Advances in Electronics and Electron Physics, Volume 29

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 29

CONTRIBUTORS TO

THISVOLUME

John P. Blewett A. Corney G. T. Konrad Rocco S. Narcisi Walter Roth J. E. Rowe Bod0 Zimmermann

Advances in

Electronics and Electron Physics EDITED BY L. MARTON Smithsonian Institution, Washington, D.C. Assistant Editor

CLAIRE MARTON EDITORIAL BOARD T. E. Allibone E. R. Piore H. B. G. Casimir M. Ponte W. G. Dow A. Rose A. 0. C. Nier L. P. Smith F. K. Willenbrock

VOLUME 29

1970

ACADEMIC PRESS

New York and London

COPYRIGHT 6 1970, BY ACADEMIC PRESS,INC, ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

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Contents CONTRIBUTORS TO VOLUME 29 F O R E W O R D . ..

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vii ix

Harmonic Generation and Multisignal Effects in Nonlinear Beam Plasma Systems G. T. KONRAD and J. E. ROWE

I. Introduction . . . . . . . . . . . . . . . . . 11. Development of the “Circuit Equation” for Plasma Wave Propagation 111. Lagrangian Formulation of the Beam-Plasma Interaction Equations . IV. Results of Solution of the Nonlinear Equations . . . . . . . V. ExperimentalStudyofBearn-PlasrnaInteractions . . . . . . Conclusions . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .

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1 12 23 36 51 72 75

The Formation of Cluster Ions in Laboratory Sources and in the Ionosphere Rocco S. NARCISIand WALTER ROTH LABORATORY SOURCES

I. Introduction . . . . 11. Experimental Techniques 111. Results . . . . .

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79 80

82

THE IONOSPHERE IV. Introduction . V. Experiment . VI. Measurements VII. Discussion . VIII. Conclusion . References . .

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95

96 99 106

112 112

The Measurement of Lifetimes of Free Atoms, Molecules, and Ions A. CORNEY I. Introduction . . . . . . . . . . . . . . . . . 11. The Direct Observation of Exponential Decays Using Analog Methods 111. Delayed Coincidence Photon Counting . . . . . . . . . V

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116 124 130

vi

C 0N TEN TS

IV. The Beam-Foil Technique . . . . . . . . . . . . V. The Phase-Shift Technique . . . . . . . . . . . . VI . The Radiation Width and Pressure Broadening of Spectral Lines . VII . Techniques Using Resonance Fluorescence . . . . . . . VIII . Miscellaneous Topics . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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159 169 185 197 218 223

Recent Advances in Particle Accelerators JOHN P . BLEWETT I . Introduction . . . . . . . . . . . . . . . . . . . 223 I1 . Electron Ring Accelerators . . . . . . . . . . . . . . . 234 111. Heavy Ion Accelerators . . . . . . . . . . . . . . . . 236 IV . Superconducting Accelerators . . . . . . . . . . . . . . 238

V. Proton Synchrotrons . . . . . . VI . Electron Accelerators . . . . . . VII . Storage Rings and Colliding Beams . . VIII . Cyclotrons . . . . . . . . . IX . Meson Factories . . . . . . . . X . R.I.P. . . . . . . . . . . . . XI . Conclusion . . . . . . . . . References . . . . . . . . .

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248 252 253 254 . 255 . 256

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Broadened Energy Distributions in Electron Beams BODOZIMMERMANN I . Introduction . . . . . . . . . . . . . . I1. Energy Spread and Internal Energy . . . . . . . I11. Estimating the Energy Spread . . . . . . . . . IV . Broadened Energy Distributions . . . . . . . . V . Different Energy Widths of the Same Distribution . . . VI . Absolute Calculation of the Energy Spread . . . . . VII . Further investigation of Beams of Constant Current Density . . . VIII . Experimental Support for the Theoretical Results Appendix . . . . . . . . . . . . . . . References . . . . . . . . . . . . . .

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AUTHORINDEX .

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SUBJECT INDEX.

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251 260 264 267 214 211 290 297

308 311 313 322

CONTRIBUTORS TO VOLUME 29 JOHNP. BLEWETT, Brookhaven National Laboratory, Upton, New York

A. CORNEY,The Clarendon Laboratory, University of Oxford, Oxford, Eng1and G. T. KONRAD,*Electron Physics Laboratory, Department of Electrical Engineering, The University of Michigan, Ann Arbor, Michigan ROCCOS. NARCISI,Air Force Cambridge Research Laboratories, Bedford, Massachusetts WALTER R O T H ,The ~ KMS Technology Center, San Diego, California J. E. ROWE,Electron Physics Laboratory, Department of Electrical Engineering, The University of Michigan, Ann Arbor, Michigan

BODO ZIMMERMANN, Krupp Research and Development Center, Essen, Germany

* Present address: Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts. t Present address: Diagnostic Instruments, Inc., San Diego, California. vii

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FOREWORD Plasma research, after a number of years of effort, remains in the forefront of activity in many laboratories. J. E. Rowe and G. T. Konrad review investigations of the interaction of a plasma column with an rf field and for this purpose the column is considered as a slow-wave circuit along which electromagnetic energy can propagate. Somewhat related to the same subject is the review written by R. S. Narcisi and W. Roth, concentrating on the role of cluster ions, both in laboratory sources and in the ionosphere. The most important are water clusters, around positive or negative ions; a few others are included in their discussion. The investigation of cluster ions is a relatively new subject and the authors review essentially the experimental evidence. Plasmas are a mixture of free electrons, ions, atoms, and molecules in different excited states. The latter decay and the various methods used for the measurement of their lifetimes are the subjects of A. Corney’s review. The information obtained is not only useful in plasma physics; spectroscopy and astrophysics benefit equally from the knowledge of decay times of excited levels. Particle accelerator technology advanced greatly since our first volume appeared with M. S. Livingston’s review on the subject. Two years ago, the subject was updated partially in E. L. Hubbard’s review of linear ion accelerators; J. P. Blewett’s review in this volume completes the updating for other accelerators. The last review, by B. Zimmermann, takes up the subject of energy distribution in electron beams, covered by P. A. Lindsay in an earlier volume. Future reviews, with the names of their authors, are as follows:

Electron Precursors Light Interaction with Plasma Information Storage in Microspace Plasma Instabilities and Turbulence Electron Polarization Energy Beams as Tools Recent Advances in Field Emission Microfabrication Using Electron Beams Recent Advances in Biological Temperature Measurements Frequency FET Noise Parameters and Approximation of the Optimum Source Admittance ix

Richard G . Fowler Heinz Raether Sterling Newberry C. Keith McLane Stephen J. Smith K. H. Steigerwald et al. Lynwood Swanson and Francis Charbonnier A. N. Broers Hardy W. Trolander M. Strutt

X

FOREWORD

The Effects of Radiation in MIS Structures Research in Solid State Electronics with Electron Microprobes Small-AngleDeflection Fields for Cathode-Ray Tubes Electromigration Failure Mechanisms in Large Scale Integrated Circuits Systems Approach to Muscle Control Recent Advances in Design of Magnetic Beta-Ray Spectrometers Scalar DiffractionTheory in Electron Optics Sputtering Interpretation of Electron Microscope Images of Defects in Crystals Galactic and Extragalactic Radio Astronomy Electromagnetic Scattering by Plasma Turbulence Problems in Satellite and Space Communications related to Physical Phenomena and Devices

Karl Zaininger D. Wittry R. G. E. Hutter and H. Dressel James R. Black George L. Schnable and Ralph S. Keen K. Tomovic Milorad Mladjenovic Jiri Komrska M. W. Thompson M. J. Whelan F. J. Kerr and Wm. C. Erickson David L. Feinstein and Victor L. Granatstein P. L. Bargellini and E. S. Rittner

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 29

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Harmonic Generation and Multisignal Effects in Nonlinear Beam-Plasma Systems G . T. KONRAD*

AND

J. E. ROWE

Electron Physics Laboratory Department of Electrical Engineering The University of Michigan Ann Arbor, Michigan

I. Introduction. . . . . . . . . . . . . . A. Historical Survey and Cr B. Theory of Beam-Plasma Interactions ................................... 11. Development of the “Circuit Equation” for Plasma Wave Propagation A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Dispersion Equation for a Plasma Column . . C. Derivation of the Equivalent Circuit Elements. . . . . . . D. Derivation of the Circuit Equation . . . . . . . . . . . . . . . . . 111. Lagrangian Formulation of the Beam-Plasma Interaction Equations.. . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Variation of the Electric Field across the Plasma Column. - . . . . . . . . . . . C. Lagrangian Formulation. ................................. D. The Circuit Equation in Lagrangian Coordinates. . . . . . . . . . . . . E. The Force Equations in Lagrangian Coordinates.. ....................... F. Solution of the Large-Signal Equations ................................. G. One-Dimensional Equations with Beam-Plasma Collisions Included . . . . . . . . IV. Results of Solution of the Nonlinear Equations . . . . . . . A. One-Dimensional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Two-Dimensional Results. . . . . . . . . . . . . . . . . V. Experimental Study of Beam-Plasma Interaction A. Description of the Experimental Apparatus.. B. rf Test Results.. ......................... VI. Conclusions ....... ............... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 12 12

23 23

30 33 35 36 36

I. INTRODUCTION A . Historical Survey and Critique It is well known that if the temperature of a gas is raised to a sufficiently high value, the thermal agitation of the gas particles may be so large that

* Present address: Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts. 1

2

G. T. KONRAD AND J. E. ROWE

electrons are stripped off by the collisions among the particles. The dynamical behavior of the gas may then become dominated by the long-range electromagnetic forces acting on the free ions and electrons. The behavior of such a fluid differs appreciably from that of an ordinary gas and it is generally referred to as a “plasma.” A plasma is thus an assembly of charged particles with additional characteristics to be described more fully in the following paragraphs. The electric and magnetic fields of such an assembly of charged particles may add together in a coherent way, provided the density of the charged particles is sufficiently great so that space-charge effects dominate. This is a very general proviso and does not necessarily depend upon the degree of ionization of the assembly of charged particles or its neutrality. As an example, it is well known that an unneutralized electron beam behaves in many respects like a plasma. Ordinarily, when one speaks of a plasma, however, gross charge neutrality is assumed to exist. The tendency toward quasineutrality is very strong because the forces on the individual particles are always in a direction to reduce the space-charge density. Thus a more limited definition of a plasma includes quasi-neutrality. Statistically speaking, a shielding cloud forms around each ionized particle in the plasma so as to cancel out the electric potential of that particle at a sufficiently large distance from it. This distance is known as the Debye length, a concept developed in the study of electrolytes by P. Debye, and is given by

AD2 = kTE,/ng2, (1) where T is the temperature in degrees Kelvin characterizing the motion of the particles, n is the particle density in particles per cubic meter, q is the charge of the particles, k is Boltzmann’s constant, and E,, is the permittivity of free space. Thus, the Debye length is a fundamental parameter in plasma physics because any collection of charged particles can be called a plasma only if its dimensions are much larger than a Debye length. The discussion above is sensible only if there are many charged particles in a Debye sphere. This implies that

where ND is the number of particles in a Debye sphere. In highly ionized gases, which are of interest in this discussion, Eq. (2) is always satisfied. Taking the spacing between the ionized particles to be L 2 n p 1 l 3 ,then the average potential energy is given by

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

3

while the mean kinetic energy is kT.From Eqs. (1)-(3) one obtains

and so for a plasma,

(KE) (PE)

-

=

1.

It appears reasonable that if an ionized gas is not to recombine, the kinetic energy should be greater than the potential energy. Plasma physics became a recognizable field of study with the work of Tonks and Langmuir ( I ) , who observed oscillations in a plasma of electrons and positive ions. These authors neglected the thermal velocities of the electrons and described the oscillation observed as displacements of groups of electrons with respect to a background of positive ions. They established the well-known relation for these characteristic oscillations or “ plasma oscillations ” as

where copis the frequency of oscillation in radians per second, n, is the electron density in electrons per cubic meter, and e and me are the charge and mass, respectively, of the electron. The modern theory of plasma oscillations and beam-plasma interactions had its real beginning around 1948-1950 with the work of Pierce (2) who discussed fluctuations in electron beams due to ions and Bohm and Gross (3-5) who discussed electron oscillations associated with the presence of electrostatic potential waves in a plasma. Bohm and Gross considered the effects of random thermal motion, collisions, boundaries of their plasma model, and the presence of electron beams. A prediction of the Bohm and Gross theory was that an electron beam sent through a homogeneous plasma should excite plasma oscillations. The experimental results of (6) described by Merrill and Webb in 1939 and also those of Looney and Brown (7), the latter appearing to be in disagreement with the Bohm and Gross theory, in fact substantiate the theory of Bohm and Gross. The apparent paradox between the Looney and Brown results and the Bohm and Gross theory was resolved by Sturrock (8), who was among the first to work out a systematic procedure for interpreting dispersion laws for plasma models such as those analyzed by Bohm and Gross. In addition to the work of Pierce (2) another early contribution applying microwave tube concepts to beam-plasma interactions was that of Smullin and Chorney (9).

4

G. T. KONRAD AND J. E. ROWE

I . Wave Propagation through Plasmas

In an elementary plasma model, the small disturbances, which always oscillate at the plasma frequency, do not propagate away from their point of origin. Since the convection current of the electrons is exactly cancelled by the displacement current, there is no magnetic field associated with the oscillatory motion ; hence the disturbances remain localized. If the plasma electrons have a finite temperature, however, these disturbances do propagate away from their point of origin with a velocity comparable to their thermal speed. If the disturbance has a fixed wavelength initially, then the energy associated with this disturbance can diffuse into other wavelengths by a process of collisionless damping, generally known as Landau damping (10). Physically, this damping process is due to the fact that there is a decreasing number of particles in higher velocity classes. Hence, there are slightly more particles traveling more slowly compared with the original disturbance than there are particles traveling faster, resulting in a decrease of the amplitude of the original disturbance. The electromagnetic waves that can propagate in an ionized medium are frequently studied by choosing a simple zero-temperature plasma model. The response to an oscillating rf field is accounted for in terms of an equivalent charge-free frequency-dependent permittivity E given by E =

EO(l- o,‘/o’)

(7)

for a homogeneous, isotropic plasma, where o is the radian rf frequency. This equivalent permittivity is employed in the usual way in the solution of the Maxwell field equations. When the plasma is immersed in a magnetic field, the permittivity becomes both anisotropic and frequency-dependent, so that it is then a tensor quantity. Solving Maxwell’s equations, a dispersion equation may be found which shows that electromagnetic waves do not propagate in an ionized gas below the plasma frequency and that the plasma has little effect on the waves at frequencies much higher than the plasma frequency. Many models of analysis assume that the collision frequency is negligible compared with the rf wave frequency so that an electron can oscillate many cycles before it makes a collision. In those models where this assumption is not valid, the effects of collisions on the damping of the waves can be taken into account by assuming the frequency to be complex and replacing w z by w ( o -jv), where v is the radian collision frequency. The propagation characteristics have been worked out in considerable detail for many such models. Regarding the plasma as a dielectric and solving the resulting field equations, Trivelpiece and Gould (11) found that slow

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

5

waves propagating on a plasma of finite transverse dimensions can exist even in the absence of a drift motion or thermal velocities. Their use of the quasistatic assumption was justified because their model did not include fast waves. Bevc and Everhart (12) and Bevc (13) considered wave propagation in plasmafilled waveguides without this restriction and also considered transverse magnetic waves. Cyclotron waves, which are due to the cyclotron frequency resonance (or gyro-resonance), were also included in their analyses. It is at the cyclotron radian frequency w , that the electrons spiral about the magnetic field lines. For electrons 0,

