g V 5  S and therefore these three structures should be included into the classical action in order to provide renormalizability. Therefore, the essential non-minimal interactions with torsion are the ones which contain the torsion pseudotrace S . If the space–time possesses torsion, the non-minimal parameters "1 and $4 have the same status as the $1 parameter has for the torsionless theory. Of course, $1 remains to be essential—independent of whether torsion is present. The special role of the two parameters "1 ; $4 , as compared to others: "2 and $2; 3; 5 , is due to the fact that minimally only S -component of torsion interacts with matter *elds. It is remarkable that not only spinors but also scalars have to interact with torsion if we are going to have a renormalizable theory.  The terms which describe interaction of matter *elds with T and q· components of torsion, can be characterized as purely non-minimal. One can put parameters $2; 3; 4 ; "2 to be zero simultaneously without jeopardizing the renormalizability. Indeed, if the "2 -term is included, it is necessary to introduce also the $2; 3 -type terms. In the case of abelian gauge theory with complex scalars one may need to introduce some extra non-minimal terms [39] (see also Sections 2.3 and 3.3). Besides the non-minimal terms, one can meet the vacuum structures which satisfy the conditions of dimension and general covariance. The action of vacuum depends exclusively on the  . Hence, the corresponding counterterms result from the diagravitational *elds g and T· grams which have only the external tails of these *elds. The most general form of the vacuum action for gravity with torsion has been constructed in [48]. This action satis*es the conditions of covariance and dimension, but it is very bulky for it contains 168 terms constructed from curvature, torsion and their derivatives. Using the torsionless curvature, one can distinguish the terms of the types R2::: ;

R::: T 2 ;

R::: ∇T;

T 2 ∇T;

T4

plus total derivatives. It turns out, that the number of necessary terms can be essentially reduced without giving up the renormalizability. At the one-loop level, we meet just an algebraic sum of the closed loops of free vectors, fermions and scalars, and only the last two kind of *elds contribute to the torsion-dependent vacuum sector. Therefore, calculating closed scalar and spinor loops one can *x the necessary form of the classical action of vacuum, such that this action is suLcient for renormalizability but does not contain any unnecessary terms. Since we are

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considering the renormalizable theory, the divergent vertices in the matter *eld sector are local and have the same algebraic structure as the classical action. For this reason, the structures which do not emerge as the one-loop vacuum counterterms, will not show up at higher loops too. Hence, one can restrict the minimal necessary form of the vacuum action, using the one-loop calculations. 3.2. One-loop calculations in the vacuum sector In this section we derive the one-loop divergences for the free matter *elds in an external gravitational *eld with torsion. As we already learned in the previous sections, only scalar and spinor *elds couple to torsion, so we restrict the consideration by these *elds. For the purpose of one-loop calculations we shall consistently use the Schwinger–DeWitt technique. One can *nd the review of this method, its generalizations and developments and the list of many relevant references in [64,16,180,11]. Also, in Section 5, some new application of this technique will be given. Now, we need just a simplest version of the Schwinger–DeWitt (1) technique. The one-loop contribution to the e4ective action V =(i=2) Tr ln Hˆ has the following integral representation:    ∞ ∞ d s i D1=2 (x; x ) i i (1) 2  V  = − Tr exp −ism + /(x; x ) (is)k aˆk (x; x ) ; (3.3) 2 s (4=i s)n=2 2s 0 k=0

where /(x; x ) is the world function (geodesic distance between two close points, /= 12 ∇ /∇ /) and D1=2 (x; x ) is the Van Vleck–Morette determinant      92 / 1=2  D (x; x ) = det −    ; 9x 9x n is the parameter of the dimensional regularization. The details about the dimensional regularization in the Schwinger–DeWitt technique can be found in [16]. For the minimal di4erential operator

Hˆ = 1ˆ + 2hˆ ∇ + Eˆ

(3.4)

acting on the *elds of even Grassmann parity, the divergent part of the functional trace (3.3) is a factor of the coincidence limit of the trace tr lim aˆ2 (x; x )
x →x

of the second coeLcient of the Schwinger–DeWitt expansion. Direct calculation yields [64]     Hˆ  i n−4 1ˆ (1) ˆ n √ div (H ) = Tr ln − 2  = − d x −g tr (R2 − R2 + R) 2    180  div   1 ˆ 1 ˆ 1ˆ ˆ P + P · P + S  · Sˆ ; (3.5) + 6 2 12

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where  = (4=)2 (n − 4) is the parameter of dimensional regularization,  is the dimensional parameter, 1ˆ is the identity matrix in the space of the given *elds,   1ˆ Pˆ = Eˆ + R − ∇ hˆ − hˆ hˆ 6 and Sˆ  = (∇ ∇ − ∇ ∇ ) 1ˆ + ∇ hˆ − ∇ hˆ + hˆ hˆ − hˆ hˆ : One has to note that the last formula is nothing but the commutator of the covariant derivatives D = ∇ + hˆ : Of course, expression (3.5) can be written in terms of the covariant derivative with torsion ˜  , but it is useful to separate the torsion-dependent terms. The last observation is that for ∇ operator (3.4) acting on the *elds of odd Grassmann parity, expression (3.5) changes its sign. In a complicated situations with the operators of mixed Grassmann parity (like that we shall meet in Section 5) it is useful to introduce special notation Str for the supertrace. Let us *rst consider the calculation of divergences for the especially simple case of free scalar *eld. The one-loop divergences are given by Eq. (3.5), where 1 2 S0 Hˆ sc = − $i Pi : = − m2 − 2 2 ’ Here we use notation (2.26) of the previous section. Applying (3.5), one immediately obtains   Hˆ sc  i (1) div (scalar ) = − Tr ln  2 2  div    1 ˆ 1 ˆ2 n−4 1 n √ 2 2 d x −g P+ P ; =− − R + R) + (3.6) (R  180  6 2 where

1 Pˆ = R − $i Pi − m2 : 6 i

As it was already mentioned above, in order to provide renormalizability one has to include into the classical action of vacuum all the structures that can appear as counterterms. For the scalar *eld on the external background of gravity with torsion, the list of the integrands of the vacuum action consists of R2 and R2 , *ve total derivatives Pi , 10 products Pi Pj and six mass dependent terms: m4 and m2 Pi . The total number of necessary vacuum structures is 23, and 7 of them are total derivatives. This number of 23 can be compared, from one side, with the 6 terms R2 ; R2 ; R2 ;

R; m4 ; m2 R;

which emerge in the torsionless theory, and from the other side, with the 168 algebraically possible covariant terms constructed from curvature, torsion and their derivatives [48].

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It is sometimes useful to have another basis for the torsionless fourth derivative terms. We shall use the following notations: C 2 = C C  = R R − 2 R R + 13 R2 for the square of the Weyl tensor, which is conformal invariant at four dimensions and E = R R − 4 R R + R2 for the integrand of the Gauss–Bonnet topological term. The inverse relations have the form R2 = 2C 2 − E + 13 R2

and

R2 = 12 C 2 − 12 E + 13 R2 :

(3.7)

Let us now consider the fermionic determinant, which has been studied by many authors (see, for example, [87,116,142,32,53,93]). One can perform the calculation by writing the action through the covariant derivative without torsion [142,32]. So, we start from the general non-minimal action (2.35). The divergent contribution from the single fermion loop is given by the expression div [g; A; S] = −i Tr ln Hˆ |div ; (3.8) where Hˆ = i  (D − im) and

D = ∇ + ieA + i" 5 S

is generalized covariant derivative. For the massless theory this covariant derivative respects general covariance, the abelian gauge symmetry (2.36) and the additional gauge symmetry (2.37). In the massive case this last symmetry is softly broken, but the above notation is still useful. In order to calculate functional determinant (3.8), we perform the transformation which preserves covariance with respect to the derivative D . First observation is that, by dimensional reasons, the (3.8) is even in the mass m. In other words, (3.8) does not change if we replace m by −m. It proves useful to introduce the conjugate derivative, D∗ = 9 + ieA − i" 5 S and ∗ the conjugated operator Hˆ = i  (D + im). Then, we can perform the transformation: i i ∗ fermion = − Tr ln Hˆ · Hˆ = − Tr ln {−  D  D − m2 } 2 2 i = − Tr ln {−(   D∗ D + m2 )} : (3.9) 2 After a simple algebra, one can write two useful forms for the last operator: the non-covariant (with respect to D ): ∗ −Hˆ · Hˆ = ∇2 + R ∇ + E ; with (here / = ea eb /ab = (i=2)[   −   ]) R = 2ieA + 2" / S 5 ; E = ie∇ A + i" 5 9 S  − e2 A A + +

i

2

ie

2

1   F − R + m2 4

"   5 S + "2 S S  + 2ie"/ 5 A S ;

(3.10)

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where S = 9 S − 9 S , F = 9 A − 9 A and covariant ∗

−Hˆ · Hˆ = D2 + E  D + F ;

(3.11)

with 1 ie i F = m2 − R +   F + " 5   S : 4 2 2 The intermediate expressions, for the covariant version, are ie 1 Pˆ = i" 5 ∇ S  − 2"2 S S  +   F − R + m2 2 12 and E  = −2i "   5 S  ;

(3.12)

(3.13)

Sˆ  = 14 6 R6 − ieF + "/ 5 ∇ S  − "/ 5 ∇ S  + (     −     )"2 S  S  :

(3.14)

The divergent part of (3.9) can be easily evaluated using the general formula (3.5). After some algebra we arrive at the following divergences:   2 2 2 1 n−4 2 (1) n √ 2 V div (Dirac spinor) = d x −g e F + "2 S − 8 m2 "2 S S  − m2 R + 2m4  3 3 3 1 7 2 4 1 2 R − R2 − R6 − "2 (S  S ) 72 45 360 3  4 2 1     + " ∇ (S ∇ S − S ∇ S ) − R : 3 30

+

(3.15)

The form of divergences (3.15) is related to the symmetry transformation (2.37). For instance, di4eomorphism and gauge invariance (2.36) are preserved. The one-loop divergences contain the 2 -term, that indicates that in the massless theory symmetry (2.37) is also preserved. And the S appearance of the massive divergent m2 S 2 term reveals that the symmetry under transformation (2.37) is softly broken by the fermion mass. Symmetry (2.37) and the form of divergences (3.15) will be extensively used later on, in Section 5, when we try to formulate the consistent theory for the propagating torsion. Indeed, it is very important that the longitudinal (9 S )2 -term is absent in (3.15), for it would break symmetry (2.37). The one-loop divergences coming from the fermion sector, add some new necessary terms to the vacuum action. The terms which were not necessary for the scalar case, but appear from spinor loop are (remind that T is hidden inside A ): S S  ;

T T  ;

∇ (S ∇ S  − S  ∇ S ) ;

(3.16)

where we denoted, as in Section 2, T = 9 T − 9 T . Finally, for the theory including scalars, gauge vectors and fermions, the total number of necessary vacuum structures is 26, and 7 of them are surface or topological terms. One can see, that in such theory 142 of the algebraically  -component of possible counterterms [48] never show up, most of them depend on the q· torsion.

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3.3. One-loop calculations in the matter 5elds sector The one-loop calculation in the matter *elds sector in curved space–time with torsion is very important and it has been carried out for various gauge models [36,37,39,164,4]. The *rst such calculation was done in [36,37] for several SU (2) models. The renormalization of the same models in 9at space–time has been studied earlier in [183], where they were taken as examples of theories with the asymptotic freedom in all (gauge, Yukawa and scalar) couplings taking place on the special solutions of the renormalization group equations. Later on, the same models have been used in [31] for the *rst calculation of the one-loop divergences and the study of the renormalization group for the complete gauge theory in an external gravitational *eld. The one-loop renormalization in the torsion-dependent non-minimal sector [36] possesses some universality, and this motivated consequent calculations performed in [36,37,39,4] for the abelian, O(N ) and SU (N ) models with various *eld contents. Here, we present only a brief account of these calculations, so that the origin of the mentioned universality becomes clear. As we have already learned, there is some di4erence in the interaction with torsion for abelian and non-abelian gauge models. In the last case, the massless vector *eld does not interact with torsion at all, while in the *rst the non-minimal interaction (2.38) is yet possible. Therefore, besides being technically simpler, the abelian gauge theory is somehow more general and we shall use it to discuss the details of the one-loop renormalization. We mention, that the one-loop calculation for the abelian model was *rst performed in [140], but has been correctly explained only in [39]. Let us consider the abelian gauge model including gauge, Yukawa and four-scalar interactions. The classical action is some particular case of (3.1), it is given by the sum of non-minimal matter actions (2.26), (2.34) and (2.38), interaction terms and the action of vacuum with all 26 terms described in the previous section:
 √ 1 S = d 4 x −g − F F  + K  F 4 5 1  1 2 2 1 1 + g 9 ’9 ’ + m ’ + $i Pi ’2 − f ’4 2 2 2 24 i=1    + i V   ∇ + ie  A − im − i h ’ + "j Qj  + Svac :

(3.17)

j=1;2

For the sake of the one-loop calculations one can use the same formula (3.5). However, since we are going to consider, simultaneously, the *elds with di4erent Grassmann parity, this formula must be modi*ed by replacing usual traces, Tr and tr, by the supertraces, Str and str. The decomposition of the *elds into classical background A , ’, V , and quantum ones a , /, (, V ( is performed in the following way: A → A + a ;

’ → ’ + i/;

V = V + (; V

=

+ (;

i ( = − ∇ " : 2

(3.18)

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In the abelian theory, we can *x the gauge freedom by simple condition ∇ a =0. The Faddeev– Popov ghosts decouple from other quantum *elds. They contribute only to the vacuum sector, which we already studied in the previous section. The resulting operator (3.4) has the block structure which emerges from the bilinear expansion of the classical action [39]   a      √ 1  S (2) = d 4 x −g(a | / | () V Hˆ  (3.19) / : 2 " The details of the calculation can be found in [140,36]. The matrix structure of the operators Pˆ and Sˆ  is the following:     P11 P12 P13 S11;  S12;  S13;     ˆ   ; Pˆ =   P21 P22 P23  and S  = S21;  S22;  S23;  S31;  S32;  S33;  P31 P32 P33 where the terms essential for our consideration are [39] S31 = S32 = 0;

P32 = 2h ;

P31 = −2e  ;

ie 1 i P13 = − ∇6 V  6 + ieh V  ’ − e2 V  A6 6 − e V  6 "j Qj 6 ; 2 2 4 1 ih i "j Qj 6 : (3.20) P23 = ∇6 V 6 − ih2 V ’ + eh V A6 6 + h V 6 2 2 4 Now, substituting Pˆ and Sˆ  into (3.5), after some algebra we arrive at the divergent part of the one-loop e4ective action
n−4  √ 8e"2  2e2 (1) V div = − d n x −g − F F  + F ∇ T + 2h2 g 9 ’ 9 ’  3 3      2  1 2 2 2 f 1 4 4 2 2  − 2h ’ + ’ − $i Pi − R f + h R − 16h "1 S S + 8 2 6 3 i

+ i(2e2 + h2 ) V  (∇ + ieA − i"2 T ) + (2e2 − h2 ) "1 V 5  S  2 2 + (8e − 2h )h V ’ + vacuum divergent terms :

(3.21)

The above expression is in perfect agreement with the general considerations of Section 3.1. Let us comment on some particular points. 1. One can see that the non-minimal divergences of the V 5  S and ’2 S S  -types really emerge, even if the starting action includes only minimal interaction with "2 = 18 ; "2 = 0 and $2; 3; 4; 5 =0. Thus, exactly as we have supposed, the non-minimal interaction of the S -component of torsion with spinor and scalar is necessary for the renormalizability. The list of essential parameters includes $1 ; "1 ; $4 parameters of the action which are not related to T or q· .

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Another potentially important parameter is 22 , because the corresponding topological counterterm (2.38) can emerge at higher loops.  components of torsion are purely non-minimal. Furthermore, 2. All terms with T and q· the substitution eA + "2 T → eA explains all details of the renormalization of the parameter "2 . In particular, this concerns the non-minimal interaction of the gauge vector A with torsion trace T , which leads to the renormalization of the non-minimal parameter 25 in (2.38). It is worth mentioning that such a mixing is not possible for the non-abelian case [36]. 3. One can observe some simple hierarchy of the parameters. The non-minimal parameters $i ; "j ; 2k do not a4ect the renormalization of the coupling constants e; h; f and masses. One can simply look at Fig. 1 to understand why this is so. In turn, vacuum parameters do not a4ect the renormalization of neither coupling constants nor the non-minimal parameters $i ; "j ; 2k . One important consequence of this is that the renormalization group equations in the minimal matter sector are independent on external *elds so that the renormalization of the couplings and masses is exactly the same as in the 9at space–time. The renormalization group in the mixed non-minimal sector depends on the matter couplings, but does not depend on the vacuum parameters. Finally, the renormalization in the vacuum sector depends, in general, on the non-minimal parameters. 4. The last observation is the most complicated one. The contributions to the spinor sector may come only from the mixed sector of the products of operators (3.20). Now, since S31 = S32 = 0, all the fermion renormalization comes from two traces: tr (P13 · P31 ) and

tr(P23 · P32 ) :

It is easy to see, that the arrangement of the -matrices in the expressions for P13 ; P31 ; P23 ; P32 is universal in the sense that it does not depend on the gauge group. For any non-abelian theory this arrangement is the same as for the simple abelian model under discussion. Therefore, in the fermionic sector, the signs of the counterterms will always be equal to the ones we meet in the abelian model. The renormalization of the essential parameter "1 has the form   C 2 0 ; (3.22) "1 = "1 1 − h  with C = 2 in the abelian case. Using the above consideration we can conclude that the renormalization of this parameter in an arbitrary gauge model will have the very same form (3.22) with positive coeLcient C. Of course, the value of this coeLcient may depend on the gauge group. In the theory with several fermion *elds the story becomes more complicated, because there may be more than one non-minimal parameters "1 , and in general they may be di4erent for di4erent *elds. The renormalization can mix these parameters, but the consideration presented above is a useful hint to the sign universality of the -function for "1 , which really takes place [36,39,37,4]. The details of the calculations of the one-loop divergences in the variety of SU (2), SU (N ), O(N ) models including the *nite and supersymmetric models can be found in [36,39,37,4], part of them was also presented in [34]. Qualitatively, all these calculations resemble the sample

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we have just considered, so that all the complications come from the cumbersome group relations and especially from the necessity to work with many-fermion models. The results are in complete agreement with our analysis, in particular this concerns the universal sign of C in (3.22). Let us now discuss another, slightly di4erent, example. Consider, following [164], the Nambu– Jona–Lasinio (NJL) model in curved space–time with torsion. This model is regarded as an e4ective theory of the SM which is valid at some low-energy scale. If we are interested in the renormalization of the theory in an external gravitational *eld, then the NJL model may be regarded as the special case of the theory with the Higgs scalars [103,131]. Our purpose is to study the impact of torsion for this e4ective theory. Consider the theory of N -component spin 12 *eld with four-fermion interaction in an external gravitational *eld with torsion. The action is  √ Snjl = d 4 x −g{Lgb + i V  (D − i"1 5 S ) + G( V L R )2 } : (3.23) Here, Lgb is the Lagrangian of the gauge boson *eld, D is the covariant derivative with respect to both general covariance and gauge invariance, G is the dimensional coupling constant. The above Lagrangian (103) is direct generalization of the one of the paper [103] for the case of gravity with torsion. The introduction of the non-minimal interaction with torsion re9ects the relevance of such an interaction at high energies. Introducing the auxiliary Higgs *eld H , one can cast the Lagrangian in the form  √ Snjl = d 4 x −g{Lgb + i V  (D − i"1 5 S ) + ( V L R H + V R L H † ) − m2 H † H } : (3.24) It is easy to see that, in this form, theory (3.24) is not renormalizable due to the divergences in the scalar and gravitational sectors. However, our previous analysis can be successfully applied here if we add to (3.24) an appropriate action of the external *elds. First, we note that the scalar *eld is non-dynamical, so that both kind of divergences are similar to the vacuum ones in ordinary gauge theories in curved space–time. Then, one can provide the renormalizability by introducing the Lagrangian Lext of external *elds H; H † ; g ; S into action (3.24). If we do not consider surface terms, the general form of Lext is

Lext = g D H D H † + $1 RH † H + $4 H † HS S  − (H † H )2 + Lvac ; 2

(3.25)

where the form of Lvac corresponds to the possible counterterms (3.15), as it has been discussed in the previous section. Some terms in (3.24) were transferred to Lext . This re9ects their role in the renormalization. When analyzing possible divergences of theories (3.23) and (3.24), one meets a serious di4erence. Theory (3.23) is not renormalizable, while theory (3.24), (3.25) is. The reason is the use of the fermion-bubble approximation. Then, (3.24) becomes the theory of the free spinor *eld in the external background composed by gauge boson, scalar, metric and torsion *elds. All possible divergences in this theory emerge at the one-loop level, and only in the external *eld sector. In this sense, one can formulate the renormalizable NJL model in the external gravitational *eld with torsion.

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The direct calculations give the following result for the divergent part of the e4ective action (we omit the gauge boson and surface terms):   1 2N n−4 (1) n √ V d x −g g D H D H † + RH † H div = −  6  1 1 2 +  + 2   + 4"1 H HS S − (H H ) − S S + C C + ··· : (3.26) 3 20 The *rst terms in (3.26) are the same as in a purely metric theory [130,131], while others are typical for the theory with external torsion. 3.4. Renormalization group and universality in the non-minimal sector The renormalizability of the Quantum Field Theory in curved background enables one to formulate the renormalization group equation for the e4ective action and parameters of the theory. The derivation of these equations in the space–time with torsion is essentially the same as for the purely metric background. Let us outline the formulation of the renormalization group [30,34]. The renormalized e4ective action  depends on the matter *elds 4 (as before, we denote all kind of non-gravitational *elds in this way), parameters P (they include all couplings, masses, non-minimal parameters and the parameters of the vacuum action), external  , dimensional parameter  and the parameter of the dimensional regularization n. *elds g ; T· The renormalized e4ective action is equal to the bare one:   [g ; T· ; 4; P; ; n] = 0 [g ; T· ; 40 ; P0 ; n] :

(3.27)

Taking derivative with respect to  we arrive at the equation    9 9  n √   + P [g ; T· ; 4; P; ; n] = 0 : (3.28) + d x −g 4 9 9P 4 Here  and functions are de*ned in a usual way: 9P 94 P (n) =  and (n)4 =  : (3.29) 9 9 The conventional n = 4 beta- and gamma functions are de*ned through the limit n → 4. Using the dimensional homogeneity of the e4ective action we get, in addition to (3.28), the equation     9 9 9 n √  [g ; T· ; 4; P; ; n] = 0 ; (3.30) +  + dP + d4 d x −g 4 9t 9 9P 4 where dP and d4 are classical dimensions of the parameters and *elds. Combining (3.28) and (3.30), setting t = 0, replacing the operator 2g (=g )[g ; : : : ] by (9= 9t)[e−2t g ; : : : ], and taking the limit n → 4, we arrive at the *nal form of the renormalization group equation appropriate for the study of the short-distance limit:    √  9 9  − (P − dP ) − ( − d4 ) d n x −g 4 [g e−2t ; T· ; 4; P; ] = 0 : (3.31) 9t 9P 4 The general solution of this equation is   [g e−2t ; T· ; 4; P; ] = [g ; T· ; 4(t); P(t); ] ;

(3.32)

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where *elds and parameters satisfy the equations d4 4(0) = 4 ; = [ (t) − d4 ]4; dt dP (3.33) = P (t) − dP ; P(0) = P : dt In fact, torsion does not play much role in the above derivation, mainly because it does not transform under scaling. In fact, this is natural, because the physical interpretation of the UV limit in curved space–time is the limit of short distances. But, geometrically, the distance between two points does not depend on torsion, so it is not a big surprise that torsion is less important than metric here. From Eq. (3.32) follows that the investigation of the short-distance limit reduces to the analysis of Eqs. (3.33). Our main interest will be the behavior of the non-minimal and vacuum parameters related to torsion, but we shall consider the renormalization of other parameters when necessary. First of all, according to our previous discussion of the general features of the renormalization, the matter couplings and masses obey the same renormalization group equations as in the 9at space–time. Furthermore, the running of $1 and those vacuum parameters, which are not related to torsion, satisfy the same equations as in the torsionless theory. Let us concentrate our attention on the parameters related to torsion. Consider the most important equation for the non-minimal parameter "1 . Using the universality of its one-loop renormalization (3.22), and the classical dimension (in n = 4) for the Yukawa coupling dh n − 4 (2=)2 = h + h (4) ; dt 2 we can derive the universal form of the renormalization group equation at n = 4 d "1 (4=)2 (3.34) = C"1 h2 : dt Here, according to (3.22), the constant C is positive, but its magnitude depends on the gauge group. For instance, in the case of abelian theory C = 2, and for the adjoint representation of the SU (N ) group the value is C = 1 [4]. The physical interpretation of the universal running (3.34) is obvious: the interaction of fermions with torsion becomes stronger in the UV limit (short-distance limit in curved space–time). This provides, at the *rst sight, an attractive opportunity to explain very weak (if any) interaction between torsion and matter *elds. Unfortunately, the numerical e4ect of this running is not suLcient. Let us consider the simplest case of the constant Yukawa coupling h = h0 . Then, replacing, as usual, d = d t for  d = d , (3.34) gives 2   2  Ch0 =(4=) "1 () : (3.35) = "1 (UV ) UV It is easy to see, that even the 50-order change of the scale  changes "1 for less than one order. The e4ect of (3.35) is stronger for the heavy fermions with larger magnitude of the Yukawa coupling. If we suppose, that all the fermions emerge after some superstring phase transition, and that the high-energy values of the non-minimal parameters are equal for all the spinors, the low-energy values of these parameters will not di4er for more than 2–3 times. Indeed, there

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may be some risk in the above statement, related to the non-perturbative low-energy e4ects of QCD. However, since the quarks are con*ned inside the nucleus, and there is no chance to observe their interaction with an extremely weak background torsion (see the next section for the modern upper bounds for the background torsion), the e4ect of the "1 (t) running does not have much physical importance. In the non-minimal scalar sector we meet standard equation for the $1 parameter, which does not depend on torsion [36,34]   1 2 d $1 (4=) [k1 g2 + k2 h2 + k3 f] ; (3.36) = $1 − dt 6 where the magnitudes of k1 ; k2 ; k3 depend on the gauge group, but always k1 ¡ 1 and k2; 3 ¿ 0. The equation for the scalar–torsion interaction parameter $4 has the form [36] (4=)2

d $4 = [k1 g2 + k2 h2 + k3 f]$4 − k4 h2 "21 dt

(3.37)

with k4 ¿ 0. The asymptotic behavior of $1 depends on the gauge group and on the multiplet composition of the model. For some models $1 → 1=6 in the UV limit t → ∞ [31,40], and one meets the asymptotic conformal invariance. For other theories, including the minimal SU (5) GUT [149], the asymptotic behavior is the opposite: |$1 | → ∞ at UV. It is remarkable, that due to the non-homogeneous term in the beta-function (3.37) for $4 , this parameter has an universal behavior which does not depend on the gauge group. It is easy to see, that in all cases |$4 (t)| → ∞

at t → ∞ :

(3.38)

Consequently, the interaction of scalar with torsion also gets stronger at shorter distances, and weaker at long distances. Qualitatively, the result is the same as for the parameter "1 . Consider, for example, the SU (2) gauge model with one charged scalar multiplet and two sets of fermion families in the fundamental representation of the gauge group. In 9at space–time, the -functions for this model (which is quite similar to the SM) have been derived in Ref. [183]. The Lagrangian of the renormalizable theory in curved space–time with torsion has the form [36]    ig a a 1 a a  ig a a † f  † 9 & +  A & 9 & −  A & − (&† &)2 L = − G G +g 4 2 2 8   5 m + $i Pi &† & + i$0 T  (&† 9 & − 9 &† · &) + i (V(k)   ∇ + j Qj  ((k) i=1 m+n

+

k=1

 iV

(k)



  ∇ − ig  a  Aa + "j Qj 

2

j=1;2

j=1;2

k=1

(k)

m  V (k) ((k) & + &† (V(k) −h

(k)



:

k=1

(3.39) Let us write, for completeness, the full set of the -functions for the coupling constants and non-minimal parameters.

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(i) Couplings [138]: (4=)2

d g2 = −b2 g2 ; dt

(4=)2

d h2 9 = (3 + 4m)h4 − g2 h2 ; dt 2

(4=)2

df = 3f2 − 32mh4 + 9g4 + 8mfh2 − 9fg2 : dt

b2 = −

43 − 4(m + n) 4 g ; 3

(3.40)

Likewise all three SU (2) models of Voronov and Tyutin [183], this one is asymptotically free in all e4ective couplings, but only in the special regime, when they are proportional to each other. The necessary condition of the asymptotic freedom in g2 is m + n 6 10 and for h2 it is m + n ¿ 8. The asymptotic freedom in f(t) happens only for m + n = 10 and m = 1. On the special asymptotically free solutions of the renormalization group equations one meets the following behavior: g2 (t) =

g2 ; 1 + b2 g2 t=(4=)2

1 h2 (t) = g2 ; 2

0 ¡ f 6 g2 :

(3.41)

(ii) Non-minimal parameters [31,36]: d "1 = h2 ("1 + 1 ); dt  2 d "2 2 (4=) = h "2 − 2 − dt

(4=)2

(4=)2

d 1 = 2h2 "1 ; dt

 1 $0 ; 2

(4=)2

d 2 = h2 (−2"2 + $0 ) ; dt

d $0 = −4m2 h2 (2 − "2 ) ; dt   1 d $2; 5 2 d $1 (4=) · A; (4=)2 = $1 − = A$2; 5 ; dt 6 dt

(4=)2

(4=)2

d $3 = A$3 + 4mh2 ("2 − 2 )2 ; dt

(4=)2

3 2

9 2

where A = f − g2 + 4mh2 ;

d $4 = A$4 − mh2 ("1 + 1 )2 : dt

(3.42)

The solutions of these equations are rather cumbersome [165] and we will not write them here, but only mention that they completely agree with the general analysis given above (the same is true for all other known examples). Finally, the asymptotic behavior of the non-minimal parameters is the following [165,36]: $1 − 16 ; $0 ; "2 ; 2 → 0;

"1 ; 1 → ∞ sgn(1 + 2"1 );

$4 → +∞ :

A special case is the renormalization group equations for the NJL model (3.24). Since this theory can be viewed as the theory of free spinor *elds in an external scalar and gravitational *elds, the exact -functions coincide with the one-loop ones. The renormalization group equations for

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the essential e4ective couplings $4 ; "1 have the form d "21 8N 4 = " ; dt 3 1   8 2 2 d $4 (4=) = 2N $4 − "1 : dt 3

(4=)2

(3.43)

The analysis of these equations shows that the strength of the interaction of spinor and scalar *elds with torsion increases at short distances. Let us now consider the renormalization group for the parameters of the vacuum energy. If we write the vacuum action as  26 √ Svac = d 4 x −g pk Jk ; (3.44) k=1

where pk are parameters and Jk are vacuum terms (for instance, J1 =C 2 ; J2 =E; : : :), the one-loop divergences will have the form  26 √ n−4 d n x −g div = − Mk Jk : (3.45) j k=1

Here Mk are the sums of the contributions from the free scalar, vector and spinor *elds. The relations between renormalized and bare parameters have the form   M pk0 = n−4 pk + k : j The -functions are derived according to the standard rule k = dpk = d . It is important, from the technical point of view, that the one-loop vacuum divergences depend only on the non-minimal parameters $; ", but not on the couplings g; h; f. The renormalization of the non-minimal parameters does not include the n−4 factor, and that is why we can derive universal expressions for the vacuum -functions and renormalization group equations dpk Mk ; = k = (4 − n)pk − dt (4=)2

pk (0) = pk0 :

(3.46)

Standard n = 4 beta-functions can be obtained through the limit n → 4. Since the total number of vacuum terms is 26, it does not have much sense to study the details of scaling behavior for all of them. We shall just indicate some general properties. One can distinguish the Mk coeLcients which are parameter independent (like the M1 ; M2 ), the ones which are proportional to the squares of the masses of scalars or spinors m2 , the ones which are proportional to the squares of the non-minimal parameters of the torsion–matter interaction "2j or $2i . In general, all these types of terms will have distinct asymptotic behavior. Let us consider, for simplicity, only those parameters which are related to the completely antisymmetric torsion and correspond to the massless theory. For the same reason we can take some *nite model, in which the beta-functions for masses equal zero (this can be also considered as an approximation, because typically "1 runs stronger than the mass). We can notice that if the scalar mass does not run, the

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behavior of $4 is $4 (t) ∼ "22 (t). Finally, for the torsion-independent terms we get the asymptotic behavior M pk (t) − pk0 ∼ − k 2 t ; (4=) while for the "21 -type terms the behavior is 2

2

pk − pk0 ∼ e2Ch t=(4=) ∼ "21 (t) and for the $24 -type term 2

2

pk − pk0 ∼ e4Ch t=(4=) ∼ "41 (t) : Thus, the running of the torsion-dependent vacuum terms is really di4erent from the running of the torsion-independent ones. For the last ones, we meet the usual power-like scaling, which may signify asymptotic freedom in UV or (for the massless theories) in IR—this depends on the relative sign of pk0 and Mk . For the torsion-dependent terms the behavior is approximately exponential, exactly as for the non-minimal parameter "1 . Indeed, the numerical range of the running is not too large, because of the 1=(4=)2 -factor. 3.5. E=ective potential of scalar 5eld in the space–time with torsion: spontaneous symmetry breaking and phase transitions induced by curvature and torsion Let us investigate further the impact of the renormalization and renormalization group in the matter *eld sector of the e4ective action. We shall follow [33] and consider the e4ective potential of the massless scalar *eld in the curved space–time with torsion. The e4ective potential V is de*ned as a zero-order term in the derivative expansion of the e4ective action of the scalar *eld ’:    1 4 √  [’] = 0 + d x −g −V (’) + Z(’)g 9 ’9 ’ + · · · ; (3.47) 2 where the dots stand for higher derivative terms, and 0 is the vacuum e4ective action. We shall discuss the derivation of 0 in the next sections. The classical potential of the scalar *eld has the form 4

Vcl = af’ −

5

bi $i Pi ’2 ;

(3.48)

i=1

where we have used notations (2.25). If the space–time metric is non-9at and the scalar *eld couples to spinors through the Yukawa interaction, two of the non-minimal parameters $1 and $4 are necessary non-zero. Therefore, even for the 9at space–time metric the potential feels torsion through the parameter $4 . Here we consider the general case of the curved metric and take all parameters $i arbitrary for the sake of generality. The quantum corrections to the classical potential (3.48) can be obtained using the renormalization group method [54,30,34,33]. The renormalization group equation for the e4ective potential follows from the renormalization group equation (3.28) for the whole e4ective action. Since (3.28) is linear, all terms in expansion (3.47) satisfy this equation independently. It is

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supposed that the divergences were already removed by the renormalization of the parameters, and therefore in this case one can put n = 4 from the very beginning. Thus, we get     9 9 9 4 √   +  + P V (g ; T· ; ’; P; ) = 0 : (3.49) + d x −g ’ 9 9 9P ’ Here P stands, as before, for all the parameters of the theory: gauge, scalar and Yukawa couplings and non-minimal parameters $i .  is the gauge *xing parameter corresponding to the term Lgf = (1=2)(∇ A )2 and  is the renormalization group function corresponding to . We shall solve (3.49) in the approximation ’2 |Pi | for all Pi , and neglect higher order terms. Physically, this approximation corresponds to the weakly oscillating metric and weak external torsion. In the gravitational *eld without torsion this approximation has been used in [108]. Similar method can be applied to the derivation of other terms in the e4ective action, including higher order terms [41] (see also [34]). The initial step is to write the e4ective potential  in the form V =V1 +V2 , where V1 does not depend on the external *elds g ; T ·  , and V2 = 5i=1 V2i Pi . Since all Pi are linear independent, V2i must satisfy Eq. (3.49) independently. Then this equation is divided into the following set of equations for V1 ; V2i : d 4 V1 d 2 V2i =0 ; (3.50) (D − 4 ) 4 = 0; (D − 2 ) d’ d ’2 where 9 9 9 D = −(1 + ) +  + P (3.51) 9t 9 9P and t = (1=2) ln ’2 =2 . If we use the standard initial conditions [34]   d 4 V1  d 2 V2i  = 4af; = −2bi $i Pi ; d ’4 t=0 d ’2 t=0 the solution of Eq. (3.50) can be easily found in the form d 4 V1 d 2 V2i 4 = 4af(t)/ (t); = −2bi $i (t)/2 (t)Pi ; d ’4 d ’2 where    t  /(t) = exp − [P(t V )] d t  :

(3.52)

0

The e4ective charges P(t) ≡ (f(t); $i (t); : : :) satisfy the renormalization group equations [34] ˙ = V P (t); P(0) = P; where V P = P and V = P(t) : (3.53) 1+ 1+ In the one-loop approximation one has to take the linear dependence on t, so that P(t) = P + P · t. Then Eqs. (3.52) become   1 d 4 V1 ’2 = 4a f + (f − 4f ) ln 2 ; d ’4 2    1 d 2 V2i ’2 = −2bi $i + ($i − 2$i ) ln 2 : (3.54) d ’2 2 

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Integrating these equations with the renormalization (boundary) conditions    d 2 V1  d 4 V1  d 2 V2i  = 0; = 4af; = −2bi $i Pi ; d ’2 ’=0 d ’4 ’= d ’2 ’= we arrive at the *nal expression for the one-loop e4ective potential     5  ’2 25 ’2 4 4 V = af’ + A ln 2 − ’ − bi $i + Bi ln 2 − 3 Pi ’2  6 

(3.55)

i=1

with the coeLcients 1 a A = (f − 4f ) and Bi = bi ($i − 2$i ) : (3.56) 2 2 Eq. (3.55) is the general expression for the one-loop e4ective potential in the linear in Pi approximation. One can substitute the values of a and b from some classical theory, together with the corresponding  and functions, and derive the quantum corrections using (3.55) and (3.56). The gauge *xing dependence enters the e4ective potential through the -function of the scalar *eld. We remark that, in general, the problem of gauge dependence of the e4ective potential is not simple to solve. It was discussed, for instance, in [144]. In the case of f ∼ g4 and $i ∼ g2 the gauge *xing dependence goes beyond the one-loop approximation and the corresponding ambiguity disappears. These relations are direct analogs of the one which has been used in [54] for the similar e4ective potential on 9at background without torsion. Let us present, as an example, the explicit expression for the e4ective potential for theory (3.39) with m + n = 10 and m = 1. The necessary -functions are given in (3.42). The -function can be easily calculated to be () = 34 g2 − 94 g2 + 2h2 : Then, using the general formulas (3.55), (3.56) one can easily derive the e4ective potential for the theory (3.39). 10   5 f † 2 3f2 + 9g4 − 32h4 − 3fg2 † |’|2 25 † V = (’ ’) − $i Pi ’ ’ + ’ ’ ln 2 − 8 (4=)2  6 i=1     5 1 |’|2 3 1 3 9 2 † 2 2 ln − ’ ’ − 3 − g ) $ P − − P1 (f f + 4h g i i 2(4=)2 2 2 6 2 2 i=1  − h2 (2 − "2 )2 P3 − h2 (1 + "1 )2 P4

:

(3.57)

Consider, using the generalexpressions (3.55) and (3.56) the spontaneous symmetry breaking. It is easy to see that, for the 5i=1 bi $i Pi ¿ 0 case, the classical potential (3.48) has a minimum at 5 1 2 ’0 = bi $i Pi ; (3.58) 2af i=1

10

Some small misprints of [33] are corrected here.

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so that the spontaneous symmetry breaking might occur, at the tree level, without having negative mass square. It is not diLcult to take into account the quantum corrections to ’20 , just solving the equations 9V= 9’ = 0 iteratively. For instance, after the *rst step one obtains     5 ’20 11 ’20 A 2 1 2 2 ’1 = ’0 − + Bi Pi ln 2 − 2 : ’ ln 2 − af 0  3 2af  i=1  One may consider a marginal case, when bi $i Pi ¡ 0, so that there is no spontaneous symmetrybreaking at the tree level. Suppose also that the $i ≈ 0, so that the  absolute value of  the sum bi $i Pi is very small, and that bi $i Pi ¿ 0, such that the sign of bi $i Pi changes under the quantum corrections. Approximately, Bi = 12 bi $i . Then, from the equation 9V= 9’ = 0 one gets   5 5 bi $i 11 A 2 ’2 A 2 1 2 ’ − bi $i Pi : (3.59) Pi − ’ + ln 2 ’ − 2af 3 af  af 4af i=1

i=1

One can denote the positive expression ’20 =

5

1 bi $i Pi : 2af i=1

Then, after the *rst iteration Eq. (3.59) gives   5 2 ’ 11 A A 1 ’21 = ’20 + bi $i Pi : ’2 − ln 20 ’2 − 3 af 0  af 0 4af

(3.60)

i=1

It is easy to see, that in this case the spontaneous symmetry breaking emerges only due to the quantum e4ects. In all the cases: classical or quantum, the e4ect of spontaneous symmetry breaking is produced by expressions like bi $i Pi or bi $i Pi . In particular, the e4ect can be achieved only due to the torsion, without the Ricci curvature scalar P1 = R. Let us now investigate the possibility of phase transitions induced by curvature and torsion. We shall be interested in the *rst order phase transitions, when the order parameter ’ changes by jump. It proves useful to introduce the dimensionless parameters x = ’2 =2 and yi = Pi =2 . The equations for the critical parameters xc ; yic corresponding to the *rst order phase transition, are [117]:   9V  92 V  V (xc ; yic ) = 0; = 0; ¿0 : (3.61) 9x xc ;yic 9x2 xc ;yic These equations lead to the following conditions: 1=2  5 2A2 = − qi i Di ±  (Di Dj − 4A2 Bi Bj )i j qi gj  ; i=1

i; j

5 8 1 af − A + A ln x − Bi i qi ¿ 0 ; 3 2 i=1

(3.62)

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where

yic : (3.63) x Besides (3.62), the quantities qi have to satisfy the conditions 0 ¡ qi 1. The last inequality means that our approximation ’2 |Pi | is valid. In order to analyze the above conditions one has to implement such relations for the parameters that the result would be gauge *xing independent. Let us, for this end, take the relation 4 2 af = 11 3 A, as it has been done in [54]. Then f ∼ g . Consider the special case |$i |g . Then from (3.62) follows [33] Di ≈ − 92 ABi ; qi ≈ A=4:3Bi i . We notice that in the *rst of (3.62) one has to take positive sign, otherwise the qi 1 condition does not hold. In this approximation A ¿ 0; Bi ¿ 0, therefore one has to take all i ¿ 0. As we see, the theory admits the *rst order phase transition, which may be induced by curvature and (or) torsion. In fact, there are other possibilities, for instance where all or part of the non-minimal parameters satisfy opposite relations |$i |g2 [33]. We will not present the discussion of these possibilities here. An important observation is that, in the point of minimum, the e4ective potential generates the induced gravity with torsion:
   1 (ind) 4 √ 4 √ Sind = − d x −gV (’c ) = − d x −g 5ind − 2 Pi (3.64) 16=Gind i i Di = Abi $i − afBi − 56 Abi ;

with 5ind = and

af’4c

i = sign Pi ;

qi =

  ’2c 25 4 + A ln 2 − ’c



6

  5  1 ’2c (ind) − 2 = bi $i + Bi ln 2 − 3 ’2c : 16=Gind i 

(3.65)

(3.66)

i=1

It is reasonable to take 2i(ind) = 1, and then other 2i(ind) will give the coeLcients in the induced analog of the Einstein–Cartan action (2.18). Since these coeLcients depend on the non-minimal parameters $i , and these parameters have di4erent scale dependence (see, for example, (3.42)), the coeLcients of the action are, in general, di4erent from the ones in (2.18), which √ induced ˜ correspond to the −gR-type action of the Einstein–Cartan theory. 3.6. Conformal anomaly in the spaces with torsion: trace anomaly and modi5ed trace anomaly As we have already learned in Section 2.4, three di4erent types of local conformal symmetry are possible in the theory of gravity with torsion. Consequently, one meets di4erent versions of conformal anomaly, which violates these symmetries at the quantum level. In this section, we shall consider only the vacuum sector, and just note that the trace anomaly in the matter *eld sector with torsion (which requires the renormalization of composite operators similar to the one performed in [28]) has not been performed yet. On the other hand, this anomaly would not

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be very di4erent from the one in the purely metric theory, and the vacuum e4ects look much more interesting. Let us start from the anomaly corresponding to the weak conformal symmetry [32]. In this case, torsion does not transform and the Noether identity, in the vacuum sector, is just the same as in the purely metric gravity: 2 S T = − √ g =0 : −g g

(3.67)

The last identity indicates that the vacuum action can be chosen to be conformal invariant. 11 The vacuum action may include the non-conformal terms, but their renormalization is not necessary in the case of conformal invariant free massless *elds. But these terms may be, indeed, important from other points of view. In particular, one can include the Einstein–Cartan action into Svacuum , and treat the anomaly-induced e4ective action as quantum correction to the classical action of gravity with torsion at the very high energy, when the particle masses are negligible. Consider the anomaly in identity (3.67). We shall use the dimensional regularization, which is the most useful for this purpose [60,70]. The divergent part of the one-loop e4ective action that emerges after integrating over one real scalar and one Dirac spinor, have the form (3.6) and (3.15). If we restrict consideration by the case of a purely antisymmetric torsion, the result for the N0 real scalars, N1=2 Dirac spinors and N1 gauge bosons will be    N1=2 N1 n−4 N0 n √ V div (N0 ; N1=2 ; N1 ) = − d x −g C2 + +  120 20 10     11 N1=2 31 N1 N1=2 N1 N0 N0 − E+ − R + + + 360 360 180 180 30 10   2"2 N1=2 4N1=2 2 N0 N0 2  2  − (S  S ) S S + $ (S S ) + " − $ 3 2 3 6  4N1=2 2     − " ∇ (S ∇ S − S ∇ S ) 3  √ n−4 2 d n x −g aC 2 + bE + c R + d S + eS 4 + f S 2 =−  + g∇ (S ∇ S  − S  ∇ S  ) :

(3.68)

The standard arguments show that the one-loop e4ective action of vacuum is conformal invariant before the local counterterm US is introduced [70]. Consider the general expression 11 In fact, this is true (exactly as in the purely metric theory) only in the one-loop approximation. At higher loops, the non-minimal parameter $1 of the scalar–curvature interactions departs from the one-loop conformal *xed point [167]. As a result, in order to preserve the renormalizability, one needs, strictly speaking, a non-conformal vacuum action. But, as it was noticed in [167], the coeLcients in front of the non-conformal terms can be safely kept very small, and one can always consider the conformal invariant vacuum action as a very good approximation. In order to avoid the discussion of this issue, we consider, in this section, the vacuum e4ects of the free matter *elds.

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for the one-loop e4ective action  = S + V + US ;

(3.69)

where V is the quantum correction to the classical action and US is an in*nite local counterterm (1) which is called to cancel the divergent part of V . Then, the anomalous trace is    2  V  2   T = T = − √ g  = − √ g US  : (3.70) −g g n=4 −g g n=4 The most simple way of calculating this expression is to perform the local conformal transformation g = gV · e2/ ;

/ = /(x);

det(gV ) = const:

(3.71)

and use the identity

  2  1 −4/  2/  − √ g A[g ] = − √ e : A[gV e ] −g g / −gV gV →g ;/→0

(3.72)

When this operator acts on   n−4 2 US = d n x −gVe(n−4)/ (aCV + · · ·) ; j the 1=(n − 4)-factor cancels and we immediately arrive at the expression T =−

1 2 [aC 2 + bE + c R + d S + eS 4 + f S 2 + g∇ (S ∇ S  − S  ∇ S  )] (4=)2

(3.73)

with the same coeLcients a; b; : : : ; g as in (3.68). The derivation of the anomaly for the general torsion case can be done in the same way [165]. The important thing is that for the case of the  do not transform. weak conformal symmetry all components of torsion S ; T and q· The calculation of the anomaly for the case of strong conformal symmetry always reduces to the one for the weak conformal symmetry. As it was already noted in Section 2.4, the Noether identity (2.42) separates into two independent identities: one of them is (3.67), and second simply requests that the actions does not depend on the torsion trace T . As it was mentioned in Section 2.4, the second identity cannot be violated by the anomaly, and we are left with (3.67) and with the corresponding anomaly (3.73). Let us now consider the most interesting case of the compensating conformal symmetry, which will lead us to the modi*ed trace anomaly. We shall follow Ref. [102]. This version of conformal symmetry depends on the torsion trace T and on the non-minimal interaction of this trace with scalar. In the spinor sector the symmetry requires that there is no any interaction with T , so that "2 = 0. Therefore, we can restrict our consideration to the case of a single scalar *eld. It is easy to see that the Noether identity corresponding to symmetry (2.52) looks as follows: 2g

St $2 St St + 9 −’ =0 : g $3 T ’

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Then, due to the conformal invariance of the vacuum divergences (coeLcient at the n = 4 pole), the vacuum action may be chosen in such a way that √ Svac $2 Svac − −gT = 2g + 9 =0 : (3.74) g $3 T The new form (3.74) of the conformal Noether identity indicates the modi*cation of the conformal anomaly. In the theory under discussion, the anomaly would mean T = 0 instead of usual T =  0. Therefore, we have a special case here and one cannot directly use the relation between the one-loop counterterms and the conformal anomaly derived above, just because this relation does not take into account the non-trivial transformation law for the torsion *eld. One can derive this new anomaly directly, using the same method as before. However, it is possible to *nd T in a more economic way, after performing a special decomposition of the background *elds. Let us try to change the background variables in such a way that the transformation of torsion is absorbed by that of the metric. The crucial observation is that P, from (2.53), transforms, under (2.52), 12 as P = P · e−2/(x) . The non-trivial transformation of torsion is completely absorbed by P. Since P only depends on the background *elds, we can present it in any useful form. One can imagine, for instance, P to be of the form P = g E E where the vector E does not transform, exactly as the axial vector S . After that, the calculation readily reduces to the case of an antisymmetric torsion (3.73), described above. For the single scalar, the one-loop divergences have the form    1 1 2 n−4 1 2 1 1 (1) n √ div = − d x −g R+ P+ P : (3.75) C − E+ (4=)2 (n − 4) 120 360 180 6 2 Taking into account the arguments presented above, one can immediately cast the anomaly under the form   1 1 2 1 1 1 1  T = − − R+ P+ P : (3.76) C C E+ (4=)2 120  360 180 6 2 3.7. Integration of conformal anomaly and anomaly-induced e=ective actions of vacuum: application to inAationary cosmology One can use the conformal anomaly to restore the induced e4ective action of vacuum. This action can be regarded as a quantum correction to the classical gravitational action. We notice, that the induced action proved to be the best tool in the analysis of anomaly, see e.g. [14,73], including the theory with torsion [32]. The equation for the *nite part of the one-loop correction V to the e4ective action can be obtained from anomaly. Let us consider, *rst, the weak conformal symmetry. Then 2 V − √ g =T : (3.77) −g g In the case of purely metric gravity this equation has been solved in [153,77]. For the torsion theory with the weak conformal symmetry the solution has been found in [32] (see also [34]). 12

As a consequence, the action [141].

√

−gP’2 is conformal invariant. This fact has been originally discovered in

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Finally, for the most complicated case of the compensating conformal symmetry, the problem has been solved in [102]. We start from the case of purely antisymmetric torsion, corresponding to the strong conformal symmetry. The simplest possibility in solving (3.77) is to divide the metric in two parts: the conformal factor /(x) and the *ducial metric gV (x) with *xed determinant (3.71), and write the (3.80) via (3.72). Since torsion does not transform, we put S = SV  . Then we get [32,34]      1 2 V2 V 2 2 4 V V V V d x −gV a/C + b/ E − ∇ R + 2b/MV 4 / + d /SV   = Sc [gV ; S  ] + 2 (4=) 3  2     + e/(SV  SV )2 + (f + g=2)SV (∇V /)2 + g(SV ∇V  /)2 − g∇V  /(SV  ∇V  SV − SV ∇V  SV )    2 1 2  V2 2 V V V V −f∇ /∇ S − c + b [R − 6(∇/) − ( V /)] + Sc [gV ; SV  ] ; 12 3

(3.78)

where Sc [gV ; SV  ] is an unknown functional of the metric and torsion which serves as an integration constant for any solution of (3.77). Indeed, if one succeeds to rewrite (3.78) in terms of the original variables g ; S , the action Sc [gV ; SV  ] must be replaced by an arbitrary conformal-invariant functional of these variables. It is, in principle, possible to proceed and, following [153], derive the covariant form of the induced action (3.78). This action contains, exactly as in the torsionless case, the local and non-local pieces [60,153]. Action (3.78), being the quantum correction to the Einstein–Cartan theory, can serve as a basis for the non-singular cosmological model with torsion. This model has been constructed in [32] (see also [34]). Without going into technical details, we just summarize that, for the conformally 9at metric g =" a2 and isotropic torsion axial vector S =(T; 0; 0; 0), the dynamical equations have approximate classical solution of the form (in physical time) a(t) = a(0)eHt ;

T (t) = T (0)e−2Ht :

(3.79)

This solution has an obvious physical interpretation: torsion exponentially decreases during in9ation, and that is why it is so weak today. Of course, this concerns only speci*c background torsion, but the result is indeed relevant for the cosmological applications of torsion. Let us now derive the conformal non-invariant part of the e4ective action of vacuum, which is responsible for the modi*ed conformal anomaly (3.76). Taking into account our previous treatment of the compensating conformal transformation of torsion, we consider it hidden inside the quantity P of Eq. (2.53), and again imagine P to be of the form P = g E E . Then, the equation for the e4ective action [g ; E ] is 13 2  − √ g = T : (3.80) −g g In order to *nd the solution for , we can factor out the conformal piece of the metric g = gV · e2/ , where gV has *xed determinant and put P = PV · e−2/(x) , that corresponds to EV  = E . 13 We remark that this equation is valid only for the “arti*cial” e4ective action [g ; E ], while the e4ective action  would satisfy the modi*ed equation (3.74). The standard form of Eq. (3.80) for the in original variables g ; T e4ective action is achieved only through the special decomposition of the external *elds.

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The result can be obtained directly from the e4ective action derived in [32], and we get       1 1 1 V2 1 4 2 4 V  = Sc [gV ; P] − d x −g(x)R (x) + d x −gV / C 2 2 12 270(4=) (4=) 120     1 V2 1 1 V V 1 2 V 1 V  V 2 V V V V E− R + P + − ; (3.81) /U/ − (∇ /)∇ P + P(∇ /) 360 3 2 180 6 6 where Sc [gV ; PV ] is an unknown functional of the metric gV (x) and PV , which acts as an integration constant for any solution of (3.80). Now, one has to rewrite (3.81) in terms of the original *eld variables, g ; T  . Here, we meet a small problem, because we only have, for the moment, the de*nition E = EV  for the arti*cial variable E , but not for the torsion. Using the previous result (2.52), we can de*ne T  = TV  − 13 [ 9 / −  9 /] ;  where TV  is an arbitrary tensor. Also, we call TV = gV  TV  , etc. Now, we can rewrite (3.81) in terms of metric and torsion components   1 1  4 V  = Sc [gV ; T  ] − d x −g(x)R2 (x) 12 270(4=)2      1 1 4 V + 1 CV 2 / − 1 EV − 2 V RV / + d x − g V + / U/ (4=)2 180 120 360 3  2 2 $2 V 1     V V V V V V + / − R + 6$2 (∇ T ) + 6$3 T  T + 6$4 S  S + 6$5 qV qV 72 $3  2  $2 V 1 2    2  V V V V V V V + [( V / + (∇ /) ] − R + 6$2 (∇ T ) + 6$3 T  T + 6$4 S  S + 6$5 qV qV : 6 $3

(3.82) This e4ective action is nothing but the generalization of expression (3.78) for the case of general metric–torsion background and compensating conformal symmetry. The curvature dependence in the last two terms appears due to the non-trivial transformation law for torsion. The physical  = 0. interpretation of action (3.82) coincides with the one of (3.78) in case T = q· From a technical point of view, Eq. (3.82) is a very interesting example of an exact derivation of the anomaly-induced e4ective action for the case when the background includes, in addition to metric, another *eld with the non-trivial conformal transformation. 3.8. Chiral anomaly in the spaces with torsion: cancellation of anomalies Besides the conformal trace anomaly, in the theory with torsion one can meet anomalies of other Noether identities. In particular, the systematic study of chiral anomalies has been performed in Refs. [17,1,3]. In many cases, due to the special content of the gauge theory, the anomaly cancel. The most important particular example is the Standard Model of particle physics, which is a chiral theory where left and right components of the spinors emerge in

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a di4erent way. The violation of the corresponding symmetries could, in principle, lead to the inconsistency of the theory [92]. However, it does not happen in the SM, because the dangerous anomalies cancel. The history of chiral anomaly in curved space–time with torsion started simultaneously with the derivation of divergences (always related to the a2 (x; x )-coeLcient) for the corresponding fermion operator [87]. In this paper, however, the explicit result has not been achieved because of the cumbersome way of calculation. Let us indicate only some of the subsequent calculations [116,142,135,192,52,51,66] and other papers devoted to the closely related issues like index theorems, topological structures and Wess–Zumino [188] conditions [143,46,128,191,150]. We shall not go into details of these works but only present the most important and simple expression for the anomaly and give a brief review of other results. For the massless Dirac fermion, there is an exact classical symmetry (2.37), and the corresponding Noether identity is ∇ J5 = 0, where (2.61) J5 = V 5 

:

The anomaly appears due to the divergences coming from the fermion loop. The mechanism of the violation of the Noether identity can be found in many books (for instance, in [55,109]). However, the standard methods of calculating anomaly using Feynman diagrams are not very useful for the case of gravity with torsion. In principle, one can perform such calculations using the methods mentioned at the beginning of this section: either introducing external lines of the background *elds, or using the local momentum representation. The most popular are indeed the functional methods (see e.g. [79]), which provide the covariance of the divergences automatically. The investigation of the anomaly in curved space–time using the so-called analytic regularization based on the Scwinger–DeWitt (Seeley–Minakshisundaram) expansion for the elliptic operator on the compact manifolds with positive-de*ned metric has been performed in [155]. The anomaly can be calculated on the Riemann or Riemann–Cartan [142]) manifold with the Euclidean signature. The Euclidean rotation can be done in a usual way, and the result can be analytically continued to the pseudo-Euclidean signature. Then the vacuum average of the axial (spinor) current is



√ d V d J5 exp{ d 4 x g V  D }  J5 =

;

√ d V d exp{ d 4 x g V  D } where D = ∇ + i" 5 S is covariant derivative with the antisymmetric torsion. It can be easily reduced to the minimal covariant derivative ∇˜  = ∇ − (i=8) 5 S , but we will not do so, and keep " arbitrary in order to have correspondence with other sections of this section. The vacuum average of the axial current divergence can be presented as [155,142] 

4 √

  D V V d exp{ d x g( − J 9 ) } d    5  ∇ J5 = :



 √  (x) d V d exp{ d 4 x g V  D } =0 The analysis of [155,142] shows that this expression is nothing but A = ∇ J5 = 2 lim
tr 5 a2 (x; x ) ; x→x

(3.83)

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where a2 (x; x ) is the second coeLcient of the Schwinger–DeWitt expansion (3.3). Applying this formula to the theory with torsion, that is using expressions (3.5), (2.53), and (3.14), one can obtain the expression for the anomaly in the external gravitational *eld with torsion [142]   2 1 1 2    6/ ∇ K + 6/ R· · R A= + "  S S ; (3.84) (4=)2 48 6 where K = − 23 "( + 4"2 S S − 12 R)S  :

One can consider the anomalies of other Noether currents [51] and obtain more general expressions. From the physical point of view, the most important question is whether the presence of torsion preserves the cancellation of anomalies in the matter sector. If this would not be so, the introduction of torsion could face serious diLculty. This problem has been investigated in [51] and especially in [66]. The result is that the presence of an external torsion does not modify the anomalous divergence of the baryonic current JB = (1=Nc )Q†  Q while it changes the leptonic current JL = L†  L by expression (3.84). Independent on whether neutrinos are massless or massive, the cancellation of anomalies holds in the presence of external torsion. In particular, weak external torsion does not a4ect the quantization of the SM charges [66]. Therefore, the existence of the background torsion does not lead to any inconsistency. On the other hand, the leptonic current gains some additional contributions, and this could, in principle, lead to some e4ects like anisotropy of polarization of light coming from distant galaxies [65]. However, taking the current upper bound on the background torsion from various experiments (see Section 4.6) one can see that the allowed magnitude of the background torsion is insuLcient to explain the experimental data which have been discussed in the literature [138]. 4. Spinning and spinless particles and the possible e ects on the classical background of torsion The purpose of this section is to construct the non-relativistic approximation for the quantum *eld theory on torsion background and also develop the consistent formalism for the spinning and spinless particles on the torsion background. We shall follow the original papers [13,158,84]. First, we construct the non-relativistic approximation for the spinor *eld and particle, then use the path integral method to construct the action of a relativistic particle. The action of a spin 1=2 massless particle with torsion has been *rst established in [156] on the basis of global supersymmetry. The same action has been rediscovered in [154], where it was also checked using the index theorems. In [150] the action of a massive particle has been obtained through the squaring of the Dirac operator. This action does not possess supersymmetry, and does not have explicit link to the supersymmetric action of [156,154]. In [84] the action of a spinning particle on the background of torsion and electromagnetic *eld has been derived in the framework of the Berezin–Marinov path integral approach. This action possesses local supersymmetry in the standard approximation of weak torsion *eld and includes the previous actions of [156,154,150] as a limiting case.

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In the last part of this section we present a brief review of the possible physical e4ects and of the existing upper bounds on the magnitude of the background torsion from some experiments. Since our purpose is to investigate the e4ects of torsion, it is better not to consider the e4ects of the metric. For this reason, in this and next sections we shall consider the 9at metric g = " . 4.1. Generalized Pauli equation with torsion The Pauli-like equation with torsion has been (up to our knowledge) *rst discussed in [159], and derived, in a proper way, in [13]. After that, the same equation has been obtained in [95,169,120], and the derivation of higher order corrections through the Foldy–Wouthuysen transformation with torsion has been done in [158]. In [120] the Foldy–Wouthuysen transformation has been applied to derive Pauli equation. The starting point is action (2.35) of Dirac fermion in external electromagnetic and torsion *elds:  S = d 4 x{i V  (9 + i" 5 S + ieA ) + m V } : The equation of motion for the fermion can be rewritten using the standard representation of the Dirac matrices (see, for example, [19])     1 0 0 ˜/ 0 0 = = ; ˜ = ˜ = : 0 −1 ˜/ 0 Here, as before, 5 = −i 0 1 2 3 . The equation has the form 9 i˝ = [c˜p ˜ − e˜˜A − "˜˜S 5 + e4 + " 5 S0 + mc2 ] : 9t Here, the dimensional constants ˝ and c were taken into account, and we denoted A = (4; ˜A); S = (S0 ; ˜S) :

(4.1)

Following the simplest procedure of deriving the non-relativistic approximation [19] we write   ’ 2 = eimc t=˝ : (4.2) ( Within the non-relativistic approximation (’. From Eq. (4.1) it follows that   9 i˝ − "1˜/ · ˜S − e4 ’ = (c˜/ · p ˜ − e˜/ · ˜A − "S0 )( 9t and   9 2 ˜ i˝ − "˜/ · S − e4 + 2mc ( = (c˜/ · p ˜ − e˜/ · ˜A − "S0 )’ = 0 : 9t

(4.3)

(4.4)

At low energies, the term 2mc2 ( in the l.h.s. of (4.4) is dominating. Thus, one can disregard other terms and express ( from (4.4). Then, in the leading order in 1=c we meet the equation   9’ 1 2 ˜ ˜ i˝ ˜ − e˜/ · A − "S0 ) ’ : (4.5) = "˜/S + e4 + (c˜/ · p 9t 2mc2 (

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The last equation can be easily written in the SchrYoedinger form 9’ i˝ = Hˆ ’ (4.6) 9t with the Hamiltonian 1 2 ˜ ; (4.7) Hˆ = ˜= + B0 + ˜/ · Q 2m where e "1 ˜= = p ˜ − ˜A − ˜/S0 ; c c 1 B0 = e4 − 2 "2 S02 ; mc ˜ = "˜S + ˝ e H ˜ : Q (4.8) 2mc ˜ = rot ˜A is the magnetic *eld strength. This equation is the analog of the Pauli equation Here, H for the general case of external torsion and electromagnetic *elds. The above expression for the Hamiltonian can be compared to the standard one, which contains only the electromagnetic terms. Some torsion-dependent terms resemble the ones with the magnetic *eld. At the same time, the term −("1 S0 =mc)˜ p · ˜/ does not have analogies in quantum electrodynamics. 4.2. Foldy–Wouthuysen transformation with torsion One can derive the next to the leading order corrections to the non-relativistic approximation (4.6) in the framework of the Foldy–Wouthuysen transformation with torsion [158]. The initial Hamiltonian of the Dirac spinor in external electromagnetic and torsion *elds can be presented in the form H = m + E + G ;

(4.9)

where E = e4 + " 5˜˜S

and

G = ˜(˜ p − e˜A) − " 5 S0

(4.10)

are even and odd parts of the expression. From this instant, if this is not indicated explicitly, we shall use the conventional units c = ˝ = 1. Our purpose is to *nd a unitary transformation which separates “small” and “large” components of the Dirac spinor. In other words, we need to *nd a Hamiltonian which is block diagonal in the new representation. We use a conventional prescription (see, for example, [24]): H  = eiS (H − i9t )e−iS ;

(4.11)

where S has to be chosen in an appropriate way. We shall try to *nd S and H  in a form of the weak-relativistic expansion, and thus start by taking S to be of order 1=m (with ˝ = c = 1).

I.L. Shapiro / Physics Reports 357 (2002) 113–213

Then, to the usual accuracy, we arrive at the standard result 1 1 i H  = H + i[S; H ] − [S; [S; H ]] − [S; [S; [S; H ]]] + [S; [S; [S; [S; H ]]]] 2 6 24 ˙ + ··· : ˙ ] + 1 [S; [S; S]] ˙ − i [S ; S −S 2 6 One can easily see that E and G given above (anti)commute with  in a usual way E =  E;

G = −G

165

(4.12)

(4.13)

and therefore one can safely use the standard prescription for the lowest order approximation: i S=− (4.14) G : 2m This gives H  = m + E + G ;

(4.15)

where G is of order 1=m. Now one has to perform second Foldy–Wouthuysen transformation with S = −(i=2m)G . This leads to the H  = m + E + G

(4.16)

with G ≈ 1=m2 ; and then a third Foldy–Wouthuysen transformation with G = −(i=2m)G removes odd operators in the given order of the non-relativistic expansion, so that we *nally obtain the usual result   1 2 1 4 1  H = m+ G − G +E− [G; ([G; E] + iG˙ )] : (4.17) 3 2m 8m 8m2 Substituting our E and G from (4.10), after some algebra we arrive at the *nal form of the Hamiltonian   1 1 4 e  2 ˜ ˜ H = m+ p ˜ + e4 − "(˜/ · ˜S) − (˜ p − eA − "S0˜/) − ˜/ · H 3 2m 8m 2m " " e ! ˜ ˜ + 2˜/ · [E ˜ ×p + i˜/ · rot E − 2 div E ˜] + {˜/ · ∇S˙ 0 − [pi ; [pi ; (˜/ · ˜S)]+ ]+ 8m 8m2 +2 rot ˜S · p ˜ − 2i(˜/ · ∇)(˜S · p ˜ ) − 2i(∇˜S)(˜/ · p ˜ )} ;

(4.18)

˜ where we have used standard notation for the anticommutator [A; B]+ = AB + BA. As usual, E ˜ = −(1=c)9˜A= 9t − grad 4. One can proceed in denotes the strength of the external electric *eld E the same way and get separated Hamiltonian with any given accuracy in 1=m. The *rst *ve terms of (4.18) reproduce the Pauli-like equation with torsion (4.7). Other terms are the next-to-the-leading order weak-relativistic corrections and torsion-dependent corrections to the Pauli-like equation (4.6). In those terms we follow the system of approximation which is standard for the electromagnetic case [24]; that is we keep the terms linear in interactions. One can note that for the case of the constant torsion and electromagnetic *elds one can achieve the exact Foldy–Wouthuysen transformation [137]. We do not reproduce this result here, because

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torsion (if it exists) is de*nitely weak and the the leading order approximation [158] is certainly the most important one. Further simpli*cations of (4.18) are possible if we are interested in constant torsion. This version of torsion can be some kind of relic cosmological *eld or it can be generated by the vacuum quantum e4ects. In this case we have to keep only the constant components of the pseudovector S . Then the e4ects of torsion will be: (i) a small correction to the potential energy of the spinor *eld, which sometimes looks just like a correction to the mass, and (ii) the appearance of a new gauge-invariant spin–momentum interaction term in the Hamiltonian. 4.3. Non-relativistic particle in the external torsion 5eld In this section we start from the simple derivation of the action for the non-relativistic particle. This action will be used later on to test the more general relativistic expression. If we consider (4.7) as the Hamiltonian operator of some quantum particle, then the corresponding classical energy has the form 1 2 ˜ ; (4.19) ˜= + B0 + ˜/ · Q 2m ˜ are de*ned by (16) and ˜= = m˜v. Here ˜v = ˜x˙ is the velocity of the particle. The where ˜=; B0 ; Q expression for the canonically conjugated momenta p ˜ follows from (4.19). e " p ˜ = m˜v + ˜A + ˜/S0 : (4.20) c c One can consider the components of the vector ˜/ as internal degrees of freedom, corresponding to spin. Let us perform the canonical quantization of the theory. For this, we introduce the operators of coordinate xˆi , momenta pˆ i and spin /ˆ i and demand that they satisfy the equal-time commutation relations of the following form: H=

[xˆi ; pˆ j ] = i˝ ij ;

[xˆi ; /ˆ j ] = [pˆ i ; /ˆ j ] = 0;

[/ˆ i ; /ˆ j ] = 2iijk /ˆ k :

(4.21)

The Hamiltonian operator Hˆ which corresponds to energy (4.19) can be easily constructed in terms of the operators xˆi ; pˆ i ; /ˆ i . From it we may write the equations of motion i˝

d xˆi = [xˆi ; Hˆ ]; dt

i˝ i˝

d pˆ i = [pˆ i ; Hˆ ]; dt

d /ˆ i = [/ˆ i ; Hˆ ] : dt

(4.22)

After the computation of the commutators in (4.22) we arrive at the explicit form of the operator equations of motion. Now, we can omit all terms which vanish when ˝ → 0. Thus, we obtain the non-relativistic, quasi-classical equations of motion for the spinning particle in the external

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167

torsion and electromagnetic *elds. Note that the operator ordering problem is irrelevant because of the ˝ → 0 limit. The straightforward calculations lead to the equations [13]:  d˜x e " 1 p ˜ − ˜A − ˜/S0 = ˜v; (4.23) = m c c dt d˜v ˜ ] − "∇(˜/ · ˜S) ˜ + e [˜v × H = eE dt c   d S0 " "2 " + (˜v · /)∇S0 − (˜v · ∇S0 )˜/ − ˜/ + 2 ∇(S02 ) + S0 [˜/ × ˜R] c dt mc c

and d˜/ = [˜R × ˜/]; dt

  2" ˜ 1 e ˜ ˜ where R = S − ˜vS0 + H : ˝ c mc

(4.24)

(4.25)

Eqs. (4.23) – (4.25) contain the torsion-dependent terms which are similar to the magnetic terms, and also some terms which have a qualitatively new form. 4.4. Path-integral approach for the relativistic particle with torsion In this section, we are going to construct a path-integral representation for a propagator of a massive spinning particle in external electromagnetic and torsion *elds. The consistent method of constructing the path-integral representation has been developed by Berezin and Marinov [21]. Various aspects of the Berezin–Marinov approach were investigated in the consequent publications (see, for example, [29]). In this section we shall follow Ref. [84] where the path-integral representation has been generalized for the background with torsion. It was demonstrated in [75,85], that a special kind of path-integral representations for propagators of relativistic particles allow one to derive gauge-invariant pseudoclassical actions for the corresponding particles. Let us remark that in [76] some path-integral representation for massive spinning particle in the presence of the torsion was derived using the perturbative approach to path integrals. First, we consider the path-integral representation of the scalar *eld propagator in external torsion S and electromagnetic A *elds. As we already know, the scalar *eld interacts with torsion non-minimally, and this interaction is necessary for the renormalizability of scalar coupled to the fermions. Therefore, the Klein–Gordon equation in external electromagnetic and torsion *elds has the form 2

[Pˆ + m2 + $S 2 ]’(x) = 0 ;

(4.26)

where P = i9 − eA ; S 2 = S S  and $ is an arbitrary non-minimal parameter. This is the very same parameter which was called $4 in Section 2. Since in this section there are no other $, it is reasonable to omit index 4, exactly as we omitted the index in "1 . Our consideration is very similar to the one presented in [85] for the torsionless case. The propagator obeys the equation 2

[Pˆ + m2 + $S 2 ] Dc (x; y) = −(x; y) :

(4.27)

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The Schwinger representation for the propagator is c Dc (x; y) = x|Dˆ |y :

Here |x are eigenvectors for some Hermitian operators of coordinates X  and the corresponding canonically conjugated momenta operators are P . Then, the following relations hold:  |x x| d x = I; X  |x = x |x ; x|y = 4 (x − y); [P ; X  ]− = −i ; P |p = p |p ; p|p = 4 (p − p );  1 ipx |p p| dp = I; x|P |y = −i9 4 (x − y); x|p = e ; (2=)2 [E ; E ]− = −ieF (X );

E = −P − eA (X );

F (X ) = 9 A − 9 A :

(4.28)

We can write (4.27) in an operator way c ˆ where Fˆ = m2 + $S 2 − E2 ; Fˆ Dˆ = 1; and use the Schwinger proper-time representation:  ∞ −1 ˆ c Dˆ = Fˆ = i e−i (F−ij) ;

(4.29)

0

where j → 0 at the end of calculations. For the massive theory j-term can be included into the mass, and that is why we do not write it in what follows. Indeed, for the massless case j is important for it stabilizes the theory in the IR domain. In principle, for the constant torsion and $S 2 ¿ 0, torsion term can stabilize the proper time integral even in the massless case. It proves ˆ = Fˆ · , and rewrite the previous expression for the Green’s function (4.29) useful to denote H  ∞ ˆ c c ˆ ˆ D = D (xout ; xin ) = i xout |e−iH( ) |xin d 0



= i lim

N →∞ 0

∞

 d 0

+∞

−∞

d x1 : : : d xN d 1 : : : d N

N #

ˆ

xk |e−iH( k )=N |xk−1 ( k − k−1 ) ;

k=1

(4.30) where x0 = xin ; xN = xout . For large enough N one can approximate % $ i ˆ ˆ k )=N −iH( xk |e |xk−1 ≈ xk |1 − H( k )|xk−1 : N

(4.31)

ˆ ( ) as [20] Introducing the Weyl symbol H( ; x; p) of the operator H H( ; x; p) = (m2 + $S 2 − P2 );

P = −p − eA ;

we can express each of (4.31) in terms of the Weyl symbols in the middle points xVk = (xk + xk−1 )=2:    dpk H( k ; xVk ; pk ) ˆ k )=N −iH( xk |e |xk−1 ≈ exp pk (xk − xk−1 ) − : (4.32) (2=)4 N

I.L. Shapiro / Physics Reports 357 (2002) 113–213

Then the expression for the propagator becomes  ∞  +∞ # N dpk d =k Dˆ c = i lim d 0 d xk d k N →∞ 0 (2=)4 2= −∞ k=1   H( k ; xVk ; pk ) ×exp pk (xk − xk−1 ) − ; + =k ( k − k−1 ) N

169

(4.33)

where we have also used the Fourier representation for the delta-functions ( k − k−1 ) of (4.30). Eq. (4.33) is the de*nition of the Hamiltonian path integral for the propagator of scalar particle    1  xout    ∞ c 2 2 2 ˙ ˆ D =i d 0 Dx D DpD= exp i d [ (P − m − $S ) + px˙ + = ] : 0

xin

0

(4.34) The integral is taken over the path x (); p (); (); =() with *xed ends x(0) = xin ; x(1) = xout ; (0) = 0 . Integrating over the momenta p , one arrives at the Lagrangian form of the path-integral representation (where we substituted = 2=2 in order to achieve the conventional form)   xout  

1 i ∞ c ˆ D = d 20 Dx D2M (2) D=ei 0 L d ; (4.35) 2 0 xin 20 where the Lagrangian of the scalar particle has the form x˙2 2 2 − (m + $S 2 ) − ex A + =2˙ : (4.36) 22 2 It is easy to see that the presence of torsion does not make any essential changes in the derivation of the particle action. The result may be obtained by simple replacement m2 → m2 + $S 2 . Let us now consider the path-integral representation for the propagator of spinning particle. We shall follow the original paper [84], where the technique of Ref. [75] was applied to the case of the spinning particle on the torsion and electromagnetic background. One can also consult [84] for the further list of references on the subject. Consider the causal Green’s function Mc (x; y) of the equation of motion corresponding to action (2.35). Mc (x; y) is the propagator of the spinor particle. L=−

[  (Pˆ  + " 5 S ) − m]Mc (x; y) = −4 (x − y) :

(4.37)

It proves useful to introduce, along with 0 ; 1 ; 2 ; 3 ; 5 , another set of the Dirac matrices 0 ; 1 ; 2 ; 3 ; 4 . These matrices are de*ned through the relations  4 = i 5 ;

 = 4  :

(4.38)

It is easy to check that the matrices n ; n = 0; 1; : : : ; 4, form a representation of the Cli4ord algebra in *ve dimensions: [n ; m ]+ = 2"nm ;

"nm = diag(1; −1; −1; −1; −1) :

(4.39)

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In order to proceed, we need a homogeneous form of the operator (Mc )−1 . Therefore, we perform the 4 -transformation for Mc (x; y). M˜c (x; y) = Mc (x; y)4 : The new propagator M˜c (x; y) obeys the equation [ (Pˆ  − i"4 S ) − m4 ]M˜c (x; y) = 4 (x − y) :

(4.40)

Similar to the scalar case, we present M˜c (x; y) as a matrix element of an operator M˜c . M˜c (x; y) = x|M˜c |y : All further notations are those of (4.28). The formal solution for the operator M˜c is ˆ −1 ; M˜c = F

ˆ = E  − m4 − i" 4 S : F

ˆ may be written in an equivalent form The operator F ˆ = E  − m4 − i "j S     ; F 6 using the following formula:  4 = 16 j    ;

(4.41)

j0123 = 1 :

The last relation is important, for it replaces the product of an even number of ’s for the product of an odd number of ’s. Together with the 4 -transformation of the propagator, this ˆ becomes purely fermionic helps us to provide the homogeneity of the equation, so that F −1 ˆ operator. Now, F can be presented by means of an integral [75]:  ∞  ˆ2 ˆ ˆ −1 = F d ei[ (F +ij)+(F] d ( : (4.42) 0

Here ; ( are the parameters of even and odd Grassmann parity. Taken together, they can be ˆ: considered as a superproper time [75]. Indeed, commutes and ( anticommutes with F ˆ ] = 0; [ F

ˆ ]+ = 0 : [(; F

ˆ 2 we *nd Calculating F 2

ˆ = E2 − m2 − "2 S 2 − F

ie

2

F   + Kˆ    + "9 S  0 1 2 3 ;

(4.43)

where i" Kˆ  = [E ; S  ]+ j ; E2 = P 2 + e2 A2 + e[P ; A ]+ : 2 Thus, we get the integral representation for the propagator:  ∞  c ˜ ˆ ( ; ()] ; M = d ( exp[ − iH d 0

(4.44)

I.L. Shapiro / Physics Reports 357 (2002) 113–213

where ˆ ( ; () = H



m2 + "2 S 2 − E2 + 

0

1

2

−"9 S    

3



ie

2

171

F   − Kˆ    



− ( E  −

im"

6



4





j S    





:

The Green’s function M˜c (xout ; xin ) has the form  ∞  ˆ M˜c (xout ; xin ) = d xout |e−iH( ; () |xin d ( : 0

(4.45)

Now, we are going to represent the matrix element entering in expression (4.45) by means of a path integral [75,84]. The calculation goes very similar to the scalar case. We write, as

ˆ ˆ usual, e−iH = (e−iH=N )N , then insert (N − 1) identities |x x| d x = I and introduce N additional integrations over and (  ∞  # N ˆ M˜c (xout ; xin ) = lim d 0 xk |e(−i=N )H( k ; (k )M |xk−1 N →∞ 0

k=1

×( k − k−1 )((k − (k−1 ) d (0 d x1 : : : d xN −1 d 1 : : : d N d (1 : : : d (N ;

(4.46)

where x0 =xin ; xN =xout . Using approximation (4.31), and introducing the Weyl symbol H( ; (; x; p) ˆ of the symmetric operator H  ie H( ; (; x; p) = m2 + "2 S 2 − P2 + F   − K   2    im"  0 1 2 3   4    − "9 S     − ( P  − j S     6 with K = −"P S  j , we can express the matrix elements (4.31) in terms of the Weyl symbols at the middle point xVk . Then (4.31) can be replaced by the expressions    dpk xk − xk−1 exp i pk − H( k ; (k ; xVk ; pk ) U ; (4.47) (2=)4 U where U=1=N . Such expressions with di4erent values of k do not commute due to the -matrix structure and, therefore, have to be replaced into (4.46) in such a way that the numbers k increase from the right to the left. For the two -functions, accompanying each matrix element in expression (4.46), we use the integral representations  i ( k − k−1 )((k − (k−1 ) = d =k d k exp{i[=k ( k − k−1 ) + k ((k − (k−1 )]} ; 2= where k are odd variables. Then we attribute to the -matrices in (4.47) an index k. At the same time we attribute to all quantities the “time” k according to the index k they have, k = kU. Then,  ∈ [0; 1]. Introducing the T-product, which acts on -matrices, it is possible to gather all the expressions, entering in (4.46), in one exponent and postulate that at equal

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times the -matrices anticommute. Finally, we arrive at the propagator   xout       ∞ M˜c (xout ; xin ) = T d 0 d (0 Dx Dp D D( D= D 0

  ×exp i

0

1 

xin

0

P2 − m2 − "2 S 2 −

(0

ie

2

F   + K  

   im" + "9 S  0 1 2 3 + ( P  − j S  4    6   +px˙ + = ˙ + (˙ d  ;

(4.48)

where x(); p(); (); =(), are even and ((); () are odd functions. The boundary conditions are x(0) = xin , x(1) = xout ; (0) = 0 ; ((0) = (0 . The operation of T-ordering acts on the -matrices which formally depend on the time . Expression (4.48) can be transformed as follows:  ∞     xout    M˜c (xout ; xin ) = d 0 d (0 D D( Dx Dp D= D 0

  ×exp i

0

0

(0

1 

xin

P2 − m2 − "2 S 2 − 



ie

2

F

l l   + K l l 6 6 6 6

         i" +"9 S l l l l + ( P l − m l − j S  l l l 60 61 62 63 6 64 6 6 6 6     1  n ˙ 6n () d  ; + = + (˙ d  T exp 

0



px˙

6=0

where *ve odd sources 6n () are introduced. They anticommute with the -matrices by de*ni 1 tion. One can represent the quantity T exp 0 6n ()n d  via a path integral over odd trajectories [75],    1  1 n n 9l T exp 6n () d  = exp i exp ( n ˙ n − 2i6n n ) d  9Sn (0)+ (1)=S 0 0    n + n (1) (0) D  (4.49) S=0

with the modi*ed integration measure   D =D D exp (0)+ (1)=0

0

1

−1 n ˙ d  : n

Here Sn are odd variables, anticommuting with the -matrices, and n () are odd trajectories of integration. These trajectories satisfy the boundary conditions indicated below the signs of

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173

integration. Using (4.49) we get the Hamiltonian path-integral representation for the propagator:  ∞      xout    n 9l M (xout ; xin ) = exp i d d

( D D( Dx Dp D= D 0 0 9Sn

0 (0 xin 0 2
    1 i × D exp i

P +  ( + d − (m2 + "2 S 2 )

˜c

(0)+ (1)=S 

+2i eF



+ 16 "9 S



 −i

n

0

˙ n + px˙ + = ˙ + (˙

0

1

2

3



+ 2i( m

   d  + n (1) n (0)  

4

2 + 3 ;





d (4.50)

S=0

where d = −2i"j S 





:

(4.51)

Integrating over the momenta, we get the Lagrangian path-integral representation for the propagator,  ∞      xout   9 M˜c (xout ; xin ) = exp in ‘ n d 20 d (0 M(2)D2 D( Dx D= D 9S 20 (0 xin 0   1 2  z 2 − × D exp i − M 2 − x˙ (eA − d ) + i2e F   22 2 (0)+ (1)=S 0      2  4 n n ˙ ˙ + i( m + d − i n + =2 + (˙ d  + n (1) (0)  ; (4.52) 3 S=0

where 2 = 2 and the measure M(2) has the form:  M(2) =

Dp exp

  i

2

0

1

 2p2 d 

(4.53)

and M 2 = m2 + "2 S 2 − 16"9 S 

0

1

2

3

;

z  = x˙ + i(



:

(4.54)

The discussion of the role of measure (4.53) can be found in [75]. The exponential in integrand (4.52) can be considered as an e4ective non-degenerate Lagrangian action of a spinning particle in electromagnetic and torsion *elds. It consists of

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two principal parts. The term  1 SGF = (=2˙ + () ˙ d ; 0

can be treated as a gauge *xing term corresponding to the gauge conditions 2˙ = (˙ = 0. The other terms can be treated as a gauge-invariant action of a spinning particle. It has the form  1 2 z 2 S= − − M 2 − x˙ (eA − d ) + i2eF   22 2 0    2  4 n ˙ d ; + i( m + d − i n (4.55) 3 where z  ; M 2 and d have been de*ned in (4.54) and (4.51). Action (4.55) is a generalization of the Berezin–Marinov action [21,29] to the background with torsion. One can easily verify that (4.55) is invariant under reparametrizations: d (2$) d (($) x = x$; ˙ 2 = ;  n = ˙ n $ (n = 0; 1; 2; 3; 4); ( = : dt dt I could establish the explicit form of the local supersymmetry transformations, which generalize the ones for the Berezin–Marinov action, only in the linear in torsion approximation. These transformations have exactly the same form as in the case without torsion (see, for example, [86]): 1 x = i  j; 2 = i(j;   = (x˙ + i(  )j( = 0; 1; 2; 3); 22   i 5  ˙5 m  m 5 ( = j˙;  = − − ( j 2 m2 2 with j = j(). The demonstration of supersymmetry is technically non-trivial, and in particular one needs the identity: 







 9 S = 14      9 S : In the general case one can establish the supersymmetry of the action through the structure of the Hamiltonian constraints in the course of quantization [84]. Let us analyze the equations of motion for the theory with action (4.55). These equations contain some unphysical variables, that are related to the reparametrization and supersymmetry invariance. One can choose the gauge conditions ( = 0 and 2 = 1=m to simplify the analysis. Then we need only two equations r S 2i i = 2i ˙  − 2i2e F  − x˙ ( + (d + 4i"mu x˙ S     e 3 4"2 8"    9 S  ; (mu  S   − 3  3   i d x˙ d S     = + ex˙ F + i2eF;  + ex˙ 9 A + x d  2 d 2 −

+ "2S  9 S − 8"2(9 9 S  )

0

1

2

3

− x˙ (9 d ) −

 (

(4.56)

− eA + d

2i (  (9 d ) = 0 : 3

(4.57)

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Now, in order to perform the non-relativistic limit we de*ne the three-dimensional spin vector ˜/ as [21] i i j l /k = 2ijkjl l j ; = jkjl /k ; ˙ j l = jkjl /˙ k (4.58) 4 4 and consider d xi 0 ≈ 0; x˙0 ≈ 1; x˙i ≈ vi = 0 (4.59) dx as a part of the non-relativistic approximation. Furthermore, we use standard relations for the components of the stress tensor: F0i = −Ei = 90 Ai − 9i A0 and Fij = jijk H k : Substituting these formulas into (4.57) and (4.56), and disregarding the terms of higher orders in the external *elds, we arrive at the equations: ˜ + e [˜v × H ˜ ] − "∇(˜/ · ˜S) − " d S0 ˜/ + " (˜v · /)∇S0 + · · · ; m˜v˙ = eE c c dt c    e ˜ 2"S0 2" ˜v × ˜/ : ˜/˙ = (4.60) H + ˜S − mc ˝ c˝ They coincide perfectly with the classical equations of motion (4.24), (4.25) obtained from the generalized Pauli equation. This correspondence con*rms our interpretation of action (4.55). Additional arguments in favor of this interpretation were obtained in the framework of canonical quantization [84]. The quantization leads us back to the Dirac equation on torsion and electromagnetic background. Therefore, we have all reasons to consider (4.55) as the correct expression for the action of spin-1=2 particle in the external background of A and S *elds. 4.5. Space–time trajectories for the spinning and spinless particles in an external torsion 5eld In this section, we shall consider several particular examples of motion of spinless and spinning particles in an external torsion *eld. Let us start from the scalar particle with action (4.36). For the sake of simplicity we do not consider electromagnetic *eld. Using the gauge condition = = 0, and replacing the solution for the auxiliary *eld 2 back into the action, we cast it in the form  1 & S =− d  (m2 + $S S  )x˙2 : (4.61) 0

It is obvious, already from action (4.36), that the role of the constant torsion axial vector S = const: is just to change the value of the mass of the scalar particle m2 → m2 + $S 2 . Indeed, this is true only until the metric is 9at. As far as we take a curved metric, the square S S  depends on it, even for a constant torsion. Let us suppose that S S  is coordinate dependent. Performing the variation over the coordinate x , after some algebra we arrive at the equation of motion x˙2 $9 (S  S ) xY = : 2 m2 + $S 2

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Taking into account that the square of the 4-velocity is constant, x˙2 = 1, we obtain   2  1 $S xY = 9 ln1 + 2  : 2 m

(4.62)

Thus, in case of the non-constant S S  the motion of particle corresponds to the additional four-acceleration (4.62), and the torsion leads to the potential of the form & VS = −ln |m2 + $S 2 |=m2 : Indeed, this potential is constant for the scalar minimally coupled to torsion $ = 0. However, as we have seen in the previous section, for any scalar which interacts with spinors, the non-minimal interaction $ = 0 is nothing but a consistency condition. So, if torsion would really exist, and if its square is not constant, the force producing (4.62) should exist too. At the same time, the only scalar which is supposed to couple to the fermions through Yukawa interaction, is the Higgs. Since the Higgs mass is supposed to be, at least, of the order of 100 GeV, all experimental manifestations of the Higgs particle are possible only at the high-energy domain. Since torsion (if exists) is a very weak *eld, there are extremely small chances to observe torsion through acceleration (4.62). Let us now discuss the spin 1=2 case. Here we follow, mainly, Ref. [158]. The physical degrees of freedom of the particle are its coordinate ˜x and its spin ˜/. For the sake of simplicity, we shall concentrate on the non-relativistic case and consider the motion of a spinning particle in a space with constant axial torsion S = (S0 ; ˜S), but without electromagnetic *eld. In this case the equations of motion (4.24), (4.25) have the form d˜v "S0 d˜/ = −"˜S(˜v · ˜/) − ; dt c dt 2" d˜/ 2"S0 (4.63) = + [˜S × ˜/] − [˜v × ˜/] : dt ˝ ˝c Consider *rst the case when S0 = 0 so that only ˜S is present. Since ˜S = const:, we can safely put S1; 2 = 0. The solution for the spin can be easily found to be     2"S3 t 2"S3 t /3 = /30 = const:; /1 = 6 cos ; /2 = 6 sin ; (4.64) ˝ ˝ & 2 + /2 . For the *rst two components of the velocity we have v =v =const:; v = where 6= /10 1 10 2 20 v20 = const:; but the solution for v3 turns out to be complicated. For /3 = 0 the solution is   6˝(/3 v10 ˝ − 2mv20 ) −("S3 /3 =m)t 6˝ v3 (t) = v30 + e − 4m2 + ˝2 /32 4m2 + ˝2 /32      2"S3 t 2"S3 t + (/3 v20 ˝ + 2mv10 ) sin : (4.65) × (/3 v10 ˝ − 2mv20 ) cos ˝ ˝ Physically, such a solution means (i) precession of the spin around the direction of (ii) oscillation of the particle velocity in this same direction accompanied (for /3 = 0) exponential damping of the initial velocity in this direction. We remark that the value

˜S and by the of the

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relic torsion *eld should be very weak so that very precise experiments will be necessary to measure these (probably extremely slow) precession, oscillation and damping. Consider another special case ˜S = 0 and S0 = 0, which is the form of the torsion *eld motivated by the isotropic cosmological model (3.79). The equations of motion have a form: 14 d˜v 2"2 S02 [˜v × ˜/] ; = dt c˝ d˜/ 2"S0 [˜v × ˜/] : =− dt ˝

(4.66)

In order to analyze these equations we note, that the squares ˜v 2 ; ˜/2 and the product (˜v · ˜/) are integrals of motion. Consequently, the magnitudes of the two vectors, and the angle between them do not change during the motion. Another obvious integral of motion is the linear combination   2"S0 "S0 ˜v + w ˜= (4.67) ˜/ : ˝ c Therefore, the evolution of ˜v and ˜/ performs such that the plane of two vectors is rotating around the constant vector w ˜ . By elementary means one can check that the frequency of this rotation is exactly w = |w ˜ |, so that the period is T = 2==w. In this case, the magnitude and direction of the precession of spin and acceleration of the particle depends on the magnitude and mutual orientation of its spin and velocity. It is interesting that in the case [˜/;˜v] = 0 both spin and velocity are constants, but as far as ˜/ and ˜v are not exactly parallel, the period of precession depends only on the magnitude of the vector w ˜ . In other words, both vectors may be in*nitesimally non-parallel, and the frequency of the precession will not be in*nitesimal (but the amplitude of the precession will). The last observation is that, for the gas of particles with random orientation of velocities, their precession in the S0 *eld would be also random. We note that the possibility of the accelerating motion of the spinning particles in an external torsion *eld has been discussed also in [193,169,99]. The torsion *eld is supposed to act on the spin of particles but not on their angular momentum [170,193]. Therefore, a motion like the one described above will occur for individual electrons or other particles with spin as well as for macroscopic bodies with *xed spin orientation but it does not occur for the (charged or neutral) bodies with random orientations of spins. 4.6. Experimental constraints for the constant background torsion In the previous and present sections we are considering the approach in which torsion is purely background *eld. Thus, we shall describe only those possible e4ects which do not need propagating torsion. A brief discussion of the possible experimental manifestations of the propagating torsion will be given in the next section. In principle, the background torsion may produce two kinds of e4ects: the change of the trajectory for the particles with (or even without—for the case of Higgs particle) spin, or the change of the spectrum due to the torsion-dependent terms 14

The analysis of Eq. (4.66) in our paper [158] was wrong. I am very grateful to Luiz Garcia de Andrade who found this mistake and noticed me about it.

