Physics Reports vol.328

A.Ya. Ender et al. / Physics Reports 328 (2000) 1}72

COLLECTIVE DIODE DYNAMICS: AN ANALYTICAL APPROACH

A.Ya. ENDER , Heidrun KOLINSKY
, V.I. KUZNETSOV , H. SCHAMEL
Iowe Physico-Technical Institute, St. Petersburg 194021, Russia
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Physics Reports 328 (2000) 1}72

Collective diode dynamics: an analytical approach A.Ya. Ender *, Heidrun Kolinsky
, V.I. Kuznetsov , H. Schamel
Iowe Physico-Technical Institute, St. Petersburg 194021, Russia
Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany Received June 1999; editor: D.L. Mills Contents 1. Introduction and review 1.1. Introduction 1.2. Review on plasma diode theories 1.3. Experiments and applications of the Bursian and Pierce diode instabilities 2. Notation and basic equations 3. Equilibrium solutions without and with re#ection 4. Criteria for existence and stability of equilibria without re#ection in the Lagrangian representation 4.1. The Lagrangian formulation 4.2. Equivalence between Eulerian and Lagrangian representation 4.3. On the boundary of solutions without re#ection 4.4. Linear dispersion relation 5. The transformation of states 5.1. De"nition and transformation of states without re#ection

4 4 6 15 18 20

24 24 26 28 29 30 30

5.2. Transformation of states with re#ection 5.3. Some properties of the transformation 6. Classi"cation of potential distributions (PD): (g, e)-diagrams and I}< characteristics 6.1. Generalized Pierce diode 6.2. Nonneutral diode 6.3. (g, e)-diagrams and I}< characteristics 7. Stability of equilibrium solutions 7.1. Aperiodical instability boundaries for nonre#ective solutions 7.2. Aperiodical instability boundaries for solutions with partial re#ection 7.3. Aperiodical instability boundaries for solutions with total re#ection 7.4. Dispersion curves and boundaries of oscillatory instability 8. Summary and conclusions Acknowledgements References

32 33 36 36 40 42 44 45 52 55 57 67 70 70

Abstract An analytical study of the plasma states in nonneutral plasma diodes and of their stability is presented for an arbitrary neutralization parameter c, including the Pierce (c"1) and the Bursian (c"0) diode as special cases. Physically such a study is of interest, e.g. in the transport problem of an electron beam in spatially bounded electronic devices. Similarity transformations are obtained which connect equilibrium solutions of di!erent c's. This implies that by simple transformations one can infer from equilibria of the generalized

* Corresponding author. 0370-1573/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 9 2 - 7

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Pierce diode to equilibria of nonneutral diodes. The regimes with partial and total re#ection of electrons are studied in detail for the "rst time. A classi"cation of nonuniform solutions for these regions as well as for the regime without re#ection is presented. The equivalence between the Eulerian and the Lagrangian formulation of the diode dynamics is proved, and both, the aperiodical and oscillatory eigenmodes of the generalized Pierce diode are examined. New bifurcation points in the branches of dispersion relations are discovered.  2000 Elsevier Science B.V. All rights reserved. PACS: 41.75.!i; 85.30.Fg; 52.35.Qz; 85.45.!w; 84.47.#w Keywords: Plasma diode dynamics; Beam plasma device; Generalized Pierce diode; Nonneutral diode; Re#ection by virtual cathodes; Space charge limited current

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1. Introduction and review 1.1. Introduction This article reports on recent theoretical progress in the understanding of diode dynamics as a collective phenomenon. Many systems in physical science and technology, in which charge carriers are transported between two electrodes, are known to exhibit a diodic behavior, the most celebrated diodic event being the phenomenon of `space-charge-limited #owa. This #ow limitation, which in most situations severely restricts the operation conditions, indicates a bifurcation of states, i.e. a transition between two distinct stable states. An understanding of these bifurcation processes is hence of crucial importance for all systems in which a space-charge-induced #ow limitation takes place. Consequently, most of the present work is devoted to bifurcation studies including investigations on the existence of diode states and on their stability properties. Diode dynamics essentially a!ects physical and technical processes such as that found in Q-machines [1,2], thermionic converters [3}5], microwave generators [6,7], electronic switches [8], low-pressure discharges and processing [9], accelerators [10], inertial con"nement devices [11,12], xerographic technologies [13], semiconductor devices [14}16], etc., to mention some of them. Common to these devices is the fact that the particle transport can be modeled in good approximation by a one-dimensional, ballistic and monoenergetic #ow. Despite the simplicity of this model } "rst ideas and descriptions date back a long time [17}19] } it gained new interest in recent years mainly due to its potential applicability and its richness in the phenomena associated with. Focusing on the dominant collective behavior of the lighter charged species (mostly electrons), we describe the #uid-like dynamics on the fast time scale by continuity and momentum equation supplemented by Poisson's equation. At "rst glance, an investigation of this type may look simple and straightforward; however, as shown later, the dynamics shows up with several subtleties and pitfalls, some of them being described for the "rst time. The nonlinearity and time dependency of these plasma equations, the possibility of particle re#ection due to the occurrence of an internal potential minimum and the boundary conditions render the problem highly nontrivial. It is our concern in this theoretical report, to present a complete picture of the dc states of diodes under these conditions, to evaluate almost completely their linear stability behavior and to o!er for the "rst time a transformation between dc states, which yields a substantial simpli"cation of the analysis. Also, a review of previous works on diode dynamics is given. Fig. 1 shows schematically the underlying diode model. A beam of electrons of given density n> and velocity v enters the diode region at the emitter (donor) electrode at x"0. The latter is   kept at zero potential. If no re#ection occurs, all electrons leave the diode at the collector, at x"¸. A bias voltage is applied across the diode region. Electrons which are re#ected by the internal  space charge potential return to the emitter electrode and are totally absorbed. It is furthermore assumed that the diode region is occupied uniformly by in"nitely massive ions of constant density n . We allow any value of n . This implies that charge neutralization (n "n>) is only one of the  many options discussed. The ions are, hence, treated as immobile, a dynamical situation often found to be valid in lowest approximation as our focus is on the fast electron processes. (A practical example where ions do not participate in the dynamics and hence can be treated by a constant

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Fig. 1. Schematic view of the electron dynamics in the diode. Some of the electrons, all enter the diode region at x"0, are re#ected by a virtual cathode. Ions contribute by their constant density n only.

density is their perpendicular injection. If their velocity is su$ciently large they will leave the diode region essentially unchanged, i.e. without having experienced a change in their density distribution both in longitudinal and transversal direction. Hence, fresh ions with a homogeneous distribution will provide the necessary background.) The adjective ballistic implies that the #ow is assumed to be collisionless, i.e. that the electron mean free path well exceeds the diode length ¸, a situation often met. Such a regime of a plasma diode, often referred to as the Knudsen regime, can be found, for example, in Q-machines [1,2], where the su$ciently dilute and hot plasma is provided by the surface ionization of a neutral particle beam that hits a hot plate (emitter). A comparable situation is given in high-temperature thermionic energy converters [3}5]. Cold electron beam evaporation diode (or triode) systems, in which the thermal spread of the beam is negligible and where ballistic transport prevails, are found in micro- and nanoelectronic semiconductor "lm capacitors as well [15,16,20]. Field emitter arrays of the Spindt cathode type [21}23] constitute another important class of applications. Typically, these devices experience a current self-quenching due to space charge e!ects and show current}voltage characteristics with portions of a negative di!erential resistivity (see later). A monoenergetic electron #ow can also be generated in triodes where a highly accelerating voltage is applied between the thermocathode and the grid. The space between the grid and the collector constitutes a further experimental realization of the model. A current limitation is observed whenever one of the following quantities exceeds the critical value: the particle density, the current density and the gap distance. In this situation, an aperiodic instability sets in that destabilizes the diode. As a result, a virtual cathode arises internally which re#ects electrons partially or totally, depending on the parameter regime involved. In the case of charge neutrality, n>"n , this instability is known as the Pierce instability and the diode under  short-circuited conditions as the Pierce diode [19]. On the other hand, if at all ions are absent and hence if only electrons are present, the diode is called Bursian diode [18]. One reason, as mentioned before, why such an idealized con"guration exhibits such a complex dynamical pattern lies in the boundary conditions. The #ow characteristics for "xed boundary conditions are generally di!erent at injection and at exit. In contrast to periodic systems the #ow at

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injection has thus no information about the actual status of the diode. To meet the "xed boundary conditions in case of a time-dependent behavior, such as that treated by a linear stability analysis of the homogeneous state under short-circuited conditions, a Fourier analysis with complex frequencies and wavenumbers is required which has rather intricate consequences [24]. Textbooks on plasma physics and related topics, on the other hand, almost exclusively deal with unbounded or periodic plasmas } three of the few exceptions are the monographs of Llewellyn [25], Birdsall and Bridges [26] and Nezlin [27] } and are hence not suited to describe such situations theoretically. In these more simple cases either the frequency or the wavenumber can be chosen real, depending on the situation. Hence, bounded plasmas are something else possessing their own characteristic dynamics. Furthermore, the ballistic nature of the #ow suggests to bring in also the Lagrangian picture, in which the dynamics is described in a reference system comoving with a #uid element. In certain situations this description is preferred to an Eulerian description, as it allows pragmatically a solution in contrast to the latter which sometimes gets lost in analytically unsolvable equations. An example is the linear stability analysis with an arbitrary degree of neutralization and a dc bias. We hence pursue in this report both descriptions, the Eulerian and the Lagrangian. In addition, it is also helpful to consider a Dirichlet boundary value problem, namely given the potential at the electrodes, and contrast it to a Cauchy boundary value problem, where the normalized potential g and the normalized electric "eld e are prescribed at the emitter. As we shall show, the use of both concepts, the Lagrangian method, promoted in Bayreuth and the so-called (g, e)-method, developed in St. Petersburg, allow us to achieve results that have not been obtained before. With the (g, e)-method we are able to investigate completely the aperiodic instability also in situations of dc virtual cathodes, i.e. when particle re#ections occur. On the other hand, the Lagrangian description allows to solve the linear stability problem in the case of nonre#ective equilibria, encompassing oscillatory instabilities, too. These methods are hence in some sense complementary. Although a number of works have emerged in the last decades, many of them having been referred to in the adjacent review of plasma diode theories and experiments, a systematic study is lacking. In addition, the region of existence of solutions in parameter space and the interrelations between di!erent solutions have not yet been analyzed. This paper intends to "ll this gap. Before we enter into the detailed mathematical analysis we report and summarize previous works on diode dynamics. 1.2. Review on plasma diode theories 1.2.1. Bursian diode (pure electron diode) In this review, we summarize the highlights in the analytic description of plasma diodes. First we treat pure electron diodes since from these all the later developments originate. The very "rst paper on the theory of bounded collisionless plasmas was written by Child [17]. He made the "rst analytic description of the static electric potential in a vacuum diode in which an ion #ow leaves the emitter with zero velocity and an accelerating negative voltage, !<, is applied between collector and emitter. The calculated current transported through the diode gap was found to be proportional to </¸, where ¸ is the electrode distance, the celebrated Child } or `3/2-lawa, (being also called Child}Langmuir}Schottky law [28,29]). These results are valid for electrons as well if the

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signs of charge and potential are reversed and are usually referred to as the space-charge-limited current in a plasma diode. Bursian and Pavlov [18] have studied the time-independent states of a short-circuited vacuum diode with a monoenergetic electron beam entering from one side of the diode. They introduced for the "rst time a model with partial re#ection. Three solutions were found, all of them endowed with a virtual cathode (VC). The authors noted that one of these solutions was unstable and that, when the gap size reached a certain value, the regime without electron re#ection disappeared being removed by a new state, the one with partial re#ection. The threshold current for this `Bursian instabilitya was evaluated. Gill [30] has studied experimentally a vacuum diode with an electron #ow having a small velocity spread. He found that the collector current I at "rst grows with increasing emitter current I and then drops sharply to a lower level. A hysteresis type dependence of I(I ) was found. The   author has calculated this dependence analytically by assuming a VC and by adopting the Child law to the two regions between the VC and both electrodes. He obtained a hysteresis pattern for the I(I ) dependence and was able to explain the main experimental results.  Lukoshkov [31] has treated analytically a vacuum triode, where a grid is placed between the two electrodes. It serves to control the current through the triode. The grid-collector space can be considered as a planar diode with a cold electron #ow entering through the grid. The author considered "ve types of time-independent states, a potential distribution (PD) which grows or decreases monotonically, a PD with a VC without and with re#ection and a PD with total re#ection. In particular, he also obtained the Bursian}Pavlov results but did not refer to it. Fay et al. [32] have studied stationary states of a vacuum diode with a cold electron #ow with variable collector potential. They found four solutions and plotted the PD as function of the electric "eld strength at the re#ection point in case of total re#ection, as function of the re#ection ratio in case of partial re#ection, as function of the minimum potential in case of a VC solution without re#ection and as function of the emitter electric "eld strength in case of monotonic solutions without re#ections. For the latter they invented the notation normal C-#ow and C-overlap, respectively, but they overlooked the solution with partial re#ection of the second kind (see later). Implicit analytical expressions for PDs and current}voltage (I}<) curves were obtained. The authors pointed out, that `for certain conditions more than one type of PD may exista. The regions in the I}<-plane corresponding to the di!erent solutions were presented graphically. Lukoshkov's paper was not referred to. Llewellyn [25], in his monograph, has described a method for investigating the instability of a vacuum electron diode. An equation, usually called Llewellyn's equation, associating the third time derivative of the position of an electron #uid element with the total current has been derived. This paper can be viewed as one of the "rst Lagrangian-type treatments of a diode (for a more extensive historical assessment of Llewellyn's work see [26, Chapter 2]). He linearized the electron transit time with respect to small perturbations of the dc state and found diode solutions for zero injection velocities and emitter electric "eld strengths which were also veri"ed experimentally. The ideas of Llewellyn were used by Lomax [33], who studied the aperiodical instability of the dc state without re#ection of a short-circuit electron vacuum diode (in fact the Bursian diode). He pointed out the existence of three stationary solutions for certain values of the dimensionless gap values, having di!erent transition times, t (t (t . The t -solution was found to be stable, and    

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the t -solution unstable. The t -state which belongs to the regime with electron re#ection was not   investigated. Paschenko and Rutkevich [34] have studied the same problem as Lomax by a di!erent method and obtained the same result. They did not refer to Lomax. Nezlin [27], in his monograph, on the experimental conditions on the beam propagation in rare"ed plasmas, has discussed the Bursian diode, the development of various instabilities and among them the Bursian instability. Coutsias [35] has studied the e!ect of a thermal spread on the space charge limit of an electron beam in a short-circuited vacuum diode and found for the regime without re#ection that the space charge limit } the maximum gap distance at which the whole emitted current passes the diode } decreases proportional to k¹/w in comparison with the cold #uid case, where ¹ is the beam temperature and w the beam energy at the emitter. Kuznetsov et al. [36] have studied the Bursian diode with arbitrary external voltage using the (g, e)-diagram method. The region of aperiodical instability of the di!erent stationary solutions for both regimes, without and with re#ection, has been constructed and exhibited in the (e, d)-plane, where e and d are the dimensionless emitter electric "eld strength and the interelectrode distance, respectively. For the "rst time regimes with partial re#ection of the second kind, for which the re#ection point coincides with the collector position, have been found and studied. Alyeshin and Kuz'menkov [37] have investigated steady states of a planar short-circuited vacuum diode with an electron beam of "nite thermal width. They showed that with increasing m, where m is the ratio of thermal velocity and beam velocity at the emitter, the region narrows where two solutions exist. At m+0.18 this region vanishes. The interpretation is that, as long as m is less than this critical value, three steady states exist, one of which is unstable. If m exceeds this value only one state remains. Liu and Dougal [38] have studied the 1-D vacuum diode with cold electrons theoretically by investigating the initial velocity e!ect on the space-charge-limited current. In their case, a virtual cathode was considered such that the potential minimum equals the emitter electron energy, but they disregarded re#ection. Exact analytical expressions for the potential, the "eld and the charge density were obtained in terms of the gap distance. The formula found for the current coincided with Child's law. Kuznetsov and Ender [39] have treated the instability of steady states of a planar short-circuited vacuum diode with an electron beam of "nite thermal width numerically. They made use of a method developed by the authors themselves in which the electron distribution function at the emitter was chosen waterbag-like. The position and the depth of the virtual cathode were plotted for steady states as functions of the dimensionless gap distance. For gap values, where three steady states exist, the transition processes from the unstable to a stable state have been studied in detail. A perturbative form was chosen in accordance with linear theory. Time-dependent processes involving perturbations of steady states with electron re#ection were studied too. For this case a linear theory is absent, and the perturbations arose from numerical inaccuracies. They have shown that oscillatory perturbations grow when the gap distance exceeds a certain value. Frequencies and growth rates as functions of the gap distance were determined numerically. With this they could prove that the onset and the existence of an oscillatory regime in a diode with re#ection require that the minimum gap distance exceeds a certain value. This contrasts with other authors, who found by particle simulations virtual cathode oscillations for practically any gap value.

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Finally, Kolinsky and Schamel [40] studied the two di!erent states of a Bursian diode with an arbitrary applied voltage analytically, the solution with no re#ection, called type-I #ow, and the one with partial re#ection (of the "rst kind in our later notation) called type-II #ow. The Lagrangian integral formalism was used. A dispersion relation associated with perturbations of equilibria of the type-I #ow was obtained which coincides with that of Lomax [33]. Its numerical evolution shows that one of the two branches of equilibria (the one called C-overlap in the notation of Fay et al. [32]) is unstable with respect to purely growing modes. Analytical expressions were obtained for various equilibrium quantities, such as the splitting rates in case of re#ection, the transmitted current or the emitter electric "eld as functions of the applied potential. 1.2.2. Pierce diode (neutralized plasma diode) Next we call attention to a plasma diode in which the monoenergetic electron beam leaving the emitter is completely neutralized by a uniform component of immobile ions. This device is called a Pierce diode (one of the "rst persons who made use of this name was Kuhn [41]) and we distinguish between the classical Pierce diode, where no potential is applied across the diode, and the generalized Pierce diode, where a "nite collector potential is admitted. Pierce [19], in his classical paper, has studied theoretically a planar short-circuited diode with a cold electron #ow moving through the uniform background of in"nitely massive ions which neutralizes altogether the electronic space charge. He was the "rst to prove that an electron current limit exists in the time-stationary regime, beyond which no electron #ow can pass the diode. It was found that the limit current exceeds only by a factor of 5.6 the vacuum diode limit current (Bursian limit current). The existence of the limit current was shown to be due to the development of the electron aperiodic instability, subsequently called the Pierce instability, which is connected with the feedback between the electrodes through the external circuit. Pierce has derived the dispersion relation and evaluated the maximum growth rate. Shapiro and Shevchenko [42] have developed a method, which looks like a Lagrangian formalism, for constructing nonuniform steady states of a planar short-circuited Pierce diode in the regime without electron re#ection and presented a stability analysis. Kuznetsov and Ender [43] have explored the stability of stationary bounded Knudsen plasma states for an arbitrary spatial distribution of in"nitely heavy ions and for a nonzero thermal spread of the electron beam. Electron re#ection was not considered. An integro-di!erential equation for the perturbations of the potential distribution was derived and solved analytically for two cases. First, they assumed a delta-function electron distribution and an unperturbed linear potential pro"le. Boundaries of the aperiodic instability in the (<, d)-plane were plotted, where < is the collector potential and d the dimensionless gap distance. In this case, the perturbed potential pro"le consists of combinations of Bessel and Neumann functions. Second, the electron distribution function was waterbag-like and the unperturbed potential distribution was uniform. Regions of aperiodical instability have been obtained. Cary and Lemons [44] have plotted the oscillatory branches of the Pierce instability for the uniform state of a planar short-circuited diode with in"nitely massive ions. Burinskaya and Volokitin [45] have derived the dispersion relation for the aperiodic instability of nonuniform steady states of the planar, short-circuited Pierce diode, using the method of [42]. Mosiyuk [46] has studied the instability of uniform equilibria of planar, short-circuited Pierce diodes with in"nitely massive ions in the regime without electron re#ection. He obtained

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`windows of transparencya by means of a transcendental equation, i.e. gap regions in which an instability does not develop. Kuhn and HoK rhager [47] have studied external-circuit e!ects on the instability of uniform equilibria of Pierce diodes in the regime without electron re#ection. A dispersion relation was derived and the circuit e!ects on the instability branches were explored. HuK hne and Kuhn [48] have derived a dispersion relation for perturbations of a Pierce diode with a cold electron #ow and a linear unperturbed potential pro"le in the regime without electron re#ection (see [43]). The branches of instability found were compared with that of the Pierce diode. Godfrey [49] has investigated the existence and stability of uniform equilibria of the shortcircuited Pierce diode without electron re#ection and ion inertia. Equilibria were found as straight lines in the (e, d)-plane (the Pierce parameter d was in his notation denoted by a). An integral formalism was employed to treat the electron dynamics, to incorporate the boundary e!ects and to derive a dispersion relation for linear perturbations, involving aperiodic as well as oscillatory solutions. An interesting scenario of bifurcations could be detected for d being in the vicinity of 2.9p. This scenario of Feigenbaum type occurs in a narrow region just left to the point 2.9p. Lawson [50] has extended Godfrey's analysis of equilibria by the inclusion of a passive external circuit and has derived and evaluated the dispersion relation. And "nally Ender et al. [51] have used the Pierce diode as an approximation to model a Knudsen diode with surface ionisation, abbreviated by KDSI. A classi"cation of potential distributions was carried out and the e!ect of a "nite thermal spread was studied in detail. The Pierce diode was shown to be a good model for the KDSI. Especially, the ground states of the KDSI were found to be monotonic potential distributions connecting the cathode region with the quasineutral plasma region being at the collector voltage. If the normalization was chosen appropriately the plasma characteristics of the two models coincided. A linear eigenmode theory of the KDSI was developed and the pro"les of the eigenmodes for the Pierce diode and the KDSI were shown to be similar. Furthermore, the nonuniform, time-independent potential distributions of both diodes without and with electron re#ection and with an ion distribution corresponding to the ground states were studied and compared and the (g, e) diagrams were investigated. 1.2.3. The (g, e)-diagram technique Norris [52] has investigated the nature of spontaneous oscillations in a cesium diode energy converter and has come out with several criteria for the electron instability of KDSI states. One of these has been formulated as follows. Assume, the load consists of batteries in a circuit only. This means that the potential di!erence between collector and emitter is held constant. A graph was then drawn showing the collector potential in terms of the emitter electric "eld strength, the so-called (g, e)-diagram. The ion distribution was assumed to be "xed. He constructed such curves qualitatively and crossed these curves by the horizontal line which is at the level of the collector potential and is determined by the batteries in the circuit at "xed electron equilibrium conditions. He stated that if the curve crosses the load line with a negative slope, then the diode is stable, otherwise the state is unstable. He explained this criterion in the following way. Suppose one has a crossing of curves with a positive slope. If we add a small negative charge on the emitter, the electric "eld strength becomes more negative which in turn decreases the potential at the collector. The battery tries to correct the potential change by putting more negative charge on the emitter

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which increases the initial alteration and hence, one has instability. Similar arguments in case of the other crossover points and of adding a positive charge demonstrate the matter completely. This criterion is, in fact, equivalent to the (g, e)-diagram criterion which will be used intensively in the present work. The author could not investigate the instability of an actual state of the KDSI by this method. Smirnov [53] has studied the instability of nonlinear stationary potential oscillations in electron-ion beams. He recommended a criterion for an electron instability of steady states in a collisionless diode plasma with a given spatial ion distribution. His discussions resemble those of Norris. If dg /de '0, then the state is unstable otherwise it is stable, where g is the collector    potential and e the emitter electric "eld strength. The author analyzed, with this criterion, the  Bursian diode and the Pierce diode with various neutralization parameters in regimes without and with total re#ection. He did not examine the regime with partial re#ection and asserted that for certain regions ("partial re#ection in our terminology) of the emitter electric "eld steady states do not exist. Two of the present authors, Kuznetsov and Ender [54], applied the (g, e)-diagram concept for investigating states of a bounded collisionless plasma, in which the load consists of batteries only and in which the ion distribution was "xed. It was shown that the (g, e)-diagram can be constructed that way. They solved the time-independent Vlasov equation for electrons together with Poisson's equation for varying emitter electric "elds and found the corresponding potential distributions. The dependence of the collector potential on the emitter electric "eld was constructed for all possible values of the latter. The crossing with the load line shows possible states and if, as above, the slope of the curve at the crossing point is negative, the state is stable. Time-dependent processes in the KDSI, related to a relatively slow redistribution of ions in space, show up as an evolution of the (g, e)-diagram, allowing to "nd out the states at each moment, its stability property and to which state the system evolved in case of instability. An aperiodic instability of time-independent states of the KDSI with monotonic potential distributions in the overneutralized regime was studied and the picture of stability and instability regions was plotted in a (d, g )-plane where d is  the normalized gap distance and g the normalized collector potential, with the Debye length and  the emitter temperature as the reference quantities. In a further paper, already mentioned, Kuznetsov and Ender [43] have shown the equivalence between the (g, e)-diagram criterion and the aperiodic instability criterion, de"ned by a dispersion relation. They obtained an analytical expression for the potential perturbation for a linear unperturbed potential pro"le. Hence, for a diode with an arbitrary uniform unperturbed electric "eld strength one can "nd both the growth rate of the perturbation and the spatial form of the perturbed potential. Besides this, any unperturbed potential distribution in a diode may be approximated linearwise by a set of subdiodes each of which has a uniform electric "eld strength. Having an exact analytical solution for each subdiode one may construct analytical expressions for the potential in the whole diode and de"ne the growth rate of its perturbation and the spatial form of the potential perturbation for such a diode. 1.2.4. The nonneutralized Pierce diode This section is devoted to the nonneutralized Pierce diode, corresponding to a cold electron #ow in a diode with a neutralization parameter c di!erent from unity. Again planar diodes and immobile ions are of general interest here.

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Smirnov [53] has studied the nonuniform steady states for cO1 in the regimes without electron re#ection and with total re#ection. Formulas for wavy distributions without re#ections have been derived and aperiodic instabilities were studied, too. Some formulas were obtained that can be used to determine the boundaries of aperiodic instability. The line was constructed in the (c, e)-plane for 04c4R representing the boundary of aperiodically stable regions for all steady states with total re#ection. The main advantage in comparison with other papers was the use of an arbitrary external voltage. However, he did not study situations with partial re#ection, assuming that such states do not exist. Also, oscillatory instabilities were not at all discussed. Furthermore, regions of existence of potential distributions of di!erent kind in the (e, d)-plane and curve separating stability from instability regions were not shown. Shumilin [55] has studied the stability properties of nonuniform steady states of planar diodes with a positive anode potential and an electron #ow passing through a positive ion background in the regime of a space-charge-limited diode current. In this case potential distributions (PD) with a virtual cathode (VC) exist in the interelectrode space, re#ecting back to the cathode a portion of electrons. Between the VC and the anode a monotonic PD was assumed, too, accelerating the electrons. Shumilin has used a model in which the VC was placed at the emitter, so that the electrons enter into the diode space with zero velocity. By this he disallowed the VC to oscillate when a perturbation was applied. Instead of the usual Pierce boundary conditions at the emitter, namely vanishing perturbations of the potential, electron velocity and density, which were used by all authors in studying the stability of Pierce diodes, Shumilin preferred to describe a beam with a zero velocity at the emitter subject to di!erent boundary conditions: vanishing values of the perturbed potential, the electron velocity and the electric "eld strength at the emitter. He derived a dispersion relation and has studied the stability for two special cases, the vaccum diode (the ion density being zero everywhere) and a diode with constant ion density, 0(c(2. Pretending to describe the space-charge-limited diodes, i.e. the regimes with partial re#ection, the author concluded that for both cases considered the system is stable. The correctness of the assumption that the position of a VC does not change when perturbations occur is questionable because in numerical calculations [39] it was shown that in the course of an instability in diodes with electron re#ection the position of the VC oscillates. Therefore, the results obtained in Ref. [55] are doubtful. Coutsias and Sullivan [6] have investigated nonuniform steady states of planar, short-circuited diodes in the regime without electron re#ection, namely the vaccum diode (without reference to Bursian) and the nonneutralized Pierce diode with in"nitely massive ions and di!erent c's. They used the method of characteristics, referring to Whitham's book [56]. Regions (lines) of existence of such states without re#ection were studied and plotted for the Bursian diode in the (e, d)-plane. For the Pierce diode di!erent units on axes were used: ec and dc, where d is the dimensionless gap, c the neutralization parameter and e the emitter electric "eld strength. Calculations were carried out for c"0.8, 0.95, 1.0 and 1.1. The authors analyzed in detail the stability properties of those states of the Bursian diode which are located in the vicinity of the bifurcation point d"4/3. Only short-circuited diodes have been studied and lines marking stable and unstable states with respect to the aperiodic instability have been plotted. (Godfrey [49] has plotted such curves four years later and only for the case c"1.) In plotting the (e, d)-plane the authors unlike Godfrey [49] used the true physical sign of the emitter electric "eld strength e. They did not study regimes with electron re#ection at all and disregarded an external voltage.

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Vybornov [57] has studied in detail nonuniform states of the planar, short-circuited, nonneutralized Pierce diode with in"nitely massive ions and di!erent c's in the regime without re#ection. In his study, the author made use of the methods exhibited in [42,45]. He has classi"ed di!erent kinds of potential distributions for this regime and has derived their formulas. The author was the "rst to show the d(e) dependencies, i.e. regions of existence of potential distributions of di!erent kind, for cO1 at "xed collector potential <"0. He has derived the dispersion relation and has evaluated the aperiodic stability properties for various c's and d's. He asserted that for "xed c in the range 0(c(1 steady states exist which have a current in excess of the Pierce current and which are aperiodically stable for any current. For c'1 the stable steady states should exist everywhere, according to him. As will be shown later (see Section 7.12) these statements are incorrect. The curve in the (d, c)-plane separating regions of stable and unstable states has been plotted. In the next section of our paper, we will obtain formulas for wavy steady potential distributions for the regime without re#ection which essentially di!er from the formulas of his paper. The drawback of Vybornov's formulas is that they are di!erent for c(1 and for c'1 and that he has a singularity at c"1, while our formulas will be the same in both regions and have no singularity. And "nally, in a more recent paper, Chen and Lindsay [58] have investigated the short-circuited nonneutralized Pierce diode for di!erent c's making use of the Lagrangian approach. They have numerically shown the existence of oscillatory solutions without re#ection and have plotted regions of unstable, stable, oscillatory and chaotic regimes in the (c, d)-plane around d/p+3 and c+1. As in Godfrey [49] a Feigenbaum-like scenario was found for the approach to the chaotic regime. Their numerical time-dependent solutions in the oscillatory regime show after a brief transient that the model settles to a steady oscillation at a "xed amplitude and frequency. Whereas the amplitude and frequency were found to depend markedly on c, the separation of the electrodes d had only a negligible e!ect. For the use as a vircator, a virtual cathode oscillator, they advise the regime 1(c(1.3. Thus our historical review of nonneutralized Pierce diodes with arbitrary c has shown, as a rule, that the authors have studied only regimes without electron re#ection of short-circuited diodes and have not explored the full picture of regions of stable and unstable steady states in the (d, e)-plane including arbitrary collector voltages. In addition, the oscillatory branches, if at all, have not been treated exhaustively. Before we "nally report on experiments and applications of the Bursian and Pierce instabilities, we mention some further extensions, which have been performed in the past. 1.2.5. Various extensions of plasma diode theories One obvious extension is the inclusion of ion mobility. Yuan [59] has written a note on the e!ect of a "nite ion mass and an electron thermal spread on the aperiodic instability of a space-chargeneutralized electron beam in a planar diode. He has derived dispersion relations and has plotted the branches of the aperiodic Pierce instability. A "nite ion mass was shown to increase the threshold for instability, i.e. the limiting current in the case of monoenergetic electrons. The instability branches do not cross the zero level and are bounded to the left side in the d-space. For large d, (d<1), they lie below the branches obtained with immobile ions. Analytic formulas for the left boundaries of these branches were not presented. He also found that for immobile ions and for a "nite thermal spread of the electrons, an aperiodic instability can only develop if the electron #ow's Mach number M exceeds unity. The growth rate and the threshold current are found to be

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reduced in comparison with the cold beam case, i.e. the instability is weakened and the boundaries of instabilities are enlarged by a factor M/(M!1). Vladimirov et al. [60] have investigated and plotted the oscillatory branches of the Pierce instability for the uniform state of a planar short-circuited diode both for in"nitely heavy ions and for hydrogen ions in the limit of initially resting ions, v "0. They obtained the dispersion  relation, but missed the stable aperiodic branches. Kolyshkin et al. [61] have treated the aperiodic and oscillatory branches of the Pierce instability for m "R and for 1004m /m 4250 000. Each aperiodic branch is shown to split into stable and  unstable ones in the vicinity of vanishing growth rates. Both, oscillatory Pierce branches and new Pierce}Buneman branches are plotted for a wider d-region. Farouki et al. [9] have studied monotonic potential distributions in a 1-D vaccum diode with a #ow of cold ions at nonzero ion velocity and the emitter electric "eld strength in the regime without re#ection. The ion transit time through the gap was found as zeros of a polynomial of third degree, which may have three real roots. For the case of only one real root, the authors have obtained a closed-form expression for the spatial PD. Schamel and Bujarbarua [62] have generalized the Lagrangian integral formulation of [49,50] in order to investigate the plasma diode dynamics on the slow ion time scale. Cold ions were injected with "nite velocity, whereas electrons were treated kinetically described by an equation of state. Deviations from short circuitity and charge neutrality were allowed for. Existence and stability of equilibria for the case of charge neutrality (c"1) were investigated, including virtual cathode solutions involving electron re#ection. Equilibria under electron-rich injection conditions (c(1) were found, too. Schamel and Maslov [63] have applied the Lagrangian integral formulation of [62] to the electron #ow, supplemented by "nite ion #ow, in the classical short circuited Pierce diode under neutral injection conditions. The electronic diode dynamics was found to be controlled by the new system parameter k( ,m v /m v which expresses the ratio of electron and ion kinetic energy at    the injection point. Kolinsky and Schamel [64] have investigated the e!ect of a "nite ion velocity on the stability of short-circuited, neutralized Pierce diodes, making use of the generalized Lagrangian formulation of [62,63]. A generalized dispersion relation for electrostatic perturbations was found, which exhibits several new features. A new control parameter of the system was found: dK ,d(1#k( ), where d is the usual Pierce parameter. New growing as well as damped oscillatory modes were observed, which in the limit v P0 become Pierce}Buneman modes and undamped ion plasma oscillations,  respectively. The lowest threshold for aperiodic instability was found to be given by dK "p which implies dP0 in the limit v P0. In this case of initially resting ions, the diode is completely  destabilized as found earlier by Faulkner and Ware [65], Vladimirov et al. [60], Kolyshkin et al. [61] and Gedalin et al. [66]. As mentioned, in an application to inertial con"nement schemes with heavy ions, the luminosity (density) of the ion beam can be substantially enhanced by a neutralization scenario using a fast moving electron beam. For example, if n denotes the ion density just   below threshold of the Pierce instability, then about n +0.8n can be achieved by choosing   k( "4. This is a dramatic increase in comparison with a neutralization by use of comoving electrons. The latter would result only in n &n /1836 for a proton beam.   Kolinsky and Schamel [24] have extended the problem of Ref. [64] by considering counterinjection of ions, too. It was shown, that counterinjection introduces new unstable oscillatory branches

A.Ya. Ender et al. / Physics Reports 328 (2000) 1}72

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which destabilize the diode for all values of the Pierce parameter d, in contrast to coinjection where the diode is stable for su$ciently small values of d. The analysis includes Pierce}Buneman modes for initially resting ions, v "0. If "v "/v is increased, a stronger destabilization is observed for    counterstreaming ions in comparison with costreaming ions. As a representative of another interesting series of investigations we call attention to a paper of Fruchtman et al. [67], in which transverse and transient e!ect of the current conduction in plasma-"lled diodes were addressed. An oscillatory electron #ow was found in the early stage of current conduction, when ions are still resting, and the current conducted by beam electrons was found to be neutralized by a return current carried by the preexisting plasma in the steady state. This investigation employs a 2-D, electromagnetic plasma behavior and makes use of the concept of a generalized vorticity, frozen into the electron #uid. Lau and coworkers have extended the classical papers of Child, Langmuir and Schottky for the limiting current to the quantum regime [68], to a planar diode crossed by an external transverse magnetic "eld [69,70] and to a diode with an emitter, in which the emission was self-regulating [71]. In [68], for example, a diode was studied in which the gap size might scale down to tens of angstroms so that quantum mechanic tunneling can a!ect the transmitting current. In [71] a tip diode was studied in which the electron emission was controlled by the surface electric "eld that depended on the applied potential. In deriving the formulas relevant analogs of Llewellyn's equation were used. In limiting cases the solutions turned smoothly into the classical Child law or the Fowler}Nordheim relation [72], respectively. Another recent application of Llewellyn's equation was presented by Christensen and Lau [73]. They have investigated the 2-D cycloidal multistream electron #ow in a gap with crossed electron and magnetic "elds. A violent modulational instability was found in cases of a small ac voltage imposed across the gap, which is due to the formation of a virtual cathode right in front of the cathode. This paper reinforces the necessity to pay close attention to the low-energy electrons near the cathode and their careful treatment in particle codes. The role of the Pierce instability in the formation of double layers [74,75] has been considered by Raadu and Silevitch [76]. They identi"ed the Pierce instabilities in double-layer experiments where they lead to ion trapping. The e!ects of an external circuit were included in a dispersion relation that was corrected later by Kuhn. It shows that the growth rate in the unstable range is reduced by an external impedance and that in case of immobile ions the onset criteria are una!ected by the circuit. Last but not least, we mention two review articles by Siegbert Kuhn [77,78] which for the time being summarize the status of art of diode dynamics and of the physics of bounded plasma systems, in addition to his own work. Especially, the last one deals with the fundamentals and results of a kinetic treatment and of the simulation of bounded plasma systems, such as Pierce-like diodes and single-emitter plasma diodes. 1.3. Experiments and applications of the Bursian and Pierce diode instabilities In this last part of Section 1 laboratory experiments are addressed in which the two instabilities, the Bursian and the Pierce diode instabilities, play a decisive role. Nezlin and Solntzev [79] have determined experimentally the thresholds of instability which develop when an electron beam passes the plasma layer. They a$rmed in particular that a virtual cathode, limiting the electron current, is formed as a result of the Pierce instability.

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Gverdtziteli et al. [80] have studied experimentally the nonlinear current oscillations in a Knudsen diode with surface ionization (KDSI) which was a thermionic energy converter (TIC). They evaluated the time needed for an excitation of the virtual cathode and hence for the limitation of the current in the course of the nonlinear oscillations and asserted on this basis that the e!ects resulting in a current limitation were more likely connected with the Pierce instability. Babanin et al. [5] have investigated experimentally large-amplitude current oscillations in a Cs}Ba diode. They determined the threshold of oscillations in such a KDSI (TIC) with varying neutralization parameter. The results were in good agreement with the theoretical ones of Ref. [54]. The amplitude e!ect of an external transversal magnetic "eld on the oscillations was also studied. A su$ciently weak magnetic "eld (Larmor radius of the order of the gap distance) resulted in a strong reduction of the oscillation amplitudes. It converts a beam that was formed by a potential jump near the emitter into an approximately isotropic distribution function in the course of the oscillatory process and suppressed any Pierce-type instability. For the "rst time oscillations were studied in an underneutralized regime (c(1). Oscillations in Cs}Ba diode were detected at electric current densities of the order of several A/cm, which exceeded by an order of magnitude the maximum current densities in Cs "lled diodes. Iizuka et al. [81] have studied the double-layer dynamics and low-frequency oscillations of a current passing through a Q-machine, which is e!ectively a KDSI. They have measured the spatial potential distributions at di!erent times. As a result one can distinguish a slow and a fast stage of the process which develops due to a Pierce-type instability. These measurements corroborate theoretical investigations, such as Refs. [82}85]. Burger [82] as well as Braithwaite and Allen [83] have argued that there are essentially two phases in the evolution and they have noted that there exists a second dc state which is adapted by the system after a rearrangement of the electrons occurring on the fast electronic time scale during which the ions are virtually frozen. In Ref. [84], Kuznetsov and Ender have shown that the oscillatory process is closely connected to an evolution of the (g, e)-diagram which allowed to "nd out the triggering time instants of the electron instability and the plasma state from which a new slow stage of the oscillatory process starts. The ion velocity distributions during the di!erent stages was calculated for the "rst time, and ions with a spatially localized kinetic energy were found to be produced during the oscillation process. Bauer and Schamel [85] have pointed out by a PIC simulation the decisive role, kinetic phase space vortices such as electron and ion holes play during this process, which is known also as the `potential relaxation instabilitya. Also, the controlling nature of the virtual cathode was recognized which acts like a value through a periodic change of its depth. A kind of potential relaxation instability was also addressed by an investigation of Kucherov et al. [86]. These authors have studied the electric "eld distribution of a plasma diode by probing the plasma with a pulsed planar electron beam similar to that used by Ott [87] previously. The dependence of the threshold of the steady-state diode current on an applied voltage was plotted. The spatial potential distributions for various phases of the oscillation process and the velocity of the moving virtual cathode were measured and good agreement with the theoretical results of Ref. [84] was found. It was shown that sharp redistributions of the potential in the gap occurred on the electron time scale as a result of a Pierce-type instability on speci"c stages of the oscillatory process.

A.Ya. Ender et al. / Physics Reports 328 (2000) 1}72

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Rukhadze et al. [88] have reviewed the physics of an electron beam injected with high enough energy into a vacuum drift chamber. As the beam current exceeded the limiting vacuum current, the Bursian instability developed resulting in the formation of a virtual cathode (VC) and a fraction of electrons was re#ected back to the cathode. The VC began to oscillate and the electron beam interacting with the t-dependent electric "eld located between the cathode and the VC lost its energy which may have been converted into microwave radiation. Hence, in such a system, a direct conversion of e-beam energy to microwave energy may take place. Devices operating on the basis of the VC oscillations are termed vircators [89]. Here electrons are emitted by the cathode and accelerated by a positive voltage which exists between the anode and the cathode. An electron beam with a small velocity spread enters through the transparent anode into the vacuum drift chamber. Those electrons that are re#ected by the VC pass through the anode netting or foil and return by a retarded "eld again, and thus oscillate between the anode and the VC. If the oscillating electrons are moved o! such a device has been named reditron [90]. Another possibility of microwave emission was proposed analytically by Botton and Ron [91]. They have shown that the e$ciency of radiation sources by means of intense relativistic electron beams is enhanced when plasma-"lled diodes and the mechanism of self-induced distributed feedback were used. In this Cerenkov free electron maser two counterpropagating waves modulate the plasma density by the ponderomotive force causing a grating in the refractive index of the plasma. The two waves are coupled by Bragg re#ections. The distributed feedback enhances the interaction between the resonant wave and the beam and reduces the threshold for oscillations, both enhancing the e$ciency of the oscillator. In the TIC a Pierce type instability develops at the moment when the operation point switches from the retarding potential region of the I}<-characteristics to the saturation current region. As a result, a VC is created in the interelectrode gap and a strong current reduction takes place. This e!ect was proposed as a mean to generate directly an alternating current in the TIC [92,93]. If an inductance is connected in parallel to the TIC then magnetic energy is stored by the inductance during the stage of current change along with the retarding potential. When the operation point enters the saturation current region and the current is cut o!, the magnetic energy, accumulated in the inductance, is released as a voltage pulse in the external circuit. After this the process repeats itself periodically. This e!ect was also investigated experimentally: electric power was generated as 15 V voltage pulses with a period de"ned by the magnitude of the inductance. (In the dc regime the voltage generated in a TIC is only about 1 V.) The conditions for the development of a Pierce-type instability may not only be realized in the KDSI, where ions are generated at the emitter surface by contact ionization, but also in the thermionic Knudsen discharge, where an electron beam crosses an ion background generated in the volume. This e!ect was used to create a current modulation in a Cs}Ba diode [94]. Here the plasma turned out to be unstable due to the Cs atoms leaving the interelectrode gap. As a result of the Pierce-type instability the current was cut o! sharply. The diode returned to its initial state only after the neutral pressure was restored. A stable modulation of the current was observed for a Cs pressure between 1.5;10\ and 3.5;10\ Torr. An applied voltage of 10}50 V with current densities of 10}20 A/cm was modulated by frequencies of 5}20 kHz. Selemir et al. [95] have reviewed the theoretical and experimental investigations on microwave devices with VC which have been carried out in the Russian Federal Nuclear Center (Arsamas). Some unsolved problems were formulated. The bulk of them was to "nd conditions under which

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virtual cathode oscillations appear as well as to "x the region of external parameters where the electron beam converts directly its energy to the electric "eld with maximum e$ciency. Klinger et al. [96] have studied experimentally in particular strongly nonlinear relaxation oscillations of the discharge current in thermionic Knudsen discharges. They suggested the Pierce-type instability as a trigger mechanism for the formation of the plasma structures limiting the current. Last but not least, Kolinsky et al. [97] have made an attempt to understand theoretically the trigger mechanism of a Pierce-type hydrodynamic instability found experimentally in thermionic discharges at low pressure. Particle in cell simulations were used to obtain detailed information about the plasma parameters such as phasespace and potential distributions. In addition, a theoretical model was evaluated in which the simultaneous in#uence of the ion dynamics, collisions with neutrals, and the sheath capacitance was taken into account. This was done by adopting the Lagrangian description of the generalized Pierce diode developed earlier [24,63,64]. The new results obtained were threefold. First, few ion-neutral collisions prevent the coupling between the electron and ion dynamics on the electronic time scale thus removing oscillatory growing Pierce}Buneman modes. Second, the classical Pierce diode supplemented by an external capacitance to account for the damping e!ect of the sheath on the Pierce instability, explains the existence of stable equilibria. It further reveals the nature of the bifurcation at the instability threshold namely, as a Hopf bifurcation that triggers the nonlinear relaxation oscillations observed both in simulation and experiment. Third, the transient behavior prior to the onset of instability is qualitatively described within the hydrodynamic model. In summary, an extensive literature, only part of which could be mentioned, is already available dealing with the various aspects of plasma diodes. Nevertheless, this review has shown that the problem of a nonneutralized Pierce diode with a neutralization parameter deviating from unity lacks a full theoretical analysis. As a rule, authors have studied analytically only regimes without electron re#ection and without an applied potential V and have not plotted the full picture of regions (including arbitrary V-values) with stable and unstable steady states in the (e, d)-plane. In addition, the oscillatory branches of the instability were not studied either. We hence feel motivated for the following detailed analysis of the such extended diode physics.

2. Notation and basic equations We are dealing with a planar diode of length ¸ in the cold beam approximation and assume that the electrons of the beam enter the diode region with constant velocity v and constant density  n> at the emitter surface which is located at z"0 and is at zero potential. Electrons which arrive  at the electrodes will be totally absorbed. The collector is placed at z"¸ and has a potential .  Ions are assumed to be immobile and uniform with density n . Collisions and other discrete particle e!ects, such as the emission of waves, are assumed to be absent throughout the paper. We are interested in the various states the plasma diode can adopt and introduce the following parameters: the neutralization parameter c" : n /n> , 

(1)

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the normalized interelectrode distance (Pierce parameter) d" : u> ¸/v ,: ¸/j ,   " and the normalized bias between collector and emitter

(2)

<" : e /mv , (3)   where u> "(n>e/e m) is the plasma frequency corresponding to n>, and e, m are the     (positive) elementary charge and the electron mass, respectively and j is the formal Debye length " j " : v u>\. "   The set of equations governing the diode dynamics depends on whether electron re#ection takes place or not. If no electron re#ection takes place, our basic system of equations becomes R n#R (nv)"0 , (4a) O D R v#vR v"!e , (4b) O D !R e"n!c , (4c) D which are the cold electron #uid equations coupled with Poisson's equation. In Eqs. (4a)}(4c) the normalized coordinates q"u> t, f"z/j  " and #uid quantities

(5)

n"n /n>, v"v /v     are used, and the normalized electric potential and electric "eld are given by

(6)

g"e /mv , e"eEj /mv ,  "  where e"!R g holds. D The system (4) is supplemented by the Dirichlet boundary conditions

(7)

n(0, q)"1, v(0, q)"1 ,

(8a)

g(0, q)"0 ,

(8b)



g(d, q)"<"!

B

e(f, q) df . (8c)  In the time-independent case when electron re#ection takes place within the diode we distinguish three types of cold electrons, denoted by subscripts s, where s"1 stands for the transmitted, s"2> for incoming and s"2\ for the re#ected electron beam. We then have in normalized quantities R n #R (n v )"0 , O Q D Q Q R v #v R v "!e(f, q) , O Q Q D Q !R e" n (f, q)!c , D Q Q

(9a) (9b) (9c)

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with the boundary conditions n (0, q)"1!r, n > (0, q)"r ,  

(10)

v (0, q)"1, 

(11)

v > (0, q)"1 , 

where r is the re#ection parameter which may generally depend on time. In Ref. [40] the two regimes are called type-I and type-II #ow and the splitting parameter l of that paper is related to the present one by l"(1!r)/(1#r). Note that (4) (resp. (9)) contain as special cases the Bursian diode (pure electron diode), c"0, and the generalized Pierce diode, c"1.

3. Equilibrium solutions without and with re6ection The simplest equilibrium is the one where the incoming beam is completely transmitted (r"0). Dropping the time dependence, from Eqs. (4a), (4b) and (8a), (8b) for the density we easily "nd n"(1#2g)\ ,

(12)

and Poisson's equation (4c) becomes g(f)"(1#2g)\!c .

(13)

Multiplying (13) by g(f), we get by integration g(f)!e "2[(1#2g)!cg!1] , 

(14)

where e represents the electric "eld at the emitter.  By a second integration we can obtain g(f) and by applying the boundary condition (8c) we can establish a relation between the electric "eld at the emitter e and the collector potential <.  Next, we turn to equilibria with re#ection. In this case there exists a point fH in the diode where the potential energy of the electrons coincides with their energy mv /2 at the emitter, i.e. the velocity  of the electrons vanishes, and the value of the potential becomes gH"!1/2. In the case of partial re#ection this point represents the point with minimum potential, g "!1/2.

 To the left of the potential minimum the total electron density consists of the sum of the injected electrons (s"1, 2>) and of the re#ected electrons (s"2\). To the right only the transmitted electrons (s"1) contribute. We, therefore, have for the total density



n"

(1#r)(1#2g)\, f4fH , (1!r)(1#2g)\, f'fH .

(15)

Poisson's equation then becomes



n(f)"

(1#r)(1#2g)\!c, f4fH , (1!r)(1#2g)\!c, f'fH .

(16)

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Denoting by f the position of the potential minimum nearest to the emitter we obtain a "rst

 integral both for the case without re#ection (r"0) and the case with partial re#ection (0(r(1): (g(f))">c\[1!Z(g)], f4f

,

 ">c\+[Z(g)#2r>\]![Z(g )#2r>\],, f'f ,



 where we introduced the functions >" : [ce #(1#r!c)] ,  Z(g) " : [c(1#2g)!(1#r)]>\ .

(17a) (17b)

(18) (19)

In the regime without re#ection, r"0, the value of the potential minimum has to be greater than !1/2, and Eqs. (17a), (17b) equals Eq. (14). If we now use the conditions g"! and g"0 for the  minimum point then we obtain for the boundaries of this regime "e "((2!c) . (19a)  In the regime with partial re#ection, 0(r(1, we will, according to Ref. [36], distinguish between two types of potential distributions (PD): a PD of the "rst kind, where the re#ection point lies inside the diode region, and a PD of the second kind, where the re#ection point is at the collector. In the regime of partial re#ection of the "rst kind the potential of the re#ection point is equal to !1/2 and the electric "eld dg/df vanishes in this point. With these values we obtain from (17a), (18) and (19) a relation between the re#ection coe$cient r and the emitter electric "eld strength e :  "e ""[2(1#r)!c] . (20)  The boundaries of the regime of partial re#ection are given by substituting r"0 and r"1, respectively, (2!c)4"e "4(4!c) . (21)  In the considered region the typical PDs for large enough diode lengths d are of wavy type with alternating potential minima and maxima. These extremum points can be directly derived from (17a), (17b) by setting g"0. For the regime without re#ection (r"0) we obtain g

(22a) "!#+1![ce #(1!c)],/2c ,   g "!#+1#[ce #(1!c)],/2c . (22b)

   For the regime with re#ection the maximum potential nearest to the emitter for e (0 is found to  be g



"!#+1#r#>,/2c . (23)

  Note that, whereas the electric "eld strength in the point of re#ection always vanishes in the case of partial re#ection of the "rst kind, it generally di!ers from zero in the cases of re#ection of the second kind or of total re#ection. The expression for the PD is found analytically by integration of

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Eqs. (17a). To the left of the "rst extremum we obtain sgn(e )  +>[(1!Z(g))!(1!Z(0))]#(1#r)[arcsin Z(0)!arcsin Z(g)], . f" c Extremum points in the PD are given by Z"$1. For e (0 the "rst maximum is at  e (1#r) p #arcsin[(1#r!c)/>] !  , f "

 c c 2






(24)

(25)

and the "rst minimum is at






(1#r) 3p e f " #arcsin[(1#r!c)/>] !  .

 c c 2 and the PD between f and f reads



 f"f





#c\>(1!Z(g))#(1#r)c\

(26)



p !arcsin Z(g) . 2

For e '0, the location of the "rst minimum is found from (24) and becomes  (1#r) p e f " !arcsin[(1#r!c)/>] !  .

 c 2 c





(27)

(28)

The PD solution beyond the "rst minimum and the next maximum can be obtained by integration of (17b):



f #c\[p #arcsin ZI (g)![ce #(1!c)][1!ZI (g)]], r"0 ,   f"  r'0 , f #c\(1!r)[p #arcsin ZI (g)![1!ZI (g)]],

  where ZI (g) is given by



ZI (g)"

[c(1#2g)!1][ce #(1!c)]\, r"0 ,  [c(1#2g)!(1!r)]/(1!r), r'0 .

(29)

(30)

From (29) we can immediately extract the spatial period of the PD beyond the "rst minimum which is given by 2pc\(1!r) .

(31)

If all electrons are re#ected, r"1, the electric "eld strength in the re#ection point becomes according to (17)}(19) eH"(e #c!4) , (32)  which, as a rule, di!ers from zero. There are no electrons at all to the right of this point and the PD is a parabola g(f)"!!eH(f!fH)!c(f!fH)/2 . 

(33)

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These formulas allow us to carry out a complete analysis of the PD in a diode with varying external parameters. Figs. 2a}d show how the PD changes when e is varied for c"1 (the generalized  Pierce diode). Figs. 2a and b represent solutions without re#ection, Figs. 2c and d with re#ections. In Figs. 2a and c e '0 and in Figs. 2b and d e (0 hold. In the regime without re#ection, as it   follows from (22a), (22b), "e "(1 holds. In this case the potential vanishes at the points f"2mp  with m"1, 2, 3,2 independent of e . In this regime the maximum potential grows with increasing  "e " and the minimum diminishes until !1/2. In the regime with partial re#ection, Figs. 2c and d,  g stays for increasing "e " equal to !1/2 and g to the right of the re#ection point decreases

 

 until !1/2. As a result the wave amplitude in the wavy region "rst increases with increasing "e "  and then, when the transition to the regime with re#ection occurs, it begins to decrease and at "e ""3"1.732 it vanishes. In contrast to the regime without re#ection, where the period is  independent of "e ", the period in the regime with re#ection is proportional to 1!r and decreases  with increasing "e " and "nally disappears at "e ""3. Therefore, in the latter regime, as a rule, for   given diode length d and collector potential < several PD solutions exist.

Fig. 2. (a) The electric potential g of the generalized Pierce diode (c"1) as a function of space f for several values of the emitter electric "eld e . Its stability behavior is denoted by di!erent drawings: solid (dashed) lines represent aperiodically  stable (unstable) solutions whereas dotted lines mark marginally stable solutions. A transition of the stability is in addition emphasized by an open circle. Here the situation of no re#ection and e '0 is drawn. (b) Same as Fig. 2a, except  for e (0. (c) Same as Fig. 2a, except that solutions with re#ection are shown for e '0. (d) Same as Fig. 2c, except for   e (0. 

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4. Criteria for existence and stability of equilibria without re6ection in the Lagrangian representation 4.1. The Lagrangian formulation The problem, we dealt with in Section 3, namely the establishment of equilibrium solutions, can be alternatively solved in Lagrangian formulation. Restricting the analysis to equilibria without re#ection we will derive an existence criterium and show, that part of the solution can be simpli"ed by a scaling argument. Furthermore, a linear dispersion relation will be derived which solves the linear stability problem of these types of equilibria. Following essentially Ref. [40] and the papers cited therein we "rst transform the Eulerian system (4) into a Lagrangian system. De"ning the stream function q (f, q) by  R q "!n, R q "nv , (34) D  O  we get by a f-integration of the "rst equation



D

n(f, q) df . (35)  For "xed q we obtain by inversion of (35) f(q , q). It represents the position of a #uid element at time  q which was emitted at time q . Switching to Lagrangian quantities [40] and integrating Poisson's  equation with respect to q we "nd  (36) f$ (q , q)!(q!q )#cf(q , q)"!e (q) ,     where dot stands for R . In the steady state the electric "eld on the emitter is time independent. In O generality e will depend on time.  The dynamics represented by (36) is controlled by two constraints: q (f, q)"q! 

(i) the transit condition f(q , q #¹)"d ,   and

(37)

(ii) the potential condition



O\2

dq f$ (q , q)f(q , q)"< ,   

(38) O where dash stands for d  . ¹ is the transit time, the #uid element needs to reach the collector. From O Eqs. (36)}(38) we see how the external control parameters c, d and < enter into the description; d and < are introduced by the boundary conditions whereas c appears in the dynamical equation (36) only. It is worth noting that the Lagrangian formulation is superior to the Eulerian formulation in situations where two or more particle streams with "nite velocity are involved [24]. In the present paper with only one #uid (if we disregard re#ections) both methods work equally well. In the case of equilibrium without re#ection we solve (36) by the ansatz q" : q!q , s(q) " : f(q , q) ,  

(39)

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and obtain s(q)#cs(q)"q!e .  The solution for s(0)"0 and s(0)"1 is given by (c!1) q e sin(cq) . s(q)" #  [cos(cq)!1]# c c c

(40)

(41)

In addition, we have to ful"ll the two constraints (37),(38) which become s(¹)"d ,

(42)

s(¹)"(1#2<) ,

(43)

and which read after insertion of (41) as ¹ e c!1 #  [cos(c¹)!1]# sin(c¹)"d , c c c

(44)

e c!1 1 !  sin(c¹)# cos(c¹)"(1#2< . c c c

(45)

These are two transcendental equations for the two unknown quantities e and ¹. Fig. 3 demon strates the dependence of the transit time ¹ on the gap value d for the PDs presented in Figs. 2a and b for c"1 and for various values of e . It is seen that for e '0 (e (0) the deviation of ¹ from   

Fig. 3. The transit time ¹ as a function of the gap distance d for di!erent emitter electric "elds e , corresponding to the  PDs of Figs. 2a and b.

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that of the uniform solution (e "0) is positive (negative). Moreover, the deviation increases with  larger "e ". Similar results are obtained for cO1. Elimination of e from both equations yields   K[1!cos(c¹)]#(cd!¹)csin(c¹)"0 , (46) where K, the Kolinsky parameter, is given by K" : 2!c[1#(1#2<)] .

(47)

Eq. (46) suggests the following rescaling ¹I " : c¹, dI " : cd ,

(48)

in terms of which, Eq. (46) becomes K[1!cos ¹I ]#(dI !¹I )sin ¹I "0 .

(49)

For given c and < (and hence K) the rescaled transit time ¹I is a unique function of dI and K: ¹I "¹I (dI ; K), i.e. it is in some sense universal as it depends on two (derived) parameters only. By the renormalization (48) we, therefore, achieved an interesting simpli"cation of our problem, namely, a reduction of the 3-D parameter space (c, <, d) to a 2-D parameter space (dI , K). If we, hence, have solved (49) for a given set dI and K we know ¹ for a 1-D subset of the parameters (c, <, d), belonging to the same dI and K. The transcendental equation (49) can be rewritten by using trigonometric functions of the half angle ¹I /2. We "nd K tan ¹I /2"¹I !dI ,

(50)

which has to be solved numerically in generality. On the other hand, if we also introduce in (44), (45) trigonometric functions of the half angle, eliminate the terms proportional to (c!1) and use (50) we "nd 2e c(¹I !dI )"(¹I !dI )#K[K#2(c!1)] . (51)  This equation tells us that c cannot be scaled out by a simple renormalization of e . The  determination of e remains a triple parametric problem. Nevertheless, after having solved (50) it is  easy to obtain from (51) e "e (dI , K, c).   4.2. Equivalence between Eulerian and Lagrangian representation In the next step, we will show that both approaches, the Eulerian formulation of Section 3 and the Lagrangian formulation of this section, yield identical results. We introduce the quantity a " : ¹I !dI and solve (51) with respect to a and obtain a"e c!sgn e [e c!K(K#2(c!1))] , (52)    where we have selected a branch which satis"es aP0 if
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on the falling part of the potential curve then we have a plus sign, independent on the sign of e . It is  the cos in (44), (45) which changes sign at an extremum and which is represented by the square root in (52). In terms of a, Eq. (50) becomes dI "!a#2 arctan(a/K) .

(53)

One can "nd by formula (24), using (18), (19) for "xed values of c and e , the dependency f(g) and we  can thus plot the PD in the regime without re#ection. For c and e "xed, the PD can also be plotted  with the help of Eqs. (52) and (53), if we move the collector along the PD, i.e. if we put <"g and dI "f. For comparison, we set g"< and f"d in (24), with dI "cd, and obtain dI "sgn e +>([1!Z(<)]![1!Z(0)])#arcsin Z(0)!arcsin Z(<), ,  where > and Z(<) are given by

(54)

>"[ce #(1!c)] ,  Z(<)">\=(<) ,

(55a) (55b)

=(<)"c(1#2<)!1 .

(55c)

We show that there is a unique correspondence between the nontrigonometric parts in (53), (54) and between the trigonometric parts, separately. From the last equation in (55) it follows, with K given by (47), that K"(1!c)!=(<)"!=(0)!=(<)"!>[Z(0)#Z(<)] .

(56)

We then have >[1!Z(<)]"[>!=(<)]"[ce !K(K#2(c!1))]  and >[1!Z(0)]""e "c ,  so that the nontrigonometric part on the rhs of (54) coincides with !a, given by (52), as it should. To show the equivalence of the trigonometric parts, we introduce Z(<)"sin F(<), Z(0)"sin F(0) ,

(57)

and write !a as !a"sgn e >+[1!Z(<)]![1!Z(0)],"sgn e >+cos F(<)!cos F(0), .   This means that the transit time of an electron along its trajectory from the emitter up to "xed point f with a potential g is expressed by q"f!sgn e >+[1!Z(g)]![1!Z(0)],  The value of a/K then becomes with K from (56): a cos F(<)!cos F(0) F(<)!F(0) "sgn e "!sgn e tan ,  sin F(<)#sin F(0)  K 2

(57a)

(58)

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where the last equation follows from setting F(<),b/2#a/2 and F(0),b/2!a/2 and by using the addition theorem for trigonometric functions. From (58) we then "nd




2 arctan

a "!sgn e a"sgn e [F(0)!F(<)]"sgn e [arcsin Z(0)!arcsin Z(<)] ,    K

which shows the equivalence between the trigonometric parts of (53) and of (54). 4.3. On the boundary of solutions without reyection Next we undertake a "rst evaluation of the region of existence for solutions without re#ection. We know that the collector potential < cannot be below !1/2 for this case. On the other hand, for small d, < can take any positive value for solutions belonging to the class of monotonic or single maximum PDs. For large enough d (d'd , see Section 6) the wavy PDs without re#ection are  bounded by wavy PDs with re#ection. At c"1 this boundary is at "e ""1 (see (19a)). If we restrict  ourselves to d5d and to "e "(1, < will lie in the limited region in the (d, <)-plane, shown in   Fig. 4a for c"1 and in Fig. 4b for c"0.8. The region of existence is periodic, being repeated beyond the point d'4p, with a period of 2p. Part of the bounding curves is obtained from (24) by setting e "1 and !1. The curve for negative < between points A and B represents the 

Fig. 4. (a) The region of existence of solutions without re#ection, bordered by the solid lines for c"1. (b) Same as Fig. 4a, expect for c"0.8.

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envelope of solutions with di!erent e starting from e "#1 at point A and extending to   e "!1 at point B (see Section 7.1.1). The envelope equation is given by  d"3pG[(!2<)#arcsin(!2<)] . (59) 4.4. Linear dispersion relation In this section, we "nally draw the attention to the linear stability of equilibria without re#ection. We solve (36) by the ansatz e (q)"e #e exp(pq) , (60)    f(q , q)"s(q)#f (q) exp(pq) , (61)   where e and f are small quantities and e and s(q) are equilibrium quantities of order unity    and p is the sought complex growth rate (or ip the complex frequency, respectively). To lowest nontrivial order we get from (36) for f  (p#c)f #2pf #f "!e (62)     subject to the boundary conditions f (0)"0"f (0) .   The solution is



(63)



 

p e exp(!pq) cos(cq)# sin(cq) !1 . f (q)"   c p#c

(64)

In addition, the boundary conditions (37) and (38) have to be satis"ed. The deviation from equilibrium in (60) and (61) leads to a deviation in the transit time ¹"¹ #¹ exp(pq), where   ¹ represents the equilibrium value now. Inserting this expression into (37), we "nd to "rst order in  the perturbation ¹ "!f (¹ )(1#2<)\ . (65)    From the potential condition (38) we "nd with the help of (62) and(65) in "rst-order perturbation theory [64],



2

dq f (q)#e d"0 .    Finally, we obtain by insertion of (64) into (66) and by integration the dispersion relation F(p, ¹ )"0 , 

(66)

(67a)

with F(p, ¹ )"e\N2 [2p cos(c¹ )#c\(p!c) sin(c¹ )]    !d(p#c)!2p#¹ (p#c) , 

(67b)

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where ¹ "¹ (<, c, d) represents the equilibrium transit time. It yields the (generally) complex   frequency of linear perturbations of equilibria without re#ection, which in generality are inhomogeneous. Several limiting cases, such as the transition to the generalized Pierce diode (c"1), to the uniform state of the short-circuited Pierce diode (c"1, ¹ "d) or to the Bursian diode  (c"0) ([40], Eq. (45)) can easily be recovered. Eq. (67) possesses a property of c-invariance, as seen by the introduction of the rescaled quantities (48), where ¹I "c¹ , and of   p"c\p . (68) We get the explicitly c-independent equation (69) e\N 2I  [2p cos ¹I #(p!1) sin ¹I ]"dI (p#1)#2p!¹I (p#1) .    Hence, it appears as if also the dispersion relation has some universal character. However, this question is more delicate than expected, because we need in addition to know how e (c) and K(c)  (resp. <(c)) transform, where we already made use of the notation of Section 5. As we will see in this section, not only e (c) but also d(c) must experience a shift, besides the rescaling of the above type,  in order to keep either borders of regions unchanged or to maintain the value of the potential minimum in the case of re#ection. Further details of the stability properties of the nonneutral beam plasma diode will be presented in Section 7.

5. The transformation of states 5.1. Dexnition and transformation of states without reyection From the above, especially from Section 3, it is clear that the PD is uniquely determined if the electric "eld at the emitter is given. In other words, (24) is a unique solution of the Cauchy problem for any given e .  In this section, we will now show that it su$ces to know the solution for c"1 in order to deduce the solution for any other value of c. We do this by establishing simple transformations. To begin with we set r"0 and see from (24), that the PD is expressed by two functions > and =" : >Z de"ned by (18) and (19) (see also (55)). The essential step in "nding the transformation is to preserve the value of > when c changes from unity to an other value: >(c, e (c)),>(1, i) ,  where for simplicity we introduce the shorthand notation

(70)

i" : e (1) . (71)  Eq. (70) uniquely de"nes e (c), where the parametric c dependence is explicitly pointed out. There  are some restrictions on the choice of possible i's, but this will be discussed later. We, however, remind that in the regime without re#ection "i"41 (see (19a)).

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With e (c), given by (70), the new PD(c) is uniquely determined and could be found by (24).  However, we now show that the PD(c) follows directly from PD(1) by a shift and a scaling transformation of the coordinates g and f. This transformation will be denoted by g,g(1)Pg(c) ,

(72a)

f,f(1)Pf(c) .

(72b)

The "rst one will be determined by preserving the value of = =(c; g(c)),=(1; g(1)) .

(73)

From conditions (70) and (73) it follows immediately that Z(c; g(c)),Z(1; g(1)) .

(74)

The second transformation is found by the ansatz f(c)"c\[f(1)#S(c)] ,

(75)

where the shift S(c) is needed for the compensation of the g-independent terms in (24), i.e. the terms associated with Z(c; 0). The concrete expressions are then easily found. In the present case r"0, we get from (70) and (18) e (c)"c\[i!(1!c)] .  We see that the rhs must be nonnegative which leads to the inequality "i"5"1!c" .

(76)

(77)

This means that for given c only the subset of PD(1) with e (1),i satisfying (77) can be  transformed into a new PD(c). If the equality sign in (77) holds, we have e (c)"0, which means that  e (c) cannot change sign in the region given by (77). If we also take into account that e "i, we   obtain that sgn e "sgn i. From (73) (resp. (74)) and (19) we "nd  g(c)#"c\[g(1)#] . (78)   In (78) only the parametric c-dependence and not the f-dependence is considered. Substituting (75) into (24) we obtain f(1)#S(c)"sgn e (c)+>(1, i)[(1!Z(1; g(1)))!(1!Z(c; 0))]  # [arcsin Z(c; 0)!arcsin Z(1; g(1))], .

(79)

On the other hand, by setting c"1 in (24) we also have f(1)"sgn i+>(1, i)[(1!Z(1; g(1)))!(1!Z(1; 0))] # [arcsin Z(1; 0)!arcsin Z(1; g(1))], .

(80)

Subtracting (80) from (79) we obtain S(c)"sgn i+>(1, i)[(1!Z(1; 0))!(1!Z(c; 0))]#arcsin Z(c; 0)!arcsin Z(1; 0), . (81)

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Furthermore, we have >(1, i)""i",

Z(c; 0)"(c!1)/"i",

Z(1; 0)"0 ,

(82)

and from (81) we get for the shift,




 



(c!1)  c!1 S(c)"sgn i "i" 1! ! #arcsin . i "i"

(83)

If we introduce y"arcsin((c!1)/i), the rhs of (83) can be written as i(1!"cos y")#y, from which it follows that in the present case of "i"41 the sign of S(c) equals the sign of (c!1)/i. Hence, the direction of the shift in f-space is determined by sgn((c!1)/i). sgn S(c)"sgn((c!1)/i) .

(83a)

We start with an arbitrary reference point in the (f, g)-plane on the PD(1), given by (f(1), g(1)). The new reference point (f(c), g(c)) is then given by (75) together with (83) and (78). Hence both, abscissa and ordinate of the new reference point are obtained by a shift and a rescaling. In the latter case, the shift of the potential is found from (78) as (1!c). (By the way, also e (c) follows from a similar   type of transformation, as seen from (76).) If we select as an initial reference point the point of re#ection (g(1)"!1/2) then according to (78) the new reference point will again be a point of re#ection (g(c)"!1/2). 5.2. Transformation of states with reyection Now, we lift the situation of PDs without re#ection and include the ones with partial re#ection. In this case the starting equation (24) describes the solution to the left of the "rst minimum, which is the re#ection point. To obtain the appropriate transformation we can proceed as before which means that (70)}(74) are still valid. We merely have to use expressions (18), (19) and (24) with rO0. In principle, r could be a transformed quantity, too. However, we in addition demand that it has invariance properties, too: r(c)"r(1),r ,

(84)

and show its validity. From (70) and (18) we have instead of (76) e (c)"c\[i#r!(1#r!c)] , (85)  and from (73) and (19) we obtain (78) again. Eq. (85) represents >"const. If we remain within the class of PDs with partial re#ection of the "rst kind, which means that e (c) is given by (20), then for  > we have >(c, e (c))"[1#r(c)] , (86)  from which it follows that the constancy of > implies the constancy of r. For this class of PDs, (84) is hence not an additional demand but comes out automatically. However, for the class of PDs of second kind, both conditions >"const. and r"const. are independently needed.

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With rO0 the ansatz (75) can still be made, but the shift will now depend on r. We "nd analogous to the case r"0 S(c)"sgn i+>(1, i)[(1!Z(1; 0))!(1!Z(c; 0))] # (1#r)[arcsin Z(c; 0)!arcsin Z(1; 0)], ,

(87)

extending (81) with respect to rO0. The other quantities are easily obtained. We "nd >(1, i)"(i#r) ,

(88)

Z(c; 0)"[c!(1#r)]/>(1, i) ,

(89a)

Z(1; 0)"!r/>(1, i) .

(89b)

For PDs with partial re#ection of the "rst kind, where i"1#2r

(89c)

which follows from (20), we then get >(1, i)"1#r and the shift becomes S(c)"sgn i(1#r)+(1!Z(1; 0))!(1!Z(c; 0)) # arcsin Z(c; 0)!arcsin Z(1; 0), ,

(90)

where Z(c; 0)"[c!(1#r)]/(1#r) and Z(1; 0)"!r/(1#r). We expect that the transformation obtained so far for the region to the left of the re#ection point can be applied also to the region to the right of it. To show this, we have to look at the second parts of Eqs. (29) and (30) (namely that for r'0). The two conditions (78) and (84) yield ZI (c; g(c)),ZI (1; g(1)) and therefore we "nd c[f(c)!f (c)]"f(1)!f (1) ,



 which means that the transformation formula (75) together with the shift remains unchanged. A similar statement can also be made for the case of total re#ection, as seen from formula (33) as well as from the formulas for the regime with partial re#ection with r"1 as obtained in Section 5.2. 5.3. Some properties of the transformation In the case of partial re#ection of the "rst kind, while c(2, the parameter r varies from 0 to 1 and i is determined by r from Eq. (89c). For c'2 we obtain from the positiveness of the rhs of (85) with the substitution of (89c), c/2!14r41 ,

(90a)

where the second inequality is obvious. For c"4 a zone of partial re#ection of the "rst kind vanishes. The allowed range of r is depicted in Fig. 5 by the nonhatched region. In terms of i, de"ned by (71), the inequalities for r (90a) transform to max+1, (c!1),4"i"(3 .

(91)

Fig. 6 shows the allowed range in the ("i", c) plane. In the vertically hatched region there is no correspondence between the solutions for cO1 and c"1, respectively. On the boundary of this region e (c)"0 holds. 

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Fig. 5. The region of allowed values of the re#ection coe$cient r and the charge-nonneutrality parameter c, represented by the nonhatched area. Fig. 6. The region of allowed values of "i" (where i is the emitter electric "eld for c"1) and of c, represented by the nonhatched area.

The transformation of coordinates (75) and (78) can be supplemented by the transformation of the transit time. This transformation can be found by the ansatz ¹(c)"c\[¹(1)#S ] . (92a) 2A The formula for the shift S is easily obtained as follows. From (57a) we obtain for cO1 and 2A c"1, respectively, in the notation of Section 4.2, ¹(c)c!f(c)c"!sgn i>(1; i)+[1!Z(c; g(c))]![1!Z(c; 0)], , ¹(1)!f(1)"!sgn i>(1; i)+[1!Z(1; g(1))]![1!Z(1; 0)], . Subtracting the second from the "rst equation and taking into account (74), Z(1; 0)"0 and the expression for S(c) given by (81), we obtain




S "sgn i arcsin Z(c; 0)"arcsin 2A For S

c!1 . i

(92b)

we may also write 2A S "S(c)#sgn i[i!(1!c)]!i , (92c) 2A where S(c) is given by (83). To summarize this section, we have shown that the equilibrium states of the nonneutralized beam plasme diode can be obtained by a linear transformation of the states of the generalized Pierce diode (c"1; < arbitrary). The transformation is given by (75) and (78) in all cases. For di!erent situations only the shift has to be adapted since it generally depends on c and i, as seen from (81), (83), (87) and (90). Due to the linearity of the transformation extrema are mapped onto itself and each subregion, in which the potential is monotonic, has its counterpart after transformation. Each peculiarity of the

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solution is hence preserved. Furthermore, a re#ection point keeps its value and remains a re#ection point after transformation. An example of the transformation is given in Figs. 7a}f, where it is shown how a certain piece of the source PD of the generalized Pierce diode (c"1; Figs. 7a, c and e) is mapped into the corresponding PD of the nonneutralized diode for c"0.8 and c"1.2,

Fig. 7. (a) The source PD of the generalized Pierce diode (c"1) as a function of space with i"0.2 which is a boundary of the transformation in the nonre#ective regime. (b) The PDs after the transformation of the source PD for two values of c. (c) Same as Fig. 7a, except for i"0.5 lying now inside the allowed range. (d) Same as Fig. 7b. (e) The source PD in the regime with partial re#ection of the "rst kind, r"0.3 and i"!1.265. (f ) Same as Fig. 7b.

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respectively (Figs. 7b, d and f ). The transformed PDs agree with the curves calculated from Eqs. (24), (27)}(30). The beginning and the end of each piece of a PD are marked by a thick point. Note that in the case of a positive shift S(c) we need a piece of the source PD from a nonphysical region, !S(c)4f40, in order to obtain the piece of a PD of a diode with cO1 in the physical region located between the emitter and the left thick point. Figs. 7a and b demonstrate a mapping at the boundary of transformation (see Fig. 6) for the nonre#ective regime. As it follows from (77), the value of the boundary emitter "eld strength i"0.2 is the same for both c values and is transformed to e (c)"0. According to (83a) a shift of the origin turns into a positive position at  c"1.2 and into a negative one at c"0.8. The topological similarity of the PDs is conserved for all cases. Figs. 7c and d represent a mapping again for the nonre#ective regime but inside the allowed range, i"0.5. In the case of a partial re#ection of the "rst kind, Figs. 7e and f, the value of the potential minimum is conserved being equal to !1/2.

6. Classi5cation of potential distributions (PD): (g, e)-diagrams and I+V characteristics In the previous section we have shown that for the construction of all possible potential distributions (PD) for a diode with arbitrary c, it is su$cient to know the PDs of the Pierce diode (c"1) for all values of the emitter electric "eld i,e (1). All the PDs with arbitrary c can be  obtained from these solutions by a simple transformation of the coordinates, given by (75), (76) and (78). Therefore, the investigation of the Pierce diode for all values of i is a central issue. In Section 6.1 we solve this problem for the generalized Pierce diode in which both i and <, the collector potential, are generally nonzero, in contrast to the classical Pierce diode, where i"0"<. (We mention in parenthesis that the uniform equilibrium state of the classical Pierce diode is no longer attainable for diodes with arbitrary neutralization parameter c.) The results of Section 6.1 are then used to determine the classi"cation scheme for the nonneutral diode in Section 6.2 via the transformation formula. In Section 6.3 we present (g, e)-diagrams and I}< curves. 6.1. Generalized Pierce diode A classi"cation of solutions can be done in the (e , d/p)-plane and is shown in Figs. 8a and b.  Here the regions of existence of di!erent types of solutions are exhibited for e '0 in Fig. 8a and  for e (0 in Fig. 8b. (The cases for c"0.8 and c"1.2, which are dealt with later, are shown in  Figs. 9a, b and 10a, b, respectively). In these "gures we distinguish four types of solutions represented by a di!erent shadowing: solutions without re#ection (blank), solutions with partial re#ection of the "rst kind (positively inclined lines) and of the second kind (negatively inclined lines) and solutions with total re#ection (dotted). 6.1.1. The case e (0  A classi"cation of solutions for the classical Pierce diode in the lower half plane has been made earlier in Ref. [51]. Let us at "rst consider the lower half plane, e (0, and regions  without re#ection. Curve 1 in Fig. 8b serves as a boundary of monotonic PDs which lie to the left of this line. This curve corresponds to the positions of the potential maxima, obtained from

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37

Fig. 8. The classi"cation of equilibrium solutions of the generalized Pierce diode in the (e , d/p)-plane for e '0 (e (0)    upper (lower) frame. The di!erent types of solutions are exhibited by di!erent shadowing, as explained in the main text where the integer labels are also referred to (see Sections 6.1 and 6.2). Fig. 9. Same as Fig. 8 except for c"0.8. Note the di!erent scales in both "gures.

Eq. (25) for r"0: (93) d "p/2!e .   The horizontal line 2, representing e "!1, marks the boundary of PDs without re#ection  having multiple minima and maxima (the wavy-type PDs without re#ection of Fig. 2b with e "!1  and large enough d). For e (!1 and su$ciently small d-values PDs with a single maximum exist  in the region located in between lines 1 and 4 (see also Fig. 11, lowest curve and Fig. 14a, highest curve). Line 4 corresponds to PDs with <"!1/2 but nonvanishing electric "elds at the collector (the boundary of the region of solutions with partial re#ection of the second kind, Ref. [36]). The determining equation for line 4 is obtained from (25) and (27) for g"!1/2 and r"0: 1 #[e !1]!e . (94) d "p#arcsin    "e "  The position of the leftmost point of line 2, being de"ned as an intersection of lines 2 and 4 denoted by d is found from (94) for e "!1:   3p d " #1+1.8183p .  2

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A.Ya. Ender et al. / Physics Reports 328 (2000) 1}72

Fig. 10. Same as Fig. 9 except for c"1.2.