= (elm)&

(8)

where B is the magnetic field strength in webers per square meter. It should be noted that the gyro-resonance for ions is much lower in frequency than for electrons, due to the much higher mass of the ions. Consequently, the ion cyclotron waves will be of little interest in the present study. In a plasma column not bounded by a good conductor, surface waves of an incompressible nature can propagate in addition to the space-charge or body waves discussed so far. These surface waves (11, 14-16) have a finite electric field along the axis of a cylindrical model. The electric field increases radially to the plasma edge and then discontinuously falls to zero at the conducting wall. In these respects, the surface wave is similar to the wave propagating along a slow-wave structure in a traveling-wave amplifier. Surface waves on a plasma column in free space and in the absence of a magnetic field propagate at velocities less than the velocity of light and have a passband (16) extending from o = 0 to w = wp/J2; the resonance frequency is at w = wp/J2. When the medium outside the plasma column is a homogeneous isotropic dielectric with dielectric constant E , the resonance frequency is at

If a magnetic field is present, two surface wave resonances occur. For the case of the plasma column in free space with up% a,, one resonance is at w1 = w P / J 2 , as before and the second one occurs at

which is the electron-ion hybrid resonance, where wceand wCiare the electron and ion cyclotron frequencies in radians per second, respectively. 2. Description of Electron Beam-Plasma Systems

In a conventional microwave interaction device making use of an electron beam, it is necessary to pass the beam close to some form of a metallic

6

G . T. KONRAD AND J . E. ROWE

slow-wave circuit or a resqnant structure in order to convert a fraction of the energy i n the electron beam into rf energy. A schematic diagram of a particular form of such a system is shown in Fig. 1. The physical dimensions of the interaction structure must be comparable to a free-space wavelength, if the interaction is to be efficient. Thus the size of the electron beam, and hence the current that can be carried, is limited. In a beam-plasma system the rf interaction structure is replaced by a plasma, through which the beam is allowed to pass. Consequently, the beam and the plasma are intimately mixed and the rf fields do not vary appreciably over the cross-sectional area of the beam. This relieves the restriction on the beam diameter, and at the same time the size restrictions on a metallic structure in the vicinity of the electron beam may be greatly alleviated. These characteristics make the beam-plasma interaction attractive at high power levels and at very high frequencies, such as millimeter wavelengths. RF INPUT

I

RF OUTPUT

ELECTRON BEAM COLLECTOR 7

t

RF INTERACTION STRUCTURE

OR PLASMA

L

I

FIG.1. Schematic diagram of an electron beam interaction system.

The plasma necessary in a beam-plasma device may be produced by any one of a large variety of discharges, by contact ionization, or by the electron beam itself. Getty and Smullin (17) have examined the beam-generated plasma in detail. The electron beam, when turned on, ionizes the background gas and after a few microseconds the excited plasma electrons become the dominant ionization source. In their experiment an absolute (nonconvective) instability initiated rf oscillations, which first appeared at the electron cyclotron frequency. Then a convective instability at the plasma frequency generated the oscillations that sustained the discharge. Targ and Levine (18), as well as Hedvall (19), also studied beam-generated plasmas and reported on their characteristics, such as plasma density, pressure ranges, and the frequency regions of the observed oscillations. In fact, electron beam excitation is employed exclusively in many beam-plasma systems and partially in many others in order to generate the plasma. I t is extremely difficult to design an experimental beam-plasma system in which the plasma is everywhere homogeneous. In a cylindrical geometry, for example, there are usually density variations in the radial as well as the axial directions (14, 20, 21). Bicchler et al. (22) derived a typical plasms density profile for a cylindrical geometry and noted the agreement with experimental devices to be quite good. They found that in the axial direction there was a

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

7

sinusoidal variation of density, with the density dropping to zero at the metal end walls. In the radial direction the plasma extended by ambipolar diffusion into those regions where there was no ionization, with the density again dropping to zero at the conducting cylinder walls. Nonuniformities in the plasma density make theoretical analyses of realistic laboratory devices considerably more difficult, but, as will be seen below, these inhomogeneities may have beneficial effects on the coupling of rf energy to and from the beam-plasma system. B. Theory of Beam-Plasma Interactions If an electron beam is introduced into a plasma, a distributed interaction may take place between the beam and the plasma, as has already been mentioned. Boyd et al. (23) conducted the first conclusive experiments demonstrating microwave amplification when an electron beam is passed through an arc discharge plasma. They showed that amplification occurred at frequencies below the plasma frequency f,, reached a peak at f,, and extended to slightly higher frequencies depending upon collisions and velocity distributions of the electrons in the beam and the plasma. Shortly thereafter Bogdanov et a/. (24) reported on similar work in which amplifications of 20 to 40 dB in the 1-10 GHz range were observed. Berezin et al. (25, 26) measured the energy loss of the electron beam as it gave up part of its kinetic energy to the high frequency wave, yielding the efficiency of the energy exchange process. The strength of the high frequency fields was also measured.

I . Linear Beam-Plasma Theory A great many researchers, both in this country and abroad, have considered the beam-plasma interaction both theoretically and experimentally. The work to be reviewed in the following few paragraphs thus can be only a representative sample of what has been done. In a series of papers, Vlaardingerbroek et al. (27-29) derived the dispersion equations for cylindrically symmetric geometries for one or more beams of charged particles in both infinite and finite magnetic fields. The same group of investigators (30) has reported on an experiment in which 60-70 dB electronic gain was observed over a length of 20 cm of mercury plasma. They also assessed the practical value of beam-plasma amplifiers (31) on the basis of their experimental work. When the beam and plasma are infinite in extent, an infinite growth rate of oscillations occurs at the plasma frequency. In order to provide for a physically realistic finite growth rate, some authors have incorporated the effects of particle collisions or thermal velocity distributions in their models of infinite cross section.

8

G . T. KONRAD AND J. E. ROWE

Crawford (32) considered both infinite and finite beams in an infinite, uniform, warm plasma. At the beam-plasma interface the Hahn (33) rippled surface-charge model was used when the beam and plasma were both cold. For a warm plasma, where the electronic Debye length may be comparable to or greater than the rf excursions of the beam electrons, this model was not considered to be appropriate because the surface-charge effects on the plasma side of the beam-plasma interface became negligible. Therefore, the Hahn model was replaced by a continuity condition of pressure and plasma current across the beam-plasma boundary. It has been shown by Kislov and Bogdanov ( I d ) , Bogdanov et al. (24), and Vlaardingerbroek (27) that growing waves with finite growth rates occur in cold, collisionless beam-plasma systems of bounded cross section. In plasmas with parameters of the usual values attainable in the laboratory, the finite geometry has a considerably greater damping effect than do thermal velocities and collisions, as was shown by Simpson (34). He, as well as Karplyuk and Levitskii (35), Goland et al. (36) and Stover (37), considered the effects of dielectrics, such as a glass or ceramic tube containing the plasma, on the dispersion equation. Such realistic additions add extra boundary conditions to be satisfied in the solution of the field equations before the final dispersion equation is obtained. It is found that for such models the dispersion equation quickly becomes intractable for solution, even on a modern, high-speed digital computer. As an example, Simpson (34) has obtained solutions for the case of a beam and plasma of radius b completely or partially filling a metallic pipe of radius a. This model is particularly appropriate for a beam-generated plasma. Solutions to a case closely approximating a typical beam-plasma interaction are shown in Fig. 2. The imaginary parts of the propagation constant k are seen to be negative and thus represent the small-singal gain due to the beamplasma interaction. Then uo is the dc velocity of the beam electrons and the subscripts " p " and " b " on the charge density p refer to the plasma and the beam, respectively.

2. Nonlinear Beam-Plasnza Theory Many investigators have solved the dispersion equation for beam-plasma systems under various simplifying assumptions, as was pointed out in the previous section. The complex propagation constants appearing in the dispersion equation yield the small-signal gain of the system as a function of frequency. The general result of these theories is that maximum values of gain in the order of 3 to 15 dB/cm may be expected for low-level operation of a beam-plasma system. As the signal level increases, however, nonlineari-

9

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

PLASMA BACKWARD WAVE

1.0-

o/b - - - - :=: :I :< FAST

a

33

04-

SLOW CYCLOTRON WAVE

' -96

-04

-02

0

02

04

06

I 08

10

12

14

kuo/wp

FIG.2. Solution of the dispersion equation for a beam-plasma system ( w , / w , 0.316. D J D ~= lo3. beam microoerveance = 1.0) Isimpson (341.

=

ties begin to set in, limiting the power output of the device. Thus, for efficiency computations and harmonic generation analyses, it is necessary to employ a nonlinear large-signal analysis. Rowe (38,39) used a Lagrangian (ballistic) theory to calculate particle trajectories and the extent of energy conversion between the beam, the plasma, and the rf wave. Microscopic collisions within each charge group were not included in the calculations, but Rowe (40)pointed out that collision effects may be introduced simply by adding an additional electric field term appropriate to the collision model in the Lorentz force equation. Finite temperature effects were accounted for by assigning appropriate particle distribution functions in velocity-phase space. Gould and Allen (41) subdivided the electron beam into disks and determined the forces on them from potentials found by appropriate Fourier transformations, which is a permissible approach since the plasma was assumed to remain linear and superposition is valid in that case. The plasma was also assumed to be warm, thus including the electron thermal speeds and the dissipative effects of short-range collisions. Geidne (42) also used the disk model for the electron beam as well as the plasma electrons. He expanded the force equation to third order for the beam electrons and to first order for the plasma electrons (the plasma was assumed to respond linearly to the electric field in this treatment also).

10

G. T. KONRAD A N D J. E. ROWE

Brackett (43) investigated the interaction of a modulated electron beam with a uniform, cold collisionless electron plasma on the basis of a perturbation analysis which was carried through terms of third order in the depth of modulation parameter. He neglected electron crossing but did consider the effects of harmonic generation and multisignals applied to the amplifier input. I t was learned that within the assumptions made the plasma remained well described by the first-order approximation, whereas the beam was quite strongly affected by the nonlinear terms. All four of the nonlinear theories described above yield information on harmonic generation in a beam-plasma system. It is known from the smallamplitude zero-space-charge ballistic theory for klystrons (39) that i, I , = 2 J , Inl a x

= 2(0.58) =

1.16,

where i, and I , are the fundamental rf and dc beam currents, respectively and J , is the Bessel function of the first kind and first order. Similarly, for the maximum second harmonic current, i2/Zo = 0.98. There is a relatively slow diminution of the harmonic amplitude as the harmonic order increases, arising from the very sharp peaks in the current density in a tighly bunched electron stream. Large-signal calculations, including space-charge effects, yield values of i, I , in the vicinity of 1.6 under optimum conditions i n traveling-wave amplifiers and klystrons. Similar results have been found for beam-plasma interactions, with harmonic-to-dc current magnitudes being on the order of unity (38). Some experimental results to support these theories have been reported by Berezin et al. (44) and Allison and Kino (45). 3. Coupling to Beam-Plasma System

The time-varying magnetic field of space-charge waves is quite negligible and these waves do not radiate into space. The wavelength of the space-charge wave is much smaller than the free space wavelength and it is necessary to use probes, slow-wave structures or waveguides to couple effectively. These structures have external fields which may couple to the fringing fields of the space-charge wave. Another possibility is mode coupling, which is ordinarily a weak effect, however, Coupling to a plasma column without the presence of material walls, probes, or beams can also occur at steep density gradients or in certain directions in anisotropic media. An example of the former are the Tonks-Dattner (46) resonances, which are resonances due to space-charge oscillations. These were analyzed and explained theoretically by Nickel et a/. (47) as follows. When the frequency of an applied rf wave exceeds the electron plasma frequency at the plasma column edge, a wave of the type discussed by Bohm and Gross ( 3 ) can propagate in toward the column axis. As it does so, it

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

11

propagates through an increasing electron density function until a point is reached at which the applied frequency equals the local plasma frequency. Beyond this point the wave is evanescent. At a discrete number of applied frequencies, standing waves can occur between the column edge and the high-density core of the plasma. Waves at these particular frequencies thus give rise to the Tonks-Dattner resonances because the standing waves transverse to the column also represent the cutoffs of a series of modes propagating along the column. In beam-plasma amplifiers it is desirable to couple as much of the input signal as possible into the space-charge waves of the device. Unfortunately this is not a simple procedure. In most practical cases the net power output and the actual gain of the device are quite severely limited by the coupling methods so far employed (20,48), even though a considerable amount of work has been expended in improving the coupling (20,49,50) and even though many different schemes have been used, such as cavities, reduced height waveguides, coupling helices, and slow-wave structures immersed in the plasma. Unfortunately these schemes have the same size limitations that microwave tube circuits have and thus cannot make use of the inherent beamplasma advantages at high frequencies mentioned in Section A.2. A typical beam-plasma system having helices as coupling structures is depicted in Fig. 3. One very attractive coupling scheme which has been investigated fairly extensively theoretically is that proposed by Feinstein (51) who examined the conversion of longitudinal wave energy of bunched charges into transverse electromagnetic form in electron tubes and multistream plasmas. Neufeld and Doyle (52) showed that the longitudinal oscillations in a homogeneous plasma due to an electron beam may be converted into transverse oscillations when the longitudinal waves interact with the density fluctuations of the plasma electrons. Allen et al. (53)pointed out that in certain frequency regions the anisotropic dielectric “constant ” of a homogeneous plasma is such that the radial dielectric constant E,, is negative while the longitudinal dielectric ELECTRON GUN

PLASMA CATHODE ELECTRON BEAM HELICAL SLOWWAVE STRUCTURE

FIG.3. Beam-plasma interaction system employing coupled-helix couplers for beam modulation.

12

G. T. KONRAD AND J. E. ROWE

constant E,, is positive, a situation which results in real propagation constants in both the radial and the longitudinal directions. Thus the fields in the plasma do not fall off radially, as they do on conventional slow-wave circuits. When radial density variations exist, as is usually the case in practice, Stover and Kino (54) showed that it is not necessarily possible to obtain radial propagation through the plasma column. This is so because the sign of E , , / E ~ ~may change across the plasma, the density of which was assumed to vary parabolically across the column. This type of variation does not necessarily exist in practice due to sheath formations. Allen et al. (55) reported on an experiment in which the presence of a plasma increased the coupling in a system consisting of two waveguide couplers by approximately 20 dB. They cautioned, however, that the optimum plasma density for coupling is not ordinarily the same as for maximum gain. A number of Russian investigators also reported on radial propagation as a means of coupling to a beamplasma system (56-58).

1i. DEVELOPMENT OF THE “ CIRCUIT EQUATION FOR PLASMA WAVEPROPAGATION ”

A . Introduction When an electron beam is passed through a plasma, amplification of an rf signal propagating through the plasma may take place under certain conditions. In this analysis cylindrical systems, i.e., plasma columns and drifting electron beams, will be considered. Maximum use will be made of previously developed traveling-wave tube theory, which can be readily adapted to this analysis. i t is possible to consider a plasma column by itself as a slow-wave “circuit ” along which rf energy can propagate just as along a more conventional transmission line. For such a transmission line, an equivalent circuit may be found which is made up of distributed elements given in terms of the plasma parameters. This approach restricts the plasma to linear behavior. Geidne (42) and Brackett (43) have calculated the electron displacements and the rf currents in the beam and the plasma of a beam-plasma system and found that when the device approaches saturation the beam electrons are driven beyond overtaking, while the displacement of the plasma electrons is very small. Physically speaking, the charge density of a typical electron beam is much smaller than the charge density in the plasma and since the same rf fields must be supported in both the beam and the plasma, the stronger spacecharge restoring forces in the plasma then prevents the plasma particles from making large rf excursions. Thus it is justifiable to assume linear behavior for the plasma and introduce the nonlinearities in the electron beam.