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in the Dirac equation. The possible experiments with the motion of particles are quite obvious. Let the electromagnetic *eld to be absent. Then, according to the results of the previous section, the interaction with torsion twists the particle trajectory. Then, any charged particles may be the source of electromagnetic radiation. The structure of the radiation provides the opportunity to look for torsion e4ects. However, for a very feeble torsion, the electromagnetic radiation (as a second order e4ect) will be very weak and this way of detecting torsion is not really promising. Let us now comment on the spectroscopy part, using the non-relativistic approximation. We shall follow the consideration of [13]. Consider the Schroedinger equation (4.6) with the Hamiltonian operator (4.7) without electromagnetic *eld. It is evident that the e4ect of a torsion *eld can modify the particles spectrum. Some modi*cations are similar to the ones which arise in the electromagnetic *eld. At the same time, another modi*cations are possible due to the qualitatively new term ("=mc)S0 p ˜ · ˜/ in (4.7). It is natural to suppose that torsion is feeble enough and therefore one can consider it as a perturbation. This perturbation might lead to the splitting of the known spectral lines and hence one can, in principle, derive an upper bound for the background torsion using the spectral analysis experiments. 15 One can expect the splitting of the spectral lines for the hydrogen atom (such a splitting has been also discussed in [82] for torsion coupled to massive electrodynamics). Consider the constant torsion S =const: and estimate possible modi*cations of the spectrum. In this particular case the Hamiltonian operator is " ˆ ˆ 1 2 Hˆ = ˜ +p ˜ˆ Sˆ 0 ) · ˜/ˆ ; =ˆ + "˜Sˆ · ˜/ˆ − (S 0 p 2m 2mc where e ˜= = p ˜ − ˜A : c ˜ |mc and hence the second S0 deIn the framework of the non-relativistic approximation |p pendent term in the brackets can be omitted. The remaining term "˜S · ˜/ admits the standard interpretation and gives the contribution ±"S3 into the spectrum. Thus, if the S3 component of the torsion tensor is not equal to zero, the energy level is splitted into two sub-levels with the di4erence 2"S3 . If now, the week transversal magnetic *eld is switched on then the cross between the new levels will arise and the energy absorption takes place at the magnetic *eld frequency w = 2"=S3 . Note that the situation is typical for the magnetic resonance experiments, however in the present case the e4ect arises due to the torsion, but not to the magnetic *eld e4ects. There were several attempts to draw numerical bounds on the background torsion using known experiments and the torsion corrections to the SchrYoedinger or Dirac equation. One can distinguish two approaches. One of them is more traditional, it does not really distinguish between purely background and propagating torsion. It is suLcient to suppose that the torsion mass is dominating over the possible kinetic terms. In this case the e4ect of torsion is to 15 Indeed, torsion e4ects will compete with the relativistic corrections and with the *ne structure e4ects coming from QED. Therefore, this our consideration has mainly pedagogical purpose. At the end of the section we shall brie9y present the modern limits on torsion coming from the complete studies.

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provide the contact spin–spin interactions. As an examples of works done in this direction one can mention [99] (one can *nd there more references) and [95], where the Pauli-like equation (4.6), (4.8) has been applied together with the Einstein–Cartan action for torsion. The most recent and complete upper bound for torsion parameters from the contact interaction have been obtained in [18]. We shall present the corresponding results in the next section after discussing the problem of torsion mass and propagating torsion. An alternative way is to suppose the existence, in our part of the Universe, of some constant torsion axial vector S , and to look for its possible manifestations. The most recent publication with the analysis of this possibility and the derivation of the corresponding upper bound for torsion is [120], where the constraints on the space–time torsion were obtained from the data on the Hughes–Drever experiments on the basis of the Pauli-like equation. The limit on the magnitude of the space component of the antisymmetric contorsion has been obtained in [120] using the experimental data concerning violation of Lorentz invariance [50]. Besides these papers devoted to the search of the torsion e4ects, there were, in the last decades, numerous publications on the same subject, but without explicit mentioning the word “torsion”. These works were devoted to the search for the Lorentz and CPT violations coming from various odd terms in the Dirac equation. The most popular form (there are others) of such an insertion is the b axial vector. The modi*ed form of the Dirac equation is (i  D − 5  b − m) = 0 ;

(4.68)

where D = 9 − iA . It is easy to see that the b is nothing but the normalized axial vector of torsion b = "S and, fortunately, we have the possibility to use the corresponding data on the limits coming from the CPT and Lorentz anomalies. There is no reason to repeat the details of the existing numerous reviews on the subject (see, for example, [25,118] and references therein), so we shall just present the main results. The violation of the Lorentz and CPT symmetries occurs because b is a constant vector with the *xed space component. Consequently, any Lorentz boost breaks the form of the Dirac equation. The limits on the magnitude of the b *elds come from the studies of neutral-meson oscillations in the kaon system, experimental test with leptons and barions using Penning traps, comparative spectroscopy of the hydrogen and antihydrogen atoms, measurements of muon properties, clock-comparison Hughes–Drever type experiments, observation of the anomaly in the behavior of the spin-polarized torsion pendulum and tests with the spin-polarized solids [26]. The overall limits on |b| di4er and depend on the type of experiment. In particular, these limits are di4erent for di4erent fermions. These limits are typically from 10−25 to 10−30 GeV, so that the universal phenomenological bound, valid for all fermion species, is between 10−27 and 10−30 GeV. If we really associate b vector with torsion, and remember the renormalization-group based arguments (see Section 3.4) about the universality of the fermion–torsion interaction, the estimates for di4erent fermions can be put together and we arrive at the total universal limit |b| ¡ 10−30 GeV. Thus, the limits derived from numerous laboratory experiments, are very small. They leave no real chance to use torsion for the explanation of physical phenomena [65] like the anomaly in the polarization of light coming from distant galaxies [138]. The same concerns the creation of particles by external torsion *eld [157] and the helicity 9ip for the solar neutrino which could be, in principle, induced by torsion [96].

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5. The e ective quantum &eld theory approach for the dynamical torsion The theoretical description of any new *eld, including torsion, must have two important elements: the interaction of this *eld with the well established matter *elds and the proper dynamics of the new *eld. From Quantum Field Theory point of view, any classical description may be considered as an approximation to some complete theory including quantum e4ects. Following this line, one has to construct the theory of torsion in such a way that it would pass the necessary test of consistency as a quantum theory. As we have seen in the previous two sections, the interaction of torsion with matter does not lead to any diLculty, until we consider torsion as a purely background *eld. However, this semi-classical theory is de*nitely incomplete if we do not attempt to formulate torsion dynamics. The simplest approach is just to postulate the Einstein–Cartan theory (2.18) as a torsion action. As we have already learned in Section 2, in this case torsion does not propagate and leads only to the contact spin–spin interactions. Furthermore, since the torsion mass is of the Planck order of magnitude, such a contact interaction is suppressed, at low energies, by the Planck mass. As a result there are very small chances to observe torsion at low energies. The main purpose of the present section is to follow the recent papers [18,22] where we have discussed an alternative possibility for a smaller torsion mass. We start the section by making a short account of the previous works on the dynamical torsion, and then proceed by applying the ideas of e4ective quantum *eld theory to the formulation of torsion dynamics. We shall mainly concentrate on the theoretical aspects, and provide only a short review of the phenomenological bounds on the torsion mass and couplings. The interested reader is referred to the second paper in Ref. [18] for further phenomenological details. 5.1. Early works on the quantum gravity with torsion Since the early days (see [99] for a review) torsion has been considered as an object related to quantum theory. Thus, it is natural to discuss propagating torsion in terms of Feynman diagrams, Green’s functions and S-matrix instead of using the dynamical equations. As any other propagating *eld, torsion must satisfy the condition of unitarity. So, it is natural that the attempts to construct the theory of the propagating torsion started from the study of the constraints imposed by the unitarity [132,133,161,162,134]. The initial motivation of [132,133,161,162] was to construct the theory of quantum gravity which would be both unitary and renormalizable. So, let us brie9y describe the general situation in Quantum Gravity (see [163] for more extensive review). It is well known that the program of quantizing General Relativity met serious diLculty, because this quantum theory is non-renormalizable by power counting. If taking only the super*cial logarithmic divergences of the diagrams into account, the dimension (number of derivatives of the metric) of the n-loop counterterms is d = 2 + 2n. Then, with every new order of the loop expansion the dimension of the counterterms grows up and hence one needs an in*nite number of the renormalization conditions to extract a single prediction of the theory in the high-energy region. The non-renormalizability of quantum General Relativity becomes apparent already at the one-loop level for the case of gravity coupled to matter [107,61] and at two-loop level for the pure gravity [88]. The situation in supergravity (see e.g. [83]) is better in the sense that the on-shell divergences do not show up at second

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(N = 1 case) or even higher (perhaps seventh for N = 8 supergravity) loops. However, the supersymmetry does not solve the principal problem, and all known versions of supergravity generalizations of General Relativity are expected to be non-renormalizable. At the same time, it is fairly simple to construct renormalizable theory of the gravitational *eld by adding the fourth-derivative terms [175]  √ SHD = d 4 x −g(R6/ R6/ + R R + R2 +  R) (5.1) into the classical action. It is better to write the above expression in another basis, so that the algebraic properties of the terms become more explicit. Let us use (3.7), so that Eq. (5.1) can be written as  √ SHD = d 4 x −g(a1 C 2 + a2 E + a3 R2 + a4 R) : (5.2) It is well known (see e.g. [175,34]), that the contributions of the terms of Eq. (5.2) to the propagator of the gravitational perturbations are very di4erent. This propagator can be divided into irreducible parts through introducing the projectors to spin-2, spin-1 and spin-0 states. It turns out that spin-1 states can be completely removed by the gauge *xing (of course, the Faddeev–Popov ghosts must be taken into account). Furthermore, the C 2 term contributes to the gauge *xing independent spin-2 part of the propagator, while the R 2 -term contributes only to the spin-0 part and does not contribute to the spin-2 part. The term E does not contribute to the propagator at all, even if we change the dimension of the space–time from 4 to n. In n = 4 this term is topological, so it is supposed not to a4ect the vertices either (algebraically, the situation is not so simple [43], but there are no indications of the non-trivial quantum e4ect of this term). At the same time, for n = 4 all three terms contribute to all vertices: to the interactions of the metric components of all spins. Now, suppose we take a theory with the action    1 4 √ 2 2 St = d x −g − 2 R + a1 C + a2 E + a3 R (5.3)  with a1 = 0 and a3 = 0. In this case the propagators of all components, including ghosts [34], behave like k −4 in the high-energy domain. Similarly, there are vertices proportional to the fourth, second and zero power of the momenta. In this situation, the power counting shows that the super*cial degree of divergence of the IPI diagrams does not depend on the loop order, and the maximal possible dimension of the logarithmic counterterms is 4. Taking the locality and general covariance of the counterterms [175,184] into account, we see that they have the same form as the classical action (5.3). Thus, the theory is renormalizable. Still, it is not perfect. Spin-2 sector of the propagator has the form   1 1 1 1 (2) 2 G (k) ∼ 2 2 = − + m2 ; (5.4) k (k + m22 ) m22 k 2 k 2 where m2 ∼ 1=2 is the mass of the non-physical ghost. Another massive pole exists in spin-0 sector, but its position depends on the gauge *xing while spin-2 part (5.4) is gauge independent. Excluding the non-physical particles from the physical spectrum, one breaks unitarity [175]. The situation described above is quite general, because further modi*cations of the action

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do not change the situation. For instance, introducing more higher derivative terms into the action cannot provide unitarity [9]. It is quite obvious that the situation cannot be improved by changing the dynamical variables, because this operation cannot change the position of the massive gauge independent pole of the propagator. Some observation is in order. If we put, in action (5.3), the coeLcient a1 = 0, the theory will not be renormalizable (despite it still possesses higher derivatives). The point is that such a theory contains high derivative vertices of the interaction of spin-2 states with themselves and with other states, but spin-2 part of the propagator behaves like G (2) (k) ∼ k −2 . It is easy to see that the power counting in this theory would be even worst than in quantum General Relativity. Thus, the renormalizability of quantum gravity theory with high derivatives crucially depends on the k −4 UV behavior of spin-2 propagator. In turn, since spin-2 sector always contains a k −2 part due to the Einstein term, the (5.4) structure of the propagator looks as an unavoidable consequence of the renormalizability. Of course, these considerations do not have absolute sense, one can try to look for an action which could solve the problem of quantum gravity. The original idea of [132] was to use the *rst order formalism and to establish the combination of the R2·· -terms which could preserve the renormalizability and, simultaneously, provide the absence of the massive unphysical pole. As it was already explained in Section 2.5, the *rst order formalism treats the aLne connection as an object independent on the metric. Anyhow, the connection can always be presented as a sum of the Christo4el symbol and some additional tensor which includes torsion and non-metricity. Furthermore, if one is interested in the propagator of perturbations on the 9at background, it is possible to classify all the *elds by the  -component of representations of the Lorentz group. Indeed, the non-metricity tensor and q· torsion both contain spin-2 states. At least one of these states should be massless in order to provide the correct Newtonian limit. Then, if there are other, massive spin-2 states, the general structure of the complete spin-2 propagator should be equivalent to (5.4). Of course, the word “equivalent” in the last statement cannot be understood as an identity. For instance, in the *rst order formalism the propagator may be free of high derivatives at all. The equivalence signi*es that the r.h.s. of (5.4) will be restored, in the linearized theory, after elimination of an extra (one can call them auxiliary) *elds. The *rst attempt to *nd the renormalizable and ghost-free theory of Ref. [132] did not include torsion, which was implemented later in Refs. [133,161,162,134]. In all these papers,  the gravitational *eld has been parametrized by the vierbein ea and spinor connection wab , but  as we have discussed in Section 2, these variables are equivalent to another ones—(g ; T·· ): if one does not introduce a non-metricity. In principle, one could include, as we mentioned above, all curvature and torsion depending terms (there are 168 of them [48]) plus all possible terms depending on the non-metricity. The technical part of the original papers [132,133,161,162] is quite cumbersome and we will not reproduce it here. The *nal result has been achieved for the theories with torsion but without non-metricity. The number of ghost-free R2·· -type actions has been formulated, but all of them are non-renormalizable. Obviously, these actions resemble (5.3) with a1 = 0, for they possess high derivatives in the vertices but not in the propagators. If one performs the loop calculation in such theory, the ghost-free structure of the propagator will be immediately destroyed. Therefore, at the quantum level the unitarity of the classical S-matrix can be spoiled, if the theory is not renormalizable. The conclusion is that, for the consistent quantum theory, one needs both unitarity and renormalizability.

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In the next sections we will not focus on the problem of quantum gravity. Instead, we shall concentrate on the possibility to have a theory of the propagating torsion, which should be consistent at the quantum level. The application of the ideas of the e4ective *eld theories enables one to weaken the requirement of renormalizability (see e.g. [187,69]), but even in this case we shall meet serious problems and limitations in constructing the theory of the propagating torsion. As to the formulation of consistent quantum gravity, the string theory is, nowadays, the only one visible candidate. 5.2. General note about the e=ective approach to torsion In order to construct the action of a propagating torsion, we have to apply two requirements: unitarity and renormalizability. The problem of renormalizability is casted in another form if we consider it in the content of e4ective *eld theory [187,69]. In the framework of this approach one has to start with the action which includes all possible terms satisfying the symmetries of the theory. Usually, such an action contains higher derivatives at least in a vertices. However, as far as one is interested in the lower energy e4ects, those high derivative vertices are suppressed by the great massive parameter which should be introduced for this purpose. Then, those vertices and their renormalization are not visible and e4ectively at low energies one meets renormalizable and unitary theory. The gauge invariance of all the divergences is guaranteed by the corresponding theorems [182] and thus this scheme may be applied to the gauge theories including even gravity [186,68]. Within this approach, it is important that the lower-derivative counterterms have the same form as the terms included into the action. This condition, together with the symmetries and the requirement of unitarity, may help to construct the e4ective *eld theories for the new interactions such as torsion. If one starts to formulate the dynamical theory for torsion in this framework, the sequence of steps is quite de*nite. First, one has to establish the *eld content of the dynamical torsion theory and the form of the interactions between torsion and other *elds. Then, it is necessary to take into account the symmetries and formulate the action in such a way that the resulting theory is unitary and renormalizable as an e4ective *eld theory. Indeed, there is no guarantee that all these requirements are consistent with each other, but the inconsistency might indicate that some symmetries are lost or that the theory with the given particle content is impossible. In the next sections we consider how this scheme works for torsion, and then compare the situation with that of e4ective low-energy quantum gravity (see e.g. [68]). Among the torsion components (2.16) S ; T ; q· , only S is really important, because only this ingredient of torsion couples to spinors in a minimal way. Therefore, in what follows, we shall restrict the consideration to the axial vector S which parameterizes the completely antisymmetric torsion. 5.3. Torsion–fermion interaction again: softly broken symmetry associated with torsion and the unique possibility for the low-energy torsion action In this section, we consider the torsion–spinor system without scalar *elds. Thus, we start from the action (2.35) of the Dirac spinor non-minimally coupled to the vector and

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torsion *elds S1=2 = i



d 4 x V [  (9 + ieA + i" 5 S ) − im]

:

First one has to establish its symmetries. At this stage we consider the vector *eld A as an abelian one but later we will focus on the vector *elds of the SM which are non-abelian. The symmetries include the usual gauge transformation (2.36), and also softly broken symmetry (2.37):  V  = V e 5 (x) ; = e 5 (x) ; S = S − "−1 9 (x) : The massive term is not invariant under the last transformation. The symmetries of the theory have serious impact on the renormalization structure. In particular, since the symmetry under (2.37) is softly broken, it does not forbid massive counterterms in the torsion sector and hence S has to be a massive *eld. Below we consider the torsion mass as a free parameter which should be de*ned on a theoretical basis, and may be also subject of experimental restrictions. As far as torsion is taken as a dynamical *eld, one has to incorporate it into the SM along with other vector *elds. Let us discuss the form of the torsion action in the framework of the e4ective approach—that is focusing on the low-energy e4ects. The higher derivative terms may be included into the action, but they are not seen at low energies. Thus, we restrict the torsion action by the lower-derivative terms and arrive at the expression:  Stor = d 4 x{−aS S  + b(9 S  )2 + Mts2 S S  } ; (5.5) where S = 9 S − 9 S and a; b are some positive parameters. Action (5.5) contains both transverse vector mode and the longitudinal vector mode where the last one is equivalent to the scalar [45]. In particular, in the a = 0 case only the scalar mode, and for b = 0 only the vector mode propagates. It is well known [74] (see also [45] for the discussion of the theory (5.5)) that in the unitary theory of the vector *eld the longitudinal and transverse modes cannot propagate simultaneously, 16 and therefore one has to choose either a or b to be zero. In fact the only correct choice is b = 0. In order to see this one has to reveal that symmetry (2.37), which is spoiled by the massive terms only, is preserved in the renormalization of the dimensionless coupling constants of the theory (at least at the one-loop level). In other words, the divergences and corresponding local counterterms, which produce the dimensionless renormalization constants, do not depend on the dimensional parameters such as the masses of the *elds. This structure of renormalization resembles the one in the Yang–Mills theories with spontaneous symmetry breaking. As we shall see later, even the b = 0 choice is not free of problems, but at least they are not related to the leading one-loop e4ects, as in the opposite a = 0 choice. Thus, the only one possible torsion action is given by Eq. (5.26) with b = 0. In order to illustrate this, we remind that the divergences coming from fermion loop are given 16

One can easily check this without even looking into the textbooks: the kinetic terms for the transverse vector and for the “longitudinal” scalar, coming from the a (9 S  )2 -term, have opposite signs, while both *elds share massive term. As a result one of these *elds must have, depending on the signs of a; b; Mts2 , either negative mass or negative kinetic energy. Hence, the theory with both components always includes either ghost or tachyon.

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by experssion (3.15), which is in a perfect agreement with transformation (2.37). Namely, the 2 and the massive term while the (9 S  )2 term is absent. one-loop divergences contain S  It is well known that the fermion loop gives rise, in theory (2.35), to axial anomaly. But, as we have discussed in Section 3.8, the problem of anomaly does not spoil our attempts to implement torsion into the fermion sector of the SM. And so, the only possible form of the torsion action which can be coupled to the spinor *eld (2.35) is    1 4  2  Stor = d x − S S + Mts S S : (5.6) 4 In the last expression we put the conventional coeLcient −1=4 in front of the kinetic term. With respect to the renormalization this means that we (in a direct analogy with QED) can remove the kinetic counterterm by the renormalization of the *eld S and then renormalize the parameter " in action (2.35) such that the combination "S is the same for the bare and renormalized quantities. Instead one can include 1="2 into the kinetic term of (5.6), that should lead to the direct renormalization of this parameter while the interaction of torsion with spinor has minimal form (2.33) and S is not renormalized. Therefore, in the case of a propagating torsion the di4erence between minimal and non-minimal types of interactions is only a question of notations on both classical and quantum levels. 5.4. Brief review of the possible torsion e=ects in high-energy physics As we have already seen, spinor–torsion interactions enter the Standard Model as interactions of fermions with a new axial vector *eld S . Such an interaction is characterized by the new dimensionless parameter—coupling constant ". Furthermore, the mass of the torsion *eld Mts is unknown, and its value is of crucial importance for the possible experimental manifestations of the propagating torsion and *nally for the existence of torsion at all (see the discussion in the last part of this section and in Section 6). In the present section we consider " and Mts as arbitrary parameters and review the limits on their values from the known experiments [18]. Later on we shall see that the consistency of the fermion–torsion system can be achieved for the heavy torsion only, such that Mts Mfermion . However, we shall follow [18] where one can *nd the discussion of the “light” torsion with the mass of the order of 1 GeV. The strategy of [18] was to use known experiments directed to the search of the new interactions. One can regard torsion as one of those interactions and obtain the limits for the torsion parameters from the data which already *t with the phenomenology. Torsion, being a pseudovector particle interacting with fermions, might change di4erent physical observables. For instance, this speci*c type of interaction might lead to the forward– backward asymmetry. The last has been precisely measured at the LEP e+ e− collider, so the upper bounds for torsion parameters may be set from those measurements. One can consider two di4erent cases: (i) torsion is much heavier than other particles of the SM and (ii) torsion has a mass comparable to that of other particles. In the last case one meets a propagating particle which must be treated on an equal footing with other constituents of the SM. Contrary to that, the very heavy torsion leads to the e4ective contact four-fermion interactions. Let us brie9y review all mentioned possibilities.

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(i) Forward–backward asymmetry: Any parity violating interactions eventually give rise to the space asymmetry and could be revealed in the forward–backward asymmetry of the particle scattering. Axial-vector type interactions of torsion with matter *elds is this case of interactions. But the source of asymmetry also exists in the SM electroweak interactions because of the presence of the  5 structure in the interactions of Z- and W -bosons with fermions. The interactions between Z-boson and fermions can be written in general form as g i i 5 V  (gV LZff = − − gA ) Z ; (5.7) 2 cos 2W i i where, 2W is Weinberg angle, g = e= sin 2W (e—positron charge); and the vector and axial couplings are: i gV ≡ t3L (i) − 2qi sin2 2W ;

(5.8)

i gA ≡ t3L (i) :

(5.9)

Here t3L is the weak isospin of the fermion and i has the values +1=2 for ui and i while it is −1=2 for di and ei . Here i = 1; 2; 3 is the index of the fermion generation and qi is the charge of the i in units of charge of positron. The forward–backward asymmetry for e+ e− → l+ l− is de*ned as /F − /B AFB ≡ ; (5.10) /F + /B where /F (/B ) is the cross section for l− to travel forward(backward) with respect to electron direction. Such an asymmetries are measured at LEP [125]. In the SM, from asymmetries one derives the ratio gV =gA of vector and axial-vector couplings, but the presence of torsion could change the result. In fact, the measured electroweak parameters are in a good agreement with the theoretical predictions and hence one can establish the limits on the torsion parameters based on the experimental errors. The contribution from torsion exchange diagrams has been calculated in [18]. From those calculations one can establish the limits on " and Mts taking into account the mentioned error of the experimental measurements. Deviations of the asymmetry from SM predictions would be an indication of the presence of the additional torsion-like type axial-vector interactions. The analysis of [18] shows that the electron AeFB asymmetry is the best observable among others asymmetries to look for torsion. The details of the corresponding analysis can be easily found in [18] and we will not reproduce them here. (ii) Contact interactions: Since the torsion mass comparable to the mass of the fermions leads to problems (which will be explained in the following sections), it is especially important for us to consider the case of heavy torsion. Since the massive term dominates over the covariant kinetic part of the action, the last can be disregarded. Then the total action leads to the algebraic equation of motion for S . The solution of this equation can be substituted back into the action and thus produce the contact four-fermion interaction term "2 Lint = − 2 ( V 5  )( V 5  ) : (5.11) Mts As one can see the only quantity which appears in this approach is the ratio Mts =" and therefore for the very heavy torsion *eld the phenomenological consequences depend only on

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this single parameter. As it was mentioned above, the axial-vector type interactions would give rise to the forward–backward asymmetry which have been precisely measured in the e+ e− → l+ l− (qq) V scattering (here l = (; ; e) stands for the leptons and q for quarks) at LEP collider with the center-mass energy near the Z-pole. Due to the resonance production of Z-bosons the statistics is good (several million events) and it allowed to measure electroweak (EW) parameters with high precision. There are several experiments from which the constraints on the contact four-fermion interactions come [18]: (1) Experiments on polarized electron–nucleus scattering: SLAC e-D scattering experiment [152], Mainz e-Be scattering experiment [98] and bates e-C scattering experiment [173]. (2) Atomic physics parity violations measures [122] electron–quark coupling that are di4erent from those tested at high-energy experiment provides alternative constraints on new physics (see also Section 4.6). (3) e+ e− experiments—SLD, LEP1, LEP1.5 and LEP2 (see for example [124,145,126,5,121]). (4) Neutrino–Nucleon DIS experiments—CCFR collaboration obtained a model independent constraint on the e4ective qq coupling [129]. Consider the limits on the contact interactions induced by the torsion. The contact four-fermion interaction may be described by the Lagrangian [72] of the most general form: jij  V V  L

= g2 (5.12) j )2 ( i  i )( j j ) : (5 ij i; j=L;R q=u;d







Subscripts i; j refer to di4erent fermion helicities: (i ) = R;( L) =(1 ± 5 )=2 · ( ) ; where ( ) could be quark or lepton; 5ij represents the mass scale of the exchanged new particle; coupling strength is *xed by the relation: g2 =4= = 1, the sign factor jij = ±1 allows for either constructive or destructive interference with the SM and Z-boson exchange amplitudes. Formula (5.12) can be successfully used for the study of the torsion-induced contact interactions because it includes an axial–axial current interaction (5.11) as a particular case. Recently, the global study of the electron–electron–quark–quark(eeqq) interaction sector of the SM have been done using data from all mentioned experiments [15]. For the axial–axial eeqq interactions (5.12) takes the form (we put g2 = 4=): Leeqq = −

4= (e V  5 e)(q V  5 q) : (5jAA )2







(5.13)

The limit for the contact axial–axial eeqq interactions comes from the global analysis of Ref. [15]: 4= ¡ 0:36 TeV−2 : 52AA

(5.14)

Comparing the parameters of the e4ective contact four-fermion interactions of general form (5.13) and contact four-fermion interactions induced by torsion (5.11) we arrive at the following relations: "2 4= = 2 : 2 Mts 5AA

(5.15)

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From (5.14) and (5.15) one gets the following limit on torsion parameters: " ¡ 0:6 TeV−1 ⇒ Mts ¿ 1:7 TeV · " : Mts

(5.16)

The last relation puts rigid phenomenological constraints on the torsion parameters "; Mts , and one can also take into account that the modern scattering-based analysis cannot be relevant for the masses beyond the 3 TeV. It is easy to see that bound (5.16) is incompatible with the one established in [95], because in this paper the torsion mass has been taken to be of the Planck order of magnitude. An additional restriction can be obtained from the analysis of the TEVATRON data but since they concern mainly the light torsion, we will not give the details here. In [18], one can *nd the total limits on torsion from an extensive variety of experiments (see also [47] for the consequent analysis concerning the torsion coming from small extra dimensions). In conclusion, we learned that torsion, if exists, may produce some visible e4ects, while the phenomenological analysis put some limits on the torsion parameters "; Mts . Of course, these limits will improve if the experimental data and(or) the theoretical derivation of observables in the SM become more precise. In case one detects some violation of the phenomenological bounds, one can suppose that this is a manifestation of some new physics. This new physics can be torsion or something else (GUT, supersymmetry, higher dimensions and so on). In each case one has to provide the consistent quantum theory for the corresponding phenomena. Therefore, the relevance of phenomenological considerations always depends on the formal *eld-theoretical investigation. As we shall see in the next sections, in the case of torsion the demands of the theory are more restrictive than the phenomenological limits. 5.5. First test of consistency: loops in the fermion–scalar systems break unitarity To this point, we have considered only interaction between torsion and spinors. Now, in order to implement torsion into the SM, one has to include scalar and Yukawa interactions. When introducing scalar *eld we shall follow the same line as in the previous section and try *rst to construct the renormalizable theory. Hence, the *rst thing to do is to analyze the structure of the possible divergences. The divergent diagrams in the theory with a dynamical torsion include, in particular, all such diagrams with external lines of torsion and internal lines of other *elds. Those grafs are indeed the same one meets in quantum *eld theory on a classical torsion background. Therefore, one has to include into the action all terms which were necessary for the renormalizability when torsion was a purely external *eld. All such terms are already known from our investigation of quantum *eld theory on an external torsion background. Besides the non-minimal interaction with spinors, one has to introduce the non-minimal interaction ’2 S 2 between scalar *eld and torsion as in (2.26) and also the terms which played the role of the action of vacuum (see, e.g., (3.6) or (3.68)) in the form    1 1 4  2   2 Stor = d x − S S + Mts S S − W(S S ) + surface terms : (5.17) 4 24

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Fig. 3. Two-loop diagrams corresponding to torsion self-interaction and torsion–scalar interaction.

Here W is some new parameter, and the coeLcient 1=24 stands for the sake of convenience ' (2 only. The necessity of the S S  term in the classical action follows from the fact that such a term emerges from the scalar loop with a divergent coeLcient. Then, if not included into the classical action, it will appear with in*nite coeLcient as a quantum correction. On the other hand, by introducing this term into the classical action we gain the possibility to remove the corresponding divergence renormalizing the coupling W. So, if one implements torsion into the complete SM including the scalar *eld, the total action includes the following new terms: torsion action (5.17) with the self-interacting term, and non-minimal interactions between torsion and spinors (2.35) and scalars (2.26). However, at the quantum level, such a theory su4ers from a serious diLculty. The root of the problem is that the Yukawa interaction term h’ V is not invariant under transformation (2.37). Unlike the spinor mass, the Yukawa constant h is massless, and this non-invariance may a4ect the renormalization in the massless sector of the theory. In particular, the non-invariance of the Yukawa interaction causes the necessity of the non-minimal scalar– torsion interaction in (2.26) which, in turn, requires an introduction of the self-interaction term in (5.17). Those terms do not pose any problem at the one-loop level, but already at the second loop one meets two dangerous diagrams presented at Fig. 3. These diagrams are divergent and they can lead to the appearance of the (9 S  )2 -type counterterm. No symmetry is seen which forbids these divergences. Let us consider, following [18], the diagrams presented at Fig. 3 in more details. Using the actions of the scalar *eld coupled to torsion (2.26) and torsion self-interaction (5.17), we arrive at the following Feynman rules: (i) (ii) (iii) (iv)

scalar propagator: G(k) = i=(k 2 + M 2 ) where M 2 = 2Mts2 , torsion propagator: D (k) = [i=(k 2 + M 2 )]( + k k  =M 2 ), the ’2 S 2 —vertex: V  (k; p; q; r) = −2i$" ,  vertex of torsion self-interaction: V  (k; p; q; r) = (iW=3)g(4) ,

 = g g + g g + g g and k; p; q; r denote the outgoing momenta. where g(4) The only one thing that we would like to check is the violation of the transversality in the kinetic two-loop counterterms. We shall present the calculation in some details because it is quite instructive. To analyze the loop integrals we have used dimensional regularization and in particular the formulas from [123,101]. It turns out that it is suLcient to trace the 1= j2 -pole, because even this leading divergence requires the longitudinal counterterm. The contribution to the mass-operator of torsion from the second diagram from Fig. 3 is given by the following integral:  " + M −2 (k − q) (k − q) dn k dn p (2) 2 E (q) = −2$ : (5.18) (2=)4 (2=)4 (p2 + M 2 )[(k − q)2 + M 2 ][(p + k)2 + M 2 ]

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First, one has to note that (as in any local quantum *eld theory) the counterterms needed to subtract the divergences of the above integrals are local expressions, hence the divergent part of the above integral is *nite polynomial in the external momenta q . Therefore, in order to extract these divergences one can expand the factor in the integrand into the power series in q :  −1   ∞ 2 l 1 1 1 −2k · q + q2 l −2k · q + q = 1+ = 2 (−1) (k − q)2 + M 2 k 2 + M 2 k2 + M 2 k + M2 k2 + M 2 l=1

(5.19) and substitute this expansion into (5.18). It is easy to see that the divergences hold in this expansion till the order l = 8. On the other hand, each order brings some powers of q . The divergences of the above integral may be cancelled only by adding the counterterms which include high derivatives. 17 To achieve the renormalizability one has to include these high-derivative terms into action (5.17). However, since we are aiming to construct the e4ective (low-energy) *eld theory of torsion, the e4ects of the higher derivative terms are not seen and their renormalization is not interesting for us. All we need are the second derivative counterterms. Hence, for our purposes expansion (5.19) can be cut at l = 2 rather that at l = 8 and moreover only O(q2 ) terms should be kept. Thus, one arrives at the known integral [101]  dn k dn p 1 1 1 (2) 2 2 E (q) = −6$ q " + ··· 4 4 2 2 2 2 2 (2=) (2=) p + M (k + M ) (p + k)2 + M 2 =−

12$2 q2 " + (lower poles) + (higher derivative terms) : (4=)4 (n − 4)2

(5.20)

Another integral looks a bit more complicated, but its derivation can be done in a similar way. The contribution to the mass-operator of torsion from the *rst diagram from Fig. 3 is given by the integral     k k dn k dn p W2 (2)6/ (2)  1 (1) E (q) = − g " + 2 g 108 (2=)4 (2=)4 k 2 + M 2 M   (p − q)6 (p − q) 1 × "6 + 2 2 M p + M2   (p + k)/ (p + k) 1 × "/ + : (5.21) M2 (k + q)2 + M 2 Now, we perform the same expansion (5.19) and, disregarding lower poles, *nite contributions and higher derivative divergences arrive at the following leading divergence: (1) E (q) = −

17

W2 q2 " + · · · : (4=)4 (n − 4)2

(5.22)

This is a consequence of the power-counting non-renormalizability of the theory with massive vector *elds.

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Thus, we see that both diagrams from Fig. 3 really give rise to the longitudinal kinetic counterterm and no any simple cancellation of these divergences is seen. In order to understand the situation better let us compare it with the one that takes place for the usual abelian gauge transformation (2.36). In this case, the symmetry is not violated by the Yukawa coupling, and (in the abelian case) the A2 ’2 counterterm is impossible because it violates gauge invariance. The same concerns also the self-interacting A4 counterterm. The gauge invariance of the theory on quantum level is controlled by the Ward identities. In principle, the non-covariant counterterms can show up, but they can be always removed, even in the non-abelian case, by the renormalization of the gauge parameter and in some special class of (background) gauges they are completely forbidden. Generally, the renormalization can be always performed in a covariant way [182]. In the case of transformation (2.37) if the Yukawa coupling is inserted there are no reasonable gauge identities at all. Therefore, in the theory of torsion *eld coupled to the SM with scalar *eld there is a con9ict between renormalizability and unitarity. The action of the renormalizable theory has to include the (9 S  )2 term, but this term leads to the massive ghost. This con9ict between unitarity and renormalizability reminds another one which is well known—the problem of massive unphysical ghosts in the high-derivative gravity [175]. The di4erence is that in our case, unlike higher-derivative gravity, the problem appears due to the non-invariance with respect to transformation (2.37). We shall proceed with the discussion of this analogy in Section 5.8. 5.6. Second test: problems with the quantized fermion–torsion systems The problem of consistency of the fermion–torsion system has been studied in [22]. Since the result of this study had crucial importance for the theoretical possibility of torsion, we shall present many details of the investigation of [22] here. In order to understand the source of the problems, let us *rst write the fermion action with torsion using the Boulware-like parametrization. For pedagogical reasons, we *rst consider the usual vector case, that is, repeat the transformation of [27]. The action of the massive vector *eld V , in original variables, has the form    1 2 1 4    ; (5.23) Sm-vec = d x − V V + M V V + i V [ (9 − igV ) − im] 4 2 where V = 9 V − 9 V and, after the change of the *eld variables [27]:     1 ig ig V = exp · ’ · (; = (V · exp − · ’ ; v = V⊥ − 9 ’ ; M M M the new scalar, ’, is completely factored out:    1 ⊥ 2 1 2 ⊥ ⊥ 1  4  ⊥ V (9 + igV )]( : Sm-vec = d x − (V ) + M V V + 9 ’9 ’ + i([ 4 2 2

(5.24)

(5.25)

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Let us now consider the fermion–torsion system given by the action 18    1 1 2 4    V Stor-fer = d x − S S + M S S + i [ (9 + i" 5 S ) − im] : 4 2 The change of variables, similar to the one in (5.24), has the form     i" 5 i" 5 1 V = (V exp = exp ’ (; ’ ; S = S⊥ − 9 ’ ; M M M

(5.26)

(5.27)



where S⊥ and S = 9 ’ are the transverse and longitudinal parts of the axial vector, respectively, the latter being equivalent to the pseudoscalar ’. One has to notice that, contrary to (5.24), but in full accordance with (2.37), the signs of both the exponentials in (5.27) are the same. In terms of the new variables, the action becomes   1 ⊥ ⊥ 1 2 ⊥ ⊥ 4 Stor-fer = d x − S S + M S S 4 2  1   ⊥ (2i"=M ) 5 ’ +i([ V (9 + i" 5 S ) − im · e ]( + 9 ’9 ’ ; (5.28) 2 ⊥ = 9 S ⊥ − 9 S ⊥ = S . Contrary to the vector case (5.24), for the torsion axial vector where S      (5.28) the scalar mode does not decouple. Moreover, the interaction has the following unpleasant features: (i) it is Yukawa-type, resembling the problems with the ordinary scalar, (ii) besides, the interaction is exponential, which makes almost certain that the model is not power-counting renormalizable. On quantum level, symmetry (2.37) manifests itself through the Ward identities for the Green’s functions. The analysis of the Ward identities arising in the fermion–torsion system con*rms that the new scalar mode does not decouple and that one cannot control the ’-dependence on quantum level [22]. However, at *rst sight, there is a hope that the above properties would not be fatal for the theory. With respect to point (i), one can guess that the only result of the non-factorization, which can be dangerous for the consistency of the e4ective quantum theory, would be the propagation of the longitudinal mode of the torsion, and this does not directly follow from the non-factorization of the scalar degree of freedom in the classical action. On the other hand, (ii) indicates the non-renormalizability, which might mean just the appearance of higher-derivative divergences. But, this does not matter within the e4ective approach. Thus, a more detailed analysis is necessary. In particular, the one-loop and especially two-loop calculations in theory (5.26) may be especially helpful. The one-loop calculation in theory (5.26) can be performed using the generalized Schwinger–DeWitt technique developed by Barvinsky and Vilkovisky [16], but the application of this technique here is highly non-trivial. The problem has been solved in [22]. First of all, we note that the derivation of divergences in the purely torsion sector is not necessary, since the result is (3.15), that is the same as for the spinor loop on torsion background. 18

For the sake of simplicity we consider a single spinor only. Since torsion is an abelian massive vector, the results cannot depend on the gauge group.

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Of course, one has to disregard all curvature-dependent terms in (3.15), since now we work on the 9at metric background. Now, we are in a position to start the complete calculation of divergences. The use of the background *eld method supposes the shift of the *eld variables into a background and a quantum part. However, in the case of the (axial)vector–fermion system, the simple shift of the *elds leads to an enormous volume of calculations, even for a massive vector. Such a calculation becomes extremely diLcult for the axial massive vector (5.26). That is why, in [22], we have invented a simple trick combining the background *eld method with the Boulware transformation (5.27) for the quantum *elds. According to the method of [22], one has to divide the *elds into background (S ; ; V ) and quantum (t⊥ ; ’; (; () V parts, according to = ei("=M ) 5 ’ · ( + () ; V → V  = ( V + () V · ei("=M ) 5 ’ ; 1 S → S = S + t⊥ − 9 ’ : M The one-loop e4ective action depends on the quadratic (in quantum *elds) part of the total action:     1 4m"2 V (2) 4 ⊥ 2 ⊥ ⊥  V d x t ( + M )t + ’(− )’ + t (−2" 5 )( + ’ − S = ’ 2 M2      4im" 5 4i"m V 5  5 ⊥  +(( V −2" )t + (V ’+’ V i D + 2m)( : ( + ((2 M M (5.29) →



Making the usual change of the fermionic variables ( = −(i=2)(  D + im), and substituting ’ → i’, we arrive at the following expression:  ⊥ t  1   (2) 4 ⊥ ˆ d x(t ’ () S = V ·H· ’  ; 2  where the Hermitian bilinear form Hˆ has the form    2 ( + M 2) 0 2 (L 9 + M  )   Hˆ =  0 +N A 9  + B ; P 2  Q 1ˆ + R 9 + E

(5.30)

2   =   − 9 1 9 being the projector on the transverse vector states. The elements of the matrix operator (5.30) are de*ned according to (5.29): i R = 2"/ 5 S ; E = i" 5 (9 S  ) + "   5 S + "2 S S  + m2 ; 2     2 V   V L = −i" 5 ; M = " S + "m V 5  ;

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m V  m2 V m 5 ; B = 2"2 V  S − 2" 5 ; M M M m m N = 4"2 2 V ; P  = −2"  5 ; Q = −4" 5 : M M

A = 2i"

(5.31)

The operator Hˆ given above might look like the minimal second order operator ( + 2h ∇ + E); but, in fact, it is not minimal because of the projectors 2  in the axial vector–axial vector t⊥ – t⊥ sector. That is why one cannot directly apply the standard Schwinger–DeWitt expansion to derive the divergent contributions to the one-loop e4ective action, and some more sophisticated method is needed. Such a method, which can be called the generalized Schwinger– DeWitt technique in the transverse space, has been developed in [22], where the reader can *nd the complete details concerning the one-loop calculations in both theories (5.25) and (5.26). Let us, for the sake of brevity, present only the *nal result for the one-loop divergences in the theory (5.26):   n−4 2"2 3 (1) div dn x − =− S S  + 8m2 "2 S  S − 2m4 + M 4 +  3 2  2 3   4 2 2 2 8" m 2 V + 8 " m ( V )2 + 4i " m V  D∗ + − 6" m : (5.32) M2 M4 M2 It is interesting to note that the above expression (5.32) is not gauge invariant. One can trace the calculations back in order to see that the non-invariant terms come as a contribution of the scalar ’ (see, e.g., Eq. (5.28)). Thus, the non-invariant divergences emerge because there are variables, in which the violation of symmetry (2.37) is not soft. Consider the one-loop renormalization and the corresponding renormalization group. Expression (5.32) shows that theory (5.26) is not renormalizable, but the new type of divergences are suppressed if we suppose that the torsion has very big mass mMts (remember our notation Mts2 = 2M 2 ). Let us, for a while, take this relation as a working hypothesis. Then, the relations between bare and renormalized *elds and the coupling " follow from (5.32):     1 8"2 1 2"2 m2 (0) (n−4)=2 (0) (n−4)=2 S =  S 1 + · ; = 1+ · ; j 3 j M2    m2 1 8"3 (0) (4−n)=2 " = "− · · 1+ 2 : (5.33) j 3 M Similar relations for the parameter ˜ = (M 4 =m2 ) of the ( V )2 -interaction, have the form   4 ˜ 2 m2 (0) 16" 8

" − : (5.34)

˜ = 4−n ˜ + j M 2j These relations lead to a renormalization group equation for ", which contains a new term proportional to (m=M )2 :   2 m2 4 8 2 d" (4=) 1 + 2 " ; "(0) = "0 : (5.35) = dt 3 M

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Indeed, for the case mM and in the low-energy region, this equation reduces to the one presented in [18] (that is identical to the similar equation of QED). In any other case, the theory of torsion coupled to the massive spinors is inconsistent, and Eq. (5.35) is meaningless. One can also write down the renormalization group equation for the parameter ˜ de*ned above. Using (5.34), we arrive at the following equation: (4=)2

d ˜ = 16"4 : dt

(5.36)

This equation con*rms the lack of a very fast run for this parameter. Indeed, all this renormalization group consideration has meaning only under the assumption that mM . In order to provide the one-loop renormalizability, it is necessary to add the ( V )2 -term to the classical action. However, as we have learned, in Section 3, on the example of the Nambu–Jona–Lasinio model, such a term does not a4ect the one-loop renormalization of other sectors of the action. However, it becomes extremely important for the two-loop contributions. The two-loop calculation is the crucial test for the consistency of the fermion–torsion e4ective theory. The point is that there are no one-loop diagrams which include any non-symmetric vertices and can contribute to the dangerous longitudinal term in the torsion action. However, there are two-loop diagrams contributing to the propagator of the axial vector, S . The question is whether there are longitudinal divergences (9 S  )2 -type at the two-loop level. Then, it is reasonable to start from the diagrams which can exhibit 1= j2 -divergences. The leading 1= j2 -two-loop divergences of the mass operator for the axial vector S come from two distinct types of diagrams: the ones with the ( V )2 -vertex and the ones without this vertex. As we shall see, the most dangerous diagrams are those with four-fermion interaction. As we have seen above, this kind of interaction is a remarkable feature of the axial vector theory, which is absent in a massive vector theory. Now, we shall calculate divergent 1= j2 -contributions from two diagrams with the ( V )2 -vertex, using expansion (5.19); in Appendix B of [22] this calculation has been checked using Feynman parameters. Consider the *rst diagram of Fig. 4. It can be expressed, after making some commutations of the -matrices, as 1 E = − "2 tr {I · I } ;

(5.37)

where ∼ m2 =M 4 is the coupling of the four-fermion vertex, the trace is taken over the Dirac spinor space and  d n p p= − m p= − q= − m I (p) =  5 : (5.38) (2=)n p2 − m2 (p − q)2 − m2 Following [18], we can perform the expansion   ∞ 2 l 1 1 l −2p · q + q = (−1) : (p − q)2 − m2 p2 − m2 p2 − m2

(5.39)

l=0

Now, as in the previous section, one can omit the powers of q higher than 2. Besides, when performing the integrations, we trace just the divergent parts, thus arriving (using the integrals

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Fig. 4. Two-loop “dangerous” diagrams, which can contribute to the (9 S  )2 -counterterm. The *rst two give non-zero contributions and the second two cannot cancel them.

from [101]) at the expressions:   i 1 2 1 2   − q  − 2m  −    q q + mq + · · · ; I = j 6 6 where the dots stand for the *nite and higher-derivative divergent terms. Substituting this into (5.37), we obtain the leading divergences of the diagram:   28

"2 16 1; div E = − 2 +16m4 " + m2 q q − m2 q2 " + · · · : (5.40) j 3 3 This result shows that the construction of the *rst diagram contains an 1= j2 -longitudinal counterterm. 2 , The contribution of the second two-loop diagram of Fig. 4 to the polarization operator, E is written, after certain transformations, in the following way: 2 E = − "2 tr {I · J } ;

where



I = and



J=

d np

(2=)n d np

(2=)n

p= − m p= − q= + m p= − m   2 2 2 2 2 p − m (p − q) − m p − m2

k= − m : k 2 − m2

It proves useful to introduce the following de*nitions: A6 =     6 ; B = −q6   6   + m(     −     −     ) ;

(5.41) (5.42)

(5.43)

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C = m2 (    −    −    ) + mq (     +     ) ; D = −m2 q    + m3   : Then, the *rst integral can be written as  d np A6 · ppp6 + B · pp + C p + D I = : (5.44) (2=)n (p2 − m2 )2 ((p − q)2 − m2 ) Using expansion (5.39), and disregarding higher powers of q, as well as odd powers of p in the numerator of the resulting integral, one obtains, after using the integrals of [101]:   1  i 1    6 I = + ··· ; B  + (A 6 + A 6 + A 6 )q j 4 12 which gives, after some algebra,   1 1 i 2 6 I = m   + 3m" − 6 q " +  q +  q + · · · : (5.45) j 3 3 3 The divergent contribution to J is i J = − m3 + · · · : j Now, the calculation of (5.41) is straightforward:

"2 2;div = 2 8m4 " + · · · : (5.46) E j As we see, this diagram does not contribute to the kinetic counterterm (with accuracy of the higher-derivative terms), and hence the cancellation of the contributions to the longitudinal 1 do not take place. counterterm coming from E One has to note that other two-loop diagrams do not include the ( V )2 -vertex. Thus, even 1 should if those diagrams contribute to the longitudinal counterterm, the cancellation with E require some special *ne-tuning between and ". In fact, one can prove, without explicit calculation, that the remaining two-loop diagrams of Fig. 4 do not contribute to the longitudinal 1= j2 -pole. In order to see this, let us note that the leading (in our case 1= j2 ) divergence may be obtained by consequent substitution of the contributions from the subdiagrams by their local divergent components. Since the local counterterms produced by the subdiagrams of the last two graphs of Fig. 4 are minus the one-loop expression (5.32), the corresponding divergent vertices are 1= j factor classical vertices. Hence, in the leading 1= j2 -divergences of the last two diagrams of Fig. 4, one meets again the same expressions as in (5.32). The result of our consideration is, therefore, the non-cancellation of the 1= j2 -longitudinal divergence (5.40). This means that theory (5.26), without additional restrictions on the torsion mass, like mM , is inconsistent at the quantum level. 5.7. Interpretation of the results: do we have a chance to meet propagating torsion? Obviously, we have found a very pessimistic answer about the possibility of a propagating torsion. Torsion is not only helpless in constructing the renormalizable quantum gravity, as people thought [132,161], but it cannot be renormalizable by itself. Moreover, even if we give up

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the requirement of power counting renormalizability and turn to the e4ective approach, the theory remains inconsistent. In this situation one can try the following: (i) Invent, if possible, some restriction on the parameters of the theory, for which the problems disappear. (ii) Analyze the approach from the very beginning in order to look for the possible holes in the analysis. Let us start with the *rst option, and postpone the second one for the next section. The problems with the non-renormalizability and with the violation of unitarity become weakened if one input severe restrictions on the torsion mass, which has to be much greater than the mass of the heaviest fermion (say, t-quark, with a mass about 175 GeV), and continue to use an e4ective quantum *eld theory approach, investigating the low-energy observables only. This approach implies the existence of a fundamental theory which is valid at higher energies. Hence, in order to have a propagating torsion, one has to satisfy a double inequality: mfermion Mtorsion Mfundamental :

(5.47)

Usually, the fundamental scale is associated with the Planck mass, MP ≈ 1019 GeV. Indeed, the identi*cation of the torsion mass with the Planck scale means, for Mtorsion ≈ MP , that at the low (much less then MP ) energies, when the kinetic and other terms are negligible, we come back to the Einstein–Cartan theory (2.18). As we have readily noticed, in this theory torsion is not propagating, but it can mediate contact interactions. These interactions are too weak for the high-energy experiments described in Section 5.4, but maybe, in future, they can be detected in very precise experiments like the ones in the atomic systems. It is important to remark that, in principle, the e4ective quantum *eld theory approach may be used only at the energies essentially smaller than the typical mass scale of the fundamental theory. If the mass of torsion is comparable to the fundamental scale MP , all the torsion degrees of freedom should be described directly in the framework of the fundamental theory. For instance, in the low-energy e4ective actions of the available versions of string theory torsion enters with the mass 1= , which is conventionally taken to be of the Planck order. But, other degrees of freedom associated to torsion also have a mass of the same order. Thus, it is unclear why one can take only this “static” mode and neglect in*nite amount of others with the same huge mass. Later on, in Section 6, we shall discuss torsion coming from the string theory, and see that the standard approach to string forbids propagating torsion at all. Then, the Einstein–Cartan theory (2.18) becomes some kind of universal torsion theory for the low-energy domain. On the other hand, relation (5.47) still leaves a huge gap in the energy spectrum, which is not completely covered by the present theoretical consideration. In other words, (5.47) might be inconsistent with the string theory, but it does not contradict the e4ective approach. Of course, this gap cannot be closed by any experiment, because the mass of torsion is too big. Even the restrictions coming from the contact experiments [18] achieve only the region M ¡ 3 TeV. And that is not really enough to satisfy (5.47) for all the fermions of the Standard Model. It is clear that the existence of fermions with masses many orders of magnitude larger than mt (like the ones which are expected in many GUTs or SSM) can close the gap on the particle spectrum and “forbid” propagating torsion.

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5.8. What is the di=erence with metric? The situation with torsion is similar to the one with quantum gravity. In both cases, there is a con9ict between renormalizability and unitarity. In the case of quantum gravity, there are models which are unitary (General Relativity and its supersymmetric generalizations) but non-renormalizable, and other models, with higher derivatives, which are renormalizable but not unitary. For the case of torsion, there are models which are unitary at the tree level (take, for instance, all versions described in [162], or simply our action (5.6)), but non-renormalizable. At the same time, it is not diLcult to formulate renormalizable theory of torsion. For that one has to take, say, the general metric–torsion action of [48], which will provide renormalizable metric–torsion theory. The same action of [48], but with the 9at metric g = " will give a renormalizable theory of the torsion alone. However, such a theory will not be unitary, because both transverse and longitudinal components of the axial vector S will propagate. In some sense, this analogy is natural, because both metric and torsion are geometric characteristics of the space–time manifold rather than usual *elds. Therefore, one of the options is to give up the quantization of these two *elds and consider them only as a classical background. This option cures all the problems at the same time, and leaves one a great freedom in choosing the model for metric, torsion and other gravity components: there are no constraints imposed by quantum theory anymore. Indeed, there are very small number of the shortcomings in this point of view [163]. The most important of them is the quantum-mechanical inconsistency of the systems composed by quantum and classical constituents (see, for example, [185]). If one does not accept this option, it is only possible to consider both metric and torsion as e4ective low-energy interactions resulting from a more fundamental theory like string. There is, however, a great di4erence between metric and torsion. The e4ective *eld theory permits the long-distance propagation of the metric waves, because metric has massless degrees of freedom. Indeed, there may be other degrees of freedom, with the mass of the Planck order, which are non visible at low energies. But, the massless degrees of freedom “work” at very long distances, and provide the long-range gravitational force. Then, the study of an e4ective quantum *eld theory for the metric does not meet major diLculties [68,187]. In case of torsion, the massless degrees of freedom are forbidden, because symmetry (2.37) is violated by the spinor mass. It might happen, that some new symmetries will be discovered, which make the consistent quantum theory of the propagating torsion possible. However, in the framework of the well-established results, the remaining possibilities are that torsion does not exist as an independent *eld, or it has a huge mass, or it is a purely classical *eld which should not be quantized. Indeed, in the *rst two cases there are no chances of detecting torsion experimentally. 6. Alternative approaches: induced torsion As we have seen in the previous section, the formulation of the theory of propagating torsion meets serious diLculties. In this situation it is natural to remind that some analog of the space– time torsion is generated in string theory, and then try to check whether this is consistent with the situation in the e4ective *eld theory. On the other hand, we know that usual metric gravity

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can be induced not only in string theory, but also through the quantum e4ects of the matter *elds (see, for example, [2] for a review and further references). This short section is devoted to a brief description of these two approaches: torsion induced in string theory and torsion induced by the quantum e4ects of matter *elds. 6.1. Is that torsion induced in string theory? The covariant (super)string action has a geometric interpretation as a two-dimensional nonlinear sigma model. Then, the completeness of the spectrum (it can be seen as the requirement of the correspondence between string and the sigma-model) requires an additional Wess–Zumino– Witten term to be included. The action of a bosonic string has the form
  1 jab 1 ab 2    b (X )9a X  9b X  Sstr = d z |h| h g (X ) 9 X 9 X +  a b 2  |h|  +4(X )(2) R + T (X )

:

(6.1)

Here ;  = 1; 2; : : : ; D; and a; b = 1; 2: g (X ) is the background metric, b (X ) is the antisymmetric tensor background *eld, 4(X ) is the dilaton *eld, T (X ) is the tachyon background *eld, which we do not consider in what follows. 19 (2) R is the two-dimensional curvature, X  = X  (z) are string coordinates. The parameter  may be considered as the parameter of the string loop expansion. Since the dimension of  is the inverse of the mass, usually  is associated with the inverse square of the Planck mass 1=MP2 . Indeed, this choice is strongly motivated by the common belief that the (super)string theory is the theory which should unify all the interactions including gravitational, in one quantum theory. The X  are the dynamical *elds de*ned on the world sheet. At the same time they are coordinates of the D-dimensional space with the metric g . Initially, the geometry of this D-dimensional space is not known, it is generated by the quantum e4ects of the two-dimensional theory. In the sigma-model approach (see, for example, [94,178] for the general review and [114] for the technical introduction and complete list of results of the higher-loop calculations of the string e4ective action) the e4ective D-dimensional action of the background *elds g (X ); b (X ); 4(X ) appears as a result of the imposition of the Weyl invariance principle at each order of the perturbative expansion of the two-dimensional e4ective action. For example, the dilaton term in (6.1) is not Weyl invariant, but it has an extra factor of  . Therefore, after integration over the *elds X  we *nd that this classical term contributes to the trace of the energy–momentum tensor and that this contribution is of the *rst order in  . Other similar contribution comes from the one-loop e4ects, including the renormalization of the composite operators in the T expression [78,42,146,114]. Requesting the cancellation of two contributions we get a set of the one-loop e4ective equations: conditions on the background *elds g (X ); b (X ); 4(X ). The corresponding action is interpreted as a low-energy e4ective action of string. 19

Tachyon does not pose a problem for the superstring.

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The one-loop e4ective equations do not directly depend on the antisymmetric *eld b (X ), but rather on the stress tensor H = 9[ b ] :

(6.2)

The covariant calculation includes the expansion in Riemann normal coordinates [105,106,114]), and the coeLcients of the expansion for b depend only on H and its derivatives, but not directly on the b components. After all, the geometry of the D-dimensional space is characterized by three kinds of *elds: metric g , dilaton 4 and completely antisymmetric *eld H , which satis*es the constraint  9 H = 0 coming from (6.2). In principle, the *eld H can be interpreted as a space–time torsion. In order to understand better the correspondence with our previous treatment of torsion, let us parametrize the H *eld by the axial vector S  =  H . Then the constraint  9 H gives 9 S  = 0, and we arrive at the conclusion that the string-induced torsion has only transverse axial vector component—exactly the same answer which is dictated by the e4ective *eld theory approach (see the beginning of Section 5.3). In fact, on the way to this interpretation H ∼ T one has to check only one thing: how this *eld interacts to the fermions. Indeed, we are interested in the answer after the compacti*cation into the four-dimensional space–time. However, compacti*cation cannot change the general form of the interaction. If we suppose that the fermions interact to the H *eld through the axial current, then H can be identi*ed as the space–time torsion T . One can note that this is the form of the interaction which looks quite natural from the point of view of dimension and covariance. This is not the unique possible choice, indeed. If we consider the tensor b as an independent *eld (see e.g. [172]), the situation becomes quite di4erent. In four dimensions, the antisymmetric *eld is dual to the axion-speci*c kind of scalar *eld (see e.g. [34]). In a conventional supergravity theory, which might be taken as the low-energy limit of the superstring, the axion *eld is present and the formulation of its quantum theory does not pose any problem (in a sharp contrast to torsion!). Of course, the interaction of axion with fermions is quite di4erent from the one of (2.34), so in this case the antisymmetric *eld cannot be associated to the space–time torsion. On the other hand, the renormalizable and local supergravity theory is not the unique possible choice. There are some arguments [119] that the non-local e4ects can play a very important role in the string e4ective actions, and one of the popular phenomenological models for these e4ects are related to the b *eld, which we already mentioned by the end of Section 4.6. The b *eld is a close analog of torsion, and therefore the available predictions of the string theory are not completely de*nite. Let us now discuss the form of the string-induced action for torsion. At the one-loop level, this is some kind of the metric–dilaton–torsion action considered in Section 2.4. As an example of the string-induced action we reproduce the one for the bosonic string     1 D −24 2

 Se4 ∼ d x |g|e −R − 2D 4 + H  H ; (6.3) 3 where one can put 4 = const: for the analysis of the torsion dynamics. Since the torsion square enters the expression in the linear combination with the Ricci scalar, the torsion mass equals to

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the square of the Planck mass. This is, again, in a perfect agreement with our results about the e4ective approach to the propagating torsion. The expressions similar to (6.3) show up for all known versions of string theory. For superstring the one-loop result is exactly the same [56]. For the heterotic string the torsion term is the same as in (6.3), the di4erence concerns only the dilaton. No version of string is known, which would produce the zero torsion mass in the low-energy e4ective action. Perhaps, this is more than a simple coincidence, for the massless torsion should lead to problems in the e4ective framework. The propagation of torsion can be caused by the higher loop string corrections to the low-energy e4ective action. It was notices by Zweibach in 1985, that the de*nition of the higher order corrections includes an ambiguity related to the choice of the background *elds [194,62,177,111]. In particular, one can perform such a reparametrization, that the propagators of all three *elds: metric, torsion and dilaton do not depend on higher derivatives. For the theory with torsion the corresponding analysis has been done in Ref. [111]. The general form of the transformation is g → g + g ;

b → b + b ;

4 → 4 + 4 ;

(6.4)

where 2 g = x1 R + x2 Rg + x3 H + x4 ∇ ∇ 4 + x5 ∇ 4∇ 4

+x6 H 2 g + x7 g ∇2 4 + x8 (∇4)2 + · · · ;

b = x9 ∇ H· + x10 H· ∇ 4 + · · · ;

4 = x11 R + x12 H 2 + x13 ∇2 4 + x14 (∇4)2 + · · · :

(6.5)

In these transformations, the coeLcients x1 ; x2 ; : : : ; x14 ; : : : are chosen in such a way that the high-derivative terms contributing to the propagators disappear. The transformation can be continued (in Ref. [111] the proof is presented until the third order in  , but the statement is likely to hold at any order, see for example, [9]). The motivation to make a transformation (6.4) is to construct the low-energy theory of gravity free of the high-derivative ghosts. At the same time, one can make several simple observations: (i) At higher orders the transformation is not uniquely de*ned, at least if we request just a ghost-free propagator. For instance, let us take a R R R -term. This term can be easily removed by some transformation similar to (6.4), despite it is innocent—it does not contribute to the propagator of the metric perturbations. Usually, all such terms are removed in order to simplify the practical calculations, but the validity of this operation is not obvious. (ii) If we are not going to quantize the e4ective theory, the necessity of whole (6.4) is not clear. It is well known, that in many cases the physically important classical solutions are due to higher derivatives (for example, in9ation may be achieved in this way [174,127]). (iii) If we are going to quantize the e4ective theory (see corresponding examples in [68] for metric and [18,22] for torsion), one has to repeat reparametrization (6.4) after calculating

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any loop in the e4ective quantum *eld theory. This operation becomes a necessary component of the whole e4ective approach, and it substitutes the standard consideration of Ref. [186,68]. Of course, since the e4ective approach cares about lower derivatives only, the physical consequences of two de*nitions must be the same. Indeed, transformation (6.4) kills the torsion kinetic term, so that the torsion action consists of the mass and interaction terms. In the second loop, for the bosonic and heterotic strings, the interaction terms of the H 4 and RH 2 -types emerge (see, for example, [114]). These terms cannot be removed by (6.4). For the superstring, the two- and three-loop contributions cancel, and at the four-loop level the torsion terms can be “hidden” inside the R4::: -type terms. It is unlikely that this can be done at higher loops but, since we are interested in the torsion dynamics, it has no real importance. The conclusion is that string torsion does not propagate, and that in this respect the e4ective low-energy action of string agrees with the results of the e4ective quantum *eld theory. 6.2. Gravity with torsion induced by quantum e=ects of matter Despite induced torsion looks as an interesting possibility, it seems there are not many publications on this issue (except [58,32,102] devoted to the anomaly-induced action with torsion). One can note, however, that the action of gravity with torsion may be induced in the very same manner as the action for gravity without torsion. Let us repeat the consideration typical for the standard approach (see, for example, [2]), but with torsion. One has to suppose that there are gravity *elds: metric and torsion, which couple to quantized matter *elds. For instance, it can be the non-minimal interaction described in Section 2.3. The non-minimal interaction is vitally important, for otherwise the theory is non-renormalizable. The geometry of the space–time is described by some vacuum action, depending on the metric and torsion, but this action is, initially, not de*ned completely. The exact sense of the last statement  ] is a will be explained in what follows. Now, the action of gravity with torsion Sgrav [g ; T· result of the quantum e4ects of matter on an arbitrary curved background. This means the following representation for this action (cf. to (3.2)):   iSgrav [g ;T· ] e = D4 eiSmatter [4;g ;T· ] : (6.6) Here, 4 means the whole set of non-gravitational *elds: quarks, gluons, leptons, scalar and  ] vector bosons and their GUT analogs, gauge ghosts, etc. Consequently, Smatter =Smatter [4; g ; T· is the action of the *elds 4. After these non-gravitational *elds are integrated out, the remaining action can be interpreted as an action of gravity and the solutions of the dynamical equations following from this action will de*ne the space–time geometry. Then, the geometry depends on the quantum e4ects of matter, including spontaneous symmetry breaking which occurs at di4erent scales, phase transitions, loop corrections, non-perturbative e4ects and scale dependence governed by the renormalization group. An important observation is in order. As we have already indicated in Section 3, the action must also include a vacuum part. For the massive theory one has to introduce not only the R2 ; R · T 2 ; R∇T; T 2 ∇T and T 4 -terms, but also R and T::2 -type terms, similar to the ones in the Einstein–Cartan action. Then, the lower derivative action of gravity cannot be completely

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induced. The induced part will always sum up with the initial vacuum part which is subject of an independent renormalization. Of course, for the massless theory, one can choose the R2:: -type vacuum action, and then the low-energy term will be completely induced through the dimensional transmutation mechanism (see Section 3.5). Let us now comment on the induced part of the Sgrav . One can introduce various reasonable approximations and evaluate action (6.6) in the corresponding framework. We shall list the following approaches in deriving the one-loop e4ective vacuum action: (i) The anomaly-induced action (see [102] and Section 3.6 of this report for the best available result in gravity with torsion). In some known cases this looks as the best approximation [174,32,73,14]. Indeed, in all cases, except [32,102], the theory without torsion has been considered. Usually, the anomaly-induced action has been treated as a quantum correction to the Einstein action. The advantage of the anomaly-induced action is that it includes the non-local pieces. This is especially important for the cosmological applications (see, e.g. solution (3.79) as an example of such application). (ii) The induced action of gravity with torsion which emerges as a result of the phase transition induced by torsion or curvature (see Section 2.5 for the case with torsion). We remark that the derivation of the e4ective action can be continued beyond the e4ective potential, so that the next terms in the derivative expansion could be taken into account. For gravity without torsion this has been done in [41] (see also [34]), and it is technically possible to realize similar calculus for the theory with torsion. Then, after the phase transition, in the critical point one meets not only the non-minimal version (3.66) of the Einstein–Cartan theory, but also the next order corrections, including high derivative terms, terms of higher order in torsion and curvature, the ones described in [48], etc. The common property of all these terms is locality. The expansion parameter will be the inverse square of the Planck mass, therefore these terms will be negligible at low energies. (iii) Alternatively, one can simply take the contributions of the free *elds and take into account the higher orders of the Schwinger–DeWitt expansion (3.3). Starting from tr aˆ3 (x; x), all these terms will be *nite and they are indeed contributions to Sgrav . In general, these terms are not very much di4erent from the ones described in (ii). There is a possibility to sum the Schwinger–DeWitt expansion, but this has been achieved [11] only for the especially simple backgrounds without torsion. (iv) The non-local terms can be taken into account using the generalized Schwinger–DeWitt technique. Such a calculations have been performed by Vilkovisky et al. (see [180] for the review and further references). This method seems more appropriate for the low-energy region, for the non-localities are not directly related to the high-energy behavior of the massive *elds (as it is in the (i) case). The calculations for torsion gravity has not been performed yet. It may happen, that the non-local e4ects are relevant for torsion (as they are, perhaps, for gravity), and then the e4ective *eld theory approach does not give full information. In the content of string theory similar possibility led to the consideration of the (already mentioned) b -*eld, which strongly resembles torsion. (v) Another way of deriving the non-local piece is to take the renormalization-group improved action [67]. The corresponding corrections are obtained through the replacement of 1= j, in the expression for vacuum divergences, by the ln( =2 ). Indeed, these terms can be relevant only in the high-energy region.

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All mentioned possibilities concern the de*nition of the torsion action, but one can make a stronger question about inducing the torsion itself. In principle, the axial current and the interactions similar to the torsion–fermion one, can be induced in a di4erent ways. One can mention, in this respect, the old works on the anomalous magnetic *eld [137], recent works on the anomalous e4ects of the spinning 9uid [181] and the torsion-like contact interactions coming from extra dimensions [47]. The phenomenological bounds on the contact interaction derived in the last work are similar to the ones of [18]. 7. Conclusions We have considered various aspects of the space–time torsion. There is no de*nite indications, from experiments, whether torsion exists or not, but it is remarkable that purely theoretical studies can put severe limits on this hypothetical *eld which would be an important geometrical characteristic of the space–time. The main point is that the consistency of the propagating torsion is extremely restricted. Indeed, there is no problem in writing the classical action for torsion, and this can be done in many ways. Also, the quantum *eld theory on classical torsion background can be successfully formulated, and we presented many results in this area of research. However, without the dynamical theory for the torsion itself the description of this phenomena is incomplete, and one can only draw phenomenological upper bound for the background torsion from known experiments. The serious problems show up when one demands the consistency of the propagating torsion theory at the quantum level [18,22]. Then, we have found that there is no theory of torsion which could be simultaneously unitary and renormalizable. Moreover, there is no theory which can be consistent even as an e4ective theory, when we give up the requirement of the power counting renormalizability. The only one possibility is to suppose that torsion has a huge mass— much greater than the mass of the heaviest fermion. Then, if we assume that torsion couples to all fermions, its mass has to greatly exceed the TeV level. Hence, torsion cannot propagate long distances and can only produce contact spin–spin interactions. The necessity of a huge torsion mass can explain the weakness of torsion and diLculties in its observation. Up to our knowledge, this is the *rst example of rigid restrictions on the geometry of the space–time, derived from the quantum *eld theory principles. In this review, we avoided to discuss some, technically obvious, possibilities—like a spontaneous symmetry breaking which would provide, simultaneously, the mass to all the fermions and to the torsion. The reason is that such an approach would require torsion to be treated as a matter *eld, and in particular to be related to some internal symmetry group—like SU (2), for example. Besides possible problems with anomalies, this would mean that we do not consider, anymore, torsion as part of gravity, but instead take it as a matter *eld. And that certainly goes beyond the scope of the present review, devoted to the *eld-theoretical investigation of the space–time torsion. Thus, the only solution is to take a string- or matter-induced torsion. As we have seen in the previous section, both approaches do not give any de*nite information. First of all, the torsion induced in string theory has a mass of the Planck order. Thus, it cannot be really treated as an independent *eld, but rather as a string degree of freedom, integrated into the complete

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string spectrum. Furthermore, if we take supergravity as a string e4ective theory, there is no torsion, but rather there is an axion-scalar *eld with quite di4erent interaction to fermions. Equally, the investigation of the matter-induced torsion o4ers some possibilities concerning torsion action, but in all available cases the torsion mass is of the Planck order. Anyhow, the possibility of induced torsion remains, and it is still interesting to look for more precise bounds on a background torsion and to develop the formal aspects of quantum *eld theory on curved background with torsion.

Acknowledgements First of all, I would like to express my great thanks to all colleagues with whom I collaborated in the study of torsion and especially to I.L. Buchbinder, A.S. Belyaev, D.M. Gitman, J.A. Helayel-Neto, L.H. Ryder and G. de Berredo Peixoto. I am very grateful to M. Asorey, T. Kinoshita, I.B. Khriplovich, I.V. Tyutin, L. Garcia de Andrade and S.V. Ketov for discussions. I wish to thank J.A. Helayel-Neto and G. de Oliveira Neto for critical reading of the manuscript. Also I am indebted to all those participants of my seminars about torsion, who asked questions. I am grateful to the CNPq (Brazil) for permanent support of my work, and to the RFFI (Russia) for the support of the group of theoretical physics at Tomsk Pedagogical University through the project 99-02-16617.

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I.L. Shapiro / Physics Reports 357 (2002) 113–213 Dirac particles in general relativity with torsion and electromagnetism. III. Matter production in a model of torsion, 10 (1979) 647. L.H. Ryder, I.L. Shapiro, On the interaction of massive spinor particles with external electromagnetic and torsion *elds, Phys. Lett. A 247 (1998) 21. V. de Sabbata, P.I. Pronin, C. Siveram, Neutron interferometry in gravitational *eld with torsion, Int. J. Theor. Phys. 30 (1991) 1671. V. de Sabbata, C. Siveram, Torsion and in9ation, Astron. Space Sci. 176 (1991) 141. E. Sezgin, P. van Nieuwenhuizen, New ghost-free gravity Lagrangian with propagating torsion, Phys. Rev. D 21 (1981) 3269. E. Sezgin, Class of ghost-free gravity Lagrangians with massive or massless propagating torsion, Phys. Rev. D 24 (1981) 1677. I.L. Shapiro, in: P. Fayet, J. Gates, S. Duplij (Eds.), Quantum Gravity: Traditional Approach, Review article for the “Concise Encyclopedia on SUPERSYMMETRY”. Kluwer Academic Publishers, Dordrecht, to be published. I.L. Shapiro, On the interaction of torsion with matter *elds: NJL model, Modern Phys. Lett. A 9 (1994) 729. I.L. Shapiro, Renormalization and renormalization group in the models of quantum gravity, Ph.D. Dissertation, Tomsk State University, Russia, 1985, 1–158. I.L. Shapiro, G. Cognola, Back reaction of vacuum and the renormalization group 9ow from the conformal *xed point, Class. Quant. Grav. 15 (1998) 3411. I.L. Shapiro, H. Takata, Conformal transformation in gravity, Phys. Lett. B 361 (1996) 31. P. Singh, L.H. Ryder, Einstein–Cartan–Dirac theory in the low-energy limit, Class. Quant. Grav. 14 (1997) 3513. R. Skinner, D. Grigorash, Generalized Einstein–Cartan *eld equations, Phys. Rev. D 14 (1976) 3314. A.A. Slavnov, Invariant regularization of non-linear chiral theories, Nucl. Phys. B 31 (1971) 301. H.H. Soleng, I.O. Eeg, Skewons and Gravitons, Acta Phys. Pol. 23 (1992) 87. W.P.A. Souder et al., Measurement of parity violation in the elastic scattering of polarized electrons from 12 C, Phys. Rev. Lett. 65 (1990) 694. A.A. Starobinski, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91 (1980) 99. K.S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16 (1977) 953. A.A. Tseytlin, On the Poincare and de Sitter gauge theories of gravity with propagating torsion, Phys. Rev. D 26 (1982) 3327. A.A. Tseytlin, Ambiguity in the e4ective action in string theories, Phys. Lett. B 176 (1986) 92. A.A. Tseytlin, Sigma model approach to string theory, Int. J. Mod. Phys. A 4 (1989) 1257. R. Utiyama, Invariant theoretical interpretation of interaction, Phys. Rev. 101 (1956) 1597. G.A. Vilkovisky, E4ective action in quantum gravity, Class. Quant. Grav. 9 (1992) 895. G.E. Volovik, On induced CPT-odd Chern–Simons terms in the 3 + 1 e4ective action, JETP Lett. 70 (1999) 1. B.L. Voronov, P.M. Lavrov, I.V. Tyutin, Canonical transformations and the gauge dependence in general gauge theories, Yad. Fiz. (Sov. J. Nucl. Phys.) 36 (1982) 498; J. Gomis, S. Weinberg, Are nonrenormalizable gauge theories renormalizable? Nucl. Phys. B 469 (1996) 473. B.L. Voronov, I.V. Tyutin, Models of asymptotically free massive *elds, Yad. Fiz. (Sov. J. Nucl. Phys.) 23 (1976) 664. B.L. Voronov, I.V. Tyutin, On renormalization of R2 gravitation, Yad. Fiz. (Sov. J. Nucl. Phys.) 39 (1984) 998. R.M. Wald, General Relativity, University of Chicago Press, Chicago, 1984. S. Weinberg, in: S.W. Hawking, W. Israel (Eds.), General Relativity, Cambridge University Press, Cambridge, 1979. S. Weinberg, The Quantum Theory of Fields: Foundations, Cambridge Univ. Press, Cambridge, 1995. J. Wess, B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95. C. Wetterich, E4ective nonlocal Euclidean gravity, Gen. Rel. Grav. 30 (1998) 159.

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Physics Reports 357 (2002) 215–458

Electron–molecule scattering cross-sections. I. Experimental techniques and data for diatomic molecules Michael J. Brungera , Stephen J. Buckmanb; ∗ a

Physics Department, Flinders University of South Australia, GPO Box 2100, Adelaide, 5001, South Australia b Atomic and Molecular Physics Laboratories, Australian National University, Canberra 0200, Australia Received March 2001; editor: J: Eichler

Contents 1. Introduction 2. Experimental techniques—recent developments 2.1. Molecular beam sources 2.2. Single collision experiments 2.3. Swarm experiments 2.4. Sources of excited species 2.5. Magnetic rotation of the scattering geometry 2.6. Normalisation techniques 2.7. Data extraction techniques: spectral deconvolution

217 219 219 222 227 228 232 232 242

3. Experimental data 3.1. H2 3.2. N2 3.3. O2 3.4. The halogens 3.5. Carbon monoxide (CO) 3.6. Nitric oxide (NO) 3.7. Hydrogen halides 3.8. Cross-section trends amongst diatomics 4. Summary and suggested measurements Acknowledgements References

∗

Corresponding author. Tel.: +61-2-6125-2402; fax: +61-2-6125-2452. E-mail address: [email protected] (S.J. Buckman). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter  PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 3 2 - 1

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Abstract Experimental collision cross-sections for electron-molecule scattering processes at low to intermediate energies (meV–100 eV) are compiled and critically reviewed. Recent advances in experimental techniques are summarised. Of principle interest are di@erential, total and momentum transfer cross-sections for elastic scattering and rotational, vibrational and electronic excitation processes. Wherever possible, available theoretical cross-sections are also compared and discussed. Resonance e@ects, whilst not treated explicitly, are discussed in the context of the enhancement they produce in the various scattering cross-sections. c 2002 Elsevier Science B.V. All rights reserved. Scattering from excited molecules is also considered.
PACS: 34.80.Be; 34.80.Gj Keywords: Electron collisions; Diatomic molecules; Cross sections; Experimental techniques

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1. Introduction Some 18 years ago Trajmar et al. [1] published a compilation and critical assessment of available cross-sections for electron-molecule scattering. This document has become an invaluable addition to the Fles of all workers in this, and associated Felds as it provides a comprehensive coverage of all molecules for which quantitative scattering information existed and, in particular, it includes extensive tabulations of cross-section information. This article came at a time when activity in the Feld had increased dramatically and, as this level of activity has continued throughout the subsequent years with an increase in both the type and sophistication of electron collision experiments, it is perhaps appropriate that this compilation be updated. In addition there has been an increasing interest from other, allied, Felds in the products of electron-initiated collisions in a range of gases which have technological importance, and in the magnitude of the cross-sections for their production. Applied areas as divergent as plasma deposition and etching of semiconductors, gaseous high voltage switches and electrostatic precipitators for the processing of atmospheric pollutants, have all required a wide variety of cross-section information for numerical modelling studies. The active constituents in these applications are typically polyatomic molecules and as such there has been a marked increase in both experimental and theoretical studies of these complicated targets. Thus, the intention of this article is to attempt to update the data compilation of Trajmar et al. to account for measurements over the elapsed years. Once again, the emphasis will be on both di@erential and integral cross-section measurements for elastic scattering, rotational, vibrational and electronic excitation. We shall also give a brief discussion of experimental techniques, with particular emphasis on new developments in the measurement of absolute scattering cross-sections since the paper by Trajmar and colleagues. This review gave a comprehensive account of experimental techniques and practices, many of which still form the backbone of experimental activities today. Whilst temporary negative ion resonances are often of great importance in low-energy electron-molecule scattering, the treatment of resonance dynamics is beyond the scope of this article and we will only discuss them in the context of cross-section enhancement. Similarly, whilst we will attempt to discuss in a critical fashion the comparison between experimental and theoretical cross-sections, it is also beyond both the scope of this article, and the expertise of the authors, to o@er a detailed critique of the many theoretical approaches. A number of other review articles in the literature overlap with the broad aims of the present work and we strongly recommend that these be consulted in conjunction with this paper. It is not possible to mention all of them here but recent articles by Trajmar and McConkey [2], Fillipelli et al. [3], Crompton [4], Schneider [5], Itikawa [6] and Zecca et al. [7] provide an excellent summary of experimental and theoretical techniques used in the determination of electron scattering cross-sections. Also, an important guide to bibliographies and review articles in the broad Feld of collision physics has been compiled by McDaniel and Mansky [8]. This article will concentrate on those measurements of grand total scattering, di@erential (in angle), integral and momentum transfer cross-sections for electron-molecule scattering. The di@erential cross-section is typically measured in a beam–beam or beam–cell conFguration for well-deFned values of both the incident and the scattered electron energy, Ei and Es , for an

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interaction of the form e− (Ei ) + ABi → e− (Es ) + ABj ;

(1.1)

where ABi and ABj represent the initial and Fnal states of some generic diatomic or polyatomic molecule. If the energy resolution of the incident beam and the nature of the molecular target is such that the initial and Fnal states are well-deFned, the di@erential cross-section at a scattering angle is given by kf d

(1.2) (E0 ; )i→f = |fi→f (E0 ; )|2 ; ki d where ki and kf are the initial and Fnal momenta given by (in atomic units)
Ei; f ki; f = ; (1.3) ER fi→f is the scattering amplitude for the interaction and ER is the Rydberg energy. In general, state-of-the-art energy resolution of modern crossed beam spectrometers (≈10 meV) is only suJcient to resolve rotational excitation in H2 . Also, most quantitative experiments do not involve spin polarised beams, or targets which have been “dressed” in a single Fne or hyperFne level, let alone a single rotational level, resulting from the (usually) elevated temperature of the beam. Thus, the measured quantities typically involve an average over the possible initial states of the molecular target and a sum over Fnal states. The total cross-section for the excitation process i → f; i→f , is obtained by integration of the di@erential cross-section over all scattering angles   d

i→f = 2 sin d (1.4) 0 d and the Grand Total cross-section QT is the sum of the total cross-sections for all energetically available processes n  QT (E) =

n (E) : (1.5) n

Another commonly measured quantity, particularly from swarm experiments, is the momentum transfer cross-section, m , which, in terms of the di@erential cross-section for any given process i → f, is deFned as   d

m (E) = 2 (1.6) sin (1 − cos ) d : 0 d It is worth noting that in the analysis of swarm experiments using the “two-term” Boltzmann equation, the momentum transfer cross-section which arises is, in fact, not the elastic momentum transfer cross-section but rather the following quantity (which we shall refer to as the “total” momentum transfer cross-section):   k −k

m (E) = mel (E) +

m (E) +

m (E) ; (1.7) k el

m

k

k and −k are the momentum is the elastic momentum transfer cross-section and m where m transfer cross-sections for the kth inelastic and superelastic scattering processes, respectively.

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In the analysis of transport data where the two-term approach is used, the inelastic scattering is assumed to be isotropic and the inelastic momentum transfer cross-sections are equivalent to the total cross-sections for each inelastic event. Thus it is important, when making comparisons with swarm derived cross-sections at the momentum transfer level, that the appropriate cross-sections are compared. This is particularly important when the inelastic scattering cross-sections are signiFcant, when compared to the elastic, as for example in the case of gases with Ramsauer minima, such as methane, or for gases where resonance e@ects are large, such as N2 . The structure of this review is such that the molecular species are treated roughly in order of increasing structural complexity, which also roughly corresponds to a decreasing amount of experimental activity. Unfortunately, we have had to divide the work into two volumes because of the sheer amount of information that is now available in the literature. This Frst volume contains a summary of recent experimental advances and the data for diatomic molecules. Volume II will contain the data for polyatomic molecules. In both cases, where it is possible, we highlight the level of agreement, or otherwise, between experiment and contemporary theory by the use of Fgures and the new experimental data, since the publication of the Trajmar review, is tabulated. In general, we have only included data that has been published in the open literature and we have avoided using digitised curves or graphs wherever possible. There is little doubt that we have omitted or overlooked some published information and for that we apologise in advance and plead ignorance. This article also corresponds loosely, and completely fortuitously, with the retirement of Sandor Trajmar from active research at JPL and in a larger sense from the atomic physics community. It is a testament to his enduring impact on the atomic collisions community that the present article still contains many contemporary references to his work and that of his close colleagues. 2. Experimental techniques---recent developments Since the experimental methods used for absolute cross-section determination were discussed by Trajmar et al. [1] there have been signiFcant reFnements to, and advances in, our understanding of the methodology of low-energy absolute cross-section measurements. Various aspects of these advances have been discussed elsewhere [2,9,10] but it is nonetheless appropriate that an overview be given here. 2.1. Molecular beam sources Many areas of research in atomic, molecular and surface physics require intense, highly collimated beams of atoms and molecules. The wide variety of techniques available for the production of such “beams” include static gas targets, thin walled oriFce sources, simple single-capillary sources with an appropriate length to diameter (aspect) ratio, and the combination of many such miniature tubes in a capillary array source. Of particular relevance to the present review is the use of such molecular beam sources in crossed beam experiments. Crossed beam scattering experiments are a powerful method [2,9] for investigating particle– particle interactions and for determining di@erential cross-sections. In such experiments the

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cross-section is related to the measured scattered intensity by a rather complicated expression involving geometrical and instrumental functions. The cross-section averaged over the instrumental angular and energy resolution can be expressed as the product of the measured intensity and a proportionality function. With an idealised scattering geometry and instrument, this proportionality function is a constant, i.e. it is independent of scattering angle and electron energy. In practice, however, experiments involve extended scattering volumes and non-localised, non-uniform target density and incident beam Pux distributions as a consequence of the need to balance geometrical and signal requirements. These requirements, in turn, lead to an angular dependence [9] being introduced into the proportionality factor. Furthermore, the electron optics, and hence the detection eJciency, are in most cases energy dependent. Thus an angle- and energy-dependent proportionality factor, commonly known as the e@ective path length correction factor, has to be applied to the measured intensities in order to determine these cross-sections. As we shall see later (Section 2.2.3), the determination of this proportionality factor is non-trivial. As a result, a technique that avoids this problem by using the ratio of the scattered intensities for the target and standard gas and the known di@erential cross-section of the standard gas, has been developed. This technique is typically referred to as the relative Pow technique. In the accurate application of the relative Pow technique [2,9] (see Section 2.6.1), careful attention to experimental detail is required to ensure that either the measurements in both gases are carried out under identical conditions or, if this is not possible, the ramiFcations of any di@erences are well understood. The crucial requirement of the technique is that the interaction volume, usually deFned as the overlap of the electron and molecular beams, should be identical for both gases [10]. Consequently, the characterisation of the spatial proFles of molecular beams, formed from both single-capillary tubes and multichannel capillary arrays, at distances from the beam source (1–5 mm) which are typical of those used in collision experiments, is an important consideration. A signiFcant number of studies, both theoretical [11–14] and experimental [11,15 –20] have been carried out on such single and multichannel array sources in an attempt to characterise both the shape and absolute axial number densities of the beams they produce. In general, these studies have shown a reasonable level of agreement between the experimental distributions for both single and multicapillary sources and those calculated from a variety of theoretical approaches, provided the distance from the source is large [18,20]. However, most of the experimental measurements have either been conducted with relatively large (40 cm or greater), or unspeciFed, separations between the exit of the capillary source and the detector. Exceptions to this were the work of Jones et al. [16], Adamson and McGilp [18,19] and Adamson et al. [20], who measured angular distributions for a number of di@erent single capillaries at separations of 5, 6.30 and 12 mm, respectively. Nonetheless, it is clear that there was little or no information concerning the shape of such distributions close to the exit of the source, or how this shape varied as a function of both driving pressure and gas species at the higher driving pressures outside the molecular Pow regime [9], until the recent investigation of Buckman et al. [10]. In this study, spatial proFles of molecular beams formed by both single and multicapillary sources were investigated for He, Ne, Ar, Kr, H2 and N2 . The proFle measurements were undertaken at distances from the source exit and driving pressures which are typical of those used in atomic and molecular collision experiments (1–5 mm and 0.02–10 Torr, respectively). An example of the results obtained in this study is shown in Fig. 1.

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•

Fig. 1. (a) A comparison of the beam proFles obtained with a ( ) capillary array and ( ) single capillary tube at a distance of 1:5 mm from the oriFce exit for a helium beam at a driving pressure of 0:98 Torr; (b) As in (a) but at a distance of 4:5 mm.

A number of quite important conclusions, with respect to measurements of charged particle scattering cross-sections using crossed beam geometries, can be drawn from the work of Buckman et al. [10]. For example, it was apparent that in cases where a spatially well-deFned molecular beam is required, the use of a capillary array source rather than a single capillary is favoured. This was particularly true when the interaction volume for such experiments could not be located very close to the exit of the source (see Fig. 1). The measurements of Buckman et al. showed that when the interaction volume was situated more than ≈ 2 mm from a 1 mm active diameter source, there were clear advantages in using a capillary array. Furthermore, and of particular relevance to this review, they found that there was a signiFcant di@erence in the widths of the beam proFles for di@erent gases, as the driving pressure was increased. In the case of their glass capillary array they found that this di@erence was of an acceptable level (≈5%) whilst the mean free path () of the atoms and molecules was greater than or equal to about twice the diameter (d) of the individual capillaries ( ¿ 2d). However, these di@erences in width were found to increase rapidly with increasing pressure (decreasing mean free path). They observed that this e@ect appeared to be independent of the type of source used to form the beam and that it appeared to be loosely correlated with the mass of the species, helium always producing beams of narrower width. This is an important result as the standard gas

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Fig. 2. The angular variation of atomic beam proFles for He, Ar and Xe atoms as a function of mean free path. The straight line is a model prediction by Khakoo et al.

which is almost universally used to obtain absolute scattering cross-sections, via the relative Pow technique, is helium [9]. Very recently, Khakoo and colleagues [21] have extended the study of Buckman et al. by measuring, for a variety of sources and driving pressures, the angular variation of the beam proFles about the source, thereby obtaining a “full” three-Dimensional picture of the beam proFle. In this work, they found that even for a capillary array source at quite low driving pressures, the shoulders of the proFles were quite broad. In an attempt to understand this behaviour they have developed a model, which includes molecular collisions in a self-consistent method, that starts to add gas into the source. Agreement between the measured data and the results of their model calculations, even in the Clausing Pow regime, was very good with an example of this being given in Fig. 2. 2.2. Single collision experiments 2.2.1. Grand total cross-sections The principles and techniques for total electron scattering cross-section measurements have been well documented in the literature [22–26]. The electron transmission method (with or without time-of-Pight discrimination) is now well established and widely applied and whilst it is arguable that it produces cross-sections which provide the least sensitive test for the validity of the various scattering theories [27], the cross-sections from these measurements are still the most accurate and reliable. Typical error limits for these cross-sections are in the range ±3–5%, for targets that are gases at room temperatures. The cross-sections are deduced by measuring the attenuation of the electron beam passing through the target gas and utilising the Beer–Lambert attenuation law. All that is required, in principle, is the ratio of the initial and transmitted electron beam intensities as a function of target pressure. The experimental problems with this technique are associated with the accurate

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determination of the target pressure, the path length and scattering to the forward direction (which contributes to the transmitted signal). In the latter case, we highlight the particular problem that can occur in the study of some polar systems [1]. Here the molecular dipole moment of the target signiFcantly enhances scattering to forward angles so that the di@erential cross-section is very strongly peaked in the forward direction. Under these circumstances great care must be exercised in performing attenuation measurements. In the case of reactive gases it may also be necessary, for stability concerns, to have the spectrometer in a di@erentially pumped manifold so that the source optics, scattering cell and detection optics are located in independently pumped chambers. All these questions are addressed elsewhere [22–26,28] and so we do not go into any detail here. However, many of the above problems are now quite well understood for targets that are gases at room temperature such that with further improvements it is not inconceivable that the error limits on the total cross-sections may fall to the ±1–3% range or below [2]. On the other hand, for condensable (vapour) targets the pressure measurements are more diJcult and subject to larger errors, which has led to some signiFcant discrepancies amongst the cross-section data available for these targets [29,30]. Consequently, such measurements have been limited to only a few species [2]. 2.2.2. State-speci9c integral cross-sections In measurements where the scattered electrons are detected, integral cross-sections (ICS) are obtained mostly from di@erential cross-sections by integration over all scattering angles. The well-known relationship between the DCS and ICS (Eq. (1.4)) has been discussed in the introduction. Most di@erential cross-section measurements are usually limited in their angular ◦ range to, typically, min 6 measured 6 140 where, depending on the electron beam energy of the ◦ ◦ study, min ≈ 10 for elastic scattering and min ≈ 5 , or better, for inelastic processes. The upper limit on the angular range is most often set by the physical dimensions of the energy selector and energy analyser. Consequently, techniques to extrapolate the di@erential cross-sections to ◦ ◦ 0 and 180 must be employed. We note that it is precisely these un-accessed parts of the integrand which quite often make a signiFcant contribution to the integral and, as such, the extrapolation techniques which are used are important and are dealt with immediately below. There currently appear to be three techniques available to experimentalists to extrapolate ◦ ◦ their measured di@erential data to 0 and 180 and then derive integral cross-sections. The Frst, which is applicable to both elastic and inelastic scattering, is to use the shape of a reliable theory to aid the process [31]. Previously, this technique’s applicability had been limited by the lack of availability of reliable theoretical scattering data. However, with today’s computer hardware and numerical analysis techniques, there are now many quite sophisticated calculations being performed so that good theoretical data can usually be obtained. It is arguable that, whilst this technique still retains a degree of subjectivity, the use of a theory which encapsulates the physics of the scattering problem renders it less subjective than the even simpler procedure of extrapolating by eye. The second technique, which is applicable to elastic scattering measurements, was outlined by Lun et al. [32] and Sun et al. [33]. In the former case, they have generalised the complex phase-shift analysis technique for electron-atom scattering [34] to the electron-molecule case. SpeciFcally, they applied Fxed-energy inverse scattering theory to analyse di@erential cross-sections for elastic electron scattering from water (H2 O) molecules. They used both semiclassical (WKB) [35] and fully quantal [36] inversion methods with the

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H2 O data [37], which was taken in the energy range of 100 –1000 eV. Full details can be found in the paper of Lun et al. although we note that excellent Fts to the measured di@erential cross-section data were obtained in all cases. The phase-shifts derived from these Fts were then used to unambiguously extrapolate the data and derive total elastic, total absorption and elastic momentum transfer cross-sections. Finally, we note that the procedure of Lun et al. explicitly views the electron-water scattering process as a central Feld problem, an assumption which is not strictly correct. However, as the di@erential cross-section was dominated by forward angle scattering, i.e. by large values for the partial waves at large r, and as the scattering potential becomes more spherically symmetric for larger r, it is likely that the approximation of this system as a central Feld problem is a reasonable one. The other numerical extrapolation procedure, developed by Morrison and colleagues [33], was recently applied to low-energy elastic electron-N2 scattering. This technique, which was used for producing integral elastic cross-sections, utilises a non-linear least-squares Ft to the measured angular distributions subject to the following guidelines: (1) the Ftting equations should rePect the known physical properties of the system and its S-matrix; (2) the Ft should be independent of any particular theoretical calculation. Sun et al. argue that the above philosophy underlies the widely used phase shift analysis methods of electron-atom scattering [34] which, although appropriate to spherically symmetric potentials, are incorrect for electron-molecule systems. The method of Sun et al. is based on their equation for the DCS in the body-Fxed, Fxednuclear-orientation (BF-FNO) approximation, with “appropriate” expansion coeJcients. In formulating the Ftting equations they identify the “free-parameters” that will be allowed to vary in order to Ft the measured angular distribution data to the DCS equation (see Eq. (3:1) of Sun et al.). Consistent with the philosophy of constraining the Ft as much as possible, they sought the fewest number of free parameters that would allow suJcient Pexibility to Ft the data. To this end they employed the usual Frst Born approximation (FBA) expressions for the higher-order partial waves, as described below. The dynamical information of the scattering process is contained entirely in the T -matrix elements. However, these elements themselves are not convenient as parameters: they are complex, so that each element would lead to two real parameters. Sun et al. therefore chose instead the K-matrix, the elements of which were easily parameterised in terms of the “phase parameters”  l . Since the K-matrix is real, the corresponding S-matrix is guaranteed to be unitary. As Sun et al. noted, the number of K-matrix elements that contribute to the DCS may be large, and to allow all such elements to be free parameters would lead to a Ft which is severely under-constrained. Consequently, they incorporated into their procedure the fact that most of these elements, in particular diagonal elements with high partial-wave order and all non-negligible o@-diagonal elements, are accurately given in the FBA. So although these high-order and o@-diagonal elements are extremely important in determining the shape of the DCS (especially at low angles) their inclusion via the FBA is trivial. For e-N2 scattering Sun et al. found that only o@-diagonal elements with |U‘| = 2 contributed to the DCS coeJcients and so they were able to set all other such elements to zero. Thus, the Ft only needed to determine a few free parameters—those corresponding to the low-order partial waves.