The region of PDs with partial re#ection of the "rst kind (i.e. the potential minimum is inside the diode, Section 2) is bounded by lines 2, 6 and 3 (see the curves with e '!(3) in Fig. 2d and also  the curve with r"0.48 in Fig. 11). Line 3 is determined from Eq. (20) with r"1 and becomes e "!3. It separates these solutions from those with total re#ection. An example for total  re#ection is given in Fig. 2d for e "!1.735(!(3. Line 6 separates the solutions with partial  re#ection of the "rst and the second kind. This line corresponds to PDs which have the potential minimum at the collector and for which the electric "eld strength vanishes there. Therefore, it is de"ned parametrically by the two equations, by (20) with c"1 and by (26) resulting in e "!(1#2r) ,  3p r #arcsin #(1#2r . d "(1#r)  2 1#r





(95a) (95b)

Here r varies from 0 (d"d ) to 1 (d"d ). The intersection point of lines 3 and 6 is obtained   from Eq. (95) at r"1: d "10p/3#(3+3.8847p.  The region of PDs with partial re#ection of the second kind is bounded by lines 4}6. These are single maximum PDs with the minimum potential of !1/2 at the collector (dotted curves in Fig. 14a). If we "x the e -value, e.g. less than !(3), and increase d, starting from line 4, r will  grow from 0 (line 4) to 1 (line 5). When r reaches 1 the PDs with partial re#ection switch to PDs

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39

Fig. 11. Several PDs of the generalized Pierce diode for four values of the re#ection coe$cient and e "!1.4. For the  di!erent kind of drawings, see Fig. 2.

with total re#ection. This occurs at line 5, the equation of which is obtained from Eqs. (25) and (27) at g"!1/2 and r"1: d "2[p#arcsin(2(e #1)\)#arcsin((e #1)\)]#(e !3!e . (96)      Fig. 11 shows an example of how the PD changes for "xed e "!1.4 as d is varied. In Fig. 8b this  corresponds to a movement along the horizontal line with e "!1.4. If d is on the left side of line  1 a monotonic PD occurs. If d lies between lines 1 and 4 nonre#ective single maximum PDs are found (curve with r"0 in Fig. 11). When d is in the region between lines 4 and 6, PDs with partial re#ection of the second kind exist, the re#ection ratio changes from 0 (on line 4) to (e !1)/2 (on  line 6) (dotted curves in Fig. 11). Wavy-type PDs with partial re#ection occupy the region to the right of line 6. An example of this type of curve is shown in Fig. 11 for r"0.48 (dotted-dashed line). 6.1.2. The case e '0  The boundary lines in the upper half-plane (e '0) are plotted in Fig. 8a. They are denoted by  dashed numbers. Line 1 is the continuation of line 1 (note the di!erent scaling of the axis), and is described by Eq. (93). Line 1 is also the right boundary of the monotonically decreasing PDs for small d's. This line corresponds to the positions of the potential minima with zero electric "eld strength at the collector. Line 4 continues line 1. It is the right boundary of monotonic PDs for e '1 and corresponds to nonre#ective PDs which have the collector potential !1/2. The  determining equation for line 4 is obtained from Eqs. (24), (18) and (19) for g"!1/2 and r"0:




1 #(e !1!e . (97) d "arcsin   Y e  Nonre#ective wavy PDs (Fig. 2a) lie in the region bounded by lines 1 and the straight line 2 which is described by the equation e "1. The position of the leftmost point of line 2 lies at the  intersection of lines 1 and 2 and is found from Eq. (93) for e"1: d "p/2!1+0.1817p. YY Wavy PDs with re#ection (Fig. 2c) are situated in the region bounded by lines 2, 6 and 3. The straight line 3 is described by the equation e "(3). Line 6 as well as line 6, separate the PDs 

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A.Ya. Ender et al. / Physics Reports 328 (2000) 1}72

with re#ection of the "rst kind from those of the second kind. It is obtained parametrically from the two equations, (20) with c"1 and (28) resulting in e "(1#2r)  r p !(1#2r . d "(1#r) !arcsin Y (1#r) 2





(98a) (98b)

The intersection point of lines 3 and 6 is obtained from Eq. (98) for r"1: d " YY 2p/3!(3)+0.1153p. PDs with re#ection of the second kind are located at the region bounded by lines 4, 6 and 5. Line 5 separates such solutions from those with total re#ection. This line corresponds to the monotonic PDs with r"1 and the potential at the collector equal to !1/2. It is obtained from Eqs. (24), (18) and (19) for g"!1/2 and r"1: (99) d "2+arcsin[2(e #1)\]!arcsin[(e #1)\],#(e !3!e .  Y    PDs with total re#ection are in the regions bounded by lines 5 and 3. Note that all boundary lines in the upper half-plane have small value of d. It is also worth noting that all PDs located outside the region bounded by lines 3, 6 and 1 are monotonically decreasing PDs. 6.2. Nonneutral diode Now we will consider what happens when c deviates from unity. All formulas for the boundaries of di!erent types of PDs can be derived from the corresponding formulas for the case c"1 with the help of the rescaling equations (75), (76), (78), (83)}(85), (87) and(90). First we turn to lines 1 and 1 in Figs. 9 and 10 which serve as the right boundaries of the monotonic PDs (see Eq. (93)). In the case of overneutralization (c'1) we can see from Eqs. (75), (76), (83) and (93) that if e (c)P0# then  d(c)Pp/c. If e (c) tends to zero from below (e (c)P0!) we obtain that d(c) vanishes indepen  dently of c. In the underneutralized case (c(1) the opposite situation holds: if e (c)P0# then d(c)  vanishes for all c's and for the case of e (c)P0!, d(c)Pp/c. Figs. 9a,b and 10a,b demonstrate  both cases. Note that e "0 does no longer correspond to the uniform state of the short-circuited  Pierce diode g(f)"0, 04f4d, when c deviates from unity (see the previous section). Boundaries of wavy-type PDs 2, 2, 3 and 3 transform to the straight lines $(2!c) and $(4!c), respectively in accordance with Eq. (76) (i"$1) and Eq. (85) (i"$3). If we increase c, starting with c"1, the region of nonre#ective wavy PDs between lines 2 and 2 shrinks to zero at c"2, and this type of PDs disappears. For 2(c(4 the region of wavy PDs with partial re#ection between lines 3 and 3 becomes smaller, because in this c-region a lower boundary for the re#ection ratio exists c/2!1(r(1 (see Fig. 5). At c"4 wavy type PDs are absent. Increasing c from c"1 we can see the lines 4, 4, 5, 5, 6 and 6 shift to smaller d's. Let us consider the case where the neutralization ratio c tends to zero. For this case, as it follows from Eq. (77), we have 1!c4"i"41, i.e. "i"P1. For negative values of e one obtains from Eq. (83) S(c)Pp/2!1 for cP0. We see that in  Eq. (75) both terms in the sum d(1)#S(c) are positive and the sum remains "nite. Therefore, d(c) increases approximately proportional to c\. Hence, the boundary lines 1 and 4 tend to in"nity and only solutions without re#ection exist.

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41

Another situation occurs for positive e values. It is worth remembering that in this case the  boundary lines 1, 4, 5 and 6 are connected with monotonic PDs and are determined by the position of the "rst potential extremum f . As shown in Section 6.1.2 for the Pierce diode (c"1)

 the boundary line 1 (see Eq. (93)) is located more left than d"p/2, and the lines 4, 5 and 6 are located still closer to the emitter: for them d4p/2!1 holds. In order to "nd the position of the potential minima for arbitrary c one needs to evaluate the shift S(c) more accurately than for e (0. Let us consider how line 1 transforms with c. In the nonre#ecting regime one has to make  a c-power series expansion of the rhs of (83). As in the regime for c(1, the region of the emitter "eld strength 1!c4i41 in the Pierce diode is converted to the region: 04e (c)4(2!c)  (see Section 5.1), one needs to know in what way i tends to 1 as cP0. It can be shown that in the "rst approximation in c i"1![1!e (0)/2]c, holds with e (0) being the emitter "eld strength for   the Bursian diode, which in the nonre#ective regime changes from 0 to 2. Then it is obtained from Eq. (83) by using Eq. (1.641) of Ref. [98] that S(c)"!(p/2!i)#e (0)[1!e (0)]c#O(c) .   It was shown in Ref. [36] that

(100a)

1 1#r q! e (0)q#q , f" 2  6 e(f)"e (0)!(1#r)q .  Here, e(f) is the emitter "eld in the f-point, q is the electron transit time from the emitter to the f-point, and r is the re#ection parameter. At r"0 we obtain for the position of the potential minima in the Bursian diode f (0)"e (0)[1!e (0)] (100b)

   i.e. in (100a) the coe$cient of the power c coincides with line 1 in the Bursian diode. Taking into account Eqs. (93) and (100b) we obtain "nally for S(c): S(c)"!f

(1)#f (0)c#O(c) . (101a)



 Now we are able to evaluate f . Substituting (101a) into (75), we see that in the parenthesis the

 term with f(1)"f (1) cancels, and the expansion starts with the power c, i.e. the value of f is



 bounded for cP0: f (c)"f (0)#O(c) . (101b)



 Thus, the transformation of states, given in Section 5, is valid up to c"0, i.e. it includes the Bursian diode states as well and the states close to c"0 are determined by (101b). We can show that Eqs. (101a) and (100b) are also valid for the regime with partial re#ection of the 1-kind (i.e. on line 6 from Fig. 8a), and in this case f (0)"2[3e (0)]\ holds. Moreover, for c(1 the position of the

  boundaries in the existence region of the various PDs does not change strongly with c. This can be seen from Fig. 12, where the boundary lines for the Pierce diode (solid lines) and the Bursian diode (dashed lines) are drawn for positive values of e . Thus, all boundaries in the upper half of the  (e , d/p) plane remain limited for all c's. 

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A.Ya. Ender et al. / Physics Reports 328 (2000) 1}72

Fig. 12. The boundary lines of di!erent regions in the (e , d/p) plane for the Pierce diode (solid lines) and Bursian diode  (dashed lines) for e '0. The labels are explained in the text. 

Fig. 13. The (g, e) and (I, e) diagrams for the Pierce diode with d/p"2.2, where I is the transmitted current. Typical values are marked by letters. Solutions corresponding to <"0.4 are marked by numbers.

6.3. (g, e)-diagrams and I}< characteristics Next, we consider (g, e)-diagrams [52}54] for the principal case of c"1. This diagram is used for "nding all solutions for a given d-value and a "xed collector potential <. In addition, it allows to determine whether the corresponding diode state is aperiodically stable or not. In order to construct (g, e)-diagrams we choose some value of the emitter "eld strength e . From  Figs. 8a and b we can determine what type of PD exists for chosen d and e . Now we know the  analytical formula for the electron density as a function of the potential inside the diode from Eq. (15). We solve the Cauchy problem for the di!erential equation (16) and determine the potential g at the position of the collector d. Varying e we complete the (g, e)-diagram.  Fig. 13 demonstrates exemplarily such a diagram for d"2.2p. Shadowed regions in the (g, e) diagram correspond to analogous regions in Figs. 8a and b. Especially, it can be seen that in the region corresponding to the solutions with partial re#ection of the "rst kind, several PDs may exist for the same <-value. Figs. 14a and b show PDs corresponding to the points A, B,2, H in the (g, e)-diagram.

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43

Points of intersection of the load line g "< with the (g, e) curve determine all equilibrium  plasma states with "xed collector potential <. Solutions corresponding to the intersection points with negative shape of (g, e) diagram are aperiodically stable, and those with positive shape are aperiodically unstable. An example is given for <"0.4 in Fig. 13. The intersection points marked by odd numbers are stable and those with even numbers are unstable. For more details see Ref. [54]. In the next section, we will utilize this criterion of instability to build boundaries of the aperiodical instability of the diode states in the (e , d/p) plane. Note that aperiodically stable PDs  can still be oscillatory unstable. During the construction of an (g, e)-diagram the re#ection ratio r is also calculated for each PD (see Sections 6.1.1 and 6.1.2). The relative current transisting through the diode is given by I,j/j "1!r. Therefore, we can also build the current}voltage characteristic, the I}< curve. An # example for such a curve (corresponding to Fig. 13) is presented in Fig. 15. The points, marked by capital letters, correspond to those in the (g, e)-diagram (see Fig. 13). We can see an unusual form of the I}< curve. If we follow the change of the PDs (see Figs. 14a and b) for the marked points, we can see to which PD each point of the I}< curve corresponds. Of course, the vertical part of the

Fig. 14. (a) The PDs as functions of space corresponding to the points marked in Fig. 13 for e (0. (b) Same as Fig. 14a,  except for e '0. 

Fig. 15. The current voltage characteristic for the Pierce diode with d/p"2.2. The points marked by letters correspond to those in Fig. 13.

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I}< curve corresponds to the PDs of the second kind (collector potential <"!1/2). In this case, a portion of electrons returns to the emitter. The `oscillatorya part of the I}< curve corresponds to PDs of the "rst kind. We know that as r tends to 1 the wavelength of the PD diminishes proportional to 1!r according to (31), and both maximum and minimum potential tend to !1/2. As a result at r&1 many PDs may with the same potential value, and a point of thickening is observed in the I}< curve. A similar phenomenon was "rst demonstrated theoretically in Ref. [99]. Here complete selfconsistent PDs and I}< curves were obtained for the Knudsen diode with surface ionization (KDSI) in which the velocity distribution function was chosen half-Maxwellian for both electrons and ions, at the entrance of the diode region. A thickening of the I}<-curve takes place always in points where wavy PDs pass to PDs with a quasineutral plateau. The (g, e) diagrams and I}< curves for nonneutral diodes can be easily constructed, using the PDs of the generalized Pierce diode, by means of the coordinate transformation introduced in Section 5. In this case one has to use the PDs of the Pierce diode for variable i and d. This method was realized in an algorithm that calculates the I}< curves for di!erent c and d. In order to build an I}< curve for very small c's and not too large d's one has to know the PDs of the Pierce diode in the vicinity of line 6, corresponding to the position of the "rst potential minimum for PDs with partial re#ection of the 1-kind. As a result if c tends to zero then all I}< curves for di!erent c's are reconstructed very easily from one another. Let an I}< curve be built for c"c and d"d . Then for another c-value one has to choose d"d c/c, and the <(c)-points     on the I}< curve will be found from <(c)"<(c )c /c, the current I"1!r(c) being conserved.   This means that in transiting from one small c-value to another small c-value, conserving the combination dI "dc and choosing
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Fig. 16. (a) An universal I}< curve for small c's. dI /p"0.3162, c"0.1, d"10 (Curve 1); 0.01, 10(2); 0.001, 10 (3). (b) I}< curves for the Pierce diode. d/p"(0.1) (curve 1), 0.5 (2). (c) A fragment of the I}< curve for c"0.1 in the vicinity of <"!0.5, I"0 on an enlarged scale.

criterion (see Section 6.3) and the criterion based on the analysis of the dispersion relation (67) (see Section 4.4). First, the aperiodical instability of nonre#ective solutions will be investigated by both methods in Section 7.1. It will be shown that in this case both methods yield the same results. Then the aperiodical instability of the solutions with electron re#ection will be studied by the (g, e)-diagram technique. In Section 7.2 we deal with solutions with partial re#ection and in Section 7.3 with total re#ection. And, "nally, in Section 7.4 from the dispersion relation (67) the boundaries of the oscillatory instability will be constructed for the nonre#ective regime. 7.1. Aperiodical instability boundaries for nonreyective solutions 7.1.1. (g, e)-diagram method In constructing the (g, e)-diagram, the full account of which is given in Section 6.3, the gap value d is held "xed. Let us consider for example the case d"2.2p, c"1.0 (Fig. 13). Each PD in Fig. 14

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corresponds to a point in this diagram. We can see from Fig. 13 that the points A, E, F and H correspond to the aperiodically stable solutions (the slope of the curve in these points is negative), the point D is unstable (the slope is positive), and the points B, C and G relate to the marginal stable ones (the slope vanishes). It is also seen that a passage from a stable to an unstable solution and vice versa with varying e can occur in two ways: either the derivative Rg/Re vanishes   in the boundary point (see, e.g., G in Fig. 13) or the sign of such a derivative changes jump-like (see, e.g., the point at e "!1 in Fig. 13). Our investigations have shown that both in the generalized  Pierce diode and in the nonneutral diode the jump-like change in the sign of the derivative occurs at lines 2 and 2 (see Figs. 8}10) separating solutions without electron re#ection from those with re#ection of the "rst kind; remember that on these lines e "$(2!c). Note that the derivative  may also jump to zero but this occurs only at the boundaries of the solutions with partial re#ection of the second kind (see the horizontal piece of (g, e)-diagram in Fig. 13). We shall construct in the (e , d/p)-plane the boundaries of the aperiodical instability, i.e. the lines  derived from the condition of a sign change of the derivative R


Rd Rd Rd d< "! "e(<) "c\>\+ce [>#=(<)]  Re R< Re de    !sgn e(<)[>#c!1][>!=(<)], .

(102)

Here e(<) is the collector electric "eld strength, and > and =(<) are given by (55a) and (55c), respectively. Setting this derivative equal to zero and taking into account the fact that > depends only on e we obtain the connection between the potential < at the minima of the (g, e)-diagram 

Fig. 17. Fragment of the (g, e)-diagrams of the diode with c"1 in the nonre#ective region for di!erent d's: d/p"0.25 (curve 1), 0.5 (2), 0.75 (3), 1 (4), 1.25 (5), 1.5 (6), 1.75 (7), 2.0 (8). The dashed line connects diagrams minima (Eq. (105)).

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47

and the emitter "eld strength: ce [>(e )#=(< )]"sgn e(< )[>(e )#c!1][>(e )!=(< )] .  

 

After squaring both parts of Eq. (103) we "nd by simple algebra

(103)

=(< )"(e #c)\[!e Gc(e #c!1)] . (104)

   Substituting the de"nition of the function = in (104), we "nd < (e ) dependence. First we choose

 the minus sign in the square bracket of (104). Then < (e ) given the desired minima of the

 (g, e)-diagrams < "!2(e #c)\(e #c!1) . (105)

  The corresponding curve is plotted in Fig. 17 as the dashed line. The squaring of Eq. (103) can lead to spurious roots and therefore for each concrete case one has to check the signs of both parts of Eq. (103) after substitution of the function =(< ) given by (104)

into this equation. It turns out that the function < (e ) de"ned by Eq. (105) is a solution of Eq. (103)

 only if d is located between f and f "f #pc\ (f is given by (28) with r"0) at e '0







  or if d is located between f and f "f #pc\ (f is given by (25) with r"0) at e (0.







  Using the consideration stated above and substituting (105) consecutively into each Eq. (24), (27) or (29) with r"0, we obtain the equation for the boundary line separating aperiodically stable and unstable solutions in the (e , d/p)-plane:  2ce 1!c (1#c)e !c(1!c)  !sgn e arcsin  d"c\ p! #arcsin . (106)  >(e ) (e #c)>(e ) e #c     Here >(e ) is de"ned by (55a).  We shall mark this boundary line as A . Fig. 18a shows an example of such a line for c"1. The  line A intersects the line e "0 in the point d"pc\, where the collector potential < turns out   to be 2(1!c)c\. As expected, at c"1 (see Fig. 18a) the abscissa of this point coincides with the threshold of the aperiodical instability for the classical Pierce diode [19] (at e "0, <"0 and  d"p). Note that Eq. (59), which determines the envelope equation for the boundary of solution without re#ection, is easily obtained form Eqs. (105) and (106) with c"1. This is clear since these equations correspond to the position of (g, e)-diagram minimum within the range of the nonre#ective wavy PDs. The line A is located between the points  e "(2!c), d"c\[p/2!arcsin(1!c)!(2c!c)] (107)  and







e "!(2!c), d"c\[3p/2#arcsin(1!c)#(2c!c)] . (108)  These points coincide with the points d and d , respectively, i.e. the line A rests on the YY   boundary of the re#ective region.

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Fig. 18. (a) A periodically stable (S) and unstable (US) regions for the diode with c"1. Lines A and B (s"0, 1, 2,2) Q Q are boundaries of unstable solutions without and with partial re#ection of the "rst kind, respectively. Lines 4, 5, 4 and 5 are the boundaries of regions (shadowed) corresponding to solutions with partial re#ection of the second kind. Line ¹ is the boundary of unstable solutions with total re#ection. Between the line 3 and a line B , as well as line 3 and line A , Q Q there are many boundary lines not yet drawn. (b) Same as Fig. 18a, except for a smaller range of d/p.

Eq. (106) may be reduced to a more symmetric form if we go over from two arcsins to one arcsin:




  
  
 

2ce 2ce  !  c#e c#e   2ce 2ce  !  d" c\ arcsin c#e c#e   2ce 2ce   ! c\ 2p#arcsin c#e c#e   c\ p!arcsin

if "e "4c ,  if c(e 4(2!c) , 

(109)

if !(2!c)4e (!c . 

Note that the curve (109) is symmetric with respect to the replacement of (e , dc) by  (!e , 2p!dc).  Now we consider solutions corresponding to a choice of the plus sign in Eq. (104). In this case, we obtain that for all "e "4(2!c) <(e )"0 holds. These solutions correspond to the piece of the   (g, e)-diagram which at each point is not a local extremum, giving a marginally stable state (see e.g. the line corresponding to d/p"2 in Fig. 17). In this case a boundary of the aperiodical instability in the (e , d/p)-plane is the vertical line d"2pc\ (see curve B in Fig. 18a). This is clear since in the   nonre#ective regime all wavy PDs pass through the point d"2pc\, and for this d value the piece "e "((2!c) of (g, e)-diagram reduces to a part of the horizontal line <,0.  Thus, in the nonre#ective regime the aperiodical instability boundary consists of lines of two types: the line, being described by Eq. (106) or (109), and the line, being part of the vertical line d"2pc\. These two lines are repeated every 2pc\-unit, with increasing gap value. We shall mark them as A and B with s"0, 1,2 (see, e.g., Fig. 18a). Q Q At su$ciently small d-values the nonre#ective solutions can also exist for "e "'(2!c).  A stability boundary of such solutions will be analyzed in Section 7.2.

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7.1.2. Equivalence between dispersion relation and (g, e)-diagram criteria In the nonre#ective regime for a nonneutral diode a dispersion relation can be obtained (see Section 4.4). In particular, in case of the aperiodical instability this relation allows to "nd a dependence of the growth rate on the parameters c, < and e . Undoubtedly, such dependencies can be also  used to plot the boundary lines separating aperiodically stable and unstable solutions. It is of interest to compare such lines with those obtained by the (g, e)-diagram criterion in Section 7.1.1. As noted above, Eq. (67) turned into the dispersion relation if one substitutes the relation (44), connecting the transit time ¹ with d and e , into it. The aperiodical stability of solutions may   hence be studied directly from the dispersion relation (67) by taking p to be real, and the boundary of instability is found from the condition p"C"0. Then Eq. (67) reduces to (cd)!(c¹ )#sin(c¹ )"0 . (110)   Eqs. (110) and (44) give a parametric expression for the boundary of aperiodical instability in the (e , d/p)-plane. However, it is possible to "nd an explicit expression d(e ) from these equations.   Using (110) and (44) we obtain sin(c¹ )!(c\e )[1!cos(c¹ )]"0 ,    or




c¹  2

sin


cos






c¹ c¹  !(c\e )sin   2 2



"0 .

(111)

It is easily seen that Eq. (111) has two types of roots. The "rst type is found from the equation






c¹  "0 . 2

sin

In the case ¹Qc"2p(s#1) with s"0, 1, 2,2 the corresponding gap value is determined from  Eq. (110) as d "2pc\(s#1) and the obtained solutions give the lines which coincide with the Q lines B in Fig. 18a. Q The other type of roots is




tan



c¹  "ce\ .  2

In this case the transit time equals



2 arctan(ce\)#2ps if e '0 ,   ¹Qc"  2p#2 arctan(ce\)#2ps if e (0 .   Going over to d we obtain from (110) the explicit expressions for the stability boundary:



2ce #2ps 2 arctan(ce\)! if e '0 ,   c#e  d c" Q 2ce #2ps if e (0 . 2p#2 arctan(ce\)!   c#e  Here s"0, 1,2 .

(112)

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Eq. (112) is equivalent to Eq. (109). In order to prove this it is su$cient to use Eq. (1.628.1) from Ref. [98]:



!p!2 arctan x if x(!1 , 2x arcsin " 2 arctan x if "x"41 , 1#x p!2 arctan x if x'1 with x"ce\. Thus the lines (112) obtained from the dispersion relation by the condition  p"C"0, coincide with the lines A obtained by the (g, e)-diagram criterion (see Figs. 18a and b; Q the latter is a fragment of Fig. 18a for not large d-values). Earlier, the equivalence of evaluations of the aperiodical instability boundaries by means of the (g, e)-diagram criterion and by the dispersion equation under the condition of zero complex frequency was proved in [43] for the Knudsen diode with surface ionization. This made it possible to give a physical meaning to the Pierce aperiodical instability; this instability describes the fast aperiodical electron processes at the given stage of growing nonlinear oscillations in the Knudsen diode with surface ionization. Having the classi"cation of solutions and the properties of instabilities for the nonre#ective regime, we can analyze the results of other authors. We performed such a classi"cation for the generalized Pierce diode of arbitrary collector potential. Often authors have tried to obtain such a classi"cation for the Pierce diode of zero external voltage, <"0 (see e.g. [6,49,50,57]). For example, Vybornov [57] has plotted a line in the (c, d)-plane, which according to his argumentation separates regions with nonre#ective aperiodically stable and unstable solutions with <"0. At a "rst glance, it appears that it can easily be made if we "nd for "xed c the points on the line A , which correspond to <"0, and then by  varying c we "nd the c(d) dependence, i.e. if we put < "0 in (105), "nd e (c) and substitute it into

 Eq. (106) d"c\+pG2[c(1!c)#arcsin(1!c)], . This dependence has been plotted as the line a in Fig. 19. With a slightly di!erent procedure this  line was drafted by Vybornov. In fact, however, one should take into consideration that both the line A and the point on this line corresponding to <"0 are simultaneously shifted when c is  varied, and one should check whether the line a is the boundary, separating in the (c, d)-plane  aperiodically stable from unstable solutions. A more elaborate analysis shows that Vybornov made a mistake. Such an analysis is easily carried out by looking at the (g, e)-diagrams for di!erent points in the (c, d)-plane. In Fig. 19 two series of points a , l"1, 2,2, 9, for c"0.8 and b , l"1, 2,2, 5 for J J c"1.2 are presented. One of these points (point a ) lies on the line a . In Figs. 20a and b the   (g, e)-diagrams for c"0.8 are shown corresponding to all points a . Nonre#ective wavy-type J solutions exist in the region which is located between the dashed vertical lines. The curve a in  Fig. 20a has no intersection with the line g "0 and therefore no solution exists with <"0. Hence  inside the region D in Fig. 19, where on Vybornov's opinion the solutions should be unstable, no  solution exists. In the point a on the line a bounding the region D there is a solution with    vanishing growth rate, and the (g, e)-diagram does become tangential to the line <"0 (curve a in  Fig. 20a). Outside the region D (point a in Fig. 19) two nonre#ective wavy-type solutions   with <"0 exist (curve a in Fig. 20a), the solution with e "!1.0182 being stable, the other  

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Fig. 19. Regions of existence of wavy-type nonre#ective solutions with <"0 in the noneutral diode. In the regions D , D and D wavy-type nonre#ective solutions with <"0 are absent. The dashed horizontal line is the boundary of    nonre#ective wavy-type PDs. The points a }a correspond to c"0.8 and di!erent d's: d/p"1.8 (point a ), 2.165 (a ), 2.4     (a ), 2.7 (a ), 2.796 (a ), 2.9 (a ), 3.2 (a ), 3.424 (a ), 4.0 (a ). The points b }b correspond to c"1.2 and di!erent d's:          d/p"1.0 (point b ), 1.4 (b ), 1.522 (b ), 1.6 (b ), 2.0 (b ).     

Fig. 20. (a) (g, e)-diagrams of the diode with c"0.8 for di!erent d's corresponding to points a }a in Fig. 19. Dashed   vertical lines correspond to boundaries of nonre#ective wavy-type PDs. (b) Same as Fig. 20a, except for points a }a   given in Fig. 19. (c) Same as Fig. 20a, except for c"1.2 and for points b }b given in Fig. 19. (d) Same as Fig. 20c, except   for points b }b given in Fig. 19.  

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e "!0.1443 being unstable. Thus, the curve a is not a boundary of stability, but only   separates regions of existence of nonre#ective wavy-type solutions with <"0 from those of nonexistence. Note that there are other regions in the (c, d)-plane where wavy-type solutions with <"0 are absent (see e.g. regions D and D in Fig. 19, bounded by the lines a and a which are obtained in     analogy with the line a from the lines A with s'0).  Q For the point a inside the region D , the nonexistence of such solutions is seen from the related   (g, e)-diagram in Fig. 20b. For su$ciently large values of d, this can be seen also in Fig. 4b. To continue, note that apart from the lines a obtained form the boundary lines A , the lines Q Q b can be drafted in the (c, d)-plane from the other boundary lines B (the dotted lines in Fig. 19). Q Q Remember that <,0 holds on the lines B ! Consider a small vicinity of the line b for c"0.8 Q  (see points a and a in Fig. 19). One can see in the related (g, e)-diagram (Figs. 20a and b) that only   one solution exists with <"0 which corresponds to each point in a nonre#ective region "e "((2!c) situated between the dashed vertical lines, the solution left to the line b being   unstable and that right to the line being stable. While increasing d the transition from an unstable to stable solutions occurs on the line b where the solution is marginally stable.  A similar change is seen in the stability of the solutions when an intersection of the line b occurs  at c"1.2. This fact is demonstrated by the related (g, e)-diagram (see curves b in Figs. 20c and d). Q Hence, the line b does separate stable and unstable solutions. Far from the nearest vicinity of  the line b two solutions emerge again (see, e.g. points a and b in Fig. 19 and corresponding    (g, e)-diagrams in Figs. 20b and d). The absence of nonre#ective wavy-type solutions with <"0 in the regions D , D and D does    not mean that solutions with <"0 are completely absent in these regions. For example, they can contain solutions with partial re#ections of the "rst kind or solutions with a deep electron potential well. These solutions correspond to an intersection of the (g, e)-diagram with the load line <"0 outside the region bounded by the dashed vertical lines in Figs. 20a}d. 7.2. Aperiodical instability boundaries for solutions with partial reyection At present there is no dispersion relation for the re#ective regime. Therefore, the (g, e)-diagram criterion is the only chance to study the stability of equilibrium states of the Knudsen diode in such a regime. Earlier, the stability of solutions with partial re#ection of the Bursian diode (c"0) was studied in Ref. [36] and boundaries of aperiodical instability were constructed. In this section, we will construct such boundaries of solutions with partial re#ection for the nonneutral diode. In the (e , d/p)-plane the re#ective solutions satisfy "e "'(2!c) (see, e.g., Fig. 8). The   solutions with partial re#ection of the second kind lie in the region bounded by lines 4, 6 and 5 as well as by lines 4, 6, and 5. For these solutions the corresponding part of the (g, e)-diagram is the straight line g "!1/2, and hence all such solutions are marginally stable. The relevant regions in  Figs. 18a and b for negative e are shaded diagonally, indicating marginal stability. For positive e the same region is very small and is not seen in Figs. 18a and b. Note that at comparatively small d-values there are only nonre#ective solutions in the region located to the left of the lines 4, A and 4, all these solutions being aperiodically stable. This can  easily be proved because of after substitution of the solutions, corresponding to this region, in Eq. (102) one sees the derivative d
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The solutions with partial re#ection of the "rst kind are wavy PDs and lie in the (e , d/p)-plane  in the strips (2!c)(e ((4!c) to the right of the line 6 and !(4!c)(e (   !(2!c) to the right of the line 6, respectively (see, e.g., Fig. 8). From Eq. (29) by taking into account the connection between e and r through Eq. (20) we "nd for the derivative d






 

2c Rd c "!c\ pl#sgn e #arcsin 1!  (2#2r!c) Rr 1#r






!sgn e(<) arcsin ZI (<, r)!2

1#ZI (<, r)  1!ZI (<, r)

.

 (114)

In Eq. (114) the quantity l is connected with a parameter s which is the number of the wave beyond the "rst minimum minus unity (at the same time s is a number of a boundary line B ) in the Q following way: l"2s if e e(<)(0, l"2s#1 if e '0 and e(<)'0, and l"2s!1 if e (0 and    e(<)(0; s"0, 1, 22; ZI (<, r) is de"ned by (30) with r'0, and e(<)"$(1!r)c\[1!ZI (<, r)] .

(115)

The sign of e(<) is simply de"ned by the position of the collector within the wavy-type PD: it is negative if the left extremum nearest to the collector is the potential minimum and is positive in the opposite case. One can see from Fig. 13 that in the region, corresponding to the regime of partial re#ection of the "rst kind, the (g, e)-diagram has many local extrema. Each extremum will give a boundary separating stable and unstable solutions, and there will be many such lines in this zone. We shall "nd there extrema on the (g, e)-diagram from the condition d
  c

(116)

Here s"0, 1, 22 and f is determined by (28) or (26) with r'0. The lines plotted by means of

 (116) are shown in Fig. 18a and b. These lines extend the corresponding lines A (109) to the Q re#ective region, and we have kept same notation. Just as expected, lines 6 and 6 satisfy (116) and continue the line A to the re#ective region, for e '0 and e (0, respectively (see Figs. 18a    and b). For e "xed, the positions of the corresponding points on all lines A are at the same distance  Q c(r)"2p(1!r)c\ to each other (see (116)). As rP1 this distance tends to zero. This means that all boundary lines A for e P(4!c) converge to the same point d and those for Q  YY e P!(4!c) to d , respectively (see Fig. 18a).  

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Now we address the boundary lines of another type, namely, those corresponding to the condition Rd/Rr"0. An example of such lines for s"0 is shown in Fig. 18b: they are the lines located between the points a and b and e '0 and the points b and a at e (0. In order to be   more de"nite we consider the lower (e , d/p) half-plane. All conclusions will be valid for the upper  half-plane, too. The point a (r"1) is the intersection point of the lines 3, 5 and 6. In order to get the coordinates of the point b (r"0) we consider (114) at l"0. Our analysis has shown that the rhs of (114) can vanish if the collector lies within the "rst wave, beyond the "rst potential minimum, and to the right of the maximum, i.e. e(<)'0. This means that s"0. It is easily seen from (114) that the point b corresponds to the collector potential <"0; in this case ZI (0, 0)"c!1 and both expressions contained in the braces have the same value and the opposite sign. From (29) with r"0 and <"0 we "nd the corresponding gap value: dB (0)"2f (0) .



(117)

Here the position of the "rst potential minimum f (0) is given by (26) with r"0.

 In the region located between lines 6, (c, b) and (b, a) the solutions are aperiodically unstable. On the other hand, in the nonre#ective region lying above the horizontal line (c, b) the solutions are aperiodically stable. At last in order to evaluate the stability of the solutions corresponding to the line (c, b) we plotted the (g, e)-diagrams for various d's from region (c, b) and looked at the behavior of such diagrams in the vicinity of the points with e "!(2!c). We have found that for all  considered d's the derivative d
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b and from Eq. (29) dBQ (0): dBQ (0)Kf

(0)#c\[(2p#1)s!sgn e 2b ],  p 2c # #arcsin(1!c) bK2 2ps#sgn e  2 (2!c)










\

(118) .

In fact, formulas (118) represent a good estimate of the positions of the limit points for s52. We determined the stability of the horizontal parts of the lines B by the (g, e)-diagram technique. Q In the unstable case the derivative d Q  B and B in Fig. 18a for the case c"1).   For "xed d and increasing number s the pieces of the boundary lines A and B , corresponding Q Q to the solutions with electron re#ection are located more and more closely to the horizontal lines 3 and 3. Such lines for s'6 are not shown in Fig. 18a. 7.3. Aperiodical instability boundaries for solutions with total reyection The stability of solutions with total re#ection in the nonneutral diode was studied in Ref. [53]. It was noted that such solutions may be aperiodically unstable. However, there did not exist analytical formulas for the instability boundaries in that paper. From Fig. 21, exhibiting an example of the (g, e)-diagram for c"1 and d"4.5p, one can see that in the regime with total re#ection when e (0, namely in the vicinity of e "!2, a portion of PDs turn out to be unstable.   We shall show that such features of the total re#ective solutions are conserved for other c's, too. In the regime with total re#ection we have an explicit dependence of the collector potential < on e (see Eq. (33)):  <"!1/2!eH(e )[d!fH(e )]!(c/2)[d!fH(e )] .   

(119)

Here fH and eH are the position of the re#ection point and the electric "eld strength at this point. The dependence of eH on e is given by Eq. (32). Note that eH is positive. From Eqs. (24) and (27)  with r"1 at g"!1/2 we obtain for fH,



 



2 2!c c[eH(e )!e ]#2 arcsin !arcsin   >(e ) >(e )   fH(e )c"  2 2!c #arcsin c[eH(e )!e ]#2 p#arcsin   >(e ) >(e )  

if e '0 , 



if e (0 . 

(120)

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Fig. 21. An (g, e)-diagram for c"1 and d"4.5p.

Using (18) with r"1 for >(e ) and (32) we can calculate the derivative d(e )   We shall show that for e '0 the derivative d > de  Here F(e )"c[eH(e )]#2ceH(e )#4e . At large d's we can neglect the second term as compared     with the "rst one in (123). We shall show that the function F(e ) vanishes at some e (!(4!c).   Note that e "eH#4!c (see (32)) so that one can rewrite this function in the following way:  (ceH#4)(ceH!2[1!c!(1#2c)])(eH!(2/c)[1!c#(1#2c)]) . F" ceH#2ceH!4e  It is seen that only the third cofactor gives a real zero of the function F, and we easily "nd the corresponding negative e which makes F to be zero:   2 e "! 2!c# (1#(1#2c)) . (124)  c









The dependence of e on c is shown in Fig. 22 as the solid curve. It goes below the dashed line  e "!(4!c) being the upper boundary of the solutions with total re#ection. This solid line 

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Fig. 22. The dependence of e on c (solid line 2). The dashed line 1 is the boundary of solutions with total re#ection. 

gives the asymptote of the aperiodical instability boundary as dPR in the regime of total re#ection. A similar line was represented in Ref. [53]. Now we can "nd the explicit formula for the boundary of the aperiodical instability from Eq. (123): 2#[eH(e )]!e eH   [eH(e )] . d"fH(e )#   !(c[eH(e )]#2ceH(e )#4e )   

(125)

Here fH is determined by (120) for e (0 and eH by Eq. (32). In Fig. 18a this boundary line is  marked by the symbol ¹. In this picture aperiodically stable solutions with total re#ection are bounded by lines 5 and ¹, and unstable ones by lines 3 and ¹. The line ¹ starts from the intersection point of lines 3, 5 and 6. This line is constructed by Eq. (125) by varying e from  eH corresponding to the boundary line, separating solution with partial from total re#ection, up to e (124). Thus in the region with total re#ection for d'd both stable and unstable solutions   always exist. The boundary curves for the aperiodically stable and unstable regimes of the nonneutral diodes are shown in Figs. 23 (c"1.2) and Fig. 24 (c"0.8). We can see that the features of stable and unstable regions are the same as in the case c"1. Thus we have found the boundaries for aperiodical stability for all types of PDs in the nonneutral diode. 7.4. Dispersion curves and boundaries of oscillatory instability The dispersion properties of bounded plasmas driven by an electron beam were studied only for the regime without electron re#ection. The branches of instability, i.e. the dependency of the growth rate and the frequency on the interelectrode gap d, for the case of immobile ions, which is considered only in our paper, were given for the uniform states (<"0, e "0) of the Pierce diode  (c"1) (see, e.g., Refs. [44,60,61]). These dispersion curves are presented in Figs. 25a and b, taken from Ref. [61]. It was shown that the aperiodical branches (solid lines A with s"1, 2,2 in Q\ Fig. 25a) are a set of curves being positive in the regions (2s!1)p(d(2sp (regions of the

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Fig. 23. Same as Fig. 18a, except for c"1.2. Fig. 24. Same as Fig. 18a, except for c"0.8.

Fig. 25. (a) The dispersion curves for the uniform state of a diode with c"1: growth rates C as functions of d/p. (b) The dispersion curves for the uniform state of a diode with c"1: frequencies X as functions of d/p.

aperiodical instability). Each aperiodical branch has single bifurcation point at d"2ps, from which the oscillatory branch O grows (dashed curves in Figs. 25a and b), extending continuously Q up to d"R. At d'p the states of the Pierce diode are unstable (either aperiodical or oscillatory) everywhere except for narrow stable windows, located at the left of the points d"(2s#1)p [46]. Note that if one takes into account "nite ion mass, i.e. if one considers ion mobility e!ects on the diode dynamics, new bifurcation points emerge at the aperiodical branches and new oscillatory branches appear [24,40,44,59}61,65]. 7.4.1. Dispersion branches of instability The dispersion relation not only allows to clarify whether the considered state of the nonneutral diode is stable with respect to small perturbations (both aperiodical and oscillatory) but it also makes it possible to calculate the dependencies of the growth rate C and the frequency X on the diode parameters. One possibility for a systematic study of the dispersion properties of the nonneutral diode consists in studying the growth rate C(d) and the frequency X(d) for a "xed

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electric "eld e at the emitter and for a "xed neutralization parameter c. This implies that we study  the stability of the solutions not under a constant external voltage but under a "xed e (at various  voltages). In the beginning let us consider the aperiodical branches of instability (A-branches). They are de"ned by the roots of Eq. (67a) for real p's: p"C in Eq. (67b) F(C, ¹ ),exp(!C¹ ) [2C cos(c¹ )!(c!C)c\ sin(c¹ )]     !2C!d(c#C)#¹ (c#C) . 

(126)

In Eq. (126) the electron transit time ¹ of the unperturbed trajectory is expressed in terms of  e and d by Eq. (44).  Fig. 26 shows various A-branches of instability for the diode with c"1 for several values of e . It  can be seen that the most right points of all branches do not change with e . In these points,  d"2sp, C"0 holds. However, the left border of the unstable regions changes with e . For e I 0   it is left (right) to d"(2s!1)p, which is the border for e "0. If these points are plotted in the  (e , d/p)-plane (see Fig. 18a) they lie on the lines A , A ,2 in the region "e "41 which was found     as the boundaries of the aperiodical instability by the (g, e)-diagram stability criterion. Fig. 26 also shows that the maxima of C increase with e and are shifted to the left. In Fig. 27 the growth rates  as functions of d/p are depicted for various values of c with e chosen to be e "0.99(2!c), i.e.   near the boundary lines separating the solutions without re#ection from those with re#ection. Whereas the left borders of the unstable regions do not change appreciably, the right borders (not shown in Fig. 27) have a larger c-dependence, given by d "2psc\. Q Note that for the A -branches with s51 there are points of in"nite derivative, marked in the Q "gures by thick points and denoted by c (upper bifurcation point) and k (lower bifurcation point). Q Q In Fig. 26 c and k are shown for di!erent e . It generally holds that if the growth rate    corresponding to c is positive then this point represents the left border of the aperiodically Q unstable region (see e.g. the A -branches for e 50.5 in Fig. 26). If the growth rate corresponding   to c is negative (see the A -branch for e "0.2 in Fig. 26) then the intersection point of the Q  

Fig. 26. Same as Fig. 25(a), except for several values of e .  Fig. 27. Same as Fig. 25(a), except for a diode with e "0.99(2!c and several values of c. 

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A -branch with the C"0 axis serves as a border of this region. As shown later (Figs. 30a, 31a and Q 32a) at each of these points a complex conjugate pair of unstable oscillatory modes emerges. It is interesting to "nd the PD corresponding to the bifurcation points c . Fig. 28 shows the PD Q for two values of e , e "0.5 and e "0.9, in a diode with c"1. The bold points mark the d- and    g-values corresponding to c and c . It is seen that in these points the collector potential g is close   to a potential minimum independent of e .  Note that for e (0 the bifurcation points are completely absent and "rst occur for "nite,  positive values of e only, that means for a rather strong inhomogeneity. We shall denote these  e -values by eQ. For example, we get for c"1 for the family of A -branches e"0.1965 and for     the family of A -branches e"0.1145.   In order to "nd the bifurcation points for "xed e the condition RC/Rd"R is used. The  derivative RC/Rd can be easily found by using the function (126), describing implicitly the dependence of d on c, and the relation (44). The condition RC/Rd"R corresponds to the relation RF/RC"0: exp(!C¹ )[2(1!C¹ )cos(c¹ )#[2C#¹ (c!C)]c\ sin(c¹ )]!2(1!C¹ )       !4dC(c#C)"0 . (127) The curves of bifurcation points, calculated this way, are drawn in Fig. 33a showing the (e , d/p) plane as solid lines C and K . Q Q Now we turn to the oscillatory branches (O-branches). At the beginning we will consider the nonuniform states with e (0. In this case the A -branches have a single bifurcation point the  Q position of which is d "2p(s#1)c\. As an example, consider the dispersion curves for Q e "!0.5 (Fig. 29). It is seen from a comparison with Fig. 25 that the dispersion picture for the  nonuniform case for negative e is qualitatively similar to the uniform one. Each O -branch starts  Q again at the bifurcation point of the relevant A -branch and continues up to large gap values. Q\ A di!erent situation holds for the states having the new bifurcation points c , k . It is known that Q Q the appearance of the new bifurcation at the aperiodical branches can lead to the birth of new oscillatory branches being already discussed in [24,40,60,61]. An example of the dispersion curves of such a nonuniform state with e "0.5 is shown in Fig. 30. Here e is larger than e. Similar to   

Fig. 28. Two PDs for c"1 and e "0.5 and e "0.9, respectively.  

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Fig. 29. (a) The growth rates C as functions of d/p for the nonuniform state with e "!0.5. (b) The frequencies X as  functions of d/p for the nonuniform state with e "!0.5. (c) Same as Fig. 29a, except for a di!erent range of d/p.  (d) Same as Fig. 29b, except for a di!erent range of d/p.

Fig. 30. (a) Same as Fig. 29a, except for e "#0.5. (b) Same as Fig. 29b, except for e "#0.5.  

the uniform case (Fig. 25) the O -branch starts at the bifurcation point d "2psc\ of the Q Q A -branch. In this point the aperiodical branch has always C"0 and X"0. With growing d, the Q\ O -branch rather quickly connects the upper bifurcation point c of the A -branch, where X"0 Q Q Q but CO0. At this point the O -branch disappears. With a further growth of d an oscillatory branch Q

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emerges again at the low bifurcation point k of the A -branch where X"0 and C(0, which Q Q extends up to large gap values similar to one in the uniform case. For simplicity, we shall denote this branch also as O . Thus in the case considered all oscillatory branches have discontinuities Q between the points c and k . If e is chosen between eQ and eQ> then interruptions will occur for Q Q    all O branches with t's, and the branches with t(s will have no interruptions like the ones for R the uniform state. The in#uence of the neutralization parameter on the dispersion branches is demonstrated in Fig. 30 (c"1, e "i"0.5), Fig. 31 (c"1.2, e "0.418330), and Fig. 32 (c"0.8, e "0.512348).    In these "gures e for di!erent c's is chosen in accordance with formula (76). It is seen that all  relevant dispersion curves behave rather similarly. Looking at the growth rate C(d) of the nonuniform state for su$ciently e and increasing d (see, e.g., Fig. 30a for c"1) one can see that  after the "rst intersection of the A -branch with the axis C"0 (at the threshold of the aperiodical  instability d"d ) a branch with nonnegative growth rate always exists, i.e. stable windows are  absent. First, in the range d (d(d "2p we have the aperiodical branch followed by the   oscillatory branch O up to the point c . Then again an unstable aperiodic branch A exists,    followed by a second unstable oscillatory branch O and so on. Stable windows occur near  A -branch, as e is less than e@Q which is somewhat larger than eQ. As an example, the way of Q    "nding e@ and e for the diode with c"1 is shown in Fig. 33b. As seen from Fig. 26 the   bifurcation points c arise at negative C-values. With increasing e , the situation where the point Q  c "nds itself on the C"0 axis occurs. According to this the e -value just equals e@Q. All these Q   points, where C(c )"0 (they are denoted as u in Figs. 33a and b), can be found from Eqs. (126) and Q Q (127) with C"0: dc!c¹ #sin(c¹ )"0 ,   (c¹ )sin(c¹ )!2[1!cos(c¹ )]"0 . (128)    From Eqs. (128) we "nd ¹@Q and d@Q, corresponding to the points u , and then from the relation  Q (44) we get e@Q:  c¹@Q"2XQ, d@Q"[2XQ!sin(2XQ)]c\, e@Q"c[XQ]\ . (129)  

Fig. 31. (a) Same as Fig. 29a, except for c"1.2, e "0.41833. (b) Same as Fig. 29b, except for c"1.2, e "0.41833.  

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Fig. 32. (a) Same as Fig. 29a, except for c"0.8, e "0.512348. (b) Same as Fig. 29b, except for c"0.8, e "0.512348.  

Fig. 33. (a) Boundary lines of oscillatory modes for c"1 in the (e , d)-plane. The labels are explained in the text.  (b) A fragment of Fig. 33a. (c) Same as Fig. 33a, except for a di!erent range d/p.

In (129) XQ are the roots of the equation tan(x)"x, for example, X"4.493410 and X"7.725252, where s is the branch number. One can see from (129) that e@Q'0 for all c-values.  Besides that, e@Q diminishes as the branch number grows so that e@ is the maximum one.   Therefore, it is clear that if e is close to e@Q stable windows exist near all aperiodical branches   A with t4s, and there are no such windows near the branches with t's. R

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7.4.2. Boundaries of oscillatory instability Now we are interested in the boundaries of existence of oscillatory modes. Fig. 33a shows the boundary lines, concerning the oscillatory modes, for the diode with c"1 in the (e , d)-plane.  These are the lines corresponding to the solutions of the dispersion relation with C"0 and XO0 (dashed curves ¸ , M and N , s"1, 2,2), as well the lines, going through the points in which Q Q Q the oscillatory branches disappear with CO0 (solid curves C and K ). At e 'e@Q the line Q Q   C corresponds at the same time to the leftmost point of the aperiodical branch with the positive Q growth rate (see, e.g., Fig. 30a). In Fig. 33a the lines are also plotted, corresponding to the condition C"0 and X"0, i.e. for the aperiodical branches (dotted curves A and B ). The points u show Q Q Q the intersection of the lines ¸ and C . Q Q As our analysis has shown, the oscillatory branch O starts always at the right bifurcation point Q of the aperiodical branch A , and in the vicinity of this point the growth rate of the oscillatory Q\ mode turns out to be positive (see, e.g., Figs. 29a}32a). The abscissa of the right bifurcation point in the A -branch equals 2psc\, and hence the left boundaries of the oscillatory instability regions Q\ coincide with the lines B (Fig. 33a). Q\ Now we will consider the boundary lines ¸ , M and N . The equations for these lines are Q Q Q obtained from Eq. (67a) by setting p"iX(C"0) 2X cos(X¹ ) cos(c¹ )#c\(c#X) sin(X¹ ) sin(c¹ )"2X ,     c\(c#X) cos(X¹ ) sin(c¹ )!2X sin(X¹ ) cos(c¹ )     "!d(c!X)#¹ (c!X) . 

(130)

This system must be supplemented by the relation (44) connecting ¹ with e and d. After squaring   both equations of (130) and after summation we "nd



X"c



dc!¹ c$sin(¹ c)    . dc

(131)

After substituting (131) and (44) in one of the equations of (130) a single transcendental equation between ¹ and d is obtained. Keep, however, in mind that by squaring `spuriousa roots may  occur. By our special code which is intended to calculate the lines ¸ , M and N all roots were Q Q Q found and analyzed, and the spurious ones were discarded. Note that in Eq. (131) the plus sign applies to the line ¸ , and the minus sign to the lines M and N. Q Q In fact, the lines N are extension of the lines M ; however, for an easier understanding of the Q Q considered picture we shall use both names. In Fig. 33a the lines ¸ , M and N are plotted for Q Q Q c"1, and the regions which contain the oscillatory unstable solutions are shaded, horizontally. In order to elucidate this picture we choose e"!0.5 and observe how the stability of solutions changes with increasing d. Let us compare Fig. 29a with Fig. 33a. In Fig. 29a the intersection points of the oscillatory branches with the axis C"0 are denoted by the same symbols as for the corresponding dashed curves in Fig. 33a. It is seen that the solutions are oscillatory unstable in the region located between lines B and ¸ , and after crossing the line ¸ , become oscillatory    stable up to M . In the region embedded between lines M and N , the solutions become    oscillatory unstable again, and to the right of the line N they are oscillatory stable up to B . In the   second and third zones (s"2, 3) a similar alternation of the oscillatory unstable and stable regions

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65

occurs. The only di!erence is that in these zones the "rst transition into the stable region occurs at the intersection with the line M and not ¸ . The same situation also occurs for the "rst zone at Q Q su$ciently large negative values of e , approximately at e (!0.6. This is due to the fact that the   lines ¸ and M intersect. The intersection points of these lines are denoted by d . Q Q Q For the coordinates of the points d on the (e , d/p)-plane the analytic formulas were obtained. It Q  was found that the sin in Eq. (131) vanishes at the point d is reached. From this condition we "nd ¹ , and then X and d from Eq. (130) and e from Eq. (44).   ¹Q"(2s#1)pc\ , (132)  2m#1 c , (133) XQ" K 2s#1 (2s#1)p c\ , dQ" K 1!((2m#1)/(2s#1))

(134)

(2m#1)p eQ "! c\ . K 2(2s#1)[1!((2m#1)/(2s#1))]

(135)

In Eqs. (132)}(135) m is an integer being less than the oscillatory branch number s. As far as our study is concerned, the regime without electron re#ection additional restrictions on the value of m are imposed by the condition eQ 5!(2!c). K One can show that the desired points d , i.e. the intersection points of the lines ¸ and M are Q Q Q found from Eqs. (134) and (135) at m"0. It is seen from these equations that all points d lie in the Q lower (e , d/p) half-plane, and with increasing s they become closer to the axis e "0. We have also   found that at m'0, formulas (132)}(135) de"ne additional intersection points of oscillatory instability boundaries in the (e , d/p)-plane. This is owing to the fact that at large s-values besides  the lines ¸ , M and N other boundary lines exist in the crossing points in which C changes its sign. Q Q Q Our calculations by Eqs. (130) for c"1 have shown that such lines exist in the zone with s"7 (see Fig. 33c). We can see from Fig. 29c, which demonstrates the oscillatory branches occurring in this zone, that not only the O -branch but also the O -branch crosses the axis C"0. We shall denote   the new points of the O -branch connected with C"0 by the symbols P , Q , R and Q\ Q\ Q\ Q\ S (see, e.g., the points P and Q in Fig. 29c), and the corresponding dashed lines in Q\   (e , d/p)-plane } by the same symbols (see, e.g., the lines P , Q , R and S in Fig. 33c). We shall      denote by f (see, e.g., the point f in Fig. 33c) the intersection point of the lines Q and R . Q\  Q\ Q\ The coordinates of the points f just correspond to m"1 in Eqs. (134) and (135). We "nd from Q\ the condition e (s)5!(2!c) that, for example, at c"1 the parameter m becomes 1 for the K "rst time at s"7. With an increasing departure of c from 1 the values of s, at which m"1 for the "rst time, increases: for example, at c"0.5(1.5) s"8, and at c"0.1(1.9) s"16. The boundary lines Q and R can also intersect the lines ¸ and M (in Fig. 33c, for Q\ Q\ Q Q example, such intersection points are denoted by d and d ). For large values of the parameter   s the presence of new lines with C"0 and new intersection points of these lines may additionally restrict the region of oscillatory stable solutions. Let us consider, for example, the zone with s"7 (Fig. 33c). We choose e "!0.5 and observe how the stability of solutions changes with increasing  d. Here the line B is the left boundary of the "rst oscillatory unstable region. However the right 

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boundary of this region is not the line M as, for example, in the similar region for s"3 in Fig. 33a,  but the line Q . This becomes clear if we look at the dispersion branches in Fig. 29c. In the zone  with s"7 at "rst the Q -branch has positive growth rate and then with increasing d the positive  growth rate appears on the O -branch. Therefore, the solutions turn out to be unstable up to the  point Q . The stable solutions occur only in the region located between the points Q and M . At    more negative e -values the region of stable solutions is additionally restricted from the right by the  line R too.  So far we were only interested in the negative e -values. Now we turn to the region with e '0   and continue to study the boundary lines in this half-plane. As an example we consider small values of the parameter s according to Fig. 33a. With increasing e the boundary line ¸ gets into the limit  Q point u . This point is characterized by C"0 at e 'e@Q for the leftmost bifurcation point on the Q   aperiodical branch A . If e (e@Q then C(0 for the bifurcation point, and C'0 in this point for Q   the opposite case. At e 'e@Q the boundary lines ¸ are absent, and in principle there are only the   Q boundary lines M and N in this region (see, e.g., the second and the third zones in Fig. 33a). At Q Q e 'e@Q each O -branch breaks in the leftmost bifurcation point c of the A -branch, and every  Q Q Q where between the points 2ps and c the growth rate is nonnegative (see, e.g., Fig. 30a). Therefore in Q each zone 2ps(d(2p(s#2) the line C at s'0 is the right boundary of the "rst oscillatory Q unstable region (see Fig. 33a). There is no oscillatory mode O in the region bounded by the lines C and K . Both of these lines Q Q Q exist only at e 'eQ .   One can also "nd the regions in the (e , d/p)-plane corresponding to the aperiodical stable  solutions. Such regions were plotted in Section 7.1 for the nonre#ective solutions. The study of the aperiodical branches makes it possible to correct to a certain extent the boundary lines for these regions. At e (e@Q the intersections of the A -lines with the line C"0 give the boundary curve   Q separating aperiodically stable from unstable solutions. Therefore, for these e -values the aperiodi cal stable solutions lie in the regions, restricted by the line B from the left and A from the right. Q Q At e 'e@Q the line B remains the left boundary; however, now, the line C becomes the right   Q Q boundary instead of the line A . It should be noted that the growth rate is positive rather than zero Q for these e -values.  By comparing the intersections of regions containing the aperiodically stable solutions with that containing the oscillatory stable solutions we can "nd the stable windows in the (e , d/p)-plane, i.e.  the subregions in which solutions are relatively stable with respect to small perturbations (both aperiodical and oscillatory). These regions are grayish in Figs. 33a}c. The (e, d/p)-plane now serves to get an overview about the stability properties for "xed potential. For example, take the known Pierce equilibria (c"1, <"0) and draw the corresponding lines in Fig. 33a. From this we immediately get its stability properties [49]. For example, in the case of the uniform Pierce equilibrium, e "0, we see from Fig. 33a very quickly the stability regions,  including the stable window near d"3p. For the inhomogeneous Pierce equilibria as well, given by e "!(p/2)(d/p!n), n"1, 3, 5,2 the stability properties can be seen. We especially see that  for e (0 the equilibria are located completely in the stable range, passing the points d for n"3,   d for n"5 and d for n"7. The additional marginally stable equilibria coincide with the lines B .   Q This also proves, why the dispersion properties of perturbations of the uniform Pierce equilibrium, shown in Fig. 25, is relatively simple since the complicated pattern in the region e (0 in Fig. 33a is  absent. This can be done for any equilibrium as long as c"1.

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8. Summary and conclusions In the present work we analyzed beam plasma diode equilibria and their linear stability for an arbitrary charge neutralization parameter c"n /n>; in addition unlike the classical Pierce diode  the collector potential < is varied. The three main results we have achieved are: 1. a complete classi"cation of the equilibria, 2. a transformation, which allows to deduce nonneutral equilibria (cO1) from generalized Pierce diode equilibria (c"1), 3. an extended, almost complete, stability scenario. The classi"cation distinguishes solutions without and with electron re#ection and furthermore discriminates between solutions in which the potential minimum (`virtual cathodea), re#ecting a fraction of electrons back to the emitter, is located inside the gap (partial re#ection of "rst kind). In this case, the re#ection ratio r, is related to the transmitted current I by I"1!r. It varies between 0 and 1. For completeness, also the case of total re#ection has to be considered. The transformation is of self-similar type and casts a generalized Pierce diode solution onto a nonneutral equilibrium solution. Hence, it is su$cient to know the former one in order to obtain a solution for a nonneutral diode with an arbitrary c. The transformation makes use of a Cauchy boundary value problem with given injection conditions. Therefore, e (c) is found by the given  i"e (1); then, d(c) is found by d(1) and "nally <(c) uniquely by <(1). For "xed values of c and  d a relation between the collector potential < and e is given by an (g, e)-diagram, which can easily  be constructed by utilizing these transformations. An (g, e)-diagram method is used to study the stability too. It yields an aperiodic stability analysis for any kind of equilibria, hence including that of virtual cathode solutions. On the other hand, for solutions without re#ection the stability analysis could be extended, embodying oscillatory eigenmodes, too. The latter utilizes a Lagrangian description for the cold electron dynamics and arrives at a dispersion relation that could be solved generally. The equivalence between an Eulerian, Cauchy boundary values problem and a Lagrangian, Dirichlet boundary value problem was shown. To be more speci"c, we reviewed in Section 1 existing plasma diode theories and commented on their contributions. Also, laboratory experiments were addressed in which bounded plasma instabilities play an important role. The de"nition of parameters and the basic equations have been presented in Section 2. Section 3 contained the various kinds of equilibria and also the boundaries of the di!erent regimes were investigated. Section 4 was devoted to the Lagrangian description of diode solutions without re#ection. The central quantity was the position f(q , q) of an electron #uid  element at (normalized) time q which was injected at time q . Two constraints, the transit condition  and the potential condition, restricted the manifold of solutions. An interesting parameter K and a rescaling with respect to d and the transit time ¹ were found, which allowed a projection of the 3-D parameter space (c, <, d) onto a 2-D parameter space (dI , K), where dI "dc. Also the linear dispersion relation for this class of solutions was derived and a kind of c-invariance of this equation was found. Section 5 was devoted to the transformation of a generalized Pierce diode equilibrium. It was accomplished by an appropriate shift and a rescaling of the potential coordinate g and the spatial coordinate f. As shown, the transformation took di!erent forms depending on whether re#ection is

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involved or not. The allowed ranges for the re#ection coe$cient (transmitted current) and for the emitter electric "eld of the Pierce diode in terms of c were presented (Figs. 5 and 6), i.e. that subset of the solutions for the generalized Pierce diode was found from which all solutions for any "xed c can be constructed. How the transformation looks like in di!erent situations was shown in Figs. 7a}f. We proved that extrema were mapped onto itself and that each monotonic potential region had its counterpart apart transformation. Hence, the topology of the PDs remained unchanged. In Section 6 the equilibria were classi"ed in the (e , d/p) parameter space for three di!erent c's.  We recognized a rather delicate pattern in the existence region of the various equilibria. Also typical examples of PDs were exhibited. An asymptotic treatment for small c showed that the transformation of states is valid up to c"0, the Bursian diode case, and that for e '0 the  boundaries in the existence region do not change very much for c(1. For given d and c, we then constructed an (g, e) diagram which shows how the collector potential and the type of equilibrium change as e is varied. Also the re#ection ratio r and transmitted current I were plotted and the corresponding current}voltage characteristic was shown in Fig. 15. For small c and "xed d a universal I}
A.Ya. Ender et al. / Physics Reports 328 (2000) 1}72

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Therefore, the study of the stability properties of generalized Pierce diodes is directly related to an investigation of the initial stage of the TIC instability. Later in the nonlinear stage of the electron dynamics an essential reformation of the potential distribution occurs in the TIC, resulting in an ion redistribution so that the electron beam moves already through a nonuniform and incompletely neutralized ion background. An analysis has shown that at certain times during the ion redistribution, the conditions of a Pierce-type instability arise again [84,94]. Thus, the investigation of nonneutral diodes is of fundamental interest for a study of the nonlinear oscillations in a TIC. A deeper analysis of these oscillations requests, therefore, a nonuniform ion background. In Refs. [84,94] it was shown that a Dirichlet boundary value problem, i.e. the case where the potential is prescribed at the boundaries, has often several solutions. Moreover, a strong re#ection of the electrons from the virtual cathode is characteristic of some of these solutions. This means that the transition from one state to another is typically accompanied by a strong decrease or increase of the current passing through the diode. Thus, a collisionless beam plasma diode can be used as a switch element switching on and o! the current. If an initial state of the diode is chosen to be near the bifurcation point, such a switch can be controlled by applying a voltage pulse of low intensity. Because the growth rate of an aperiodic instability is always comparable to the transit time of electrons, the switching speed of a device that is based on the collisionless beam plasma diode can be rather high. So, for example, for d"1 mm and an electron beam energy of 0.2 eV (such values are typical for a TIC) this time is about 10\ s. With d decreases and beam energy increases the switching time can be still smaller. Another signi"cant advantage of such switches is the high current density which can attain tens of amperes per square centimeter [94]. The possibility of switches being based on the e!ect of several existing solutions in the Bursian diode was proposed in Ref. [36], where the nonuniqueness of the solutions for the same external voltage < was observed even when a fast electron beam moves in the vacuum. It is known further that a vacuum diode with an electron beam (the Bursian diode) can be used as a microwave generator. In such a device the generation of oscillations occurs in the virtual cathode regime and is realized through a current control by means of oscillations of the depth of the virtual cathode. Vircators and reditrons are based on this e!ect [95]. A maximum generation was observed for an interelectrode gap value which slightly exceeds the critical one (d"4/3), corresponding to a unique bifurcation point in the dependence of the transient current on the gap value. It was proved in Ref. [39] that these oscillations are connected with the oscillatory instability from which the generation regimes of the diode were found. It was also shown that highly accurate numerical methods are required to calculate these nonlinear oscillations. Many features of these oscillatory processes were discovered by utilizing such methods [100]. In particular it was established that the oscillations disappear entirely already at a quite small value of the thermal spread of the electron beam velocity distribution function of about 2}3%. It was also shown that all long-living electrons (namely those particles which were approaching the oscillating potential barrier and which were captured by the potential trap, walking in the barrier's vicinity during several oscillation periods) are born from the very narrow region in the (q , v )-plane, q and    v being the dimensionless time and velocity of the electrons at the emitter. It is of interest to  continue such calculations for diodes with various c's and "rst of all for small c-values, i.e. to study how the properties of such generators vary when a tiny ion background is present. The theoretical investigations carried out in this paper bear a direct relation also to the nonneutralized charge particle beam transport.

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Many of the enumerated problems can also be studied experimentally, and can in future be realized by means of diodes with a crossed electron and ion #ow. If su$ciently high speed ions pass across the fast electron beam and the Pierce instability develops, the ion density will not even be redistributed after the termination of the nonlinear transient electron process and during the reformation of the potential distribution in the interelectrode gap. Therefore, in such devices after the plasma transition has taken place, the new state will be conserved for in"nitely long time. The regimes of switches constructed on the base of such diodes can be controlled by a slight variation of the density of the ion #ow. An optimization of the generation regime of the microwave generators can be realized by varying the spatial pro"le of the ion #ow. Thus the investigation of the beam plasma diodes opens new opportunities for the creation of microwave generators, high e!ective switches and other devices of the plasma electronics. The most important result of this paper is the following: for all devices mentioned the search for equilibrium states, the investigation of their stability as well as the study of the initial stage of an electron instability could be carried out analytically. The main task of future investigations in the scienti"c branches mentioned above will be an extensive use of accurate numerical methods for studying the electron processes in the regime with electron re#ection both of the linear stage and of the nonlinear stage. In addition there is also a need for an analysis of beam plasma diode states with a nonuniform density distribution of ions in space. Acknowledgements We would like to thank P.V. Akimov for discussions and Mrs. Monika Glaser and F. SchmoK gner for their help in the preparation of the manuscript. This work is supported in part by NATO Collaborative Research Grant HTECH.CRG971234, in part by the Russian Fund for Fundamental Research (Project No. 97-02-18080), and in part by the German Graduiertenkolleg `Nichtlineare Spektroskopie und Dynamika. References [1] R.G. McIntyre, J. Appl. Phys. 33 (1962) 2485. [2] R.W. Motley, Q-Machines, Academic Press, New York, 1975. [3] F.G. Baksht et al., in: Moizhes, Pikus (Eds.), Thermionic Converters and Low-Temperature Plasma, Washington, DC, 1976. [4] V.I. Kuznetsov, A.Ya. Ender, Sov. Phys. Tech. Phys. 17 (1973) 1859. [5] V.I. Babanin, I.N. Kolyshkin, V.I. Kuznetsov, A.S. Mustafaev, V.I. Sitnov, A.Ya. Ender, Sov. Phys. Tech. Phys. 27 (1982) 793. [6] E.A. Coutsias, D.J. Sullivan, Phys. Rev. A 27 (1983) 1535. [7] V.L. Granatstein, I. Alexe! (Eds.), High-Power Microwave Sources, Artech House, Norwood, 1987. [8] W. Hertz, Phys. Technol. 10 (1979) 195. [9] R.T. Farouki, M. Dalvie, L.F. Pavarino, J. Appl. Phys. 68 (1990) 6106. [10] R.B. Miller, Introduction to the Physics of Intense Charged Particle Beams, Plenum Press, New York, 1982. [11] J.W. Poukey, J.P. Quintenz, C.L. Olson, Appl. Phys. Lett. 38 (1981) 20. [12] J.W. Poukey, J.P. Quintenz, C.L. Olson, J. Appl. Phys. 52 (1981) 3016. [13] Yu.N. Gartstein, P.S. Ramesch, J. Appl. Phys. 83 (1998) 2037.

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[93]

[94] [95] [96] [97] [98] [99] [100]

A.Ya. Ender et al. / Physics Reports 328 (2000) 1}72 I.N. Kolyshkin, V.I. Kuznetsov, A.Ya. Ender, Sov. Phys. Tech. Phys. 29 (1984) 882. H. Schamel, S. Bujarbarua, Phys. Fluids B 5 (1993) 2278. H. Schamel, V. Maslov, Phys. Rev. Lett. 70 (1993) 1105. H. Kolinsky, H. Schamel, Phys. Plasmas 1 (1994) 2359. J.E. Faulkner, A.A. Ware, J. Appl. Phys. 40 (1969) 366. M.E. Gedalin, V.V. Krasnosel'skikh, D.G. Lominadze, Sov. J. Plasma Phys. 11 (1985) 508. A. Fruchtman, M. Benari, A.E. Blaugrund, Phys. Plasmas 2 (1995) 1296. Y.Y. Lau, D. Chernin, D.G. Colombant, P.-T. Ho, Phys. Rev. Lett. 66 (1991) 1446. Y.Y. Lau, P.J. Christensen, D. Chernin, Phys. Fluids B 5 (1993) 4486. A.W. Hull, Phys. Rev. 18 (1921) 31. Y.Y. Lau, Y. Liu, R.K. Parker, Phys. Plasmas 1 (1994) 2082. R.H. Fowler, L. Nordheim, Proc. R. Soc. London Ser. A 119 (1928) 173. P.J. Christensen, Y.Y. Lau, Phys. Rev. Lett. 76 (1996) 3324. H. Schamel, S. Bujarbarua, Phys. Fluids 26 (1983) 190. H. Schamel, Phys. Rep. 140 (1986) 161. M.A. Raadu, M.B. Silevitch, J. Appl. Phys. 54 (1983) 7192. S. Kuhn, International Conference on Plasma Physics, Kiev, World Scienti"c, Singapore, 1987, Inv. Papers 2, p. 954. S. Kuhn, Contrib. Plasma Phys. 34 (1994) 495. M.V. Nezlin, A.M. Solntzev, Sov. Phys. JETP 26 (1968) 290. G. Gverdtziteli, V.Ya. Karakhanov, E.A. Kashirskii, R.Ya. Kucherov, Z.A. Oganezov, Sov. Phys. Tech. Phys. 17 (1972) 78. S. Iizuka, P. Michelsen, J.J. Rasmussen, R. Schrittwieser, R. Hatakeyama, K. Saeki, N. Sato, J. Phys. Soc. Japan 54 (1985) 2516. P. Burger, J. Appl. Phys. 36 (1965) 1938. N.St.J. Braithwaite, J.E. Allen, Int. J. Electronics 51 (1981) 637; J. Phys. Coll. C 7 (Suppl. 7) (1979) C7-491. V.I. Kuznetsov, A.Ya. Ender, Sov. Phys. Tech. Phys. 28 (1983) 1431. F. Bauer, H. Schamel, Physica D 54 (1992) 235. R.Ya. Kucherov, Z.A. Oganezov, L.S. Timoshenko, V.K. Tskhakaya, Sov. J. Plasma Phys. 15 (1989) 766. W. Ott, Z. Naturforsch. 22a (1967) 1057. A.A. Rukhadze, S.D. Stolbetzov, V.P. Tarakanov, Radiotekhnika Elektronika 37 (1992) 385 (in Russian). D. Sullivan, Proceedings Vth International Top. Conference on High Power Electron and Ion Beam Research and Technology, San Francisco, 1983, p. 557. T. Kwan, Phys. Rev. Lett. 57 (1986) 1895. M. Botton, A. Ron, in: H.E. Brand (Ed.), Intense Microwave and Particle Beams III, SPIE Proc., Vol. 1629, 1992, p. 188. A.Ya. Ender et al., in: H.S. El-Genk (Ed.), Proceedings of the 11th Symposium on Space Nuclear Power and Propulsion, CONF-940101, AIP Conf. Proc., no. 301, vol. 2, 1994, p. 861; Amer. Inst. Phys., NY, V.I. Babanin, A.Ya. Ender, I.N. Kolyshkin, V.I. Kuznetsov, V.I. Sitnov, D.V. Paramonov, Next generation solar bimodal systems, Proc. 32-IECEC, AIChE, 1997, p. 427. V.I. Babanin, I.N. Kolyshkin, V.I. Kuznetsov, V.I. Sitnov, A.Ya. Ender, Proceedings of the second Intersociety Conference on Nuclear Power Engineering in Space, Physics of Thermionic Energy Converters, Sukhumi, USSR, Ministry of Atomic Energy, 1991, p. 95. V.I. Babanin, I.N. Kolyshkin, V.I. Kuznetsov, A.S. Pashchina, V.I. Sitnov, A.Ya. Ender, Tech. Phys. 39 (1994) 552. V.D. Selemir, B.V. Alekhinin, V.E. Vatrunin, A.E. Dubinov, N.V. Stepanov, O.A. Shamro, K.V. Shibalko, Plasma Phys. Rep. 20 (1994) 621. T. Klinger, F. Greiner, A. Rohde, A. Piel, Phys. Plasmas 2 (1995) 1822. H. Kolinsky, F. Greiner, T. Klinger, J. Phys. D: Appl. Phys. 30 (1997) 2979. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980. V.I. Kuznetsov, A.Ya. Ender, Proceedings of the Third International Conference on Thermionic Conversion, JuK lich, Germany, 1972. P.V. Akimov, H. Kolinsky, V.I. Kuznetsov, H. Schamel, A.Ya. Ender, Proceedings of the Conference on Low Temperature Plasma, Vol. 1, Petrozavodsk State Univ. Publ., Petrozavodsk (Russia), 1998, p. 513.

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ELECTRON AND LIGHT EMISSION FROM ISLAND METAL FILMS AND GENERATION OF HOT ELECTRONS IN NANOPARTICLES

R.D. FEDOROVICH, A.G. NAUMOVETS, P.M. TOMCHUK Institute of Physics, National Academy of Sciences of Ukraine, 46 Prospect Nauki, UA-03039, Kiev 39, Ukraine

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Electron and light emission from island metal "lms and generation of hot electrons in nanoparticles R.D. Fedorovich, A.G. Naumovets*, P.M. Tomchuk Institute of Physics, National Academy of Sciences of Ukraine, 46 Prospect Nauki, UA-03039, Kiev 39, Ukraine Received July 1999; editor: G. Comsa Contents 1. Introduction and outline 2. Island metal "lms: preparation and major experimental "ndings 2.1. Substrates and contact electrodes 2.2. Preparation of IMFs 2.3. Electroforming of IMFs 2.4. Emission centers 2.5. Chain island "lms 3. Electrical conductivity and electron emission properties of IMFs 3.1. Major experimental "ndings in brief 3.2. Electrical conductivity of IMFs 3.3. Electron emission from IMFs under conduction current excitation: integral characteristics 3.4. Electron emission from a single emitting center 3.5. E!ect of overlayers on the conductivity and electron emission from IMFs 3.6. Electron emission from silicon island "lms 3.7. Electron emission from IMFs under infra-red laser excitation 4. Light emission from island metal "lms 4.1. Light emission from clean IMFs 4.2. E!ect of overlayers on light emission from IMFs

76 77 77 78 81 83 85 86 86 87

92 93 96 101 102 104 104 107

5. Hot electrons in metal nanoparticles 5.1. Introductory remarks 5.2. Heating up of electrons 5.3. A model of heating of the electron gas in IMFs 5.4. Phenomena caused by hot electrons in IMFs 5.5. Mechanisms of light emission from IMFs 5.6. Summary of Section 5 6. Electron}lattice energy exchange in small metal particles 6.1. Introductory remarks 6.2. Peculiarities of the electron}lattice energy transfer in island metal "lms 6.3. Surface vibrations of small particles 6.4. Surface electron}phonon energy exchange 6.5. Derivation of the equation describing the sound generation by hot electrons 6.6. Concluding remarks about electron} lattice energy exchange 7. Optical absorption by small metal particles 7.1. Introductory remarks 7.2. Statement of the problem 7.3. Local "elds 7.4. Electron distribution function

* Corresponding author. Fax: #(380) 44 265 15 89. E-mail address: [email protected] (A.G. Naumovets) 0370-1573/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 9 4 - 0

108 108 109 112 114 115 121 122 122 122 125 127 130 132 133 133 134 136 139

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Electric absorption Magnetic absorption Quantum kinetic approach Resonance plasma absorption of light in IMFs 7.9. Conclusions about optical absorption of small particles 8. Examples of applications of island metal "lms 8.1. IMF cathodes 8.2. A gold IMF microcathode 8.3. Electron emission from island "lms of LaB  8.4. IMF cathodes with large emitting area 8.5. SnO island "lm cathodes 

141 146 150 154 156 156 157 157 157 158 161

8.6. Island "lm cathodes for #at information displays 8.7. IMF cathodes for IR electron-optical converters 8.8. Tensometric sensors 8.9. A microsource of light 8.10. Hot electrons beyond IMFs 9. Conclusions Acknowledgements Abbreviations Appendix A Appendix B References

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162 163 164 166 166 167 168 168 168 171 174

Abstract We review experimental and theoretical works devoted to electron and photon emission from island metal "lms (IMFs) representing ensembles of small metal particles deposited onto a dielectric substrate and coupled via penetrable potential barriers. Electrons and photons are emitted when the "lms are energized by passage of current through them or by laser irradiation. In either case the primary recipient of the energy is the electron gas, which can be heated up to temperatures much higher than the particle lattice temperature. A theoretical substantiation of the model of hot electrons in nanoparticles is presented. The major physical factor that permits generation of hot electrons in IMFs is the dramatic reduction (by orders of magnitude) of the electron}lattice energy transfer in the particles whose size is smaller than the mean free path of electrons in the volume. In such particles with a ballistic motion of electrons, the energy is being lost mainly in surface scattering acts which are less e!ective in energy transfer than generation of volume phonons. Thus, the electron temperature can become substantially higher than the lattice temperature provided the absorbed power density is high enough and the lattice of the island is intensively cooled by the substrate. The model of hot electrons is used to interpret experimental data. Non-equilibrium electron heating in IMFs can be observed even under stationary conditions, so the island metal "lms basically di!er in their electronic properties from continuous metal "lms and bulk metals where hot electrons can be obtained only for very short times (410\ s). Thus, the island metal "lms represent an important variety of nanomaterials having rather unusual physical properties. IMFs can be utilized to fabricate cathodes having interesting application potentialities in vacuum microelectronics, information display technologies and infrared image conversion. Hot electrons generated in nanoparticles may also play a signi"cant role in various dispersed systems exposed to energy #uxes.  2000 Elsevier Science B.V. All rights reserved. PACS: 73.50.Fq; 79.60.Jv; 78.66.Vs; 73.61.Tm; 36.40.Vz; 79.40.#z Keywords: Island "lms; Nanoparticles; Hot electrons; Electron emission; Photon emission; Optical absorption

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1. Introduction and outline Various kinds of dispersed systems have attracted intense interest for many decades due to their peculiar properties caused by the fact that these objects represent ensembles of small particles. As the size of the particles R is being reduced, the number of surface atoms in relation to that of volume atoms is growing as R\. Thus, the speci"c surface area of dispersed matter can attain huge values, and many aspects of behaviour of such systems are known to be determined by surface processes. This is actually a geometric size e!ect. However, there are a number of other size e!ects which come into play whenever the particle size becomes equal to some characteristic physical length like the electric "eld screening length, mean free path of electrons, de Broglie wavelength, etc. An interesting example of dispersed systems are island metal "lms (IMFs), representing quasitwo-dimensional ensembles of small particles. Such "lms can easily be prepared, e.g. by evaporating a small amount of metal upon a dielectric substrate which is unwettable by the metal. If the mean distance between the metal islands is small enough, the islands are coupled by electron tunneling and thus represent an entity. Nevertheless, the properties of such an object carry an imprint of the properties of the small particles making up the "lm. Important factors that determine the peculiar characteristics of IMFs are: size e!ects in properties of small particles; the ability of such particles to pass high-energy #uxes without destruction due to strong cooling e!ect of the substrate; the tunnel coupling between the particles; possible existence of high local "elds around them. Interest in IMFs was sparked more than 30 years ago when Borziak, Fedorovich and Sarbei discovered that passing a current through an IMF gives rise to electron and light emission originating from some small (41 lm) spots called emission centers [1]. Later on, the same e!ect was observed in the "lms being irradiated by a su$ciently powerful infrared laser beam [2}4] or microwave radiation [5]. This "eld has a comparatively extensive literature, including some monographs and reviews. The "rst 15 years of investigations on IMFs were summarized in Borziak and Kulyupin's monograph [6]. The subject was also partially covered in Nepijko's book [7] published in 1985. Switching phenomena in IMFs were reviewed by Pagnia and Sotnik ten years ago [8]. There is also our brief review which summarizes more recent works [9]. Meanwhile, a considerable progress has been achieved quite recently both in understanding the physics of hot electrons in small metal particles and in applications of the island "lms. This problem starts to attract renewed interest in the context of the rapid advancement of nanophysics and nanotechnologies. Therefore we hope that a review giving a more or less self-contained coverage of the problem may be timely and useful, especially when one considers that many essential works are scattered over not easily accessible journals and proceedings. This article is devoted mainly to phenomena of electron and light emission from island metal "lms and to nonequilibrium heating of electrons in small metal particles. The review is organized as follows. After this introductory section, we shall consider methods of preparation of IMFs (Section 2). Experimental data on the conductivity of IMFs and electron emission from them are presented in Section 3 which also discusses the main concepts suggested for interpretation of these results. Section 4 is concerned with light emission from the "lms. Sections 5}7 give a rigorous theoretical substantiation of the model of hot electrons in small metal particles. The readers who are not interested in the theoretical details, may skip these sections or just read their summaries. Examples of practical applications of IMFs are considered in Section 8. Finally, Section 9 contains

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general conclusions as well as some remarks about outlooks for further studies in the "eld and possible implications of hot electrons in various dispersed systems. It should be noted that "gures are numbered independently in each section. Let us start with a closer look at the objects whose properties are discussed in this review: the island metal "lms (IMFs).