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

13

Initially the plasma characteristics will be considered and subsequently an electron stream will be introduced and the interaction of the waves in the plasma with the spacexharge waves on the electron stream will be analyzed.

B. The Dispersion Equation for a Plasma Column Consider the geometry for a plasma column coaxial with a conducting cylinder, as shown in Fig. 4. The following assumptions are made about the plasma column : (1) The plasma is cold, stationary, uniform, and neutral. (2) A static magnetic field B, = B, is assumed to exist along the zdirection. (3) Nonrelativistic mechanics applies. (4) A quasi-static analysis is appropriate. (5) A one-dimensional treatment is appropriate for the present. The one-dimensional treatment limits the motion of the electrons in the plasma “circuit” to the axial direction. This approach is of course a good one in the case of an infinite or very strong axial magnetic field. For a more moderate magnetic field it is reasonable to assume that in a plasma column of fairly small transverse dimensions this restriction on the motion can be satisfied regardless of the magnitude of the magnetic field by assuming that the plasma is in a region where the axial rf electric field is strong compared with the transverse electric field. In most microwave beam-type amplifiers the rf wave is slowed sufficiently so that it travels at approximately the same velocity as the beam electrons.

r PLASMA COLUMN

CONDUCTINGJ CYLINDER WALL

LDIELECTRIC TUBE WITH DIELECTRIC CONSTANT K~

FIG.4. Geometry of a plasma column slow-wave circuit.

14

G . T. KONRAD A N D J . E. ROWE

The velocity of the beam electrons In turn is usually substantially less than the speed of light. Thus the wavelength of the disturbances in the slow-wave system is much less than a free-space wavelength. This fact permits the use of a quasi-static analysis, in which Maxwell’s equations describing the model under consideration in this study are simplified considerably by replacing the fidl set of field equations by the equations of electrostatics. It is possible to find a simple relation between the electric field in the z-direction and the “circuit” voltage. In terms of the magnetic vector potential A given by

(12)

B=VxA,

where B is the magnetic field strength, and the scalar potential V‘ the electric field may be written as

E = -VV‘

-

(aA/dt).

(13)

In the quasi-static analysis B is time independent, so that one of Maxwell’s curl equations becomes

VxE

=

-dB/dt

= 0,

(14)

and E is directly derivable from the scalar potential. Assume that the rf quantities in general vary as exp[j(nwt -1,q - k,z)]. Note that k is a complex propagation constant, in general given by

or

kn

= Pn

-j a n ,

where pn is the phase constant and zn is the attenuation constant for the nth harmonic signal. A negative value of a, implies a growth in the wave amplitude. Utilizing the fact that there is no radial component of current and no azimuthal variation of A in the present analysis, one obtains from the usual Lorentz gauge condition that

c‘A,,/dz

=

-jk,, A,,

=

-jnop,

E,

V”‘.

(16)

Since k,, = nw(,u, E,)”’ is the propagation constant of free space and y,,is the radial propagation constant, defined by yn2 2 ! k,,’

-

k,:,

one obtains

E,,

2jk,

V,’,

(17)

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

15

because for slow waves yn2 kn2. If V’ at r = b is the “circuit voltage” V of the plasma column, then

(19) From Maxwell’s equations, and using V x E z 0, one obtains the wave equation for propagation in a plasma column, Ezbn

=Jkn Vn .

Here the radial propagation constant in the plasma Tpnis related to the longitudinal propagation constant k, by

The

K’S

are components of the specific inductive capacity tensor (59)

opis the radian plasma frequency given by op2

e2n, A -, m e go

w, is the radian electron cyclotron frequency given by w,

VBO ,

(24)

where 9 is the absolute value of the charge-to-mass ratio of an electron, and v is the radian collision frequency. I n solving the boundary value problem for Fig. 3 and Eq. (20), the following dispersion equation is obtained:

1

K: =

T . JI(T e Ab ) Ko(k, b)Zo(k, a ) - Ko(k, a)lo(k, b ) Jo(Tpn 6) I,(kna)Kl(kn ’1 + Ko(kna)11(knb) ’

(25)

Ke k n

which is a relation connecting o with k , . For the case of free space between the plasma column and the metallic wall, K , = 1. In many cases the collision frequency v is so small compared with o that one may set v / o = 0. Then k, = p, and complex argument Bessel functions may be avoided in Eq. (25). The solution of the dispersion equation is then

16

G. T. KONRAD AND J. E. ROWE

fairly straightforward. Figures 5 and 6 show the solution for a typical set of values encountered in a plasma column useful in beam-plasma interaction devices. It will be shown later that the case of w , < wp is of particular interest in relation to some experimental studies. Thus Fig. 6 shows this case for dielectric materials of several values of K , (see Fig. 4). The forward-wave branch is seen to depend on K, , while the backward-wave branch does not. For the forward-wave branch, the frequency om,where p,, approaches infinity, can be found by noting that Tpnis imaginary for w , < w < wp and w, < wp (provided v = 0). Thus Ke2 = K I I n K l n (26) as

on

--f

co, which may be solved to yield

(27)

For the cases of w , = 0 or K, = 1 much simpler expressions may be obtained from Eq. (27). Thus for K , = 1 and w, # 0,

and for w,

=0

and

K,

# 1, w, =

U P

(1 -t-

K,)1’2

’

C. Derivation of the Equivalent Circuit Elements

The equation of motion for a plasma electron is given by the following force equation :

where v1 is the rf velocity, and the dc velocity is zero for a stationary plasma. I n the one-dimensional case, with B = B, along the z-direction, and retaining first-order terms only, Eq. (30) reduces to

where t’, is the axial rf velocity and E,, is the total axial electric field in the plasma. All rf quantities are assumed to vary as exp[j(nwt - k , z ) ] . Then

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

17

FIG.5. Forward wave passband characteristicsfor the lowest order axially symmetric mode in a dielectric-lined metallic cylinder filled with plasma. ( 4 6 = 1.2, K , = 6.0.)

FIG.6. Phase characteristics for the lowest order axially symmetric mode in a dielectric-lined metallic cylinder. (alb = 1.2, w,/wc = 2.0.)

18

G. T. KONRAD AND J. E. ROWE

The convection current density is given by = (Po

Jz,,

+ Pl)%l>

(33)

where p o and p1 are the dc and rf charge densities, respectively, in the plasma. Neglecting the second-order term, Eq. (33) becomes Jzcn

= Po V l n .

(34)

The total convection current for the plasma column may be expressed as

which yields, after combination with Eqs. (32) and (34) and introduction of the proper expression for the electric field, the following transmission-line equation :

at r

From Maxwell's curl H equation and Stokes' theorem one finds that = b, 2 n b H d b ) = I,," + I z d n (37)

where I z d n is the plasma column displacement current through the crosssectional area of the plasma and may be expressed by

jo lo 2n

Izdn

=

b

jWE0

EAYP d r d~

(38)

which yields the second transmission-line equation after some algebraic manipulations. Thus __

aZ

+ 2rrb~,k, ice( I d k n a ) K , ( k , b ) + Ko(kn a)I,(k, b) KO(k" b)lo(k, a> - K d k " a)lo(kn b )

Equations (36) and (39) describe the transmission line pictured in Fig. 7, if they are written as follows:

and

19

; ; Condz 7; Condz ; ; Condz ; ; Condz 7; CondZ 7; Condz ;;Condz

where

and

(43) Inspection of the last two equations reveals that the collision terms introduce resistive components in the equivalent circuit. In Fig. 7 the lossy elements are not shown separately, but are included in Lonand C o n .If collisions are negligible, vjw +O and Lonand Conare pure inductances and capacitances per unit length, respectively. For the transmission line of Fig. 7 the phase velocity vOnis given by vOn = nwlBn.

(44)

In general,

which reduces to

, with v where Lbn and C& are Lonand C o nrespectively,

= 0.

k,Z = w2 LonC o n ;

From Eq. (45) (47)

which is identical with the dispersion equation, Eq. (25). D . Derivation of the Circuit Equation

If an electron beam with convection current - izn is allowed to pass near or through a plasma, one has in effect a beam-loaded plasma column

20

G. T. KONRAD AND J. E. ROWE

transmission line as depicted in Fig. 8. The impressed current in the plasma due to the motion of the electrons in the beam is ai,,/az. One thus modifies Eqs. (40) and (41) to include the presence of the beam as follows:

and

Introducing the charge density p n in favor of the current, differentiating with respect to z and t , and combining so as to eliminate Zzc(z, t), one obtains a wave equation whch will be designated the " circuit " equation. Thus a2vn(z, t ) a2Vn(z, t ) a2pn(z, t ) = -Lon - LonCon at2 az2 at2 *

I

I

I

FIG.8. Electron beam interacting with a plasma column.

Recalling Eqs. (44) and (45) and noting that the characteristic impedance of the transmission line of Fig. 7 is given by

(50)

20, = (Lon/Con>"2.

The final form of the circuit equation is a2vn(z, t ) a22

- -1 D;n

@v,(z, t ) at2

ZOn

a2p,(z, t )

Don

at2

'

(51)

provided a, 4 p, so that pn g k,. In a typical laboratory plasma column useful for beam-plasma interaction this requirement is readily satisfied, as will be shown later. Figures 9 and 10 show the real part of the characteristic impedance as a function of b for v 4 o.The modulus of Zo, is chosen to correspond to the

an

21

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

v)

E 1 0

X N

0 V W

z

B w a

z

0 tE W IL:

tV

a IL: r a V

6 I-

I

I

I

I

I

P"b

FIG.9. Real part of the characteristic impedance of a plasma column. (w,/w = 0.152 cm.)

= 2.49,

w,/w = 0.428, wp = 2.65 x 1O'O rad/sec, a/b = 6.26, b

I

I

C W V

z a

0 W

a

E

L!

2 LL W

V I-

a LL

a I V

FIG.10. Real part of the characteristic impedance of a plasma column. ( w , 10'' rad/sec, a/b = 1.2, b = 0.792 cm, K , = 6.0, w,/w, = 5.0.)

= 4.4 x

22

G . T. KONRAD AND J. E. ROWE

interaction impedance used in traveling-wave tube theory, while the imaginary part of Z,,, which is usually small compared to the real part by virtue of v -4 w, gives rise to the " circuit " loss. The parameters chosen for the computation of 2," in Figs. 9 and I0 apply to a plasma column with a particle density on the order of 3 x 10'l cm-3 and a cyclotron frequency on the order of 1.0 GHz, with w, < w < wp . Figure 9 is for a/b = 6.26 and thus is appropriate for a typical beam-generated plasma. Figure I0 is for a/b = 1.2 and would thus apply to a plasma filling the entire dielectric cylinder. Such a plasma could for example be generated by a Penning discharge. Comparison of the figures shows that the former type of plasma column has a substantially higher characteristic impedance and would thus result in better beam-plasma interaction. This is borne out by experiment. In Fig. 11 both the real and the 1000

r R E A L PART

i c $! z

loo-

a

0

FIG. 11. Characteristic impedance of a plasma column. (w,/w= 2.49,wJw = 0 . 4 2 8 , = ~~

2.65 x loio rad/sec, a/b = 6.26, b = 0.152 cm, K, = 2.0.)

W

a

E

I

NEGATIVE IMAGINARY = 0.006) PART (

5

10

NEGATIVE IMAGINARY PART [ = 0.0012)

8

101 0

I 4

I

8 Bnb

imaginary parts of the complex characteristic impedance are shown for the K , = 2.0 case of Fig. 9. Note that the imaginary part is quite small compared to the real part and is negative. This is so because the imaginary part of the equivalent inductance Lo, is small but negative, while the imaginary part of the equivalent capacitance Co, is positive and in general has a slightly larger phase angle than the inductance. The value of vlw = 0.0012 is typical for beam-plasma interaction devices.

HARMONIC GENERATION AND MULTJSIGNAL EFFECTS 111.

23

LAGRANGIAN FORMULATION OF THE BEAM-PLASMA INTERACTIONEQUATIONS A . Introduction

In the previous section, the plasma column was analyzed in a onedimensional fashion. From traveling-wave tube theory it is known, however, that the radial variations of the circuit and space-charge fields tend to reduce the gain and the conversion efficiency. It is therefore desirable to take these effects into account theoretically in order to obtain better agreement with experimental observations. In this section, the plasma “circuit ” equation will be considered to be quasi-two-dimensional in nature. The circuit field term will be used directly from the one-dimensional analysis, but the radial variation in the electric field will be taken into account by defining a weighting function, which is proportional to the radial variation of the longitudinal electric field. It will be shown that, for the frequency rate of interest, the effective circuit field at the electron stream is reduced from its value at the plasma edge. Since the electron density and hence the plasma frequency in a typical electron stream used in amplifiers of the type considered in this study is very much less than in the plasma, the electron cyclotron frequency due to the applied magnetic field needed to focus the stream is usually considerably greater than the plasma frequency of the beam electrons. Thus many analyses assume to a fairly good approximation that the magnetic field is infinite as far as the beam electrons are concerned (60). This in effect neglects the transverse motion of the stream electrons. In the present analysis this restriction will not be made, and the electron stream will be treated essentially in two dimensions, i.e., the circuit rf potential and the space-charge potential are assumed to vary in the radial direction but are axisymmetric. The action of a finite axial magnetic focusing field will be included by considering electron

FIG.12. Model for anelectron stream passing through a plasma column. ELECTRON STREAM PLASMA COLUMN

24

G. T. KONRAD AND J. E. ROWE

motion, without bunching, around the axis. This inclusion of the magnetic field does not complicate the equations unduly; it merely introduces the angular velocity u 4 . The coordinates of the model to be considered are shown in Fig. 12 and the analysis follows that of Rowe (39). B. Variation of the Electric Field across the Plasma Column

The longitudinal electric field variation inside a plasma column of uniform density is given by Ezn(r) = An Jo(Tpn (52) which may be obtained from the development in Section 1I.B. Here, J o is the zero-order Bessel function of the first kind, Tpnis the radial propagation constant for the nth harmonic, and A, is an arbitrary constant. Solution of the dispersion equation for a plasma column yields the w-p diagram shown in Fig. 13. The cyclotron frequency is chosen to be below opbecause in a

i z

V W 3

0 W

nI -PLASMA COLUMN IN FREE SPACE

PLASMA FILLED DRIFT TUBE

PROPAGATION CONSTANT, P

FIG.13. Dispersion curves for a plasma column (see text)

typical device operating in the 10-cm wavelength region a magnetic field ranging from a few hundred gauss to at most one kilogauss ( f c 2.8 GHz) is usually adequate for focusing the electron stream. For frequencies in regions I and I1 of Fig. 13 a forward wave can propagate along the plasma column so that a traveling-wave tube type of interaction may take place between the wave on the plasma column and the electron beam space-charge waves. For frequencies in region 1V a backward wave can propagate along the plasma column, which may result in backward-wave amplification or

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

25

backward-wave oscillation in the presence of an electron beam. The plasma waveguide is cut off for frequencies in region 111 and does not support a wave. The dielectric constant is negative in this range. Therefore, the plasma presents an inductive impedance to the electron beam. The electron beam traveling through the plasma column acts in cooperation with the plasma “circuit,” which alone cannot support a propagating slow wave. Thus any charge bunching on the beam is enhanced, as in a multicavity klystron or i n an Easitron (61) amplifier. This enhancement takes place through the induced charges formed in the plasma by the alternating fields of the modulated electron beam. The beam electrons are slowed down by the fields of these charges giving up some of their energy. From Eq. (52) it follows that Ezn has the distributions shown in Fig. 14 for different values of o.In the region between the plasma boundary at r = b

d -1 FIG.14. Axial electric field

variation as a function of radius for the case of w , < wp(see text).

w lL

u

[L

I-

B

-1 w

and the conducting wall at r = a the fields go to zero exponentially in all cases, as shown. In the plasma region the field distribution is uniform for T,, = 0, which occurs at W , = M W . For v = 0 and k = b, Tpnis either a positive real or a positive imaginary number. For Tpnreal, the rf field is highest along the axis of the plasma column and decreases as the Bessel function J , . This corresponds to region I of Fig. 13. If Tpnis imaginary, the rf field is lowest along the axis and increases as the modified Bessel function I, due to the fact that Jo(jT,, 1.1 = Zo(Tpnr ) .