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Operationally, the procedure of Sun et al. determined the free parameters using the Levenberg– Marquardt non-linear least-squares Ft. As a measure of the quality of the Ft they adopted the usual deFnition of the reduced 2 function. Thus, a value of 2 near 1 indicated that the model and the data (with the given error estimates of the data) were consistent, and the Ftted parameters were validly determined. The elastic integral cross-sections and momentum transfer cross-sections obtained from this procedure were then derived. Historically, integral electron impact excitation cross-sections for molecules have been primarily obtained by measuring optical excitation functions where the direct+cascade emission at a particular wavelength is measured as a function of the electron impact energy. Photoemission from the particular state excited by electron impact is monitored in these experiments. The optical excitation function can be corrected to various degrees of approximation to yield the required electron impact excitation cross-sections. The experimental determination of the excitation cross-section requires knowledge of the absolute measurement of the photon emission rate, branching ratios for decay paths of the excited states and the scattering geometry. Further complications arise due to cascade contributions to the population of the excited state under study, polarisation e@ects and radiation trapping for high target densities. A recent application of this technique to the C 3 Vu electronic state of N2 was reported by Zubek [38], giving more details of the measurement procedures. We note that if not appropriately corrected for, cascade contributions from the decay of energetically higher states into the excited state of interest can lead to an overestimation of the magnitude of the excitation cross-section under study. Finally we note that under certain, somewhat limited, circumstances integral cross-sections for a particular state can be deduced from the total yield of metastable atoms as a function of the incident electron energy. This procedure was applied by Brunger et al. [39] to determine the near-threshold integral cross-section for the E 3 W+ g state of N2 , and by Furlong and Newell [40] to determine the integral cross-section for the a3 V state of CO. 2.2.3. Di:erential scattering cross-sections The relationship between the measured signal and the corresponding cross-section in electron beam–molecular beam scattering experiments has been discussed in detail previously [9,41]. However, as a precise deFnition and interpretation of this relation is important for deriving cross-sections from the experimental data and assessing their reliability (error limits), a brief discussion is now presented. The relationship, for process n, between the di@erential cross-section and the number Nn of scattered particles detected per second with nominal energy EX n = EX 0 − UE where EX 0 is the impact energy and UE is the energy loss corresponding to process n, at the X ) X can be written as [41] X ; angle (     d 3 (E0 ; En ; ) X X X Nn (E 0 ; E n ; ) = (r)f(E0 ; r)(En ; r) (2.1) d dE0 dEn d 3 r ; dEn d r E0 En  where (r) is the density distribution of the target molecules, f(E0 ; r) is the energy and spatial distribution of the incident electron Pux, and (En ; r) is the response function of the detector for electrons of energy En scattered by a target molecule at position r. In Eq. (2.1) we assume that the background contribution to the scattered signal has been removed. This can be achieved by taking the true scattering signal Nn to be the di@erence between the measured signal with

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the target beam on and o@. Note that for this procedure to be valid the scattering conditions in the beam on and o@ modes must be identical. To ensure that this latter criterion is met, one usually introduces the proper leakage of gas into the background that was eliminated by shutting o@ the target beam. A detailed knowledge of the instrumental function and the target density and electron beam Pux distributions is required to extract the cross-section from (2.1). In general, however, some simplifying assumptions can be made. For excitation of a discrete Fnal state n it is usually possible to integrate over the energy loss proFle, assuming constant system response over the natural line width. Furthermore, if the cross-section does not change signiFcantly over the energy and angular resolution of the experiment it can be removed from within the integral so that Eq. (2.1) simpliFes to give X = K(r; E0 ; EX n ; ) Nn (EX 0 ; EX n ; ) where K(r; E0 ; EX n ; ) =

   r

E0



d (E0 ; ) X ; = K(r; E0 ; EX n ; )DCSn (EX 0 ; ) d

(r)f(E0 ; r)(EX n ; r) dE0 d d 3 r

(2.2)

(2.3)

includes all the instrumental and geometrical factors. The instrument-averaged singly di@erential cross-section is indicated as DCS. This is the quantity obtained from the measurement (n may now refer to a collection of experimentally indistinguishable processes). Following Tramjar and McConkey [2], we have denoted the quantity in (2.3), which is often referred to as the “e@ective scattering volume” [1,9,41], as K (instead of Ve@ ) to highlight the fact that it does not have dimensions of cm3 . In general, relation (2.3) can be simpliFed by assuming that the energy distribution of electrons is independent of r and that the detector response depends only on En and not on r so that f(E0 ; r) = Ie f(E0 )f(r)

(2.4)

and (En ; r) = (En ) ; where Ie is the incident electron beam current. In this case, the integrals over the energy distributions and coordinates can be separated to give K(EX n ; ; ) = Ie C(EX n )V˜e@ ( ; ) with V˜e@ ( ; ) = and C(EX n ) =

  r

(r)

(r)f(r) d(r) d 3 r

(2.5) (2.6)

  En

E0

f(E0 )(En ) dEn dE0 ;

(2.7)

where C(EX n ) denotes the energy dependence of the detector response function and is, in general, a quite complex function. A further simpliFcation occurs in that for conventional scattering measurements the molecules are randomly oriented (or the scattering is orientation independent)

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so that the  dependence disappears and  can be replaced by . In this case, Eqs. (2.2), (2.5) and (2.6) become X = K(EX n ; )DCS X X ; X 0 ; ) Nn (EX 0 ; EX n ; ) (2.8) n (E X = Ie C(EX n )V˜e@ ( ) X ; K(EX n ; )   X ˜ Ve@ ( ) = 2 (r)f(r) sin  (r) d  (r) d 3 r ; r

 (r)

(2.9) (2.10)

where V˜e@ can be calculated from a knowledge of the density distribution of the intersecting beams and the acceptance angles of the detector, as done by Brinkman and Trajmar [41]. However, the practical determination of absolute values for the overlap integral V˜e@ is quite diJcult to obtain since it is necessary to determine both the molecular beam density and electron beam Puxes. Furthermore, even if V˜e@ could be determined absolutely, one would still need to characterise C(EX n ) to obtain the absolute cross-sections. Consequently, the usual practice X is to apply a normalisation procedure [27], which in Fxing the absolute scale for DCSn (EX 0 ; ) we small discuss later in Section 2.6. 2.3. Swarm experiments A detailed discussion of the drift and di@usion of electrons in gases under the inPuence of external electric and magnetic Felds is beyond the scope of this review and only a brief summary will be given. For a full description of the earlier experimental methods and analysis procedures, the book by Huxley and Crompton [42] should be consulted. An update of these techniques, particularly for the low E=N regime (where E is the applied uniform electric Feld and N is the gas number density), can be found in the articles of Schmidt and colleagues [43,44] and Crompton [4]. The technique involves high precision measurements of characteristic transport properties, the transport coeJcients, of an ensemble or swarm of electrons as they drift and di@use through a gas under the inPuence of an E Feld at pressures ranging from a few torr to many atmospheres. The most commonly measured transport parameters are the drift velocity W , which is deFned as the velocity of the centroid of the swarm in the direction of E; the ratio DT =&, where DT is the di@usion coeJcient perpendicular to the electric Feld and & is the electron mobility; and, when a magnetic Feld B transverse to E is present, the ratio W⊥ =W , where W⊥ is the drift velocity at right angles to E and B. Techniques for the measurement of these transport coeJcients include the Bradbury–Nielsen [42] and photon-Pux techniques [45,46]. For a given gas all these coeJcients are functions only of the ratio E=N , the gas temperature T , and when a magnetic Feld is present, of B=N . The macroscopic transport coeJcients are related to the desired microscopic quantities (integral cross-sections as a function of energy) through an energy distribution function which is usually non-Maxwellian and often complex in form. The microscopic properties have to be determined in a complicated and cumbersome [42] unfolding procedure by seeking a self-consistent set of cross-sections which produce the experimental transport coeJcients via a solution of an “adequate” formulation of the Maxwell–Boltzmann transport equation [47].

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The major experimental development in swarm experiments since 1983 has been the accurate measurement of electron transport coeJcients in crossed electric and magnetic Felds by Schmidt and colleagues [44]. In this case, a laser-driven swarm source is used to produce a line-shaped electron cloud at variable drift distances from the anode. The experiment of Schmidt et al. [43] was designed to measure both components of the drift velocity in the E × B Feld, the directly measurable quantities being W and the tangent of the magnetic dePection angle ('): tan ' = W⊥ =W . To determine these quantities the experiment could be run in either of two distinct modes with di@erent orientations of the ionisation line. In the Frst mode, the line-shaped charge swarm is orientated parallel to the magnetic Feld and tan ' is determined from the charge distribution at the anode as measured along the x-axis. In the second orientation the experiment is used to measure the drift velocity component parallel to z. Here the arrival time distribution of the charge is collected at the grid shielded section of the anode to determine both the mean time of arrival, tm , of the swarm and the longitudinal width, z , of the swarm from which W can be deduced. For B=N = 0; W was measured to an uncertainty of about 0.2%, whilst for B=N = 0, an additional uncertainty of 0:3% tan ' had to be added due to a possible calibration error in the angle between the directions of the incident UV beam and the electric Feld. The apparatus of Schmidt et al. could be operated over a wide range of reduced electric ◦ ◦ and magnetic Felds with dePection angles from 0 to 50 without signiFcant loss of accuracy. Thus, it was able to furnish interesting new information which, in principle, should enhance the reliability of swarm-derived cross-sections. When applied to atomic systems at low “mean energies” swarm experiments can provide very accurate elastic momentum transfer cross-sections. However, in molecular systems there are particular problems in obtaining a unique set of cross-sections from swarm experiments [45,48,49], as is evidenced by a recently reported v = 0–1 rovibrational integral cross-section for low-energy electron–H2 scattering [44]. In this case, Schmidt et al. [44] have derived, in a somewhat more sophisticated experiment than previously reported at low E=N , a rovibrational cross-section for the v = 0–1 excitation which is nearly 14% di@erent from that which had hitherto been considered the deFnitive swarm result for this process [50]. The point here is not that the cross-section of Schmidt et al. is arguably in better agreement with theory [51,52] and beam experiments [53] than was the earlier swarm data [50] but that two quite similar analyses of swarm data produced two signiFcantly di@erent results for the H2 rovibrational (0 –1) integral cross-section. This problem of a lack of uniqueness with the swarm technique, which increases as more inelastic channels become open, is a major limitation in its application [42] to the determination of low-energy electron–molecule scattering cross-sections. Nonetheless, the swarm technique does have an important role to play in the checking for self-consistency of a given cross-section set. In this case, the “known” cross-section set can be used in conjunction with the Boltzmann equation to derive transport parameters which are then compared against those measured in the swarm experiments. In this way the self-consistency of this cross-section set can be assessed. 2.4. Sources of excited species An excellent review of electron collisions with excited atoms, including a detailed description of the techniques for the production of excited species, was recently provided by Trajmar and

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Nickel [54]. On the other hand, the study of electron interactions with excited-state molecules (vibrational and=or electronic-state) is still in its infancy in spite of its fundamental importance in laser and plasma physics, chemistry, and the life sciences [55]. Indeed, the fact that very little quantitative experimental work has been done in this Feld is a rePection of the diJculties in producing and characterising excited molecular targets with suJcient number densities for scattering experiments. One of the earliest reported quantitative experiments for electron scattering from an excited target species was conducted by Hall and Trajmar [56], who measured di@erential and integral cross-sections for scattering of 4:5 eV electrons from the metastable a1 Ug state of molecular oxygen (O2 ). In some sense, the a1 Ug state was a natural choice for study as it has a long lifetime with respect to both radiative and collisional deactivation. In this work they utilised a microwave discharge to generate an O2 molecular-beam target which contained about 7% O2 (a 1 Ug ). Subsequent to this, Armentrout et al. [57] used the charge transfer neutralization technique [58] to measure the electron impact ionisation cross-section of the metastable A 3 W state of N2 from threshold to 240 eV. In this application, a 1 keV beam of N was neutralised by N2 and by nitric oxide (NO). The ionisation cross-section for an N2 beam formed by charge transfer with N2 was found to match literature data for ionisation of the ground molecular state (X 1 W) state of N2 . The beam formed in N+NO neutralisation was found to contain ∼50% ground state and ∼50% excited (A 3 W) state N2 . By measuring the ionisation for this mixed state beam and subtracting the contribution from ionisation of the ground state, they deduced the ionisation cross-section for the A3 W state over an extended electron energy range. Further studies with excited molecular species include, amongst others, the work of Spence and Schulz [59], White and Ross [60], Huetz et al. [61], Belic and Hall [62] and Srivastava and Orient [63]. In several of these experiments [60 – 62], a microwave discharge was used to generate the excited state species. SpeciFcally, Spence and Schulz studied the temperature dependence of dissociative attachment in O2 and CO2 , White and Ross considered the vibrationally excited population of ground-state N2 in a Powing afterglow, Huetz et al. considered the initial vibrational state dependence of resonant excitation and dissociative attachment in electron–N2 scattering, whilst Belic and Hall studied dissociative electron attachment to the a1 Ug state of O2 . Srivastava and Orient, on the other hand, used two beams of electrons to study the process of dissociative attachment with vibrationally excited carbon dioxide (CO2 ). Here the CO2 is initially excited to the vibrational levels by the Frst electron beam and then these excited state species are crossed with the second electron beam. More recently, there has been some interest in electron scattering by vibrationally excited molecules [55]. Here the vibrational levels may be populated by thermal processes, the population of an upper level of a molecule being given by the well-known Boltzmann distribution PR =

gR e−ER =kT ; /gR e−ER =kT

(2.11)

where PR is the percentage population, gR the statistical weight of the level with energy ER , the energy relative to the ground state, k is Boltzmann’s constant and T the ambient temperature of the surrounding environment. For simple diatomic molecules ER is high (¿150 meV), so that at room temperature only a small percentage of the molecules will be in excited vibrational

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states. However, for even the simplest polyatomic molecules ER is much smaller and hence the lowest vibrational states will be populated, even at room temperature. The Frst quantitative experimental study of low-energy electron scattering from vibrationally excited molecules was that of Buckman et al. [64] in which low-energy total electron scattering cross-sections were measured for CO2 molecules as a function of temperature using a linear attenuation, time-of-Pight (TOF) spectrometer. In this work, the scattering cell was heated resistively by passing a current through a number of coaxial heating elements mounted on its outer surface. At T = 310 K only about 5% of the CO2 targets are in the lowest lying (83 meV) (010) bending mode whilst at 573 K the ground state population is reduced from 95% to 80% and the population of the (010) mode increases to about 15%. Care was taken to ensure that the heaters and their associated wiring within the vacuum system were arranged so that no induced magnetic Feld resulted during their operation. Also, a feedback system ensured the temperature ◦ of the scattering cell was maintained to within 0:1 C of the desired set point. Buckman et al. observed a substantial increase in the total cross-section at energies below 2 eV which they attributed to enhanced scattering due to the non-zero average electric dipole moment of the (010) mode. No signiFcant change in the cross-section was observed at higher energies. Ferch et al. [65] repeated this experiment using a time-of-Pight spectrometer and conFrmed the earlier result [64] of direct dipole scattering but found, in contrast, a pronounced change in the total cross-section at energies from about 3–5 eV, where the 2 Vu resonant state of CO2 is formed. The resonance contribution from the vibrationally excited molecules, again mostly in the (010) mode, was observed to shift to lower energies by 0:3 eV and broaden somewhat. In an alternative approach to the above for the study of the di@erence in the energy dependence of the cross-sections, for the ground state vis aZ vis excited states, Stricklett and Burrow [66] used a commercial CO2 infrared laser to pump vibrational transitions in sulphur hexaPuoride (SF6 ). The SF6 was expanded from a nozzle, pumped by the laser, and then crossed downstream by an electron beam. Here the attenuated beam current was measured and, by modulating the laser light and detecting the AC component of the transmitted electron current, they engineered a technique that provided a sensitive measure of the changes in the cross-section upon vibrational excitation. They found that electron scattering was strongly altered below 2 eV and at electron energies associated with the formation of temporary negative ions. Johnstone et al. [67] reported the Frst di@erential cross-sections for electron scattering from ◦ vibrationally excited CO2 at 4 eV and a scattering angle of 30 . They found that the elastic cross-section for scattering from the (010) mode was 15% higher than that from the ground state, whilst the inelastic cross-section (000 → 010) was almost twice that of the corresponding superelastic cross-section (010 → 000). A typical energy loss spectrum from this work is given in Fig. 3. Johnstone et al. employed two heating methods to thermally excite the CO2 molecules. In the Frst, the radiative heat from a quartz-halogen lamp was used to heat the tube and the enclosing interaction region. In the second method, the output from a 280 W CW carbon dioxide laser was focussed onto the top of the tube, where the direction of the laser beam was perpendicular to the electron scattering plane. Using the Frst method, temperatures ranging from 313 to 493 K were easily obtained, where they measured the temperature using a copper-constantan thermocouple, mounted 5 mm from the tube exit. Using the laser, temperatures up to 773 K were achieved. This work was recently extended by Johnstone et al. [68], ◦ but at the slightly di@erent energy of 3:8 eV and for scattering angles between 20 –80 .

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Fig. 3. Energy loss spectra for electron scattering from CO2 at 4 eV and 30 ◦ ◦ (a) 40 C, (b) 400 C.

◦

231

at two di@erent temperatures:

Similar experiments [69] for electron scattering from the isoelectronic partner of carbon dioxide, nitrous oxide (N2 O), indicated less dramatic changes in the inelastic features as a function of temperature, compared to those observed in CO2 . As N2 O has a permanent dipole moment in its ground state due to its asymmetric bond length, the induced dipole for the (010) mode would play a smaller relative role than in CO2 , and accordingly less dramatic scattering phenomena were observed. The measurement of cross-sections for electron impact excitation of electronic states from vibrationally excited targets is virtually unexplored at this time. Celiberto and Rescigno [70] reported the dependence of the electron impact cross-sections for H2 (X 1 W → B 1 W) and H2 (X 1 W → C 1 Vu ) upon the initial vibrational quantum number of the ground molecular state (X 1 W). Their calculation shows a dramatic dependence of the former process on the initial vibrational

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mode; the cross-section steadily increasing for v = 0–10 then decreasing for v = 11–14. On the other hand, the X 1 W → C 1 Vu cross-section rises gently for v = 0–12 whereafter it is constant. Presently, no experimental information exists for such phenomena. 2.5. Magnetic rotation of the scattering geometry A major experimental advance has been made in recent years by the Manchester group [71,72], who have developed a highly innovative technique for extending scattering measurements to the full range of backward scattering angles. In a departure from the normal mindset of high-resolution electron spectroscopists, who shun magnetic Felds as a curse, they have imposed a localised magnetic Feld upon the collision region of an electron spectrometer in order to rotate the incident and scattered electron trajectories. The magnetic Feld is applied by two pairs of coaxial solenoids with opposing currents in each pair such that the incident electron beam trajectory can be rotated in the scattering plane to an arbitrary angle with respect to the un-rotated trajectory. The Felds are arranged such that the incident electrons always pass through the atomic beam, which is directed along the axis of the solenoid pairs. The Felds are also arranged such that they are highly localised, dropping to negligible strength in the regions beyond the interaction region where the sensitive, low energy, electrostatic electron optics are operated. The details of the design are provided in the paper of Read and Channing [72]. If, for instance, the Felds are set up such that the incident electron beam is rotated through an angle ◦ of 45 , then those electrons which are elastically scattered in the backward direction will also ◦ su@er a similar rotation on leaving the interaction volume and emerge at 90 with respect to the initial, un-rotated, incident direction. As a result, an analyser placed at a conventional scattering ◦ angle of 90 can record elastic backscattering events. For inelastic scattering the rotation on leaving the interaction volume is greater but this can be readily calculated from the detailed formulae provided by Read and Channing. To date, the technique has not seen extensive use in the measurement of absolute cross-sections for electron–molecule scattering, but the obvious beneFts have been shown in a number of recent papers [73–75] and those that are of relevance to the present review are discussed in later sections. However, there is little doubt that it has the potential to make a major contribution to the Feld in providing important information for probing short-range aspects of the interaction potential such as exchange and correlation, and in providing DCS measurements over a larger angular range from which integral cross-sections of higher reliability can be derived. 2.6. Normalisation techniques 2.6.1. Elastic scattering: the relative =ow technique The most widely accepted strategy for the measurement of elastic DCSs, at the present time, is to utilise a set of established elastic DCSs as “standards” for eliminating the response function C(EX n ) and the integral represented by V˜e@ ( ) (Section 2.2.3). This is achieved by the relative Pow technique which relates the di@erential cross-section of a particular target-gas, at a speciFc energy and scattering angle, to that of a reference gas under identical experimental conditions. For low-energy electron scattering the “standard” cross-section which is normally used is that for electron–helium elastic scattering. This cross-section has been measured extensively [76 –78],

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233

Fig. 4. Di@erential cross-section (in units of 10−16 cm2 =sr) for elastic e-He scattering at 40 eV. (- - -) 62-state convergent close coupling calculation by Fursa and Bray, ( ) Register et al. and ( ) Brunger et al.

•

the absolute values being obtained from the application of phaseshift analysis techniques. This cross-section has also been calculated in an ab initio fashion by several groups [79 –81], and is widely regarded as being known to within a few percent for energies below the Frst inelastic threshold at 19:8 eV. The extension by Fursa and Bray [82,83] of their convergent close-coupling model, to two-electron atoms, along with the experiments of Register et al. [77] and Brunger et al. [78], has also established the case for helium to be considered as a standard cross-section up to 50 eV, at least to the 7% level (see Fig. 4). Boesten and Tanaka [84] have also derived a set of parameters for generating a “preferred” helium elastic scattering cross-section over a broad range of energies. In the light of several low-energy (¡10 eV) experimental and theoretical studies in neon [85 –90] which are generally in excellent agreement with one another, it has also been proposed [87] that neon may be used as a “secondary standard” for low-energy electron scattering. As such it was not intended that it would usurp the role of helium but rather serve as a means for checking the operation of any scattering apparatus which uses the helium cross-section as a reference standard. Furthermore, the work of Buckman et al. [10] (see Section 2.1) on the size and shape of pseudo-e@usive beams indicated that at elevated driving pressures, the width of

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a helium beam was smaller than that of most of the other gases they studied, even when the driving pressures were adjusted such that the mean free paths of the gases were identical. As the relative Pow technique (see below) is based on experimental measurements carried out under identical scattering conditions for the target and standard gases, the smaller widths observed for helium beams is a matter of some concern. This served to strengthen the case for establishing a secondary standard for low-energy scattering. The fact that Buckman et al. did not observe any signiFcant di@erences in the size of neon beams, as a function of pressure, compared with a range of other atomic and molecular gases (with the exception of helium) only added further weight to the case for its use as a secondary standard. The relative Pow procedure requires carrying out elastic scattering measurements for the target (T) and standard (S) gases under identical experimental conditions at Fxed values of E0 and (see Fig. 5 for a schematic diagram of the gas handling system for a relative Pow experiment). Under these circumstances, we have from Eqs. (2.8) – (2.10), the relative scattering intensity given by   T N0T DCS0T IeT C T (EX n )V˜e@ ( ) = : (2.12) N0S DCS0S I S C S (EX n )V˜ S ( ) e@ e The index zero on N and DCS refers to elastic scattering. If for a given E0 and , the factors in the bracket on the right-hand side of Eq. (2.12) do not change when one switches from the target gas to the standard gas then the DCS is simply obtained from Eq. (2.12). However, to establish this condition is rather diJcult and not really necessary as, unless we deal with highly reactive gases, to a very good approximation C T (EX n=0 ) = C S (EX n=0 ). To handle V˜e@ ( ), we follow the prescription of Nickel et al. and factor the target density function as (r) = nb n(r) ;

(2.13)

where nb is the total number of molecules in the beam and n(r) refers to the spatial distribution function of the density. If the driving pressures of the two gases behind the beam-forming device are chosen such that the mean free paths of the two gases are identical, then nT (r) = nS (r). Under these circumstances Eq. (2.12) simpliFes to   N0T DCS0T IeT nTb = : (2.14) N0S DCS0S IeS nSb Nickel et al. have further shown that the ratio nTb =nSb can be related to the ratio of the Pow rates ˙ of the gases through the capillary array via (F) T nTb F˙ m1=2 T = ; nSb F˙ S m1=2 S

(2.15)

so that Eq. (2.14) reduces to its now familiar form (upon minor further rearrangement) IS DCS0T = eT Ie

S T F˙ m1=2 S N0 DCS0S ; T N0S F˙ m1=2 T

(2.16)

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235

Fig. 5. A schematic diagram of the gas handling system for a relative Pow experiment (from Gulley, 1994).

where the electron beam currents are easily measured via a Faraday Cup, the scattered count rates are measured with, for example, standard channel electron multiplier detectors and pulse ampliFcation=counting techniques, and the relative Pow rates are conveniently determined in a separate series of experiments, usually as a function of the capillary driving pressures which are

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in turn monitored by an accurate pressure gauge. We note that in more recent measurements [87] both the target and standard gases are brought into the scattering chamber continuously. One is introduced to the scattering volume whilst the other is introduced to the general chamber at another point. This procedure, particularly in the case of polar and reactive molecules, avoids the shifts in contact potential which can be encountered when one gas is removed from the vacuum chamber and another introduced. ˙ 1=2 , (see Fig. 5 for an The technique employed to measure the normalised Pow rates, Fm apparatus schematic) is very similar to that outlined in Khakoo and Trajmar [91]. Here it is not necessary to actually determine the absolute Pow rates but rather the ratio of the normalised Pow rate for the target gas to the standard gas. To do this the rate of change of pressure behind the capillary array, as measured by a pressure gauge e.g. a Baratron, is recorded for both the target gas and the standard gas for various settings of their leak valves. For the purpose of accurately measuring the Pow rates an in-line volume, as shown in Fig. 5, is opened to reduce the pressure gradient. Data acquisition is achieved under computer control. The procedure to obtain the normalised Pow rates for both the target gas and the standard gas is as follows: (i) the gas is routed to the chamber (via the capillary array) and the initial driving pressure (the pressure behind the capillary array) is set manually with the associated leak valve. (ii) the initial Baratron pressure reading Pb is recorded and the pneumatic valve to the capillary V7 is closed. (iii) the rate of change of the Baratron pressure is recorded for a short period. (iv) the pneumatic valve V7 is opened and the Baratron pressure readings are displayed by the computer where they are checked for self-consistency before the relevant Pow rate is calculated. (v) the initial driving pressure is manually set to the next value. Once stability in the Baratron pressure reading is achieved, the procedure (ii) – (v) is repeated until measurements for the desired range of driving pressures has been covered. A typical result [92] for sulphur dioxide (SO2 ) and He is given in Fig. 6. For gases with high viscosity a time period of 5 –10 min was not uncommon before stability in the driving pressure was achieved. A technique employed by Khakoo et al. [93] to overcome lengthy periods between setting subsequent driving pressures, was to determine a pressure versus time stabilisation curve and extrapolate this curve to obtain the value of the steady-state driving pressure. For an ideal gas, kinetic theory predicts that for collisionless Pow the normalised Pow rate 1=2 ˙ (Fm ), should be independent of gas species. Thus, if the normalised Pow rates are plotted as a function of driving pressure for both the standard gas, i.e. He, and the target gas of interest, and no species dependence was found, then these terms in Eq. (2.16) could be conveniently replaced by their respective driving pressures. In practice, however, it was found [94] that this was not the case for most gases. Fig. 7 shows a summary of measurements of the ratio of normalised Pow rates for a number of gases to that of He to illustrate this point. In some cases, di@erences of over 20%, from the expected ratio of 1, have been observed. We note that even though the extent of these di@erences will almost certainly be apparatus speciFc, they indicate that care should be taken in establishing the true relationship between the gas Pow rates. Failure

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237

•

Fig. 6. Normalised relative Pow rates for ( ) SO2 and ( ) He as a function of their capillary driving pressures.

◦

•

Fig. 7. A comparison of the ratios (to helium) of the normalised Pow rates for ( ) Ne; ( ) N2 ; ( ) NH3 ; (4) H2 S and (×) SO2 as a function of capillary driving pressure.

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to do so would see potentially signiFcant errors being carried over into the determination of the absolute cross-section. Thus, in practice, in a typical experiment the ratio of the driving pressures for the two gases will be determined from their mean free paths. The absolute value of the pressures to be used are then determined by considerations based on the size of the capillary source used for the beam-forming device. For instance, Buckman et al. established that for their own capillary array it was necessary to impose a maximum limit on the driving pressure such that the mean free path  ¿ 2d, where d is the diameter of the individual capillaries. Once the absolute gas pressures to be used are established, then the relative Pow rates for each gas at these pressures must be determined as described above. One approach is to determine the Powrate at a range of driving pressures about the ideal value and to Ft the curve of normalised Powrates vs. driving pressure. The Ftting parameters or their equivalent then enable the appropriate normalised Pow rate ratio to be substituted directly into the relative Pow equation (2.16) for any value of the ratio of the capillary driving pressures. Consequently, all the terms on the right-hand side of Eq. (2.16) are known and the required absolute elastic di@erential cross-section is thus determined for a given energy and angle. The entire procedure, with the exception of the calibration curve, is then repeated for all angles until the angular dependency of the DCS at a given energy is known. 2.6.2. Inelastic scattering At this time, it appears that the most logical and eJcient way for determining absolute inelastic cross-sections, for the target gas of interest, is to measure relative inelastic to elastic scattering intensities and then achieve normalisation through the target gas elastic di@erential cross-section (DCS). The target gas elastic DCS has already been obtained by utilisation of the elastic DCS of He or Ne and the relative Pow technique as discussed in the preceding section. Normalisation of inelastic to elastic signal in the target gas can be achieved by two methods. In the Frst method, the angular distribution of the inelastic process is Frst measured and, subsequently, the inelastic angular distribution is normalised to the elastic cross-section at one or two angles [53]. In the second method, energy loss spectra, encompassing both the elastic and inelastic features, are obtained at each scattering angle (for a given energy) and the inelastic to elastic DCS normalisation is performed at each angle. The Frst method requires a knowledge of the overlap geometry of the gas beam, detector viewcone and electron beam in order to correct for possible angular dependence of the e@ective scattering volume. Such an e@ect can be greatly minimised [41] by ensuring that the gas beam is well collimated but usually any angle-dependent e@ects are calibrated out by comparison to the known angular variation of a standard cross-section, at the energy of interest. The second method is insensitive to an e@ective path length correction factor [41] since both the elastic and inelastic signals are a@ected equally. However, at small angles the detector viewcone begins to overlap with the incident electron beam and care must be taken to ensure that the signiFcant elastic background is accurately subtracted. Note this problem is avoided with the Frst method by carrying out the normalisation at suJciently large scattering angles. In any event, the major problem here is the establishment of the relative detector response function for the residual-energy (or equivalently, energy loss) range corresponding to the elastic and inelastic events. From Eqs. (2.8) – (2.10) it follows that the expression for the measured

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inelastic to elastic scattering ratio at Fxed E0 and can be written as NnT (E0 ; ) DCSnT (E0 ; ) C T (En ) : = N0T (E0 ; ) DCS0T (E0 ; ) C T (En=0 )

(2.17)

Thus from Eq. (2.17), one only needs the relative response function of the detector (relative to that corresponding to elastic signal detection), C T (En )rel , to obtain absolute inelastic di@erential cross-sections. At high impact energies (compared with the energy loss UE), the instrumental response can usually be considered to be independent of En [2], particularly if potentials on the lens elements are continuously adjusted during the energy loss sweep to maintain the focal properties of the electron optics [95 –97]. In this case C T (En )rel = 1. The situation, however, rapidly becomes more complicated as the incident energy is lowered as under these circumstances the response function may change drastically with residual energy. One can, in principle, correct for this effect by designing and maintaining the electron optics achromatic. However, one cannot just rely on calculations that are based on Gaussian optics [98] and neglect space charge and surface condition e@ects to ensure that this condition is met. The relative response function C T (En )rel must be determined experimentally for the actual system under experimental tuning and operating conditions. A popular [9,99] procedure for determining the relative response function relies on measuring the yield of electrons (ejected and scattered) from helium, following near-threshold ionisation [100]. As a consequence of the Wannier theory for near-threshold ionisation [101], the yield of detected electrons is expected to be independent of energy for values of excess energy (Eexc = E0 − ionisation potential) up to some value, Emax , above the ionisation threshold. Furthermore, this criterion should be met for any angle of observation. Thus in a measurement of the yield of scattered and ejected electrons resulting from electron impact ionisation at an appropriately low energy, one may expect to observe a Pat spectrum in all situations where the excess energy is less than Emax . Hence, any deviations from Patness directly rePect the distortion introduced by the non-uniform response function of the analyser. Experimental veriFcation of the value of Emax is still uncertain and, to some extent, controversial. The range of validity of the Wannier theory has been tested in both di@erential and integral measurements of the ionisation cross-section for excess energies up to 8 eV [100,102–104]. Some measurements [102,105] indicate that even when Eexc = 6 eV the Wannier theory appears to hold, whilst others [106] show that there are departures from this theory, of the order of 5%, at excess energies as low as 0:075 eV. Measurements by Hawley–Jones et al. [104] indicated that the Wannier model was valid in (e,2e) experiments to an excess energy of ≈ 1:5 eV for coplanar scattering, but could be valid to perhaps as high as 6 eV excess energy for scattering in the perpendicular plane. An excellent summary of the state of this Feld can be found in the article of Lubell [107]. However, we can only conclude that the present situation regarding the range of validity of the Wannier model is not well established and, furthermore, it is also not well established how serious the discrepancies from the predictions of the Wannier theory actually are above Emax . In addition, issues such as the e@ect on the analyser response of (i) changing from the standard gas to the target gas of interest and (ii) changing the beam energy from that of the calibration energy to the desired incident energy of the experiments, need to be addressed. The crucial aspect in minimising such potential problems is

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to ensure that in all cases the focal properties (object distance (P), image distance (Q) and magniFcation (M ), etc.) of both the Fnal pre-target zoom lens stack and the initial post-collision zoom lens stack are maintained. All electron optical surfaces should be clean and care and consideration should be given to the pumping of the interaction region. Whilst great care is commonly taken by most groups to ensure these conditions are met, it is doubtful that the analyser response can be determined to an accuracy better than 10%. Consequently, we agree with the opinion of Tramjar and McConkey [2] that the reliable and convenient calibration of the analyser response, for inelastic cross-sections, remains an outstanding problem at the present time. Calibration of the analyser response function can also be achieved by relying on known elastic di@erential cross-sections and measuring elastic and inelastic scattering signals with the same residual energy. That is, assuming that the relative detector response function for elastic scattering at E0 and for inelastic scattering at impact energy E0 + UE will be the same. This approach was utilised by Hall et al. [108], Williams and Willis [109], Williams [110,111] and Shyn and Sweeney [112] in their various measurements. In changing the impact energy from that required for elastic to that required for the inelastic measurement (to keep En constant), they corrected for the change in the electron beam current but assumed that the geometry and detector response function remained unchanged. The validity of these latter assumptions was tested by checking the procedure on “known” cross-sections whilst the focussing and magnitude of the electron beam current was maintained with a Faraday Cup. In their work on di@erential cross-sections for electron impact excitation of the C 3 Vu ; E 3 W+ g and a 1 W+ g electronic states of N2 , Zubek and King [113] used a novel approach to determine the response function of their analyser over the residual energy range En = 5:2–10 eV. They made a comparison of an energy loss spectrum, where the incident energy is Fxed, and a constant residual energy spectrum where the collection energy of the analyser is held Fxed, so that the incident and collection energies in each spectra were adjusted to ensure that for one particular vibrational level, E0 (and also the collection energy) had the same value in both spectra. SpeciFcally, they chose the v = 1 level of the C 1 Vu state at UE = 13:206 eV for this purpose. The intensities of the peaks corresponding to other states were expected and found to be di@erent in both spectra because they were excited at di@erent values of incident energy and because of the di@erence in the transmission of the analyser at the two values of electron residual energy. This di@erence in the transmission was deduced from the observed intensity ratio of the C 3 Vu ; v = 0 peak in the two spectra and using a recently determined excitation cross-section for this state [38]. Zubek and King then repeated this procedure for various values of residual energy, En , that covered the range 5.2–10 eV. An uncertainty of 4% was placed on their derived response function with this largely being due to the uncertainty in the excitation function (3.5%) of Zubek [38]. Both these error estimates are perhaps a little optimistic but, nonetheless, the overall procedure they adopted to calibrate their response function seems sound. For completeness, we also note that at high impact energies, for excitation of optically allowed states from the ground molecular state, normalisation of the inelastic DCS can be achieved by the use of the optical oscillator strength f. This procedure is based on the “limit theorem” originally described by Bethe [114], later generalised by Lassettre et al. [115] and discussed in detail by Inokuti [116]. For further details we advise the reader to consult reference [2] as this technique is not strictly applicable in the energy regime of this review.

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Fig. 8. ToF-derived energy loss spectra for N2 from the di@erential electron scattering experiment of LeClair et al.

Another technique which has recently been used to e@ectively overcome the problems associated with determining the analyser transmission function for inelastic scattering measurements is the time-of-Pight approach developed by LeClair et al. [117]. In principle, this technique enables the direct measurement of the ratios of elastic to inelastic scattering intensities, even over a broad energy range of energy loss values. A pulsed electron beam is incident on a well-deFned atomic or molecular beam. Electrons of di@erent energies which arise from the collision process are temporally dispersed over a long (∼20 cm) drift tube and detected with a large area channelplate detector. The result is a timing spectrum consisting of a number of peaks due to either elastic or inelastic scattering. In the absence of any electric or magnetic Felds, the integrated intensity under each peak is directly proportional to the scattering cross-section for the particular process. With careful attention to detail, LeClair et al. have demonstrated how this approach can be used to measure near-threshold electronic excitation cross-sections by measuring ratios of elastic to inelastic scattering and using the relative Pow technique to establish the relevant elastic cross-section. In their apparatus its application was somewhat limited by the fact that they used an electron beam with a broad, thermal energy distribution. However, they successfully demonstrated the technique for measurements in a number of atoms and molecules [118] and these results are addressed in later sections of this review. For a given drift length and timing resolution this technique is most applicable at near-threshold energies. However, this is just the region where problems associated with electron analyser transmission functions can be at their worst, so the technique would appear to have a wide level of applicability to future measurements. An example of a time-of-Pight spectrum for N2 using this technique is shown in Fig. 8. We note one Fnal diJculty associated with the measurement of di@erential crosssections for inelastic processes. SpeciFcally, we refer to the electron impact excitation of electronic states from the ground molecular state, where even for most simple homonuclear diatomics

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Fig. 9. Schematic diagram of the energy levels (En ¡ 12:4 eV) for the electronic states of N2 . For clarity the singlet and triplet states have been partitioned.

there can be appreciable overlap in the vibrational bands of two or more of these excited electronic states e.g. Schumann–Runge continuum in O2 . The “unraveling” of these overlapping states requires the use of spectral deconvolution techniques which are discussed in the next section. 2.7. Data extraction techniques: spectral deconvolution In the measurement of absolute di@erential cross-sections for electronic states in molecules one is confronted with the very real problem of how to extract, uniquely, a cross-section for a particular electronic state of interest from its near-neighbours, when in many cases there is appreciable overlap in the vibrational sub-levels of these adjacent electronic states. As just one example, if we consider Fig. 9, a schematic diagram of the electronic states of N2 , it is transparent that the B 3 Vg state (excitation energy = 7:353 eV) and the W 3 Uu state (excitation energy = 7:355 eV) have strongly overlapping energy loss features. Consequently, to extract these data, spectral deconvolution techniques have been developed. In these techniques each measured energy-loss spectrum is analysed using a computer least-squares-Ftting technique which was Frst described in Trajmar et al. [119] and subsequently reFned by Cartwright and co-workers [96,120]. Here the separation energies of the electronic transitions are assumed known as are the Franck– Condon factors connecting each vibrational level of the excited electronic states with the lowest vibrational level of the ground-electronic state. The Ftting procedure [96,121] then yields

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243

relative intensities for inelastic scattering for the various excited electronic states. These intensities when summed appropriately are proportional to the required di@erential cross-section. Following Trajmar et al. [119] and Brunger and Teubner [99], the general form of the Ftting function is given by N (E0 ; ; En ; ; Ie ) =

N 

M (n )

Xn (E0 ; )

n =0



qv v F(Wn v − En ) + B(E0 ; ; En ; ; Ie ) ;

(2.18)

v =0

where N (E0 ; ; En ; ; Ie ) is the signal as measured by the experimental apparatus in the energy loss spectrum, Xn is proportional to the di@erential cross-section for the electronic state n ; N is the total number of electronic transitions, M (n ) is the number of vibrational bands within a given electronic state, qv v is the Franck–Condon factor, B represents any background contribution to the measured signal, Wn v is the energy of each vibrational sub-level relative to the v = 0 vibrational sub-level of the ground electronic state, F(Wn v − En ) is a function which characterises the experimental proFle of the analyser and, as deFned previously, ; Ie and En are, respectively, the target molecular beam density, electron beam current and energy loss. In general, previous studies [96,99,122] have shown that the term B(E0 ; ; En ; ; Ie ) can be well represented by the polynomial function B=

4 

ai Eni−1 :

(2.19)

i=1

The form of the background, as given by Eq. (2.19), was found by Cartwright et al. [96,120] and Teubner and co-workers [99,122] to be an excellent representation of that measured experimentally throughout their studies which rePected the excellent signal-to-noise ratio across the entire energy loss range of their work. The experimental resolution function F(Wn v − En ) was also found by these groups to be Gaussian in form so that 1 2 2 F(x) = √ e−x =28 ; 8 2

(2.20)

where 8 is related to the full-width half-maximum energy resolution and is a parameter to be determined in the Ft. The quantities Xn (E0 ; ); B and 8 are determined, for a given energy loss spectrum, from Eqs. (2.18) and (2.20) by requiring the di@erence between the measured and calculated spectra be a minimum in a least-squares sense [96]. In particular, having obtained Xn (E0 ; ) ∀n = 0; 1; 2; : : : ; N from the Ft we have seen from our discussion in Section 2.6.2 that the required electronicstate cross-sections are then, in principle, obtained from Xn (E0 ; ) DCSn (E0 ; ) = X0 (E0 ; ) DCS0 (E0 ; )

∀n = 1; : : : ; N ;

(2.21)

where the absolute elastic di@erential cross-section, DCS0 (E0 ; ) is assumed known. There are now two further considerations which we need to address: (i) the validity of the calculated Wn v and qv v as obtained from spectroscopic data and (ii) the degree of uniqueness for Xn (E0 ; ) as obtained in the spectral deconvolution.

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In the technique described immediately above, we have assumed a knowledge of the energy of each vibrational sublevel relative to the v = 0 vibrational sublevel of the molecular ground electronic state and the corresponding Franck–Condon factors. However, such information is not always readily available in the literature. For example, whilst for their work in N2 , Cartwright et al. [96,120] and Brunger and Teubner [99] were able to readily obtain the necessary information from Krupenie and Lohftus [123], Benesch et al. [124] and Cartwright [125] no such detailed information was available in the literature for their studies on CO [122,126,127] and NO [121]. For these systems, extensive tables of energy levels and Franck–Condon factors were not available and even the necessary spectroscopic data with which to apply the Rydberg–Klein– Rees (RKR) formalism to generate the required energy levels and Franck–Condon factors was not easily found. Cartwright [125] has discussed the applicability of the Franck–Condon approximation for processes originating from the v = 0 vibrational level of the ground electronic state. He showed for the speciFc case of N2 that, with the exception of the near-threshold region, the Franck–Condon approximation was quite good. He proposed that this result was due to the fact that the initial vibrational wavefunction is Gaussian in character and centred at the equilibrium separation distance Re of the initial electronic state. Consequently, the major contribution to the excitation amplitude must occur at this same Re and so the Franck–Condon approximation should be reasonably valid. However, we note that Zubek and King [113] found in their stud 1 + ies on N2 for the C 3 Vu ; E 3 W+ g and a Wg electronic states that deviations from “idealised” Franck–Condon behaviour, even away from threshold, were observed. Such behaviour had also already been observed for CO by Trajmar et al. [126] and more recently for NO by Campbell et al. Indeed, it is probably fair to state that for strongly perturbed molecular systems, molecular systems with avoided level crossings and for near-threshold investigations of molecular systems, one should expect a breakdown from “idealised” Franck–Condon behaviour to be the norm rather than the exception. In addition, we would also note that for strongly perturbed systems, such as NO, one must also be careful in placing too much reliance on the values of Wn v calculated from spectroscopic data. For example, Campbell et al. [121] found that shifts of up to 100 meV, in the energy loss position of a particular feature, compared with that predicted by theory, were observed. To combat the potential breakdown of the Franck–Condon approximation Nickel et al. [9] introduced the concept of “Pux-factor corrected” Franck–Condon factors in which, after the Xn (E0 ; ) were determined by the Frst iterative round, the calculated Franck–Condon factors qv v were scaled initially by the Pux term kf =ki and renormalised so that  (kf =ki )qv v = 1 : (2.22) v

This had the e@ect of increasing the Franck–Condon factors for the lower v levels at the expense of the higher v states in the progression. These “Pux-factor corrected” Franck–Condon factors were themselves allowed to vary (the Xn are  held Fxed at their optimal values), subject to the constraint that for a given electronic state all qv v = 1, to improve the quality of the Ft to the measured data. This procedure is useful both in determining the extent of any deviations from idealised Franck–Condon behaviour for a particular electronic state and, by then continuing the iterative least-squares minimisation process, in correctly determining the required Xn (E0 ; ).

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245

Fig. 10. Di@erential cross-sections for the electron impact excitation of (a) B3 Vg , (b) W3 Uu and (c) B3 Vg + W3 Uu electronic states of N2 by 15 eV electrons. ( ) Zetner and Trajmar, (×) Cartwright et al., ( ) Trajmar et al. and ( ) Brunger and Teubner.

•

◦

Such a procedure was found to be essential for the near-threshold studies in CO by Trajmar et al. [127]. Finally, the question of uniqueness in the determination of the Xn (E0 ; ) from the Ft must be addressed. An excellent example of a possible manifestation of this can be found by considering the 15 eV di@erential cross-sections [99,128] for the B 3 Vg and W 3 Uu states of N2 (see Fig. 10). Also shown in Fig. 10 is the sum of these cross-sections. It is immediately apparent from this Fgure that whilst there are some di@erences between the individual B 3 Vg and W 3 Uu cross-sections of Brunger and Teubner [99] and Zetner and Trajmar [128] the level of agreement between the experiments for the summed B 3 Vg + W 3 Uu cross-sections is good. One possible explanation for this observation is that, given the highly overlapping nature of these states, it is an indication of a lack of uniqueness in the spectral deconvolution process. On the other hand, it could also be due to both Zetner and Tramjar and Brunger and Teubner being optimistic in the errors that they estimated for the deconvolution process and subsequently incorporated into their overall uncertainties for the B 3 Vg and W 3 Uu cross-sections. This latter point was recently investigated by Campbell et al., who incorporated into their spectral deconvolution algorithm, the method of Bevington and Robinson [129] in order to have available a mathematically

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rigorous procedure for extracting “true” conFdence limits from multiparameter least-squares Fts of energy loss data. They found that when this procedure was applied to the N2 data of Brunger and Teubner the values of Xn for the B 3 Vg and W 3 Uu states were consistent with those derived by Brunger and Teubner but the deconvolution errors, for these highly overlapping states, were found to be signiFcantly larger than what was previously estimated by Brunger and Teubner. A similar observation is almost certain to be applicable for the work of Cartwright et al. [96] and Zetner and Trajmar. Consequently, we are of the view that Fig. 10 does not necessarily represent a problem with the uniqueness of the values of Xn derived previously but is rather an indication of an overly optimistic view of the errors associated with the spectral unfolding process. Further, the adaptation of the U2 = 1 procedure of Bevington and Robinson to obtain rigorous conFdence intervals on the coeJcients derived from the multiparameter least-squares Fts of energy loss spectra, goes some way in addressing concerns with respect to the uniqueness of the derived coeJcients. 3. Experimental data 3.1. H2 Molecular hydrogen, as the most fundamental of all electron-diatomic molecule scattering systems, has been the subject of numerous experimental studies both before and since the review article of Trajmar and colleagues [1]. There has also been a recent, speciFc review article on H2 by McConkey et al. [130] dealing mostly with excitation cross-sections, and the paper by Morrison et al. [131] provides an excellent summary of the comparison between experiment and theory at low energies. Also, a set of recommended cross-sections for elastic and inelastic processes has been given by Tawara et al. [132]. 3.1.1. Grand total cross-section Grand total cross-sections have been recently measured by Deuring et al. [133], Jones [134], Subramanian and Kumar [135], Nickel et al. [136] and Randell et al. [137], in addition to the measurements [138–141] which were summarised and tabulated in the Trajmar review [1]. The recent measurements of Randell et al. extend to very low energies (10 –175 meV), and thus overlap with the low-energy data of Ferch et al. [138], whilst the other experiments provide additional, accurate, absolute cross-sections at energies between 0.2 and 300 eV. These recent data are compared in Fig. 11 where it can be seen that the level of agreement for the total scattering cross-section for H2 between the various experiments is extremely good. The cross-section of Randell et al. is derived from measurements of total backward scattering, in conjunction with the “known” momentum transfer cross-section, using a high-resolution synchrotron radiation photoionization source. The momentum transfer cross-section used was that of England et al. [50] and they suggest that their backward scattering measurements are sensitive enough to imply that the m is too low at very low energies. Their calculated total cross-sections are in reasonable agreement with the data of Ferch et al. although they exhibit a little higher scatter. With the exception of their lowest energy data point at 0:21 eV, the cross-section of Subramanian and Kumar is in good agreement with that of Ferch et al. below 1 eV. Between 1 and 10 eV the

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◦

247

•

Fig. 11. Grand total cross-section for electron scattering from H2 . ( ) Ferch et al.; ( ) Deuring et al.; ( ) Jones; (4) Nickel et al.; (×) Subramanian and Kumar; () Randell et al.

data of Subramanian and Kumar, Jones and Nickel et al. are in excellent agreement, as are the values of the latter two at energies above 10 eV where they overlap. In general, the data of Deuring et al. lie a little lower in absolute magnitude than the other measurements at energies between 6 and 20 eV. At higher energies there is excellent agreement between Deuring et al., Jones and Nickel et al. A general summary of the recent grand total cross-section measurements for H2 appears in Table 1. 3.1.2. Elastic scattering With a few exceptions (e.g. [142]) which will be discussed later, the di@erential elastic scattering data for H2 comprise cross-sections which are rotationally averaged. Since the publication of the Trajmar review, there have been measurements of rotationally averaged elastic cross-sections by Furst et al. [143], Nishimura et al. [144], Khakoo and Trajmar [145], Buckman et al. [51] and Brunger et al. [31,53]. Furst et al. covered an energy range from 1 to 19 eV, Nishimura et al. from 2.5 to 200 eV, Khakoo and Trajmar from 15 to 100 eV and Brunger et al. from 1.0 to 5 eV. Previous measurements have been well summarised in the Trajmar review [1] and will not be repeated in detail here. The cross-sections from these most recent studies are presented in Table 2. Each of these cross-section sets has been placed on an absolute scale by use of the relative Pow technique, although the comparison between them is slightly confused by the fact that each has used a di@erent set of helium cross-sections as their “standard”. Furst et al. used their own helium cross-sections, Nishimura et al. used the cross-section set of Register et al. [77] as did Khakoo and Trajmar. Buckman et al. and Brunger et al. used the theoretical cross-sections of Nesbet [79]. Nonetheless, this does not

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Table 1 Grand total cross-sections for electron scattering from H2 in units of 10−16 cm2 Energy (eV) 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.1 0.12 0.15 0.2 0.21 0.25 0.3 0.35 0.4 0.42 0.45 0.5 0.6 0.64 0.73 0.8 0.88 1 1.09 1.14 1.2 1.4 1.5 1.6 1.8 2 2.18 2.2 2.4 2.5 2.6 2.66 2.8 2.85 2.9 3 3.2 3.23 3.41

Ferch et al.

Deuring et al.

Jones

Nickel et al.

Subramanian and Kumar

7.41 7.77 8.05

Randell et al.

7.24 8.44 8.94 8.84 8.89

8.52 8.79 9.19

9.7 8.97 8.64

9.75 10.05

11.47

10.45 10.71 10.95 11.21

11.28

11.36 11.62 11.9

11.54 12.39

12.4 12.8

13.26

13.2 13.5 13.7

13.85 14.4 14.62 14.86 15.26 15.58 15.99 16.14 16.27 16.3 16.4 16.46 16.46 16.48

12.66 13.67 13.64

15.59 16.03

16.18 16.46

16.3 16.34

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249

Table 1 (Continued) Energy (eV) 3.5 3.8 4 4.38 4.5 4.58 4.77 5 5.01 5.24 5.5 6 6.5 6.55 7 7.22 7.5 7.78 8 8.5 9 9.14 9.5 10 11 12 13 14 15 16 17.5 18 20 25 30 35 40 45 50 60 70 75 80 90 100

Ferch et al.

Deuring et al.

Jones

Nickel et al.

16.36 16.2 16.1

16.07

15.67 15.2

15.36

13.01

14.7 14.16 13.66

14.28

12.48

13.14

10.8 10.49 9.8 8.79 8.18 8.13 8.12 6.67 5.22 4.75 4.06 3.42 3.2 2.87 2.71 2.44

12.18 11.76 11.33 10.98 10.6 9.88 9.32 8.81 8.5 7.91 7.52 6.9 6.38 5.41 4.78 4.3 3.94 3.64 3.4

15.65 15.75 15.22 15.28 14.85 14.57 13.67 13.05

12.68 12.02

Subramanian and Kumar

12.38

12.22

11.23 10.73 9.49 8.49 7.72 7.12 6.54 5.59 4.98 4.2 3.7 3.34 3.08 2.87 2.69 2.53

Randell et al.

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Table 2 Di@erential cross-sections for elastic scattering (rotationally summed) from H2 in units of 10−16 cm2 sr −1 (a–c) and 10−18 cm2 sr −1 (d) Energy (eV) (a) Angle 20 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 130

1.0 Brunger et al. 0.60 0.58 0.53 0.51 0.54 0.57 0.63 0.74 0.90 1.08 1.28 1.45

Furst et al. 0.50 0.61 0.60 0.80 0.76 0.77 0.79 0.86 0.86 0.96 0.97 1.18 1.05 1.18 1.19 1.31 1.36 1.40

2.0

1.25 Brunger et al. 0.60 0.56

1.5 Brunger et al. 0.78 0.72

Brunger et al. 1.26 1.20

0.51

0.66

1.04

0.43

0.61

0.94

0.47

0.62

0.81

0.57

0.70

0.81

0.75

0.79

0.76

0.86

0.92

0.86

1.04

1.12

1.08

1.25

1.28

1.20

1.51 1.66

1.46 1.63

1.35 1.49

Furst et al. 1.30 1.27 1.24 1.08 1.02 0.91 0.84 0.80 0.80 0.84 0.89 0.88 0.98 0.99 1.00 1.09 1.19 1.25

Energy (eV) (b) Angle 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 130

2.5 Brunger et al. 1.69

3.0 Brunger et al. 1.97

4.0 Furst et al.

1.50

1.70

1.12

1.37

0.95

1.21

0.79

1.07

0.74

0.84

0.75

0.75

0.83

0.79

0.93

0.99

1.09

1.18

1.31 1.43

1.28 1.41

2.12 1.98 1.73 1.49 1.30 1.24 1.11 1.05 1.02 0.89 0.85 0.84 0.82 0.81 0.78 0.78 0.86 0.91

5.0 Brunger et al. 2.81 2.25 1.78 1.42 1.14 0.94 0.79 0.71 0.71 0.73 0.77 0.83

Furst et al.

6.0 Furst et al.

2.65 2.54 2.12 2.04 1.78 1.67 1.36 1.30 1.10 1.00 0.92 0.91 0.79 0.74 0.73 0.76 0.78 0.79 0.80

2.47 2.40 1.88 1.78 1.66 1.05 1.29 1.22 1.03 0.97 0.87 0.81 0.77 0.70 0.62 0.63 0.64 0.64 0.66

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Table 2 (Continued) Energy (eV) (c) Angle 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125

8.0 Furst et al. 2.4 2.26 1.97 1.76 1.58 1.33 1.24 1.11 0.99 0.89 0.72 0.69 0.63 0.59 0.51 0.48 0.44 0.44 0.41

10 Furst et al. 2.99 2.62 2.31 2.03 1.79 1.55 1.24 1.15 1.04 0.91 0.78 0.66 0.58 0.53 0.49 0.41 0.38 0.34 0.31 0.30

12 Furst et al. 3.53 3.10 2.66 2.30 1.91 1.76 1.40 1.26 0.99 0.85 0.71 0.61 0.54 0.49 0.41 0.35 0.32 0.30 0.26 0.26

(d)a Angle 20 30 10 15 251 20 207 184 25 177 134 30 156 102 35 128 80.9 40 107 61.3 45 86.1 46.2 50 73.8 35.4 55 59.9 28.6 60 48.2 22.1 65 40.9 18.3 70 35.3 15.0 75 29.3 12.5 80 24.8 10.2 85 21.1 8.62 90 18.0 7.46 95 15.5 6.58 100 13.2 5.81 105 11.3 5.13 110 9.9 4.83 115 8.65 4.45 120 8.35 4.36 125 8.11 4.18 a The data are those of Khakoo and Trajmar.

15 Khakoo and Trajmar 2.32 1.98 1.74 1.47 1.30 1.14 0.95 0.78 0.67 0.56 0.48 0.41 0.34 0.30 0.27 0.24 0.21 0.19 0.18 0.17 0.16 0.15

17.5 Khakoo and Trajmar 2.20 1.80 1.62 1.34 1.14 1.00 0.85 0.72 0.61 0.52 0.45 0.36 0.32 0.27 0.24 0.20 0.17 0.15 0.13 0.11 0.10 0.094

19 Furst et al. 2.84 2.61 2.26 1.81 1.38 1.12 0.91 0.79 0.63 0.51 0.39 0.32 0.26 0.23 0.22 0.17 0.14 0.13 0.12 0.12

Energy (eV) 40 180 142 102 72.4 52.0 39.6 30.3 23.4 18.3 14.4 11.3 9.45 7.66 6.40 5.40 4.69 4.17 3.68 3.40 3.09 2.87 2.75 2.67

60 116 91.3 67.8 47.2 33.0 22.8 16.7 12.0 9.25 7.24 5.53 4.38 3.50 2.91 2.47 2.10 1.90 1.66 1.55 1.39 1.25 1.15 1.08 1.02

100 122 84.2 57.6 37.4 23.7 16.0 11.1 7.1 5.28 3.94 2.99 2.37 1.94 1.63 1.31 1.19 1.06 0.97 0.90 0.81 0.72 0.68 0.66 0.64

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Fig. 12. Di@erential cross-sections for rotationally summed elastic electron scattering from H2 at (a) 1:0 eV and (b) 2:5 eV. (4) Linder and Schmidt; (+) Furst et al.; () Nishimura et al.; ( ) Shyn and Sharp; ( ) Brunger et al.; (—); Morrison et al.

•

substantially e@ect the overall qualitative comparison that can be made between these and other measurements. Comparisons between these results and some of the earlier measurements are given in Figs. 12–14. In Fig. 12 we show the elastic DCS at incident energies of 1.0 and 2:5 eV. At 1:0 eV comparing the more recent results of Furst et al. and Brunger et al. with the earlier results of Linder et al. [142] there is clearly good qualitative, and reasonably good quantitative, agreement between these cross-sections. Also shown is the result of a vibrational close coupling calculation by Morrison and colleagues which is in good overall agreement with experiment. At an energy of 2:5 eV, we show a comparison between a number of results [53,142,144,146]. Once again there is good overall agreement, both in shape and magnitude, between the DCS of Linder and Schmidt, Shyn and Sharp and Brunger et al. The DCS of Nishimura et al. has both a di@erent overall shape, and is much larger (40 – 60%) in absolute magnitude at forward scattering angles. The theory of Morrison et al. is in good general accord with all the DCS measurements, with the exception of that of Nishimura et al. Although we do not illustrate it here, similar trends are observed in a comparison of the available data at an energy of 3 eV. In Fig. 13, we show a comparison between a number of results [91,142–144,146,147] for incident energies of 10 and 20 eV. At 10 eV there are substantial di@erences in these DCS,

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Fig. 13. Di@erential cross-sections for rotationally summed elastic electron scattering from H2 at (a) 10 eV and (b) 20 eV. (4) Linder and Schmidt; ( ) Srivastava et al.; (+) Furst et al.; () Nishimura et al.; ( ) Shyn and Sharp; ( ) Khakoo and Trajmar.

◦

•

particularly at forward and backward scattering angles. For example, the cross-section of ◦ ◦ Furst et al. is almost 50% larger than that of Srivastava et al. at 20 , whilst at 110 these two measurements are in good agreement but about 30% lower than that of Shyn and Sharp. It would appear that the weight of experimental evidence supports the larger cross-section in the forward direction as the data of Nishimura et al. and Shyn and Sharp also show strongly enhanced scattering at small angles. At 20 eV there is relatively good agreement between the DCS of Shyn and Sharp, Nishimura et al. and Khakoo and Trajmar over the whole angular range. The cross-section of Furst et al. (at 19 eV) is higher at forward angles whilst that of Srivastava et al. is lower at all scattering angles. Finally, we show the elastic DCS for incident energies of 40 and 100 eV in Fig. 14. At 40 eV we again see good agreement between the data sets of Shyn and Sharp, Nishimura et al. ◦ ◦ and Khakoo and Trajmar at scattering angles less than about 90 , whilst at 125 the latter is about 30% larger in magnitude. The cross-section of Srivastava et al. is once again smaller across the entire angular range. At 100 eV, we make a comparison between several measurements [140,144 –146,148]. In general, there are di@erences as large as 40% between the earlier cross-section measurements [1]. The more recent data of Khakoo and Trajmar are in good

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Fig. 14. Di@erential cross-sections for rotationally summed elastic electron scattering from H2 at (a) 40 eV and (b) 100 eV. ( ) Srivastava et al.; (4) Fink et al.; ( ) van Wingerden et al.; () Nishimura et al.; ( ) Shyn and Sharp; ( ) Khakoo and Trajmar.

◦

•

agreement with the cross-sections of both Fink et al. and van Wingerden et al., whilst that of Nishimura et al. lie well below most of the other measurements at all scattering angles. As this small sample of data indicates, the situation regarding the DCS for elastic scattering from H2 at energies below 100 eV is not as good as one might hope for this simplest of molecular species. In particular the largest di@erences appear to be at the higher energies (10 eV and above) where, in principal, the experimental diJculties are less, notwithstanding the relatively strong forward scattering that occurs. Some of the above di@erential elastic measurements have been extrapolated and integrated to provide total elastic and elastic momentum transfer cross-sections. Elastic momentum transfer cross-sections are also available from the swarm experiments of Crompton and colleagues [50,149,150] as well as from those of Schmidt et al. [44]. A sample of the total elastic cross-section data is given in Fig. 15 and tabulated in Table 3. As can be seen from the Fgure, there is reasonably good agreement between most of the measurements at energies below 3 eV and above about 15 eV, whilst between these two limits there are substantial di@erences. In general, the values of Srivastava et al. and Nishimura et al. are higher than most other derived cross-sections which is consistent with their results for the DCS. The elastic momentum transfer cross-section is shown in Fig. 16 and also tabulated in Table 4. There is generally excellent agreement between the two swarm results with the exception of the region around 1 eV where

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•

Fig. 15. Integral elastic cross-section for H2 between 0.25 and 100 eV. (N) Linder and Schmidt; ( ) Srivastava et al.; ( ) Shyn and Sharp; () Nishimura et al.; (+) Furst et al.; ( ) van Wingerden et al.; ( ) Khakoo and Trajmar; () Brunger et al.

◦

they di@er by up to 5%. Given the uncertainties in the other cross-sections which have been derived from DCS measurements, the level of agreement is acceptable. There has been a report by Freeman and Wada [151] of a Ramsauer–Townsend minimum in the momentum transfer cross-section for H2 at an energy of 9 meV, based on measurements of electron mobilities. This has been challenged by Crompton and Morrison [152], who argue that the e@ect is due to a systematic error in the measurements of Freeman and Wada. 3.1.3. Rotational excitation As mentioned above, there have been several measurements of rotationally resolved crosssections for H2 . This is one of the few molecules for which such measurements can be unambiguously performed as the rotational spacing in the ground X 1 W+ g state is of the order of 45 meV and thus the excitation of rotational states can be readily resolved in a high-resolution experiment. The analysis of room temperature rotational scattering data for naturally occurring H2 , a mixture of para- and ortho-hydrogen, is further complicated by the rotational population being distributed amongst the J = 0; 1; 2; 3 levels. However, the UJ = ± 2 selection rule for electron impact excitation limits the number of possible excitation=de-excitation channels that need to be considered and a number of studies have been carried out. Such measurements include the early di@erential measurements [142,147,153], the recent DCS measurements of Jung et al. [154] and Sohn et al. [155] and the swarm-derived total cross-sections of Crompton and co-workers [50,149]. A good summary of most of this data, and a detailed comparison of the level of agreement with theory, has been given [131], so we will only provide a brief discussion here.

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Table 3 Integral elastic cross-sections for H2 in units of 10−16 cm2 Energy (eV) 1.0 1.25 1.5 2.0 2.5 3.0 4.0 5.0 6.0 7.0 8.0 10 12 15 17.5 19 20 25 30 40 50 60 70 80 90 100

Shyn and Sharp

14.4 15.8 15.9 13.6

Nishimura et al.

17.1 16.8 16.2 15.4

11.3

12.2 10.1

7.6

7.2

5.6

5.7 4.3 3.3 2.2 1.6 1.4 1.2 0.87 0.83 0.65

3.4 2.5 1.3

0.77

Furst et al.

Brunger et al.

14.2

11.7 13.0 13.9 14.6 15.1 15.5

15.7 15.1 15.2 13.3 11.3 10.2 9.8 6.8

Khakoo and Trajmar

13.9

7.04 6.27 5.55 3.63 2.52 1.15

0.834

Linder and Schmidt [142] measured an extensive array of DCS for pure elastic scattering (UJ = 0), rotational excitation for J = 1–3, pure vibrational excitation and rovibrational excitation over an energy range from 0.3 to 15 eV. Wong and Schulz measured rotational excitation DCS for J = 1–3 at 4:5 eV and Srivastava et al. also measured J = 1–3 excitation between 3 and 100 eV at selected scattering angles. This data has been tabulated in the Trajmar review [1]. More recently, Sohn et al. [155] have measured J = 1–3 rotational excitation DCS at incident energies of 0.2 and 0:6 eV. In general, all these di@erential data are in good agreement at those energies where a comparison is possible. There is also good agreement with theory for the pure rotational J = 1–3 DCS and for the ratio of pure rotational to rovibrational excitation as a function of both energy and angle, the latter providing a more meaningful test in many respects as it avoids the considerable normalisation problems involved in the determination of the absolute experimental cross-sections (see [131]). The other point of comparison for rotational excitation comes at the integral cross-section level where the swarm technique can supply accurate values at energies below the threshold for vibrational excitation (≈ 0:5 eV). There have been several measurements of the J = 0–2 and

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◦

Fig. 16. Elastic Momentum transfer cross-section for H2 . ( ) Shyn and Sharp; ( ) Khakoo and Trajmar; () Nishimura et al.; ( ) Brunger et al.; ( – – ) England et al.; (- - -) Schmidt et al.

•

1–3 rotational excitation cross-sections by Crompton and co-workers [50,149,150]. In particular, Crompton et al. [149] measured transport parameters in para-hydrogen and the reduction in excitation channels enabled them to unambiguously determine the cross-section for the J = 0–2 excitation below 0:5 eV, without having to resort to theory or other experiment as a guide for the strength or energy dependence of other collision processes. A comparison of this cross-section with the close-coupling theory of Morrison et al. is shown in Fig. 17. This in turn enabled the determination of the J = 1–3 cross-section over the same energy region and a comparison between these derived cross-sections, the close-coupling calculation and the integral J = 1–3 rotational cross-section derived from the crossed beam measurements of Linder and Schmidt [142] is shown in Fig. 18. Recent values for the rotational excitation cross-sections are tabulated in Table 5. 3.1.4. Vibrational excitation The magnitude of the near-threshold vibrational excitation cross-sections, particularly that for the v = 0–1 excitation, has been the subject of considerable uncertainty and debate for many years. This uncertainty arose initially due to the disagreement between total cross-sections for the Frst vibrational excitation derived from swarm [50,149] and single collision [142,156] experiments. At energies below a few eV these cross-sections disagreed with each other by as much as 60% with the “beam” cross-sections being larger than the swarm. This situation, although important, received little further attention until the 1980s when advances in electron– molecule scattering theory were such that attention was focussed on rotational and vibrational excitation of this fundamental diatomic system. The experimental work prior to 1983 has been well summarised and tabulated [1]. Since that time, there has been further experimental and

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Table 4 Elastic momentum transfer cross sections for H2 in units of 10−16 cm2 Energy England et al. Schmidt et al. Shyn and Sharp Nishimura et al. Khakoo and Trajmar Brunger et al. (eV) 0.01 0.02 0.03 0.04 0.046 0.05 0.06 0.07 0.08 0.09 0.1 0.13 0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.9 1 1.1 1.25 1.4 1.5 1.6 1.8 2 2.5 3 4 5 6 7 8 10 15 20 25

7.26 7.95 8.45 8.91 9.05 9.22 9.5 9.79 10.04 10.24 10.44 10.93 11.33 11.93 12.92 13.82 14.61 15.51 16.2 17 17.3 17.59

7.719 8.138 8.558 8.978 9.103 9.267 9.525 9.791 10.04 10.25 10.47 11.08 11.43 12.06 13 13.71 14.45 15.19 15.64 16.14 16.46 16.84

18.09 18.14 18.19 18.09 17.89 17.69 16.9 14.71 12.92 11.93

18.1 18.42 18.59 18.15 17.9 17.35 16.62 13.7

7.6 1.49

14.59 16.84 16.41 17.1 17 14 10.14 7.07 3.29 2.11

16.51 17.11 16.33

16.5 16.8 14.1

11.45

12.7 8.03 6.14 3.2 2.27 1.54

3.26 2.21

theoretical e@ort on this problem. Firstly Morrison and co-workers [131] carried out a number of vibrational close-coupling calculations on electron–hydrogen scattering and, whilst we will not discuss the theory in any detail here, their calculations for the integral v = 0–1 cross-section largely agreed with the crossed beam experiments and not the swarm-derived values. A similar situation is apparent from the recent complex Kohn calculations of Rescigno et al. [52].

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Fig. 17. Total cross-section for the J = 0–2 rotational excitation of H2 . (—) close-coupling theory of Morrison et al., ( ) swarm-derived result of England et al.

•

Fig. 18. Total cross-section for the J = 1–3 rotational excitation of H2 . Comparison of (—) close-coupling theory of Morrison et al., ( ) the swarm-derived result of England et al. and ( ) the crossed beam measurements of Linder and Schmidt.

•

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Table 5 Rotational excitation cross sections for H2 in units of 10−16 cm2 Energy (eV)

England et al. 0 –2

0.044 0.047 0.05 0.055 0.06 0.065 0.07 0.0727 0.075 0.08 0.085 0.09 0.095 0.1 0.11 0.12 0.13 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.6 0.7 0.8 0.9 1

0 0.0185 0.027 0.035 0.042 0.048 0.053 0.062 0.068 0.074 0.079 0.088 0.097 0.115 0.132 0.152 0.175 0.2 0.228 0.26 0.323 0.394 0.469 0.555 0.636

England et al. 1–3

0 0.007 0.014 0.0198 0.0237 0.0265 0.028 0.033 0.0364 0.0394 0.045 0.058 0.0719 0.086 0.1 0.114 0.1285 0.1439 0.1776 0.2135 0.2518 0.2919 0.333

Nishimura et al. [144] measured di@erential and derived integral cross-sections for the v = 0–1 excitation at energies above 2:5 eV, and whilst these measurements, by virtue of the normalisation process used, inherit the apparent problems discussed previously for elastic scattering by this group, the integral cross-sections were in reasonably good agreement with those of Ehrhardt et al. [156] at 2.5, 3.0 and 4:0 eV. Given the (apparently) substantial problems associated with these measurements for the elastic cross-section at low energies, they did not really serve to resolve the problem. Further swarm studies were undertaken by Crompton and colleagues during the 1980s using techniques involving the measurement of transport parameters in gas mixtures. Although these experiments resulted in small di@erences in the Fnal swarm-cross-section set of England et al. [50], the overall conclusion regarding the above discrepancy between beam and swarm measurements was unchanged. Also, the use of a numerical optimisation technique to Fnd the optimal hydrogen cross-section set which was compatible with the transport parameters [50] did not reveal any major di@erences in the cross-sections [157].

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Fig. 19. Di@erential cross-sections for the rovibrational excitation of H2 (; = 0–1) at incident energies of (a) 1.0 and (b) 1:5 eV. (4) Linder and Schmidt; ( ) Brunger et al.; (—) Morrison et al.; ( – – ) Rescigno et al.

•

A series of absolute elastic and vibrational excitation measurements and scattering calculations were undertaken by Buckman et al. [51] and Brunger et al. [31,53] with the speciFc aim of addressing this discrepancy. These experiments, which involved measurements of the ratios of vibrational excitation to elastic scattering, were placed on an absolute scale by the use of the relative Pow technique for the elastic channel and a careful characterisation of the relative analyser transmission for elastic and inelastic electrons. At those energies where the discrepancy between the former measurements was largest, around 1:5 eV, the integral cross-sections derived from the new crossed beam results were in good agreement with the older crossed beam studies and with the vibrational close-coupling calculations, but substantially (≈ 60%) larger than the swarm-derived cross-section. Importantly, the di@erential scattering cross-sections were also in good agreement with the theory in both magnitude and shape. These di@erential results are summarised in Figs. 19 and 20 where we illustrate the cross-sections at a number of energies between 1.0 and 5:0 eV. A tabulation of recent DCS values is also provided in Table 6. There have been several other recent experimental and theoretical investigations of the ; = 0–1 excitation cross-section and although not all of them have involved absolute crosssections, given the extent of the discrepancy it is worth commenting upon them in a little detail. Firstly, it is worth noting the experiments of Allan [158], who measured the energy

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Fig. 20. Di@erential cross-sections for the rovibrational excitation of H2 (; = 0–1) at incident energies of (a) 2.5 and (b) 5:0 eV. (4) Linder and Schmidt; () Nishimura et al.; ( ) Brunger et al.; (—) Morrison et al.; ( – – ) Rescigno et al.

•

dependence of the forward plus backward di@erential vibrational excitation cross-sections (; = 0– 1; 2; 3; 4; 5; 6) using a trochoidal spectrometer. Whilst it is not entirely clear that the comparison is a valid one, he found that the energy dependence of the cross-sections for the Frst three vibrational modes were in good agreement with those of Ehrhardt et al. [156]. Secondly, a new generation of swarm experiments has been implemented by Schmidt and colleagues [44] (see Section 2.3 for details) to measure transport parameters (drift velocity and both transverse and longitudinal di@usion coeJcients) in both pure electric and crossed electric and magnetic Felds. The vibrational excitation cross-section derived from these measurements is considerably higher than the previous swarm result above 0:6 eV, e@ectively bisecting the di@erence between the beam and swarm cross-sections outlined above between 0.6 and 2:0 eV. In this region, the largest di@erence between the beam and new swarm results is now about 30%. Finally, the Kohn variational technique has recently been applied to the near-threshold vibrational (; = 0–1) excitation problem and the calculated cross-sections are in excellent agreement with the beam measurements of Brunger et al. [53] for both di@erential and total cross-sections at energies between 1.0 and 5 eV. All of these recent results for the total vibrational cross-section at energies between threshold and 5 eV are illustrated in Fig. 21 where they are also contrasted with the original swarm and beam cross-sections. Tabulated values are provided in Table 7.

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Table 6 Vibrational excitation cross sections (0–1) for H2 in units of 10−18 cm2 sr −1 (a)a Angle 5 10 20 30 40 50 60 70 80 90 100 110 120 130 (b)b Angle 15 20 25 30 40 50 60 70 80 90 100 110 120

Energy (eV) 1.0 — — 1.22 0.99 0.74 0.65 0.57 0.40 0.32 0.32 0.40 0.46 0.57 0.78

1.25 — 2.77 2.49 2.15 1.62 1.17 0.91 0.87 0.81 0.78 0.89 1.11 1.24 1.62

1.50 4.96 4.74 4.3 3.65 3.23 2.79 2.39 1.91 1.61 1.45 1.36 1.44 1.59 1.78

2.0 6.86 6.45 5.72 4.98 4.06 3.37 2.79 2.13 1.80 1.66 1.64 1.64 1.91 2.21

2.5 9.14 8.51 7.65 6.18 4.94 4.00 3.22 2.36 1.99 1.78 1.78 1.90 2.26 2.81

3.0 9.59 8.73 7.79 6.28 5.29 4.47 3.52 2.88 2.13 1.76 2.17 2.65 2.85 3.48

5.0 8.19 7.18 6.39 4.91 3.65 2.65 2.22 1.68 1.44 1.37 1.41 1.59 1.94 2.19

15.0 0.999 0.807 — 0.507 0.326 0.208 0.167 0.147 0.194 0.221 0.308 0.442 0.540

20.0 — 0.518 — 0.217 0.141 0.124 0.104 0.117 0.135 0.168 0.201 0.270 0.322

Energy (eV) 2.5 — 5.59 5.47 5.07 4.15 3.21 3.07 2.82 2.92 2.82 2.90 3.06 3.34

3.0 — 6.20 5.64 5.48 4.01 3.24 3.16 2.40 2.31 2.10 2.13 2.64 2.92

4.0 — 6.99 — 6.25 5.50 4.06 3.71 2.92 2.53 2.67 2.54 2.74 3.04

6.0 — 3.51 — 3.23 2.93 2.32 1.88 1.68 1.56 1.39 1.46 2.11 3.39

8.0 — 2.99 — 2.31 1.87 1.54 1.30 1.18 1.09 1.02 1.17 1.26 1.64

Energy (eV) (c)b Angle 25 30 40 50 10 1.08 — 0.726 — 15 0.669 0.523 0.254 0.195 20 0.333 0.254 0.049 0.073 30 0.112 0.073 0.055 0.054 40 0.074 0.073 0.080 0.076 50 0.075 0.078 0.085 0.082 60 0.078 0.099 0.108 0.010 70 0.093 0.084 0.081 0.092 80 0.108 0.089 0.092 0.095 90 0.124 0.095 0.090 0.078 100 0.150 0.096 0.076 0.061 110 0.195 0.125 0.069 0.057 120 0.230 0.146 0.064 0.045 a The data are the di@erential scattering measurements of Brunger et al. b The data are the di@erential scattering measurements of Nishimura et al.

10.0 2.13 1.70 — 1.41 1.19 0.950 0.682 0.676 0.526 0.639 0.699 1.01 1.50

60 0.350 0.113 0.065 0.072 0.084 0.103 0.099 0.092 0.074 0.060 0.043 0.036 0.029

80 0.249 — 0.048 0063 0.077 0.085 0.079 0.072 0.054 0.043 0.032 0.031 0.018

100 0.167 — 0.037 0.045 0.062 0.069 0.062 0.052 0.044 0.030 0.023 0.020 0.015

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◦

Fig. 21. Integral cross-section for rovibrational excitation (; = 0–1) of H2 from threshold to 7 eV. ( ) Ehrhardt et al.; (4) Linder and Schmidt; () Nishimura et al.; ( ) Brunger et al.; (—) Morrison et al.; ( – – ) England et al.; (- - -) Rescigno et al.; (- - -) Schmidt et al.

•

The new experimental and theoretical values would seem to favour a vibrational excitation cross-section which is higher than that provided by the swarm analysis of Crompton and colleagues (e.g. [50]). However the reasons for this discrepancy, which has been extensively studied over the past ten years, still remain elusive. 3.1.5. Electronic excitation At the time of the Trajmar review, there were only fragmentary measurements of absolute cross-sections for the electronic excitation of molecular hydrogen by electron impact. Some of these were actual measurements of scattered electron intensities whilst most consisted of measurements of optical excitation functions. The main drawback of the latter technique is the, at times, unknown role that cascade contributions from higher lying states can make to the intensity of a given optical line. These early measurements have been described and tabulated by Trajmar et al. and we shall not discuss them further here. Since the early 1980s there has been, relatively speaking, a marked increase in the number of measurements of cross-sections for the electron impact excitation of H2 . Hall and Andric [159] measured near threshold DCS for the b3 W+ u state, Khakoo and Trajmar [145] measured DCS for 3 + 1 + 3 1 the a Wg ; B Wu ; c Vu , and C Vu states, Nishimura and Danjo [160] measured the DCS for the excitation of the b3 W+ u state as did Khakoo et al. [161] and Khakoo and Segura [162]. Finally, the total excitation cross-section for the c3 Vu state was measured by Mason and Newell [163]. In the experiments of Khakoo and collaborators, the spectral deconvolution techniques, which were outlined in Section 2.7 were used to great advantage to unravel the complex excitation spectrum.

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Table 7 Integral vibrational excitation cross-sections (0 –1) for H2 in units of 10−18 cm2 Energy Ehrhardt et al. (eV) 0.535 0.55 0.56 0.60 0.65 0.66 0.74 0.75 0.80 0.85 0.90 0.95 1.0 1.1 1.2 1.25 1.30 1.40 1.48 1.50 1.6 1.8 2.0 2.2 2.4 2.46 2.5 2.6 2.80 3.0 3.1 3.4 3.5 3.8 4.0 4.5 4.6 4.7 5.0 5.3 5.6 6.0 6.1 7.0 8.0 10.0

Linder and Schmidt

Nishimura et al.

Brunger et al.

1.0 1.7 2.7 3.8 4.7 5.9 9.4

8.0

17.2

16.9

24.0

29.0

25.0

40.3 44.8

33.2

48.3 50.6 51.1 49.8 46.1 38.0 32.4 30.2 27.0 21.0

45.0

40.1

47.9

45.9

43.0 37.2

46.2

31.1 28.2

36.9 20.4 15.2

England et al.

Schmidt et al.

0.3 0.7 1.1

0.021 0.033 0.045 0.80 0.99

2.2

1.64

3.4

2.80

4.8 5.7 7.4 9.4

4.83 6.07 8.20 10.22

11.4 13.4

12.53 14.32

17.6 21.8

19.6 25.5

30.3 34.3

35.3 36.8

38.3 41.2 44.1 45.1 46.8 46.8 45.5 44.0 39.0

36.3

37.3 35.0 30.7 28.0 21.0

32.6

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Table 8 Di@erential and integral cross-sections (in units of 10−18 cm2 sr −1 ) for the excitation of the B1 W+ u state of H2 . The data are those of Khakoo and Trajmar. The uncertainty on the DCS values varies between 18% and 20% and that for the ICS between 19% and 21%. Integral cross-sections (in units of 10−18 cm2 ) are given at the foot of each column Energy (eV) Angle

20

30

40

60

10 20 30 40 50 60 70 80 90 100 110 120

18.6 8.99 4.26 2.77 2.05 1.55 1.20 0.927 0.783 0.621 0.508 0.490

41.8 10.7 3.61 2.23 1.59 1.15 0.920 0.755 0.623 0.504 0.372 0.316

66.8 19.8 5.02 1.83 0.918 0.644 0.545 0.433 0.350 0.252 0.197 0.151

86.3 7.22 1.86 0.888 0.564 0.424 0.218 0.204 0.133 0.110 0.100 0.093

Qi

21.24

24.37

30.37

29.51

The B1 /u+ state. The “B state” is the lowest lying excited singlet state in H2 , the ground vibrational level occurring at 11:19 eV. The Frst measurements of electron scattering cross-sections for this state were made by Weingartshofer et al. [164] and Srivastava and Jensen [165]. These data have been described and tabulated in the review of Trajmar et al. and the latter set has also been renormalised by these authors. To our knowledge the only recent di@erential scattering measurements for this excited state are those of Khakoo and Trajmar [145] at incident energies ◦ ◦ of 20; 30; 40 and 60 eV for a range of electron scattering angles from 10 to 120 . This data is presented in Table 8 and two examples of the DCS, at 20 and 60 eV, are presented in Figs. 22(a) and (b), respectively. At 20 eV there is reasonably good agreement between the experimental DCS of Khakoo and Trajmar and Srivastava and Jensen. The exception appears to be at backward angles where the more recent results appear to be substantially lower in magnitude. This cross-section has also been calculated using a number of theoretical approaches, including the multichannel Schwinger variational technique of Gibson et al. [166] and the R-matrix technique of Branchett et al. [167], and comparison with experiment can be found in these publications. At 60 eV, a similar comparison is shown in Fig. 22(b) between the experimental results [145,165] and once again the level of agreement is quite good. Comparisons at other energies can be found in Khakoo and Trajmar [145]. Integral cross-sections for the excitation of this state are available via the integration of the above DCS, from optical excitation function measurements, and from a variety of theoretical calculations. The more recent experimental values are given in Table 8 and a comparison of some of the available data and theory is given in Fig. 22(c).

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267

•

Fig. 22. Di@erential cross-section for the excitation of the B1 W state of H2 at (a) 20 eV: (b) 60 eV; ( ) Khakoo and Trajmar: ( ) Srivastava and Jensen. (c) Integral cross-section for the B1 W state of H2 , ( ) Khakoo and Trajmar: ( ) Srivastava and Jensen, (—) Gibson et al.

•

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Table 9 Di@erential cross-sections (in units of 10−18 cm2 sr −1 ) for the excitation of the b3 W+ u state of H2 . The data of various authors are given. The integral cross-sections (10−18 cm2 ) derived from the di@erential measurements are shown at the foot of each column Energy (eV)

Angle 20 30 40 50 60 70 80 90 100 110 120 125 130 Qi

9.2 Khakoo and Segura 0.18 0.15 0.19 0.28 0.27 0.31 0.35 0.63 0.97 1.55

12.1

10.2 Khakoo and Segura

10.5 Hall and Andric

12 Hall and Andric

12 Nishimura and Danjo

0.28 0.33 0.32 0.45 0.71 0.91 1.00 1.09 1.41 2.18 2.62

20.80

1.52 1.62 2.17 2.05 2.29 2.34 2.45 3.12 3.47 3.4 5.0 5.46 39.0

54.0

12.2 Khakoo and Segura 1.33 1.25 1.47 1.36 1.61 1.81 2.13 2.67 3.02 5.01 6.53

13 Nishimura and Danjo 1.68 1.51 1.68 1.68 2.74 2.49 2.93 3.12 3.64 4.70 5.45 6.91

40.0

49.4

51.8

Energy (eV)

Angle 10 15 20 25 30 35 40 45 50 60 70 80 85 90 100 110 120 125 130 Qi

15 Nishimura and Danjo

15.2 Khakoo and Segura

17 Nishimura and Danjo

17.2 Khakoo and Segura

20 Khakoo et al.

6.08

4.59 4.24

5.03

4.62

4.33

3.65

4.32

4.57

4.04

2.68

4.09

3.61

3.56

2.89 2.85 3.02 3.50

3.87 3.44 3.53 3.68

3.42 3.37 3.09 3.55

3.14 3.18 2.94 3.99

3.79 4.04 4.60 5.75

4.19 5.55 6.57 8.63

3.44 3.60 3.53 3.45

3.35 4.26 5.17 5.65

6.25 5.72 5.26 4.87 4.13 3.82 3.44 3.36 3.27 3.49 3.49 3.85 3.68 3.78

6.94 60.4

20 Nishimura and Danjo

20.2 Khakoo and Segura

5.60 4.23 4.54

4.44

3.44

3.92

3.09

3.49

2.86 2.86 2.46 2.23

2.82 2.35 2.09 1.86

2.92 3.28 4.14 4.56

2.35 2.45 2.98 3.32

1.27 83.0

53.7

57.3

49.3

48.4

39.2

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269

Table 9 (Continued) Energy (eV)

Angle 10 15 20 30 40 50 60 70 80 90 100 110 120 130 Qi

30 Nishimura and Danjo 3.26 2.93 2.68 2.37 2.26 1.81 2.02 1.67 1.72 1.68 1.44 1.40 1.46 1.27 21.1

40 Nishimura and Danjo

60 Nishimura and Danjo

1.50 1.32 1.21 1.47 1.46 1.33 1.19 0.858 0.614 0.436 0.360 0.256 0.229 0.218

1.08 1.02 0.876 0.980 0.898 0.626 0.464 0.356 0.192 0.158 0.099 0.0826 0.0678

8.21

3.80

The b3 /u+ state. One of the main dissociation pathways for the H2 molecule is via excitation of the lowest lying, repulsive, “b-state” which consists of a broad continuum occupying the excitation energy interval between about 6 and 15 eV. Prior to the Trajmar et al. review there was only one study of absolute excitation cross-sections for this state, by Weingartshofer et al. and these values have been tabulated by Trajmar et al. Since that time there have been several other studies which have yielded absolute scattering cross-sections. Hall and Andric [159] measured the near-threshold cross-section for the b-state with a view to investigating the role of resonances in this scattering channel. They measured DCS at four energies, 10:5; 11:0; 11:6 and 12:0 eV and also derived integral cross-sections at each of these energies. Unfortunately, they do not provide tabulated cross-sections. Nishimura and Danjo [160] measured DCS for the b-state ◦ ◦ at incident energies between 12 and 60 eV and at scattering angles from 10 to 130 . Their technique involved integrating the scattered electron signal over the energy loss range from 7 to 10:38 eV and absolute values were obtained by normalising to the elastic electron signal, the absolute cross-section for which had been obtained by normalising to the helium cross-section using the relative Pow technique. Khakoo et al. [161] also measured DCS in the 20 –100 eV energy range and Ftted the proFle of the b3 W+ u state, extrapolating the Ftted Franck–Condon envelope to an energy loss value of 15 eV. Their measurements cover the angular range of ◦ 15 –120 . More recently, Khakoo and Segura [162] have extended the earlier measurements of Khakoo and Trajmar to energies below 20 eV. These data are summarised in Table 9, along with the integral cross-sections that have been derived from them. An example of the DCS for the b-state excitation at an incident energy of 20 eV is shown in Fig. 23(a). Here we compare the experimental results of Nishimura and Danjo, Khakoo et al. and Khakoo and Segura with the calculated cross-sections of Lima et al. [168], who use

270

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•

Fig. 23. (a) Di@erential cross-section for the excitation of the b3 W state of H2 at 20 eV; ( ) Khakoo et al., ( ) Nishimura and Danjo, (×) Khakoo and Segura, (—) Lima et al. (b) Integral cross-section for the excitation of the b3 W state of H2 : ( ) Khakoo et al., ( ) Nishimura and Danjo, (×) Khakoo and Segura, (4) Hall and Andric, (—) Lima et al.