2. Island metal 5lms: preparation and major experimental 5ndings 2.1. Substrates and contact electrodes Substrates for preparation of IMFs are made from various dielectric materials, most often from glass, quartz and mica. Their surfaces are cleaned by rinsing in standard chemical solutions and distilled water. In some studies the "lms were deposited on alkali-halide crystal surfaces which were obtained by cleaving the crystals immediately before the vacuum evaporation of the "lm. The contact electrodes whose typical geometry is illustrated in Fig. 2.1 represent continuous "lms with thickness of &100 nm which are prepared either by thermal evaporation in vacuum through a mask or by standard photolithographic technique. In the former case, there is a transition (`penumbrala) region near the boundary of the continuous contact "lm where the "lm has an island structure with variable mass thickness (Fig. 2.2a). Its existence can substantially a!ect the growth and structure of the island "lm which is subsequently prepared in the gap between the contacts (see Section 2.2). For structural studies by transmission electron microscopy, the island "lms were evaporated onto carbon "lms 10}30 nm thick deposited on meshes by the standard method. When structure and electrical characteristics were investigated in parallel, special substrates were used representing thin mica plates with a hole in the middle for passing the electron beam (Fig. 2.3). Each plate was "rst entirely (including the hole) coated with a Formvar "lm about 20 nm thick, and then a 50 nm SiO "lm was deposited over the Formvar sublayer. In the next step this sublayer was removed V from the hole by heating at 150}2003C leaving there only the SiO "lm. This "lm served as the V substrate for deposition of Au contact electrodes (by evaporation through a mask) and of an island "lm to be studied in the electron microscope. To prevent electrostatic charging of the SiO "lm by V the probing electron beam, its rear side was coated with a carbon layer 5}10 nm thick.

Fig. 2.1. Schematic of the sample with an island "lm between two contact electrodes.

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Fig. 2.2. (A) A `penumbrala region near the edge of a contact electrode. (B) Structure of an Au island "lm with a mass thickness of 5 nm near the contact with the `penumbrala edge.

Fig. 2.3. Schematic of a sample for in situ electron microscopic experiments with IMFs.

2.2. Preparation of IMFs The simplest way of preparation of IMFs is vacuum evaporation of a metal onto a dielectric substrate [10]. One can also apply cathode sputtering in a noble gas atmosphere [11], electrodeposition from solutions [12] as well as spraying of a suspension containing "ne solid particles

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[13]. Up to now, the most widely used method has been vacuum evaporation. The metals utilized for fabricating IMFs have been Au, Ag, Pd, Pt, Cu, Cr, Mo and some others. Graphite island "lms also show very good characteristics. It is seen that metals with su$ciently high melting points, whose atoms have a comparatively low mobility at room temperature, are deposited. This ensures a reasonably good stability of the "lm properties. Prior to preparing the island "lm, a pair of contact electrodes is deposited on the substrate as considered in the previous section. Then a smaller amount of a metal is additionally evaporated upon the whole surface which allows one to create an island "lm within the gap between the contacts (Fig. 2.2b). Typically the gap is about 10 lm. Since metals normally do not wet dielectric surfaces, the equilibrium growth mode corresponds in this case to the Volmer}Weber mechanism [14]. Thus, if the mass thickness of the deposited "lm is small enough, the metal atoms coalesce during deposition or subsequent annealing into islands to minimize the surface and interface free energy [15}17]. The nuclei appearing at the early stage of the "lm growth have an atomic size. They grow three-dimensionally as the metal deposition is continuing, but their dimension parallel to the surface increases much faster than normal to the surface (Fig. 2.4). The average size of the islands and their distribution over the substrate depend on many factors such as temperature of the substrate, evaporation rate, the mass thickness of the evaporated "lm, the chemical nature of the adsorbate and the substrate as well as on the temperature and duration of subsequent annealing. It is also well known that the spatial arrangement of the islands is sensitive to the presence and nature of surface defects (the decoration of the defects by metal atoms is routinely used in electron microscopy to visualize atomic steps and other surface imperfections). By choosing all the factors listed above, one can vary the parameters of IMFs within broad limits. The late stages of deposition result in coalescence of adjacent islands, then in the attainment of the percolation limit at some critical coverage, and "nally in the formation of a continuous "lm. Typically, the mass thickness of IMFs described in this article is about 4}10 nm. The evolution of the structure of growing island "lms could be investigated simultaneously with measurements of their electrical properties immediately in an electron microscope equipped with a built in vacuum chamber [18]. A pressure of &10\ Torr around the sample under study was provided by di!erential pumping. In other works, investigations of this type have been performed in a high-vacuum electron microscope [19]. To study the structure of the "lms deposited on thick substrates, it is necessary either to transfer the "lm onto another substrate which is transparent for electrons in the transmission electron microscope or to use the well-known method of replicas. In any case the "lm must be protected from changes which may occur while carrying the sample through atmosphere. This is achieved by coating the "lm, prior to its exposure in air, with a layer of SiO or carbon about 30}40 nm thick. The "lms obtained on 50 nm SiO substrates are the most V V convenient objects, since all the measurements can be carried out in situ. The method of preparation of such substrates has been described in Section 2.1. The electron microscopic investigations have shown that the size distribution of the islands is rather wide: there are relatively few islands about 10 nm in size and at the same time a large number of nanosized islands in between. It is also evident that the structure of the deposited island "lm can be peculiar in the `penumbrala region of the contacts where these electrodes themselves have an island structure. Such peculiarities do occur and are manifested both in a nonuniform potential drop across the "lm when a voltage is applied to it [20] and even in a speci"c optical

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Fig. 2.4. Structure of thin Bi "lms on a carbon substrate at various mass thicknesses (in nm). Substrate temperature during Bi evaporation was 1003C [21].

absorption in the near-contact areas [21]. If the edges of the contact electrodes are sharp, as e.g. in the case when the gap is obtained by scratching a continuous "lm (Fig. 2.5), the structure of the island "lm forming in the gap is found to be statistically homogeneous. Although our review is focused on the properties of metal island "lms, it is appropriate to mention here the possibilities of preparation of semiconductor island "lms, since some experiments reveal similarity in emission properties of metal and semiconductor "lms (see Section 3.6). In principle, there are no obstacles to fabrication of semiconductor island "lms. Actually the corresponding methods have been (and are being) intensively developed in connection with investigations of quantum dots [22}24]. In Section 3.6, we describe a method in which a Si island "lm is obtained as a result of controlled evaporation of thin Si single crystal epitaxially grown on sapphire.

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Fig. 2.5. Structure of an Au island "lm on glass substrate near the contact electrode with a sharp edge. The mass thickness of the "lm is 4 nm.

An important physical e!ect that probably can be used to better control the properties of island "lms is a pronounced self-organization which occurs, under proper conditions, in the course of thin "lm growth. This phenomenon has been much investigated in recent years (see e.g. [22}33]). The process starts at submonolayer coverages and continues at later stages when mesoscopic islands are being formed. Diverse mechanisms are predicted to drive the self-organization [34}36]. The most universal of them seems to be connected with elastic strains induced in the substrate by the growing "lm. Due to this e!ect, both the distances between the islands and the size of the islands can have rather narrow distributions around their average values. For example, the size of the islands can be obtained uniform to within $10% [22,23], supposedly due to reduction in adatom attachment probability caused by strain around the islands. 2.3. Electroforming of IMFs The current channels with emission centers in them can be electroformed in "lms of various thickness (from island "lms to semicontinuous ones) [6,8]. It has been found that the #ow of a conduction current through the "lm is a necessary condition of its electroforming. Indeed, experiments carried out at rather high electric "elds (up to &5;10 V/cm parallel to the substrate surface) have shown that the presence of such a "eld alone (without current) does not produce any appreciable e!ect on the island "lm properties [37]. Some other authors [15] reported on sensitivity of the "lm structure to the electric "elds as low as &10 V/cm, but their experiments were made at "lm thicknesses when a considerable current could pass through their "lms [38]. To electroform a gold or silver island "lm of `standarda geometry (see Section 2.1), it is usually su$cient to raise the voltage applied to the "lm up to about 20}30 V for a time of 0.5}2 min. The island "lms prepared from refractory metals are electroformed at somewhat higher voltages. The presence of overlayers such as BaO or some organic species on IMFs facilitates the electroforming process, probably due to reduction of the surface free energy. The electroforming can also be carried out by applying a pulsed voltage of about 100 V at a pulse width of the order of milliseconds.

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The electroforming proceeds more controllably when the contact electrodes have a special geometry which favours the emergence of the current channels in some particular places. The same e!ect is attained in chain island "lms (see Section 2.5). The current #ow, which "rst sets in at increased voltages only, seems to induce the process of electromigration in the "lm facilitated by some Joule heating, which results in the formation of clear-cut tunneling percolation paths (`current channelsa) and, as a consequence, in a sharp and irreversible decrease of the "lm resistance [39]. The current channels can be visualized with the aid of a thermosensitive "lm deposited on the surface of a sample through which a current is passed [40]. The "lm becomes decolorized over the channels, which can be seen under an optical microscope. The current channels were also investigated at a higher resolution by electron microscopy [41]. With this aim, the island "lms were prepared on thin SiO substrates transparent V to probing electrons (see Section 2.1). The evaporation of the "lms as well as the investigation of their structure, conductivity and emission characteristics were performed in situ. To reveal the current channels, the "lms deposited between the contacts were decorated by a very thin metal layer [42]. When a current is passed through the "lm, the previously deposited decorating metal atoms drift along the channels (and maybe are partially evaporated from them, since the current density in the channels can amount to 10}10 A/cm). If the decoration is being made while passing the current, one obtains the impression that the decorating atoms are not at all adsorbed by the channels. Thus the channels appear as winding light lines (Fig. 2.6) [41]. High-resolution electron microscopy has shown that the current channel consists of an ultradispersed system of nano-islands separated by nm distances. They are coupled with each other by a tunneling process and in this way provide a continuous conduction path in the "lm. The channels contain also a small number of larger islands which are shaped in the process of electroforming. Thus, after irreversible switching of the "lm to the low-resistance state, an appreciable conduction current #ows even at voltages 5}10 V, much lower than those used for electroforming, and this current is concentrated predominantly within the individual channels. It is just in the electroformed

Fig. 2.6. An Au-decorated island "lm of Au on SiO with a region previously identi"ed as an emission center. The V decoration with Au was carried out while a current passed through the "lm. The current channel shown by an arrow appears as a light path, since Au being evaporated was carried away from it.

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Fig. 2.7. A device for simultaneous observation of electron and light emission centers. 1, a glass bulb, 2, the sample with an island "lm, 3, cathodoluminescent screen, 4, window, and 5, an optical microscope. Fig. 2.8. Centers of light emission in an IMF (a) and the corresponding image of electron emission centers in the same "lm on a cathodoluminescent screen (b).

state of IMFs that Borziak et al. [1] discovered the electron and light emission which accompany the passage of the conduction current through the "lms. In the following sections, we shall give and discuss the emission characteristics in closer detail. At this point we only note that electrons and photons are emitted not from the whole "lm, but emerge from small areas (spots) sized 41 lm and termed emission centers. This was found in a device where the spatial distribution of emitted electrons was visualized on a cathodoluminescent screen [43] (Fig. 2.7). It was also stated that, within the resolution provided by optical microscopy, the emission of electrons and photons originates from the same centers (Fig. 2.8). Under an optical microscope, the centers appear as spots somewhat di!erent in colour and scattered rather chaotically over the "lm. A colour micrograph of the luminous centers in a gold island "lm was published in the "rst work on the emission e!ects in IMFs [1]. As the voltage applied between the contact electrodes is increased, the emission current and light intensity grow both due to increase of the emissivity of each center and due to possible appearance of new centers. However, some centers can be extinguished if the applied voltage becomes too high and induces irreversible changes in the "lm structure. 2.4. Emission centers Electron microscopic investigations have shown that the emission centers are located within the current channels, one center per channel [44,45]. The location of the emission centers is actually unpredictable on the substrates with #at surfaces prepared by ordinary methods such as mechanical polishing or cleavage. This is caused by the uncontrolled con"guration of the current channels which re#ects the statistical character of the nucleation of the islands. A typical emission center is a structure consisting of a relatively large (&100 nm) island surrounded by nano-sized islands (Fig. 2.9) [44]. Such con"gurations are created in the course of electroforming. It is well known that

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Fig. 2.9. Structure of an Au island "lm in the vicinity of an emission center. The substrate is a SiO "lm 50 nm thick. V

in the process of the Ostwald ripening of dispersed systems the large particles are growing at the sacri"ce of smaller ones due to the di!erence of the saturation vapour pressures over them. Thus, the emergence of local con"gurations `a large island#nano-islands around ita should be quite probable, and such structures have indeed been documented in many works on thin "lms (see e.g. a recent paper [31] and references therein). However, as noted above, usually a single emission center appears within a current channel so the number of the centers in a "lm cannot be larger than the number of the channels in it. There is a depletion zone around the large island and due to this a considerable potential drop occurs at this place. As will be shown in Section 3.4, this plays an important role in the electron and photon emission from IMFs, since a substantial part of the power fed into the current channel is released within the center. The process of electromigration in IMFs is so far poorly investigated. Generally, the phenomenon of electromigration is rather complicated and highly sensitive to the chemical nature and the structure of systems under study. For example, electromigration in a number of bulk metals (Al, Ag, Au, In, Sn) and in their continuous thin "lms is driven by the scattering of electrons (`electron winda), i.e. occurs towards the anode [46}50]. The "lms of the same metals being deposited on a semiconductor surface grow by the Stranski}Krastanov mechanism, i.e. form a continuous monolayer with three-dimensional islands on it, and often exhibit electromigration towards the cathode (see e.g. review [51]). Probably, the emission centers arise as a result of an intricate interplay between the structure of the "lm and the processes of electro- and thermal migration. In particular, electromigration depends on the local current density and resistivity, depending in turn on the "lm structure. If a considerable enhancement of the current density appears in some place in the current channel, it can be anticipated that the electromigration, perhaps combined with somewhat enhanced Joule heating, may cause a progressing destruction (`burninga) of this segment

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of the channel. This results in a growth of the voltage drop across this segment at the cost of the potential drop across the remaining parts of the channel. This avalanche-like process appears as a kind of instability of the uniform potential distribution which, however, does not lead to a complete burning of the channel if the voltage applied to the "lm is not too high. The result is that only one emission center survives in a channel. Probably, at some stage of the process there occurs a stabilization of the "lm structure through coming into play of a mechanism of self-limitation. This is mirrored in the fact that electroformed IMFs show practically stable conductivity and emission properties for thousands of hours under appropriate exploitation conditions (see Section 8). We have dwelt so much on this point, because it is just the segment in the current channel with a sharp potential drop across it that gives the electron and light emission. The mechanisms of this e!ect will be discussed in the following sections.

2.5. Chain island xlms As noted in the previous section, the electroforming of the current channels and emission centers in them is actually a poorly controlled process on the substrates with #at surfaces prepared by standard methods. The electrons in the "lm are seeking paths with the lowest resistance where the electromigration forms eventually a well conducting percolation channel. Since the arrangement of the islands re#ects a chaotic distribution of the nucleation centers (various defects), the shape of the current channels is usually very winding and their position is unpredictable. Correspondingly, the emission centers are also scattered rather randomly over the "lm. It was found, however, that the centers appear more frequently in the vicinity of the contact electrodes [52] which may re#ect some speci"cs of the "lm structure in the `penumbrala regions of the contacts (see Section 2.1). For this reason the emission centers appear as a discrete and irregular chain stretched more or less parallel to the contacts. There is, however, a possibility of making the process of the generation of emission centers in IMFs more controllable. To this end, the "lm is evaporated at a grazing angle onto a substrate whose surface is grooved (the incidence plane of the atomic beam is orthogonal to the grooves) (Fig. 2.10). Under such an evaporation geometry, the "lm has the largest mass thickness on the walls of the grooves whose orientation with respect to the atomic beam axis is closest to normal. After annealing, one obtains a "lm consisting of the chains of islands located in the grooves. A micrograph of such a "lm is given in Fig. 2.11. Such `chaina island "lms exhibit a high anisotropy

Fig. 2.10. Schematic of a grooved substrate with a chain island "lm on it.

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Fig. 2.11. Electron microscopic images of chain island "lms on (a) glass substrate with a di!raction grating and (b) a corrugated surface prepared by photolithography.

of conductivity: the ratio of the conductivities along and across the grooves (chains) amounts to &10 [53]. An island chain appears as an almost ready current channel, and actually no special electroforming of the chain island "lm is needed for obtaining the electron and light emission from it. The emission is observed at once as a `normala (not increased) operating voltage is applied to the "lm. It is interesting, however, that the potential drop along the chains nevertheless appears nonuniform. The region of an enhanced potential drop emerges in all the chains, and this region, which just marks the position of the emission center, is located for all chains at approximately the same distance from one of the contacts. As a result, the position of the emission centers in such a "lm is more controllable. Instead of randomly scattered luminous spots, one sees a narrow luminous strip formed by the individual centers located in all the grooves (chains). The strip is stretched across the grooves. Probably, the emergence of the emission centers in the chain "lms occurs faster and at lower voltages (below or equal to the operating voltage).

3. Electrical conductivity and electron emission properties of IMFs 3.1. Major experimental xndings in brief Experiments carried out in [1] and in later works on emission properties of IMFs have revealed the following main features. (1) The conduction current}voltage characteristics of IMFs are linear at low voltages, but become superlinear at higher voltages. (2) Electron emission and light emission set in at voltages at which the conduction current} voltage characteristics of IMFs exhibit deviation from Ohm's law. The emission is observed in both the continuous and pulse regimes. In the latter case, the measurements were made in the range of pulse durations q"10\}10\ s and frequencies f"1}10 Hz.

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(3) The conduction current grows substantially as an island "lm is covered with an overlayer that lowers the work function. For instance, the deposition of a BaO monolayer increases the conduction current by a factor of 10}15. At the same time, the emission current is enhanced by 3}4 orders of magnitude. Some organic overlayers cause the occurrence of a section of negative di!erential resistance in the conduction current}voltage characteristics of IMFs (see Section 3.5). (4) The local density of the emission current estimated at the emission centers can amount to 10}10 A/cm. The total emission current can reach up to 10\ of the total conduction current. The energy spectrum of the emitted electrons is rather broad. In the cases when it appears to be Maxwellian, the electron temperature amounts to &10 K. (5) The emitted light spectra usually have maxima in the red region, but deviate from the Planck law in the sense that they contain more energetic photons. The spectra are considerably narrower for the chain island "lms than for `irregulara ones. (6) After the electroforming procedure in which the IMF structure and conductivity undergo substantial and rapid changes, the IMF properties are stabilized at a constant level. Under moderate exploitation conditions, they can be maintained for 10}10 h. (7) The electron and light emission from IMFs can also be observed when the "lm is exposed to infrared laser radiation. In this case, previous electroforming of the island "lm is not required. (8) The emission phenomena observed in IMFs occur also in the island "lms of semiconductors. In Sections 3.2}3.7 we shall consider these points in detail. Let us start with the discussion of the conduction current}voltage characteristics which are very important for understanding the peculiar properties of IMFs. 3.2. Electrical conductivity of IMFs The electrical conductivity of IMFs has been addressed in many papers and a number of monographs [54}58]. Their message is that the conductivity of IMFs on dielectric substrates is basically distinct from that of bulk metals and resembles to some extent the conductivity of semiconductors. The main questions are the mechanisms of charge transfer between the islands, the explanation of the non-ohmic conduction current}voltage characteristics and the exponential temperature dependence of the conductivity, the in#uence of adsorbates on the conductivity, and aging e!ects. As mentioned above, the dependence of the conduction current versus voltage applied to an IMF obeys Ohm's law at weak "elds, but deviates from it at E&10}10 V/cm (E is the average electric "eld in the "lm) (Fig. 3.1). Within a narrow interval of thicknesses of IMFs, the temperature resistance coe$cient is found to be negative. So a possible explanation for the superlinear conduction current}voltage characteristics could be the Joule heating of the islands. However, this hypothesis was discounted by the experimental data [59]. Since the dependence ln I versus 1/¹  appears linear in a given temperature range, some authors introduced the concept of the activation energy of conductivity, E . The value of E decreases with increasing mass thickness of the "lm and in the limiting case, as the metal "lm becomes continuous, the temperature resistance coe$cient turns positive. In terms of E , the non-ohmic current}voltage characteristics were attributed to a change in the activation energy with increasing "eld and to a mechanism of `activated tunnelinga [60]. Below, we shall describe another model, which substantiates the possibility of nonequilibrium

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Fig. 3.1. Dependences of the conduction current (1) and electron temperature (2) on the average "eld strength in a 4 nm gold island "lm.

heating of electron gas in small particles and allows a consistent explanation of all unique electronic properties of IMFs. Whatever the mechanism of the conductivity may be, the islands must exchange electrons and in the general case the current from the ith to the (i#1)th island can be written as



(E, ¹)D (E ) dE . I "e U GG> GG> V GG>

(1)

Here U (E, ¹) is a function that determines the number of electrons with energy E which are GG> able to pass from island i to island (i#1) at temperature ¹, and D (E ) is the probability of GG> V the transition for an electron having an energy E connected with the motion along direction x. V If an idealized system of identical islands located at equal distances from each other is placed in an external "eld, the total current will be equal to the di!erence of currents between the adjacent islands:



!U ]D(E ) dE . I"e [U GG> G>G V

(2)

To obtain an explicit dependence of the current on voltage, temperature, the "lm structure and electronic properties of the islands, it is necessary to specify the model. In principle, charge transfer between the islands can occur by means of thermionic emission into vacuum [61] or into the substrate [62], by tunneling through barriers metal}vacuum}metal or metal}substrate}metal [57,60,63}67], or by transport through impurity levels in the substrate [68]. For the "lms consisting of islands &5 nm in size separated by distances 2}5 nm, the most probable mechanism seems to be the tunneling of electrons between the islands. Judging from the strong e!ect of adsorbed BaO molecules on the conduction current [69] and the weak sensitivity of this current to the substrate material [6], the tunneling should occur via vacuum. The non-ohmic conduction current}voltage characteristics can be explained in terms of nonequilibrium heating of electrons in the IMFs at high electric "elds [64,70]. One further

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Fig. 3.2. A diagram of potential energy of an electron in an island metal "lm in the absence (solid line) and in presence (dashed line) of electric "eld. See text for explanations.

consequence of electron heating is electron emission from IMFs. Let us consider the concepts of this model in greater detail. Suppose we have a linear chain of identical islands with equal spacings between them. The potential barriers separating the islands are assumed to be parabolic (Fig. 3.2): ; (z)"(gz /2)!(g/2)(z!z ) ,   

(3)

where ; (z) is the barrier height in the absence of electric "eld, 2z is the distance between the   islands and g is a parameter. In the presence of an electric "eld the barrier is lowered and its maximum is displaced: ; (z)"(g/2)[z !(z!z )]!eEz"(g/2)[z !(eE/g)]!(g/2)+z![z !(eE/g)], . (4) #     Here ; (z) is the height of the barrier in point z in the presence of the "eld and e is the electron # charge. Usually, the lowering of the barrier is not very strong, so one has z <eE/g and therefore  ; (z)"(gz /2)!eEz !+z![z !(eE/g)], . #   

(5)

It can be seen that under the above assumptions the barrier is lowered by eEz "*;/2, where *;  is the potential di!erence between the adjacent islands. The transparency of such a barrier is

 

 

\ E#(*;/2) , #1 D(E)" exp ! e 

(6)

where e "(h/2p)(g/m) is a parameter characterizing the barrier shape and m is the electron  mass. Since the Fermi levels of the adjacent islands are shifted by *;, the electrons tunneling to an island from its neighbour are `hota with respect to other electrons in this island. The redistribution of the excess energy between all electrons in the island results in a raising of the electron temperature. In principle, the functions U (E, ¹) in (1) can be distinct in di!erent GG> islands. The distinction can be caused, for instance, by di!erent heating of the electron gas. For

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simplicity, we assume the electron temperature to be equal in all the islands. Then the function U (E, ¹) can be taken as a Fermi function with an e!ective temperature h (expressed in energy GG>  units), which depends on the power pumped into the island:




f (E)" exp

 

\ E#u . #1 h 

(7)

Substituting (6) and (7) into Eq. (2) gives I"A

  



E#(*;/2) 1#exp ! e  \

  \




1#exp

E#u h 



\

dE

(8)

with A"4pme*;/(2p ). To obtain an explicit dependence of the conduction current on voltage, temperature and other parameters, it is necessary to "nd the relationship between the electron temperature, the lattice temperature and the power fed into the "lm by passing current. The peculiarities of the dissipation of the energy of electrons in small particles will be treated at length in Sections 5 and 6. Here we note only that the motion of electrons is ballistic in the particles whose size is smaller than the electron mean free path in the bulk metal. For this reason electrons lose their energy in collisions with defects and with the island as a whole when they are re#ected from the potential barrier. The re#ection is almost elastic and the transferred momentum equals *p"2p, where p is the electron momentum. Thus the transferred energy is *E"Em/M, where m and E are respectively the mass and energy of electron and M is the mass of the atom (or the island). It should be noted that the factor m/M, which determines the part of the energy lost by electron in a collision, remains somewhat uncertain. However, this fact need not concern us, since the factor enters into a phenomenological coe$cient which may be determined from experiment. The number of collisions per unit time equals 1/q"v/a, where v is the electron velocity and a is the island dimension. Hence the power transferred from an electron to the lattice is *P"*E/q&((m/M)(E/a) .

(9)

The total power transferred by electrons to the whole "lm can be obtained by averaging (9) over the nonequilibrium component of the electron energy distribution and multiplying the result by the volume of the islands. Under steady-state conditions, this power must be equal to the power fed into the "lm P"I;, where ; is the applied voltage. Hence the following expression can be obtained for the electron temperature: (10) h "k¹ "(h #aHI; .    Here h is the lattice temperature in energy units and aH is a coe$cient which is independent of the  "eld and temperature and determines the e$ciency of electron heating. In the case e 'h , the   current is carried mainly by electrons tunneling near the Fermi level, because the barrier is rather steep and its transparency depends on energy less sharply than the electron energy distribution function. Then the expression for the current becomes I"Ae exp(!u/e )+1#(p/6)(h /e ), exp(*;/2e ) ,     

(11)

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where u is the work function. In the opposite case e (h , when the transparency depends on   energy stronger than the distribution function, the current is carried predominantly by electrons tunneling near the top of the barrier and we obtain I"Ah exp(!u/e )+1#(p/6)(h /e ), exp(*;/2e ) . (12)      Thus, the maximum contribution to the tunneling current belongs to the group of electrons which are located at some distance from the Fermi energy, depending on the shape of the barrier (characterized by the parameter e ) and electron temperature h . If h is low, the tunneling occurs    just above the Fermi level. As h increases, the maximum of the tunneling #ux is shifted to the top of  the barrier. Some contribution to the conduction current can also originate from overbarrier electrons, but their part seems to be comparatively small (since the emission current is found to be much smaller than the conduction current, see Section 3.3). All these factors cause the deviation from the Ohm law. In work [64], we proposed a method for determination of the model parameters e and aH, and  hence of the electron temperature, from the dependences of the conduction current on the applied voltage ("eld) and temperature. The typical electron temperature corresponding to beginning of the deviation of conduction current}voltage characteristics from linearity was found to be about 0.15 eV (+2;10 K) (Fig. 3.1). This model not only explains the variation of the conduction current versus electric "eld, temperature and work function, but also allows one to understand the main regularities of electron emission from the island "lms, which will be discussed in the following sections. To conclude, a few additional comments are in order. Proposing the model of electron heating described above, we were guided by the following considerations. The deviation of the conduction current from Ohm's law and the occurrence of the electron and light emission (discussed in detail below) set in at approximately the same voltage applied to the "lm. This suggests that these phenomena should have a common cause. Such a cause may be the generation of hot electrons. Taking into consideration the weak temperature dependence of the conduction current in the Ohmic region, we have assumed the tunnelling conduction mechanism. In the case of the thermionic mechanism, the conduction current would be strongly dependent on temperature, which is not observed. The next comment is connected with the work function of the islands. We have assumed that the work function as well as the Fermi energy are the same in all the islands. However, this assumption can be not valid for very small islands where the e!ects of the Coulomb blockade can be essential. Generally, such e!ects should not be pronounced in the conductivity of two-dimensional ensembles of islands at room temperature. It is known (see e.g. [71]) that two major prerequisites must exist for the observation of the Coulomb blockade: (1) one should provide the condition e/2C
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3.3. Electron emission from IMFs under conduction current excitation: integral characteristics Electron emission from IMFs is always observed in the non-Ohmic region of the conduction current}voltage characteristic. Typical dependences of the emission current on the voltage applied to the contact electrodes of the "lm and on the anode (extracting) voltage are depicted in Fig. 3.3. Such dependences have been used by many authors to consider the mechanism of the electron emission. Two alternative mechanisms have been discussed: "eld emission and emission of hot electrons. The former model [73,74] considers that, despite the low value of the voltage applied to the "lm, rather high electric "elds can exist in the narrow gaps between the islands. The "elds can have elevated values near the smallest islands and in the emission centers where the potential drop is found to be enhanced [73,75}78]. These "elds can give rise to "eld emission currents #owing from one island to another (in other words, the conduction current should be "eld emission by its nature), and it may be imagined that a part of this current can be sucked o! by the extracting electrode (anode). Qualitatively, this model may seem rather reasonable. In support of this interpretation its adherents argue that in a number of cases the dependences of the emission current, I , on the voltage applied to the "lm, ;, appear linear when plotted in the  Fowler}Nordheim (`"eld-emissiona) coordinates lg (I /;) versus 1/; (see e.g. Fig. 3.4). Notice  that ; is the voltage between the contact electrodes of the "lm and not the anode voltage ; , which is kept constant in the experiment. (Incidentally, I as a function of ; is nonexponential and  exhibits a tendency to saturation at high values of ; (Fig. 3.3).) It is therefore tacitly assumed that

Fig. 3.3. Dependences of the conduction current I and electron emission current I on the voltage applied to an Au   IMF. Inset: Dependences of electron emission current I on the average electric "eld applied between the IMF (cathode)  and anode. The average electric "elds within the "lm are respectively 10 V/cm for curve 1 and 4;10 V/cm for curve 2. Fig. 3.4. Dependences of the electron emission current I on the power fed into "lm (P"I ;) and the voltage applied   across the "lm (;). In plots 1 and 2, the data for two di!erent "lms are presented in the coordinates ln I versus P\. In  plots 1 and 2, the same data are presented in the coordinates ln I /; versus 1/; (from Ref. [6]). 

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the part of the conduction current branched o! to the anode remains constant as the current varies. It has to be admitted, however, that results similar to those depicted in Fig. 3.4 can hardly be considered as a convincing proof of the "eld emission model. The range of the voltages in which such dependences can be recorded is usually rather narrow, so it is not very surprising that a straight line can be "tted more or less satisfactorily to log I /; within this range. Further still,  this model is in poor agreement with experimental data on the e!ect of adsorbed layers: as one covers an IMF with an overlayer reducing its work function, the conduction current increases several-fold (or by an order of magnitude at most) whereas the emission current grows by a few orders of magnitude (see Section 3.5 for more details). This "nding is in evident contradiction with the expectations of the "eld emission model that the electron emission current should vary proportionally with the conduction current (or at least to its "eld-emission component). The results [75] about the high sensitivity of the electron emission current to the deformation of the "lm substrate (`the tensoe!ecta) are also poorly compatible with the "eld emission model [73] which suggests the emission from nanosized (:10 nm) island `cathodesa to large islands serving as `anodesa and located at a distance of &1 lm from the `cathodesa. It is well known from the "eld emission experiments that in such a geometry the emission current should be weakly dependent on the cathode}anode spacing and therefore on the substrate deformation. There are also discrepancies of the data on the temperature dependence of the electron emission from IMFs [79] with predictions of the "eld emission model. Let us turn now to the model of hot electrons. Recall that it also considers the tunneling of electrons between the islands, but, contrary to the "eld emission model, the su$cient transparency of the potential barriers is attributed just to small inter-island spacings rather than to the e!ect of the external electric "eld. This model predicts [70] (see also Section 3.2) that the emission current should be a linear function in the coordinates log I versus 1/(P), where P is the power pumped  into the emission center. Some experimental results obtained in the case of the conduction current excitation of thin "lms are consistent with this prediction (Fig. 3.5) [6] while others are not [73,74]. (It is important to note that actually one plots the dependence of log I versus reciprocal square  root of the total power fed into the "lm, which is believed to be proportional to the power fed into individual emission centers.) Obviously, a vulnerable point of the above experimental arguments is that characteristics of the phenomena occurring in the emission centers, which are 41 lm in size and take just a minor part of the total "lm surface area, are correlated with integral values such as the voltage or the power applied to the whole "lm. It is therefore necessary to test both the models on a larger set of experimental data including the electron energy distributions, the electron emission stimulated by infra-red irradiation and the light emission from IMFs. This will be done in the following sections. 3.4. Electron emission from a single emitting center For typical geometries of IMFs, the electroforming procedure usually results in emergence of many emission centers. It is clear that examination of a single center is much more informative from the physical standpoint than analysis of integral characteristics which contain contributions from a set of the centers. (In addition, their number can change if the voltage is varied within a broad range.) The investigation of individual centers is especially important when it is necessary to analyze the electron energy distribution. If the integral retarding "eld characteristics of the

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Fig. 3.5. Dependence of the electron emission current I from an Au island "lm on P\, where P"I ; is the power fed   into the "lm [6].

Fig. 3.6. Retarding-"eld curve of electron emission current in a device with plane-parallel geometry. Fig. 3.7. Schematic of a device for recording the retarding-"eld curves of electron emission from a single emitting center.

emission current are measured in a device of the #at capacitor type, one usually records curves with two saturation terraces (Fig. 3.6). This indicates that the current comes from two groups of emission centers located in di!erent areas of the "lm (typically near the contacts) having di!erent potentials. In the case when the centers are grouped near one of the contacts only, the retarding "eld curves have an ordinary shape with one saturation level. To obtain such a curve for an

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Fig. 3.8. Retarding-"eld curves of electron emission current from a single emitting center at various voltages applied to the "lm: 1}10 V; 2}15 V; 3}17.5 V; 4}20 V; 5}24 V; 6}30 V. Fig. 3.9. Energy spectra of secondary electrons emitted from the contact electrodes (1,2) and of electrons emitted from a center in an Au "lm (3). The energy of the bombarding electrons is 25 keV and the voltage applied to the "lm is 2.5 V [82]. Inset: schematic of the sample under study (1 and 2 are contact electrodes).

individual center, a device of spherical geometry (`a spherical capacitora) was used which allowed the electrons emitted from a selected center to be directed into the energy analyzer (Fig. 3.7) [43,80]. The device represented a spherical glass bulb 150 mm in diameter whose inner wall was coated with a transparent conducting layer and a cathodoluminescent screen. The bulb had a probe hole in the screen and a retarding "eld analyzer placed behind it. The "lm studied was deposited onto a glass sphere 2 mm in diameter located at the end of a glass rod which could be moved with a manipulator. Fig. 3.8 depicts examples of the retarding "eld curves recorded for an emission center at various voltages applied to the "lm. The width of the steep (`retarding "elda) section of the curve, *<, is determined by the spread in electron energies, the potential drop within the center and the resolution of the analyzer. *< was found to be only weakly dependent on the voltage applied to the "lm which means that only a small part of this voltage drops across the center. Anyway, the retarding "eld curves showed that the sum of the electron energy spread and the voltage drop within the center does not exceed &3 V. The minimum value of *< recorded experimentally was &1 V. The electron energy distribution was also studied in a scanning electron microscope equipped with a retarding "eld analyzer having an immersion objective [81,82]. The experiments were carried out with gold IMFs which were prepared within a 20 lm gap between two gold contact electrodes. The secondary electron emission from the contacts was used for calibration of the energy scale of the analyzer. To minimize the e!ect of electric micro"elds existing over the islands [76] on the energy spectra recorded, the measurements were made at voltages on the "lm equal to 2}3.5 V (the emission current was at the level &10\ A). Fig. 3.9 shows the energy distributions for electrons originating from an emission center and for secondary electrons emitted from the contacts. The position of the energy spectrum of electrons between the curves corresponding to the secondary emission from the contacts is determined by the potential drop across the "lm. The high energy wing of the energy distribution was found to be Maxwellian, and the electron gas

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temperature within the emission center was estimated at ¹ &3400 K [82]. This value agrees by  order of magnitude with the electron temperatures evaluated by other methods [83}85]. 3.5. Ewect of overlayers on the conductivity and electron emission from IMFs Let us start with electropositive overlayers. The evaporation of BaO on island "lms leads to a signi"cant increase of the conduction current (Fig. 3.10, curves 2}4) and to a drastic, by several orders of magnitude, growth of the emission current (Fig. 3.10, curves 2}4). Both the e!ects can be ascribed to the reduction of the work function. The variation of the work function was monitored by recording the shift of the red boundary of photoemission from a continuous Au "lm deposited adjacent to the island "lm and exposed to the same BaO molecular beam. The model of nonequilibrium heating of electrons in small particles (Sections 3.2, 5 and 6) interprets the observed electron emission from IMFs as the Richardson emission of the hot electrons. With this assumption we used the work function dependence of the emission current to estimate the electron temperature ¹ and obtained its value at +2000 K. Contrary to this, the  lattice temperature of the islands remains much lower which ensures the long-term operation of such IMF cathodes. If the BaO coating over an IMF is only about one monolayer thick, the emission current, which is initially very strongly enhanced, decreases with time to a new steady-state level which still remains much higher than in the absence of the BaO layer. The current can be increased again by repeated evaporation of BaO. The measurements of the emission current}voltage curves for individual emission centers performed in a quasi-spherical energy analyzer have shown that the decay of the emission current is caused by some work function increase [86]. Probably, this can be associated with the "eld induced drift of BaO molecules out of the emitting center. It should be recalled that in the case of a polar adsorption bond, the direction of the "eld drift is dependent on the mutual orientation of the electric "eld and the dipole moment [87,88]. To ensure a longterm e!ective electron emission, with an emission-to-conduction current ratio of +10}15%, it is necessary to provide a continuous supply of BaO to the emitting center. This e!ect can be achieved by deposition of island "lms onto a substrate previously coated with a thick BaO layer (see Section 8).

Fig. 3.10. Current}voltage curves for conduction current (1}4) and emission current (l}4). Curves 1 and 1: clean Au "lm; curves 2}4 and 2}4: Au "lm with various BaO overlayers.

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Fig. 3.11. Current}voltage curves of conduction current (1) and electron emission (2) for an Au island "lm. (1) and (2) are the same characteristics after deposition of organic molecules.

A similar e!ect of BaO as well as Ba overlayers on the electron emission was also found for graphite island "lms. These "lms were excited both by the conduction current and by pulsed CO  laser irradiation (j"10.6 lm, q"200 ls) (Section 3.7). Adsorption of BaO molecules represents a comparatively simple example of the e!ect of overlayers on properties of IMFs. There are, however, more complicated cases. Early experiments with IMFs detected that quite often the conductivity current becomes unstable when the "lm is kept in vacuum obtained using oil di!usion pumps [89,90]. In particular, under certain conditions the conduction current}voltage characteristics were found to become nonmonotonic (N-shaped), i.e. to reveal a voltage-controlled negative resistance (VCNR). An example of such characteristics is given in Fig. 3.11. It is known that in MIM (metal}insulator}metal) structures, the VCNR and switching from low-resistanse to high-resistance state and vice versa are frequently observed e!ects (see e.g. [91}93]). Obviously, the systems considered (IMFs and MIM structures) bear some similarity to each other. This similarity becomes even more far-reaching if one considers that electroforming of the MIM structures can produce current channels which contain small metal particles arising due to the partial destruction of one of the electrodes [94,95]. A detailed review of the VCNR and switching e!ects in the island "lms (planar MIM diodes) and in sandwich MIM diodes was given by Pagnia and Sotnik [8], so we shall discuss here only the points which received less attention in [8] as well as some results of later works. Major experimental "ndings regarding VCNR in island metal "lms can be summarized as follows. 1. VCNR e!ects have not been found for the island "lms prepared in ultrahigh vacuum on well outgassed substrates. A prerequisite to the observation of such e!ects is the presence of speci"c kinds of adsorbates on the "lm surface, in particular of some organic compounds or of water containing their traces [96]. There are data suggesting that at least in some cases it is just the hydrated molecules of these compounds that may cause the occurrence of VCNR [41]. It is also important to add that, in order to prepare an island "lm exhibiting VCNR, it is necessary to apply an appropriate voltage to the "lm either during or immediately after the deposition of the overlayer, i.e. to `polarizea it.

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2. Electron microscopic investigations performed with a resolution of 2}3 nm did not reveal any structure changes in the "lm during its transition from low-resistance to high-resistance state or in the opposite direction [41]. The "lm structure starts to show visible changes only when the applied voltage exceeds several-fold its value corresponding to the current maximum in the current}voltage curve. However, in this case the "lm is irreversibly destroyed. 3. Scanning electron microscopy was used to visualize the potential distribution within the "lms displaying VCNR behaviour [77]. The distribution was found to be uniform in the low-resistance state, i.e. in the voltage range preceding the negative di!erential resistance. In the high-resistance state, potential jumps localized within the regions of 40.1}1.0 lm were observed. Later STM investigations [78] con"rmed the existence of the potential jumps and showed that regions of their localization are even narrower. 4. In the low-resistance state of the "lms with VCNR, the temperature resistance coe$cient in the interval 20}300 K is positive and close to its values typical of metals. This coe$cient becomes negative in the high-resistance state and its value is characteristic of the clean IMFs [97]. 5. The VCNR behaviour disappears if the "lm coated with the overlayer is annealed at 500}600 K. However, this behaviour is restored after repeated adsorption. 6. The return trace of the I-U-hysteresis loop (Fig. 3.12), segment BA' is sluggish. If this trace is passed fast (in a time (0.1 s), the current changes along the path BCO, and the high-resistance state can keep inde"nitely long (a `"eld memorya). To restore the low-resistance state, it is su$cient to increase the voltage by 0.1}0.2 V with respect to point C. Then the further change of the current proceeds along path ODA. 7. The shape of the I-U-curve depends on the number of the current channels in the "lm (equal to the number of the emission centers } see Section 2.4) and on the spread in their properties. If these properties are su$ciently uniform or if only one current channel is present, the transition from the high- to low-resistance state becomes extremely sharp indicative of a switching process [72].

Fig. 3.12. Conduction current}voltage curves for an Au island "lm covered with a naphthalene layer. ODA: lowresistance state. BCO: high-resistance state. OC: region of "eld memory.

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8. If in the low-resistance state the temperature of the "lm is lowered to (100 K, the negative resistance segment is observed only once and further changes of the current proceed along path OCB (Fig. 3.12). The VCNR behaviour is restored as temperature is raised above 100 K. Therefore such "lms possess a `temperature memorya. 9. The shape of the I-U-curve and the height of the maximum are sensitive to the pressure of residual gases [8]. If an IMF kept in the low-resistance state is coated with a thin dielectric layer, the VCNR segment OAB can be passed only once. The dielectric layer irreversibly prevents the restoration of the low-resistance state and the return trace proceeds along BO (Fig. 3.12). These results demonstrate that the VCNR behaviour is determined by surface electronic processes and not by processes in the substrate as was supposed in early studies [57,98]. 10. The position of the maximum in the current}voltage curve shifts to lower voltages when a pulsed instead of a constant voltage is applied. 11. In the VCNR region, one observes electron and light emission from the "lm. Here we shall exemplify the above regularities with data obtained recently for gold IMFs coated with stearone overlayers [69]. Stearone [(C H ) CO] was evaporated onto island "lms with    previously formed current channels and emission centers. The evaporation was performed while a voltage of +10}15 V was applied to the "lm. Stearone molecules adsorbed under such conditions induce an enhancement of the conduction current, and, what is more important, a VCNR section appears in the conduction current}voltage curves at voltages 4}6 V (Fig. 3.13a, curve I ). Simultaneously the characteristics of electron and light emission are also changed  (Fig. 3.13, curves I and I ). Quite analogous e!ects were observed with naphthalene (C H )  *   overlayers evaporated on the island "lms [99]. As noted above, organic admolecules increase the conduction current and bring about the VCNR behaviour only upon holding the "lm under a voltage of 10}15 V either in the course of evaporation of the organics or after it. (By contrast, BaO enhances the conduction current without such procedure). It has been suggested that organic species are polarized in the high electric "eld existing near the islands (estimated at 10}10 V/cm) and pulled into the emission centers where

Fig. 3.13. Current}voltage curves of conduction current I (1), electron emission current I (2) and light emission   intensity I (3) for an Au island "lm covered with a stearon layer. Curve 4 shows a conduction current}voltage curve * I versus ; for a chain island "lm covered with an organic adsorbate. Units at the ordinate axis are di!erent for all  curves.

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they self-assemble into thin quasi-polymeric "laments spanning the gaps between the islands [45]. It should be recalled that the formation of molecular waveguides for electrons was observed a few decades ago in experiments with organic molecules adsorbed on metal tips in the "eld emission microscope [100,101]. The average diameter of the polymeric "laments, oriented along the "eld lines, was estimated at several nm [102], i.e. the same as the size of nano-islands. If similar organic bridges are formed in IMFs, the current density in them should be at least &10 A/cm. Probably, such a high current density could be passed without destroying the "laments because of the very e$cient withdrawal of the Joule heat to the substrate. However, at higher current densities the conducting "laments can be melted or even evaporated which may cause a decrease in the conduction with increasing voltage. When the voltage is reduced, the temperature decreases and organic conducting bridges can be restored (replenished) by migration of the adsorbate from adjacent areas to the emission centers. This `desorption/di!usiona model might, in principle, explain qualitatively both the occurrence of the N-shaped I !; curve and its repeated reproduc tion ('10 times) while cycling the voltage [39]. However, the data on the light emission from island "lms covered with organics (typical for electroluminescence in semiconductor structures } see Section 4) impel consideration of an alternative model. We suppose that organic admolecules arrange themselves into semiconducting bridges (which may be poorly ordered) between the islands. As a consequence, the VCNR region in the conduction current}voltage characteristics can arise through one of the mechanisms suggested for switching phenomena in semiconductors: thermal (structural), electronic or combinations of these. This interesting topic goes beyond the scope of our paper and we refer the reader to a recent review focused on switching in semiconductor thin "lms [103]. It should be noted that the slope of the VCNR section is an integral characteristic of the whole "lm which contains many current channels with distinct properties. Therefore, the peculiarities seen in the VCNR region (see small steps and terraces in curves shown in Fig. 4.6, Section 4) may originate from contributions of di!erent channels. In the cases when the current channels have rather close parameters, e.g. in the chain island "lms, the VCNR section appears very steep so that one actually observes a switching regime [39,72]. A few works have been devoted to electron emission from the island "lms showing VCNR behaviour [72,79,96,104,105]. For example, Blessing and Pagnia [104] utilized a photoemission microscope with a spatial resolution of 100 nm and a Moelenstedt energy analyzer with an energy resolution of 0.5 eV (at acceleration voltage 20 kV) to study spatial and energy distribution of electrons emitted from Au IMFs. According to their interpretation, electron emission in the VCNR region is provided by hot electrons (perhaps with some contribution from microplasmas), while at higher voltages they argued for the "eld emission mechanism (see Section 3.3 for a discussion of this model). In later works of this group [105}108], a model of carbon islands has been suggested. They can arise in an oil-pumped vacuum due to cracking of residual hydrocarbons and subsequent graphitization. Owing to injection of fast electrons through tunneling junctions, the electron temperature in the carbon islands can be elevated up to &4000 K (as estimated from photon emission spectra [106]) which results in thermionic emission of hot electrons. (See also work [79] on the emission properties of carbon island "lms.) The carbon islands can be destroyed at high temperatures by residual oxygen and again be built up at lower temperatures. However, in the context of the experimental results listed above and obtained for various `coateda island "lms showing VCNR, the model of carbon bridges (islands or "laments) seems to be appropriate only in

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some particular cases. The general conclusion is that it cannot be accepted as a universal explanation of the VCNR behaviour of the island "lms. The results presented in this section show that deposition of various overlayers on nanodispersed thin "lms allows a wide-range modi"cation of their properties. In particular, using this approach one can obtain planar composite structures whose characteristics combine the peculiar properties of nanoparticles and of the substance "lling the gaps between them. 3.6. Electron emission from silicon island xlms In this section, we present some data which demonstrate that in semiconductor island "lms phenomena can occur analogous to those observed in the case of metal "lms. If the model of hot electrons which is discussed in this review in application to metals is true, electron and light emission from small semiconductor particles caused by pumping energy into them should also be expected. Indeed, nonequilibrium heating of electrons in semiconductors (contrary to bulk metals) is a common phenomenon [109,110]. The experiments have been performed with Si. To obtain a Si island "lm, we "rst deposited epitaxially a continuous single-crystal n-Si "lm (100 nm thick, 10 ) cm) on a sapphire substrate. The continuous "lm had in the middle a section 50 lm wide where its thickness was 50 nm. The contacts were evaporated on the thicker parts of the "lm. The "lm was "rst outgassed at &10\ Torr and then a su$ciently strong current was passed through it which led to the formation of a Si island "lm in the thinner part of the sample. For comparison, some experiments were made with a porous Si sample prepared by electrochemical etching. The conduction current I!; curves for clean silicon island "lms are quite similar to those recorded for metal island "lms (Fig. 3.14), but neither electron nor light emission is observed until the work function of Si islands is reduced by some appropriate overlayer (e.g. BaO). Thus, the hot electrons which can be generated in Si islands cannot pass into vacuum if the Si surface is clean. This result resembles the experiments carried out in an attempt to obtain electron emission from Si p}n-junctions cleaved in high vacuum [111]. In that case, no emission was observed, too, which was attributed to the unfavorable relation between the impact ionization energy (2.25 eV) and the electron a$nity (3.6 eV) in Si. Due to the low impact ionization energy, the electrons cannot be

Fig. 3.14. (a) Structure with the island Si "lm; (b) structure for observation of electroluminescence of porous silicon; (c) I!; curves for conduction current of clean silicon island "lm (1), BaO/Si island "lm (2) and for emission current from BaO/Si island "lm (3). (A) and (B) are light emission spectra for structures a and b.

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heated enough to overcome the surface barrier. However, the emission emerged upon reducing the Si work function by a few eVs. In the case of Si island "lms, the evaporation of a BaO overlayer lowering the work function by 2.2}2.5 eV resulted in a substantial enhancement of the conduction current and in the appearance of electron and light emission (Fig. 3.14; curve 3). We have compared these results with data on electroluminescence and electron emission from a porous Si sample prepared by electrochemical etching of p-Si (0.01 ) cm) plates. The size of the pores and the thickness of the walls between them was 5}10 nm. A sandwich test structure was used (Fig. 3.14b). The electroluminescence and electron emission were observed for as-prepared samples (no activating coating was needed). The spectrum of the electroluminescence of the porous Si is rather similar to that recorded for the Si island "lm covered with BaO (Fig. 3.14; curves A, B). It was found, however, that the electroluminescence, photoluminescence and the electron emission disappear after outgassing the porous Si sample. They could be restored after H O adsorption  [112}114], which demonstrates that surface conditions strongly a!ect the behavior of porous Si. It should be recalled that there is a number of reports on electron emission from porous Si (see e.g. [115,116]). The photon emission from porous Si has also been recorded in the STM [117]. 3.7. Electron emission from IMFs under infra-red laser excitation As noted in Section 3.2, the physical interpretation of emission experiments with the island "lms excited by passing current cannot be accepted as conclusive. It was therefore desirable to obtain additional data on this phenomenon, in particular, to examine the electron emission from the "lms subjected to an alternative (currentless) excitation. Such an excitation has been e!ected by infra-red laser irradiation [2}4]. In Section 7, we give detailed physical arguments that using a su$ciently powerful IR laser one can very e$ciently pump energy into small metal particles having a special shape. A pulsed CO TEA-laser (j"10.6 lm, q"0.2}1.0 ls, f"1}30 Hz) was used to irradiate Au, Cu  and graphite island "lms on Si [4,118]. The laser beam intensity was measured with a pyroelectric detector. Neutral "lters were utilized to attenuate the beam irradiating the "lm. In the experiment, the same area of the "lm was exposed to the beam whose intensity was progressively increased. The structure of the "lm was investigated at di!erent stages of the exposure in an electron microscope. The electron and photon emission set in as the power density P reaches a value of about 10 W/cm. It is known that in the case of bulk metals similar e!ects appear at P'10 W/cm and are ascribed to various mechanisms (thermionic and "eld emission, many-photon photoe!ect) [119,120]. Within the range of 10}10 W/cm, the emission current from the island "lms is stable and the shape of the emission pulses reproduces quite closely that of the laser pulses. The delay time of the emission is estimated at (2;10\ s. The work functions of the materials under study (+4.5}5.0 eV) are by a factor of &40 higher than the quantum energy hl"0.12 eV at j"10.6 lm. Thus, the photoemission, both one-photon and many-photon, can be excluded as a possible emission mechanism. The "eld emission seems also improbable due to low electric "elds (&10 V/cm) in the laser beams used. Taking into account the above work functions and the saturation vapour pressures of the materials, one is led to the conclusion that the thermionic mechanism, too, cannot ensure the observed stable current densities of &10\ A/cm, at least for Au and Cu "lms. The thermionic emission does come into e!ect, but at much higher power

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Fig. 3.15. Logarithm of the electron emission current from a carbon island "lm on Si excited by a CO laser as a function  of P\.

densities (&5;10 W/cm) at which the emission pulses become signi"cantly lengthened and the islands are quickly destroyed thereby causing the emission to decay. Fig. 3.15 shows the emission current as a function of the power density in the laser beam [121]. When plotted as log I versus P\, this dependence is seen to be linear in agreement with the  theory [122] (see Section 3.3). We recall that such a behaviour of the emission current follows from the calculations which predict that, at large power densities fed into an island and independent of the mechanism of excitation, the electron temperature should be proportional to P. In other words, one obtains a Richardson dependence of emission current on the hot electron temperature. (The lattice temperature is much lower } see Section 5.) Energy distributions of electrons emitted from IMFs under IR laser irradiation were investigated in work [123]. The gold IMFs were prepared on a Si wafer coated with a thin layer of silicon nitride. The sample was placed into a scanning electron microscope and the same energy analyzer was employed which had been used earlier to measure the energy spectra of electrons emitted from IMFs under current excitation (Section 3.4). The half-width of the energy distribution of electrons in the case of laser excitation was found to be +0.6 eV. This value was considerably larger than the estimated instrumental width and therefore corresponded, roughly speaking, to an electron temperature of &10 K. However, the distribution was non-Maxwellian. Discussing the data in terms of the model of hot electrons, the authors [123] supposed that the non-Maxwellian character of the distribution may be caused by the fact that the measured emission current contains contributions from many islands which have di!erent electron temperatures. Thus the whole set of data suggests that the electron emission observed at moderate irradiation intensities is most probably due to nonequilibrium heating of electrons in metal islands. This model is also corroborated by the following "ndings: (1) the energy spectra of the emitted electrons reveal the existence of electrons with energies up to +3 eV (at P"5;10 W/cm); (2) the spectra of the concomitant light emission, if they were roughly ascribed to the equilibrium radiation from the islands, would correspond to ¹+1000 K which is too low to provide any appreciable thermionic (equilibrium) emission. Strictly speaking, however, the light emission spectra of the island "lms do not at all obey Planck's law and show instead two or three more or less pronounced maxima [124,125] (see also Section 4). A detailed theoretical analysis has revealed that the light emission

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may be caused by (i) bremsstrahlung, (ii) the inverse of surface photoelectron emission, (iii) inelastic tunneling of electrons and (iv) radiative decay of surface plasmons [126]. From the practical viewpoint, the electron emission from the island "lms is quite stable at moderate power densities (up to 10 W/cm for Au and 10 W/cm for graphite "lms). Thus such "lms can be utilized as photocathodes to visualize the intensity distribution in IR laser beams (see Section 8).

4. Light emission from island metal 5lms 4.1. Light emission from clean IMFs Here we shall "rst consider the light emission from IMFs which occurs under conduction current excitation. We recall that both electrons and light are emitted from the same centers which has been demonstrated in experiments comparing the relative position of the luminous spots on the "lm with the pattern produced by emitted electrons on a screen (see Section 3). This statement has been convincingly corroborated by measurements which have detected the existence of almost full correlation between the low-frequency #uctuations of the electron emission current and light intensity [127] (Fig. 4.1). The light emission as well as the electron emission starts at the voltages where a deviation from Ohm's law becomes apparent (Fig. 4.2). The light emission appears without perceptible time lag on applying voltage to an electroformed "lm. The emitting centers are readily visible and may be coloured di!erently within the same "lm. Fig. 4.3 shows typical spectra of light recorded from two simultaneously emitting centers in an Au island "lm (curves 1 and 2) [127]. It is seen that the spectra are rather broad and sometimes consist of distinct bands. A similar shape of the spectra has been found for silver island "lms (Fig. 4.4). The existence of several intensity maxima may suggest that a few light emission mechanisms are operating in parallel. Theoretically, one can predict that the light emitted by IMFs contains contributions from intraband quantum transitions, bremsstrahlung, inverse surface photoe!ect and plasmon-mediated radiation (the plasmons being generated both in inelastic tunneling and by hot electrons). We shall address this question in more detail in Sections 5 and 7. For a chain island "lm (Fig. 4.3, curve 3), the light emission spectrum is considerably narrower. Now, let us compare the above data with light emission spectra measured for other related systems. In particular, one can readily see an obvious similarity between a pair of adjacent metal islands on a dielectric substrate and a metal}insulator}metal (MIM) structure as well as a tunnel gap in STM. The di!erence is that the island structure is more open than the MIM structure, and two islands are separated from each other both by a gap on the insulator surface and by a vacuum gap. Therefore, the comparison of the characteristics of the objects listed above may be meaningful. The light emission from MIM tunnel junctions has been investigated in many works (see e.g. [67,128}135]). The spectra obtained have been attributed mainly to the radiative decay of surface plasmon-polariton modes excited by inelastic electron tunneling. There are also experimental arguments in favor of the radiative decay of surface plasmons that are generated by hot electrons injected into one of the metal electrodes without energy loss in the tunneling gap [132]. In Fig. 4.3 we compare spectra of the light emission from an Au IMF (curves 1 and 2) with the spectra recorded by Berndt et al. from a tunnel gap between an Au (110) surface and W tip in

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Fig. 4.1. Time #uctuations of the electron emission current (I ) and of integral light intensity (I ) recorded at various   values of power fed into a gold island "lm: (a) 0.02 W; (b) 0.04 W; (c) 0.2 W [6,127]. Fig. 4.2. The current}voltage curves of the conduction current (I ), electron emission current (I ) and light intensity (I )    on the voltage applied to the "lm.

Fig. 4.3. Light emission spectra: (1) and (2) for two emitting centers in the same Au "lm; (3) for a chain Au island "lm; (4) and (5) for an Au/W tunnel gap in STM [136,137]. Fig. 4.4. Light emission spectra: (1) for an individual emitting center in a silver IMF excited by conduction current; (2) for the same "lm excited by electron bombardment (50}200 eV); (3) for a Ag-tip/Ag(111) tunnel gap [138].

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a scanning tunneling microscope (curves 4 and 5) [136,137]. The STM spectra were taken at various tip bias voltages, and one can see some resemblance between them and the IMF spectra, although the latter are more complex in shape. A spectrum for an Ag IMF having a single light emitting center is juxtaposed in Fig. 4.4 with a spectrum of an Ag-Ag(111) tunnel gap in a STM [138]. This comparison is of special interest, because the STM data relate to the case when both the tip and the sample were made of the same material (Ag). A rather close similarity in some features of the spectra is evident. The photon emission spectra recorded in a STM are interpreted in terms of radiative decay of tip-induced plasmon modes excited in inelastic electron tunneling [136}147]. There are good reasons to believe that similar localized plasmons can form and decay in the tunnel gap between the adjacent nano-islands. However, the spectrum for the IMF has a few peaks (with wave lengths corresponding to photon energies 3.5, 2.5 and 1.9 eV). In particular, the ultraviolet peak (j+350 nm, hl"3.5 eV) coincides with the maximum in the photon radiation observed under electron bombardment of the Ag island "lm from an external electron source. Thus, we hold to the view that several maxima seen in the photon spectra of IMFs may indicate the operation of diverse physical mechanisms of light emission from these "lms. This possibility can originate from the generation of hot electrons which are able to induce electromagnetic radiation due to various e!ects (see Section 5). As mentioned above, the spectra of the emitting centers in the chain island "lms are usually considerably narrower and exhibit only one maximum [69] (Fig. 4.3, curve 3). This behaviour seems to re#ect the fact that conditions of electron and light emission in the chain "lms are more uniform than in the `irregulara "lms. The light emission spectra were also investigated for IMFs prepared of Ag, Cu, Mo, Bi, Pd and some other metals. The "lms were excited both by conduction current [6,148] and by low-energy electron bombardment (`cathodoluminescence regimea) (Fig. 4.4) [6,149}151]. Recently, such measurements for Ag IMFs were carried out in a wider frequency range using a more sensitive CCD detector system [152}154]. The Ag island "lms were prepared by vacuum evaporation and by a gas aggregation method [155]. It was found that the light emission spectra are independent of the manner of preparation of IMFs and the total intensity of radiation is growing with increasing voltage applied to the "lm ;. The most pronounced intensity maxima are located in the red region (j+600}730 nm). It is important to note that the light emission in this region (hl+1.7!2.1 eV) occurs even in the case when the voltage is as low as 1 eV. This result is readily explicable in the framework of the model of hot electrons (see Sections 3, 5 and 6). Starting from ;"12 eV, new features (in addition to the intensity maxima in the range 600}730 nm) appear in the ranges 300}460 and 900}950 nm. In particular, the peak emerging at j+320 nm (hl+3.9 eV) corresponds to the energy of plasma oscillations. It is supposed [152,153] that the number of the hot electrons with energies su$cient to excite the plasmons, which generate photons in their radiative decay, may be too small at ;(12 eV. The absolute and relative intensities of the high energy radiation maxima increase with increasing ;. The features observed in the near infra-red region are more intense than that recorded at j+320 nm and are ascribed to the processes of inelastic tunneling and re#ection of hot electrons. The dependence of the light emission spectra on the size of the Pd particles bombarded by electrons at E"10}10 eV was recently investigated in [150]. The average size of Pd particles was varied in the range 0.5}7 nm. The spectra were recorded in the range hl"1.2}6.2 eV, so the possible light emission stemming from the radiative decay of plasmons (hl"7.3 eV for Pd) could not be detected in these experiments.

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4.2. Ewect of overlayers on light emission from IMFs The BaO deposition within the limits of a monolayer a!ects (increases) mainly the intensity of the emitted light while its spectrum remains almost unchanged. However, as noted in Section 8.1, the long-term functioning of IMF cathodes can be achieved by their fabrication on substrates previously coated by a thick BaO layer. In such a case, especially when large emission currents are extracted, one observes signi"cant changes in the spectrum of the emitted light due to electroluminescence of BaO itself. In experiments with stearone overlayers (see Section 3.5), the enhancement of the conduction and electron emission currents and formation of N-shaped conduction current}voltage curves were accompanied by substantial changes in the characteristics of light emission (Fig. 4.5, curve I in the * inset). There is a considerable di!erence between the light spectra corresponding to the rising and falling (VCNR) segments of the conduction current}voltage curves. In the former case the spectrum is nearly structureless, but shows a steep intensity growth in the red and infrared region (j'600 nm). In the latter case there are two broad maxima at j+510 and 610 nm, and the intensity strongly increases at j'700 nm [156]. Both the spectra are distinct from the spectrum for clean gold island "lms (Fig. 4.3) which shows a number of pronounced maxima in the visible region. As noted in Section 3.5, there are arguments suggesting that organic molecules evaporated in vacuum onto IMFs can self-assemble into thin quasi-polymeric bridges spanning the gaps between the islands. Fig. 4.6 displays a close correlation between variations in the conduction

Fig. 4.5. Inset: conduction current I , emission current I and light intensity I versus AC voltage applied to an Au island   * "lm covered with stearone. Scales for I , I and I are di!erent. (1) and (2) are light emission spectra in rising and falling   * section of the I !; curve, respectively.  Fig. 4.6. Hysteresis loops for the conduction current (1) and light intensity at j"615 nm (2) recorded for an Au "lm covered with stearone. Voltage frequency: 500 Hz. Sweep time: 2 min.

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current and light intensity (j"615 nm). These results recorded in the alternating voltage regime show that the hysteresis loop for the light intensity mirrors rather closely the modulus of the derivative of the conduction current loop. Such a behaviour is quite typical for electroluminescence of semiconductors (see e.g. Refs. [157}159]). Thus we suppose that what is observed in our case may be the electroluminescence of organic bridges which are excited by hot electrons injected into them from adjacent metal islands. The results presented in this section demonstrate the close correlation between the intensities of the electron and light emission from IMFs and thus evidence that a uni"ed explanation should exist for both phenomena. Taking into account the nonequilibrium heating of the electron gas it is possible to explain the major origins of the electron and light emission from the island "lms.

5. Hot electrons in metal nanoparticles 5.1. Introductory remarks Sections 5}7 are devoted to theoretical treatment of electronic kinetics and optical absorption in small metal particles and their ensembles. The peculiarities of these characteristics are caused both by the speci"c properties of individual particles and by interaction between the particles. The condition of the onset of a size e!ect is the comparability of the particle size to a physical value with the dimension of length (the de Broglie wavelength, the electron mean free path, the depth of the skin layer, the length of an electromagnetic wave, etc.). In what follows, we will show in detail that in the case when the particle size becomes smaller than the electron mean free path, the intensity of the electron}lattice energy exchange is strongly suppressed in comparison with its bulk value. Furthermore, in this situation the absorption of light by free electrons is determined by their scattering at the surface rather than in the volume. The optical properties of small particles as well as of their ensembles are also di!erent depending on whether the particle size is smaller or larger than the length of the electromagnetic wave and the depth of the skin layer. In the island metal "lms, additional features appear due to interaction between the metal particles. In the "rst place, the system of metal particles on a dielectric substrate can be coupled by electron tunneling and therefore can be conducting. However, the temperature and "eld dependence of the conductivity is di!erent from that in continuous metal "lms and in bulk metals. Secondly, the dipoles which are induced in the particles by external alternating electromagnetic "elds interact with each other giving rise to local "elds. The local "elds determine a diversity of phenomena in IMFs such as optical absorption and re#ection, second harmonic generation, opto-acoustical e!ect etc. Thus there is a very broad scope of e!ects in which peculiar properties of small metal particles and their ensembles can come into play. We will concentrate on the phenomena which occur in the process of feeding power into the electronic subsystem of IMFs. The power can be pumped by passing a current through the "lm and by laser or electron irradiation. In any case we are dealing with nonequilibrium electron}phonon systems. The most spectacular and unexpected peculiarity of IMFs is the possibility of generation of hot electrons in them under stationary (or quasistationary) conditions: the electron temperature can exceed by one to two orders of magnitude the lattice temperature of the islands which are sitting on a substrate with a good thermal conductivity. This e!ect appears unexpected, since the

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hot electrons in continuous metal "lms and in bulk metals can be obtained just for very short times, when e.g. these objects are exposed to ultrashort laser pulses or to ultrashort electron pulses generated in explosion emission. Under such conditions, electrons are heated up during the time of &10\ s. The lattice receives energy from the electron gas in &10\ s, and the metal can be melted and evaporated after that if the pulse is powerful enough. By contrast, the hot electrons in IMFs can be generated under stationary conditions and without destruction of the "lm. In this section we will try to explain why this is possible. The generation of hot electrons in IMFs allows one to understand the whole body of phenomena which occur when the "lms are excited by the conduction current as well as laser or electron irradiation. In particular, the model of hot electrons seems now to be the only model which accounts for the features of electron and photon emission from IMFs exposed to IR laser beams. It should be recalled that the electron emission is induced by photons whose energy is lower by a factor of 30}40 than the work function. This occurs at such laser beam intensities that the many-photon processes are unessential and the "lm remains stable for 10}10 h. No emission is observed under such irradiation intensities in bulk metals and continuous metal "lms. As will be shown below, the heating of electrons in IMFs depends on the power pumped into the islands, on the special features of electron}lattice energy exchange in small particles and, "nally, on the conditions of the heat transfer from the particles to the substrate. A strict analytical treatment of these factors will be given in Sections 6 and 7. In this section, we will begin with a graphic model which explains why the conditions in the small particles are favourable for nonequilibrium electron heating and then will discuss in more detail the consequences of the heating of the electron subsystem. 5.2. Heating up of electrons Suppose power is fed into the electron subsystem of an IMF. This can be achieved by passing a current through the islands, which are coupled by transparent potential barriers, as well as by irradiation of the "lm by a laser or electron beam. Owing to existence of the electron}phonon interaction, the power absorbed by the electron subsystem is transferred to the phonon system of the islands and then is carried o! to the substrate. It is evident from general considerations that the electron temperature should be higher than the lattice temperature. The point is how large this temperature di!erence can be and whether or not the IMF will be destroyed (melted and evaporated) through the power injection. Let us proceed from the system of equations that determine the electron (¹ ) and phonon (¹) temperature in a metal island:  R (C ¹ )"div(K ¹ )!a(¹ !¹)#Q , (13)    Rt   R (C¹)"div(K¹)#a(¹ !¹) .  Rt

(14)

Here C and C are electron and phonon speci"c heats, K and K are electron and phonon heat   conductivities, and Q is the speci"c power absorbed in the island. The coe$cient a characterizes the intensity of electron}phonon interaction and correspondingly the product a(¹ !¹) determines  the power transferred from electrons to phonons. To avoid misunderstanding, it should be stressed

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that the power is proportional to (¹ !¹) only in the case when this di!erence is not too large. In  principle, the values Q, ¹ and ¹ can di!er in di!erent islands. The system of equations (13) and  (14) is to be complemented by a boundary condition which describes the process of heat transfer from the metal island to the substrate. But before discussing the speci"c behaviour of electrons in IMFs, it is appropriate to recall some basic data about the heating up of electrons in bulk metals [160]. Since the speci"c heat of the electron subsystem (C ) is much smaller than that of phonons (C), the characteristic relaxation time  for the electron temperature q &C /a is much shorter than the corresponding time for phonons.   To an order of magnitude, one usually has q &10\ s and q &10\ s. For this reason   the phonon temperature cannot change substantially in a time t(q , and during such a time the  electrons behave as if they were thermally isolated. The result is that, in feeding power into the metal, the electron temperature grows very steeply until the energy #ux from electrons to phonons becomes equal to the absorbed power, i.e. a(¹ !¹)+Q . 

(15)

During a time of t&q the phonon temperature in the bulk metal lines up with the electron  temperature. This is the reason why, in bulk metals, the hot electrons can be observed only for times t(q , e.g. by using short, but powerful laser pulses for irradiation [161]. The "rst basic  di!erence between a bulk metal and small metal particles placed on a well heat-conducting substrate is that the lattice of the particles remains cold even when the electrons in them are strongly heated. The electron and phonon temperatures are not lined up even under steady-state conditions. Of course, the metal islands can be thermally destroyed, but only at power #uxes that are orders of magnitude higher than those sustained by bulk metals. Let us dwell on this point in more detail. We shall consider the times t'q when the electron  gas has already been heated up and condition (15) is obeyed. This means that all the power received by the electrons is being transferred to the lattice, so the variation of the electron temperature with time is connected only with the heating of the lattice. Then Eq. (14) with consideration of condition (15) transforms to R (C¹)"div(K¹)#Q . Rt

(16)

Instead of solving this equation with a boundary condition corresponding to an island on the substrate surface, we shall further simplify the problem and consider a small metal sphere inside a dielectric. Furthermore, let us assume for a while that C and K are the same for the metal sphere and its dielectric surroundings. The value Q is non-zero only inside the sphere. Such a model was exploited earlier in a study of the optical durability of laser glasses [162]. The solution of (16) under such conditions reads

 





Q(r, t) (r!r) R i exp ! . dt dr ¹(r, t)" [4i(t!t)] 4i(t!t) pK 

(17)

Here and below the phonon temperature is reckoned from its equilibrium value (at Q"0), < is the volume of the metal sphere and i"K/C. It is assumed that a power #ux Q constant in the whole

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volume of the sphere is turned on at a moment t"0, i.e.



Q "const, r(a, t50 ,  0, r'a ,

Q(r, t)"

(18)

where a is the radius of the sphere. Under such an assumption the integration of (17) can be readily performed to give the result



¹(r, t)"

 



 



Q 4a r#a r!a  #g ; !;  g  g 4K 3r

r'a ,

,

Q r#a r!a 2  2a! r#g ; !;  g  g 4K 3

(19) , r(a .

The following notations are used here:










2 1 g e\V 1 2 g ; (x)" x! x! # x# ! x U(x) ,  3 2 r (p 2 3 r

(20)

g"(4it and U(x) is the probability integral. It can be seen that for g


4 a , r'a , Q ¹(r, t)+  3 r (21) 4K 2a!r, r(a .  The steady-state regime is reached in a time t'a/4i. This stationary solution can be obtained directly from Eq. (16) without using a rather cumbersome equation (19). In particular, by solution of Eq. (16) inside and outside the sphere and by joining the temperatures and heat #uxes at the boundary (r"a), one can easily "nd the temperature distribution in the situation when the parameter K is di!erent for the sphere and its surroundings. For example, in the case when



K , r(a ,  (22) K , r'a ,  one obtains instead of solution (21) the following expression for the temperature inside the sphere: K"





a Q a!r # , r(a . (23) ¹(r)"  2K K 3   By the way, this formula shows that, if the thermal conductivity of the sphere is substantially higher than that of the matrix, the temperature within the sphere is nearly constant. Such a situation is always valid for the metal particles whose size is smaller than the mean free path of electrons in the bulk metal. In this case the temperature everywhere within the sphere is constant and equal to Q a ¹K  . 3 K 

(24)

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It is seen that the smaller the particle and the higher the thermal conductivity of the matrix, the lower is the temperature of the particle and hence the higher is its endurance against the injection of power. Expression (24) has been obtained under the condition that the sphere absorbs a power Q <"paQ which is conveyed to the matrix through the sphere surface with the area S"4pa.    For a small metal particle of an arbitrary shape which has a volume < and a contact area with a substrate S , one easily arrives at an approximate formula  Q < (25) ¹+  R . 4S K   Here R is a characteristic dimension of the particle. It should be noted that the thermal endurance of small metal particles on a heat-conducting substrate can be substantially higher when the power Q is fed in a pulsed regime. Suppose the pulse duration is q. During this time the heat from the particle spreads within the substrate over a distance l &(4Kq/C. For typical heat-conducting insulators and q&10\ s, one will have O l &10\}10\ cm. Thus, if a metal particle is about 10\ cm in size, the energy Q
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Fig. 5.1. A model of communicating vessels illustrating the electron}lattice energy transfer and nonequilibrium heating of the electron gas in a small particle on a substrate. See text for explanations.

obviously must be low enough to avoid the melting and evaporation of the island. As can be seen from this model and Eq. (15), the gap between the temperatures of electrons and phonons is wider the larger is the energy #ux Q and the smaller is the electron}phonon coupling coe$cient a: (¹ !¹)+Q/a .  The high-energy-transmitting endurance of IMFs has already been discussed in Section 5.2. Let us now point out further two unique features of these "lms. (a) The intensity of the electron}lattice energy exchange, denoted above as a, can be by a few orders of magnitude lower in a small metal island than in the bulk metal. (b) Much higher energy #uxes can be injected into small metal islands without their destruction than into bulk metals. These peculiarities will be considered at length in Sections 6 and 7. Here we shall give only some introductory explanations. Concerning point (a), it should be noted that the main energy losses of hot electrons in bulk metals are due to generation of acoustic phonons by Cherenkov's mechanism [163,164]. In a small metal particle whose dimensions are smaller than the mean free path of electrons in the volume, the character of electron scattering drastically changes. Electrons now execute mainly a quasi-periodic motion from one wall of the particle to another, and the Cherenkov mechanism of energy dissipation becomes ine$cient under such conditions. The electron energy losses are caused in this case by the surface scattering and the coe$cient a can be lower by orders of magnitude than in the bulk (see Section 6). As to point (b), the injection of high-energy #uxes into small metal islands can be implemented, e.g. through laser irradiation or through conduction current excitation. In the former case, large Q values are attained owing to

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peculiarities of light absorption by small metal particles in the range of infra-red CO laser  generation. The underlying physics is detailed in Section 7. In the conduction current excitation, the high-energy #uxes are provided by formation of the conduction channels with a high current density (see Section 3). The islands with hot electrons arise just within these channels. We have dealt with this issue in Section 2.4. To summarize, we list again the main physical factors that can ensure the appearance of hot electrons in IMFs in stationary or quasi-stationary conditions. These are the high thermal endurance of small metal particles on the heat-conducting substrate, the suppression of the electron}lattice energy exchange in such particles, and the possibility of pumping high energy #uxes into them. 5.4. Phenomena caused by hot electrons in IMFs Here we shall discuss some important phenomena attributed to the generation of hot electrons in IMFs: (a) deviations from Ohm's law, i.e. the occurrence of non-linear conduction current}voltage characteristics; (b) electron emission from IMFs; (c) photon emission from IMFs. The explanation of e!ect (a) is fairly evident. As noted above, the conduction current through an IMF is e!ected by electrons tunneling from one island to another. The tunneling occurs mainly at energies close to the Fermi level. At low electric "elds, the tunneling current between two adjacent islands is proportional to the o!set of their Fermi levels, so Ohm's law is obeyed. At higher "elds, the heating of electrons sets in and a group of electrons is generated with energies above the Fermi level. The contribution of the hot electrons depends on the electron temperature, which in its turn depends on the applied "eld. This results in a nonlinear current}voltage characteristics. Their shape was speci"ed in more detail in Section 3.2 where the experimental "ndings were discussed. As considered above, the electron emission from IMFs can be induced by the conduction current as well as by laser irradiation of the "lms. In both the cases the hot electrons are generated. A theoretical estimation given in Section 3 and experimental measurements show that the electron temperature in small metal islands can amount to &10 K while the lattice remains virtually cold. Therefore, what is observed is interpreted as a Richardson emission of the hot (nonequilibrium) electrons with the current I Je\PI 2 . 

(26)

Here u is the work function of the island and k is the Boltzmann constant. As indicated in Section 3.3, under current excitation of the "lms the electron emission arises when the conduction current}voltage characteristics start to deviate from the linearity. This seems to be evident, since both the e!ects have the same origin: generation of hot electrons. In the case of laser excitation, the emission current represents a kind of photoresponse of the "lm to the IR irradiation. However, there is no way to interpret this phenomenon as a one-photon or many-photon photoemission. The one-photon photoe!ect is impossible because a CO infra-red  laser generates quanta whose energy (&0.12 eV) is lower by a factor of 30}40 than the work

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function of the islands. On the other hand, the intensity of the laser beams used in the experiments was far too low to bring about a many-photon electron emission. Contrary to these mechanisms, we ascribe the e!ect to the absorption of the laser beam power by the whole ensemble of electrons of the island. The electrons are heated up in a time q &10\ s, so again the Richardson emission  of the hot electrons becomes possible. The inference about the central role of the hot electrons in the peculiar properties of IMFs appears all the more credible when one considers the whole body of experimental data on the nonlinear conduction current}voltage characteristics, and the electron and light emission. Let us now address the mechanisms of the light emission. 5.5. Mechanisms of light emission from IMFs Recall that the photons are emitted from the spots about 1 lm in size. The spots are visible to the naked eye which means that at least some of the emitted quanta have energies which are about 20 times as high as the energy of exciting quanta in the case of the infra-red laser irradiation. The possibility of generation of hot electrons in IMFs, the existence of numerous tunnel junctions in the "lms, the rough surface geometry and, in some cases, the presence of microscopic connecting bridges between the islands, all can be responsible for occurrence of a variety of light radiation mechnisms in IMFs. A list of possible one-electron and many-electron radiation mechanisms in IMFs can include bremsstrahlung of the hot electrons; radiation generated in the inelastic tunneling of electrons from one island to another; inverse photoe!ect; transitions between discrete energy levels in the connecting bridges; radiative decay of collective electron excitations. For small particles, the plasma frequency is substantially dependent on the shape and relative position of the particles, the existence of bridges between them and other factors [165,166]. This diversity of mechanisms is the reason why the spectrum of the light radiation from IMFs can extend from infra-red frequencies to the frequencies of volume plasmons. The necessity to satisfy simultaneously the conservation laws for energy and momentum makes impossible both radiation and absorption of photons by free-moving electrons. However, such a possibility arises when an electron interacts with phonons or a surface. If the island dimension is larger than the mean free path of electrons, they are scattered mainly on phonons. In the opposite case the surface scattering is dominant. The scattering can be both elastic and inelastic, but photons can be radiated or absorbed in the latter case only. If the electron subsystem is nonequilibrium (i.e. hot electrons are present), the generation of photons prevails. Below we shall consider the radiative transitions in inelastic surface scattering and inelastic electron tunneling between adjacent islands [167]. The time-dependent Schroedinger equation for an electron moving inside the particle in the "eld of an electromagnetic wave reads








e  1 Rt p( ! A #;(x) t . i " c 2m Rt

(27)

Here A is the vector potential of the electromagnetic "eld inside the particle. In the case when one calculates the energy absorbed by the particle, this "eld should be determined from the external electromagnetic "eld. This point will be addressed in Section 7. Here we shall focus mainly on the radiation mechanisms.