26

G . T. KONRAD AND J . E. ROWE

This corresponds to region I1 of Fig. 13. The wave propagating along the plasma column i n this case is known as a surface wave, which has a resonance ( p + 00) in the range

as was discussed i n Section I1 (62). The resonance frequency in this range is determined by the geometry and the effective dielectric constant of the material surrounding the plasma column. If the electric field distribution across the plasma column is known, it is straightforward to determine an appropriate field weighting function $, . At this point, several alternatives present themselves on how to handle the weighting function in the calculations to be described subsequently. Of course, in a full two-dimensional treatment $,, is a function of radius and an expression for $,,(r) obtained from a field solution is used. As an alternative, a two-dimensional treatment could be employed for the stream but with the stream subdivided into layers. A value of $,(r) that is an average for that layer could then be used. One would then have to assume that the radial positions of the layers do not appreciably change during saturation. This is probably justified as long as the device is operated = ./20p,); with a magnetic field greater than 50 % above the Brillouin field (0, in practice this is usually the case. A third alternative is the use of an effective value for the field across the beam. The same restrictions as in the previous case apply. This case will be carried through in the derivation of the equations to follow. For the case of a solid electron stream with radius rm1equal to or smaller than the radius b of the plasma column, the weighting function taking account of radial circuit field variations is denoted by $,,(rm1).Normalizing with respect to the electric field at the plasma column edge one obtains

On the other hand, for a thin, hollow electron stream with mean radius r,,,z < b, the weighting function taking radial field variations into account is denoted by $,(rmz)

= JdTp, rmz)/Jo(Tpn

b).

(54)

C. Lugrangiun Formulation

In a Lagrangian analysis one follows individual charges or charge groups through the interaction region. In this way, it is possible to account for over-

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

27

taking of one charge group by another as strong nonlinearities develop in the device. Thus for the one-dimensional case the conservation of charge equation in Lagrangian coordinates is

where I , is the dc current in the electron stream moving at a velocity u,, and zo is the initial position of a charge group, while z is the position at a later time. The absolute value sign is introduced due to the multivalued nature of the charge density. Physically, this accounts for electron crossing between zo and z. While in an Eulerian analysis the independent variables are usually distance and time, it is convenient to choose the distance and the entry phase @, of the fundamental rf component as the independent variables in a Lagrangian treatment. With an appropriate normalization the distance variables are expressed in terms of the fundamental frequency as follows:

where Be is the stream phase constant, ro is the mean radius of a charge ring at z = 0, and C,is a beam-circuit coupling parameter, or “gain parameter,” for the fundamental frequency given by

Cl3 =

Iz,

l10/4h5

(59)

and N , 2 z / & is the number of stream wavelengths at the fundamental frequency. In Eq. (59), Z , , is the interaction impedance obtainable from Eq. (50), I , is the dc current in the stream, and V, is the stream voltage. Note that the modulus of Z , , is used in the definition of the gain parameter. This is necessary when v # 0 (and hence Z , , is complex) in order to keep C, a real number, which ensures that quantities such as y , and some of the other normalizations in the following remain physically meaningful quantities. The entrance phase for the fundamental frequency is defined by QOl 4

wzo/uo= -at,.

(60)

It is convenient to define dependent variables normalized with respect to the initial average electron velocity u,. In a nonlinear theory the average

28

G. T. KONRAD AND J. E. ROWE

stream velocity is a function of displacement and so the dependent velocity variables are

The dependent variable x is a function of y,, xo, and Qol. A general expression for x may be written as

dx

xo

+

Jo

dZ

z

"OC1

+ 2C,U,(YI,

xo Q o 1 ) l ' 9

which becomes, when Eq. (62) is substituted under the integral sign,

Equation (64) constitutes one of the working equations in the Lagrangian analysis. Let On represent the phase of the nth harmonic component of the rf plasma wave relative to the phase at the entrance to the interaction region. Then On is based on a coordinate system which moves with the dc stream velocity uo . In a beam-wave interaction device the actual wave traveling along the circuit has a phase shift (a phase lag for a typical amplifier) with respect to a hypothetical wave traveling at the stream velocity u o . This phase shift en(yl)is due to the beam loading as energy is given to the wave. The phases of the j t h charge group at any value of [y,, Qn(y,, xo, @o,,jj] denote phase positions relative to the wave at that yl-plane. Hence at any displacement plane, @ol,j

+ ~ ( Y ~ I C-I )

en(Y1)

= @01,j

+ nwt +

Qn(~1,

xo

7

@01,j)>

or, dropping the particle label .j Bfl(Y1)

= n(JJI/Cl>- n o t

-

@fl(Yl>

xo, @ o J

(65)

This may be written as

L', = (C1/n)(@nOJ1,xo >

@0l)

+ e n ( ~ l >+ nwt).

(66)

29

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

Following Rowe (39), it is possible to find a second large-signal working equation by combining the above expressions for the velocity and the phase. Thus after simplification

D. The Circuit Equation in Lugrungiun Coordinates In a two-dimensional analysis the circuit equation (cf. Section 1I.D) is

where on is a linear charge density at a given position z and expressed by

where b' is the radius of the solid cylindrical electron stream and $, is given by Eq. (53) in the present quasi-two-dimensional analysis. Following Rowe (39), Eqs. (68) and (69) are written in terms of Lagrangian coordinates and the charge density and voltage are expanded in Fourier series. The following normalization is introduced to define a voltage amplitude A , ( y , ) : a)

V(yl, x, (Do,)

(u

1 c,

= Ren = l

A,,(y1)$,,(x)e-j@").

Equating real and imaginary parts in the circuit equation obtained by this procedure results in two circuit equations, which can be put into the following final form after considerable algebraic manipulation :

(70)

and

30

G. T. KONRAD AND J. E. ROWE

Here the primes on the independent variables merely denote variables of integration and L is an integer that ensures that the fundamental signal frequency and its harmonics have commensurate periods in the Fourier expansion. Note that Z,, is, in general, complex with the small imaginary component representing a loss term. It is due to plasma particle collisions and is related to the well-known traveling-wave tube loss parameter d by dn = -(1/2C1)(zoni/lzo~ I),

(72)

where the subscript “ i ” signifies the imaginary part of the complex interaction impedance. The plasma dispersion is contained in the factors u,ji~,, and Z,,jZ,, on the right-hand sides of the equations. E. The Force Equations in Lagrangian Coordinates

The Lorentz force equations are to be formulated now so as to include the effects of the electron stream and the space-charge forces within the stream on the waves propagating along the plasma column. As usual, low enough velocities are assumed so that relativistic effects are negligible and hence rf magnetic fields need not be considered. In this two-dimensional analysis angular bunching of the beam is neglected, but the action of a finite axial magnetic field is included. The radial circuit field has been neglected in accordance with the discussion of Section 1II.B. The various components of the Lorentz force equation are written as follows if the magnetic field B, is assumed to be entirely axially directed in the rf interaction region :

and

where the subscript ‘‘ c ” denotes circuit terms and the subscript “ sc ” denotes space-charge terms. Thus Vnsc-,(z,r, t ) includes the radial field due to the dc space-charge. From a solution of Poisson’s equation in cylindrical coordinates it may be found that the dc space-charge field is given by Eo-r

=

Po ro2 _

2 ~ , ,r

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

31

Thus

where wbis the plasma frequency for the beam electrons. When the normalized variables y, and x from Eqs. (56) and (57) are introduced and the cyclotron frequency is defined by w, 4 yBo as before, the force equations become

and I d dx - - (2C,2wxu4) = w, -. x dt dt Following Rowe (39), expressions may be obtained for the two-dimensional space-charge field terms Esc.

and

Here the primes on the independent variables denote variables of integration as before. The two-dimensional space4harge field weighting functions, which relate the influence of a charge group at the primed position variable on a charge group at the unprimed position variable in determining the spacecharge field forces, are defined by

and

32

G. T. KONRAD AND J. E. ROWE

Equations (77) and (78) may be substituted into the force equations, which may be simplified to the following:

and

(83)

It is possible to obtain a simpler form of the angular force equation from Busch's theorem (63). It was assumed above that the magnetic field Bo is entirely axially directed throughout the interaction region. If the magnetic field at the electron beam cathode of radius r, is B, , then one may write

where K is defined by

If Eqs. (63) and (84) are combined the following angular force equation is obtained:

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

33

For the special case of constant magnetic field everywhere (immersed flow) K = 1. For shielded Brillouin flow, for which dqldt = qBo12, (87) there is assumed to be no magnetic field at the beam cathode and K = 0. If a fraction of the focusing field threads the cathode region, the beam electron trajectories ripple in going through the interaction region. In practice this problem is overcome and balanced flow is obtained by increasing the focusing field above the Brillouin value and placing a bucking coil around the gun region so as to reduce the fringing field. In that case K = 1- ~(W,/W,)~.

(88)

When Eq. (86) is solved for u4 and substituted into Eq. (82), one obtains the following simplified form of the radial force equation:

The left-hand sides of the final forms of the force equations are the accelerations and the right-hand sides are proportional to the electric and magnetic field forces acting on the drifting beam particles. The integral terms are specifically the rf space-charge field forces.

F. Solution of the Large-Signal Equations In the two-dimensional case it is, in general, necessary to solve seven equations [Eqs. (64), (67), (70), (71), and (81)-(83)] in seven unknowns. The seven unknown quantities are the dependent variables A&), O,(yl), @"(yl, X O , @od, x, u,(yl, x o , moll, ux(yl, x o , Qo1), and u&, x o , Qol). These are given in terms of the independent variables y,, xo, and Qo1. Due to the complexity of the equations, it is necessary to solve them on a digital computer. The two circuit-equations for each harmonic can be used to eliminate the voltage amplitude in favor of the phase variable. The force equations and the velocity-phase equations allow the calculation of the electron velocities and the phases. Once the phases are known, the amplitudes can then be calculated readily. Since the wave amplitude at the end of a beam-plasma interaction region is not a priori known for a given input signal, this nonlinear analysis is

34

G. T. KONRAD A N D J. E. ROWE

treated as an initial value problem. The working equations must be written as difference equations to be suitable for machine computation. A set of discrete m0,, are chosen so that they represent the entering phases of a set of representative charge groups at the beginning of the interaction region. Numerical integration then proceeds in the y-direction in finite steps. An important factor that affects the accuracy of computation and the execution time is the magnitude of the integration increment Ay. This quantity must be chosen to be small enough so that the change of any signal quantity over Ay is a small fraction of its amplitude in that region. A difficulty with very small Ay is that in addition to increasing the computing time the roundoff errors increase due to the increased number of iterations and the limited number of significant figures used in the computer arithmetic. For the results to be presented below, a value of Ay = 0.01 was found to give reasonable accuracy. In general, Ay should be small compared to the shortest wavelength in the system. For a solution of the equations it is necessary to specify the initial conditions on the dependent variables. Thus at y, = 0 the following conditions are required : (1) An(()) = An,

(90)

*

where bnis the velocity parameter and is a measure of the beam-wave relative velocity. By definition

for an unbunched stream. For a stream bunched in the longitudinal direction, but with negligible bunching in the radial direction, dAn(y,)ldy, l Y 1 = o = 4Zonr/IZolI) sin a n ,

(95)

where an is the bunch injection phase angle.

(5)

@n(O,

xo >

@oI,j)

=

2nj

( j = O , 1,2, ..., m ) ,

(96)

where j denotes a particular charge group and the stream is unbunched so that the charge groups are injected uniformly distributed in phase over one cycle of the rf wave. Other distributions may be specified when taking prebunched streams into account. In the two-dimensional analysis one must not

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

only take the entering charge per ring as distributed over but one must also consider layers of rings of charge. ~ y ( 0 ,xo 9 @o 1 ,j ) = 0 ,

( 6 ) uAO, xo @o 1,j ) 2

= 0,

35

mn into account, (97) (98)

q o , &QIO,1 , j ) = 0. (99) These expressions state that the rf velocity components of the j t h charge group are zero for an unmodulated stream. The input data to the computer consists of values of n, C1, bl, u ~ ~ / L ~ ~ ~ Z o f l / ~ Z1, oP,b’ l o,/o,and w,/w. In addition, the equivalent circuit radius must be given by b/b‘ and the radial circuit field function, $,,(b’), must be specified. G. One-Dimensional Equations with Beam-Plasma Collisions Included

In the two-dimensional analysis, collision effects between the beam electrons and the plasma particles were neglected. They can be readily included, however, by adding a term proportional to the collision frequency v, to the usual Lorentz force equation. This was carried out for a one-dimensional model (64).The analysis proceeds in the same way as for the two-dimensional case and the final equations are

36

G. T. KONRAD AND J . E. ROWE

Here Fl -= is the one-dimensional space-charge field weighting function relating the influence of a charge group at @,' on a charge group at @,, in determining the space-charge field forces. It is given by

where R,, is the plasma frequency reduction factor (65), a quantity that is dependent only upon the geometry. It should be noted that the effect of collisions between the beam electrons and the plasma is to introduce a viscous damping term in the axial force equation. The collision frequency 17, must be treated as a constant parameter of the system. In the equations of the one-dimensional analysis, Eqs. (loo)-( 103), there are four unknowns. They are the dependent variables A,(y,), O,(yl), @,(yl, @,,), and u ( y , , a,,), and are given in terms of the two independent variables y , and @,,. Solution of these equations proceeds i n the same way as in the two-dimensional case. IV. RESULTSOF SOLUTION OF T H E NONLINEAR EQUATIONS

In a practical situation, only a finite number of harmonics can be included i n the analysis, due to the finite number of charge groups (500 in the onedimensional case) distributed over phase that can be tracked through the interaction region. Thus in the working equations the summation is terminated at a value i n accordance with this restriction. In particular. the onedimensional program was set up to handle a fundamental input signal and up to three harmonics, while the two-dimensional program could handle the fundamental and four harmonics. A . One-Dimensional Results

The one-dimensional program was written i n FORTRAN I V and was solved on an IBM 360 67 digital computer. A typical run with one signal present required an execution time of 2 min if 16 charge groups distributed over phase were injected into the interaction region. A four-signal run usually required 64 charge groups for reasonable accuracy and a computing time of approximately 8 min. 1. N o Collisions

Figure 15 shows the voltage amplitude and power level relative to C , I , V, along the beam-plasma interaction region when only one signal, the fundamental frequency, is assumed to interact with the beam space-charge waves.