•

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271

Table 10 Di@erential cross-sections (in units of 10−18 cm sr −1 ) for the excitation of the c3 >u state of H2 . The data are those of Khakoo and Trajmar Energy (eV) Angle

20

30

40

60

20 30 40 50 60 70 80 90 100 110 120

2.69 2.65 2.40 1.85 1.52 1.32 1.10 0.927 0.743 0.656 0.601

1.91 1.15 0.661 0.540 0.282 0.230 0.166 0.170 0.175 0.182 0.193

0.927 0.755 0.336 0.178 0.132 0.088 0.056 0.060 0.058 0.078 0.085

0.263 0.200 0.184 0.103 0.064 0.045 0.024 0.026 0.024 0.030 0.036

the multichannel Schwinger variational technique. For reasons of clarity we do not show the error estimates on the data of Nishimura and Danjo but they are about twice those illustrated on the data of Khakoo et al. Thus, there is reasonable agreement between the three experiments as to the shape and magnitude of the cross-section. There is also reasonable agreement between the experiments and the calculation of Lima et al. Comparisons at other energies up to 60 eV can be found in previous publications [160,161]. In Fig. 23(b) we show the integral cross-sections for the b-state at energies between 10 and 100 eV. There is clearly quite good agreement between experiment and theory for this excited state cross-section. The c3 >u state. There have been several measurements of the excitation cross-section for the c-state in H2 since the Trajmar et al. review. Khakoo et al. measured DCS at energies between 20 and 60 eV using electron energy loss spectroscopy and their spectral deconvolution technique. Mason and Newell [163] took advantage of the metastability of this excited state to measure total excitation cross-sections using a time-of-Pight recoil technique. This work did not provide absolute values, the authors normalising to the distorted wave calculations of Lee et al. [169]. Nonetheless, we shall discuss it here as it provides a good description of the energy dependence of the cross-section from the threshold to about 60 eV. The measured di@erential cross-sections [145] are given in Table 10. Integral cross-sections are presented in Table 11 and here we have taken the liberty to normalise the relative values of Mason and Newell to the integral cross-sections of Khakoo et al., at an energy of 20 eV. Note also that the relative values of Mason and Newell were digitised from their Fgure. In Fig. 24(a), we illustrate an example of the c-state DCS at an energy of 30 eV where we compare the data of Khakoo et al. with the calculation of Lima et al. [168]. The cross-sections are similar in shape but there is a discrepancy in absolute magnitude of between a factor of two and three across the entire angular range. In Fig. 24(b) we show a comparison of some of the available integral cross-sections for the c-state. Absolute experimental cross-sections have been determined by Khakoo and Trajmar, by

272

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Table 11 Integral cross-sections (in units of 10−18 cm2 ) for the excitation of the c3 Vu state of H2 . The relative data of Mason and Newell have been normalised to that of Khakoo and Trajmar at an energy of 20 eV Energy (eV)

Khakoo and Trajmar

Energy (eV)

Mason and Newell

20 30 40 50

14.69 4.56 2.13 0.818

11.52 11.91 12.28 12.57 13.41 15.66 16.63 17.39 18.62 19.54 20.27 21.34 22.17 23.20 24.27 25.16 27.14 29.05 31.25 33.34 35.11 37.34 39.48 41.29 43.10 46.31 49.02 50.93 53.92 57.29

0.119 2.06 8.03 15.37 16.21 17.38 17.75 17.31 16.41 15.54 14.69 13.23 12.35 11.54 10.11 9.17 8.24 7.67 7.22 6.73 6.33 6.09 5.88 5.94 5.89 5.75 5.86 5.77 5.61 5.62

extrapolating and integrating the above DCS data. Also shown are the metastable time-of-Pight measurements of Mason and Newell and the theoretical estimate of Lima et al. The cross-section of Khakoo and Trajmar is lower than that of Mason and Newell at energies above 20 eV, perhaps due to the inPuence of cascade contributions in the latter. There is good agreement in shape between the cross-section of Lima et al. and that of Khakoo and Trajmar, although there is roughly a factor of four di@erence in magnitude between the two. The a3 /g+ state. Electron impact excitation studies of the a-state are rather limited. The only DCS studies are those of Khakoo and Trajmar and there are only two determinations of the integral excitation cross-section, one from Khakoo and Trajmar and the other, an optical measurement, by Ajello et al. [170]. The di@erential data, at energies in the range from

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• •

273

Fig. 24. (a) Di@erential cross-section for the excitation of the c3 V state of H2 at 30 eV; ( ) Khakoo and Trajmar, (—) Lima et al. (b) Integral cross-section for the excitation of the c3 V state of H2 : ( ) Khakoo and Trajmar, ( ) Mason and Newell, (—) Lima et al. (×0:25).

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Table 12 Di@erential cross-sections (in units of 10−18 cm2 sr −1 ) for the excitation of the a3 W+ g state of H2 . The data of Khakoo and Trajmar are shown. The integral cross-sections (10−18 cm2 ) derived from these di@erential measurements are shown at the foot of each column Energy (eV) Angle 10 20 30 40 50 60 70 80 90 100 110 120 Qi

20 2.15 2.41 1.85 1.18 1.772 0.654 0.580 0.495 0.431 0.432 0.450 10.06

30

40

60

1.04 0.780 0.353 0.214 0.192 0.211 0.190 0.211 0.238 0.262 0.287 0.296

0.399 0.342 0.146 0.070 0.076 0.060 0.063 0.064 0.081 0.062 0.071 0.076

0.159 0.088 0.045 0.041 0.031 0.034 0.026 0.020 0.015 0.011 0.012 0.009

3.56

1.11

0.305

◦

◦

20 –60 eV and at scattering angles between 10 and 130 , are given in Table 12 and the integral cross-sections in Table 13. In Fig. 25(a) we illustrate the DCS for excitation of the a3 W+ g state at an incident energy of 20 eV. The measured cross-section of Khakoo and Trajmar is in excellent agreement with the calculations of Lima et al. across the entire angular range. This apparently is not the case at all energies (e.g. 30 eV) and the reader is referred to the paper of Khakoo and Trajmar where comparisons at other energies are shown. The integral cross-sections at energies between 10 and 60 eV are shown in Fig. 25(b). There is rather good agreement between the cross-sections derived from the DCS measurements of Khakoo et al. and the optical excitation function of Ajello et al. The close-coupling calculations of Lima et al. are also shown for comparison. The C 1 >u state. Once again studies of this excited state are limited and the only di@erential scattering measurements in the literature are those of Khakoo and Trajmar. At the integral cross-section level there are values derived from the above DCS measurements and optical excitation function measurements by Shemansky et al. [171]. The di@erential and integral cross-sections of Khakoo and Trajmar are given in Table 14. Khakoo and Trajmar also provide a comparison of their measured DCS with various theoretical approaches and Fnd best agreement with the distorted wave calculation of Lee et al. The DCS-derived integral cross-sections are in reasonably good agreement with the optical measurements of Shemansky et al. but in poor agreement with most of the theoretical calculations. HD and D2 . There have been surprisingly few measurements of absolute cross-sections for either of the isotopes, D2 and HD. Becker and McConkey [172] measured optical emission cross-sections for the Lyman and Werner bands in D2 and found no signiFcant isotopic effect. Buckman and Phelps [173] published a cross-section set for D2 based on a Boltzmann

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275

Table 13 Integral cross-sections (in units of 10−17 cm2 ) for the excitation of the a3 W+ g state of H2 . The data are those of Ajello et al. and have been placed on an absolute scale by normalisation to the cross-section of Khakoo and Trajmar at an energy of 20 eV (see Table 12) Energy (eV)

ICS

Energy (eV)

ICS

11.29 11.52 11.75 11.87 12.10 12.33 12.45 12.68 12.80 13.03 13.15 13.26 13.38 13.50 13.61 13.73 13.96 14.19 14.31 14.66 14.77 15.12 15.24 15.47 15.59 15.82 16.17 16.40 16.75 17.10 17.33 17.68 18.15 18.49 18.84 19.42 19.66 20.00

0.007 0.021 0.050 0.085 0.127 0.170 0.283 0.453 0.559 0.701 0.871 1.013 1.098 1.183 1.282 1.409 1.523 1.629 1.671 1.771 1.841 1.898 1.926 1.933 1.926 1.898 1.834 1.778 1.693 1.608 1.530 1.452 1.346 1.289 1.197 1.098 1.048 0.999

20.47 21.05 21.63 22.45 23.61 24.54 25.35 25.82 26.75 27.79 29.19 30.00 31.16 31.86 32.79 33.84 35.58 36.74 37.90 39.18 41.16 41.97 43.60 44.99 46.62 48.48 50.57 52.32 54.18 56.38 58.13 60.0

0.921 0.836 0.758 0.680 0.595 0.538 0.496 0.467 0.425 0.390 0.347 0.326 0.290 0.276 0.255 0.241 0.205 0.191 0.177 0.163 0.149 0.142 0.135 0.127 0.120 0.113 0.106 0.099 0.092 0.092 0.085 0.085

equation analysis of their own infra-red excitation coeJcients and previously published transport, excitation and ionisation coeJcients. These authors also list the source of several previous cross-section sets for D2 . Petrovic and Crompton [174] derived a momentum transfer cross-section for D2 based on an analysis of new measurements of the transverse di@usion

276

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• •

Fig. 25. (a) Di@erential cross-section for the excitation of the a3 W state of H2 at 20 eV; ( ) Khakoo and Trajmar, (—) Lima et al. (b) Integral cross-section for the excitation of the a3 W state of H2 : ( ) Khakoo and Trajmar, (—) Lima et al., (- -) Ajello et al.

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277

Table 14 Di@erential cross-sections (in units of 10−18 cm2 sr −1 ) for the excitation of the C1 Vu state of H2 a Angle

Energy (eV) 20

30

40

60

10 20 30 40 50 60 70 80 90 100 110 120

13.4 7.23 3.64 1.45 0.930 0.743 0.676 0.660 0.581 0.539 0.579 0.611

25.2 5.77 1.69 1.04 0.722 0.580 0.543 0.469 0.407 0.370 0.348 0.328

48.1 8.16 1.05 0.613 0.466 0.369 0.312 0.282 0.260 0.255 0.231 0.237

114.0 4.72 0.622 0.298 0.203 0.180 0.145 0.135 0.131 0.130 0.140 0.122

Qi

15.58

17.60

19.61

22.20

a

The data of Khakoo and Trajmar are given. Integral cross-sections (10 measurements are shown at the foot of each column.

−18

2

cm ) derived from the di@erential

coeJcient and drift velocity. Their initial reference cross-section was that of Buckman and Phelps; however, they found that slight modiFcations were needed to obtain better self-consistency between calculated and measured swarm parameters. As there are no substantive di@erences in the cross-sections found in these cases from those in H2 we do not tabulate any of these cross-sections here but refer the interested reader to the above articles. 3.2. N2 Molecular nitrogen has been the subject of more electron scattering studies than any other molecular species. This is not surprising given that it is the most abundant molecule in our atmosphere and plays a vital role in many atmospheric and technological processes. It is also due, in part, to the presence of a formidable negative ion resonance structure at low energies which completely dominates the scattering cross-sections for rotational and vibrational excitation of the ground state. This resonance has also been of particular fascination to many experimental and theoretical research groups as a result of its relatively short lifetime, which is comparable to the vibrational period of the negative ion. This leads to unusual behaviour in the observed quasi-vibrational peaks which vary in energy depending on the Fnal state which is being observed and the scattering angle at which the measurement is made. This very feature has been the subject of a large number of studies but, to some extent, the itinerant nature of the resonance proFle between 2 and 4 eV has led to many erroneous and misleading comparisons between experiments and between experiment and theory.

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◦

Fig. 26. Grand total cross-section for electron scattering by N2 at energies below 1 eV. ( ) Sun et al.—ToF data, ( ) Sun et al. from DCS data, ( ) Kennerly, (4) Ferch et al., (×) Jost et al., (- - -) R-matrix calculation, (— —) Hybrid theory, (- - -) Weatherford, (—) vibrational close coupling.

•

3.2.1. Grand total cross-sections Measurements of the grand total cross-section have been well summarised [1]. The time-ofPight (ToF) measurements of Kennerly [175] between the energies of 0.5 and 50 eV perfectly illustrate the dominant role of the 2 Vg resonance in the scattering at low energies. These measurements are still the most accurate in this energy region and have been reproduced by several recent experiments. At energies below 0:5 eV there has been considerable uncertainty in the magnitude of the total cross-section in both experiment and theory. Measurements by Jost et al. [176], Ferch et al. [177] and Sohn et al. [178] largely disagree with each other and with the early data of Baldwin [179] with, in some cases, the discrepancies being of the order of 20%, well outside the stated uncertainties on the transmission measurements which are typically less than 5%. In this case, theory was of little practical guidance as the di@erences between various R-matrix [180], hybrid approaches [181] and close-coupling calculations [182] were considerably larger. A summary of this situation can be found in a recent paper by Sun et al. [33] where their recent ToF measurements are also presented. As the situation regarding the grand total cross-section has not altered substantially for energies above 0:5 eV since the Trajmar review [1], we only show the status of measurement and theory below 1 eV in Fig. 26 and tabulate recent measurements in this energy range in Table 15. 3.2.2. Elastic scattering Once again, with a few exceptions that will be considered in the following section, most of the recent information on elastic di@erential cross-sections from the N2 X1 W+ g ground electronic state stem from measurements which do not resolve individual rotational transitions. Given that

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279

Table 15 Grand total cross-sections (in units of 10−16 cm2 ) for electrons scattered from N2 at energies below 1:5 eV Energy

Ferch et al.

Jost et al.

Sun et al.

0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.14 0.15 0.16 0.18 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.0 1.1 1.2 1.3 1.4 1.5

— — — — — 5.14 — — 5.73 — — 6.37 6.92 7.37 — 8.17 — 8.76 — 9.12 — 9.41 — 9.63 — 9.81 — 9.98 10.10 10.21 10.49 — —

4.20 4.55 4.85 5.15 5.40 5.60 — — 6.55 — — 7.20 7.73 8.07 8.35 8.60 8.90 9.10 9.28 9.42 9.57 9.68 9.80 9.88 9.97 10.03 10.10 10.14 10.23 10.41 10.57 10.77 11.20

— — — 4.47 — 4.78 4.96 5.32 — 5.56 5.86 6.10 6.49 7.19 7.74 8.04 8.34 8.56 8.76 8.95 9.11 9.12 9.26 9.32 9.46 9.59 9.67 9.73 9.93 10.15 10.29 10.53 10.91

the rotational energy spacing is only a few meV, most room temperature elastic scattering and vibrational excitation measurements involve a sum over all possible rotational excitations, including those from molecules which are not initially in the ground rotational state. Since the publication of the Trajmar review there have been elastic DCS measurements by Sohn et al. [178], Brennan et al. [183], Shi et al. [184], Sun et al. [33], Nickel et al. [185] and Gote and Ehrhardt [186] with most of these measurements concentrating on the energy region below 10 eV, and on the inPuence of the 2 Vg resonance on the scattering cross-sections in particular. These experiments have highlighted a number of inconsistencies between the various data sets, both within and outside the region of inPuence of the resonance. Within the resonance region, a series of close comparisons between experiments and between experiment and theory

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•

Fig. 27. Di@erential cross-sections for elastic scattering from N2 at (a) 0:55 eV and (b) 1:5 eV. ( ) Sun et al., ( ) Sohn et al., () Shi et al., (×) Brennan et al., (+) Shyn and Carignan, (— —) Hybrid theory, (—) vibrational close coupling, ( – - – ) Schwinger variational method.

(e.g. [183,184,187,33]) indicate that to a large extent the discrepancies may well be due to inappropriate comparisons rather than fundamental di@erences between the data. This arises because the cross-sections exhibit pronounced energy dependence in the resonance region and small absolute energy di@erences between experiments or experiment and theory may lead to erroneous comparisons of the resulting cross-sections. Two recent papers [33,184] have proposed a comparison protocol based on the location of well-deFned structures within the resonance rather than at some predetermined energy. Di@erential cross-sections for low-energy electrons scattered from N2 are provided in Table 16 and a selection of elastic DCS measurements and theory at energies below the resonance are shown in Fig. 27. At 0:55 eV (Fig. 27(a)) there is excellent agreement between the measurements of Shi et al. and Sun et al. which in turn are both 20 –30% higher than the measurements of Sohn et al. The close-coupling calculation of Sun et al. is also in good agreement with the two most recent experiments. At 1:5 eV (Fig. 27(b)) there is less accord between the various experiments and, in fact, it was the lack of agreement at this energy, which is still well isolated from the e@ects of the resonance, which has been the motivation for many of the recent studies. Although there are some di@erences between the recent measurements of Brennan et al., Shi et al. and Sun et al. which are larger than the collective uncertainties, they all lie substantially

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higher (≈ 50%) than the earlier measurements of Shyn and Carignan and Sohn et al. Recent theoretical calculations [33,187] also seem to favour a larger cross-section as does the comparison of total elastic cross-sections derived from the recent DCS measurements with the grand total cross-section of Kennerly. The 2 Vg resonance begins to have a major inPuence on the elastic scattering cross-sections at energies above about 1:8 eV. As discussed above, recent studies [183,184,33] have been carried out in this region primarily with a view to resolving long-standing discrepancies within the various experiments, and between experiment and theory, on the absolute values of the DCS in this energy region. Shi et al., amongst others, have pointed out the danger in the comparison of experiments, and of experiment and theory, in this region where the cross-sections change so rapidly, both in shape and magnitude, as a function of incident energy. Not only do the quasi-vibrational resonance features manifest themselves at di@erent energies depending on the observation channel, they also appear at di@erent energies when observed at di@erent scattering angles in the same observation channel. For example, the low-lying resonance peaks shift by ◦ ◦ almost 40 meV when the observation angle for elastic scattering is changed from 20 to 130 . Given that most DCS experiments have an absolute energy scale and energy resolution that is at best uncertain by at least this amount, it is perhaps not surprising that there was such a substantial level of disagreement between the existing DCS measurements through the resonance region. Shi et al. proposed a comparison protocol whereby the di@erential cross-section is measured at the energy corresponding to the position of a resonance peak at each angle, in this way giving an “energy optimised” DCS for the resonance process. They give an example of this technique for the second peak in the resonance proFle near 2:1 eV. In a joint approach to this problem, Sun et al. have combined experiment and theory and proposed a slightly di@erent protocol for comparison. In this work, experimental and theoretical DCS were measured and calculated at “energies” corresponding to the positions of particular features (e.g. peaks and dips in the ◦ energy dependence of the cross-sections) at a speciFed scattering angle of 60 . Examples of this approach are shown in Fig. 28(a and b) for elastic DCS at energies corresponding to the Frst and ◦ third peaks in the elastic excitation function at 60 . The level of agreement is extremely good in both cases and it is surprising that the relative importance of the resonant d-wave contribution to the cross-section appears to wane from peak one to three. Fig. 28(c) illustrates the elastic DCS measured and calculated at the position corresponding to the Frst peak in the ; = 0–1 excitation ◦ function, again at a scattering angle of 60 . This is at an energy which (experimentally) is 60 meV higher than that in Fig. 28(a) and yet the di@erence in both the shape and magnitude of the cross-section is substantial, highlighting the importance of the comparison protocol if an accurate level of comparison between experiment and theory is required. Tabulated values of the DCS, ICS and momentum transfer cross-sections in the resonance region are given in Table 16. Above the energy region of the 2 Vg resonance there are several new sets of elastic scattering measurements since the publication of the Trajmar review [1]. These include the experiments of Sun et al. who measured cross-sections at energies up to 10 eV, Nickel et al. [185], who measured cross-sections at energies between 20 and 100 eV and Gote and Ehrhardt [186], who covered the energy range from 10 to 200 eV. Tabulated values for di@erential elastic scattering cross-sections at energies above the resonance are presented in Table 16. Also included in these

282

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Fig. 28. Di@erential cross-sections for electrons elastically scattered from N2 at energies which correspond to certain ◦ quasi-vibrational 2 V resonance peaks observed at 60 . (a) the Frst peak in the elastic channel, (b) the third peak in the elastic channel and (c) the Frst peak in the 0 –1 vibrational excitation channel. ( ) Sun et al., (+) Shyn and Carignan, (— —) Hybrid theory, (—) vibrational close coupling.

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Table 16 Absolute di@erential cross-sections for electrons elastically scattered from N2 at incident energies below 10 eV (in units of 10−16 cm2 sr −1 ). Integral and momentum transfer cross-sections (both in units of 10−16 cm2 ) derived from the DCS measurements are shown at the foot of each column. Energy (eV)

Angle 10 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 138 Qi Qm

0.1 Sohn et al.

0.55

1.0

0.35 Sohn et al.

Sohn et al.

Shi et al.

0.121

0.221

0.175

0.106

0.133

0.255

0.280

0.306

0.205

0.123

0.164

0.280

0.393

0.385

0.277

0.157

0.223

0.377

0.490

0.461

0.393

0.203

0.301

0.449

0.590

0.580

0.522

0.254

0.379

0.551

0.706

0.685

0.625

0.286

0.456

0.592

0.803

0.761

0.702

0.322

0.520

0.626

0.875

0.819

0.722

0.369

0.564

0.670

0.898

0.868

0.753

0.396

0.638

0.674

0.921

0.888

0.727

0.422

0.692

0.674

0.935

0.896

0.689

0.459 0.488

0.724 0.746

0.614 0.593

0.924

0.632 0.615

4.44

6.15

6.81

Sun et al.

Sohn et al. 0.121 0.149

9.12 10.68

7.30

Sun et al. 0.227 0.256 0.324 0.378 0.466 0.524 0.576 0.669 0.731 0.811 0.864 0.913 0.921 0.978 0.970 1.006 0.978 0.970 0.965 0.965 0.928 0.905 0.856 9.84 10.78

Energy (eV) 1.5 Angle 15 20 25 30 35 40 45 50 55 60 65 70

Sohn et al.

Shi et al.

Sun et al.

1.92 Sun et al.

1.98 Sun et al.

2.46 Sun et al.

2.61 Sun et al.

0.365

0.310

0.297

0.424

0.299

0.566

0.403

0.788

0.554

0.981

0.681

1.228

0.336 0.354 0.386 0.432 0.525 0.597 0.670 0.794 0.890 1.001 1.041

— 2.046 1.732 1.479 1.299 1.168 1.111 1.093 1.139 1.22 1.345 1.479

— 2.304 1.978 1.771 1.511 1.348 1.252 1.207 1.171 1.169 1.217 1.269

4.96 4.522 4.114 3.604 3.194 2.74 2.334 1.975 1.735 1.53 1.403 1.307

2.829 2.637 2.508 2.311 2.12 1.892 1.691 1.520 1.336 1.189 1.083 0.978

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Table 16 (Continued) Energy (eV) 1.5 Angle 75 80 85 90 95 100 105 110 115 120 125 130 138 Qi Qm

Sohn et al.

Shi et al.

0.834

1.337

0.846

1.371

0.791

1.327

0.692

1.193

0.567

0.945

0.452 0.424 7.43

Sun et al.

1.92 Sun et al.

1.98 Sun et al.

2.46 Sun et al.

2.61 Sun et al.

1.139 1.188 1.176 1.213 1.186 1.187 1.105 1.048 0.997 0.914 0.853 0.782

1.58 1.687 1.775 1.776 1.722 1.65 1.536 1.437 1.301 1.184 1.101 1.022

1.385 1.430 1.450 1.450 1.419 1.348 1.274 1.178 1.092 1.021 1.001 0.994

1.301 1.332 1.286 1.303 1.275 1.218 1.155 1.088 1.056 1.095 1.150 1.335

0.941 0.883 0.841 0.800 0.758 0.742 0.710 0.696 0.714 0.741 0.815 0.912

10.83 11.12

18.00 17.40

19.93 18.03

21.10 16.65

Energy (eV) (The data are those of Sun et al.) Angle 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 Qi Qm

4.0 1.613 1.705 1.781 1.782 1.742 1.686 1.625 1.521 1.405 1.239 1.138 0.986 0.878 0.784 0.697 0.649 0.625 0.609 0.608 0.586 0.594 0.601 0.606 0.634 12.01 9.64

5.0 — 1.572 1.605 1.617 1.678 1.599 1.509 1.400 1.291 1.115 0.998 0.882 0.789 0.704 0.649 0.600 0.551 0.543 0.520 0.521 0.527 0.536 0.551 0.558 10.90 8.64

6.0 1.481 1.495 1.547 1.567 1.582 1.557 1.510 1.417 1.305 1.200 1.051 0.926 0.800 0.705 0.629 0.565 0.522 0.516 0.513 0.508 0.520 0.534 0.540 0.570 10.69 8.29

7.0 1.565 1.552 1.572 1.557 1.567 1.540 1.452 1.357 1.221 1.101 0.954 0.847 0.714 0.647 0.559 0.512 0.480 0.465 0.462 0.469 0.492 0.514 0.535 0.568 10.44 8.31

8.0 1.719 1.641 1.603 1.564 1.535 1.484 1.442 1.357 1.262 1.133 1.000 0.858 0.693 0.607 0.529 0.472 0.436 0.433 0.438 0.448 0.492 0.511 0.551 0.588 10.60 8.53

10.0 2.342 2.036 1.875 1.774 1.654 1.587 1.454 1.299 1.154 0.990 0.842 0.704 0.573 0.471 0.402 0.354 0.351 0.352 0.378 0.410 0.465 0.515 0.561 0.609 10.55 8.40

14.78 12.38

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Table 16 (Continued) Energy (eV)

Angle

10 Gote and Ehrhardt

20 Gote and Ehrhardt

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

2.914 2.509 2.086 1.820 1.501 1.169 0.841 0.585 0.449 0.430 0.497 0.594 0.744 0.942 1.086 1.195

7.281 4.692 2.737 1.637 1.058 0.658 0.405 0.239 0.189 0.234 0.331 0.452 0.549 0.654 0.786 0.923

20 Nickel et al. 4.93 2.60 1.61 1.04 0.672 0.383 0.217 0.200 0.271 0.393 0.511

30 Gote and Ehrhardt 8.294 5.304 2.732 1.312 0.685 0.355 0.218 0.149 0.120 0.121 0.131 0.222 0.307 0.447 0.602 0.774

30 Nickel et al. 5.04 2.59 1.29 0.669 0.355 0.221 0.150 0.131 0.146 0.208 0.298

40 Gote and Ehrhardt 8.092 4.475 1.990 0.893 0.442 0.228 0.15 0.109 0.084 0.093 0.134 0.225 0.352 0.541 0.766 1.003

40 Nickel et al. 4.63 2.02 0.896 0.420 0.223 0.138 0.923 0.0731 0.0832 0.130 0.219

Energy (eV)

Angle

50 Gote and Ehrhardt

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

8.368 4.125 1.682 0.673 0.312 0.169 0.111 0.083 0.068 0.071 0.111 0.193 0.312 0.447 0.614 0.757

50 Nickel et al. 4.21 1.63 0.653 0.302 0.171 0.112 0.0759 0.0636 0.0779 0.131 0.216

60 Gote and Ehrhardt 8.517 3.796 1.437 0.533 0.240 0.131 0.089 0.069 0.063 0.076 0.117 0.177 0.265 0.333 0.403 0.453

60 Nickel et al. 3.77 1.35 0.515 0.239 0.132 0.0887 0.0660 0.0623 0.0796 0.117 0.185

70 Gote and Ehrhardt 8.307 3.713 1.219 0.433 0.194 0.113 0.081 0.066 0.067 0.081 0.116 0.170 0.222 0.284 0.331 0.368

70 Nickel et al.

80 Nickel et al.

90 Nickel et al.

3.32 1.13 0.415 0.192 0.112 0.0827 0.0667 0.0696 0.0830 0.116 0.170

3.00 0.924 0.354 0.171 0.103 0.0775 0.0695 0.0696 0.0816 0.102 0.138

2.82 0.793 0.295 0.160 0.104 0.0804 0.0710 0.0709 0.0776 0.0869 0.124

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Table 16 (Continued) Energy (eV)

Angle

100 Gote and Ehrhardt

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

7.407 2.910 0.864 0.286 0.141 0.100 0.083 0.080 0.085 0.094 0.106 0.119 0.147 0.178 0.221 0.261

100 Nickel et al.

2.70 0.765 0.269 0.146 0.102 0.0785 0.0778 0.0760 0.0766 0.0837 0.101

150 Gote and Ehrhardt

200 Gote and Ehrhardt

6.885 2.440 0.616 0.198 0.107 0.085 0.071 0.063 0.056 0.055 0.062 0.073 0.085 0.098 0.105 0.103

5.886 2.035 0.462 0.157 0.099 0.081 0.063 0.048 0.040 0.042 0.050 0.055 0.058 0.059 0.062 0.069

tables are the elastic integral and momentum transfer cross-sections of Sun et al. Examples of several of the DCS are given in Fig. 29. At 5 eV, Fig. 29(a), comparison can be drawn between the recent data of Sun et al. and a number of previous measurements. In general, the agreement between them is rather good although there are some di@erences at forward scattering angles. At 7 eV, Fig. 29(b), there is also good agreement between recent experiment and theory ◦ at scattering angles larger than about 60 . At forward angles, the most recent experimental results [33] lie about 25% lower than previous measurements [188,189]. The shape of the elastic cross-section at forward angles in the 5 –7 eV energy region, just above that where the resonance dominates, is also interesting. At 5 eV, as observed in a number of experiments, the ◦ DCS is decreasing in magnitude below about 30 . This behaviour slowly changes as the energy is increased until at 7 eV the forward angle behaviour is essentially Pat. Finally by 10 eV, Fig. 29(c), the DCS rises quite steeply at small scattering angles and the overall level of agreement between the experiments has improved somewhat. At energies above 10 eV the data sets of Nickel et al. [185] and Gote and Ehrhardt [186] are in excellent agreement with one another, an example of which is shown in Fig. 29(d). 3.2.3. Rotational excitation To our knowledge, the only new measurements of discrete rotational excitation cross-sections for the ground state of N2 , since the publication of the Trajmar review, have been carried out by Gote and Ehrhardt [186]. In an impressive series of measurements, they have used a combination of high energy resolution (10 meV) and a deconvolution technique (see Section 2.7) to unravel the individual rotational state-to-state contributions to the vibrationally elastic cross-section over ◦ an energy range of 10 –200 eV and an angular range of 10 –160 . These cross-sections are given

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287

Fig. 29. Di@erential cross-sections for electrons elastically scattered from N2 at energies of (a) 5.0, (b) 7.0, (c) 10 eV and (d) 50 eV. ( ) Sun et al., (+) Shyn and Carignan, () Srivastava et al., ( ) Gote and Ehrhardt, (4) Nickel et al., (— —) Hybrid theory, (—) vibrational close coupling, ( – - – ) Schwinger variational, (- - -) R-matrix.

•

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Table 17 The transfer of rotational angular momentum (J = 0; 2; 4; 6; 8) by electron impact on the ground state of N2 , expressed as a percentage of the rotationally summed elastic di@erential cross-section (see Table 16) at incident energies between 10 and 200 eV. The data are those of Gote and Ehrhardt Energy (eV) Angle 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Angle 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

0 85.6 81.3 80.6 79.1 74.2 66.1 50.1 33.5 21.1 15.5 22.8 28.9 33.1 31.3 33.7 31.5

0 98.9 99.1 95.9 89 73.7 60.7 34.8 19.8 32.4 7.6 1.1 ¡1 ¡1 1.3 5.8 7.8

2 3.8 7.8 8.1 10 14.2 21 34.8 46.7 52.8 59.7 54.2 49.9 49.2 48.5 47.3 49.6

10 4 10.6 10.9 11.3 10.9 11.4 11.4 10.6 14.7 26.1 21.2 20.9 19.6 17.7 20.2 17.9 18.7

2 ¡1 ¡1 1.7 8.5 20 39.3 50 44.5 29.4 50.3 52.2 58.3 51.1 31.1 11.8 10

50 4 ¡1 ¡1 1.5 ¡1 ¡1 ¡1 11 11.9 23.7 36.5 37.5 40.6 36.6 61.2 79.5 72.5

6 ¡1 ¡1 ¡1 ¡1 ¡1 1.5 3.6 4.9 ¡1 1.3 2 1.6 ¡1 ¡1 1.1 ¡1

6 ¡1 ¡1 ¡1 1.2 3.7 ¡1 ¡1 21.4 14 5.5 9.1 ¡1 8.9 4.8 ¡1 8.5

8 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1

8 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 1.7 ¡1 ¡1 ¡1

0 95.4 91.9 91.4 86.7 78.2 70.6 49.7 24.5 13.6 36.6 54.6 49.5 34 22.7 13.3 10.3

0 95.9 95 92.4 77.6 59.3 43 29 35 17.7 17.1 11 ¡1 ¡1 4.7 ¡1 9.3

2 2.9 7.2 5.5 8.5 18.2 22 41.5 54.3 58.3 46.8 31.8 35.6 47.3 62.9 70 73.5

20 4 1.4 ¡1 2.1 1.3 ¡1 5.7 2.8 20 20 13.8 13.5 13.8 14.8 10.6 14.1 14.5

2 3.9 4.5 5.5 19.5 36.8 48.8 51.2 38.3 36.5 39.3 47.5 55.2 44 26.5 20.7 18.5

60 4 ¡1 ¡1 1.6 ¡1 ¡1 ¡1 6.4 9 20.6 34.3 41.5 42.1 53.3 67.8 77.2 68.7

6 ¡1 ¡1 ¡1 1.6 2.2 1.7 3.8 2.2 4.4 2.7 ¡1 ¡1 3.9 2.4 ¡1 ¡1

8 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 1.2 ¡1 1.5 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1

0 83.3 95.6 92.2 84.2 78.1 57.3 35.6 25 26.1 26.5 26.3 20.7 11.3 12.3 6.5 3.2

2 1.1 2.6 5.4 12.7 14.3 32.7 51 54.1 42.1 42.5 44.8 51.9 54.4 46.6 49.8 50.6

30 4 ¡1 1.2 1.2 ¡1 4.2 1.8 ¡1 5.4 22.1 21.7 22.6 26.1 31.8 40.3 40.6 41.7

6 1.2 ¡1 ¡1 2.2 1.7 4.6 9.3 10.8 9.7 7.3 ¡1 ¡1 ¡1 ¡1 3.2 4.4

8 1.6 ¡1 ¡1 ¡1 ¡1 ¡1 2.9 3.2 ¡1 1.5 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1

0 99.4 97.8 92.3 82.3 69.1 49.9 34.9 27.8 28.5 24.8 7.2 4.5 2.5 3.4 7.3 8.6

2 ¡1 1.4 6.3 15.4 25.9 40.5 50.7 44.4 33.1 33.6 61.2 61.2 52.5 50.2 50 44

40 4 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 2.1 15.3 19.8 29.2 29.1 30.1 44.2 42.4 41.1 44.6

6 ¡1 ¡1 1.3 1.4 4.1 8.1 9.5 9.9 15.9 7.9 2.2 3.4 ¡1 3.4 1.4 2.7

8 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 2.2 1.7 2.3 3.1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1

2 ¡1 2.1 2.2 19.3 28.8 57.7 66.1 56.2 47.9 53.3 42.4 35.2 31.7 20.6 11.5 5.6

70 4 1.1 ¡1 2.2 ¡1 7.5 6.7 5.7 17.1 37.3 37.3 39.6 47.4 47.6 69.2 75.5 83.6

6 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 1.8 3.3 9.3 4.8 7.3 13.7 12.8 10.2 11.5 10.7

8 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 2 1.5 1.5 ¡1 8.7 3.7 2.2 ¡1 ¡1 ¡1

0 99 95.3 81 51.6 30.2 14.5 4.6 9.9 8 5.1 9.8 ¡1 6 ¡1 ¡1 4.1

2 ¡1 3.9 17.7 44.7 64.4 69.2 71.5 68 60.1 31.2 17.1 10.9 8 ¡1 ¡1 15.6

100 4 ¡1 ¡1 ¡1 ¡1 ¡1 10 13.7 15.6 21.1 47.2 65.3 73.1 70.6 71.5 60.9 59.5

6 ¡1 ¡1 ¡1 1.2 2.8 5.7 10.2 6.7 10.6 15.8 7.9 11.5 15.4 17.5 30.9 20.8

8 ¡1 ¡1 ¡1 ¡1 2 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 4.5 ¡1 7.2 6.5 ¡1

Energy (eV) 6 ¡1 ¡1 ¡1 1.9 3.3 5.9 13.3 17.8 17.1 9.5 ¡1 ¡1 ¡1 ¡1 1.9 2.5

8 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 1

0 98.4 96.5 94.5 79 61.1 33.2 15.9 9.2 3 3 1.9 ¡1 ¡1 ¡1 1.5 ¡1

in Table 17 and a few examples are shown graphically in Fig. 30. Perhaps the most surprising outcome of these measurements is the observation that signiFcant rotational excitation occurs at all the energies studied, although as the energy increases the rotational inelasticity is mainly conFned to larger scattering angles.

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Table 17 (Continued) Energy (eV) Angle 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

0 96.8 95.3 83.3 53 26.8 21.7 14.1 17 14.9 9.5 1.4 4.4 4.4 5.7 ¡1 1.9

2 2.2 3.8 15.7 44.7 69.8 68.6 69.2 34.9 23 9.3 12.7 7.7 4 18 7.1 6.2

150 4 ¡1 ¡1 ¡1 ¡1 1.1 7.1 13.2 42.9 52 56 62.1 58.9 57.6 29.5 44.6 20.2

6 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 3.6 10 19.2 23.8 29 34 37.8 45.2 67.2

8 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 1.5 ¡1 ¡1 ¡1 4.7 ¡1 4

0 97.6 94.5 43.3 5.5 7.8 7.8 5.3 8 ¡1 ¡1 6.2 6 ¡1 6.1 ¡1 ¡1

2 ¡1 1.1 51.5 82.6 80.5 54.2 43.5 12.6 1.7 ¡1 16.2 15.6 16.3 8.7 1.8 ¡1

200 4 1.9 4.2 2.9 12.4 20.4 32.5 44.4 61.8 62.6 50.7 36.1 17.3 16.1 11.1 8.1 5.2

6 ¡1 ¡1 2.1 ¡1 ¡1 5.3 1.3 17.4 30.9 42.9 30.5 50.9 54.7 56.3 54.3 63.9

8 ¡1 ¡1 ¡1 ¡1 ¡1 ¡1 2.7 ¡1 4.7 5.0 8.2 9.8 11.1 18 29.2 27.7

3.2.4. Vibrational excitation On the other hand, there have been a substantial number of new measurements for the vibrational excitation of the ground state of N2 since the Trajmar review. At incident energies below the inPuence of the N2− 2 Vg resonance, Sohn et al. [178] have measured ; = 0–1 excitation cross-sections at 0:5; 1:0 and 1:5 eV. Absolute rovibrational excitation cross-sections in the energy region around the resonance (2–5 eV), whose formation and decay is responsible for the overwhelming majority of the magnitude of the vibrational excitation cross-section, have been measured by Brunger et al. [190], Brennan et al. [183] and Sun et al. [33]. Allan [191] has also measured the relative energy dependence of the Frst 17 vibrational excitation channels, within a few electron volts of their thresholds, and established an absolute scale for these processes by normalising to the measurements of Jung et al. [192]. At higher incident energies (20 –50 eV) the DCS for 0 –1 vibrational excitation has been measured by Middleton et al. [193]. When attempting to ascertain the extent to which a particular scattering cross-section is understood, one usually relies, wherever possible, on a close comparison between experiments and between experiment and theory. In this respect, an assessment of the situation for resonant vibrational excitation of N2 presents the same problems as we described above for elastic scattering because of the rapid variation of the cross-sections as a function of both energy and angle. This variation is so rapid and large that typical experimental uncertainties in the incident electron energy (≈ 50 meV) may lead to large di@erences in the measured rovibrational cross-sections between any two experiments ostensibly performed at the same energy. To some extent this problem has masked and confused the true level of accord between experiments and experiment and theory, and a good example of a particular case (for elastic scattering at around 2:1 eV) is given by Shi et al. [184]. Recent experiments and theory have also attempted to

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Fig. 30. Di@erential cross-sections for rotational excitation of N2 at (a) 10 eV and (b) 200 eV. The rotationally summed cross-section is shown ( ), as well as the cross-sections for the transfer of ( ) 0, (+) 2, () 4, ( ) 6 and () 8 rotational quanta.

•

◦

circumvent this particular problem by careful attention to the energy (position) at which any given DCS measurement is carried out, with respect to the dominant resonance structures. Absolute DCS for rovibrational excitation (; = 0–1; 2; 3; 4) of N2 at energies below 5 eV are given in Table 18(a) – (d) and selected examples are shown in Figs. 31 and 32. In Fig. 31(a) we plot the DCS of Sohn et al. [178] at an energy of 0:5 eV which is, to our knowledge the only recent absolute measurement at such low incident energy. At 1:5 eV in Fig. 31(b), the DCS data of Sohn et al. are compared with that of Brennan et al. [183]. Whilst the two experiments agree reasonably well in the general features of the cross-section, the data of Brennan et al. are substantially higher at backward angles. In the region of the resonance, recent experimental [33,184] and theoretical [33,187] approaches on these cross-sections have proposed the adoption of a careful comparison protocol which will remove some of the arbitrariness from the process. Rather than propagate what in our view are most likely erroneous comparisons between experiments which may have, inadvertantly, been performed at slightly di@erent energies, the present comparisons shall concentrate only on those data where the energies (position) are unambiguous. In Fig. 32, we compare the recent experimental and theoretical close-coupling results of Sun et al. at three di@erent energies near the middle of the resonance. The Frst energy (Fig. 32(a)) corresponds to the position of the Frst resonance peak in the vibrational (0 –1) ◦ excitation function as determined at a scattering angle of 60 . The absolute value of the energy

M.J. Brunger, S.J. Buckman / Physics Reports 357 (2002) 215–458

291

Table 18 Absolute di@erential cross-sections for rovibrational excitation (a) 0 –1, (b) 0 –2, (c) 0 –3, (d) 0 –4 and (e) 0 –1 (at higher energies) of N2 (in units of 10−18 cm2 sr −1 ) at incident energies between 0.5 and 5:0 eV. Integral cross-sections (10−18 cm2 ) are shown at the foot of each column Energy (eV) 1.5

(a) 0.5 Angle Sohn et al.

1.0 Sohn et al.

Sohn et al.

Brennan et al.

2.1 Brennan et al.

3.0 Brennan et al.

5.0 Brennan et al.

5 10 15 20 30 40 50 60 70 80 90 100 110 120 130 138

— — — 0.075 0.046 0.036 — 0.029 — — 0.02 0.015 — 0.014 0.014 0.013

— — — 0.178 0.176 0.136 0.105 0.093 0.083 0.065 0.057 0.043 0.040 0.026 0.033 0.046

— — 1.82 1.41 0.90 0.519 0.342 0.265 0.259 0.339 0.395 0.381 0.334 0.239 0.199 0.154

— — 1.68 1.42 1.09 0.773 0.527 0.579 0.643 0.671 0.708 0.635 0.565 0.503 — —

84.6 70.5 — 51.0 29.8 15.0 11.2 10.5 13.5 14.2 12.4 10.8 8.43 7.54 8.70 —

41.7 37.2 — 30.0 21.9 13.6 9.12 6.41 7.23 9.12 9.80 8.72 7.48 5.91 5.56 —

2.03 1.91 — 1.52 1.05 0.733 0.526 0.409 0.546 0.597 0.609 0.496 0.394 0.306 0.377 —

Qi

0.48

0.93

5.2

8.9

197

137

8.0

Energy (eV) (a) Angle

1.98 Sun et al.

2.46 Sun et al.

2.61 Sun et al.

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130

— 0.749 0.621 0.523 0.420 0.333 0.267 0.231 0.208 0.208 0.230 0.261 0.307 0.357 0.379 0.396 0.398 0.374 0.352 0.315 0.272 0.246 0.214 0.199

0.314 0.269 0.241 0.187 0.156 0.128 0.097 0.080 0.072 0.080 0.086 0.101 0.109 0.126 0.132 0.142 0.141 0.135 0.123 0.111 0.098 0.088 0.079 0.075

0.849 0.716 0.623 0.514 0.415 0.328 0.267 0.223 0.199 0.200 0.226 0.255 0.299 0.331 0.360 0.368 0.365 0.361 0.323 0.288 0.256 0.230 0.211 0.206

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Table 18 (Continued) Energy (eV) (b) Angle 5 10 20 30 40 50 60 70 80 90 100 110 120 130

2.1 Brunger et al.

Brennan et al. 48.1 41.6 27.9 17.8 10.4 6.75 7.47 8.46 9.16 8.67 7.37 5.84 5.69 6.90

26.1 16.7 10.5 6.40 7.03 7.78 7.41 5.96

2.4 Brunger et al.

38.4 22.5 11.0 9.29 8.84 9.33 9.83 9.75

3.0 Brunger et al.

Brennan et al. 20.5 18.9 15.2 10.5 7.39 4.94 4.29 5.58 6.81 7.27 6.86 5.91 4.68 4.07

7.21 7.05 5.43 3.75 4.51 5.99 7.89 7.54

132.0

Qi

90.0 Energy (eV)

(c) Angle 5 10 20 30 40 50 60 70 80 90 100 110 120 130

2.1 Brunger et al.

Brennan et al. 11.0 9.50 6.60 4.10 2.51 1.71 1.82 2.37 2.44 2.23 1.66 1.50 1.62 1.75

4.88 4.98 2.92 1.81 1.67 2.08 4.24 5.07

2.4 Brunger et al.

32.2 16.5 6.72 8.50 7.56 6.97 7.19 7.30

3.0 Brunger et al.

5.56 5.38 4.32 3.76 2.51 1.45 1.08 1.53 1.81 2.13 1.77 1.59 1.16 1.16

4.67 3.60 2.27 1.30 2.27 2.58 3.78 3.72

34.0

Qi

Brennan et al.

26.0 Energy (eV) (data of Brunger et al.)

(d) Angle

2.1

2.4

3.0

20 30 40 50 60 70 80 90

6.63 6.83 4.71 2.99 2.76 3.28 1.15 2.49

14.3 8.33 4.10 3.51 2.93 2.95 2.70 3.06

3.43 2.72 0.98 0.30 0.14 0.74 1.73 2.09

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293

Table 18 (Continued) Energy (eV) (data of Middleton et al.)

(e) Angle

20

30

50

10 20 30 40 50 60 70 80 90

3.16 2.63 2.01 1.46 1.01 1.19 0.78 0.74 0.83

0.89 0.56 0.44 0.32 0.24 0.25 0.21 0.19 0.16

0.42 0.14 0.10 0.07 0.08 0.08 0.06 0.06 0.05

Fig. 31. Di@erential cross-sections for rovibrational excitation of N2 at an incident energy of (a) 0:5 eV, (b) 1:5 eV. ( ) Sohn et al., ( ) Brennan et al.

•

is of secondary importance in this case as the comparison is made with the DCS calculated at the corresponding energy where the theory predicts the Frst peak in the 0 –1 excitation ◦ function at 60 . The agreement between experiment and the vibrational close-coupling calculation is clearly excellent. The remaining two Fgures in this sequence demonstrate a similarly

294

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Fig. 32. Di@erential cross-section for rovibrational excitation of N2 at the positions of (a) the Frst quasi-vibrational resonance peak in the 0 –1 channel, (b) the third peak in the 0 – 0 channel and (c) the third peak in the 0 –1 channel. ◦ All positions being determined at a scattering angle of 60 . ( ) Sun et al., (— —) Hybrid theory, (—) vibrational close coupling.

•

M.J. Brunger, S.J. Buckman / Physics Reports 357 (2002) 215–458

295

Table 19 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the A3 W+ u state of N2 . Integral cross-sections (10−18 cm2 ) are shown at the foot of each column. In the case of the Brunger and Teubner data, the integral cross-sections were derived by Campbell et al. Energy (eV)

Angle 6 10 15 20 25 30 34 40 50 52 60 70 71 80 89 90 108 126 135 Q

15 Zetner and Trajmar 2.13 2.05 1.54 1.59

1.94

2.91 2.13 2.09 1.06 1.22 1.96 24.4

15 Brunger and Teubner

15 LeClair and Trajmar

17.5 Brunger and Teubner

20 Brunger and Teubner

20 LeClair and Trajmar

30 Brunger and Teubner

50 Brunger and Teubner

2.097

1.110

0.092

0.416

0.547

1.706

1.060

0.138

0.306

0.278

1.621

0.949

0.199

0.244

0.179

2.025 1.821

0.739 0.947

0.258 0.265

0.152 0.175

0.124 0.123

1.326 1.083

1.304 1.630

0.293 0.361

0.253 0.275

0.110 0.129

0.827

1.590

0.268

0.383

0.132

1.703

0.229

0.444

0.109

7.8

5.5

0.744

14.2

1.45

16.0

11.1

1.03

formulated comparison, but at the third peak in the elastic channel (Fig. 32(b)) and the third peak in the vibrational (0 –1) channel (Fig. 32(c)), respectively. Once again, the level of agreement is excellent. These examples also serve to demonstrate the dominant role that is played by the d
symmetry in the excitation process. Even where the contribution of resonant scattering is minimised (the third peak in the 0 – 0 channel corresponds to a minimum between peaks 2 and 3 in the 0 –1 channel—Fig. 32(b))—the DCS still displays an almost pure d-wave character, albeit with a considerably reduced magnitude. The experiments of Brennan et al. also indicate that this dominance extends at least to an energy of 5 eV. The experimental DCS for vibrational excitation (0 –1) of N2 at signiFcantly higher energies (20; 30 and 50 eV), are given in Table 18(e). 3.2.5. Electronic excitation Excitation of the electronic states of N2 has been studied by a number of techniques. Given that several of these states are metastable and have relatively large excitation cross-sections, their study is important in a number of Felds including applied gaseous electronics and atmospheric physics. Much of the work in this area was done prior to the Trajmar review and this

296

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Fig. 33. Di@erential cross-section for the excitation of the A3 W+ u valence state of N2 at an energy of 15 eV. ( ) Zetner and Trajmar, ( ) Brunger and Teubner, (4) LeClair and Trajmar, (—) R-matrix.

•

data has been compiled in that document. However there have been several recent measurements of di@erential and total cross-sections and whilst in some cases the level of agreement is reasonably good, there still appear to be some outstanding problems regarding some of the electronic state cross-sections. A3 /u+ state. The lowest lying electronic excited state of N2 is the A3 W+ u valence state which has an excitation threshold just above 6 eV. The only recent experimental work on this excited state consists of the studies of Zetner and Trajmar [128], who measured near-threshold ◦ ◦ di@erential cross-sections at angles between 7 and 127 , Brunger and Teubner [99], who measured di@erential cross-sections at a number of incident energies between 15 and 50 eV and for ◦ scattering angles in the range 10 –90 and LeClair and Trajmar [118], who used a di@erential time-of-Pight technique for measurements at energies of 15 and 20 eV for a scattering angle of ◦ 90 . Tabulated values of these cross-sections are given in Table 19 and a comparison, at the only energy at which they overlap (15 eV), is given in Fig. 33. The level of agreement between ◦ these measurements is reasonably good at forward angles (¡ 60 ) but there are some di@erences ◦ at large angles. The single 90 point of LeClair and Trajmar lies midway between the data of Brunger and Teubner and Zetner and Trajmar, which di@er by more than a factor of two at this angle. All of these cross-sections show signiFcant di@erences to the earlier measurements of Cartwright et al. [96] and the renormalised versions of these cross-sections given in the Trajmar

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297

Table 20 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the B3 Vg state of N2 . Integral cross-sections (10−18 cm2 ) are shown at the foot of each column. In the case of the Brunger and Teubner data, the integral cross-sections were derived by Campbell et al. Energy (eV)

Angle 6 10 15 20 25 30 34 40 50 52 60 70 71 80 89 90 108 126 135 Q

15 Zetner and Trajmar 0.803 0.954 1.01 1.31 1.79 2.87 2.12 2.68 2.49 2.53 3.03 28.5

15 Brunger and Teubner

17.5 Brunger and Teubner

20 Brunger and Teubner

30 Brunger and Teubner

50 Brunger and Teubner

1.526

1.129

0.398

0.312

0.220

1.935

2.160

0.507

0.560

0.217

2.633

3.285

0.794

0.685

0.208

4.049 6.455

2.246 2.085

0.674 0.464

0.525 0.488

0.190 0.199

5.134 4.270

2.495 2.685

0.392 0.386

0.508 0.537

0.182 0.194

4.173

3.127

0.369

0.669

0.207

5.535

4.126

0.434

0.838

0.230

8.7

4.9

45.1

28.2

13.4

review. A recent R-matrix calculation by Gillan et al. [194] is also shown in this Fgure and it demonstrates a better level of agreement with the DCS of Brunger and Teubner. Campbell et al. [195] have recently derived integral cross-sections for this state by Ftting and extrapolating the DCS data of Brunger and Teubner. These values are given at the foot of each column in Table 19, along with integral cross-sections derived by Zetner and Trajmar. B3 >g state. This electronic state has a threshold energy of 7:35 eV and it has also been studied in recent times by Zetner and Trajmar and Brunger and Teubner. These new DCS measurements are presented in Table 20 and a comparison of these cross-sections and the earlier cross-section of Trajmar et al., at an energy of 15 eV, is shown in Fig. 34. The agreement here is rather poor with, in some instances, a factor of 4 –5 di@erence in the absolute values of the DCS, the values of Brunger and Teubner being larger at all scattering angles. Integral cross-section derived from the DCS values of Brunger and Teubner by Campbell et al. are given at the foot of each column of Table 20. W 3 8u state. The excitation threshold for this state is almost coincident with that of the B state above and the ability to extract cross-section information from energy loss spectra is an excellent example of the utility of the spectral deconvolution technique developed by

298

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•

Fig. 34. Di@erential cross-section for the excitation of the B3 Vg state of N2 at an energy of 15 eV. ( ) Zetner and Trajmar, ( ) Brunger and Teubner, (+) Cartwright et al.

Cartwright et al., and further implemented by Brunger and Teubner. The DCS measurements of Brunger and Teubner at energies between 15 and 50 eV and those of Zetner and Trajmar at 15 eV. are given in Table 21. We also show a comparison of these results at 15 eV, with the earlier values of Cartwright et al., which have been renormalised and tabulated by Trajmar et al., in Fig. 35. In this case, the data of Brunger and Teubner is in reasonably good agreement with the previous measurements of Cartwright et al., although it is somewhat lower than the cross-section of Zetner and Trajmar. At most other energies the results of Brunger and Teubner are smaller in magnitude than the earlier data although the shapes of the DCS are generally in good agreement. B3 /u− state. The threshold for this triplet state is at 8:16 eV and it has only been studied by Brunger and Teubner in the years since the Trajmar review. This data is presented in Table 22. At higher energies (30 and 50 eV) they Fnd excellent agreement with the earlier results given in the Trajmar review whilst closer to threshold their results are signiFcantly larger than the previous measurements. a1 /u− state. The lowest lying member of the singlet manifold of excited states in N2 is the a1 W− u state with an excitation threshold just below 8:4 eV. Once again the only recent work is that of Brunger and Teubner, who found good agreement with the previous cross-sections summarised by Trajmar et al. The recent cross-sections for this state are given in Table 23.

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299

Table 21 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the W3 Uu state of N2 . Integral cross-sections (10−18 cm2 ) are shown at the foot of each column. In the case of the Brunger and Teubner data, the integral cross-sections were derived by Campbell et al. Energy (eV)

Angle 6 10 15 20 25 30 34 40 50 52 60 70 71 80 89 90 108 126 135 Q

15 Zetner and Trajmar 1.84 1.95 1.64 1.94 2.59 4.68 2.82 3.45 3.05 3.75 4.93 42.8

15 Brunger and Teubner

17.5 Brunger and Teubner

20 Brunger and Teubner

30 Brunger and Teubner

50 Brunger and Teubner

1.296

0.859

0.440

0.003

0.035

1.174

1.384

0.380

0.069

0.036

1.211

1.398

0.312

0.102

0.040

1.353 1.472

1.146 1.192

0.265 0.192

0.115 0.126

0.042 0.044

1.465 1.506

1.453 1.659

0.149 0.165

0.138 0.130

0.056 0.059

1.332

2.116

0.179

0.163

0.065

1.490

2.468

0.209

0.208

0.057

6.0

2.0

19.2

21.1

10.8

a1 >g state. The threshold energy of the a1 Vg state is just below 8:55 eV. This state is metastable with a measured lifetime of around 115 s and its optical emission in the ultraviolet is responsible for the well-known Lyman–Birge–HopFeld (LBH) band. There have been several recent measurements for this state, the di@erential scattering data of both Zetner and Trajmar and Brunger and Teubner, the total cross-section results of Mason and Newell and the optical emission measurements of Ajello and Shemansky. The measurements of Mason and Newell were made by detecting the yield of metastable molecules produced by electron impact. These relative cross-sections were normalised to a value of 3:5 × 10−17 cm2 at an energy of 17 eV, based on previous values available in the literature. The di@erential and integral data are shown in Tables 24 and 25, respectively. We note that the data of Mason and Newell and Ajello and Shemansky potentially include the e@ects of cascade contributions. The only common energy for the two most recent DCS measurements is 15 eV and, as we see from Fig. 36(a), they are in excellent agreement across the entire range of scattering angles although both are substantially larger at forward angles than the cross-section of Cartwright et al. At higher energies, the data of Brunger and Teubner are in good agreement with the earlier measurements. The total scattering cross-section is shown in Fig. 36(b). Here the data of Mason and Newell

300

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•

Fig. 35. Di@erential cross-section for the excitation of the W3 U state of N2 at an energy of 15 eV. ( ) Zetner and Trajmar, ( ) Brunger and Teubner, (+) Cartwright et al.

Table 22 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the B3 W− u state of N2 . The data are those of Brunger and Teubner. Integral cross-sections (10−18 cm2 ) derived by Campbell et al. are shown at the foot of each column Energy (eV) Angle 10 20 30 40 50 60 70 80 90 Qi

15 1.191 0.793 0.586 0.730 1.619 1.906 1.481 1.360 1.805 20.4

17.5 0.686 0.787 0.817 0.829 0.922 1.176 1.285 1.476 1.630 12.4

20

30

50

0.727 0.656 0.596 0.390 0.238 0.210 0.209 0.238 0.358

0.767 0.479 0.445 0.306 0.238 0.163 0.134 0.153 0.204

0.589 0.202 0.147 0.075 0.095 0.076 0.066 0.040 0.041

7.1

3.8

2.7

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301

Table 23 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the a1 W− u state of N2 . The data are those of Brunger and Teubner. Integral cross-sections (10−18 cm2 ) derived by Campbell et al. are shown at the foot of each column Energy (eV) Angle

15

10 20 30 40 50 60 70 80 90

0.109 0.213 0.454 0.644 1.236 1.697 1.421 1.040 1.477 12.3

Qi

17.5

20

30

50

0.067 0.184 0.170 0.150 0.174 0.196 0.181 0.171 0.254

0.275 0.329 0.387 0.296 0.194 0.114 0.103 0.105 0.127

0.205 0.323 0.372 0.279 0.186 0.134 0.101 0.108 0.139

0.587 0.434 0.291 0.130 0.084 0.056 0.054 0.056 0.059

4.2

3.4

1.8

1.2

Table 24 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the a1 Vg state of N2 Energy (eV)

Angle 6 10 15 20 25 30 34 40 50 52 60 70 71 80 89 90 108 126 135

15 Zetner and Trajmar 21.9 18.5 11.9 10.0 4.24 3.20 1.43 1.95 1.32 1.07 1.02

15 Brunger and Teubner

17.5 Brunger and Teubner

20 Brunger and Teubner

30 Brunger and Teubner

50 Brunger and Teubner

16.99

31.8

14.0

12.5

22.05

13.65

26.6

12.0

11.34

20.7

9.04 6.71

13.2 5.31

9.38

7.87

7.55

5.87

2.46

4.17 1.470

2.19 1.083

0.729 0.427

4.18 2.81

1.501 0.847

0.775 0.608

0.639 0.542

0.393 0.348

2.04

0.672

0.625

0.824

0.304

2.29

0.711

0.859

1.058

0.257

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Table 25 Integral cross-sections (10−17 cm2 ) for the excitation of the a1 Vg state of N2 Energy (eV) 9 10 11 12 13 14 15 16 17 17.5 18 19 20 23 25 27 29 30 31 33 35 38 40 41 43 47 50 55 60 65 70 71 80 81 90 91 100 101 111 121 131 141 150 200

Ajello and Shemansky

Zetner and Trajmar

0.246 1.36 2.36 2.88 2.99 3.02 2.93 2.47 2.07 1.78 1.56

1.24 1.04 0.889 0.778 0.691 0.622

0.415 0.311

4.29

Mason and Newell 0.48 1.27 2.00 2.54 3.18 3.26 3.34 3.45 3.50 3.44 3.34 3.25 3.02 2.86 2.70 2.58 2.48 2.29 2.10 1.97 1.88 1.75 1.56 1.56 1.24 1.14 1.11 1.05 0.99 0.89 0.84 0.76 0.72 0.67 0.65

Campbell et al.

4.32 4.69 2.66

2.25

1.32

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303

•

Fig. 36. (a) Di@erential cross-section for the excitation of the a1 Vg state of N2 at an energy of 15 eV. ( ) Zetner and Trajmar, ( ) Brunger and Teubner, (+) Cartwright et al. (b) Integral excitation cross-section for the a1 Vg state of N2 . ( ) Ajello and Shemansky, ( ) Mason and Newell, (4) Campbell et al.

◦

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Table 26 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the w1 Uu state of N2 . The data are those of Brunger and Teubner. Integral cross-sections (10−18 cm2 ) derived by Campbell et al. are shown at the foot of each column Energy (eV) Angle

15

17.5

20

30

50

10 20 30 40 50 60 70 80 90

1.201 0.903 0.560 0.415 0.211 0.132 0.115 0.169 0.352

1.447 0.943 0.638 0.273 0.181 0.142 0.134 0.248 0.491

1.492 0.605 0.236 0.217 0.181 0.166 0.183 0.191 0.206

0.224 0.141 0.095 0.098 0.081 0.069 0.056 0.059 0.061

0.327 0.102 0.126 0.058 0.028 0.023 0.023 0.025 0.022

Qi

7.8

5.7

3.8

1.08

0.52

is compared with that of Ajello and Shemansky and the values of Campbell et al. which have been derived from the DCS measurements of Brunger and Teubner. In general, the level of agreement between them all is reasonably good. w1 8u state. The measurements of Brunger and Teubner for this state, with an excitation threshold of 8:89 eV, show some discrepancies at low energies with the only earlier work, that of Cartwright et al. which has been tabulated in the Trajmar review. At higher energies there is good agreement between the measurements. The recent measurements of Brunger and Teubner are given in Table 26. C 3 >u state. The C-state of nitrogen, with a threshold at just above 11 eV, is of signiFcant importance in a range of atmospheric and applied Felds. It has a relatively large cross-section at near-threshold energies and its excitation, and subsequent cascading, provide a signiFcant mechanism for the population of the low-lying metastable A3 W+ u state. The intermediate step in the cascade process, the population of the B3 Vg state, proceeds via the emission of visible radiation which gives rise to the so-called second positive system. There have been several recent measurements of the excitation cross-section for this state. Brunger and Teubner and Zubek and King [113] have measured DCS at a range of incident energies up to 50 eV while LeClair and Trajmar [118] have reported a single measurement ◦ at 15 eV and 90 . Zubek [38] has measured total excitation cross-sections by observing the energy dependence of the emission from the second positive band and Poparic et al. [196] have also derived the near-threshold, total scattering cross-section from measurements of the energy ◦ dependence of vibrational excitation at a scattering angle of 0 . The DCS data are presented in Table 27 and the total cross-sections in Table 28. A comparison of the DCS data is provided in Fig. 37(a) at an energy of 17:5 eV. There are obviously serious discrepancies (∼ a factor of 4) between the various measurements at this energy, particularly at forward scattering angles with the DCS of Brunger and Teubner being larger in magnitude than that of Zubek and King. The level of agreement at higher energies between the data of Brunger and Teubner and the earlier

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305

Table 27 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the C3 Vu state of N2 Energy (eV) 15 15 17.5 17.5 20 20 30 50 Brunger and LeClair and Brunger and Zubek and Brunger and Zubek and Brunger and Brunger and Angle Teubner Trajmar Teubner King Teubner King Teubner Teubner 10 20 30 40 50 60 70 80 90 100

5.62 5.737 5.132 5.060 5.063 4.680 4.464 4.218 4.597

2.21

4.230 4.474 3.943 2.610 2.329 2.395 2.593 3.746 3.600

0.956 1.06 1.18 1.19 1.34 1.47 1.61 1.82 1.86 1.96

0.968 1.75 1.615 1.500 1.289 1.139 1.196 1.105 1.076

0.787 0.858 0.892 0.925 1.05 1.13 1.20 1.33 1.29 1.39

0.268 0.415 0.455 0.525 0.499 0.553 0.583 0.614 0.680

Table 28 Integral cross-sections for the excitation of the C3 Vu state of N2 Energy (eV)

Zubek (10−18 cm2 )

11.25 11.5 11.75 12 12.25 12.5 12.75 13 13.25 13.5 13.75 14 14.25 14.5 14.75 15 15.5 16 16.5 17 17.5 20 30 50

0.74 1.95 2.99 4.55 5.85 7.98 11.0 16.0 22.9 30.5 36.9 40.8 42.4 41.9 40.1 37.9 33.3 29.9 27.4 25.6 24.0

Zubek and King (10−18 cm2 )

Campbell et al. (10−18 cm2 )

51.8

24.5 18.9

30.4 16.1 9.5 3.4

0.167 0.227 0.246 0.167 0.163 0.169 0.180 0.186 0.190

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Table 28 (Continued) Energy (eV)

Poparic et al. (10−17 cm2 )

Energy (eV)

Poparic et al. (10−17 cm2 )

11.044 11.206 11.313 11.420 11.474 11.528 11.635 11.797 12.011 12.173 12.388 12.549 12.764 12.871 12.979 13.032 13.086 13.140 13.193 13.247 13.301 13.355 13.408 13.462 13.570 13.677 13.731 13.784 13.838 13.892 13.999

0.113 0.230 0.406 0.582 0.817 0.758 0.875 1.110 1.462 1.814 2.166 2.576 2.987 3.104 3.104 3.339 3.691 3.632 3.632 3.867 4.160 4.160 4.101 4.336 4.746 4.746 4.922

14.053 14.107 14.161 14.214 14.268 14.322 14.375 14.429 14.537 14.644 14.805 14.966 15.074 15.343 15.504 15.719 15.987 16.202 16.363 16.578 16.740 16.901

5.802 5.743 5.743 5.861 5.978 5.919 5.861 5.919 5.861 5.685 5.450 5.216 4.981 4.629 4.336 4.101 3.925 3.749 3.515 3.397 3.280 3.221

measurements of Cartwright et al. is somewhat better. The total cross-section for this state is shown in Fig. 37(b). Here we see reasonable agreement between the various measurements with the possible exception of the magnitude of the cross-section peak in the near-threshold region. E 3 /g+ state. The E-state is yet another metastable level of N2 with a lifetime of the order of 200 s and an excitation energy of 11:88 eV. There have been several studies of the excitation cross-section for this state since the publication of the Trajmar review. These include the total cross-section measurements of Brunger et al. [39] and Poparic et al. [197] and the DCS measurements of Brunger and Teubner and of Zubek and King. The cross-sections for this state are typically very small although a strong negative ion resonance located just above threshold appears to preferentially decay via the E-state, enhancing the near-threshold total cross-section signiFcantly. The recent DCS data for the E-state cross-sections is shown in Table 29 and the

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307

Fig. 37. (a) Di@erential cross-section for the excitation of the C3 Vu state of N2 at an energy of 17:5 eV. ( ) Brunger and Teubner, ( ) Zubek and King, (+) Cartwright et al. (b) Integral excitation cross-section for the C3 Vu state of N2 . (◦) Poparic et al., ( ) Zubek, ( ) Zubek and King, (4) Campbell et al.

•

•

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Table 29 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the E3 W+ g state of N2 Energy (eV)

Angle 10 20 30 40 50 60 70 80 90 100

15 Brunger and Teubner

17.5 Brunger and Teubner

17.5 Zubek and King

20 Brunger and Teubner

20 Zubek and King

30 Brunger and Teubner

50 Brunger and Teubner

0.027 0.030 0.024 0.029 0.034 0.038 0.030 0.024 0.022

0.174 0.151 0.129 0.079 0.054 0.025 0.014 0.016 0.022

0.157 0.143 0.106 0.0704 0.0481 0.0342 0.0235 0.0225 0.0371 0.0577

0.669 0.450 0.238 0.117 0.059 0.038 0.037 0.038 0.056

0.353 0.256 0.138 0.0706 0.0412 0.0319 0.0272 0.0319 0.0606 0.0728

0.416 0.219 0.108 0.089 0.103 0.098 0.069 0.059 0.048

0.220 0.032 0.055 0.028 0.014 0.009 0.006 0.005 0.006

integral cross-section data in Table 30. A comparison of the DCS measurements for an incident energy of 17:5 eV is shown in Fig. 38(a). Here we see that there is excellent agreement between the cross-sections of Brunger and Teubner and those of Zubek and King. The recent total cross-section measurements are shown in Fig. 38(b). The near-threshold resonance is clearly evident in the cross-sections of both Brunger et al. and Poparic et al. and there is reasonably good agreement between the values of Campbell et al. and Zubek and King at higher energies. a1 /g+ state. This state has an excitation energy of 12:25 eV and has been the subject of recent di@erential cross-section measurements by Brunger and Teubner and Zubek and King, again at energies between 15 and 50 eV. The tabulated DCS are shown in Table 31. At the energies where these measurements overlap there is reasonably good agreement between them at most scattering angles. Integral cross-sections, derived by Campbell et al. from the DCS values of Brunger and Teubner, are shown at the foot of each column of Table 31. 3.3. O2 An understanding of low-energy, electron impact excitation of O2 is desirable because of its important role in the physics and chemistry of the earth’s atmosphere. In particular, transitions to the lowest lying excited electronic states (a1 Ug ; b1 W) from the ground molecular state (X3 W) give rise to the atmospheric infrared (Herzberg) and red (Babcock) bands, respectively. The metastable a1 Ug state is also important in the O2 –I2 mixing laser [198]. Furthermore, because of its long lifetime it permits the production of an adequate number density to enable the study of scattering from an excited molecular species by means of a collision experiment [199]. From a theoretical perspective O2 , as a quite small homonuclear diatomic, represents a logical extension from closed-shell target calculations to that for an open-shell system [200]. Consequently, given the scientiFc endeavour of the last decade or so, it is something of a paradox that a knowledge of its pertinent cross-sections remains quite fragmentary.

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309

Table 30 Integral cross-sections (10−18 cm2 ) for the excitation of the E3 W+ g state of N2 Energy (eV)

Brunger et al.

11.868 11.898 11.927 11.931 11.956 11.986 12.015 12.044 12.073 12.103 12.132 12.136 12.161 12.190 12.220 12.249 12.278 12.307 12.337 12.366 12.395 12.424 12.454 12.483 12.512 12.541 12.571 12.600 12.629 12.659 12.688 15.0 17.5 20.0 30 50

0.1 5.9 13.3

Zubek and King

Poparic et al.

Campbell et al.

13.65

10.3 8.8 7.8 7.5 7.1 6.5 6.1

9.50

5.9 4.2 4.2 3.3 2.0 1.5 1.6 1.7 2.1 2.2 2.2 3.1 3.6 4.9 3.4 3.0 3.2 2.8 2.5 0.61 0.84

0.31 0.57 1.7 1.1 0.2

3.3.1. Grand total cross-sections Early experiments on electron scattering cross-sections for O2 were carried out by Bruche [201] and Ramsauer and Kollath [202]. Later measurements include the work carried out by Sunshine et al. [203], Salop and Nakano [204], Dalba et al. [205] and GriJth et al. [206]. More recently Zecca et al. [207], Dababneh et al. [208], Subramanian and Kumar [209] and Kanik et al. [210] have reported e− + O2 grand total cross-sections. The results of these latter four studies, along with the proposed optimum cross-section set of Kanik et al. [211], are illustrated in Fig. 39 and tabulated at selected energies in Table 32. The optimum cross-section set of

310

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Fig. 38. (a) Di@erential cross-section for the excitation of the E3 W state of N2 at an energy of 17:5 eV. ( ) Brunger and Teubner, ( ) Zubek and King. (b) Integral excitation cross-section for the E3 W state of N2 . (—) Brunger et al., ( ) Zubek and King, ( ) Poparic et al., (4) Campbell et al.

•

•

◦

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311

Table 31 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for the excitation of the a1 W+ g state of N2 . Integral cross-sections (10−18 cm2 ), derived by Campbell et al. for the data of Brunger and Teubner, are shown at the foot of each column Energy (eV) 15 Brunger and Teubner

17.5 Brunger and Teubner

17.5 Zubek and King

20 Brunger and Teubner

20 Zubek and King

30 Brunger and Teubner

50 Brunger and Teubner

10 20 30 40 50 60 70 80 90 100

2.180 0.816 0.444 0.334 0.294 0.261 0.279 0.251 0.283

1.841 0.959 0.500 0.255 0.189 0.174 0.167 0.179 0.217

2.58 1.29 0.534 0.351 0.306 0.279 0.207 0.211 0.232 0.278

3.835 2.303 1.324 0.758 0.441 0.291 0.254 0.213 0.228

3.39 1.15 0.372 0.380 0.364 0.271 0.183 0.182 0.202 0.216

2.600 0.577 0.473 0.456 0.344 0.277 0.194 0.191 0.189

1.840 0.237 0.365 0.226 0.176 0.129 0.104 0.087 0.068

Qi

4.2

5.2

3.1

1.75

Angle

6.3

◦

Fig. 39. Grand total cross-section for e + O2 scattering. ( ) Subramanian and Kumar, ( ) Zecca et al., (×) Kanik et al., () Dababneh et al., (—) the preferred cross-section of Kanik et al.

Kanik et al. [211] was constructed from the (then) currently available grand total measurements, including that of Kanik et al. [210]. In Fig. 39 we see that in the energy range 0–1 eV, there is fair agreement between the data of Zecca et al. and that of Subramanian and Kumar. The cross-sections of Sunshine et al. are higher,

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Table 32 Grand total cross-section (in units of 10−16 cm2 ) for electron scattering from O2 Energy (eV)

Zecca et al.

Dababneh et al.

Subramanian and Kumar

Kanik et al.

Kanik et al.

0.15 0.23 0.26 0.38 0.5 0.6 0.75 0.84 0.92 0.95 1 2 3 4 5 6 7 8 9 10 12 15 20 30 40 50 60 70 80 90 100

— 4.27 — — 4.96 5.21 — 5.59 — 5.78 — 6.98 7.40 7.54 7.83 8.17 8.70 9.48 10.15 10.63 10.90 10.70 10.70 10.90 10.40 10.20 9.79 9.52 9.32 9.21 9.08

— — — — — — — — — — — — — — 8.60 — — — — 10.80 — 11.65 12.20 11.90 11.70 11.80 10.90 10.60 9.40 9.15 —

4.96 — 4.94 5.07 5.27 — 5.77 — 5.85 — — 6.16 5.94 6.11 6.79 7.54 7.89 8.53 9.15 — — — — — — — — — — — —

— — — — — — — — — — — — — — 7.40 7.96 — 8.70 — 10.14 10.62 — 10.79 11.12 10.83 10.36 9.77 9.41 9.07 8.76 8.26

— — — — — — — — — — 5.8 6.2 6.6 6.9 7.4 8.06 8.70 9.09 10.15 10.38 10.76 10.70 10.75 11.10 10.71 10.28 9.78 9.46 9.19 8.98 8.67

by between 7% and 13%, than both those measurements. From 1–5 eV there is considerable scatter in the data, the optimum cross-section set of Kanik et al. seemingly placed midway between those of Zecca et al. and Subramanian and Kumar. At impact energies above 5 eV, the reported cross-sections [203–211] are generally in agreement with one another, to within the stated error limits, except for those of Sunshine et al. and Dababneh et al. SpeciFcally, the results of Kanik et al. agree fairly well with those of Zecca et al., but are lower by ∼5% below 12 eV and higher by ∼2% above 14 eV. The results of GriJth et al. were lower than those of Kanik et al. by ∼6% in the full 5 –50 eV energy range. This discrepancy may be due to an ine@ective discrimination against electrons inelastically scattered in the forward direction in the experimental arrangement of GriJth et al. In the peak region, the result of Dababneh et al. is appreciably higher than all the others, although the reason for this discrepancy is not clear.

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313

Below 20 eV the agreement between Kanik et al. and Dababneh et al. is fair, with the Kanik et al. data being ∼5% lower across this range. 3.3.2. Elastic cross-sections 3.3.2.1. Di:erential cross-sections. The early measurements of elastic electron scattering from O2 were discussed in detail by Trajmar et al. [1]. These included the work of Linder and Schmidt [212], Trajmar et al. [119], Dehmel et al. [213], Wakiya [214] and Shyn and Sharp [215]. Each of these investigations used di@erent techniques to establish the absolute scale for their measured angular distributions and there are obvious [1] di@erences between them below ◦ about 10 eV impact energy, e.g. by as much as 40 –50% at 5 eV and a scattering angle of 40 . These di@erences cannot be due to the di@erent normalisation techniques alone as there are also signiFcant di@erences in the shape of the DCS. The angular distributions of both Linder and Schmidt and Trajmar et al. were placed on an absolute scale by normalising to the grand total cross-sections of Salop and Nakano [204]. For the low-energy (¡ 2 eV) results of Linder and Schmidt, the grand total cross-sections of Salop and Nakano had to be extrapolated. We note that the normalisation of Trajmar et al. also took into account contributions for electronic excitation. Dehmel et al. employed the same technique as that just described for Linder and Schmidt and Trajmar et al. except that in their case they used the grand total cross-sections of Sunshine et al. [203] as well as those of Salop and Nakano. Shyn and Sharp relied upon measurements of relative scattering intensities for O2 and helium. In this application, the scattering chamber was Pooded with each gas and the relative gas number densities determined from measurements with a calibrated ionisation gauge. The only studies reported in the literature subsequent to this early work were those of W\oste et al. [216], Sullivan et al. [217] and Green et al. [218]. W\oste et al. measured relative angular distributions between 5 and 20 eV and then placed them on an absolute scale by normalisation ◦ ◦ to the data of Shyn and Sharp at a scattering angle of either 50 or 60 . Sullivan et al. applied a full relative Pow technique (see Section 2.6) for the absolute cross-section determination. The data of W\oste et al. is given in Table 33(a) whilst that of Sullivan et al. is given in Table 33(b). At lower energies there is good agreement, in terms of shape, between the data of Linder and Schmidt and Sullivan et al. The agreement in terms of magnitude for these data is also good, as illustrated in Fig. 40 at an energy of 3:0 eV. On the other hand, the data of Shyn and Sharp at 3 eV is in poor agreement with both the above measurements, particularly at an◦ gles ¿ 100 . The comparison of all the available DCS data at 5 eV (Fig. 41) provides a good illustration of the discrepancies alluded to above between 5 and 10 eV impact energy, as all ◦ the Fve data sets show di@erent shapes and absolute magnitudes. For angles greater than 50 there is a reasonable level of agreement between the data of Trajmar et al., Dehmel et al., ◦ W\oste et al. and Sullivan et al. However, at scattering angles ¡ 50 the situation is poor and this led to the measurements of Green et al., who endeavoured to resolve these discrepancies. The data of Green et al. were measured at three energies, 5; 7 and 10 eV and this data is given in Table 33(c) and illustrated in Fig. 41. Here we see that there is excellent agreement between their measurements and those of Sullivan et al. For impact energies of 10 eV and above, the discrepancy between the data sets has largely vanished with all the measured DCS showing the same overall angular behaviour and absolute magnitude. A good illustration of this

314

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Table 33 (a) Absolute di@erential cross-sections (10−16 cm2 sr −1 ) for elastic electron scattering from O2 . The data are those of W\oste et al.a Energy (eV) Angle 10 20 30 40 50 60 70 80 90 Error (%)

5

7

8

9

9.5

10

11

15

20

0.991 0.982 0.965 0.960 0.941 0.870 0.785 0.680 0.603

1.56 1.44 1.407 1.324 1.20 0.902 0.776 0.609 0.499

1.61 1.42 1.30 1.14 1.07 0.950 0.759 0.593 0.462

2.18 1.65 1.40 1.28 1.14 0.969 0.709 0.549 0.426

2.42 1.78 1.48 1.31 1.20 0.980 0.806 0.639 0.501

2.63 2.15 1.78 1.45 1.20 0.97 0.77 0.61 0.52

3.11 2.21 1.66 1.45 1.27 0.999 0.762 0.578 0.445

3.142 2.605 1.969 1.555 1.150 0.837 0.534 0.344 0.293

4.43 2.99 2.00 1.31 0.920 0.640 0.420 0.281 0.213

15

14

14

14

14

15

14

15

16

(b) Di@erential elastic scattering cross-sections for O2 (10−16 cm2 sr −1 ). The data is that of Sullivan et al.b Energy (eV) Angle

1.0

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130

— 0.165 — 0.253 — 0.329 — 0.398 — 0.459 — 0.505 — 0.518 — 0.540 — 0.562 — 0.567 — 0.556 — 0.561

Qi Qm

6.1 6.7

2.0 (11) (9) (8) (8) (8) (7) (7) (8) (10) (8) (8) (7)

— 0.303 — 0.406 — 0.517 — 0.604 — 0.650 — 0.667 — 0.651 — 0.625 — 0.583 — 0.544 — 0.509 — 0.468 6.7 6.5

3.0 (9) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

— 0.418 0.480 0.552 0.594 0.644 0.699 0.728 0.752 0.770 0.760 0.743 0.709 0.685 0.651 0.605 0.569 0.536 0.503 0.470 0.448 0.424 0.406 0.386 6.9 6.1

4.0 (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

0.487 0.514 0.576 0.636 0.699 0.758 0.806 0.838 0.854 0.856 0.828 0.809 0.749 0.698 0.651 0.589 0.537 0.489 0.470 0.426 0.392 0.372 0.346 0.342 7.0 6.0

5.0 (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

0.585 0.603 0.642 0.706 0.765 0.824 0.858 0.886 0.899 0.914 — 0.815 — 0.677 — 0.565 — 0.464 — 0.391 — 0.343 — 0.323 7.1 5.7

7.0 (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

0.877 0.836 — 0.853 — 0.910 — 0.931 0.912 0.881 — 0.780 — 0.620 — 0.500 — 0.401 — 0.359 — 0.335 — 0.338 7.3 5.7

8.0 (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

1.23 (9) 1.03 (9) — 0.985 (9) — 0.991 (9) — 0.959 (7) — 0.920 (7) — 0.780 (9) — 0.594 (8) — 0.461 (7) — 0.390 (8) — 0.359 (9) — 0.360 (8) — 0.383 (7) 7.8 6.2

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315

Table 33 (Continued) Energy (eV) Angle

9.0

10

15

20

30

12 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130

— 1.45 (7) 1.30 (7) 1.18 (8) 1.12 (7) 1.07 (8) 1.06 (7) 1.03 (8) 0.988 (7) 0.942 (7) 0.872 (7) — 0.720 (7) — 0.541 (7) — 0.430 (7) — 0.379 (7) — 0.369 (7) — 0.375 (7) — 0.391 (7)

— 1.85 (7) 1.58 (7) — 1.30 (7) — 1.15 (7) — 1.04 (7) — 0.894 (7) — 0.714 (7) — 0.554 (7) — 0.453 (7) — 0.390 (8) — 0.397 (8) — 0.410 (7) — 0.431 (7)

3.03 (8) 2.76 (7) 2.29 (7) — 1.71 (7) — 1.38 (7) — 1.11 (7) — 0.844 (7) — 0.599 (7) — 0.423 (7) — 0.333 (7) — 0.298 (7) — 0.304 (7) — 0.347 (7) — 0.399 (8)

— 3.44 (8) 2.73 (8) — 1.77 (7) — 1.30 (8) — 0.958 (8) — 0.639 (8) — 0.431 (8) — 0.282 (8) — 0.218 (8) — 0.218 (8) — 0.248 (8) — 0.303 (9) — 0.392 (8)

— 5.08 (8) 3.84 (8) — 2.02 (7) — 1.11 (11) — 0.654 (10) — 0.420 (7) — 0.270 (8) — 0.184 (8) — 0.137 (8) — 0.130 (10) — 0.166 (9) — 0.257 (10) — 0.425 (10)

Qt Qm

7.9 5.9

8.6 6.4

8.8 5.9

8.7 6.1

8.8 6.0

(c) Di@erential elastic scattering cross-sections for O2 (10−16 cm2 sr −1 ). The data is that of Green et al.c Energy (eV) Angle 20 25 30 40 50 60 70 80 90

5

7

10

— 0.857 (6.0) 1.611 (19) 0.564 (6.4) — — 0.641 (8.1) 0.880 (6.2) 1.395 (11) 0.773 (7.6) 0.864 (5.6) 1.167 (9.4) 0.797 (7.4) 0.999 (6.2) 1.014 (8.8) 0.854 (7.5) 0.826 (7.2) 0.824 (7.6) 0.692 (7.0) 0.713 (6.2) 0.704 (8.7) 0.591 (8.1) 0.561 (5.0) 0.626 (9.0) 0.514 (6.7) 0.512 (6.5) 0.525 (10) a Figures at the foot of each column indicate the percentage uncertainty for that energy. b Figures in parentheses indicate the percentage uncertainty in the absolute values. Integral elastic (Qi ) and elastic momentum transfer cross-sections (Qm ) are shown at the foot of each column. c Figures in parentheses indicate the percentage uncertainty in the absolute values.

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Fig. 40. Di@erential cross-sections for elastic electron scattering from O2 at 3 eV. () Linder and Schmidt, ( ) Shyn and Sharp ( ) Sullivan et al.

•

Fig. 41. Di@erential cross-sections for elastic electron scattering from O2 at 5 eV. ( ) Shyn and Sharp, (×) Trajmar et al., (♦) Dehmel et al., (◦) W\oste et al., ( ) Sullivan et al., (—) Ribeiro et al., () Green et al.

•

(see Fig. 42) occurs at 20 eV where there is excellent agreement between the results of Sullivan et al. and those of Trajmar et al., Shyn and Sharp and W\oeste et al. We note that in Figs. 41 and 42 we have also included the results of the Schwinger variational method calculation of Ribeiro et al. [219]. Agreement between theory and experiment is only marginal at 5 eV, in this case, although it clearly improves as one goes to higher incident electron energies. 3.3.2.2. Integral elastic cross-sections. Experimental integral elastic cross-sections have been reported by Trajmar et al., Kanik et al. and Sullivan et al. whilst theoretically there are the calculations of Noble and Burke [200] and Higgins et al. [220]. All the experimental data are listed in Table 34, the early work being included here because they were not done so previously in the Trajmar review. In Fig. 43 we plot and compare the experimental data and in this case the calculation of Higgins et al. is also included. It is clear that the results of Trajmar et al. and Sullivan et al. are in good accord over their common energy range. This is in spite of the fact that for energies less than about 10 eV there were quite signiFcant di@erences in the shapes of their di@erential cross-sections. This latter

M.J. Brunger, S.J. Buckman / Physics Reports 357 (2002) 215–458

317

Fig. 42. Di@erential cross-sections for elastic electron scattering from O2 at 20 eV. ( ) Shyn and Sharp, (×) Trajmar et al., ( ) W\oste et al., ( ) Sullivan et al., (—) Ribeiro et al.

◦

•

observation is consistent with our earlier remarks (Section 2) concerning the sensitivity of differential cross-section versus integral cross-section measurements. We also note that the data of both Trajmar et al. and Sullivan et al. are in good accord with the optimum integral elastic cross-sections proposed by Kanik et al. On the other hand, the data of Shyn and Sharp for impact energies between 3 and 10 eV, clearly overestimate the strength of the elastic integral cross-sections. This we believe is a rePection of some unphysical behaviour in their di@erential ◦ cross-section measurements at backward angles (¿ 110 ) in this energy range. At low energies the calculation of Higgins et al. signiFcantly overestimates the magnitude of the integral elastic cross-section e.g. below 4 eV the calculated values are nearly twice as large as the experimental measurements. However, at higher energies, e.g. 10 eV, the agreement with all of the experiments is good. Noble and Burke observed that the discrepancy below 4 eV is probably due to the omission of long-range polarisation e@ects in the expansion of the scattering wavefunction. The omission of these e@ects will be especially signiFcant in the low-energy elastic scattering cross-section, particularly as large contributions to the cross-section came from symmetries with ‘ = 0 components [220]. 3.3.2.3. Elastic momentum transfer cross-sections. Elastic momentum transfer cross-sections have been reported by Hake and Phelps [221], Shyn and Sharp and most recently by Sullivan et al. For completeness, we tabulate the results of Shyn and Sharp and Sullivan et al. in

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Table 34 Integral cross-sections (10−16 cm2 ) for elastic electron scattering from O2 Energy (eV)

Trajmar et al.

Kanik et al.

Shyn and Sharp

Sullivan et al.

1 2 3 4 5 6 7 8 9 10 12 15 20 30 40 50 60 70 80 90 100

— — — 6.78 6.95 — 7.60 — — 9.03 — 7.89 8.02 — — — — — — — —

5.8 6.2 6.6 6.8 7.3 7.9 8.3 8.6 8.8 8.9 9.1 9.0 8.6 7.8 7.1 6.5 6.1 5.7 5.4 5.0 4.8

— 6.5 8.0 — 9.3 — 9.9 — — 10.9 — 10.0 9.1 8.2 7.5 6.6 — 5.3 — — 3.8

6.1 6.7 6.9 7.0 7.1 — 7.3 7.8 7.9 8.6 — 8.8 8.7 8.8 — — — — — — — ±20%

Fig. 43. Integral cross-sections for elastic scattering from O2 . ( ) Shyn and Sharp, (×) Trajmar et al., (+) Kanik et al., ( ) Sullivan et al., ( – – ) Higgins et al.

•

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Table 35 Elastic momentum transfer cross-sections (10−16 cm2 ) for O2 Momentum transfer cross-section Energy (eV)

Shyn and Sharp

1 2 3 4 5 6 7 8 9 10 15 20 30 40 50 60 70 100

— 6.7 7.7 — 8.0 — 8.2 — — 8.6 7.2 6.1 5.2 4.5 3.5 — 2.5 1.5

Sullivan et al. 6.7 6.5 6.1 6.0 5.7 — 5.7 6.2 5.9 6.4 5.9 6.1 6.0 — — — — — ±20%

Table 35. Again, we see a signiFcant di@erence between these values which we believe is simply a rePection of the di@erences in the backward angle behaviour of their DCS. 3.3.3. Rovibrational excitation 3.3.3.1. Di:erential cross-sections. Previous experimental investigations into excitation of the rovibrational states of the electronic ground state of O2 have been limited. Linder and Schmidt presented relative data for vibrational excitation to the v = 1; 2; 3 and 4 levels by electrons with energies ranging from threshold to 4 eV. They observed that in the region of the 2 Vg resonance the energy dependence of the vibrational cross-section consisted of a series of spikes. This behaviour is qualitatively similar to that observed in N2 by Allan [191], who measured, in an extremely sensitive experiment, excitation functions for vibrational levels up to v = 17 in the energy region 0.25 –5:0 eV. The Frst report of absolute di@erential cross-sections for vibrational excitation in O2 was provided by Wong et al. [222]. Here they detailed data for excitation of ◦ the v = 0; 1; 2; 3; and 4 levels at an electron scattering angle of 25 and with electrons of energy in the range 4 –15 eV. They observed enhancement of the vibrational cross-sections which they attributed as being caused by the 4 W resonance. Further they found the energy dependence of these cross-sections to be bell-shaped with a broad peak of maximum near 9:5 eV. The work of Wong et al. was the Frst time that a shape resonance, connected with the ground electronic state, was experimentally shown to make a large contribution to the rovibrational cross-section. The most up-to-date experimental studies of rovibrational excitation of the O2 molecule have been

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Table 36 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for rovibrational (0 –1) excitation of O2 Energy (eV)

(a)a Angle 12 24 36 48 60 72 84 96 108 120 132 144 156

(b)b Angle 10 20 30 40 50 60 70 80 90

5

7

10

15

1.5 1.4 1.1 0.83 0.72 0.64 0.52 0.65 0.60 0.64 0.73 0.85 0.86 17%

4.2 3.5 3.3 2.8 2.3 2.2 1.7 1.6 2.0 2.3 2.5 2.9 3.3 17%

3.9 3.4 2.6 2.2 1.9 1.9 2.0 2.2 2.1 2.5 2.6 3.2 4.1 15%

1.1 0.86 0.62 0.38 0.35 0.29 0.32 0.35 0.38 0.41 0.38 0.48 0.64 17%

Energy (eV) 5

6

7

8

9

9.5

1.03 1.44 0.699 0.238 6.01 5.23 0.922 1.31 1.08 1.179 5.57 5.12 0.894 1.00 1.54 3.120 4.41 4.52 0.806 0.90 1.93 2.305 3.49 3.76 0.637 1.06 1.96 2.082 3.39 3.15 0.476 1.02 1.79 1.843 3.29 3.37 0.407 1.02 1.70 2.087 3.09 3.56 0.323 0.94 1.73 2.058 3.15 3.83 0.263 0.84 1.60 2.125 3.02 3.76 a The data are those of Shyn and Sweeney and Fgures at the foot of uncertainty. b The data are those of Brunger et al.

10

11

15

20

7.98 2.014 1.77 0.598 6.67 4.354 1.17 0.347 3.90 3.447 1.01 0.234 4.14 3.181 0.816 0.163 4.56 3.358 0.684 0.145 4.32 3.177 0.674 0.110 4.07 3.177 0.761 0.082 4.01 3.199 0.798 0.081 4.12 3.135 0.721 0.092 each column represent the percentage

performed by Shyn and Sweeney [223], Noble et al. [224], Brunger et al. [225] and Allan [226]. ◦ Shyn and Sweeney measured DCS for v = 0–1; 2; 3; 4 over the angular range 12–156 for the incident electron energies 5; 7; 10 and 15 eV using a crossed beam apparatus. Brunger et al. also reported DCS data for excitation of the v = 1; 2; 3; 4 vibrational states of the ground electronic ◦ state of O2 although, in this case, their angular range was 10–90 while their incident electron energies were 5; 6; 7; 8; 9; 9:5; 10; 11; 15 and 20 eV. Allan reported absolute excitation function ◦ measurements at 90 , from threshold to 17 eV for the 0 → 1; 2; 3; 4; 5; 6; 7 and 8 rovibrational excitation processes. Absolute DCS, i.e. angular distribution data, were only reported by Allan for the v = 0 → 1 process at 3.8 and 9:0 eV and for electron scattering angles in the range ◦ 5–135 . The DCS of Shyn and Sweeney and Brunger et al. are given in Tables 36 –39, whilst that of Allan is given in Table 40.

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Table 37 Absolute di@erential cross-sections (10−18 cm−2 sr −1 ) for rovibrational (0 –2) excitation of O2 Energy (eV)

(a)a Angle 12 24 36 48 60 72 84 96 108 120 132 144 156

(b)b Angle 10 20 30 40 50 60 70 80 90

5

7

10

15

— 0.64 0.50 0.30 0.27 0.29 0.20 0.19 0.14 0.19 0.21 0.24 0.29 17%

1.9 1.5 1.2 0.8 0.78 0.79 0.63 0.58 0.77 0.74 0.99 1.2 1.4 17%

2.2 1.8 1.4 1.3 1.1 1.0 1.1 1.2 1.2 1.3 1.3 1.6 1.9 15%

0.37 0.24 0.16 0.13 0.13 0.09 0.07 0.07 0.09 0.12 0.13 0.13 0.16 17%

Energy (eV) 6

7

8

9

9.5

0.223 0.169 0.067 2.05 1.83 0.246 0.246 0.382 2.07 1.94 0.217 0.390 1.075 1.70 1.80 0.183 0.579 0.811 1.36 1.57 0.196 0.540 0.759 1.38 1.29 0.157 0.459 0.646 1.38 1.50 0.165 0.473 0.744 1.26 1.60 0.171 0.494 0.759 1.33 1.67 0.168 0.499 0.818 1.23 1.65 a The data are those of Shyn and Sweeney and the Fgures at the foot of each b The data are those of Brunger et al.

10

11

15

3.20 0.772 0.806 2.77 1.788 0.473 1.37 1.448 0.408 1.34 1.346 0.336 1.98 1.508 0.321 2.26 1.439 0.282 1.81 1.415 0.270 1.73 1.394 0.264 1.67 1.336 0.245 column represent the uncertainty.