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A model surface potential barrier with a "nite width will be considered later, but "rst we suppose that a one-dimensional barrier has a step-like shape



;(x)"

0,

x40 ,

(28)

;, x'0 .

The time dependence of A(t) is assumed to be A(t)"A+e SR#e\ SR, ,

(29)

where u is the frequency and A a constant vector amplitude. Considering A as a weak perturbation, Eq. (27) can be linearized against A and its solution can be represented as follows: 2e AU!(r)e\  C! SR . (30)

>\ Here e is the electron energy, t(r, t) is the solution of Eq. (27) at A"0 and U! is a vector function to be determined. The equations necessary for "nding U!(r), U!(r), U!(r) can be V W X obtained by substitution of solution (30) into a linearized equation (27) and by equating the coe$cients standing at identical components of the vector A. In particular, for the function U!(r) V one obtains t(r, t)"t(r)e\ C R#i





2m R D# (e$ u!;(x)) U!(r)" t(r) . V

 Rx

(31)

Assuming that A in (29) is a constant vector we have neglected the spatial dispersion. In this approximation the terms in (30) proportional to U!(r) and U!(r) do not contribute to the W X inelastic current. For this reason we shall not write down equations for them. The problem of the coordinate dependence of local "elds within small metal particles will be analyzed in more detail in Section 7. It is clear from Eq. (30) that the sign `#a in the function U!(r) corresponds to absorption and V the sign `!a to the radiation induced by an external "eld. We are interested in a spontaneous rather than induced radiation. Let us show how the probability of the spontaneous radiation can be found from the probability of induced radiation. To this end we shall write down the function t(r) in its explicit form: t(r)"t(x)e IW W>IX X .

(32)

Here k are the components of the wave vector and the function t(x) for the step-like barrier (28) G has a standard form (33) t(x)"e IV V#R e\ IV V at x40 ,  Fe G V, e 5;, x'0 , V (34) t(x)" Fe\G V, e 4;, x'0 . V In (34), e is the component of energy corresponding to the motion of an electron normal to the V barrier:



e "(1/2m)( k ) . V V

(35)

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We also use the notation i "+(2m/ )";!e ", . (36)  V The unknown coe$cients R , F and F are determined from the smooth joining of functions (33)  and (34) at x"0. It can be readily proved that the solution of Eq. (31) can be represented as

R t(r)#u!(x)e IW W>IX X , U!(r)"$ V 2mu Rx

(37)

where the function u!(x) has the following form:



de\ OV, x40,

x50, e $ u5; , V ge\AV, x50, e $ u4; . V In (38) we have used the notations u!(x)" ge AV,

(38)

2m (e $ u) ,

 V



(39)



2m "e $ u!;" .

 V

(40)

q"

c"

Since we are considering the radiation, it is natural to assume that e ' u. The unknown V coe$cients d, g, g in Eq. (38) can be determined from the continuity condition for the function U!(r) and its derivative at x"0 [122]. In doing so we can "nd an explicit expression for the V function U!(r) and hence calculate the required probabilities of the inelastic transitions. Indeed, V taking into account (37) and (38) and substituting function (30) into the expression for the current density





Rt RtH e

tH !t , I " V i2m Rx Rx

(41)

one can calculate the elastic and inelastic (i.e. connected with u!(x)) component of the current density. Relating then the inelastic component to the incident #ux (I" k /m), we obtain the V V probability of the inelastic transitions






IL 2e q D!(e , u)" V " A "d" . V V I c

k V V After the determination of d and its substitution into (42), we arrive at the result



(e (e $ u), V V 2 2eA  (eV (eV # u)C(eV # u), V D!(e , u)" V mu c

(e (e ! u)C(e ), V V V (e (e $ u)C(e $ u), V V V






e (;, V e 4;, V e 5;, V e 5;, V

e $ u4; V e # u5; V e ! u4; V e $ u5; V

(42)

, , , .

(43)

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The function C(e ) has here the form V ; C(e )" . V ((e #(e !;) V V

(44)

In (43), the case e 4;, e $ u4; describes the usual bremsstrahlung for the `#a sign and V V the inverse e!ect for the `!a sign. The case e 4;, e # u'; corresponds to the surface V V photoe!ect and the case e 5;, e ! u(; to the inverse surface photoe!ect. Finally, the V V situation e 5;, e $ u5; represents a usual inelastic scattering at the potential barrier. V V Before addressing the problem of the spectral density of the radiation, let us brie#y turn to the inelastic tunneling between the islands. Instead of Eq. (16), one has in this case



x40 ,

0,

;(x)" ;, 04x(a ,  0, x'a . 

(45)

The solution procedure remains the same with the di!erence that, instead of Eqs. (33) and (34), one uses the functions corresponding to potential (45) and the functions u!(x) from (38) must be replaced by



x40 ,

de\ OV,

u!(x)" f eAV#f e\AV, 04x4a ,    ge OV, x5a . 

(46)

The details of the calculations can be found in Appendix A. In addition, the procedure of joining the functions and their derivatives should be carried out not only at x"0, but also at x"a . In this case the probability of the inelastic scattering with  absorption (the sign `#a) and radiation (the sign `!a) of photons is given by






2e q D!( e , u)" A "g" . V c V k V

(47)

Here




"g""4

;  k V +ci (ch ca !ch i a )#(i qsh ca !ck sh i a ),       V  

u G(i , k )G(c, q)  V

(48)

and sh x, ch x are respectively hyperbolic sine and cosine. We also have introduced the notation G(i k )"(i !k)shi a #4i k chi a .  V  V    V  

(49)

The above method of calculation, within a uni"ed approach, of all processes connected with the inelastic electron re#ection from the barrier and the inelastic tunneling was "rst used to this end in [122].

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The radiative transitions considered up to this point correspond to transitions induced by the external "eld. To obtain the probabilities of spontaneous transitions which we need, we are using the following arti"cial expedient [168]. Let us choose such a normalization of the vector potential of the electromagnetic "eld that the "eld quantization volume < will contain in the average  N photons characterized by the energy u and a given wave vector and polarization. Under such  conditions one obtains






2p N   . (50) < u  To calculate the probability of spontaneous transitions, it is necessary to substitute (50) into (42) (or (43)) and (47) and then to set N "1. In this way we shall obtain the probabilities of  spontaneous transitions into the "eld state with a given polarization, frequency u and wave vector q . To "nd the full probability of radiation in a unit frequency interval and a solid angle dX,  the probabilities obtained above should be multiplied by the density of the "nal state of the "eld equal to A"c

< < q dq do(u)"    dX"  u dX . (2pc) (2p )du

(51)

The total probability of spontaneous radiation in inelastic electron re#ection from the barrier or the inelastic tunneling is given by (52) = "[D\(e , u)]  ) do(u) .  V ,  The spectral density of radiation of all electrons into a solid angle dX from the surface area S equals



  

2 uS    dE(u, X)" dp v = (e , u) dp dp f (e)[1!f (e! u)] V V  V W X (2p ) V

CYS \ \  2mh  Su[exp( u/h )!1]\ ) de =(e , u)Z(e u) . (53) " V V V  (2p ) CV Y S In this expression, f (e) is the Fermi function with an e!ective electron temperature h "k ¹   and Z(e , u) is V 1#exp((k# u!e )/h ) V  , (54) Z(e , u)"ln V 1#exp((k!e )/h ) V  where k is the Fermi energy. Having the explicit expressions for the probability of inelastic electron re#ection from the barrier (43) and of the inelastic tunneling (47), it is easy to calculate, using formula (53), the spectral density of radiation in any speci"c case of barrier parameters, frequency range etc. Unfortunately, a simple analytical expression for the general case does not exist. It can be obtained only in various limiting cases. For instance, in the frequency interval h ( u(u, k ,  the total spectral density of radiation in all directions is





2e 2 E(u)"S

ue\ SF k(k# u)! ( u) . 3(2c ) 3

(55)

(56)

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In the case speci"ed by condition (55) the main contribution to the radiation stems from bremsstrahlung of hot electrons occurring in their collision with the surface barrier. It is instructive to compare the above quantum mechanical treatment of the bremsstrahlung with its classical description. The total (to all directions) power of bremsstrahlung from a charge moving with an acceleration v is known to be dE 2 e " v  . dt 3 c

(57)

Hence the total energy radiated during the time of the acceleration is



 



 dE   2 e  . (58) dt" du v e\ SR dt dt 3p c \  \ Since the change of the velocity of an electron in its collision with a step-like barrier occurs jumpwise, the value ut in the exponent of formula (58) can be neglected, which gives a simple expression for the spectral density of radiation: E"

2 e "*v" . E(u)" 3p c

(59)

Here *v is the change in the velocity of an electron in its re#ection from the barrier (*v"2v ). Thus V we arrive at the result 16 e 8 e v " e . E(u)" 3p c V 3p mc V

(60)

In the quantum mechanical treatment, expressions (52), (43) and (50) give for the contribution of an electron to the spectral density of the bremsstrahlung 2e (e (e ! u) dX V V .

u= "(cos h)  m pc

(61)

Here h is the angle between the vector of the electric "eld in the radiated wave and the normal to the surface. The integration of (61) over all angles under the condition e < u gives the expression V that is exactly coincident with (60). In closing this section let us address brie#y the radiation generated in the inelastic tunneling. In the case i a '1 , (62)   relations (47) and (52) predict that the probability of the tunneling accompanied by the radiation of a quantum u equals 4e

(e (e ! u)i e\? G cos h dX . ="   pmc; V V

(63)

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Hence, the spectral density of this radiation into a solid angle dX is



2eh u 3  cos hdX de (e (e ! u)eI\CV F ) i e\? G . dE(u, X)+S (64)  V V V pm c;

S It should be kept in mind that the integrand in (64) is not quite accurate at both the lower and the upper limit of integration. The inaccuracy at the lower limit is due to the necessity of taking into account Pauli's exclusion principle for the states with e (k. The inaccuracy at the upper limit is V caused by the fact that condition (62) is not obeyed at e P;. However, these inaccuracies are V partially compensated by other physical factors and for this reason cannot signi"cantly in#uence the value of integral (64). Indeed, the inaccuracy of the approximation at the lower integration limit is partially counterbalanced by the very low tunneling probability (because of the inequality a i <1). At the upper limit, there are very few electrons with high energies (for which one has   a i (1). Therefore, no substantial error will be made if one assumes u(k at (e ) +k and   V  determines (e ) from the condition (e ) +;! /2ma . This procedure gives V  V   3 de (e (e ! u)eI\CV F ) i e\? G  V V V

S





3\ K?

(65) de (e (e ! u)eI\CV F ) i e\? i .   V V V I Using the theorem of the mean we can take outside the integral sign the value of the function +

(e ! u at a point e "e . The remaining integral does not depend on the frequency. As follows V V  from (64) and (65), the spectral density of the radiation generated in the inelastic tunneling can decrease very slowly with growing frequency. This may explain why many photons emitted from IMFs have energies which exceed the energy of excitation photons (in the case of IR laser irradiation) by a factor of ten and more. The radiation mechanisms connected with the decay of plasmon excitations in IMFs were discussed in Section 4. 5.6. Summary of Section 5 In this section, we formulated and discussed on a qualitative level three factors which ensure the generation of hot electrons in IMFs: the strong attenuation of the electron}lattice energy exchange in small metal particles; the high power density throughput of the small particles sitting on a substrate with a good thermal conductivity; the favorable conditions which exist for pumping high power densities into small metal particles. We have also presented a simple model which illustrates, using a hydrodynamic analogy, the interplay of these factors in the nonequilibrium heating of electrons in IMFs. Then we substantiated in detail one of the three factors listed above, namely, the high throughput capacity of the small metal particles which enables them to pass high energy #uxes without destruction. We also considered the consequences of the nonequilibrium heating of electrons: the electron and photon emission from IMFs. In what follows we shall treat at length theoretically the remaining two factors: the electron}lattice energy exchange in small metal particles (Section 6) and the possibility of feeding high power densities into them using IR laser irradiation (Section 7).

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6. Electron}lattice energy exchange in small metal particles 6.1. Introductory remarks As noted in Sections 3 and 5, one of the reasons for the generation of hot electrons in IMFs is the sharp attenuation of the electron}lattice energy exchange in particles with dimensions smaller than the electron mean free path. Contrary to the situation in bulk metals where the motion of electrons is mostly rectilinear and is only occasionally interrupted by scattering acts, their motion in small particles is oscillation-like, because electrons `#ya ballistically from one wall to another changing very frequently the direction of the motion. We will show below that this change in the character of the motion can strongly suppress the intensity of the electron}lattice energy exchange. It should be recalled that the lower this value, the larger will be the di!erence between the electron and lattice temperatures at a given power absorbed by the electron subsystem. Atomic vibrations in a small metal particle and interaction of electrons with them can be treated either classically or quantum-mechanically, the choice being the matter of convenience. In Section 6.2, we shall apply the former approach to describe the generation of acoustic phonons by a moving electron. The corresponding classical equation is derived from "rst principles in Section 6.6. The main losses of energy of hot electrons in a bulk metal are due to Cherenkov generation of acoustic waves by electrons whose velocity is much larger than the sound speed. This mechanism is highly e$cient, but it vanishes in small particles where the motion of electrons becomes oscillation-like [126,169]. In the latter case, the energy losses are determined mainly by mechanisms of surface scattering of electrons. These processes can be treated more conveniently in the quantum-kinetical approach. The corresponding results are presented in Sections 6.3, 6.4 and 6.6. 6.2. Peculiarities of the electron}lattice energy transfer in island metal xlms As noted above, the power gained from the "eld, static under the current heating or highfrequency under the laser irradiation, is transferred by electrons to lattice vibrations (phonons) and then drained to the substrate via heat conduction. In the framework of the classical approach [163,164] the electron energy losses are treated as the Cherenkov generation of acoustic waves by the moving electrons whose energies exceed the Fermi energy. Both classical and quantum kinetic approaches were shown [164] to yield the same expression for the electron energy losses. The classical approach suits us better because it o!ers a simpler description of the peculiarities of electron motion in metal islands with characteristic dimensions smaller than the mean free path. The Cherenkov generation of acoustic waves is known to be the dominant mechanism of hot-electron energy losses in the bulk of metals. In what follows emphasis is given to the proof of the assertion that this dissipation mechanism may vanish in the metal island whose dimensions are smaller than some critical value. The longitudinal acoustic vibrations generated by the moving electron (i.e. the longitudinal vibrations that are responsible for the losses under consideration) are known to be described by the equation [164] (Ru/Rt)!s*u"!(K/o) d(r!r(t)) .

(66)

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Here u is the longitudinal component of the displacement vector, s is the sound velocity in the metal, o is the density, K is the electron}lattice interaction constant, r(t) is classical electron path (trajectory), and r and t are the coordinate and time. In an in"nite medium r(t)"*t, where * is the electron velocity. The Fourier expansion of the force proportional to d(r!*t) in Eq. (66) shows that there are harmonics for all frequencies and, consequently, there exist relevant conditions for the resonant excitation of all lattice vibrations. A distinguishing feature of metal particles smaller than the electron mean free path, which is governed by the scattering by lattice vibrations, is that there occur harmonic oscillations with frequencies of the order of v/¸, where ¸ is the characteristic size of the metal particle. In this case, the driving force does not contain all harmonics and, as soon as electron oscillation frequency exceeds the limiting Debye frequency, resonant interaction between a moving electron and lattice vibrations becomes impossible and hence, the Cherenkov losses vanish. A brief mathematical substantiation of the above arguments may be to the point. To take into account the peculiarities of a "nite system, it is su$cient, as will be shown later, to study the periodical electron motion only in one direction, i.e., to assume that r(t)"+r "const, z" (t), , ,

(67)

where (t) is a periodic function of time. The Fourier expansion of Eq. (66) is given by



 dk s(k , k )e k,  r, >IJ X . u, s" , , J IJ \ Substitution of (68) in (66) yields the following equation for s: 1 K Rs #u(k)s"! e\ IJ (R , 8p o¸ Rt

(68)

(69)

where u(k)"s(k #k). , J The electron energy losses due to the generation of lattice vibrations are given by [164]





Rs Ru de "K

d(r!r(t))dr"K dk (k #k) e\ IJ (R , , , J Rt Rt dt IJ  R "0 . u Rt X X!*



(70)

(71)

Here 2¸ is the size of the system along the z-axis. The solution of Eq. (69) can be written as



R K dt e\ IJ (RY sin[u(k)(t!t)] . s"! 8po¸u(k) 

(72)

Substituting this solution into (70), we get





de R K "! dk (k #k) dt cos [u(k)(t!t)]e IJ (R\(RY . , , J dt 8po¸ 

(73)

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Assuming now that the electron oscillating motion is described by (t) and taking it in the simplest form




(t)"¸ sin

v t , ¸

(74)

we can easily estimate, within the context of (73), how the "nite size of the system in#uences the electron}lattice energy exchange. First of all, we shall show that for ¸PR, (73) yields the known equation for hot-electron energy losses in an in"nite medium [164]. Note that for ¸PR, tPR, we have



R

p dt cos [u(k)(t!t)]cos [k ( (t)! (t))]" +d[k v#u(k)]#d[k v!u(k)], . J J J 2

 When deriving the latter equation, it is important to keep a proper order in proceeding to the limits ("rst ¸PR, then tPR). As a result, we obtain from (73) (for ¸PR, tPR):



K de "! dk ku(k)d[(kv)!u(k)] . 8po¸ dt

(75)

The result, given in Ref. [164], follows from (75) immediately. Now let us consider a "nite system. We employ the following expansion:  e\ IJ (R"e\ IJ *  TR*" J (k ¸)e\ LTR , L J L\ where J is the Bessel function. This expansion makes in possible to reduce (73) to the form L J (k ¸) de K L J "! dk (k #k) , , J n v/¸$u(k) dt 8po¸ IJ  L >\ v ; sin n t!k (t) # sin[$u(k)t#k (t)] . (76) J J ¸

 







Hence, we see that de/dt is a rapidly oscillating function for tPR. If v/¸'max u(k),u (u " " is the Debye frequency), i.e. in the absence of resonances, then the mean value of this oscillating function tends to zero since the averaging procedure implies the operation 1/tR dtde/dt for tPR.  Thus, we have shown that the electron motion under consideration is not accompanied by the energy losses associated with the Cherenkov generation of acoustic lattice vibrations, though the latter mechanism of energy dissipation is dominant in bulk metals. In the case of an in"nite metal, the expression for the total losses (due to all hot electrons) can be derived by multiplying (76) by the number of electrons with energies exceeding the Fermi energy (only such electrons can generate lattice vibrations). As a result, this expression can be reduced (see Ref. [164]) to the form a(¹ !¹) where  p ms a" nl (77) 6 ¹ (n is the electron concentration).

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For small metal particles, whose characteristic size ¸ satis"es the inequality v /¸'u $ "

(78)

(where v is electron velocity on the Fermi surface), these bulk losses have been shown to vanish. $ 6.3. Surface vibrations of small particles Since we will be discussing metal particles with dimensions on the order of or less than the mean free path, we can use the following model to calculate the surface energy exchange. The electron gas is in a spherical potential well of radius R (if thermal vibrations are ignored)  and height < . This model was used in Ref. [170] to study optical absorption in island metal "lms.  As we have already mentioned, the reason for the energy exchange is an interaction of the electron with thermal vibrations of the surface. These vibrations can be classi"ed somewhat crudely as either shape vibrations (so-called capillary vibrations), in the course of which the volume does not change, or surface vibrations, which are accompanied by a change in density (acoustic vibrations). A theory for the surface vibrations of a spherical particle is set forth in detail (for the case of vibrations of the surface of an atomic nucleus) in Ref. [171]. We begin our analysis with the capillary vibrations. We expand the radius of the vibrating surface in spherical harmonics > (h, u): HI R(h, u)"R







1# a > (h, u) . HI HI HI

(79)

The Hamiltonian of the capillary vibrations can then be written in the following form, in accordance with Ref. [171]:





1 1 "p " HI #D u "a " . H" +D "a "#C "a ",, H HI H HI H H HI 2 2 D H HI HI

(80)

Here p "D aH is a generalized momentum, and u "(C D ) is the frequency of the HI H HI H H H capillary vibrations. The constants D and C depend on the island dimensions in di!erent ways. H H According to Ref. [171], they are given by D "MnR /j, H 

C "p R (j!1)(j#2) . H  

(81)

Here M is the mass of the atom, n is the density, and p is the surface energy.  It can be seen from (81) that the frequency of the shape vibrations depends strongly on the radius of the metal island, R : 








C  (j!1)j(j#2)  " p . u " H  H D MnR H 

(82)

For the discussion below we will take a quantum-mechanical approach in which p and a HI HI are replaced by corresponding operators, which are related to the operators of creation (b> ) and HI

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annihilation (b ) of surface phonons by HI p( "i(D u /2)(b !b> ) , (83) HI H H HI HI

 (b #b> ) . (84) a( " HI HI HI 2D u H H After this replacement, the Hamiltonian of the capillary vibrations takes the standard form:











1 (85) HK " u b> b # . H HI HI 2 HI To "nd the electron}phonon energy exchange, we need an explicit expression for the corresponding Hamiltonian. According to the model adopted above, the potential energy of an electron in a metal island is =(r)"< *(r!R(h, u)) ,  where 1, x'0 , *(x)" 0, x(0 .

(86)



(87)

Using expansion (79) for R(h, u), we "nd from (86) =(r)+< *(r!R )#d(r!R )< R a > (h, u) . (88)      HI HI HI The second term in (88) describes the energy of the electron}phonon interaction associated with the surface vibrations. Writing this term in the second-quantization representation [using (84)], we "nd






 2(b #b> )a> a . (89) HK "< R 1tH "d(r!R )>H (h, u)"t JYLYKY HI HI JLK JYLYKY JLK  HI    2D u H H The operators a> and a in (89) create and annihilate an electron in the corresponding state. JLK JYLYKY The meaning of the subscripts on these operators becomes clear when we recall that the electron wave function in a `spherical potential square wella is 1 t (r)" R (r)> (h, u) . (90) JLK JK C J JL Here C is a normalization factor, and the radial wave function is JL j (k r), for r(R ,  R (r)" J JL (91) J h(iK r) for r'R . J JL  The quantity j (x) in (91) is the spherical Bessel function, and h(x) is the spherical Hankel J J function. In addition,








2m  e , k " JL

 JL





2m   (< !e ) K " . JL JL

 

(92)

Here m is the mass of an electron, and e is the energy of the electronic levels in a spherical  JL square potential well. These conditions are found from the condition for the joining of the electron

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wave function and its derivative at the point r"R In view of the rapid decay of the electron wave  function inside the barrier, we write k in the following form, as in Ref. [170]: JL k "k #*k , (93) JL JL JL where k are the roots of the equation JL j (k R )"0 . (94) J JL  Eq. (94) corresponds to the case of an in"nitely deep potential well. Assuming that *k is small in JL comparison with k (this point is easily checked), we "nd the following result from the condition for JL the joining of the wave function and its derivative at the point R :  *k "!k /R K . (95) JL JL  JL Here we have used the asymptotic expression





ip 1 exp !Kr! (l#1) , r'R . R (r)+  JL 2 iKr

(96)

Since we will be interested below in the electron levels near (and above) the Fermi energy, we can use the method of Ref. [170], "nding approximate solutions of (94) through the use of the asymptotic representation of the spherical Bessel function: p (2n#l) . (97) k " JL 2R  Now, in accordance with (93), (95), and (97), we have an explicit expression for k . Consequently, JL the electron wave functions in (90) and (91) have been determined completely. Using them, we can put the Hamiltonian for the electron}photon interaction, (89), in the form



  

2

e e  p p JL JYLY H "< du dh sin h> ) (h, u)   D u (< !e )(< !e ) JL H H  JL  JYLY   ;> (h, u)(b #b> )a> a . JYLY HI HI JLK JYLYKY

(98)

6.4. Surface electron}phonon energy exchange Now that we have explicit expressions for the electron and photon spectra and also for the Hamiltonian of the electron}photon interaction, we can move to the problem of determining the electron}photon exchange. This exchange can be taken into account systematically by a kineticequation approach. For brevity, we will be using the notation l"+l, n, m,,

q"+j, k, .

(99)

The change per unit time in the distribution of electrons among states caused by the scattering of electrons by phonons is then given by Rf JJ ,If " = +[(N #1) f (1!f )!N f (1!f )]d[e !e # u ] JJ JJYO O JJ JYJY O JYJY JJ JY J O Rt JYO #[N f (1!f )!(N #1) f (1!f )]d[e !e ! u ], . O JJ JYJY O JYJY JJ JY J O

(100)

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Here f "1a>a 2 is the electron distribution function, and N "1b>b 2 the phonon distribuJJ J J O O O tion function. As usual, the angle brackets mean an average over the statistical operator. Furthermore, in our case we have e "e , u "u , i.e., the spectrum is degenerate. It is simple matter to J JL O H derive an explicit expression for the transition probabilities, by "rst writing the interaction Hamiltonian (98) in the compact form HK " C [b #b>]a>a .  JJYO O O J JY JJYO

(101)

Then 2p = " "C " . JJYO

JJYO

(102)

The energy transferred from the electrons to the phonons per unit time is RE R " e f " e If . J JJ Rt Rt J JJ

(103)

We note that the electron distribution in a metal island, f , depends on only the electron energy: JJ F "f (e ). JJ J Treating the phonon system as a heat reservoir (with respect to the electron subsystem), we take the phonon distribution function N to be Planckian with a temperature ¹. Expanding collision O integral (100) in a series in the small quantity u (i.e., actually expanding in the ratio of the phonon O energy to the Fermi energy), we "nd the following result for expression (103):





f (e )[1!f (e )] RE J JY d(e !e ) . + = N ( u ) (104) JJYO O O JY J k ¹ Rt JJYO We can now write an explicit expression for the electron distribution function. Because of the intense electron}electron interaction, the power acquired by the electron subsystem from the external source becomes distributed among many electrons rapidly. As a result, a Fermi distribution with some e!ective electron temperature ¹ , is established:  e !e \ $ #1 , (105) f (e )" exp T T k ¹  where e is the Fermi energy. Substituting (105) into (104), we "nd $ ¹ RE "  !1 = N ( u )(e !e )d(e !e ) . (106) JJYO O O JY J J $ ¹ Rt JJYO To pursue the calculations we need to use the explicit expression for = which follows from JJYO (102) and from a comparison of (98) and (101):

 




 



4p = " JJYO (< !e )(< !e ) M JL M JYLY

 



p p  du dh sin h> (h, u)> (h, u)> (h, u) . JL JYLY HI  

(107)

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Substituting (107) into (106) we "nd that an explicit dependence of the integrand on the indices characterizing the electron states remains only in the spherical harmonic, because of the presence of the function d(e !e ),d(e !e ) in the integral. We can thus sum over the electron indices (96). J $ JL $ In doing so, we make use of the orthogonality of the spherical harmonics:  J d(u!u)d(h!h) . (108) > (h, u)> (h, u)" JK JK sin h J K\J In our case the summation over l is bounded by the condition e (e . This circumstance does JL $ not introduce any signi"cant error, however, since the maximum value of l is large, l &10. This

 estimate of l follows from the relation



pl  1

 e " . (109) $ 2R 2m   As a result of these calculations, we "nd from (106)













RE ¹ R m e  <  u N  $  H H . +  !1  (110) D Rt ¹ 2p

u H  HI Here u "< !e is the work function of the metal, and N "N(u ) is the Planckian   $ H H distribution function of the capillary vibrations. We are left with the task of evaluating the phonon sum in (110): H  H u N(u ) k ¹ H  2j#1 H + H . (111) D

D H H H \H H In (111) we have recognized that the energy corresponding to the Debye frequency of the capillary vibrations is considerably smaller than k ¹ (at room temperature). It follows from (82) in this case that






p   u "u + j . " H 

 MnR M As a result of these calculations we "nd




(112)




RE 4pR 3 v m u <   k (¹ !¹) $ n  "  . (113) +  Rt 3 u 16p R p    In the literature, the power transferred from the electrons to the phonons is customarily written in the form




RE 4pR  a(¹ !¹) . "  Rt 3

(114)

Here we have assumed that the particle is a sphere in our case. The constant a, which is a measure of the rate of the electron-phonon energy exchange, is given in our case by




3 v m u <  a" . k $n  "  16p R u p M  

(115)

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Let us evalute this quantity for a gold particle (a sphere) with the following parameter values: n"6;10 cm\, v "10 cm/s, R "10\ cm, p "10 erg/cm, u "3;10 s\ [165], and $   " (< /u )"5. We "nd a"2;10 erg/(cm s deg).   The value found for a is two orders of magnitude lower than the corresponding value in bulk metals. An experiment carried out to determine a in small particles has con"rmed this estimate [172]. A reduction in the intensity of electron}phonon interaction in small particles has also been found experimentally in another recent work [173]. In this work, a somewhat di!erent system has been studied than in our case (a dielectric core surrounded by an ultrathin Au shell). As we mentioned earlier, in addition to the shape vibrations (the capillary vibrations) of the particles there are surface vibrations which do involve a change in density (acoustic vibrations). The dispersion relation for these phonons is u "k s , (116) LH LH where s is the sound velocity, and the wave vector k is determined by the roots of the equation LH j (k R )"0 . (117) H LH  The interaction of the electrons with these vibrations can be dealt with by an approach like that taken above. As a result we "nd the following expression for the value of a determined by the surface acoustic vibrations:





1 nv m u  <   . (118) k $  " a+ u 16p  R o s   Here u is the Debye frequency of the acoustic vibrations, and o is the density of the material. " An estimate of a from (52) for the same gold particles as discussed above yields a value an order of magnitude smaller than the result in (49). Consequently, the interaction with capillary waves is predominant for these particles. We would simply like to point out that the idea of classifying the vibrations as either capillary or acoustic is valid only if u and u are substantially di!erent. This " " condition is satis"ed in the case under consideration here. 6.5. Derivation of the equation describing the sound generation by hot electrons The Hamiltonian of the interaction of an electron, residing in point r, with atoms of the lattice can be written as R H " +<(r!Rn !u(n))!<(r!Rn ),+! u(n) <(r!Rn ) .  Rr n n

(119)

Here < is an atomic potential, Rn is the radius vector of the nth lattice point and u(n) is a small displacement of the nth atom from its equilibrium position due to lattice vibrations. For a simple cubic lattice we have  R n " n a , G G  G

(120)

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where a is the translation vector. The potential energy of the lattice vibrations can be expressed in G a standard way in terms of small displacements of the atoms from their equilibrium positions: 1 (121) P" AK (n!n)u(n)u(n) , 2n n Y where AK (n) is a force matrix. Taking into account the kinetic energy of atoms with a mass M 1 K" M u (n) 2 n

(122)

and the energy of their interaction with an electron, one obtains an equation for lattice vibrations: R R MuK (n)# AK (n!n)u(n)"! H " <(r!R ) . (123)  L Rr Ru(n) n Y In most cases the electron}lattice interaction can be treated in a continuum approximation where the discrete vector is considered as a continuous one. Bearing in mind the short-range character of interatomic interactions we can also use the expansion




u(n)Ku(n)# l




R 1 R  u(n)#2 . u(n)# l Rn 2 Rn

(124)

Here l,n!n. Besides, let us take into account that AK (l )"0, AK (l)l"0 . l

(125)

l

The former of these conditions considers that the elastic energy of the lattice does not change if the crystal is displaced as a whole while the latter one allows for the symmetry A(l)"A(!l). The substitution of (124) into (123) with conditions (125) gives R R 1 MuK (n)# A (l)l l u (n)" <(r!Rn ) . (126) G Rx 2 l GH ? @ Rn Rn H ? @ G Taking into account Eq. (120), expression (126) can be written for the simple cubic lattice as






1 R R MuK # A (l)R (l)R (l) u " <(r!R) . (127) G ? @ 2 l GH RR RR H Rx ? @ G We have thus obtained an equation known from the theory of elasticity of continuous media which has a general form ouK !j u "g . (128) G GIJK I J K G Here o is the density of the medium and g is a component of the external force. In the G approximation of the isotropic continuum, an elastic medium is characterized by two elastic constants: modulus of dilatation (K ) and shear modulus (k ). For such a medium, instead of (127)   or (128), one obtains R 1 u! Rt o








k 1 K #   ( u)#k  u "  <(r!R) .  0 0  0 3 k P

(129)

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The full displacement vector u can be represented as the sum of its longitudinal and transverse components: u"u #u ; ;u "0;  u "0 . , , , ,

(130)

Combining (129) and (130), we have R 1 u !s  u "  <(r!R) . 0 , k P Rt ,

(131)

Here s"(K #k /3)/o is the square of the longitudinal speed of sound. The expression for   u will not be given here because its right side does not contain the force which could generate such , displacements. Now it remains to carry out the last step in obtaining the equation that has been used in [164] to treat the electron}lattice energy exchange in bulk metals. Let us again take into account that the atomic potential is short-range in character. This allows the following approximation in the description of the long waves in lattice vibrations: KX K 1 <(r!R)+  d(r!R)" d(r!R) , o k k

(132)

where X is the unit cell volume and K is a constant having the dimensions of energy. To clarify the  physical meaning of K, turn again to Eq. (119):



R 1 R dR u(R)<(r!R) H "! u(n) <(r!Rn )+!  Rr Rr X n



R dR u(R)d(r!R)"!K div u(r) . +K Rr

(133)

It is seen that under the assumptions used, H is reduced to the known Bardeen and Shockley  deformation potential. Hence the constant K represents the constant in this potential. In (131) and (132), r is a radius vector which determines the electron position. If the electron moves along a trajectory, one should write r(t) instead of r. The vector r(t) determines the coordinates of the electron in a moment t. Besides, in Eq. (119) the notation r is used instead of R. In other words, Eq. (119) follows from Eq. (131) under the change of notations rPr(t) and RPr. 6.6. Concluding remarks about electron}lattice energy exchange The results presented above demonstrate that the main mechanism of electron-lattice energy exchange operating in bulk metals does not function in small metal particles. In the latter case, hot electrons lose their energy in surface collisions. The "rst crude estimations of the surface energy exchange in small metal particles were made in [70]. The fraction of the energy transferred by an electron in its collision with a surface atom was taken to be proportional to the ratio of masses of the electron and atom. In [174], this part was assumed to be proportional to the ratio of the

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electron mass to the total mass of surface atoms within a circle with a diameter j , where j is   the electron wavelength. The number of these atoms equals (p/4)jn (n is the concentration of the    surface atoms). Since n +n and j "2p /m v K2/n, the result agrees with the result [70] to    $ within a factor of the order of unity. Such an estimate has been used in Section 3. However, this approximation can be justi"ed only for electrons with the energy considerably higher than the Fermi energy. Electrons having low energies interact with the surface as a whole rather than with individual atoms. A theory of such an interaction has been considered above. The theory gives an analytical expression for the constant of the surface electron}lattice energy exchange. The value of this constant appears lower by two orders of magnitude than the analogous constant in the volume. This estimate has been corroborated experimentally [172]. Therefore, the suppression of the electron}lattice energy exchange in small metal particles does occur and this e!ect must favour the generation of hot electrons.

7. Optical absorption by small metal particles 7.1. Introductory remarks In this section we shall concentrate on the peculiarities of optical absorption by small metal particles and their ensembles. The particle dimensions are assumed to be smaller than the wave length of the electromagnetic "eld. On the other hand, the dimensions may be either a smaller or larger than the electron mean free path. Depending on this inequality, either a surface or volume absorption mechanism is dominant. If the particles are small in the sense just speci"ed, the most characteristic feature is the extremely high sensitivity of their optical absorbance in the IR range to the particle shape and to wave polarization. We shall see that under equal conditions, the power absorbed by particles which have the same volume but distinct shapes can di!er by orders of magnitude. The same is true for absorption of the electromagnetic waves with di!erent polarizations. These peculiarities play the decisive role in our model that explains electron and photon emission from IMFs exposed to a powerful IR laser irradiation (see Sections 3 and 5). These emissions depend exponentially on the electron temperature which, in turn, is determined by the power being absorbed. This is the reason why strongly nonuniform emission currents are usually observed from an IMF which, on the average, is uniform. In other words, the emission is determined not by the average, but rather by an extremal absorption of individual islands comprising the "lm. It should be noted that optical absorption by small metal particles has been investigated for long, and many relevant results have been published in monographs (see e.g. [165,175,176]). However, a number of important regularities of absorption as a function of particle shape and wave polarization have been elucidated only quite recently [177,178]. To better understand the di$culties emerging in the development of the all-embracing theory of optical absorption by small metal particles, let us remind the absorption mechanisms. A substantial role in the absorption by small particles is played both by the electric and magnetic components of the incident electromagnetic wave. The electric component induces a local potential "eld inside the metal particle, and this brings about the so-called electric absorption. The magnetic component of the incident wave induces inside the particle an eddy electric "eld and eddy (Foucault's) currents. The corresponding

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absorption is named the magnetic absorption. Depending on the size and shape of the particles, either electric or magnetic absorption can prevail. Therefore, a well-founded theory must involve simultaneous evaluation of both electric and magnetic components of absorption for particles whose dimensions can be either smaller or larger than the electron mean free path. For the particles smaller than the electron mean free path, the particle shape a!ects not only the values of the local "elds, but also the character of optical conductivity, which characterizes the response of the system to these "elds. The optical conductivity of asymmetric particles is a tensor with components dependent on particle shape. Thus, the construction of a coherent theory of total absorption by small particles requires the calculation of local potential and eddy "elds corresponding to speci"c particle shapes, as well as currents induced by these "elds. However, the implementation of this program encounters formidable di$culties. In particular, one fails in calculating the magnetic absorption for the nonspherical particles shaped as a "nite cylinder or a parallelepiped for which one can carry out an exact evaluation of the electric absorption. Considerable attention has also been given to ellipsoidal particles (see e.g. [165]), but only the particles with dimensions larger than the electron mean free path have been considered and the tensor of optical conductivity for them has not been calculated. The above-mentioned and many other works have been concerned with the resonance (plasmon) absorption and with the e!ect of the interaction between the small particles on the absorption. In recent years, a method has been developed which allows, for the "rst time, a uni"ed approach to calculation of electric and magnetic absorption by ellipsoidal particles whose dimensions can be both smaller and larger than the electron mean free path [177,178]. The present section describes this method and some results derived by it. Since the theory of electric absorption by metal particles shaped as a sphere [170], cylinder [179] or parallelepiped [180] and having dimensions smaller than the electron mean free path was based on a quantum mechanical approach, this method will also be reviewed in Section 7.7. Section 7.8 is devoted to discussion of the resonance (plasmon) optical absorption in IMFs. Let us brie#y dwell on the practical importance of this problem. Obviously, the pronounced dependence of the optical absorption by small particles on their shape, wave polarization and other parameters can be exploited to tailor the re#ectance and absorbance of surfaces by deposition of coatings which contain appropriate metal particles. In space, small particles can be dangerous for satellites, but a laser beam can be used to destroy the particles if one knows how to tune the beam to their maximum absorbance. Ensembles of small particles deposited onto surfaces or embedded into matrices can exhibit a speci"c opto-acoustic e!ect [181], generate second harmonics [182] and manifest other nonlinear e!ects. In recent years, a considerable attention has also been focused on dusty gaseous plasmas containing clusters and small particles, including metal ones (see e.g. [183,184]). Evidently, the radiative and absorbing ability of such plasmas can substantially be determined by the properties of the small particles. 7.2. Statement of the problem Consider a metal particle exposed to an electromagnetic wave



 E

E

"

H

H

exp+i(k ' r)!ut, ,

(134)

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where E and H are the electric and magnetic vectors of the wave, u and k are its frequency and wave vector, and r and t denote the spatial coordinates and time, respectively. The wavelength j"2p/k is assumed to be considerably larger than the size of the particle, so the metal particle resides actually in spatially uniform, but time-varying, electric and magnetic "elds. The external electric "eld Ee\ SR induces a local potential electric "eld E inside the particle, which in turn gives rise  to electric current with the density j . The magnetic "eld He\ SR induces in the particle a vortex  (eddy) electric "eld E , which gives rise to an eddy electric current with the density j . 

As a result, the total dissipation of the wave energy, i.e. absorption by the particle, is



1 ="= #= " Re 

2

4

dr ( j ) EH #j ) EH ) ,  



(135)

where < is the particle volume. The "rst term in Eq. (135) corresponds to electric absorption and the second to magnetic absorption. To calculate the total absorption, it is necessary to "nd the potential electric "eld E ,  the eddy electric "eld E , and the corresponding current densities j and j . For spherical particles  

that are either larger or smaller than the electron mean free path, the formulas determining simultaneous electric and magnetic absorptions were obtained earlier (see, e.g., [185]). The total energy absorbed by a spherical particle is





1 uR 9 # "E" , ="
(136)

where e and e are, respectively, the real and imaginary parts of the dielectric constant, R is the particle radius, and c is the speed of light. The "rst term in Eq. (136) describes electric absorption and the second term magnetic absorption. For the particle larger than the mean free path, the bulk scattering is dominant and the dielectric constant of the metal has its standard form u l u   #i , e "e#ie"1!  u u#l u#l

(137)

u being the plasma frequency and l the collision rate.  Eqs. (136) and (137) can be used to estimate the relative contributions of electric and magnetic absorption to the total absorption. For instance, for a gold particle one has u +5;10 s\ and  l+10 s\. Assume that R"3;10\ cm and that u is the frequency of a CO laser, i.e.,  u+2;10 s\. Then Eq. (136) yields e+!600 and e+30, and the magnetic-to-electric absorption ratio is




1 uR  =

" "e "+2 .  90 c = 

(138)

Thus, for the given set of parameters magnetic absorption is twice as large as electric. Obviously, for di!erent parameters of the particle and a di!erent frequency range electric absorption can be either larger or smaller than magnetic absorption. Hence, when studying the shape dependence of

136

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optical absorption by a small metal particle, we must take into account both electric and magnetic absorption. To our knowledge, the total absorbed power for asymmetric particles smaller than the mean free path was "rst calculated in [177,178]. The only quantity studied earlier had been the light-induced conductivity p(u), de"ned by j "pE , (139)   for spherical [170] and cylindrical [179] particles, and for particles shaped like parallelepipeds [180]. All these works use a quantum mechanical approach, and the shape of the particles is chosen in such a way that the Schoedinger equation can be solved analytically for the potential well corresponding to such a symmetry. For particles larger than the mean free path, the light-induced conductivity is known (p"ue/4p), and the calculation of the e!ect of particle shape on absorption reduces to "nding E and E . The local electric "eld in the case of an ellipsoidal particle is independent of the   coordinates and can easily be expressed in terms of the components of the depolarization tensor. This feature has been used to consider the dependence of electric absorption on the particle shape in the case where bulk scattering is dominant (see, e.g., Ref. [165] and references therein). For perfect conductivity e<"e", magnetic absorption by ellipsoidal metallic particles larger than the electron mean free path was calculated by Levin and Muratov [186]. In most theoretical works treating optical absorption by island metal "lms, attention is focused on the e!ect of the interaction between the metallic particles on E and hence on electric  absorption (see, e.g., Ref. [187] and references therein). In some cases, this interaction can indeed lead to a signi"cant change in the absorbed power [188]. However, the e!ect of particle shape on electric and magnetic absorption, and the polarization dependence of the electric-to-magnetic absorption ratio have not been studied. Meanwhile, these factors can change the absorbed power in the IR region not just severalfold, but by several orders of magnitude. 7.3. Local xelds In what follows we shall examine ellipsoidal metal particles. Such an assumption has several advantages. First, by considering ellipsoids of di!erent oblateness and elongation, one can simulate the majority of real particle shapes (from `pancakea to needle-like). Second, the potential (E ) and  eddy (E ) local "elds for such particles can easily be calculated.  For ellipsoidal particles, the potential local electric "eld E induced by a uniform external  electric "eld E is known to be coordinate-independent [185]. The "eld E can be linearly  expressed in terms of E by employing the depolarization tensor. In terms of the principal axes of the depolarization tensor, which coincide with the principal axes of the ellipsoid, one has (E ) "E!4p¸ P "E!¸ (e!1)(E ) , (140)  H H H H H H  H Here ¸ are the principal values of the components of the depolarization tensor, and P is the H polarization vector. As we shall see below, the light-induced conductivity becomes a tensor for asymmetric particles smaller than the mean free path, so Eq. (140) needs to be modi"ed. This will be done somewhat later. It should also be noted that in the case of a particle ensemble, the polarization vector in a given particle is induced not only by the external "eld, but also by the

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dipoles induced by this "eld in other particles [126,189]. Here we neglect such e!ects, but they can easily be incorporated into the picture [188]. Calculating (E ) from (140), we get  H (E) H "(E ) "" . (141)  H [1#¸ (e!1)]#(¸ e) H H When the light-induced conductivity of the small particles becomes a tensor (we consider this case later), we must substitute 4pp /u for e in Eq. (141) (p is the jth diagonal component of the JJ HH light-induced conductivity tensor). Let us now "nd the eddy local "eld E which obeys Maxwell's equations  u ;E "!i H ,  c (142) ;E "0 ,  augmented by the boundary condition E ) n " "0 , (143)   1 where n is a unit vector normal to the surface S.  Here the following remark is in order. On the right-hand side of the "rst equation in (142), we take the external (spatially uniform) magnetic "eld H for the magnetic "eld inside the particle. Such an approximation is justi"ed if the depth d of the skin layer is much larger than the & characteristic particle size R: d ,(u/c Im(e )\





yH iu zH X W ! R . (145) (E ) "  V c R#R R#R V V W X V The other components of E can be obtained via cyclic permutations. In Eq. (145), R , R , and  V W R are the semi-axes of the ellipsoid. X Knowing E and E for particles whose characteristic size is larger than the electron mean free   path, it is easy to derive from Eq. (135) a formula for the absorbed power. Note that in this case the currents are related to the "eld through relationships of type (139). If for the sake of simplicity we consider an ellipsoid of revolution (with the z-axis chosen as the axis of revolution), then combining (135), (141), (145), and (139) we get ="<





(E) uR u R R ue , , (H) , H , (H)# # [1#¸ (e!1)]#(¸ e) 10c , 5c R #R , 8p , , H H

(146)

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where H"H, H"((H)#(H), , X , V W R "R "R , R "R . V W , X , Eq. (146) generalizes formula (136) to the case of ellipsoidal particles. Note that in (146) E"H, and we do not express H in terms of E (as we did in (136)), so that the dependence of the absorbed power on wave polarization can be more graphic. The principal values of the components of the depolarization tensor for particles which are ellipsoids of revolution read ¸ "¸ "(1!¸ ) , X V W  1!e 1#e N !2e , R 'R , N ln N , , 1!e 2e N N ¸ " (147) X 1#e N [e !arctan e ], R (R , N N , , e N where e""1!R /R ". N , , Fig. 7.1 shows ¸ as a function of the ellipsoid semi-axis ratio R /R . The components ¸ of the X , , H depolarization tensor can vary between zero and unity. The denominator in (141) is seen to contain ¸ as a cofactor of (e!1). As follows from the above estimates, one has e+4;10 in the H frequency range of a CO laser. Consequently, "E "can strongly depend on particle shape. Fig. 7.2   depicts the dependence of "E "/"E" on the semi-axis ratio R /R for the case where E is  , , directed along the axis of revolution. It is seen that the ratio of the square of the local "eld to the square of the external "eld can vary by several orders of magnitude. According to (146), this means that electric absorption by metallic particles can vary by several orders of magnitude, depending on particle shape and wave polarization. This fact entails important consequences. As we noted earlier (see Sections 3 and 5), the electron gas of the metal particles heats up when an island metal "lm is illuminated by laser light. This leads







Fig. 7.1. Dependence of the factor of depolarization along the ellipsoid's axis of revolution on the ellipsoid's semiaxis ratio.

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139

Fig. 7.2. Dependence of the ratio of the square of the local electric "eld inside the particle to the square of the wave's electric "eld on the ellipsoid's semiaxis ratio for the chosen wave polarization.

to electron and photon emission. In particular, electron emission is here actually thermionic and, according to Richardson's law, is proportional to exp+!u/k¹ ,, where u is the work function and  ¹ is the electron temperature. The electron temperature ¹ is determined by the energy absorbed   by a particle, which in turn is strongly dependent on particle shape and wave polarization. In this situation, introducing average (e!ective) absorption cross sections makes no sense. It is just the particles absorbing the most light that are the emitters of electrons. In other words, these phenomena are determined primarily by particles with maximum absorption and not by particles with some average (`e!ectivea) absorption, which constitute the majority in an island "lm. 7.4. Electron distribution function In Section 7.3 we determined the local "elds and derived a general expression for the power absorbed by an ellipsoidal metal particle in the case of bulk scattering (i.e., for particles larger than the mean free path). Here we address the case of particles smaller than the mean free path. The more general method we now develop can also be applied to particles larger than the mean free path. To determine the absorbed power, it is necessary to derive expressions for the high-frequency currents induced in the particle by the known local potential and eddy electric "elds. By de"nition, the current density is





2me 2e *f d(mv)" *f (*) dv , j" (2p ) (2p )

(148)

where f (*) is the electron velocity distribution function, and e and m are electron charge and mass. In the presence of local "elds, the distribution function can be represented as a sum of two terms, f (*)"f (e)#f (*) ,  

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where f (e) is the Fermi distribution function, depending only on energy e"mv/2, and f (*) is   a correction generated by the local "elds. f (*) can be found by solving the appropriate kinetic  equation. In the linear approximation, the kinetic equation in the electromagnetic "eld assumes the form Rf Rf (l!iu) f #* )  #eF ) *  "0 .  Re Rr

(149)

Here we have taken into account that f Je SR; l is the bulk collision rate and  F"E #E . (150)   Eq. (149) must be augmented by boundary conditions for f . We assume, as is often done, the  electron scattering at the boundary to be di!use, i.e., f (r, *)" "0 for v (0 , (151)  1 L where v is the velocity component normal to the surface S. L Based upon these assumptions, Lesskin et al. [190] studied the magnetic scattering by a spherical metal particle. To solve Eq. (149) with the boundary conditions (151), we employ the method of characteristic curves, which demonstrated its e!ectiveness in Ref. [190]. However, for ellipsoidal particles the method used in Ref. [190] needs to be modi"ed. The essence of this modi"cation is as follows. We transform to a deformed system of coordinates in which the original ellipsoid particle,  x G "1 R G G becomes a sphere of radius R. In other words, we assume that

(152)

x "x /c , c "R/R , R"(R R R ), c c c "1 . (153) G G G G G       Under such a deformation the shape of the particle changes, but its volume is conserved. This means that the electron number density remains unchanged, and so does the normalization of the function f. In the new system of coordinates, Eq. (149) and the boundary conditions (151) acquire the form Rf Rf (l!iu) f #* )  #eF(r) ) *  "0 ,  Re Rr

(154)

f (r, *)" "0 for r ) *(0 .  PY0 In (154) and (155) we also introduced the `deformeda velocity components

(155)

v "c v . G G G Eq. (154) for the characteristic curves has the form

(156)

dx /v "!df /lf "dt, l,l!iu , G G  

(157)

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141

which implies that r"*t#R ,

(158)

where R is the radius vector whose tip is at the given point of the sphere from which the trajectory begins. Here the parameter t can formally be considered as the `timea of motion of the electron along the trajectory. If we transfer vt in (158) to the left-hand side and square the resulting equation, the solution of this new (scalar) equation can be written as 1 t" [r ) *#((R!r)v#(r ) *)] . v

(159)

The characteristic curve (159) depends only on the absolute value of R and not on the orientation of R. Such independence of the characteristic curve from the position of a point on the surface was achieved by transforming to the coordinates (153). From (159) we also see that t"0 at r"R. Bearing this in mind, we can use (157) to "nd an f that satis"es Eq. (154) and the boundary condition (155): 



Rf R f "!  dq exp+!v (t!q),e* ) F(r!*(t!q)) .  Re 

(160)

Taking into account the coordinate dependence of F (see (150) and (145)), from (160) we obtain





Rf  x R f "!e  * ) E # a v H #v   GH G H Re c Rl H GH



1!exp+!lt, . l

(161)

If initially the particle is spherical, then a "!a and the last term in (161) vanishes. GH HG 7.5. Electric absorption Combining (161), (148), and (135), we obtain the following expression for the electric absorption:





em 1 =" Re "* ) E "d(e!k)(1!exp+!lt,) dr dv ,  (2p )  l

(162)

where k is the Fermi energy, and where we considered the fact that Rf /Re"!d(e!k) . Allowing for the form of t (according to (159)), it is convenient to integrate (162) with respect to r by directing the z-axis along the vector * and introducing two new variables, v r f" , g" t . R R

(163)

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The result reads





(1!exp+lt,) dr"2p

0



dr r



p



dh sinh(1!exp+!lt,)

  


"2pR



df f

>D

dg






g!f#1 lRg 1!exp ! 2gf v



 \D dg  lRg df f(g!f#1) . "pR 1!exp ! g v  E\ Further calculation of the integral is easy, and as a result from (162) we obtain







pemR 1 =" Re dv "* ) E "d(e!k)t(q) ,   (2p ) l

(164)

where we have introduced




4 4 1 4 2 1# exp+!q, , t(q)" ! # ! q 3 q q q (165)

2u 2l q,q !iq " R!i R .   v v Eq. (164) gives the general form of the electric absorption by an ellipsoidal metal particle for an arbitrary bulk-to-surface scattering ratio. The above case of bulk scattering (the Drude case) follows from (164) when q<1. Then, according to (165), t(q)"4/3, and from Eq. (164) we obtain for the electric absorption l "E " en "E "  ,
(166)

where n is the electron concentration, which can be expressed in terms of the Fermi velocity v$ or the Fermi energy k:



2k 8p (mv ) $ , v " n" . $ m 3 (2p )

(167)

Obviously, Eq. (166) corresponds to the "rst term in (146). We now analyze the situation when the particle is smaller than the mean free path, and hence surface scattering is dominant. This corresponds to q "(2l/v)R;1 . (168)  As for q ,2uR/v, when surface scattering is dominant, this parameter can be either larger or  smaller than unity. It is of interest then to study the two limits q "(2u/v)R<1 ,  q ;1 . 

(169) (170)

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143

The case (169) corresponds to high-frequency surface scattering, and (170) to low-frequency surface scattering. If we ignore bulk scattering (q P0) and assume that q is arbitrary, then (165) yields   1 1 2 4 4 Re t(q) + ! sin q # (1!cos q ) . (171)   v u q q q   This expression is present in (164). We see that terms which are oscillating functions of particle size have emerged. Such oscillation e!ects in spherical particles were studied by Austin and Wilkinson [191] for electric absorption, and by Lesskin et al. [190] for magnetic absorption. These e!ects, which are moderate by themselves, are even less important for asymmetric particles. The reason is that the `deformeda velocity v, which enters into the expression for q , is angle dependent. In view of this, the integration over angles smooths out the oscillation e!ects. Furthermore, Eq. (171) implies that these oscillations can be essential only when q "2Ru/v+1,  i.e., when the electron transit time from wall to wall, 2R/v, coincides with the period of the electromagnetic wave. In the limiting cases given by (170), these e!ects are negligible. Let us turn to the high-frequency case (169). For q <1 and q ;1 we have   2 1 + , Re uq lt(q)  which in accordance with (164) gives

  



 



pemR =+ dv v"* ) E "d(e!k) .  (2p )u 

(172)

To study the dependence of absorption on particle shape, it is su$cient to consider an ellipsoid of revolution. In this case



v v ,# , , (173) R R , , where v and v are the electron velocity components perpendicular and parallel to the axis of , , revolution. With allowance for (173), the integral in (172) can easily be calculated: v"R

ne v 9 $ = "< [u "E "#u "E "] . (174)  , ,  mu R 16 , ,  , Here u and u are functions of the ellipsoid eccentricity (we note once more that , , e""1!R /R "): N , , 1 1 1 1 1# (1!e# 1! arcsin e , R (R , N e N , , 2e 4e 1 2 N N N u " , 2 1 1 1 1 (175) 1! (1#e# 1# ln((1#e #e ), R 'R , N e N N , , 2e 4e 2 N N N







 





 

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1 1 1 1! (1!e# arcsin e , R (R , N N , , 2 2e 4e N N u " 1 1# 1 (1#e! 1 ln((1#e#e ) , , N 4e N N 2 2e N N

(176)

R 'R . , ,

Having the general expression (174) for the electric absorption of an ellipsoidal metal particle in the case of high-frequency scattering, we can easily obtain the components of the light-induced conductivity tensor. To this end, we write the expression for electric absorption in terms of the principal values p of the conductivity tensor, HH 1  = "< p "E " , HH H   2 H and compare it with (174). As a result we have ne v 9 $ u , p "p ,p " VV WW , mu R 8 , , ne v 9 $ u . p "p " XX , mu R 8 , , The case of a spherical particle follows from (174)}(177) as e P0. N Allowing for (175) and (179), we "nd that ne 3 v $ . p "p " , , mu 4 R

(177)

(178)

(179)

Comparing (178) and (179), we can see that the light-induced conductivity of metallic particles smaller than the mean free path is a scalar quantity only if the particles are symmetric. In the general case of asymmetric particles, the light-induced conductivity becomes a tensor whose components depend on particle shape. Fig. 7.3 illustrates the dependence of p /p on the , , ellipsoid semiaxis ratio R /R . Eq. (178) was used to plot the curves. We see that the components , , of the light-induced conductivity tensor di!er considerably, depending on the degree of particle asymmetry. Comparing (179) with the expression for the conductivity that follows from (166) for u
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145

Fig. 7.3. Dependence of the ratio of conductivity normal to the ellipsoid's axis of revolution (p ) to conductivity parallel , to this axis (p ) on the ellipsoid's semiaxis ratio in the case u;v /R , v /R . , $ , $ ,

ellipsoids:











9p ne v 3 $ "(E ) "#"(E ) " , < =+  ,  128 mu R 2  , , R ;R , , , 9 ne v 1 $ "(E ) "#"(E ) " , =+ <  ,  32 mu R 2  , , R
(180)

(181)

The factors
(182)

Here the frequency of the electromagnetic wave is much higher than the bulk collision rate, but is much lower than the frequency of transit from wall to wall. If condition (182) is satis"ed, we have t(q)+q/2, and from (164) we obtain



pemR dv "* ) E "d(e!k) . =+   v (2p )

(183)

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After evaluating the integral in (183) we can transform the result to (177), where instead of (178) we will have 1 1 1 ! (1#e# 1# ln((1#e #e ) , N e N N 2e 2e N N N 9 ne R R, 'R, , , (184) p " 1 1 1 , 8 m v (1!e # 1! arcsin e , $ N e N 2e 2e N N N R (R , , ,















1 1 (1#e! ln((1#e#e ) , N N N e e N N ne R , R 'R , p " , , , m v $ 1 1 ! (1!e # arcsin e , R (R . N e N , , e N N

(185)

Formula (183) for the absorbed power assumes a simple analytic form for the limits of highly #attened and highly elongated ellipsoids: 9 ne R , =+ <  16 m v $ R

 






R 1 ln 2 , ! "E "#"E " , ,  ,  R 2 ,

(186)



ne R 1 , "E "#"E " , R ;R . (187) = +< , ,  ,  m v 2 ,  $ In addition, at R "R , Eq. (183) yields the well-known result for a spherical particle: , , 3 ne R "E " . =+ < (188)  8 m v  $ Fig. 7.4 depicts the dependence of p /p on the ellipsoid's semiaxis ratio R /R constructed , , , , from Eqs. (184) and (185). Comparing Figs. 7.3 and 7.4, we see that the e!ect of particle asymmetry on the ratio of the components of the conductivity tensor di!ers not only quantitatively, but also qualitatively in the high- and low-frequency cases (provided that surface scattering is dominant). 7.6. Magnetic absorption Magnetic absorption is given by the second term in (135). Combining (144), (148) and (161), we obtain an expression for it:



 aH a 1!exp+!lt, em Re dr dv d(e!k) JI GH v x v x , = "

(2p ) cc cc J I G H l  J I G H

(189)

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Fig. 7.4. Dependence of the ratio of conductivity normal to the ellipsoid's axis of revolution (p ) to conductivity parallel , to this axis (p ) on the ellipsoid's semiaxis ratio in the case u;v /R , v /R . , $ , $ ,

where the summation is performed over all indices from 1 to 3. To calculate the integral with respect to r, we direct the z-axis along the vector r. Then, according to (159), t is independent of the angle u (in the plane perpendicular to v). Hence in (189) we can "rst integrate with respect to u. It can be shown that



p










v v sinh v v d !3 H I dux x "2pr H I # HI H I v 2 v



, (190)  where h is the angle between r and v . After (190) is inserted into (189), the calculation becomes similar to one used in calculating expression (162) for the electric absorption. As a result, Eq. (189) becomes



1  "a "  "a #a " v v  pemR HG G H t (q) Re dv d(e!k) GH v t (q)#2 GH = "

cc G  cc v v  2(2p ) G H  G H  where 1 8 1 4 24 3 3 t " ! # ! !8 # # exp+!q, ,  q q q 15 q q q




t 








1 2 1 8 6 32 5 16 16 3 " ! # ! # !2 # # # exp+!q,! t (q) . q q q q 5 q 3q q q 4 

,

(191)

(192)

Eq. (191) determines the magnetic absorption by a particle in general form for an arbitrary ratio of the bulk and surface contributions. For spherical particles, the last term on the right-hand side of Eq. (191) vanishes, since in this case a "!a . GH JG Information about the scattering mechanism is contained in the quantities q"2Rl/v and l,l!iu. From (191) we can obtain simple analytic expressions in the limits of pure bulk

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scattering and pure surface scattering. In the "rst case ("q"<1), we arrive at the known expression for magnetic absorption determined by the second and third terms on the right-hand side of Eq. (146). When surface scattering is dominant, the high-frequency case is the most interesting, since the eddy "eld E is proportional to the frequency, with the result that the relative role of  magnetic absorption grows with frequency. We therefore assume that q ;1 and q <1. Then   from (191) it follows that






 "a " vv pemR 1  = + dv d(e!k) GH vv# "a #a " G H GH HG v

4(2p )u c G 2  G 



.

(193)

Again, in the case of an ellipsoid of revolution, the integrals in (193) can easily be evaluated and we obtain





R 9 env $ R u (H)# , < U(H) . =

,

128 mc , , , (R #R ) , ,

(194)

Here, in addition to the function u de"ned in (175), we have introduced a new function U: ,



U"





 





 

1 1 1 2! !e (1!e# 1# arcsin e , R (R , N N 2e N , , 4e 2e N N N 1 1 1 2# #e (1#e# 1! ln((1#e#e ), R 'R . N N N N , , 4e 2e 2e N N N

(195)

For spherical particles (i.e., as e P0), the result obtained by Lesskin et al. [190] follows from N (194): 3 env $ R(H) . = + <

64 mc

(196)

Eq. (194) acquires a simple analytic form for the limits of highly elongated and highly #attened ellipsoids:





9 3p env 1 $R = + < (H)#(H) , R ;R ,

128 8 , , , mc , 2 , 9 env $R = + <

128 mc ,






R  , (H)#(H) , R
(197)

(198)

From (197) it follows that in the case of a highly elongated ellipsoid, the magnetic absorption is twice as high when the magnetic "eld is perpendicular to the axis of revolution than when it is parallel to it. The situation is similar for bulk scattering. Earlier we estimated the relative contribution of the electric and magnetic terms of spherical particles to absorption (see Eq. (138)). Now, having the expressions for the electric (Eq. (174)) and

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Fig. 7.5. Dependence of the magnetic-to-electric absorption ratio on the ellipsoids semiaxis ratio for the case when H is parallel to major semiaxis.

Fig. 7.6. Dependence of the magnetic-to-electric absorption ratio on the ellipsoids semiaxis ratio for the case when E is parallel to major semiaxis.

magnetic (Eq. (194)) absorption by asymmetric particles, we can return to that problem. For an asymmetric particle, the ratio of the electric and magnetic contributions to absorption (at "xed frequency) is strongly dependent on the degree of particle asymmetry and wave polarization. Figs. 7.5 and 7.6 show the dependence of = /= on the ellipsoid semi-axial ratio for two di!erent

 polarizations. It can be seen that these curves di!er strongly not only quantitatively but also qualitatively.

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Optical absorption in small metal particles and their arrays has also been investigated in a recent work [192]. A substantial dependence of the absorption on the size and shape of the particles has been found. 7.7. Quantum kinetic approach As mentioned above, the majority of studies concerning the dependence of the optical conductivity on the shapes of metal particles have been performed in terms of quantum mechanics. This approach is necessary only for very small (1 nm) particles, when quantization of the electron spectrum must be taken into account. In all other cases, the quantum kinetic approach is only convenient when the solutions of the Schrodinger equation with the relevant potential wells are known. Here we shall consider this approach brie#y. It provides an opportunity to treat the absorption by small metal particles from a new point of view and to analyze the compatibility of the two treatments. Thus, we start from the equation for the statistical operator o( : i Ro( /Rt"[HK o( ] .

(199)

The Hamiltonian HK for an electron in the electromagnetic wave "eld described by the vector potential A is given by the standard expression






e  1 p( ! A #;(r),HK #HK , HK "   c 2m

(200)

where e p(  (p( A#Ap( ) . HK " #;(r), HK !   2m 2mc

(201)

Here HK is written in the linear approximation with respect to A; ;(r) is the electron potential  energy; p( is the momentum operator. If one considers only electric absorption, i.e., disregards the eddy currents, then the electric "eld within the metal particle may be treated with su$cient accuracy as spatially homogeneous (for wavelengths much greater than particle dimensions). The vector potential may be written as A"Ae\ SR>ER ,

(202)

where g is the adiabatic parameter. Operator o( may be written as a sum of two terms o( "o( #o( , (203)   where o( is the statistical operator of the system without electromagnetic wave, o( is the correction   to A linear with respect to o( .  We linearize Eq. (199) with respect to A, rewrite it on the proper basis of HK and thus obtain  (ig#u)1a"o( "b2"1a"HK o( "b2!1a"o( HK "b2#1a"HK o( "b2!1a"o( HK "b2 . (204)         

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On this basis we have o( "b2"f "b2, 1a"HK o( "b2"1a"HK "b2f ,  ?    @ where f is the occupation function of the ath state, i.e. ? f "2f (e ),2+exp[(e !k)/k¹]#1,\ . (205) ? ? ? In Eq. (205), f is the Fermi function, ¹ is the temperature. We have taken into account that each state with energy e can be occupied by two electrons with opposite spins. ? Eq. (204) yields f !f ( f !f )1a"HK "b2 e ? @ @  1a"o( "b2" ? "! A1a"p( "b2 .  e !e ! u!ig mc e !e ! u!ig ? @ ? @ The current density operator may be written as






e e p( e e ! A ,JK #JK , JK " *" +r( , HK ,"   < < m m <

(206)

(207)

and the statistical average of the current density is given by J"Sp(o( JK )+Sp+o( JK #o( JK ,     f !f e e ? @ 1a"Ap( "b2 1b"p( "a2 . (208) "! nA! mc< e !e ! u!ig mc @ ?@ ? The last equality was derived making use of Eq. (206) and the normalization relation for f , i.e., ? f "



(210)



( f !f )1a"p( "b2 1b"p( "a2 en e @ I J d # ? . (211) p (u)"i IJ e !e ! u!ig mu IJ <mu ? @ ?@ The relation between the complex conductivity tensor and the complex dielectric permittivity tensor is given by p (u)"d #i(4p/u)e (u) . IJ IJ IJ Making use of the relation




(212)



1 1 "P $ipd(e !e ! u) ? @ e !e ! uGig e !e ! u ? @ ? @

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(for gP0), we "nd from Eqs. (135), (210), and (211) that the electric absorption is described by pe = "! ( f !f )"1a"p( E "b2"d(e !e ! u)  ? @  ? @ 2mu ?@ <  (213) , (Re p )"E " . II I  2 I The last relation is written in the coordinate system associated with the principal axes of the conductivity tensor. It may be applied to calculate the principal values of this tensor (real values of the complex conductivity tensor). To do this, we have to calculate the matrix elements 1a"p( E "b2  on the proper basis of the Hamiltonian HK , i.e., to specify the model.  We shall consider the simplest model of a parallelepiped particle that was proposed in Ref. [180]. The wave functions and energy spectrum of an electron in a parallelepiped potential well with sides a , a , a are given by the simple expressions    8 pn pn pn  x sin  y sin z , a"2"t    " sin (214) LLL n n n a a a      

p  n (215) H . e " ? a 2m H H The matrix elements of interest are readily calculated in terms of the wave functions (214). If the state 1a" is described by the set of quantum numbers +n ,, and the state "b2 by the set +n ,, H H then we "nd that








8  nn  I I +1!(!1)LI >LI ,d H H , jOk . "1a"p( "b2"" L L I a (n!n ) I I I Eq. (216) was derived within the context of the identity

(216)

+(!1)LI >LI !1,"2+1!(!1)LI >LI , . Inasmuch as Eqs. (216) and (213) contain, respectively, the unit tensor d H H and the d-function, we LL "nd that

p (n!n )" u . (217) e !e " I ? @ 2ma I I Substituting Eq. (216) into Eq. (213) and making use of Eq. (217), we obtain



2pe  "E " V  n n [1!(!1)L >L ] ="    a mu  L  L



[ f (e )!f (e # u)]d(e !e ! u)#2 . @ @ ? @

(218)

Formula (218) gives only the absorption component proportional to "E ". The other V  components may be obtained by obvious interchanges of subscripts. If the electron spectrum quantization is unessential, then we may employ the relation p " (pn /a ) and replace the sum entering (218) by an integral over the quasi-continuous I I I

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momentum. Thus, we obtain

 

p <e2(2m  "E " 1 I  I # u[ f (e)!f (e# u)] dp p =" I 2m  a

(pmu) 8 I I en 3  v $ "E " .

< (219) a I  mu 4 I I The factor 1/8 before the integral, has appeared because integration over the region p 50 I was extended to all values of the vector p. Moreover, when deriving Eq. (219) from Eq. (218), we replaced the periodic function +1!(!1)L, of the index n by its average value which is equal to one. The case of essential quantization of the electron spectrum, as well as other details of the parallelepiped model, are considered in Ref. [180]. Let us now compare the approaches and solutions of this model with the results of other models. First, the optical conductivity in this model is also a tensor quantity (see Eq. (219)). Second, as follows from Eq. (166), the replacement of the frequency l by the frequency tensor (v /a , v /a , v /a ) in the high-frequency Drude case (u
(220)

where signs (#) and (!), respectively, correspond to the absorption and emission of a quantum. (To avoid misunderstanding, we note that the authors of [126] employ a real vector potential, whereas the treatment in this section involves a complex vector potential.) Making use of Eq. (220) we may write the energy absorbed by an electron gas scattered on a surface of area S : V   2 uS  V =" dp v D>(e ) dp dp f (e)[1!f (e# u)] V V V V W X (2p ) V N Y \ \   dp v D\(e ) dp dp f (e)[1!f (e! u)] . (221) 2 uS V V V W X V ! \ \ (2p ) V

CYS NV 













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In the second term of Eq. (221) we introduce a shift e Pe ! u and take into account that dp V V V v "de , f (e) f (e! u)Pf (e# u) f (e); D\(e# u)"D>(e ). V V V Then the two terms of Eq. (221) may be combined and reduced to the form



uS V dp v D>(e )[ f (e)!f (e# u)] . =" V V V V (2p )

(222)

This expression, with the accuracy of the factor 1/2, reproduces formula (222) for k"1. But since the parallelepiped has two surfaces S (the front and the back walls), the coincidence of the results V obtained by means of the two approaches is obvious. Inasmuch as only one surface was taken into account in Eq. (222), this formula could not contain the collision frequency of the type v /a . This is another argument that supports the above $  interpretation of Eqs. (180) and (181). It is also instructive to compare the transversal conductivity of a strongly elongated ellipsoid (R 'R ), calculated in this section, and of a cylindrical sample, calculated in [179]. According to , , (180) and (177), one has for the ellipsoid




9 27p ne v e k  $" < p + . , 128 mu R 32p R  u , , For the cylindrical sample at k< u, calculations [179] give

(223)




k  3e . (224) p K , pR  u , It is seen that expressions (223) and (224) coincide to within a factor of 3p/32+1, as should be expected in this limiting case (R
In a more general case when e O1, formula (141) turns to

(E) H . (225) "(E ) "" * H "e #¸ (e!1)"#(¸ e) K H H

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This expression must be substituted for the "rst summand in (146), which characterizes the electric absorption. In the case when the particle dimension is smaller than the electron mean free path, it is also necessary to make the change eP(4p/u)p (u) , HH where p (u) are the components of the optical conductivity tensor. HH The plasma frequency is determined from the resonance condition

(226)

e #¸ (e!1)"0 . (227)

H The value e is given by formula (137), so condition (227) results in the following expression for the plasma frequencies: u  . (228) uH"  [1!e (1!1/¸ )]

H This expression shows that an ellipsoidal metal particle is characterized by three plasma frequencies corresponding to charge oscillations along three principal axes of the ellipsoid. As noted above, the electric absorption is given by the "rst summand in (146), which in the case e O1

and with the account of (226) and (227) assumes the shape (E) ue  H = "<  8p "e #¸ (e!1)"#(¸ e) H H H  (E )(uuH)/u¸ ue  H   H "< . (229) 8p (u!uH)#e (uuH)/u H    It is seen that the electric absorption has maxima at the plasma frequencies (228). For particles smaller than the electron mean free path, the change (226) additionally modi"es the shape of the plasma absorption peaks. As one passes over from a single metal particle to an ensemble of such particles (e.g. an IMF), the absorption acquires some new important features. The localized electron density oscillations in separate particles possess oscillating dipole moments, which interact with each other and arrange themselves into collective modes. The simplest way of taking into account the interaction of the particles is to change the external "eld E in (140) by the sum of this "eld with the "elds of all dipoles induced by it in the rest of the particles. The summation of the dipole "elds can be easily carried out in various model geometries, e.g. for a system of identical ellipsoidal particles located in the points of a square lattice [189,193,194]. It has been shown that in such a case the Coulomb interaction can be formally considered through the change of the depolarization factor ¸ in H formulas (225) and (227)}(229) by an e!ective factor ¸ P¸ !(
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It should be noted that a huge literature is devoted to plasma resonance absorption in dispersed systems (see e.g. [165,175,176,195] and references therein). We have brie#y dwelt on this mechanism only to stress that a strong optical absorption by small metal particles is possible not only in the IR range considered in detail above, but in the visible range as well. Correspondingly, the nonequilibrium heating of electrons and phenomena accompanying it are also possible in this case. It should also be recalled that radiative decay of plasmons excited in the inelastic tunneling of electrons or by hot electrons seems to be one of the mechanisms responsible for the light emission from IMFs (Section 4). 7.9. Conclusions about optical absorption of small particles In this section we have given expressions for electric and magnetic absorption by nonspherical particles smaller than the electron mean free path [178]. We have found that for small asymmetric particles, their electric and magnetic absorption can vary by several orders of magnitude under particle shape variations with the volume remaining constant. Such drastic variations in absorption can also occur under variations of wave polarization. Simple analytic formulas have been derived for highly elongated and highly #attened particle shapes. We have also established that for nonspherical metal particles smaller than the electron mean free path, the light-induced conductivity is a tensor, in contrast to the Drude case. The components of the conductivity tensor have been found for particles in the form of an ellipsoid of revolution. We have studied the dependence of these components on the degree of particle asymmetry. It should be emphasized that most theoretical investigations of the optical properties of island metal "lms were devoted to the mutual e!ect of the particles on local "elds and electric absorption. In recent years the re#ection of IR radiation from a layer of small metallic particles has also been studied (see, e.g., Ref. [196]). Allowance for the mutual e!ect of particles can indeed strongly in#uence the values of the local "elds. Estimates have shown [188] that in favorable cases, allowance for this mutual e!ect can alter the local "eld inside a given particle severalfold. However, our results presented in this section show that allowing for particle shape together with considering the electric and magnetic absorption, can change the total absorption by several orders of magnitude. These factors, therefore, must be taken into account from the outset. These features of absorption are even more important in such phenomena as electron and phonon emission from island metal "lms illuminated by laser light. (The fact that absorption by small spherical particles is proportional to their surface area was reported earlier by Manykin et al. [174].) It is in such phenomena that the extreme cross sections of absorption by an ensemble of metal islands play a much more important role than the average (e!ective) cross sections.

8. Examples of applications of island metal 5lms Various phenomena and properties speci"c to IMFs have long been attracting many investigators who tried to develop new cathodes and sensors using these "lms. As usual in practical applications, high stability, reproducibility and economical e$ciency of performance are the most important operational features to be attained in devices based on IMFs. The technological experience accumulated to date shows that these stringent requirements can be satis"ed.

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8.1. IMF cathodes The phenomenon of electron emission induced by passing current through IMFs has been utilized to develop `colda cathodes for vacuum microelectronic devices, which combine advantages of vacuum electronics (such as high thermal and radiation stability) and solid-state electronics with its as yet inexhausted potentialities of miniaturization. A considerable e!ort, especially in the Saratov plant of receiver-ampli"er electron tubes, has been undertaken in the development of gold IMF cathodes activated by barium oxide [74,197]. A technology has been elaborated allowing the fabrication of such cathodes with an e$ciency of 2}5 mA/W and the emission to conduction current ratio c up to 5%. The emission current from a cathode cell sized 0.5 cm;20 lm at the working voltage 10}15 V amounted to tens of lA and was stable in the continuous operation mode for about 1000 h [74]. Such cathodes were exploited in indicating displays. A still higher e$ciency has been achieved with IMF cathodes based on refractory metals [79,198,199]. For example, a Mo thin "lm cathode emitter sized 15 lm; 3 mm produced a stable current up to 100 lA at c"10}15% [198]. However, a disadvantage of this cathode is a comparatively high working voltage of 50}80 V. It should be noted that IMF cathodes can be fed both by direct and alternating current. A few types of triode cells using IMF cathodes have been proposed [200,201]. Quite good results have been obtained with comparatively simple cells depicted in Fig. 8.1a and b. A controlling electrode (`gatea) and an anode were evaporated on a dielectric interlayer previously deposited on the contacts of the cathode. To reduce the leakage currents between the electrodes, the interlayer can be shaped by etching as shown in Fig. 8.1a. The anode current}voltage characteristic exhibits "rst a fast growth of the current with increasing voltage and then a leveling-o! at the electric "elds comparable to the average "eld within the "lm (Fig. 3.3). An e!ective control of the current can be achieved when the operating point is chosen in the steep section of the anode current}voltage curve. 8.2. A gold IMF microcathode As noted above (Sections 2 and 3), electrons are not emitted uniformly from the whole island "lm, but rather come out from the emission centers having size 41 lm and scattered over the "lm. This may complicate the focusing of electron beams obtained from the cathodes representing long and narrow "lms. A more expedient choice in this case is a point-like IMF microcathode [202]. An example of such a cathode fabricated on a pyroceram substrate with Mo contacts is shown in Fig. 8.2. The gap between the contacts was 20;300 lm, and an Au "lm was evaporated into the gap as a strip only 20}30 lm wide. The work function of the "lm was reduced by a BaO overlayer. This cathode gave a stable electron emission at the level of 0.1 lA for a few thousand hours under technical vacuum. The half-width of the electron energy distribution was about 0.5 eV. 8.3. Electron emission from island xlms of LaB  Lanthanum hexaboride is a high-melting compound with metallic conductivity widely used as an e!ective electron emitter. LaB "lms were evaporated in vacuum using the laser ablation  technique. The substrates were glass plates with previously deposited Pt contacts. The gap between

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Fig. 8.1. Schematic of two triode cells with IMF cathodes. ; is the voltage apllied to the "lm, ; the anode voltage and  ; the regulating voltage.  Fig. 8.2. Electron emission current as a function of applied voltage for an Au "lm (1) and the same "lm covered with BaO (2). Inset (a) shows a typical con"guration of the contacts and "lm in an IMF microcathode.

the contacts was 10 lm. The "lms had an island structure which rearranged in the course of electroforming and after this procedure their conduction current}voltage curves assumed a strongly pronounced nonlinear character [203], as in the case of island metal "lms. Simultaneously, electron emission set in. The examination of the LaB "lm cathodes in an emission microscope  showed that two distinct working regimes are possible, depending on the cathode structure. In one case the emission stems from small spots, as from IMFs. In another case it is more uniform. The reason for this is so far not clear. It is interesting that after a prolonged period of operation, the emission from the region near the negative contact considerably increases, probably due to electromigration of lanthanum toward this contact. On the whole, the LaB "lm cathodes show  a good stability. Their e$ciency is an order of magnitude higher than that of clean Au IMF cathodes, but is considerably worse than the e$ciency of the Au cathodes coated with BaO [74]. 8.4. IMF cathodes with large emitting area The emitting area of most cold cathodes (such as tip and MIM emitters, cathodes based on p}n-junctions, etc.) ranges from small fractions of lm to mm. In order to increase the area, it is possible either to extend the emitter itself (e.g. by fabricating a larger MIM sandwich structure) or to create emitter arrays [204}206]. Each of these ways involves some speci"c di$culties, especially with regard to providing a su$cient uniformity of emission. The problem of fabricating large-area IMF emitters is resolved rather easily. Recall that usually the IMF emitters represent a structure that consists of two contacts separated by a gap +10}20 lm wide and +5}10 mm long where an island "lm is formed. The emitting centers are

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Fig. 8.3. A schematic of contact electrodes of an IMF cathode (a) and a cathodoluminescent image of emission distribution in this cathode (b). Fig. 8.4. The current}voltage curves of conduction current I and emission current I for the cathode shown in Fig. 8.3.  

located within a rather narrow (41 lm) region which occupies only a small part of the gap. For this reason a mere widening of the gap cannot increase the emitting area. There are two ways of creating large-area IMF emitters. The "rst of them is the fabrication of a long zig-zag-shaped cathode between two comb-like electrodes inserted into each other (Fig. 8.3a). The second possibility is to place a large number of microcathodes on one substrate, i.e. to make a cathode matrix. In the former case one actually has a long IMF emitter which is put into a zig-zag form to make it compact and `two-dimensionala. To this end, one "rst deposits the comb-like electrodes on a substrate and then the island "lm. After that the "lm is electroformed by applying a voltage about 15}20 V. The distribution of the emission centers can be judged from the luminescence of the cathode during its operation, because the centers of electron and light emission are known to coincide (see Section 2). The luminescence "rst appears in a few points of the cathode and then, as the voltage is increased up to 15}20 V, propagates along the entire length of the gap between the electrodes. An image of the emitting area of such a cathode obtained on a cathodoluminescent screen is given in Fig. 8.3b. Fig. 8.4 shows the conduction and emission current}voltage curves of

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Fig. 8.5. An example of comb-like contact electrodes, prepared by photolithography, for large-area IMF cathodes. The gap between the `teetha equals 12 lm. Fig. 8.6. The current}voltage curves of conduction current I and emission current I for the large-area cathode with   contact electrodes shown in Fig. 8.5. Inset: a cathodoluminescent image of emission distribution in this cathode.

a large-area IMF cathode in which the zigzag gap was prepared by scratching (using a needle) of a gold "lm evaporated on a glass substrate. Such cathodes can easily be fabricated in various shapes, which is convenient e.g. for purposes of pictorial indication. The emitting regions in an electroformed IMF do not change their positions as the cathode is exposed to the atmosphere. The characteristics presented above related to the large-area IMF cathodes prepared on transparent glass substrates, which are convenient for parallel observation of electron and light emission. However, the emitting strips in these cathodes were arranged at comparatively large distances apart (& 0.1}0.3 mm). A more dense zigzag emitting structures were obtained on ceramic (pyroceram) substrates with Mo comb-like contacts fabricated by photolithography (Fig. 8.5) [207]. In this case the width of the gap was equal to 12 lm. The current}voltage curves of such cathodes are given in Fig. 8.6 and an example of the cathodoluminescent image of the emitting surface is reproduced in the inset. Let us now consider some realizations of matrix IMF emitters. They consist of many elementary cathodes working in parallel. In an example illustrated in Fig. 8.7a and b each elementary cathode represents an island "lm "lling a circular gap (50}60 lm wide) between two concentric electrodes [208]. The conduction and emission current}voltage characteristics of matrix IMF cathodes are similar to those of the zigzag cathodes (Fig. 8.4). Typically, the emission centers take up to 10}25% of the cathode geometrical surface. Fig. 8.7c shows the emission image of a matrix cathode having

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Fig. 8.7. (a) A schematic of a fragment of the matrix IMF cathode. (b) A micrograph of a section of such a cathode. (c) A cathodoluminescent image of emission distribution in this cathode.

a total diameter of about 25 mm. The total emission current extracted from this cathode amounted to 500}100 lA at the e$ciency c"0.6}0.7%. 8.5. SnO island xlm cathodes  A number of works have been devoted to the investigation of electron emission from SnO  island "lms [209}211]. In [209], SnO "lms were deposited onto a polished quartz substrate using  pyrolytic dissociation of SnO at 4503C. The conductivity of the "lms was controlled through the  addition of Sb and NH F. The geometry of the "lms is sketched in Fig. 8.8. To obtain electron  emission, a procedure of electroforming (passing a current through the "lm) was necessary. Later experiments with such "lms showed that in the course of the electroforming the narrow part of the "lm was strongly heated and partially destroyed so that it "nally acquired an island structure [211]. An important advantage of the SnO "lms is the possibility of carrying out their electro forming in air. Typical dependences of the conduction and emission currents on the voltage applied to the "lm are depicted in Fig. 8.8. For the conduction current, this dependence can vary from close-to-linear to close-to-exponential, whereas for the emission current it is always close to exponential. The ratio of the emission current to the conduction current, i.e. the cathode e$ciency c, varies in the range from 1 to 50%. Thermionic emission from the "lm seems improbable, since the emission current is independent of the duration of the pulses applied to the "lm in the range from 1 to 1000 ls. The authors [210,211] supposed that the main physical mechanism in such cathodes may be "eld emission from the SnO islands, although they did not exclude also that some  contribution can stem from hot electrons in the islands. The cathodes were shown to give stable currents densities in the range from 1 to 10 A/cm for hundreds of hours and to retain their emission characteristics after exposure to air. Quite recently, island cathodes based on small particles of metal oxides have been utilized in #at information displays.