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

I2-

10

5

-.--. %

-

37

10-

gP

-

0

-

O

4

G

0 k

00-

W

t

-5

a J

5

y

2

t W 1

06-

-10

a

[L _I

W

>

5 0 z

w

_I

-15

04-

[L

W

3 a 0

- 20 02-25

- 30 AXIAL DISTANCE VARIABLE, y

FIG.15. Voltage amplitude (solid line) and Power level (dashed line) of the fundamental as a function of distance along a beam-plasma amplifier. ( C , = 0.1 11, bl = 0.8, ZOI= 324 0, vC/w= 0,f, = 4.24 G H z , ~= , 728 M H z , ~= , 1.79 GHz.)

The plasma column is assumed to be generated by the electron beam, which corresponds to a large portion of the experimental data to be presented below. The input level of the fundamental signal is normalized with respect to the beam power and is chosen to be 30 dB below C,Z, V, , as is done in all subsequent cases. This value corresponds approximately to the gain obtained in the beam-plasma interaction and therefore is convenient for comparing the saturation lengths of the various theoretical and experimental cases. Figure 15 shows that the device saturates at a distance y = 6.1, or 8.75 electronic wavelengths, corresponding to an actual distance of 7.23 cm, with a gain of 35 dB. The initial amplitude of the harmonics cannot be set identically equal to zero because singularities would then result in the difference equations; however, the initial harmonic amplitudes are chosen sufficiently small so as to have no effect on the manner in which saturation occurs. It is found that, if the input level of the harmonics is at least 80 dB below C,Z, V, , the voltage amplitudes as a function of axial distance do not depend on the initial levels

38

G. T. KONRAD AND J. E. ROWE

AXIAL DISTANCE VARIABLE, y (a1

(bl

FIG.16. (a) Voltage amplitude ( A l , solid line; A z , dashed line) and (b) power level (fundamental, solid line) of the fundamental and second harmonic as a function of distance v,/w = 0, f, = 4.24 along a beam-plasma amplifier. ( C , = 0.111, b, = 0.8, Zol = 324 GHz,fc = 728 MHz, fs = 1.79 GHz.)

a,

'

m

U

P

08

,+

-10

u 0

c W

2

-20

5 W I1I _I

W

>

-30

n W 3

- 40 2 (a1

AXIAL DISTANCE VARIABLE, y

4

g

6

(bl

FIG.17. (a) Voltage amplitude ( A l , solid line; A z , dashed line; A 3 , dotted line) and (b) power level (fundamental, solid line; 2nd, dashed line; 3rd, dotted line) of the fundamental and first two harmonics as a function of distance along a beam-plasma amplifier. (C1 = 0.111, bl=0.8,Zoi =324R,~,/w=O,f,=4.24GH~,f,=728MH~,f,=l.79GH~.)

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

39

AXIAL DISTANCE VARIABLE, y (0)

(b)

FIG.18. (a) Voftage amplitude and (b) signal level of the fundamental and first three harmonics as a function of distance along a beam-plasma amplifier. (a): A , (-); A Z (--); A 3 (---); A4 ( - . -). (b): fundamental (-); 2nd harmonic (--); 3rd (---); 4th (-’-). (C1=0.111, b 1 = 0 . 8 , Zo1=324sZ, v , / w = O , f , = 4 . 2 4 G H ~ , f , = 7 2 8 MHz, f, = 1.79 GHz.)

of the harmonics. Thus a value of 90 dB below C,Zo V, is ordinarily used. In Figs. 16-18 the results for conditions identical to those for Fig. 15 are shown, but with harmonics through the second, third, and fourth, respectively, assumed to be coupled out of the device. Note that the presence of higher harmonics reduces the amplitude of the fundamental because a fraction of the beam power converted to rf power is converted into harmonic power. For example, the figures show that the presence of three harmonics reduces the power level in the the fundamental signal by approximately 6 dB. For a uniform plasma column calculations indicate that the fundamental and only the second harmonic should exist, because the other harmonics are well above the plasma frequency. In actual practice, as the experimental results show, higher harmonics could be detected. This is believed to be due to nonuniformities in the plasma column, such that the plasma frequency near the center of the column is considerably greater than the estimated value of 4.24 GHz Figures 19 and 20 give an overview of the variation of the maximum signal amplitudes and the saturation length as the velocity and gain parameters are varied. These results are obtained from a large number of computer runs. The fundamental and the next two harmonics only are assumed to exist in the beam-plasma device. The range of values for b and C is reasonable for a

40

G. T. KONRAD AND J. E. ROWE

VELOCITY PARAMETER, b,

FIG. 19. Saturation characteristics for a beam-plasma interaction as a function of the velocity parameter. ( C , = 0.1 13, Z,, = 324 s2, v,/w = 0. Solid line: voltage amplitude and signal level; dashed line: distance to first maximum.)

GAIN PARAMETER, C I

FIG.20. Saturation characteristics for a beam-plasma interaction as a function of the gain parameter. ( b l = 1.0, zol = 324 s2, v , / w = 0. Solid line: voltage amplitude and signal level; dashed line: distance to first maximum.)

HARMONIC GENERATION AND MULTlSIGNAL EFFECTS

41

typical beam-plasma amplifier. Figure 19 indicates that the minimum distance for saturation occurs for b , N 0.7, even though more power output can be obtained when a larger value of b, and a longer interaction distance is used. For values of b, > 2.0 the output from the device falls off very sharply due to the electron stream being too far out of synchronism with respect to the rf wave. In Fig. 20 the signal level at the output decreases with an increase in the gain parameter. This is so because the value of b, is kept fixed. If b, would be increased appropriately as C, is increased in order to maximize the output, then there would be a rise in the voltage amplitude curves as C, increases.

2. Eflect of Beam-Plasma Particle Collisions Figures 21 and 22 show the voltage amplitude and the signal level of the fundamental for various collision frequency values. Figure 21 is for synchronism between beam and wave, b, = 0, while Fig. 22 is for b, = 1.4. In the latter figure an increase in the collision frequency causes a more rapid rise in the voltage amplitude and considerably earlier saturation. This is due to

AXIAL DISTANCE VARIABLE, y

FIG.21. Voltage amplitude and signal level of the fundamental as a function of distance along a beam-plasma amplifier, with beam-plasma collisions included. ( C , = 0.1 13, bl=O,Zo1=34On,f,=4.24GHz,f,=728 MHz,f,=1.70GHz.)

I

m v

10

<-

I

-

V

20

0 + W

2

G1 W

30 cc J W

>

<

W J

40

P v) 50

2

0

4

6

60

8

AXIAL DISTANCE VARIABLE. y

FIG. 22. Voltage amplitude and signal level of the fundamental as a function of distance along a beam-plasma amplifier, with beam-plasma collisions included. ( C , = 0.110,

b,

= 1.4,Zol = 213

n,fp

4.24GHz,~, = 728 MHz,f= 1.85 GHz.)

(a)

AXIAL DISTANCE VARIABLE, y

(b)

FIG. 23. (a) Voltage amplitude and (b) signal level of the fundamental and first three harmonics as a function of distance along a beam-plasma amplifier. (a): A l (-); A 2 (- -); A B(- --); A4 (- . -). (b): fundamental(-); 2nd (--); 3rd(-- -); 4th(-. -). (CI = 0.111, bl = 0.8,Zol = 324 a, v,/w = 0.001,f, = 4.24 GHz,~, = 728 MHz,f, = 1.79 GHz.)

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

43

the fact that for a value of b , = 1.4 the beam average velocity is considerably greater than the wave phase velocity and maximum gain occurs at longer distances than for lower values of b,. The collision term in the force equation, however, represents a slowing contribution to the velocity. Thus a lower value of b, actually applies when beam collision effects are included, resulting in the more rapid saturation seen in Fig. 22. When the fundamental and the next three harmonics are assumed to exist in the device, the voltage amplitudes and signal levels shown in Fig. 23 are calculated for the case b, = 0.8 and v,lw = 0.001. Comparing this with the corresponding no-collision case of Fig. 19 a reduction in saturation gain of nearly 1 dB is found due to beam collision effects. 3. EfSect of Plasma Collisions The effect of plasma collisions on the gain of a beam-plasma amplifier when only the fundamental signal is present is shown in Fig. 24. For a collision

AXIAL DISTANCE VARIABLE, y

FIG.24. Voltage amplitude and signal level of the fundamental as a function of distance along a beam-plasma amplifier showing the effect of plasma collisions. ( C , = 0.1 11, bl = 0.8, Z o i r = 324 VJW 0, f, = 4.24 GHz, 728 MHz, ,L = 1.79 GHz. (-) Z o i J l Z o i l - 0 ( v / w = ~ ) ; (--) Zoi,/lZoil = -0.1; ( - - - ) Z o i i / l Z o i l = -0.0303 ( v / w = 0.0012); (- ’ -) Zo,i/~zo,~ = 0.)

a,

44

G. T. KONRAD AND J. E. ROW€

frequency estimated to be typical of an experimental device (v/w E 0.0012), an impedance ratio Zoli/lZ,,I = -0.0303 is obtained, where ZOliis the imaginary part of the interaction impedance for the fundamental frequency component. The reduction in gain due to plasma particle collisions is found to be less than 0.1 dB. 4 . Two-Signal Operation The computer program used for the one-dimensional calculations was written in a sufficiently general way so that not only harmonic frequencies, but also fairly closely spaced input signals could be analyzed. When two signals are introduced in a beam-plasma device and the dispersion of the plasma column is sufficiently small, intermodulation products at mf, - nfb, where in = 2, 3, . . . and n = 1, 2, 3, . . . , (m> n), may be generated. In addition, cross-modulation resulting in interference between the input signals, f , and fb , may take place. One of the problems encountered when multisignal operation is to be analyzed is the increase of the number of charge groups that must be followed through the interaction region, and the resulting increase in computer time. This increase is due to the fact that the integration over the entering charge groups has to be carried out over a complete period of the input, which is generally longer than the period of each single signal. For example, for two input signals with frequencies spaced 5 % apart, the period of the combined input is 20 times the period of a single signal. Thus it was not possible to make calculations for input signals spaced appreciably closer than 100 MHz apart. For the case of Fig. 25, wheref, = 1.7 and& = 1.8 GHz, 305 charge groups I

I

1

AXIAL DISTANCE VARIABLE, y

FIG.25. Two-signal operation of a beam-plasma amplifier showing two intermodulation components. tf(-); fib(-'-); 2fb-f. (---); 2f-fb(--).] (C.=O.111, b.= 0.8, Zoo == 338 Q,fp =4.24 G H z , ~ = , 728 M H z , ~ = , 1.7 GHz,fh = 1.8 GHz.)

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

45

I

I

AXIAL DISTANCE VARIABLE, y

FIG.26. Two-signal operation of a beam-plasma amplifier showing two intermodulation components. [f(-);fb(. -); 2f -fb (- - -1; 2fb -fa (- -).I (c,= 0.111, b. = 0.8, Zoo = 338 a,f, ~4.24 GHz,f, = 728 M H z , = ~ 1.7 G H z , ~= , 1.8 GHz.)

had to be tracked through the interaction region for reasonable accuracy. The program was set up to handle at most 500. On the other hand, for a spacing of 6 MHz betweenf, and fb,more than 4600 charge groups would have been required. This would have made a computation entirely impractical from a standpoint of computing time used. The two input signals of equal magnitude shown in Fig. 25 were chosen to be approximately 6 dB below the saturation value in order to ensure a short interaction length and thus conserve computing time. An execution time of approximately 7 min was required for the case shown. Signalf, is seen to grow more rapidly, even though it is the higher frequency signal, because its velocity parameter is closer to the value corresponding to maximum saturation gain than is the case for f,. Thus the dominance of the lower frequency signal frequently seen in less dispersive devices, such as a helix-type travelingwave tube, is masked in this case. In Fig. 26 the lower frequency signal does dominate, but this is due to the fact that it is approximately 6 dB larger at the input. In general, it can be said that the signal with the largest initial strength or the most rapid growth rate is likely to dominate, unless the circuit is quite nondispersive. In that case, for signals that have an approximately equal initial magnitude, the lower frequency signal may dominate. This is found to be true in the experimental part of this study, where the frequency difference between the input signals was only a few megahertz.

46

G . T. KONRAD AND J. E. ROWE

B. Two-Dimensional Results

In the two-dimensional analysis, the entering electron stream is divided radially into three annular layers, and within each layer 32 charge groups are injected into the interaction region. Because charge groups are followed at various radial positions, the axial velocity can vary over the beam cross section. The computer program was set up to handle only one signal, namely the fundamental. Thus only A , ( y ) can be calculated in the two-dimensional case; however, the normalized rf currents &/I,, for the fundamental and harmonics through the fifth are computed. If I , and the interaction impedance at the harmonics are known, the harmonic output power can easily be estimated. The program for the two-dimensional case was written directly in machine language for the IBM 7090 computer. A typical run neglecting beam spacecharge effects requires an execution time of 8 to 12 min, depending on the interaction length. When rf space-charge effects are included, the execution time for a similar run is 14 to 2 hr. For that reason only a few space-charge runs were completed. In Fig. 27, the results obtained from the one-dimensional model are compared with those for two-dimensional theory, both with and without space-charge effects included. The gain at saturation is approximately 0.6 dB less when two-dimensional effects, i.e., radial electric field variations are

AXIAL DISTANCE VARIABLE, y

FIG.27. Comparison of one-dimensional and two-dimensional analyses of a beamplasmasystem,(C1 = O . l l l , h l =0.8,ZoI = 324n,vc/w = O , & =4.24GHz,f, =259MHz, fc = 728 MHz, = 1.79 GHz.) (-) one-dimensional; (- - -) two-dimensional without space charge; (- -) two-dimensional with space charge.

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

47

included. Saturation occurs at a position approximately 33 % farther along the device when compared with the one-dimensional case. In particular, it should be noted that beam space-charge effects do not make a very drastic difference in the gain and power output level for the systems studied, justifying the neglect of space-charge effects in many of the calculations. The fundamental and the harmonic rf currents are shown in Figs. 28-30. Figure 28 shows the total normalized rf currents, while Figs. 29 and 30 show the rf layer currents normalized with respect to their respective dc layer currents for the fundamental and second harmonic, respectively. It should be noted that each harmonic has its first peak at a slightly larger interaction length than the next higher harmonic. There are subsequent peaks that may be higher than the first one or even those of lower harmonics. The significant observation from Figs. 29 and 30 is that the largest portion of the rf currents is near the surface of the electron beam until well past saturation. Figure 31 shows the saturation gain and the conversion efficiency expected in a typical beam-plasma amplifier over a frequency range determined by the dispersiveness of the plasma volume. The effects of beam space charge are also shown. In Fig. 32 the results of many computer runs similar to that for Fig. 28 are summarized. The harmonic currents for the higher values of n peak 1.2

-

0

\"

._

10-

-I

z

k!

0.8-

3 Lz

0

I! 8 B

0.6-

9

LL

04-

[L

02-

0

I

2

3

4

5

6

7

0

9

AXIAL DISTANCE VARIABLE, y

FIG.28. rf harmonic currents as a function of distance along a beam-plasma amplifier with beam space4harge effects included [fundamental (-), 2nd (- - -), 3rd (- . -), 4th (- -), 5th (- - -)I. (Ci = 0.11 1 , bi = 0.8, Zol = 324 a,V J W = 0, f, = 424 GHz, fb = 259 MHz,f, = 728 M H z , ~= , 1.79 GHz.)