To date, there have been no theoretical DCS calculations for rovibrational excitation in O2 by electron impact. In principle, however, Noble and Burke [200], Teillet-Billy et al. [227] and Ribeiro et al. [219] should be able to extend their models, by the inclusion of nuclear motion, to this scattering system. Indeed, Higgins et al. have already shown its feasibility with an R-matrix approach. In Fig. 44(a) we compare the data of Shyn and Sweeney, Brunger et al., Allan, and Wong et al. at 5 eV for the v = 0–1 transition. Agreement between Brunger et al. and Wong et al. is good, the absolute magnitude of the data of Shyn and Sweeney being somewhat larger in ◦ value. The most recent data of Allan, at 90 , lies between the absolute values of Brunger et al. and Shyn and Sweeney. On the other hand, agreement in terms of the shape of the DCS is quite good between the date of Shyn and Sweeney and Brunger et al. Although not shown, the level of agreement between the data of Brunger et al. and the recent data of Allan at 9 eV for the v = 0–1 transition, is excellent over the entire common angular range of measurement.

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Table 38 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for rovibrational (0 –3) excitation of O2 Energy (eV)

(a)a Angle 12 24 36 48 60 72 84 96 108 120 132 144 156

(b)b Angle

7

10

15

0.98 0.74 0.53 0.35 0.23 0.21 0.25 0.22 0.33 0.33 0.40 0.43 0.48 17%

1.1 0.78 0.60 0.55 0.54 0.46 0.53 0.52 0.59 0.65 0.64 0.65 0.78 15%

0.18 0.11 0.07 0.05 0.05 0.05 0.03 0.03 0.04 0.04 0.05 0.06 0.07 17%

Energy (eV) 7

8

9

9.5

10

11

15

10 20 30 40 50 60 70 80 90

0.074 0.032 1.525 1.419 1.59 0.405 0.414 0.123 0.178 1.03 0.973 1.37 0.926 0.236 0.183 0.472 0.893 0.900 0.78 0.734 0.196 0.253 0.356 0.711 0.799 0.72 0.675 0.161 0.293 0.336 0.684 0.628 1.02 0.782 0.184 0.241 0.280 0.694 0.751 1.15 0.751 0.183 0.204 0.330 0.635 0.804 1.01 0.737 0.180 0.220 0.322 0.644 0.869 0.92 0.711 0.125 0.206 0.341 0.600 0.827 0.83 0.686 0.135 a The data are those of Shyn and Sweeney and the Fgures at the foot of each column represent the uncertainty. b The data are those of Brunger et al.

At 15 eV (see Fig. 44(b)), on the other hand, the data of Brunger et al. is, in fact, larger in magnitude by about a factor of 2, for the v = 0–1 excitation, than that found by Shyn and ◦ ◦ Sweeney for all angles ¿ 40 . The data of Allan, at the single angle of 90 , for this excitation process lies between the data of Brunger et al. and Shyn and Sweeny. It is interesting to note that the angular dependence of the rovibrational cross-sections of Brunger et al., between 5 and 10 eV, are similar in form to those predicted for the elastic channel by Sullivan et al. An alternative approach in the presentation of this data is given in Fig. 44(c) where we illustrate the DCS data, again for the v = 0–1 rovibrational excitation, as a function of incident ◦ electron energy for a scattering angle around 30 . It is clear from this Fgure that agreement between the data of Wong et al. and Brunger et al. is quite good. The level of agreement between that of Shyn and Sweeney and Brunger et al. at higher incident energies is also good, although at lower energies Shyn and Sweeney measure a larger cross-section. We note that both

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323

Table 39 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for rovibrational (0 – 4) excitation of O2 Energy (eV)

(a)a Angle 12 24 36 48 60 72 84 96 108 120 132 144 156

(b)b Angle 10 20 30 40 50 60 70 80 90

7

10

15

0.76 0.55 0.41 0.24 0.22 0.21 0.22 0.21 0.21 0.17 0.24 0.29 0.34 17%

0.76 0.42 0.31 0.27 0.24 0.23 0.27 0.26 0.30 0.33 0.37 0.36 0.48 15%

0.08 0.05 0.04 0.03 0.03 0.02 0.02 0.01 0.02 0.02 0.02 0.02 0.03 17%

Energy (eV) 7

8

0.048 0.0181 0.032 0.100 0.081 0.256 0.106 0.197 0.123 0.188 0.147 0.153 0.143 0.176 0.102 0.174 0.077 0.194 a The data are those of Shyn and Sweeney uncertainty. b The data are those of Brunger et al.

9

9.5

10

0.601 0.540 0.853 0.577 0.541 0.814 0.491 0.517 0.470 0.454 0.454 0.428 0.395 0.364 0.559 0.399 0.441 0.641 0.357 0.478 0.504 0.384 0.486 0.539 0.354 0.474 0.482 Figures and the Fgures at the foot of each

11

15

0.338 0.273 0.535 0.143 0.447 0.116 0.414 0.109 0.445 0.087 0.443 0.059 0.425 0.062 0.425 0.086 0.401 0.081 column represent the

the data of Wong et al. and Brunger et al. show a resonance enhancement in the cross-section which peaks at an incident energy of about 9.5 –10 eV. For the v = 0–2; 3; 4 DCS, similar trends ◦ to that described immediately above are also found in their energy dependence at 30 . In all cases there is a clear resonance enhancement of the DCS. 3.3.3.2. Integral cross-sections. Integral rovibrational cross-sections have been derived from di@erential measurements by Shyn and Sweeney and Noble et al. for incident electron energies between 5 and 20 eV and for the 0 → 1; 2; 3; and 4 transitions. These data are summarised in Table 41. Noble et al. also reported theoretical (R-matrix) calculations for all of these integral rovibrational excitation cross-sections and highlighted the role of several resonance states in the vibrational excitation process.

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Table 40 Di@erential cross-sections (10−19 cm2 sr −1 ) for rovibrational excitation (0–1) of O2 . The data are those of Allan and the uncertainty is ±35% Energy (eV) Angle

3.8

9.0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130

5.55 5.50 4.75 4.25 3.95 3.60 3.45 3.25 3.15 3.15 3.05 3.00 2.99 2.85 2.75 2.60 2.55 2.50 2.48 2.48 2.45 2.40 2.30 2.20 2.12 2.05

80.0 72.5 67.5 60.0 52.5 46.5 42.0 38.0 35.0 34.5 34.0 33.5 34.0 35.0 35.0 35.0 34.0 33.5 33.0 32.0 30.0 29.0 28.0 27.0 25.0 24.5

3.3.4. Excitation of electronic states 3.3.4.1. Di:erential cross-sections. There has recently been renewed interest in the study of electron impact excitation of the lowest-lying excited electronic states (a1 Ug ; b1 W) of O2 . These crossed beam investigations include the experiments of Doering [228], Shyn and Sweeney [112], Allan [229] and Middleton et al. [230]. The work of Middleton et al. also included some R-matrix calculations. The data of Doering for the a1 Ug state, were collected at incident electron ◦ energies of 2:6; 4:6; 7:6; 11:1; 16:1 and 28:6 eV over the angular range 20 –134 . These data are summarised in Table 42(a). Shyn and Sweeney performed di@erential cross-section experiments ◦ for both the a1 Ug and b1 W states at 5; 7; 10; 15 and 20 eV and over the angular range 12–156 . These data are given in Tables 42(b) and 43(a), respectively. Allan recently reported absolute excitation function measurements for both the a1 Ug and b1 W states at the electron scattering ◦ ◦ angles 30 and 90 and with a stated uncertainty of ±35%. DCS data were also reported for both these states at impact energies of 4.5 and 7:6 eV. Allan’s a1 Ug data are summarised in Tables 42(c) and (d), and the b1 W data in Tables 43(b) and (c). Finally, Middleton et al.

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325

Fig. 44. Di@erential cross-sections for v = 0–1 rovibrational excitation in O2 at (a) 5 eV and (b) 15 eV. ( ) Shyn and Sweeney, ( ) Brunger et al., () Allan, () Wong et al. (c) The energy dependence of the DCS at an angle ◦ near 30 ( ) Brunger et al., ( ) Wong et al., () Shyn and Sweeney.

•

•

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Table 41 Integral rovibrational excitation cross-sections (10−18 cm2 ) for excitation of O2 Energy (eV)

(a)a Transition

5

7

10

15

0 –1 0 –2 0 –3 0–4

9.5 3.4 — —

30.5 11.4 4.5 3.5

31.2 16.5 7.5 4.0

5.7 1.5 0.65 0.31

(b)b Transition

Energy (eV) 5

6

7

8

9

9.5

10

0 –1 5.21 11.55 20.24 25.62 40.26 46.36 53.63 0 –2 — 2.26 6.04 9.60 16.32 20.11 25.52 0 –3 — — 2.64 4.07 8.11 10.12 11.45 0–4 — — 1.15 2.28 4.73 5.82 6.53 a The data are those of Shyn and Sweeney. b The data are those of Noble et al. The uncertainties in the cross-sections are 20% 0 –2, 24% for the 0 –3 and 26% for the 0 – 4.

11

15

20

40.56 17.47 9.02 5.30

11.57 3.89 2.15 1.20

1.75 — — —

for the 0 –1, 22% for the

reported an extensive series of measurements at ten incident energies between 5 and 20 eV, for ◦ excitation of both the a1 Ug and b1 W states. The angular range of this work was 10–90 with the uncertainties on their data ranging from 17 to 25% for the a1 Ug state and 19 –30% for the b1 W state. These data are listed in Tables 42(e) and 43(d), respectively. In the early studies, Schulz and Dowell [231] reported upper limits on the integral crosssections for near threshold excitation of the a1 Ug and b1 W states of 3 × 10−20 and 6 × 10−21 cm2 , respectively, using the trapped electron method. Skerbele et al. [232] found that the relative intensities of the a1 Ug state referred to the elastic peak remained constant in the ◦ scattering range 3–12 for electrons whose incident energy was 45 eV. Konishi et al. [233] reported integral cross-sections for collision energies in the range 20–70 eV while Linder and Schmidt [212] detailed di@erential and integral cross-sections for excitation of both the a1 Ug and b1 W states from threshold to 4 eV. At about the same time as Linder and Schmidt, Trajmar et al. [119] measured di@erential and integral cross-sections, in the energy range 4 –45 eV, for excitation of the a1 Ug and b1 Wg states. The di@erential cross-sections were measured with a ◦ crossed beam conFguration over the scattered angular range 10 –90 . Absolute values were then assigned by applying a novel deconvolution procedure to the measured energy loss spectra and assuming a knowledge of the absolute elastic di@erential cross-section. Wong et al. [222] measured di@erential cross-sections for the a1 Ug and b1 W electronic states at ◦ an electron scattering angle of 25 and incident beam energies in the range 4 –15 eV. Subsequent to this Wakiya [214,234] reported both di@erential and integral cross-sections for the a1 Ug and ◦ b1 W states for energies in the range 20 –200 eV and at scattering angles from 10–130 . Tabulated values of all of these earlier cross-sections can be found in the Trajmar review. From a theoretical perspective, only the R-matrix method calculations of Middleton et al. are available in the literature for di@erential cross-sections of the a1 Ug and b1 W states. This calculation was performed with a model including nine electronic states of the target molecule

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Table 42 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for electron impact excitation of the a1 Ug state of O2 Energy (eV)

(a)a Angle

2.6

4.6

7.6

11.1

16.1

28.6

20 30 40 50 60 70 90 110 120 134

— — 0.324 0.460 0.537 0.389 0.218 0.217 — 0.207

— — 0.502 0.308 — 0.576 0.226 0.480 0.545 0.421

— 1.236 1.056 0.680 — 0.651 0.497 0.689 — 0.297

— — — 0.323 — 0.395 0.489 0.398 0.376 0.383

— 0.174 — 0.171 — 0.283 0.261 0.420 — —

0.127 0.103 — 0.142 — 0.118 0.118 0.177 0.271 0.416

Energy (eV)

(b)b Angle

5

7

10

15

20

12 24 36 48 60 72 84 96 108 120 132 144 156

0.99 0.95 0.78 0.58 0.53 0.57 0.57 0.62 0.61 0.49 0.53 0.63 0.61

1.23 1.1 0.93 0.69 0.58 0.64 0.65 0.74 0.79 0.86 0.99 1.1 1.2

1.4 1.1 0.82 0.74 0.63 0.62 0.67 0.70 0.63 0.69 0.72 0.76 0.76

0.44 0.29 0.27 0.24 0.27 0.29 0.28 0.32 0.35 0.37 0.42 0.43 0.43

0.22 0.17 0.15 0.16 0.17 0.17 0.17 0.17 0.19 0.19 0.21 0.21 0.24

Angle

(c)c Energy (eV)

30

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

0.67 1.5 2.6 3.8 4.5 4.8 4.8 4.5 4.1 3.9 4.8 6.3 7.7 8.3 8.4 8.1 7.5 7.1

◦

90

◦

0.62 1.4 2.3 3.1 3.7 4.1 4.4 4.5 4.5 4.8 5.4 6.1 6.5 6.4 6.2 5.9 5.7 5.4

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Table 42 (Continued) Angle

(c)c Energy (eV)

30

10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 16 17

6.5 5.9 5.3 4.7 4.2 3.7 3.3 2.9 2.6 2.4 2.1 1.7

◦

90

◦

5.3 5.1 5.0 4.8 4.7 4.5 4.3 4.2 3.9 3.7 3.3 3.0 Energy (eV)

(d)d Angle

4.5

7.6

5 10 20 30 40 50 60 70 80 90 100 110 120 130 135

6.7 6.4 5.4 4.7 3.9 3.6 3.7 4.0 4.2 4.4 4.3 4.0 3.6 3.3 3.1

11.8 11.3 9.6 8.1 7.1 6.5 6.3 6.5 6.6 6.4 6.1 5.6 5.1 4.6 4.7

(e)e Angle 10 20 30 40 50 60 70 80 90

Energy (eV) 5

6

7

8

9

9.5

0.657 0.699 0.184 0.048 0.701 0.998 0.795 0.623 0.245 0.324 1.361 1.155 0.794 0.536 0.367 0.879 1.090 1.095 0.791 0.530 0.444 0.675 0.961 0.887 0.667 0.550 0.452 0.649 0.786 0.741 0.538 0.592 0.481 0.543 0.758 0.777 0.554 0.578 0.517 0.625 0.693 0.789 0.534 0.526 0.482 0.623 0.687 0.799 0.588 0.610 0.586 0.619 0.635 0.767 a The data are those of Doering and the uncertainty is ±25%. b The data are those of Shyn and Sweeney and the uncertainty is ±16%. ◦ c Excitation function for the a1 Ug state of O2 at scattering angles of 30 and the data are those of Allan. d The data are those of Allan. Units are 10−19 cm2 . e The data are those of Middleton et al.

10

11

15

20

1.833 1.369 1.102 1.027 0.996 1.009 0.878 0.843 0.747

0.614 0.857 0.699 0.643 0.641 0.598 0.618 0.583 0.552

0.591 0.479 0.459 0.379 0.334 0.343 0.452 0.461 0.378

0.221 0.216 0.188 0.172 0.188 0.200 0.162 0.166 0.211

◦

and 90 . The units are 10−19 cm2 sr −1

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329

Table 43 Absolute di@erential cross-sections (in units of 10−18 cm2 sr −1 ) for electron impact excitation of the b1 W+ g state of O2 Energy (eV)

(a)a Angle

5

7

10

15

20

12 24 36 48 60 72 84 96 108 120 132 144 156

0.094 0.15 0.18 0.17 0.14 0.16 0.14 0.15 0.17 0.19 0.16 0.19 0.16

0.088 0.23 0.29 0.31 0.28 0.27 0.29 0.26 0.27 0.30 0.27 0.25 0.20

0.093 0.16 0.21 0.24 0.18 0.19 0.14 0.14 0.13 0.11 0.11 0.11 0.091

0.039 0.085 0.089 0.10 0.079 0.074 0.062 0.055 0.053 0.045 0.047 0.035 0.031

0.026 0.047 0.06 0.061 0.058 0.057 0.055 0.050 0.042 0.033 0.025 0.018 0.009

Angle

(b)b Energy (eV)

30

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15 16 17

0.57 1.6 3.4 5.3 6.1 6.0 5.7 5.1 6.0 9.1 12.7 17.3 20.2 21.1 21.4 21.0 20.1 19.3 18.0 16.6 15.1 13.8 12.3 11.5 10.7 10.0 9.2 7.6 6.6

◦

90

◦

0.7 2.6 4.2 6.1 7.9 9.2 10.2 12.0 15.1 19.2 22.4 25.2 25.6 24.7 23.8 23.0 21.8 20.6 19.7 18.3 16.8 15.4 14.3 13.3 12.5 11.7 10.6 9.4 8.5

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Table 43 (Continued) Energy (eV)

(c)c Angle

4.5

7.8

10 20 30 40 50 60 70 80 90 100 110 120 130 135

1.5 3.4 5.6 7.4 8.9 9.5 9.1 8.7 9.2 8.8 9.6 9.3 9.0 7.9

5.6 11.9 19.5 25.4 29.8 29.7 28.4 26.5 24.0 19.3 18.0 14.1 10.4 11.1

(d)d Angle

Energy (eV) 6

7

8

9

9.5

10

11

15

20

10 20 30 40 50 60 70 80 90

0.061 0.062 0.0012 0.048 0.053 0.243 0.043 0.120 0.047 0.080 0.101 0.030 0.202 0.153 0.307 0.162 0.110 0.049 0.110 0.125 0.143 0.256 0.241 0.311 0.219 0.115 0.059 0.099 0.161 0.183 0.338 0.295 0.357 0.272 0.127 0.061 0.065 0.192 0.241 0.364 0.321 0.341 0.330 0.158 0.072 0.056 0.170 0.218 0.310 0.360 0.348 0.301 0.138 0.062 0.070 0.159 0.236 0.302 0.310 0.296 0.287 0.135 0.060 0.089 0.164 0.201 0.252 0.284 0.264 0.237 0.117 0.072 0.138 0.181 0.194 0.205 0.257 0.267 0.217 0.101 0.077 a The uncertainty on the data is ±18%. The data are those of Shyn and Sweeney. ◦ ◦ b Excitation function data for the b1 W+ and 90 . The units are g state of O2 at scattering angles of 30 10−20 cm2 sr −1 and the data are those of Allan. c Absolute di@erential cross-sections (10−20 cm2 sr −1 ) for electron impact excitation of the b1 W+ g electronic state of O2 . The data are those of Allan. d The data are those of Middleton et al.

and conFguration interaction (CI) representations of the target wavefunctions. It represented one of the Frst ab initio calculations of electron scattering from an open-shell molecule. For the purpose of a critical review of the a1 Ug and b1 W data we consider each electronic state in turn, starting with the a1 Ug state. At 5; 6; 15 and 20 eV; i.e. in the regions away from the e@ects of the 2 Vu resonance, it is clear [235] that the level of agreement between the data and calculation of Middleton et al. is good. This is indicated by Figs. 45(a) and (b) which show the di@erential cross-sections for the a1 Ug state at 6 and 15 eV. Although not shown in detail, the measurement of Middleton et al. is found to be in fair agreement with those of Trajmar et al., Wong et al. and Shyn and Sweeney [112] in the range from 5 –20 eV. In particular, we note that there is very good agreement between the data of Middleton et al. and Shyn and

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Fig. 45. Di@erential cross-sections for electron impact excitation of the a1 Ug state of O2 (a) 6 eV, (b) 15 eV. ( ) Middleton et al., ( ) Wong et al., () Shyn and Sweeney, (×) Trajmar et al. (—) R-matrix calculation.

•

Sweeney at 10 eV. At 20 eV we can also compare some of the aforementioned experiments with the data of Wakiya. Again, whilst this is not shown in the Fgures, the absolute magnitude of the data of Wakiya is in better agreement with the data of Middleton et al. than with that of Trajmar et al., although the overall level of agreement between all of the data sets is fair. As we shall see in the next section, Middleton et al. [235] found that the integral cross-section for the resonance enhanced a1 Ug state excitation peaked at about 9.5 –10 eV whilst Noble and Burke [200] found this same peak to be at 8 eV. This discrepancy in the position of the maximum of the integral cross-section is also rePected in the level of agreement between the Middleton et al. theoretical and experimental di@erential cross-sections in the resonance region, i.e. 7–11 eV. In this energy regime, the general trend in the level of agreement is most easily assessed by comparing DCS for Fxed angles as a function of the scattering energy. The energy ◦ ◦ ◦ dependence of the DCS at scattering angles of 30 ; 60 and 90 are illustrated in Figs. 46(a) – (c), respectively. These Fgures demonstrate that despite the discrepancy Middleton et al. found between their theory and experiment concerning the position of the peak in the cross-section, there was otherwise good agreement in the overall trends and on the general magnitude of ◦ the cross-section. With the exception of 7 eV, the DCSs of Wong et al. at 25 were in very good agreement with the measurement of Middleton et al. over the entire range 7–11 eV. Both Trajmar et al. and Shyn and Sweeney only reported data at 7 and 10 eV in the energy region we are now discussing. At 10 eV the data of Trajmar et al. was in poor agreement with both the experiment and calculation of Middleton et al. and the experiment of Shyn and Sweeney, although at 7 eV the agreement of the data of Trajmar et al. with the calculation of Middleton et ◦ al. is fair. Only at 90 does Doering report enough measurements for a quantitative comparison to be made (see Fig. 46(c)) to the other data, the level of agreement being generally quite good

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Fig. 46. Di@erential cross-sections for electron impact excitation of the a1 Ug state of O2 plotted as a function of ◦ ◦ ◦ the incident energy for Fxed scattering angles of (a) 30 , (b) 60 , (c) 90 . ( ) Middleton et al., () Shyn and Sweeney, (♦) Doering, (×) Trajmar et al., (− − −) Allan, (—) R-matrix result.

•

◦

◦

for this case. The data of Allan [229] for angles of 30 and 90 are included in Figs. 46(a) and 46(c). This data is in quite good agreement with that of Middleton et al., particularly when one considers that it was gathered in a manner di@erent from that used by Middleton et al. The data of Allan also provides further conFrmation for resonance enhancement of the excitation process via the 2 Vu resonance, albeit at a slightly lower energy for the resonance peak compared to that found by Middleton et al. Middleton et al. had previously noted that the inclusion of nuclear motion into the R-matrix calculation of Noble and Burke could well lead to better agreement between experiment and theory with respect to the position of Qmax . This suggestion was taken up by Higgins et al. [220], who found that whilst there was some evidence to support the hypothesis of Middleton et al., it was more likely that the discrepancy between theory and experiment would be removed if a more complete treatment of the electronic motion in the scattering wavefunction was made. The level of agreement between the experiment and calculation of Middleton et al. for the excitation of the b1 W state was also good. This is illustrated in Figs. 47(a) – (c) where comparisons are made of all the experimental data and the R-matrix-calculated DCS at energies of 7; 10 and 11 eV, respectively. The form of this cross-section is largely determined by the “selection rule” discussed by Cartwright et al. [236]. They showed that for W+ ↔ W− transitions there are symmetry constraints which imply that the di@erential cross-section should tend ◦ ◦ to zero when the scattering angle approaches 0 or 180 . Indeed the experiments of Middleton et al., Trajmar et al. and Shyn and Sweeney are all consistent with this expectation at forward angles. With the exception of the data at 10 eV, Middleton et al. and Trajmar et al. are in good agreement, at the common energies of both studies, in terms of both the shapes and magnitudes

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Fig. 47. Di@erential cross-sections for electron impact excitation of the b1 W state of O2 at (a) 7 eV, (b) 10 eV and (c) 11 eV. ( ) Middleton et al., ( ) Wong et al., () Shyn and Sweeney, (×) Trajmar et al., (—) R-matrix result.

•

of the di@erential cross-sections. The recent measurements of Shyn and Sweeney at 10 eV were in better agreement (see Fig. 47(b)) with those of Middleton et al. than that of the work of Trajmar et al. although it too underestimated the magnitude of the b1 W cross-section to some extent. In general, the level of agreement between Middleton et al. and Shyn and Sweeney is ◦ reasonably good for the common energies 7; 10; 15 and 20 eV. The data of Wong et al. at 25 is also found to be in good agreement with that of Middleton et al. over the entire energy range of interest. Similarly, the work of Wakiya et al. [214,234] at 20 eV is in fair agreement with both Middleton et al. and Trajmar et al. The most recent data of Allan [229] (not shown) is in very good agreement with that of Middleton et al. for electron impact excitation of the b1 W+ g electronic state of O2 (see Fig. 5 of Allan). Consequently, we conclude that there exists a signiFcant body of experimental DCS data for the b1 W+ g state which are in reasonably good accord with each other. Electron impact excitation of the three excited “Herzberg” electronic states of O2 (c1 W− u + 3 A Uu + A3 W+ ) has been studied by Campbell et al. [237] and Green et al. [238]. The di@eru ential and integral cross-sections of these studies were measured at 9; 10; 12; 15 and 20 eV. The ◦ di@erential cross-sections were investigated over the scattered electron angular range 10 –90 and these measurements are summarised in Table 44. As there is no theory at the DCS level for these states we do not show them in graphical form. We also note that there is an excitation ◦ function measurement of the Herzberg states at 90 by Allan [229] although no tabulated values were published. Shyn et al. [239] have reported absolute di@erential cross-sections for a deconvolution of the Schumann–Runge (SR) continuum into three constituent states. These states are B3 W; I3 Vg

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Table 44 3 3 + Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the (c1 W− u + A Uu + A Wu ) states of O2 . The data are those of Green et al. Energy (eV) Angle

9 eV

10 eV

12 eV

15 eV

20 eV

10 15 20 30 40 50 60 70 80 90

1:52 ± 0:51 — 0:84 ± 0:31 0:62 ± 0:29 0:29 ± 0:12 0:37 ± 0:16 0:50 ± 0:17 0:47 ± 0:25 0:51 ± 0:26 0:46 ± 0:20

1:54 ± 0:35 — 1:26 ± 0:35 0:82 ± 0:28 0:64 ± 0:24 0:56 ± 0:35 0:69 ± 0:45 0:67 ± 0:38 0:58 ± 0:27 0:56 ± 0:28

— 0:92 ± 0:66 1:30 ± 0:79 1:20 ± 0:71 0:90 ± 0:53 0:83 ± 0:54 0:83 ± 0:43 0:64 ± 0:28 0:51 ± 0:17 0:48 ± 0:18

0:28 ± 0:09 — 0:32 ± 0:12 0:42 ± 0:13 0:45 ± 0:17 0:42 ± 0:16 0:48 ± 0:23 0:57 ± 0:21 0:43 ± 0:10 0:47 ± 0:16

0:20 ± 0:12 — 0:31 ± 0:20 0:45 ± 0:25 0:37 ± 0:18 0:33 ± 0:17 0:34 ± 0:16 0:35 ± 0:17 0:33 ± 0:13 0:35 ± 0:13

Table 45 Absolute di@erential cross-sections (10−18 cm2 sr −1 ) for electron impact excitation of the I3 Vg state of O2 . The data are those of Shyn et al. and the uncertainty is ±23% Energy (eV) Angle

15

20

30

50

12 24 36 48 60 72 84 96 108 120 132 144 156

4.4 2.0 1.0 0.45 0.21 0.11 0.093 0.11 0.20 0.30 0.42 0.52 0.64

4.0 2.0 1.1 0.38 0.19 0.15 0.21 0.36 0.36 0.40 0.43 0.47 0.54

3.0 1.1 0.41 0.12 0.03 0.041 0.075 0.11 0.11 0.081 0.052 0.040 0.074

1.3 0.28 0.084 0.031 — — 0.023 0.031 0.025 0.026 0.018 — —

and an as yet unclassiFed state, with tabulated values of their di@erential cross-sections being given in Tables 45 – 47, respectively. The unclassiFed state in the spectrum of Shyn et al. may simply be an artifact of their assumption of Gaussian line proFles for the states in the deconvolution process, whereas in fact the Schumann–Runge line proFles are all highly asymmetric. Consequently, we note that it is entirely possible that the unclassiFed state in Shyn et al. is ◦ actually a component of the I3 Vg level. The angular range of this study was 12–156 , whilst the incident electron energies were 15; 20; 30 and 50 eV. Previous studies [240,1] had indicated that substantial contributions to the SR continuum were made by four valence states and so we are unsure as to the validity and uniqueness of the deconvolution of Shyn et al., although we note that they speciFcally address this point in their paper. All the measured DCS for the

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335

Table 46 Absolute di@erential cross-sections (in units of 10−18 cm2 sr −1 ) for electron impact excitation of the B3 W− u state of O2 . The data are those of Shyn et al. and the uncertainty is ±18% Energy (eV) Angle

15

20

30

50

12 24 36 48 60 72 84 96 108 120 132 144 156

43 19 8.2 3.6 2.5 2.3 2.2 1.9 1.7 1.6 1.7 2.3 2.9

64 20 5.8 2.4 2.5 2.5 2.1 1.7 1.4 1.5 2.0 3.0 4.0

71 13 2.6 1.4 1.4 1.3 0.89 0.72 0.68 0.99 1.40 2.3 3.3

39 4 1.1 0.8 0.53 0.32 0.19 0.13 0.16 0.31 0.51 1.2 2.0

Table 47 Absolute di@erential cross-sections (in units of 10−18 cm2 sr −1 ) for electron impact excitation of the unclassiFed SR continuum state of O2 . The data are those of Shyn et al. and the uncertainty is ±20% Energy (eV) Angle

15

20

30

50

12 24 36 48 60 72 84 96 108 120 132 144 156

6.60 2.70 1.00 0.48 0.42 0.45 0.41 0.37 0.37 0.39 0.45 0.62 0.78

18 5.20 1.10 0.44 0.55 0.57 0.45 0.36 0.24 0.26 0.40 0.61 0.90

26 3.40 0.48 0.41 0.48 0.37 0.19 0.14 0.15 0.22 0.37 0.67 0.85

15 1.00 0.29 0.27 0.17 0.09 0.06 0.04 0.05 0.09 0.14 0.31 0.49

three states are forward peaked with the degree of this e@ect increasing as the incident beam energy was increased. Details of the earlier DCS data for the SR continuum can be found in Wakiya [234] and Trajmar et al. [119] and in the Trajmar review. Similarly, Shyn et al. [241] have also recently reported DCS for the 9:97 eV “longest” band (LB) transition and 10:29 eV “second” band (2B) transition. These DCS data are given in Table 48. The angular range of these ◦ measurements was 12–156 , whilst the incident electron energies were 15; 20; 30 and 50 eV. Two minima were apparent in both their LB and 2B distributions, indicating d-wave character.

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Table 48 Absolute di@erential cross-sections (in units of 10−18 cm2 sr −1 ) for electron impact excitation of the (a) LB and (b) 2B of O2 . The data are those of Shyn et al. and the uncertainty is (a) ±20% and (b) 23% Energy (eV) Angle

15

20

30

50

(a)

12 24 36 48 60 72 84 96 108 120 132 144 156

2.460 1.740 0.897 0.400 0.273 0.184 0.274 0.270 0.255 0.314 0.329 0.352 0.432

5.140 2.310 0.856 0.336 0.242 0.239 0.286 0.255 0.223 0.210 0.282 0.401 0.545

7.250 2.070 0.403 0.132 0.209 0.253 0.192 0.138 0.126 0.165 0.266 0.428 0.595

3.510 0.368 0.061 0.105 0.102 0.062 0.031 0.012 0.022 0.047 0.099 0.211 0.357

(b)

12 24 36 48 60 72 84 96 108 120 132 144 156

0.391 0.323 0.131 0.066 0.024 0.021 0.021 0.019 0.022 0.025 0.028 0.033 0.049

1.360 0.400 0.100 0.029 0.020 0.018 0.020 0.020 0.012 0.012 0.016

1.500 0.328 0.031 0.014 0.0167 0.0182 0.0116 0.0087 0.0089 0.0076 0.0143

1.21 0.061 0.013 0.016 0.011 0.007 0.003 0.0014 0.003 0.006 0.102

It is noteworthy that both minima approached smaller angles as the incident energy increased. Additionally, both showed strong forward scattering and relatively isotropic scattering at middle and high angles, an e@ect Shyn et al. described as indicative of both electric-dipoleallowed and dipole-forbidden transitions being involved in the production of the longest and second bands. 3.3.4.2. Integral cross-sections. Experimental integral cross-sections for the a1 Ug and b1 W electronic states have been reported by Linder and Schmidt [212], Trajmar et al. [119], Konishi et al. [233], Wakiya [234] and more recently by Doering [228], Shyn and Sweeney [112] and Middleton et al. [235]. From a theoretical perspective we note the R-matrix calculations of Noble and Burke [200,242] and Higgins et al. [220] and the e@ective range theory (ERT) results of Teillet-Billy et al. [227]. Tabulated values of the a1 Ug and b1 W integral cross-sections are given in Tables 49 and 50.

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Table 49 Integral cross-sections (10−18 cm2 ) for electron impact excitation of the a1 Ug state of O2 . Figures in parentheses represent the percentage uncertainty Energy (eV)

Doering

Shyn and Sweeney

Middleton et al.

2.6 4.6 5 6 7 7.6 8 9 9.5 10 11 11.1 15 16.1 20 28.6

3.56 5.51 — — — 7.88 — — — — — 4.90 — 4.14 — 2.58

— — 7.6 — 10.4 — — — — 7.7 — — 4.2 — 2.3 —

— — 7.79 (20) 7.28 5.71 (25) — 7.80 (21) 9.82 11.1 (16) 11.7 (15) 8.13 — 6.25 (19) — 5.70 (23) —

(25) (25)

(25)

(25) (25) (25)

Table 50 Absolute integral cross-sections (in units of 10−18 cm2 ) for excitation of the b1 W+ g state of O2 . Figures in parentheses represent the percentage uncertainty Energy (eV)

Shyn and Sweeney

Middleton et al.

5 6 7 8 9 9.5 10 11 15 20

2.0 — 3.3 — — — 1.9 — 0.78 0.55

— 1.39 2.30 3.17 3.30 3.57 3.73 3.05 1.49 1.40

(29) (27) (23) (21) (20) (19) (16) (25) (42)

In Fig. 48 we plot the recent integral cross-sections for the a1 Ug state, where we see that the data of Middleton et al. are generally larger in magnitude than both the ERT and the experimental results of Trajmar et al. However, within the experimental uncertainties on the data of Middleton et al. and Trajmar et al., there is little di@erence between them at most of the energies studied away from the resonance region. The data of Doering and Shyn and Sweeney favour the data of Middleton et al. at some energies and the data of Trajmar et al. at others. The level of agreement between the most recent R-matrix calculation [200] and the data of Middleton et al. in the energy regions away from the resonance peak

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•

Fig. 48. Integral cross-sections for electron impact excitation of the a1 Ug state of O2 . ( ) Middleton et al., (×) Trajmar et al., (♦) Doering, () Shyn and Sweeney, (- -) Noble and Burke [239] and (—) Noble and Burke [197] are both R-matrix results, and ( – – ) an ERT calculation.

•

Fig. 49. Integral cross-sections for electron impact excitation of the b1 W state of O2 . ( ) Middleton et al., (×) Trajmar et al., () Shyn and Sweeney, (- -) and (—) are both R-matrix results and ( – – ) an ERT calculation.

is good. Middleton et al. clearly measured the resonance peak to be at an incident energy ∼9:5– 10 eV whilst the calculation predicted it to be at 8 eV. The more sophisticated calculation by Higgins et al., which included both nuclear motion and a more exact CI description of the target than that of Noble and Burke, improved the level of agreement between theory and experiment with respect to the position of the resonance peak in the a1 Ug cross-section. The integral cross-sections for the b1 W state are given in Fig. 49. The data of Middleton et al. is again seen to be in quite good agreement with that of Trajmar et al. away from the resonance region, whilst the results of Shyn and Sweeney generally underestimate the strength of this cross-section at the higher energies. Agreement with the R-matrix theory is less impressive in this case with the predicted magnitude of the resonance peak being signiFcantly larger than that measured by Middleton et al. and the peak position again being lower than that seen in the

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339

Table 51 3 Absolute integral cross-sections (in units of 10−18 cm2 ) for the excitation of the A3 W + A U + c W sates of O2 . The uncertainty on the data of Teillet-Billy is estimated to be ±50% Energy (eV) 9 10 12 14 15 16 18 20

Teillet-Billy et al. 12 10 8 8 6.5 5

Green et al. 9:2 ± 3:8 13:2 ± 4:1 12:0 ± 4:0 10:7 ± 4:0 7:6 ± 4:0

experiment. Nonetheless, the data of Middleton et al. clearly conFrmed the prediction of Noble and Burke that there is a resonance enhancement in both the b1 W and a1 Ug cross-sections. Middleton et al. noted that the coupling of the O− (2 P) state and the 3 P; 1 D and 1 S states gives rise to some 46 negative ion states [222]. Of these Krauss et al. [243] found that, aside from the 2 Vg ion ground state, only the potential energy curves for the 2 Vu ; 2 W and 4 W states would extend to the shorter internuclear separations to be able to intersect the Franck–Condon region of the v = 0 vibrational sublevel of the neutral ground state. Of these states, the 2 W state is not accessible for attaching an electron to the target X3 W state because this would require a change in rePection symmetry (− → +) during the attachment process. Consequently, in addition to the 2 V ground ion state, only the 2 V and 4 W states, which Krauss et al. characterised as valence g u u Feshbach and shape resonances, respectively, are likely to play a role in low-energy resonance attachment to the neutral ground state. However as quartet states cannot decay into either of the singlet excited states [222], Middleton et al. concluded, in agreement with Noble and Burke, that the observed resonance enhancement in both the a1 Ug and b1 W integral cross-sections must be due to the decay from the 2 Vu core-excited resonance. Teillet-Billy et al. [227] have reported experimental integral cross-sections for the A3 W + 3 A Uu + c1 W groups of states. As discussed above, these Herzberg states have also been investigated recently by Campbell et al. [237] and Green et al. [238]. Teillet-Billy’s results were ◦ obtained by measuring the DCS at a scattering angle of 90 and, assumimg isotropic scattering, they multiplied this by 4
to obtain the ICS. Green et al. have subsequently shown this assumption to be incorrect. These data are listed in Table 51 and plotted in Fig. 50 along with the ERT results of Teillet-Billy et al. and the R-matrix results of Noble and Burke and Higgins et al. It is clear that the ERT calculation is in somewhat better agreement with experiment than the R-matrix result, although the R-matrix result is only slightly above the upper limit of the error bars in most cases. Noble and Burke have argued that the level of agreement between the ERT and experiment must be regarded as fortuitous as the R-matrix results indicated that there were signiFcant contributions to the scattering process from symmetries other than the 2 Vg , which is the only symmetry considered in the ERT approach. The existence of the strong resonance feature, predicted by theory in the integral cross-section, remains an open question as the data of Green et al. have not been performed at a Fne enough energy mesh to answer this deFnitively.

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•

Fig. 50. Integral cross-sections for electron impact excitation of the A3 W + A3 U + c1 W states of O2 . ( ) Green et al., (×) Teillet-Billy et al. (- - -) ERT result of Teillet-Billy et al., (—) and ( – – ) R-matrix results.

Table 52 Absolute integral cross-sections (in units of 10−18 cm2 ) for the excitation of the I3 Vg ; B3 W− u and unclassiFed (??) states of O2 . The data are those of Shyn et al. and Fgures in parentheses indicate the percentage uncertainty Energy (eV)

I 3 Vg

B 3 W− g

?? state

15 20 30 50

6.5 (23) 7.0 (23) 2.8 (23) 0.94 (23)

53 58 44 20

9.2 (21) 14 (21) 13 (21) 6.7 (21)

Shyn et al. [239] have also reported integral cross-sections for the electron impact excitation of the I3 Vg ; B3 W− u and unclassiFed states of the SR continuum. These data are summarised in Table 52. The I3 Vg and B3 W state excitation cross-sections were found to exhibit nearly the same energy dependence, both having a maximum at about 20 eV. The I3 Vg state cross-sections, however, have a substantially smaller magnitude than the B3 W cross-sections and diminish more rapidly that those of the B3 W state as the impact energy increases. Bearing in mind our caveat of Section 3.3.4(1), the integral cross-sections of the unclassiFed state have a magnitude between those of the other two and a maximum which appears to occur at around 30 eV. Shyn et al. [241] have also reported integral cross-sections for the LB and 2B excitations. These data are summarised in Table 53. Both cross-sections were seen to exhibit a broad maximum around 20 eV, with the LB integral cross-section being about a factor of 6 larger in magnitude, across the entire beam energy range, than the 2B integral cross-section.

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Table 53 Absolute integral cross-sections (10−18 cm2 ) for the longest band (LB) and second band (2B) transitions of O2 . The data are those of Shyn et al. and Fgures in parentheses represent the percentage uncertainty Energy (eV)

LB

15 20 30 50

5.75 6.69 6.22 2.36

2B (20) (20) (20) (20)

0.72 (23) 0.915 (23) 0.871 (23) 0.472 (23)

Fig. 51. The grand total scattering cross-section for Cl2 at incident energies between 20 meV and 10 eV. The data are those of Gulley et al.

3.4. The halogens Electron scattering from the halogens presents substantial experimental diJculties and consequently the amount of data available for absolute scattering cross-sections is very small. Nonetheless, since the Trajmar review the situation has improved somewhat although, to the best of our knowledge, the only experimental studies have been those on molecular chlorine. There is a similar situation on the theoretical front with the only recent calculation being that of Rescigno [244], who determined integral elastic and momentum transfer cross-sections as well as excitation and dissociation cross-sections using the Kohn variational approach. 3.4.1. Cl2 Total scattering cross-section. To our knowledge the only published measurement of a total scattering cross-section for molecular chlorine is that of Gulley et al. [245]. They used photons from the Daresbury Synchrotron Radiation Source to photoionise argon atoms at near-threshold

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energies, providing a source of highly monochromatic, low-energy electrons. The absolute total cross-sections were measured using an attenuation cell and applying the Beer–Lambert law. The measurements covered the energy range from 20 meV to 10 eV. The total cross-section is presented in Table 54 and shown in Fig. 51. As can be seen from the Fgure the cross-section is characterised by a very large absolute magnitude at both low (¡ 50 meV) and high (∼7 eV) energies and sharp resonance structure in the vicinity of 0:1 eV. This doublet structure is attributed to the intervention of the excited
-states of Cl− 2 . Whilst the comparison is not a completely valid one, there is also reasonable agreement in both shape and magnitude between the total cross-section of Gulley et al. and the calculated total elastic cross-section of Rescigno. Elastic scattering. Di@erential elastic scattering measurements have recently been carried out by Gote and Ehrhardt [186] using a high-resolution electron spectrometer in a crossed Table 54 Grand total cross-section for electron scattering from Cl2 (in units of 10−16 cm2 ). The data are those of Gulley et al. Energy (eV)

Cross-section

Energy (eV)

Cross-section

Energy (eV)

0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.034 0.036 0.038 0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.058 0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.08 0.082

39.97 40.62 38.69 39.24 36.94 34.25 33.83 32.01 30.41 28.69 26.75 24.80 22.56 21.05 18.57 18.01 15.04 13.67 12.32 11.49 10.71 9.72 8.74 8.12 7.96 6.98 7.36 7.79 8.08 7.98 8.38 8.94

0.084 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1 0.102 0.104 0.106 0.108 0.11 0.112 0.114 0.116 0.118 0.12 0.12 0.122 0.124 0.126 0.128 0.13 0.132 0.134 0.136 0.138 0.14 0.142 0.144

9.46 9.50 10.49 10.64 9.99 10.29 9.65 10.02 9.68 9.79 9.39 9.15 9.27 8.65 9.30 8.72 9.06 8.78 9.06 9.10 8.77 8.88 9.54 9.58 9.76 9.56 9.89 9.74 9.78 9.89 9.74 9.38

0.146 0.148 0.15 0.152 0.154 0.156 0.158 0.16 0.162 0.164 0.166 0.168 0.17 0.172 0.174 0.176 0.178 0.18 0.182 0.184 0.186 0.188 0.19 0.192 0.194 0.196 0.198 0.2 0.21 0.22 0.23 0.24

Cross-section 9.68 9.16 8.90 8.90 8.83 8.39 8.22 7.96 8.36 7.94 7.89 7.41 7.19 6.75 7.01 6.67 6.42 6.44 6.29 5.94 5.62 5.52 5.37 5.47 5.31 4.93 5.27 5.09 4.84 4.44 4.43 3.70

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343

Table 54 (Continued) Energy (eV)

Cross-section

Energy (eV)

Cross-section

Energy (eV)

Cross-section

0.25 0.275 0.3 0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

3.80 4.12 4.04 3.55 3.72 3.62 3.75 4.17 4.15 4.36 4.32 4.81 5.00 5.24 5.83 6.00 6.55 7.03 7.36 7.53 7.97 8.38 9.06 9.58 10.37 11.14 11.36 12.03 12.59 13.32 13.93 14.20

2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8

14.83 15.29 15.84 16.04 16.31 16.86 16.93 17.44 18.45 19.07 20.66 21.25 21.92 23.11 23.68 24.49 25.97 26.84 28.09 29.85 30.98 32.23 34.53 35.83 37.51 38.70 39.72 41.22 41.23 42.64 42.67 41.88

8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.5

42.77 42.46 41.34 41.74 41.37 40.97 40.60 40.64 40.89

beam conFguration. In an impressive series of measurements on a range of diatomic molecules, their energy resolution was high enough (∼10 meV) to enable them to employ spectral deconvolution techniques to isolate contributions from speciFc rotational transitions to the vibrationally elastic scattering. These measurements covered the energy range from 2 to 200 eV and the tabulated cross-sections are presented in Table 55 and selected examples are shown in Fig. 52(a) – (c). An elastic momentum transfer cross-section has been derived by Rogo@ et al. [246] from a Boltzmann analysis of an RF Chlorine discharge. They do not provide tabulated cross-sections but illustrate a cross-section set with a momentum transfer cross-section which has a value of ^ 2 between 0.01 and 1:0 eV and then rises to a maximum of approximately 40 A ^ 2 at an energy 2A of about 6 eV. This is roughly consistent with the total scattering cross-section of Gulley et al.

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Table 55 Absolute di@erential cross sections for elastic scattering from Cl2 (10−16 cm2 sr −1 ). The data are those of Gote and Ehrhardt Energy (eV) Angle

2

5

10

20

30

50

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

1.591 1.129 0.856 1.017 1.318 1.528 1.615 1.577 1.453 1.291 1.029 0.723 0.540 0.510 0.600 0.744

5.819 4.584 3.349 2.984 2.542 2.023 1.720 1.506 1.381 1.177 1.115 1.196 1.146 1.156 1.225 1.294

21.283 14.387 7.491 4.615 1.740 1.411 1.082 0.963 0.843 0.858 0.874 0.977 1.080 1.569 2.058 3.124

31.605 19.320 9.406 3.970 1.834 1.316 0.961 0.754 0.799 0.886 0.893 0.831 0.651 0.548 0.571 0.886

31.838 15.896 5.860 2.263 1.180 0.687 0.406 0.431 0.621 0.712 0.593 0.393 0.218 0.111 0.107 0.198

28.722 8.607 2.409 1.012 0.505 0.196 0.158 0.304 0.440 0.485 0.435 0.288 0.179 0.154 0.327 0.688

Energy (eV) Angle

70

100

150

200

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

22.238 5.434 1.457 0.592 0.262 0.131 0.152 0.219 0.295 0.320 0.237 0.117 0.093 0.177 0.435 1.049

15.695 3.498 0.912 0.415 0.188 0.133 0.148 0.201 0.192 0.148 0.076 0.036 0.073 0.153 0.590 0.909

14.123 2.397 0.778 0.345 0.185 0.154 0.154 0.126 0.086 0.054 0.026 0.030 0.084 0.154 0.328 0.547

12.915 2.518 0.831 0.357 0.230 0.190 0.132 0.094 0.064 0.035 0.026 0.048 0.110 0.239 0.339 0.580

described above, as well as with the calculated momentum transfer cross-section of Rescigno (see [244]). Electronic excitation. Information on electronic excitation of molecular chlorine is sparse with, to our knowledge, the only available data being that derived from the Boltzmann analysis of Rogo@ et al. They obtained a summed cross-section for the 1 P and 1 S Rydberg states

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345

Fig. 52. Di@erential cross-sections for elastic scattering from Cl2 at energies of (a) 2 eV (b) 5 eV (c) 50 eV. The data are those of Gote and Ehrhardt.

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and this was shown [244] to be in reasonable agreement with Rescigno’s Kohn variational calculation. 3.5. Carbon monoxide (CO) Cross-sections for the interaction of electrons with carbon monoxide (CO) are needed in areas such as the modelling of various plasmas for which CO is an important component, the studies of CO adsorbed on surfaces and re-entry physics. SpeciFc examples of applications of measurements for CO are the dominance of the Cameron bands (a3 V → X1 W+ ) in the emission spectrum of the Martian atmosphere and the importance of CO in atmospheric convection on Jupiter. In general there is a deFnite need for a detailed study of electron impact excitation of CO in order to obtain a broader understanding of our own atmosphere and the astrophysics of the interstellar medium. 3.5.1. Grand total cross-sections Grand total cross-sections for CO were Frst measured in the 1930s by Bruche [201] and by Ramsauer and Kollath [202]. More recently, measurements by Gus’kov et al. [247] Szmytkowski and Zubek [248], Hasted et al. [249], Buckman and Lohmann [250], Kwan et al. [251], Sueoka and Mori [252], Garcia et al. [253], Kanik et al. [210], Karwasz et al. [254] and Xing et al. [255] have been reported in the literature. We also note the recommended grand total cross-section, for energies between 1 and 1000 eV of Kanik et al. [211]. With the exception of Sueoka and Mori, who did not provide a table of their measured values, a selection of these data, from 1980 onwards, are tabulated in Table 56(a–c) and plotted in Figs. 53(a) and (b). The majority of the above measurements employed either a modiFed Ramsauer-type technique [254] or the linear transmission technique, either with [250,252] or without [251,253,210,255] time-of-Pight analysis. The energy ranges of the post-1980 measurements were 1.2–403 eV for Sueoka and Mori, 0.5 –4:9 eV for Buckman and Lohmann, 1–500 eV for Kwan et al., 380 –5200 eV for Garcia et al., 5 –300 eV for Kanik et al., 80 –400 eV for Karwasz et al. and 400 –2600 eV for the work of Xing et al. Consequently, the available data covers a wide range of overlapping energy regimes thereby allowing Kanik et al. to construct their set of recommended values. From a theoretical perspective Chandra [256] reported results of calculations for low-energy grand total scattering cross-sections. These data were obtained by using the frame-transformation theory with a single-centre pseudo-potential method. This work was previously shown to be inadequate [250] and so we do not reproduce it in Fig. 53. Jain and Baluja [257] reported grand total cross-sections in the 10 –5000 eV energy range using a model-complex-optical potential in the variable-phase approach. This yielded the complex scattering phase-shifts from which the grand total cross-sections were derived in the usual manner. This theoretical result is shown in Fig. 53(b) where it is compared with the experimental data. The only other calculation we are aware of at this point in time was reported by Jain and Norcross [258]. In this work, grand total cross-sections were calculated in the low-energy region (0.001–10 eV) using an exact exchange (via the separable-exchange formulation) plus a parameter-free correlation–polarisation model in the Fxed-nuclei approximation (FNA).

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347

Table 56 Grand total cross sections (in units of 10−16 cm2 ) for electron scattering from CO at energies (a) below 5 eV, (b) between 5 and 80 eV and (c) 80 and 1000 eV (a) Energy (eV)

Buckman and Lohmann

Kwan et al.

Kanik et al. (recommended)

0.5 0.533 0.575 0.625 0.67 0.717 0.763 0.805 0.854 0.905 0.945 0.995 1 1.064 1.08 1.158 1.25 1.253 1.346 1.38 1.405 1.43 1.5 1.57 1.6 1.68 1.7 1.75 1.8 1.81 1.82 1.84 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2

10.69 11.03 11.45 11.73 11.96 12.19 12.46 12.74 13.06 13.44 13.77 14.19 — 14.9 — 16.45 — 17.89 19.78 — 24.02 — 29.25 — 34.54 — 37.95 — 41.61 — 41.94 42.32 42.73 43.02 43.52 43.03 43.44 43.17 43.24 43.19

— — — — — — — — — — — — — — 20.4 — — — — 30.8 — 35.4 — 39.9 — 43.2 44.3 — — 44.7 — — — — — — — 45.2 — 44.1

— — — — — — — — — — — — 14.2 — — — — — — — — — — — — — — — — — — — — — — — — — — 43.2

2.1 2.2 2.26 2.3 2.4

42.3 40.38 — 38.24 36.02

— — 38.4 — 34.2

— — — — —

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Table 56 (Continued) (a) Energy (eV)

Buckman and Lohmann

Kwan et al.

Kanik et al. (recommended)

2.5 2.6 2.7 2.8 2.9 3 3.1 3.15 3.3 3.4 3.9 4 4.4 4.8 4.9

33.56 31.45 29.01 27.07 25.62 — — 22.07 — 19.62 16.9 — 15.12 — 14.44

— 28.6 — — 25.4 — 22.7 — 20.7 — 17.5 — — 15.1 —

— — — — — 23.7 — — — — — 17 — — —

(b) Energy (eV)

Kwan et al.

Kanik et al.

Kanik et al. (recommended)

5 5.9 6 7 7.9 8 9 9.8 10 11 11.7 12 13 13.7 14 15 15.7 16 17 17.7 18 19 19.8 20 25 30 35 40 50 60 70 75

— 13.8 — — 13.65 — — 13.36 — — 13.1 — — 13.85 — — 13.92 — — 14.3 — — 14.15 — 13.94 13.42 12.8 12.45 11.72 — — 10.18

14.73 — 13.63 13.61 — 13.88 13.62 — 13.27 13.18 — 13.34 — — 13.75 — — 14.07 — — 14.26 — — 14.46 14.06 13.54 — 12.67 11.9 11.19 10.62 —

15 — 13.9 13.4 — 13.1 13.05 — 13 — — 13.2 — — — 13.8 — — — — — — — 14.1 — 13.5 — 12.7 11.9 11.1 10.5 —

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Table 56 (Continued) (c) Energy (eV)

Kwan et al.

Kanik et al.

Kanik et al. (recommended)

Karwasz et al.

Garcia et al.

Xing et al.

80 90 100 125 144 150 175 196 200 250 256 300 324 350 381 400 484 500 569 600 700 763 800 900 968 1000

— — 9.01 — — 7.43 — — 6.3 — — 4.85 — — — 4.12 — 3.56 — — — — — — — —

10.11 9.69 9.29 8.41 — 7.65 — — 6.6 5.79 — 5.23 — — — — — — — — — — — — — —

10 9.5 9 — — 7.5 — — 6.3 — — 4.95 — — — 4.1 — 3.55 — 3.14 2.8 — 2.56 2.35 — 2.18

10.1 9.74 9.27 8.18 — 7.55 7.05 — 6.49 5.63 — 4.98 — 4.5 — 4.14 — 3.56 — 3.16 2.8 — 2.52 2.29 — 2.1

— — — — — — — — — — — — — — 4.312 — — — 3.28 — — 2.64 — — 2.22 —

— — — — — — — — — — — — — — — 4.2 — 3.61 — 3.16 2.88 — 2.55 2.37 — 2.18

In Figs. 53(a) and (b) it is apparent that from about 2 to 1000 eV the reported values are in quite good agreement. The exception to this general statement is the cross-section of Sueoka and Mori which is considerably lower (∼13%) than the other experiments. We note that Karwasz et al. attributed this discrepancy to the normalisation procedure used by Sueoka and Mori. At the higher energies studied, the results of both Garcia et al. and Xing et al. are consistent in all cases to within their combined experimental uncertainties. On the other hand, at the very low energies studied (¡ 2 eV), where the excitation process is dominated by the 2 V shape resonance, there is a signiFcant disagreement between the results of Kwan et al. and Buckman and Lohmann. Kanik et al. in their table of recommended grand total cross-sections for CO, preferred the more recent values of Buckman and Lohmann over those of Kwan et al. although we note that no rationale for this choice was provided. The earlier measurement of Szmytkowski and Zubek sheds no light on this situation as it is typically about 25% lower, across the entire energy range, than the data of Buckman and Lohmann. Clearly, another independent measurement for energies below about 2 eV may be desirable in order to resolve this discrepancy.

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•

Fig. 53. The grand total scattering cross-section for CO (a) 0 –10 eV: (b) 10 –1000 eV. ( ) Buckman and Lohmann, (×) Kwan et al., (+) Garcia et al., ( ) Kanik et al., ( ) Karwasz et al., () Xing et al., (—) Kanik et al., (recommended), ( – - – ) Jain and Baluja, (− − −) Jain and Norcross.

With regard to a comparison between theory and experiment the calculation of Jain and Baluja is in very poor agreement with the experimental data for energies less than about 600 eV (see Fig. 53(b)) and only in fair agreement thereafter. We consequently conclude that the approach employed in this case does not provide a satisfactory description for the scattering process. Contrary to this, the low-energy calculation of Jain and Norcross is in

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351

Table 57 Di@erential cross sections (in units of 10−16 cm2 sr −1 ) for elastic scattering from CO. The data are those of Nickel et al. and the uncertainty in each measurement is ±20% Energy (eV) Angle

20

30

40

50

60

70

80

90

100

20 30 40 50 60 70 80 90 100 110 120

4.93 2.83 1.67 1.04 0.645 0.391 0.263 0.242 0.301 0.393 0.48

5.14 2.51 1.24 0.629 0.344 0.214 0.147 0.128 0.145 0.198 0.292

4.68 2.00 0.869 0.412 0.221 0.138 0.096 0.076 0.084 0.131 0.219

4.13 1.58 0.633 0.299 0.171 0.114 0.082 0.068 0.081 0.131 0.214

3.66 1.30 0.50 0.237 0.137 0.094 0.072 0.066 0.078 0.117 0.179

3.32 1.08 0.403 0.196 0.119 0.091 0.074 0.071 0.083 0.113 0.158

2.94 0.906 0.343 0.174 0.111 0.086 0.073 0.071 0.080 0.100 0.130

2.79 0.785 0.295 0.163 0.112 0.086 0.076 0.073 0.076 0.086 0.113

2.70 0.742 0.272 0.150 0.108 0.084 0.077 0.073 0.072 0.082 0.096

fair agreement with the experimental results of Buckman and Lohmann and Kwan et al., although, at the peak in the resonance, the theory overestimates the magnitude of the cross-section by about 25%. One possible explanation for this discrepancy was the neglect of nuclear motion in the calculation, although we note that Jain and Norcross specifically tried to compensate for the limitations of the FNA by employing the multipoleextracted-adiabatic-nuclei approximation. 3.5.2. Elastic scattering 3.5.2.1. Di:erential cross-sections. Previous investigations into elastic scattering of electrons from CO have been surprisingly limited, especially since Ehrhardt et al. [156] established, as long ago as 1968, the interesting resonance phenomena that could be studied in the elastic and rovibrational channels. Indeed, if one considers the volume of e@ort directed towards the isoelectronic “partner” of CO (N2 – Section 3.2) then its neglect is quite remarkable. Furthermore, from a more applied perspective, public concern over the consequences of air pollution mitigates for a serious experimental and theoretical study into its properties. Di@erential elastic scattering cross-sections were reported by Bromberg [259], Du Bois and Rudd [260], Tanaka et al. [261] and Jung et al. [192]. We also note the low energy, relative angular distributions of Ehrhardt et al. at the energies 1; 1:45; 2 and 3:5 eV. All this earlier data was discussed previously in the Trajmar review and so we do not discuss them in detail again here. Subsequent to this review there have been measurements by Sohn et al. [262] at ◦ the single energy of 0:165 eV and for scattering angles between 10 and 110 , and by Nickel ◦ et al. [185], who measured data for the energy range 20 –100 eV and angular range 20 –120 . Nickel et al. set their absolute scale via the use of the relative Pow technique with helium as the standard gas. These data are presented in Table 57. Middleton et al. [263] measured ◦ angular distributions at energies of 20; 30; 40 and 50 eV for angles between 10 and 100 . These ◦ data were normalised, at each energy, to the corresponding value of Nickel et al. at 50 . They

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Table 58 Di@erential cross-sections (in units of 10−16 cm2 sr −1 ) for elastic scattering from CO. The uncertainty for each energy is indicated at the foot of each column. The data are those of Middleton et al. Energy (eV) Angle

20

30

40

50

10 20 30 40 50 60 70 80 85 90 95 100

8.48 4.85 2.79 1.55 1.04 0.599 0.372 0.274 0.262 0.267 0.311 — 22%

7.49 4.36 2.24 1.18 0.622 0.379 0.214 0.161 — 0.131 — 0.151 23%

14.41 5.94 2.27 0.927 0.412 0.202 0.129 0.0958 — 0.0639 — 0.0799 23%

12.60 5.31 1.89 0.697 0.299 0.172 0.0919 0.0819 — 0.0631 — 0.0698 23%

Table 59 Di@erential cross-sections (10−16 cm2 sr −1 ) for elastic electron scattering from CO. The data are those of Gote and Ehrhardt Energy (eV) Angle

5

10

25

50

75

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

1.808 2.045 2.133 2.057 1.925 1.693 1.417 1.145 0.930 0.809 0.708 0.683 0.705 0.750 0.851 0.955

2.542 2.117 1.787 1.464 1.180 0.969 0.674 0.488 0.384 0.391 0.459 0.550 0.619 0.712 0.791 0.806

6.915 4.247 2.171 1.136 0.649 0.396 0.264 0.188 0.162 0.183 0.222 0.302 0.390 0.496 0.634 0.751

9.718 4.195 1.444 0.545 0.244 0.150 0.096 0.067 0.053 0.058 0.094 0.149 0.243 0.321 0.425 0.529

8.649 3.064 0.995 0.303 0.141 0.092 0.068 0.055 0.050 0.053 0.075 0.104 0.135 0.167 0.208 0.240

are given in Table 58. The most recent investigations were two extensive studies reported by Gote and Ehrhardt [186] and Gibson et al. [264]. In the Frst case, the energy range was very ◦ broad, covering the region between 5 and 200 eV and for scattering angles between 10 and ◦ 160 . Gibson et al. performed measurements at energies between 1–30 eV, whilst the scattered ◦ angular range was 15 –130 . In both of these works the absolute cross-section scale was set

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353

using the relative Pow technique, again employing helium as the standard gas. Following on from the protocol that the latter group established for N2 [33] the experimental beam energies in the region of the 2 V resonance were chosen to correspond to the energies for the structures in the resonance proFle, in either the 0 – 0 or 0 –1 channels. Tabulated values of these DCS are given in Table 59 (Gote and Ehrhardt) and Table 60 (Gibson et al.). From a theoretical perspective there have been even fewer investigations than those described immediately above for the experiments. Onda and Truhlar [265] reported the result of a close-coupling calculation at 10 eV impact energy. The CO molecule was treated as a rigid rotator and the calculations were based on a static charge distribution that had the correct dipole and quadrapole moments, cusps at the nuclei, and was augmented by an SCF treatment of polarisation and a local approximation for exchange scattering. A more sophisticated approach was adopted by Jain and Norcross who carried out ab initio calculations at the exact-static-exchange plus polarisation (ESEP) levels under the Fxed-nuclei approximation (FNA) at the energies of 0:165; 1:8; 2:1; 3; 5; 7:5 and 10 eV. The most detailed theoretical e@ort, to date, were the R-matrix calculations of Morgan [264] which were based on the work of Morgan and Tennyson [266]. These calculations, conducted with both SCF- and CI-level wavefunctions provided DCS at most of the low energies investigated by Gibson et al. Note that the more recent R-matrix calculation of Morgan was published along with the experimental measurements of Gibson et al. In Figs. 54(a) – (d) we compare representative examples of the available data and calculations. At 1:91 eV (Fig. 54(a)) the data of Gibson et al. and Jung et al. are in quite poor agreement, although this may partly be due to the mismatch in energies in a region where the 2 V resonance does have some e@ect in the elastic channel. Both the ESEP calculation of Jain and Norcross and the R-matrix calculations of Morgan and co-workers overestimate the magnitude of this DCS, although we note that they are in better accord with the data of Gibson et al. compared to the earlier data of Jung et al. Both theories have similar shapes, with the R-matrix result being in marginally better agreement with the data of Gibson et al. than the ESEP prediction. We note that in this case the earlier calculation of Morgan and Tennyson (not shown) appears to be in better agreement, with a poorer wavefunction, with the DCS of Gibson et al. than the more sophisticated later work. In Fig. 54(b) we see there is quite poor agreement between the recent measurement of Gibson et al. and the earlier result of Tanaka et al. [261] at 3 eV impact energy. We note that the shape of the relative angular distribution of Ehrhardt et al. [156] is in fair accord with the measurement of Gibson et al. Neither the ESEP or R-matrix results are able to quantitatively reproduce our preferred experimental result of Gibson et al., although at this energy the ESEP, particularly with regard to the shape of the DCS, is clearly the better of the calculations. Indeed we make the general observation that at the lower energies, ¡ 5 eV, the level of agreement between theory and experiment, and between the available experiments, is still quite unsatisfactory with further studies being required. At 7.5 and 9:9 eV (Figs. 54(c) and (d), respectively) there is better accord between Tanaka et al. and Gibson et al. and, in the latter case, the data of Gote and Ehrhardt. We also highlight the very good agreement between the R-matrix result and the data of Gibson et al. at both these energies. The ESEP calculation is also quite successful in predicting the shape of the DCS at both energies but clearly overestimates the magnitude. At 10 eV the CC result of Onda and Truhlar was found to be inadequate across the entire angular range.

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Table 60 Di@erential cross-sections (in units of 10−16 cm2 sr −1 ) for elastic scattering from CO. The percentage uncertainty for each measurement is indicated in parentheses. The data are those of Gibson et al. Energy (eV) Angle

1.0

1.25

15 17 20 25 30 35 40 45 50 55 60 70 80 90 100 110 120 130

— — — 0.591 0.542 — 0.480 — 0.540 — 0.670 0.850 1.041 1.240 1.415 1.573 1.701 1.802

— 1.370 1.260 — 0.967 — 0.831 — 0.804 — 0.884 1.048 1.219 1.430 1.603 1.761 1.920 2.078

(14) (7) (7) (6) (6) (6) (7) (7) (6) (6) (6) (6)

1.50 (9) (7) (7) (7) (7) (6) (7) (7) (6) (7) (6) (6) (6)

3.362 — 3.041 2.733 2.526 2.294 2.073 1.937 1.841 1.745 1.677 1.659 1.741 1.827 1.937 2.055 2.219 2.445

1.91 (7) (6) (7) (6) (7) (7) (6) (7) (6) (6) (6) (6) (6) (6) (6) (6) (6)

5.974 — 5.757 5.510 5.232 4.928 4.619 4.285 3.951 3.560 3.278 2.789 2.329 2.000 1.716 1.551 1.506 1.634

2.45 (7) (7) (8) (8) (8) (8) (7) (8) (9) (7) (7) (7) (7) (7) (7) (7) (7)

5.389 — 5.267 5.318 5.097 4.980 4.625 4.392 4.018 3.742 3.364 2.956 2.287 1.575 1.180 0.924 0.822 0.880

3.0 (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

3.789 (7) — 3.853(7) 3.917 (7) 3.846 (7) 3.745 (7) 3.600 (7) 3.407 (7) 3.190 (7) 2.953 (7) 2.651 (7) 2.144 (7) 1.619 (7) 1.228 (7) 0.916 (7) 0.720 (7) 0.621 (7) 0.674 (7)

Energy (eV) Angle

5.0

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130

— 1.956 2.066 2.071 2.113 2.119 2.064 2.018 1.896 1.777 1.641 1.507 1.312 1.175 1.018 0.897 0.798 0.715 0.640 0.588 0.565 0.538 0.553 0.549 0.553

6.0 (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

— 2.028 1.903 1.887 1.911 1.911 1.883 1.815 1.720 1.598 1.466 1.345 1.176 1.029 0.916 0.799 0.710 0.628 0.582 0.544 0.528 0.526 0.548 0.564 0.590

7.5 (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

— 2.314 2.130 2.039 1.947 1.871 1.826 1.705 1.606 1.455 1.318 1.183 1.006 0.861 0.732 0.622 0.543 0.486 0.463 0.460 0.465 0.488 0.521 0.560 0.607

9.9 (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

— 2.813 2.465 2.282 2.079 1.958 1.824 1.623 1.464 1.314 1.119 0.939 0.774 0.625 0.521 0.441 0.391 0.382 0.387 0.406 0.440 0.484 0.539 0.575 0.624

(7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7) (7)

20

30

7.60 (12) — 4.504 (7) 3.509 (7) 2.569 (7) 2.015 (7) 1.572 (7) 1.239 (7) 0.995 (7) 0.788 (7) 0.611 (8) — 0.383 (7) — 0.244 (7) — 0.222 (8) — 0.273 (7) — 0.362 (7) — 0.447 (7) — 0.554 (7)

10.616 (10) — 5.292 (7) — 2.544 (7) — 1.221 (7) — 0.636 (8) — 0.348 (7) — 0.208 (7) — 0.142 (7) — 0.123 (8) — 0.141 (7) — 0.199 (8) — 0.297 (8) — 0.441 (7)

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Fig. 54. Di@erential cross-section for elastic electron scattering from CO at (a) 1:91 eV, (b) 3 eV, (c) 7:5 eV and (d) 9:9 eV. ( ) Gibson et al., ( ) Tanaka et al., () Jung et al., (+) Ehrhardt et al., () Gote and Ehrhardt, (− −) Jain and Norcross, (—) Morgan, (- - -) Onda and Truhlar.

•

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•

Fig. 55. Di@erential cross-section for elastic electron scattering from CO at (a) 20 eV and (b) 50 eV. ( ) Nickel et al., ( ) Middleton et al., () Gote and Ehrhardt.

In the 20 –100 eV impact energy range there is only the “two-potential” coherent approach calculation of Jain et al. [267] at some energies in the range 50 –800 eV, against which we can compare the data of Nickel et al., Middleton et al., Gote and Ehrhardt and Gibson et al. This is done in Figs. 55(a) and (b) at impact energies of 20 and 50 eV, respectively. It is apparent that the data of Nickel et al., Middleton et al. and Gote and Ehrhardt are in excellent agreement, a trend found throughout the 20 –50 eV impact energy range. On the other hand, in the same energy region, the data of Tanaka et al. (not shown) were found to be somewhat smaller in absolute value and less forward peaked. This same trend was also typically observed at the lower energies by Gibson et al., when comparing to the results of Tanaka et al., suggesting perhaps that Tanaka et al. did not “correctly” apply their relative Pow procedure. With the exception of the region of the minimum in the DCS, the theoretical results of Jain et al. are typically 20 –50% higher than the data of Nickel et al. and Gote and Ehrhardt although, somewhat surprisingly, the agreement seems best at the lowest (50 eV) calculated energy. 3.5.2.2. Integral cross-sections. Integral elastic cross-sections have been reported by Tanaka et al., Nickel et al. and Gibson et al. In addition Kanik et al. have published a set of recommended elastic ICS. These data are presented in Table 61(a). We note that for each of Tanaka et al.,

M.J. Brunger, S.J. Buckman / Physics Reports 357 (2002) 215–458 Table 61 (a) Integral elastic scattering cross-sections (in units of 10−16 cm2 ) for CO Energy (eV)

Tanaka et al.

Nickel et al.

Gibson et al.

Kanik et al.

1 1.25 1.50 1.91 2 2.45 3 4 5 6 7 7.5 8 9 9.9 10 12 15 20 30 40 50 60 70 75 80 90 100 150 200 300 400 500 600 700 800 900 1000

— — — — — — 18 — 15 — — 11 — — 12 — — 11 9.9 6.6 — 5.5 — — 4 — — 3 — — — — — — — — — —

— — — — — — — — — — — — — — — — — — 10.5 9.2 7.6 6.15 5.35 4.71 — 4 4.05 3.25 — — — — — — — — — —

15.4 20.4 26.8 35.8 — 30.2 22.7 — 14.0 12.9 — 12.0 — — 11.4 — — — 11.0 10.3 — — — — — — — — — — — — — — — — — —

13.56 — — — 37.6 — 23.2 16.84 14.9 13 12.2 — 12 11.7 — 11.5 11.2 11 9.4 7.8 6.8 6 5.5 5.1 — 4.7 4.4 4.2 3.3 2.8 2.21 1.9 1.65 1.5 1.38 1.28 1.2 1.1

(b) Elastic momentum transfer cross-sections for CO. Units are 10−16 cm2 Energy (eV) 0.4 0.7 1.0 1.2 1.25 1.3

Tanaka et al.

Gibson et al.

Haddad and Milloy

— — — — — —

— — 19.1 — 25.4 —

13.5 14.5 16.0 21.5 — 26.5

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Table 61 (Continued) Energy (eV)

Tanaka et al.

Gibson et al.

Haddad and Milloy

1.5 1.7 1.9 1.91 2.1 2.4 2.45 3 4 5 6 7.5 9.9 15 20 30 50 75 100

— — — — — — — 13 — 12 — 9.7 10 8.5 6.7 4.1 3.2 1.6 1.1

27.8 — — 30.1 — — 21.4 15.8 — 10.3 9.8 8.9 8.3 — 6.6 5.6 — — —

38.0 43.0 43.0 — 35 25.0 — 18.5 13.0 — — — — — — — — — —

Nickel et al. and Gibson et al. the ICS were derived from their measured DCS using an appropriate extrapolation technique (see Section 2.2.2). From a theoretical perspective the results from the calculations of Onda and Truhlar, Jain and Baluja, Jain et al., Morgan, and Morgan and Tennyson are available at this time. These calculations and the above experimental data are compared in Fig. 56 where it is obvious that there is a very good level of agreement between the data of Gibson et al. and the recommended set of Kanik et al. We also note the quite good level of agreement between the most recent R-matrix result of Morgan [264] and both Gibson et al. and Kanik et al. for impact energies ¿ 2 eV. On the other hand, the calculation of Morgan and Tennyson clearly predicts a lower energy for the resonance peak, although its width seems in reasonable accord with the experimental result. In fact, within the combined uncertainties on the data sets, all of the experimental results are in reasonable agreement. Contrary to this, the higher energy theoretical results of Jain et al. and Jain and Baluja signiFcantly overestimate the magnitude of the ICS up to energies of 1000 eV. Clearly both these theories are inadequate although we note that the result of Jain et al., using a “two-potential” coherent approach, is in better agreement with the experimental data than the model complex optical potential, variable-phase approach of Jain and Baluja. While we do not show it in Fig. 56, at the single energy of 10 eV the calculation of Onda and Truhlar is about 30% higher than the recommended value for the ICS of Kanik et al. [211]. In summary it seems that at the ICS level there is a volume of experimental data, over a wide range of energies, that is largely in good agreement with one another and with the recommended set of Kanik et al. On the other hand, more theoretical e@ort is required to

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359

◦

Fig. 56. Integral cross-section for elastic electron scattering from CO. ( ) Gibson et al., ( ) Tanaka et al., ( ) Nickel et al., (—) Kanik et al., (− − −) Morgan, ( – - – ) Morgan and Tennyson, (— —) Jain et al., ( – – ) Jain and Baluja.

understand the scattering process involved in this system, particularly at intermediate energies. 3.5.2.3. Elastic momentum transfer cross-sections. Momentum transfer cross-sections have been derived from swarm measurements by Haddad and Milloy [268], from di@erential scattering measurements by Tanaka et al. and Gibson et al. while calculated results have been reported by Jain and Norcross, Jain et al., Morgan, and Morgan and Tennyson. All these data are illustrated in Fig. 57 with the experimental data being listed in Table 61(b). Although we do not depict them on the plot or list them in the table we note, for completeness, the earlier swarm-derived result of Land [269] and Hake and Phelps [221]. In Fig. 57 we see that the data of Gibson et al. and Haddad and Milloy are in quite good agreement, particularly within the uncertainty on the beam data. Similarly, Gibson et al. are also in reasonable agreement with the cross-sections of Tanaka et al. The R-matrix calculation of Morgan [264] is in good accord with the data of Gibson et al. for energies greater than 1:9 eV. The earlier result of Morgan and Tennyson places the resonance peak at too low an energy, similar to the observation made for the elastic ICS. The work of Jain and Norcross clearly reproduces the gross features of the momentum transfer cross-section, although there are some di@erences in the Fne detail when this work is compared to Haddad and Milloy and Gibson et al. At higher energies, the calculation of Jain et al. appears to be in reasonable accord with the data of Tanaka et al. Given our observations with regard to this work in the preceding section for the elastic ICS, this is consistent with the conclusion that this work is more accurate for backward scattering angles.

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Fig. 57. Elastic momentum transfer cross-sections for electron scattering from CO. ( ) Haddad and Milloy, ( ) Tanaka et al., ( ) Gibson et al., (- - -) Jain and Norcross, ( — — ) Jain et al., (− −) Morgan and Tennyson, (—) Morgan.

◦

•

Table 62 Integral rotationally inelastic cross-sections (10−16 cm2 ) for electron impact excitation of CO. The data are those of Randell et al. Energy (meV)

J = 0–1

J = 1–2

2 3 4 5 10 30 50 100

34.6 31.5 30.2 29.4 23.8 11.4 7.5 4.2

15.7 15.7 15.7 15.6 13.3 6.6 4.4 2.5

3.5.3. Rotational excitation Integral cross-sections for rotationally inelastic (J = 0–1 and J = 1–2) collisions of electrons with the ground state of CO have been published by Randell et al. [270]. The energy range of this work was E0 = 2–100 meV and the data are summarised in Table 62. 3.5.4. Rovibrational excitation 3.5.4.1. Di:erential cross-sections. Di@erential cross-sections for excitation of the Frst vibrational quantum (v = 0 → 1) have been reported by Land, as a result of a swarm experiment, by Chutjian and Tanaka [271] at energies from 3–100 eV and by Sohn et al. [262] at energies in the range 0.37–1:26 eV. More recently, Middleton et al. [193] and Gibson et al. have also

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Table 63 Di@erential cross-sections (10−18 cm2 sr −1 ) for vibrational excitation (0 –1) of CO. The data is that of Chutjian and Tanaka. The numbers in the Fnal row are the percentage uncertainties on the measurements Energy (eV) Angle

3

15 20 25 30 35 40 50 60 70 80 90 100 110 120 130

— 7.20 5.85 5.03 4.37 4.06 3.46 3.03 3.08 2.80 2.51 2.08 1.69 1.77 1.98 27%

5

9

4.26 3.70 2.68 1.72 1.27 1.01 0.654 0.540 0.573 0.530 0.467 0.414 0.389 0.438 0.564 17%

20

0.914 0.484 0.336 0.210 0.188 0.175 0.160 0.162 0.144 0.128 0.104 0.091 0.0868 0.0894 0.148 17%

1.44 1.16 0.92 0.804 0.778 0.810 0.785 0.793 0.738 0.678 0.651 0.604 0.673 0.726 0.778 20%

30 0.310 0.204 0.143 0.121 0.108 0.102 0.112 0.124 0.157 0.161 0.159 0.142 0.134 0.163 0.193 20%

50 0.134 0.068 0.0413 0.0374 0.0460 0.0492 0.0514 0.0492 0.0450 0.0424 0.0448 0.0489 0.0510 0.0546 0.0651 20%

75

100

0.133 0.084 0.0708 0.0895 0.107 0.0813 0.0534 0.0391 0.0333 0.0324 0.0281 0.0259 0.0246 0.0286 0.0326 20%

— 0.0954 0.0686 0.106 0.103 0.0946 0.0704 0.0433 0.0305 0.0245 0.0293 0.0381 0.0443 0.0542 0.0608 20%

Table 64 Di@erential cross-sections (10−18 cm2 sr −1 ) for vibrational excitation (0 –1) of CO. The data is that of Middleton et al. The numbers in parentheses are the percentage uncertainties on the measurements Energy (eV) Angle

20

10 20 30 40 50 60 70 80 90

4.52 2.69 2.37 2.10 1.64 1.45 1.17 0.88 0.82

(24) (22) (22) (22) (23) (22) (22) (23) (22)

30

40

50

1.45 (22) 0.162 (22) 0.334 (22) 0.272 (22) 0.193 (22) 0.210 (22) 0.204 (22) 0.198 (22) 0.165 (22)

1.48 (22) 0.49 (22) 0.20 (20) 0.13 (23) 0.095 (22) 0.086 (22) 0.062 (21) 0.063 (22) 0.065 (22)

0.532 0.268 0.106 0.089 0.074 0.066 0.064 0.061 0.054

(21) (22) (22) (21) (22) (21) (22) (21) (22)

reported DCS for this transition at energies between 1 and 50 eV. These data are given in Tables 63– 65. From a theoretical perspective there have been low-energy Born Dipole Approximation (BDA) calculations by Sohn et al., the R-matrix calculations of Morgan and Tennyson (1–3 eV) and the more recent R-matrix results of Morgan (1–10 eV). Without wishing to prejudge the validity of these theoretical approaches and also without underestimating the diJculty of such calculations for vibrational motion, it is quite clear that more theoretical attention needs to be paid to electron impact excitation of the vibrational quanta of CO, particularly at intermediate energies.

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Table 65 Di@erential cross-sections (10−18 cm2 sr −1 ) for vibrational excitation (0 –1) of CO. The data is that of Gibson et al. The numbers in parentheses are the percentage uncertainties on the measurements Energy (eV) Angle 10 15 17 20 25 30 35 40 45 50 55 60 70 80 90 100 110 120 130

1.0

8.4 (12) 5.7 (16) 5.0 (17)

1.25

1.50

1.91

2.45

3.0

18.8 (23) 15.9 (14) 14.0 (13)

42.9 (12)

73.2 (15)

49.7 (12)

20.6 (12)

39.5 36.4 33.5 30.0 27.0 24.5 22.4 19.0 16.3 12.4 10.0 9.2 9.7 11.8 15.8 20.4

69.1 64.8 61.0 56.1 50.7 45.4 41.1 35.9 33.6 27.8 24.7 23.3 24.9 28.3 34.7 43.9

45.4 43.9 39.6 36.1 32.5 29.7 26.8 24.2 22.5 19.1 18.0 16.6 17.4 18.0 21.0 25.3

19.0 17.8 15.8 14.8 13.2 11.3 10.5 10.1 9.2 7.6 6.9 6.7 7.1 6.8 6.7 8.8

10.4 (17)

3.3 (16)

9.1 (13)

2.8 (14)

6.6 (13)

2.0 (18) 1.4 (19) 0.99 (22) 0.94 (18) 0.83 (31) 0.87 (37) 1.1 (28) 1.2 (23)

5.1 3.5 2.6 2.3 2.2 2.8 3.7 4.7

(14) (14) (13) (14) (15) (15) (14) (14)

(12) (12) (12) (12) (12) (12) (12) (12) (12) (12) (12) (13) (12) (12) (13) (12)

(14) (14) (14) (14) (13) (14) (14) (16) (14) (14) (14) (13) (14) (13) (13) (13)

(12) (12) (12) (12) (12) (12) (12) (12) (12) (12) (12) (12) (12) (13) (12) (13)

(12) (13) (13) (14) (15) (13) (13) (15) (14) (12) (12) (12) (12) (14) (12) (12)

20

30

2.3 (14)

1.4 (16)

1.3 (13) 1.1 (16) 1.0 (17) 0.98 (20) 0.88 (16) 0.86 (15) 0.88 (14) 0.91 (17) 0.79 (14) 0.72 (15) 0.62 (18) 0.58 (13) 0.63 (17) 0.68 (16) 0.81 (16) 0.92 (13)

0.48 (16) 0.40 (36) 0.30 (29) 0.24 (18) 0.24 0.25 0.24 0.25 0.20 0.28 0.27 0.23

(24) (20) (15) (29) (27) (29) (14) (18)

In Figs. 58(a) – (d) we consider a selection of the available low-energy DCS data and theory for the 0 –1 excitation process. At 1 eV (Fig. 58(a)) the only experimental DCS available is that of Gibson et al. which we compare to the result of a BDA calculation and the R-matrix result of Morgan. It is apparent that whilst there are some overall similarities in the shapes of the data of Gibson et al. and the BDA result, the BDA theory underestimates the magnitude of the measured DCS over the entire angular range. The somewhat unusual behaviour of the ◦ DCS of Morgan, for angles 6 30 , is believed to be a convergence problem at that energy. At 1:25 eV (see Fig. 58(b)) we compare the data of Gibson et al. with the earlier 1:26 eV result of Sohn et al. The overall level of agreement in terms of both shape and magnitude of these DCS is quite good. On the other hand, the BDA is seen to be inadequate, severely underestimating the magnitude of the DCS and not reproducing the backward angle behaviour. Similar to the 1 eV case, the calculation of Morgan again su@ers from a problem of insuJcient partial waves in the R-matrix. In Fig. 58(c), at 1:91 eV, the data of Gibson et al. can now be compared to the relative data of Ehrhardt et al. [156] and Tronc et al. [272]. In both cases, the relative data has ◦ been normalised to that of Gibson et al. at a scattering angle of 90 . There is good agreement in the shape of the DCS between all three measurements. The experimental data is also compared to the R-matrix result of Morgan (see Fig. 58(c)) and outstanding agreement is found. At 3 eV (Fig. 58(d)), we again Fnd that there is good agreement in shape between the DCS of Gibson et al. and Ehrhardt et al. On the other hand there is a signiFcant di@erence between the absolute value of the DCS, at all common angles, between the data of Chutjian and Tanaka and Gibson

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Fig. 58. Di@erential cross-sections for rovibrational (0 –1) excitation of CO by electron impact at (a) 1 eV, (b) 1:25 eV, (c) 1:91 eV and (d) 3 eV. () Sohn et al., () Ehrhardt et al., ( ) Jung et al., (×) Tronc et al., ( ) Chutjian and Tanaka, ( ) Gibson et al., (—) Morgan, (- - -) Born Dipole.

◦

•

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Fig. 59. Di@erential cross-sections for rovibrational (0 –1) excitation of CO by electron impact at (a) 20 eV and (b) 30 eV. ( ) Gibson et al., ( ) Middleton et al., ( ) Chutjian and Tanaka, (×) Tronc et al.

•

◦

et al., the latter being as much as a factor of 3 larger at some angles. At this energy, there is a severe mismatch in the results of the two R-matrix calculations with the earlier one (not shown) favouring the most recent measurement of Gibson et al. and the latter, intrinsically more exact calculation, favouring the data of Chutjian and Tanaka. This mismatch between the data of Gibson et al. and the result of Morgan, particularly after the good agreement at 1:91 eV, is possibly indicates that the calculated width and=or the peak energy of the resonance are incorrect. In Figs. 59(a) and (b) we consider some of the available data in the energy range between 20 and 50 eV for 0 → 1 excitation process. This work includes the measurements of Gibson et al., Middleton et al., Chutjian and Tanaka and Tronc et al. (at 19:5 eV). We note that in this same energy region there is currently no theory against which we can compare the experiments. This is clearly a situation that we believe should be rectiFed. At 20 eV (Fig. 59(a)) the earlier measurement of Chutjian and Tanaka and the recent measurement of Gibson et al. are in good agreement over the common angular range of measurement. This agreement is somewhat fortuitous in that the elastic DCS measurements of these groups (see Section 3.6.2.1) did not agree very well at 20 eV and the ratios of their inelastic to elastic intensities were also di@erent. It is probably a cancellation of these e@ects which leads to the result seen in Fig. 59(a). Nonetheless, the data of Tronc et al., at 19:5 eV exhibits the

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same angular behaviour as that of Chutjian and Tanaka and Gibson et al. providing further support for these determinations of the DCS. The other available measurement for the 0 → 1 DCS at 20 eV is due to Middleton et al. [193]. At all angles this data is larger in magnitude than those just discussed and, given the good level of agreement between the elastic scattering data of Middleton et al. [263], Nickel et al. and Gibson et al., it is a rePection of their ratio for the intensity of inelastic to elastic scattering being larger than the other data. We do not believe this discrepancy is due, at this energy, to any untoward transmission e@ects with the post-collision analyser optics; rather, it is possibly due to a mismatch in the incident energies of the various groups. These energy “mismatches” are caused by contact potential e@ects in calibrating against the 2 S resonance in helium and then performing the experiment in CO. Such e@ects can lead to a 0:5 eV o@set in the “true” energy. We note that in their experiments Gibson et al. have CO and He running into the chamber simultaneously throughout, so that such an e@ect is minimised, while Middleton et al. did not. Of course this would all be of largely academic interest (as we shall see in Fig. 59(b) for 30 eV) unless considering an energy region where resonance enhancement of the cross-section is occurring. This is precisely what is happening in CO at around 20 eV where the well-known [273] CO W− resonance is found. Even though this shape resonance is quite broad, an energy mismatch of the order of 0:5 eV could quite easily account for some of observed discrepancy between Middleton et al. and Gibson et al. Supporting evidence for this hypothesis is found from the 30 eV DCS data (Fig. 59(b)), in a region away from the W− shape resonance, where the data of Gibson et al. and Middleton et al. is in good accord. Note that this level of agreement at 30 eV is found for both the elastic DCS, the intensity ratio of vibrational excitation to elastic scattering and thus the 0 –1 DCS. At this energy, similar to what we have seen previously at 3 eV (Fig. 58(d)), the DCS of Chutjian and Tanaka underestimates the magnitude of the DCS across the entire angular range, although the shape is in fair agreement with both Middleton et al. and Gibson et al. 3.5.4.2. Integral cross-sections. Experimental integral rovibrational cross-sections for the v = 0–1 excitation have been reported in the literature by Sohn et al., Land, Chutjian and Tanaka and Gibson et al. These data are presented in Table 66. Theoretical calculations are due to Morgan and Morgan and Tennyson, employing an R-matrix formalism, while at very low energies BDA level results are also available. All the experimental data and theoretical calculations are depicted in Fig. 60. From this Fgure, it is apparent that there is not a great deal of overlap between the various experimental determinations of the ICS and that the theory results are restricted to the low-energy region. Where overlap does exist between data sets, the level of agreement between them is only marginal. For example at energies ¡ 0:5 eV the swarm derived ICS of Land and the beam values of Sohn et al. are in quite poor agreement, the BDA calculation favouring the swarm result. On the other hand, at energies between 1 and 1:5 eV, the data of Sohn et al., Land and Gibson et al. appear to be in fair accord. In general, the ICS data of Chutjian and Tanaka are smaller in magnitude than that found by Gibson et al., the exception to this being at 20 eV may be fortuitous. However, as we discussed earlier in the DCS case, this agreement at 20 eV is somewhat fortuitous. The R-matrix calculations of Morgan and Tennyson, who employed a SCF level wavefunction, and Morgan, who utilised a CI wavefunction, are in fair accord with the ICS data of Gibson et al., albeit over a rather limited energy range. Indeed

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Table 66 Integral cross-sections (10−16 cm2 ) for vibrational excitation (0 –1) of CO Energy (eV)

Sohn et al.

Land

Chutjian and Tanaka

Gibson et al.

0.37 0.45 0.64 0.85 1 1.25 1.26 1.5 1.91 2.45 3 5 9 20 30 50 75 100

0.274 0.276 0.189 0.215 — — 0.576 — — — — — — — — — — —

0.149 0.157 0.156 0.224 — — 1.689 — — — — — — — — — — —

— — — — — — — — — — 0.365 0.104 0.024 0.097 0.021 0.008 0.005 0.007

— — — — 0.240 0.654 — 2.361 4.874 3.131 1.200 — — 1.107 0.035 — — —

•

Fig. 60. Integral cross-section for rovibrational (0 –1) excitation in CO by electron impact. () Sohn et al., ( ) Gibson et al., (×) Land, ( ) Chutjian and Tanaka, (—) Morgan, (− −) Morgan and Tennyson, (- - -) Born Dipole Approximation.