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Fig. 8.8. The current}voltage curves of conduction current I and emission current I for a SnO island cathode. Inset:    geometry of the cathode [211].

8.6. Island xlm cathodes for yat information displays Considerable attention has been focused in recent years on the development of #at cathodoluminescent displays. They possess high ergonomic characteristics in comparison with liquid crystal and other types of displays and are expected to take the place of bulky traditional kinescopes in many applications. Up to now, the main hopes have been pinned on arrays of "eld emitters of various con"gurations as the most promising cathodes for such displays (see e.g. [205,206]). However, quite recent publications of Japanese workers from Canon Research Center have presented a #at (9.6 mm thick) cathodoluminescent display using a cathode which they name `the surface conduction emittera [13,212,213]. The emitter was fabricated of "ne PdO particles, and its micrographs show that the size of the particles is about 5}10 nm, so it actually is an island "lm emitter (Fig. 8.9). To fabricate such emitters, an ink-jet printing process is carried out in air. A movable ink-jet head generates very small droplets of a PdO `inka which are programmably placed onto a glass plate. An elementary emitter is &10 nm thick and &100 lm in size (Fig. 8.9). To activate the electron emission from as-deposited "lms, they are subjected to a forming procedure in which the voltage applied to the "lm is increased until the conduction current is almost broken irreversibly [212]. The authors suppose that this occurs due to partial melting of the PdO "lm. The result is a lowering of the driving voltage which must be applied to the cathode to obtain the electron emission. The ratio of the emission current to the conduction current is typically +0.2% at the driving voltage 15 V and anode voltage 1 kV. In the range of the driving voltages from +13 to 17 V, the dependences of both emission and conduction current for one of the cathodes were found to be approximately linear when plotted in the Fowler}Nordheim coordinates. However, as noted in Section 3, such an observation cannot be considered as a su$cient proof of the "eld emission mechanism for these cathodes. Actually, the authors [213] consider a model in which electrons tunnel from one particle to another and then, after multiple elastic re#ections from its surface, travel to the anode. However, the broad electron energy

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Fig. 8.9. The conduction current I and emission current I as functions of the voltage applied to the "lm consisting of   ultra"ne PdO particles. Insets: (a) schematic structure of the cathode, (b) an SEM image of the PdO "lm "red after ink-jet printing [13,212]. Fig. 8.10. A schematic of electron-optical image converter with an IMF cathode for recording and visualizing IR laser radiation.

distribution (with a width of +2 eV) recorded in the work can hardly be explained in the framework of this mechanism. In our opinion, the experimental results [213] are better compatible with the model of hot electrons discussed in this review. Contrary to [213], the model of hot electrons assumes that electrons after tunneling are not elastically re#ected from the adjacent particle, but enter it and share their excess energy with other electrons heating them up. The glass plate serving as a substrate for fabrication of a cathode matrix is previously coated with two systems of parallel metal wires (`column linesa and `row linesa), which are isolated from each other, perpendicular to each other and allow switching on of all the elementary emitters in turn. In this way an image can be displayed on the cathodoluminescent screen placed in vacuum parallel to the cathode matrix. The whole technology has been claimed to be quite simple, reproducible and economical. The authors [13] have succeeded to fabricate with this cathode a 10-in #at display which, in their words, `shows full color images as good as CRTsa. The coming years will probably show if the island "lm cathodes can rival other types of electron emitters in the new generation of #at information displays. 8.7. IMF cathodes for IR electron-optical converters Island thin "lms have been used to develop a cathode for visualization and measurement of the spatial power distribution of pulse infrared laser radiation [9,214]. The cathode represents a gold IMF coating a dielectric substrate which is transparent to IR radiation (Fig. 8.10). The "lm consists of two subsystems of islands: the larger ones (0.1}0.5 lm) ensure an e!ective absorption of IR radiation (see Section 7) while the nanosized islands provide the conductivity in the cathode.

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Fig. 8.11. Laser beam intensity distributions visualized by a carbon IMF (a) and a gold IMF (b). The dark strip in (b) corresponds to a cathode section where gold islands were absent. The laser power density was P"5;10 W/cm.

The known emitters used for the visualization of IR radiation represent matrices consisting of individual IR-sensitive elements [215]. In our case, each island is a functional analogue of the individual element of the matrix. In the island, there occurs a conversion of the energy of incident IR radiation into the energy of an electron gas, which results in electron emission. The spatial power distribution in the laser beam is mirrored in the distribution of the electron emission current density over the IMF cathode. Such an Au cathode prepared on a Si substrate has been tested with a pulsed CO laser (j"10.6 lm, q"1 ls). The density of the emission current amounted to  10}10 A/cm at the laser power density 5;10 W/cm. The cathode reproduces the shape of nanosecond laser pulses [3]. Fig. 8.10 shows an electron-optical converter for visualization of IR radiation with the aid of an Au IMF emitter deposited on a Si plate. The IR beam from a CO laser is introduced into the  vacuum device through a Si window and is incident on the back side of the cathode. On passing the Si plate, it excites the electron gas in the islands and induces the electron emission. The emitted electrons are accelerated with a voltage of &1 kV toward the cathodoluminescent screen to produce a visible image. Examples of such images are given in Fig. 8.11. It has been found that Au IMF emitters allow the visualization and characterization of IR laser beams at power densities of 5;10 to 10 W/cm. Utilizing a microchannel plate, it is possible to detect lower power densities and record photographically single IR radiation pulses. The spatial resolution of the IMF electron-optical converter is limited by the structure of the island "lm. The "lms prepared from refractory metals have been shown to sustain high power densities and provide a good reproducibility even after reception of &10 powerful laser pulses. 8.8. Tensometric sensors The high sensitivity of the resistance of IMFs to deformation of the substrate has been known since the 1960s [216,217]. Later on, a strong e!ect of the deformation of substrate on electron and photon emission characteristics was also found [75,154]. It is generally agreed that the high stress

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Fig. 8.12. Schematic of an IMF tensometric sensor. 1, a steel substrate, 2, an insulating sublayer, 3, contact electrodes and 4, an IMF.

sensitivity of IMFs is due to the tunnel mechanism of their conductivity. Thus the choice of appropriate "lm structure is very important to fabricate tensometric sensors with good performance. The coe$cient of tensosensitivity cH, or piezoresistance coe$cient, is de"ned as cH"(*R/R)/(*l/l) , where *R/R is the relative change of resistance and *l/l is the strain of the substrate. In the Ohmic region of the conduction current}voltage characteristic (up to the mean "elds in the "lm of &10}10 V/cm), cH is practically independent on the applied voltage. cH is very sensitive to the "lm structure and passes through a maximum at the mass thickness of 3}4 nm which in the case of gold IMFs corresponds to islands with an average size &10 nm and the interisland spacings of &2}3 nm [218]. An example of the design of an IMF tensometer is depicted in Fig. 8.12. The substrate was made of a 40 lm steel foil having a small residual strain and coated with a layer of polyurethane lacquer a few lm thick. Two contact electrodes with a 20 lm gap between them were deposited onto the lacquer coating and then an island "lm was evaporated in vacuum. To speed up the stabilization of the "lm structure, it is recommended to evaporate the IMF onto a heated substrate. Then the sample was exposed to air and coated with a protective dielectric "lm. The conduction current}voltage curves of such sensors have a shape typical of the island "lms. The working point was chosen at 1}2 V which corresponded to the Ohmic segment of the curve. The resistance was 1}1.5 M for sensors with the best sensitivity (cH"80}100 for Au "lms and cH"10}30 for Ta, Cr, Mo and Pt "lms). The values of cH remain practically stable in time, but the resistance of the sensors increased by 15}20% over a few years. Similar sensors shaped as membranes were fabricated on a 10 lm Lavsan substrate. They provided a reliable measurement of an excess pressure of a few millimeters of water column. Over 5 years, their resistance increased by not more than 20%. To obtain the IMF tensometers with a higher sensitivity, one can use the e!ect of substrate deformation on the electron emission current [75] and light emission [154]. Such sensors were prepared on thin mica sheets, and cH values of 250}300 were attained.

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Fig. 8.13. Schematic of an IMF-based microsource of light. 1, a substrate, 2, contact electrodes, 3, a BaO "lm (12 nm thick), 4, a protecting SiO coating, 5, an IMF. V

8.9. A microsource of light The light emission from IMFs (see Section 4) has been utilized to fabricate a miniature source of light [219}221] (Fig. 8.13). The width of the contact electrodes on a dielectric substrate was taken 100 lm and the width of the gap between them 10 lm. An island "lm (of Au, Ag, Pt, Cu, Cr, Mo, or Bi) with a mass thickness of 6}8 nm was evaporated into the gap. Its electroforming was carried out in vacuum to obtain electron and light emission centers. Then all the centers save one were burnt by applying voltage pulses with an amplitude which is about twice as high as the working voltage. Then, to lower the working voltage, a thin layer of BaO was evaporated on the "lm and "nally the whole structure was covered by a transparent protective SiO coating. Such a point-like light  source is quick-response, economical, save and can be exploited in air. The size of the luminous area is about 0.5 lm and its radiation power is &10\ W within the spectral range 500}780 nm. 8.10. Hot electrons beyond IMFs At this point it seems appropriate to discuss possible manifestations of hot electrons in other low-dimensional and dispersed systems. First of all, it should be recalled that in many semiconductors, such as Ge, Si, InSb and others, the electron}phonon interaction is weak enough to ensure favorable conditions for intense generation of hot electrons in bulk materials. There is an abundant literature on this topic (see e.g. [109,110]). Taking into account the size e!ect in the electron}phonon interaction considered above, it can be anticipated that the conditions for nonequilibrium heating of electrons in small semiconductor particles should be even more favorable. Hot electrons are known to play an important part in semiconductor nanostructures [222]. The situation in metals is basically di!erent and, as noted above, hot electrons can be generated under stationary conditions only in metal nanoparticles. In making the comparison of metals and semiconductors, it should be remembered that the concentration of free electrons in metals is by many orders of magnitude higher than in semiconductors. Correspondingly, various e!ects which can be stimulated by hot electrons should be much more intense in the case of metals. Since the physical regularities underlying the heating of electrons in small particles seem to be rather general, e!ects similar to those observed in IMFs can be expected in other dispersed systems fed with energy. As an example in this context, let us mention recent works on `hot-electron

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femtochemistry at surfacesa (see e.g. [223}228]). This term was coined for chemical reactions stimulated at surfaces by hot electrons that are generated in the substrate with the aid of powerful femtosecond laser pulses. Alternatively, it was proposed to obtain hot electrons for this purpose in metal}insulator}metal structures [228]. In the latter case one has a possibility to tune the energy of hot electrons to the energy of short-lived negative ion state of adsorbed species which is an important intermediate stage in the chemical reaction. Obviously, hot electrons generated in small particles can manifest themselves in similar chemical reactions which occur on dispersed surfaces such as e.g. supported catalysts. This is an example which demonstrates that exploitation of hot electrons may appear useful in the targeted development of various nanomaterials.

9. Conclusions The whole body of currently available evidence on electron and photon emission phenomena in IMFs seems to be consistent with the model which predicts a strong nonequilibrium heating of the electron gas in nanoparticles. The size of such particles is smaller than the mean free path of electrons in the volume, so the scattering of electrons occurs mainly at the particle surface and the channel of electron energy losses due to generation of volume phonons is essentially cut o!. The result is a strong reduction in the electron-lattice energy transfer, which provides a favorable possibility for generation of hot electrons when a su$cient power is fed into the particle. This can easily be realized by passing a current through the island "lm or by its laser irradiation. In the former case, the energy is delivered to the emission centers through narrow percolation channels where the current density is very high. In the latter case, the incident electromagnetic energy is absorbed very intensively when the particle has a special shape. The comparison of the theoretical predictions with experimental results shows that the electron emission from IMFs can be interpreted as Richardson emission of the hot electrons which have their own temperature di!erent from that of the lattice. The light emission from IMFs can also have its origin in the appearance of hot electrons. Its possible mechanisms can be the bremsstrahlung, inverse surface photoe!ect, radiation generated by inelastic tunneling of electrons between the islands and the radiative decay of plasma excitations. Thus, from the point of view of electronic kinetics, the metal island "lms are more similar to semiconductors and gas plasmas, where hot electrons occur universally, than to bulk metals where they can be generated only for very short times. The application potential of IMFs, used so far only to a minor extent, ranges from microcathodes and microsources of light to large-area cathodes which seem promising for novel information displays. It is also appropriate to mention that hot electrons have been shown to play an important role in stimulation of surface chemical reactions (`hot-electron femtochemistry at surfacesa). Thus, a deep insight into mechanisms of generation of hot electrons in nanoparticles may appear useful to better understand the known chemical behaviour and potentialities of various dispersed systems (such as supported catalysts). Hot electrons have also an important impact in semiconductor nanostructures. Hopefully, advances in nanotechnologies will open new possibilities in the preparation of the island "lms with more controllable parameters which in turn will ensure further progress in this interesting "eld of nanoscale science.

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Acknowledgements We are grateful to Professor G. Comsa for the invitation to write this review. This work was supported in part by the Ministry of Science and Technologies of Ukraine (grant C 2.4/790). A.G.N. also gratefully acknowledges the support in the framework of the International Soros Science Education Program through grant CSPU072041. We are indebted to Mrs. O.L. Fedorovich, Dr. M.V. Paliy, Dr. V.N. Bykov and Mr. O.E. Kiyayev for their help in preparation of the typescript. Abbreviations IMF } Island metal "lm IR } Infra-red MIM } Metal-insulator-metal STM } Scanning tunneling microscopy VCNR } Voltage-controlled negative resistance

Appendix A The solution of the SchroK dinger equation linearized with respect to the amplitude of the electromagnetic wave is represented as t(r, t)"t(r)e\ C R#t(r, t) ,

(A.1)

where the "rst term gives the solution in the absence of the wave and the second term accounts for the contribution of the wave "eld. The function W(r, t) can be found from the equation





Rt e p(  i

! #;(x) t(z, t)"! (Ap( )t(r)e\ C R . Rt mc 2m

(A.2)

Taking into account the time dependence of A(t) (29), we can write !(Ap( )t(r)e\ C R"i (A)t(r)+e\  C\ SR#e\  C> SR, .

(A.3)

This relationship allows Eq. (A.1) to be represented as 2e AU!(r)e . (A.4)

>\ Equations for functions U! can be obtained by substituting (A.4) into (A.2) and equating the G terms that contain the same components A. In particular, the equation for U! is G G t(r, t)"i





2m R D# (e$ u!; ) U!" t(r) . V V

 Rx

(A.5)

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Consider now in more detail the case of a rectangular barrier along x having a width a and  height ; (see Eq. (45)). With this barrier, one obtains t(r)"t(x)e IW W>IX X



e IV V#R e\ IV V, x40 ,  ,e IW W>IX X B eG V#B e\G V, 04x4a ,    x5a . C e I V,  

(A.6)

The coe$cients R , B , B and C are determined from conditions of joining the function (A.6)     and its derivatives in the points x"0 and x"a . The procedure of the joining gives a system of  algebraic equations which determine the unknown coe$cients: 1#R "B #B ,    ik (1!R )"i (B !B ) , V     B eG ?#B e\G ? "C e IV ? ,    i (B eG ? !B e\G ? )"ik C e IV ? .  V   

(A.7)

Considering the structure of function (A.6), the dependence of U (r) on the variables y and z can V be separated into an isolated factor: U (r)"u(x)e IW W>IX X . V

(A.8)

The substitution of (A.8) into Eq. (A.5) gives du 2m # (e $ -)u(x)"ik (e IV V!R e\ IV V), x40 , V 

 V dx du 2m # (e $ -!;)u(x)"i (B eG V!B e\G V), 04x4a ,     dx

 V

(A.9)

du 2m # (e $ -)u(x)"ik C e IV V, x5a .    dx

 V It can be easily checked by immediate substitution into (A.9) that its solution can be represented as

Rt(x) u(x)"$ #u!(x) . 2mu Rx

(A.10)

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Here the shape of u!(x) is given by formula (46). The "rst term in (A.10) is a partial solution of the nonuniform system of equations (A.9) and u!(x) gives a solution of the uniform system. The conditions of joining the function (A.10) and its derivative in the points x"0 and x"a  give a system of algebraic equations which determine the coe$cients d, f , f , and g:   d"f #f ,   f eA? #f e\A? "ge O? ,  

(A.11) ; (c#iq) f #(!c#iq) f "G (B #B ) ,   

u  ; (c!iq)eA? f !(c#iq)e\A? f "G  e IV ? C .   

u The probability of inelastic tunneling is determined by the coe$cient g and that of the inelastic re#ection from the barrier by the coe$cient d. Let us illustrate this statement for the case of tunneling. As can be seen from Eqs. (A.1), (A.4), (A.6),(A.8) and (A.10), the electron wave function at xPR is given by




2e A t(r, t)"C e> IV e IWW>IXXe\ C R#i  c V








$ C ik e IV V#ge OV 2mu  V

;e IW W>IX Xe\  C! SR .

(A.12)

E It should be recalled that the summands U!(r) and U!(r) do not contribute to the inelastic W X current if the dispersion of the electromagnetic wave is not considered. By substitution of function (A.12) into expression (41) for the current I we obtain V








 

e k e 2eA  1 k

k V V $ V #1 "C " . I " V "C "# q"g"# V   m m mc 4 mu mu

(A.13)

The cross-contribution of the "rst and second terms in (A.12) to the current vanishes when averaged over the wave period. The "rst summand in (A.13) determines the probability of the elastic tunneling. In the braces, the "rst term accounts for the probability of the inelastic tunneling while the second one gives a correction to the probability of the elastic tunneling due to the presence of the wave "eld. The intelastically re#ected current can be calculated similarly to (A.13). In this case, "d" appears instead of "g" in the expression for the current. The coe$cients g and d are found from (A.11) with allowance for (A.7). The expression for "g" is given by (48). For "d", we obtain




;  k V +[ci !ci ch ca ch i a #qk sh ca sh i a ]      V   

u G(i , k )G(c, q)  V # [ck ch ca sh i a #(i q ch i a sh ca )], . (A.14) V        The calculation method presented above can easily be generalized for the case of an asymmetric barrier. "d""4

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Appendix B As suggested in Section 7, the mutual e!ect of metal islands on the local "eld strength in a given island can be formally accounted for by replacing the depolarization factor ¸ with ¸ !*¸ . H H H Below, we will "nd the explicit form of *¸ for a particular model. H It should be noted that the problem of the exact determination of the local "eld presents considerable di$culties even in the simplest case of two identical spherical particles (see e.g. [229,230]). Usually, an approximation is applied in which the electrostatic potential induced by an island is expanded as a power series in multipoles. As a rule one restricts the consideration to the lowest multipoles (most often, the dipoles). Such an approximation is not very appropriate for an island "lm in which the distance between the islands is of the same order as their size. In what follows we describe a method of the calculation of the local "eld which does not apply the multipole expansion [231]. Consider a linear periodic chain consisting of identical metal islands. The mutual e!ect of the islands on each other will be maximum when the external electric "eld E is parallel to the chain axis. We assume just such an orientation of the "eld. Since both the size of the islands and spacings between them are in our case much smaller than the length of the incident wave, the problem of the determination of the resulting "eld reduces to the solution of the Laplace equation with appropriate boundary conditions: *u (r)"0 ,  u>(r)" "u\(r)" ,     (B.1) Ru> Ru\   " . e Rn Rn   The signs (#) and (!) correspond here to the limiting values of the function u inside and  outside the surface S, respectively, and n is the outward normal to it. The dielectric susceptibility of the medium is taken equal to 1, and that of the islands to e. The solution of Eq. (B.1) can be represented as







ds o (r)   . (B.2) u (r)"!Er#  "r!r!ak" I\ Here o (r) is the surface charge density which, with the account for the assumed periodicity of the  chain, obeys the following integral equation [232]:



cos h  1 1!e 1!e I ds o (r) "! En . (B.3) o (r)#   2p"r!r!ak" 2p 1#e 1#e I\ h is the angle between the vectors r!r!ak and n , where n is the outward normal to the I PY PY surface in the point r and a is a vector connecting the centers of two adjacent islands. The direction of the vector a to one or another side along the chain plays no role, since the summation is carried out over all k's.

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Eq. (B.3) has an exact solution for the case of a sungle island, i.e. for k"0 (or aPR): 3 e!1 o" En .  4p e#2

(B.4)

The substitution of (B.4) into (B.2) with retaining only k"0 gives the known result for a dielectric sphere in a uniform external electric "eld. It will be recalled once again that at frequencies much lower than the plasma frequency, a conducting metal island behaves as a dielectric. In a linear chain of identical spherical islands, the surface charge density o (r) depends only on  the angle h between the "eld E and the radius vector r in the point on the sphere surface. Therefore o can be expanded in the general case into a series   o " C P (cos h) .  K K K

(B.5)

As it is clear from the symmetry considerations, Eq. (B.5) contains only odd Legendre's polynomials. By substitution of (B.5) into (B.3), multiplying the result by P (cos h) sin h and then by L integration over all h's, one obtains a system of algebraic equations which determine the unknown coe$cients C : K




 
 



1 4 (n#m)! R L>K>   C kL>K> K (2m#1)(2n#1) n!m! a I K 1 # n(2n#1)

1 E 1#e # C "! d . 2n#1 L 6p L 1!e

(B.6)

Here R is the island radius; besides, the indices m and n pass over odd numbers only. The sums over k's in (B.6) converge rapidly to unity. For example,  1/k+1.2;  1/k+1.04. I I It can be easily found from Eq. (B.6) that when a"3R, i.e. when the gap between the islands equals the island radius, the coe$cient C and all the subsequent ones are much smaller than C .   The value of C can be comparable to that of C only in the case if the islands almost touch   each other. For a known charge distribution, the electrostatic potential can be found with the aid of Eq. (B.2). However, our problem is substantially simpli"ed by the circumstance that it is su$cient to determine only the component of the local "eld inside the island normal to its surface. If the surface charge density is known, this component of the local "elds is easily calculated from the boundary conditions: (E !E )n "4po (r) , * P  E n "eE n . P * P

(B.7)

Here E is the "eld at the outward side of the island. It follows from (B.7) that E n "4po /(e!1) . * P 

(B.8)

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The normal component of the local "eld will have its maximum value, equal to the full local "eld, when its direction coincides with that of the external "eld. It is just in this direction that the surface charge density induced by the "eld is maximum. This direction corresponds to h"0 in expansion (B.5). In such a case o" " C . (B.9)  F K By orienting n in Eq. (B.8) along the external "eld (i.e. taking h"0), we obtain  4p 4p  E " C . (B.10) o" " * e!1  F e!1 K K As noted above, the coe$cient C is much larger than C and all the following coe$cients even   in the case when the gap between the adjacent islands is equal to the island radius. At larger gaps this tendency is all the more pronounced. Thus, one may retain in sum (B.10) only the coe$cient C which is determined by Eq. (B.6). As a result. one obtains from Eq. (B.10) the local "eld inside  a metal island comprising a part of the periodic linear chain of identical islands:








1 4 R  1 \ . (B.11) E "E 1#(e!1) ! * 3 3 a k I As the local "eld E is rewritten in a standard shape  E "E+1#(e!1)[¸"*¸],\ , (B.12) * its comparison with (B.11) gives ¸"1/3 for a spherical island. The parameter D¸ for this situation is *¸+(R/a) . (B.13)  In a similar way one can consider a periodic chain of ellipsoidal islands. Suppose the islands are identical ellipsoids of revolution with their major axes oriented along the chain. Taking into account the explicit form of the right-hand side of (B.3), it is convenient to present the charge distribution over the ellipsoid surface as  (R R ) , , C P (cos h) . (B.14) o" K K  (R cos h#R sin h) , , K Here h is the angle between the radius vector to a point at the surface and the major axis of the ellipsoid which coincides with the direction of the external "elds; R and R are the major and , , minor semiaxes of the ellipsoid. The coe$cients C can also be found from a system of algebraic K equations, similar to (B.6), which can be obtained in the same way as described above. In the same approximation as that applied to derive Eq. (B.11), we "nd for the chain of ellipsoidal particles:








R  1 4 E "E 1#(e!1) ¸ ! (1!e ) , , 3 N a * k I



\

,

(B.15)

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where ¸ is the depolarization factor, determined by formula (147), and e "1!R /R . The , N , , value a stands as before for the distance between the centers of the islands. For *¸ we obtain the , following expression from (B.15): (B.16) *¸ +(1!e)(R /a) . N , ,  If the gaps between the islands are very small, the values of the local "elds can easily be determined more exactly by using (B.10) and retaining not only C , but also C (and possibly C ).    References [1] P.G. Borziak, O.G. Sarbej, R.D. Fedorovich, Phys. Stat. Sol. 8 (1965) 55. *** [2] L.I. Andreeva, A.A. Benditsky, L.V. Viduta, A.B. Granovskii, Yu.A. Kulyupin, M.A. Makedontsev, G.I. Rukman, B.M. Stepanov, R.D. Fedorovich, M.A. Shoitov, A.I. Yuzhin, Fizika Tverdogo Tela 26 (1984) 1519 (in Russian). [3] A.A. Benditskii, L.V. Veduta, Yu.A. Kulyupin, A.P. Ostranitsa, P.M. Tomchuk, R.D. Fedorovich, V.A. Yakovlev, Izvestiya Akademii Nauk SSSR, Ser. Fiz. 50 (1986) 1634 (in Russian). [4] A.A. Benditskii, L.V. Veduta, V.I. Konov, S.M. Pimenov, A.M. Prokhorov, P.M. Tomchuk, R.D. Fedorovich, N.I. Chapliev, V.A. Yakovlev, Poverkhnost Fiz. Khim. Mekh. No. 10 (1988) 48 (in Russian). *** [5] D.A. Ganichev, V.S. Dokuchaev, S.A. Fridrikhov, Pisma v ZhTF 8 (1975) 386 (in Russian). [6] P.G. Borziak, Yu.A. Kulyupin, Elektronnye Processy v Ostrovkovykh Metallicheskikh Plenkakh (Electron Processes in Island Metal Films), Naukova Dumka, Kiev, 1980 (in Russian). *** [7] S.A. Nepijko, Fizicheskiye Svoistva Malykh Metallicheskikh Chastits (Physical Properties of Small Metal Particles), Naukova Dumka, Kiev, 1985 (in Russian). ** [8] H. Pagnia, N. Sotnik, Phys. Stat. Sol. (a) 108 (1988) 11. ** [9] R.D. Fedorovich, A.G. Naumovets, P.M. Tomchuk, Prog. Surf. Sci. 42 (1993) 189. * [10] L.I. Maissel, R. Gland (Eds.), Handbook of Thin Film Technology, McGraw-Hill, New York, 1970. [11] L.I. Maissel, in: Physics of Thin Films, Vol. 3, Academic Press, New York, 1966, p. 61. [12] K.R. Lawless, in: G. Hass, R. Thun (Eds.), Physics of Thin Films, Vol. 4, Academic Press, New York, 1967. [13] E. Yamaguchi, K. Sakai, I. Nomura, T. Ono, M. Yamanobe, N. Abe, T. Hara, K. Hatanaka, Y. Osada, H. Yamamoto, T. Nakagiri, J. Soc. Inform. Display 5 (1997) 345. ** [14] E. Bauer, H. Poppa, Thin Sol. Films 12 (1972) 167. ** [15] K.L. Chopra, Thin Film Phenomena, McGraw-Hill, New York, 1969. [16] V.M. Ievlev, L.I. Trusov, V.A. Kholmiansky, Strukturnye prevraschenia v tonkikh plenkakh (Structure Transformations in Thin Films), Metallurgia, Moscow, 1982 (in Russian). [17] L.I. Trusov, V.A. Kholmiansky, Ostrovkovye metallicheskiye plenki (Island Metal Films), Metallurgia, Moscow, 1973 (in Russian). * [18] A. Barna, P. Barna, J. Pocza, Vacuum 17 (1967) 219. [19] G. Honjo, K. Takeyanagi, K. Yagi, K. Kobayashi, Jpn. J. Appl. Phys. 2 (1974) 539. [20] S.A. Nepijko, Mikroelektronika 5 (1976) 86 (in Russian). [21] A. Barna, P. Barna, R. Fedorovich, H. Sugawara, D. Radnoczi, Thin Solid Films 36 (1976) 75. [22] D. Leonard, M. Krishnamurthy, C.M. Reaves, S.P. Denbaars, P.M. Petro!, Appl. Phys. Lett. 63 (1993) 3203. [23] J.M. Moison, F. Houzay, F. Barthe, L. Leprince, F. Andre, O. Vatel, Appl. Phys. Lett. 64 (1994) 196. [24] P. Tognini, L.C. Andreani, M. Geddo, A. Stella, P. Cheyssac, R. Kofman, A. Miglori, Phys. Rev. B 53 (1996) 6992. * [25] R. Kern, H. Niehus, A. Schatz, P. Zeppenfeld, J. George, G. Comsa, Phys. Rev. Lett. 67 (1991) 855. ** [26] E. Sondergard, R. Kofman, P. Cheyssac, A. Stella, Surf. Sci. 364 (1996) 467. [27] M. Zinke-Allmang, in: M. Tringides (Eds.), Surface Di!usion: Atomistic and Collective Processes, Plenum Press, New York, 1997, p. 389. * [28] G.R. Carlow, R.J. Barel, M. Zinke-Allmang, Phys. Rev. B 56 (1997) 12519. [29] G. Rosenfeld, M. Esser, K. Morgenstern, G. Comsa, Mat. Res. Soc. Symp. Proc. 528 (1998) 111. [30] G. Rosenfeld, K. Morgenstern, I. Beckmann, W. Wulfhekel, E. Laegsgaard, F. Besenbacher, G. Comsa, Surf. Sci. 401 (1998) 402}404.

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[199] H. Araki, T. Hanawa, Thin Solid Films 158 (1988) 207. [200] A.S. Sukharier, S.V. Zagrebneva, E.N. Petrov, V.M. Suchilin, V.M. Trusakov, Bulletin of Inventions of USSR, No 8, Certi"cate No 296179 (1971). [201] A.S. Sukharier, S.V. Zagrebneva, V.A. Osipov, E.N. Petrov, V.M. Trusakov, B.L. Serebrjakov, Bulletin of Inventions of USSR, No 25, Certi"cate No 349044 (1972). [202] R.D. Fedorovich, A.G. Naumovets, P.M. Tomchuk, 9th Intern. Conf. on Vacuum Microelectronics Digest, St Petersburg, 1996, p. 179. [203] S.S. Ivanets, N.G. Nakhodkin, A.I. Novoselskaja, R.D. Fedorovich, Bulletin of Inventions of USSR, Certi"cate No 1271278 (1985). [204] G.A. Vorobyov, V.V. E"mov, L.A. Troyan, S. Lubsanov, Abstr. 5th Symp. on nonheated cathodes, Tomsk, 1985, p. 240. [205] C.A. Spindt, I. Brodie, L. Humphrey, E.R. Westerberg, J. Appl. Phys. 47 (1976) 5248. * [206] P.R. Schwoebel, I. Brodie, J. Vac. Sci. Technol. B 13 (1995) 1391. [207] R.D. Fedorovich, A.G. Naumovets, P.M. Tomchuk, Condensed Matter Physics, 7 (1996) 5. * [208] L.V. Viduta, R.D. Fedorovich, Abstracts of the 16th All-Union Conference on Emission Elecronics, Makhachkala, 1976, p. 40 (in Russian). [209] M.I. Elinson, A.G. Zhdan, G.A. Kudintseva, M.Ye. Chugunova, Radiotekhnika i Electronika 10 (1965) 1500 (in Russian). ** [210] V.V. Nikulov, G.A. Kudintseva, M.I. Elinson, L.A. Kosulnikova, Radiotekhnika i Electronika 17 (1972) 1471 (in Russian). [211] G.A. Kudintseva, M.I. Elinson, in: M.I. Elinson (Ed.), Nenakalivayemye Katody (Nonheated Cathodes), Sovietskoye Radio, Moscow, 1974, p. 29 (in Russian). ** [212] K. Sakai, I. Nomura, E. Yamaguchi, M. Yamanobe, S. Ikeda, T. Hara, K. Hatanaka, Y. Osada, H. Yamamoto, T. Nakagiri, Proc. 16th Int. Display Res. Conf. (Euro Display 96) (1996) 569. [213] A. Asai, M. Okuda, S. Matsutani, K. Shinjo, N. Nakamura, K. Hatanaka, Y. Osada, T. Nakagiri, Society of Information Display, Int. Symp. Digest Tech. Papers (1997) 127. [214] A. Benditskii, D. Danko, R. Fedorovich, S. Nepijko, L. Viduta, Int. J. Electron. 77 (1994) 985. [215] B.M. Singer, Patent USA No 3919555 (1975). [216] R.L. Parker, A. Krinsky, J. Appl. Phys. 14 (1972) 2700. [217] J.F. Morris, Thin Solid Films 11 (1972) 259. [218] L.V. Viduta, A.P. Ostranitsa, R.D. Fedorovich, S. Chumak, Ultradispersnye chastitsy i ikh ansambli (Ultrdispersed Particles and Their Ensembles), Naukova Dumka, Kiev, 1982, p. 110 (in Russian). [219] Yu.A. Kulyupin, K.N. Pilipchak, R.D. Fedorovich, Bulletin of Inventions of USSR, No 43, Certi"cate No 1193843 (1985). [220] Yu.A. Kulyupin, K.N. Pilipchak, R.D. Fedorovich, Bulletin of Inventions of USSR, No 40, Certi"cate No 1279433 (1986). [221] Yu.A. Kulyupin, K.N. Pilipchak, in: Dispergirovannye Metallicheskiye Plenki (Dispersed Metal Films), Institut Fiziki AN Ukr SSR, Kiev, 1972, p. 238 (in Russian). [222] Jagdeep Shah (Ed.), Hot Carriers in Semiconductor Nanostructures: Physics and Applications, Academic, San Diego, 1992. ** [223] J.W. Gadzuk, L.J. Richter, S.A. Buntin, D.S. King, R.R. Cavanagh, Surf. Sci. 235 (1990) 317. * [224] R.R. Cavanagh, D.S. King, J.C. Stephenson, T.F. Heinz, J. Phys. Chem. 97 (1993) 786. [225] M. Brandbyge, P. Hedegard, T.F. Heinz, J.A. Misewich, D.M. Newns, Phys. Rev. B 52 (1995) 6042. [226] R.G. Sharpe, St.J. Dixon-Warren, P.J. Durston, R.E. Palmer, Chem. Phys. Lett. 234 (1995) 354. [227] J.W. Gadzuk, Surf. Sci. 342 (1995) 345. [228] J.W. Gadzuk, J. Vac. Sci. Technol. A 15 (1997) 1520. * [229] A. Goyette, N. Navon, Phys. Rev. B 13 (1976) 4320. [230] R. Ruppin, J. Phys. Soc. Japan 58 (1989) 1125. [231] E.D. Belotskii, P.M. Tomchuk, Int. J. Electronics 73 (1992) 915. [232] D.Ya. Petrina, Zhurn. Vychislitelnoi matematiki i mat. "ziki 24 (1984) 709.

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BARYON RESONANCE EXTRACTION FROM nN DATA USING A UNITARY MULTICHANNEL MODEL

T.P. VRANA , S.A. DYTMAN , T.-S.H. LEE
Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA
Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Baryon resonance extraction from nN data using a unitary multichannel model T.P. Vrana , S.A. Dytman *, T.-S.H. Lee
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA
Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Received 12 July 1999; editor: G.E. Brown Contents 1. Introduction 2. The CMB unitary multi-channel model 2.1. Representation of three-body "nal states 2.2. Model details 2.3. Resonance parameter extraction 2.4. Relationship to other models 3. Database 3.1. nN elastic data 3.2 nNPnnN data 3.3. Description of the nNPgN analysis 4. Illustrative examples } P and S   partial waves 4.1. Single-channel}single-resonance case 4.2. Two-channel}two-resonance case

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4.3. Cusp structure in the S partial wave  4.4. Model dependence in analysis of the S  partial wave 4.5. Elastic data dependence 4.6. The non-resonant amplitude 5. Results and discussion 5.1. Details of "tting 5.2. General results 5.3. Detailed discussion } D , D , and S    partial waves 5.4. Observables 6. Conclusions Acknowledgements References

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Abstract A unitary multi-channel approach, "rst developed by the Carnegie-Mellon Berkeley group, is applied to extract the pole positions, masses, and partial decay widths of nucleon resonances from the partial wave amplitudes for the transitions from nN to eight possible "nal baryon}meson states. Results of single energy analyses of the VPI group using the most current database are used in this analysis. A proper treatment of threshold e!ects and channel coupling within the unitarity constraint is shown to be crucial in extracting resonant parameters, especially for the resonance states, such as S (1535), which have decay thresholds very 

* Corresponding author. Tel.: 412-624-9244; fax: 412-624-9163.  Present Address: Fisher Scienti"c Corp., Pittsburgh, PA. E-mail address: [email protected] (S.A. Dytman) 0370-1573/99/$ - see front matter  2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 0 8 - 8

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close to the resonance pole position. The extracted masses and partial decay widths of baryon resonances up to about 2 GeV mass are listed and compared with the results from previous analyses. In many cases, the new results agree with previous analyses. However, some signi"cant di!erences, in particular for the resonances that are weakly excited in nN reactions, are found.  2000 Published by Elsevier Science B.V. All rights reserved. PACS: 13.30.!a; 13.75.Gx; 13.30.Eg; 14.20.!c

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1. Introduction The interest in the study of baryon resonances began many years ago and led to the important discovery of SU(3) symmetry. Although many states were discovered, the quality and scope of the data limited the analyses. Interest has grown signi"cantly in the last few years because of the prospects for new data of high quality at facilities such as the Thomas Je!erson National Accelerator Facility, the Bonn synchrotron, and Brookhaven National Laboratory. These are excited states of the nucleon with excitation energy of about 0.3}1 GeV. Since most of these states were discovered in amplitude analyses of nNPnN scattering data, they are labeled by the approximate mass and the nN quantum numbers: the relative orbital angular momentum ¸, the total isospin ¹, and the total angular momentum J. For example, the D (1520)  resonance has a mass of about 1520 MeV, isospin ¹", total angular momentum J", and   decays into a ¸"2 nN state. Information about the various baryon resonances is evaluated and tabulated in biennial publications of the Particle Data Group [1]. In their 1998 publication, they list about 20NH (¹") and 20D (¹") states. Some states are well established in the data and   most analyses agree on their properties. On the other hand, some very large discrepancies exist between di!erent analyses. For example, the extracted full widths for the S (1535) state (A 4H  state of PDG [1]) are 66 MeV [2], 120$20 MeV [3], 151$27 MeV [4], 151}198 MeV [5], and 270$50 MeV [6]. This state has a number of striking properties } unusual strong decay width to gN [1] and an unusually #at transition form factor [7] } that have made it a very interesting state to understand. Although there are a number of strongly excited states (rated 4H or 3H by PDG), there are many weakly excited states whose properties are poorly determined with the existing data. The di!erences between various amplitude analyses originate for various reasons, including handling of data, method of parameterizing non-resonant background, and handling of the many NH decay channels. We emphasize the fact that the baryon resonances always decay into a baryon and one or more of various mesons; the existing analyses di!er from each other signi"cantly in the methods used to describe this intrinsically multi-channel problem. Thus, model dependence in extraction of resonance properties makes it di$cult to test predictions of theoretical models with existing data. With the new experimental facilities, the situation will soon be greatly improved when more exclusive data for di!erent "nal states, such as gN, nD, and oN, become available. One of the major goals of the new experiments is to obtain data for a large number of reactions. This greatly increases the probability of seeing new resonances, e.g. those that couple weakly to the nN channel, but puts signi"cant demands on models to interpret the data consistently. The objective of this work is to revive the multi-channel analysis of partial wave amplitudes into resonance parameters of the Carnegie-Mellon Berkeley (CMB) group. Although this is meant to be a signi"cant step toward a full analysis of all contributing reactions, the equally important problem of extracting partial wave amplitudes from the observables is not considered in this paper. The baryon resonances (called NH states from now on in this paper) extracted from amplitude analyses are thought to be predominantly composed of 3 valence quarks because of the SU(3) symmetry seen in the spectrum of hadrons of low total angular momentum. This notion has been the basis for developing various quark models, ranging from the well-studied Constituent Quark Models [8}11], Chiral Bag Models [12,13], NJL models [14], Soliton models [15], to the most recent Chiral Constituent Quark Models [16]. All of these hadron models are motivated by QCD

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and are constructed phenomenologically to describe the hadron spectrum in terms of suitable parameterizations of quark con"nement mechanisms and quark}quark interactions. With some additional assumptions on the decay mechanisms, these models can use the constructed wave functions to predict the decay widths of the NH states. For example, the one-body form of quark currents is used [10,17] in most of the constituent quark models in predicting the NHPcN decay width. A P model is assumed in Ref. [18] for creation of q q pairs in the  calculation of the decay of an NH into nucleons and mesons. The comparison of these predictions with the decay width data, such as that listed by PDG, is clearly a more detailed test of hadron models. In recent years, lattice QCD calculations have been extended to predict masses of the low-lying baryons [19]. Although the lattice QCD calculations are far superior to the empirical quark models, they are much more di$cult to apply to the decay widths. In any case, precise data of resonance parameters such as masses and partial decay widths for various "nal states are needed to distinguish QCD-inspired hadron models and, ultimately, understand the non-perturbative aspects of QCD. The determination of resonance parameters is almost always a two-step process. First, phase shift analyses (elastic data) and isobar analyses (inelastic data) are used to separate the cross section and polarization observable data into partial wave amplitudes, often in the form of ¹ matrices. With a large data set, this determination has less model dependence than the extraction of resonance properties in the second step. If non-resonant e!ects are small, resonances will show up as counterclockwise rotations in the complex energy plane (Argand diagram) as the energy dependence of these ¹ matrices is plotted. Partial wave amplitudes have been determined by various groups for elastic scattering and much less often for inelastic scattering. Older elastic scattering analyses of Carnegie-Mellon Berkeley (CMB) [20] and KarlsruK he Helsinki (KH) [3,21] stressed the importance of theoretical constraints such as dispersion relations to ensure the uniqueness of the "t. The data situation has improved signi"cantly since then; the more recent work of the Virginia Polytechnic Institute and State University (commonly called VPI) group is the most visible recent e!ort [2]. They have regularly updated the data base and attempted to cull out older less viable results. They determine single-energy (also called energy-independent) ¹ matrices for partial waves up to ¸"6 and satisfy a smaller set of dispersion relations than the earlier work. All these analyses are available in the VPI repository [2]. We use the VPI single-energy elastic partial wave amplitudes from the 1995 analysis (SM95) in this work. At the same time, we recognize the point made by HoK hler [22] that this analysis is not as well constrained as the older analyses. We use the single-energy solutions rather than the more debatable smoothed solutions. The second step is to extract the resonance parameters from the partial wave amplitudes. In a simple picture, each transition amplitude in a nN reaction would be parameterized as the product of the excitation strength of the incident channel to a given resonance and the decay strength of the resonance into the allowed "nal states with a resonance propagator for the intermediate state. However, this s-channel resonant mechanism is far from complete since the u-channel and t-channel mechanisms, as implied by crossing symmetry or meson-exchange mechanisms, are known to be important. Thus, the parameterization of the amplitude in each partial wave must contain a resonant part and a non-resonant part (called the background term in most of the literature). Furthermore, the threshold e!ects associated with each decay channel (nN, gN, cN, nD, oN, uN, nNH(1440) and others) must be treated correctly within the multi-channel

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unitarity condition. Thus, resonance extraction requires a signi"cant calculational e!ort and many articles have presented various ways to proceed in practice. The PDG mostly bases its recommended values for baryons on a few works that study the full resonance region (1170 MeV( =(2200 MeV). These include older work by the CMB group [6] and the KH group [3], and more recent work by the Kent State University (KSU) group [4], and the VPI group [2]. A very recent work of Feuster and Mosel [5] "ts data for =(1900 MeV. All of these e!orts use the data of nN reactions. All maintain unitarity, though the methods employed are quite di!erent. This is reasonable since there is more than one way to implement the unitarity condition. However, these analyses di!er signi"cantly from each other in handling the multichannel character of the nN reactions. The CMB and KSU groups use a formalism that allows for many channels, while the KH and VPI groups focus on the nN elastic channel. KH accounts for all inelasticity in absorption parameters and VPI uses a dummy channel to account for all inelasticity. Feuster and Mosel [5] "t elastic scattering and inelastic scattering cross-section data with asymptotic two-body "nal states directly, and account for all the remaining components of the total inelastic cross section with a dummy inelastic channel. For most strongly excited, isolated states, these "ve analyses tend to agree within the assigned errors. However, as mentioned above, signi"cant di!erences between them exist in many cases. This paper will revive the CMB approach [6] and apply it to extract baryon resonance parameters from partial wave amplitudes of nN reactions with a large variety of "nal states. This approach emphasizes the analytic properties of scattering amplitudes in the complex energy plane that are consistent with the dispersion-relation approach and potential scattering theory. Other methods of handling multi-channel unitarity are through the K-matrix approximation [2,5,23] and the KSU model [4]. Although the transition amplitude is parameterized in a form similar to these models, there are distinct and important di!erences. In particular, the threshold for each of eight possible channels is treated correctly with two- and three-body unitarity requirements imposed. Thus, resonances can be found as poles in the ¹-matrix, a feature missing in most K-matrix models. Obviously, the CMB approach is most suitable for extracting resonances that are close to inelastic channel thresholds. Although the KSU approach by Manley and Saleski and more recent K matrix models [5] also account for the structure due to channel openings, their multi-channel parameterization is completely di!erent from the CMB model in realizing the unitarity condition. In the CMB model, dispersion relations are used to guarantee analyticity in the amplitudes. The KSU parameterization is an extension of a K-matrix formulation. It does not allow the analytic continuation into the complex energy plane, which is required to "nd the resonance pole. Feuster and Mosel present poles from a speed plot analysis, stating that technical problems associated with their partial wave decomposition prevent direct determination of poles from their K matrix analysis. There is the additional problem that resonances are associated with poles in the ¹ matrix rather than the K matrix. In the CMB model, the non-resonant t-channel and u-channel mechanisms are simulated by including the transitions to sub-threshold one-particle states with masses below the nN threshold and with the possibility of producing either an attractive or a repulsive potential. Existing models treat these e!ects in a variety of ways. The empirical method used here is rather di!erent from the polynomial parameterization often used in previous amplitude analyses. The consequence of this di!erence is very signi"cant in practice since the intrinsic `structurea of the non-resonant term due to the channel opening is built in correctly in the CMB approach, but can be easily missed if the

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`smoothnessa of the background term is the only criterion. Modern K matrix methods [5] "t coupling strengths in various diagrams. These have the advantage of "tting fewer parameters than the more empirical approaches and have the correct partial wave decomposition of the diagrams include. The disadvantages include ignoring all o!-shell intermediate state scattering and the inability to describe resonances of total angular momentum larger than .  The CMB approach was published in 1979. Although inelastic data were used in the analysis, the elastic nN scattering data were emphasized. In this work, we follow closely their theoretical approach, making only small changes in the parameterization of the amplitudes. However, we make signi"cant changes in the data set used. We include all nNPgN data, some of which was unavailable then. (They depended on a separate analysis of backward n\p elastic scattering and other data [24] to model the eta cusp. This analysis would give different results with the present data set.) We also directly "t nNPnnN inelastic data, represented by the amplitudes determined in the isobar model "t of Manley et al. [25]. By including 30% more data, the Manley et al. inelastic amplitudes are more accurate than those used in the original CMB analysis. Furthermore, we will use the most recent VPI energy-independent amplitude [2] as the nN elastic scattering input. This amplitude is signi"cantly di!erent from the one used by the CMB group and is also more accurate than the KH amplitude in the S channel, as  discussed in Ref. [26]. These di!erences in the input data make our results for some cases (in particular the S (1535) state) signi"cantly di!erent from the CMB values listed in Particle Data  Table. Batinic et al. [27] have applied the CMB model to perform an analysis with only 3 channels: nN, gN, and a dummy channel meant to represent the complex set of nnN channels. Their focus is on the dynamics associated with the gN channel. They "t the KH80 [3] energy-independent amplitudes of the nN elastic data and the nNPgN data. The gNPgN amplitudes are the predictions of the model. The Brown et al. data [28] is the largest body of nNPgN data in the energy region close to threshold, but it is felt to have signi"cant systematic errors [27,29]. Batinic et al. give it a weight 5 times smaller than the other data to deemphasize it. We will use the same nNPgN data with the same weights in our analysis. This is an important part of the determination of the parameters associated with the S (1535) resonance.  The present work is the "rst step in an ongoing program to develop a model appropriate for the new generation of data coming from Je!erson Lab and other labs. Although we anticipate further development of the model, the goal of this work is to apply the CMB model in a form very similar to its original implementation. We closely follow the procedures of Ref. [6], but use it to determine the baryon spectrum and decay branching fractions from the modern data. Future publications will address interesting but nontrivial issues such as how to determine the `besta baryon spectrum and how to di!erentiate between resonance poles, bound state or virtual poles, and poles that have changed Riemann sheet by moving across an inelastic threshold cut to the sheet most directly reached from the physical data. In Section 2, we present a detailed account of the CMB multi-channel unitary model. To illustrate the main features of the model, we discuss in Section 3 the examples of an isolated resonance and the two-resonance}two-channel situation. The full results are presented and discussed in Section 4. A summary and outlook are given in Section 5. A more complete discussion of the methods and results is given in Ref. [30].

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2. The CMB unitary multi-channel model 2.1. Representation of 3-body xnal states The "rst step of a multi-channel analysis to determine NH parameters is to extract from the nNPnnN data a set of partial-wave amplitudes for the transitions from a nN state to various quasi-two-particle channels in which one of the two particles is either a nN or a nn resonant state: nD, oN, nNH(1440) and a very broad (nn) N channel. For convenience, we will use the (2 notation p to label the S-wave, isoscalar nn state. This procedure was introduced by the CarnegieMellon Berkeley (CMB) group in the 1970s. A similar procedure was used later by the Virginia Polytechnic Institute and State University (VPI) and Kent State University (KSU) groups [25] to obtain partial wave amplitudes from all of the nNPnnN data available in 1984. These amplitudes, called VPI}KSU amplitudes, will be used in this work. A more precise analysis should start with the original data of nNPnnN reactions. This event data has been stored by the VPI group. The raw nNPnnN data was not received from them in time for the present analysis. 2.2. Model details To extract resonance parameters from VPI}KSU amplitudes, it is necessary to employ a multichannel formulation of the nN reaction. This is accomplished in the CMB approach by assuming that the transition amplitudes of the nN reaction can be written in the center-of-mass (c.m.) frame as , (1) ¹ " f (s)(o (s)c G (s)c (o (s) f (s) , @ @ ?@ ? ? ?G GH H@ GH where s is the total center-of-mass energy squared, indices a, b denote the asymptotic channels which can be either a stable two-particle state (nN, gN, KK,2) or a quasi-two-particle state (nD, oN,2). These asymptotic channels are coupled to a set of intermediate states (resonances) denoted by indices i, j. The scattering matrix de"ned by Eq. (1) is related to the S-matrix via S"1#2i¹ with SRS"1. Hence we have the following unitarity condition: + Im(¹ )" ¹H ¹ (2) ?@ ?A A@ A The crucial step of the CMB model is to choose a parameterization of various quantities in Eq. (1) such that Eq. (2) is satis"ed. Furthermore, the resulting analytic structure of the scattering amplitudes is consistent with the well-developed dispersion relations for nN elastic scattering and multi-channel potential scattering theories. This is accomplished by using the following prescription. First, it is assumed that the ith resonance to be found is identi"ed with the bare particle, i, with a bare mass squared, s . The strength constant c and form factor f (s) in Eq. (1) de"ne the decay G ?G ? of the ith resonance into an asymptotic channel a. The form factor is de"ned by






p J? ? , f (s)" ? Q #(p#Q  ? 

(3)

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Fig. 1. Schematic diagram for the Dyson equation iteration.

where Q and Q are empirical constants de"ning how quickly a channel, with orbital angular   momentum l , opens up. p is the center of mass momentum for channel a. In this analysis, we set ? ? Q and Q equal to the pion mass. These are the same values as were used by CMB; they chose this   parameterization and these values as a result of a study of the best way to model the left-hand cut. The right-hand or unitarity cut is set through the channel propagators, Im (s)"f o (s). The ? ? ? phase space factor o in Eq. (1) is de"ned by ? (4) o "p /(s ? ? for a stable two-particle state. For quasi-two-particle channels, o clearly must be de"ned consis? tently with the phase space factors used in de"ning the propagation of this resonant two-body state during the collision, as required by the unitary condition Eq. (2). In the CMB model, this is achieved by assuming that the only interaction during collisions is a vertex interaction which converts intermediate states into asymptotic states. Then the propagator G in Eq. (1) can be GH graphically depicted in Fig. 1 and is de"ned by the Dyson equation: G "G #G R G . (5) GH GH GJ JI IH Eq. (5) is an iterative equation. A sum over repeated indices is implicit. G, G , and R all vary with  s and are N;N matrices, where N is the number of intermediate resonant and non-resonant states considered. Each bare intermediate state has a propagator, G , de"ned by  G (s)"d e /(s!s ) , (6) GH GH G  G where e "#1 for states that correspond to the resonances that will be "t. To simulate the G t-channel and u-channel mechanisms, two subthreshold bare states with a mass s below the nN  G threshold are introduced. These subthreshold states will simulate an attractive background potential for e "!1, and a repulsive potential for e "#1. In principle, this prescription can G G simulate any t- and u-channel mechanisms if a su$ciently large number of bare states are included. This is well known in potential scattering theory. The actual number of the subthreshold states needed is not intrinsically known. We use two subthreshold states and one state at very high energy for every partial wave. In the original work, CMB [6] found negligible di!erences between "ts using a (smaller) number of subthreshold states and "ts using actual potentials that simulate the left-hand cut. They are allowed to couple to nN and gN asymptotic states. The self-energy, R , describes the dressing of bare particles by the coupling with two-particle GH channels, as depicted in Fig. 2. It therefore must depend on the strengths (c 's) and form factors G? ( f 's), and is assumed to take the following form: ? + R (s)" c U (s)c (7) GH AG A AH A

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Fig. 2. Schematic diagram for the self-energy (R) iteration. Each resonance is allowed to couple to all open channels.

For the contributions from stable two-particle channels, we have U (s)" (s) A A

(8)

with



1  dsf H(s)g (s; s) f (s) . (9)

(s)" A A A A p A Q Here s is the threshold for channel c, and g (s) is the propagator of the two-particle channel c. A A The task now is to choose g (s; s) such that the unitarity condition Eq. (2) is satis"ed and the A desired analytic structure of the amplitude can be generated. It can be shown that the (s) of the A CMB model is obtained by assuming that





o (s) o (s) 1 A !P A , (10) g (s; s)" A s!s p s!s#ie  where the density of states o (s) has been de"ned in Eq. (4), and P means taking the principle-value A part of the propagator. Substituting Eq. (10) into Eq. (9), we then obtain the following dispersion relation for the auxiliary function : A Im( (s))"f (s)o (s) , A A A (11)  Im (s) s!s A ds . Re( (s))"Re( (s ))# A A  p Q (s!s)(s!s ) We see that Eq. (11) is a subtracted dispersion-relation which has a form similar to what has been established in many studies of nN scattering. In the complex-s plane, one can choose the subtraction point s such that the resulting scattering amplitude has a pole on the left-hand side  and a branch cut from s "(m #m ) to #R. For each channel, we choose the value at L, L , threshold for s , and we set the subtraction constant, Re( (s )), such that Re( (s)) is 0 at threshold.   A This arbitrary choice has the primary e!ect of shifting the value of the bare mass squared, s , for G each resonance in a partial wave, but does not a!ect the physical mass of the `dresseda resonance. Here we assume that this dynamical assumption is valid for all stable two-particle channels like gN, and KK. For a quasi-two-particle channel c, the function U (s) in Eq. (7) must account for the mass A distribution of one of the two particles which is itself a resonance state. To be speci"c, let m be the 



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mass of the stable particle in the channel c and m and m be the masses of the two daughter   particles from the decay of the resonant subsystem into a channel r. Then the form assumed by the CMB model can be more explicitly written as



U (s)" A

(Q\K  

K >K 

ds p(s ) (s ) P P A P

(12)

where the mass distribution of the quasi-particle was taken to be c Im( (s))/n P P p(s)" . (M!s)#cIm( (s)) P P P

(13)

Here (s) is again de"ned by the subtracted dispersion relation Eq. (11) for the appropriate P resonant subsystem. Eq. (13) has the commonly used Breit}Wigner resonance form. The coupling strength c is related to the width C of the considered resonance state by C "[cIm( (M))]/M . P P P P P P In this work r is either a nN state or a nn state. From the empirical values of the widths for D and o, mass distribution functions for the nD and oN channels can be "xed. For the other quasi-twoparticle channels, pN and nNH(1440), the width C is "xed at a standard value in the "t [4]. The P above formalism for the quasi-two-body channels provides a unitarity cut along the real s-axis from the three-body nnN threshold to in"nity. Furthermore, it makes the resulting scattering amplitudes satisfy the unitarity condition Eq. (2). This is how the three-body unitarity is implemented in CMB model. The Dyson equation, Eq. (5), is algebraic and can be solved by inverting an N;N matrix. Schematically, we have G (s),[H\(s)] GH GH

(14)

with the matrix element of H de"ned by H (s)"[(s!s )/e ]d !R (s) . GH G G GH GH

(15)

Now all of the ingredients needed for calculating the ¹-matrix elements of Eq. (1) are in place. The variable parameters (i.e. couplings c and poles s ) are then adjusted to "t the VPI}KSU partial GA G wave ¹-matrix elements. 2.3. Resonance parameter extraction Following Ref. [6] a resonance position is identi"ed with a pole of the scattering ¹ matrix in the complex energy plane. This can only be done for models that can be evaluated for complex values of s, i.e. models that have the correct analyticity structure. In the CMB model, the determinant of the H matrix de"ned by Eq. (15) equals zero at the pole position, s"s , in the complex s-plane.   Only the poles located close to the real axis are interpreted as resonances. This procedure involves an analytic continuation of (s) into the complex s-plane for Im(s)(0. Clearly, the analyticity, A de"ned by the dispersion relation, Eq. (11), plays a `dynamicala role in "nding the resonance parameters. This is one of the main di!erences between our approach and the KSU approach.

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To proceed, we need to evaluate Eqs. (11) and (12) for complex s. Each ¹-matrix element has a branch cut beginning at the elastic and each inelastic threshold. The branch cut can have a strong e!ect on the amplitude, in extreme cases producing a cusp, e.g. in the nNPnN S partial wave.  Above each threshold, the amplitude is multi-valued. This is traditionally described by a Riemann sheet structure [31]. The amplitude is continuous when analytically continuing to the appropriate new sheet as the value of s crosses the branch cut, but discontinuous when staying on the same sheet. The function, (s) de"ned by Eqs. (9) and(10) is the channel propagator for the `"rst sheeta A of the complex s-plane, labeled . At s5s , it has a discontinuity in its imaginary part A' A determined by unitarity as s crosses the real axis. The resonance pole is on the `second sheeta in which (s)" (s); it has the same discontinuity A A'' as the "rst sheet except for the opposite sign,

(s#ie)! (s!ie)" (s!ie)! (s#ie) for s's . (16) A'' A'' A' A'  It is (s) which is used in the search of the resonance pole positions. Using the same strategy as A'' CMB [6], we search the nNPnN ¹ matrix for poles on the sheet most directly reached from the physical region. As discussed by Cutkosky and Wang [32] and elsewhere in the literature, each resonance has additional poles on other Riemann sheets associated with each inelastic threshold. However, the pole closest to the physical region is most closely associated with the physical characteristics of the resonance. The formalism presented here can be extended to search for poles on other sheets and try to distinguish between resonance poles and poles that arise from bound states of composite particles. A resonance pole is found by searching for a zero in the determinant of the H matrix de"ned in Eq. (15). Once a pole is found, H(s) is diagonalized at the pole position s . This is done to   eliminate resonance}resonance interference e!ects when multiple resonances are present in a partial wave. By using the resulting eigenfunctions s , the ¹-matrix in the vicinity of the pole can be G written as [6] (B (Im( )g g (Im( )(B B !d ?C C C D D D@ . ?@ # (17) ¹ " ?@ ?@ D(s) 2i CD The g describe the coupling of the resonance to channel c, as de"ned in Eq. (19). This form can be A shown to be equivalent to the full CMB model. Eq. (17) is a general form for a Breit}Wigner resonance shifted by non-resonant (background) reaction mechanisms. To determine the resonance properties, we look at the ¹ matrix in the vicinity of the pole with a simplifying assumption. The form of the background part, B , is assumed to be smooth in the immediate vicinity of a pole. This ?@ allows us to use the denominator, D(s), to de"ne the Breit}Wigner resonance parameters at the pole, ignoring the background. The denominator of the above expression is then matched to a relativistic Breit}Wigner form. The full denominator can be written as D(s)"r!s!v y , A A A where





,  y ""g "" c s . GA G A A G

(18)

(19)

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The real constants, r and v, are de"ned by equating D(s )"0. The resonance mass comes from   the real part of the denominator and the width comes from the imaginary part. The general form of the Breit}Wigner denominator for a multi-channel situation is "(= !s)!i= C (20)    A A where = is the mass of the resonance and C is the decay width of the resonance to asymptotic  A state c. Thus, shifts in the resonance mass can be identi"ed with the real parts of the denominator and the imaginary part can be identi"ed with a sum over the partial decay widths of the resonance to various channels, c. To obtain numerical values, we use a linear approximation for the real part of the summation in Eq. (18). Using the above de"nitions, the following qualitative statements about the relationship between the resonance parameters and the model parameters can be made. The resonance width is approximately equal to twice the imaginary part of the denominator at resonance. Because the self-energy term in the denominator has strong energy dependence, the physical pole position is shifted from the bare pole by an amount that depends of the value of the coupling parameter (c) and both the value and shape of Re (s) in the vicinity of the pole. The mass of the resonance is further shifted from the physical pole because the mass is determined on the real axis of the complex energy plane. Shifts from the real part of the bare pole to the mass can be positive or negative; the size of the shift depends on many factors and can be quite large (more information can be found in Section 5). Eq. (17) has an energy-dependent pole shift due to Re (s). We wish to remove this energy dependence to make a connection between the full ¹-matrix in Eq. (17) and a standard relativistic Breit}Wigner shape. Making the assumption that the `reala part of the term y is linear close A A A to the resonance pole: D

5

v Re y +a#bs , (21) A A A the ¹-matrix can be re-written in the form of a relativistic Breit}Wigner resonance, without an energy-dependent pole shift: (B Im g (B Im g D@ D D ?C C C (1#b (1#b B !d ?@ # . (22) ¹ " ?@ ?@ ( P\? )!s!iv y Im /(1#b) 2i A A A CD >@ In Eq. (22), the quantitative expressions for the resonance parameters can then be identi"ed in terms only of quantities evaluated at the resonance mass: Re D(M )"0 de"nes resonance mass PM ,   Im D(M )  de"nes resonance width , C" M Re D(M )   y Im

A C de"nes branching fraction into channel c . C" A A y Im

? ? ?

(23)

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In the above equations, D(s) is the derivative of D with respect to s, Re D(s)"!(1#b). This term accounts for the fact that there is an energy dependence to Re (s), which shifts the pole position. This formulation is identical to the generalized Breit}Wigner form that is the basis of most "tting and theoretical models. However, we only use this form in the immediate vicinity of the pole to determine the resonance parameters. The data are still "t with the full model and the pole position is then determined from that "t. This type of prescription for obtaining the mass, width and partial widths of a resonance is model dependent. It is important to realize that all de"nitions of baryon resonance parameters are model dependent. For a highly elastic and isolated resonance, the model is not very important. However, very few resonances "t this description and we are trying to present a formulation that minimizes the model dependence. 2.4. Relationship to other models The CMB model has a number of features that are not included in commonly used models. Here, we present a discussion of formulations similar to standard models by making approximations to the full CMB model. These simpler formulas will be used in Section 4.4 to show the corresponding model dependence in the extracted resonance properties. The full CMB model contains a dispersion relation which guarantees the analyticity. The imaginary part of the channel propagator function (Eq. (11)) is the relativistic phase space function and the real part is then calculated from a dispersion integral. Models which are not analytic [4] include only the phase space, so we set Re (s)"0 for all s to simulate them. The full CMB model uses a Dyson equation to allow for conversion into open intermediate states (resonant or non-resonant) and open asymptotic channels. The bare propagator (Eq. (6)) is `dresseda by all the open intermediate and asymptotic states. The K-matrix formulation [2,4,5,23] uses the bare propagator of the CMB model in place of the dressed propagator as a K-matrix rather than a ¹-matrix and identi"es the resonance properties with its parameters. A non-analytic unitary multichannel K-matrix using a relativistic Breit}Wigner form [23] can then be constructed for the contribution of resonance R to the reaction between initial state i and "nal state j: (Im (s) c c (Im (s) 0G 0H H , G K0 (s)" GH s !s 0

(24)

where Im (s) is the product of the form factor and phase space for channel i as de"ned in Eq. (11). G M and C are the mass and partial width for decay to channel i of resonance R; the total width is 0 0G C . These three physical quantities are de"ned by 0 M "(s , 0 0 Im (s )(c ) G 0 0G , C " 0G M 0 C " C . 0 0G G

(25)

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For the non-relativistic Breit}Wigner case, (Im (s) c c (Im (s) 0G 0H H , G K0 (s)" GH M != 0 where C is de"ned by 0

(26)

C "2 Im (s )(c ) . (27) 0 G 0 0G G In either case, the corresponding ¹-matrix can then be found through the standard de"nition: ¹"K*(I!iK)\ .

(28)

Since both K and ¹ are matrices, there is no simple closed expression for ¹ corresponding to Eqs. (24) and (26) except in the single-channel case. Non-resonant K-matrices must also be de"ned. To maintain unitarity, the full K-matrix is obtained by adding all the resonant and non-resonant K-matrices (e.g. [38]). The corresponding ¹-matrix includes e!ects of resonance interference and coupling to non-resonant processes, but only on-shell. Various theoretical schemes have been built on the K-matrix method. At low energies, the characteristics of the D (P (1232)) and the S (1535) have been determined with an e!ective   Lagrangian method [38]. More complete formulations have been developed [5,33]. In these models, the mesons and baryons are each fundamental particles. Although the number of parameters is reduced from what is required for the more empirical models, there are still a number of ambiguities in the construction of the Lagrangian and in the proper development of a multiresonance, multi-channel model.

3. Database We devote a separate section to the database because it plays a critical role in the results we obtain. There are many possible asymptotic states that can couple to each resonance. There are also a number of resonances which couple weakly to the nN channel so that the state is only seen in the inelastic data. Just as it is important to include various inelastic channels with proper threshold e!ects in the theoretical model, it is also important to include as much of the relevant data as possible with appropriate error bars. Although it is best to use the original data, various partial wave analyses producing ¹ matrix representations of the data are available. This kind of analysis can be done with much less model dependence than is found in the analysis used for determination of baryon resonance properties. Nevertheless, these analyses make choices in the data used in the "ts and their absolute normalizations since not all data sets are consistent with each other; these choices can add error beyond what was in the original data, thus adding uncertainty to the "t. On the other hand, "tting partial wave amplitudes allows a simpler "tting strategy } separate "ts can be made for each partial wave and less computer time is required. This is the same procedure chosen by KSU [4]. We choose to "t the single-energy partial wave amplitudes of the VPI group for elastic scattering [2] and the isobar

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model "ts of KSU}VPI for inelastic pion production [25]. We also make a separate partial wave analysis of the nNPgN data. 3.1. nN elastic data Data in this channel is easiest to measure. Therefore, the data are more complete and of higher quality than in the other channels. In many resonance parameter analyses, these data have a dominant role in the results. This takes advantage of using the best data. However, the inelastic data must be included to "nd resonances that would not be seen in nN elastic scattering. The Constituent Quark Model [8,18] predicts the existence of a large number of states (roughly a number equal to the number of states seen to date) and "nds that many of them couple weakly to the nN channel. There have been numerous analyses of the elastic data. Since complete experimental results are not yet available, theoretical constraints must be employed to get unambiguous "ts. Older analyses of CMB [20] and KH [3] had strong reliance on dispersion relations to generate unique "ts. KH80 used "xed-t and "xed-h dispersion relations at many angles. The real parts of these results were later compared with partial-wave "xed-t dispersion relation predictions by Koch [21]. Although KH80 results are quite noisy close to threshold in the high ¸ partial waves, there is good qualitative agreement with the additional constraints. CMB80 uses hyperbolic constraints in the Mandelstam variables. Although the older works have the best theoretical underpinnings, we use the latest nN elastic partial wave amplitudes of VPI [2]. The VPI analysis uses a signi"cantly larger data set than was available for the earlier CMB80 and KH80 elastic analyses. Most of the new points are at =(1600 MeV, but there are also many new results at higher energies [2]. Thus, a number of older data points with signi"cantly larger estimated errors could be dropped from the "t, resulting in an improved "t. These more recent results are consistent with "xed-t dispersion relations at h"0 and low =. Although improvements in the VPI analysis are expected, we use this somewhat debatable [22] approach in this work since this provides a way to include all of the data published since the older work. Desire for an update of the 1980s work has been expressed at conferences for several years, but no such work is in progress. 3.2. nNPnnN data For the best available representation of inelastic data, the choices are much more limited than for elastic data. Since there are no model-independent methods for the 3-body "nal states, isobar models are employed. Since no recent interpretation of the nNPnnN data is presently available, we use the quasi-two-body channel decomposition of Manley et al. [25]. That work "t an isobar model to the nNPnnN data, isolating the contributions of nD, oN, pN (with p representing the nn strength in an isoscalar s-wave), and nNH(1440) channels. Although the pN channel will absorb some of the non-resonant nnN strength, these choices ignore some of the non-resonant nnN strength and states such as nNH(1520) that might be expected to share strength. They used a data sample of about 241,000 events spread over 1320}1910 MeV in =; statistical accuracy is much poorer than for the elastic channel and there is no data at the highest energies where resonances are found. All data published after this analysis were very close to threshold. It will be clear from our analysis that these data need augmentation and that the isobar analysis should be repeated.

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As with the choice in elastic data amplitude, this choice is a compromise between including all the existing data in a simple form and using a more proper analysis. An analysis the nN inelastic data without the assumption of the isobar model would be extremely di$cult and unjusti"ed with the quality of the present data set. In the CMB analysis [6,32], the nNPnnN data were weighted by a factor of  smaller than the  results of Manley et al. [25] because they felt the errors were understated. We have a similar attitude and weight the inelastic data by a factor of . Otherwise, the "ts to the higher quality elastic  data are degraded. 3.3. Description of the nNPgN analysis The most signi"cant inelastic channel at low = is the gN state; data involving this channel is thought to be crucial in analyzing S (1535) because the s-wave nNPgN cross section is large and  rapidly changing close to the resonance mass. Unfortunately, the data in this channel is both limited and of uncertain quality [29]. The data set with the most points close to threshold was published by Brown et al. [28]. Clajus and Nefkens [29] argue that these data have unknown errors in the assigned values of =, making them unusable. As mentioned earlier, Batinic et al. [27] put a very small weight on these data points in their partial wave "t to the nNPgN data. To produce partial wave amplitudes for this reaction, we reproduce the Batinic et al. "t to the nNPgN data, e!ectively leaving out the controversial Brown et al. [28] data set. Since there is very little data available for the reaction nNPgN, we use a simpli"ed version of the full CMB model to simultaneously "t the nN elastic ¹ matrices and the nNPgN data to provide a parameterization for each of the partial waves with ¸44 contributing to the g production cross section. The partial wave analysis followed the procedures used by Ref. [27], but used the newest VPI nN elastic partial wave amplitudes [2] instead of the older KarlsruK he Helsinki (KH80) amplitudes. The channels used in the analysis are nN, gN, and a dummy channel consisting of a "ctitious meson with mass chosen so that the dummy channel opens at about the energy where nN inelasticity due to channels such as nnN start. The mass of that "ctitious meson changes from partial wave to partial wave. The values used in the analysis are given in Ref. [27]. The partial waves used are the I" partial waves through G . All partial wave parameters were varied   simultaneously. More details on this procedure given in this section can be found in the Refs. [27,30]. For this process, the S (1535) resonance makes up most of the total cross section. Therefore, the  S partial wave is the most accurately extracted. The other partial wave amplitudes are smaller  and less accurately determined. The results of this "t are ¹-matrices that best model the partial wave data. The best "t is shown as an error band in the "gures of the results section. For the "nal "ts, 40 data points between threshold and 2.3 GeV were used for each partial wave. This insures that these data provide the appropriate contribution to the total chisquare. 4. Illustrative examples ⴚ P33 and S11 partial waves The purpose of this section is to introduce features of the model through the examination of speci"c partial waves. The P and S partial waves are chosen for this purpose. The P (1232) or   

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D is the best example of an isolated and elastic, i.e. simple, resonance; its characteristics are largely well-established. On the other hand, the S partial wave has some of the most interesting structure  of all the partial waves contributing to nN scattering. Because of this structure, extra care must be taken when extracting resonance parameters for this partial wave. Some of the interesting features exhibited in this partial wave are listed below. 1. There are 2 PDG 4* resonances (S (1535) and S (1650)) which overlap signi"cantly.   2. The S (1535) has strong coupling to both the nN and gN channels and is very near the gN  threshold (+1487 MeV). This produces a strong cusp in the nN elastic S ¹-matrix element.  3. There are 7 decay channels that have measurable coupling to the S states, each of which has  di!erent phase space which can cause structure in all of the other channels via unitarity. All of the above features are adequately handled in the CMB Model. This section will help in understanding the unitarity, analyticity, and other properties of the model, which are especially important in the S partial wave. A discussion of the cusp associated  with the S partial wave will also be given. Furthermore, this section will present a systematic  study of the model dependence of resonance parameters extraction by examining results obtained when leaving out various features of the full CMB model. 4.1. Single-channel}single-resonance case The equations from Section 2 give expressions for a reaction ¹-matrix element for any number of open scattering channels or asymptotic states (nN, gN, etc.) and any number of intermediate states (S (1535), P (1232), etc.). The ¹-matrix elements for one asymptotic state and one intermediate   state are much simpler. The resulting equations can then be applied with good success to the P partial wave near the D(1232) resonance. Despite the complexity of the CMB formulation, the  ¹-matrices for an isolated, single-channel resonance have a form similar to that found in the Breit}Wigner shapes commonly used. The one asymptotic state-one intermediate state case requires two parameters (using D to label the P (1232) intermediate state and nN as the asymptotic state), one coupling cD and one bare L,  pole energy, sD . Since there is only one intermediate state the matrix equations are reduced to scalar equations and the math is simple. There is only one-channel propagator which is determined L, analytically with a contour integral. (There is a functional form for the channel propagator only in the S-wave case. The results for the nN channel in an S-wave are shown in Fig. 3.) There is also one self-energy term according to Eq. (9): RD D (s)"cD (s) .  L, L, Since the function varies with energy, R does also. The H matrix is then de"ned by

(29)

H"(sD !s)!RD D "(sD !s)!cD . (30)  L, L, The G matrix (the dressed propagator) is then calculated from H as de"ned by Eqs. (14) and (15) in Section 2: 1 G" , (sD !s)!cD

L, L,

(31)

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Fig. 3. (s) (channel propagator) distributions for the nN and gN channels in S partial wave. These functions are used  in the two-channel}two-resonance test case and in the "nal "ts.

¹ is de"ned using Eq. (1) and the above equations: L,L, cD Im

L, L, ¹ " . L,L, (sD !s)!cD

L, L,

(32)

The results of a model calculation for the ¹ matrix is shown in Fig. 4 with a solid dot at the physical resonance mass. The equations produce the characteristic shape of a resonance with a maximum in the imaginary part and a zero in the real part at the resonance mass. A similar signal should be seen for strong resonances, but there will be shifts in the real data when underlying backgrounds of varying smoothness are included. Eq. (32) has the usual form of a relativistic Breit}Wigner resonance as expected, but there are some important new features. The real part of the pole is shifted from the bare pole energy (sD ) by cRe (s) and the pole gains an imaginary part, cIm (s) due to couplings of the resonance to the asymptotic state as it propagates. In this model, these shifts are energy dependent and come from both unitarity and analyticity requirements. The analyticity condition, Eq. (11) then allows an analytical continuation of the above expressions to the complex s plane where the physical pole position, mass, and width of the resonance must then be determined by a search discussed in Section 2. In other contexts, unitarity requirements are satis"ed through inclusion of "nal state interactions. These methods add terms to the denominator similar to what comes from the CMB model. However, these models are not necessarily analytic. Near the resonance peak, Eq. (32) can be expressed in terms of a mass and width identical to a generalized Breit}Wigner shape. Thus, the features of the CMB model can be absorbed into e!ective constants for the case of an isolated resonance.

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Fig. 4. One-resonance}one-channel test case ¹ matrix (real and imaginary parts) using parameters appropriate for the P (1232) resonance. Solid dots are placed at the energy of the resonance mass. 