AXIAL DISTANCE VARIABLE, y

FIG.29. Fundamental rf currents in the three beam layers as a function of distance along a beam-plasma amplifier with beam space-charge effects included. [Cl = 0.11 1 , bl = 0.8,Zoi = 324 a, V,/W = O , f , = 4.24 GHz, fb 1259 MHz, fe = 728 MHz, h = 1.79 GHz. (--) outer layer; (- -) inner layer; (- - -) middle layer.]

AXIAL DISTANCE VARIABLE, y

FIG.30. Second harmonic currents in the three beam layers as a function of distance along a beam-plasma amplifier with beam spacexharge effects included. [ C , = 0.11 1, 4.24 GHz, fb = 259 MHz, fc = 728 MHz, fs = 1.79 bl = 0.8,20, = 324 R, V,/W = O , f , GHz. (-) outer layer; (- -) inner layer; (- - -) middle layer.]

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

49

451

4 0-

z ku W

ar

w

n.

35-

i

z V

w

u k W

30-

n

z a

m

-

n

z

25-

a I . ! ) z 0

gi

20-

u

3

trn I5 -

‘

10 15

I

I

I

I

17 18 FREQUENCY, GHz

16

2

19

)

FIG.31. Saturation gain and efficiency as a function of frequency in a beam-plasma amplifier. (vJw = O , f , = 4.24 GHz,fb = 259 MHz,f, = 728 MHz.)

%

FIG.32. Position of first maxima in the voltage amplitude A l ( y ) and the harmonic currents as a function of b ~( C . , = 0.113, 2 0 1 = 340s2, v,/w = 0, f , = 4.24 G H z , ~= , 259MHz,f,= 728MH2, fs = 1.7 GHz.) [(-), voltage amplitude; (- . - . -) 1st harmonic;(- - -) 2nd; (. - . -) 3rd: (- -) 4th; (- - -) 5th.l

c

i

10

5-

E

9-

c

g

G

8-

2 w

e

7-

a 6-

5

\.

--

-

.\ .b /-= .-S=-/e’ I

I

I

I

I

50

G. T. KONRAD AND J . E. ROWE

within a shorter interaction distance; however, all rf currents peak before the circuit voltage amplitude does. This is explained by the fact that saturation of the device actually occurs when, in addition to the initial electron bunch which gives energy to the wave, a second bunch is formed, which is in the accelerating phase of the circuit wave, taking energy from the wave. At saturation the transfer of energy between these bunches and the wave is balanced so that there is no net transfer of energy between the stream and the wave. At that point the circuit voltage amplitude has its maximum. The current maxima, however, are expected to be at the position where the initial bunch is tightest, which occurs somewhat sooner. Figure 33 is again a composite of a number of computer runs. It shows that the rf currents are highest in the vicinity of beam-wave synchronism, where the tightest bunches occur. As the beam voltage is increased above synchronism, the rf current maxima are seen to fall off markedly. This is directly connected with the explanation given in the previous paragraph, where it was pointed out that the maximum rf current amplitude occurs for tightest electron bunching and not necessarily for maximum stream nonlinearity. As Fig. 34 shows, the voltage amplitude, gain, and efficiency increase with b, due to the fact that the initial bunch can slow down more and, hence, give up more energy before it reaches the optimum position in the decelerating phase, if it initially traveled somewhat faster than the wave. When b, is increased, however, the individual stream electrons stay out of synchronism

-10

-05

0

05

10

15

20

VELOCITY PARAMETER, b,

FIG.33. Maximum values of the rf currents as a function of b l . ( C , = 0.1 13, Z O ,= 340 = O,f, = 4.24 GHz,f, = 259 MHz,f, = 728 MHz, = 1.7 GHz.) [(-) fundamental; (- - -) 2nd; (- . -) 3rd; (- -) 4th; (- - -) 5th.l

R,

V,/W

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

51

45

; m I-

40

W

a

35

t; z

30

$

w

0 n

z

a 25

20

g za W

o_

, I5

0.4 -1.0

tLT

3

5

I0

-05

0

0.5

I0

1.5

VELOCITY PARAMETER, b,

FIG.34. Maximum values of the voltage amplitude, saturation gain, and efficiency as a function of b , . ( C , = 0.113, Z , , = 340 a, VJW = 0, f, = 4.24 GHz, fb = 259 MHz, fc = 728 MHz, f, = 1.7 GHz.)

longer. Consequently, the bunching is not as efficient and the rf currents are not as high for large values of b,, as Fig. 33 shows. In comparing these results with those for the one-dimensional calculations two facts should be noted. First, maximum values of gain and efficiency are predicted for lower values of b, in the two-dimensional analysis. This is attributed to the importance of radial effects in the model analyzed. Second, the rf currents shown in Fig. 33 are seen to decrease as b, is raised, while the voltage amplitudes shown in Figs. 19 and 34 increase as b, is raised. It should be noted that the maxima in the rf currents do not occur at the same y-position as the maxima in the voltage amplitudes. Thus these results are to be expected in view of the explanations given in the previous two paragraphs.

V. EXPERIMENTAL STUDYOF BEAM-PLASMA INTERACTIONS A . Description of the Experimental Apparatus In the theoretical considerations described so far, many simplifying assumptions were made, such as the equivalent circuit representation of a plasma, linearity of the plasma, neglect of angular or radial variations, and

G. T. KONRAD AND J. T. ROWE

52

,-BEAM COLLECTOR

/""""'

HELIX TUBE r

PLASMA CATHODE HOUSING

a ' / -

PLASMA DISCHARGE TUBE

"il

ELEC

UN

H

1

FIG.36. Photograph of the complete beam-plasma device.

neglect of nonuniformities. The only reliable check on the validity of these assumptions is an experimental study. To this end a fairly versatile test vehicle was constructed and will be described in the following discussion. Figure 35 is a diagram of the beam-plasma device, while Fig. 36 shows a photograph of the completely assembled device ; metal-ceramic construction was used. The electron gun is a conventional Pierce-type gun with a dispenser cathode and can operate up to voltages of several kilovolts with unity microperveance. The design diameter of the electron beam is 2.5 mm. The gun envelope and the pole piece in front of the gun are made of kovar in order to exclude the magnetic field from the gun region. Despite this, some flux did apparently penetrate, and no compensation could be made. As a result of this penetration and mechanical misalignments, the electron beam was found to ripple and the interception on subsequent electrodes was found to be substantial under some conditions. In order to premodulate the electron beam before entering the plasma region, a short section of helical rf structure is placed immediately after the electron gun. This helix is of the type commonly used in traveling-wave tubes. Some of its characteristics are shown in Table I. The dielectric loading factor rf

TABLE I PARAMETERS

HELIX

~

i.d. o.d. Pitch DLF Synchronous voltage A, at 2 GHz Interaction length

3.91 mm 4.93 mm 1.37 mm 0.65 -1o0o0v 1 .O cm 5.0 cm

‘-COLLECTOR

POLE PIECE

COLLECTOR PUMP-OUT TUBE

FIG.35. Diagram of the beam-plasma device.

54

G . T. KONRAD AND J. E. ROWE

(DLF) is quite low because of the fact that the helix is placed in direct contact with the ceramic cylinder in which it is contained. Even though this method of mounting the helix would not be desirable in a traveling-wave amplifier, due to the high loss, it is acceptable here where little gain is desired in the helix sections. The plasma cathodes are dispenser cathodes with a fairly porous tungsten matrix (Semicon Associates, Type S-75). The high porosity ensures a copious supply of barium at the emitting surface, even when poisoning might be expected to be quite high at the higher pressures. At a temperature of 1050°C a hot-cathode Penning discharge of several hundred milliamperes is created by a voltage of 25 to 30 V applied between the cathode and the ground electrode adjacent to it with the gas pressure in the lo-’ Torr region. A double-ended discharge is used to ensure a more uniform axial plasma density. The plasma region is approximately 10.5 cm long. An output helix identical to the input helix is employed for demodulating the electron beam after emerging from the plasma interaction region. A beam collector surrounded by a water jacket is used for dissipating the spent electron beam. 1. rf Coupling Two means of coupling rf energy into and out of the beam-plasma device are employed. One makes use of coupled-helix couplers (66), which are employed extensively in traveling-wave tubes. The coupler is a short section of helix, about two to three turns long and wound in the direction opposite to that of the helix in the beam-plasma device. It is essentially an impedance transformer that can be adjusted to match the 50-0 impedance of a coaxial line to the substantially higher impedance of the interaction helix. The other means of coupling consists of elliptic cavity couplers that may be placed directly around the plasma discharge region of Fig. 35. The rf energy launched by an antenna at one focus of an elliptic cavity is transferred phase coherently to the other focus, where the plasma column is located. A cross-sectional drawing of such a cavity is shown in Fig. 31. This type of coupler has been used to launch rf energy on other slow-wave structures, such as a helix, with a coupling loss of only a few decibels (67). Elliptic cavity couplers may be analyzed by considering them as a radial transmission line, consisting of two parallel conductors with a spacing b between them as depicted in Fig. 38. Solving the wave equation for such a system one finds that the dominant E-type mode is a TEM mode (68). The excitation geometry and guide dimensions are chosen so that the dominant mode characterizes the field everywhere. The nonvanishing field components of this mode are the E,-component,

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

55

COUPLING ANTENNA

INPUT CONNECTOR

FIG.37. Diagram of an elliptic cavity coupler. OPENING FOR PLASMA TUBE

b

REFLECTING r WALL

(C)

(d)

FIG.38. Development of an elliptic cavity coupler. (a) radial transmission line coordinates, (b) terminated radial transmission line, (c) top view of terminated radial transmission line, (d) elliptic development of radial transmission line (top view).

56

G. T. KONRAD AND J. E. ROWE

which has no variation in the z-direction, and the H4-component, which corresponds to a total radial current 27crH4, directed outward in one plate and inward in the other. The component E, corresponds to a total voltage E, b between the plates. Thus

E,

= - V(r)/b

(105)

and

The corresponding characteristic impedance is then

where

when the space between the radial waveguide planes is empty. For computation of power flow the input impedance of a matched line is not in general equal to its characteristic impedance. In a radial waveguide the concept of guide wavelength loses its customary meaning because of the nonperiodic nature of the field variation in the transmission direction. Consequently the usual relation between guide wavelength and cutoff wavelength is not valid. However, the cutoff wavelength, defined as the wavelength at which the propagation constant is zero, is useful as an indication of the propagating or nonpropagating character of a mode. For an E-type mode the cutoff wavelength is given by (68)

Thus for the dominant mode, for which m = n = 0, there is no lower frequency cutoff. Instead of letting the radial transmission line extend to infinity, it is assumed to be terminated in an impedance Z, uniformly distributed around a radius rL , as shown in Fig. 38b. Assume also that an input signal is applied at ri . For such a case Ram0 and Whinnery (69) derive an expression for the input impedance Z i defined by

For arbitrary values of Z , the result is

zi = zoiZ , cos(Oi - $), + j Z , , ZoLCOS($~ - e,) + jZ,

sin(Oi - eL) sin(Gi - $,)’

HARMONIC GENERATION A N D MULTISIGNAL EFFECTS

57

where

e(ko r) = tan-'[N,(k,

r)/Jo(kor)],

(113)

Go(kor) = [ ~ , ' ( k , r)

+ ~ , ' ( k , r)]'/',

(1 15)

Gl(ko r)

+ N 1 2 ( k or)]'/'.

(1 16)

and = [J,'(k,

r)

Here k , is the propagation constant of free space and the J's and N's are Bessel functions of the first and second kind, respectively. Choosing Z, = 50 0 for a reason which will be apparent shortly, Eq. (1 11) is used to calculate the input impedance as a function of k , rL with ko ri = 0.41 6. The results are shown in Fig. 39.

2

t

X N

2 W 0

z

dW

a

z

FIG.39. Input impedance of a radial transmission line terminated in 50 0. [kori = 0.416; 0 , resistive component ; 0 ,reactive component.]

If the characteristics of a terminated radial transmission line are known the final step in the development of an elliptic cavity coupler is depicted in Figs. 38c and d. The transmission line of height b and radius rL is replaced by an elliptic line of height b such that c d = r L . A perfectly reflecting surface is assumed to be located at A' and the 50-0 load is at B. The foci of the ellipse are at A and B. The final geometry of the elliptic cavity is shown

+

58

G . T. KONRAD AND J . E. ROWE

in Fig. 40. It is a characteristic of an ellipse that the lengths of the rays between one focus, the wall, and the other focus are all equal. In particular, in Fig. 40, FlAF2 - F,BF, = rL. (117)

A transition to a 50-0 coaxial transmission line may now be placed at F2 and a plasma column of radius ri at F l . Then rf energy launched in the cavity at F2 is transmitted in phase to the plasma at F, due to the equal path lengths between the two foci. The electric field in the cavity is such that it lies along the longitudinal direction of the plasma column, as is desired for a beam-plasma interaction. Phase errors are introduced due to the fact that the path lengths are not all exactly equal since the plasma column and the

,

LINE

.=50n)

FIG.40. Concept of an elliptic cavity coupler. ( r L = F I A $-AF, .)

coaxial line transition have finite dimensions. This problem may in part be compensated for in an actual cavity by making trial and error adjustments on the matching antenna between the coaxial line and the cavity. To be suitable for the experimental device described here the cavity dimensions are taken as shown in Table 11. This choice of dimensions for the cavity places the operating point of 2.5 GHz near the center of one of the TABLE 11 CAVITY DIMENSIONS Major axis of ellipse Minor axis of ellipse Eccentricity of ellipse Cavity height Design center frequency ri

ko ri ko r L (from Fig. 39)

6.25 cm 5.70 cm 0.407 1.06 cm 2.5 GHz 0.793 cm 0.416 6.545

HARMONIC GENERATION GENERATION AND AND MULTISIGNAL MULTISIGNAL EFFECTS EFFECTS HARMONIC

59 59

component of of the the input input impedance impedance shown shown in in broad valleys in the resistive component component is is near near zero zero for for those those conditions. conditions. Thus Thus Fig. 39. The reactive component band of of frequencies frequencies around around 2.5 2.5 GHz. GHz. operation should be possible over aa band input impedance is is in the the range range of of 20 20 to to 100 100 R, R, requiring requiring aa Note that the input plasma column corresponding to to Fig. Fig. 10 10 rather rather than than Figs. Figs. 99 and and 11. 11. This This plasma that the plasma plasma column is is required required to to fill fill the the tube tube necessitating necessitating aa hothotimplies that plasma, where where the the plasma plasma is is essentially essentially cathode discharge. A beam-generated plasma, confined to the beam beam region, cannot cannot be be expected expected to to be be matched matched to to the the cavity. cavity. These points will be discussed discussed more more fully fully below. below. The results of a coupling experiment experiment are are shown in in Fig. Fig. 41. 41. The The cold cold loss loss through two cavities and the plasma plasma tube with no no plasma plasma present present is is in in the the 60-80 dB range from from 1.3 to 2.6 GHz. When a plasma plasma is is present present the the cold cold loss loss drops markedly markedly over this this range, particularly for aa discharge discharge current current of of 400 400 mA/cathode. In the upper L-band the insertion loss varies from 30 to 40 dB mA/cathode. loss varies from 30 to 40 dB while in the lower S-band it is in the vicinity of 25 to 30 30 dB. dB. Just Just below below the the design frequency of 2.5 GHz the insertion loss is is only only 21 21 dB. dB. Since Since the the rf rf plasma column may may be be expected expected to to be be small small losses in the cavities and the plasma loss, most of the 21 dB dB may be be attributed attributed to to the the compared to the coupling loss, observed over over aa fairly fairly wide wide coupling. Thus a loss of 10 to 15 dB/coupler is observed interaction yields yields aa gain gain of of range of frequencies. Since the beam-plasma interaction gain should should be be possible possible with with this this more than 30 dB in the device tested, net gain method of coupling.