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367

we highlight that the shape and magnitude of the ICS calculation, in the region of the 2 V resonance, of Morgan and Tennyson is in quite good agreement with that found by Gibson et al., although the width of the resonance is somewhat broader than the experimental observation. In conclusion, the overall picture for the excitation of the 0 –1 vibrational transition is at best “patchy” with further experimental and theoretical studies being required before a clearer picture of this process becomes apparent. 3.5.5. Electronic excitation 3.5.5.1. Di:erential cross-sections. Previous experimental studies into electronic excitation di@erential cross-sections have been limited and, until quite recently, intrinsic diJculties in performing the experiments and data analysis had restricted their measurement to the very early work of Rice [274] and Trajmar et al. [126]. In their study, Trajmar et al. did not report di@erential cross-sections for all the possible electronic states of CO with energy loss 6 12 eV, but limited their work to the a3 V; a3 W+ ; A1 V; b3 W+ ; B1 W+ ; C1 W+ and E1 V electronic states. Furthermore, this study was conducted only at the single electron energy of 20 eV ◦ and over the angular range 10 –90 . Subsequent to this work, Zetner and Trajmar [128] reported a preliminary series of low-energy measurements (12:5 and 15 eV), in this case for ◦ angles in the region 5–135 , and for the a3 V, a3 W+ , d 3 U and A3 V electronic states. This data has been recently updated by Zetner et al. [275]. The most comprehensive experimental study to date for electron impact excitation of the electronic states of CO is due to Middleton et al. [122]. In this work, DCS for the a3 V, a3 W+ , (d 3 Ui + e3 W− + I1 W− + D1 U), A3 V; b3 W+ , B1 W+ , j3 W+ , (C1 W+ + c3 V) and E1 V electronic states were reported for incident electron energies of 20, 30, 40 and 50 eV. These DCS were obtained for the scattered electron ◦ angle range 10–90 by analysing electron-energy-loss spectra, at a number of Fxed scattering angles within that range, using a deconvolution technique as described in Section 2.7 of this review. We also note the higher energy study of Kanik et al. [276], who measured DCS for the B1 W+ , C1 W+ and c3 V states at 100 eV impact energy for scattering angles in the ◦ range 5–120 . The experimental study by Zobel et al. [277] dealt with near threshold excitation function measurements for the a3 V electronic state of CO. This work was conducted for impact energies ◦ in the range ∼6–10:5 eV, scattering angles between 30 and 140 and for the v = 0; 1; 2; 3 and 4 vibrational sublevels of the a3 V state. Absolute data were reported and can be extracted from the relevant graphs in their paper. This preliminary study by Zobel et al. has been extended [278,279] to include the electron impact excitation of other electronic states, i.e. the a3 W; A1 V, b3 W, B1 W, C1 W and E1 V as well as the a3 V state. This work was carried out at incident energies ◦ ◦ in the range 6:5–15:2 and for scattering angles between 20 and 140 . LeClair and Trajmar [118] applied the new ToF technique developed by LeClair et al. [117], which was discussed in Section 2.5, to the measurement of absolute excitation cross-sections ◦ for the a3 V and A3 V state of CO at a scattering angle of 90 and for incident electron energies between 6 and 15 eV. This experiment involves the measurement of ratios of elastic to inelastic scattering and assumes a constant transmission function for the ToF analyser at the elastic and inelastic energies. The inelastic scattering intensities were placed on an absolute scale by using the absolute elastic scattering cross-section of Gibson et al. [264]. Also, Zetner et al. [275] have

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used the measurements of LeClair et al. to normalise their measurements of the a3 V, a3 W+ , d 3 U and A3 V electronic states at energies of 10, 12.5 and 15 eV. Integral cross-sections for electron impact excitation of the v = 0 level of the b3 W+ electronic (Rydberg) state were measured by Zubek et al. [280]. The energy range of this study was from threshold to 17 eV. Their procedure involved measuring the optical excitation function for the b3 W+ –a3 V(0; 1) transition ( = 297:5 nm), and then normalising this to the emission cross-section of the C3 Vu –B3 Vg (0; 0) band of N2 [38]. An error of 20% on the b3 W+ integral cross-section is quoted. Mason and Newell [281] measured ICS for the metastable I1 W state and, in a similar experiment, Furlong and Newell [282] reported ICS for the a3 V state. Both Kanik ◦ ◦ et al. and Zetner et al. extrapolated their DCS to 0 and 180 to derive ICS. Similarly, Zobel 3 et al. [277] also reported ICS for the a V state, as determined from the DCS measurements. The above experimental values of DCS and ICS published since the Trajmar review for excitation of the electronic states of CO can be found in Tables 67–79. From a theoretical perspective, the complicated nature of the electron–target interaction potential (static, non-local exchange, polarisation and correlation) has prevented rapid progress. Lee and McKoy [283] employed a distorted-wave model to calculate DCS for the A1 V; a3 V, a3 W+ , D1 U and d 3 Ui states by electrons with energies in the range 20–50 eV. Weatherford and Huo [284] have applied a Schwinger multichannel (SMC) formulation to calculate DCS for excitation of the b3 W+ state in the 10 –20 eV energy range, while Sun et al. [285] have employed their Schwinger multichannel variational method to report DCS for the a3 V, A1 V; a3 W+ , e3 W− , d 3 Ui , I1 W− and D1 U states at speciFc energies in the range 6:5–30 eV. We also note the R-matrix calculation of Morgan and Tennyson [266] for energies in the range 6–18 eV, who in principle reported DCS results for the a3 V, a3 W+ , d 3 Ui , A1 V, e3 W− ; I1 W− and D1 U electronic states, but in practice actually only plotted a single DCS for the a3 W+ state at 10:4 eV. An extensive series of distorted-wave calculations (both ICS and DCS values are tabulated) for electron impact excitation of the a3 V, b3 W+ , A1 V and B1 W+ electronic states was published by Lee et al. [286]. The energy range investigated was 20–100 eV. Good qualitative agreement with previous experimental data was claimed although, as we shall see, the magnitude of the theory generally overestimated that of the experiment. Interestingly, they also found that the description of the target wavefunction had a strong inPuence on the calculated DCS and ICS. a3 > state (excitation energy = 6:010 eV ). Di@erential cross-sections for the electron impact excitation of the a3 V state of CO are given in Table 67. The data of Zetner and Trajmar and Middleton et al. are given in Table 67(a), that of Zobel et al. in Table 67(b) and that of LeClair and Trajmar in Table 67(c). In Fig. 61 the DCS data of Zobel et al., LeClair and Trajmar and Zetner et al. are compared, at energies of (a) 8 eV and (b) 15 eV, with the results of the SMC calculation of Sun et al. It is clear from Fig. 61(a) that the agreement in the shape of the DCS between theory and experiment is quite good at 8 eV, and there is only a relatively small di@erence in terms of the absolute values. There is also good agreement between the two experimental cross-sections. At 15 eV, the situation is not so good with a di@erence of a factor of two in magnitude between experiment and theory. In Fig. 62(a) the DCS of Middleton et al. at 20 eV is compared with the earlier experimental data of Trajmar et al., the distorted wave (DW) theory of Lee and McKoy, the SMC calculation of Sun et al. and the more recent DW calculation of Lee et al. Whilst the two sets of experimental data are, in general, in accord as to the order of magnitude of the DCS, the measurements of Trajmar et al. were larger in

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369

Table 67 Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the a3 V state of CO Energy (eV) (a)a Angle

12:5 Zetner et al.

9 10 14 20 24 30 34 39 40 44 50 54 60 64 69 70 74 80 84 90 94 99 104 114 124 129 134

1.17 (16)

Qi

28.45 (17)

15 Zetner et al.

1.30

0.89 (20)

1.45

1.10

1.80

1.17

1.82

1.27

2.17

1.17

2.51

1.43 1.95

3.14

2.22

3.55 3.78

2.38

4.78 5.29 5.23

20 Middleton et al.

30 Middleton et al.

40 Middleton et al.

50 Middleton et al.

0.58 (26)

0.58 (26)

0.33 (26)

0.30 (26)

1.36

0.69

0.53

0.43

1.49

0.69

0.48

0.40

1.61

0.87

0.44

0.32

1.47

0.84

0.36

0.29

1.44

0.68

0.36

0.26

1.47

0.65

0.37

0.24

1.49

0.75

0.33

0.24

1.51

0.83

0.36

0.23

2.26 2.42 3.07 3.54 13.06 (17) Energy (eV)

(b)b Angle

6.5

7.0

8.0

9.0

9.5

20 30 40 50 60 70 80 90 100 110 120 130 140

3.4 2.8 2.5 2.6 2.9 2.4 2.6 2.5 2.5 2.5 — — 3.2

5.5 4.6 4.3 4.2 4.2 4.2 4.3 3.9 4.3 4.5 — — 4.7

4.4 4.1 3.8 3.6 5.2 5.7 6.2 6.0 7.4 8.3 — — 10.0

— — — — 5.3 6.4 7.4 8.1 9.9 11.6 — — 14.3

— — — — 4.9 6.1 7.1 7.9 9.6 11.0 — — 13.2

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Table 67 (Continued) (c)c E (eV)

DCS

6 0 6.5 2.66 (10) 7 4.55 (7) 7.5 6.24 (6.7) 8 7.77 (7.1) 8.5 8.66 (7.5) 9 8.97 (7.6) 9.5 8.24 (8) 10 7.12 (8) 10.5 6.17 (8.3) 11 5.23 (7.8) 11.5 4.26 (8.2) 12 3.83 (8.1) 12.5 3.34 (8.7) 13 3.00 (8.3) 13.5 2.79 (9) 14 2.59 (9.7) 14.5 2.47 (11) 15 2.35 (12) a Numbers in parentheses at the top of each column indicate the percentage uncertainty for that column. Where determined, integral cross-sections (10−18 cm2 ) are given at the foot of each column. b The data are those of Zobel et al. ◦ c Electron impact excitation of the a3 V state of CO at 90 . The data are those of LeClair et al. The numbers in parentheses are the estimated percentage uncertainties on the absolute cross-sections.

magnitude at forward angles and smaller at backward angles. The earlier DW theory clearly overestimates the magnitude of the DCS across the entire angular range but it shows some similarities in shape with the data of Middleton et al. The SMC calculation and the later DW result both show less structure in the DCS compared with the DW result but in terms of the magnitude of the DCS they are both in reasonable agreement with the result of Middleton et al. Lee et al. noted that the improvement in agreement with experiment in their calculation over the earlier DW model is that the more recent calculation has a more accurate description of the target wavefunction. The 30 eV DCS is shown in Fig. 62(b). In this case a comparison is only possible between the data of Middleton et al., the SMC calculation and the DW theory of Lee et al. It is seen that there is only reasonable qualitative agreement between theory and experiment in the shape of the DCS. a3 /+ state (excitation energy = 6:863 eV ). Di@erential cross-sections for the electron impact excitation of the a3 W+ state of CO are given in Table 68. Table 68(a) contains the data of Zetner et al. and Middleton et al. whilst Table 68(b) contains the data of Zobel et al. In Fig. 63, the 12:5 eV DCS is illustrated and the data of Zetner et al. are compared to the SMC calculation of Sun et al. Whilst the theory is larger in magnitude, there is reasonable agreement as to the shape of the DCS. At 20 eV (Fig. 64), the data of Middleton et al. exhibits a distinctive dip ◦ in the angular region 30–40 which is not observed in the earlier experiment of Trajmar et al. On the other hand, the calculation of Sun et al. does indicate a dip in the DCS, albeit at the

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371

Fig. 61. Di@erential cross-sections for electron impact excitation of the a3 V electronic state of CO at (a) 8 eV and (b) 15 eV. ( ) Zobel et al., ( ) LeClair and Trajmar ( ) Zetner et al., (− − −) Sun et al.

•

◦

◦

slightly larger angle of 50 . The calculation of Sun et al. at this energy, clearly overestimates the magnitude of the DCS although we note that it is in better agreement with the result of Middleton et al. than that of Trajmar et al. (d3 8i + e3 /− + I1 /− + D1 8) states (excitation energies = 7:516, 7.820, 8.003 and 8:108 eV ). During the initial stages of their analysis of the energy loss spectra for CO, Middleton et al. carried out their Ftting procedure (see Section 2.7 of this article) without including these non-Franck–Condon region electronic states in the analysis, and then once more with them included. From the value of 2 obtained in each case, it was clear that a better Ft to the data was obtained with these states included in the Ftting programme. The combined DCS of Middleton et al. for these electronic states of CO in the non-Franck–Condon region (d 3 Ui + e3 W− + I1 W− + D1 U) are given in Table 69. This work of Middleton et al. represented the Frst time that any absolute experimental data for these states were published in the literature. Subsequent to this study, Zetner et al. [275] reported DCS at 10, 12:5 and 15 eV for the d 3 Ui electronic state. These data are given separately in Table 70. Sun et al. have published individual DCS for each of these states using their SMC method and these individual DCS were summed by Middleton et al., so that a comparison between their data and the summed calculation could be made. Fig. 65(a) shows the DCS measured by Middleton et al., along with the SMC result, at an incident energy of 20 eV. It can be seen that the DCS for the non-Franck–Condon region

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Fig. 62. Di@erential cross-sections for electron impact excitation of the a3 V electronic state of CO at (a) 20 eV and (b) 30 eV. () Trajmar et al., () Middleton et al., (− − −) Sun et al., (- - -) Lee and McKoy, (—) Lee et al.

states is relatively Pat but the theory is an order of magnitude larger than the experimental result. A similar trend can also be seen for 30 eV in Fig. 65(b), except that the experimental data is slightly more forward peaked. For the incident energies 40 and 50 eV (not shown) the DCS becomes even more forward peaked which demonstrates the emerging dominance of the singlet states over the triplet states in the scattering process. A1 > state (excitation energy = 8:028 eV ). Tabulated DCS data for the excitation of the A1 V state of CO are given in Table 71. Table 71(a) contains the data of Zetner et al. and Middleton et al. and 71(b) the data of Zobel et al. It can be seen from these tables that the cross-sections are forward peaked at the energies shown, as is expected for a dipole-allowed, singlet–singlet transition. The strong forward scattering by the A1 V state was also observed to increase with increasing incident electron energy. The 10 eV data of Zetner et al. [275] and Zobel et al. are shown in Fig. 66. Also shown in these Fgures are the SMC results of Sun et al. The experimental data is signiFcantly more forward peaked than that predicted by ◦ the calculation. However, for angles larger than 60 , the overall level of agreement between them is fair. The two experimental results are in good agreement at all angles. The data of Middleton et al. for incident energies of 20 and 30 eV are plotted in Figs. 67(a) and (b), respectively. Comparison is also made in these Fgures with the earlier experimental results of Trajmar et al. at 20 eV, the DW theories of Lee and co-workers and the SMC calculation of Sun et al. In

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373

Table 68 Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the (a) a 3 W+ state of CO and 3 (b) a W+ # (; = 6; 7; 9) electronic state of CO a

(a) Angle 9 10 14 20 24 30 34 39 40 44 50 54 60 64 69 70 74 80 84 90 94 99 104 114 124 129 134 Qi

12:5 Zetner et al. 2.51 (20) 2.23 — 2.18 — 1.12 1.09 — 0.82 — 0.69 — — 0.89 — 0.83 0.91 — 0.84 0.77 0.69 12.47 (21)

15:0 Zetner et al. 1.35 (20) 1.39 0.85 0.74 0.56 0.80 1.13 1.22 1.06

Energy (eV) 20 30 Middleton et al. Middleton et al.

40 Middleton et al.

50 Middleton et al.

1.61 (28)

—

—

0.45 (28)

0.62

—

—

0.34

0.19

0.17 (28)

0.06 (28)

0.15

0.35

0.19

0.11

0.13

0.38

0.26

0.12

0.13

0.49

0.20

0.15

0.18

0.58

0.22

0.19

0.21

0.59

0.36

0.20

0.25

0.46

0.49

0.25

0.21

0.72 0.61 0.83 1.17 13.06 (21)

Energy (eV) (b)b Angle 9.0 10.0 11.0 20 0.594 — — 30 0.459 — — 40 0.278 — — 50 0.261 — — 60 0.178 0.272 0.211 70 0.120 0.223 0.181 80 0.106 0.172 0.148 90 0.092 0.168 0.156 100 0.104 0.156 0.180 110 0.113 0.149 0.176 120 0.148 0.174 0.188 130 0.166 0.184 0.177 140 0.160 0.189 0.167 a The numbers in parentheses at the top of each column indicate the percentage uncertainty for that column. Where determined, integral cross-sections (10−18 cm2 ) are given at the foot of each column. b The data are those of Zobel et al.

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Fig. 63. Di@erential cross-sections for electron impact excitation of the a3 W+ electronic state of CO at 12:5 eV. ( ) Zetner et al., (– – –) Sun et al.

◦

Fig. 64. Di@erential cross-sections for electron impact excitation of the a3 W+ electronic state of CO at 20 eV. () Trajmar et al., () Middleton et al., (− − −) Sun et al.

Fig. 67(a), the level of agreement between the experimental measurements is quite good, particularly at backward scattering angles. At forward angles the data of Trajmar et al. is substantially more forward peaked, overestimating the size of the DCS by a factor of about 5. The distorted wave theory of Lee and McKoy [283] overestimates the magnitude of the DCS across the entire angular range whilst the SMC theory shows good agreement with the data of Middleton et al. At 30 eV (Fig. 67(b)) the agreement between experiment and theory is even better. b3 /+ state (excitation energy = 10:394 eV ). Tabulated DCS and ICS data for the excitation of the b3 W state of CO are given in Table 72. Table 72(a) contains the DCS data of Middleton et al., Table 72(b) the DCS data of Zobel et al. and Table 72(c) the ICS data of Zubek et al. [280]. At an incident energy of 20 eV, a Schwinger two-channel calculation by Weatherford and Huo [284] using two di@erent basis sets (which we denote by A and B—see [284] for full details) has been carried out. We note that whilst their investigation was aimed primarily at investigating resonance phenomena in the b3 W+ channel from the threshold to 20 eV, DCS data at 20 eV were reported. In addition there are also results from the DW calculation of Lee et al. [286]. This theoretical data is compared to the recent measurements of Middleton et al. and the

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375

Table 69 Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the (d 3 Ui + e3 W− + I1 W− + D1 U) states of CO. The data are those of Middleton et al. The uncertainties on the data are estimated to be about 30% Energy (eV) Angle

20

30

40

50

10 20 30 40 50 60 70 80 90

0.07 0.08 0.11 0.17 0.14 0.17 0.15 0.17 0.14

0.17 0.16 0.12 0.15 0.10 0.07 0.09 0.10 0.12

1.21 0.38 0.14 0.11 0.10 0.07 0.06 0.06 0.07

0.90 0.22 0.16 0.11 0.07 0.05 0.06 0.06 0.06

Table 70 Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the d 3 Ui state of COa Energy (eV) Angle 9 14 24 34 39 44 54 64 69 74 84 94 99 104 114 124 129 134

12:5 2.68 1.58 2.11 1.81 1.78 2.52 1.96 1.58 1.65 1.94 1.83 1.66 1.48

15:0 2.95 2.32 2.01 2.00 1.93 1.99 2.66 2.60 2.72 2.80 2.85 2.75 2.80

21.90 32.02 The data are from Zetner et al. and the estimated error in the DCS values are ±28%. Integral cross-sections (10−18 cm2 ) are given at the foot of each column and the estimated uncertainties are ±29%. Qi

a

earlier data of Trajmar et al. in Fig. 68. Clearly, the calculations overestimate the magnitude of the DCS, when compared to our preferred data set of Middleton et al. Not surprisingly, the more complete basis, set A, of Weatherford and Huo gives better agreement with the experiment then did the more limited one. The calculation of Lee et al. seems to reproduce the

376

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Fig. 65. Di@erential cross-sections for electron impact excitation of the (d 3 Ui + e3 W− + I1 W− + D1 U) electronic states of CO at (a) 20 eV and (b) 30 eV. ( ) Middleton et al., (− − −) Sun et al.

•

Fig. 66. Di@erential cross-sections for electron impact excitation of the A1 V electronic state of CO at 10 eV. ( ) Zobel et al., ( ) Zetner et al., (- - -) Sun et al.

•

◦

shape of the experimental DCS very well, but it is substantially larger in magnitude. Trajmar et al. again found the DCS to be more forward peaked than did Middleton et al., a trend which suggests the possibility of an e@ective path length error in the data of Trajmar et al. B1 /+ state (excitation energy = 10:776 eV ). Tabulated experimental data for the B1 W+ state of CO are given in Table 73. The data of Middleton et al. and Kanik et al. are given in Table 73(a) and those of Zobel et al. in Table 73(b). Fig. 69 shows a comparison of the data of Middleton et al. with the earlier data of Trajmar et al. at 20 eV incident electron energy. At backward scattering angles the agreement between the two sets of data is very good but at forward angles the DCS of Trajmar et al. is substantially more forward peaked (about a factor of 10) than that of Middleton et al. This is somewhat surprising given that the result of Trajmar et al. at 20 eV is even more forward peaked than the 100 eV result of Kanik et al. The usual trend for a 1 W+ →1W+ excitation process is that the degree of forward peaking increases with increasing energy. The 100 eV data of Kanik et al. were measured using techniques consistent with the procedures outlined in Section 2.7 of this review and so we are conFdent that this later data does not su@er from any angle-dependent e@ects. On the other hand, and as mentioned

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377

Table 71 Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the A1 V state of CO (a)a Angle 9 10 14 20 24 30 34 39 40 44 50 54 60 64 69 70 74 80 84 90 94 99 104 114 124 129 134

12.5 Zetner et al. 26.52 (16)

15 Zetner et al.

20.68

39.80 (16)

14.25

23.56

Qi

45.30 (17)

6.18

9.55

4.98

5.65

3.63

3.26

2.50

2.96 3.21

2.29

3.56

2.34 2.76

3.48

2.44 2.15 1.91

Energy (eV) 20 30 Middleton et al. Middleton et al.

40 Middleton et al.

50 Middleton et al.

36.13 (25)

80.61 (25)

161.39 (25)

94.14 (25)

21.33

31.17

38.99

30.46

10.92

11.87

12.78

9.36

5.99

6.54

5.11

4.02

3.37

3.98

2.60

2.14

2.72

2.31

1.67

1.32

2.23

1.80

1.27

0.92

1.77

1.71

0.97

0.68

1.66

1.73

0.80

0.58

2.73 2.19 2.41 2.26 64.95 (17)

Energy (eV) (b)b Angle 8.5 9.0 10.0 11.0 11.5 20 0.30 1.72 3.43 — — 30 0.32 1.34 2.77 — — 40 0.24 0.86 2.18 — — 50 0.23 0.81 1.97 — — 60 0.16 0.57 1.86 2.35 2.89 70 0.16 0.51 1.56 1.89 2.32 80 0.18 0.55 1.52 1.76 2.26 90 0.16 0.50 1.39 1.76 2.23 100 0.15 0.49 1.44 1.72 2.38 110 0.13 0.44 1.49 1.71 2.19 120 0.13 0.59 1.67 1.75 2.12 130 0.17 0.67 1.77 1.80 2.14 140 0.20 0.73 1.92 1.78 2.06 a The Fgures in parentheses indicate the estimated percentage uncertainty for the data in that column. Where determined, integral cross-sections (10−18 cm2 ) are given at the foot of each column. b The data are those of Zobel et al.

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Fig. 67. Di@erential cross-sections for electron impact excitation of the A1 V electronic state of CO at (a) 20 eV, and (b) 30 eV. () Trajmar et al., () Middleton et al., (− − −) Sun et al., (- - -) Lee and McKoy, (—) Lee et al.

previously, the earlier 20 eV data of Trajmar et al. must be considered with some caution. Also shown in Fig. 69 is the result of the calculation of Lee et al. It is in good agreement with the ◦ data of Middleton et al. at small angles (¡ 40 ). j 3 /+ state (excitation energy = 11:265 eV ). The actual threshold energy for the j 3 W+ ; ; = 0 state of CO has been the cause of much controversy in the literature. Tilford and Simmons [287] reported the energy of the j 3 W+ ; ; = 0 state to be at 11:281 eV. This is consistent with the results of several trapped electron experiments [288–290] which measure the threshold excitation probability and show a peak in the 11:25–11:30 eV region. Alternatively there are the near-threshold, energy-loss experiments of Swanson et al. [291], who discovered a prominent peak, some 16 meV lower than Tilford and Simmons, at 11:265 eV. Further, in a separate series of excitation function measurements at ∼11:26 eV they found that this state was strongly excited near the threshold and gradually decreased in intensity thereafter. This behaviour indicated that the state at 11:265 eV was triplet in character. Swanson et al. speculated that this state was a previously unobserved triplet (or otherwise forbidden) state close to the j 3 W+ state, on the basis that neither Skerbele and Lassetre [292] nor Trajmar et al. saw a peak at 11:265 eV in their energy loss spectra. In all the energy loss measurements of Middleton et al. the best Ft to the data was obtained by assigning the relevant peak energy to be E = 11:265 ± 0:01 eV,

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379

Table 72 Di@erential (a, b) and integral (c) cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the b3 W state of CO Energy (eV)

(a)a Angle

20

30

40

50

10 20 30 40 50 60 70 80 90

0.27 0.29 0.23 0.15 0.07 0.06 0.06 0.07 0.08

0.34 0.19 0.10 0.08 0.07 0.07 0.05 0.04 0.04

0.66 0.20 0.07 0.04 0.05 0.04 0.03 0.01 0.01

0.33 0.09 0.05 0.04 0.03 0.03 0.02 0.02 0.02

Energy (eV)

(b)b Angle

10.74

11.2

12.2

13.14

14.1

20 30 40 50 60 70 80 90 100 110 120 130 140

0.95 0.97 0.66 0.47 0.35 0.37 0.21 0.18 0.14 0.11 0.20 0.40 0.47

0.30 0.31 0.28 0.29 0.31 0.34 0.36 0.39 0.34 0.28 0.27 0.23 0.19

0.28 0.28 0.27 0.26 0.24 0.21 0.20 0.17 0.17 0.16 0.16 0.16 0.15

— — — — 0.32 0.27 0.21 0.19 0.14 0.12 0.13 0.12 0.13

0.39 0.39 0.34 0.32 0.29 0.25 0.19 0.16 0.13 0.12 0.13 0.15 0.19

(c)c Energy (eV)

ICS

9.97 10.17 10.26 10.36 10.45 10.56 10.65 10.75 10.85 10.88 10.95 11.04 11.14 11.24 11.43 11.63 11.83 12.05

0.00 0.16 0.59 2.40 5.64 7.66 8.19 6.75 4.96 4.50 4.64 4.66 4.53 4.16 3.55 3.22 3.04 2.97

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Table 72 (Continued) (c)c Energy (eV)

ICS

12.24 12.34 12.53 12.73 12.83 13.03 13.23 13.43 13.63 13.82 14.02 14.21 14.4 14.6 14.8 15.0 15.2 15.41 15.61 15.8 16.0 16.19 16.39 16.58 16.78 17.0 17.2

2.98 2.86 2.70 2.50 2.46 2.47 2.46 2.45 2.44 2.43 2.42 2.40 2.38 2.37 2.36 2.35 2.35 2.36 2.36 2.37 2.38 2.40 2.42 2.44 2.45 2.46 2.46

The data are those of Middleton et al. (10−18 cm2 sr −1 ). The estimated uncertainty in the DCS is ±27%. The data are those of Zobel et al. for ; = 0 (10−18 cm2 sr −1 ). c The integral cross-section (10−18 cm2 ) data are those of Zubek et al. and the uncertainty is ±20%. a

b

which is entirely consistent with that found by Swanson et al. It is believed that this is the true threshold energy for the j 3 W+ , ; = 0 state because it is much more likely for Tilford and Simmons to slightly err in the j 3 W+ threshold energy than for them to completely miss the existence of another electronic state. The DCS data of Middleton et al. is given in Table 74. This data represents the Frst absolute DCS measurements to have been reported in the literature for the j 3 W+ state. (C 1 /+ + c3 >) states (excitation energies = 11:396 eV; 11:416 eV ). The energy resolution in the early measurements of Trajmar et al. was clearly inadequate to resolve these states while the energy resolution of Middleton et al. was only just suJcient to partially resolve them in their CO energy loss spectra. Consequently, Middleton et al. chose to combine their DCS data for the C1 W+ and c3 V electronic states. The most recent study by Kanik et al. did not state their energy resolution but did provide an energy loss spectrum from which we can glean that it is likely that the cross-section they quote for the C1 W+ state is actually a sum of the C1 W+ and c3 V states. However as this latter study was conducted at an energy of 100 eV, the C1 W+

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381

Fig. 68. Di@erential cross-section for electron impact excitation of the b3 W+ electronic state of CO at 20 eV. () Trajmar et al., () Middleton et al., (− − −) Weatherford and Huo (A), (- - -) Weatherford and Huo (B), (—) Lee et al. Fig. 69. Di@erential cross-sections for electron impact excitation of the B1 W+ electronic state of CO at 20 eV. () Trajmar et al., () Middleton et al., (—) Lee et al.

state would be expected to dominate in the excitation process. This is also the likely scenario for the more recent measurements of Zobel et al., who quote a cross-section only for the C1 W+ state at near-threshold energies. The cross-sections for these states are given in Tables 75(a) and 75(b) and in Fig. 70 we compare the 20 eV DCS of Middleton et al. and Trajmar et al. Also shown in this Fgure is the cross-section of Zobel et al. at an energy of 15:1 eV. Note the absence of any theoretical calculations for these states at this time. It can be seen that the earlier data of Trajmar et al. overestimates the magnitude of the DCS across the whole angular range of measurement and is again substantially more forward peaked than that of Middleton et al. The general trend in the data of Middleton et al. indicates that the DCS become more forward peaked as the energy increased, consistent, for example, with our previous observation for the (d 3 Ui + e3 W− + I1 W− + D1 U) states. This perhaps reveals the increasing dominance of the excitation of the C1 W+ state over the c3 V state in the scattering process as the energy is increased. The result of Kanik et al., at 100 eV, Fts neatly into this trend. E1 > state (excitation energy = 11:522 eV ). The DCS data for the electron impact excitation of the E1 V electronic state are summarised in Tables 76(a) and (b). In Fig. 71 we compare

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Table 73 Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the (a) B1 W+ and (b) B1 W+ (; = 0) state of CO Energy (eV) a

(a) Angle 5 10 15 20 30 40 50 60 70 75 80 90 100 110 120 (b)b Angle

20 Middleton et al.

30 Middleton et al.

40 Middleton et al.

50 Middleton et al.

0.95

3.47

8.32

4.98

0.63 0.44 0.38 0.28 0.23 0.21

0.87 0.50 0.51 0.40 0.24 0.20

1.24 0.70 0.50 0.32 0.24 0.20

1.03 0.73 0.52 0.34 0.24 0.18

0.16 0.17

0.19 0.18

0.13 0.12

0.16 0.14

100 Kanik et al. 19.34 3.41 1.04 0.34 0.49 0.28 0.16 0.12 0.084 0.073 0.064 0.065 0.065 0.072 0.093

Energy (eV) 10.88

11.08

11.58

20 0.059 0.28 1.21 30 0.064 0.26 0.94 40 0.066 0.22 0.62 50 0.064 0.18 0.40 60 0.068 0.14 0.24 70 0.064 0.10 0.15 80 0.061 0.07 0.11 90 0.053 0.05 0.12 100 0.042 0.04 0.12 110 0.035 0.05 0.13 120 0.033 0.06 0.14 130 0.047 0.09 0.17 140 0.037 0.11 0.15 a The estimated uncertainty in the data of Middleton et al. b The data are those of Zobel et al.

12.08 1.36 1.03 0.66 0.42 0.28 0.22 0.20 0.21 0.26 0.24 0.24 0.24 0.21 is ±27%.

12.78

13.78

14.48

1.46 0.98 0.58 0.39 0.30 0.28 0.26 0.28 0.29 0.21 0.31 0.20 0.27

1.88 1.04 0.59 0.44 0.40 0.37 0.25 0.26 0.39 0.42 0.43 0.41 0.37

1.92 0.98 0.57 0.49 0.44 0.41 0.29 0.38 0.43 0.46 0.47 0.44 0.42

the DCS of Trajmar et al. with that of Middleton et al. and the level of agreement is generally fairly poor. Once again the data of Trajmar et al. was also seen to be signiFcantly more forward peaked than that found by Middleton et al. which, we reiterate, is probably due to an angle-dependent problem with the earlier data. 3.5.5.2. Integral cross-sections. Measurements of ICS for electronic-state excitation of CO have been limited and mainly restricted to those states that can be measured by the detection of

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383

Table 74 Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the j3 W+ state of CO. The data are those of Middleton et al. and the uncertainty is ±29% Energy (eV) Angle

20

30

40

50

10 20 30 40 50 60 70 80 90

0.05 0.07 0.06 0.04 0.02 0.03 0.02 0.02 0.01

0.13 0.09 0.07 0.04 0.02 0.02 0.02 0.02 0.01

0.35 0.13 0.07 0.04 0.02 0.01 0.01 0.01 0.01

0.20 0.08 0.03 0.02 0.01 0.01 0.01 0.01 0.01

emitted photons, or those that are metastable states. Emission measurements have been carried out by Ajello [293] for the fourth positive (A1 V → X1 W+ ) and the Cameron (a3 V → X1 W+ ) system, by Mumma et al. [294] for the A1 V → X1 W+ transition, by Aarts and de Heer [295] for the A1 V → X1 W+ , B1 W+ → X1 W+ and C1 W+ → X1 W+ transitions, by Skubenich [296] for the b3 W+ and B1 W+ states and most recently by James et al. [297] for extreme ultraviolet emission cross-sections in CO produced by electron impact at 20 and 200 eV. Cross-sections for the excitation of the a3 W+ state were also obtained by Land [269] from an analysis of swarm experiments. ICS for excitation of the metastable I1 W− state have been measured by Borst and Zipf [298], Wells et al. [299] and by Mason and Newell [281], with this latter data being given in Table 77. These relative measurements of Mason and Newell were placed on an absolute scale by normalisation to the result of Wells et al. at 16 eV impact energy. More recently, Furlong and Newell [282,300,301] reported a total metastable cross-section for the a3 V electronic state of CO from threshold to above 70 eV electron impact energy. This data is summarised in Table 78. Kanik et al. [276] extrapolated their measured 100 eV DCS for the ◦ B1 W+ , C1 W+ and E1 V states to 0 at 180 and integrated them to generate the relevant ICS. These data are given in Table 79. The most recent experimental ICS measurements for CO were by Zobel et al. [279], who investigated the a3 V electronic state of CO at energies from about 6–9:5 eV, Zetner et al. and Middleton et al., who derived ICS from their DCS measurements, and Furlong and Newell. This data is shown in Fig. 72 along with the DW calculation of Lee et al., the R-matrix result of Morgan and Tennyson [266] and the result of the calculation of Sun et al. [285]. The overall level of agreement between these various studies is reasonable, with perhaps agreement being best between Zobel et al. and Sun et al. in the near-threshold region. The cross-section of Furlong and Newell is larger at all energies than the other experimental results. Great care was taken in the measurement of both the DCS and ICS in Zobel et al. and as such this data represents a stringent test of scattering theory for the a3 V state at near-threshold energies. Theoretical studies have also been reported by Chung and Lin [302], who calculated the ICS using the Born–Ochkur–Rudge (BOR) approximation for electron impact excitation of a large number of electronic states of CO. These included the singlet states A1 V, E1 V, B1 W+ ,

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Table 75 Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the (a) (C1 W+ + c3 V) and (b) C1 W+ (; = 0) states of CO Energy (eV) a

(a) Angle 5 10 15 20 30 40 50 60 70 75 80 90 100 110 120 (b)b Angle

20 Middleton et al.

30 Middleton et al.

40 Middleton et al.

50 Middleton et al.

0.52

3.60

12.75

10.11

0.29 0.17 0.13 0.10 0.08 0.06

1.01 0.35 0.29 0.27 0.12 0.09

1.78 0.71 0.44 0.25 0.15 0.09

1.82 0.83 0.49 0.27 0.14 0.09

0.05 0.04

0.07 0.07

0.07 0.06

0.07 0.06

100 Kanik et al. 141.5 15.5 3.6 1.38 0.62 0.24 0.12 0.077 0.048 0.050 0.043 0.041 0.046 0.051 0.57

Energy (eV) 11.60

20 0.054 30 0.045 40 0.041 50 0.033 60 0.041 70 0.052 80 0.052 90 0.049 100 0.053 110 0.056 120 0.068 130 0.085 140 0.071 a The uncertainty on the data of Middleton et al. is ±27%. b The data are those of Zobel et al.

13.2

15.1

0.47 0.32 0.28 0.23 0.24 0.23 0.23 0.21 0.21 0.19 0.18 0.19 0.15

1.15 0.64 0.41 0.37 0.38 0.35 0.35 0.29 0.28 0.22 0.21 0.17 0.17

C1 W+ and D1 U, in the energy range from threshold to 1000 eV, and the triplet states a3 V, c3 V, b3 W+ , j3 W+ , a3 W+ and d 3 Ui in the energy range from threshold to 100 eV. Although the BOR may have some validity at high impact energies, the Born theory is not capable of predicting resonance behaviour and is generally found to be in poor agreement with the experiment at low and intermediate energies. Lee and McKoy [283] reported DW calculations for the ICS of the A1 V, D1 U, a3 V, a3 W+ and d 3 Ui states in the energy range 20–50 eV. These DW calculations are also expected to overestimate the magnitudes of the cross-sections because of the e@ects of shape resonances although they give qualitatively correct shapes for the DCS. Lee et al. repeated

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385

Fig. 70. Di@erential cross-sections for electron impact excitation of the C1 W+ + c3 V electronic states of CO at 20 eV. () Trajmar et al., () Middleton et al., ( ) Zobel et al. at 15:1 eV.

•

Fig. 71. Di@erential cross-section for electron impact excitation of the E1 V electronic state of CO at 20 eV. () Trajmar et al., () Middleton et al., ( ) Zobel et al. at 15:1 eV.

•

these studies over a more extended energy range and with a more sophisticated wavefunction. Weatherford and Huo calculated ICS using the SMC approach for the b3 W+ state from threshold to 20 eV. This calculation indicated the existence of many sharp structures due to resonance states of CO− . Sun et al. also gave results from their SMC variational method calculation, which employed single conFguration wavefunctions, for the electron impact excitation of the seven lowest electronically excited states of CO. Morgan and Tennyson reported ICS for these same seven states using the R-matrix method and conFguration interaction wavefunctions, in the somewhat smaller energy range from threshold to 18 eV. The ICS obtained using these two methods agree only broadly in the magnitudes they predict and di@er in the shapes. In principle, the CI wavefunctions utilised by Morgan and Tennyson should be superior to the SCF wavefunctions of Sun et al., except for the optically allowed transition to the A1 V state for which the CI target wavefunction was particularly poor. Furthermore, Morgan and Tennyson also treated the e@ects of correlation and polarisation in a more exact fashion than did Sun et al. However, the cross-sections obtained using the R-matrix method are unreliable above 15 eV due to the unavailability of CI target wavefunctions for the higher excited states which couple strongly to the states of interest. Consequently, the SMC method, which is much less a@ected by

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Table 76 Di@erential cross-sections (10−18 cm2 sr −1 ) for the electron impact excitation of the (a) E1 V and (b) E1 V (; = 0) state of CO Energy (eV) (a)a Angle 5 10 15 20 30 40 50 60 70 75 80 90 100 110 115 (b)b Angle

20 Middleton et al.

30 Middleton et al.

40 Middleton et al.

50 Middleton et al.

0.40

2.52

10.21

8.23

0.29 0.19 0.14 0.10 0.08 0.06

0.78 0.33 0.32 0.19 0.12 0.08

1.54 0.69 0.41 0.18 0.12 0.09

1.65 0.86 0.41 0.17 0.11 0.08

0.04 0.03

0.08 0.07

0.08 0.06

0.07 0.05

100 Kanik et al. 62.5 6.5 1.56 0.72 0.28 0.088 0.042 0.028 0.015 0.015 0.012 0.011 0.012 0.012 0.013

Energy (eV) 11.77

20 0.052 30 0.043 40 0.048 50 0.038 60 0.051 70 0.064 80 0.072 90 0.056 100 0.044 110 0.045 120 0.051 130 0.059 140 0.052 a The uncertainty on the data of b The data are those of Zobel et

13.32

15.22

0.41 0.76 0.21 0.48 0.29 0.35 0.24 0.22 0.22 0.29 0.20 0.25 0.19 0.23 0.15 0.19 0.16 0.20 0.15 0.18 0.16 0.18 0.17 0.16 0.16 0.14 Middleton et al. is ±26%. al.

omitting target states and which also improves in reliability above 15 eV where correlation and polarisation e@ects are less important, probably remains the most reliable calculation available currently in the literature. Finally, we note the recent compilation of electronic state integral cross-sections by Liu and Victor [303], who considered electron energy deposition in carbon monoxide gas. These data are illustrated in Fig. 73 and whilst it is beyond the scope of the present article to discuss how this set was compiled we note that a full description is given by these authors in Section 2 of their article. By using this cross-section set, Liu and Victor were able to calculate a value

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387

Table 77 Integral cross-sections (10−18 cm2 ) for the electron impact excitation of the I1 W− state of CO. The data are those of Mason and Newell Energy (eV)

Cross-section

10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 27 29 31 33 35 37 39 42 47 52 57 60

0:06 ± 0:02 0:31 ± 0:05 0:90 ± 0:07 2:00 ± 0:13 2:66 ± 0:16 2:95 ± 0:18 3:00 ± 0:18 2:81 ± 0:17 2:50 ± 0:16 2:34 ± 0:16 2:21 ± 0:16 2:05 ± 0:14 1:95 ± 0:14 1:89 ± 0:13 1:79 ± 0:13 1:56 ± 0:12 1:38 ± 0:12 1:30 ± 0:11 1:19 ± 0:11 1:14 ± 0:11 1:10 ± 0:10 1:06 ± 0:10 1:00 ± 0:10 0:95 ± 0:10 0:94 ± 0:10 0:90 ± 0:10 0:90 ± 0:10

for the mean energy per ion pair which was in very good agreement with that measured in an independent experiment [304]. 3.6. Nitric oxide (NO) There has been renewed interest in nitric oxide (NO) as a scattering system since its important role in the catalytic destruction of ozone in the stratosphere was conFrmed [305,306]. From a more fundamental perspective, electron scattering by NO represents a considerable challenge to theory in that it is an open-shell molecule and it possesses a small, but signiFcant, permanent dipole moment of 0.157D. In the former case, particularly at low energies, Tennyson and Noble [307] have shown that for elastic scattering by open-shell molecules, polarisation e@ects are particularly important because of the presence of low-lying electronically excited states. In the latter case, the very forward angle behaviour of the elastic di@erential cross-section (DCS) is expected to be divergent [308] within a Fxed-nuclei treatment. This e@ect can only be treated exactly by the inclusion of nuclear motion in the Hamiltonian.

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Table 78 Integral cross-sections (10−16 cm2 ) for the electron impact excitation of the a3 V electronic state of CO. The data are those of Furlong and Newell E0 (eV)

Cross-section

6.14 6.55 7.46 8.49 9.65 10.56 11.55 12.59 13.50 14.50 15.49 16.53 17.53 18.63 19.56 20.77 22.72 24.67 26.98 29.01 30.57 35.54 40.68 45.66 50.70 60.69 71.09

0:425 ± 0:022 0:543 ± 0:023 1:070 ± 0:054 1:696 ± 0:086 1:892 ± 0:096 1:793 ± 0:091 1:629 ± 0:083 1:454 ± 0:074 1:455 ± 0:074 1:389 ± 0:071 1:299 ± 0:065 1:225 ± 0:062 1:272 ± 0:065 1:052 ± 0:054 1:022 ± 0:052 0:910 ± 0:046 0:745 ± 0:038 0:681 ± 0:035 0:566 ± 0:029 0:471 ± 0:024 0:453 ± 0:023 0:317 ± 0:016 0:231 ± 0:012 0:208 ± 0:011 0:192 ± 0:009 0:172 ± 0:001 0:136 ± 0:001

Table 79 Integral cross-sections (10−18 cm2 ) for electron impact excitation of the B1 W+ ; C1 W+ and E1 V electronic states of CO at an energy of 100 eV. The data are those of Kanik et al. State

ICS

B1 W + C1 W + E1 V

3.83 12.34 4.43

The pioneering experimental study of electron scattering from NO was undertaken by Br\uche [201], who used a Ramsauer technique to measure the absolute total cross-section in the energy range 1–49 eV. Subsequently, there has been very little work on absolute cross-section determinations. Most of the experimental studies having concentrated on the ionisation process [309 –314] and on dissociative electron attachment [315 –320]. Strong resonance e@ects in e-NO scattering have been observed in low-energy transmission functions [321–326] and in the

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389

◦

Fig. 72. Integral cross-section for electron impact excitation of the a3 V electronic state of CO. ( ) Zetner et al., () Middleton et al., ( ) Zobel et al., (+) Furlong and Newell, (− − −) Sun et al., ( – - – ) Morgan and Tennyson, (—) Lee et al.

•

backscattered current [327]. We also note the higher-energy resonance (structure) behaviour observed in a 0 –3 vibrational excitation function measurement [328]. More detailed experimental studies of the resonant scattering below 3 eV were performed by Spence and Schulz [329], Tronc et al. [330] and Alle et al. [331] with the observed oscillatory structure being attributed to a vibrational series of NO− shape resonances. In particular, these short-lived negative ions were found by Tronc et al. to belong to a three-member multiplet of 3 W− , 1 U and 1 W+ symmetries. Using a simple formalism, Teillet-Billy and Fiquet-Fayard [332] analysed these structures and estimated the spectroscopic constants of the two lowest resonances. The resonance positions and widths calculated by Tennyson and Noble [307], using an R-matrix formalism, were found to be in quite good agreement with the measurements of Tronc et al. More recently, Alle et al. have extended the measurements, at the grand total cross-section level, and analysis to even lower energies (E ∼0:2 eV). For completeness, we also note the semi-empirical calculation of Koike [333] on low-energy resonances in the elastic and vibrational channels. Further resonance e@ects at impact energies up to about 20 eV were observed in transmission spectroscopy by Sanche and Schulz [334], in high-resolution experiments by Gresteau et al. [335 –337], in excitation function measurements for the 0 → 3 vibrational channel by Malegat et al. [328] and in the 0 →1 vibrational excitation function measurement of Mojarrabi et al. [338]. In particular, we highlight the work of Gresteau et al., who obtained signiFcant results on the decay of Feshbach resonances. More recently, Tronc et al. [339], King et al. [340] and Camilloni et al. [341] reported studies on the inner-shell excited states of NO. Experiments on radiative emission following electron impact excitation of NO have been reported by Stone and Zipf [342], Povch et al. [343], Imami and Borst [344], Fukui et al.

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Fig. 73. Integral cross-section for electron impact excitation of the (a) singlet and (b) triplet electronic states of CO. The data are those of Liu and Victor.

[345], Skubenich et al. [346] and van Sprang et al. [347]. These were partly stimulated by the role of NO in auroral displays. The work of Skubenich et al. is very interesting in that their integral electronic state excitation cross-section measurements for the A2 W+ ; B2 V; B2 U, F2 U and b4 W− electronic states are, typically, at least an order of magnitude less in strength than the corresponding N2 electronic states. Middleton [348] and Mojarrabi et al. [349] have measured DCS and ICS for the excitation of a number of electronic states in NO. These results have recently been extended by Brunger et al. [350,351] to cover a further 23 excited states. In each case, the results were analysed using the spectral deconvolution technique discussed in Section 2.7. From a theoretical perspective the situation regarding recent work is is even worse with only the Born-closure Swinger variational method (BCSVM) calculation of Lee et al. [352], the optical model approach of Jain et al. [353], the Schwinger multichannel method (SMC) of da Paixao et al. [354], the BCSVM-AN calculation for vibrational excitation (v = 0 → 1; 2) of Lee et al. [355] and the extension of both the distorted wave (DW) and SMC models [356,357] by these latter workers to electron impact excitation of the A2 W+ , C2 V and D2 W+ electronic states being available. The study of Lee et al. reported DCS’s for elastic scattering at selected energies in the range 5–500 eV. Where possible, comparison was made with the data

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391

Fig. 74. Grand total cross-section for electron scattering from NO (a) at energies below 2 eV, (b) for energies – ) Alle et al. between 2–1000 eV. ( ) Zecca et al., (U) Szmytkowski and Maciag, () Dalba et al., ( –

◦

•

of Kubo et al., although with uncertainties of the order of ±30% on this data it did not provide a particularly stringent test of the BCSVM calculation. The elastic DCS of Mojarrabi et al. rectiFed this problem by reporting data with errors typically in the range 6–12%. Lee extended the BCSVM model to allow for nuclear motion (in the adiabatic nuclei (AN) approximation), thereby enabling calculations of rovibrational DCS at certain energies in the range 5–30 eV. The calculation by Jain et al., based on the work of Gianturco and Jain [358], employs an

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optical potential approach where the interaction potential consists of the static, polarisation and exchange terms. Results were presented at 5, 10 and 20 eV and were generally found to be in marginal agreement with the experimental results of Kubo et al. The SMC calculation of da Paixao et al. [354] reported only a single result at 10 eV and agreement with Kubo et al. seemed fair, except at forward angles. However, as neither Jain et al. or da Paixao et al. gave tables of their DCS we will not discuss them further, except at the integral elastic and elastic momentum transfer level where tabulated values in the range 5–100 eV were provided. Machado et al. have reported calculations for excitation of the A2 W+ , C2 V and D2 W+ Rydberg electronic states for 6 energies in the range 12:5–50 eV. This work is compared against the results of Brunger et al. [350,351] later in this section. Similarly, the more exact SMC calculations, again performed at energies corresponding to those of Brunger et al., are detailed later. 3.6.1. Grand total cross-sections The more recent absolute grand total cross-sections (GTS) for e− +NO scattering are displayed in Figs. 74(a) and (b) and listed in Table 80. The pioneering measurements of Br\uche are not included and only those values of Szmytkowski and Maciag which were tabulated in their paper are included in the Fgures for the sake of clarity. The most striking feature of the measured GTS is the oscillatory nature of the cross-section below about 2 eV impact energy. This structure was attributed by Tronc et al. to a vibrational series belonging to a multiplet of 3 W− , 1 U and 1 W+ scattering symmetries of NO− shape resonant states. This interpretation was somewhat tentatively supported by the calculation of Tennyson and Noble and conFrmed by the more exact calculation, which employed a larger and more sophisticated CI basis set, of Morgan and Gillan [359]. The observed variations in the intensities of the oscillations, peak spacings and, to some extent, the shape of the direct background on which the oscillatory structure is superimposed, may be partly associated with interference e@ects between overlapping vibrational series of the successive resonant states. From Fig. 74(a), it is apparent that there is reasonable accord, with regard to the positions of the maxima of the oscillatory structures, in the measurements of Zecca et al., Szmytkowski and Maciag (not speciFcally shown) and Alle et al. The magnitude of the oscillations in Zecca et al. (for the peak-to-dip ratio) is noticeably smaller than that found by Szmytkowski and Maciag, which is in turn smaller than those in the data reported by Alle et al. The discrepancy between Zecca et al. and Szmytkowksi and Maciag is somewhat intriguing as it cannot be explained by the di@erence in the energy resolution of these experiments which were similar (∼60 meV). On the other hand, the di@erence in the peak-to-dip ratio of the oscillatory structures between the work of Szmytkowski and Maciag and Alle et al. can almost certainly be ascribed to the superior energy resolution (¡ 25 meV at 0:75 eV) of the most recent study. Furthermore, within the combined uncertainties on the data of Szmytkowski and Maciag and Alle et al., the overall level of agreement between them for the GTS can be characterised as being quite good. At this time we also note the observation of Alle et al. that diJculties in maintaining electron optical stability, when cycling reactive gases through their apparatus, limit the determination of the absolute energy scale to no better than about ±20 meV. The direct or non-oscillatory part of the GTS of Zecca et al. is also found (see Figs. 74(a),(b)) to be lower in magnitude than that determined by both Szmytkowski and Maciag and

M.J. Brunger, S.J. Buckman / Physics Reports 357 (2002) 215–458 Table 80 Selected grand total cross-sections for scattering from NO (units are 10−16 cm2 ) Energy (eV)

Zecca et al.

Alle et al.

0.037 0.039 0.040 0.050 0.060 0.070 0.080 0.090 0.100 0.110 0.120 0.130 0.140 0.150 0.152 0.160 0.170 0.180 0.190 0.200 0.210 0.220 0.230 0.240 0.250 0.260 0.265 0.270 0.280 0.290 0.300 0.305 0.310 0.320 0.330 0.340 0.350 0.360 0.370 0.380 0.390 0.400 0.410 0.420 0.430 0.440 0.450 0.460

17.51 13.41 12.65 12.13 10.68 9.7583 9.0410 8.3253 7.9364 7.6997 7.5913 7.4725 7.3367 7.1673 — 7.1410 7.1250 7.200 7.3750 7.750 8.125 8.675 9.125 9.600 9.750 9.750 — 9.700 9.500 9.125 8.8750 — 8.5000 8.250 8.119 8.250 8.400 8.750 9.125 9.400 9.800 10.125 10.375 10.400 10.400 10.170 9.8750 9.625

— — — — — — — — — — — — — — 9.3795 7.1980 6.0019 5.7184 5.6163 6.0477 6.3581 6.3989 6.6547 7.2060 8.5480 10.307 11.650 13.138 14.505 16.678 16.230 15.050 13.838 9.6632 7.6849 7.2333 7.2698 7.3686 7.4866 7.6192 7.9653 8.5642 9.5532 10.966 12.778 14.667 16.314 17.061

393

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Table 80 (Continued) Energy (eV)

Zecca et al.

Alle et al.

0.465 0.470 0.480 0.490 0.500 0.510 0.520 0.530 0.540 0.550 0.560 0.570 0.580 0.590 0.600 0.610 0.615 0.620 0.630 0.640 0.650 0.660 0.670 0.680 0.690 0.700 0.710 0.720 0.730 0.740 0.750 0.760 0.765 0.770 0.780 0.790 0.800 0.810

— 9.250 8.875 8.834 8.800 8.810 9.000 9.375 9.600 9.875 10.250 10.500 10.700 10.750 10.800 10.750 — 10.605 10.375 10.245 10.1375 10.250 10.375 10.625 10.825 11.000 11.161 11.250 11.262 11.250 11.161 11.000 — 10.875 10.650 10.620 10.501 10.501

16.609 15.711 12.696 9.6088 8.3003 7.9269 7.8744 7.8269 7.9700 8.2320 8.6909 9.3869 10.486 11.755 13.110 14.433 14.911 15.150 14.619 13.092 11.364 10.351 10.010 10.089 10.416 10.915 11.609 12.410 13.277 14.146 14.804 15.240 15.364 15.375 15.031 14.121 12.866 11.616

Energy (eV)

Zecca et al.

Alle et al.

Szmytkowski and Maciag

0.820 0.830 0.840 0.850 0.860 0.870 0.880 0.890

10.625 10.875 11.125 11.370 11.625 11.759 11.882 11.995

10.766 10.361 10.285 10.523 10.995 11.723 12.648 13.737

— — — 11.800 — — — —

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395

Table 80 (Continued) Energy (eV)

Zecca et al.

Alle et al.

Szmytkowski and Maciag

0.900 0.910 0.920 0.930 0.935 0.940 0.950 0.960 0.970 0.980 0.990 1.000 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.100 1.110 1.120 1.130 1.140 1.150 1.160 1.170 1.180 1.190 1.200 1.220 1.240 1.250 1.260 1.270 1.280 1.290 1.300 1.320 1.340 1.360 1.380 1.390 1.400 1.410 1.420

12.000 11.995 11.750 11.375 — 11.250 10.750 10.620 10.620 10.750 11.000 11.125 11.250 11.500 11.750 11.875 11.895 11.914 11.915 11.875 11.750 11.625 11.416 11.250 11.114 11.103 11.114 11.125 11.250 11.416 11.563 11.630 11.700 11.625 11.500 11.375 11.250 11.130 11.120 11.110 11.110 11.130 11.325 11.375 11.370 11.360 11.265 11.249

14.913 16.047 16.981 17.432 17.405 17.198 16.263 14.950 13.589 12.463 11.679 11.152 10.927 10.940 11.194 11.664 12.434 13.300 14.272 15.304 16.010 16.351 16.316 15.827 15.061 14.208 13.513 12.856 12.418 12.195 12.151 12.282 13.133 14.271 — 15.046 15.201 15.146 14.886 14.422 13.526 12.888 12.790 13.140 13.451 13.736 13.999 14.205

— — — — — — — — — — — — — — — — 12.600 — — — — — — — — — — — — — — — — — 13.100 — — — — — — — — — — — — —

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Table 80 (Continued) Energy (eV)

Zecca et al.

Alle et al.

Szmytkowski and Maciag

1.430 1.440 1.450 1.460 1.470 1.480 1.490 1.500 1.520 1.540 1.560 1.580 1.600 1.620 1.640 1.650 1.660 1.680 1.700 1.720 1.740 1.760 1.780 1.800 1.820 1.840 1.850 1.860 1.880 1.900 1.920 1.940 1.960 1.980 2.000 2.050 2.2 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5

11.130 11.005 10.887 10.875 10.827 10.801 10.773 10.765 10.801 10.875 10.889 10.855 10.750 10.625 10.510 10.500 10.450 10.382 10.382 10.382 10.383 10.375 10.3125 10.250 10.200 10.120 10.115 10.110 10.110 10.115 10.050 10.000 9.951 9.916 9.840 9.750 9.570 9.375 9.134 9.0625 9.000 9.000 8.950 9.000 9.000 9.100 9.130 9.200

14.371 14.391 14.304 14.116 13.880 13.599 13.342 13.109 12.795 12.780 13.014 13.232 13.404 13.318 13.117 — 12.792 12.530 12.406 12.343 12.394 12.457 12.470 12.417 12.273 12.127 — 11.984 11.900 11.855 11.770 11.734 11.824 11.747 11.697 11.430 10.992 10.416 — — — — — — — — — —

— — 12.500 — — — — — — — — — — — — 11.900 — — — — — — — — — — 11.100 — — — — — — — 10.700 — 10.200 9.750 9.450 9.380 9.300 9.210 9.220 9.23 9.21 9.41 9.46 9.52

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397

Table 80 (Continued) Energy (eV)

Zecca et al.

Dalba et al.

Szmytkowski and Maciag

8.0 8.5 9.0 9.5 10 11 12 14 16 18 20 25 30 35 40 45 50 55 60 65 70 75 80 90 100 110 120 121 140 144 160 169 196 225 256 289 324 400 484 576 676 784 900 1024 1156 1296 1444

9.250 9.344 9.437 9.500 — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

— — — — — — — — — — — — — — — — — — — — — — — — — — — 8.465 — 8.211 — 7.223 6.713 6.092 5.823 5.286 5.099 4.251 3.781 3.369 2.891 2.650 2.337 2.118 1.925 1.687 1.576

9.65 9.68 9.81 9.92 10.1 10.5 11.0 11.4 11.7 11.5 11.4 11.1 10.8 10.5 10.3 9.79 9.59 9.40 9.30 9.12 8.96 8.80 8.75 8.58 8.48 8.29 8.13 — 7.76 — 7.44 — — — — — — — — — — — — — — — —

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Alle et al. over the whole energy region of mutual overlap. In this case, we can hypothesise that the reason for this discrepancy could be the normalisation procedure applied by Zecca et al. in their experiment. The next prominent feature of the GTS (Fig. 74(b)) is the very broad, structureless maximum centred at around 16 eV electron impact energy. This result is consistent with that observed by ◦ Mojarrabi et al., in their 0 → 1 vibrational excitation function measurement at 10 . They reported a broad structure in the rovibrational cross-section centred at about 15 eV. Theoretical support for this was further provided by the BCSVM calculation of Lee, although in that work the broad peak was centred at about 20 eV (see Section 3.7.3), which is probably indicative of an insuJcient number of channels being incorporated into the calculation. Evidence for the existence of resonance phenomena in the 12–20 eV energy range was originally provided by Sanche and Schulz in their transmission spectra, where they identiFed several structures associated with both core-excited Feshbach resonances and core-excited shape resonances. In the former case they identiFed resonances associated with the grandparent b3 V, A1 V and B1 V excited states of NO+ . These states were originally considered as likely candidates for compound-state formation from the ground molecular state on the basis that their internuclear separations were similar to that of the NO ground state [360]. In the latter case, the core-excited shape resonances arise when singly or doubly excited states of the neutral molecule bind an extra electron by the centrifugal barrier. More detail on the precise nature of the physical process causing the observed structure in this 12–20 eV energy regime was provided by Malegat et al., who concluded that it should be interpreted in terms of a ( u ) shape resonance. For energies above 16 eV, the GTS is observed (Fig. 74(b)) to decrease in magnitude with increasing energy. There is a small overlap in the higher energy measurements of Szmytkowski and Maciag with the lower energy measurements of Dalba et al. In general, the data of Szmytkowski and Maciag are somewhat lower in magnitude than those of Dalba et al., although these di@erences were all well within the combined uncertainties of the two experiments and, in general, the trend in the data is that the GTS of Szmytkowski and Maciag maps rather nicely onto that of Dalba et al. 3.6.2. Elastic scattering 3.6.2.1. Di:erential cross-sections. The recent, absolute elastic di@erential cross-sections for electron scattering from NO of Mojarrabi et al. [338] are presented in Table 81 and shown, at selected energies, in Figs. 75 and 76, where they are also compared to the only other available experiment [361,362] and theory [352]. The work of Mojarrabi et al. contains data measured on two di@erent spectrometers (one at the Australian National University (ANU) and the other at Flinders University). The data presented in Table 81 are those from the ANU spectrometer taken using the relative Pow technique. The cross-sections for energies below 10 eV are shown in Fig. 75. The most surprising and unusual aspect of the DCSs at these energies is the forward angle behaviour. At 1:5 eV ◦ (Fig. 75(a)), the DCS exhibits a clear maximum at around 60 , dropping o@ in magnitude towards both forward and backward scattering angles. This behaviour is even more marked at an incident energy of 3 eV (Fig. 75(b)) where the cross-section drops rapidly from its ◦ maximum value of about 1:1 × 10−16 cm2 sr −1 at around 50–60 . Unfortunately, there are no

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399

Table 81 Di@erential cross-sections (10−16 cm2 sr −1 ) for elastic electron scattering from nitric oxide. The data are those of Mojarrabi et al. and the Fgures in brackets indicate the percentage uncertainty Energy (eV) Angle

1.5

3.0

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130

— 0.799 — 0.843 — 0.910 — 0.955 — 0.960 — 0.947 — 0.895 — 0.838 — 0.805 — 0.774 — 0.761 — 0.750

(7.1) (6.6) (6.8) (6.6) (7.1) (6.7) (6.6) (6.9) (6.5) (6.9) (6.9) (6.5)

0.812 0.850 0.908 0.942 0.993 1.028 1.056 1.081 1.064 1.074 1.054 1.004 0.928 0.864 0.810 0.742 0.696 0.622 0.625 0.597 0.568 0.538 0.526 0.517

5.0 (6.9) (6.5) (7.4) (7.3) (6.5) (6.4) (6.8) (6.6) (6.6) (6.8) (6.4) (6.6) (6.6) (6.7) (6.6) (7.7) (7.5) (6.8) (7.1) (7.5) (7.6) (6.4) (6.8) (6.7)

0.941 0.949 1.010 1.095 1.104 1.121 1.174 1.148 — 1.114 — 0.976 — 0.802 — 0.642 — 0.534 — 0.472 — 0.443 — 0.441

7.5 (7.1) (6.5) (6.4) (6.4) (6.9) (6.9) (7.1) (6.4)

1.330 1.234 1.186 1.202 1.209 1.221 1.193 1.207 1.130 1.067 — 0.873 — 0.647 — 0.496 — 0.409 — 0.395 — 0.421 — 0.470

(6.5) (6.4) (6.5) (6.5) (6.4) (6.5) (6.3) (6.5)

10.0 (6.7) (6.7) (6.8) (6.6) (6.4) (7.1) (6.5) (6.8) (6.5) (6.6) (6.7) (6.9) (6.6) (6.5) (6.4) (6.4) (6.5)

1.870 1.629 1.475 1.375 1.333 1.292 1.230 1.172 1.086 0.996 0.920 0.752 0.631 0.517 0.439 0.373 0.345 0.333 0.355 0.366 0.404 0.448 0.473 0.519

(7.6) (6.7) (6.4) (6.4) (6.5) (6.5) (6.8) (6.5) (6.4) (7.1) (8.5) (6.9) (8.3) (7.2) (7.5) (7.3) (7.5) (6.4) (7.1) (6.7) (7.2) (6.9) (6.4) (6.7)

Energy (eV) Angle

15.0

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

— 3.600 2.907 2.424 1.922 1.639 1.409 1.264 1.109 0.910 0.739 0.622 0.529 0.428 0.343 0.284 0.278 0.274

20.0 (6.9) (9.7) (12.1) (9.6) (7.2) (6.7) (7.6) (8.6) (9.3) (8.4) (6.7) (6.4) (9.1) (11.3) (7.6) (7.7) (7.8)

— 4.236 3.444 2.968 2.132 1.764 1.502 1.231 1.036 0.737 0.567 0.481 0.361 0.285 0.233 0.202 0.186 0.183

30.0 (11.1) (13.6) (8.7) (8.0) (7.4) (12.5) (12.0) (10.8) (8.5) (8.0) (9.2) (9.4) (14.1) (11.1) (13.0) (10.2) (12.9)

— 6.484 4.490 3.387 2.290 1.671 1.323 0.987 0.739 0.576 0.417 0.296 0.249 0.193 0.157 0.148 0.138 0.137

40.0 (10.4) (10.9) (8.0) (8.5) (9.6) (11.1) (10.7) (15.2) (13.7) (11.0) (14.6) (15.9) (15.1) (19.7) (17.3) (12.1) (15.2)

9.874 6.594 4.364 3.042 1.958 1.275 0.885 0.578 0.410 0.318 0.245 0.189 0.149 0.121 0.101 0.080 0.070 0.066

(7.5) (7.4) (7.3) (7.3) (7.3) (7.3) (7.4) (8.3) (8.0) (8.2) (7.5) (7.8) (8.0) (8.7) (7.7) (8.7) (8.6) (8.2)

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Table 81 (Continued) Energy (eV) Angle

15.0

100 105 110 115 120 125 130

0.305 0.326 0.382 0.436 0.472 0.511 0.536

20.0 (7.4) (7.6) (6.5) (9.3) (7.5) (8.4) (7.4)

0.197 0.213 0.246 0.294 0.326 0.382 0.454

30.0 (10.5) (8.1) (9.6) (7.9) (11.4) (13.3) (13.7)

0.136 0.166 0.206 0.262 0.315 0.417 0.498

40.0 (12.1) (12.7) (10.0) (10.8) (18.3) (15.3) (18.6)

0.068 0.082 0.110 0.152 0.206 0.277 0.345

(8.8) (7.6) (7.8) (8.1) (9.9) (7.6) (8.7)

other experimental or theoretical results at either of these energies with which we can compare the data of Mojarrabi et al. At 5 eV (Fig. 75(c)) the decrease in the cross-section in the forward direction is once again evident, but not so pronounced, and the cross-section maximum ◦ of around 1:2 × 10−16 cm2 sr −1 has again shifted to a lower angle, in this case to about 40–50 . Here, however, there are other experimental and theoretical results available for comparison. The experimental cross-section of Kubo et al. is in good general agreement with that of Mojarrabi ◦ et al., exhibiting the same overall shape with a maximum at around 50 and a minimum at ◦ 130 , although there are some small di@erences between them in absolute magnitude over this range. The theoretical curve of Lee et al., whilst exhibiting similarities to the experimental ◦ results at scattering angles above about 60 , is substantially larger in magnitude than both the experiments at small angles. Whilst the y-axis of the graph has been truncated the BCSVM result does exhibit a local maximum at small angles. Finally, at 7:5 eV (Fig. 75(d)), we can compare the two sets of data which comprise the work of Mojarrabi et al. The level of agreement between them is quite good over their common angular range. Both the DCS measurements of Mojarrabi et al. exhibit an angular behaviour similar to that observed previously for lower energies (Figs. 75(a) – (c)), although the ANU measurement perhaps suggests a Patter shoulder ◦ between 20 and 50 than seen previously at the lower energies. This behaviour at forward angles, where the elastic DCS decreases as a function of angle over a range of fairly low electron impact energies, is not unique to the elastic cross-section for NO. Indeed, we refer to you to the earlier sections in this review pertaining to N2 (3:2), O2 (3:3) and CO (3:5). We can only speculate at this time upon the reason for this behaviour but it appears to be associated with long-range interactions which are mediated, principally, by the polarisation potential. We note that these molecules (N2 , O2 , CO and NO) all have similar static dipole polarisabilities and as a result one might naively expect similar trends in the forward angle scattering behaviour. A fuller discussion of this point can be found later in Section 3.8 of this article. At higher energies the trends in the shape of the DCS continue to develop, with the scattering becoming more enhanced in the forward direction and the cross-section minimum becoming deeper (see Fig. 76). At 10 eV (Fig. 76(a)) the elastic DCS of Mojarrabi et al., from both Flinders and the ANU, are compared with the data of Kubo et al. The agreement between the two measurements reported by Mojarrabi et al., particularly with respect to the shape of the DCS, is quite good, as is their overall agreement with Kubo et al. The theoretical calculation

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401

Fig. 75. Di@erential cross-sections for the elastic scattering from NO at (a) 1:5 eV, (b) 3 eV, (c) 5 eV and (d) 7:5 eV. ( ) and () Mojarrabi et al., ( ) Kubo et al., ( — ) BCSVM calculation of Lee et al.

•

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Fig. 76. Di@erential cross-sections for elastic scattering from NO at (a) 10 eV, (b) 15 eV, (c) 20 eV and (d) 30 eV. ( ) and () Mojarrabi et al., ( ) Kubo et al., ( — ) BCSVM calculation of Lee et al., ( – – ) SVIM calculation of Lee et al.

•

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403

reproduces the gross features of the DCS but not the detailed structure, particularly the shoulder at forward angles. At 15 eV (Fig. 76(b)) both the data sets of Mojarrabi et al. are in excellent agreement with one another but are uniformly lower in magnitude and di@erent in shape from the DCS of Kubo et al. The theory, in this case, however, is in considerably better agreement with the data of Mojarrabi et al. than found previously at the lower energies. In Fig. 76(c), for 20 eV impact energy, we see a similar situation with the ANU and Flinders DCS data of Mojarrabi et al. being in excellent agreement but lower in magnitude than those of Kubo et al., particularly at large scattering angles. Finally, at 30 eV (Fig. 76(d)) the two data sets of Mojarrabi et al. are again in excellent agreement with each other over their common angular ◦ range and, at angles greater than 70 they are also in excellent agreement with Kubo et al. ◦ On the other hand, for angles greater than 70 all the experimental cross-sections are lower in ◦ magnitude than the BCSVM result. At smaller scattering angles (6 70 ) the results of Mojarrabi et al. lie above those of Kubo et al. but are in good agreement with the theory. In summary, the recent study of Mojarrabi et al. supercedes that of Kubo et al. in both the breadth of its energy range and in its accuracy. Thus it provides an extensive set of elastic DCS against which theoretical workers can compare the results of their methods. To date only the BCSVM model has been tested against the data of Mojarrabi et al. with the general trend being that this model gives a reasonable picture of the elastic scattering process for energies above about 20 eV. However, below 20 eV impact energy there is a clear need for more sophisticated theoretical calculations to be performed. 3.6.2.2. Integral cross-sections. The elastic DCS measurements of Mojarrabi et al. were ex◦ ◦ trapolated by those authors to 0 and 180 and then integrated to yield both integral elastic cross-sections and elastic momentum transfer cross-sections. The extrapolation to forward and backward angles by Mojarrabi et al. was done largely by eye although in those cases where there was theory available for comparison, the shape of the theory was utilised. The integral cross-sections arising from this procedure are given in Table 82 and plotted in Fig. 77. We also note that the original ICS values in the paper of Mojarrabi et al. were incorrectly reported but that this has been remedied in the paper of Brunger et al. [351] and in the present tabulated values. Also, plotted in this Fgure are the BCSVM result of Lee et al. and the static exchange plus polarisation (SEP) result of Jain et al. [353]. It is apparent from Fig. 77 that, for energies greater than 15 eV, the BCSVM calculation is in fair agreement with the revised ICS data of Mojarrabi et al. For energies less than 15 eV the BCSVM result diverges from the experimental ICS values in that it tends to overestimate the magnitude of the cross-section. These observations for the ICS are entirely consistent with what one might expect on the basis of the level of comparison made in the preceding section for the DCS. With respect to the SEP calculation of Jain et al. then it is quite clear this model overestimates the magnitude of the ICS at all the energies considered and that it also predicts a structure in the ICS not seen experimentally. Consequently, we conclude that of the two available calculations the BCSVM method is clearly superior in the energy range shown. Once again the only experimental data for momentum transfer cross-sections in NO are those derived from the DCS measurements of Mojarrabi et al. These are also listed in Table 82 and shown in Fig. 78 where the experimental momentum transfer cross-section demonstrates

404

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Table 82 Integral ( i ) and momentum ( m ) transfer cross-sections for elastic electron scattering from NO (10−16 cm2 ). The (revised) data of Mojarrabi et al. are given (Figures in brackets indicate the percentage uncertainty.) Energy (eV)

i

1.5 3 5 7.5 10 15 20 30 40

10.47 9.60 9.24 9.10 9.24 7.71 9.71 9.31 8.21 —

m (30) (30) (20) (20) (20) (20) (20) (20) (20)

7.03 6.15 5.51 4.88 4.50 4.04 3.09 2.64 1.67 —

(30) (30) (20) (20) (20) (20) (20) (20) (20)

a gradual decrease in magnitude from 1 to 40 eV. In this Fgure, the data of Mojarrabi is also compared to the BCSVM and SEP calculations. The experimental data is uniformly lower in magnitude than the BCSVM theory, as again might be expected from the previous comparison at the DCS level, but both display the same energy dependence. On the other hand, for energies less than 30 eV the energy dependence of the SEP elastic momentum transfer cross-section is quite di@erent from that of both Mojarrabi et al. and Lee et al. It also signiFcantly overestimates the magnitude of the elastic momentum transfer cross-section at energies less than 30 eV, although thereafter it is in good accord with the BCSVM result. As the momentum transfer cross-section is a measure, in part, of the probability of backward angle scattering, we can only presume that for energies less than 30 eV the SEP result is indicative of this model failing quite severely in predicting the large angle scattering behaviour of the elastic DCS. 3.6.3. Rovibrational excitation 3.6.3.1. Di:erential cross-sections. Experimental di@erential cross-sections for electron impact excitation of the v = 0 → 1; 0 → 2; 0 → 3 and 0→4 vibrationally inelastic transitions in the NO ground state were reported by Mojarrabi et al. [338] These data are given in Tables 83–86. For completeness we also note the absolute v = 0 → 3 excitation function measurement, taken ◦ at 65 , of Malegat et al. [328]. An interesting and quite informative way of viewing the rovibrational data is depicted in ◦ Fig. 79, where we look at the energy dependence of the v = 0 → 1 transition at 10 . Two points are immediately apparent from this Fgure. Firstly, there is a broad structure in the experimental rovibrational cross-section, centred at about 15 eV, and secondly that the BCSVM theory of Lee et al. [355] whilst also predicting a broad structure, does not predict the same energy at which the peak occurs. As we have previously discussed the evidence for the existence of resonance phenomena in the 12–20 eV energy range we do not do so again here, except to recall that we believe the structure in Fig. 79 is associated with a shape resonance of the NO negative ion, consistent with the interpretation of Malegat et al. Note that the structure in Fig. 79 was

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405

•

Fig. 77. Integral cross-section for elastic electron scattering from NO. ( ) Mojarrabi et al., ( — ) BCSVM calculation of Lee et al., (- - -) SEP calculation of Jain et al.

•

Fig. 78. Elastic momentum transfer cross-section for electron scattering from NO. ( ) Mojarrabi et al., (—) BCSVM calculation of Lee et al., (- - -) SEP calculation of Jain et al.

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Table 83 Di@erential cross-sections for rovibrational (0 –1) excitation of NO (10−18 cm2 sr −1 ). The data are those of Mojarrabi et al. (Figures in the last row indicate the percentage uncertainty.) Energy (eV) Angle

7.5

10.0

15.0

20.0

30.0

40.0

10 20 30 40 50 60 70 80 90

— 0.755 0.457 0.320 0.221 0.214 0.169 0.119 0.089 17%

2.277 0.689 0.427 0.343 0.324 0.378 0.356 0.350 0.365 15.3%

3.673 2.541 2.168 1.590 1.685 1.766 1.713 1.496 1.430 16%

2.017 1.337 0.895 0.725 0.672 0.649 0.659 0.540 0.477 16.5%

0.758 0.292 0.233 0.128 0.132 0.095 0.110 0.104 0.103 17%

1.224 0.238 0.116 0.108 0.100 0.085 0.077 0.053 0.043 18.7%

Table 84 Di@erential cross-sections for rovibrational (0 –2) excitation of NO (10−18 cm2 sr −1 ). The data are those of Mojarrabi et al. (Figures in the last row indicate the percentage uncertainty.) Energy (eV) Angle

10.0

15.0

20.0

10 20 30 40 50 60 70 80 90

0.086 0.063 0.062 0.064 0.059 0.075 0.074 0.081 0.085 16.5%

0.689 0.568 0.520 0.427 0.495 0.509 0.503 0.437 0.421 18%

0.408 0.352 0.240 0.224 0.191 0.180 0.166 0.132 0.096 18%

also observed at all other angles for the v = 0 → 1 transition and, similarly for the v = 0 → 2 transition. In the latter case, the experimental cross-section was always larger in magnitude than that predicted by the theory, indicating that the theory is not accurately reproducing the correct branching ratio for these processes. Further note that the lack of agreement between the BCSVM calculation and the data of Mojarrabi et al., with respect to the position of the structure, is likely to be due to the calculation not including a suJcient number of channels. This result has important ramiFcations for the level of agreement between theory and experiment for the angular dependence of the DCS, as we now discuss. Representative examples of the DCS data of Mojarrabi et al. and the results of the BCSVM calculation of Lee et al. are illustrated in Figs. 80 –84. The 0–1 rovibrational DCS of Mojarrabi

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407

Table 85 Di@erential cross-sections for rovibrational (0 –3) excitation of NO (10−18 cm2 sr −1 ). The data are those of Mojarrabi et al. (Figures in the last row indicate the percentage uncertainty.) Energy (eV) Angle

10.0

15.0

20.0

10 20 30 40 50 60 70 80 90

0.051 0.046 0.040 0.036 0.031 0.034 0.033 0.030 0.034 20%

0.390 0.219 0.207 0.165 0.189 0.201 0.198 0.168 0.153 21%

0.143 0.145 0.095 0.080 0.076 0.071 0.058 0.045 0.035 23%

Table 86 Di@erential cross-sections for rovibrational (0 – 4) excitation of NO (10−18 cm2 sr −1 ). The data are those of Mojarrabi et al. (Figures in the last row indicate the percentage uncertainty.) Energy (eV) Angle

15.0

20.0

10 20 30 40 50 60 70 80 90

0.154 0.107 0.084 0.080 0.085 0.090 0.089 0.072 0.065 25%

0.067 0.068 0.052 0.036 0.029 0.028 0.025 0.018 0.017 27%

et al. at 7:5 eV is given in Fig. 80. The level of agreement, in terms of both magnitude and shape, between theory and experiment is fair across the entire common angular range, the theory in general slightly overestimating the strength of the 0–1 excitation process. The data of Mojarrabi et al. at this energy also indicates the possible existence of a shallow structure in the ◦ ◦ range between 40 and 70 , which the theory fails to predict. In Fig. 81 we illustrate the experimental and theoretical 0–1 and 0–2 rovibrational DCSs, as measured at an energy of 10 eV. For the 0 → 1 cross-section, the data of Mojarrabi ◦ et al. are in very good agreement with the theory for angles less than 50 but thereafter the theory somewhat underestimates the magnitude of the cross-section. On the other hand, the level of agreement between theory and experiment for the 0–2 excitation is generally poor

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◦

Fig. 79. Energy dependence of the rovibrational (0 –1) excitation cross-section for NO at a scattering angle of 10 . ( ) Mojarrabi et al., ( — ) BCSVM calculation of Lee et al.

•

at all angles. The theory again underestimates the magnitude of the DCS and, in this case, does not reproduce the shape of the experimental DCS which is essentially isotropic. At 15 eV (Fig. 82) the BCSVM calculation underestimates the magnitude of both the 0–1 and 0–2 experimental DCSs across the measured angular range, although agreement in terms of the shape is fair. This may be due to the fact that the maximum of the resonance enhancement e@ect on both the 0–1 and 0–2 cross-sections is found to be at about 15 eV in the experiment, whilst the theory predicts it to occur at a somewhat higher energy (∼ 20 eV). Consequently, it is not surprising to Fnd that at 20 eV (Fig. 83) the BCSVM calculation somewhat overestimates the magnitude of the experimental 0–1 DCS. However, the overall level of agreement between theory and experiment is still fair. For the 0–2 transition at 20 eV (again see Fig. 83) there is also a reasonable level of agreement between theory and experiment for the shape of the angular distribution, although the theory seriously underestimates the magnitude of the DCS. This is consistent with what we saw previously at both 10 and 15 eV and establishes the clear trend that the calculation does not reproduce the correct branching ratio between the 0–2 and 0–1 transition at the energies studied by Mojarrabi et al. In Fig. 84, the 0–1 DCS of Mojarrabi et al. for an energy of 30 eV is plotted with the result of the BCSVM calculation. In this case, the theory signiFcantly overestimates the magnitude of the cross-section although, as before, the shapes of the experimental and theoretical angular distributions are in quite good accord. As there is no theory available at 40 eV we do not plot the experimental cross-section. Like the data at 15, 20 and 30 eV, it is strongly peaked in the forward direction, rePecting the e@ect of the permanent dipole moment of the NO molecule.

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•

Fig. 80. Di@erential cross-section for rovibrational (0 –1) excitation in NO at 7:5 eV. ( ) Mojarrabi et al., ( — ) BCSVM calculation of Lee et al.

•

Fig. 81. Di@erential cross-section for rovibrational (0 –1 and 0 –2) excitation in NO at 10 eV. ( ) 0 –1 and ( ) 0 –2 data of Mojarrabi et al., ( — ) 0 –1 and (- - -) 0 –2 BCSVM calculations of Lee et al.

3.6.3.2. Integral cross-sections. The only experimental integral rovibrational cross-sections available in the literature are those derived from the DCS measurements of Mojarrabi et al., ◦ ◦ using the BCSVM theory of Lee et al. as a guide to extrapolate to 0 and 180 . Due to the quite large uncertainties associated with their extrapolation procedure Mojarrabi et al. estimated that the errors on their 0–1 and 0–2 integral cross-sections varied from 25% to 30%. These data are given in Table 87. Theoretical ICS for the 0–1 and 0–2 transitions have been calculated within the BCSVM-AN framework by Lee et al. 3.6.4. Electronic-state excitation 3.6.4.1. Di:erential cross-sections. The spectral properties of the NO ground state and excited states have been the subject [363] of experimental study since 1926, but it is only in the past 15 years that the high-lying excited states have become accessible for study through the use of multi-photon techniques. However, in spite of the application of new techniques, there is still considerable uncertainty in the detailed shapes of the Rydberg and valence potential energy curves, and the location of the excited state vibrational levels. This is particularly true for states with excitation energies above about 7:5 eV, because of strong Rydberg-valence perturbations.

410

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•

Fig. 82. Di@erential cross-section for rovibrational (0 –1 and 0 –2) excitation in NO at 15 eV. ( ) 0 –1 and ( ) 0 –2 data of Mojarrabi et al., ( – ) 0 –1 and (- - -) 0 –2 BCSVM calculations of Lee et al.

•

Fig. 83. Di@erential cross-section for rovibrational (0 –1 and 0 –2) excitation in NO at 20 eV. ( ) 0 –1 and ( ) 0 –2 data of Mojarrabi et al., ( – ) 0 –1 and (- - -) 0−2 BCSVM calculations of Lee et al.

This is a very important point (see Section 2.7 for a full discussion) when it comes to obtaining accurate excitation cross-sections by deconvolving partially resolved electron energy loss spectra. The most important spectral parameters are the excited state vibrational energy levels and the Franck–Condon factors [364] relative to the v = 0 vibrational sub-level of the X2 Vr NO ground electronic state. These vibrational energy levels, and associated Franck–Condon factors, for all the excited electronic states of NO were only recently determined by Campbell et al. [121] by applying well-developed computational methods [365] to the best available spectroscopic data. Consequently, attempts to deconvolute the highly overlapping electronic states of NO (see Fig. 85) have only recently been made by Middleton [348], Mojarrabi et al. [349], and very recently by Brunger et al. [350,351]. The analysis of Middleton concentrated on electronic-state processes at energy loss values less than about 7:5 eV. Furthermore, this work also assumed that the cross-sections for the a4 V and B2 V states in this energy loss range were so small that they could be neglected in the deconvolution procedure. Mojarrabi et al. extended this analysis signiFcantly by including in their spectral deconvolution all possible electronic states in the energy loss range between 4.5 and 10:5 eV and as such it superseded the work of Middleton. This later study showed that the

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411

•

Fig. 84. Di@erential cross-section for rovibrational (0 –1) excitation in NO at 30 eV. ( ) Mojarrabi et al., ( — ) BCSVM calculation of Lee et al. Table 87 Integral rovibrational cross-sections (10−16 cm2 ) for electron scattering from NO. The data are those of Mojarrabi et al. (Figures in brackets indicate the percentage uncertainty.) Energy (eV)

0→1

7.5 10 15 20 30 40

0.028 0.074 0.270 0.097 0.022 0.014

0→2 (25) (25) (25) (25) (25) (25)

— 0.014 (30) 0.073 (30) 0.022 (30) — —

assumption used by Middleton of omitting in his analysis the a4 V state, in the energy range 15–50 eV, was basically sound. On the other hand, Mojarrabi et al. clearly demonstrated the necessity of including the B2 V electronic state in the analysis in this energy regime. The other major advance in the deconvolution procedure of Mojarrabi et al. over that of Middleton is that (see Section 2.7 for further details) they have incorporated a numerically valid procedure for determining the errors on the Ftted parameters, i.e. in essence the errors on the electronic-state di@erential cross-sections. Typical examples of the quality of the Ft to their data, and the

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Fig. 85. Potential energy curves for NO− , NO and NO+ .

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413

Fig. 86. Fitted spectra and spectral deconvolution for the energy loss spectrum of Mojarrabi et al. at an incident ◦ energy of 30 eV and a scattering angle of 10 .

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Fig. 87. Fitted spectra and spectral deconvolution for the energy loss spectrum of Mojarrabi et al. at an incident ◦ energy of 30 eV and a scattering angle of 50 .

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415

Fig. 88. Fitted spectra and spectral deconvolution for the energy loss spectrum of Mojarrabi et al. at an incident ◦ energy of 30 eV and a scattering angle of 90 .

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spectral components they found from the deconvolution process, are given in Figs. 86 –88. ◦ ◦ ◦ These examples are for an energy of 30 eV, at electron scattering angles of 10 , 50 and 90 . It is obvious from these Fgures that the Fts to the measured energy loss data are not perfect, although in most cases this is largely due to the assumption of Mojarrabi et al. that the line proFle is of pure Gaussian form, which is in fact not necessarily the case. In a further, considerable extension to the work of Mojarrabi et al. a series of di@erential cross-section measurements for electron impact excitation of 18 excited electronic states and 4 composite excited electronic states in NO was reported by Brunger et al. [350]. The incident electron energies in this study were 15, 20, 30, 40 and 50 eV and the scattered electron angular ◦ range was 10–90 . Tabulated values of their measured DCS for the A2 W+ , E2 W+ , S2 W+ , C2 Vr , 2 2 2 K V, Q V, D W+ , M2 W+ , H2 V, H2 W+ , F2 U, a4 V, b4 W− , B2 V, L2 a, B2 U, L2 V, O2 V + O2 W+ , W2 V + Y2 W+ , T2 W+ + U2 U + 5f and Z2 W+ + 6d + 6f electronic states are given in Tables 88–91. This work also employed the spectral deconvolution technique which has been described fully by Campbell et al. [121]. According to Brunger et al. [350] these latest values supersede the previous cross-sections of both Middleton and Mojarrabi et al., although there is little di@erence between their cross-sections and the results for the A2 W+ , b4 W− , C2 V and D2 W+ states which were covered in the earlier work. Using a variety of techniques, including a molecular phase shift analysis (MPSA) procedure, ◦ ◦ Brunger et al. [351] extrapolated their measured DCS to 0 and 180 and derived the corresponding ICS for each excited electronic state at each energy they studied. These ICS are listed in Table 93. Machado et al. reported DCS and ICS for the X2 V → A2 W+ , C2 V and D2 W+ electronic-state transitions in NO for six energies in the energy range 12:5–50 eV. In this work they found it convenient to represent the e− -molecule scattering DCS in the jt -basis [366] and the SCF wavefunction of NO was constructed from a (6s4p1d) contracted Gaussian basis for both the N and O atomic centres. With this basis set, the resulting SCF energy was −129:01 a:u:, to be compared with the near-Hartree–Fock energy [367] of −129:2953 a:u. Their resulting excitation energies were 5.403, 6.383 and 6.400 eV, for the X2 Vr → A2 W+ , C2 V and D2 W+ transitions, as compared to the corresponding experimental values of 5.484, 6.499 and 6:617 eV [349]. The scattering wavefunctions and the reactance K-matrices were obtained using the SVIM [368]. Machado et al. limited the partial-wave expansion of the scattering functions at lmax = 10 and Mmax = 2. In addition, all matrix elements appearing in those calculations were computed by a single-centre expansion technique. These expansions were truncated at lmax = 20. For the radial integrations they employed a grid of 900 points which extended to 78:25a0 with the step sizes varying from 0.005 to 0:24a0 . We now speciFcally discuss the results for the A2 W+ , b4 W− , C2 V and D2 W+ electronically excited states in more detail, as in most of these cases experiment and theory can be compared. A2 /+ state. The experimental DCS of Brunger et al. [350] for the electron impact excitation of the A2 W+ state at energies of 15, 20, 30, 40 and 50 eV are given in Table 88(a). A selection of these data are also plotted in Figs. 89(a) – (d) where they are compared against the results of the DW calculations of Machado et al. The experimental DCS for the A2 W+ electronic state are all forward peaked with the degree of forward peaking apparently increasing with increasing energy. The theoretical DCS are also all peaked in the forward direction and, in general, are in quite good accord with the shape

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417

Table 88 Di@erential cross-sections (10−19 cm2 =sr) for the electron impact excitation of NO. The data are those of Brunger et al. Numbers in parentheses are the absolute uncertainties on the data Energy (eV) Angle

15

20

30

(1.1) (0.56) (0.97) (0.2) (0.16) (0.17) (0.17) (0.17) (0.19)

19.3 (3.7) 5.15 (1.05) 1.90 (0.49) 1.13 (0.32) 0.76 (0.23) 0.76 (0.17) 0.65 (0.15) 0.56 (0.20) 0.54 (0.13)

31.1 (4.5) 9.38 (1.55) 2.72 (0.46) 1.36 (0.33) 1.10 (0.32) 0.73 (0.23) 0.53 (0.19) 0.38 (0.16) 0.46 (0.17)

40.6 (5.3) 6.99 (1.08) 1.85 (0.43) 0.81 (0.22) 0.50 (0.10) 0.41 (0.14) 0.29 (0.12) 0.23 (0.15) 0.20 (0.12)

20.1 (2.8) 4.63 (0.94) 1.44 (0.43) 0.75 (0.38) 0.39 (0.32) 0.20 (0.19) 0.12 (0.11) 0.17 (0.14) 0.14 (0.21)

(b) X2 Vr → C2 Vr 10 21.6 (4.1) 20 8.38 (1.21) 30 3.24 (0.59) 40 1.52 (0.28) 50 1.06 (0.21) 60 0.97 (0.21) 70 0.96 (0.25) 80 1.01 (0.25) 90 1.08 (0.29)

60.0 (10.9) 10.1 (2.05) 2.38 (0.67) 1.74 (0.48) 1.22 (0.30) 1.08 (0.25) 0.96 (0.23) 0.76 (0.30) 0.74 (0.21)

80.3 (11.5) 14.1 (2.5) 3.69 (0.73) 2.11 (0.53) 1.91 (0.48) 1.45 (0.40) 0.80 (0.30) 0.61 (0.25) 0.84 (0.28)

130.7 (16.8) 9.65 (1.65) 3.02 (0.67) 1.65 (0.37) 1.01 (0.19) 0.68 (0.21) 0.53 (0.19) 0.39 (0.22) 0.34 (0.18)

43.1 (5.8) 5.43 (1.03) 2.71 (0.65) 1.38 (0.45) 0.88 (0.50) 0.65 (0.32) 0.37 (0.21) 0.40 (0.36) 0.30 (0.22)

(c) X2 Vr → D2 W+ 10 14.0 (2.65) 20 5.33 (0.79) 30 2.43 (0.49) 40 1.18 (0.24) 50 0.83 (0.19) 60 0.68 (0.17) 70 0.82 (0.22) 80 0.78 (0.21) 90 0.88 (0.19)

38.3 (7.04) 6.60 (1.30) 2.07 (0.57) 1.23 (0.33) 0.83 (0.36) 0.57 (0.18) 0.61 (0.18) 0.51 (0.26) 0.55 (0.18)

55.4 (8.1) 9.70 (1.9) 2.95 (0.63) 1.46 (0.42) 0.98 (0.32) 0.69 (0.29) 0.42 (0.23) 0.45 (0.22) 0.62 (0.25)

93.3 (12.4) 6.27 (1.2) 1.84 (0.51) 0.91 (0.27) 0.43 (0.13) 0.29 (0.16) 0.21 (0.15) 0.27 (0.20) 0.22 (0.16)

28.9 (4.1) 4.12 (0.88) 1.52 (0.48) 0.74 (0.47) 0.40 (0.26) 0.34 (0.22) 0.20 (0.29) 0.32 (0.32) 0.25 (0.21)

(a) X2 Vr → A2 W+ 10 9.40 20 3.85 30 1.96 40 1.10 50 0.77 60 0.78 70 0.86 80 0.75 90 0.74

(d) X2 Vr → a4 V 20 0.13 30 0.30 40 0.35 50 0.68 60 0.72 70 1.03 80 0.66 90 0.71

(0.16) (0.23) (0.37) (0.31) (0.35) (0.44) (0.41) (0.54)

0.13 0.23 0.45 0.67 0.60 0.58 0.64 0.72

(0.10) (0.41) (0.54) (0.61) (0.31) (0.33) (0.55) (0.36)

— 0.15 0.20 0.53 0.59 0.78 1.29 1.24

40

(0.33) (0.29) (0.46) (0.55) (0.53) (0.54) (0.59)

— — — 0.09 0.45 0.63 0.78 0.86

50

(0.25) (0.26) (0.39) (0.43) (0.39)

— 0.39 0.71 0.68 0.74 1.46 1.18 1.59

(0.84) (0.56) (0.28) (0.42) (0.71) (0.10) (1.42)

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Table 88 (Continued) Energy (eV) Angle 2

(e) X Vr → b 20 30 40 50 60 70 80 90

15 4

20

30

40

50

−

W 0.45 0.51 1.27 1.43 1.88 2.37 2.72 3.53

(0.34) (0.23) (0.27) (0.27) (0.32) (0.39) (0.48) (0.57)

0.28 0.37 0.93 1.23 1.08 1.45 1.70 2.20

(0.16) (0.40) (0.36) (0.27) (0.26) (0.34) (0.43) (0.39)

(f) X2 Vr → L2 a 10 — 20 1.47 30 1.34 40 0.88 50 1.32 60 1.90 70 2.12 80 2.72 90 2.45

(0.68) (0.90) (1.15) (0.55) (1.03) (2.6) (1.74) (1.0)

9.17 3.40 1.65 1.13 0.46 0.15 0.06 — 0.11

(2.6) (2.6) (1.4) (0.97) (0.91) (0.32) (0.07) (0.45)

0.34 0.90 1.37 1.62 1.94 1.96 2.61 2.92

(0.47) (0.39) (0.40) (0.43) (0.45) (0.48) (0.53) (0.58)

19.4 (5.6) 6.26 (4.4) 5.05 (1.5) 1.41 (0.97) 0.88 (0.78) 0.25 (0.89) 0.20 (0.50) 0.17 (0.24) 0.38 (0.24)

0.11 0.22 0.43 0.55 0.83 1.06 0.99 1.22

(0.11) (0.20) (0.25) (0.13) (0.22) (0.24) (0.31) (0.30)

28.4 (8.8) 8.29 (2.8) 4.41 (1.52) 2.04 (0.89) 0.60 (0.35) 0.64 (0.44) 0.54 (0.40) 0.64 (0.71) 0.51 (0.49)

0.47 0.59 1.09 0.10 0.88 0.59 1.02 0.98

(0.29) (0.23) (0.28) (0.56) (0.52) (0.22) (0.63) (0.60)

23.7 (4.4) 10.1 (2.6) 4.27 (2.1) 3.68 (1.48) 2.39 (1.63) 1.07 (0.74) 1.26 (0.94) 1.69 (2.05) 1.24 (1.29)

of the experimental cross-sections. At each energy considered in Fig. 89 the magnitude of the theoretical DCS was signiFcantly larger than that found by the experiment. This is not an unexpected result, given the DW formalism of the calculation. As anticipated, we found that the level of scaling needed to bring the theory into accord with the experiment decreased with increasing energy. b4 /− state. The b4 W− state is interesting because of the four NO electronic states we consider in this section it is the only one which is non-Rydberg in nature. Indeed, this valence state in fact arises from the excitation of one of the closed-shell electrons into the antibonding ∗ orbital. The experimental DCS of Brunger et al. [350] for this state, again at 15, 20, 30, 40 and 50 eV, are given in Table 88(e). In this case, however, there is unfortunately no theory against which we can compare these experimental results and consequently no plots are given for it. Nevertheless, it is clear that unlike the A2 W+ state, the DCS are anything but forward peaked and it is also apparent from Table 88(e) that the magnitude of the b4 W− DCS is always larger ◦ ◦ at 90 than at 10 . This is perhaps not surprising given the 2 V →4W− nature of the excitation process, which technically makes it a “spin-forbidden” transition. From a spectroscopic point of view an understanding of the population mechanisms for the b4 W− state is important. This is particularly true with respect to the quartet–quartet b4 W− → a4 V transition, as this gives rise to the well-known Ogawa 2 bands. C 2 > state. Experimental DCS for the electron impact excitation of the C2 V state are given in Table 88(b). A selection of these data are also plotted in Figs. 90(a) – (d) where they are compared against the results of the DW calculation of Machado et al.