4.2. Two-channel}two-resonance case The equations get rapidly more complicated as the number of resonances and open channels in a partial wave increases. The two-channel (or asymptotic states) two-resonance (or intermediate states) case is still instructive. The relevant equations are given below for the ¸"0 isospin (S )   partial wave. In reality, this partial wave has two strong channels and two strong resonances. The results shown below use realistic parameters for these states, but are not meant to be an accurate representation of data because non-resonant processes and other channels are ignored. For clarity, channel labels nN and gN and resonance labels of 1535 (referring to S (1535)) and 1650 (referring  to S (1650)) are used. Parameters that must be determined include four coupling strengths  (c ,c ,c , and c ) and two bare poles (s and s ). L, E, L, E,   The terms in the R self-energy matrix are constructed from Eq. (7): R "c

#c

,  L, L, E, E, R "c c

#c c

,  L, L, L, E, E, E, R "c

#c

.  L, L, E, E, The R matrix is symmetric by construction so it takes on the simple form




R

R



 . (34)   Fig. 5 shows the R (s) function. It is a weighted sum of and , shown in Fig. 3, and  L, E, thus carries the threshold behavior of both the gN and nN channels. The other elements have R(s)"

R



(33)

R

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201

Fig. 5. Self energy function, R for the test case where only nN and gN channels are included. 

similar qualitative behavior, but varying weights of because the two resonances couple to the two asymptotic channels with di!erent strength. The H matrix becomes






s !s!R !R   . H(s)"(s !s)!R(s)"  G !R s !s!R    The dressed propagator, the G matrix, has a simple form since it is a 2;2 matrix:




1 s !s!R G"H\" "H" R  where "H" is the determinant of H:



R  , s !s!R  

(35)

(36)

"H""(s !s!R )(s !s!R )!R . (37)      The ¹ matrix elements are then de"ned in Section 2 in Eq. (1). Explicitly, the nN elastic ¹-matrix element is c Im

L, L, R  !s!R !   s !s!R   c Im

L, L, # R  s !s!R !   s !s!R   2c c R Im

L, L,  L, # . (s !s!R )(s !s!R )!R     

¹ " L,L, s

(38)

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T.P. Vrana et al. / Physics Reports 328 (2000) 181}236 Table 1 Model parameters for a test calculation in the S partial wave. These are the bare mass, the physical mass and  width, and the dressed pole location for the two resonances included in the calculation. No background contributions are included State

Bare mass (MeV)

Physical mass (MeV)

Full-width (MeV)

Physical pole (GeV)

S (1535)  S (1650) 

1500 1650

1518 1680

83 200

1.507}0.032i 1.676}0.101i

Table 2 Model parameters for test calculation in the S partial wave. These are the coupling  parameters and branching fractions (b.f.) between the two asymptotic channels and the two resonances included in the calculation. No background contributions are included State

cnN (GeV)

S (1535)  S (1650) 

0.5 0.9

cgN (GeV)

0.85 !0.04

nN b.f. (%)

gN b.f. (%)

42 96

58 4

Note that all the functions used in this expression depend on s. The above expression is the sum of two resonances with an interference term. (If there is only one resonance, this formula simpli"es to the same as Eq. (32).) The terms that look like single resonances are more complicated than in the previous example. The self-energy terms have the necessary analytic cuts from the gN as well as the nN channels and also make the amplitude unitary. There are analogous ¹-matrix elements for the processes: nNPgN and gNPgN. In this case, two poles must be found in the complex energy plane. Since the two resonances have signi"cant interference, the non-diagonal elements of the G matrix are large. This means the second resonance has a contribution to the propagator of the "rst resonance at the pole of the "rst resonance. Referring to Eq. (38), there are shifts to the mass and width of each resonance due to the presence of the other in addition to those due to the asymptotic channel couplings which were not seen in the one-channel}one-resonance case. Here, the H matrix (the inverse of G or the `denominatora matrix) must then be diagonalized at each pole to isolate the contributions from each individual resonance. The nN elastic and nNPgN ¹ matrices in this partial wave corresponding to the above equation for representative parameters (see Tables 1 and 2 for a complete listing of the relevant values) are shown in Fig. 6. The eta threshold has a signi"cant e!ect on the observables. Therefore, the nN elastic ¹-matrix element peaks at the gN threshold rather than at the peak of the S (1535)  resonance. The physical masses are shown as solid dots which are near the peak of the cross section and the imaginary part of ¹ for the higher state, but above those positions for the lower state. For all channels, the peak of the ¹-matrix is shifted by a few dozen MeV from the bare mass by the

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Fig. 6. ¹ matrices for two-channel}two-resonance test case. The real and imaginary parts are shown for the nNPnN and nNPgN reactions. Solid dots are placed at the values of the resonance masses obtained from these ¹ matrices.

self-energy terms. (The S (1535) resonance shape is also modi"ed by the presence of a strong  inelastic threshold.) There is also signi"cant interference between the two resonances. Both cross-section bumps are unusually narrow compared to the physical widths because of interference e!ects. We "nd that the S (1535) properties are signi"cantly altered by the interference with  S (1650), similar to the "ndings of Sauermann et al. [33] in a K-matrix photoproduction  calculation. 4.3. Cusp structure in the S partial wave  There are two main causes of the strong cusp structure observed in nN elastic scattering di!erential cross section, as well as the S partial wave amplitudes. The "rst is that the S (1535)   resonance couples strongly to both the nN and gN channels and the gN channel threshold (+1487 MeV) is just below the 1535 pole. The second is that the orbital angular momentum is zero (nN S-wave); thus, the cross section in this partial wave increases linearly with momentum. Therefore, through analyticity and unitarity, a cusp structure appears in the nN elastic channel (Fig. 7). Fig. 5 shows the self-energy term labeled R for the two-channel}two-resonance model  of Section 4.2. This self-energy has the appropriate analytic phase space factors and therefore shows a cusp. Since R enters directly into the ¹-matrix elements, the cusp shows up there as well. Fig. 8 shows the ¹-matrix elements for elastic scattering and o production. Although the cusp structure is apparent in the ¹ matrices for all channels other than that of eta production, it is most evident in the nN elastic channel because most of the decay width of the S (1535) is split roughly equally  between the nN and gN channels. All other inelastic channel openings have the potential of

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Fig. 7. Best "t calculation for S partial wave at the gN threshold. ¹ matrices for nNPnN and nNPo N are shown.  

creating a cusp, but this is by far the strongest case. At present, the nN data is not of high enough quality to see any other cusp structures. 4.4. Model dependence in analysis of the S



partial wave

The variation between this and other models can best be seen in an analysis of the data in the nN S channel, as discussed in previous sections. Because of the peculiarities of this channel, data for  both nNPnN and nNPgN reactions would be required for a high quality determination of the S (1535) parameters and data for nNPnnN would be required for a good determination of  properties for the higher energy states. Presently, the inelastic data is of much less quality than for the elastic channel. To test model dependence in the resonance parameters, we have analyzed all available and subsets of the data for the S channel with the full CMB model and various approximations that  simulate the models employed by various other groups. Although the "nal "ts were of similar quality for all cases, there can be large di!erences in the extracted resonance parameters. In Tables 4 and 5, masses, widths and branching fractions are given for two of the three S resonances used in this analysis for a number of di!erent model types using various data sets to  constrain the "ts. The four columns on the left of the table describe features of the model used and what data was used in the "t. The "ve columns on the right give the results of the "t for the resonance parameters } mass, width, and branching fractions. The Unitarity column labels how unitarity was imposed in the "ts. K-matrix means that unitarity was imposed by commonly used K-matrix methods of Moorhouse et al. [23]. For the results shown in the tables, we recreate the K-matrix "t of Ref. [23] with the appropriate phase space factors ( (s)) from our model as discussed in Section 2.4. Dyson equation means that unitarity was imposed using the Dyson-equation approach of the CMB model. Channels can interact any

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Fig. 8. ¹-matrices for S partial wave for elastic scattering and "nal states of o N, pN, and nNH(1440) calculated with   the full model using "nal "t parameters. The dotted line is calculated with only resonance couplings turned on and the dashed line is calculated with only non-resonant couplings enabled. The full calculation, which cannot be a sum of the dotted and dashed lines because the resonant and non-resonant diagrams interfere, is shown as a solid line.

number of times in various forms before "nally decaying because the Dyson equation is iterative. The column labeled Disp Rel refers to whether or not the dispersion relation was used to make the phase space factors in the self-energy terms analytic functions or not. If the dispersion relation is evaluated, the phase space factors and hence the self-energies are analytic functions of the square of

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Table 3 Channels included in this analysis. Since isospin symmetry is assumed to simplify the analysis, all charge states of a particle have the same mass. We choose channels very similar to those used by KSU with identical nomenclature. All channels used in "ts have "xed spin, orbital, and total angular momentum. In the text, the oN channel is denoted by (o N)l with s equal to twice the total spin of the oN system and l giving the orbital angular momentum Q in spectroscopic notation. The orbital angular momentum of the nD is also ambiguous, so it is also given as a subscript Channel

Baryon mass

nN gN oN pN uN nD nN* KK

939 939 939 939 939 1232 1440 1116

Baryon width

Meson mass

Meson width

139 549 770 800 782 139 139 498

115 200

153 800

Table 4 Model dependence for the S (1535). See text for details  Unitarity

Disp. Rel.

Res. Type

Channels in "t

Mass (MeV)

Width (MeV)

nN (%)

gN (%)

nnN (%)

K-Matrix K-Matrix K-Matrix K-Matrix K-Matrix K-Matrix Dyson eq. Dyson eq. Dyson eq.

NO NO NO NO NO NO YES YES YES

NRBW NRBW NRBW RBW RBW RBW RBW RBW RBW

nN nN, gN ALL nN nN, gN ALL nN nN, gN ALL

1518 1532 1535 1514 1533 1534 1531 1526 1542

87 108 126 84 110 125 72 114 112

43 45 42 35 44 42 16 36 35

6 39 44 0 40 43 62 41 51

51 16 14 65 16 15 22 23 14

the CM energy, s; otherwise they are not. When using the Dyson equation and not evaluating the dispersion relation (in the "ts, we set Re "0.0), all intermediate interactions in the scattering process occur on-shell. Note that this does not a!ect unitarity. If unitarity is achieved through K-matrix methods, the mass and widths are direct parameters in the "t. The column Res Type describes the form of resonance used. NRBW refers to a non-relativistic Breit}Wigner shape and RBW means a relativistic Breit}Wigner shape was used. For the NRBW case, a resonance has a c/(w !w!ic) form and the c couplings have units of (energy. For the  RBW case, a resonance has a c/(s !s!ic) form; the c couplings have units of energy.  Finally, the column labeled Channels in xt describes the types of data used in the "t. For the nN case, only the VPI nN elastic S ¹-matrix elements were used in the "t. For the nN, gN cases both  the VPI elastic data as well as constraints from a partial wave analysis of nNPgN done by this

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Table 5 Model dependence for the S (1650). See text for details  Unitarity

Disp. Rel.

Res. Type

Channels in "t

Mass (MeV)

Width (MeV)

nN (%)

gN (%)

nnN (%)

K-Matrix K-Matrix K-Matrix K-Matrix K-Matrix K-Matrix Dyson eq. Dyson eq. Dyson eq.

NO NO NO NO NO NO YES YES YES

NRBW NRBW NRBW RBW RBW RBW RBW RBW RBW

nN nN, gN ALL nN nN, gN ALL nN nN, gN ALL

1645 1689 1694 1682 1692 1690 1692 1676 1689

233 225 259 161 233 229 138 104 202

35 67 72 78 75 65 65 54 74

49 31 16 5 15 25 35 45 6

16 2 12 17 10 10 0 1 20

group were used. For cases labeled All, nNP+nN, gN, nnN, are all included in the "t. The nnN channel is composed of quasi-two-body channels (o N) , (o N) , (nD) , pN, and nNH(1440).  1  " " Partial wave quasi-two-body ¹-matrix elements of Manley et al. [25] are used for these channels. The last row in each table contains results for our full model. All channels are used in the "t, unitarity comes from use of the Dyson equation, analytic phase space factors are used, and the bare resonance has a RBW dependence. Signi"cant model dependence is seen along with sensitivity to the data used. The lower state (S (1535)) has signi"cant model dependence as noted above while S (1650) tends to have poorer   quality data, i.e. missing data or data with error bars that are too small. We do not show any results for the third resonance (S (2090)) because the data quality dominates the "tting, causing wide  variation in "t results, e.g. masses vary between 1509 and 2028 MeV. The data quality for this partial wave is discussed in detail in Section 5.3. We note that without the dispersion relation, the ¹ matrices for the K-matrix model and the model using the Dyson equation for the resonance propagator are equivalent. Therefore, only the K-matrix results without the dispersion relation are given in the table. In general, the choice of relativistic vs. non-relativistic shape for the bare Breit}Wigner resonance does not have a strong in#uence for an isolated resonance such as the 1535 MeV state. However, the 1650 MeV state has a weaker signal (in part because of poorer data quality) and the two shapes can produce larger di!erences. Even there, the agreement in the case where all data is used (line 3 for NRB= vs. line 6 for RB=) is very good. More important di!erences are found when comparing K-matrix vs. Dyson equation results with the dispersion relation included (e.g. line 6 vs. line 9). The former is close to the model employed by Manley and Saleski [4]. These two models have di!erences of about 10% in the total width and up to 50% in the branching fractions. The most important deviation from the full result comes from the use of a truncated data set. For the 1535 MeV state, ignoring the interference with the gN "nal state causes the model to "t the Breit}Wigner shape to the cusp at the gN threshold. The VPI work [2] has a very small width for the 1535 MeV state; although the gN channel is mocked up, none of the actual data is used. For

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even the full model, leaving out the nnN "nal state data (such as was done by Batinic et al. [27]) produces 20% deviations in the branching fractions. We reproduce the updated results of the Batinic et al. paper [27]. Both the mass and the width of the S (1535) tend to increase as more data channels are added  into the "t. It is interesting to note that the small widths (i.e. (100 MeV) are in situations where only the nN elastic data is used in the "t. Also, the branching fraction of gN tends to be smaller than that of nN unless the analyticity is taken into account in the phase space factor. In other words, when the cusp is handled appropriately, the gN channel becomes the dominant decay mode. The Manley}Saleski [4] analysis, which does not address the analyticity issue, is unable to match the cusp well and concludes that nN is the dominant decay mode. The S (1650) and the third S resonance have more random shifts in their resonance   parameters. At energies in the region of the excitation of the third S state, only elastic and  g production data exist and the signal is not strong. Therefore, the "t parameters for the third S state are largely determined by "tting non-statistical #uctuations of the data. The "tted masses  vary widely from case to case and we feel the results with the di!erent models do not give information about features of the models. 4.5. Elastic data dependence A major question is the balance between model dependence and data dependence. The previous section showed the model dependence using various subsets of the data used in this analysis. In this section, we present results for "ts of the S channel using two di!erent sets of elastic partial wave  amplitudes. In Table 6, we compare our standard results with the results of a "t using the combination of CMB80 and KH80 data sets. This "t uses the same elastic data as Manley and Saleski [4] except for the additional nNPgN data. The di!erences seen in the table are somewhat similar (in total width) to but somewhat smaller (in branching fraction) than those presented in the previous section, verifying that the major di!erence between the present work and Manley and Saleski is likely due to model dependence. The changes in the results are larger for the second state than for the lowest state, but the di!erences are signi"cant for both states. The di!erences are much larger for the third S , but are not shown because of the weak evidence for this state.  Table 6 S Resonance parameters from "ts to di!erent elastic data sets. All results use the nNPgN and nNPnnN  partial wave amplitudes as in the full analysis. Comparisons are made for results using the VPI and a mixture of the CMB80 and KH80 elastic partial wave amplitudes (as was done in Ref. [4]). The "rst two lines in the table give results for S (1535) and the last two lines give results for S (1650). Although all channels were used in the "t, only   the total nnN branching fraction is given Elastic data set

Mass (MeV)

Width (MeV)

nN b.f. (%)

gN b.f. (%)

nnN b.f. (%)

VPI CMB/KH80 VPI CMB/KH80

1542 1535 1689 1691

112 137 202 222

35 35 74 58

51 53 6 15

14 12 20 27

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209

The balance between the branching fraction into nN and gN for the lower state are very similar in the two cases, but the total width found using the older data is 22% larger. This gives more evidence for the "ndings of the previous section that di!erences in branching fractions are a primary result of the di!erence between this model and that used by Manley and Saleski. 4.6. The non-resonant amplitude A major problem in any extraction of resonance parameters is in the careful separation of resonant and non-resonant mechanisms. Ideally, an independent calculation would be used for the non-resonant diagrams. Here, we choose a more objective strategy and use a smooth background that can then be `dresseda to contain the correct threshold behavior. To do this, we add two subthreshold resonances (one repulsive and one attractive) and one very high-energy resonance in each partial wave as &bare' propagators. These propagators are then dressed identically to the true resonances. More details are given in Section 2.2. The separation into resonant and non-resonant components is shown in Fig. 8 for various reactions in the S partial wave. We show the magnitude of the ¹-matrix for four di!erent "nal  states calculated with "nal "t parameters. The three lines shown correspond to including only non-resonant couplings, only resonant couplings, and all couplings. Since the non-resonant and resonant processes are intermixed in the Dyson equation, there is no way to sum them to get the full result. While some reactions are dominated by resonant processes (e.g. elastic scattering), others are dominated by the non-resonant processes (e.g. nNPo N). The resonance excitation must be  sampled through a variety of channels to provide the full picture. At very high energies (=&1.9 GeV), the lack of data allows the non-resonant processes to dominate. Resonance extraction at these high energies is very unclear with the data presently available.

5. Results and discussion We have applied the CMB model to the database presented in Section 3 } nN single-energy elastic ¹ matrices of VPI [2], the inelastic ¹ matrices of Manley et al. [25], and our own partial wave analysis of the nNPgN data. The nNPnnN raw data was not available in time for the present analysis. A reanalysis of the nNPnnN data is in progress [34] by our group; a more complete analysis can be presented when that work is "nished. The analysis presented here contains features of CMB [6] and KSU [4] since we use the formalism of the former and a data set similar to that used by the latter. However, the present analysis goes beyond any previously published. We present general results and a detailed discussion of the D , D , and S partial waves. D is an excellent example of an isolated     resonance, but has strong inelastic couplings. The D and S partial waves each have a strong   state (well understood for the former and poorly understood for the latter) along with less well-understood states; the interpretation of most of these states are sensitive to the features of the present model. We will compare our results to previous work with an emphasis on works that treat many channels. We also compare to the composite results of PDG [1].

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5.1. Details of xtting Although the nNPnnN data set used here is identical to that used by KSU, they do not include the nNPgN data and use older elastic analyses. KSU includes an gN channel in the S partial  wave "t, using the requirements of unitarity to "x its coupling strength to the resonances. For the elastic channel, they use CMB and KH80 elastic amplitudes simultaneously. For the inelastic channels involving two pions, we de"ne channels identical to those chosen in Ref. [4]. They are listed in Table 3. Each channel has a speci"c orbital and spin angular momentum; the nomenclature used is given in the table. These nNPnnN channels are almost identical to those chosen in Ref. [25]. The only di!erence is in the choice of the mass and width of the "ctitious isoscalar-spin 0 meson, taken as 1 GeV for each parameter in Ref. [25]; like Ref. [4], we choose a mass and width of 800 MeV. We agree with their conclusion that the results are not sensitive to this choice. The number of states sought in each partial wave was the same as used by KSU [4]. We also included the same number of open channels in each partial as KSU because of the choices in the amplitudes "tted in Ref. [25]. If no amplitudes are provided by Ref. [25] for a partial wave, e.g. for F ,  an appropriate dummy channel is used to absorb the #ux. Table 8 labels rows of this type as &Flux'. The number of parameters depends on the number of resonances to be "t and the number of open channels included in the "t. For example, the S partial wave has three resonances and we "t  values for each bare pole and its couplings to as many as 8 channels. There are also two subthreshold and one high energy `statesa used to simulate background. The masses of these states are constrained to be far away from any of the actual resonances. Three parameters, a pole and coupling strengths to nN and gN are "t for each subthreshold background pole. The high-energy state is allowed to couple to all open channels. Thus, there are 38 parameters "t in this partial wave. For the D partial wave, only one resonance (with coupling to four channels) and three  background poles were "t, a total of 15 real parameters. In the elastic channel, the data quality is reasonable at values of = from threshold to about 2.0 GeV. At values of = larger than roughly 1.8 GeV, no inelastic data is available for the nnN "nal state. The inelastic data has signi"cant #uctuations. If the data are to be represented by a smooth function (assumed in all analyses), the error bars are underrepresented. In fact, the Manley et al. paper [25] states that only diagonal errors were included in their output. Correlations should be signi"cant in the analysis and can only add to the uncertainties. Although the elastic data were able to be "t well, the inelastic data were not. Since the elastic data is of much higher quality than the inelastic data, the inelastic error bars were weighted by a factor of 2 lower than the elastic error bars in order to ensure a reasonable "t to the elastic data. (The original Cutkosky et al. paper [6,32] used a factor of 3 to weight the inelastic data.) For elastic data, values of s/datapoint were 1.7 and 1.6 for the S and D waves, respectively. For the inelastic amplitudes, s/datapoint values were   9.2 and 22.2. The s values are given for the partial wave amplitude values without the extra weighting factor. (We do not quote s per degree of freedom because the parameters are shared between the elastic and inelastic "ts.) Although we get qualitatively better "ts to the elastic data than Manley and Saleski [4] in most cases, similar quality "ts are found for the inelastic amplitudes. (No values of s are given by KSU.) However, the shapes of our inelastic ¹ matrices are qualitatively di!erent in many cases. Speci"c partial waves will be discussed in Section 5.3 (Fig. 9).

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211

Fig. 9. ¹ matrices according to the best "t for the D partial wave. The VPI partial wave amplitudes are shown by data  points with error bars. The KSU "t of Manley and Saleski is shown as dashed lines and the "t of this work is shown as solid lines.

For a complicated multi-parameter "t, errors are di$cult to determine because correlations can be signi"cant. The partial wave data we use as input quotes only diagonal errors. We include error estimates in all extracted quantities due to propagation of errors quoted in the partial wave data. In addition, we add contributions determined from additional "ts where the background parameterization is varied. To allow a full error analysis, separate "ts were made to data over a limited energy range to isolate single resonances. A 2;2 K matrix was used to model 2 channels at a time and a number of "ts were made for each resonance to determine errors on each of the extracted quantities. In one "t, the errors on the resonance mass and width and the error on the largest branching ratio were

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Fig. 9. (Continued.)

determined. With nN as one channel and the channel with the largest remaining branching fraction (often nD) for the resonance under consideration as the second channel, the resonance mass, width, and branching fraction together with a simple parameterization of the non-resonant amplitude were "t to the data. Errors for the branching fraction to the channels with less coupling strength were determined in "ts where the mass and width of the resonance were "xed at the "nal "t value of this work and only the branching fraction and background were "t. For the K matrix in these cases, the channel for which the branching fraction was being determined was one K-matrix channel and the nN channel or the channel with the largest branching fraction was chosen as the second K-matrix channel. In all cases, two "ts were done with di!erent simple assumptions for the background dependence, either #at vs. linear or linear vs. quadratic. The "rst type was most often

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213

Fig. 9. (Continued.)

used. Two components to the error for each parameter under study were determined from each pair of "ts, a relative error for each "t parameter from the "t with the largest s and the absolute di!erence between the two determinations of the "t parameter. The "rst component is representative of the statistical error in the data points; the second error is due to systematic e!ects between the di!erent background parameterizations. When two reasonable "ts were obtained, the components were added in quadrature. For prominent states, the "ts were easily made. However, most states required many trials to "nd "ts of the appropriate quality. Background shapes, channel choices, = ranges, and weighting of the inelastic channel in the s determination were all varied until two good "ts were obtained. All error bars quoted in the tables discussed below were determined from these "ts.

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5.2. General results A list of resonances found in this analysis is given in Tables 7 and 8 and compared to the results of KSU [4], those of CMB [6] and the latest recommended values given by PDG [1]. For various states, many other results exist. We make direct comparisons only with the previous results that use multi-channel models and provide error bars for their determinations. We show masses and widths in Table 7 and branching fractions to various inelastic channels in Table 8. These are the results of the analysis discussed in Section 2.3. Poles corresponding to most of the states found by KSU are found in this analysis, although the properties can be di!erent; states for which there is no evidence in the present analysis are P (1900), F (2000), and D (1940), all of    which are 1H states according to PDG [1]. Although we obtain "t results for many weak states, a detailed study of the validity of the "t for weak states was not attempted. Discussion in this section will be limited to PDG 3H and 4H resonances. In Table 9, we show the results for pole positions of this analysis. The pole positions have less model dependence than the parameters in Tables 7 and 8. The analysis presented here is not identical to any previous analysis. The data base used is similar to KSU, but the formalism has signi"cant di!erences. The formalism is identical to CMB, but the data base used is quite di!erent. CMB used the best representation of data at the time } their own elastic amplitude analysis [20] and the inelastic quasi-two-body amplitudes of the SLAC-Berkeley [35] and Imperial College [36]. Batinic et al. [27] use a truncated version of the formalism used here and still another data set. Strong isolated resonances that have a strong elastic coupling are "t well with all models and results for the resonance parameters, such as the D (1675) and F (1680) masses and widths, tend   to have close agreement between previous results and the new results. PDG gives a range of 15 MeV for both masses and a range of 40 and 20 MeV for the D and F full-widths and our   values are within these intervals. For the elastic branching fraction, PDG suggests a range of 10%; our results are inside the range for F and just outside it for D .   The bene"t of the multichannel analysis is readily apparent for states with a very small elastic branching fraction. For example, D (1700) and P (1710) (both PDG 3H resonances) are not seen   in the VPI elastic analysis [2] because there are no strong signs of the resonance in the elastic ¹ matrix. However, there is a strong resonance signal in the nNPnD ¹ matrix in each case (Fig. 10). For the cases where KSU di!ers signi"cantly from the consensus of previous results, the S (1620) mass and the S (1650) elasticity, the new results tend to agree with the older values. To   test for the dependence on the elastic data used in the "t, we re-did "ts with the same elastic data used by KSU and found results qualitatively similar to our "nal analysis. An unusual feature of the CMB analysis is the large value for the S (1535) width, about 40%  larger than any other analysis. This analysis obtains a width in closer agreement with KSU and KH than CMB. We have been unable to reproduce the large width. We note that CMB based their "t on a subsidiary analysis of Bhandari and Chao [37] with a compatible model. Although the nNPgN data has changed little since then, the elastic data we use is a global "t to all data rather than the single data set they used. The VPI SM95 single energy solution is systematically larger and somewhat #atter than the data used in that 1977 work. The most confusing aspect is that Bhandari and Chao quote a full-width of 139$33 MeV, much more compatible with the present results than with the full CMB results.

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236

215

Table 7 Results for masses, full widths, and elastic fractions for all resonances found in this analysis. All resonances found in the KSU analysis were searched for, but not all were found. No attempt was made to "nd new resonances because then data quality is not good enough for a new search. See text for more details Resonance

Mass (MeV)

Width (MeV)

Elasticity (%)

Reference

S (1535)  ****

1542(3) 1534(7) 1520}1555 1550(40) 1689(12) 1659(9) 1640}1680 1650(30) 1822(43) 1928(59) +2090 2180(80) 1479(80) 1462(10) 1430}1470 1440(30) 1699(65) 1717(28) 1680}1740 1700(50) 2084(93) 1885(30) +2100 2125(75) 1716(112) 1717(31) 1650}1750 1700(50) 1518(3) 1524(4) 1515}1530 1525(10) 1736(33) 1737(44) 1650}1750 1675(25) 2003(18) 1804(55) +2080 2060(80) 1685(4) 1676(2) 1670}1685 1675(10)

112(19) 151(27) 100}250 240(80) 202(40) 173(12) 145}190 150(40) 248(185) 414(157)

35(8) 51(5) 35}55 50(10) 74(2) 89(7) 55}90 65(10) 17(3) 10(10)

350(100) 490(120) 391(34) 250}450 340(70) 143(100) 478(226) 50}250 90(30) 1077(643) 113(44)

18(8) 72(5) 69(3) 60}70 68(4) 27(13) 9(4) 10}20 20(4) 2(5) 15(6)

260(100) 121(39) 383(179) 100}200 125(70) 124(4) 124(8) 110}135 120(15) 175(133) 249(218) 50}150 90(40) 1070(858) 447(185)

12(3) 5(5) 13(5) 10}20 10(4) 63(2) 59(3) 50}60 58(3) 4(2) 1(2) 5}15 11(5) 13(3) 23(3)

300(100) 131(10) 159(7) 140}180 160(20)

14(7) 35(1) 47(2) 40}50 38(5)

Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB

S (1650)  ****

S (2090)  *

P (1440)  ****

P (1710)  ***

P (2100)  *

P (1720)  ****

D (1520)  ****

D (1700)  ***

D (2080)  **

D (1675)  ****

(¹able continued on next page)

216

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236 Table 7 (Continued) Resonance

Mass (MeV)

Width (MeV)

Elasticity (%)

Reference

F (1680)  ****

1679(3) 1684(4) 1675}1690 1680(10) 2311(16) 2086(28) +1990 1970(50) 2168(18) 2127(9) 2100}2200 2200(70) 1617(15) 1672(7) 1615}1675 1620(20) 1802(87) 1920(24) 1850}1950 1890(50) 1721(61) 1744(36) +1750 1995(12) 1882(10) 1870}1920 1910(40) 1234(5) 1231(1) 1230}1234 1232(3) 1687(44) 1706(10) 1550}1700 1600(50) 1889(100) 2014(16) 1900}1970 1920(80) 1732(23) 1762(44) 1670}1770 1710(30) 1932(100) 1956(22) 1920}1970 1940(30)

128(9) 139(8) 120}140 120(10) 205(72) 535(117)

69(2) 70(3) 60}70 62(5) 22(11) 6(2)

350(120) 453(101) 547(48) 350}550 500(150) 143(42) 154(37) 120}180 140(20) 48(45) 263(39) 140}240 170(50) 70(50) 299(118)

6(2) 20(4) 22(1) 10}20 12(6) 45(5) 9(2) 20}30 25(3) 33(10) 41(4) 10}30 10(3) 6(9) 8(3)

713(465) 239(25) 190}270 225(50) 112(18) 118(4) 115}125 120(5) 493(75) 430(73) 250}450 300(100) 123(53) 152(55) 150}300 300(100) 119(70) 599(248) 200}400 280(80) 316(237) 526(142) 250}450 320(60)

29(21) 23(8) 15}30 19(3) 100(1) 100(0) 98}100 100(0) 28(5) 12(2) 10}25 18(4) 5(4) 2(2) 5}20 20(5) 5(1) 14(6) 10}20 12(3) 9(8) 18(2) 5}20 14(4)

Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB

F (1990)  **

G (2190)  ****

S (1620)  ****

S (1900)  ***

P (1750)  * P (1910)  ****

P (1232)  ****

P (1600)  ***

P (1920)  ***

D (1700)  ****

D (1930)  ***

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217

Table 7 (Continued) Resonance

Mass (MeV)

Width (MeV)

Elasticity (%)

Reference

D (2350)  *

2459(100) 2171(18) +2350 2400(125) 1724(61) 1752(32) 1873(77) 1881(18) 1870}1920 1910(30) 1936(4.5) 1945(2) 1940}1960 1950(15)

480(360) 264(51)

7(14) 2(0)

400(150) 138(68) 251(93) 461(111) 327(51) 280}440 400(100) 245(12) 300(7) 290}350 340(50)

20(10) 0(1) 2(1) 9(1) 12(3) 5}15 8(3) 44(1) 38(1) 35}40 39(4)

Pitt-ANL KSU PDG CMB Pitt-ANL KSU Pitt-ANL KSU PDG CMB Pitt-ANL KSU PDG CMB

F (1752)  * F (1905)  ****

F (1950)  ****

There has been signi"cant interest and controversy in the properties of states in the P partial  wave. In large part this is because the data in this partial wave has always been poor. In the present "t, s/datapoint for this partial wave is 3.8 for the elastic channel and ranges from 8.5 to 17.6 for the inelastic channels. With the low quality of the existing inelastic data, no e!ort was made to determine the correct number of P states. We "nd three P states with properties somewhat   di!erent than those previously obtained, although all of our values are within the suggested ranges of PDG. In our results, P (1440) has a mass that just overlaps the PDG window and one of the  largest widths obtained. In our "t, this state has a signi"cant contribution from non-resonant interactions; that together with the low quality of the present data produces a large estimated error for the values given. KSU results for this state show a somewhat smaller width and a much smaller error bar. We can only comment that the KSU model handles background quite di!erently than the present model and they used older nN elastic data. For the P (1710), we "nd mass and width  values well within the large PDG ranges; however, KSU has a very large width with a large estimated error. Since this state sits on the tail of the P (1440) and does not have a strong signal in  any channel, it is clear that the properties of the 2 lowest P states are closely coupled and  multichannel analyses are very appropriate. A later analysis of Cutkosky and Wang [32] of this partial wave using the CMB model compared results obtained with the VPI (SM89) and CMB80 partial wave analyses. They "nd qualitatitively similar results (large width for the Roper and small width for the 1700 MeV state) to those obtained here. Evidence for the highest energy P state in  the present data is very poor because only elastic data exists in the appropriate energy range. Other unusual cases found in this new analysis include the P (1910) and S (1900) states. The   P (1910) is found at signi"cantly higher mass and has signi"cantly larger width than previous  determinations. Since this state is at high mass, the inelastic data is very important in determining its properties. However, there is almost no existing inelastic data in this partial wave other than a few points in the nNPnNH(1440) reaction. PDG has given this state a 4H rating, but with the present data this rating should be downgraded. Although VPI does not "nd the S (1900) state, it is  very prominent in KSU and PDG. We "nd the mass at 1802 MeV and a width of 48 MeV (with

218

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236 Table 8 Results for decay branching ratios of all resonances found in this analysis. Fractions are expressed as a percentage of the full width found in Table 7 Resonance

Channel

Pitt-ANL

KSU

PDG

S (1535) 

nN gN o N  (o N)  " (n*) " (pN) . nNH(1440) nN gN o N  (o N)  " (n*) " (pN) . nNH(1440) nN gN o N  (o N)  " (n*) " (pN) . nNH(1440) nN gN o N  (n*) . (pN) 1 K" nN gN o N  (n*) . (pN) 1 K" nN gN o N  (n*) . (pN) 1 K" nN gN o N  Flux nN gN

35(4) 51(5) 2(1) 0(1) 1(1) 2(1) 10(9) 74(2) 6(1) 1(1) 13(3) 2(1) 1(1) 3(1) 17(7) 41(4) 36(1) 1(1) 1(1) 2(1) 2(1) 72(2) 0(1) 0(1) 16(1) 12(1) 0(1) 27(4) 6(1) 17(1) 39(8) 1(1) 10(10) 2(1) 61(61) 4(1) 2(1) 10(1) 21(20) 5(5) 4(1) 91(1) 0(1) 63(1) 0(1)

51(5) 43(6) 2(1) 1(1) 0(0) 1(1) 2(2) 89(7) 3(5) 0(0) 3(2) 2(1) 2(2) 1(1) 10(10) 0(3) 49(22) 0(1) 6(14) 4(10) 30(22) 69(3)

35}55 30}55 0}4

22(3) 9(2)

20}30 5}10

9(4)

10}20

3(7) 49(10) 2(4) 37(10) 15(6)

5}25 15}40 10}40 5}25

S (1650) 

S (2090) 

P (1440) 

P (1710) 

P (2100) 

P (1720) 

D (1520) 

0}1 0}3 0}7 55}90 3}10 4}14 3}7 0}4 0}5

60}70 0}8

27(79) 24(18) 32(71) 2(6) 13(5)

10}20

87(5)

70}85

59(3)

50}60

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236

219

Table 8 (Continued) Resonance

D (1700) 

D (2080) 

D (1675) 

F (1680) 

F (1990)  G (2190) 

S (1620) 

S (1900) 

P (1750) 

Channel

Pitt-ANL

KSU

PDG

(o N)  1 (n*) " (n*) 1 (pN) . nN gN (o N)  1 (n*) " (n*) 1 (pN) . nN gN (o N)  1 (n*) " (n*) 1 (pN) . nN gN o N  (o N)  " (n*) " nN gN (o N)  $ (o N)  . (n*) $ (n*) . (pN) " nN gN Flux nN gN (o N)  " (u N)  " nN o N  (o N)  " (n*) " nNH(1440) nN o N  (o N)  " (n*) " nNH(1440) nN nNH(1440) Flux

9(1) 11(2) 15(2) 1(1) 4(1) 0(1) 7(1) 79(56) 11(1) 0(1) 13(2) 0(2) 6(6) 17(10) 40(10) 24(24) 35(2) 0(1) 0(1) 1(1) 63(2) 69(1) 0(1) 3(1) 5(1) 1(1) 14(3) 9(1) 22(3) 0(1) 77(77) 20(1) 0(1) 29(28) 51(51) 45(1) 14(3) 2(1) 39(2) 0(1) 33(6) 30(2) 5(1) 28(1) 4(1) 6(6) 83(1) 11(11)

21(4) 15(4) 5(3)

15}25 5}12 10}14 0}8 5}15

1(2) 13(17) 80(19) 5(10) 2(4) 23(3) 26(14) 21(14) 3(7) 27(12) 47(2)

0}35

40}50

0(0) 0(0) 53(2) 70(3)

1}3

2(1) 5(3) 1(1) 10(3) 12(3) 6(2) 94(2)

1}5 0}12 0}2 6}14 5}20

22(1)

10}20

29(6) 49(7) 9(2) 25(6) 4(3) 62(6)

20}30 7}25

41(4) 5(7) 33(10) 16(8) 6(9) 8(3) 28(9) 64(9)

50}60 60}70

30}60 10}30

(¹able continued on next page)

220

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236 Table 8 (Continued) Resonance

Channel

Pitt-ANL

KSU

PDG

P (1910) 

nN nNH(1440) Flux nN (n*) . nNH(1440) nN (n*) . nNH(1440) nN (n*) . nNH(1440) nN (o N)  1 (n*) " (n*) 1 nN K" Flux nN K" Flux nN (o N)  . (n*) $ (n*) . nN (o N)  . (n*) $ (n*) . nN (o N)  $ (n*) $ Flux

29(29) 56(7) 15(15) 100(1) 0(1) 0(1) 28(5) 59(10) 13(4?1) 5(61?) 41(2.9) 53(8.2) 5(1.6) 1(1) 4(1) 90(1.7) 9(8) 91(11)

23(8) 67(10) 10(1) 100(0) 0(0) 0(0) 12(2) 67(5) 20(4) 2(2) 83(26) 15(24) 14(6) 8(4) 4(3) 74(7) 18(2)

15}30

P (1232)  P (1600)  P (1920)  D (1700) 

D (1930)  D (2350)  F (1752) 

F (1905) 

F (1950) 

7(14) 93(15) 0(1) 60(60) 40(1) 0(1) 9(2.2) 24(1) 44(1) 23(1) 44(1) 36(1) 20(20)

98}100

10}25 40}70 10}35 5}20

10}20 5}20 1}7 25}50 5}20

82(2) 2(0) 98(0) 2(1) 22(14) 48(16) 28(18) 12(3) 86(3) 0(1) 1(3) 38(1) 43(1) 18(3)

5}15 0}60 0}25 35}40 0}10 20}30

a large estimated error) while KSU "nds 1950 MeV and 263 MeV. This signi"cant di!erence is largely due to the elastic data sets used. There is a strong bump at about 1900 MeV in the elastic ¹ matrices used by KSU which has vanished in the VPI partial wave amplitudes. For the branching fractions presented in Table 8, the only recent result is KSU, which is presumably weighted heavily in the PDG listings (also shown in the table). We divide the oN and nD channels into the appropriate spin channels since the component angular momenta can sometimes have more than one value. For D , the oN spin can be  with orbital angular   momentum 2 or  with orbital angular momentum 0 or 2. Since only the spin  orbital angular   momentum 0 case was found to be important in the nNPnnN isobar analysis [25], this is the only

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236

221

Table 9 Pole positions. The complex energy of the pole for each state is given along with the physical mass Resonance

Res mass (MeV)

Pole position (MeV)

S (1535)  S (1650)  S (2090)  P (1440)  P (1710)  P (2100)  P (1720)  D (1520)  D (1700)  D (2080)  D (1675)  F (1680)  F (1990)  G (2190)  S (1620)  S (1900)  P (1750)  P (1910)  P (1232)  P (1600)  P (1920)  D (1700)  D (1930)  D (2350)  F (1752)  F (1905)  F (1950) 

1545 1693 1822 1479 1699 2083 1716 1520 1729 2002 1687 1679 2311 2168 1633 1798 1721 1995 1234 1687 1889 1732 1932 2459 1724 1873 1936

1525}51i 1663}120i 1795}110i 1383}158i 1679}66i 1810}311i 1692}47i 1504}56i 1704}78i 1824}307i 1674}60i 1667}61i 2301}101i 2107}190i 1607}74i 1795}29i 1714}34i 1880}248i 1217}48i 1599}156i 1880}60i 1726}59i 1883}125i 2427}229i 1697}56i 1793}151i 1910}115i

oN channel we include for this partial wave. The nD channel can couple with orbital angular momentum of 0 or 2 in this partial wave and both possibilities are included in the "ts. As with KSU, uncertain "ts due to underestimated error bars and/or missing data make interpretation di$cult in some cases. We quote values for this analysis and give estimated errors for each quantity. Since we use the same inelastic ¹ matrices as KSU for input to the "t, the results should be qualitatively similar. Based on our study in Section 4.4, we feel there is roughly 20% di!erence between the KSU and Pitt-ANL results due to model dependence in the most sensitive quantities. The poor "ts make this true less often than might be expected. For F (1680), the elastic branching  fraction is &70% in both analyses and the largest inelastic channels are nD P-wave and (nn) N in Q both; agreement is within errors for the largest values. For D (1700), the elastic branching fraction  is small and the inelastic strength in concentrated in the nD s-wave, so we are in close agreement with KSU. However, S (1620) is a strong state where agreement is not good. There are 2 strong 

222

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236

Fig. 10. ¹ matrices for D



partial wave. The same labeling is used as in Fig. 9.

inelastic channels, o N and nD D-wave. Although there is agreement in the strength of o N, the   elastic strength is much smaller for KSU and the nD branching fraction is of course larger. The full-widths seen are only 15% di!erent and the estimated errors overlap. The s per datapoint for the S partial wave in the elastic channel is 2.1 and ranges from 4.3 to 17.2 for the inelastic  channels. The calculation of Capstick and Roberts uses a relativized quark model to calculate resonance masses and a P model for creation of qq pairs [18]. They "t the two quark level coupling  parameters to the nN decay amplitudes of the non-strange resonances rated by PDG as 2H or better and then predict the remaining decay amplitudes for various meson#baryon "nal states (including many "nal states for which there are no data). Although their primary purpose was to

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236

223

Fig. 10. (Continued.)

look for signatures of `missinga quark model resonances, we note that the qualitative agreement with our analysis is satisfactory. There are notable successes such as D (1520) where quantitative  agreement comes with all signi"cant decay channels and notable failures such as S (1535) where  both nN and gN decay widths (and the full-width) are overestimated by a factor of 4. 5.3. Detailed discussion } D , D , and S partial waves    Presented in this section are detailed results for three representative partial waves. It includes "gures of all the channel ¹ matrices and a discussion of the results. These partial waves have

224

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236

Fig. 10. (Continued.)

reasonable quality data and contain both prominent and less prominent states. Numerical data can be found in Tables 4, 5, 8, and 7. The ¹ matrices are found in Fig. 9 for the D partial wave, 10 for  D , and 11 for S . In each we show the relevant phase shift amplitudes [2,25] along with the "t of   Manley and Saleski [4] (dashed lines) and our "nal "t (solid line). Only data up to ="2.15 GeV were used in the "ts because the data at higher values of = are of diminished quality. Since all the data in each partial wave were simultaneously "t with the requirements of unitary, not all channel data shown in the "gures is "t equally well. In general, the "ts to the elastic data are good (s/datapoint &2) and the "ts to the inelastic data are poor (s/datapoint'10) for both the KSU "ts and the present results, as discussed above.

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236

225

Fig. 10. (Continued.)

The D partial wave contains a strong (PDG [1] 4H), isolated resonance for which the results of  the one resonance example (see Section 4.1) apply fairly well. In this case, both the elastic and nNPnD (Dwave) inelastic channels are prominent in the D partial wave. Since neither threshold is  nearby and there is no interfering resonance, the ¹ matrix in each of these channels has the characteristic shape of an isolated resonance with a peak in the elastic and inelastic cross section near the resonance mass. The non-resonant amplitude is a smooth function. The KSU and present "ts are similar in the ability to match features in the data in the two strong channels near the cross section peak at =K1680 MeV. The two analyses di!er in their description of the high energy data (=51.8 GeV), but the data are very sparse there. Neither model "ts the smaller o production channels well. The o N channel shows that the model expectations (based on the requirements of  unitarity and the data in the other channels) do not match the data in this channel and a poor "t results. Essentially all analyses for the D (1680) resonance give similar values for the mass, full-width,  and elastic fraction (see Table 7). Even though VPI [2] accounts for the inelasticity by using a dummy channel, the "t parameters for this state are similar to those obtained in the present "t. PDG [1] gives quite small error bars for the mass and width of this state, re#ecting the unanimity of the "tting results for this state. It is encouraging that the complicated features of the present model are not important for the simple case presented in this partial wave. The D partial wave contains a 4H state at 1520 MeV (PDG standard value, the actual mass  is slightly di!erent), a 3H state at 1700 MeV, and a 2H state at 2080 MeV. Since the lowest two resonances have signi"cant overlap in energy, the features of the CMB model are important for this partial wave. The lowest state is highly elastic, but also shows up prominently in (o N)  1 and (nD) "nal states. Note that the peak in the imaginary ¹ matrix is inverted. As already 1 mentioned, the second state is barely seen in the elastic channel, thus is not found in the VPI [2] analysis. Both KSU and the present analysis "nd most of the decay strength in the nnN "nal states with less than 10% of the decay strength to the elastic channel. KSU di!ers with this model in the

226

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236

Fig. 11. ¹ matrices for S partial wave. The same labeling is used as in Fig. 9. 

distribution of the inelastic strength. While KSU "nds most of the strength in oN, this model "nds it in nD. The KSU model "ts available inelastic data better. Feuster and Mosel [5] "nd a systematically lower width than other models for the lowest state, but their width for the 2nd state is unusually large. Evidence for the third state is very weak in this analysis; there is only elastic data at =51.9 GeV and there is very little evidence of structure. It is best "t with a very large width. The three resonances have varying strength in the di!erent channels. As a result, the "tted curve also changes signi"cantly from channel to channel. In the nD channels, the shapes are complicated because the 1700 MeV state is important. The imaginary part of the ¹ matrix has a dip at the 1520 MeV state and a peak at the 1700 state. Nevertheless, this model and the KSU model "t the features well and the "tted parameters for the higher energy state agree within stated errors.

T.P. Vrana et al. / Physics Reports 328 (2000) 181}236

227

Fig. 11. (Continued.)

The S partial wave was already discussed in Section 4 with regard to the signi"cant model  dependence seen. As a result of this model dependence, the S (1535) full-width has been quoted as  66}250 MeV based on di!erent analyses of similar data (almost all "ts are based prominently on the elastic nN channel). The relatively small width of the S (1535) in this analysis (112$30 MeV)  is determined in large part by a signi"cant overlap of the S (1535) with the S (1650). This overlap   causes a large interference e!ect. Although a similar e!ect was seen in one description of the cpPgp data that uses a formalism similar to the Pitt-ANL model [33], many models based on a ¹-matrix formalism (e.g. [38}40]) have a formalism where it is di$cult to account for more than one resonance in a partial wave.

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Fig. 11. (Continued.)

The multichannel K-matrix analysis of Feuster and Mosel [5] gets values of the full-width in the range of 151}198 MeV for di!erent form factors. However, their model does not handle the gN cusp as well because it lacks the analytic phase space, does not model the nnN inelasticity well because a zero width particle is used, and does not include o!-shell intermediate state scattering e!ects because of the K-matrix approximation. Although 8 channels are included in the "t, many of them turn out to have small coupling to S (1535) (in agreement with PDG [1]). The two major channel couplings for S (1535) are nN   and gN. Therefore, another determining factor in the full-width is the total cross section for the

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Fig. 11. (Continued.)

n\pPgn in Fig. 11 and the corresponding ¹ matrix in Fig. 11b. The cross section of Fig. 12 is made up of nNPgN ¹-matrix amplitudes for S through G , however its main component is   S . Although the shape of the cusp (see Fig. 7) does carry information about the gN channel, the  overall data quality isn't satisfactory. We await higher quality data for a more conclusive determination of this amplitude. The branching fraction of the S (1535) to the nN channel is at the low end of the PDG range.  A surprising feature of the S (1535) has been its unusually large decay width to gN. We "nd strong  evidence for a value at the high end of the PDG range. We believe our result because our model fully accounts for the threshold enhancement (cusp e!ect) due to the gN channel opening and the

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Fig. 12. Total cross section for the n\pPgn reaction. The data was taken from sources in Ref. [27], leaving out the low energy Brown et al. data. The solid line shows the total cross section calculated with the "nal "t result of this work using all partial waves. The contribution from the S partial wave would be very similar to the full result. 

interference of the two overlapping resonances. Since no other model has such a complete formulation, they can have model-dependent systematic errors as a result. The properties of S (1650) are largely determined through the prominent bump in the elastic  channel. The elastic ¹ matrix has a peak in the imaginary part at about the right energy and the real part is decreasing at that energy; a strong resonance signal. The strongest inelastic signal comes in the o N channel. The decay width to gN is small in this analysis, in agreement with previous  analyses. The striking di!erence between branching fractions to gN has been another unusual feature of these states. The existence of the third S resonance is very weakly supported by the  existing data. In our analysis, this state is most often determined by the need to "t non-statistical features in the partial wave amplitudes. Further data will be required to sort out the question of whether or not it exists. The role of the inelastic gN channel is very important in the determination of resonance parameters for the S (1535) state, as shown in some detail in Section 4. Unfortunately, the existing  data for nNPgp is sparse and of uncertain quality [29] (see Fig. 12). Recently published data [41] for cpPgN has been interpreted as evidence for a very large width (K200 MeV). We do not include the Krusche et al. data in this analysis. However, very simple resonance models [41,42] were used to determine the large width. Since the large body of nN data was not "tted with their model, possible inconsistencies exist in their parameterization. Although the simple model has validity for the gN coupling to S (1535), we feel there will still be signi"cant e!ects  from interference with S (1650) and coupled channel e!ects (e.g. cNPS (1650)PnNP   S (1535)PgN). 

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Fig. 13. Pitt-ANL model calculation using "nal fit compared with the total cross section for the process n\pPnn. The "t was to ¹-matrix amplitudes derived in part from the data shown here. Data shown is from the VPI SAID database.

5.4. Observables ¹-matrix amplitudes were "t using the Pitt-ANL model. We do not "t directly to data; rather, we "t the partial wave amplitudes. It is then interesting to compare the actual experimental data to our calculated observables to verify that nothing has been lost by not "tting to the actual data points. The f and g spin non-#ip and spin #ip amplitudes are de"ned in terms of the Pitt-ANL ¹-matrix elements for nN elastic scattering by [43] 1  f (h)" [(l#1)¹l #l¹l ]Pl (cos h) , > \ ql  1  g(h)" (¹l !¹l )Pl (cos h) , (39) > \ ql  where Pl and Pl are the Legendre polynomials and their "rst derivative with respect to cos h, respectively. The ¹l are the amplitudes for a particular nN scattering charge channel, where the ! l$ refers to the total spin J"l$, and q is the center of mass momentum of the "nal state pion.  The ¹l are de"ned by Eq. (1) for the nNPnN channel. ! The di!erential cross section (dp/dX) and polarization P are: dp "" f (h)"#"sin h g(h)" , dX dp P"2Im [ f (h)gH(h)] sin h . dX

(40)

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Fig. 14. Pitt-ANL model calculation using "nal fit parameters compared to the di!erential cross section for the process n>pPn>p. Data from the VPI SAID database is shown.

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Fig. 15. Pitt-ANL model calculation using "nal fit parameters of the polarization, P, for the process n\pPn\p. Data from the VPI SAID database is shown.

Fig. 13 shows a plot of the total cross section vs. the Pitt-ANL calculation for the process n\pPnn. Fig. 14 shows plots of the di!erential cross section, dp/dX, at a number of di!erent energies for the process n>pPn>p. Fig. 15 shows plots of the nucleon recoil polarization, P, for a number of di!erent energies for the process n\pPn\p. In each case, the calculation using "nal "t parameters is in good agreement with the data. For the P data, the error bars are sometimes large. For the plot in Figs. 13}15 the data is taken from VPI's SAID database. It is a large collection of scattering data for a number of di!erent reaction channels, including nN elastic and charge exchange reactions. Based on our studies, we can conclude that our resonance parameters are very close to what would have been obtained if the "t had been made to the original data.

6. Conclusions We have presented a new study of excitation of baryon resonances through pion-nucleon interactions. An expanded version of the Carnegie-Mellon Berkeley (CMB) was used in this work.

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We added the gN channel to the calculation and used the latest data sets. The model satis"es many of the desirable theoretical constraints } two- and three-body unitarity, time reversal invariance, and analyticity. Because the model is based on a separable interaction, it is linked to a Hamiltonian approach, unlike many of the models previously used [2,4]. The model has a key feature of allowing large numbers of asymptotic channels and more than one resonance per partial wave. Since analyticity is satis"ed and all inelastic thresholds are included, structure in the physical observables due to non-resonant e!ects are carefully treated. Resonances can then be found as poles in the complex s plane. As in the original CMB model, non-resonant processes are included as very wide resonances at very high or very low energies which are then subject to the same threshold e!ects as the resonances. This empirical approach is well established in potential scattering theory, but could be replaced later by theoretical calculations based on meson exchange models, e.g. [44]. Rather than deal with the complexities of a direct "t to the cross section and polarization data, we use the partial wave amplitudes of the VPI and KSU groups [2,25] and our own "t to the nNPgN data. A re-evaluation of the nNPnnN data is in progress by our group. The features of this model are expected to be very important in the S partial wave where there  are two overlapping resonances at 1535 and 1650 MeV and a strong threshold opening. The lower state pole and the gN channel opening are close together and the quantum mechanical interference of the two states is found to be signi"cant. We show how the interpretation of these states has very strong model dependence and how relaxing the features of the present model can skew the "tting results for the width and branching fractions to nN and gN. We get a new result for the branching fraction to gN at the high end of what has been previously published for S (1535), making its  characteristics even more unusual. The general "tting results are presented in Tables 7 and 8. We searched for the same states seen in the previous analysis of Manley and Saleski [4]. This simpli"cation was made because the data quality was not high enough for a valid search for weakly excited states. Although values of s/datapoint are roughly 2 for elastic "ts, they are 10 or more for "ts to the nNPnnN ¹ matrix data. This is perhaps due to the lack of a treatment of correlations in the error analysis of the pion production data. Modern experimental techniques could greatly improve the inelastic data set. We strongly encourage new measurements of the inelastic channels. A signi"cant e!ort was made to determine error bars re#ecting both the estimated errors in the ¹ matrices used in the "ts and the di!ering choices of background energy dependence. We "nd strong and isolated states (e.g. P (1232) and D (1675)) with very similar parameters as   previous analyses [2}4,6]. States with strong model dependence such as S (1535) or with  signi"cant changes in the data set such as S (1900) and D (1700) get quite di!erent results. We   "nd a full width of S (1535) at the low end of the range of previous values, especially with respect  to the interpretations of recent cpPgp data [41]. The results presented here do not include the photoproduction data; rather, the present results are dependent on a fairly weak data set for n\pPgn. Nevertheless, the large widths obtained in analyses highly dependent on the Mainz eta photoproduction data are only loosely coupled to the large data set used in this study. We are in the process of further extending the model to include photoproduction and electroproduction reactions. The Born terms are included for production of pions and etas and the resonance spectrum is being re"t. Results of the present paper will then be subject to change.

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Acknowledgements We are grateful to R. Arndt, R. Workman, and M. Manley for sharing their partial wave data and their analysis results with us. We have also bene"ted greatly from the programming and database work of D. Ciarletta, J. DeMartino, J. Greenwald, D. Kokales, M. Mihalcin, and K. Bordonaro at the University of Pittsburgh. Special thanks go to C. Tanase for investigating the analytic structure of this model in detail. References [1] Particle Data Group, Phys. Rev. D 54 (1996) 1. [2] Richard A. Arndt, Igor I. Strakovsky, Ron L. Workman, Marcello M. Pavan, Phys. Rev. C 52 (1995) 2120. [3] G. HoK hler, F. Kaiser, R. Koch, E. Pietarinen, Handbook of Pion}Nucleon Scattering, [Physics Data No. 12-1 (1979)]. [4] D.M. Manley, E.M. Saleski, Phys. Rev. D 45 (1992) 4002. [5] T. Feuster, U. Mosel, Phys. Rev. C 58 (1998) 457. [6] R.E. Cutkosky, C.P. Forsyth, R.E. Hendrick, R.L. Kelly, Phys. Rev. D 20 (1979) 2839; R.E. Cutkosky, C.P. Forsyth, J.B. Babcock, R.L. Kelly, R.E. Hendrick, in: N. Isgur (Ed.), Baryons 1980, Proceedings of the IV International Conference on Baryon Resonances, University of Toronto, 1980, p. 19. [7] F. Foster, G. Hughes, Rep. Prog. Phys. 46 (1983) 1445. [8] N. Isgur, G. Karl, Phys. Rev. D 18 (1978) 4187; D 19 (1979) 2653. [9] S. Capstick, N. Isgur, Phys. Rev. D 34 (1986) 2809, S. Capstick, B.D. Keister, Phys. Rev. D 51 (1995) 3598. [10] R. Bijker, F. Iachello, A. Leviatan, Ann. Phys. (N.Y.) 236 (1994) 69. [11] F. Cardarelli, E. Pace, G. Salme, S. Simula, Phys. Lett. B 357 (1995) 267. [12] G.E. Brown, M. Rho, Phys. Lett. B 82 (1979) 177; G.E. Brown, M. Rho, V. Vento, Phys. Lett. B 94 (1979) 383. [13] S. Theberg, A.W. Thomas, G.A. Miller, Phys. Rev. D 22 (1980) 2838; 24 (1981) 216; D.H. Liu, A.W. Thomas, A.G. Williams, Phys. Rev. C 55 (1997) 3108. [14] Chr. V. Christov et al., Prog, Part. Nucl. Phys. 37 (1996) 91; R. Alkofer, H. Reinhart, H. Weigel, Phys. Rep. 265 (1996) 139. [15] See review by L. Wilets, Non-Topological Solitons, World Scienti"c, Singapore, 1989. [16] Zhenping Li, Phys. Rev. D 50 (1994) 5639; A. Glozman, D.O. Riska, Phys. Rep. 268 (1996) 263; P.-N. Shen, Y.-B. Dong, Z.-Y. Zhang, Y.-W. Yu, T.-S.H. Lee, Phys. Rev. C 55 (1997) 2024. [17] S. Capstick, Phys. Rev. D 46 (1992) 2864. [18] S. Capstick, W. Roberts, Phys. Rev. D 47 (1993) 1994; Phys. Rev. D 49 (1994) 4570. [19] F. Butler, H. Chen, J. Sexton, A. Vaccarino, D. Weingarten, Phys. Rev. Lett. 70 (1993) 2849. [20] R.E. Cutkosky, R.E. Hendrick, J.W. Alcock, Y.A. Chao, R.G. Lipes, J.C. Sandusky, R.L. Kelly, Phys. Rev. D 20 (1979) 2804; R. Kelly, R.E. Cutkosky, Phys. Rev. D 20 (1979) 2782. [21] R. Koch, Z. Phys. C 29 (1985) 597; Nucl. Phys. A 448 (1986) 707; R. Koch, E. Pietarinen, Nucl. Phys. A 336 (1980) 331. [22] G. HoK hler, nN Newsletter, G. HoK hler, W. Kluge, B.M.K. Nefkens (Eds.), 13 (1997) 320. [23] R.G. Moorhouse, H. Oberlack, A.H. Rosenfeld, Phys. Rev. D 9 (1979) 1. [24] R. Bhandari, Y.-A. Chao, Phys. Rev. D 15 (1977) 192. [25] D.M. Manley, R.A. Arndt, Y. Goradia, V.L. Teplitz, Phys. Rev. D 30 (1984) 904. [26] G. HoK hler, nN Newsletter, G. HoK hler, W. Kluge, B.M.K. Nefkens (Eds.), 9 (1993) 1. [27] M. Batinic, I. Slaus, A. Svarc, B.M.K. Nefkens, Phys. Rev. C 51 (1995) 2310. [28] R.M. Brown et al., Nucl. Phys. B 153 (1979) 89. [29] R. Clajus, B.M.K. Nefkens, n N Newsletter, G. HoK hler, W. Kluge, B.M.K. Nefkens (Eds.), 7 (1991) 76. [30] T.P. Vrana, University of Pittsburgh Ph.D. Thesis, unpublished. [31] William R. Frazer, Archibald W. Henry, Phys. Rev. 134 (1964) B1307. [32] R.E. Cutkosky, S. Wang, Phys. Rev. D 42 (1990) 235.

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T.P. Vrana et al. / Physics Reports 328 (2000) 181}236 C. Deutsch-Sauermann, B. Friman, W. NoK renberg, Phys. Lett. B 341 (1995) 261; Phys. Lett. B 409 (1977) 51. M. Mulhearn, J. Mueller, private communication. D.J. Herndon et al., Phys. Rev. D 11 (1975) 3165; R.L. Longacre et al., Phys. Rev. D 17 (1978) 1795. K.W.J. Barnham, in: Proceedings of the Topical Conference on Baryon Resonances, R.T. Ross, D.H. Saxon (Eds.), Univ. Press, London, 1976. Ramesh Bhandari, Yung-An Chao, Phys. Rev. D 15 (1977) 192. M. Benmerrouche, Nimai C. Mukhopadhyay, J.F. Zhang, Phys. Rev. D 51 (1995) 3237. L. Tiator, C. Bennhold, S.S. Kamalov, Nucl. Phys. A 580 (1994) 455. Franz Gross, Yohanes Surya, Phys. Rev. C 47 (1993) 703. B. Krusche et al., Phys. Rev. Lett. 74 (1995) 3736. B. Krusche, N. Mukhopadhyay, J.F. Zhang, M. Benmerrouche, Phys. Lett. B 397 (1997) 171. M.L. Goldberger, K.M. Watson, Collision Theory, Wiley, New York, 1967. T. Sato, T.-S.H. Lee, Phys. Rev. C 54 (1996) 2660.

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PHASES OF DENSE MATTER IN NEUTRON STARS

Henning HEISELBERG , Morten HJORTH-JENSEN
NORDITA, Blegdamsvej 17, DK-2100 K~benhavn }, Denmark
Department of Physics, University of Oslo, N-0316 Oslo, Norway

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

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Phases of dense matter in neutron stars Henning Heiselberg *, Morten Hjorth-Jensen
NORDITA, Blegdamsvej 17, DK-2100 K~benhavn }, Denmark
Department of Physics, University of Oslo, N-0316 Oslo, Norway Received August 1999; editor: M. Kamionkowski Contents 1. Introduction 1.1. The past, present and future of neutron stars 1.2. Physics of neutron stars 2. Phases of dense matter 2.1. Prerequisites and de"nitions 2.2. Nucleonic degrees of freedom 2.3. A causal parametrization of the nuclear matter EoS 2.4. Hyperonic matter 2.5. Kaon condensation 2.6. Pion condensation 2.7. Super#uidity in baryonic matter 2.8. Quark matter 3. Thermodynamics of multi-component phase transitions 3.1. Maxwell construction for one-component systems 3.2. Two-component systems in a mixed phase 4. Structure of neutron stars 4.1. Screening lengths 4.2. Surface and Coulomb energies of the mixed phase 4.3. Is the mixed phase energetically favored?

240 240 241 243 245 248 265 269 275 276 277 281 283 283 285 285 286 287 289

4.4. Melting temperatures 4.5. Funny phases 4.6. Summary of neutron star structures 5. Observational consequences for neutron stars 5.1. Masses from radio pulsars, X-ray binaries and QPO's 5.2. TOV and Hartle's equations 5.3. Neutron star properties from various equations of state 5.4. Maximum masses 5.5. Phase transitions in rotating neutron stars 5.6. Core quakes and glitches 5.7. Backbending and giant glitches 5.8. Cooling and temperature measurements 5.9. Supernovae 5.10. Gamma-ray bursters 6. Conclusions 6.1. Many-body approaches to the equation of state 6.2. Phase transitions and sti!ness of EoS from masses of neutron stars Acknowledgements References

*Corresponding author. E-mail addresses: [email protected] (H. Heiselberg), [email protected] (M. Hjorth-Jensen) 0370-1573/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 1 0 - 6

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Abstract Recent equations of state for dense nuclear matter are discussed with possible phase transitions arising in neutron stars such as pion, kaon and hyperon condensation, super#uidity and quark matter. Speci"cally, we treat the nuclear to quark matter phase transition, the possible mixed phase and its structure. A number of numerical calculations of rotating neutron stars with and without phase transitions are given and compared to observed masses, radii, temperatures and glitches.  2000 Elsevier Science B.V. All rights reserved. PACS: 12.38.Mh; 21.30.!x; 21.65.#f; 26.60.#c; 97.60.Gb; 97.60.Jd Keywords: Neutron star properties; Phase transitions; Equation of state for dense neutron star matter

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1. Introduction 1.1. The past, present and future of neutron stars The discovery of the neutron by Chadwick in 1932 prompted Landau to predict the existence of neutron stars. The birth of such stars in supernovae explosions was suggested by Baade and Zwicky in 1934. First theoretical neutron star calculations were performed by Tolman, Oppenheimer and Volko! in 1939 and Wheeler around 1960. Bell and Hewish were the "rst to discover a neutron star in 1967 as a radio pulsar. The discovery of the rapidly rotating Crab pulsar in the remnant of the Crab supernova observed by the Chinese in 1054 A.D. con"rmed the link to supernovae. Radio pulsars are rapidly rotating with periods in the range 0.033 s4P44.0 s. They are believed to be powered by rotational energy loss and are rapidly spinning down with period derivatives of order PQ &10\}10\. Their high magnetic "eld B leads to dipole magnetic braking radiation proportional to the magnetic "eld squared. One estimates magnetic "elds of the order of B&10}10 G. The total number of pulsars discovered so far has just exceeded 1000 before the turn of the millenium and the number is increasing rapidly. A distinct subclass of radio pulsars are millisecond pulsars with periods between 1.56 ms4 P4100 ms. The period derivatives are very small corresponding to very small magnetic "elds B&10}10 G. They are believed to be recycled pulsars, i.e. old pulsars with low magnetic "elds that have been spun up by accretion preserving their low magnetic "eld and therefore only slowly spinning down. About 20 } almost half of the millisecond pulsars } are found in binaries where the companion is either a white dwarf or a neutron star. Six double neutron stars are known so far including the Hulse}Taylor PSR 1913#16. The "rst binary pulsar was found by Hulse and Taylor in 1973 and by measuring the general relativistic corrections to Newtonian gravity one could determine all parameters in the binary system as both masses, orbital periods and period derivatives, orbital distances and inclination. Parameters are overdetermined and thus provide a test of general relativity. Inward spiralling or orbital decay is an additional test of general relativity to an unprecedented accuracy. The binary neutron stars all have masses in the narrow interval 1.3}1.5M , which may either be due to the creation process or that heavier neutron stars > are unstable. With X-ray detectors on board satellites since 1971 almost two hundred X-ray pulsars and bursters have been found of which the orbital period has been determined for about sixty. The X-ray pulsars and bursters are believed to be accreting neutrons stars from high (M910M ) and > low mass (M:1.2M ) companions, respectively. The X-ray pulses are most probably due to > strong accretion on the magnetic poles emitting X-ray (as northern lights) with orbital frequency. The X-ray bursts are due to slow accretion spreading all over the neutron star surface before igniting in a thermonuclear #ash. The resulting (irregular) bursts have periods depending on accretion rates rather than orbital periods. Recently, bursters and pulsars have been linked by observations of X-ray pulsations in bursts from several low mass X-ray bursters [1]. The pulsations are with spin frequency 300}400 Hz and increase by a few Hz only during the burst. The small increase is expected from cooling after a thermonuclear explosion, which leads to smaller size and moment of inertia and, conserving angular momentum, to larger frequency. The radiation from X-ray bursters is not blackbody and therefore only upper limits on temperatures can be extracted from observed luminosities in most cases. Masses are less accurately measured than for binary

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pulsars. We mention recent mass determinations for the X-ray pulsar Vela X-1: M"(1.9$0.1)M , and the burster Cygnus X-2: M"(1.8$0.2)M , which will be discussed > > later. A subclass of six anomalous X-ray pulsars are slowly rotating but rapidly spinning down indicating that they are young with enormous magnetic "elds, B&10 G, and thus named Magnetars [3]. Recently, quasi-periodic oscillations (QPO) have been found in 12 low mass X-ray binaries. The QPO's set strict limits on masses and radii of neutrons stars, but if the periodic oscillations arrive from the innermost stable orbit [4], it implies de"nite neutron star masses up to MK2.3M . > Non-rotating and non-accreting neutron stars are virtually undetectable. With the Hubble space telescope one single thermally radiating neutron star has been found [5]. Its distance is only 160 pc from Earth and its surface temperature is ¹K60 eV. From its luminosity one deduces a radius of the neutron star R414 km. In our galaxy astrophysicists expect a large abundance &10 of neutron stars. At least as many supernova explosions have occurred since Big Bang which are responsible for all heavier elements present in the Universe today. The scarcity of neutron stars in the solar neighborhood may be due to a high initial velocity (asymmetric `kicka) during their birth in supernovae. Recently, many neutron stars have been found far away from their supernova remnants. Future gravitational microlensing observation may determine the population of such `invisiblea neutron stars as dark matter objects in the galactic halo. From the view of physicists (and mass extinctionists) supernova explosions are unfortunately rare in our and neighboring galaxies. The predicted rate is 1}3 per century in our galaxy but the most recent one was 1987A in LMC. With luck we may observe one in the near future which produces a rapidly rotating pulsar. Light curves and neutrino counts will test supernova and neutron star models. The rapid spin down may be exploited to test the structure and possible phase transitions in the cores of neutron stars [6}8]. The recent discovery of afterglow in gamma ray bursters (GRB) allows determination of the very high redshifts (z51) and thus the enormous distance and energy output E&10 ergs in GRB if isotropically emitted. Very recently evidence for beaming or jets has been found [2] corresponding to `onlya E&10 ergs. Candidates for such violent events neutron star mergers or a special class of type Ic supernova (hypernovae) where cores collapse to black holes. The latter is con"rmed by recent observations of a bright supernova coinciding with GRB 980326. The marvelous discoveries made in the past few decades will continue as numerous earth-based and satellite experiments are running at present and more will be launched. History tells us that the future will bring great surprises and discoveries in this "eld.

1.2. Physics of neutron stars The physics of compact objects like neutron stars o!ers an intriguing interplay between nuclear processes and astrophysical observables. Neutron stars exhibit conditions far from those encountered on earth; typically, expected densities o of a neutron star interior are of the order of 10 or more times the density o +4;10 g/cm at &neutron drip', the density at which nuclei begin to  dissolve and merge together. Thus, the determination of an equation of state (EoS) for dense matter is essential to calculations of neutron star properties. The EoS determines properties such as the mass range, the mass}radius relationship, the crust thickness and the cooling rate. The same EoS

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Fig. 1. Possible structure of a neutron star.

is also crucial in calculating the energy released in a supernova explosion. Clearly, the relevant degrees of freedom will not be the same in the crust region of a neutron star, where the density is much smaller than the saturation density of nuclear matter, and in the center of the star, where density is so high that models based solely on interacting nucleons are questionable. These features are pictorially displayed in Fig. 1. Neutron star models including various so-called realistic equations of state result in the following general picture of the interior of a neutron star. The surface region, with typical densities o(10 g/cm, is a region in which temperatures and magnetic "elds may a!ect the equation of state. The outer crust for 10 g/cm(o(4;10 g/cm is a solid region where a Coulomb lattice of heavy nuclei coexist in b-equilibrium with a relativistic degenerate electron gas. The inner crust for 4;10 g/cm(o(2;10 g/cm consists of a lattice of neutron-rich nuclei together with a super#uid neutron gas and an electron gas. The neutron liquid for 2;10 g/cm(o(;10 g/cm contains mainly super#uid neutrons with a smaller concentration of superconducting protons and normal electrons [9]. At higher densities, typically 2}3 times nuclear matter saturation density, interesting phase transitions from a phase with just nucleonic degrees of freedom to quark matter may take place [10]. Furthermore, one may have a mixed phase of quark and nuclear matter [6,11], kaon [12] or pion condensates [13,14], hyperonic matter [6,15}20], strong magnetic "elds in young stars [21,22], etc. The "rst aim of this work is therefore to attempt at a review of various approaches to the equation of state for dense neutron star matter relevant for stars which have achieved thermal equilibrium. Various approaches to the EoS and phases which may occur in a neutron star are discussed in Section 2 while an overview of the thermodynamical properties of the mixed phase and possible phases in neutron stars are presented in Sections 3 and 4. Our second aim is to discuss the relation between the EoS and various neutron star observables when a phase transition in the interior of the star occurs. Astronomical observations leading to global neutron star parameters such as the total mass, radius, or moment of inertia, are important since they are sensitive to microscopic model calculations. The mass, together with the moment of inertia, are also the gross structural parameters of a neutron star which are most accessible to observation. It is the mass which controls the gravitational interaction of the star with other systems such as a binary companion. The moment of inertia controls the energy stored in rotation

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and thereby the energy available to the pulsar emission mechanism. Determining the possible ranges of neutron star is not only important in constraining the EoS, but has important theoretical consequences for the observational prediction of black holes in the universe. Examples are the galactic black hole candidates Cyg X-1 [23] and LMC X-3 [24], which are massive X-ray binaries. Their masses (0.25M and 2.3M ) are, however, smaller than for some low-mass X-ray binaries > > like A0620-00 [25] and V404 Cyg [26], which make better black hole candidates with mass functions in excess of three solar masses. There is a maximum mass a non-rotating neutron star can have. There is however no upper limit on the mass a black hole can have. If, therefore, one can "nd a dense, highly compact object and can argue that its rotation is slow, and can deduce that its mass is greater than the allowed maximum mass allowed to non-rotating (or slowly rotating) neutron stars, then one has a candidate for a black hole. Since neutron stars are objects of highly compressed matter, this means that the geometry of space}time is changed considerably from that of a #at space. Stellar models must therefore be based on Einstein's theory of general relativity. Based on several of the theoretical equations of state and possible phases of matter discussed in Sections 2}4, various properties of non-rotating and rotating neutron stars are presented in Section 5. The relevant equations needed for the study of the structure of a neutron star are summarized in this section as well, for both non-rotating and rotating stellar structures. There we also discuss the observational implications when phase transitions occur in the interior of the star. In addition to studies of the mass}radius relationship and the moment of inertia, we also extract analytical properties of quantities like the braking index and the rate of slowdown near the critical angular velocity where the pressure inside the star just exceeds that needed to make a phase transition. The observational properties for "rst- and second-order phase transitions are also discussed. Other properties like glitches and cooling of stars are also discussed in Section 5. Summary and perspectives are given in Section 6. Finally, we mention several excellent and recent review articles covering various aspects of neutron stars properties in the literature addressing the interesting physics of the neutron star crust [27], the nuclear equation of state [14], hot neutron star matter [28] in connection with protoneutron stars, and cooling calculations [29]. However, as previously mentioned, our aim will be to focus on the connection between the various possible phases of dense neutron star matter in chemical equilibrium and the implications of "rst- and second-order phase transitions for various observables.

2. Phases of dense matter Several theoretical approaches to the EoS for the interior of a neutron star have been considered. Over the past two decades many authors [10] have considered the existence of quark matter in neutron stars. Assuming a "rst-order phase transition one has, depending on the equation of states, found either complete strange quark matter stars or neutron stars with a core of quark matter surrounded by a mantle of nuclear matter and a crust on top. Recently, the possibility of a mixed phase of quark and nuclear matter was considered [6] and found to be energetically favorable. Including surface and Coulomb energies this mixed phase was still found to be favored for reasonable bulk and interface properties [11]. The structure of the mixed phase of quark matter

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embedded in nuclear matter with a uniform background of electrons was studied and resembles that in the neutron drip region in the crust. Starting from the outside, the crust consists of the outer layer, which is a dense solid of neutron-rich nuclei, and the inner layer in which neutrons have dripped and form a neutron gas coexisting with the nuclei. The structure of the latter mixed phase has recently been calculated in detail [27] and is found to exhibit rod-, plate- and bubble-like structures. At nuclear saturation density o K2.8;10 g/cm there is only one phase of uniform  nuclear matter consisting of mainly neutrons, a small fraction of protons and the same amount of electrons to achieve charge neutrality. A mixed phase of quark matter (QM) and nuclear matter (NM) appears already around a few times nuclear saturation density } lower than the phase transition in hybrid stars. In the beginning only few droplets of quark matter appear but at higher densities their number increase and they merge into: QM rods, QM plates, NM rods, NM bubbles, and "nally pure QM at very high densities if the neutron stars have not become unstable towards gravitational collapse. In this section we review various attempts at describing the above possible phases of dense neutron star matter. For the part of the neutron star that can be described in terms of nucleonic degrees of freedom only, i.e. b-stable matter with protons, neutrons and electrons (and also muons), we will try to shed light on recent advances within the framework of various many-body approaches. This review is presented in Section 2.2. For the more exotic states of matter such as hyperonic degrees of freedom we will point to recent studies of hyperonic matter in terms of more microscopic models in Section 2.4. The problem however with e.g. hyperonic degrees of freedom is that knowledge of the hyperon}nucleon or hyperon}hyperon interactions has not yet reached the level of sophistication encountered in the nucleon}nucleon sector. Mean "elds methods have however been much favoured in studies of hyperonic matter. A discussion of pion and kaon condensation will also be presented in the two subsequent subsections. Super#uidity is addressed in Section 2.7. In general, we will avoid a discussion of non-relativistic and relativistic mean "elds methods of relevance for neutron matter studies, mainly since such aspects have been covered in depth in the literature, see e.g. Refs. [17,18,30,31]. Moreover, as pointed out by Akmal et al. [14], albeit exhibiting valuable tutorial features, the main problem with relativistic mean "eld methods is that they rely on the approximation kr;1, with k the inverse Compton wavelength of the meson and r the interparticle spacing. For nuclear and neutron matter densities ranging from saturation density to "ve times saturation density, kr is in the range 1.4 to 0.8 for the pion and 7.8 to 4.7 for vector mesons. Clearly, these values are far from being small. The relativistic mean "eld approximation can however be based on e!ective values for the coupling constants, taking thereby into account correlation e!ects. These coupling constants have however a density dependence and a more microscopic theory is needed to calculate them. Our knowledge of quark matter is however limited, and we will resort to phenomenological models in Section 2.8 in our description of this phase of matter. Typical models are the so-called Bag model [32] or the Color-Dielectric model [33]. However, before proceeding with the above more speci"c aspects of neutron star matter, we need to introduce some general properties and features which will enter our description of dense matter. These are introduced in the "rst subsection. The reader should also note that we will omit a discussion of the properties of matter in the crust of the star since this is covered in depth by the review of Ravenhall and Pethick [27]. Morever, for neutron stars with masses +1.4M or greater, >

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the mass fraction contained in the crust of the star is less than about 2%. We will therefore in our "nal EoS employ results from earlier works [34,35] for matter at densities 40.05 fm\. 2.1. Prerequisites and dexnitions At densities of 0.1 fm\ and greater, we will in this work require properties of charge neutral uniform matter to be made of mainly neutrons, protons, electrons and muons in beta equilibrium, although the presence of other baryons will be discussed as well. In this section we will merely focus on distinct phases of matter, such as pure baryonic matter or quark matter. The composition of matter is then determined by the requirements of chemical and electrical equilibrium. Furthermore, we will also consider matter at temperatures much lower than the typical Fermi energies. The equilibrium conditions are governed by the weak processes (normally referred to as the processes for b-equilibrium) b Pb #l#l ,   J

b #lPb #l ,   J

(1)

where b and b refer to e.g. the baryons being a neutron and a proton, respectively, l is either an   electron or a muon and l and l their respective anti-neutrinos and neutrinos. Muons typically J J appear at a density close to nuclear matter saturation density, the latter being n +0.16$0.02 fm\ ,  with a corresponding binding energy E for symmetric nuclear matter (SNM) at saturation density  of E "B/A"!15.6$0.2 MeV .  In this work the energy per baryon E will always be in units of MeV, while the energy density e will be in units of MeV fm\ and the number density n in units of fm\. The pressure P is de"ned through the relation P"n RE/Rn"n Re/Rn!e ,

(2)

with dimension MeV fm\. Similarly, the chemical potential for particle species i is given by k "(Re/Rn ) , G G

(3)

with dimension MeV. In our calculations of properties of neutron star matter in b-equilibrium, we will need to calculate the energy per baryon E for e.g. several proton fractions x , which N corresponds to the ratio of protons as compared to the total nucleon number (Z/A), de"ned as x "n /n , N N  In this work we will also set G"c" "1, where G is the gravitational constant.  We will often loosely just use density in our discussions.