8

20 -

%

s5

\ 40-

F

\

\

u ul w

z

60 -

00

FIG.41. Insertion loss through two elliptic cavity couplers and 2.36-cm-long plasma column created by a dual hot-cathode discharge for various values of discharge current. [ O ,discharge current = 400 mA/cathode; .,discharge current = 300 mA/cathode; A,disTorr, Bo = 322 G,fc= 934 charge current = 200 mA/cathode; A,cold loss.] (P= 3 x MHz.)

60

G. T. KONRAD AND J. E. ROWE

Unfortunately the electron beam emission could not be maintained at pressures in the lo-’ Torr range due to ion bombardment of the cathode. When the pressure was lowered sufficiently so that an electron beam with rated current could be obtained, the hot-cathode discharge became unstable due to an anomalous diffusion and extinguished due to sputter pumping in the discharge region. The plasma that could be maintained at lower pressures was the beamTorr and for generated plasma, which existed at pressures below 2 x which the best experimental gain was obtained with the coupled-helix couplers; it had a characteristic impedance of several hundred ohms, as shown in Figs. 9 and 11 (see pp. 21 and 22). Thus it was possible to match the elliptic cavity coupler impedance to the impedance of the beam-generated plasma column over a sufficiently wide frequency region. These results are in general agreement with the work at Microwave Associates (55), where it was found that the best plasma density for coupling does not ordinarily correspond to the plasma density for optimum gain. To overcome this difficulty it may be necessary to taper the density and possibly the magnetic field (70) in order to obtain better radial propagation and hence tighter coupling to the system.

2. Characteristics of the Plasma Column The gas used in the beam-plasma experiments was chosen to be research grade xenon with a purity of 99.99 % by volume. The reason for this choice is that xenon is a noble gas with a quite high ionization cross section and may be expected to yield a sufficiently dense plasma at relatively low pressures. Thus cathode poisoning is minimized. In the course of the experimental work two methods of generating a plasma were employed : hot-cathode discharge and beam-generated plasma. It was anticipated at the beginning of the experimental work that the hotcathode discharge plasma would be used almost exclusively. During the course of the experimental work it was found, however, that the beamgenerated plasma had certain features that recommended it for extensive use in many of the large-signal measurements. One of the important considerations was that the beam-generated plasma could be maintained at lower gas pressures. This in turn yielded considerably better rf performance with the coupled-helix couplers. For a fixed beam voltage the beam-generated plasma can exist in two distinct modes for different beam currents or pressures. The mode for lower values of pressure or beam current has been called the “ beam confined mode ” by Dunn et al. (74, while the mode at higher values of pressure or beam current has been called the “total glow mode.” The former is so called

61

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

because the plasma column is confined to the approximate size of the electron beam, with the plasma density at the beam edge having dropped to approximately half of its value at the center, as some of Hedvall’s (29) measurements indicate. Above a critical pressure the total glow mode exists. It fills the entire diameter of the tube and in that respect it is similar to the hot-cathode discharge plasma. The total glow mode is accompanied by a larger noise level at the output couplers of the device, as may be detected with a spectrum analyzer. The transition from one mode to the other is quite abrupt and occurred in the pressure range of 2 x to 4 x Torr for typical operating conditions of the beam-plasma device described here. The plasma density in a hot-cathode discharge plasma could be measured in a separately constructed device but there was no provision to measure the density of the beam-generated plasma in the final test vehicle. For that reason, it was necessary to calculate it from a knowledge of the pressure and the beam density. Cavity measurements by Hedvall (29) indicate that plasma densities of many times the beam density can be obtained in the to Torr region, for example. At lower pressures there is very little scattering of the beam electrons and it appears that the predominant ionization source is the electron beam itself rather than secondary ionization by the plasma electrons. It has been found experimentally (19, 72) that the plasma density varies slowly with pressure but is proportional to the square of the longitudinal magnetic field. Above a pressure of approximately Torr, with the exact value depending on the type of gas, secondary ionization by the plasma electrons becomes predominant. Under those conditions the density is nearly independent of magnetic field for high enough values of B, but is proportional to the square of the pressure. Frey (73) has carried out a one-dimensional analysis that yields the ratio of plasma density to beam density in a beam-generated plasma. He assumes an infinite magnetic field so that the beam-generated ions and electrons can leave the system by recombination or one-dimensional ambipolar diffusion only. His results are for hydrogen, nitrogen, and argon, but TABLE 111

CHARACTERISTICS OF A BEAM-GENERATED PLASMA IN XENON Beam voltage (V)

Beam current (mA)

400 600 600 600

6 10 10 6

Pressure (Torr) 2 2 2 2

x 10-4 x 10-4 x 10-3 x 10-3

Beam electron density, neb (particles/cm3) 6.2 x lo8 8.6 x lo8 8.6 x los 7.7 x 108

nep/neb 300 250 1500 1500

Plasma frequency (GHz) 4.0 4.25 10.1 9.5

62

G. T. KONRAD AND I. E. ROWE

may easily be extended to .xenon. His agreement with Hedvall's experimental results is excellent in the region where the density varies with the square of the gas pressure. On the basis of Frey's calculations along with the experimental results of Hedvall the data for Table 111 was calculated for beam parameters appropriate to the experiments described here. B. rf Test Resirlts

The rf test data to be presented in this section were obtained with the beam-plasma interaction device described above. The coupled- helix couplers and short sections of the helical slow-wave structure were used to couple rf energy into and out of the system. A spectrum analyzer was used for all rf measurements so that the various signals, harmonics, and spurious oscillations could be ascertained simultaneously. In Fig. 42, the small-signal electronic gain as a function of beam voltage may be seen. The maximum value of electronic gain was found to be 36 dB for the parameters indicated. The beam-generated plasma operated in the beam-confined mode in this case. It should be noted that there was no net gain from the device due to cold loss values in the vicinity of 70 dB. The large variations in the gain curve can be explained as follows. Since there is an abrupt discontinuity in the plasma at the ends of the 40

30 m U

3

20-

a

z P u)

'

J

10-

a 0"

J -10 0

I

I

I00

200

I..

I

I

I

I

300

400

500

600

700

BEAM VOLTAGE, V

FIG.42. Small-signal gain versus beam voltage for a beam-plasma device using coupled helix couplers. ( B o = 260 G , = 728 MHz, fs = 2.0 GHz, f, 2 4.3 GHz, P = 2 X Torr.)

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

63

interaction region, rf waves set up in the plasma are almost totally reflected at both ends of the plasma unless good coupling to the plasma column can be achieved. A wave propagating back and forth along the plasma column can grow whenever the round-trip gain from end to end along the column is greater than unity. Thus a gain curve with sharp peaks, as the voltage or the frequency are varied, may be expected. Figure 43 shows some spectrum analyzer patterns of the output from the beam-plasma device operating under the same conditions as those for Fig. 42, except that the drive power was sufficient to nearly saturate the device. In Figs. 43a and b the fundamental is shown, with the frequency scale expanded ten times in Fig. 43b. The noise level is seen to be at least 20 dB below the

1

E

"

\

m

' I )

2 i 3 a t-

3

0

-

FREQUENCY, 3MHz/cm ( 0 )

-

-I

t-cm FREQUENCY, 3OOKHz/cm (b)

-

-i

t-cm FREQUENCY, 300KHz/cm ( C )

FIG.43. Typical spectrum analyzer patterns of the output from a beam-plasma device. (f,= 2.0 GHz, drive power = 250 mW, P = 2 x Torr, fF = 728 MHz, f, 2 4.3 GHz, 20 = 10 mA, VO = 585 V). (a) fundamental, (b) fundamental, (c) second harmonic.

64

G . T. KONRAD AND J. E. ROWE

signal. The second harmonic output is shown in Fig. 43c and is seen to be approximately 10 dB below the fundamental. Note that in these, as well as in all subsequent spectrum analyzer patterns, the frequency increases from right to left. In Figs. 44-47 the performance of the beam-plasma device operating in the beam-confined mode is summarized. Figure 44 shows the small-signal electronic gain at a number of frequencies for two different voltages and with the magnetic focusing field set at 260 G. The maximum of 36 dB was found at 2.0 GHz for 520 V. When these results are compared with Fig. 31, it should be noted that experimentally gain occurred over a considerably wider frequency range. This is believed to be due to nonuniformities in the density across the plasma column. The saturation characteristics of the device are shown in Figs. 45 and 46, where the output of the fundamental signal and the harmonics that could be detected on the spectrum analyzer are plotted. At a frequency of 1.045 GHz the device is obviously saturating, but for a frequency of 1.7 GHz sufficient drive power was available just to approach saturation. In Fig. 46 the second harmonic comes to within 5 dB of the fundamental at saturation. At no time was it possible to detect harmonic signals above 6.0 GHz. Harmonic signals between 5.0 and 6.0 GHz were usually so small that they could be seen only in a few instances. This was due to the rapid deterioration of the coupler voltage standing wave ratio (VSWR) above 5.0 GHz and the very low value of interaction impedance near the plasma frequency. The plasma frequency

10

I

I

I

I

I

14

18

2 2

26

30

34

FREQUENCY, GHz

FIG.44. Small-signal electronic gain in a beam-plasma system using coupled-helix couplers. Solid line: beam voltage = 520 V, beam current = 8.7 mA. Dashed line: beam voltage = 600 V, beam current = 10.2 mA. (Bo = 260 G , fc = 728 MHz, f, 2 4.3 GHz, P =2 x Torr.)

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

65

I

100

10

1000

DRIVE POWER, rnW

FIG.45. Saturation data for a beam-plasma device using coupled-helix couplers.

+,

0 , fundamental; 0 , 2 n d harmonic; A,3rd harmonic; A,4th harmonic; 5th harmonic. (fs = 1.045 GHz, Vo= 770 V, l o = 7.1 mA, Bo = 260 G , fc = 728 MHz, f, z 4.3 GHz,

P =2 x

Torr, small-signal gain

10

=6

dB.),

100

1000

DRIVE POWER. rnW

FIG.46. Saturation data for a beam-plasma device using coupled-helix couplers. 0 , fundamental; 0, 2nd harmonic; A 3rd harmonic. (fs = 1.7 GHz, Vo= 600 V, lo= 10.2 mA, Bo = 260 G , f. = 728,f, z 4.3 GHz, P = 2 x Torr, small-signal gain = 24 dB.)

66

G. T. KONRAD AND J. E. ROWE

1

-60

10

12

I.

I

I

I

I

I

I

14

16

18

20

22

24

26

28

FUNDAMENTAL FREQUENCY. GHz

FIG.47. Harmonic level below the fundamental at saturation in a beam-plasma amplifier. 0 , 2nd harmonic; 0, 3rd harmonic; A, 4th harmonic. ( V o 600 V, 1, g 10.2 mA, Bo = 260 G,f, g 4.3 GHz, P = 2 x Torr.)

was estimated to be slightly above 4.0 GHz for these conditions, as explained above; however, due to nonuniformities in the plasma column, harmonics above this frequency were observed experimentally. Figure 47 summarizes a large amount of experimental data, such as is shown in Figs. 45 and 46. The harmonic power output below the fundamental power level is shown at saturation. Table IV lists the gain, interaction length, and first two harmonic levels obtained experimentally and by use of the various theoretical models. For the theoretical calculations, the input level of the fundamental signal was chosen to be 30 dB below C,I, V, , which ensures that small-signal conditions prevail at the input and that the experimental and theoretical values of gain are comparable for an easy comparison of interaction length. The gain predicted by the two-dimensional analysis is too high because beam-collision effects were neglected. The distance for saturation from the one-dimensional theory is too low because radial effects are not negligible. In addition, nonuniformities in an actual plasma may result in a larger saturation length. The discrepancies between experiment and theory in the harmonic levels can be easily understood from an inspection of any of the graphs of Section IV showing the circuit voltage amplitudes or rf currents as a function of axial distance. It may be noted that these quantities vary substantially within a short axial distance in the vicinity of the saturation plane. Thus small position

67

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

TABLE IV COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS'

One-dimensional theory Experiment Electronic gain Interaction length Second harmonic Level below fundamental at saturation Third harmonic level below fundamental at saturation a

bl

VJW = 0

V,/W =

0.003

TWOdimensional theory without space charge (VJW = 0)

TWOdimensional theory with space charge ( V J W = 0)

25 dB 10.5 cm

35.5 dB 6.2 cm

26 dB 6.4 cm

32.3 dB 8.6 cm

32.1 dB 9.4 cm

5dB

14.4dB

15.8dB

16.2 dB

13.1 dB

25 dB

11.3 dB

20.2 dB

18.6 dB

18.7 dB

= 0, C1= 0.113, V , =

520 V,Io = 8.8mA,fp = 4.24GHz,f,

=

728MHz,f,

=

1.7 GHz.

changes of the output coupler of the device could result in large variations of the harmonic power output. It is also likely that the plasma extended into the helix regions in a highly nonuniform fashion so that the coupling could be enhanced, resulting in the unexpectedly high harmonic output at the second harmonic. Deterioration of the coupler VSWR on the other hand could prevent some of the higher harmonic components from being coupled out effectively. This may have been the case for the third harmonic. In view of these sources of error the agreement of the results shown in Table IV is seen to be quite reasonable. It has been found experimentally that the magnitudes of the harmonic components in this beam-plasma amplifier are comparable to those in a conventional traveling-wave amplifier with similar operating parameters, except in those cases where plasma nonuniformities enhance the coupling at certain frequencies. Operation of the beam-plasma device in the beam-confined mode is compared with operation in the total glow mode in Fig. 48. Here ZK is the cathode current and I,, is the current through the plasma region. Transition from one mode to the other occurred quite suddenly in the region between 2 x and 3 x Torr. Note the higher noise level in the total glow mode and the reduced power output. The reduction in power is attributed to: (1) reduced beam current through the plasma region at higher pressure and (2) increased collision frequency, which increases the loss through the plasma. In Fig. 49 operation in the total glow mode is displayed in more detail with the rf input signal separated from the cyclotron frequency by 90 MHz. The

68

G. T. KONRAD ALND J. E. ROWE

--I -

i-cm IOMHz/cm (a)

i -

l-cm IOMHz /cm (b)

FIG.48. Comparison of beam-confined and total glow modes. (f,= 2.013 GHz, fc = 1.357 GHz). (a) beam-confined mode (Vo== 553 v, I K = 16.3 mA, lo= 8.0 mA, P = 2x Torr) (b) total glow mode (Vo = 553 V, IK = 16.2 mA, lo= 6.0 mA,P = 3 x Torr).

FIG.49. Output characteristics of a beam-plasma device operating in the total glow mode with the input signal close to the cyclotron frequency. (Vo= 580 V, I. = 7.2 mA, = 1260 MHz, = 1150 MHz, f, E 9.56 GHz.) (a) output without rf drive power, (b) output with 250-mW-rf drive power, (c) output with 375-mW-rf drive power.