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419

Table 89 Di@erential cross-sections (10−19 cm2 =sr) for electron impact excitation of NO. The data are those of Brunger et al. Numbers in parentheses are the absolute uncertainties on the data Energy (eV) Angle 2

15

20

30

40

50

2

(a) X Vr → B V 10 20 1.54 30 0.99 40 0.38 50 0.65 60 0.86 70 1.42 80 1.58 90 1.62

(0.4) (0.59) (0.34) (0.28) (0.27) (0.36) (0.36) (0.42)

(b) X2 Vr → B2 U 10 6.83 20 5.40 30 4.92 40 4.08 50 2.95 60 1.89 70 1.83 80 1.41 90 1.86 (c) X2 Vr → L2 V 10 8.82 20 5.72 30 4.63 40 2.91 50 2.52 60 1.87 70 1.44 80 1.76 90 1.79 (d) X2 Vr → E2 W+ 10 1.82 20 0.75 30 0.11 40 50 0.06 60 0.13 70 0.19 80 0.20 90 0.18

(1.92) (0.59) (0.30) (0.55) (0.27) (0.22) (0.28) (0.30)

18.7 (3.7) 6.83 (2.18) 2.41 (0.87) 0.96 (0.43) 0.31 (0.50) 0.68 (0.49) 0.77 (0.42) 0.88 (0.39) 1.14 (0.49)

(1.89) (1.30) (0.88) (0.69) (0.47) (0.36) (0.39) (0.35) (0.45)

25.8 (5.05) 12.9 (2.91) 7.66 (1.53) 5.21 (1.13) 2.95 (0.66) 1.64 (0.40) 1.26 (0.33) 1.39 (0.45) 1.97 (0.40)

32.7 (5.5) 28.4 (4.89) 13.7 (2.16) 6.41 (1.26) 3.58 (0.85) 3.25 (0.75) 2.61 (0.68) 3.04 (0.82) 5.29 (1.0)

48.2 (8.42) 21.6 (3.59) 9.06 (1.63) 3.55 (0.74) 1.38 (0.30) 1.11 (0.35) 1.75 (0.40) 2.03 (0.53) 1.82 (0.47)

(8.21) (1.70) (1.84) (1.33) (1.09) (1.90) (1.21) (1.05) (2.07)

40.6 (10.1) 15.0 (7.1) 6.53 (3.8) 3.51 (2.6) 2.93 (2.3) 1.13 (1.14) 1.49 (1.0) 0.98 (1.5) 1.86 (0.99)

136 (22.4) 52.6 (14.2) 17.5 (4.7) 7.73 (3.9) 6.08 (2.23) 5.53 (2.3) 2.47 (1.82) 2.08 (1.7) 3.17 (1.97)

297 (46) 47.7 (11.2) 14.5 (4.9) 10.8 (2.8) 5.46 (1.24) 2.98 (1.21) 2.70 (1.28) 1.92 (1.69) 1.98 (1.37)

(0.82) (0.26) (0.29) (0.26) (0.22) (0.14) (0.2) (0.10)

2.41 1.16 0.66 0.26 0.48 0.76 0.67 0.98

3.23 0.68 0.10 0.07

(1.3) (1.04) (0.1) (0.05)

0.10 0.14 0.13 0.16

(0.26) (0.15) (0.24) (0.13)

5.34 0.90 0.33 0.16 0.11 0.04 0.12 0.12

(1.74) (0.89) (0.41) (0.37) (0.41) (0.22) (0.16) (0.15)

0.20 0.30 0.52 0.67 0.66 0.59

(0.44) (0.16) (0.22) (0.27) (0.39) (0.31)

7.55 0.94 0.45 0.15

(3.38) (0.80) (0.64) (0.26)

0.17 0.22 0.15 0.17

(0.13) (0.16) (0.13) (0.12)

0.86 0.99 1.93 1.99 1.59 1.61 1.49 1.26

(0.75) (0.44) (0.67) (0.83) (0.59) (0.53) (0.96) (0.96)

24.6 (4.2) 15.6 (2.7) 5.56 (1.29) 1.47 (0.42) 0.57 (0.36) 0.53 (0.50) 0.63 (0.58) 0.73 (0.85) 0.73 (0.49) 199 (28) 48.2 (9.5) 27.1 (6.1) 13.6 (3.5) 6.62 (3.8) 3.33 (1.5) 1.89 (2.2) 4.51 (2.9) 3.17 (3.0) 3.86 (1.87) 0.99 (0.74) 0.20 (0.34) 0.33 (0.40) 0.06 (0.32) 0.27 (0.38)

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Table 89 (Continued) Energy (eV) Angle 2

(e) X Vr → S 10 20 30 40 50 60 70 80 90

15 2

20

30

40

50

+

W 2.01 0.83 0.71 0.51 0.34 0.30 0.33 0.24 0.27

(0.66) (0.22) (0.25) (0.19) (0.14) (0.13) (0.15) (0.10) (0.18)

4.75 2.34 1.32 0.93 0.45 0.30 0.23 0.15 0.19

(1.19) (0.95) (0.57) (0.37) (0.24) (0.22) (0.15) (0.09) (0.12)

(f) X2 Vr → K 2 V 10 2.87 20 1.73 30 0.78 40 0.40 50 0.33 60 0.30 70 0.35 80 0.37 90 0.44

(0.94) (0.36) (0.34) (0.25) (0.19) (0.17) (0.20) (0.19) (0.25)

9.92 3.02 1.07 0.79 0.47 0.26 0.23 0.28 0.21

(3.03) (1.29) (0.73) (0.47) (0.31) (0.18) (0.17) (0.20) (0.14)

7.87 5.34 2.81 1.30 0.58 0.48 0.42 0.54 0.75

(1.77) (1.48) (0.64) (0.40) (0.28) (0.27) (0.19) (0.21) (0.27)

13.6 (4.2) 4.79 (1.29) 2.14 (0.67) 0.68 (0.37) 0.22 (0.12) 0.26 (0.15) 0.31 (0.16) 0.36 (0.22) 0.23 (0.16)

21.9 (3.75) 5.77 (1.97) 2.43 (0.76) 0.98 (0.49) 0.51 (0.34) 0.46 (0.34) 0.42 (0.29) 0.39 (0.28) 0.49 (0.33)

39.9 (6.84) 6.51 (1.78) 2.57 (0.81) 0.73 (0.38) 0.35 (0.16) 0.26 (0.21) 0.26 (0.20) 0.21 (0.16) 0.24 (0.19)

6.18 2.86 0.98 0.42 0.20 0.12 0.18 0.23 0.21

(2.34) (0.95) (0.57) (0.31) (0.20) (0.13) (0.16) (0.19) (0.24)

18.7 (3.42) 3.84 (1.2) 1.91 (0.72) 0.69 (0.49) 0.40 (0.38) 0.11 (0.23) 0.14 (0.20) 0.20 (0.24)

Similar to the situation for the A2 W+ electronic-state, the experimental DCS for the C2 V state are all forward peaked with the degree of forward scattering appearing to increase with increasing energy. Again, the theoretical DCS are also all peaked in the forward direction and, in general, are in good accord with the shape of the experimental cross-sections. This is speciFcally illustrated in Figs. 90(b) – (d) where the DW results are clearly much larger in magnitude than the experiment. Note that as the energy is increased this di@erence in magnitude is seen to decrease, as one would hope with a DW-based calculation. Also note that, compared to the A2 W+ state, the di@erences in magnitude between experiment and theory at each energy for the C2 V state are always smaller, indicating that within their DW model the calculation of Machado et al. gave a better representation for the 2 V →2V excitation process than the 2 V →2 W+ excitation process. D2 /+ state. Our comments made earlier for the A2 W+ state are, in general, equally applicable for the electron impact excitation of the D2 W+ electronic-state of NO. The experimental data of Brunger et al. are compared to the theoretical DW results of Machado et al. in Figs. 91(a) – (d) and are tabulated in Table 88(c). At the risk of being repetitive, Figs. 91(a) – (d) illustrates two major points: (i) the theory overestimates the magnitude of the DCS across the entire angular range with the extent of this discrepancy decreasing with increasing beam energy. (ii) the shapes of the DW-DCSs are in good accord with the experimental measurements.

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421

Table 90 Di@erential cross-sections (10−19 cm2 =sr) for electron impact excitation of NO. The data are those of Brunger et al. Numbers in parentheses are the absolute uncertainties on the data Energy (eV) Angle 2

(a) X Vr → Q 10 20 30 40 50 60 70 80 90

15 2

20

30

V 1.91 1.10 0.92 0.54 0.25 0.28 0.29 0.35 0.24

(0.79) (0.29) (0.36) (0.22) (0.15) (0.14) (0.18) (0.21) (0.22)

4.47 2.71 1.40 0.95 0.46 0.20 0.17 0.18 0.20

(1.24) (1.06) (0.58) (0.34) (0.35) (0.15) (0.14) (0.20) (0.15)

(b) X2 Vr → M2 W+ 10 1.30 20 0.86 30 0.78 40 0.51 50 0.37 60 0.28 70 0.21 80 0.19 90 0.36

(0.96) (0.34) (0.40) (0.27) (0.20) (0.18) (0.22) (0.20) (0.27)

5.37 2.55 1.37 0.93 0.41 0.21 0.23 0.30 0.23

(1.88) (1.35) (0.86) (0.55) (0.36) (0.17) (0.17) (0.28) (0.20)

(c) X2 Vr → H2 V 10 2.20 20 1.05 30 0.62 40 0.31 50 0.34 60 0.26 70 0.14 80 0.17 90 0.25

(1.1) (0.32) (0.42) (0.28) (0.18) (0.22) (0.30) (0.18) (0.25)

3.07 1.54 0.70 0.26 0.13 0.17 0.16 0.15 0.13

(1.45) (0.90) (0.65) (0.46) (0.32) (0.17) (0.15) (0.17) (0.16)

(d) X2 Vr → H2 W+ 10 2.13 20 0.79 30 0.56 40 0.29 50 0.30 60 0.39 70 0.50 80 0.52 90 0.51

(0.97) (0.33) (0.45) (0.27) (0.26) (0.18) (0.22) (0.21) (0.27)

8.20 2.58 0.91 0.89 0.61 0.40 0.40 0.33 0.42

(2.2) (1.40) (0.82) (0.53) (0.48) (0.22) (0.18) (0.28) (0.21)

9.56 6.96 2.35 0.81 0.39 0.29 0.31 0.40 0.71

40 (2.2) (1.79) (0.64) (0.43) (0.30) (0.24) (0.19) (0.25) (0.29)

19.0 (4.9) 5.66 (1.49) 1.99 (0.86) 0.51 (0.35) 0.36 (0.12) 0.29 (0.23) 0.22 (0.19) 0.25 (0.24) 0.34 (0.20)

10.6 (2.9) 5.70 (2.2) 1.95 (0.89) 1.09 (0.59) 0.49 (0.38) 0.39 (0.38) 0.34 (0.26) 0.34 (0.30) 0.76 (0.39)

15.5 (5.3) 4.70 (1.68) 2.22 (0.91) 0.51 (0.43) 0.27 (0.18) 0.25 (0.14) 0.24 (0.22) 0.29 (0.20) 0.38 (0.25)

9.35 2.91 1.09 0.40 0.25 0.16 0.15 0.18 0.37

(2.3) (1.71) (0.76) (0.51) (0.26) (0.38) (0.26) (0.17) (0.20)

13.1 (2.9) 5.20 (2.0) 2.26 (0.89) 1.07 (0.57) 0.92 (0.43) 0.62 (0.39) 0.66 (0.35) 0.56 (0.33) 0.43 (0.39)

50 8.82 3.71 1.34 0.26 0.29 0.12 0.27 0.23 0.33

(2.6) (1.10) (0.64) (0.23) (0.30) (0.19) (0.37) (0.33) (0.26)

10.9 (3.1) 2.95 (1.26) 0.80 (0.65) 0.37 (0.43) 0.27 (0.33) 0.14 (0.17) 0.16 (0.14) 0.19 (0.19) 0.17 (0.37)

10.6 (4.3) 2.57 (0.148) 1.46 (0.80) 0.47 (0.22) 0.23 (0.15) 0.30 (0.20) 0.23 (0.21) 0.20 (0.36) 0.23 (0.23)

7.41 1.99 1.68 0.69 0.20 0.21 0.12 0.10 0.07

(1.95) (1.03) (0.70) (0.49) (0.32) (0.33) (0.16) (0.38) (0.35)

24.4 (5.5) 4.75 (1.78) 2.00 (0.95) 0.91 (0.41) 0.52 (0.22) 0.38 (0.21) 0.33 (0.22) 0.22 (0.14) 0.24 (0.27)

8.47 2.40 0.65 0.81 0.53 0.27 0.34 0.45 0.53

(1.96) (1.14) (0.44) (0.85) (0.48) (0.22) (0.21) (0.45) (0.64)

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Table 90 (Continued) Energy (eV) Angle 2

(e) X Vr → F 10 20 30 40 50 60 70 80 90

15 2

U 3.47 0.79 0.66 0.24 0.16 0.16 0.17 0.30 0.33

(f) X2 Vr → N2 U 10 2.67 20 1.12 30 0.83 40 0.63 50 0.29 60 0.12 70 80 0.05 90 0.05

20 (1.12) (0.38) (0.46) (0.27) (0.22) (0.18) (0.21) (0.16) (0.24) (1.18) (0.34) (0.58) (0.64) (0.22) (0.39) (0.05) (0.04)

9.55 2.60 0.75 0.23 0.23 0.21 0.25 0.13 0.20

(2.3) (1.40) (0.49) (0.30) (0.23) (0.19) (0.18) (0.27) (0.10)

30

40

50

19.9 (3.7) 4.58 (2.0) 1.15 (0.80) 0.32 (0.53) 0.42 (0.37) 0.25 (0.27) 0.15 (0.17) 0.10 (0.42) 0.12 (0.10)

34.9 (6.2) 3.77 (1.69) 1.04 (0.80) 0.38 (0.39) 0.27 (0.17) 0.21 (0.26) 0.18 (0.16) 0.12 (0.21) 0.11 (0.09)

11.4 (2.9) 2.81 (1.08) 1.30 (0.68) 0.93 (0.53) 0.76 (0.61) 0.21 (0.32) 0.14 (0.17) 0.18 (0.17) 0.48 (0.40)

3.65 (2.4) 1.42 (1.35) 0.40 (0.80)

1.45 (1.57) 0.64 (1.17) 0.28 (0.17)

0.31 (0.36)

0.18 (0.35) 0.18 (0.30)

4.32 2.29 2.29 0.92 0.15 0.12 0.08 0.10 0.32

(3.0) (1.28) (0.87) (0.39) (0.93) (0.18) (0.31) (0.27) (0.54)

3.6.4.2. Integral cross-sections. Absolute integral cross-section measurements for electron impact excitation of the electronic states of NO have been limited to the optical study of Skubenich et al. [346] for the A2 W+ , B2 Vr , b4 W− , B2 U and F2 U states, and those derived from the DCS measurements of Mojarrabi et al. and Brunger et al. [351]. On the theoretical side the only calculations currently available in the literature are those of Machado et al. for the A2 W+ , C2 V and D2 W+ electronic states. The data of Skubenich et al. are given in Table 92, whilst the data of Brunger et al. are given in Table 93. In Fig. 92 we plot the ICS data for the electron impact excitation of the A2 W+ electronic state. The data of Brunger et al. is found to be in quite good accord with the optical measurements of Skubenich et al., both displaying a similar energy dependence for the ICS over the range of common energies. In addition both have similar absolute magnitudes. Skubenich et al. attempted to correct their data for cascade e@ects. However it is quite plausible, given the sparcity of information on the NO scattering system (particularly at the time of their study), that the di@erence in absolute magnitude between Skubenich et al. and Brunger et al. is due to an inexact cascade correction by Skubenich and co-workers. Nonetheless both experiments are clearly of the same order of magnitude, which is an interesting result as they indicate that the excitation cross-section for the A2 W+ state in NO is at least an order of magnitude smaller than the corresponding electronic state of N2 . The physical explanation for what leads to such a dramatic drop in the ICS, in going from a 15 electron hetero-nuclear diatomic molecule with a permanent dipole moment to a 14 electron homo-nuclear diatomic molecule with a closed-shell, is unclear at this time but it deserves further study. Given our previous discussion at the

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423

Table 91 Di@erential cross-sections (10−19 cm2 =sr) for electron impact excitation of NO. The data are those of Brunger et al. Numbers in parentheses are the absolute uncertainties on the data Energy (eV) Angle 2

(a) X Vr → O 10 20 30 40 50 60 70 80 90

15 

2

20 2

30

40

50

18.2 (4.5) 5.89 (1.61) 2.77 (0.83) 0.84 (0.38) 0.35 (0.16) 0.39 (0.23) 0.35 (0.20) 0.30 (0.26) 0.26 (0.21)

11.9 (2.3) 3.28 (1.15) 0.65 (0.55) 0.41 (0.41) 0.24 (0.36) 0.16 (0.20) 0.35 (0.27) 0.29 (0.30) 0.30 (0.27)

+

V+O W 2.76 (0.85) 1.25 (0.28) 0.93 (0.38) 0.59 (0.23) 0.29 (0.17) 0.32 (0.16) 0.35 (0.18) 0.24 (0.16) 0.29 (0.22)

6.20 2.82 1.42 1.11 0.46 0.25 0.24 0.16 0.20

(1.51) (1.19) (0.69) (0.48) (0.30) (0.16) (0.14) (0.17) (0.14)

10.9 (2.3) 6.56 (1.90) 2.82 (0.74) 1.47 (0.50) 0.83 (0.35) 0.56 (0.32) 0.46 (0.28) 0.59 (0.27) 0.76 (0.32)

(b) X2 Vr → W2 V + Y2 W+ 10 3.34 (1.63) 20 2.35 (0.74) 30 1.31 (0.65) 40 1.02 (0.49) 50 0.91 (0.38) 60 0.45 (0.33) 70 0.44 (0.40) 80 0.42 (0.56) 90 0.55 (0.50)

8.17 4.02 2.30 1.73 0.68 0.58 0.40 0.39 0.51

(2.7) (2.5) (1.48) (1.0) (0.64) (0.37) (0.31) (0.36) (0.41)

19.1 (4.8) 7.56 (3.7) 4.07 (1.56) 2.03 (1.04) 1.31 (0.74) 1.16 (0.68) 0.82 (0.61) 0.70 (0.57) 1.23 (0.68)

(c) X2 Vr → T2 W+ + U2 U + 5f 10 0.40 (0.33) 20 0.28 (0.11) 30 0.28 (0.15) 40 0.18 (0.10) 50 0.10 (0.07) 60 0.11 (0.07) 70 0.10 (0.8) 80 0.06 (0.07) 90 0.14 (0.09)

1.99 0.82 0.47 0.29 0.19 0.12 0.10 0.08 0.09

(0.58) (0.48) (0.29) (0.19) (0.12) (0.07) (0.07) (0.07) (0.07)

2.72 2.05 0.73 0.23 0.09 0.11 0.13 0.11 0.25

(d) X2 Vr → Z2 W+ + 6d + 6f 10 0.36 (0.24) 20 0.24 (0.08) 30 0.16 (0.11) 40 0.12 (0.06) 50 0.06 (0.05) 60 0.06 (0.05) 70 0.09 (0.06) 80 0.05 (0.05) 90 0.08 (0.08)

0.89 0.40 0.29 0.17 0.10 0.04 0.03 0.03 0.04

(0.39) (0.31) (0.20) (0.13) (0.12) (0.05) (0.04) (0.04) (0.03)

1.46 1.40 0.45 0.18 0.06 0.10 0.07 0.08 0.16

28.9 8.07 4.59 1.13 0.55 0.29 0.47 0.70 0.40

(11.5) (3.4) (3.4) (0.81) (0.35) (0.26) (0.41) (0.58) (0.46)

13.0 (5.2) 3.31 (2.3) 1.86 (0.54) 0.59 (0.88) 0.28 (0.11)

(0.83) (0.75) (0.30) (0.19) (0.08) (0.13) (0.11) (0.09) (0.13)

7.75 1.97 0.68 0.21 0.06 0.07 0.07 0.11 0.08

(2.4) (0.68) (0.32) (0.16) (0.06) (0.09) (0.06) (0.10) (0.09)

0.71 0.89 0.37 0.09 0.09 0.04 0.08 0.06 0.05

(0.68) (0.46) (0.29) (0.39) (0.13) (0.10) (0.24) (0.34) (0.13)

(0.69) (0.50) (0.20) (0.13) (0.06) (0.07) (0.08) (0.07) (0.10)

1.17 0.90 0.40 0.07 0.01 0.04 0.05 0.04 0.05

(1.23) (0.43) (0.22) (0.11) (0.06) (0.07) (0.06) (0.04) (0.03)

0.56 0.49 0.29 0.10

(0.68) (0.34) (0.21) (0.15)

0.08 0.10 0.06 0.08

(0.05) (0.19) (0.18) (0.14)

0.52 (0.45) 0.64 (0.68) 0.32 (0.89)

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Fig. 89. Di@erential cross-sections for electron impact excitation of the A2 W+ electronic state of NO at (a) 15 eV, (b) 20 eV, (c) 30 eV and (d) 40 eV. ( ) Brunger et al., (—) DW calculation of Machado et al.

•

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425

Fig. 90. Di@erential cross-sections for electron impact excitation of the C2 V electronic state of NO at (a) 15 eV, (b) 20 eV, (c) 30 eV and (d) 40 eV. ( ) Brunger et al., (—) DW calculation of Machado et al.

•

426

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Fig. 91. Di@erential cross-sections for electron impact excitation of the D2 W+ electronic state of NO at (a) 15 eV, (b) 20 eV, (c) 30 eV and (d) 40 eV. ( ) Brunger et al., (—) DW calculation of Machado et al.

•

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427

Table 92 Integral cross-sections (10−19 cm2 ) for electron impact excitation of the A2 W+ , B2 Vr , b4 W− , B2 U and F2 U states of NO. The data are those of Skubenich et al. Energy (eV)

A 2 W+

B 2 Vr

b4 W−

B 2 U

F2 U

6 7 7.5 8 9 9.6 9.7 10 11 12 13 14 15 15.3 16 17 17.5 20 25 30 40 50

3.75 13.4 — 21.3 23.5 23.6 — 23.0 21.6 20.8 20.8 21.6 22.5 — — — 22.0 23.1 26.2 28.0 33.0 38.6

0.5 1.5 — 2.3 2.4 — 2.5 2.4 2.3 2.2 2.5 3.05 3.2 — 3.1 — 3.0 3.2 5.25 5.6 6.7 7.8

4.0 26 35 25 20 — — 18.25 16.3 15.2 13.9 — — — — — — — — — — —

— — — 0.31 0.5 — — 0.625 0.85 1.05 1.21 1.35 1.43 1.46 — — 1.40 1.50 1.60 1.51 1.41 1.39

— — — 0.125 0.280 — — 0.37 0.38 0.43 0.52 0.60 0.625 — — 0.635 0.640 0.73 0.76 0.73 0.65 0.625

DCS level, it comes as no surprise to Fnd that the theoretical cross-section of Machado et al. signiFcantly overestimates the magnitude of the ICS over the energy range considered, although agreement with the experimental results is seen to improve as one goes to higher energies. For the b4 W− state the data of Brunger et al. and Skubenich et al. do not overlap, although the trend in terms of the magnitude of their ICS measurements is encouraging (see Fig. 93). In this case, the errors on the derived ICS data of Brunger et al. are ∼40–50%, signiFcantly larger than the ∼25% uncertainty on their A2 W+ , C2 V and D2 W+ integral cross-sections. This is simply a rePection of the fact that the ICS for the b4 W− state is particularly sensitive to the type of the extrapolation procedure adopted. In Fig. 94, where we plot the ICS for electron impact excitation of the C2 V electronic state, we are only able to compare the experimental ICS of Brunger et al. with the results of the DW calculation of Machado et al. Consistent with the behaviour observed at the DCS level, it is again immediately apparent that the theoretical ICS is substantially for the X2 V → C2 V excitation process than the experiment. Unlike the case for the A2 W+ state, here we Fnd that the shape of the theory and experimental ICS are in quite good accord. It is also worth noting at this time that the discrepancy between the theory and the experiment, in terms of the magnitude of the ICS, is less extreme here than we saw earlier for the A2 W+ state. Indeed on the basis of the

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Table 93 Integral cross-sections (10−19 cm2 ) for the electron impact excitation of the (a) Rydberg and (b) valence states of NO. The data are those of Brunger et al. Numbers in parentheses are the absolute uncertainties on the data (a) State

Energy (eV) 15

20

A W E 2 W+ S2 W+ C2 Vr K2 V Q2 V D 2 W+ M 2 W+  H 2V H 2 W+ F2 U N2 U  O 2 V + O2 W + W 2 V + Y 2 W+ T2 W+ + U2 U + 5f Z2 W+ + 6d + 6f

12.2 (4.3) 2.27 (1.47) 4.11 (2.3) 23.1 (6.2) 6.73 (2.6) 4.71 (2.0) 14.4 (5.0) 4.22 (1.8) 4.22 (2.4) 5.75 (3.3) 4.40 (2.0) 2.55 (3.0) 4.64 (2.1) 9.43 (3.4) 1.53 (0.99) 0.98 (0.68)

14.8 (4.9) 2.67 (1.68) 8.76 (4.6) 32.7 (8.2) 8.09 (2.9) 6.63 (2.7) 18.5 (6.1) 5.63 (2.3) 5.89 (3.4) 8.70 (4.7) 6.07 (2.7)

(b) a4 V b4 W− B2 V  L 2a  B 2U L2 V

7.63 (5.0) 47.1 (21.2) 11.9 (6.6) 27.3 (15.6) 32.5 (11.4) 36.1 (15.1)

9.8 46.5 15.0 23.0 65.3 54.7

2

+

30

6.49 (3.0) 13.4 (4.4) 2.31 (1.46) 1.10 (0.75) (6.2) (22.8) (8.2) (14.2) (21.5) (21)

40

20.2 (6.0) 3.37 (2.0) 15.4 (7.7) 41.3 (9.5) 16.3 (5.5) 12.1 (4.9) 25.4 (7.6) 9.70 (3.9) 6.92 (3.7) 13.8 (7.2) 9.29 (4.1) 5.30 (5.3) 15.4 (6.6) 22.7 (6.8) 4.72 (2.8) 2.11 (1.37) 13.5 41.4 21.3 18.4 84.2 118.4

(8.8) (18.6) (11.2) (10.1) (25.3) (45)

50

17.6 (5.5) 3.93 (2.4) 10.5 (5.2) 46.8 (10.8) 18.9 (6.4) 14.6 (5.9) 29.7 (9.2) 10.6 (4.2) 9.32 (4.9) 13.7 (6.8) 12.8 (5.5) 5.84 (5.8) 13.1 (5.5) 21.7 (6.5) 4.34 (2.6) 1.53 (0.99) 9.5 19.6 14.7 21.2 57.8 139.2

(6.2) (9.2) (8.5) (11.0) (17.9) (54)

12.3 (4.3) 2.66 (1.65) 6.68 (3.4) 32.7 (8.5) 10.3 (3.7) 9.18 (3.9) 22.0 (7.5) 6.28 (2.6) 5.90 (3.3) 8.38 (4.6) 8.00 (3.6) 5.64 (5.1) 8.69 (3.8) 13.4 (4.3) 1.74 (1.06) 1.51 (0.98) 6.7 9.4 12.4 20.3 21.6 118.4

(4.3) (4.6) (7.4) (11.6) (7.3) (50)

result shown in Fig. 94, Lee and co-workers have speculated that, on going to higher energies in the range 80–100 eV, there would be substantive agreement between a DW-based calculation and any reliable experimental data. We believe it would be very useful if this hypothesis was investigated in more detail by the measurement of experimental DCS and ICS in this energy region. Finally, in Fig. 95, we show the ICS data and theory for the electron impact excitation of the D2 W+ electronic state. As before this is limited to the work of Brunger et al. and the DW-level calculations of Machado et al. The comments we made previously with respect to the A2 W+ state in comparing the experimental and theoretical data, are almost equally applicable here. We emphasize that the theoretical cross-section for the D2 W+ state is signiFcantly larger in magnitude than that found by the experiment and also it appears to have a somewhat di@erent shape to that given by the experiment.

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429

•

Fig. 92. Integral cross-section for electron impact excitation of the A2 W+ electronic state of NO. ( ) Brunger et al., ( – - – ) Skubenich et al., (—) DW calculation of Machado et al.

•

Fig. 93. Integral cross-section for electron impact excitation of the b4 W− electronic state of NO. ( ) Brunger et al., ( – - – ) Skubenich et al.

3.7. Hydrogen halides Despite their toxic and corrosive nature there has been considerable experimental and theoretical interest in electron scattering by the hydrogen halides. This is because they represent reasonably simple molecular systems which are highly polar and which have revealed

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•

Fig. 94. Integral cross-section for electron impact excitation of the C2 V electronic state of NO. ( ) Brunger et al., (—) DW calculation of Machado et al.

•

Fig. 95. Integral cross-sections for electron impact excitation of the D2 W+ electronic state of NO. ( ) Brunger et al., (—) DW calculation of Machado et al.

interesting, and controversial, near-threshold structure in their vibrational excitation cross-sections. Most of the interest in these molecules has centred around the nature, and dynamics of formation, of the near-threshold structures and the measurement of accurate scattering cross-sections

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431

Table 94 Di@erential cross-sections for elastic electron-HF scattering. The column marked “sum” represents the rotationally summed cross-section (10−16 cm2 sr −1 ). The partial rotational cross-sections are given by their percentage contribution to the summed Fgure. The data are those of R\adle et al.

15 30 45 60 75 90 105 120 135

15 30 45 60 75 90 105 120 135

15 30 45 60 75 90 105 120 135

0 18 18 25 29 32 35 38 34 37

0 — 13 15 18 19 18 — 20 34

1 60 62 61 58 50 47 43 39 35

0.63 2 3+4 22 0 20 0 14 0 13 0 18 0 18 0 19 0 27 0 26 2

1 — 87 85 82 77 71 — 49 42

2.0 3+4 — 0 0 0 1 3 — 1 2

2 — 0 0 0 3 8 — 30 22

0 21 32 38 42 39 26 12 9 16

Sum — 16.3 5.98 3.81 2.79 1.98 1.30 1.00 1.09

Sum — 5.20 2.19 1.22 0.83 0.52 0.52 0.42 0.44

0 — 16 15 13 18 20 29 30 30

0 12 14 14 14 12 12 7 11 18

1 67 50 41 34 28 28 27 23 26

1 — 71 63 57 59 57 55 50 45

Energy (eV) 1.2 2 3+4 — — 13 0 22 0 30 0 21 2 19 4 11 5 16 4 24 1

1 83 76 70 68 67 63 57 52 43

Energy (eV) 3.0 2 3+4 5 0 10 0 16 0 18 0 16 5 22 3 30 6 31 6 33 6

Sum — 7.99 3.49 2.13 1.53 1.03 0.75 0.63 0.66

0 — — — 22 19 13 14 19 25

1 — — — 77 66 55 53 52 54

1.5 2 — — — 1 13 28 29 25 21

3+4 — — — 0 2 4 4 4 0

Sum — 7.17 2.97 1.55 1.04 0.72 0.65 0.53 0.56

3+4 0 0 0 1 5 8 15 20 20

Sum 10.2 2.79 1.23 0.79 0.43 0.29 0.28 0.26 0.29

6.0 Sum 13.4 3.81 1.68 0.91 0.57 0.37 0.33 0.30 0.33

Energy 10 eV 2 12 18 19 19 24 31 37 35 29

0 13 20 24 26 26 19 9 8 21

1 71 62 55 49 43 40 39 33 25

3+4 0 0 2 5 9 15 24 33 29

2 16 18 21 24 26 33 37 39 34

Sum 10.3 2.23 1.20 0.77 0.42 0.29 0.30 0.30 0.38

has been an almost incidental feature of this work. Nonetheless, as we discuss below, some cross-section information does exist. For completeness, we also note that there have been a number of recent review articles on the nature of the near-threshold features [131,369 –372].

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3.7.1. Hydrogen =uoride (HF) The pioneering studies on this molecule, which has a dipole moment of 1.84 D, were carried out by Rohr and Linder [373] and have been summarised in the Trajmar review. They measured di@erential excitation functions for the 0–1 and 0–2 vibrational excitation channels over a range of scattering angles. These were then integrated to yield total vibrational excitation functions from the threshold to a few eV above. The absolute scale was set by reference to values for HCl elastic scattering which in turn were obtained from comparison to the very early integral cross-section measurements of Bruche [374]. These measurements revealed a strong threshold ^ 2 , respectively, whilst the 0–1 structure in each channel with peak cross-sections of 7 and 0:12 A ^ 2 for several eV above threshold. cross-section maintained a non-resonant value of almost 1 A Furthermore, Rohr and Linder also demonstrated that the angular distributions at the energies corresponding to the peaks in both the 0–1 and 0–2 channels were isotropic. To a large extent these scattering features were reproduced in numerous measurements of dissociative electron attachment, although a detailed discussion of such work lies outside the scope of this review. However, they were revisted in an extensive and impressive series of measurements of absolute elastic scattering and rotational, vibrational and rovibrational excitation by the Kaiserslautern group [375 –378]. For HF they have measured rotationally resolved elastic scattering and rovibrational (0–1; 2; 3) excitation over a broad range of incident energies ◦ (0:5–10 eV) and scattering angles (15–135 ). They employed a high resolution spectrometer and a deconvolution procedure was used to perform a lineshape analysis of the electron energy loss proFles in order to extract information on individual rotational branches. Elastic scattering and rotational excitation. Elastic scattering and rotational excitation results for the ground vibrational state of HF have been presented by Radle et al. [377]. These results have been presented both as di@erential vibrationally elastic cross-sections, which involve a sum over all rotational transitions, and di@erential cross-sections for transitions which have involved a change in rotational quantum number of 0; 1; 2 and 3 + 4. All of these cross-sections are given in Table 94. The work of R\adle et al. represents cross-sections which are impressively large, particularly at forward scattering angles and, to our knowledge, there is no other experimental data available for comparison. The situation from a theoretical point of view is not much better, although there are several calculations for elastic scattering and rotational excitation including a close-coupling treatment by Padial and Norcross [379,380], an R-matrix approach by Th\ummel et al. [381] with which experiment can been compared. The latter authors also provide useful tabulated results from other calculations. We show some of these comparisons in Figs. 96 and 97. Firstly in Fig. 96(a), the rotationally summed di@erential cross-sections of R\adle et al. are compared with the various calculations (R-matrix, dipole close-coupling (DCC) and Frst Born), of Th\ummel et al., at an energy of 1:2 eV. The agreement is clearly very good with the marginal improvement o@ered by the ab initio calculations over the DCC and FBA indicating the dominance of the dipole interaction. A similar comparison is made in Fig. 96(b) for an energy of 6 eV. Here the full ab initio calculation shows substantially better agreement with the experiment than either of the DCC or FBA calculations, apparently as a result of the failure of the latter models to describe the 0–1 rotational excitation correctly [381]. In Fig. 96(c) we show the contributions of the various rotational branches to the elastic di@erential cross-section at an energy of 3:0 eV, and compare the experimental results with the R-matrix calculation of Th\ummel et al. and the close-coupling calculations of Padial and

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433

Fig. 96. Di@erential cross-sections for elastic scattering from HF. (a) at 1:2 eV and (b) at 6 eV. (c) shows the contribution of the various rotational branches to the elastic di@erential cross-section at 3 eV. In each case comparisons are made between the experimental data of R\adle et al. and results from the Born–Dipole approximation, the R-matrix calculation of Th\ummel et al. and the close coupling calculation of Padial and Norcross.

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•

Fig. 97. Di@erential cross-sections for rovibrational excitation (0 –1) in HF at 0:63 eV. ( ) Knoth et al., ( ) Rohr and Linder.

Norcross. The agreement is generally quite good (note the logarithmic scale in this plot) although both calculations underestimate the magnitude of the forward scattering for the 0–2 transition. Rovibrational excitation. There has been substantially more interest in vibrational excitation of the hydrogen halides compared to either elastic scattering or rotational excitation. This is mainly due to the e@ect of the near-threshold resonances which have been the subject of some controversy over the years since their initial observation by Rohr and Linder. Since the Trajmar review vibrational and rovibrational excitation of HF has been investigated by Knoth et al. [375,376,378] and R\adle et al. [377]. Knoth et al. [375] measured absolute di@erential cross-sections for vibrational (0–1) excitation of HF and employed a lineshape analysis of the vibrational energy loss peaks to determine the contribution of individual rovibrational transitions. They used the relative Pow technique to place these measurements on an absolute scale and the resulting cross-sections are tabulated in Table 95. Their rotationally summed cross-sections show similar behaviour, as a function of energy, to those of Rohr and Linder with a prominent near-threshold peak. Whilst the cross-section magnitudes are largely consistent, the latter measurements do not show isotropic angular distributions in the region of the strong near-threshold resonance, as found by Rohr and Linder, but rather a distribution which decreases in magnitude at forward scattering angles. These DCS are shown in Fig. 97, the measurement of Rohr and Linder is at an energy of 0:51 eV and that of Knoth et al. at an energy of 0:63 eV, rePecting

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435

Table 95 Di@erential cross-sections for rovibrational (; = 0 − 1) electron-HF scattering. The column marked “Sum” represents the summed rotational cross-section in units of 10−18 cm2 sr −1 . The partial rotational cross-sections for the transfer of 0, 1, 2 and 3 + 4 rotational quanta are also given. The data are those of Knoth et al. 0.63 15 30 45 60 75 90 105 120 135

0 15.2 18.3 23.2 21.0 21.3 20.1 14.4 11.0 13.2

1 22.8 24.5 30.2 36.6 34.2 31.7 26.4 29.6 31.2

2 — 1.4 — — 4.9 6.1 13.8 11.0 9.0

3+4 2.0 2.8 0.6 2.4 0.6 3.1 5.4 6.4 6.6

Sum 40 47 54 60 61 61 60 58 60

0 — 6.1 2.7 3.3 3.8 4.5 4.7 4.8 5.4

Energy (eV) 0.90 1 2 3+4 — — — 3.0 — 2.9 3.7 — 1.9 2.9 0.5 1.1 3.5 0.4 0.9 4.3 0.1 1.1 4.0 0.6 0.7 5.0 — 1.2 5.4 0.4 0.8

Sum — 12 8.3 7.8 8.6 10 10 11 12

0 — 1.7 1.4 1.5 1.5 1.6 1.7 2.2 2.7

1 — 3.0 1.9 1.8 1.9 2.1 2.4 2.6 2.6

1.2 2 3+4 — — 0.2 1.8 0.5 0.7 0.4 0.6 0.5 0.5 0.5 0.6 0.8 0.5 0.8 0.4 0.6 0.5

Sum — 6.7 4.5 4.3 4.4 4.8 5.4 6.0 6.4

the di@erence in the energy at which they Fnd the resonance peak. This was of signiFcant interest as the originally measured distribution, indicative of s-wave behaviour, was not consistent with the near-threshold shape resonance. The more recent measurements also found that if the excitation function is not measured by summation over rotational channels, but rather presented as a measure of a single state-to-state rotational transition, the width of the near-threshold resonance is considerably smaller and the peak shifted to lower energies. This is believed to be due to each rotational threshold supporting a resonance and the small (∼5 meV) di@erences in the rotational excitation thresholds. In either case, the threshold peak of Knoth et al. occurs at a signiFcantly higher energy than that observed by Rohr and Linder and its width is substantially larger. These observations also apply in comparison with the many close-coupling calculations, all of which predict a peak within a few meV of threshold (for example, see Fig. 11 of Th\ummel et al. [382]). Whilst these sorts of discrepancies have proved to be a major motivation in such studies, their detailed discussion is outside the scope of the present review. At energies away from the vibrational threshold the agreement between experiment and theory is generally quite good (for example, see Fig. 14 of Th\ummel et al. or Fig. 16 of Knoth et al.). Absolute di@erential cross-sections for the 0–2 and 0–3 vibrational transitions of HF have been measured by Knoth et al. [376,378] in addition to the earlier 0–2 measurements of Rohr and Linder. The recent results were once again performed with high-energy resolution so that a deconvolution of the energy loss proFles, into their various rotational branches, could be performed to reveal partial rotational excitation cross-sections for each vibrational level. These cross-sections are given in Tables 96 and 97, respectively. For the 0–2 excitation functions Knoth et al. show a strong near-threshold feature, similar to that observed by Rohr and Linder, with a peak magnitude of around 1:2 × 10−18 cm2 sr −1 at 1:07 eV. The angular distribution at this energy, for the summed rotational distributions, is essentially isotropic. On the other hand, as we see in Fig. 98, the partial rotational branches all show somewhat more variation in intensity with scattering angle. The near-threshold behaviour

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Table 96 Di@erential cross-sections for rovibrational (; = 0–2) electron-HF scattering. The column marked “Sum” represents the summed rotational cross-section in units of 10−18 cm2 sr −1 . The partial rotational cross-sections are given by their percentage contribution to the summed Fgure. The data are those of Knoth et al. 15 30 45 60 75 90 105 120 135

0 22 16 13 11 9 9 14 26 36

Energy 1.07 eV 2 5 18 33 40 45 42 36 28 15

1 61 57 45 41 38 41 45 42 44

3+4 12 9 9 8 8 8 5 4 5

Sum 1.02 1.13 1.16 1.19 1.19 1.28 1.22 1.24 1.14

Table 97 Di@erential cross-sections for rovibrational (; = 0–3) electron-HF scattering at an energy of 1:53 eV. The cross-section represents the summed rotational cross-section in units of 10−18 cm2 sr −1 . The data are those of Knoth et al.

DCS

15 30 45 60 75 90 105 120 135

0.048 0.046 0.053 0.064 0.066 0.080 0.086 0.090 0.101

of the excitation functions also show similar behaviour to that observed in the 0–1 excitation cross-section. The rotationally integrated results reveal a much broader threshold peak than those for a single rotational branch (jt = 0), and once again this is believed to be due to the occurrence of threshold structures at each rotational threshold which di@er in energy by around 5 meV. This conclusion is also consistent with the observation that the threshold peak shifts to lower energy for the un-integrated spectrum. Knoth et al. compare their derived integral excitation cross-sections with the close-coupling theories of Rudge [383], the R-matrix calculations of Morgan and Burke [384] and with the e@ective range theory of Gauyacq [385]. The substantive di@erence between all of the theoretical calculations and the experiment is that, like the 0–1 case, they predict a vertical onset at the threshold whereas the experiment predicts a “threshold peak” approximately 100 meV above the threshold. Cross-sections for the rotationally summed 0–3 excitation in HF have also been measured by Knoth et al. [378]. These data are given in Table 97. They once again Fnd a near-threshold peak

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437

Fig. 98. Di@erential cross-sections for rovibrational excitation (0 –2) in HF at 1:07 eV. The contributions of the various rotational branches are also shown. The data are those of Knoth et al.

about 110 meV above the onset for the 0–3 channel (at 1:411 eV) which has a peak magnitude ◦ varying monotonically from about 0.05 to 0:10 × 10−18 cm2 sr −1 between 45 and 135 . Another resonance structure and cusp activity are found in these excitation functions below and around the opening of the 0–4 channel. Indeed, the additional resonance structure observed in these excitation functions below the 0–4 threshold has been interpreted by Knoth et al. as deFnitive evidence of the existence of nuclear-excited Feshbach resonances. However, a discussion of these interesting phenomena is outside the scope of this review. 3.7.2. Hydrogen chloride (HCl) Once again the pioneering studies on this molecule were carried out by Rohr and Linder [386], who measured vibrational excitation cross-sections for the 0–1 and 0–2 transitions in the near-threshold region. These results have been summarised in the Trajmar review. Since then there have been measurements of both elastic scattering and vibrational excitation, with resolution of rotational contributions, by Knoth et al. [375,376,378], R\adle et al. [377], Schafer and Allan [387] and by Gote and Ehrhardt [186]. There have also been numerous calculations but, to our knowledge, none of recent vintage for elastic scattering. Elastic scattering and rotational excitation. Rotationally summed, di@erential elastic scattering cross-sections and the partial contributions from a number of rotational branches have been measured by R\adle et al. These measurements were made for a number of incident energies

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between 0.5 and 10 eV. The technique employed has been outlined above and the absolute cross-sections are presented in Table 98. A similar series of measurements has been carried out by Gote and Ehrhardt but at signiFcantly higher energies (5–200 eV), and these data are presented in Table 99. Some examples of both these cross-sections are shown in Figs. 99 and 100. In Fig. 99(a) and (b) we show the rotational branch cross-sections for vibrationally elastic scattering at 0.5 and 2:5 eV. This energy corresponds to the centre (as observed in vibrational excitation) of a higher-lying shape resonance. The rotationally summed cross-section rises steeply at forward angles, principally due to the dipole contribution to the 0–1 rotational ◦ branch, which compares well with the BDA at scattering angles below 50 (see R\adle et al.). Although we do not show them here, the level of agreement between the CC calculation of Padial and Norcross [379,380] and the experimental values is reasonably good although there are substantial di@erences for the 0–1 branch across the whole angular range. R\adle et al. also demonstrate that the pure elastic (J = 0–0) cross-section shows a pronounced minimum at an energy of about 1:5 eV and note that as this process is only sensitive to the spherically symmetric part of the interaction potential it can be likened to a Ramsauer minimum. At energies above a few eV, the cross-sections for the 0–2 and 0 − 3+4 rotational transitions show evidence of a broad shape resonance, which is also seen clearly in the vibrational excitation functions (see below), and both are in reasonable agreement with the CC calculation. Below 2 eV, the 0–2 cross-section increases rapidly (similar increases are also observed in the 0–0 and 0 − 3 + 4 transitions) leading R\adle et al. to speculate that it provides a hint for the existence of a threshold resonance in the elastic channel. This rise in the 0–2 cross-section (e.g. it reaches a value ◦ of 1:0 × 10−16 cm2 sr −1 at 0:5 eV and 60 ) is not predicted by the theory. In Figs. 100(a) and (b) we compare the elastic scattering cross-section measurements of R\adle et al. and Gote and Ehrhardt at 5 and 10 eV. The comparison is clearly very good at both of these energies. R\adle et al. also extrapolated and integrated their low energy DCS to obtain integral elastic cross-sections. These results were compared with the close-coupling calculations of Itikawa and Takayanagi [388] and Padial and Norcross [380]. Good agreement was found with the calculation of Padial and Norcross but they found that their ICS di@ered substantially from that of Itikawa and Takayanagi. The ICS data of R\adle et al. also show a Ramsauer-like minimum at an energy of 1:5–2 eV. Whilst R\adle et al. do not provide tabulated values for this integral elastic cross-section, its magnitude at the minimum appears to be around 6 × 10−16 cm2 and it reaches a value of about 28 × 10−16 cm2 at 10 eV. Rovibrational excitation. Since the publication of the Trajmar review, vibrational excitation cross-sections for HCl have been measured by Knoth et al. [389], Cvejanovic and Jureta [390], R\adle et al. [377], Knoth et al. [375,376,378] and Schafer and Allan [387]. These are in addition to the earlier measurements of Rohr and Linder [373,386]. For the 0–1 excitation Rohr and Linder measured an integral vibrational excitation function with a strong peak just above threshold (Eth = 0:358 eV). The magnitude of this peak was about 17 × 10−16 cm2 sr −1 and there was an obvious break in the cross-section at the threshold for the ; = 2 vibrational excitation (Eth = 0:702 eV). In addition they noted a broad shape resonance that was centred at around 2:5 eV. They also found that the angular distributions for vibrational excitation were essentially isotropic throughout the whole of the resonance region. These overall features were largely conFrmed by Knoth et al. [375,389] and R\adle et al., although there are some notable di@erences. The latter measurements Fnd a threshold peak at an energy of 0:5 eV which is

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439

Table 98 Di@erential cross-sections for elastic electron-HCl scattering. The column marked “Sum” represents the summed rotational cross-section in units of 10−16 cm2 sr −1 . The partial rotational cross-sections are given by their percentage contribution to the summed Fgure. The data are those of R\adle et al.

15 30 45 60 75 90 105 120 135

15 30 45 60 75 90 105 120 135

15 30 45 60 75 90 105 120 135

0 34 35 35 18 19 20 22 25 29

0 34 26 21 20 19 13 5 18 42

0 53 57 53 42 35 24 10 14 41

1 66 61 56 56 49 45 45 45 44

1 63 60 54 47 40 35 43 30 20

1 42 24 19 17 15 17 17 19 10

2 0 4 9 22 25 28 26 22 17

2 3 12 17 32 36 50 49 48 33

2 0 8 19 28 36 43 50 43 31

0.5 3+4 0 0 0 4 7 7 7 8 10 2.0 3+4 0 2 8 1 5 2 3 4 5 4.0 3+4 5 11 9 13 14 16 23 24 18

Sum 28.0 11.8 8.0 4.5 3.4 2.8 2.6 2.5 2.3

Sum 12.3 3.58 1.77 1.15 1.00 0.94 0.94 0.81 1.09

Sum 9.7 3.42 2.31 2.09 1.88 1.56 1.37 1.10 1.46

0 31 34 31 12 16 20 17 23 27

0 36 29 22 23 21 18 4 14 36

0 53 58 52 48 45 25 14 17 47

1 69 58 57 51 51 48 46 44 42

Energy (eV) 1.0 2 3+4 0 0 5 3 9 3 36 1 30 3 29 3 29 8 28 5 27 4

Sum 20.7 4.7 2.6 1.62 1.35 1.28 1.21 1.14 1.40

0 31 21 19 12 10 13 16 21 32

1 67 66 54 53 44 53 46 41 36

2

1 60 54 47 38 36 38 35 34 17

Energy (eV) 2.5 2 3+4 4 0 14 3 25 6 30 9 35 8 38 6 53 8 51 1 44 3

Sum 10.7 3.39 1.78 1.48 1.30 1.10 1.03 0.98 1.13

0 38 44 36 32 27 16 13 12 34

1 62 42 39 31 31 33 32 32 14

2

1 37 26 16 18 19 19 25 17 10

Energy (eV) 4.4 2 3+4 6 4 8 8 18 14 20 14 25 11 40 16 37 24 39 27 29 14

Sum 9.6 3.52 2.44 2.23 1.96 1.66 1.35 1.19 1.48

0 54 59 54 53 40 29 17 22 49

1 38 22 18 16 15 17 19 19 10

1.5 3+4 2 0 7 5 15 6 3 3 2

Sum 15.3 3.99 2.05 1.23 0.96 0.91 0.83 0.84 1.15

0 11 18 29 37 39 50 47 40

3+4 0 3 7 8 5 12 5 9 12

Sum 9.9 3.36 1.92 1.64 1.47 1.27 1.15 0.94 1.22

5.0 2 3 9 17 18 28 35 38 32 20

3+4 5 10 11 13 17 19 26 27 21

Sum 9.5 3.73 2.57 2.22 2.04 1.66 1.33 1.20 1.51

0 13 20 30 31 28 35 35 30 2.9

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Table 98 (Continued)

15 30 45 60 75 90 105 120 135 15 30 45 60 75 90 105 120 135

0 66 72 67 58 54 38 27 24 53

1 28 16 13 14 12 11 13 16 10

2 0 5 6 17 25 33 33 30 22

6.0 3+4 6 7 14 11 9 18 27 30 15 0 89 88 87 79 64 54 46 35 65

Sum 10.8 4.05 2.99 2.18 1.97 1.60 1.31 1.15 1.65

0 73 74 70 63 57 50 39 23 57 1 7 9 5 6 8 9 9 21 13

1 24 15 9 8 9 9 11 15 9

Energy (eV) 7.0 2 3+4 0 3 0 11 3 18 12 17 15 19 20 21 19 31 24 38 15 19

8.5 Sum 11.9 4.5 3.17 2.12 1.91 1.56 1.15 1.09 1.69

Energy 10 eV 2 0 0 2 7 18 27 31 29 11

0 82 78 76 66 55 45 33 30 58

1 15 14 10 12 14 11 14 21 11 3+4 4 3 6 8 10 10 14 15 11

2 0 2

5 8 13 23 24 23 10

3+4 3 6 9 14 18 21 29 26 21

Sum 17.8 5.5 3.44 2.02 1.69 1.50 0.94 0.91 1.41 Sum 21.3 6.6 3.71 2.0 1.44 1.33 0.84 0.77 1.24

narrower and of generally smaller magnitude than that of Rohr and Linder. They also indicate DCS in the resonance region with substantial deviations from isotropic behaviour. Tabulated cross-sections from the measurements of R\adle et al. and Knoth et al. are given in Table 100. An example of the measured DCS, at the energy of the threshold peak, is given in Fig. 101 ◦ where the substantial anisotropy at angles below about 60 is evident in both the rotationally summed cross-section and the individual rotational branch cross-sections. This observation led Knoth et al. to again question the simple picture of pure s-wave scattering, suggesting instead that the origin of this resonance is due to a virtual state at threshold. The measurements of Schafer and Allan provide no information on vibrational excitation cross-sections per se. Nonetheless, they are of considerable interest as they do address some of the discrepancies that have arisen between the various experiments and between almost all experiments and theory concerning the position and shape of the near-threshold structures. The measurements of Schafer and Allan were performed with a trochoidal monochromator and ◦ represent a superposition of the DCS at 0 and 180 . They were taken with an energy resolution of 35 meV and under circumstances where pure vibrational excitation is emphasised. As a result some caution should be exercised in comparison with true di@erential results such as those outlined above. However, the results discussed above indicate that there is little change in the shape of the excitation functions for this channel with angle. Also, the rotational selection available from the measurements of Knoth et al. does not appear to have a substantial e@ect on the shape of the near-threshold behaviour and as such the results of Schafer and Allan

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441

Table 99 Di@erential cross-sections (10−16 cm2 sr −1 ) for elastic electron-HCl scattering. The data are those of Gote and Ehrhardt Energy (eV) Angle

5

10

20

30

50

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

9.838 4.993 3.694 2.794 2.413 2.182 2.071 1.882 1.636 1.345 1.105 1.034 1.337 1.785 2.678 3.405

17.386 12.515 7.572 5.057 3.109 1.935 1.354 1.148 1.047 0.921 0.757 0.730 1.054 1.794 2.875 4.222

21.041 14.327 8.021 4.448 2.015 0.655 0.226 0.292 0.495 0.644 0.618 0.456 0.316 0.279 0.404 0.600

20.176 10.727 5.258 2.350 0.886 0.231 0.117 0.253 0.426 0.496 0.430 0.280 0.138 0.051 0.060 0.134

17.215 6.748 2.308 0.846 0.277 0.076 0.081 0.184 0.287 0.331 0.285 0.181 0.094 0.076 0.177 0.344

Energy (eV) Angle

70

100

150

200

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

14.909 5.014 1.329 0.430 0.153 0.067 0.085 0.147 0.198 0.202 0.150 0.076 0.041 0.102 0.282 0.520

14.054 3.477 0.861 0.263 0.115 0.083 0.097 0.120 0.119 0.092 0.043 0.009 0.023 0.109 0.273 0.449

10.126 2.461 0.537 0.213 0.119 0.096 0.093 0.079 0.054 0.029 0.009 0.009 0.043 0.107 0.209 0.315

9.285 1.999 0.451 0.192 0.117 0.092 0.075 0.050 0.029 0.017 0.011 0.019 0.048 0.092 0.183 0.240

can provide some qualitative comparison. The results of Schafer and Allan indicate a threshold structure in both the 0–1 and 0–2 excitation functions which is only 30 meV above the true threshold energies. These values may well be resolution and=or transmission limited such that the actual threshold peak is even closer to the threshold. This observation is in much better accord with most of the theoretical predictions. Finally, R\adle et al. have measured rotationally resolved DCS at higher energies, in particular, in the region of the broad shape resonance at 2:5 eV. These show that the individual rotational

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Fig. 99. Di@erential cross-sections for elastic scattering in HCl at energies of (a) 0.5 and (b) 2:5 eV. The contributions of the various rotational branches are also shown. The data are those of R\adle et al.

DCS and the rotationally summed DCS are essentially isotropic, as observed by Rohr and Linder. They further argue that rather than implying a simple s-wave scattering process the observed Pat cross-sections are also consistent with interference from higher partial waves, in particular p and d contributions. This is also consistent with conclusions drawn from close-coupling calculations [379,380]. Excitation of higher vibrational levels of HCl has been studied by Knoth et al. [378] and Schafer and Allan, since the early work by Rohr and Linder. Once again the majority of the absolute cross-section information, which is of most relevance to this review, is provided by the measurements from the Kaiserslautern group [378]. In the 0–2 excitation function they Fnd a near-threshold peak in this channel some 100 meV above the threshold, followed by a broad shape resonance centred at around 2:5 eV. There are some small di@erences in detail between these results and those of Rohr and Linder, with the latest measurements indicating cross-sections of smaller magnitude and both the rotationally summed, and rotational branch DCS, being non-isotropic. An example of these DCS for an incident energy of 0:80 eV is given in Fig. 102. Tabulated values of the cross-sections at this and several other energies are found in Table 101. This excitation channel has also been studied by Schafer and Allan. They Fnd a similar threshold structure but once again it is closer to the threshold than in the Kaiserslautern measurements.

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443

•

Fig. 100. Di@erential cross-sections for elastic scattering in HCl at energies of (a) 5 and (b) 10 eV. ( ) R\adle et al., ( ) Gote and Ehrhardt.

Knoth et al. have also measured absolute cross-sections for the 0–3 excitation in HCl. They once again Fnd evidence of a strong near-threshold structure, particularly at backward scattering angles, and a shape resonance between 2 and 4 eV. The integrated, rotationally summed cross-section shows a near-threshold peak which reaches a maximum value of about 0:02 × 10−16 cm2 at an energy of 1:15 eV. Tabulated, rotationally integrated di@erential cross-sections for a number of incident energies are given in Table 102. 3.7.3. Hydrogen bromide (HBr) An extensive series of measurements of elastic scattering and vibrational excitation of HBr were carried out by Rohr [391,392]. These measurements have been summarised by Trajmar et al. [1] and, to our knowledge, there have been no further low-energy scattering cross-section measurements on this molecule. 3.8. Cross-section trends amongst diatomics In the compilation of a review such as this a number of trends in the scattering cross-sections become evident. These are sometimes obvious and expected, such as the behaviour of the di@erential scattering cross-section for polar molecules or its behaviour near the energy of a resonance.

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Table 100 Di@erential cross-sections for rovibrational (; = 0–1) electron-HCl scattering. The column marked “Sum” represents the summed rotational cross-section in units of 10−16 cm2 sr −1 . The partial rotational cross-sections are given by their percentage contribution to the summed Fgure. The 0.5 and 1:5 eV data are those of Knoth et al. and the 2.5 and 5 eV data are those of R\adle et al. 0.5 15 30 45 60 75 90 105 120 135 15 30 45 60 75 90 105 120 135

0 25 27 28 30 31 32 30 25 22

1 55 51 54 56 56 56 59 58 52

2 14 17 15 10 9 9 8 14 23 0 7 4 7 11 17 21 14 12 5

3+4 6 5 3 4 4 3 3 3 3

Sum 0.34 0.57 0.64 0.89 0.93 0.91 0.85 0.87 0.90

0 27 29 26 25 24 23 26 25 29 1 23 18 13 15 15 16 25 17 15

Energy (eV) 1.5 1 2 3+4 23 31 19 21 29 21 26 27 21 27 29 19 25 30 21 31 28 18 30 35 9 34 33 8 35 26 10

2.5 Sum 0.13 0.18 0.17 0.15 0.16 0.18 0.20 0.20 0.21

Energy 5.0 eV 2 27 24 20 35 31 26 27 23 32

0 39 35 32 28 29 29 34 28 27

1 22 22 24 28 26 23 25 22 25 3+4 43 54 60 39 37 37 34 48 48

2 34 36 36 33 37 34 39 28 30

3+4 5 7 8 11 8 14 2 22 18

Sum 0.25 0.24 0.21 0.20 0.21 0.21 0.20 0.22 0.23 Sum 0.063 0.059 0.052 0.050 0.059 0.073 0.072 0.077 0.075

There have been several papers which have investigated the trends that may be expected, for example, in isoelectronic molecules such as N2 and CO. Nickel et al. [185] measured directly the ratio of the elastic scattering cross-sections for CO and N2 and over a wide range of ◦ energies (20–100 eV) and scattering angles (20–120 ). The biggest di@erence from unity was about 20%, with the overwhelming majority of the ratios being within 5% of unity. This leads one to conclude that the weak dipole moment of the CO molecule (N2 is non-polar) plays little role in the scattering process at these energies. These Fndings are illustrated in Fig. 103 where we show a comparison of the elastic DCS for CO and N2 at an energy of 20 eV. A number of data sets are shown here including the measurements of Nickel et al. and data from the later studies of Gibson et al. [264] for CO and Gote and Ehrhardt [186] for N2 . The four cross-sections for the two gases are clearly in excellent agreement over the whole angular range. In a later study, Middleton et al. [193] reached a similar conclusion based on measurements of vibrational excitation (0–1) in these molecules at energies between 20 and 50 eV. An example of their measurements at an energy of 30 eV is given in Fig. 104. At lower energies the situation is somewhat di@erent. The two molecules support low-energy 2 V shape resonances, but the dynamics of the decay of the two resonances are di@erent and

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445

Fig. 101. Di@erential cross-sections (10−16 cm2 sr −1 ) for rovibrational excitation (0 –1) in HCl at 0:5 eV. The contributions of the various rotational branches are also shown. The data are those of Knoth et al.

thus one may not expect any striking similarities in the elastic or vibrational excitation DCS in the resonance region. This is, in fact, the case as we can see from Fig. 105 where we compare DCS for elastic scattering at energies corresponding to structures within the 2 V shape resonance envelope in both molecules. In the case of N2 we see a clear inPuence of the autodetached d-wave electron on the elastic DCS, although there is also a substantial amount of direct (non-resonant) scattering. For CO, the autodetached electron is in the p-wave and this clearly manifests itself in the shape of the DCS at 1:91 eV. However, the two cross-sections are substantially di@erent in shape and magnitude. At 5 eV, an energy which is outside the dominant resonance region in both molecules, there are still some signiFcant di@erences in the elastic DCS, as we can see from Fig. 106. Whilst the two DCS have remarkably similar shapes (see below) the CO cross-section is larger at forward angles, no doubt due to the inPuence of the weak dipole moment. Outside the region of inPuence of the long-range dipole interaction, i.e. at large scattering angles, the two cross-sections are seen to merge. Another trend which has emerged across many of the molecules studied in this review is the behaviour of the di@erential elastic scattering cross-section at energies above the low-lying shape resonance but below about 10 eV. In N2 , O2 , CO and NO, measurements at a Fne energy grid have revealed an interesting feature which develops at forward scattering angles and displays a rather rapid energy dependence. At energies just above the position of the shape resonance

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Fig. 102. Di@erential cross-sections (10−18 cm2 sr −1 ) for rovibrational excitation (0 –2) in HCl at 0:8 eV. The contributions of the various rotational branches are also shown. The data are those of Knoth et al.

◦

in these molecules, the elastic DCS shows a local maximum at angles around 50–60 and then decreases towards smaller scattering angles (see for example Fig. 106 discussed above). Unfortunately, the presence of the primary beam precludes measurements which would enable ◦ us to say whether or not this trend continues all the way to 0 , or if there is a local minimum ◦ at angles less than 15 . As the energy increases, a minimum or shoulder develops at angles ◦ ◦ around 20–30 and the local maximum still persists near 50 . At higher energies, closer to 10 eV, the more familiar DCS shape, where strong forward scattering is exhibited, has evolved. The remarkable thing about this “regularity” in the DCS for these molecules is that, whilst they are all clearly similar in size, they represent a wide range of molecular structures—homonuclear, heteronuclear, closed shell, open shell, non-polar and mildly polar. Although it is outside the scope of this particular section of this review, studies of several polyatomic molecules (e.g. CO2 , N2 O, SF6 ) have revealed similar angular structures, suggesting that the behaviour is more general amongst non-polar, or mildly polar, molecular systems. Whilst there has been some speculation as to the origins of these structures contemporary scattering theory has not yet been able to successfully predict this behaviour. Consequently, the fundamental reasons for the occurrence of this angular behaviour remain largely elusive and clearly a challenge for modern scattering theory.

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447

Table 101 Di@erential cross-sections for rovibrational (; = 0–2) electron-HCl scattering. The column marked “Sum” represents the summed rotational cross-section in units of 10−8 cm2 sr −1 . The partial rotational cross-sections are given by their percentage contribution to the summed Fgure. The data are those of Knoth et al. 15 30 45 60 75 90 105 120 135

0 33 25 25 16 18 14 15 15 18

Energy 0.8 eV 2 7 9 11 12 13 11 10 7 6

1 57 62 60 68 66 72 72 74 72

3+4 3 4 4 4 3 3 3 4 4

Sum 2.69 3.53 5.82 6.58 6.74 6.83 6.88 7.0 7.28

Energy (eV)

1.5

2.5

4.0

5.0

15 30 45 60 75 90 105 120 135

2.18 2.74 2.97 2.66 2.63 3.08 3.22 2.99 3.19

4.76 4.00 4.39 3.64 4.11 3.83 3.67 4.28 3.89

2.44 2.24 2.18 2.74 2.72 2.77 2.94 2.63 2.27

0.76 0.84 0.73 0.70 0.70 0.92 0.73 0.76 0.84

Table 102 Di@erential cross-sections for rovibrational (; = 0–3) excitation of HCl in units of 10−8 cm2 sr −1 . The data are those of Knoth et al. Energy (eV)

1.15

2.5

4.0

5.0

15 30 45 60 75 90 105 120 135

1.05 0.87 1.23 1.29 1.23 1.48 1.71 1.6 2.02

1.29 1.12 1.26 1.01 1.20 1.18 1.15 1.48 1.26

0.59 0.64 0.64 0.90 0.84 0.81 0.87 0.90 0.67

0.15 0.27 0.32 0.24 0.29 0.27 0.28 0.28 0.29

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Fig. 103. Di@erential cross-sections for elastic scattering in CO and N2 at 20 eV. Fig. 104. Di@erential cross-sections for vibrational excitation (0 –1) in CO and N2 at 30 eV.

4. Summary and suggested measurements In this paper we have attempted to collate and critically review all of the available data on electron-diatomic molecule scattering at low-to-intermediate energies. We have limited our investigation to non-rearrangement collisions, i.e. elastic scattering, rotational and vibrational excitation and electronic excitation. This information is important in a number of both fundamental and applied Felds and we hope that the way in which it is presented here is of use to workers in these Felds. One of the few advantages that fall to the authors of such a document is the opportunity to assess those areas in their Feld where future work would be of an advantage, either in terms of furthering our understanding of the fundamental aspects of the collision processes involved or in providing cross-sections which are of relevance to technological processes. We have attempted to highlight such areas during the course of this review but it is perhaps helpful if we amplify two areas where we see future studies being of critical importance. • Electronic excitation. In several of the molecules we have dealt with, there is only one systematic study for many of the electronically excited states (e.g. NO, H2 ) and for those where more than one study exists (e.g. N2 , O2 ) the agreement between the experimental results is often patchy. It is also clear that the agreement between contemporary theory and experiment is less than satisfactory. Benchmark measurements in a number of diatomic molecular systems

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449

Fig. 105. Di@erential cross-sections for elastic scattering in CO at 1:91 eV and N2 at 1:98 eV. Fig. 106. Di@erential cross-sections for elastic scattering in CO and N2 at 5 eV.

are required both as a standard which experimentalists can use, for example, to calibrate the energy dependence of their spectrometers and as a clear test for theory. The time-of-Pight techniques developed by LeClair, Trajmar and co-workers show a great promise in this regard. We also note the complete absence of electronic excitation cross-section measurements for the halogens or hydrogen halides. • Vibrational excitation. Whilst the situation regarding vibrational excitation at high energies and in the region of resonances has been rather well studied, there is still a pressing need for benchmark absolute cross-sections for near-threshold (0–2 eV) vibrational excitation in most of the diatomic molecules we have considered. This is particularly true for NO where low-energy total cross-section measurements have shown the dominance of a series of negative ion resonances, but little or nothing is known about the role that these resonances play in near-threshold vibrational excitation. Acknowledgements This article, and its sequel on polyatomic molecules which is (hopefully) soon to follow, has been many years in production. We wish to thank all of our colleagues who have supplied

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information in the form of tabulated data and Fgures, many of whom also provided great encouragement for us Frstly to start this project, but more importantly, to complete it. Both authors wish to acknowledge the considerable input of their colleagues and students in their own institutions over the years and we trust that this article will be of some use to them in the future. MJB wishes to thank the Australian Research Council for the provision of a Queen Elizabeth II Fellowship and both authors, but particularly MJB, are indebted to Marilyn Mitchell for her typographical services, o@ered graciously in the face of considerable adversity. SJB wishes to acknowledge the support of a Fulbright Senior Fellowship during the Fnal stages of the preparation of this manuscript and the hospitality and support of the Physics department of the University of California, San Diego where the article was Fnalised. References [1] S. Trajmar, D.F. Register, A. Chutjian, Phys. Rep. 97 (1983) 219. [2] S. Trajmar, J.W. McConkey, in: B. Bederson, H. Walther (Eds.), Advances in Atomic, Molecular and Optical Physics, Academic Press, New York, 1994, p. 63. [3] A.R. Fillipelli, C.C. Lin, L.W. Anderson, J.W. McConkey, in: B. Bederson, H. Walther (Eds.), Advances in Atomic Molecular and Optical Physics, Academic Press, New York, 1994, p. 2. [4] R.W. Crompton, in: B. Bederson, H. Walther (Eds.), Advances in Atomic, Molecular and Optical Physics, Academic Press, New York, 1994, p. 97. [5] B. Schneider, in: B. Bederson, H. Walther (Eds.), Advances in Atomic, Molecular and Optical Physics, Academic Press, New York, 1994, p. 183. [6] Y. Itikawa, in: B. Bederson, H. Walther (Eds.), Advances in Atomic Molecular and Optical Physics, Academic Press, New York, 1994, p. 253. [7] A. Zecca, G.P. Karwasz, R.S. Brusa, Riv. Nuovo Cimento 19 (1996) 1. [8] E.W. McDaniel, E. Manskey, in: B. Bederson, H. Walther (Eds.), Advances in Atomic, Molecular and Optical Physics, Academic Press, New York, 1994, p. 390. [9] J.C. Nickel, P.W. Zetner, G. Shen, S. Trajmar, J. Phys. E 22 (1989) 730. [10] S.J. Buckman, R.J. Gulley, M. Moghbelalhossein, S.J. Bennett, Meas. Sci. Technol. 4 (1993) 1143. [11] J.A. Giordmaine, T.C. Wang, J. Appl. Phys. 31 (1960) 463. [12] D.R. Olander, V. Kruger, J. Appl. Phys. 41 (1970) 2769. [13] C.B. Lucas, Vacuum 23 (1973) 395. [14] D.M. Murphy, J. Vac. Sci. Technol. A 7 (1989) 3075. [15] G.R. Hanes, J. Appl. Phys. 31 (1960) 2171. [16] R.H. Jones, D.R. Olander, V.R. Kruger, J. Appl. Phys. 40 (1969) 4641. [17] H.P. Steinr\uck, K.D. Rendulic, Vacuum 36 (1986) 213. [18] S. Adamson, J.F. McGilp, Vacuum 36 (1986) 227. [19] S. Adamson, J.F. McGilp, Vacuum 38 (1988) 463. [20] S. Adamson, C. O’Carroll, J.F. McGilp, Vacuum 38 (1988) 341. [21] M.A. Khakoo, private communication, 1997. [22] B. Bederson, L.J. Kie@er, Rev. Mod. Phys. 43 (1971) 601. [23] W. Raith, Adv. At. Mol. Phys. 12 (1976) 281. [24] R.E. Kennerly, R.A. Bonham, Phys. Rev. A 17 (1978) 1844. [25] S. Trajmar, D.F. Register, in: I. Shimamura, K. Takayanagi (Eds.), Electron–Molecule Collisions, Plenum Press, New York, 1994 (Chapter 4). [26] S.J. Buckman, B. Lohmann, J. Phys. B 19 (1986) 2547. [27] I.E. McCarthy, E. Weigold, Electron–Atom Collisions, Cambridge University Press, Cambridge, 1995. [28] C. Ma, P.B. Liescheski, R.A. Bonham, Rev. Sci. Instrum. 60 (1989) 3661. [29] Z. Saglam, N. Aktekin, J. Phys. B 23 (1990) 1529.

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AB-INITIO STUDY OF THE ELECTROMAGNETIC RESPONSE AND POLARIZABILITY PROPERTIES OF CARBON CHAINS

M. BIANCHETTI, P.F. BUONSANTE, F. GINELLI, H.E. ROMAN, R.A. BROGLIA, F. ALASIA

AMSTERDAM – LONDON – NEW YORK – OXFORD – PARIS – SHANNON – TOKYO

Physics Reports 357 (2002) 459–513

Ab-initio study of the electromagnetic response and polarizability properties of carbon chains M. Bianchettia; b ; 1 , P.F. Buonsantea; b ; 2 , F. Ginellia; b ; 3 , H.E. Romana; b , R.A. Brogliaa; b; c ; ∗ , F. Alasiaa ; 4 a

Dipartimento di Fisica, Universita di Milano, Via Celoria 16, 20133 Milano, Italy b I.N.F.N., Sezione di Milano, Via Celoria 16, 20133 Milano, Italy c The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen *, Denmark Received March 2001; editor : J: Eichler

Contents 1. Introduction 2. Experimental issues 3. Theoretical and numerical issues 3.1. Local density approximation (LDA) 3.2. Time-dependent local density approximation (TDLDA) 4. Results 4.1. Electronic structure 4.2. Electromagnetic response 4.3. Static linear polarizabilities 4.4. Criticism to the model 4.5. Comparison between chain and ring isomers

461 468 470 470 471 473 473 479 492 496

4.6. Relation to DIB 5. Conclusions Acknowledgements Appendix A. Cylindrical LDA (cLDA) Appendix B. Cylindrical TDLDA (cTDLDA) B.1. TDLDA equations B.2. Numerical tests to the cTDLDA Appendix C. Classical polarizability of elongated metallic particles References

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498

∗

Corresponding author. Dipartimento di Fisica, Universit>a di Milano, INFN, Sezione di Milano, Via Celoria 16 20133 Milano, Italy. Fax: +39-2-58357487. E-mail address: [email protected] (R.A. Broglia). 1 Present address: Banca IntesaBci, Via Clerici 4, 20121 Milano, Italy. 2 Present address: Dipartimento di Fisica, Universit>a di Parma, and I.N.F.M., Parco Area delle Scienze 7A, 43100 Parma, Italy. 3 Present address: Dipartimento di Fisica, Universit>a di Firenze, and I.N.F.M., Via Largo E. Fermi 2, 50125 Firenze, Italy. 4 Present address: Twinsoft Italia S.r.l., Via Fulvio Testi 280/6, 20126 Milano, Italy. c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/02/$ - see front matter  PII: S 0 3 7 0 - 1 5 7 3 ( 0 1 ) 0 0 0 5 9 - X

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Abstract We present the most complete set of calculations to date of the ground state electronic properties and of the optical=UV response function of linear carbon chains CN , using ab-initio methods based on local density and on time-dependent local density approximations (LDA and TDLDA). Making use of the associated transition densities and wavefunctions we are able to provide microscopic insight into the collectivity of the corresponding plasmon spectrum in terms of correlated particle–hole excitations. From this analysis it is found that the (one-dimensional 1-D) delocalization of  (valence) electrons is responsible for the conspicuous values of the static dipole polarizability and of the high value of the exponent describing its dependence with the number of carbon atoms. Within this framework the electronic structure and linear response function of a carbon ring is also calculated. Although many properties of this function are similar to that associated with linear chains of the same number of atoms, the corresponding polarizabilities diFer appreciably, providing a reliable method to distinguish between linear and close structures. The Grst principle results of the properties of linear carbon chains are compared with both theoretical and experimental results available in the literature, and constitute the basis for a systematic study of these 1-D sp-bonded systems, which have been found to be involved in such seemingly disparate phenomena as fullerene growth mechanism and diFuse interstellar bands. c 2002 Elsevier Science B.V. All rights reserved.
PACS: 36.40.−c; 36.40.Vz; 61.46.+w

M. Bianchetti et al. / Physics Reports 357 (2002) 459–513

461

1. Introduction Scientists and non-scientists alike have found it remarkable that a new form of carbon could be discovered late in the 20th century, like fullerenes, only pure, Gnite form of carbon. The other two, diamond and graphite being actually inGnite networks [1]. There is also a general feeling that fullerene-based materials and nanotubes (a closed cylinder made out by wrapping a graphite sheet, thus resembling chicken wire, with capped ends; Figs. 1 and 2) will be important in the science and technology of the 21st century. In fact, nanotubes and their spin-oF are turning out to be some of today’s most fascinating objects within the realm of physics, as the late Richard Feynman had suggested would happen almost half a century ago. When Caltech physicists and Nobel laureate Richard P. Feynman stood to address the Annual Meeting of the American Physical Society on 29 December 1959, it is likely that nobody was prepared to connect with words like: “When we get to the very, very small world—say circuits of seven atoms—we have a lot of new things that would happen that represent completely new opportunities for design. Atoms on a small scale behave like nothing else on a large scale, for they satisfy the laws of quantum mechanics [2].” Within this context it is interesting to read the abstract of the paper of Smalley et al. with the suggestive title “Unraveling nanotubes: Field emission from an Atomic Wire” [3]: “Field emission of electrons from individually mounted carbon nanotubes has been found to be dramatically enhanced when the nanotube tips are opened by laser evaporation or oxidative etching. Emission currents of 0.1–1
A were readily obtained at room temperature with bias voltages less than 80 V. The emitting sources are calculated to be linear chain of carbon atoms, Cn (n = 10–100), pulled out from the graphene wall layer of

Fig. 1. Molecular models of icosahedral C60 and derived fullerene nanotubes C70 ; C90 ; C110 ; C150 and C190 . The ratios R⊥ : R between the shortest and the longest radii R⊥ and R , respectively, are also shown.

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Fig. 2. By rolling a graphene sheet (a single layer from a 3-D graphene crystal) into a cylinder and capping each end of the cylinder with half of a fullerene molecule, a “fullerene-derived tubule”, is formed. Shown here is a schematic theoretical model for a single-wall carbon tubule. If the C60 molecule is bisected normal to a Gve-fold axis, the “armchair” tubule shown in (a) is formed, while if the C60 molecule is bisected normal to a three-fold axis, the “zigzag” tubule shown in (b) is produced. The names of armchair and zigzag arise from the shape of the cross-sectional ring, as is shown at the edge of the nanotubes in (a) and (b), respectively. The primary symmetry classiGcation of a carbon nanotube is as either being “achiral” or “chiral”. An achiral carbon nanotube is one whose mirror image has an identical structure to that of the original system. Armchair and zigzag carbon nanotubes are the only two cases of achiral nanotubes, and are associated with (two component) chiral vectors characterized by the pair of integer numbers (n; m), with m = 0 (armchair) and m = n (zigzag). Chiral nanotubes (n = m = 0), as the one shown in (c), exhibit a spiral symmetry whose mirror image cannot be superposed on the original one. The indices (n; m) are particularly important for the electronic structure of the nanotubes: tubes for which n − m = 3i, where i is an integer, are metallic. All others are semiconducting. The values of the (n; m) parameters considered are: (a) (5; 5), (b) (9; 0) and (c) (10; 5) (from Ref. [1]).

the nanotube by the force of the Geld, in a process that resembles unraveling the sleeve of a sweater”. 5 In other words, a “wire” (connected to an electron reservoir) made out of few 5

It is still an open question what the energy diFerence between the open-ended and the closed conGgurations of the multi-wall nanotubes (MWN) used in Ref. [3] is under the applied bias. It has been shown that this diFerence decreases with increasing Geld [4], although it is unlikely it may be suPcient to lower the open end energy below that associated with the capped structure. How the situation is modiGed in multi-wall nanotubes is an open question.

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Fig. 3. Model of the tip of a multiwall nanotube showing a single Cn “atomic wire” extending out from the inner layer, held taut and straight by the electric Geld (after [3]).

carbon atoms and of length of the order of a nanometer (see Fig. 3), emits, under the inQuence of an electric Geld, a current as intense as a microampere. First principle calculations indicate [5] that, in fact, already a linear chain made out of eight atoms can emit a current of few microamperes, the emitted current varying linearly with the bias potential as in the case of metals. Consequently, and as hypothesized in Ref. [1], linear carbon chains behave as metallic wires, even a few atom carbon chain. In order to see how linear carbon chains look like and make all these things possible, one can just look at them, i.e., one can shine photons on them and determine which frequencies they absorb. It turns out that when the polarization vector of the photons is in a plane which contains the symmetry axis of the linear chain, the system absorbs energy of essentially one frequency [5 –7]. This means that at such frequency, all valence electrons vibrate in phase (as in the case of an antenna) with the electric Geld, the frequency becoming lower as the number of carbon atoms forming the chain is increased. This behavior can be reproduced, with a single parameter, making use of an expression given in standard textbooks [8] which relates the absorption frequency of a macroscopic metallic wire (having the same scaled shape of the linear carbon chain) to the length of the wire, and testiGes to the fact that linear carbon chains are likely to be the ultimately atomic quantum wires. The observation, in the laboratory, of the absorption frequency of linear chains also revealed a cosmic connection of these systems, indicating their presence in the interstellar media (IM)

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[9,10]. Just as dust particles in the earth atmosphere produce a red sunset, so starlight is reddened as it passes through clouds of interstellar dust. The visible–ultraviolet spectrum from distant stars is like the spectrum of the sun, our nearby star, crossed by a series of narrow, dark lines. But in the spectrum of distant stars, aside from these lines, which are associated with the absorption of light by atoms present in the outer layers of the sun, there are others which are much broader (or more diFuse). For this reason they are referred to as bands rather than lines. The quest for the origin of these diFuse interstellar bands (DIB), dubbed by Kroto (co-discover together with Smalley and Curl of C60 fullerene) “the last great problem in astronomy” [11], is closely related to the presence of carbon in the Universe [9]. In particular, as it has been agreed since 1997, of linear carbon chains [6,12–15]. One now knows that the answer to the question of the origin of DIB is more articulated than that proposed in Ref. [12] (cf. also Refs. [9,10] and refs. therein) and that it requires the presence of onion-like nanometer-sized carbon particules (cf. Fig. 4). In any case, a number of optical gas-phase electronic transitions (plasmons) of linear chains anions (i.e., charged chains), recently observed in the laboratory, have been found to coincide with some of the DIB [18] (Fig. 4). We shall show in Section 3:6 that Grst principle calculations of the spectrum of linear neutral carbon chains as well as experimental data of these chains in inert gas matrices [14], indicate also the coincidence with selected DIB. Because the spectrum of a quantal system is as revealing of its identity as Gngerprints are of individuals, these Gndings reopens the case of linear carbon chains in connection to the question of DIB. At the basis of the ubiquitous role played by carbon not only on earth, but in the universe, one Gnds a very rich chemistry. In fact, carbon has the ability to form sp1 , sp2 , sp3 hybridized orbitals, to which correspond triple, double and single bonds, respectively. These bonds are, in average, very stable and are at the basis of a discrete number of carbon allotropes and a large number of carbon-based molecules. The “classical” allotropic forms of carbon, diamond (sp3 ) and graphite (sp2 ) (cf. standard textbooks, e.g. [19]), were enriched in recent times by the discovery of fullerene-based materials (sp2+x ) (cf. e.g. Refs. [1] and refs. therein), and a chain-like, polyyne allotrope (sp1 ) has been proposed, although it still needs further experimental conGrmation [20]. Carbon molecules and clusters, both pure and impure, have engaged a great interest for many decades. One of the Grst descriptions is due to Hahn et al. [21] (for reviews cf. e.g. Refs. [1,22,23] and refs. therein). In recent times, an “inQationary” phase of researches has begun, which also resulted in the discovery of fullerenes [24 –27] and nanotubes [28,29], based on the improvement and=or development of sophisticated synthesis, storage, manipulation → Fig. 4. In 150 B.C. the Greek astronomer Hipparchus grouped the sky’s stars into six brightness categories called magnitudes, Grst magnitudes representing the brightest stars he saw, sixth the faintest. The apparent magnitude, m, of a star—the magnitude as seen by Hipparchus and by us—depends on the stars real luminosity (that is, on the amount of energy it radiates in Watts) and on its distance. A star may seem bright to us either because it is nearby (even though faint) or because, though far away, it is very luminous. If we know the star’s distance, d, we can Gnd its luminosity from its apparent brightness. In astronomy, luminosity is expressed by a star’s absolute magnitude, M , the value the apparent magnitude would have at a standard distance of 10 parsec, i.e., of 32.6 light years (the light year is the distance light travels in vacuum in a year). The two kinds of magnitude are related by the magnitude equation M = m + 5 − 5 log d, where d is in parsec. To apply the method of spectroscopic distances

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we must correct the apparent magnitude of a star for the dimming by the interstellar dust. This means knowing the degree of extinction, A, the magnitude by which the apparent visual magnitude must be brightened before it is entered into the magnitude equation. One can Gnd A because it scales with the degree of reddening. (a) The extinction function—the dependence of interstellar extinction to wavelength —has a rough −1 dependence in the optical and goes to zero at long wavelength (allowing correlated far-infrared and radio waves to penetrate interstellar T bumps have been space with penache). In the ultraviolet, interstellar space becomes remarkably opaque; the 2000 A found to be correlated with the laboratory spectrum of matrix-isolated nanometer-sized carbon particles (after Ref. [16]). (b) Two typical structures of carbon particles taken with high-resolution transmission electron microscope: (left side) carbon particle with randomly oriented basic structural units, (right side) onion-type carbon particle with several condensation seeds (cf. Ref. [9]). Theoretical calculations indicate the multishell fullerenes (carbon onions) T bumps [17]. (c) DIB observed as broad depressions in stellar spectra are to be the likely candidates for the 2200 A seen from the ultraviolet through the near-infrared [16]. (d) Gas-phase electronic transitions of C7− (charged linear carbon chain containing seven atoms of carbonium) recorded in the laboratory [18] (upper trace). The bottom trace shows a Gaussian Gt to the tabulated DIB. Those labeled with an asterisk coincide with the measured bands of C7− above.

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and characterization techniques, such as laser or arc-discharge sources, rare-gas matrix isolation, mass spectrometry, laser spectroscopy, ion mobility and reactivity measurements. Under present investigation are the structural, electronic, vibrational and rotational properties of the ground and excited states. Of particular interest are the transport mechanisms, relevant, e.g. for superconductivity (cf. e.g. Ref. [30]) and Geld emission eFects (cf. e.g. Refs. [5,31–39] and refs. therein). An interesting open question concerning the physics of carbon is that of the evolution of physical properties as a function of the number of carbon atoms and their geometrical arrangement. This topic is intimately connected with the fullerene growth mechanism, nowadays believed to proceed along a chain–ring–sphere–nanotube path (cf. e.g. Refs. [40 – 42], refs. therein, and Section 2). The above scheme needs further investigation in order to clarify the role played by the various classes of structures. In any case, it is clear that small carbon clusters are of great importance for the understanding of such a complex mechanism. A fundamental role is played by linear structures which have become a well-established Geld of research. A classical reference is in this sense Ref. [22], while more recent reviews are Ref. [23], focused onto structural and spectroscopy properties, and Refs. [43,44], dedicated to electronic spectroscopy. There are further reasons supporting investigations of carbon chains. Their one-dimensional 1-D, sp bonding structure results in peculiar properties. The mass distribution displays an odd– even eFect, which favors odd N chains, recently synthesized up to 21 atoms, across the region of stability of closed fullerene cages [45], with respect to even N chains, which are stable only up to N = 10. A subtle competition exists between linear chains and ring isomers, which cannot be separated with standard mass spectrometry methods [46] (cf. also Section 2). Also the electronic structure displays an odd–even eFect, being odd N chains closed shell systems, with a Gnite HOMO–LUMO gap, while even N chains are open shell systems with no gap. 1-D Van Hove singularities [1,19,47] are expected in the electronic density of states. The sp1 bonding character results in highly delocalized wavefunctions, with valence electrons able to move almost freely in 1-D. As a consequence, very high static polarizabilities and strong collective longitudinal excitations are expected, showing scaling properties as a function of the chain length typical of 1-D systems. As mentioned above, transport properties also show even–odd eFects (cf. e.g. Ref. [48] and refs. therein). Carbon clusters, with their own physical and chemical properties, their variety of diFerent geometries, and the subtle interplay among diFerent degrees of freedom, pose a serious challenge to theorists. Large systems are, in general, beyond the capabilities of present ab-initio, parameter-free, methods, with limiting sizes being around 10 –15 atoms for the quantum-chemical conGguration interaction (CI) method (cf. e.g. Refs. [49 –51]). As has already been found in the case of C60 , in particular in the study of its linear response function [52–54], tight-binding approximation is not particularly accurate to describe carbon structures. The ground state properties of structures up to some hundred of atoms can be investigated with calculations based on density-functional theory (DFT). The excited states can be obtained within its time-dependent counterpart, the time-dependent density functional theory (TDDFT). Both these schemes, within their various practical implementations, in particular the local density approximation for the exchange–correlation interaction (LDA and TDLDA, respectively) are well-established methods in condensed matter physics as predictive ab-initio theories that can be applied to systems containing a large number of electrons (cf. Section 3 for details).

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While it is possible to use already existing computer codes (like e.g. the Stuttgart Car-Parrinello code, CPMD [55,56]), their computational cost becomes prohibitive when large, isolated systems are treated without making use of any symmetry-based reduction of the size of the problem. Periodic boundary conditions and a plane-wave basis are particularly ePcient to treat periodic structures. On the other hand, the systems we are interested in are isolated and have a built-in axial symmetry. Consequently, the proper choice of the cell in which to carry out the calculations and of the orientation of the system in it, so as to avoid the eFect of the images of the original system, become important steps of the calculation, as well as limiting factors on the dimensions of the systems which can be treated accurately. These are the reasons why, in order to study the static and the dynamic electronic and structural properties of elongated, isolated axially symmetric molecules we have developed LDA and TDLDA codes in a cylindrical basis, named cLDA and cTDLDA, in which one system can be treated at a time, avoiding periodic boundary conditions. 6 This is also in keeping with the experience accumulated in our group, of the advantages found in the use of a basis displaying, as close as possible, the symmetries of the system under study. In particular, of a spherical basis to study the electronic properties and the linear response function of almost spherical fullerenes like C60 ; C28 and C20 , and how this advantage is gradually lost in the case of systems which depart from quasi-sphericity like in the case of the quadrupole-deformed fullerene C70 and of the incipient nanotubes C90 and C110 [52,61– 66]. We have applied our method to the systematic calculation of the ground and excited states of linear carbon chains CN in the range of available experimental data. Ab-initio calculations have previously been published, both by members of our group [5,34 –36] as well as by other research groups (cf. e.g. Refs. [7,41,42,50,67] and refs. therein). Aside from Gnding general agreement with other calculations carried out with diFerent methods and codes and from producing systematic results for the electronic structure and the electromagnetic response of neutral carbon chains, we have also further investigated a number of interesting properties of these systems. In particular, we have obtained the wavefunctions and the transition densities associated with the dipole photoabsorption resonances of chains CN (N = 3–21), which demonstrate the strong collectivity of these (plasmon) states. Also the static dipole polarizabilities, which display a rather fast scaling as a function of the chain length, a consequence of the strong  delocalization of sp hybridized  valence electrons, could eventually be used to separate isomers characterized by diFerent geometries on the basis of their expected diFerent polarizabilities. Our theoretical results are compared with both a classical model for ellipsoidal metallic particles and with the data reported in the experimental and theoretical literature, thus constituting the basis for a complete and systematic description of the static and dynamic properties of the linear carbon chains, which eventually can be extended to more complex systems, such as hydrogenated chains [68]. This work is organized as follows: in Section 2 we address the experimental observations relevant to the subject of carbon chains. In Section 3 we brieQy describe the cLDA and cTDLDA equations. In Section 4 we illustrate the results obtained and we discuss the 6 While another group have reported LDA calculations making use of an axially symmetric basis in a cylindrical cell, these calculations employ periodic boundary conditions along the z direction and have been carried out for small systems only (dimers and their anions, cf. Refs. [57– 60]).