(4)

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where n"n #n , the total baryonic density if neutrons and protons are the only baryons present. N L In that case, the total Fermi momentum k and the Fermi momenta k , k for protons and $ $N $L neutrons are related to the total nucleon density n by n"(2/3p)k "x n#(1!x )n"(1/3p)k #(1/3p)k . (5) $ N N $N $L The energy per baryon will thus be labelled as E(n, x ). E(n, 0) will then refer to the energy N per baryon for pure neutron matter (PNM) while E(n, ) is the corresponding value for SNM.  Furthermore, in this work, subscripts n, p, e, k will always refer to neutrons, protons, electrons and muons, respectively. Since the mean free path of a neutrino in a neutron star is bigger than the typical radius of such a star (&10 km), we will assume throughout that neutrinos escape freely from the neutron star, see e.g. the work of Prakash et al. in Ref. [28] for a discussion on trapped neutrinos. Eq. (1) yields then the following conditions for matter in b equilibrium with e.g. nucleonic degrees freedom only k "k #k , L N C

(6)

and n "n , (7) N C where k and n refer to the chemical potential and number density in fm\ of particle species i. If G G muons are present as well, we need to modify the equation for charge conservation, Eq. (7), to read n "n #n , N C I and require that k "k . With more particles present, the equations read C I (n>G #n>G )" (n\G #n\G ) , J @ J @ G G

(8)

and k "b k #q k , (9) L G G G J where b is the baryon number, q the lepton charge and the superscripts ($) on number densities G G n represent particles with positive or negative charge. To give an example, it is possible to have baryonic matter with hyperons like K and R\> and isobars D\>>> as well in addition to the nucleonic degrees of freedom. In this case the chemical equilibrium condition of Eq. (9) becomes, excluding muons, kR\ "kD\ "k #k , L C kK "kR "kD "k , L kR> "kD> "k "k !k , N L C kD>> "k !2k . L C

(10)

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A transition from hadronic to quark matter is expected at high densities. The high-density quark matter phase in the interior of neutron stars is also described by requiring the system to be locally neutral n !n !n !n "0 , C  S  B  Q

(11)

where n are the densities of the u, d and s quarks and of the electrons (eventually muons as SBQC well), respectively. Moreover, the system must be in b-equilibrium, i.e. the chemical potentials have to satisfy the following equations: k "k #k , B S C

(12)

k "k #k . Q S C

(13)

and

Eqs. (11)}(13) have to be solved self-consistently together with e.g. the "eld equations for quarks at a "xed density n"n #n #n . In this section we will mainly deal with distinct phases of matter, S B Q the additional constraints coming from the existence of a mixed phase of hadrons and quarks and the related thermodynamics will be discussed in Section 3. An important ingredient in the discussion of the EoS and the criteria for matter in b-equilibrium is the so-called symmetry energy S(n), de"ned as the di!erence in energy for symmetric nuclear matter and pure neutron matter S(n)"E(n, x "0)!E(n, x ") . N N 

(14)

If we expand the energy per baryon in the case of nucleonic degrees of freedom only in the proton concentration x about the value of the energy for SNM (x "), we obtain, N N  E(n, x )"E(n, x ")# (dE/dx)(n)(x !)#2 ,  N N  N N 

(15)

where the term dE/dx is to be associated with the symmetry energy S(n) in the empirical mass N formula. If we assume that higher-order derivatives in the above expansion are small (we will see examples of this in the next subsection), then through the conditions for b-equilbrium of Eqs. (6) and (7) and Eq. (3) we can de"ne the proton fraction by the symmetry energy as

c(3pnx )"4S(n)(1!2x ) , N N

(16)

where the electron chemical potential is given by k " ck , i.e. ultrarelativistic electrons are C $ assumed. Thus, the symmetry energy is of paramount importance for studies of neutron star matter in b-equilibrium. One can extract information about the value of the symmetry energy at saturation density n from systematic studies of the masses of atomic nuclei. However, these results  are limited to densities around n and for proton fractions close to . Typical values for S(n) at   n are in the range 27}38 MeV. For densities greater than n it is more di$cult to get a reliable   information on the symmetry energy, and thereby the related proton fraction. We will shed more light on this topic in the next subsection.

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Finally, another property of interest in the discussion of the various equations of state is the incompressibility modulus K at non-zero pressure K"9 RP/Rn .

(17)

The sound speed v depends as well on the density of the nuclear medium through the relation 









v  dP dP dn K  " " " . dn de c de 9(m c#E#P/n) L

(18)

It is important to keep track of the dependence on density of v since a superluminal behavior can  occur at higher densities for most non-relativistic EoS. Superluminal behavior would not occur with a fully relativistic theory, and it is necessary to gauge the magnitude of the e!ect it introduces at the higher densities. This will be discussed at the end of this section. The adiabatic constant C can also be extracted from the EoS by C"(n/P) RP/Rn .

(19)

2.2. Nucleonic degrees of freedom A major part of the densities inside neutron stars can be well represented by nucleonic degrees of freedom only, namely the inner part of the crust to the outer part of the core, i.e. densities ranging from 0.5 to 2}3 times nuclear matter saturation density. There is a wealth of experimental and theoretical data, see e.g. Ref. [36] for an overview, which lend support to the assumption that nucleons do not loose their individuality in dense matter, i.e. that properties of the nucleon at such densities are rather close to those of free nucleons. The above density range would correspond to internucleon distances of the order of &1 fm. At such interparticle distances there is little overlap between the various nucleons and we may therefore assume that they still behave as individual nucleons and that one can absorb the e!ects of overlap into the two nucleon interaction. The latter, when embedded in a nuclear medium, is also di!erent from the free nucleon}nucleon interaction. In the medium there are interaction mechanisms which are obviously absent in vacuum. As an example, the one-pion exchange potential is modi"ed in nuclear matter due to `softeninga of pion degrees of freedom in matter. In order to illustrate how the nucleon}nucleon interaction is renormalized in a nuclear medium, we will start with the simplest possible many-body approach, namely the so-called Brueckner} Hartree}Fock (BHF) approach. This is done since the Lippmann}Schwinger equation used to construct the scattering matrix ¹, which in turn relates to the phase shifts, is rather similar to the G-matrix which enters the BHF approach. The di!erence resides in the introduction of a Pauliblocking operator in order to prevent scattering to intermediate particle states prohibited by the Pauli principle. In addition, the single-particle energies of the interacting particle are no longer given by kinetic energies only. However, several of the features seen at the level of the scattering matrix, pertain to the G-matrix as well. Therefore, if one employs di!erent nucleon}nucleon interactions in the calculation of the energy per baryon in pure neutron matter with the BHF G-matrix, eventual di!erences can be retraced at the level of the ¹-matrix. We will illustrate these

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aspects in the next subsection. More complicated many-body terms and relativistic e!ects will be discussed in Sections 2.2.2 and 2.2.3. 2.2.1. From the NN interaction to the nuclear G-matrix The NN interactions we will employ here are the recent models of the Nijmegen group [37], the Argonne < potential [38] and the charge-dependent Bonn interaction (CD}Bonn [39]). In 1993,  the Nijmegen group presented a phase-shift analysis of all proton}proton and neutron}proton data below 350 MeV with a s per datum of 0.99 for 4301 data entries. The above potentials have all been constructed based on these data. The CD}Bonn interaction has a s per datum of 1.03 and the same is true for the Nijm-I, Nijm-II and Reid93 potential versions of the Nijmegen group [37]. The new Argonne potential < [38] has a s per datum of 1.09.  Although all these potentials predict almost identical phase shifts, their mathematical structure is quite di!erent. The Argonne potential, the Nijm-II and the Reid93 potentials are non-relativistic potential models de"ned in terms of local potential functions, which are attached to various (non-relativistic) operators of the spin, isospin and/or angular momentum operators of the interacting pair of nucleons. Such approaches to the NN interaction have traditionally been quite popular since they are numerically easy to use in con"guration space calculations. The Nijm-I model is similar to the Nijm-II model, but it includes also a p term, see Eq. (13) of Ref. [37], which may be interpreted as a non-local contribution to the central force. The CD}Bonn potential is based on the relativistic meson-exchange model of Ref. [40] which is non-local and cannot be described correctly in terms of local potential functions. For a given NN interaction <, the R-matrix (or K-matrix) for free-space two-nucleon scattering is obtained from the Lippmann}Schwinger equation, which reads in the center-of-mass (c.m.) system and in a partial-wave decomposition



1 dq (20)
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Following the conventional many-body approach, we divide the full Hamiltonian H"¹#<, with ¹ being the kinetic energy and < the bare NN interaction, into an unperturbed part H "¹#; and an interacting part H "
(21)

where u is the unperturbed energy of the interacting nucleons and Q is the Pauli operator which prevents scattering into occupied states. The Pauli operator is given by



1, k 'kOK , k 'kOL , K $ L $ Q(k q , k q )" K K L L 0 otherwise,

(22)

in the laboratory system, where kOG de"nes the Fermi momenta of the proton (q ") and neutron $ G  (q "!). For notational economy, we set "k ""k . K K G  The above expression for the Pauli operator is in the laboratory frame. In the calculations of the G-matrix, we will employ a Pauli operator in the center-of-mass and relative coordinate system. Further, this Pauli operator will be given by the so-called angle-average approximation, for details see Ref. [43]. Eq. (21) reads then (in a partial wave representation)



Q2X (q, K) dq
(23)

The variable K is the momentum of the center-of-mass motion. Since we are going to use an angular average for the Pauli operator, the G-matrix is diagonal in total angular momentum J. Further, the G-matrix is diagonal in the center-of-mass orbital momentum ¸ and the total spin S, all three variables represented by the index a. The variable a di!ers therefore from the de"nition of the R-matrix, where K"0. Three di!erent G-matrices have to be evaluated, depending on the individual isospins (q q ) of the interacting nucleons ( , !! and ! ). These quantities are       represented by the total isospin projection ¹ in Eq. (23). The di!erent G-matrices originate from X the discrimination between protons and neutrons in Eq. (22). The term H in the denominator of  Eq. (23) is the unperturbed energy of the intermediate states and depends on k, K and the individual isospin of the interacting particles. Only ladder diagrams with intermediate two-particle states are included in Eq. (21). The structure of the G-matrix equation in Eq. (23) can then be directly compared to the R-matrix for free NN scattering, Eq. (20). Therefore, as discussed below, eventual di!erences between various potentials in a "nite medium should be easily retraced to the structure of the R-matrix. It is also obvious that one expects the matrix elements of G to be rather close to

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those of R with only small deviations. These deviations originate from two e!ects which reduce the contributions of second and higher order in < to the G-matrix as compared to their contributions to R. One is the above-mentioned Pauli quenching e!ect: the Pauli operator Q in Eq. (23) restricts the intermediate particle states to states above the Fermi energy. The second one is the dispersive e!ect: the energy denominators in Eq. (23) are de"ned in terms of the single-particle energies of nucleons in the medium while the corresponding denominators of Eq. (20) are di!erences between the energies of free nucleons. Since the absolute values for the energy di!erences between nucleons, which feel the mean "eld of the nuclear system, are larger than the energy di!erences between the kinetic energies, also this dispersive correction reduces the attractive contributions of the non-Born terms. As a result, the matrix elements of G tend to be less attractive than the corresponding matrix elements of R, see e.g. Refs. [41,42] for further details. We use a continuous single-particle (sp) spectrum advocated by Mahaux et al. [44]. It is de"ned by the self-consistent solution of the following equations: e "t #u "(k/2m)#u , G G G G G

(24)

where m is the bare nucleon mass, and u " 1ih "G(E"e #e ) "ih2 . G F 1 G FXI$

(25)

In Eqs. (24) and (25), the subscripts i and h represent the quantum numbers of the single-particle states, such as isospin projections q and q , momenta k and k , etc. The sp kinetic energy is given G F G F by t and similarly the sp potential by u . G G Finally, the non-relativistic energy per nucleon E is formally given as k 1 1 1hh"G(E"e #e )"hh2 . E" F # F FY 1 2m 2A A $ $ $ FXI FXI  FYXI

(26)

In this equation we have suppressed the isospin indices for the Fermi momenta. Eq. (26) is actually calculated for various proton fractions x , and is thereby a function of both density n and x . We N N will therefore in the following discussion always label the energy per particle as E(n, x ). N In the limit of pure neutron matter only those partial waves contribute where the pair of interacting nucleons is coupled to isospin ¹"1. Due to the antisymmetry of the matrix elements this implies that only partial waves with even values for the sum l#S, like S , P , etc. need to be   considered in this case. For proton fractions di!erent from zero, in particular the case of symmetric nuclear matter, also the other partial waves, like S }D and P contribute. In a BHF    calculation the kinetic energy is independent of the NN interaction chosen. We will then restrict the following discussion to the potential energy per nucleon U, the second term in the RHS of Eq. (26). Putting the contributions from various channels together, one obtains the total potential energy per nucleon U for symmetric nuclear matter and neutron matter. These results are displayed in

 In our calculations we include all partial waves with l(10.

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Fig. 2. Potential energy per particle U for symmetric nuclear matter as function of total baryonic density n. Fig. 3. Potential energy per particle U for pure neutron matter.

Figs. 2 and 3 as functions of density n for the CD}Bonn interaction [39] (solid line), the three Nijmegen potentials, Nijm-I (long dashes), Nijm-II (short dashes) and Reid93 (dotted line) [37] and the Argonne < [38] (dot-dashed line).  The di!erences between the various potentials are larger for the energy in nuclear matter. This is mainly due to the importance of the S }D contribution which is absent in pure neutron matter.   This is in line with previous investigations, which showed that the predicted binding energy of nuclear matter is correlated with the strength of the tensor force, expressed in terms of the D-state probability obtained for the deuteron (see e.g. [43]). This importance of the strength of the tensor force is also seen in the calculation of the binding energy of the triton in Refs. [39,45]. The CD}Bonn interaction yields a binding energy of 8.00 MeV, the Nijm-I potential gives 7.72 MeV while the Nijm-II yields 7.62 MeV, the same as does the new Argonne potential [38]. The experimental value is 8.48 MeV. Typically, potentials with a smaller D-state probability have a weaker tensor force and exhibit therefore a smaller quenching of the non-born terms in Eq. (23). Moreover, the fact that the CD}Bonn interaction and the Nijm I potential include e!ects of non-localities, yields also a further attraction from the central force both in the S }D and the   singlet S channels. The latter explains the additional di!erence in nuclear matter between the  Nijm I and the Nijm II, Reid93 and Argonne < potentials as well as part of the di!erence seen in  Fig. 3 for pure neutron matter (PNM). For both SNM and PNM there are also additional di!erences arising from P waves, notably for the Argonne potential in PNM, where the di!erence between the Argonne < interaction, the Reid93 and the Nijm II potentials is mainly due to more  repulsive contributions from P-waves. For higher partial waves, the di!erences are rather small, typically of the order of few per cent. The di!erences seen in Figs. 2 and 3 should also be re#ected in the symmetry energy de"ned in Eq. (14). This is seen in Fig. 4 where we display the symmetry energy for the above potentials as function of density n. From the di!erences in symmetry energies one would then expect that properties like proton fractions in b-stable matter will be in#uenced. This in turn has important consequences for the composition of matter in a neutron star and thereby eventual phases present in dense matter. In Fig. 5 we display the corresponding proton

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Fig. 4. Symmetry energy S as function of density n. Fig. 5. Proton fraction x for b-stable matter as function of density n. N

Fig. 6. Fermi momenta for neutrons, protons, electrons and muons in b-stable matter with the CD}Bonn interaction.

fractions obtained by calculating the energy per particle through the G-matrix of Eq. (23) and imposing the equilibrium conditions of Eq. (6) and including muons, see again Ref. [46] for further details. From Fig. 5 one notices that the potential with the largest symmetry energy, the CD}Bonn interaction, is also the one which gives the largest proton fractions. This means in turn that the so-called direct Urca process can occcur at lower densities. For the CD}Bonn interaction this happens at 0.88 fm\, for the Nijm I it starts at 1.25 fm\ while for the Reid93 interaction one reaches the critical density at 1.36 fm\. The Argonne potential allows for the direct Urca process at a density of 1.05 fm\. For the Nijm II we were not able to get the direct Urca process for densities below 1.5 fm\. It is also interesting to notice that the symmetry energy increases rather monotonously for all potentials. This means that the higher-order derivatives in Eq. (15) can be neglected and that we can, to a good approximation, associate the second derivative dE/dx with the symmetry energy N S(n) in the empirical mass formula. With this in mind, one can calculate the proton fraction

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Fig. 7. Upper panel: BHF energy per nucleon for pure neutron matter and b-stable matter obtained with the CD}Bonn interaction. Lower panel: the corresponding pressure P.

employing the theoretically derived symmetry energy shown in Fig. 4 using the simple formula of Eq. (16). A good agreement is in general obtained with the above simple formula, see e.g. Ref. [46] and the parametrization in Section 2.3. With the results for the proton fractions of Fig. 5 in mind, we plot in Fig. 6 the Fermi momenta for electrons, muons and nucleons obtained with the CD}Bonn interaction, the other potentials yield qualitatively similar results although the proton fraction is slightly smaller. From this "gure one notices that muons appear at a density close to the saturation density of nuclear matter, as expected. This can easily be seen if one were to calculate b-stable matter with non-interacting particles only, see e.g. Ref. [47]. Of further interest is the di!erence in energy per particle E(n, x ) for pure neutron matter (PNM) N and matter in b-equilibrium. In Fig. 7 we display E(n, x ) for PNM and b-stable matter for results N with the CD}Bonn interaction only since the other potentials yield qualitatively similar results. Obviously, as seen from Fig. 7 the energy per particle for b-stable matter yields a softer EoS, since the repulsive energy per particle in PNM receives attractive contributions from the ¹ "0 channel. X This is re#ected in the corresponding pressure as well (lower panel of the same "gure) and will in turn result in neutron stars with smaller total masses compared with the PNM case. We end this subsection by plotting in Fig. 8 the energy density per baryon E (including the contribution from leptons) for b-stable matter. Since the proton fractions are not too large, see Fig. 5, the most important contribution to e and E stems from the ¹ "!1 channel and the X

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Fig. 8. The total energy in b-stable matter as function of density n. Contributions from leptons are included.

contribution from the nuclear tensor force, especially via the S and S }D contributions with    ¹ "0, plays a less signi"cant role than what seen for the potential energy in symmetric nuclear X matter in Fig. 2. Thus, the main contribution to the di!erences between the various potentials arises from the ¹ "1 channel. For the ¹ "1 channel, the new potentials yield also results for X 8 b-stable matter in close agreement. It is thus gratifying that the new NN interactions yield similar energies per particle in neutron star matter. As we will see in the next subsection also, more sophisticated many-body calculations at the two-body level yield rather similar results for neutron matter, since the strong nuclear tensor force in the ¹ "0 channel is not present in pure neutron matter or in a less important way in X b-stable matter. However, contributions coming from real three-body interaction or relativistic e!ects, may alter this picture. This is the topic of the next two subsections. 2.2.2. Higher-order many-body calculations The previous subsection served to establish the connection between the nucleon}nucleon interaction and the simplest many-body approach possible, namely the summation of so-called ladder diagrams by the Brueckner}Hartree}Fock (BHF) method. We will label these calculations as lowest-order Brueckner (LOB) theory. This allowed us to see how the free NN interaction gets modi"ed in a nuclear medium. Eventual di!erences in, for example, the EoS for neutron star matter could then be retraced to properties of the various NN interactions. The new high-quality NN interactions have also narrowed the di!erences at higher densities observed in the literature for older interaction models, see e.g. the discussion in Ref. [41]. However, it is well-known that the BHF method in its simplest form is not fully appropriate for a description of dense matter. More complicated many-body terms arising from core-polarization e!ects, e!ective three-body and many-body diagrams and eventually the inclusion of three-body forces are expected to be important at densities above n . Moreover, LOB theory su!ers from other pathologies like the lack  of conservation of number of particles [48]. The need to include e.g. three-body interactions is seen already at the level of the triton since all the above potentials underbind the triton, albeit the discrepancies which existed earlier have been reduced. To give an example, the old Reid potential [49] gave a binding energy of !7.35 MeV while a precursor to the CD}Bonn interaction, the

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Fig. 9. First-order contribution to the ground-state energy shift *E in BHF theory.  Fig. 10. Goldstone diagrams contained in the Brueckner}Hartree}Fock G-matrix.

Bonn A interaction [40] resulted in !8.35 MeV. The modern potentials yield results in a range from !7.6 to !8.0 MeV. Here we will therefore review three possible improvements to LOB theory and discuss the resulting equations of state in detail. The "rst improvement will be to consider the summation to in"nite order of the chain of particle}particle hole}hole diagrams (PPHH) [50}53]. Thereafter, we will discuss the recent three-hole line results of Baldo and co-workers [54}56] and "nally the calculations with three-body forces as well by Akmal et al. [13,14]. In brief, the summation of PPHH diagrams means that the Pauli operator in Eq. (22) is extended in order to prevent scattering into intermediate hole}hole states as well. This means that in addition to summing up to in"nite-order diagrams with particle}particle intermediate states, we will now sum, still to in"nite order, a larger class of diagrams containing hole}hole intermediate states as well. For LOB, as discussed in the previous subsection, the e!ective two-body interaction in nuclear matter is given by the G-matrix, which includes all ladder-type diagrams with particle}particle intermediate states to in"nite order. The ground-state energy shift *E in terms of the G-matrix is  represented by the "rst-order diagram of Fig. 9 and reads *E*- " n n 1ab"G(u"e #e )"ab2 . (27)  ? @ ? @ ?@ In Eq. (27) the n's are the unperturbed Fermi}Dirac distribution functions, namely n "1 if k4k I $ and "0 if k'k where k is the Fermi momentum. $ $ The Brueckner}Hartree}Fock G-matrix contains repeated interactions between a pair of `particlea lines, as illustrated by the diagrams of Fig. 10. Note that they are so-called Goldstone diagrams, with an explicit time ordering. The third-order diagram (a) of Fig. 10 is given by




1  < < < ?@KL KLPQ PQ?@ . (28) 2 (e #e !e !e )(e #e !e !e ) ? @ K L ? @ P Q Here m, n, r, s are all particle lines, and < represents the anti-symmetrized matrix elements of the GHIJ NN interaction V. Diag.(a)"

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Fig. 11. Goldstone diagram with repeated interactions between hole lines. Fig. 12. The contributions from the bubble (BUB), ring (RING), and higher-order three hole-line diagrams (HIGH) employing the continuous choice. Results are for symmetric nuclear matter, employing the Argonne < model for the  NN interaction.

To the same order, there is also a diagram with hole}hole interactions, as shown by diagram (b) in Fig. 11, where a, b, r, s are all hole lines. This diagram is not included in standard LOB calculations of nuclear matter, for the following reason. In earlier times, nuclear-matter calculations were based on, by and large, the so-called hole-line-expansion. The essence of the hole-line approach is that diagrams with (n#1) hole lines are generally much smaller than those with n hole lines. With this criterion, diagram (b) which has 3 (independent) hole lines would be negligible compared with diagram (a) which has 2 hole lines. Thus the former could be neglected. To investigate the validity of this criterion, it may be useful to actually calculate diagrams like (b). A motivation behind the PPHH-diagram method of nuclear matter is thus to include diagrams with hole}hole correlations like diagram (b) to in"nite order. For further details on how to obtain an e!ective interaction and the energy per particle in neutron star matter, see e.g. Refs. [51,53,57]. The next set of diagrams which can be included is the summation of so-called three-hole line diagrams through the solution of the Bethe}Fadeev equations, originally pionereed by Day [58] and recently taken up again by Baldo and co-workers [54,55]. The whole set of three hole-line diagrams can be grouped into three main sets of diagrams. The so-called ring diagrams, the bubble diagram, and the so-called higher-order diagrams, with an arbitrary number of particle lines. Each of these three-hole line contributions are quite large, see Refs. [54,55] and Fig. 12, but there is a strong degree of cancellation among the various terms. Thus, the total three-hole line contribution turns out to be substantially smaller than the two-body (i.e. two hole-line) contribution. This can be seen in Fig. 12 where we plot the various three hole-line contributions and their total contribution, all with the continuous choice for the single-particle energies. The interesting feature of the calculations of Baldo et al. [55], is that the results with the continuous single-particle choice lead to three hole-line results which are rather close to the total two hole-line results. Diagrams of the PPHH type are however not included. In Ref. [59] these were estimated to be of the same sign in both the standard and the continuum choice. In spite of this methodological progress in perturbative approaches, there are still classes of diagrams which need to be summed up. As

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discussed by Jackson in Ref. [60], one can prove that there is a minimal set of diagrams which need to be summed up in order to get the physics of a many-body system right. This set of diagrams is the so-called Parquet class of diagrams [61] where ring diagrams and ladder diagrams are summed up to all orders in a self-consistent way. Since it is rather hard to sum this set of diagrams in a practical way, other many-body methods like the coupled-cluster ansatz [62] and optimized hypernetted-chain theory [63] provide a systematic approximation method to sets of Feynman diagrams that cannot be calculated exactly. In the remainder of this section we will therefore focus on a variational method based on the hypernetted chain summation techniques developed by Pandharipande, Wiringa and co-workers. The approach was developed in the 1970s [64], particularly to include the e!ects of many-body correlations, presumably important in dense neutron star matter. Calculations performed since then have con"rmed that many-body clusters make signi"cant contributions to the binding energies of equilibrium nuclear matter and light nuclei [58,64}67]. We will base our discussion of the EoS on the recent results of Akmal, Pandharipande and Ravenhall [13,14,68]. We will refer the reader to the latter references and Ref. [64] for more details. In brief, the variational wavefunction has the form






W " S“ F U , GH T GH

(29)

consisting of a symmetrized product of pair correlation operators F operating on the Fermi gas GH wavefunction U. In symmetric nuclear matter, the function F includes eight terms: GH F " f N(r )ON , GH GH GH N

(30)

representing central, spin}spin, tensor and spin}orbit correlations with and without isospin factors. In pure neutron matter, the F reduce to a sum of four terms with only odd p47. The correlation GH operators F are determined from Euler}Lagrange equations [69] that minimize the two-body GH cluster contribution of an interaction (
(31)

j " jN(r )ON . GH GH GH N

(32)

The variational parameters aN are meant to simulate the quenching of the spin}isospin interaction between particles i and j, due to #ipping of the spin and/or isospin of particle i or j via interaction with other particles in matter. The NN interaction used in Refs. [13,14,68] is the recent parametrization of the Argonne group, the so-called Argonne < two-nucleon interaction discussed above. It  has the form < "
(33)

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The electromagnetic part < consists of Coulomb and magnetic interactions in the nn, np and pp CK pairs, and it is omitted from all nuclear matter studies. The strong interaction part of the potential includes fourteen isoscalar terms with operators among these the central, spin}spin, spin}orbit and tensor operators. In addition, a phenomenological three-body force < is included, represented by GHI the Urbana models of < containing two isoscalar terms: GHI < "<p #<0 . (34) GHI GHI GHI The "rst term represents the Fujita}Miyazawa two-pion exchange interaction: <p " A (+q ) q , q ) q ,+X , X ,#[q ) q , q ) q ][X , X ]) , (35) GHI L G H G I GH GI  G H G I GH GI AWA X "S ¹ (r )#p ) p > (r ) , (36) GH GH L GH G H L GH with strength A and where the matrices p and q are the Pauli matrices for spin and isospin, p respectively. The functions ¹ (r ) and > (r ) describe the radial shapes of the one-pion exchange L GH L GH tensor and Yukawa potentials. These functions are calculated using the average value of the pion mass and include the short-range cuto!s used in the Argonne < NN interaction. The term  denoted by <0 is purely phenomenological, and has the form GHI <0 "; ¹(r )¹(r ) . (37) GHI  L GH L GI AWA This term is meant to represent the modi"cation of ND- and DD-contributions in the two-body interaction by other particles in the medium, and also accounts for relativistic e!ects. The spin}isospin dependence of these e!ects is neglected. The two parameters A and ; are chosen to p  yield the observed energy of H and the equilibrium density of nuclear matter, o "0.16 fm\.  Obviously, this "tting procedure will yield di!erent parameters if another NN interaction is employed, e.g. the CD}Bonn interaction since, see discussion above, the various potentials yield slightly di!erent binding energies for H at the two-body level. The inclusion of many-body clusters is described in Ref. [64]. Relativistic boost corrections were also evaluated in Refs. [13,14,68] but the latter will be included in our discussion of relativistic e!ects in Section 2.2.3 below. In the remainder of this subsection we will henceforth discuss the consequences for the EoS from the above many-body corrections. In Fig. 13 we plot the results for PNM and SNM obtained with two-body interactions including only PPHH diagrams or higher-order diagrams stemming from the variational cluster approach. It is noteworthy to observe that in PNM the energy per particle up to 2}3 times n is rather similar  for all calculations at the two-body level. To a certain extent this is expected since the strong nuclear tensor force contribution from the ¹"0 channel is absent. This means in turn that more complicated many-body terms are not so important in PNM. We will see this also in connection with the three-hole line discussion below. For symmetric nuclear matter the situation is however di!erent due to the strong tensor force in the np channel, leading to larger higher-order corrections. This is clearly seen in the lower panel of Fig. 13. In e.g. the PPHH calculation, where the correlations are due to hole}hole and particle}particle ladders, the tensor force plays the main role. Again this is similar to the situation for the e!ective interaction in "nite nuclei [43]. There, to e.g. second order in the interaction, diagrams with hole}hole and particle}particle intermediate states

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Fig. 13. Upper panel: Energy per particle in PNM for various many-body approaches with two-body interactions only, i.e. PPHH results with the CD}Bonn interaction (stable results were obtained up to densities 0.4 fm\ only), LOB with the Argonne < interaction and variational cluster (VCS) results for the Argonne < interaction model. Lower panel:   The corresponding results for SNM. Fig. 14. Upper panel: Energy per particle in PNM for variational calculations with the Argonne < interaction without  (VCS) and with three-body interactions (< #UIX). The result from the three-hole line expansion of Baldo and  co-workers employing the < interaction is included as well. Lower panel: The corresponding results for SNM. 

tend to be bigger than screening diagrams. In general however, none of the calculations at the two-body level reproduce properly the saturation properties of SNM. In Fig. 14 we have then included results from calculations with e!ective three-body terms and the phenomenological three-body forces. The results are compared with those from LOB with the Argonne < interaction. With the inclusion of the phenomenological three-body force described in  Eq. (34), which was "tted to reproduce the binding energy of the triton and the alpha particle, a clear change is seen both in PNM and SNM. The energy per particle gets more repulsive at higher densities. Three hole-line diagrams however, with the continuous choice are rather close to LOB with the continuous choice in SNM, while they are almost negligible in PNM, in line with our observation above about the tensor force component in the ¹"0 channel. Thus, as a summary, the inclusion of phenomenological three-body forces in non-relativistic calculations are needed in order to improve the saturation properties of the microscopically calculated EoS, whereas the inclusion of additional many-body e!ects, either three-hole line

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diagrams or through VCS calculations yield results at low densities similar to those from LOB. However, even with the three-body force, one is not able to reproduce properly the binding energy. In the next subsection we will thence discuss further corrections stemming from relativistic e!ects. 2.2.3. Relativistic ewects The properties of neutron stars depend on the equation of state at densities up to an order of magnitude higher than those observed in ordinary nuclei. At such densities, relativistic e!ects certainly prevail. Among relativistic approaches to the nuclear many-body problem, the so-called Dirac}Hartree and Dirac}Hartree}Fock approaches have received much interest [31,70,71]. One of the early successes of these approaches was the quantitative reproduction of spin observables, which are only poorly described by the non-relativistic theory. Important to these methods was the introduction of a strongly attractive scalar component and a repulsive vector component in the nucleon self-energy [72,31]. Inspired by the successes of the Dirac}Hartree}Fock method, a relativistic extension of Brueckner theory was proposed by Celenza and Shakin [73], known as the Dirac}Brueckner theory. One of the appealing features of the Dirac}Brueckner approach is the selfconsistent determination of the relativistic sp energies and wave functions. The Dirac}Brueckner approach di!ers from the Dirac}Hartree}Fock one in the sense that in the former one starts from the free NN interaction which is only constrained by a "t to the NN data, whereas the Dirac} Hartree-Fock method pursues a line where the parameters of the theory are determined so as to reproduce the bulk properties of nuclear matter. It ought, however, to be stressed that the Dirac} Brueckner approach, which starts from NN interactions based on meson exchange, is a nonrenormalizable theory where the short-range part of the potential depends on additional parameters like vertex cut-o!s, clearly minimizing the sensitivity of calculated results to shortdistance inputs, see e.g. Refs. [73}75]. The description presented here for the Dirac}Brueckner approach follows closely that of Brockmann and Machleidt [74]. We will thus use the mesonexchange models of the Bonn group, de"ned in Table A.2 of Ref. [40]. There the three-dimensional reduction of the Bethe}Salpeter equation as given by the Thompson equation is used to solve the equation for the scattering matrix [76]. Hence, including the necessary medium e!ects like the Pauli operators discussed in the previous subsection and the starting energy, we shall rewrite Eq. (23) departing from the Thompson equation. Then, in a self-consistent way, we determine the above-mentioned scalar and vector components which de"ne the nucleon selfenergy. Note that negative energy solutions are not included. An account of these can be found in the recent work of de Jong and Lenske [77]. In order to introduce the relativistic nomenclature, we consider "rst the Dirac equation for a free nucleon, i.e. (iR. !m)t(x)"0 , where m is the free nucleon mass and t(x) is the nucleon "eld operator (x is a four-point) which is conventionally expanded in terms of plane wave states and the Dirac spinors u(p, s), and v(p, s), where p"(p, p) is a four momentum and s is the spin projection.

 Further notation is as given in Itzykson and Zuber [78].

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The positive energy Dirac spinors are (with u u"1)



E(p)#m u(p, s)" 2m






s Q , r)p s E(p)#m Q

(38)

where s is the Pauli spinor and E(p)"(m#"p". To account for medium modi"cations to the Q free Dirac equation, we introduce the notion of the self-energy R(p). As we assume parity to be a good quantum number, the self-energy of a nucleon can be formally written as R(p)"R (p)!c R(p)#cpR4(p) . 1  The momentum dependence of R and R is rather weak (Serot and Walecka, 1986). Moreover, 1 R4;1, such that the features of the Dirac}Brueckner}Hartree}Fock procedure can be discussed within the framework of the phenomenological Dirac}Hartree ansatz, i.e. we approximate R+R c R"; #; , 1  1 4 where ; is an attractive scalar "eld, and ; is the time-like component of a repulsive vector "eld. 1 4 The "nite self-energy modi"es the free Dirac spinors of Eq. (38) as



EI (p)#m u (p, s)" 2m






s Q , r)p s EI (p)#m Q

where we let the terms with tilde represent the medium modi"ed quantities. Here we have de"ned m "m#; , 1 and EI "EI (p )"(m #p . (39) G G G G As in the previous subsection, the subscripts i, and h below, represent the quantum numbers of the single-particle states, such as isospin projections q and q , momenta k and k , etc. G F G F The sp energy is e "EI #;G , G G 4 and the sp potential is given by the G-matrix as m m u " G F 1ih"GI (EI "EI #EI )"ih2 , G F 1 G EI EI FXI$ G F or, if we wish to express it in terms of the constants ; and ; , we have 1 4 m u " G ;G #;G . 1 4 G EI G

(40)

(41)

(42)

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In Eq. (41), we have introduced the relativistic GI -matrix. If the two interacting particles, with isospins q and q , give a total isospin projection ¹ , the relativistic GI -matrix in a partial wave   X representation is given by GI ?2X (kkKEI )"


dq Q2X (q, K) m m I),
(43)

where the relativistic starting energy is de"ned according to Eq. (39) as EI "EI ((k#K/4, q )#EI ((k#K/4, q ) .   and EI O "EI ((q#K/4, q ) .   Eqs. (40)}(43) are solved self-consistently, starting with adequate values for the scalar and vector components ; and ; . This iterative scheme is continued until these parameters show little 1 4 variation. Finally, the relativistic version of Eq. (26) reads m m#k m m 1 1 F F# F FY 1hh"GI (EI "EI #EI )"hh2 !m . E/A" F FY 1 EI EI EI A 2A F FXI$ FXI$ FYXI$ F FY

(44)

An alternative approach which we will also discuss is the inclusion of relativistic boost corrections to non-relativistic NN interactions in the VCS calculations of Akmal et al. [14,68]. In all analyses, the NN scattering data are reduced to the center-of-mass frame and "tted using phase shifts calculated from the nucleon}nucleon interaction, <, in that frame. The interaction obtained by this procedure describes the NN interaction in the c.m. frame, in which the total momentum P "p #p , is zero. In general, the interaction between particles depends upon their total GH G H momentum, and can be written as v(P )"v #dv(P ) , GH GH GH

(45)

where v is the interaction for P "0, and dv(P ) is the boost interaction [79] which is zero when GH GH GH P "0. GH Following the work of Krajcik and Foldy [80], Friar [81] obtained the following equation relating the boost interaction of order P to the interaction in the center-of-mass frame: 1 1 P v# [P ) r P ) , v]# [(p !p );P ) , v] . dv(P)"! H 8m 8m G 8m

(46)

The general validity of this equation in relativistic mechanics and "eld theory was recently discussed [79]. Incorporating the boost into the interaction yields a non-relativistic Hamiltonian of

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the form p HH " G # (v #dv(P ))#
(47)

where the ellipsis denotes the three-body boost, and four and higher body interactions. This HH contains all terms quadratic in the particle velocities, and is therefore suitable for complete ,0 studies in the non-relativistic limit. The authors of Refs. [66,82] "nd that the contribution of the two-body boost interaction to the energy is repulsive, with a magnitude which is 37% of the <0 contribution. The boost interGHI action thus accounts for a signi"cant part of the <0 in Hamiltonians which "t nuclear energies GHI neglecting dv. In this work we will follow Refs. [13,14,68] and only keep the terms of the boost interaction associated with the static part of <, and neglect the last term in Eq. (46). That term is responsible for Thomas precession and quantum contributions that are negligibly small here [83]. The correction dv is then given by 1 P vQ# P ) r P ) vQ . dv(P)"! 8m 8m

(48)

The two terms are due to the relativistic energy expression and Lorentz contraction, and are denoted dv0# and dv*!, respectively. The three-nucleon interaction used in the HH of Eq. (47) is ,0 denoted by
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Fig. 15. Upper panel: DBHF and BHF energy per nucleon for PNM. The results with boost corrections of Akmal et al. [14] with (< #dv#UIXH) and without (< #dv) three-body forces. Lower panel: the corresponding results for SNM.   Fig. 16. Upper panel: Energy per nucleon in b-stable matter for the DBHF approach and the results of Akmal et al. with boost corrections and three-body forces (< #dv#UIXH). Lower panel: the corresponding proton fraction x . For the  N DBHF calculation both electrons and muons are included.

2.3. A causal parametrization of the nuclear matter EoS Since three-body forces are expected to be important (it su$ces to mention studies of the Triton [45]), we will in our discussions of the mixed phase in the next section and in connection with the structure of a neutron star, employ the recent EoS of Akmal et al. [14]. A non-relativistic EoS (although with the inclusion of boost corrections) is preferred here. The EoS for nuclear matter is thus known to some accuracy for densities up to a few times nuclear saturation density, n "0.16 fm\. Detailed knowledge of the EoS is crucial for the existence of, e.g. pion condensates  [13,14] or the delicate structures in the inner crust of neutron stars [27]. However, for the gross properties and our discussion of properties of neutron stars we will adopt a simple form for the binding energy per nucleon in nuclear matter consisting of a compressional term and a symmetry term u!2!d #S uA(1!2x ) . E"E (n)#S(n)(1!2x )"E u  N   N  1#du

(49)

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Fig. 17. Upper panel: Comparison of the parametrized EoS of Eq. (49) and the results of Akmal et al. [14] with boost corrections and three-body forces (< #dv#UIXH) for PNM. Lower panel: the corresponding results for SNM. 

Here u"n/n is the ratio of the baryon density to nuclear saturation density and we have de"ned  the proton fraction x "n /n. The compressional term is in (49) parametrized by a simple form N N which reproduces the saturation density, binding energy and compressibility. The binding energy per nucleon at saturation density excluding Coulomb energies is E "!15.8 MeV and the  parameter d"0.2 is determined by "tting the energy per nucleon at high density to the EoS of Akmal et al. [14] with three-body forces and boost corrections, but taking the corrected values from Table 6 of Ref. [14]. The reason behind the choice of d"0.2 will be explained below in connection with the discussion of Figs. 18 and 19. Further discussions will be presented in Section 5. The corresponding compressibility is K "18E /(1#d)K200 MeV in agreement with   the experimental value. For the symmetry term we obtain S "32 MeV and c"0.6 for the best "t.  The quality of this simple functional exhibits a s per datum close to 1 and is compared with the results of Akmal et al. [14] for both PNM and SNM in Fig. 17. As can been from this "gure, the agreement is rather good except at the very high densities where the EoS of Akmal et al. [14] becomes superluminal and therefore anyway must be wrong. A much more sophisticated "t which reproduces the data in terms of Skyrme functional approach is given by the Akmal et al. [14]. However, it is amazing that such a simple quadratic formula "ts so well the data coming from a microscopic calculation. In view of the uncertainties which pertain to the EoS at higher densities, we feel that our parametrization is within present error margins. The agreement between the

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microscopic calculation of Akmal et al. [14] and the simple parametrization of Eq. (49) may imply that the essential many-body physics close to the equilibrium density arises from two-body and three-body terms to E only. The reason being that three-body terms are proportional with n while the two-body terms are proportional with n. With three-body terms we obviously intend both e!ective interactions and contributions from real three-body forces. The evaluation of the latter is, as discussed above, still an unsettled problem. If we now restrict the attention to matter with electrons only, one can easily obtain an analytic equation for the proton fraction through the asymmetry parameter x. Recalling the equilibrium conditions for b-stable matter of Eqs. (6) and (7) and using the de"nitions of the chemical potentials for particle species i of Eq. (3) one "nds that k "(1/n) Re/Rx , (50) C N and from the latter it is rather easy to show that the proton fraction is given by (assuming ultra-relativistic electrons) (4S uA(1!2x )) N . nx "  N 3p

(51)

De"ning a"2(4S uA)/pn ,  Eq. (51) reduces to

(52)

3x#ax!a"0 ,

(53)

where x"1!2x . Since we will always look at solutions for densities greater than zero, the cubic N equation for x has actually an analytical solution which is real and given by x"!2(a/tan(2t) ,

(54)

with tan t"(tan () and tan "!2(a/3. Note well that x depends on the total baryon density  n only. This means in turn that our parametrization of the EoS can now be rewritten for b-stable matter as






2(a  u!2!d . #S uA E"E u  tan(2t)  1#du

(55)

and is an analytical function of density only. The quality of our approximation to the EoS of Akmal et al. [14] for other observables than the energy per particle is shown in Fig. 18 for the proton fractions derived from the simple expression in Eq. (54). In the same "gure we display also the resulting energy per nucleon in b-stable matter and compare it with the results of Akmal et al. [14] for various values of d. Note well that the

 This argument is for the energy density, i.e. e"En.

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Fig. 18. Upper panel: Energy per nucleon (no leptonic contribution) in b-stable matter for the parametrized EoS of Eq. (49) for d"0, 0.13, 0.2, 0.3 and the results of Akmal et al. [14] with boost corrections and three-body forces (< #dv#UIXH). Lower panel: the corresponding proton fraction x .  N

Fig. 19. (v /c) for d"0, 0.1, 0.2, 0.3, the results of Akmal et al. [14], and for Baym and Kalogera's [85] patched EoS  which shows a discontinuous (v /c). 

proton fraction does not depend on the value of d, see Eq. (51). As can be seen from this picture, the EoS with d"0 yields the sti!est EoS, and as a consequence it results in a superluminal behavior at densities greater than n+1.0 fm\. This is seen in Fig. 19 where we plot the sound speed (v /c) for  various d values and that resulting from the microscopic calculation of Akmal et al. [14]. The form

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of (49), with the inclusion of the parameter d, provides therefore a smooth extrapolation from small and large densities with the correct behavior in both limits, i.e. the binding energy per nucleon E/A"E is linear in number density. In the dilute limit this is the Lenz (optical) potential. At high densities the linearity is required by the condition that the sound speed v"RP/Re does not exceed  the speed of light. This justi"es the introduction of the parameter d in our parametrization and explains also our deviation from the results of Akmal et al. at densities greater than 0.6}0.7 fm\, see Figs. 17 and 18. For d"0.1 the EoS becomes superluminal at densities of the order of 6 fm\. From the de"nition of (v /c) in Eq. (18) and the EoS of Eq. (49) that for  d5(E /m K0.13 ,  L

(56)

the EoS remains causal for all densities. The EoS of Akmal et al. becomes superluminous at n+1.1 fm\. With this caveat we have an EoS that reproduces the data of Akmal et al. at densities up to 0.6}0.7 fm\ and has the right causal behavior at higher densities. Furthermore, the di!erences at higher densities will also not be of importance in our analysis of the dynamics and structure of neutrons stars, since the mixed baryon-quark phase, with realistic values for the bag parameter and the coupling constant a starts at densities around 0.5}0.8 fm\. Q Finally, in Fig. 19 we have also plotted the sound speed following the approach of Baym and Kalogera [85], where the sound speed is allowed to jump discontinuously at a chosen density in order to keep the EoS causal. With this prescription, Baym and Kalogera were also able to obtain an upper bound for neutron star masses of 2.9M . The approach of Baym and Kalogera di!ers > from ours since their EoS is discontinuously sti!ened by taking v "c at densities above a certain  value n which, however, is lower than n "5n where their nuclear EoS becomes superluminous. A   This sti!ens the nuclear EoS for densities n (n(n but softens it at higher densities. Their A  resulting maximum masses lie in the range 2.2M (M(2.9M . Our approach incorporates > > causality by reducing the sound speed smoothly towards the speed of light at high densities. Our approach will most likely not yield an absolute upper bound on the maximum mass of a neutron star. Therefore our maximum mass never exceeds that of nuclear EoS of Akmal et al. [14]. In fact, one may argue that at very high densities particles become relativistic and the sound speed should be even lower, vKc/3, and therefore the softening we get from incorporating causality is even on  the low side. 2.4. Hyperonic matter At nuclear matter density the electron chemical potential is &110 MeV, see e.g. Fig. 6. Once the rest mass of the muon is exceeded, it becomes energetically favorable for an electron at the top of the e\ Fermi surface to decay into a k\. We then develop a Fermi sea of degenerate negative muons, see again Fig. 6. In a similar way, as soon as the chemical potential of the neutron becomes su$ciently large, energetic neutrons can decay via weak strangeness non-conserving interactions into K hyperons leading to a K Fermi sea with kK "k . However, if we neglect interactions, or L assume that their e!ects are small, one would expect the R\ to appear via e\#nPR\#l , C

(57)

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Fig. 20. The neutron and electron chemical potentials in beta stable matter according to models < #dv#UIXH (full  line) and < #dv (dashed line). Threshold densities for the appearance of non-interacting hyperons are marked by  horizontal line segments. Taken from Ref. [14].

at lower densities than the K, even though R\ is more massive the reason being that the above process removes both an energetic neutron and an energetic electron, whereas the decay to a K, being neutral, removes only an energetic neutron. Stated di!erently, the negatively charged hyperons appear in the ground state of matter when their masses equal k #k , while the neutral C L hyperon K appears when k equals its mass. Since the electron chemical potential in matter is L larger than the mass di!erence mR\ !mK "81.76 MeV, the R\ will appear at lower densities than the K. We show this in Fig. 20 where we plot the chemical potentials for electrons and neutrons in b-stable matter. The threshold densities for R\, K and the isobar D\ are indicated by the horizontal lines. Since this work has an emphasis on many-body approaches, we will try to delineate here as well how to obtain properties of hyperons in dense nuclear matter within the framework of microscopic theories. The main problem we have to face in our case is that the hyperon}nucleon interaction and especially the hyperon}hyperon interaction are less constrained by the data as is the case in the nucleonic sector. Our many-body scheme starts thus with the most recent parametrization of the free baryon}baryon potentials for the complete baryon octet as de"ned by Stoks and Rijken in Ref. [19]. This potential model, which aims at describing all interaction channels with strangeness from S"0 to S"!4, is based on SU(3) extensions of the Nijmegen potential models [86] for the S"0 and S"!1 channels, which are "tted to the available body of experimental data and constrain all free parameters in the model. In our discussion we employ the interaction version NSC97e of Ref. [19], although we have also carried out calculations with version NSC97a. Since the results for b-stable matter are not signi"cantly altered, we will present results with version NSC97e only. The next step is to introduce e!ects from the nuclear medium. Here we will construct the G-matrix, which takes into account short-range correlations for all strangeness sectors, and solve the equations for the single-particle energies of the various baryons self-consistently. The G-matrix

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is formally given by 1B B "G(u)"B B 2"1B B "<"B B 2         1 1B B "G(u)"B B 2 . # 1B B "<"B B 2     u!e !e #ng      

(58)

 

Here B represents all possible baryons n, p, K, R\, R, R>, N\ and N and their quantum numbers G such as spin, isospin, strangeness, linear momenta and orbital momenta. The intermediate states B B are those which are allowed by the Pauli principle and the energy variable u is the starting   energy de"ned by the single-particle energies of the incoming external particles B B . The   G-matrix is solved using relative and centre-of-mass coordinates, see e.g., Refs. [20,89] for computational details. The single-particle energies are given by e G "t G #u G #m

(59)

G

where t G is the kinetic energy and m de"ned by

G

the mass of baryon B . The single-particle potential u G

G

is

(60) u G "Re 1B B "G(u"e H #e G )"B B 2 . G H G H H X$H The linear momentum of the intermediate single-particle state B is limited by the size of the Fermi H surface F for particle species B . The last equation is displayed in terms of Goldstone diagrams in H H Fig. 21. Diagram (a) represents contributions from nucleons only as hole states, while diagram (b) has only hyperons as holes states in case we have a "nite hyperon fraction in b-stable neutron star matter. The external legs represent nucleons and hyperons. In order to satisfy the equations for b-stable matter summarized in Eq. (10), we need to solve Eqs. (58) and (59) to obtain the single-particle energies of the particles involved at the corresponding Fermi momenta. Typically, for every total baryonic density n"n #n , the density of , 7 nucleons plus hyperons, Eqs. (58) and (59) were solved for "ve nucleon fractions and "ve hyperons fractions and, for every nucleon and hyperon fraction, we computed three proton fractions and three fractions for the relevant hyperons. The set of equations in Eq. (10) were then solved by interpolating between di!erent nucleon and hyperon fractions.

Fig. 21. Goldstone diagrams for the single-particle potential u. (a) represents the contribution from nucleons only as hole states while (b) includes only hyperons as hole states. The wavy line represents the G-matrix.

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The many-body approach outlined above is the lowest-order Brueckner}Hartree}Fock (BHF) method extended to the hyperon sector. This means also that we consider only two-body interactions. However, it is well-known from studies of nuclear matter and neutron star matter with nucleonic degrees of freedom only that three-body forces are important in order to reproduce the saturation properties of nuclear matter, see e.g., Ref. [14] for the most recent approach. In order to include such e!ects, we replace the contributions to the proton and neutron self-energies arising from intermediate nucleonic states only, see diagram (a) of Fig. 21, with those derived from Ref. [14] where the Argonne < nucleon}nucleon interaction [38] is used with relativistic boost  corrections and a "tted three-body interaction, model. Here we employ the parametrization of Eq. (49) with d"0.2. In the discussions below we will thus present two sets of results for b-stable matter, one where the nucleonic contributions to the self-energy of nucleons is derived from the baryon}baryon potential model of Stoks and Rijken [19] and one where the nucleonic contributions are replaced with the results from Ref. [14] following the parametrization discussed in Eq. (49). In the discussion in this subsection we will label these results with APR98. Hyperonic contributions will however all be calculated with the baryon}baryon interaction of Stoks and Rijken [19]. These models for the pure nucleonic part combined with the hyperon contribution yield the composition of b-stable matter, up to total baryonic number density n"1.2 fm\, shown in Fig. 22. The corresponding energies per baryon are shown in Fig. 23 for both pure nucleonic (BHF and APR98 pn-matter) and hyperonic matter (BHF and APR98 with hyperons) in b-equilibrium for the same baryonic densities as in Fig. 22. For both types of calculations R\ appears at densities &2}3n . Since the EoS of APR98 for  nucleonic matter yields a sti!er EoS than the corresponding BHF calculation, R\ appears at n"0.27 fm\ for the APR98 EoS and n"0.35 fm\ for the BHF EoS. These results are in fair agreement with results obtained from mean "eld calculations, see e.g., Refs. [6,17,18,28,30,87,88]. The introduction of hyperons leads to a considerable softening of the EoS. Moreover, as soon as hyperons appear, the leptons tend to disappear, totally in the APR98 case whereas in the BHF calculation only muons disappear. This result is related to the fact that K does not appear at the densities considered here for the BHF EoS. For the APR98 EoS, K appears at a density n"0.67 fm\. Recalling kK "k "k #k and that the APR98 EoS is sti!er due to the inclusion L N C of three-body forces, this clearly enhances the possibility of creating a K with the APR98 EoS. However, the fact that K does not appear in the BHF calculation can also, in addition to the softer EoS, be retraced to a delicate balance between the nucleonic and hyperonic hole state contributions (and thereby to features of the baryon}baryon interaction) to the self-energy of the baryons considered here, see diagrams (a) and (b) in Fig. 21. Stated di!erently, the contributions from R\, proton and neutron hole states to the K chemical potential are not attractive enough to lower the chemical potential of the K so that it equals that of the neutron. Furthermore, the chemical potential of the neutron does not increase enough since contributions from R\ hole states to the neutron self-energy are attractive. We illustrate the role played by the two di!erent choices for nucleonic EoS in Fig. 24 in terms of the chemical potentials for various baryons for matter in b-equilibrium. We also note that, using the criteria in Eq. (10), neither the R nor R> do appear for both the BHF and the APR98 equations of state. This is due to the fact that none of the R-baryon and R>-baryon interactions are attractive enough. A similar argument applies to N and N\. In the latter case the mass of the particle is &1315 MeV and almost 200 MeV in attraction is needed in

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Fig. 22. Particle densities in b-stable neutron star matter as functions of the total baryonic density n. The upper panel represents the results obtained at the Brueckner}Hartree}Fock level with the potential of Stoks and Rijken [19]. In the lower panel the nucleonic part of the self-energy of the nucleons has been replaced with the EoS of Eq. (49) with d"0.2.

Fig. 23. Energy per baryon in b-stable neutron star matter for di!erent approaches as function of the total baryonic density n. See text for further details.

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Fig. 24. Chemical potentials in b-stable neutron star matter as functions of the total baryonic density n. The upper panel represents the results obtained at the Brueckner}Hartree}Fock level with the potential of Stoks and Rijken [19]. The lower panel includes results obtained with the EoS of Eq. (49) with d"0.2.

order to full"l e.g., the condition kK "kN "k . From the bottom panel of Fig. 24 we see however L that R could appear at densities close to 1.2 fm\. In summary, using the parametrized EoS of Akmal et al. [14] from Eq. (49) for the nucleonic sector and including hyperons through the most recent model for the baryon}baryon interaction of the Nijmegen group [19], we "nd through a many-body calculation for matter in b-equilibrium that R\ appears at a density of n"0.27 fm\ while K appears at n"0.67 fm\. Due to the formation of hyperons, the matter is deleptonized at a density of n"0.85 fm\. Within our many-body approach, no other hyperons appear at densities below n"1.2 fm\. Although the EoS of Akmal et al. [14] may be viewed as the currently most realistic approach to the nucleonic EoS, our results have to be gauged with the uncertainty in the hyperon}hyperon and nucleon} hyperon interactions. Especially, if the hyperon}hyperon interactions tend to be more attractive, this may lead to the formation of hyperons such as the K, R, R>, N\ and N at lower densities. The hyperon}hyperon interaction and the sti!ness of the nucleonic contribution play crucial roles in the formation of various hyperons. These results di!er from present mean "eld

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calculations [6,17,18,28,30,87,88], where all kinds of hyperons can appear at the densities considered here. In Section 5.3 we will discuss the implications on neutron star observables, however in our discussions on phase transitions in neutron stars, we will make life somewhat easier by just including quark degrees of freedom through a simpli"ed model, namely the bag model, in order to account for degrees of freedom beyond the nucleonic ones. This is discussed in Section 2.8. 2.5. Kaon condensation Kaon condensation in dense matter was suggested by Kaplan and Nelson [12], and has been discussed in many recent publications [91,92]. Due to the attraction between K\ and nucleons its energy decreases with increasing density, and eventually if it drops below the electron chemical potential in neutron star matter in b-equilibrium, a Bose condensate of K\ will appear. It is found that K\'s condense at densities above &3}4o , where o "0.16 fm\ is normal nuclear matter   density. This is to be compared to the central density of &4o for a neutron star of mass 1.4M  > according to the estimates of Wiringa et al. [90] using realistic models of nuclear forces. In neutron matter at low densities, when the interparticle spacing is much larger than the range of the interaction, r
(61)

which is the optical potential obtained in the impulse approximation. If hadron masses furthermore decrease with density the condensation will occur at lower densities [91]. At high densities when the interparticle spacing is much less than the range of the interaction, r ;R, the kaon will interact with many nucleons on a distance scale much less the range of the  interaction. The kaon thus experiences the "eld from many nucleons and the kaon energy deviates from its rest mass by the Hartree potential:



u "m #n < (r) dr , &  ) ,+ )\,

(62)

As shown in Ref. [93], the Hartree potential is considerably less attractive than the Lenz potential. Already at rather low densities, when the interparticle distance is comparable to the range of the KN interaction, the kaon}nucleon and nucleon}nucleon correlations conspire to reduce the K\N attraction signi"cantly [94]. This is also evident from Fig. 25 where the transition from the low density Lenz potential to the high-density Hartree potential is calculated by solving the Klein}Gordon equation for kaons in neutron matter in the Wigner}Seitz cell approximation. Results are for square well K\N-potentials of various ranges R. Kaon}nucleon correlations reduce the K\N interaction signi"cantly when its range is comparable to or larger than the nucleon}nucleon interparticle spacing. The transition from the Lenz

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Fig. 25. Kaon energy as function of neutron density. Including nuclear correlations in the Wigner}Seitz cell approximation is shown by full curves for various ranges of the K\n potentials R"0.4 fm, R"0.7 fm and R"1.0 fm. At low densities they approach the Lenz result (Eq. (61), dotted curve) and at high densities they approach the Hartree result (Eq. (62), dashed curves). The electron chemical potential k of our EoS Eq. (49) with d"0.2 is shown with and without C (lower and upper dotted curve) a transition to a mixed phase of quark matter for a Bag constant of B"100 MeV fm\.

potential at low densities to the Hartree potential at high densities occurs already well below nuclear matter densities. For the measured K\n scattering lengths and reasonable ranges of interactions the attraction is reduced by about a factor of 2}3 in cores of neutron stars. Relativistic e!ects further reduce the attraction at high densities. Consequently, a kaon condensate is less likely in neutron stars due to nuclear correlations. However, if kaon masses drop with densities [91] condensation will set in at lower densities. If the kaon condensate occurs a mixed phase of kaon condensates and ordinary nuclear matter may coexist in a mixed phase [95] depending on surface and Coulomb energies involved. The structures would be much like the quark and nuclear matter mixed phases described above. 2.6. Pion condensation Pion condensation is like kaon condensation possible in dense neutron star matter. For an in depth survey see e.g. Refs. [36,96] and references therein. If we "rst neglect the e!ect of strong correlations of pions with the matter in modifying the pion self-energy, one "nds it is favorable for a neutron on the top of the Fermi sea to turn into a proton and a p\ when k !k "k 'm , L N C p

(63)

where m "139.6 MeV is the p\ rest mass. As discussed in the previous subsection, at nuclear p matter saturation density the electron chemical potential is &100 MeV and one might therefore expect the appeareance of p\ at a slightly higher density. One can however not neglect the interaction of the pion with the background matter. Such interactions can enhance the pion

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self-energy and thereby the pion threshold density, and depending on the chosen parameters, see again Ref. [36], the critical density for pion condensation may vary from n to 4n . These matters   are however not yet settled in a satisfying way, and models with strong nucleon}nucleon correlations tend to suppress both the pNN and p*N interaction vertices so that a pion condensation in neutron star matter does not occur. However, in addition to a charged pion condensate, one may also form a p condensate through the reaction nPn#p, if the p e!ective mass in the medium is zero. The recent analysis, based on the < interaction model [38] of Akmal et al. [13,14,68]  suggests such a pion condensate. The e!ects were partly discussed in Section 2.2 and the impact on the proton fraction was shown in Fig. 16. The p condensation of Akmal et al. [14] for pure neutron matter appears at a density of &0.2 fm\ when the three-body interaction is included, whereas without < it appears at much higher densities, i.e. &0.5 fm\. Although it is a robust GHI mechanism in the variational calculation of Ref. [14], the conclusion relies on both the model of the NN and three-body interactions adopted in the calculations. As noted in Ref. [41] for pure neutron matter, the < interaction resulted in a slightly di!erent energy per particle at densities greater  than 0.4 fm\ when compared with the CD}Bonn and Nijmegen interactions. This topic was also discussed in Section 2.2 and shown in Fig. 3. Thus, before a "rm conclusion can be reached about p condensation, it is our belief that it should also be obtained at the two-body level with the other phase-shift equivalent NN interactions. That would lend strong support to the conclusions reached in Ref. [14]. The inclusion of three-body interactions introduces a further model dependence. Due to these uncertainties, we will refrain in this work from presenting a thorough discussion of pion condensation. Rather, we will take the liberty to refer to, e.g. Refs. [14,29,36,96]. 2.7. Superyuidity in baryonic matter The presence of neutron super#uidity in the crust and the inner part of neutron stars are considered well established in the physics of these compact stellar objects. In the low-density outer part of a neutron star, the neutron super#uidity is expected mainly in the attractive S channel. At  higher density, the nuclei in the crust dissolve, and one expects a region consisting of a quantum liquid of neutrons and protons in beta equilibrium. The proton contaminant should be super#uid in the S channel, while neutron super#uidity is expected to occur mainly in the coupled P }F    two-neutron channel. In the core of the star any super#uid phase should "nally disappear. The presence of two di!erent super#uid regimes is suggested by the known trend of the nucleon}nucleon (NN) phase shifts in each scattering channel. In both the S and P }F    channels the phase shifts indicate that the NN interaction is attractive. In particular for the S channel, the occurrence of the well known virtual state in the neutron}neutron channel  strongly suggests the possibility of a pairing condensate at low density, while for the P }F channel   the interaction becomes strongly attractive only at higher energy, which therefore suggests a possible pairing condensate in this channel at higher densities. In recent years the BCS gap equation has actually been solved with realistic interactions, and the results con"rm these expectations. The S neutron super#uid is relevant for phenomena that can occur in the inner crust of  neutron stars, like the formation of glitches, which may to be related to vortex pinning of the super#uid phase in the solid crust [97]. The results of di!erent groups are in close agreement on the S pairing gap values and on its density dependence, which shows a peak value of about 3 MeV at  a Fermi momentum close to k +0.8 fm\ [98}101]. All these calculations adopt the bare NN $

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Table 1 Collection of P }F energy gaps (in MeV) for the various potentials discussed in Section 2.2. BHF single-particle   energies have been used. In case of no results, a vanishing gap was found k (fm\) $

CD}Bonn

< 

Nijm I

Nijm II

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

0.04 0.10 0.18 0.25 0.29 0.29 0.27 0.21 0.17 0.11

0.04 0.10 0.17 0.23 0.22 0.16 0.07

0.04 0.10 0.18 0.26 0.34 0.40 0.46 0.47 0.49 0.43

0.04 0.10 0.18 0.26 0.36 0.47 0.67 0.99 1.74 3.14

interaction as the pairing force, and it has been pointed out that the screening by the medium of the interaction could strongly reduce the pairing strength in this channel [101}103]. However, the issue of the many-body calculation of the pairing e!ective interaction is a complex one and still far from a satisfactory solution. The precise knowledge of the P }F pairing gap is of paramount relevance for, e.g. the cooling   of neutron stars, and di!erent values correspond to drastically di!erent scenarios for the cooling process [29]. Generally, the gap suppresses the cooling by a factor &exp(!D/¹), see e.g. Ref. [47], which is severe for temperatures well below the gap energy. Unfortunately, only few and partly contradictory calculations of the pairing gap exist in the literature, even at the level of the bare NN interaction [104}108]. However, when comparing the results, one should note that the NN interactions used in these calculations are not phase-shift equivalent, i.e. they do not predict exactly the same NN phase shifts. Furthermore, for the interactions used in Refs. [104}107] the predicted phase shifts do not agree accurately with modern phase shift analyses, and the "t of the NN data has typically s/datum+3. As we discussed in Section 2.2, progress has been made not only in the accuracy and the consistency of the phase-shift analysis, but also in the "t of realistic NN interactions to these data. As a result, several new NN interactions have been constructed which "t the world data for pp and np scattering below 350 MeV with high precision. Potentials like the recent Argonne < [38], the CD}Bonn [39] or the new Nijmegen potentials [37] yield  a s/datum of about 1 and may be called phase-shift equivalent. In Table 1 we show the recent non-relativistic pairing gaps for the P }F partial waves, where e!ective nucleon masses from   the lowest-order Brueckner}Hartree}Fock calculation of Section 2.2 have been employed, see Ref. [109] for more details. These results are for pure neutron matter and we observe that up to k &2 fm\, the various potentials give more or less the same pairing gap. Above this Fermi $ momentum, which corresponds to a lab energy of &350 MeV, the results start to di!er. This is simply due to the fact that the potentials are basically "t to reproduce scattering data up to this lab energy. Beyond this energy, the potentials predict rather di!erent phase shifts for the P }F   partial waves, see e.g. Ref. [109]. Thus, before a precise calculation of P }F energy gaps can be  

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made, one needs NN interactions that "t the scattering data up to lab energies of &1 GeV. This means in turn that the interaction models have to account for the opening of inelasticities above 350 MeV due to the ND channel. The reader should however note that the above results are for pure neutron matter. We end therefore this subsection with a discussion of the pairing gap for b-stable matter of relevance for the neutron star cooling discussed in Section 5. We will also omit a discussion on neutron pairing gaps in the S channel, since these appear at densities corresponding to the crust of the neutron star.  The gap in the crustal material is unlikely to have any signi"cant e!ect on cooling processes [27], though it is expected to be important in the explanation of glitch phenomena. Therefore, the relevant pairing gaps for neutron star cooling should stem from the the proton contaminant in the S channel, and super#uid neutrons yielding energy gaps in the coupled P }F two-neutron    channel. If in addition one studies closely the phase shifts for various higher partial waves of the NN interaction, one notices that at the densities which will correspond to the core of the star, any super#uid phase should eventually disappear. This is due to the fact that an attractive NN interaction is needed in order to obtain a positive energy gap. Since the relevant total baryonic densities for these types of pairing will be higher than the saturation density of nuclear matter, we will account for relativistic e!ects as well in the calculation of the pairing gaps. To do so, we resort to the Dirac}Brueckner}Hartree}Fock (DBHF) formalism discussed in Section 2.2.3. As an example, consider the evaluation of the proton S pairing gap using the DBHF approach.  To evaluate the pairing gap we follow the scheme of Baldo et al. [98]. These authors introduced an e!ective interaction
(64)

where the energy E is given by E "((e !e )#D, e being the single-particle energy at the I I I $ I $ Fermi surface, < is the free nucleon}nucleon potential in momentum space, de"ned by the IIY three-momenta k, k. The renormalized potential
(65)

These equations are solved self-consistently in order to obtain the pairing gap D for protons and neutrons for di!erent partial waves. In Fig. 26 we plot as function of the total baryonic density the pairing gap for protons in the S state, together with the results from the non-relativistic approach discussed in Refs. [107,113].  The results in the latter references were also obtained with the Bonn A potential of Ref. [40]. These results are all for matter in b-equilibrium. In Fig. 27 we plot the corresponding relativistic results

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Fig. 26. Proton pairing in b-stable matter for the S partial wave. The non-relativistic results are taken from Ref. [112].  Fig. 27. Neutron pairing in b-stable matter for the P partial wave. The non-relativistic results are taken from  Ref. [112].

for the neutron energy gap in the P channel. For the D channel we found both the non  relativistic and the relativistic energy gaps to vanish. The non-relativistic results for the Bonn A potential are taken from Ref. [112]. As can be seen from Fig. 26, there are only small di!erences (except for higher densities) between the non-relativistic and relativistic proton gaps in the S wave. This is expected since the proton  fractions (and their respective Fermi momenta) are rather small, see Fig. 6. For neutrons however, the Fermi momenta are larger, and we would expect relativistic e!ects to be important. At Fermi momenta which correspond to the saturation point of nuclear matter, k "1.36 fm\, the lowest relativistic correction to the kinetic energy per particle is of the order of $ 2 MeV. At densities higher than the saturation point, relativistic e!ects should be even more important, as can clearly be seen in the calculations of Ref. [74]. Since we are dealing with very small proton fractions, a Fermi momentum of k "1.36 fm\, would correspond to a total $ baryonic density &0.09 fm\. Thus, at larger densities relativistic e!ects for neutrons should be

 Even smaller di!erences are obtained for neutrons in the S channel. 

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important. This is also re#ected in Fig. 27 for the pairing gap in the P channel. The relativistic  P gap is less than half the corresponding non-relativistic one, and the density region is also much  smaller. This is mainly due to the inclusion of relativistic single-particle energies in the energy denominator of Eq. (65) and the normalization factors for the Dirac spinors in the NN interaction. As an example, at a neutron Fermi momentum k "1.5 fm\, the gap has a value of 0.17 MeV $ when one uses free single-particle energies and a bare NN potential. Including the normalization factors in the NN interaction, but employing free single-particle energies reduces the gap to 0.08 MeV. If we employ only DBHF single-particle energies and the bare NN interaction, the gap drops from 0.17 MeV to 0.04 MeV. Thus, the largest e!ect stems from the change in the singleparticle energies, although, the combined action of both mechanisms reduce the gap from 0.17 MeV to 0.015 at k "1.5 fm\. The NN interaction in the P channel depends also strongly on the $  spin}orbit force, see e.g. Fig. 3.3 in Ref. [40], and relativistic e!ects tend to make the NN spin}orbit interaction from the u-meson in P-waves more repulsive [40]. This leads to a less attractive NN interaction in the P channel and a smaller pairing gap.  The present results can be summarized as follows: } The S proton gap in b-stable matter is 41 MeV, and if polarization e!ects were taken into  account [101], it could be further reduced by a factor 2}3. } The P gap is also small, of the order of &0.1 MeV in b-stable matter. If relativistic e!ects are  taken into account, it is almost vanishing. However, there is quite some uncertainty with the value for this pairing gap for densities above &0.3 fm\ due to the fact that the NN interactions are not "tted for the corresponding lab energies. } Higher partial waves give essentially vanishing pairing gaps in b-stable matter. } We have omitted a discussion of hyperon pairing, due to the uncertainties in the determination of the hyperon}hyperon interaction. We refer the reader here to Ref. [114] for a discussion of these gaps. Consequences for cooling histories will be discussed in Section 5. 2.8. Quark matter When nuclear matter is compressed to densities so high that the nucleon cores substantially overlap, one expects the nucleons to merge and undergo a phase transition to chiral symmetric and/or decon"ned quark matter. Rephrased in terms of the relevant "eld excitations, we expect a transition from hadronic to quark degrees of freedom at high densities. Knowledge of the EOS of both hadronic and quark matter is necessary to estimate the possible e!ects of this transition in neutron stars. Recent advances in the QCD phase diagram include improved lattice QCD calculations, random matrix models [115], and models addressing the possibility of color superconductivity at "nite density [116]. Lattice QCD can only treat the case of zero baryon chemical potential and is therefore not useful for neutron stars. Lattice calculations suggest that QCD has a "rst order transition at "nite temperature and zero chemical potential, provided that the strange quark is su$ciently light [117]. The transition weakens and might change to second order for large strange quark masses in the limit of QCD with two massless #avors.

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Recent work using chiral random matrix models (chRMM) [115] suggests an e!ective thermodynamic potential of the form X( ; k, ¹)/N "X (k, ¹)# !ln+[ !(k#i¹)] ) [ !(k!i¹)], . D  

(66)

Here X (k, ¹) is unspeci"ed and independent of the chiral mean "eld . The scale of dimensional  quantities cannot be determined within the chRMM, and must be estimated by relating the variance of the Gaussian random matrix ensemble to the vacuum expectation value of the "eld via the Casher}Banks relation [115]. The value of at the minima of X( ; k, ¹) is related to the quark condensate, 1tM t2, which is the order parameter for chiral symmetry breaking. Minimization of Eq. (66) leads to a "fth-order polynomial equation for which is identical in form to the results of Landau-Ginzberg theory using a  potential. One solution to this equation corresponds to the restored symmetry phase with "0. This model predicts a second-order transition for k"0 at a temperature, ¹ , which is generally agreed to be in the range 140}170 MeV. For ¹"0, A a "rst-order transition occurs at some k . Since the phases in which chiral symmetry is broken and  restored must be separated in the (k, ¹) plane by an unbroken line of phase transitions, this implies the existence of a tricritical point in the theory of massless quarks. There has recently been speculation regarding color superconductivity at medium densities resulting from non-perturbative attraction between quarks. At "nite chemical potential, this invariably leads to the possibility of a diquark condensate which breaks global color invariance [116]. The associated color gap is &100 MeV and may become the thermodynamically favorable phase at high baryon densities. When the strange quark is taken into account many di!erent phases may exist [118] and such e!ects require more analysis. 2.8.1. Bag models Since we do not have a fully reliable theory for the quark matter phase, we will for simplicity employ the simple Bag model in our actual studies of the mixed phase and neutron start properties. In the bag model the quarks in the hadrons are assumed to be con"ned to a "nite region of space, the so-called `baga, by a vacuum pressure B. The pressure from the quarks inside the bag is provided by the Fermi pressure and interactions computed to order a "g/4p where g is the QCD  coupling constant. The pressure for quarks of #avor f, with f"u, d or s is [119]






k #k 1 D k k (k !2.5m )#1.5m ln D P " D D D 4p D D D m D







a 3 k #k D !  k k !m ln D D p 2 D D m D







!k . D

(67)

The Fermi momentum is k "(k !m ). The total pressure, including the bag constant D D D B simulating con"nement is P"P # P !B . C D D

(68)

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The electron pressure is P "k/12p . (69) C C A Fermi gas of quarks of #avor i has density n "k /p, due to the three color states. There is no G $G one-gluon exchange interaction energy between quarks of di!erent #avor, while that between quarks of #avour i is given by (2a/3p)E per quark i [120]. Here E is the average kinetic energy per G G quark, and a is the strong interaction coupling constant, assumed to have a value of 0.5. The u and d quarks are taken to be massless, and s quarks to have a mass of 150 MeV. Typical quark chemical potentials k 9m /3 are generally much larger. The value of the bag constant B is poorly O , known, and we present results using three representative values, B"100 MeV fm\ [121], B"150 MeV fm\ and B"200 MeV fm\ [122]. Another possible model which has been applied to neutron star studies, and which di!ers from the Bag-model is a massive quark model, the so-called Color-Dielectric model (CDM) [123}125]. The CDM is a con"nement model which has been used with success to study properties of single nucleons, such as structure functions [126] and form factors [127], or to describe the interaction potential between two nucleons [128], or to investigate quark matter [125,129]. In particular, it is possible, using the same set of parameters, both to describe the single nucleon properties and to obtain meaningful results for the decon"nement phase transition [125]. The latter happens at a density of the order of 2-3 times n when  symmetric nuclear matter is considered, and at even smaller densities for matter in b-equilibrium, as discussed below in this work. Another important feature is that e!ective quark masses in the CDM are always larger than a value of the order of 100 MeV. Hence chiral symmetry is broken and the Goldstone bosons are relevant degrees of freedom. This is to be contrasted with models like the MIT bag, where quarks have masses of a few MeV. We therefore expect the CDM to be relevant for computing the cooling rate of neutron stars via the Urca mechanism, as suggested by Iwamoto [130]. We will however stick to the Bag-model in our discussion of properties of neutron stars.

3. Thermodynamics of multi-component phase transitions Numerous phase transitions may occur in neutron stars, e.g. the nuclear liquid}gas transition in the inner crust and in the interior quark matter and/or condensates of kaons, pions, hyperons, etc. may be present. We shall in this section brie#y describe this curious phenomenon, the thermodynamics of multi-component systems and the corresponding mixed phases. It will be employed for neutron stars in the subsequent section. 3.1. Maxwell construction for one-component systems In the usual picture the transition between two phases occurs at a unique pressure, temperature and chemical potential. Consequently, the density is expected to jump discontinuously at the boundary between the two phases. This is not only true for systems of one component as in the everyday example of water freezing or evaporating. It is also the case for some two component systems as, e.g. electrically neutral nuclear matter in b-equilibrium. Electric neutrality requires that

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Fig. 28. Sketch of a neutron, hybrid, mixed and quark star. Fig. 29. Pressure as function of density. Maxwell construction and mixed phase for bag parameter B"200 MeV fm\. The arrow indicates where the Maxwell construction starts.

the proton and electron densities are the same in bulk n "n . N C

(70)

b-equilibrium requires that the chemical potentials of neutrons and protons only di!er by that of the electrons k "k #k,+ , L N C

(71)

in nuclear matter. These two conditions restrict two of the three components leaving only one independent variable, the baryon density. Likewise, in quark matter charge neutrality implies that n !n !n "n .  S  B   C

(72)

b-equilibrium requires analogously k "k "k #k/+ . B Q S C

(73)

Over the past two decades many authors have considered the properties of neutron stars with a core of quark matter [10]. In such `hybrida stars (see Fig. 28) it is assumed that each of the two phases are electrically neutral separately as in Eqs. (70) and (72) and in b-equilibrium separately as in Eqs. (71) and (73). Gibb's conditions P "P and k,+"k/+ (the temperatures are vanishing ,+ /+ L L in both phases) then determine a unique density at which the two bulk neutral phases coexist. This is the standard Maxwell construction and is seen in Fig. 29 as the double-tangent. In a gravitational "eld the denser phase (QM) will sink to the center whereas the lighter phase (NM) will #oat on top as a mantle as icebergs in the sea. At the interface of the phase transition there is a sharp density discontinuity and generally k,+Ok/+ so that the electron densities n "k/3p are C C C C

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diwerent in the two phases. This assumes that the sizes of QM structures are larger than electron screening lengths which, as discussed in [11], is not always the case. For small values of the bag constants the phase transition occurs at densities lower than n and  the whole neutron star is a quark star except possibly for a hadronic crust [131]. Finally, to avoid confusion we emphasize that a mixed phase may have continuous pressures and (average) densities over large scales &100 fm to kilometers and the phase transition is therefore second order in a thermodynamic sense. However, microscopically on length scales of order a few Fermi's a "rst-order phase transition between QM and NM is assumed and densities are discontinuous between the structures containing the QM and NM phases. 3.2. Two-component systems in a mixed phase For several coexisting components as, e.g. dissolved chemicals, there may exist a mixed phase where the various chemical potentials of the solvents vary continuously } as does the pressure and densities. A similar phenomenon was predicted for the nuclear liquid}gas phase transitions in the inner crust of neutron stars con"rmed by recent detailed numerical calculations [34,35]. In the inner crust the nuclei are surrounded by a neutron gas with an interpenetrating constant background of electrons, i.e. the nuclear matter and neutron gas form a mixed phase. Going a few hundred meters down in the neutron star crust the density of nuclei and thus also the average density increase continuously. Thus there is no sharp density discontinuity in bulk, i.e. over macroscopical distances of more than hundreds of Fermi's and up to several meters (see Fig. 28). However, on microscopical distances of a few Fermi's the density varies rapidly. Another example "rst considered by Glendenning [6] is that of a mixed phase of nuclear and quark matter. Contrary to the Maxwell construction described in the previous subsection, where the condition of charge neutrality applies to both phases, Eqs. (70) and (72), it is relaxed to overall charge neutrality only. Thus two conditions are relaxed to one } allowing for one new variable quantity, which is usually taken as the xlling fraction f of one of the phases in their coexisting mixture. For the nuclear and quark matter mixed phase the "lling fraction is de"ned as the fraction of the volume which is in the quark phase f,< /(< #< ) , /+ /+ ,+

(74)

and so (1!f ) is the "lling fraction of nuclear matter. The overall charge neutrality requirement is fn #(1!f )(n !n !n )"n . C N  S  B  

(75)

A number of requirements must be met in order to form such a mixed phase as addressed in [11]. These will be discussed in the following section for the mixed phase of nuclear and quark matter.

4. Structure of neutron stars The structure of the neutron star is seriously a!ected by phase transitions, the order of the phase transition and whether mixed phases can occur over a signi"cant part of the star. The important

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questions to be addressed are numerous. When is it legitimate to regard the electron density as uniform, what is the spatial structure of the new phase, and is it energetically favorable? In order to answer these questions one must "rst investigate screening lengths of the various charged particles and compare to typical size scales of structures. For that matter Coulomb and surface energies must be calculated. We will mainly discuss the mixed phase of nuclear and quark matter in cores of neutron stars. The mixed phase of nuclei and a neutron gas is in many ways similar and has been calculated in detail in Refs. [34,35,133]. 4.1. Screening lengths As described in [11] the mixed phase of quark and nuclear matter may be regarded as droplets of quark matter immersed in nuclear matter at lower densities, usually referred to as the droplet phase, even though at higher densities its structure is more complicated. If droplet sizes and separations are small compared with Debye screening lengths, the electron density will be uniform to a good approximation. The Debye screening length, j is given by "




Rn G , 1/j "4p Q " G Rk H L H$G G G

(76)

where n , k , and Q are the number density, chemical potential, and charge of particle species i. G G G Considering only electrons gives a screening length (p/4a , jC" " k $C

(77)

where aK1/137 and the Fermi momentum k "k since the electrons are always relativistic at $C C these densities. For k :150 MeV we thus obtain jC913 fm. The screening length for protons C " alone jN, is given by (pv /c4a(1#F ))/k , where F is the Landau parameter which gives the " $N  $N  energy for proton density variations. At the saturation density for symmetric nuclear matter, F K0, whereas at higher densities F &1 [132]. Since k &m, the nucleon mass, we "nd   N jN910 fm, somewhat shorter than the electron screening length, and therefore in the nuclear " matter phase, protons are the particles most e!ective at screening. The screening length for quarks is jOK7/k where q"u, d, and s refer to up, down and strange quarks. It depends only slightly " $O on whether or not s-quarks are present, so for k Km/3 we "nd jOK5 fm. O " In a composite system, such as the one we consider, screening cannot be described using a single screening length, but it is clear from our estimates, that if the characteristic spatial scales of structures are less than about 10 fm for the nuclear phase, and less than about 5 fm for the quark phase, screening e!ects will be unimportant, and the electron density will be essentially uniform. In the opposite case, when screening lengths are short compared with spatial scales, the total charge densities in bulk nuclear matter and quark matter will both vanish.

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4.2. Surface and Coulomb energies of the mixed phase When screening, lengths are much larger than the spatial scale of structures. This condition implies that the electron density is uniform everywhere, and all other particle densities are uniform within a given phase. The problem is essentially identical to that of matter at subnuclear densities [133], and the structure is determined by competition between Coulomb and interface energies. When quark matter occupies a small fraction, f, of the total volume, it will form spherical droplets immersed in nuclear matter. For higher "lling fractions, the quark matter will adopt shapes more like rods (`spaghettia) and plates (`lasagnaa), rather than spheres. For f50.5, the structures expected are the same as for a "lling factor 1!f, but with the roles of nuclear matter and quark matter reversed. Thus, one expects for increasing f that there will be regions with nuclear matter in rod-like structures, and roughly spherical droplets. To estimate characteristic dimensions, some special cases can be considered. The intricate structures in the general case will be discussed in Section 4.5. When f is small or close to unity, the minority phase will form spherical droplets. The surface energy per droplet is given by E "p4pR , 1

(78)

where p is the surface tension, and the Coulomb energy is (79) E "Ze/R"(16p/15)(o !o )R . /+ ,+ !  Here Z is the excess charge of the droplet compared with the surrounding medium, Ze" (o !o )< where < "(4p/3)R is the droplet volume and o and o are the total charge /+ ,+ " " /+ ,+ densities in bulk quark and nuclear matter, respectively. Minimizing the energy density with respect to R one obtain the usual result that E "2E and "nd a droplet radius 1 ! 15 p  p  o !o \ /+ ,+ K5.0 fm . (80) R" 8p (o !o ) p o /+ ,+   In the second formula we have introduced the quantities o "e 0.4 fm\ and p "50 MeV fm\   which, as we shall argue below, are typical scales for the quantities. (A droplet of symmetric nuclear matter in vacuum has a surface tension p"1 MeV fm\ for which (80) gives RK4 fm, which agrees with the fact that nuclei like Fe are the most stable form of matter for roughly symmetric nuclear matter at low density.) The form of Eq. (80) re#ects the fact that on dimensional grounds, the characteristic length scale is (p/(o !o )) times a function of f. /+ ,+ The total Coulomb and surface energy per unit volume is given for small f by












e "f 9((p/15)p(o !o )) 1>! /+ ,+ p o !o  /+ ,+ . (81) K44 MeV fm\ f p o   The result for f close to unity is given by replacing f by 1!f. In the case when the volumes of quark and nuclear matter are equal, f", the structure can be approximated as alternating layers,  of quark and nuclear matter and was considered in [11].






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To estimate length scales and energy densities, one needs the surface tension of quark matter and the charge densities in the two phases. A rough estimate of the surface tension is the bag constant, B, times a typical hadronic length scale &1 fm. Estimates of the bag constant range from 50 to 450 MeV fm\ [134]. The kinetic contribution to the surface tension at zero temperature has been calculated in the bag model in Ref. [135]. Only massive quarks contribute because relativistic particles, unlike non-relativistic ones, are not excluded near the surface due to the boundary conditions. The kinetic contribution to p from a quark species depends strongly on its mass and chemical potential. For m ;k it behaves as (3/4p)km , and it vanishes as m approaches k . If Q Q Q Q Q Q we adopt for the strange quark mass the value m K150 MeV, and for the quark chemical Q potentials one-third of the baryon chemical potential, which generally is slightly larger than the nucleon mass, k K k /39m/3, one obtains from Ref. [135] pK10 MeV fm\, which is Q close to the maximum value it can attain for any choice of m . We conclude that the surface Q tension for quark matter is poorly known, but lies most probably in the range 10}100 MeV fm\. (Lattice gauge theory estimates of p at high temperatures and zero quark chemical potentials lie in the range pK0.14!0.28¹&10}60 MeV fm\ [136] for ¹ &150}200 MeV, compara  ble to our estimates for cold quark matter, but it is unclear to what extent this agreement is accidental.) To estimate charge densities, we consider quark matter immersed in a uniform background of electrons. b-equilibrium insures that k "k "k #k , and therefore in the absence of B Q S C quark}quark interactions, one "nds the total electric charge density in the quark matter phase is given for k ;k &k ,k and m ;k by C S B O Q O e e o " (2n !n !n !3n )K (mk !2k k) . B Q C C O /+ 3 S p  Q O

(82)

Assuming m K150 MeV and k Km/3 the second term dominates except for small k and so the Q O C droplet is negatively charged and for k K170 MeV the density is about !0.4e fm\, the C characteristic scale of densities adopted in making estimates above. Due to the high quark density, o is small compared with o in Eq. (80) when quark matter ,+ /+ occupies a small fraction of the volume. The electron chemical potential in neutron stars depends strongly on the model for the nuclear equation of state, but generally one "nds k :170 MeV. C Consequently, for pK10 MeV fm\ we "nd from Eq. (80) a radius of R93.1 fm, whereas pK100 MeV fm\ gives R96.6 fm. For f close to unity one "nds nuclear bubble radii which are comparable with those for quark droplets, and for the layer-like structures expected for fK0.5, half the layer thickness is of comparable size. Estimates of characteristic scales for rod-like structures give similar values. Detailed calculations show that the e!ects of nonuniformity of the charge distribution a!ect estimates of Coulomb energies signi"cantly if the characteristic lengths, R and a, exceed the Debye screening length. The estimates of screening lengths made above show that screening will not be dominant for surface tensions below about 100 MeV, if the charge density di!erence is o , but for  higher values the simple picture of coexisting uniform bulk phases would become invalid, and the droplet phase would resemble increasingly two electrically neutral phases in equilibrium. For the smallest droplets of size R&5 fm the charge Z"(4p/3)Ro and baryon number /+ A"(4p/3)Rn are typically a few hundreds. /+

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4.3. Is the mixed phase energetically favored? If the bulk energy gained by going to the mixed phase is larger than the costs of the associated Coulomb and surface energies, then the mixed phase is energetically favored. Before going on to estimate these crucial energies we point out the basic physical reason why the bulk energy is lower in the mixed phase. As mentioned in connection with Eq. (82) the QM droplets are negatively charged. By immersing QM in the positively charged NM we can either remove some of the electrons from the top of the Fermi levels with energy k , or we can increase the proton fraction in C NM by which the symmetry energy is lowered. In equilibrium a combination of both will occur and in both cases bulk energy is saved and a lower energy density is achieved as seen in Fig. 29. To calculate the bulk energy we adopt a simple form for the energy density of nuclear matter consisting of a compressional term, a symmetry term, and an electron energy density as discussed in Eq. (49) e "nE"n[m#E (n)#S(n)x]#e #e , ,+   C I

(83)

where, e.g., e "k/4n. For quark matter the bag model equation of state gives an energy density C C




2a e " 1!  /+ n






3k O #B#e #e C I 4p OSBQ

(84)

with the QCD "ne structure constant a K0.4 and bag constant BK120 MeV fm\. We have  taken all quark masses to be zero. In the absence of surface and Coulomb e!ects the equilibrium conditions for the droplet phase are that the quark and nuclear matter should have equal pressures, and that it should cost no energy to convert a neutron or a proton in nuclear matter into quarks in quark matter. The last condition amounts to k "2k #k and k "k #2k . The electron L B S N B S density is the same in quark and nuclear matter, and we assume that matter is electrically neutral and in b-equilibrium, that is k "k #k and k "k #k . The chemical potentials are related to L N C B S C the Fermi momenta by k "p (1!2a /p)\. Electrons contribute little to pressures, but they O $O Q play an important role through the b-equilibrium and charge neutrality conditions. Fig. 30(a) shows the density dependence of the energy density of the droplet phase calculated neglecting surface and Coulomb energies (p"0) for a simple quadratic EoS for nuclear matter [11]. The energy of uniform, electrically neutral, bulk nuclear matter in b-equilibrium is also shown, together with the corresponding result for quark matter. The double-tangent construction gives the energy density for densities at which the two bulk neutral phases coexist. This corresponds to the standard treatment of the phase transition between nuclear matter and quark matter, in which the pressure remains constant throughout the transition, and consequently neutron stars have a core of quark matter and a mantle of nuclear matter, with a sharp density discontinuity at the phase transition. As one sees, if surface and Coulomb e!ects may be ignored, the transition from nuclear matter to the droplet phase occurs at a lower density than the transition to two bulk neutral phases, a feature also apparent in Ref. [6]. In addition, droplets of nuclear matter survive up to densities above those at which bulk neutral phases can coexist. We also observe that bulk contributions to the energy density of the droplet phase are always lower than those for coexisting bulk neutral phases. While detailed properties of the droplet phase depend strongly on the bulk

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Fig. 30. (a) The full line gives the energy density of the droplet phase without surface and Coulomb energies (p"0). Also shown are the energy densities of electrically neutral bulk nuclear matter, quark matter in b-equilibrium, and the double tangent construction (dashed line) corresponding to the coexistence of bulk electrically neutral phases. (b) Energy densities of the droplet phase relative to its value for p"0 for p"10, 50, and 90 MeV fm\. When the energy density of the droplet phase falls within the hatched area it is energetically favored. From [11].

energies, the qualitative picture we "nd persists over a wide range of possible bulk matter properties. When quark matter occupies a small fraction of space, f, one observes in Fig. 30 that the di!erence in energy between the droplet phase and bulk neutral nuclear matter varies as f . This is a general result for mixed phases which follows directly from the 1st law of thermodynamics. Consider the energy in a "xed volume at zero temperature. The change in energy by converting some of the quarks into nucleons at "xed volume is dE" k dN "dN (k !k ) . (85) G G ,+ ,+ /+ G In the latter equation we have used that at constant average density and "xed volume, the number of extra nucleons is exactly the baryon number removed from the quark phase (a third of the number of quarks removed), dN "!dN . This applies to both neutrons and protons. To ,+ /+ obtain the energy di!erence between the mixed phase and NM at the same density, we need to convert all QM into nucleons. For small "lling fraction f this number is small and we can apply Eq. (85). In fact both dN and the di!erence between chemical potentials, k !k is small as , ,+ /+ they both vanish right at the onset of the mixed phase, fK0. Therefore, they must both grow

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linearly with f and thus the di!erence between the mixed and NM phase at the same density must scale with f  according to Eq. (85). By inversion the same results apply for f close to unity. In contrast to the quadratic dependence on f of the bulk energy di!erence, the contributions to the energy density from surface and Coulomb energies are linear in f, see Eq. (81). This shows that the transitions to the droplet phase must occur via a "rst-order transition after all. Pressures and densities are continuous in the mixed phase only when surface and Coulomb energies can be completely neglected. Even for a small surface tension they will change the second order to multiple (small) "rst-order phase transitions whenever the dimensionality of the structures change. If the surface and Coulomb energies are su$ciently large, the droplet phase may never be favorable. The energy-density di!erence between the droplet phase, neglecting surface and Coulomb e!ects, and two coexisting neutral phases is at most 10 MeV fm\, as may be seen from Fig. 30. This is very small compared with characteristic energy densities which are of order 1000 MeV fm\. In Fig. 30(b) we show the energy density of the droplet phase for various values of the surface tension, relative to the value for p"0. In these calculations the geometry of the droplets was characterized by a continuous dimensionality, d, as described in Ref. [133], with d"3, 2 and 1 corresponding to spheres, rods and plates, respectively. For the droplet phase to be favorable, its energy density must lie below those of nuclear matter, quark matter, and coexisting electrically neutral phases of nuclear and quark matter. That is the droplet phase will be favored if its energy lies within the hatched region in Fig. 30(a) and (b). We see that whether or not the droplet phase is energetically favorable depends crucially on properties of quark matter and nuclear matter. For our model the droplet phase is energetically favorable at some densities provided p:70 MeV fm\. However, given the large uncertainties in estimates of bulk and surface properties one cannot at present claim that the droplet phase is de"nitely favored energetically. 4.4. Melting temperatures Should the quark-droplet phase exist in neutron stars, it could have important observational consequences. First, the pressure di!erence across the droplet phase can be large, of order 250 MeV fm\ as seen from Fig. 30(a), since the pressure is the negative intercept of the tangent to the curve. Consequently, a large portion of a neutron star could consist of matter in the droplet phase. Secondly, phases with isolated droplets would be expected to be solid. The melting temperature for a body centered cubic lattice is [137] ¹ &(Ze/170R) f  , (86)  where R/f  is the lattice spacing. The melting temperature is typically some tens of MeV for the smallest lattices and decreasing with increasing lattice spacing ( fP0). While spaghetti- and lasagna-like structures would exhibit anisotropic elastic properties, being rigid to some shear strains but not others in much the same way as liquid crystals. This could be important for quake phenomena, which have been invoked to explain observations in a number of di!erent contexts. Third, neutrino generation, and hence cooling of neutron stars could be in#uenced. This could come about because nuclear matter in the droplet phase has a higher proton concentration than bulk, neutral nuclear matter, and this could make it easier to attain the threshold condition for the nucleon direct Urca process [138]. Another is that the presence of the spatial structure of the droplet phase might allow processes to occur which would be forbidden in a translationally

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invariant system. Finally, one should bear in mind the possibility that even if the droplet phase were favored energetically, it would not be realized in practice if the time required to nucleate is too long. 4.5. Funny phases Surface and Coulomb energies determine the topology and length scales of the structures. Denoting the dimensionality of the structures by d (d"3 for droplets and bubbles, d"2 for rods and d"1 for plates) the surface and Coulomb energies are generally [27,34] E "dp(4p/3)R 1 2 d 8n (o !o )R 1! f \B #f , E " ,+ ! 3(d#2) /+ d!2 2




 

(87) (88)

where o and o are the total charge densities in bulk QM and NM, respectively. For droplets /+ ,+ ( fK0) or bubbles ( fK1) d"3 and the Coulomb energies reduce to the usual term E "(3/5)Ze/R where Z is the excess charge of the droplet compared with the surrounding ! medium, Ze"(4n/3)(o !o )R. Minimizing the energy density with respect to R we obtain /+ ,+ the usual result that E "2E . Minimizing with respect to the continuous dimensionality as well 1 ! thus determines both R and d. For the Walecka model the droplet phase is energetically favorable at some densities provided p:20 MeV/fm [139]. For comparison, using a quadratic EoS for NM [11], one "nds instead the more favorable condition p:70 MeV/fm. Given the large uncertainties in estimates of bulk and surface properties one cannot at present claim that the droplet phase is de"nitely favored energetically. The mixed phase in the inner crust of neutron stars consists of nuclear matter and a neutron gas in b-equilibrium with a background of electrons such that the matter is overall electrically neutral [34,35]. Likewise, quark and nuclear matter can have a mixed phase [6] and possible also nuclear matter with and without condensate of any negatively charged particles such as K\ [95], p\, R\, etc. The quarks are con"ned in droplet, rod- and plate-like structures [11] (see Fig. 31) analogous to the nuclear matter and neutron gas structures in the inner crust of neutron stars [34,35]. Depending on the equation of state, normal nuclear matter exists only at moderate densities, o&1!2o . With increasing density, droplets of quark matter form in nuclear matter and may  merge into rod- and later plate-like structures. At even higher densities the structures invert forming plates, rods and droplets of nuclear matter in quark matter. Finally, pure quark matter is formed at very high densities unless the star already has exceeded its maximum mass. A necessary condition for forming these structures and the mixed phase is that the additional surface and Coulomb energies of these structures are su$ciently small. Excluding them makes the mixed phase energetically favored [6]. That is also the case when surface energies are small (see [11] for a quantitative condition). If they are too large the neutron star will have a core of pure quark matter with a mantle of nuclear matter surrounding and the two phases are coexisting by an ordinary "rst-order phase transition.  The condition for "ssion instability is contrarily: 2E 4E . 1 !

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Fig. 31. Nuclear and quark matter structures in a &1.4M neutron star. Typical sizes of structures are &10\ m but > have been scaled up to be seen.

The quark and nuclear matter mixed phase has continuous pressures and densities [6] when surface and Coulomb energies are excluded. There are at most two second-order phase transitions. Namely, at a lower density, where quark matter "rst appears in nuclear matter, and at a very high density, where all nucleons are "nally dissolved into quark matter, if the star is gravitationally stable at such high central densities. However, due to the "nite Coulomb and surface energies associated with forming these structures, the transitions change from second to "rst order at each topological change in structure [11]. If the surface and Coulomb energies are very small the transitions will be only weakly "rst order but there may be several of them.

4.6. Summary of neutron star structures To summarize, we have shown that whether or not a droplet phase consisting of quark matter and nuclear matter can exist in neutron stars depends not only on bulk properties, but also on the surface tension. In order to make better estimates, it is important to improve our understanding of the transition between bulk nuclear matter and bulk quark matter. For the droplet phase to be possible, this must be "rst-order. If the transition is indeed "rst-order, better estimates of the surface tension are needed to determine whether the droplet phase is favored energetically. In these analyses several restrictions were made: the interfaces were sharp, the charge densities constant in both NM and QM and the background electron density was also assumed constant. Relaxing these restrictions generally allow the system to minimize its energy further. Constant charge densities may be a good approximation when screening lengths are much larger than spatial length scales of structures but since they are only slightly larger [11] the system may save signi"cant energy by rearranging the charges.

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5. Observational consequences for neutron stars In this section we "rst brie#y review the observational status of neutron star masses from binary pulsars and X-ray binaries. Subsequently, we revisit the equations for calculating the structure of rotating neutron stars and calculate masses, radii, moments of inertia, etc. for rotating neutron stars with the equations of state described in the previous sections with and without phase transitions. Glitches are then discussed in Sections 5.6 and 5.7 while a discussion on neutron star cooling is given in Section 5.8. In the last two subsections we discuss supernovae and gamma-raybursters. 5.1. Masses from radio pulsars, X-ray binaries and QPO's The measurements of masses and radii of neutron stars (as well as detailed study of their cooling histories and rotational instabilities) may provide a unique window on the behavior of matter at densities well above that found in atomic nuclei. The most precisely measured physical parameter of any rotating neutron star, or pulsar, is its spin frequency. The frequencies of the fastest observed pulsars (PSRB1937#21 at 641.9 Hz and B1957#20 at 622.1 Hz) have already been used to set constraints on the nuclear equation of state at high densities under the assumption that these pulsars are near their maximum (breakup) spin frequency. However, the fastest observed spin frequencies may be limited by complex accretion physics rather than fundamental nuclear and gravitational physics. A quantity more directly useful for comparison with physical theories is the neutron star mass. In Fig. 32 we show the latest compilation of Thorsett and Chakrabarty [140] of

Fig. 32. Neutron star masses from observations of radio pulsar systems. Error bars indicate central 68% con"dence limits, except upper limits are one-sided 95% con"dence limits. The vertical lines are drawn at a mass 1.35$0.04M . > Taken from Ref. [140].

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neutron star masses in binary radio pulsar systems. As can be seen from this "gure, most of observed binary pulsars exhibit masses around 1.4M . One exception may be PSR J1012#5307 > with mass M"(2.1$0.8)M [141]. However, its mass is less well determined as the system is > non-relativistic. The masses are determined from pulse delays of the millisecond pulsar as well as radial-velocity curves and spectral lines of the white dwarf companion. The recent discovery of high-frequency brightness oscillations from 11 neutron stars in low-mass X-ray binaries may provide us with a new promising method for determining masses and radii of neutron stars. These quasi-periodic oscillations (QPO) are observed in both the persistent X-ray emission and in bursts. According to the most successful model of Miller et al. [142] the QPO's are most likely the orbital frequencies of accreting gas in Keplerian orbits around neutron stars. The orbital frequency of the gas at distance r from the neutron star is l "(1/2p)(M/r) . (89) /.As the QPO last many periods, the gas has to be in a stable orbit. The innermost stable orbit R for a slowly rotating neutron star is related to its mass as

 M"(c/6G)R . (90)

 Thus we obtain the limits on the non-rotating neutron star masses and radii: M42.2M kHz/l , (91) > /.R420 km kHz/l . (92) /.For example, the 1220 Hz QPO observed in the atoll source 4U 1636-536 limits the neutron star mass to M41.8M and its radius to R416 km. It was predicted by Miller et al. [142] that as the > accretion #ux increases towards the innermost stable orbit the QPO frequency should stop increasing. This was subsequently observed by Zhang et al. [4] and Kaaret et al. [143] for 4U 1820-30 which has l "0.8}0.9 kHz. The resulting neutron star mass is MK2.3M when /.> rotation is included. The deduced mass is consistent with the hypothesis that these neutron stars were born with MK1.4M and have been accreting matter at a fraction of the Eddington limit for > 10 yr. The QPO's provide an important tool for determining neutron star masses or at least restricting them and for limiting radii. Due to accretion they are expected to be heavier than X-ray binaries and therefore potentially more interesting for restricting maximum masses and the EoS. In Fig. 33 we show the mass}radius relations for the various equations of state to be discussed in Section 5.3 (see also [144]). The shaded area represents the allowed masses and radii for l "1060 Hz of 4U 1820-30. Generally, /.GM  , (93) 2GM(R( 4pl /.where the lower limit ensures that the star is not a black hole, and the upper limit that the accreting matter orbits outside the star, R(R . Furthermore, for the matter to be outside the innermost 
stable orbit, R'R "6GM, sets an upper limit on the mass, Eq. (91). The mass}radius relations

 for the speci"c EoS's will be discussed in the following subsections. A new determination of the mass of the accretion powered X-ray pulsar Vela X-1 [145] gives the mass (95% con"dence limit)




M"1.93>  M . \  >



(94)

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Fig. 33. Neutron star masses vs. radius for the EoS of Eq. (49) with softness d"0.13, 0.2, 0.3, 04, with increasing values of d from top to bottom for the full curves. Phase transitions decrease the maximum mass whereas rotation increases it. The shaded area represents the neutron star radii and masses allowed (see Eq. (93)) for orbital QPO frequencies 1060 Hz of 4U 1820-30.

The mass could be determined from the velocities of the binaries from pulse timing of the neutron star and doppler shifts in the spectral lines of its companion as well as the inclination from orbital #ux variations. Recently, also the mass of the low mass X-ray binary Cygnus X-2 has been estimated from U}B}V light curves [146] M"(1.8$0.4)M

>

.

(95)

The existence of such large neutron star masses will require a rather sti! EoS for nuclear matter and restricts the softening due to phase transitions severely, as will be discussed in the following subsection. 5.2. TOV and Hartle's equations The theoretical description of a neutron star is governed by conditions imposed by general relativity. General relativity has to be taken into account for the determination of the gross properties of a star with approximately one solar mass M and a radius R of approximately 10 km, > since relativistic e!ects are of the order [47] M/R&0.1}0.2 .

(96)

The starting point for such studies is how to determine Einstein's curvature tensor G for IJ a massive star (R , g , and R denote the Ricci tensor, metric tensor, and Ricci scalar, respectively). IJ IJ G ,R !g R"8p ¹ (e, P(e)) . IJ IJ  IJ IJ

(97)

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A necessary ingredient for solving this equation is the energy-momentum tensor density ¹ , for IJ which knowledge of the EoS, i.e. pressure P as function of the energy density e is necessary. For a spherically symmetric and static star, the metric has the Schwarzschild form ds"!e(P dt#eKP dr#r(dh#sin h d ) ,

(98)

where the metric functions are given by eKP"(1!c(r))\ , e(P"e\KP"(1!c(r)) for r'R ,  

(99) (100)

with



2M(r)/r, r4R ,  (101) 2M /r, r5R .   Einstein's equations for a static star reduce then to the familiar Tolman}Oppenheimer}Volko! equation (TOV) [47,147,148]: c(r)"

dP(r)/dr"!(1/r)(e(r)#P(r))(M(r)#4prP(r))eKP ,

(102)

where the gravitational mass M(r) contained in a sphere with radius r is determined via the energy-density e(r) by



M(r)"4p

P

e(r)r dr .

(103)

 The metric function (r) obeys the di!erential equation dP 1 d

"! , e(r)#P(r) dr dr

(104)

with the boundary condition

(r"R )" ln(1!c(R )) . (105)    For a given EoS, i.e. P(e), one can now solve the TOV equation by integrating them for a given central energy density e from the star's centre to the star's radius, de"ned by P(R )"0.   More complicated is the case of rotating stars, where due to the rotation changes occur in the pressure, energy density, etc. The energy-momentum density tensor ¹ takes the form IJ (gIJu u "!1) [131,149,150]: I J ¹ "¹ #*¹ , (106) IJ I IJ with ¹ "(e#P)u u #Pg , IJ I J IJ *¹ "(*e#*P)u u #*Pg . IJ I J IJ

(107) (108)

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P, e, and o are quantities in a local inertial frame comoving with the #uid at the instant of measurement. For the rotationally deformed, axially symmetric con"gurations one assumes a multipole expansion up to second order (P denotes the Legrendre polynomial):  *P"(e#P)(p #p P (cos h)) , (109)    Re , (110) *e"*P RP *o"*P

Ro . RP

(111)

For the rotating and deformed star with the rotational frequency X one has now to deal with a generalized Schwarzschild metric, given by [151,152] ds"!eJPF( dt#eRPFX(d !u(r, X) dt)#eIPF( dh#eHPF( dr#O(X). (112) Here, u(r) denotes the angular velocity of the local inertial frame, which, due to the dragging of the local system is proportional to X. The metric functions of Eq. (112) which correspond to stationary rotation and axial symmetry with respect to the axis of rotation are expanded up to second order as (independent of and t) eJPFX"e(P[1#2(h (r, X)#h (r, X)P (cos ))] ,    X eRP( "r sin h[1#2(v (r, X)!h (r, X))P (cos h)] ,    X eIPF "r[1#2(v (r, X)!h (r, X))P (cos h)] ,    2 m (r, X)G#m (r, X)P (cos h)   eHPFX"e;P 1#  . 1!c(r) r





(113) (114) (115) (116)

The angular velocity in the local inertial frame is determined by the di!erential equation






d du dj(r) rj(r) #4r u(r)"0, r(R ,  dr dr dr

(117)

where u(r) is regular for r"0 with du/dr"0. j(r) abbreviates j(r),e\(P(1!c(r) .

(118)

Outside the star u(r, X) is given by 2 u(r, X)"X! J(X), r'R .  r

(119)

The total angular momentum is de"ned by




R du J(X)"  6 dr

.

P0



(120)

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From the last two equations one obtains then an angular frequency X as a function of central angular velocity u "u (r"0) (starting value for the iteration):  2 X(u )"u(R )# J(X) .   R 

(121)

Due to the linearity of Eq. (117) for u(r) new values for u(r) emerge simply by rescaling of u . The  momentum of inertia, de"ned by I"J/X, is given by



e#P u!X J(X) 8p 0 dv r " e\( . I" 3 X (1!c(r) X 

(122)

Relativistic changes from the Newtonian value are caused by the dragging of the local systems, i.e. u/X, the redshift (e\(), and the space-curvature ((1!c(r))\). For slowly rotating stars with low masses, one can neglect the dragging ((u/X)P1) and rotational deformations, but we would like to emphasize that the described treatment is not restricted by low masses and/or slow rotations. With u(r), one can also solve the coupled mass monopole equations (l"0) for m , p , where the   latter represents the monopole pressure perturbation, and h , Refs. [131,149]. The quadrupole  distortions h and v (l"2) determine the star's shape (see Refs. [131,149]). We will here just   state the equations for the monopole functions m and p in order to obtain the corrections to the   mass due to rotation. We will not deal with quadrupole corrections in this work. The equations read




du  8p Re 1 e#P dm  "4p (e#P)p # jr # rj u ,  12 dr 3 RP 1!c dr

(123)

and









dp 1#8pP (e#P) 1 jr du  1 d rju  "! # m !4p p # . dr 3 dr 1!c r(1!c)  1!c  12 1!c dr

(124)

The boundary conditions are m P0 and p P0 when rP0. Outside the star one has   m "*M!(1/r)J(X), r'R ,   with *M being the rotational correction to the gravitational mass. This corrections is given by *M"m #(1/R)J(X) ,  

(125)

at the surface of the star. Thus, when we solve the monopole equations we know also the correction to the gravitational mass. These two equations, together with Eqs. (102)}(104) and (117) form the starting point for our numerical procedure for obtaining the total mass, radius, moment of Inertia and rotational mass. Results for various approaches to the EoS are discussed in the next subsection.

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5.3. Neutron star properties from various equations of state For a given EoS we obtain the mass and radii of the neutron star by solving the equations for a weakly rotating neutron star as given by Hartle [147] and discussed in Eqs. (102)}(104), (117), (123) and (124). Various results with and without rotational corrections are displayed in Figs. 34}42, where we show total masses, mass}radius relations and moments of inertia for various approximations to the EoS. The following possible properties pertain to the various approximations to the EoS. E For the EoS parametrization of Akmal et al. [14] with just pn degrees of freedom, the EoS with d"0.13 gives the sti!est EoS and thereby the largest neutron star mass. For d(0.13 the EoS is superluminal. See also the discussion in connection with Figs. 18 and 19. For d"0.3 or d"0.4

Fig. 34. Total mass M for various values of d. See text for further details. Fig. 35. Mass}radius relation for various values of d.

Fig. 36. Total mass M for various values of the bag parameter B for the mixed phase EoS. For comparison we include also the results from the pn-matter EoS for b-stable with d"0.2. Fig. 37. Mass}radius relation for various values of the bag parameter B for the mixed phase EoS. For comparison we include also the results from the pn-matter EoS for b-stable with d"0.2.

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Fig. 38. Total mass M for various values of the bag parameter B for the Maxwell constructed EoS. For B"100 MeV fm\, the pure pn phase ends at 0.58 fm\ and the pure quark phase starts at 0.67 fm\. For B"150 MeV fm\, the numbers are 0.92 and 1.215 fm\, while the corresponding numbers for B"200 MeV fm\ are 1.04 and 1.57 fm\. In the density regions where the two phases coexist, the pressure is constant, a fact re#ected in the constant value of the neutron star mass. Fig. 39. Radius as function of central density n for various values of the bag parameter B for the Maxwell constructed  EoS. For comparison we include also the results from the pn-matter EoS for b-stable with d"0.2. The region where R is constant re#ects the density region where the pressure is constant in the Maxwell construction. See also the "gure caption to the previous "gure.

Fig. 40. Rotational mass M and gravitational mass for the pure pn EoS with d"0.2 and equations of state based on the mixed phase construction for di!erent values of B. Fig. 41. Moment of Inertia I in units of M km as function of M for the pure pn EoS with d"0.2 and for the mixed > > phase construction with B"200 MeV fm\.

the EoS di!ers from that with d"0.2 or d"0.13 at densities below n"0.3 fm\. This explains the di!erences in masses seen at low central densities in Fig. 34. The reader should also recall that in Section 2, the best "t to the results of Ref. [14] was obtained with d"0.2. E We have selected three representative values for the Bag-model parameter B, namely, 100, 150 and 200 MeV fm\. For B"100 MeV fm\, the mixed phase starts already at 0.22 fm\ and

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Fig. 42. Moment of Inertia I in units of M km as function of central density n for the pn EoS with d"0.2 and with >  quark degrees of freedom with B"200 MeV fm\ for the mixed phase and Maxwell constructions. Note well that the Maxwell construction yields a constant I since the pressure is constant in this case in the density region from 1.04 to 1.57 fm\.

the pure quark phase starts at 1.54 fm\. For B"150 MeV fm\, the mixed phase begins at 0.51 fm\ and the pure quark matter phase begins at 1.89 fm\. Finally, for B"200 MeV fm\, the mixed phase starts at 0.72 MeV fm\ while the pure quark phase starts at 2.11 fm\. E In case of a Maxwell construction, in order to link the hadronic and the quark matter EoS, we obtain for B"100 MeV fm\ that the pure pn phase ends at 0.58 fm\ and that the pure quark phase starts at 0.67 fm\. For B"150 MeV fm\, the numbers are 0.92 and 1.215 fm\, while the corresponding numbers for B"200 MeV fm\ are 1.04 and 1.57 fm\. As can be seen from Figs. 34, 36, 38, 40 and 42, none of the equations of state from either the pure pn phase or with a mixed phase construction with quark degrees of freedom, result in stable con"gurations for densities above +10n , implying thereby, see e.g. Figs. 36 and 40, that none of  the stars have cores with a pure quark phase. The EoS with pn degrees of freedom only results in the largest mass +2.2M when the rotational correction of Eq. (125) is accounted for, see Fig. 40. > With the inclusion of the mixed phase, the total mass is reduced since the EoS is softer. Several interesting conclusions can be inferred from the results displayed in Figs. 34}42. Firstly, to obtain neutron star masses of the order M&2.2M as may now have been observed in QPO's, > we need the sti!est EoS allowed by causality (i.e. dK0.13}0.2) and to include rotation, see Figs. 34 and 40. Furthermore, a phase transition to quark matter below densities of order &5n can be  excluded, corresponding to restricting the Bag constant to B9200 MeV fm\. This can be seen in Fig. 36 where we plot star masses as function of the central density n and bag-model parameter B.  These results di!er from those of Akmal et al. and Kalogera and Baym [14,85] due to the very di!erent recipes we use to incorporate causality at high densities. In Refs. [14,85] the EoS is discontinuously sti!ened by taking v "c at densities above a certain value n which, however, is   lower than n "5n where their nuclear EoS becomes superluminal. This sti!ens the nuclear EoS   for densities n (n(n but softens it at higher densities. Their resulting maximum masses are in   the range 2.2M (M(3M . Our approach incorporates causality by reducing the sound speed > > smoothly towards the speed of light at high densities. Therefore, our maximum mass never exceeds

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Table 2 Maximum gravitational mass in M and the corresponding radius R, in units of km, for the given central density n , in >  units of fm\, for various equations of state. The maximum moment of Inertia I, in units of M km, is also listed. Note > that this occurs for another central density than listed below. All results are for b-stable matter and rotational corrections have not been included in the total mass EoS d"0.13 d"0.2 d"0.3 d"0.4

pn pn pn pn

Mixed phase with d"0.2 B"200 MeV fm\ B"150 MeV fm\ B"100 MeV fm\

Max mass

Max I

R

n 

2.07 1.94 1.72 1.58

110.1 98.0 78.7 66.9

11.0 10.8 10.3 10.0

1.1 1.2 1.4 1.6

1.84 1.77 1.71

91.5 84.7 79.9

11.6 11.5 10.9

1.1 1.2 1.3

that of the nuclear EoS of Akmal et al. [14]. In fact, one may argue that at very high densities particles become relativistic and the sound speed should be even lower, vKc/3. On the other  hand, if it turns out that the QPOs are not from the innermost stable orbits and that even accreting neutron stars have small masses, say like the binary pulsars M:1.5M , this may indicate that > heavier neutron stars are not stable. Therefore, the EoS is soft at high densities d90.4 or that a phase transition occurs at a few times nuclear matter densities. For the nuclear to quark matter transition this would require B:80 MeV fm\ for d"0.2. For such small Bag parameters there is an appreciable quark and nuclear matter mixed phase in the neutron star interior but even in these extreme cases a pure quark matter core is not obtained for stable neutron star con"gurations. Finally, we end this subsection by listing in Table 2 the maximum values for masses, radii and moments of inertia for several of the equations of state discussed in Figs. 34}42. From this table we see that the pn EoS with the lowest value of d gives also the sti!est EoS, and thereby largest mass and smallest central density. Similarly, the largest value for the bag constant results also in the sti!est EoS. In connection with the discussion of QPO's, it is worth pointing out that in Kerr space the relation between the Keplerian orbital frequency l and the mass of the star is ) 2.198M (l kHz)\(1!0.748j)\ with j"Iu/M a dimensionless measure of the angular mo> ) mentum of the star. Following Ref. [143] and inserting the 1171 Hz QPO from 4U 1636-536, a rotational frequency u/2p"272 Hz and an assumed moment of inertia of &100M km results > in a mass of 2.02M and a radius of 9.6$0.6 km. From the above table, we see that these results > are fairly close to those which we get for the pure pn EoS with d"0.2, i.e. for the d value which gave the best "t to the EoS of Akmal et al. [14]. Thus, if QPO's occur near the innermost stable orbits, then neutron star masses are MK2.2M . This constrains the nuclear EoS including >

 Recall that in all equations G"c" "1.

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causality in a smooth way to only the sti!est ones } speci"cally d:0.2 in the EoS of Eq. (49). Phase transitions in cores of neutron stars softens the EoS and strong transitions can therefore be ruled out except at very high densities n95n . On the other hand, if it turns out that the QPO are not  from the innermost stable orbits and that even accreting neutron stars have small masses, say like the binary pulsars M:1.5M , this indicates that heavier neutron stars are not stable. Therefore, > the EoS must be soft at high densities, i.e. d90.4 or that a phase transition occurs at a few times nuclear matter densities. We end this subsection with a discussion on hyperon degrees of freedom, see Section 2.4. Compared with the EoS of Eq. (49) with d"0.2, the softening of the EoS due to the presence of hyperons lowers the maximum mass of a neutron star from &2M to 1.3M . Including > > rotational corrections to the total mass leads to a maximum mass of 1.4M . The hyperon > formation mechanisms is perhaps the most robust one and is likely to occur in the interior of a neutron star, unless the hyperon self-energies are strongly repulsive due to repulsive hyperon} nucleon and hyperon}hyperon interactions, a repulsion which would contradict present data on hypernuclei [153]. The EoS with hyperons yields however neutron star masses without rotational corrections which are even below &1.4M . This means that our EoS with hyperons needs to be > sti!er, a fact which may in turn imply that more complicated many-body terms not included in our calculations, such as three-body forces between nucleons and hyperons and/or relativistic e!ects, are needed. The role of phase transitions and its possible link with observation will be discussed in the following subsections. 5.4. Maximum masses A stellar object of mass &1.4M can either be an ordinary star of type F, a white dwarf, > a neutron star, a black hole, or possibly a quark star, see Fig. 43. As shown by Fechner and Joss [10] a second branch of quark stars are possible for certain equation of states (EoS) } speci"cally

Fig. 43. Mass vs. central density for the Bethe}Johnson polytrope (C"2.54) [157] with "rst-order phase transitions at e "3.2e to e . A region of instability occurs when Eq. (129) is ful"lled.   

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for some parameter values of the Bag constant. The stars considered were hybrid stars consisting of a quark matter core with a mantle of nuclear matter around. A double maximum mass in the mass-density plot for neutron and mixed phase quark stars have also been found in [154] (see also [155,156]). The occurrence of a second maximum is a curious phenomenon that occurs under speci"c conditions that can be quanti"ed. For that purpose we "rst consider a simple model that can be solved analytically, namely an EoS consisting of two incompressible #uids with a "rst-order phase transition between energy density e and e (e (e ) coexisting at a pressure P . We shall also "rst      ignore e!ects of general relativity, i.e. take the Newtonian limit. The mass, M(r)"4pP e(r)r dr, is very simple in the Newtonian limit and the boundary  condition M(R)"M relates the star mass M and radius R to the radius of the dense core, R , as  M(R)"(4p/3)(e R#(e !e )R ) ,    

(126)

where R is the radius of the dense core. From Newton's equation for hydrostatic equilibrium  dP/dr"!e M(r)/r ,

(127)

the pressure is easily obtained. From the boundary condition P(R)"0 we obtain P "(4p/3) e (R#(2 e /e !3)R !(e /e !1)R /R) .        

(128)

When the dense core is small, R ;R, the R terms in Eqs. (126) and (128) can be ignored. From   Eq. (128) we therefore observe that when 2e '3e , the radius R of the star decreases with   increasing size of the dense core R . Correspondingly, its mass M(R) of Eq. (126) decreases. In other  words, as the average density of the star increases, its mass decreases and a stability analysis reveals that the star is unstable. It will contract until R is comparable to R such that the R terms in   Eqs. (126) and (128) stabilizes the star and its mass again increases with increasing size. We can therefore conclude that when e 5e Ninstability region ,  

(129)

a second maximum mass appears. Another case, that can be solved analytically, is the C"2 polytropic EoS, PJe. In this case the Newtonian version of the TOV equation is equivalent to the SchroK dinger equation in a square well (or the Klein}Gordon equation) with solution e(r)Jsin(pr/R)/r. Including a "rst-order phase transition leads to a phase shift in the corresponding sine solution for the outer mantle at densities e4e . Curiously, one "nds exactly the same instability criteria as Eq. (129). However, varying  the polytropic index around C"2 does change this condition slightly. Also, including general relativity a!ects the instability condition of Eq. (129) when e is close to the maximum central  density where the star becomes unstable with respect to a collapse to a black hole, see Fig. 43. A second-order phase transition as, e.g. the mixed nuclear and quark matter phase, can also lead to instabilities and second maximum masses when the EoS is su$ciently softened [154].

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5.5. Phase transitions in rotating neutron stars During the last years, and as discussed in the preceding sections, interesting phase transitions in nuclear matter to quark matter, mixed phases of quark and nuclear matter [6,11], kaon [12] or pion condensates [14], neutron and proton super#uidity [9], hyperonic matter [15,17,18] crystalline nuclear matter, magnetized matter, etc., have been considered. Recently, Glendenning et al. [7] have considered rapidly rotating neutron stars and what happens as they slow down when the decreasing centrifugal force leads to increasing core pressures. They "nd that a drastic softening of the equation of state, e.g. by a phase transition to quark matter, can lead to a sudden contraction of the neutron star at a critical angular velocity and shows up in a backbending moment of inertia as function of frequency. Here we consider another interesting phenomenon namely how the star and in particular its moment of inertia behaves near the critical angular velocity where the core pressure just exceeds that needed to make a phase transition. We calculate the moment of inertia, angular velocities, braking index, etc. near the critical angular velocity and discuss observational consequences for "rst- and second-order phase transitions. Here we will make the standard approximation of slowly rotating stars, i.e. the rotational angular velocity is X;M/R. For neutron stars with mass M"1.4M and radius R&10 km > their period should thus be larger than a few milliseconds, a fact which applies to all measured pulsars insofar. The general relativistic equations for slowly rotating stars were presented by Hartle [147] and reviewed in Section 5.2. Hartle's equations are quite elaborate to solve as they consist of six coupled di!erential equations as compared to the single Tolman}Oppenheimer}Volko! equation in Eq. (102) in the non-rotating case. In order to be able to analytically extract the qualitative behavior near the critical angular velocity X , where a phase transition occurs in the center, we will  "rst solve the Newtonian equations for a simple equation of state. This will allow us to make general predictions on properties of rotating neutrons stars when phase transitions occur in the interior of a star. The corrections from general relativity are typically of order M/RK10}20% for neutron stars of mass MK1.4M . The extracted analytical properties of a rotating star are then > checked below by actually solving Hartle's equations numerically for a realistic equation of state. The simple Newtonian equation of motion expresses the balance between the pressure gradient and the gravitational and centrifugal forces

P"!e( <#X;X;r) .

(130)

Here, <(r) is the gravitational potential for the deformed star and e the energy (&mass) density. We assume that friction in the (non-super#uid) matter insures that the star is uniformly rotating. Since cold neutron stars are barotropes, i.e. the pressure is a function of density, the pressure, density and e!ective gravitational potential, U"
(131)

where P (cos h) is the 2nd Legendre polynomial and e(a) is the deformation of the star from  spherical symmetry.

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Inserting Eq. (131) in Eq. (130) one obtains for small deformations [147] the l"0 Newtonian hydrostatic equation (1/e)dP/da"!M(a)/a#Xa , (132)  where M(a)"4p ? e(a)ada is the mass contained inside the mean radius a. The factor  in the   centrifugal force arises because it only acts in two of the three directions. The equation (l"2) for the deformation e(a) is [158]






5 1 de 0 de(a) M(a)!4p o(a) da" X . 3 a da da ?

(133)

The deformation generally increases with decreasing density, i.e. the star is more deformed in its outer layers. In order to discuss the qualitative behavior near critical angular velocities we "rst consider a simple EoS with phase transitions for which Eq. (132) can be solved analytically namely that of two incompressible #uids with a "rst-order phase transition between energy density e and  e (e (e ) coexisting at a pressure P . The mass M(a) is very simple in the Newtonian limit and     the boundary condition M(R)"M relates the star radius R to the radius of the dense core, R , as  R"(RM !(e /e !1)R ) ,   

(134)

where RM "(3M/4pe ) is the star radius in the absence of a dense core. Solving Eq. (132) gives the  pressure P(a)"P #(R !a)e ((4p/3)e !X) ,      

(135)

for 04a4R and  P(a)"P #(R !a)e ((4p/3)e !X)!(4p/3)R (e !e )e (1!(R /a)) ,           

(136)

for R 4a4R [8]. The boundary condition at the surface P(R)"0 in Eq. (136) gives 







X 3 P  # u, "1!2 2pe 4p e R  





e R  !1  (1!R /R) (1!R /R)\ .   e R 

(137)

The phase transition occurs right at the center when R "0 corresponding to the critical angular  velocity X "u (2pe , where [8]    u "1!2 P RM /M . (138)   Generally, for any EoS the critical angular velocity depends on P , M, and e but not on e .    For angular velocities just below u very little of the high-density phase exists and R ;R.   Expanding (137) we obtain



u !u R  K . 3!2e /e !u RM   

(139)

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For u5u the dense phase disappears and R "0. Generally, one can interpret R as an order    parameter in analogy to, e.g. magnetization, the BCS gap, or the Higgs "eld in the standard model, however, as function of angular velocity instead of temperature. Note, that for large density di!erences, e /e 5(3!u )/2, Eq. (139) is not valid. This is related to an instability (see Eq. (129))    and will be discussed in the following subsection. The corresponding moment of inertia is for R ;R  I"(4p/5)(e R #e (R!R ))(1#e)      KMRM (1!((e /e )!1)R /RM )(1#u) ,      

(140)

where we used that the deformation from Eq. (133) is e" u in the low density phase [158].  However, for the qualitative behavior near X only the contraction of the star radius R with  the appearance of the dense core R is important whereas the deformations can be ignored.  The contraction is responsible for the term in the moment of inertia and is proportional to R J(u !u) near the critical angular velocity. Consequently, the derivative dI/du displays   the same non-analytic square root dependence as R (see Eq. (139)).  Latent heat is generated in the phase transition can be ignored because of rapid neutrino cooling which will be even faster than in supernova explosions. Thus temperatures will drop below &1 MeV in seconds. Such temperatures are negligible compared to typical Fermi energies of nucleons or quarks and the timescales are also much smaller than t .  Let us subsequently consider a more realistic EoS for dense nuclear matter at high densities such as the Bethe}Johnson EoS [157]. At high densities it can be approximated by a polytropic relation between the pressure and energy density: P"K e , where K "0.021e\  and    e "m 0.15 fm\ is normal nuclear matter mass density. As we are only interested in the dense  L core we will for simplicity employ this Bethe}Johnson polytrope (BJP) EoS. The central density of a non-rotating 1.4M mass neutron star with the BJP EoS is &3.4e . Furthermore, we assume >  that a "rst-order phase transition occurs at density e "3.2e to a high-density phase of density   e "4e with a similar polytropic EoS P"K e . From the Maxwell construction the pressure    is the same at the interface, P , which determines K "K (e /e ) . We now generalize Eq. (132)      by including e!ects of general relativity. From Einstein's "eld equations for the metric we obtain from the l"0 part M(a)#4paP 2 1 dP "! # Xa , a(1!2M(a)/a) 3 e#P da

(141)

where m(a)"4p? e(a)a da. In the centrifugal force term we have ignored frame dragging and  other corrections of order XM/R&0.1X for simplicity and since they have only minor e!ects in our case. By expanding the pressure, mass and gravitational potential in the di!erence between the rotating and non-rotating case, Eq. (141) reduces to the l"0 part of Hartle's equations (cf. Eq. (100) in [147]). Note also that Hartle's full equations cannot be used in our case because the "rst-order phase transition causes discontinuities in densities so that changes are not small locally. This shows up, for example, in the divergent thermodynamic derivate de/dP. The rotating version of the Tolman}Oppenheimer}Volko! Equation (141) is now solved for a rotating neutron star of mass M"1.4M with the BJP EoS including a "rst-order phase >

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Fig. 44. Central density (in units of e ), radii of the neutron star R and its dense core R , moment of inertia, its derivative   I/I"dI/du/I and the braking index are shown as function of the scaled angular velocity u"X/(2pe ). The rotating  neutron star has mass 1.4M and a Bethe}Johnson-like polytropic equation of state with a "rst-order phase transition > taking place at density e "3.2e to e "4e .    

transition. In Fig. 44 we show the central density, moment of inertia, braking index, star radius and radius of the interface (R ) as function of the scaled angular velocity. It is important to note that  R J(X !X for angular velocities just below the critical value X . The qualitative behavior of    the neutron star with the BJP EoS and a "rst-order phase transition is the same as for our simple analytic example of two incompressible #uids examined above. Generally, it is the "nite density di!erence between the phases that is important and leads to a term in the moment of inertia proportional to (X !X) as in Eq. (140).  The moment of inertia increases with angular velocity. Generally, for a "rst-order phase transition we "nd for X:X (see also Eq. (140) and Fig. 44)  I"I (1#c X/X !c (1!(X/X ))#2) .       

(142)

For the two incompressible #uids with momentum of inertia given by Eq. (140), the small expansion parameters are c "u and c "(5/2)u (e /e !1)/(3!2e /e !u ); for X'X           the c term is absent. For the BJP we "nd from Fig. 1 that c K0.07K2.2u . Generally, we "nd    that the coe$cient c is proportional to the density di!erence between the two coexisting phases  and to the critical angular velocity to the third power, c &(e /e !1)u . The scaled critical     angular velocity u can at most reach unity for submillisecond pulsars.  To make contact with observation we consider the temporal behavior of angular velocities of pulsars. The pulsars slow down at a rate given by the loss of rotational energy which we shall assume is proportional to the rotational angular velocity to some power (for dipole radiation n"3) (d/dt)(IX)"!CXL> . 

(143)

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With the moment of inertia given by Eq. (142) the angular velocity will then decrease with time as





 


CXL\ XQ X "! 1!c !c   I X X   1 K! 1!c  (n!1)t

X 1! X 

 

t L\ 1!  #2 , t

(144)

for t*t . Here, the time after formation of the pulsar is, using Eq. (143), related to the angular  velocity as tKt (X /X)L\ and t "I /((n!1)CXL\) for n'1, is the critical time where a phase      transition occurs in the center. For earlier times t4t there is no dense core and Eq. (144) applies  when setting c "0. The critical angular velocity is X "u (2pe K6 kHz for the BJP EoS, i.e.     comparable to a millisecond binary pulsar. Applying these numbers to, for example, the Crab pulsar we "nd that it would have been spinning with critical angular velocity approximately a decade after the Crab supernova explosion, i.e. t &10 years for the Crab. Generally, t JX\L    and the timescale for the transients in XQ as given by Eq. (144) may be months or centuries. In any case, it would not require continuous monitoring which would help a dedicated observational program. The braking index depends on the second derivative I"dI/dX of the moment of inertia and thus diverges (see Fig. 44) as X approaches X from below  X X/X X$ X  #c . n(X), Kn!2c  X  (1!X/X XQ   

(145)

For X5X the term with c is absent. The observational braking index n(X) should be distin  guished from the theoretical exponent n appearing in Eq. (143). Although the results in Eqs. (144) and (145) were derived for the pulsar slow down assumed in Eq. (143) both XQ and n(X) will generally display the (t!t behavior for t5t as long as the rotational energy loss is a smooth function of   X. The singular behavior will, however, be smeared on the pulsar glitch `healinga time which in the case of the Crab pulsar is of order weeks only. We now discuss possible phase transitions in interiors of neutron stars. The quark and nuclear matter mixed phase described in [6] has continuous pressures and densities. There are no "rst-order phase transitions but at most two second-order phase transitions. Namely, at a lower density, where quark matter "rst appears in nuclear matter, and at a very high density (if gravitationally stable), where all nucleons are "nally dissolved into quark matter. In second-order phase transitions the pressure is a continuous function of density and we "nd a continuous braking index. This mixed phase does, however, not include local surface and Coulomb energies of the quark and nuclear matter structures. As shown in [11,139] there can be an appreciable surface and Coulomb energy associated with forming these structures and if the interface tension between quark and nuclear matter is too large, the mixed phase is not favored energetically. The neutron star will then have a core of pure quark matter with a mantle of nuclear matter surrounding it and the two phases are coexisting by a "rst order phase transition. For a small or moderate interface

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tension the quarks are con"ned in droplet, rod- and plate-like structures [11,139] as found in the inner crust of neutron stars [34]. Due to the "nite Coulomb and surface energies associated with forming these structures, the transitions change from second order to "rst order at each topological change in structure. If a Kaon condensate appears it may also have such structures [95]. Pion condensates [161], crystalline nuclear matter [13,14], hyperonic or magnetized matter, etc. may provide other "rst-order phase transitions. If a neutron star cools continuously, the temperature will decrease with time and the phase transition boundary will move inwards. The two phases could, e.g. be quark-gluon/nuclear matter or a melted/solid phase. In the latter case the size of the hot (melted) matter in the core is slowly reduced as the temperature drops freezing the #uid into the solid mantle. Melting temperatures have been estimated in [34,137] for the crust and in [11] for the quark matter mixed phase. When the very core freezes we have a similar situation as when the star slows down to the critical angular velocity, i.e. a "rst-order phase transition occurs right at the center. Consequently, a similar behavior of moment of inertia, angular velocities, braking index may occur as in Eqs. (142), (144) and (145) replacing X(t) with ¹(t). Thus, if a "rst-order phase transitions is present at central densities of neutron stars, it will show up in moments of inertia and consequently also in angular velocities in a characteristic way. For example, the slow down of the angular velocity has a characteristic behavior XQ &c (1!t/t and   the braking index diverges as n(X)&c /(1!X/X (see Eqs. (144) and (145)). The magnitude of   the signal generally depends on the density di!erence between the two phases and the critical angular velocity u "X /(2pe such that c &(e /e !1)u . The observational consequences        depend very much on the critical angular velocity X , which depends on the equation of state  employed, at which density the phase transition occurs and the mass of the neutron star. We encourage a dedicated search for the characteristic transients discussed above. As the pulsar slows down over a million years, its central densities spans a wide range of order 1n (see Fig. 44).  As we are interested in time scales of years, we must instead study all &1000 pulsars available. By studying the corresponding range of angular velocities for the sample of di!erent star masses, the chance for encountering a critical angular velocity increases. Eventually, one may be able to cover the full range of central densities and "nd all "rst-order phase transitions up to a certain size determined by the experimental resolution. Since the size of the signal scales with X the transition  may be best observed in rapidly rotating pulsars such as binary pulsars or pulsars recently formed in supernova explosion and which are rapidly slowing down. Carefully monitoring such pulsars may reveal the characteristic behavior of the angular velocity or braking index as described above which is a signal of a "rst-order phase transition in dense matter. 5.6. Core quakes and glitches The glitches observed in the Crab, Vela, and a few other pulsars are probably due to quakes occurring in solid structures such as the crust, super#uid vortices or possibly the quark matter lattice in the core [139]. As the rotating neutron star gradually slows down and becomes less deformed, the rigid component is strained and eventually cracks/quakes and changes its structure towards being more spherical. The moment of inertia of the rigid component, I , decreases abruptly and its rotation and pulsar  frequency increases due to angular momentum conservation resulting in a glitch. The observed

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glitches are very small *X/X&10\. The two components slowly relaxate to a common rotational frequency on a timescale of days (healing time) due to super#uidity of the other component (the neutron liquid). The healing parameter Q"I /I measured in glitches reveals that for the Vela   and Crab pulsar about &3% and &96% of the moment of inertia is in the rigid component respectively. If the crust were the only rigid component the Vela neutron star should be almost all crust. This would require that the Vela is a very light neutron star } much smaller than the observed ones which all are compatible with &1.4M . If we by the lattice component include not only the solid > crust but also the protons in nuclear matter (NM) (which is locked to the crust due to magnetic "elds), super#uid vortices pinned to the crust [162] and the solid QM mixed phase I "I #I #I #I ,   N QT /+

(146)

we can better explain the large I for the Crab. The moment of inertia of the mixed phase is  sensitive to the EoS's used. For example, for a quadratic NM EoS [11] decreasing the Bag constant from 110 to 95 MeV fm\ increases I /I from &20% to &70% for a 1.4M neutron star } not   > including possible vortex pinning. The structures in the mixed phase would exhibit anisotropic elastic properties, being rigid to some shear strains but not others in much the same way as liquid crystals. Therefore, the whole mixed phase might not be rigid. The energy released in glitches every few years are too large to be stored in the crust only. The recurrence time for large quakes, t , is inversely proportional to the strain energy [162], which A again is proportional to the lattice density and the Coulomb energy t\J(1/a)Ze/a . 

(147)

Since the lattice distance a is smaller for the quark matter droplets and their charge larger than for atoms in the crust, the recurrence time is shorter in better agreement with measurements of large glitches. Detecting core and crust quakes separately or other signs of three components in glitches, indicating the existence of a crust, super#uid neutrons and a solid core, would support the idea of the mixed quark and nuclear matter mixed phase. However, magnetic "eld attenuation is expected to be small in neutron stars and therefore magnetic "elds penetrate through the core. Thus, the crust and core lattices as well as the proton liquid should be strongly coupled and glitch simultaneously. 5.7. Backbending and giant glitches In [7] the moment of inertia is found to `backbenda as function of angular velocity. The moment of inertia of some deformed nuclei [159,160] may also backbend when the coriolis force exceeds the pairing force breaking the pairing whereby the nucleus reverts from partial super#uidity to a rigid rotor. However, in the limit of large nuclear mass number such backbending would disappear. Instead pairing may lead to super#uidity in bulk [162]. A backbending phenomenon in neutron stars, that appears to be similar to backbending in nuclei, can occur in neutron stars although the physics behind is entirely di!erent. If we soften the EoS signi"cantly at a density near the central

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density of the neutron star, a non-rotating neutron star can have most of its core at high densities where the soft EoS determines the pro"le. A rapidly rotating star may instead have lower central densities only probing the hard part of the EoS. Thus, the star may at a certain angular velocity revert from the dense phase to a more dilute one and at the same time change its structure and moment of inertia discontinuously. Such a drastic change in moment of inertia at some angular velocity will cause a giant glitch as found in [7]. The phenomenon of neutron star backbending is related to the double maximum mass for a neutron and quark star respectively as shown in Fig. 44. The instabilities given in Eq. (129) are also evident when rotation is included as seen from Eq. (139) e 5e (3!u )NGiant glitch when uKu .    

(148)

The Bethe}Johnson EoS discussed in Section 5.4 also has this discontinuity in the moment of inertia when o /o 93/2. The neutron star may continue to slow down in its unstable structure, i.e.   `super-rotatea, before reverting to its stable con"guration with a dense core. As for the instabilities of Eq. (129) this condition changes slightly for a more general EoS and when general relativity is included. Neutron stars with a mixed phase do not have a "rst-order phase transition but may soften their EoS su$ciently that a similar phenomenon occurs [7]. We emphasize, however, that the discontinuous jump in moment of inertia is due to the drastic and sudden softening of the EoS near the central density of neutron stars. It is not important whether it is a phase transition or another phenomenon that causes the softening. 5.8. Cooling and temperature measurements The thermal evolution of a neutron star may provide information about the interiors of the star, and in recent years much e!ort has been devoted to measuring neutron star temperatures, especially with the Einstein Observatory and ROSAT, see e.g. Ref. [163]. Neutron stars are born with interior temperatures of the order 20}50 MeV, but cool rapidly via neutrino emission to temperatures of the order 1 MeV within minutes. The only information on neutron star temperatures stems from surface temperatures, which typically are of the order 10 K for about 10 yr, observed in X-ray or UV bands. However, the thermal radiation from a neutron star has yet to be identi"ed unambiguously. Most observations are for pulsars, and it is unclear how much of the observed radiation is due to pulsar phenomena, to a synchotron-emitting nebula or to the neutron star itself. Surface temperatures of neutron stars have been measured in a few cases or upper limits have been set. Table 3 collects some of these. The cooling history of the star, and energy loss mechanisms from the interior are thus to be determined through various theoretical models. The generally accepted picture is that the longterm cooling of a neutron star consists of two periods: a neutrino cooling epoch which can last until 10 yr and a photon cooling period. If we now assume that the main cooling mechanism in the early life of a neutron star is believed to go through neutrino emissions in the core, the most powerful energy losses are expected to be given by the so-called direct Urca mechanism nPp#l#l , p#lPn#l , J J

(149)

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Table 3 Luminosities, ¸, and spin-down ages, q, of pulsars Pulsar

Name

1706!44 1823!13 2334#61 0531#21 1509!58 0540!69 1951#32 1929#10 0950#08 J0437!47 0833!45 0656#14 0630#18 1055!52

Crab SNR MSH 15-52 SNR CTB 80

Vela Geminga

log q (yr)

log ¸ (erg/s)

4.25 4.50 4.61 3.09 3.19 3.22 5.02 6.49 7.24 8.88 4.05 5.04 5.51 5.73

32.8$0.7 33.2$0.6 33.1$0.4 33.9$0.2 33.6$0.4 36.2$0.2 33.8$0.5 28.9$0.5 29.6$1.0 30.6$0.4 32.9$0.2 32.6$0.3 31.8$0.4 33.0$0.6

[164] [165] [166] [167] [168] [169] [170] [171,163] [172] [173] [174] [175] [176] [177]

as rediscussed recently by several authors [178}180]. The label l refers to the leptons considered here, electrons and muons. However, in order to full"l the momentum conservation kL (kN #kC $ $ $ and energy conservation requirements, the process can only start at densities n several times nuclear matter saturation density n "0.16 fm\, see e.g. Fig. 18, where the proton fraction  exceeds x 90.14, see e.g. [47,178}180]. N Thus, for long time the dominant processes for neutrino emission have been the so-called modi"ed Urca processes "rst discussed by Chiu and Salpeter [181], in which the two reactions n#nPp#n#l#l , J

p#n#lPn#n#l , J

(150)

occur in equal numbers. These reactions are just the usual processes of neutron b-decay and electron and muon capture on protons of Eq. (149), with the addition of an extra bystander neutron. They produce neutrino-antineutrino pairs, but leave the composition of matter constant on average. Eq. (150) is referred to as the neutron branch of the modi"ed Urca process. Another branch is the proton branch n#pPp#p#l#l , J

p#p#lPn#p#l , J

(151)

pointed out by Itoh and Tsuneto [182] and recently reanalyzed by Yakovlev and Leven"sh [183]. The latter authors showed that this process is as e$cient as Eq. (150). In addition one also has the possibility of neutrino-pair bremsstrahlung. These processes form the basis for what is normally called the standard cooling scenario, i.e. no direct Urca processes are allowed. A fast cooling scenario would involve the direct Urca process, similar direct processes with baryons more massive than the nucleon participating, such as isobars or hyperons [179,180], or neutrino emission from more exotic states like pion and kaon condensates [91,36,96] or quark

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matter [30,130]. Actually, Prakash et al. [179] showed that the hyperon direct Urca processes gave a considerable contribution to the emissivity, without invoking exotic states or the large proton fractions needed in Eq. (149). If we now consider the neutron stars discussed in Section 5.3, we note that a typical star with mass &1.4}2.0M has central densities ranging from &0.5 to 1.2 fm\. Depending on the value > of the bag constant, the mixed phase could start already at &0.2}0.3 fm\. No pure quark phase was found with the bag-model for stable neutron star con"gurations. If one also recalls that our b-stable EoS allows for the direct Urca process at densities starting from 0.8 fm\, see Fig. 18, one clearly sees that, depending on the EoS and the adopted model for dense matter, there is a considerable model dependence. To give an example, the cooling could be strongly in#uenced by the mixed phase. This could come about because nuclear matter in the droplet phase has a higher proton concentration than bulk neutral nuclear matter and this could make it easier to attain the threshold condition for the nucleon direct Urca process. Another possibility is that the presence of the spatial structure of the droplet phase might allow processes to occur which would be forbidden in a translationally invariant system. Also, the mere presence of quark matter can lead to fast cooling [130] when a O0. All these mechanisms would lead to faster cooling. Q However, in order to compare with observation, the structure of the star has to be computed in detail. In particular the possible presence of super#uidity in the interior has to be considered. The super#uid would suppress the l, l emissivity and would allow for reheating through friction with the crust. In the analysis of Page as well [184], it is hard to discriminate between fast and slow cooling scenarios, though in both cases agreement with the observed temperature of Geminga is obtained if baryon pairing is present in most, if not all of the core of the star. The recent analyses of Schaab et al. [185,186] also seem to con"rm the importance of super#uidity in the interior of stars. There are also indications [187,188] that temperatures of young (&10 years old) neutron stars lie below that obtained through the so-called modi"ed Urca processes. One has also to note [184] that the modi"ed Urca processes are weakly dependent on the mass of the star, i.e. on the central density, while faster cooling mechanisms like the above direct Urca processes are in general strongly dependent on it. Thus, the detection of two coeval stars, whose temperatures di!er by a factor of the order of 2 or larger would allow to distinguish between traditional cooling scenarios, like those discussed by Page [184], and more exotic ones. Such a huge variation in the temperature of coeval stars could indicate the presence of a threshold in the cooling mechanism, triggered by the density of the star. In Fig. 45 we display results from various cooling calculations by Schaab et al. [186]. All calculations employ a super#uidity scenario which is described in Section 2.7. This corresponds to label 4 in the above "gure, while the letters A, B, C, D represent di!erent equations of state employed to calculate the mass of the star. Label A corresponds to the non-relativistic equations of state from Wiringa et al. [90], and is similar to that of Akmal et al. [14] in Section 2, and non-relativistic equations of state based on two-body interactions only. The models encompassed by class A allow for the standard cooling scenario only at higher densities. These densities however, are beyond the central density of a 1.4M neutron star. Models B and C allow for faster cooling > scenarios and the equations of state are based on relativistic mean "eld models. Typically, see also the discussion of relativistic e!ects and the proton fractions in Fig. 16, these models allow for the direct Urca for nucleons at lower densities than the non-relativistic models. Pion and kaon condensantion could also lead to faster cooling scenarios. Model D allows for direct hyperon

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Fig. 45. Thermal evolution of a 1.4M neutron star with super#uidity. See text for further details. Taken from Ref. [186]. >

cooling as well but does not include hyperon pairing, which reduces strongly the direct Urca for hyperons, see Ref. [114]. The problem with most cooling calculations is that there is no consistent calculation of properties entering the neutrino emissivities within the framework of say one given EoS and many-body approach. Typically, see e.g. Refs. [185,186] and Fig. 45, the EoS is taken from one source, while the pairing gap is taken from another calculation, with even entirely di!erent NN interactions or many-body approaches. In addition, the expressions for the emissivities of e.g. the modi"ed Urca processes calculated by Friman and Maxwell [189] treat in a rather cavalier way the role of many-body correlations. Considering also the fact that other severe approximations are made, these expressions, which enter typically various cooling codes, could introduce errors at the level of orders of magnitude. Thus, our message is that, before one attempts at a cooling calculation, little can be learned unless the various neutrino emissivities are reevaluated within the framework of a given manybody scheme for dense matter. We conclude this subsection with a demonstration of the role of super#uidity for the processes of Eqs. (150)}(151) at densities corresponding to the outer core of massive neutron stars or the core of not too massive neutron stars when we have a super#uid phase. Here we limit ourselves to study the role of superconducting protons in the core of the star employing the gap for protons in the

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S and e!ective masses from lowest-order Brueckner}Hartree}Fock calculations discussed in  Sections 2, 2.2 and 2.7. The proton superconductivity reduces the energy losses considerably in the above reactions [183], and may have important consequences for the cooling of young neutron stars. The expressions for the processes in Eqs. (150) and (151) were derived by Friman and Maxwell [189] and read [183]






mH  mH N Q +8.5;10 L L m m L N

n  C ¹ a b ,  L L n 

(152)

in units of ergs cm\ s\ where ¹ is the temperature in units of 10 K, and according to Friman  and Maxwell a describes the momentum transfer dependence of the squared matrix element in the L Born approximation for the production rate in the neutron branch. Similarly, b includes the L non-Born corrections and corrections due to the nucleon}nucleon interaction not described by one-pion exchange. Friman and Maxwell [189] used a +1.13 at nuclear matter saturation density L and b "0.68. In the results presented below, we will not include a and b . For the reaction of L L L Eq. (150) with muons, one has to replace n with n and add a factor (1#X) with X"kI /kC C I $ $ [190,191]. For the proton branch of Eq. (151) with electrons we have the approximate equation [183] Q +8.5;10(mH/m )(mH/m )(n /n )¹ a b F , N N N L L C   N N

(153)

with F"(1!(kC /4kN ))H, where H"1 if kL (3kN #kC and zero elsewhere. Yakovlev and $ $ $ $ $ Leven"sh [183] put a "a and b "b . We will, due to the uncertainty in the determination of L N N L these coe$cients, omit them in our calculations of the reaction rates. With muons, the same changes as in Eq. (152) are made. The reaction rates for the Urca processes are reduced due to the superconducting protons. Here we adopt the results from Yakovlev and Leven"sh [183], their Eqs. (31) and (32) for the neutron branch of Eq. (152) and Eqs. (35) and (37) for the proton branch of Eq. (153). We single out proton singlet-superconductivity only, employing the approximation k ¹ "*(0)/1.76, where ¹ is ! ! the critical temperature and *(0) is the pairing gap at zero temperature discussed in Section 2. The critical temperature is then used to obtain the temperature dependence of the corrections to the neutrino reaction rates due to superconducting protons. To achieve that we employ Eq. (23) of [183]. We present results for the neutrino energy rates at a density n"0.3 fm\. The critical temperature is ¹ "3.993;10 K, whereas without muons we have ¹ "4.375;10. The im! ! plications for the "nal neutrino rates are shown in Fig. 46, where we show the results for the full case with both muons and electrons for the processes of Eqs. (150) and (151). In addition, we also display the results when there is no reduction due to superconducting protons. At the density considered, n"0.3 fm\, we see that the processes of Eqs. (150) and (151) with muons are comparable in size to those of Eqs. (150) and (151) with electrons. However, the proton pairing gap is still sizable at densities up to 0.4 fm\, and yields a signi"cant suppression of the modi"ed Urca processes discussed here, as seen in Fig. 46. We have omitted any discussion on neutron pairing in the P state. For this channel we "nd the pairing gap to be rather small, see again the discussion  in Section 2.7, less than 0.1 MeV and close to that obtained in Ref. [106]. We expect therefore that the major reductions of the neutrino rates in the core come from superconducting protons in the

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Fig. 46. Temperature dependence of neutrino energy loss rates in a neutron star core at a total baryonic density of 0.3 fm\ with muons and electrons. Solid line represents Eq. (2) with electrons, dotted line is Eq. (2) with muons, dashed line is Eq. (3) with electrons while the dash-dotted line is the processes of Eq. (3) with muons. The corresponding results with no pairing are also shown.

S state. The contribution from neutrino-pair bremsstrahlung in nucleon}nucleon collisions in  the core is also, for most temperature ranges relevant for neutron stars, smaller than the contribution from modi"ed Urca processes [183]. Possible candidates are then direct Urca processes due to hyperons and isobars, as suggested in Refs. [114,179], or neutrino production through exotic states of matter, like kaon or pion condensation [36,91,96] or quark matter [30,130]. In conclusion, super#uidity reduces the neutrino emissivities considerably for the standard cooling model. However, before a "rm conclusion from cooling calculations can be obtained, one needs much more reliable estimates of various emissivity processes in dense matter. In addition, such processes should be calculated consistently within the same many-body approach and EoS model for the interior of a star. 5.9. Supernovae Neutron stars are born in type II or Ib supernovae at a rate of 1}3 per century in our galaxy. Once the iron core in massive stars exceed the corresponding Chandrasekhar mass, it collapses adiabatically until it is stopped by the incompressibility of nuclear matter, bounces, creates shock waves, stalls by infalling matter, and presumably explodes by neutrinos blowing the infalling matter o! after a few tens of seconds. Measurements of isotope abundances of various elements (r- and s-processes) give some insight in densities and temperatures during certain stages of the explosions. Neutrinos were also detected from SN1987A in the Large Magellan Cloud. Neutron stars are normally treated as an input to supernova calculations of the complicated transport processes of baryons, photons, leptons and neutrinoes going on during the explosion. Thus supernovae do not provide much information on details on neutron star structure presently. Recent supernova models include convection and spherical asymmetries (see, e.g. [192]).

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The softer the equation of state the denser the matter is compressed before it bounces and the deeper into the gravitational well the star has fallen. Also, a softer EoS creates a more coherent shock wave that excites the matter less. The additional gravitational energy available can be transferred to neutrino generation which is believed to power the supernova explosion. Besides softening the EoS a "rst-order phase transition would store latent heat which also could a!ect supernovae. With luck we may observe a supernova nearby in the near future which produces a rapidly rotating pulsar. Light curves and neutrino counts will test supernova and neutron star models. The rapid spin down may be exploited to test the structure and possible phase transitions in the cores of neutron stars [6}8]. 5.10. Gamma-ray bursters In the late 1960s the Vela satellite was launched carrying a gamma-ray detector in order to check the nuclear test ban treaty in space. It was a huge success because numerous gamma-ray bursts (GRB) were observed [193] and (more importantly) they did not come from Earth. Later (when declassi"ed) the Russians con"rmed the GRB. Numerous observations by the Pioneer Venus Orbiter, Compton Gamma Ray Observatory, Burst And Transient Source Experiment, and later gamma-ray detectors now "nd bursts every day. The GRB do not repeat (except for a few soft gamma-ray repeaters). Their duration varies from milliseconds to minutes. The short bursts imply a small source size :c ) 1 ms&100 km, which points towards neutron stars or black holes. The gamma-rays have energy in the range 30 keV}2 MeV. The high-energy gamma rays are above the threshold for cPe>e\ photo production which implies that the radiation probably is relativistically expanding. The thousands of GRB observed show a high isotropy. Thus GRB cannot be produced in the very anisotropic discs of our solar system or galaxy or even an extended galactic halo. The "nal kill to such (and many other) models came from the Bepposax observations on May 8th, 1997, where a burst could be pin-pointed on the sky within an arcminut, which subsequently allowed ground based observations of an afterglow in optical and radio wavelengths. Fe-II and Mg-II absorption lines were found at high redshifts of z"0.835. Assuming isotropic #ux gives an energy output of E 910 ergs within seconds, which is about the same energy of the optical display and kinetic A energy of the ejecta in a supernova. The total energy release in a supernova is &10 erg's of which &99% is carried away by neutrinos and antineutrinos. In comparison a few kilometer size asteroid impacting with Earth's orbital velocity of 30 km/s releases &10 ergs and the world's nuclear arsenal contains (only) &10 megatons&10 ergs of explosive energy. At the time of writing, two dozens of GRB with afterglow in X-ray, one dozen in optical and half a dozen in radio waves have been discovered. Merging neutron stars may be responsible for gamma-ray bursts [192]. Binary pulsars are rapidly spiralling inwards and will eventually merge and create a gigantic explosion and perhaps collapse to a black hole. From the number of binary pulsars and spiral rates, one estimates about one merger per million year per galaxy. With &10 galaxies at cosmological distances z:1 this gives about the observed rate of GRB. However, detailed calculations have problems with baryon contamination, i.e. the baryons ejected absorb photons in the relativistically expanding photosphere. Accreting black holes have also been suggested to act as beaming GRB. Recently, evidence

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for beaming or jets has been found [2] corresponding to E&10 ergs. Also, a bright supernova coinciding with GRB 980326 has been found which points toward that GRB are a special class of type Ic supernova where cores collapse to black holes. So far, the physics producing these GRB is not understood. The time scales and the enormous power output points towards neutron star or black hole objects. We hope to learn more about these objects and maybe someday they can be used to gain information on neutron stars. 6. Conclusions The aim of this work has been to give a survey of recent progresses in the construction of the equation of state for dense neutron star matter and possible implications for phase transitions inside neutron stars and the connections to neutron star observables. We will here try to recapitulate several of the arguments presented. 6.1. Many-body approaches to the equation of state In Section 2 we attempted at a review of the present status of microscopic many-body approaches to dense neutron star matter. Several features emerged: E Within non-relativistic lowest-order Brueckner theory (LOB), all the new phase-shift equivalent nucleon}nucleon potentials yield essentially similar equations of state up to densities of 3}4n  for both pure neutron matter and b-stable matter. Other properties like the symmetry energy and proton fractions do also show a similar quantitative agreement. E The inclusion of more complicated many-body terms at the two-body level does not alter this picture and even the recent summation of three-hole line diagrams of Baldo and co-workers [54,55] results in an EoS which is close to LOB when a continuous choice is used for the single-particle energies in matter. E At densities up to nuclear matter saturation density n , basically all many-body approaches  discussed here give very similar equations of state. E In symmetric nuclear matter the situation is however di!erent, since there the nuclear tensor force is much more dominant due to the presence of the S and D partial waves. This has   also consequences for the three-body interactions, arising from both e!ective and real threebody force terms. This is expected since the strong isospin ¹"0 channel is present in the three-body nnp and npp clusters. In neutron matter however, such clusters are in general small at densities up to n . Similar behaviors are also seen in recent works on the energy of pure neutron  drops [194] and larger-scale shell-model calculations of Sn isotopes including e!ective threebody interactions [195]. However, for PNM and densities greater than n di!erences do  however occur when one introduces real three-body forces and/or includes relativistic corrections. In a similar way, relativistic BHF calculations, yield also signi"cant corrections above saturation density. E Based on the microscopic calculation of Akmal et al. [14], a simple parametrization of the EoS was given where the energy per particle could be written as u!2!d #S uAx , E"E u   1#du

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with E "15.8 MeV, dK0.2, symmetry energy S "32 MeV and c"0.6. Causality is ensured   by d50.13. In the case of non-nucleonic degrees of freedom we discussed also recent progresses in the construction of hyperon*hyperon interactions [19,20] and applications to neutron star matter [15]. Recent results for b-stable matter with these hyperon-hyperon interactions were also discussed [89]. In this work however, we chose to focus on the Bag model in order to deal with non-nucleonic degrees of freedom in our discussion on phase transitions. For studies of super#uidity in neutron star matter, one needs still a careful analysis of polarization e!ects, especially for the S proton contaminant. For super#uid neutrons in the P partial   wave, one needs nucleon}nucleon interactions which are "tted up to at least 1 GeV in lab energy before "rm conclusions can be reached. These points leave clearly a large uncertainty for the role of super#uidity in cooling studies. 6.2. Phase transitions and stiwness of EoS from masses of neutron stars We have discussed a number of possible phase transitions in dense nuclear matter such as pion, kaon and hyperon condensation, super#uidity and quark matter. We have speci"cally treated the nuclear to quark matter phase transition and the possible mixed phase that can occur in more than one component systems and replace the standard Maxwell construction. The structure of the mixed phase is similar to the mixed phase in the inner crust of neutron stars of nuclei and the neutron gas. However, as the mixed quark and nuclear matter phase can occur already at a few normal nuclear matter densities, it can soften the EoS in cores of neutron stars signi"cantly and lower the maximum mass. A number of numerical calculations of rotating neutron stars with and without phase transitions were given. The calculated maximum masses were discussed with the observed ones and leave two natural options: Case I: The large masses of the neutron stars in QPO 4; 1820-30 (M"2.3M ), PSR > J1012#5307 (M"2.1$0.4M ), Vela X!1 (M"1.9$0.1M ), and Cygnus X!2 > > (M"1.8$0.4M ), turn out to be correct and are complemented by other neutron stars with masses > around &2M . > As a consequence, the EoS of dense nuclear matter is severely restricted and only the sti!est EoS allowed by causality are allowed (i.e. d&0.2). Also any signi"cant phase transition can be excluded at densities below :5n .  That the radio binary pulsars all have masses around 1.4M is then probably due to the > formation mechanism in supernovae and related to the Chandrasekhar mass M K1.4M of  > white dwarfs. Neutron stars can subsequently acquire larger masses by accretion. Case II: The heavy neutron stars proves erroneous by more detailed observations and only the &1.4M masses are found. > If even in accreting neutron stars does not produce neutron stars heavier than say 91.5M , this > indicates that heavier neutron stars simply are not stable which in turn implies a soft EoS, either d90.4 or a signi"cant phase transition.

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Either way, the result provides important information on neutrons stars, the EoS of dense hadronic matter and possible phase transitions. Surface temperatures can be estimated from spectra and from the measured #uxes and known distances, one can extract the surface area of the emitting spot. This gives unfortunately only a lower limit on the neutron star size, R. If it becomes possible to measure both mass and radii of neutron stars, one can plot an observational (M, R) curve in Fig. 33, which uniquely determines the EoS for strongly interacting matter at zero temperature. Pulsar rotation frequencies and glitches are other promising signals that could reveal phase transitions. Besides the standard glitches also giant glitches were mentioned and in particular the characteristic behavior of angular velocities when a "rst-order phase transition occurs right in the center of the star. We impatiently await future observations and determinations of neutron star masses, radii, angular velocities, surface temperatures and luminosities, magnetic "elds, etc., that will answer these questions.

Acknowledgements Needless to say, we have bene"tted immensely from umpteenth interactions with many colleagues. Especially, we would like to thank "ystein Elgar+y and Lars Engvik for providing us with several inputs to various "gures and many invaluable comments on the present work. Moreover, we are much indebted to Marcello Baldo for sending us the data for Fig. 12, to Fred Lamb for keeping us updated on QPO's, and to Vijay Pandharipande and Geo! Ravenhall for providing us with the data for the equation of state from Ref. [14] and for numerous discussions on nuclear many-body theory. In addition, we wish to thank, Gordon Baym, Fabio de Blasio, Greg Carter, Alessandro Drago, Lex Dieperink, Jens Hjorth, Andy Jackson, Gianluca Lazzari, Ruprecht Machleidt, Larry McLerran, Ben Mottelson, Herbert MuK ther, Eivind Osnes, Erlend "stgaard, Chris Pethick, Artur Polls, Angels Ramos, Hans-Joseph Schulze, Ubaldo Tambini and Isaac Vidan a for the many discussions on the nuclear many-body problem and neutron star physics.

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CONTENTS VOLUME 328 A.Ya. Ender, H. Kolinsky, V.I. Kuznetsov, H. Schamel. Collective diode dynamics: an analytical approach

1

R.D. Fedorovich, A.G. Naumovets, P.M. Tomchuk. Electron and light emission from island metal "lms and generation of hot electrons in nanoparticles

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T.P. Vrana, S.A. Dytman, T.-S.H. Lee. Baryon resonance extraction from nN data using a unitary multichannel model

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H. Heiselberg, M. Hjorth-Jensen. Phases of dense matter in neutron stars

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