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connection with both other theoretical results and the experimental observations. Details are reported in Appendices A–C. 2. Experimental issues One of the major problems encountered in the experimental investigation of small carbon clusters is their high reactivity. Carbon clusters smaller than C60 cannot, in general, be isolated in macroscopic quantities and analyzed by traditional spectroscopic means. As a consequence all the studies about the electronic and structural properties of carbon clusters have been conducted in molecular beams or on clusters isolated in an inert matrix (for a review see [23], also [44,69] and refs. therein). The production of small carbon clusters can be obtained by vaporizing with a laser or an ion beam a graphite target. The plasma produced condenses in clusters whose dimensions and charge state can be controlled by quenching the plasma in an inert gas and eventually expanding the mixture in vacuum to form a supersonic beam [70,71]. Information about the cluster structures can be inferred from the analysis of cluster mass abundances by time-of-Qight mass spectrometry [72]. While this method has provided the Grst evidence of the peculiarity of C60 and C70 , it must be used very carefully to extract information about smaller clusters [70]. This is particularly true for neutral clusters which must be ionized to be detected. Due to their high ionization potential, multiphoton ionization (MPI) is used for their detection. Since fragmentation is a consequence of MPI the detected abundances may not reQect the original mass distribution and hence the original stabilities [23]. Recently, using single-photon ionization of neutral aggregates, a cluster size distribution with magic numbers diFerent from those traditionally obtained from MPI has been found [73]. Conventional cluster sources can produce anion and cation clusters, the possibility of manipulating these species and, in particular, of mass selecting them using electric and magnetic Gelds, has favored a large number of spectroscopic investigations. Cluster structures can be deduced from mass abundances [74], reactivity [75] and photoelectron spectroscopy [76,77]. Optical emission studies of the carbon particle produced from a carbon plasma were reported by RohlGng [78]. He observed the emission of C and C2 species with vibrational and rotational temperature estimated at Tvib ≈ 10 400 K and Trot ≈ 1000 K . The average particle density in the laser plume was estimated to be about 1017 particles cm−3 . Reactivity studies showed that the presence of hydrogen during the vaporization can strongly aFect the Gnal cluster mass distribution. Odd number clusters always show a dramatic preference for the hydrogenation in the mass range with N ¿ 40, whereas smaller clusters form long chain polyacetylenes CN H2 . Hydrogenation using diFerent source geometries, to vary the annealing conditions, showed that cluster growth and cluster hydrogenation are partially separate processes, with growth preceding hydrogenation. Clusters grow in the high-temperature, high-density region of the plasma plume where hydrogen concentration is relatively small [78]. Carbon clusters up to 70 atoms have been investigated by Bowers and co-workers [79,80] and Jarrold and co-workers [81–83] with gas-phase ion chromatography. This technique spatially and temporally separate diFerent clusters and isomers due to their diFerent mobility in an inert

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gas. The absolute value of ionic mobility together with numerical simulation give information of the particle structure. A pulse of mass selected cluster ions are injected into a high-pressure drift cell Glled with 2–5 Torr of helium. After traveling across the drift tube, the ions exit and they are focused into a mass analyzer (quadrupole) for identiGcation and detection. The mobility measurements are performed by recording the arrival time distribution of the ions at the detector; cluster shapes can be derived from collisional cross-sections. By controlling the energy of the ions entering the scattering cell, collisional heating and annealing of the clusters is also possible. These studies suggest that small clusters are linear. For clusters with 7–10 atoms, linear and + monocyclic rings are observed, whereas between C+ 11 and C20 only monocyclic structures are + + detected. Between C20 and C30 also bicyclic isomers are present [79]. At C+ 29 3-D structure are detected, for C+ the monocyclic isomers disappear and another with a tricyclic planar structure 36 appears. Above C+ bicyclic and tricyclic forms are both present. Fullerenic isomers appear at 40 C30 and dominate by C50 [80]. In the region C30 –C40 , Bowers and co-workers Gnd no evidence of graphitic curved fullerene precursors which can grow towards C60 by simple addition of small fragments. By injecting these ions in the gas cell at fairly high energy (tens of eV) the collisional energy cause a profound rearrangement of the ionic structure and the formation of fullerenic structure is observed. An annealing of carbon cluster ions in the range C30 –C70 was performed with the same technique by Jarrold and co-workers [81–83]. They showed that clusters in this mass range can undergo structural interconversion upon annealing. The change in geometry to the fullerene form is associated to fragmentation and loss of C3 units [82]. From C40 the predominant fragment becomes C2 . They found that the conversion ePciency from ring into fullerene structures is not very high and that polycyclic ring can also rearrange to form large monocyclic rings in competition with fullerenes. The situation changes in the range C50 –C70 where the interconversion into fullerene structure becomes very ePcient (80%) [83]. A similar technique for cluster ion selection based on quadrupole Glters has been extensively used by the group of the University of Basel in order to deposit size-selected clusters into a rare-gas matrix and to perform absorption spectroscopy characterization. From these studies one has information on the electronic structure and optical properties of small carbon clusters [44,84,85]. This technique has the obvious advantage of allowing the preparation of a “bulk” sample, however, it presents also some limitations. In order to perform the deposition of a sample in a reasonable amount of time, a high cluster current (nanoamperes) must be extracted from the quadrupole assembly, implying the availability of very intense cluster ion sources [84]. The type of sources utilized in Basel produce reasonable intensities only for clusters up to 15 atoms, so that the spectroscopic characterization has been limited to this mass range. Moreover, this technique works with charged particles and neutralization during deposition is needed in order to characterize neutral species. The ePciency of neutralization is not 100% so that in the same matrix can coexist charged and neutral species, rendering the data interpretation diPcult. Quadrupole size selection cannot distinguish between isomers with the same mass. This makes the matrix isolation technique the ideal tool to distinguish between the optical response of diFerent isomers.

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As we have already discussed, carbon clusters smaller than 10 atoms have the linear conGguration as energetically preferred whereas for larger clusters cyclic conGguration is preferred. In a cluster beam diFerent isomers coexist and, in principle, it is not possible to form enough linear clusters in the mass range between 10 and 25 atoms to perform a size-selected deposition. Attempts to characterize matrix-isolated carbon vapors without mass selection have been plagued by the incertitude on the attribution of the observed spectral features [86]. Recently it has been shown that long neutral carbon chains up to 21 atoms can be produced in rare-gas matrices through a process involving the diFusion of carbon atoms [45,87]. A graphite target is vaporized with a laser and the vapor deposited in a neon matrix, this process produce a large amount of carbon atoms that are trapped in the matrix. A slight thermal annealing of the matrix allows an eFective chain-lengthening process and the subsequent formation of long chains, which would not be favored in the gas phase. Of course this preparation method does not require a mass selection of the species thus allowing the use of a relatively simple experimental set-up; the assignment of the spectral features requires the comparison with previous identiGed electronic transition on mass-selected systems. 3. Theoretical and numerical issues In this section we brieQy discuss our implementation of the LDA and the TDLDA models on a basis of cylindrical wavefunctions, supported by numerical tests. More details are reported in Appendices A–C, while a complete description can be found in Ref. [88] and will be published elsewhere [89]. 3.1. Local density approximation (LDA) The density functional theory (DFT) [90,91] of many fermion systems is a well-established method in condensed matter physics as a predictive ab-initio theory that can be applied to systems containing a large number of electrons. Nowadays, many modern electronic structure calculations start from its powerful implications (cf. e.g. Refs. [92–95]). Here, the ground state properties of the carbon chains are obtained within the Kohn–Sham (KS) density functional formalism, using the LDA for exchange and correlation interactions [90 –95], following the parameterization of Perdew and Zunger [96] and norm-conserving pseudopotentials for the carbon ions [97–99]. In our numerical calculations, we have used a code (cLDA) developed by some of us working in a cylindrical basis to deal with isolated systems displaying axial symmetry, like e.g. linear chains and nanotubes [88,89]. The LDA equations are solved in cylindrical coordinates (; z; ) by expanding the KS wavefunctions on a basis of cylindrical wavefunctions
(; z; ) = cn; ‘; m n; ‘; m (; z; ) ; (1) n;‘; m

where the n; ‘; m (; z; ) are solutions of (−˝2 =2me )∇2 n; ‘; m = n; ‘; m n; ‘; m , for a free particle within a cylindrical box of radius 0 and length 2z0 , subject to the boundary conditions

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n; ‘; m (0 ; z; ) = 0 and n; ‘; m (; ±z0 ; ) = 0, where 0 6  6 0 ; −z0 6 z 6 z0 and 0 6  6 2 (cf. Ref. [88,89] for details). We have tested our cLDA calculations in a number of ways, in particular against diFerent implementations of the DFT–LDA scheme: (i) a spherical basis implementation, already used for short linear carbon chains [5,34 –36] and fullerenes [52,61– 66], and (ii) the Car–Parrinello molecular dynamics (CPMD) [55,56]. 7 We have also tested diFerent pseudopotentials, box sizes, and linear geometries (see Appendix A for details). As a result, we have found stable numerical results for cylindrical boxes only slightly larger than the molecule inside, at variance with that needed when using periodic boundary conditions, as e.g., in CPMD, where one is forced to use very large boxes, and=or some approximation to reduce the interaction among the periodic replicas. T is suPciently In addition, we found that the use of a constant interatomic distance of 1:3 A accurate for all our purposes (see Appendix A for further details). This value is very close to the experiment and similar to the values assumed in various theoretical works (from 1:29 to T see e.g. [23] and refs. therein, and also Refs. [5,40 – 42,50,65,67,101–103]). 1:31 A, A similar implementation of LDA equations in a cylindrical basis can be found in Refs. [57– 60], within the context of CPMD [55], with a cylindrical cell with periodic boundary conditions along the z direction, which has been applied only to the case of dimers. 3.2. Time-dependent local density approximation (TDLDA) The DFT, described in Section 3.1 above, is a ground state theory, based on the two Hohenberg–Kohn theorems [90]. To go beyond this limit, and to calculate the properties of excited systems, such as the response to a time-dependent external Geld, a step further in the theory is needed. Accurate calculations of excited state properties require sophisticated techniques, such as the CI method (cf. e.g. Refs. [49 –51]), which are computationally demanding. A possible alternative is the extension of DFT to time-dependent Gelds, known as time-dependent density functional theory (TDDFT). Its origin can be traced to the pioneering work of Zangwill and Soven [104], while rigorous theoretical treatments are more recent (cf. e.g. Refs. [105 –107] and refs. therein). Within TDDFT the action of the time-dependent external Geld in the small amplitude limit is treated self-consistently within the linear response approximation [108], where the residual interaction is derived as the functional derivative of the DFT Hamiltonian with respect to the density. TDLDA is obtained when the exchange–correlation term is treated within the adiabatic approximation [106]. DiFerent implementations of TDLDA have been formulated and applied to various physical systems, ranging from nuclei to atoms, molecules, clusters and solids: the real-time method [7,67,109 –113], the Green function method in coordinate space (see e.g. Refs. [5,34 –36,52,61– 66,92,104,106,114 –121]), the matrix formulation in particle–hole conGgurational space (see e.g. [115,118,122–124]). The domain of applicability and the accuracy of the TDLDA is presently under investigation, with promising results (cf. e.g. Refs. [109 –113,124]). The use of pseudopotentials in TDLDA is discussed in Ref. [110]. 7

The parallel version of CPMD runs on the CRAY-T3E supercomputer at CINECA [100].

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Because of the absence within TDLDA of the degrees of freedom necessary to describe excitations with a more complex character than single electron–hole excitations, a question arises when one uses the TDLDA in connection with LDA single-particle spectra, being the Kohn– Sham LDA single-particle states not directly related to the electronic states of the system. But unlike the Hartree–Fock energies, the LDA single-particle energy diFerences usually provide a good approximation to particle–hole excitation energies [111], and the collective states calculated within TDLDA are rather insensitive to the non-locality of the exchange. The experimental spectra are more complex than those obtained in TDLDA, due to both the mixing of states of diFerent particle–hole character (e.g. two particle–two hole states) and the coupling with other degrees of freedom (e.g. vibrational states). However, the TDLDA is far superior than the static single electron theory, because it includes the dynamical screening eFect, which arises directly from the time dependence of the mean Geld potential, shifting the transition strength associated to the lower particle–hole excitations towards states of higher energies in the case of a repulsive residual interaction. Furthermore, the TDLDA conserves the energy-weighted sum rule (EWSR) [115,118]. Here, we choose the Green’s function method in coordinate space, because of our previous experience [5,34 –36,52,61– 66,115] and its slight simplicity and ePciency for frequency-dependent response within the LDA [67,114,117]. A central role in the coordinate-space TDLDA is played by the retarded density–density correlation function (x; x ; !) (cf. Appendix B). It obeys the Dyson-type integral equation displayed by Eq. (B.9) and depends on seven continuous variables. In the case of elongated systems it is useful to switch to a matrix form via projection onto the cylindrical basis n; ‘; m (; z; ), Eq. (1), where it depends on three continuous variables, ;  ; !, and on four discrete quantum numbers, l; m; l ; m . The cylindrical multipoles of the polarization propagator are deGned as follows:  ∗ (2) l1 m1 l2 m2 (1 ; 2 ; !) = d ˜ 1 d ˜ 2 Y˜ l1 m1 (˜ 1 )(x1 ; x2 ; )Y˜ l2 m2 (˜ 2 ) ; ˜ are the cylindrical coordinates, and Y˜ lm () ˜ the (; z) basis wavewhere x = (; ; z) = (; ) functions (cylindrical harmonics, cf. Refs. [88,89] for details). The free polarization propagator lLDA (1 ; 2 ; !) is deGned in a similar way (cf. Appendix B). The TDLDA equation (B.9), 1 m 1 l2 m 2 becomes 
 LDA l1 m1 l2 m2 (1 ; 2 ; !) =  l1 l4 m1 m4 (1 − 4 ) − d 3 d 4 3 4 l3 m 3 l4 m 4



× lLDA (1 ; 3 ; !) Kl3 m3 l4 m4 (3 ; 4 ; !) l4 m4 l2 m2 (4 ; 2 ; !) ; 1 m 1 l3 m 3

where the multipoles of the kernel are given by  ∗ Kl1 m1 l2 m2 (1 ; 2 ) = d ˜ 1 d ˜ 2 Y˜ l1 m1 (˜ 1 )K(x1 ; x2 )Y˜ l2 m2 (˜ 2 ) :

(3)

(4)

For a numerical solution the  integrals are discretized in a grid of Np integration points [125], and the propagators and the kernel become Np × Np matrices. The cylindrical TDLDA Eq. (3)

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takes the matrix form Eq. (B.10), and can be solved by matrix inversion as in Eq. (B.11). The explicit form of the cylindrical multipoles of the free polarization propagator lLDA (1 ; 2 ; !) 1 m 1 l2 m 2 and of the kernel Kl1 m1 l2 m2 (1 ; 2 ) are given elsewhere [88,89]. Once the TDLDA Eq. (3) has been solved for the polarization propagator l1 m1 l2 m2 (1 ; 2 ; !), one can calculate the dynamical linear polarizability, Eq. (B.12), as a function of the cylindrical multipoles of the polarization propagator Eq. (2) and of the external Geld, as
 0 "V (!) = d 1 d 2 1 2 Vl1 m1 (1 )l1 m1 l2 m2 (1 ; 2 ; !)Vl2 m2 (2 ) (5) l1 m 1 l 2 m 2

0

from which the physical information can be extracted as described in Appendix B. The symmetries of the system can be used in order to simplify the calculations. In our numerical TDLDA calculations, we have used a code developed by us (cTDLDA [88,89]). Based on the cLDA calculations described in Section 3.1, our cTDLDA results for the electromagnetic response have been carefully tested. We have found agreement with a diFerent implementation in a spherical basis (sLDA), already applied to the case of short linear carbon chains [5,34 –36], fullerene C60 and incipient carbon nanotubes [52,61– 66]. Details are given in Appendix B. In particular, we have tested that the particle–hole basis necessary to fulGll 90% of the EWSR, Eq. (B.19), can be sensibly reduced in size without aFecting the results for the electromagnetic response. Stability has been also found changing from optimized to ideal geometries (cf. Section 3.1 above). 8 4. Results In what follows, we discuss the electronic spectrum (Section 4.1), the electromagnetic response (Section 4.2) and the static polarizability (Section 4.3) of neutral linear chains. In Section 4.4 the LDA and TDLDA models employed here are critically discussed in the light of the present results and of the available literature. We also compare our results with those for carbon rings (Section 4.5) and with DIB (Section 4.6). 4.1. Electronic structure We Grst discuss two typical examples for odd and even N chains in Fig. 5, where we compare the electronic structures of C5 and C6 . Due to the axial symmetry of the chains, the azimuthal quantum number m is a good quantum number. States having m = 0 are denoted as $-states (admitting up to two electrons), while states having m = ± 1 are denoted as -states and are doubly degenerate (admitting up to four electrons). These states are displayed by the short and long lines for the $- and the -states, respectively. (m = 2) states, or states with m ¿ 2 are not occupied and do not appear among the lowest unoccupied states. In addition, the states are either symmetric (+) or antisymmetric (−) under the transformation z → −z, and are denoted as $± or ± . Thus, their parity (±) is a 8

Ideal geometries for the calculations of TDLDA response of carbon chains have already been used e.g. in Refs. [5,34 –36,67].

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Fig. 5. Energy spectrum of carbon chains C5 (left side) and C6 (right side) calculated within the cLDA near the HOMO–LUMO region. Full=dashed lines for occupied=empty states. Long=short lines for =$ states (doubly=singly degenerate; see text). The even chain C6 has an open-shell structure with the HOMO being a half-Glled + -state at −7:2 eV. The vertical lines show the main longitudinal dipole-allowed particle–hole transitions. To be noted is that each line connecting double degenerate -states corresponds, actually, to two-particle–hole transitions, fulGlling the dipole selection rules, Eq. (6), with the exception of the transitions involving the HOMO state in C6 , where the partial occupation forbids one of the two HOMO → LUMO transitions and allows a HOMO-1 → HOMO transition. Hence we are showing a total of Gve transitions on each panel.

good quantum number. For any symmetric=antisymmetric pair of states, the symmetric one lies always below the antisymmetric state. The structure shown in Fig. 5 indicates that the numerical calculation fulGlls this expected behavior. There are (N + 1) occupied $-states corresponding to the (N − 1) carbon–carbon bonds, plus two dangling bonds at the tips of the chain. The Grst (N − 1) $-states turn out to be the lowest in energy and well separated from the remaining levels. They are not shown in Fig. 5. The two $ dangling states always occur at higher energy values, near=at the HOMO. Regarding the -states, there are n = (2N − 2)=4 completely occupied levels for odd N carbon chains, while for even N chains there are N=2 − 1 fully occupied levels plus a remaining half-occupied -state. In the case of C5 , there are two fully occupied -states and for C6 there are two fully occupied and one half-occupied -states. In Fig. 5 also the Grst, low-lying, longitudinal dipole single particle–hole transitions are displayed. In this case the external Geld is proportional to the operator Y1; 0 (cf. Section 4.2)

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Fig. 6. (Upper panel) Probability density (| (; z;  = 0)|2 ) of the highest occupied $-state (HOMO) in C9 . The vertical axis corresponds to the radial variable , while the horizontal one to the z variable. The tick marks on the horizontal axis indicate the locations of the carbon atoms; (lower panel) same for the highest occupied -state (HOMO-2).

and the selection rules Zm = 0;

i × f = − 1

(6)

apply. 9 In the case of even N chains, because of the partial occupancy of the HOMO state, there are two low-lying single-particle transition approximately of the same energy: one is due to transitions between the HOMO and the Grst unoccupied -state, and the other corresponds to transitions from the Grst lower-lying -state and the HOMO. In Fig. 6 we show the probability densities for the highest occupied $ (upper panel) and  (lower panel) states of C9 in the  = 0 plane. For the $-state the localization of the charge density at the tips of the chain is clear, while for the -state the corresponding density is delocalized along the chain. 10 In Fig. 7, a 3-D plot of the total charge density of C9 is shown. The cumulenic structure of the chain is clearly seen. Ten peaks are evident, corresponding to the eight bonds among the nine atoms of the chain, located between the peaks, plus the two dangling bonds at the tips of the chain. 9 Here i; f denotes the parity of the initial or Gnal state, and must not be confused with the same notation used to label single-particle -states with m = ± 1. 10 In the case of C9 there is an accidental degeneracy of the higher occupied -state with the two $ tip states.

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T Fig. 7. 3-D plot of the electron number density for C9 (A

−3

T 2. ), in the region [ − 3:5 6  6 3:5] × [ − 7 6 z 6 7] A

Fig. 8. Overview of cLDA electronic spectra for CN chains with N = 3 to 21 and for N = 23; 25; 27; 29 and 30. Thick (thin) lines for occupied (empty) states. Long (short) lines for doubly (singly) degenerate states. Only states below 10 eV are reported. The characteristics of these spectra are discussed in the text.

We have calculated the cLDA electronic structure of linear carbon chains CN for odd N up to N = 29, and for even N up to N = 20 and additionally for N = 30. Experiments have established the existence of such systems for odd N 6 21 and for even N 6 10 (cf. Section 2). We show in Fig. 8 the entire set of our calculations in order to appreciate the systematic. The evolution of the electronic structure as a function of N clearly indicates the formation of an occupied $ “band” in the range (−22; −15) eV. The  “bands” are found above a “gap” which tends to about 4 eV. The two remaining occupied $-states (dangling bonds) are always located within the  “bands” in the neighborhood of the HOMO. The HOMO characteristics alternates between even and odd N chains. For odd N chains, the HOMO is always a $− -state when N 6 9, then it becomes a -state with parity (−)n +1 , where n is the number of occupied -states. For even N , the HOMO is always a half-Glled -state with parity (−)N=2+1 . Regarding the HOMO energy values, they alternate between even and odd

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N chains, such that the values for even N are always higher than for the neighboring odd N ones. Both values tend to about −6 eV for large N . The LUMO coincides with the HOMO for even N (open-shell) chains, while for odd N it is always a -state with parity opposite to the HOMOs. The HOMO–LUMO gap for odd N chains decreases as a function of N and reaches the value of 0:6 eV for C29 . In Fig. 9, we plot the density of states (DOS) for three chains of growing length, C9 (Fig. 9a), C21 (Fig. 9b), and C29 (Fig. 9c). For convenience, we have convoluted the discrete cLDA states with a Lorentzian function of the type DOS(E; ') =

M

'
nk ;  (E − Ek )2 + '2

(7)

k=1

where M is the number of LDA states. We have chosen ' = 0:1 eV in order to appreciate the whole DOS, aside Gne details, e.g., the HOMO–LUMO positions, indicated in the Ggure as H and L. The diFerent heights between $- and -states are just due to the different degeneracies (nk in Eq. (7)). The high DOS at the HOMO of C9 (Fig. 9a) is due to the accidental degeneracy of the higher occupied -state with the two $ tip states. The same accidental degeneracy happens in the case of C29 at 6:9 eV, just below the HOMO panel (Fig. 9c). 11 One can observe, in going from C9 to C29 , the formation of $ and  bands, displaying the reminiscence of van Hove singularities typical of 1-D systems [19,1,47]. In order to distinguish the contribution to the DOS of the $- and -states, we have plotted them separately, for the longer chain C29 , in Fig. 10a and b, respectively. One can see how both $- and -states tend to form two bands, corresponding to bonding and anti-bonding states. While the $ (m = 0) bands (Fig. 10a) are separated by a large gap (∼15 eV), with the two quasi-degenerate tip states isolated in the middle, the  (m ± 1) bands (Fig. 10b) are close to each other, with the higher one lying in the same (continuum) region of the antibonding $ band. The features discussed above have direct consequences on the dipole electromagnetic response of the system, which will be addressed in Section 4.2. Longitudinal dipole single-particle transitions obey the selection rule Zm = 0 (cf. Eq. (6) and Fig. 5). Hence, they are allowed only between $–$ and – states, separately, the energies of the particle–hole excitations associated with -states being smaller than those associated with $-states. Furthermore, the lower-lying $ band is completely Glled, while the anti-bonding $ band is completely empty, thus allowing inter-band transitions only. On the other hand, the lower-lying  band is semi-occupied, lying across the Fermi energy, thus allowing both intra-band and inter-band transitions. As a consequence, $–$ and – transitions are expected to contribute to diFerent parts of the response of the system. In particular, the (UV=optical) plasmon resonance should be due to – transitions. Concerning the transverse dipole response, associated to the operator Y1; ±1 , the selection rules Zm = ± 1; i × f = − 1 (cf. Eq. (6)) allow only $– transitions. Lying the occupied $-states below −15 eV (with the exception of the two tip states, 11 The existence of the C9 and C21 chains has been established (cf. Refs. [85,45], respectively, and Section 2), while the longer chain C29 has not been observed. We have introduced it exclusively in order to better appreciate the evolution of the DOS with increasing chain length.

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Fig. 9. Density of states (DOS) for: (a) C9 , (b) C21 and (c) C29 . The corresponding cLDA spectra are shown in Fig. 8, and were convoluted according to Eq. (7) with ' = 0:1 eV, including all the cLDA states in the range (−25; 10) eV. HOMO–LUMO states are located at (−7:5; −5:9); (−6:5; −5:8), and (−6:3; −5:7) eV for C9 , C21 and C29 , respectively. The existence of the C9 and C21 chains has been established (cf. Refs. [85,45], respectively), while the longer chain C29 has not been observed yet. We have introduced it exclusively in order to better appreciate the evolution of the DOS with increasing chain length (see text). Fig. 10. Density of state for C29 : (a) density of $-states; (b) density of -states and (c) total density of states. See caption of Fig. 9 and the text.

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T 3) Fig. 11. Illustrative example of cTDLDA calculations for C5 and C6 . (a) Longitudinal dipole strength S(E) (A for C5 as a function of energy, shown up to 30 eV, both for single particle (cLDA) and full response (cTDLDA). The cTDLDA strength has been multiplied by a factor 2 for clarity. (b) Accumulated fraction of the dipole EWSR M1 (E)=M1 as a function of energy up to 60 eV, both for longitudinal and transverse responses. (c) Longitudinal dipole strength for C6 . In the inset the cTDLDA response is enlarged and compared with CI calculations (short dashed) and experimental results (long dashed) from Ref. [46].

cf. also Fig. 8), the lower transitions to unoccupied states have energies of the order of 10 eV or more and the associated response is expected to show no features below this energy. These qualitative considerations will be conGrmed by the quantitative results described in Section 4.2. 4.2. Electromagnetic response We discuss here the electromagnetic response of linear carbon chains, in particular the plasmon resonances and their associated microscopic wavefunctions and transition densities. 4.2.1. Plasmon resonances In Fig. 11 we show a representative plot of the electromagnetic response of the linear carbon chains C5 and C6 . In Fig. 11a we show the dipole longitudinal strength function S(E) of C5 ,

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both for single-particle (cLDA) 12 and full cTDLDA response. The strength was smeared out with a Lorentzian parameter ' = 0:1 eV (cf. Eqs. (B.3) and (B.5)). In Fig. 11b the ratio between E   the accumulated strength, M1 (E) = 0 d E E S(E  ), and the EWSR deGned in Eq. (B.19) is displayed as a function of the energy, for both the cLDA and the full cTDLDA response. In addition, the transverse response is also shown. The longitudinal dipole cLDA response shows a single resonance, peaked around 2:75 eV, and other very small features at higher energies, (in the range 10 –15 eV and around 20 eV). From Fig. 11b one can see that the resonance of the C5 chain exhausts about 34% of the total EWSR M1 , while the other peaks, due to their higher-energy weights in M1 , provide the remaining contribution, bringing it up to 91.6% when integrating up to 60 eV. The inclusion of the residual interaction in the response shifts the LDA peak to higher energies, testifying to its repulsive character. The TDLDA peak is now located at 6:4 eV, taking 30% of M1 , while the remaining strength has been shifted to higher energies, up to about 45 eV, where the percentage of exhausted M1 reaches the LDA one Fig. 11b, continuous and long-dashed lines). The exchange–correlation interaction is responsible for a minor red-shift of the TDLDA peak, typically of the order of 1% or less of the total shift, testifying its weak attractive character. From Fig. 11b it is clear that the dipole transverse response of C5 is completely diFerent. It is smeared out along the spectrum and it shows no distinct features. In fact, both the cLDA (dotted line) and the cTDLDA (dot–dashed line) responses do not display sharp rises, and increase considerably slower than the longitudinal response. In particular, the cTDLDA transverse response of C5 shows no excitations below 10 eV, coinciding with the cLDA response at higher energies, beyond 100 eV (not shown in the Ggure). Again, the residual interaction has an important overall eFect in shifting the TDLDA dipole response to higher energies as compared to the unperturbed (LDA) response. The diFerent features of the longitudinal and transverse dipole responses can be understood as a consequence of the diFerent mobility along the longitudinal and transverse directions of the delocalized valence electrons of the chain in this quasi-1-D system, and on the distribution of the DOS, as discussed in Section 4.1 in connection with Figs. 9 and 10. In what follows, we shall discuss only the longitudinal response. Our results for C5 agree very well with the TDLDA calculations reported in Ref. [67]. The experimental literature reports for C5 a resonance at 5:4 eV, but it is not assigned to a longitudinal dipole transition [85]. Concerning the even chain C6 , Fig. 11c, the dipole response shows the same characteristics discussed for C5 , but, because of the partial occupancy of the HOMO state, there are two low-lying single-particle transitions approximately of the same energy (cf. Fig. 5), and the unperturbed cLDA response displays a double peak with similar strengths. The residual interaction shifts the strength almost entirely to the state at higher energy, leaving to the lower-lying state a very small fraction of the EWSR. This is better appreciated in the inset of Fig. 11c, where the low-energy response is enlarged and compared with recent experimental and theoretical results [46]. Our cTDLDA calculations predict two states at 2.9 and 6:0 eV, with 0.04 and 7.7 units of oscillator strengths, respectively. For the higher lying state, the same theoretical result is reported in Ref. [67]. The CI calculations of Ref. [46] predict two peaks at 2.7 and 5:7 eV, 12

The single-particle LDA strength is also denoted as “unperturbed” or “uncorrelated”, in the sense that the screening eFects due to the residual interaction present in the TDLDA are not included.

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Fig. 12. The dipole response of carbon chains CN ; (full circles) cTDLDA peak energies (eV) versus the number of carbon atoms N ; (full diamonds) experimental data (from Refs. [126,85,45,46]. The value of 5:4 eV for C5 is not assigned to a longitudinal dipole transition in Ref. [85]). The values at lower energies for N = 6; 8; 10 corresponds to the lowest dipole transitions, having small oscillator strengths (see text and Table 1). Inset: the corresponding wavelengths  (nm). The lines are Gts as explained in the text, Eqs. (8) – (14). Both theoretical and experimental wavelengths display a nonlinear scaling.

with 0.023 units of oscillator strength for the lower-lying one. 13 The experimental values are 2.42 and 5:22 eV. We see that the theoretical values are all appreciably blue-shifted. We have studied systematically the longitudinal dipole response of linear carbon chains CN with odd N 6 21 and even N 6 10, where a comparison with presently available experimental data is possible, partly from very recent experiments [45,46] (cf. Section 2). The associated response functions show features similar to those illustrated above in the case of C5 and C6 . In Fig. 12 we show the cTDLDA peak energies of the strength function for carbon chains CN versus the number of carbon atoms N , compared to the experimental results (from Refs. [45,46,85,126]). For even N chains (N = 6; 8; 10) the lowest energy states are also reported. In the inset the corresponding cTDLDA peak wavelength  are shown. The same data are reported in Table 1. The new experimental results for even N chains [46] agree with similar systematics of the older odd N data [85,126]. Our results can be compared with those reported in Ref. [67] for N 6 15, where only very minor diFerences are found, of the order of 0:1 eV, probably due to the use of diFerent numerical methods. 14 This fact shows that the systematic diFerence between the theoretical and the experimental values, which are always lower in energy, is a consequence of the TDLDA itself and not an artifact due to the numerical implementation. The 13 In Ref. [46] the oscillator strength for the higher-lying state at 5:7 eV is not reported, probably because of the high computational cost required by the CI calculation. 14 The authors of Ref. [67] implemented the numerical solution of TDLDA equations within the real-time method, and used a Troullier–Martins pseudopotential [98,99].

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Table 1 Dipole response of the carbon chains CN a N 3 5 6 7 8 9 10 11 13 15 17 19 21

exp (nm)

cTDLDA (nm)

Eexp (eV)

EcTDLDA (eV)

EcLDA (eV)

M1 (%)

f

191 230 238 511 253 277 640 295 333 770 336 379 420 461 504 544

151 194 207 427 221 244 539 259 273 636 290 326 356 394 427 459

6.55 5.40 5.22 2.42 4.90 4.47 1.94 4.20 3.72 1.61 3.69 3.26 2.95 2.69 2.46 2.28

8.2 6.4 6.0 2.9 5.6 5.1 2.3 4.8 4.5 1.9 4.3 3.8 3.5 3.1 2.9 2.7

4.0 2.7 2.2,2.9

25 30 32 10−3 31 34 10−3 35 33 10−3 36 36 37 37 37 37

3.0 6.0 7.7 0.04 8.7 10.9 0.05 12.6 13.2 0.04 15.8 18.7 22.2 25.2 28.1 31.1

2.1 1.8,2.2 1.7 1.5,1.8 1.4 1.2 1.05 0.95 0.85 0.78

a

(Column 2) The experimental values for the wavelength of the main absorption peaks from Ref. [126] for N = 3, [85] for odd N ∈ [5–15] (The value for C5 is not assigned to a longitudinal dipole transition), [45] for odd N ∈ [17– 21], and [46] for even N ∈ [6–10]; (column 3) the theoretical cTDLDA values; (column 4) the corresponding experimental energies; (column 5) the theoretical cTDLDA energies; (column 6) the theoretical cLDA single-particle energies. Even N chains display two low-lying single-particle states with similar energies and strengths (cf. Section 4.2); (column 7) the partial EWSR M1 associated to the cTDLDA peaks; (column 8) the corresponding oscillator strengths, Eq. (B.21). Both the experimental and the theoretical spectrum of even N chains display two dipole absorption peaks with very diFerent oscillator strengths f (the largest f in the UV, the other in the IR) [46].

percentual diFerence in wavelength tends to 17% for longer chains. Such a systematic deviation of the TDLDA results remains to be understood, and has also been found in connection with the study of the optical response of other systems, like alkali metal clusters (cf. e.g. Ref. [127] and refs. therein). We shall return to this subject below in Section 4.4. In connection with this diFerence, we note that no absolute value of the cross-section is reported in the experimental literature. This is an unfortunate fact, because a proper understanding of this quantity is as important as the centroid energy of the resonance. In fact, a systematic discussion of this quantity in comparison with the experiments has been instrumental in the understanding of the giant resonances in atomic nuclei [128]. The theoretical cTDLDA oscillator strengths are shown in Table 1. In column 7 we display the percentage of the EWSR M1 exhausted by the resonance. One can see how it grows from 25% and stabilizes around 37% already at N = 11. In column 8 we display the values of the f-sum rule, that is, the number of electrons involved in the resonance. They are proportional to the dipole cross-section (cf. Eq. (B.14)). In Fig. 12 we display the energy centroids of the photoabsortion peaks as a function of N . A simple parametrization of this dependence can be made in terms of the (classical) expression

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[8] for the Mie plasmon resonance [129] of an ellipsoidal metallic particle, 15 applicable also in the case of rods √ E = ˝!0 L ; (8) 

 1 1+e 1 − e2 − L= 1 + log ; (9) e2 2e 1−e

R⊥ 2 2 e =1 − ; (10) R where ˝!0 is the bulk plasmon energy, L is the axial depolarization factor, and e is the ellipsoid geometrical parameter, depending on the ratio between the short (R⊥ ) and long (R ) radii. We are able to accurately parametrize both the cTDLDA and the experimental results, shown by the lines in Fig. 12, using T ; R⊥ = 1:2 A

(11)

1 R = (N − 1)abond + aspill ; (12) 2 T is the bond length in our carbon chains and aspill = 1:45 A T is a parameter where abond = 1:3 A describing the spill-out of the electron density at the tips of the chains, with the only parameter exp ˝!0 , adjusted to ˝!0cTDLDA = 21:7 eV and ˝!0 = 18:6 eV. A value of this order can be obtained 2 2 from the standard plasmon relation, ˝ !0 = 4(˝2 e2 =m), inserting the density f = ; (13) V 4 (14) V = R⊥ R2 ; 3 where f is the number of active electrons participating to the plasmon excitation, given by the oscillator strength of column 8 in Table 1, and V is the volume of the ellipsoid. The classical relations given by Eqs. (8) – (14) reproduce well the whole set of data, for carbon T in length. This result is diFerent from what is generally chains ranging from 2.3 to 20 A found in the experimental literature (cf. e.g. Refs. [44,45,85]), where a linear N dependence of the plasmon wavelength is generally assumed, and used to identify unknown transitions in homologous series (cf. also Section 2). For large values of N the observed scaling can be approximately parametrized in terms of a linear behavior: cTDLDA = 108:6 + 16:7N and Exp = 123:0 + 19:9N . Note however that this marginal N dependence has no physical basis. 4.2.2. Plasmon wavefunctions An important question regarding the electromagnetic response of the molecules is to elucidate the degree of collectivity of the excited states. The Green’s function method in coordinate space 15

The same formulas used in the following have been applied to a smaller set of data in Ref. [5] (cf. also Ref. [67]).

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described above in Section 3.2 allows the direct calculation of the overall, energy-dependent strength function, avoiding a microscopic treatment in terms of particle–hole excitations. This is in fact a peculiar advantage of the matrix formulation of TDLDA, but its computational cost is high for many electron systems, where the dimension of the particle–hole basis grows very fast. An exception is the case in which the response of the system is mainly concentrated in a single resonance |f of energy Ef [115]. In this case one can view the excitation as a one-phonon state (plasmon) of energy Ef . The associated wavefunction
f f ∗ ,RPA (r) = [Xph p∗ (r) h (r) − Yph (15) h (r ) p (r )] ph

being a linear combination of particle–hole excitations [115,130]. The X , Y amplitudes, known as the forward- and backward-going amplitudes in the random-phase approximation (RPA), equivalent to TDLDA, can be extracted from the TDLDA strength function (B.13) as [115] f Xph = Nf

p| Hˆ |h ; ph − Ef

f Yph = Nf

h| Hˆ |p

ph + Ef

:

(16)

(17)

Here p| Hˆ |h are the particle–hole matrix elements of the induced interaction (cf. Appendix B, Eq. (B.6)), ph = p − h are the particle–hole excitation energies of LDA states, and Nf is a normalization constant to be determined by the condition
f 2 f (|Xph | − |Yph |2 ) = 1 : (18) ph

Once the X , Y amplitudes have been obtained, the transition strength (B.13) from the ground state |0 to the excited state |f can be expressed as 2
f f ˆ S(Ef ) = (Xph + Yph ) p| H |h : (19) ph We observe that this quantity is built up as sums over particle–hole states of single-particle matrix elements weighted with the TDLDA amplitudes X , Y . One can calculate the X , Y amplitudes introduced in Eqs. (16) and (17) in an approximate fashion by resorting to a simpliGed model characterized by an eFective two-body separable interaction between particle–hole states of the type K(r; r ) = 0F(r)F(r )

(20)

(cf. Eqs. (B.7), (B.8)). In keeping with the TDLDA calculation of the dipole longitudinal transition strength, we use in Eq. (20) the longitudinal dipole operator F(r) = z :

(21)

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485

Fig. 13. Longitudinal dipole response of the carbon chain C13 . (a) M1 (E)=M1 versus energy E for the cTDLDA (continuous line) and cLDA (dashed line) results. The inset displays the corresponding values for E up to 60 eV. T 3 ) versus energy E. The cTDLDA values have been multiplied by a factor 3 for (b) Strength function S(E) (A clarity. The experimental value of 3:27 eV [85] is also displayed (dotted vertical line, arbitrary units). (c) Xph (full circles) and Yph (crosses) amplitudes of the particle–hole transitions contributing to the cTDLDA peak in (b). The main transitions are indicated by the vertical lines in Fig. 5, and their X , Y amplitudes are reported in Table 2. Note that – transitions are doubly degenerate and hence must be counted twice (cf. caption to Fig. 5).

This approximation should be good when the response is mainly concentrated in a single resonance, exhausting the largest fraction of the strength [115]. The soundness of the approximation can be checked by recalculating the EWSR exhausted by the state |f as Ef S(Ef ), with S(Ef ) from Eq. (19) and by comparing it with the TDLDA value. The longitudinal dipole response of the carbon chain C13 is displayed in Fig. 13 as an illustrative example. 16 The fraction of the accumulated EWSR, M1 (E)=M1 , is shown in Fig. 13a up to 10 eV, indicating that about 36% of the 52 valence electrons of the system are involved in the main resonance, corresponding to an oscillator strength f = 18:7 (cf. Eq. (B.21)). The inset in Fig. 13a shows the same fraction up to 60 eV, where about 90% of the M1 EWSR 16

The deGnitive structural characterization of this molecule in 1994 [131] provided the Grst direct evidence for the existence of low-energy linear isomers for neutral carbon chains larger than C11 .

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Fig. 14. Accumulated fraction of the EWSR M1 exhausted by the cTDLDA longitudinal dipole resonance in C13 as a function of the particle–hole states, recalculated from the X; Y amplitudes as described in the text. The cTDLDA value (36%) is recovered within 3% only if all the particle–hole states up to 17 eV are included.

deGned in Eq. (B.19) is found. Fig. 13b displays the strength function S(E), calculated both in the cTDLDA and the cLDA. The experimental value of the resonance at 3:27 eV [85] is also reported. In Fig. 13c we display the composition of the wavefunction of the resonance in terms of its X , Y amplitudes, obtained as described above, versus the particle–hole conGguration energies. The largest (¿ 0:05) X , Y amplitudes corresponds to the Grst four (doubly degenerate) single-particle states. The other 33 states lying below 10 eV in the Ggure have smaller amplitudes, along with the remaining states up to 60 eV, the total number of single-particle states below 60 eV being 1098. In principle, one should then look only at these components. However, this would be a completely misleading description of the resonance. This can be better appreciated from Fig. 14, where we show the accumulated EWSR M1 as a function of the particle–hole states, calculated as Ef S(Ef ), with S(Ef ) obtained from Eq. (19) through the X; Y amplitudes. One can see that the fraction of M1 exhausted by the cTDLDA state (36%) is recovered only if all the particle–hole states up to 25 eV are taken into account. There are 274 of such states for the C13 chain. If only the Grst eight main components were included, a sensible fraction ( 10%) of the EWSR would be lost. Hence, in order to obtain an accurate microscopic description of the calculated 3:8 eV (plasmon) state, let us say, within 3% of the EWSR exhausted by it, one needs to include at least all the particle–hole components below 17 eV, the associated wavefunction containing of the order of 100 components. In Table 2 we display the larger ones. This result demonstrates the collective character of the plasmon modes, in which the valence electrons slosh back and forth with respect to the ions. It also shows how misleading can be a discussion (cf. e.g. Refs. [44,46]) of the properties of these states in terms of single-particle–hole excitations.

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Table 2 Detailed composition of the wavefunction of the longitudinal dipole resonance in the linear chain C13 , in terms of its X; Y amplitudes, calculated as described in Section 4.2a ph

m

ph (eV)

f Xph

1 3 5 7 9 11 13 15 18 21 23 26 27 32 33 36

          $ $ $ $ $ $

1.20 3.47 4.02 4.55 5.01 5.94 6.81 7.07 7.58 8.14 8.23 8.87 9.28 9.59 9.94 10.34

−0:636

0.289 0.247 −0:220 0.021 0.011 0.005 −0:001 0.012 −0:013 −0:003 −0:006 −0:012 −0:007 −0:018 0.005

f Yph

0.330

−0:019

0.011

−0:020

0.003 0.003 0.002 −0:000 0.004 −0:005 −0:001 −0:002 −0:005 −0:003 −0:008 0.002

ph

m

ph (eV)

37 41 44 48 49 54 55 74 75 101 103 111 112 113 116 117

   $ $ $ $ $ $ $ $ $ $ $ $ $

10.41 10.84 10.90 11.17 11.22 11.83 11.89 13.53 13.57 16.16 16.35 17.27 17.30 17.34 17.37 17.38

f Xph

0.003

−0:002 −0:003 −0:002 −0:011

0.005 0.002 0.014 0.014 0.001 −0:001 −0:001 0.001 0.002 0.003 0.002

f Yph

0.001

−0:001 −0:001 −0:001 −0:006

0.002 0.001 0.008 0.008 0.001 −0:001 −0:001 0.001 0.002 0.002 0.001

a The (basis) particle–hole components of the wavefunction are sorted according to their increasing energies (columns 3 and 8). In columns 1 and 6 we report the corresponding index, while in columns 2 and 7 we show f values larger than 10−3 are displayed their character (– or $–$ transitions). Only the particle–hole states with Xph f in columns 4 and 9. The corresponding Yph amplitudes are given in columns 5 and 10.  transitions are doubly degenerate (see text and caption to Fig. 5).

The approach from above to the asymptotic value shown in Fig. 14 can be analytically understood if we rewrite Eq. (19) as 17 2 2 Nph
Nph 2
f | p | z ˆ | h | f : S(Nph ; E)|E=Ef = (Xph + Yph ) p|zˆ|h ≈ (22) ph=1 ph=1 ph − Ef As one can see from Fig. 13c and from Table 2, the Grst two (actually four) particle–hole states are below the energy of the cTDLDA resonance at 3:8 eV, and contribute to the sum in Eq. (22) with a sign opposite to that of the other states at higher energies. The fact that the fraction of M1 exhausted by the peak, recalculated from the X; Y amplitudes (Eq. (22)), equals the value obtained directly from the cTDLDA strength, conGrms the soundness of the approximation used for the microscopic analysis of the resonances in such systems, where the strength is concentrated mainly in one single state.

17

We take into account that, in general, Xph Yph , as can be seen from Table 2, with the exception of the Grst state.

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From Figs. 13, 14 and Table 2 the following picture of the dipole longitudinal mode in the

C13 carbon chain emerges: the unperturbed cLDA peak is associated with a single-particle–hole state, the lowest in energy at 1:2 eV between the HOMO–LUMO levels (see Fig. 5), to which

it corresponds the largest single-particle dipole matrix element. Actually there are two of such states, because the – transitions must be counted twice (cf. caption to Fig. 5). The cTDLDA resonance is pushed up to 3:8 eV by the repulsive character of the residual interaction, and results from the interference of many elementary particle–hole excitations. One of them is still the lowest conGguration, but a large contribution to the wavefunction also comes from three other conGgurations below 5 eV. These four particle–hole states correspond to the four transitions already displayed in Fig. 5 (from left to right in order of increasing energy). From the values in Table 2 and Eq. (18), their contribution to the wavefunction is calculated to be about 97% (provided the right counting of the transitions: cf. the caption to Table 2), while, from Fig. 14 their contribution to the EWSR M1 exhausted by the resonance amounts to ≈ 90%. Hence, the wavefunction of the main longitudinal dipole resonance in the linear chain C13 can be written as |f − 0:64|1 + 0:29|2 + 0:25|3 − 0:22|4 + · · ·

+ 0:33|1\ − 0:02|2\ + 0:01|3\ − 0:02|4\ + · · ·

(23)

where |i denotes the ith particle–hole created from the ground state, and |i\ denotes the ith particle–hole destroyed in the ground state. Summarizing, the dipole resonance in the C13 chain has all the characteristics of a collective state. We have also systematically calculated the wavefunctions of these plasmon states. In Fig. 15 we show the main |Xph |2 (¿ 0:01) components of the single-particle–hole excitations contributing to the cTDLDA plasmons in Fig. 12 versus the corresponding particle–hole transition energies ph below 6 eV, for a subset of odd N chains. The amplitudes for the longer chains are similar and were not drawn in the Ggure for clarity. The same holds for the amplitudes associated to even N chains. The results are reported in Tables 3 and 4. For the shorter chain, C5 , we see that only one single-particle conGguration has a forward-going probability |Xph |2 ¿ 10−2 (actually, two, being a – transition) below 6 eV. This is precisely the conGguration contributing to the single-particle LDA response at 2:74 eV. Actually, there are other 10 conGgurations (mainly $–$) that give appreciable contributions, located in the range 8–20 eV. Three of them are shown in Fig. 5, left side, and all of them are reported in Table 3. For C7 the situation is quite similar, with 10 particle–hole conGgurations, two of which (actually, four) below 6 eV, are displayed in Fig. 15 (open squares). All of them are shown in Table 3. For odd N ¿ 7 the situation does stabilize, with eight conGgurations gradually shifting to lower energies. They fall below 6 eV (displayed in the Ggure, to be counted twice) for odd N ¿ 9, and are reported in Table 4. For even N there are always two low-lying states with large amplitudes. 4.2.3. Transition densities The collectivity of the resonances can be further clariGed through the study of the transition densities. When an external Geld acts on a system inducing an excitation from the ground state

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Fig. 15. Illustrative picture of the main |Xph |2 (¿ 0:01) components of the particle–hole transitions contributing to the cTDLDA peaks in Fig. 12 versus the corresponding particle–hole transition energies ph , for 5 6 N 6 13, and below 6 eV. For N = 5; 7; 9 there are, respectively, 10, 9, 4 other states contributing more than 5% located above 6 eV (not included in the Ggure). The amplitudes corresponding to longer chains are similar to those shown here, and are not reported for clarity. The lines are drawn as a guide. Note that – transitions must be counted twice (all the particle–hole states shown in the Ggure have – character).

Table 3 Main X; Y amplitudes (larger than 0.05) contributing to the wavefunction of the cTDLDA resonance in C5 and C7 chainsa N

m

ph

Xph

Yph

5

 $ $ $  $ $ $ $

2.74 8.14 9.59 10.25 10.42 11.29 13.03 14.12 20.18

−0:75 −0:06 −0:08

−0:01 −0:01

−0:06

−0:02

    $ $

2.07 5.39 6.97 8.16 9.57 10.41

0.61 0.44 0.05 −0:08 −0:07 0.06

−0:28 −0:01

7

a

0.18

0.09 0.09 0.07 −0:05

0.32

0.04

0.03 0.03 0.03 −0:03

0.01

−0:01 −0:02

0.02

(Column 2) Character of the particle–hole transition. – transitions contribute twice; (column 3) particle–hole energy; (column 3) Xph forward-going amplitude; (column 4) Yph backward-going amplitude.

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Table 4 Same as in Table 3, for the Grst four – transitions in CN (N = 9 − 21) N 9

11

13

15

17

19

21

m

ph

Xph

   

1.67 4.56 5.60 6.50

−0:56

0.49 0.09 −0:01

−0:01

   

1.40 3.94 4.68 5.36

−0:61 −0:39 −0:21

0.31 0.02 −0:01 0.02

   

1.20 3.47 4.02 4.54

−0:64

   

1.06 3.10 3.53 3.94

−0:66

0.21 −0:29 0.23

−0:02 −0:01

   

0.94 2.80 3.14 3.47

−0:55 −0:32

0.30 0.02 −0:01 0.02

   

0.85 2.55 2.82 3.10

−0:58

0.31 0.27 0.31

−0:02 −0:01

   

0.77 2.34 2.57 2.80

0.58 0.28 −0:20 −0:35

−0:32 −0:02

0.16

0.29 0.25 −0:22

0.35 0.24

Yph 0.27

0.01

−0:01

0.33

−0:02

0.01

−0:02

0.35

0.02

0.31

0.02

0.01

−0:02

|0 to an excited state |f , the transition matrix element can be written as the spatial convolution of the Geld with the transition density, f (r) [115]  ˆ

f|V |0 = dr V (r ) f (r ) : (24)

Equivalently, the transition density can be deGned as the matrix element of the density operator between the states |0 and |f , f (r) = f|ˆ|0 :

(25)

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491

−1

T ) for the carbon chain C13 , associated to the cTDLDA plasmon Fig. 16. Longitudinal dipole transition density (A T along the chain. The continuous line represents cTDLDA transition density, state at 3:8 eV, versus position z (A) the dashed line the cLDA transition density and the black dots the carbon atoms. Only half of the chain is shown, being the longitudinal dipole transition density an odd function of z. Note that the curves go to zero well before the T beyond the tip atom, i.e., at z = 11:8 A. T wall of the cylindrical box used in the numerical calculation, located 4 A T for a subset of carbon chains CN , associated T −1 ) versus z (A) Fig. 17. Longitudinal dipole transition densities (A to the plasmon states calculated within the cTDLDA and shown in Table 1.

Hence, it represents the density variations associated to the transition |0 → |f , and contains all the information regarding the spatial response of the system to the external Geld. In the case of a dipole axial transition density we have, by integrating over the ;  variables in Eq. (24),  2 2me xi f ff = 2 2 Ef d z z  (z) : (26) ˝ e In terms of X; Y amplitudes, Eqs. (16), (17) the transition density Eq. (24) is given by
f f f (r) = (Xph + Yph ) p∗ (r) h (r) (27) ph

f f built up as a sum over particle–hole states weighted with the TDLDA amplitudes Xph , Yph . The cLDA and cTDLDA longitudinal dipole transition densities for the C13 chain are shown in Fig. 16 (dashed line and continuous line, respectively). The unperturbed cLDA transition density corresponds to the two degenerate HOMO–LUMO transitions. Hence, its spatial (axial) dependence reQects that of the HOMO–LUMO wavefunctions. The longitudinal cTDLDA transition density depends on many single-particle wavefunctions, weighted by the cTDLDA X , Y amplitudes of the resonance, with eight main contributions (cf. the composition of the plasmon wavefunction, Eq. (23)). The continuous line in Fig. 16 shows how the contribution of the wavefunctions other than the HOMO–LUMO involved in the longitudinal dipole response redistributes the spatial response of the system over its length. Furthermore, we note a tail in

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the cTDLDA transition density, extending outwards from the last atom further than the cLDA one. This shows the presence of an increased spill-out of the electrons excited collectively in the plasmon state with respect to the cLDA spatial response, 18 and is responsible for the large static polarizabilities displayed by these systems. The comparison among the transition densities of various chains is given in Fig. 17. We note how the collective resonance is spatially distributed along the chains, in particular for longer chains, which display also an increasing number of oscillations in the transition density, due to the periodicity of the wavefunctions involved in the response. The latter correspond mainly to the states showing the larger X , Y amplitudes in Tables 3 and 4. Also, all the curves show a spill-out of the transition density from the tip of the chain. This spill-out is larger than the unperturbed cLDA results, as observed in Fig. 16, and decreases with the increasing length of the chain. For example, the C5 chain shows a cTDLDA spill-out ∼50 times larger than the cLDA one, while for the C9 chains it is reduced at ∼10 times. This reQects the evolution towards a classical behavior with increasing chain length. In principle, the spatial response of the system excited in the plasmon state could be measured in electron scattering experiments, as it enters the associated reaction cross-section (cf. e.g. Refs. [115,132,133]). It is a well studied quantity in nuclear physics (for a review see e.g. Ref. [133]). 4.3. Static linear polarizabilities The static polarizability measures the ability of the valence electrons to Gnd an equilibrium conGguration which screens a static external Geld. Hence, molecules with many delocalized valence electrons should display large values of static dipole polarizabilities. This is expected in particular for carbon chains, whose sp structure is rich of  bonds delocalized along the entire body of the system (cf. Fig. 6). Analogous 1-D carbon systems, like polyacetylene molecules (cf. e.g. Refs. [111,134 –137] and refs. therein), should display smaller static dipole polarizabilities, due to their sp2 bonding character and, hence, smaller degree of  delocalization. Also the high 1-D character of the chains is expected to have interesting consequences. For example, carbon rings have the same sp bonding character and degree of  delocalization, but their 2-D geometry makes more diPcult the redistribution of their valence electrons to screen an external electric Geld, both in the in-plane and in the perpendicular direction to the plane. Hence, they are expected to display a static polarizability smaller than linear carbon chains of equivalent mass number (cf. Section 4.5). These peculiarities of the polarizability of carbon chains could be used to characterize and separate structures with the same mass number. As described in Section 2, one of the major problems in the production, storage and characterization of small carbon structures in the range N 6 20 is the contemporaneous presence of linear and ring structures of the same mass and competing stability, which cannot be separated out in a mass spectrometer, and require sophisticated techniques, such as ion mobility measurements. A sensible diFerence in polarizabilities could be used to separate diFerent geometries with the same mass within an inhomogeneous electric Geld, in a “Stern–Gerlach-like” device in sequence with a mass spectrometer. This is 18

This is a consequence of Eq. (26), when the integral is calculated outwards from the last node in the transition T in the case of C13 . density, at ∼8 A

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Table 5 T 3 ) of the carbon chains CaN Static dipole polarizabilities (A N

zz "cLDA

zz "cTDLDA

zz "cTDLDA

"cTDLDA =N

"emp =N

3 4 5 6 7 8 9 10 11 13 15 17 19 21

26.7 47.0 106.0 128.6 267.9 332.9 536.9 619.1 942.2 1518.7 2261.4 3220.8 4385.5 5907.7

7.0 12.3 20.2 28.3 38.0 54.0 67.4 84.8 104.6 154.9 216.2 293.8 384.6 493.4

6.8 11.8 19.4 27.5 37.0 51.7 65.7 82.9 102.3 151.7 211.9 288.3 377.8 485.4

0.8 1.0 1.3 1.6 1.8 2.2 2.5 2.9 3.2 4.0 4.8 5.8 6.7 7.8

1.2 — 1.8 2.1 2.3 2.9 3.3 4.1 4.3 5.4 6.7 7.9 9.3 10.9

a

(Column 2) cLDA values for the longitudinal component "zz ; (column 3) cTDLDA values; (column 4) cTDLDA values without the contribution of the exchange–correlation term; (column 5) cTDLDA values for the total polarizability per atom, obtained from column 3 as described in the text; (column 6) empirical cTDLDA values corrected as described in connection with the discussion carried out in Eq. (30). The value for C4 is not reported due to the lack of the experimental plasmon energy. We report its calculated polarizability for completeness.

a well-known technique, used, e.g. to measure static polarizabilities of metal clusters (cf. e.g. Refs. [138,139]). To our knowledge there are no experimental data on the polarizability of linear carbon chains (cf. e.g. Linde [140]), probably because the main experimental activity, in particular for long chains, has been focused on the spectroscopic properties, in connection with astrophysical observations and the interpretation of DIB [44]. In TDLDA, the static polarizability is found as the limit of the dynamical linear polarizability to zero energy. The TDLDA polarizability contains the screening eFects which are not included in the LDA (cf. e.g. Ref. [118]). We have calculated the longitudinal dipole polarizabilities for linear carbon chains CN with zz zz odd N 6 21 and even N 6 10. In Table 5 we show the values (indicated as "cLDA and "cTDLDA ) calculated both in cLDA and in cTDLDA, as a function of the number of carbon atoms N . In order to test the TDLDA approximated treatment of exchange and correlation eFects, we have also calculated the static polarizabilities by excluding the exchange–correlation contribution to the TDLDA residual interaction (reported in column 4 of Table 5). This contribution is sizable, of the order of (2– 4)% (cf. the discussion below in connection with Eq. (30)). In column 5 of Table 5 we report the total polarizabilities per atom. Being the dipole transverse response very small (cf. Fig. 11), the corresponding transverse polarizabilities are negligible, in particular for long chains, and we obtained the values given in column 5 of Table 5 simply by multiplying the values given in column 3 for the longitudinal polarizabilities by a factor 13 , and dividing by the number of atoms in the chain.

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T 3 ) of carbon chains CN versus N obtained within the cTDLDA Fig. 18. Static longitudinal dipole polarizability "zz (A (full circles), compared to the independent electron (cLDA) polarizability (full squares). The lines are Gts as explained in the text. The data are taken from Table 5, columns 2 and 3. The open diamond corresponds to the T 3 [140]. calculated value of " for the isolated carbon atom C1 ; " = 1:76 A

We have also taken into account the possible source of error due to the TDLDA overestimation of the plasmon energy discussed above. The conservation of the EWSR M1 given in Eq. (B.16) requires, for the plasmon, a decreasing strength for an increasing energy. But the M−1 sum rule (Eq. (B.17)) requires an increasing polarizability for a decreasing energy. Hence, a blue-shift in the calculated plasmon energy should result in an underestimated polarizability. From the knowledge of the experimental plasmon energies, given in Table 1, we can derive a corrected polarizability that we denote as "emp . Being the longitudinal dipole strength mainly concentrated in the plasmon resonance, we can approximate the M−1 and M1 sum rules as  +∞ S(E) Splasmon M−1 = dE ; (28) E Eplasmon  0 M1 =

+∞

0

d E E S(E) Splasmon Eplasmon ;

(29)

TDLDA )2 . Assuming that which, using Eq. (B.17) in Appendix B, yields "TDLDA ≈ 2M1 =(Eplasmon exp )2 , we obtain the relation "emp = 2M1 =(Eplasmon

TDLDA 2 Eplasmon : (30) "emp "TDLDA exp Eplasmon

The empirical polarizabilities obtained from Eq. (30) are shown in the last column of Table 5. We expect that these empirically derived static polarizabilities should provide an account of the experimental ones. The results of Table 5 (columns 2 and 3) are displayed in Fig. 18, in a linear log scale. Also the value for the isolated carbon atom is reported. 19 19

So far, the static dipole polarizability has been measured for less than 20 elements. For the C atom Ref. [140] reports a theoretical value.

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495

As already done above for the plasmon energies in Fig. 12, we can parametrize the values of " in terms of the classical longitudinal dipole polarizability for a metallic ellipsoidal particle [5,8] V −1 "zz (N ) = ; (31) 4 1 + L( − 1) T V = 43 R R2⊥ is the volume of the where L is given in Eq. (9), R in Eq. (12), R⊥ = 1:2 A; ellipsoid, and  is the dielectric constant of the chain. We are able to reproduce the cTDLDA values, shown by the continuous line in Fig. 18, with the parameterization cTDLDA = 1+1:2N 1:34 . In the case of very elongated metallic particles, with R R⊥ , one Gnds the asymptotic behavior ∼ R3 =[3 log(R =R⊥ )], as derived analytically in Appendix C. 20 Using the above values "zz (N ) =  for the radii R and R⊥ , our calculated cTDLDA values in Fig. 18 are well reproduced by this asymptotic expression already for N larger than ∼5. The dashed line in Fig. 18 reproduces well the free (cLDA) polarizabilities via the formula zz "cLDA = V ( − 1)=(4), where  is the static dielectric constant of a free electron gas in a Gnite zz cylinder (cf. Ref. [19] and Appendix C), and shows a scaling behavior "cLDA (N )∼N 3 . This scaling law can be understood also as follows: for 1-D systems we have [141]  (z)∼ F (z) ; (32) F (z) = F0 − eEz ;  L=2 zz d z z(z) = "cLDA E; Pz = −L=2

(33) (34)

where F (z) is the Fermi energy of the system in an external static electric Geld and P is the z component of the polarization vector. By inserting Eqs. (32) and (33) into Eq. (34) we obtain zz the scaling law "cLDA (N )∼N 3 . To our knowledge, no experimental information regarding the polarizability of linear chains is available (see e.g. Ref. [140]). Such measurements would be of interest in order to further test the predictions of the TDLDA and other theoretical models, and also, to carry out a comparison with the ring isomers and with polyconjugated carbon chains. The latter are the closest systems to our carbon chains, characterized by sp2 hybridization, showing large static linear polarizabilities, as can be seen from Table 3 of Ref. [137]. We report here those results and compare them in Fig. 19 with the empirical values of the last column of Table 5, obtained from Eq. (30). In order to compare the polarizabilities of polyacetylenes and linear carbon chains, we must specify how to consider diFerent types of atoms in the molecules. A sensible criterion seems to be comparing structures with the same number of valence electrons. 21 As one can see from Fig. 19, for small systems, such as N = 3, the values of the calculated polarizabilities for the cumulene C3 and the polyene C2 H4 almost coincide. Similar values are still observed for C5 20

A more general derivation of the same expression can be found in Ref. [67]. A more reGned criterion could be to count the number of  electrons, since the electrons localized in the $ orbitals are much more rigid against the inQuence of an external polarizing electric Geld, and contribute much less to ". 21

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T 3 ) of carbon chains CN versus the number of carbon atoms N obtained Fig. 19. Total static dipole polarizability " (A within the cTDLDA (full circles, cf. Table 5, column 6), compared with the theoretical values for polyconjugated carbon chains (polyenes, open squares) reported in Table 3 of Ref. [137] and with the experimental value for C4 H6 (open diamond) taken from Ref. [140]. In order to compare both types of systems, we refer to the number of valence electrons. For the polyenes, having 10n + 2 valence electrons with n ¿ 1, the eFective value of N is given by N = (5n + 1)2. Note that for C3 and C2 H4 the values of the polarizabilities almost coincide.

and C4 H6 , which are very close to the experimental value for the latter [140]. For larger N , the polarizabilities of the cumulenes grow faster than for the polyenes indicating a higher degree of valence electron delocalization. Indeed, in the case of polyenes, the sp2 hybridization due to the presence of hydrogen atoms, yields alternating single and double bonds along the backbone of the chain. This charge distribution is clearly much less polarizable than for cumulenes for large number of carbon atoms, which for N = 21 already amounts to about a factor of four. 4.4. Criticism to the model Systematic calculations of the linear response of alkaline clusters carried out both within the framework of the jellium model and the RPA formalism (cf. e.g. Ref. [142]) and of LDA plus TDLDA (cf. e.g. Refs. [118,127] and refs. therein) lead to centroids of the photoabsorption cross-section which are (10 –15)% blue-shifted as compared with the experimental values. From Fig. 12 and Table 1, it is seen that the dipole resonances of linear carbon chains from C5 up to C21 calculated in the cTDLDA display similar blue-shifts as compared to the experimental data. Such a systematic deviation had already been reported [5,67] for linear chains up to N = 15. In Ref. [124] the same eFect is observed for sodium clusters. 22 The authors of Ref. [67] conclude that the results obtained are consistent with the precision assumed for the TDLDA model, that they estimate to be about 10% for the frequency of the strong transition. The authors of Ref. [124] (paper devoted to a test of the TDLDA model), concluded as well that the TDLDA is 22

In the case of Na8 , similar discrepancies are found in quantum chemical calculations (cf. e.g. Ref. [143]).

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Table 6 Test of the cTDLDA exchange–correlation interaction on the C5 plasmon energy (column 2) and polarizability (column 3), as a function of a multiplicative factor  × Vxc , displayed in column 1a 

Eplasmon (eV)

T 3) " (A

E =E=1

0 1 2 4 6

6.5 6.4 6.2 5.9 5.5

19.4 20.2 21.1 23.3 26.5

1.02 1.0 0.98 0.92 0.86

(+1:6%) (−2:3%) (−7:8%) (−14:1%)

(−4:0%) (+4:4%) (+15:3%) (+31:2%)



"=1 ="

1.02 1.0 0.98 0.93 0.87

a

In parenthesis the corresponding percentual variations with respect to the full Vxc ( = 1) are reported. In columns 4 and 5 we display also the ratios between the energies and the square root polarizabilities (with respect to  = 1). They well fulGll Eq. (30).

in “very good agreement with the experiment”. However, a closer view of Fig. 1 in Ref. [124] for the dipole response of the Na8 cluster reveals deviations comparable with the ones we Gnd for carbon chains. It is an open question whether the systematic blue-shift is connected with the limitations of the TDLDA or to its parametrization. An important eFect which is neglected in TDLDA are core polarization processes, eFects which have been found to play an important role in the case of plasmons (giant dipole resonances) in other Gnite many-body systems like atomic nuclei [128]. In this case, however, the large eFects associated with on the energy shell transtions, and thus to a damping width of the mode, masks the eFect of virtual transitions on the energy centroid of the resonance. In any case, core polarization eFects can produce a softening of the linear photoabsorption response which in the nuclear case can amount to ≈ (7–10)% of the energy centroid, smaller but within a factor of 2 equal to that observed in the case of linear chains (17%). Another source of discrepancy may be related to the parameterization of the diFerent terms entering in the TDLDA equations. The residual interaction used in the TDLDA consists of two terms, Hartree and exchange–correlation. Within the linear response approximation, they are derived as the functional derivative of the corresponding terms of the Kohn–Sham Hamiltonian. No new approximations are introduced in the largest, repulsive Hartree term. Instead, the smaller, attractive exchange–correlation term is treated within a double local density approximation [106]. To which extent the local energy functionals commonly used to describe exchange–correlation properties of the ground state within LDA, maintain a good level of accuracy when used for the calculation of excited states within TDLDA is a debated subject (cf. e.g. the recent CECAM workshops [148,149]). In Table 5 (column 4) we display the eFect of excluding the exchange–correlation term from the cTDLDA calculation. It leads to a decrease of the polarizabilities in column 4 of the order of (2– 4)%. To further investigate the importance of the exchange–correlation term in the linear response of the system, at least at a qualitative level, we show in Table 6 results obtained for the plasmon energy and the static polarizability of the C5 chain (columns 2 and 3, respectively) as a function of a multiplicative factor  of Vxc . The full Vxc interaction ( = 1) locates the plasmon resonance at 6:4 eV, with a 18% blue-shift with respect to the experimental

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value of 5:4 eV reported in the literature (cf. Table 1). If Vxc is turned oF ( = 0), we have a further blue-shift to Eplasmon = 6:5 eV, in keeping with the attractive character of the exchange– correlation interaction. For increasing values of  the plasmon energy is lowered towards the experimental value. Assuming  = 2 plus a 10% softening arising from core polarization, brings theory (≈ 5:6 eV) in essential agreement with the experimental data. Similar results are obtained for longer chains. 4.5. Comparison between chain and ring isomers We report here also a comparison of the electronic structure and the electromagnetic response for linear and ring carbon isomers. We study in particular the case of C10 , whose ring isomer has been recently observed [46]. We have relaxed the ring structure with a CPMD calculation, and have found a D10h geometry, consistent with both the experimental observations [46] and with previous theoretical calculations T [40 – 42]. The resulting interatomic distance was found to be 1:28 A. In Fig. 20, we show the cLDA electronic structure of both ring (left-hand side) and chain (right-hand side) isomers. In the case of the ring, the wavefunctions are characterized by deGnite parity (for the transformation z → −z) and by azimuthal quantum numbers m ± 10k (for integer k). To be noted is that the ring has a closed-shell structure, yielding a robust HOMO–LUMO gap of 3:6 eV. This fact is connected to the reciprocal saturation of the two dangling bonds at the tips of the linear chain (cf. Fig. 6) taking place in the monocyclic structure. In Fig. 21, we show the calculated dipole responses for the ring. In Fig. 21a, we display the in-plane strength function S(E) as a function of energy E, for both the unperturbed (cLDA) and the full (cTDLDA) cases. The strength function displays characteristics similar to those of the longitudinal response of linear even N chains (cf. Fig. 11). The in-plane cLDA response (for a Geld parallel to the molecular plane) shows two peaks of similar energies (3.7 and 4:2 eV) and oscillator strengths (about f = 4:8 for each), corresponding to the HOMO → LUMO and HOMO-1 → LUMO + 1 single-particle transitions, respectively, which are allowed by the in-plane dipole selection rules Zm = ± 1; i × f = 1. As expected, the residual interaction shifts the full cTDLDA strength to higher energies, yielding a main resonance at about 7:2 eV, with f = 6:4 and a peak exhausting a small fraction of the oscillator strength (f = 0:16) at 4 eV: 23 In Ref. [67] a TDLDA value of 7:2 eV is reported for the higher resonance, associated with an oscillator strength f = 6:9, in good agreement with our result. Experimentally, a resonance at 3:92 eV is reported in Ref. [46], and seems to correspond to the cTDLDA lower resonance at 4 eV. The accumulated fractions of the EWSR for the in-plane cLDA and cTDLDA responses are shown in Fig. 21b, along with the corresponding quantity in the case of a Geld orthogonal to the molecular plane. The latter displays, in turn, characteristics similar to those of the transverse response of linear even N chains (cf. Fig. 11b), being essentially structureless and further shifted towards higher energies. These results can be understood qualitatively in terms of the diFerent 23 To be noted is that the strength function corresponding to the linear C10 isomer displays the plasmon peak at about 4:5 eV with an oscillator strength of f = 13:2, larger than that of the ring, and a lower peak at 1:9 eV carrying a smaller oscillator strength of 0:04 (cf. Table 1).

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Fig. 20. Energy spectrum of C10 carbon ring (left side) and carbon chain (right side) for negative energies. Full=dashed lines represent occupied=empty states, and long=short lines double=single degeneracies. For the ring, also the parity and the azimuthal quantum numbers corresponding to the D10h symmetry are shown for each state (see text). The two quasi-degenerate tip states of the chain structure at about −7:4 eV saturate each other in the ring conGguration, thus resulting in a closed shell structure with a HOMO–LUMO gap of 3:6 eV. The HOMO states have similar energies for both structures, around −6:6 eV. Fig. 21. Electromagnetic response of carbon ring C10 . (a) In-plane dipole strength function S(E) as a function of energy E. The unperturbed (cLDA, dashed line) and the full (cTDLDA, continuous line) responses are reported up to 20 eV. The experimental result reported in Ref. [46] at 3:92 eV is shown (long-dashed vertical line). (b) Accumulated fraction of the in-plane dipole EWSR M1 (E)=M1 as a function of energy E up to 35 eV for both the cLDA (dashed line) and the cTDLDA (continuous line) responses, together with the corresponding transverse EWSR (dot–dashed).

“mobilities” that can be expected for the 40 valence electrons in the two geometries under the action of an external dipole Geld. We have also calculated the polarizability of the ring isomer C10 , and obtained the values T 3 for the in-plane and orthogonal polarizabilities, respectively, yielding a total 24.1 and 7:4 A T 3 . These values can be compared with the longitudinal and transverse polarizability " = 18:5 A T 3 , respectively, corresponding to a polarizabilities of the linear chain isomer, 84.8 and ≈ 2 A T 3 . These diFerent polarizabilities, which are in a ratio of about 2 for C10 , total value " ≈ 30 A suggest a practical method for the separation of small carbon isomers in electric Gelds, in a sort of Stern–Gerlach apparatus (cf. Fig. 22), which is already available within the present technology [138]. 4.6. Relation to DIB We address now the question of the possible relation between the plasmon resonances of linear carbon chains CN and DIB. In Fig. 23 we have plotted the observed values for DIBs,

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Fig. 22. Illustrative procedure for the separation of small carbon cluster isomers in a sort of Stern–Gerlach apparatus. The clusters CN produced by a source are mass-selected yielding a current of isomers with a deGnite value of N . The latter go through a region where a non-homogeneous electric Geld is present. The diFerent static dipole polarizabilities of the chain and the ring isomers induced by the Geld allow for their further separation. A similar apparatus has been employed, e.g. in Ref. [138] for the measure of the electric dipole polarizability of the Al atom T 3 , respectively). and dimer (6:8 ± 0:3 and 19 ± 2 A

T reported in Refs. [150 –152]. The corresponding Fig. 23. Observed DIB (full lines) in the range 4050 –5550 A peak heights (denoted as central depths Ac [150]) have been multiplied by a factor 200 for comparison, and the FWHMs have been discarded. Superimposed we show the plasmon peaks for carbon chains CN (dashed lines) built using the experimental energies (Table 1, column 2) and the theoretical cTDLDA oscillator strengths (Table 1, column 8).

taken from the compilation of Ref. [150 –152], by considering only the centroid of the peaks, disregarding their full-width at half-maximum (FWHM). We have then superimposed the plasmon peaks for the known carbon chains CN ; 3 6 N 6 21, using the experimental energies (Table 1, column 2) and the theoretical cTDLDA oscillator strengths (Table 1, column 8). T (with a few isolated peaks up to about While the DIB fall essentially in the range 4000 –8000 A T the plasmon resonances of CN chains occur in the range 2000 –5500 A. T As a conse13000 A), T quence, a superposition is possible in the range 4000 –5500 A, corresponding to 15 6 N 6 21. As can be seen from Fig. 23, and from Table 8, there are a couple of peaks which coincide, within the experimental errors, in much the same way in which transitions associated with

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carbon anion chains do (cf. Ref. [18]). However, the present comparison should be considered with caution, since the experimental values for the energies of the plasmon resonances are measured in inert gas matrices, and cannot be directly compared with DIB [44]. On the other hand, our cTDLDA calculations are able to predict the relative cross-sections of the chains, which are not available experimentally (cf. the discussion above). Hence, gas-phase measurements are needed to callibrate the calculated strengths as well as to be able to establish a better correspondence between some of the DIB and the carbon chain resonances. In any case, the present theoretical calculations should prove useful in the quest of determining the relative abundances of the diFerent CN -species present in the interstellar medium. 5. Conclusions We have solved the DFT and TDDFT equations within the local density approximation (LDA, TDLDA, respectively). The numerical solution has been implemented in a cylindrical basis, well suited for elongated and isolated molecules. The two numerical codes (cLDA, cTDLDA) have been tested against the results of diFerent available codes, in particular, those which make use of a spherical and a plane wave basis. We are now able to calculate the ground state electronic structure and electromagnetic response of elongated molecular structures of various types, such as linear chains, rings, and short nanotubes. We have presented here systematic and detailed results for pure linear carbon chains CN . The cLDA calculation of the ground state electronic structure has allowed us to characterize the energy spectra, wavefunctions and polarizabilities as a function of N . The latter grows with the number of C atoms fast towards large values, sensibly larger than those calculated in the literature for rather similar systems like polyacetylenes. Carbon rings, on the contrary, display sensibly smaller polarizabilities, suggesting an experimental method, based on already existing techniques, to identify diFerent structures with the same mass number. Regarding the electromagnetic response, we have calculated the cTDLDA absorption spectra of odd chains up to C21 , for which recent experimental data are available. We conGrm and extend the results of previous calculations, in particular the fact that the dipole longitudinal response is always dominated by a single mode, carrying a large fraction (about 35%) of the EWSR, and that this mode displays a non-linear scaling of the wavelength with the number of carbon atoms N . The study of the wavefunctions associated with these excitations has shown that the number of particle–hole conGgurations participating in the mode with an amplitude larger than ∼5% is of the order of 20. However, to recover the EWSR exhausted by the mode (resonance), of the order of 100 particle–hole conGgurations with energies up to about 20 eV must be taken into account, a result that demonstrates the collectivity of the modes. The calculated longitudinal dipole resonances of linear carbon chains up to C21 turn out to be systematically blue-shifted by an amount of about 15%, conGrming results previously obtained for N 6 15. Such deviations have also been observed for other systems, in particular alkaline metal clusters, as well as in RPA calculations based on the jellium model and in the results of quantum chemistry calculations. The origin of such blue-shift remains still an open question, which eventually points to the limitations of the TDLDA (i.e., lack of core excitations) and of its parametrizations.

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Acknowledgements We acknowledge Nicola Breda, Gianluca Col>o, Paolo Milani, Giovanni Onida, and Davide Provasi for fruitful discussions. We are grateful to the MURST (Programmi di Ricerca ScientiGca) for partial Gnancial support. Appendix A. Cylindrical LDA (cLDA) In this appendix we report details regarding the diFerent tests to our cLDA calculations. The Bachelet–Hamman–Schl]uter (BHS) [97] and Troullier–Martins (TM) [98,99] pseudopotentials were found to lead to essentially identical results. This can be clearly seen in Fig. 24 in the case of C5 . Due to the insensitivity of the results to the pseudopotentials, we routinely used the BHS one. T Stable numerical results were obtained for cylindrical boxes only slightly larger (at least 4 A between the box and any atom) than the molecule inside. An example is given in Fig. 24, where T 3 , corresponding the CPMD calculation (column 1) required an orthorombic box of 8 × 8 × 20 A to a basis set of 18 000 plane waves, instead of the 7500 cylindrical wavefunctions of the T 2 cylindrical box (column 2). 24 Furthermore, we found that cLDA calculation in a 4 × 13:2 A the dimension of the basis grows roughly linearly with the linear dimension of the box. This feature turns out to be useful for the calculation of elongated structures, as in the present case.

Fig. 24. LDA energy spectrum for C5 , tested in Gve diFerent ways. We refer to Fig. 8 for detailed explanation of the spectrum: (column 1) CPMD calculation, with optimized geometry and BHS pseudopotential; (column 2) cylindrical LDA calculation (cLDA) with ideal geometry and BHS; (column 3) as in column 2, but with TM pseudopotential (cTM); (column 4) as in column 2, but with CPMD relaxed positions (cRP); (column 5) as in T The spectra show no appreciable diFerences. column 2, but with an enlarged box (cEB) (atom-box distance 5 A). Details are discussed in the text. 24 For completeness, we must remind that the plane wave basis set of CPMD allows a strong analytical simpliGcation of the LDA algebra and hence a fast computation, also with large basis sets. Actually, the true limit of large basis sets is the high memory requirement, at least for CPMD.

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In Refs. [57– 60], the large basis sets required in order to separate the original system from its neighboring replicas limited the calculations to small systems (IIA, IB, IIB dimers and their T 2 cylindrical box was necessary in order anions). For example, for Hg dimers [57], a 7:4 × 14:8 A to isolate the system. Always for dimers [58], a cylindrical basis of ∼16 000 wavefunctions were employed in order to ensure a converged computation. Finally, we discuss the choice of the geometry of the molecule. Experiments are able to measure the geometry of carbon chains in some cases (cf. Section 2). On the other hand, the ab-initio LDA relaxation of the geometry of the system is a very expensive calculation, that, with the exception of small molecules, requires an implementation on parallel architectures. In the present work we follow the strategy (cf. also Refs. [46,5,67,65]) where the geometry is chosen from extrapolation (or inference) from available experimental and theoretical results. A test for the short chain C5 is shown in Fig. 24. No appreciable diFerences can be found between the fully relaxed CPMD and the cLDA spectrum. The CPMD total energy resulted to be only slightly smaller (0:5%). Hence, for the linear C5 chain, we conclude that the LDA spectrum is only slightly sensible to small diFerences in the C–C bond lengths, and we choose an average T for the C–C bond length in carbon chains. value of 1:3 A To further test this picture, and also in view of cTDLDA calculations of the dipole response (cf. Section 4.2), we have fully relaxed the geometry of some longer chains, C15 and C21 in particular, via CPMD calculations on the 128 CPU CRAY-T3E supercomputer at CINECA [100]. 25 In Table 7 we report the resulting bond length structures. The calculated geometries ◦ are substantially symmetric and linear, being the larger angular deviation from 180 smaller ◦ than 1 . The bond length structures are essentially cumulenic, with only a small “dimerization” T The CPMD electronic spectra do not diFer with bond lengths between 1.266 and 1:291 A. substantially from those calculated in cLDA with the “ideal” (constant interatomic distance) geometry (see also Table 8). Appendix B. Cylindrical TDLDA (cTDLDA) In this appendix we report some details regarding the TDLDA, its implementation in a cylindrical basis (cTDLDA) and the numerical tests. B.1. TDLDA equations The linear response function (x; t; x ; t  ) of a system of electrons to an external perturbation V (x; t) is deGned as [108,114,115]  +∞   d 3 x (x; t; x ; t  )V (x ; t  ) ; dt (B.1) (x; t) = −∞

where (x; t) is the variation of the density with respect to the ground state density (x) induced by V (x; t). Within time-dependent perturbation theory, the linear response function 25

C21 is interesting because it is the longest pure carbon chain experimentally observed so far [45]. For longer chains, for instance C25 , the linear geometry could not be relaxed to fully convergence within 3 h using 128 CPUs.

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Table 7 T for C5 , C15 and C21 . The bond angles were not reported because they are all very close CPMD bond lengths (A) ◦ to 180 Bond

C5

1–2 2–3 3– 4 4 –5 5–6 6 –7 7–8 8–9 9 –1 10 –11 11–12 12–13 13–14 14 –15 15 –16 16 –17 17–18 18–19 19 –20 20 –21

1.289 1.279 1.280 1.289

C15

C21

1.283 1.290 1.266 1.279 1.271 1.276 1.273 1.273 1.276 1.271 1.279 1.266 1.290 1.283

1.283 1.291 1.266 1.270 1.270 1.277 1.272 1.276 1.273 1.275 1.274 1.273 1.276 1.272 1.277 1.270 1.280 1.266 1.291 1.282

Table 8 DIB with wavelength comparable to the main absorption peaks of carbon chains C15 and C21 a T DIB (A)

T (N ) Carbon chains (A)

f

4180 5414.8 5420.2 5450.3

4200 (15)

22.2

5440 (21)

31.1

a (Column 1) DIB wavelength [150 –152]; (column 2) experimental values of the main absorption peaks of carbon chains C15 and C21 [45]; (column 3) cTDLDA oscillator strengths, Eq. (B.21).

coincides with the retarded density–density correlation function [108], also called polarization propagator. In frequency space, Eq. (B.1) reduces to  (x; !) = d 3 x (x; x ; !)V (x ; !) : (B.2) For a system of free, non-interacting, electrons, one has
∗ 4ph (x)Kph (!)4ph (x  ) ; LDA (x; x ; !) = ph

(B.3)

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where 4ph (x) = Kph (!) =

∗ LDA (x) hLDA (x) p

;

1 1 − ; ˝! + (Ep − Eh ) + i' ˝! − (Ep − Ep ) + i'

(B.4) (B.5)

and p (h) indicates the particle (hole) state of energy Ep (Eh ) [108]. Now, an interacting electron system reacts to the external Geld V (x; !) with a dynamical polarization, partially screening the external Geld and modifying the free response Eq. (B.3). From the mean Geld point of view, the external perturbation causes a dynamical rearrangement, that is, a Quctuation of the electron density of the system. The corresponding Quctuation of the mean Geld is called induced (or residual) interaction, and must be calculated self-consistently. The explicit form of the induced interaction becomes  ˆ H LDA [(x; !)] = d 3 x K(x; x ) (x ; !) ; (B.6) where the kernel function K(x; x ) 26 is given by a Hartree contribution KH and an exchange– LDA , correlation part Kxc vH [(x)] 1 KH (x; x ) = ; (B.7) =  (x ) x − x LDA (x; x ) = Kxc

LDA [(x)] LDA vxc d vxc = [(x)] (x − x ) : (x ) d

(B.8)

In this way one obtains a Dyson-type equation for the interacting linear response function (x; x ; !);      3 3 LDA (x; x ; !) = d x (z − x) − d y LDA (x; y; !)K(y; z ) (z ; x ; !) : (B.9) Formally, Eq. (B.9) is an operator equation of the form LDA = (1 + LDA :K)

(B.10)

and can be solved via operator inversion  = (1 + LDA :K)−1 LDA :

(B.11)

From the response function (x; x ; !) the dynamical linear polarizability can be extracted as the ratio between the induced moment and the external Geld strength [104]. Making use of Eq. (B.2) one obtains  "V (!) = d 3 x d 3 x V (x; !)(x; x ; !)V ∗ (x ; !) : (B.12) 26 Here the kernel K is a static, frequency independent, function, and the frequency dependence of the induced Geld is carried out by the Quctuating density (x; !). A more reGned theory would require a frequency-dependent kernel [104].

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The strength function SV (E) associated to the response of the system to the external Geld V is given by [115,118]
1 SV (E) = | 0|Vˆ |m |2 (E − Em ) = − I"V (E) ; (B.13)  m=0

where |0 denotes the ground state of the chain, |m the excited state, and Em its corresponding excitation energy. The strength function is related to the cross-section for photon absorption by [115,118] 42 4 $V (E) = (B.14) E SV (E) = − E I"V (E) : ˝c ˝c Some global properties of the response of the many-particle system to the external Geld V can be analyzed and understood in a transparent way in terms of sum rules (cf. e.g. Refs. [115,118,144,145]). The sum rules are deGned as the k-moments of the strength function  +∞
Mk (V ) = d E E k SV (E) = Emk | 0|Vˆ |m |2 : (B.15) 0

m=0

This quantity can be also expressed in terms of the ground state expectation value of a combination of commutators of the Geld V and the Hamiltonian H [118,144,145]. This fact is useful because the expectation value can be calculated using the LDA ground state, instead of the TDLDA ground state, 27 without the knowledge of the strength function SV (E). Particularly interesting is the Grst moment (k = 1), also called EWSR.  +∞  1 ˝2 ˆ ˆ ˆ M1 (V ) = d 3 x|∇V (x)|2 (x) ; d E E SV (E) = 0|[V ; [H ; V ]]|0 = (B.16) 2 2m 0 which has a simple physical interpretation: on the left-hand side we have the excitation energy of the system, on the right-hand side the average energy given to it by the external interaction Geld [115]. From Eq. (B.16) it is clear that the most important application of the EWSR is to the case of linear force Gelds, with a constant gradient. This is exactly the case of the electromagnetic Geld in the dipole approximation. Other interesting moments are the cases k = − 1; 0; 3 [118,145]. In particular, the moment M−1 (V ) is simply proportional to the static polarizability [118]  +∞ SV (E) 1 dE (B.17) M−1 (V ) = = "V (0) : E 2 0 This is a consequence of the Kramers–Kronig relations [147] between the real and imaginary part of the dynamical polarizability. A particular case is a dipole Geld, of the form V (x) ˙ x ; (B.18) suitable for describing an external electromagnetic Geld of wavelength much larger than the size of the system. In this case, the EWSR Eq. (B.16) can be evaluated analytically. Inserting the ith component of a dipole electromagnetic Geld V1 = V (xi ) = exi in Eq. (B.16) one has ˝2 2 M1 (xi ) = (B.19) e Ne ∀i ; 2m 27

This is the Thouless theorem [146], generalized to density-dependent Hamiltonians [118,144,145].

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where Ne is just the number of electrons in the system. This is also known as the Thomas– Reiche–Kuhn (TRK) or f-sum rule. The latter is model independent, and it constitutes an important constraint which must be satisGed by the numerical calculation. Its physical meaning is clearer when it is written in terms of the dipole absorption cross-section [115,118],  22 e2 ˝ d E $(E) = (B.20) Ne : mc Eq. (B.20) tells us that the total cross-section (averaged over its spatial components) just measures the total number of electrons taking part in the excitation of the system. Hence an experimental measure of the cross-section can be used to determine the collective nature of a resonance. Another useful quantity is the oscillator strength, deGned as 2m fmxi = 2 2 | 0|xˆi |m |2 Em : (B.21) ˝ e From the deGnition of EWSR, Eq. (B.15), one has
2m fmxi = 2 2 M1 (exi ) = Ne ∀i : (B.22) ˝ e m=0

That is, the dipole oscillator strength fm associated to the excited state |m is equal to the number of electrons participating in it. When the dipole strength is concentrated into a single peak, its centroid can be well approximated by   M1 ˝2 e2 Ne E1 = = (B.23) M−1 m" and can be used to predict the dipole resonance peak energy from the measured static dipole polarizability ", or vice versa [118]. 28 B.2. Numerical tests to the cTDLDA The choice of the particle–hole basis to be used for the calculation of the linear response is an important issue for the computational cost of TDLDA. This choice is directly connected with the amount of EWSR (cf. Eq. (B.19)) that the theoretical cTDLDA results are required to exhaust (note that TDLDA preserves the LDA EWSR). From the dipole selection rules discussed in Section 4.1, one can see that in the case of linear chains only opposite parity, $–$ and – particle–hole states are allowed in the particle–hole basis. 29 The simpler criteria used to deGne the unperturbed particle–hole basis is to select all the allowed particle–hole states below a certain energy, which is chosen high enough to exhaust a large fraction of the EWSR M1 . This is the criterion used here. In the case of the C5 chain, including all particle–hole excitations with energy equal or below 60 eV, we are able to exhaust 91.6% of the EWSR M1 , the basis containing 231 conGgurations. For longer chains, the density of allowed particle–hole transitions 28 This shows us that, in this case, there is no distinction between adiabatic and diabatic motion of the electrons with respect to the ionic background. 29 In fact, the occupied (hole) states of carbon chains have m = 0; ±1 only.

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grows very fast, as a consequence of the evolution of the DOS (cf. Section 4.1 and Figs. 9 and 10). In fact, the same 60 eV cutoF of the particle–hole basis in the case of C13 gives a very large fraction of M1 (92.6%), but the basis contains 1098 states, an order of magnitude larger than for C5 , making the numerical calculation heavier. An analysis of the contribution of each of the particle–hole basis state to M1 reveals that, actually, the major part of them contributes with a very small fraction. High-energy states in particular, are associated to small particle–hole matrix elements, due to the small overlap of the corresponding wavefunctions. Hence, we can introduce a further cutoF, disregarding all the particle–hole states which contributes less than a certain (very small) percentage to M1 , taking care in keeping it at 90% of the theoretical value Eq. (B.19). For example, for C13 we obtain 90.3% of M1 with only 220 particle–hole states, the other 876 contributing only to the remaining 2.3% below 60 eV. We used this combined criteria for all the calculations described in the following, except for the case of short chains as C5 , where the full 60 eV basis was small enough to allow calculations with reasonable computational resources. A check of the reduced basis versus the full basis result displayed no appreciable diFerences in the response, as expected. Another important parameter entering the cTDLDA calculations is the number of cylindrical multipoles l; m used for the solution of the TDLDA Eq. (3). For dipole longitudinal Gelds we have the selection rules m=0 ;

(B.24)

l even :

(B.25)

We used two criteria to Gnd a sensible value of Lmax : (i) by increasing Lmax , the calculated static polarizability must be stable, and (ii) the calculated cLDA M1 , obtained from the unperturbed cLDA strength function, must coincide with the single-particle value, obtained directly from Eq. (B.15). 30 Finally, in Fig. 25 we show the stability of the cTDLDA response function in the case of C5 and C15 for both the “ideal” and the optimized geometry of the linear chain (cf. Section above). There are no signiGcant diFerences in the electromagnetic response of an ideal and a relaxed structure, in keeping with the fact that the corresponding bond lengths diFer minimally. Hence, we used the ideal geometry for all our cTDLDA calculations. Appendix C. Classical polarizability of elongated metallic particles In this appendix we derive two expressions given in Section 4.3, for the asymptotic behavior of the static dipole longitudinal polarizability of elongated ellipsoidal metallic particles and for the free electron gas conGned in a cylinder. √ For ellipsoidal metallic particles the plasmon energy obeys Eq. (8), ˝! = ˝!0 L, where L is the axial depolarization factor, Eq. (9), and ˝!0 is the bulk plasmon energy. For very elongated 30

They are, in fact, the same thing, but the unperturbed strength function is calculated from the multipoles of the free polarization propagator, Eq. (B.3). Hence, it is sensible to the cylindrical multipole expansion.

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Fig. 25. Longitudinal dipole strength function for C5 and C15 : (continuous line) “ideal” geometry; (dashed line) “relaxed” geometry. The main resonance is substantially unchanged both in strength and in energy.

particles, with R R⊥ ; L can be written as

R R⊥ 2 ∼ log : L= R R⊥

(C.1)

In the case in which the dipole strength is concentrated into a single peak, one has approximately (cf. Eq. (B.23)) 2 2 ∼ ˝ e f ; (C.2) ˝2 !2 = ˝2 !02 L = me "zz (N ) where "zz (N ) is the static dipole longitudinal polarizability and f is the eFective number of valence electrons participating to the plasmon excitation, given by the oscillator strength of column 8 in Table 1. Using the standard plasmon relation, ˝2 !02 = 4(˝2 e2 =m), together with Eqs. (13), (14) and the above expressions, we obtain the expression given in Section 4.3 R3 zz ∼ " (N ) = : (C.3) 3 log(R =R⊥ ) In order to describe the free (cLDA) static dipole polarizabilities given in Fig. 18 we have zz = V ( − 1)=(4), where  used in Section 4.3 the formula for the 1-D free electron gas "cLDA is the static dielectric constant and is given by [19] 4me2 kF 1 (q) = 1 + : (C.4)  ˝2 q 2 Here ˜q is the wavevector and kF the Fermi wavenumber. In a Gnite cylinder of volume V = R2⊥ 2R , we have q = =R and kF = qNe =2, where Ne = 4N is the number of electrons. Therefore we Gnd 8NR  − 1= 2 ; (C.5)  aB

510

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CONTENTS VOLUME 357 A. Wacker. Semiconductor superlattices: a model system for nonlinear transport

1

I.L. Shapiro. Physical aspects of the space–time torsion

113

M.J. Brunger, S.J. Buckman. Electron–molecule scattering cross-sections. I. Experimental techniques and data for diatomic molecules

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Forthcoming issues

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FORTHCOMING ISSUES S.Y. Wu, C.S. Jayanthi. Order-N methodologies and their applications V. Barone, A. Drago, P. Ratcliffe. Transverse polarisation of quarks in hadrons M. Baer. Introduction to the theory of electronic non-adiabatic coupling terms in molecular systems S.-T. Hong, Y.-J. Park. Static properties of chiral models with SU(3) group structure W.M. Alberico, S.M. Bilenky, C. Maieron. Strangeness in the nucleon: neutrino–nucleon and polarized electron–nucleon scattering C.M. Varma, Z. Nussinov, W. van Saarloos. Singular Fermi liquids J.-P. Blaizot, E. Iancu. The quark–gluon plasma: collective dynamics and hard thermal loops A. Sopczak. Higgs physics at LEP-1 I.L. Aleiner, P.W. Brouwer, L.I. Glazman. Quantum effect in Coulomb blockade P. Tabeling. Two-dimensional turbulence: a physicist approach A. Altland, B.D. Simons, M. Zirnbauer. Theories of low-energy quasi-particle states in disordered d-wave superconductors J.A. Krommes. Fundamental descriptions of plasma turbulence in magnetic fields J.D. Vergados. The neutrinoless double beta decay from a modern perspective C.-I. Um, K.-H. Yeon, T.F. George. The quantum damped harmonic oscillator

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