-

200 M Hz / c m (c 1

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

69

first spectrum analyzer photograph shows the background noise level at the output of the device along with oscillations at the electron cyclotron frequency with no input signal applied. The second photograph shows an output signal with the input drive level slightly below saturation. When the device is driven well into saturation, as shown in the third photograph of Fig. 49, the cyclotron oscillations are completely suppressed and the second harmonic is just barely visible at a level approximately 15 dB below the fundamental. In Fig. 50 the saturation characteristics for the beam-plasma device operating in the total glow mode with a small-signal electronic gain of 20 dB are shown. Note that in this case the third harmonic is greater than the second by as much as 8 dB at saturation. The one-dimensional computer program used for calculating the harmonic components was general enough to handle closely spaced input signals as discussed in Section IV.A.4. A few results showing two-signal opzration of the experimental beam-plasma device are therefore presented for completeness. All data were obtained for operation in the beam-confined mode at a pressure of 2 x Torr. Letf, be the lower frequency signal andf, be the higher frequency signal. The two intermodulation products which are closest to the input signals are 2f, -fb and 2fb - f , . Holding the input signal level o f f , fixed, Fig. 51 shows a drop in fb at the output of the device due to cross-modulation as f, is increased to saturation and slightly beyond. The small-signal gain of the device was approximately 33 dB. The drive level was so chosen that when fa of comparable magnitude was applied the device was saturating, even though

-40 I

I

I

10

I00

DRIVE POWER, m W

FIG.50. Output as a function of the input drive level for the beam-plasma device operating in the total glow mode at a frequency of 745 MHz. ( V , = 580 V, 20 = 7.8 mA, Torr, f, z 9.5 GHz.) Bo = 280 G, fc = 780 MHz, P = 3 x

70

G . T. KONRAD AND J. E. ROWE

-301

-20

I

1

I

-10 -5 0 5 LOWER FREQUENCY OUTPUT POWER RELATIVE TO HIGHER FREQUENCY OUTPUT, dB -15

FIG.51. Intermodulation components and higher frequency output as the lower frequency input level is varied. 0,fb; 0 , 2 f b -f. ; 2f. -fb. [fa = 1.994 GHz, fb = 2.000 GHz (140-mW-drive power), small-signal gain = 33 dB, Vo = 573 V, lo= 8.9 mA, Bo = 485 Torr.] G,f, z 4.2 GHz, P = 2 x

+,

fb alone was not sufficient for saturation. The magnitude of the intermodulation components is also shown in Fig. 51. Their level comes to within approximately 15 dB of the drive signals. For the frequency spacing of 6 MHz chosen for this run only two modulation components were detected. Figure 52 shows the corresponding results for the case when the lower frequency input signal fa was held fixed while the upper frequency signal f b was increased. The intermodulation components were comparable in magnitude to the previous case, but there was a negligible decrease in the output of f a . This shows that the lower frequency signal dominates when closely spaced signals of approximately equal strength are applied to the device. Similar results are obtained for a conventional traveling-wave amplifier, but were not apparent in the theoretical calculations for the beam-plasma amplifier. This is due to the fact that, for the frequency difference of 100 MHz that had to be used in the calculations, the variation in the velocity parameter and hence the gain completely masked this effect. Figure 53 shows a series of spectrum analyzer

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

71

1

-- 20

-03-

-15 -10 -5 0 5 HIGHER FREQUENCY OUTPUT POWER RELATIVE TO LOWER FREQUENCY OUTPUT, dB

FIG.52. Intermodulation components and lower frequency output as the higher frequency input level is varied. 0 fo ; 0 ,2fb -fa ; f, 2f. -fb. [fa = 1.994 GHz (140-mWdrive power), X = 2.000 GHz, small-signal gain = 33 dB, Vo = 573 V, Zo = 8.9 mA, Bo = 485 G , f , 4.2 GHz, P = 2 x Torr.]

photographs of the behavior of the intermodulation components as fb was held fixed while f a was increased until the device saturated. All parameters are the same as for the two previous figures. Note that the intermodulation components are approximately 15 dB below the primary output signals. The next two figures are similar to the previous ones, except that the input signals are more closely spaced. They are only 1 MHz apart in Figs. 54 and 55. Consequently, two additional intermodulation components could be seen at 3S, - 2fb and 3fb - 2fa. The output from the device is shown in Fig. 54 as the higher frequency signal amplitude was held fixed, while the lower frequency amplitude was increased until saturation set in. Figure 55 shows a spectrum analyzer display of the various signals in the saturation region. Note that the higher frequency output signal has been slightly suppressed by the lower frequency one. Signals 2f, -fb and 2fb - f a appear approximately 20 dB below the primary signals, 3fb - 2fa appears approximately 25 dB below the primary signals, while 3fa - 2fb is just barely visible.

72

G. T. KONRAD AND J. E. ROWE

-3MHz/cmA

-3MHzkmd

(d 1

( C )

FIG.53. Spectrum analyzer display of the output of a beam-plasma device with two input signals. (fa = 1.994 GHz, f, = 2.000 GHz, Vo = 573 V, I. = 8.9 mA, Bo = 485 G . ) (a) output off. is 20 dB below the output offb, (b) output off, is 10 d B below the output of f b , (c) output off. is 5 dB below the output offb, (d) outputs off. an d f , are equal.

In summary, it can be said that the experimentally observed intermodulation components agreed within a few dB with the theoretical results. A detailed comparison is not possible because a far greater frequency separation had to be used for the computations than for the experiments. This was necessitated by the limit in the number of charge groups that could be tracked through the interaction region.

VI. CONCLUSIONS The equivalent circuit representation of a plasma column was found to be an effective and convenient method of modeling the plasma in a nonlinear beam-plasma interaction. The characteristic impedance found in this way included a loss term due to plasma collisions. Collisions due to the beam

HARMONIC G E N E R A T I O N A N D M U L T I S I G N A L EFFECTS

-5

-I0

-I5

73

tt

t

-20

-15

-10

-5

5

0

LOWER FREQUENCY OUTPUT POWER RELATIVE TO HIGHER FREQUENCY OUTPUT, dB

.,

FIG.54. Intermodulation components and higher frequency output as the lower fre2f. - f b ; 0 , 3 h - 2f. ; 3f. - 2fb. quency input level is varied. 0 , f . b ; 0 , 2 h - f. ; f, [f. = 2.009 GHz, f b = 2.010 GHz (1 50-mW-drive power), small-signal gain = 33 dB, VO = 573 V, IO = 8.9 mA, Bo = 485 G , f , z 4.2 GHz, P = 2 x Tort-.]

-C ) N

FIG.55. Spectrum analyzer display of the first two sets of intermodulation components at the output of a beam-plasma device. (fa = 2.009 GHz, fb = 2.010 GHz, Vo = 573 V, lo= 8.9 mA, Bo = 4856.)

2 5

4k +

MHz/cm

c m

74

G. T. KONRAD AND J. E. ROWE

electrons with the plasma were also accounted for by an additional term in the beam force equation. One of the chief stipulations in the development of the equivalent circuit concept for a plasma column was that the plasma remain linear under all conditions. It was shown that under ordinary circumstances this requirement is easily satisfied. In other applications such as plasma heating or plasma containment in high fields this may not be the case and would have to be investigated before plasma linearity may be assumed. In the present case nonlinearities were confined to the electron beam. Even though the one-dimensional interaction model resulted in a good qualitative understanding of the interaction process, it was generally optimistic in predicting the output levels from the device. In particular, the interaction distance required for saturation to occur was consistently too low. The two-dimensional model, which took radial variations into account corrected these shortcomings, so that fairly good agreement could be obtained with the experimental measurements. The beam-plasma device was found to be considerably more dispersive than a conventional helix-type traveling-wave amplifier, for example. For this reason the appropriate dispersion equation had to be solved in order to find the w-p characteristics before the characteristic impedance could be calculated. Even though the phase velocity along the plasma column was appropriate for beam-plasma interaction over a bandwidth of approximately 20 %, the characteristic impedance remained at a substantial level over a frequency band of several octaves. This suggests that a beam-plasma device might be rich in harmonics, provided they can be coupled out. This was, in fact, found to be the case. Experimental results indicated that in some cases the second harmonic was as little as 3-6 dB below the fundamental. In other cases, when the second harmonic was 12-15 dB below the fundamental, the third harmonic was actually several decibels larger than the second. Nonuniformities in the plasma density were not taken into account in the theoretical development, but they did play a major role in the exprrimental device. For one thing, they resulted in a considerable broadening of the frequency region over which gain could be observed. Second, the interaction length required for saturation to take place was increased by nonuniformities. Most significantly, however, it appears that nonuniformities permitted some of the higher harmonics to be generated that otherwise would not have existed. Since the density varied across the plasma column, the plasma frequency varied also. Thus the increase in plasma density toward the center of the column broadened the interaction and resulted in the Droduction of additional harmonics. Two methods of coupling into and out of the experimental device were employed. One made use of coupled-helix couplers and short sections of

HARMONIC GENERATION AND MULTISIGNAL EFFECTS

75

helical slow-wave structure at both ends of the plasma region. This method worked quite well and was used exclusively in the gain and harmonic generation tests. The main shortcoming of this method of coupling is that a slow-wave structure is required, just as in a conventional microwave tube. As the operating frequency becomes high this slow-wave circuit becomes very small, is difficult to fabricate, and is not able to handle large amounts of rf power. For that reason elliptic cavity couplers were built and placed directly around the plasma column. Being a quasi-optical method of coupling, it did not suffer from size limitations, as the coupled helices did. A coupling loss of 10 to 15 dB/cavity could be obtained over a substantial frequency region. Since high rf gain per unit length is possible in a beam-plasma interaction, this method of coupling holds great promise to yield net gain in a device similar to the one used for the experimental work described here. The difficulties encountered can be alleviated by operating the hot-cathode discharge at a lower pressure by ion trapping in the electron gun region or by optimizing the impedance match between the cavities and the plasma column.

REFERENCES 1 . L. Tonks and I. Langmuir, Oscillations in ionized gases, Phys. Rev. 33, 195-210 (1929). 2. J. R . Pierce, Possible fluctuations in electron streams due to ions, J. Appl. Phys. 19, 231-236 (1948).

3. D . Bohm and E. P. Gross, Theory of plasma oscillations. A. Origin of medium-like behavior, Phys. Rev. 75, 1851-1864 (1949). 4 . D. Bohm and E. P. Gross, Theory of plasma oscillations. B. Excitation and damping of oscillations, Phys. Rev. 75,1864-1876 (1949). 5. D. Bohm and E. P. Gross, Effects of plasma boundaries in plasma oscillations, Phys. Rev. 79,992-1001 (1950). 6. J. H . Merrill and H. W. Webb, Electron scattering and plasma oscillations, Phys. Rev. 55, 1191-1198 (1939). 7 . D. H. Looney and S. C. Brown, The excitation of plasma oscillations, Phys. Rev. 93, 965-969 (1954). 8. P. A. Sturrock, Excitation of plasma oscillations, Phys. Rev. 117, 1426-1429 (1960). 9. L. D . Smullin and P. Chorney, Propagation in ion-loaded waveguides, Proc. Symp.

Electron. Waveguides, New York, pp. 229-247 (1958). 10. J. Dawson, On Landau damping, Phys. Fluids 4,869-874 (1961). 11. A. V. Trivelpiece and R. W. Gould, Space charge waves in cylindrical plasma columns, Appl. Phys. 30, 1784-1793 (1959). 12. V. Bevc and T. E. Everhart, Fast-wave propagationin plasma-filled waveguides, J . Electron. Control 13, 185-212 (1962). 13. V. Bevc, Power flow in plasma-filled waveguides, J . Appl. Phys. 37, 3128-3137 (1966). 14. V. J. Kislov and E. V. Bogdanov, Interaction between slow plasma waves and an electron stream, in “ Electromagnetics and Fluid Dynamics of Gaseous Plasma,” Microwave Res. Inst. Symp. Ser. Vol. 1 1 , pp. 249-268. Polytechnic Press, Polytechnic Inst. Brooklyn, New York, 1962.

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15. P. J. Crepeau and T. Keegan, A beam plasma surface wave interaction, IEEE Trans. Microwave Theory Tech. 10, 391-392 (1962). 16. W. P. Allis, S. J. Buchsbaum, and A. Bers, “Waves in Anisotropic Plasmas.” M.I.T. Press, Cambridge, Massachusetts, 1963. 17. W. D. Getty and L. D. Smullin, Beam-plasma discharge: Buildup of oscillations, J . Appl. Phys. 34, 3421-3429 (1963). 18. R. Targ and L. P. Levin, Backward-wave microwave oscillations in a system composed of an electron beam and a hydrogen gas plasma, J. Appl. Phys. 32, 731-737 (1961). 19. P. Hedvall, Properties of a plasma created by an electron beam, J . Appl. Phys. 33, 2426-2429 (1962). 20. M . A. Allen, C. S. Biechler, and P. Chorney, Beam-plasma amplification for high power density applications, Proc. 5th Intern. Conf. Tubes High Frequency, Paris, pp. 435-438 (1964). 21. R. L. Ferrari, An electron beam-plasma amplifier at microwave frequencies, J . Electron. Control 17, 49-65 (1964). 22. C . S. Biechler, P. Chorney, H. S. Maddix, and R. J. Madore, Generation of plasmas for beam-plasma amplifiers, Proc. 5th Intern. Conf. Tubes High Frequency, Paris, pp. 441444 (1964). 23. G. D . Boyd, L. M. Field, and R. W. Could, Interactions between an electron stream and an arc discharge plasma, in “ Electronic Waveguides,” Microwaae. Res. Inst. Symp. Ser. Vol. 8, pp. 367-377. Polytechnic Press, Polytechnic Inst. Brooklyn, New York, 1958. 24. E. V. Bogdanov, V. J. Kislov, and Z . S. Tchernov, Interaction between an electron stream and plasma, in ‘‘ Millimeter Waves,” Microwave Inst. Res. Symp. Ser. Vol. 9, pp. 57-67. Polytechnic Press, Polytechnic Inst. Brooklyn, New York, 1960. 25. A. K . Berezin, Ya. B. Fainberg, G. P. Berezina, L. I. Bolotin, and V. G . Stupak, The interaction of strong electron beams with a plasma, J . Nitcl. Energy: P t . C 4, No. 4, 291-295 (1962). 26. A. K . Berezin, G . P. Berezina, L. I. Bolotin, and Ya. B. Fainberg, The interaction of pulsed high-current beams with a plasma in a magnetic field, J . Nucl. Energy: Pt C 6 , NO.2, 173-180 (1964). 27. M . T . Vlaardingerbroek, K. R. U. Weimer, and H. J. C. A. Nunnink. On wave propagation in beam-plasma systems, Philips Res. Rept. 17, 344-362 (1962). 28. M . T. Vlaardingerbroek and K. R. U. Weimer, On wave propagation in beam-plasma systems in a finite magnetic field, Philips Res. Rept. 18, 95-108 (1963). 29. H . Groendijk, M . T. Vlaardingerbroek, and K. R. U. Weinier, Waves in cylindrical beam-plasma systems immersed in a longitudinal magnetic field, Philips Res. Rept. 20, 485-504 (1965). 30. K. R. U. Weimer, H. Bodt, and M. T. Vlaardingerbroek, Interaction of an electron beam-plasma system with slow-wave structures, in “ Microwave Tubes,” Proc. 5th Intern. Congr. Paris, 1964, pp. 465468. Academic Press, New York, 1965. 31. M. T. Vlaardingerbroek and K. R. U. Weimer, beam-plasma amplifier tubes, Philips Tech. Rev. 27, 275-284 (1966)l 32. F . W. Crawford, Beam-plasma interaction in a warm plasma, Intern. J . Electrot? 19, 217-232 (1965). 33, W. C. Hahn, Small signal theory of velocity-modulated electron beams, Gen. Elec. Rev. 42, No. 6, 258-270 (1939). 34. J . E . Simpson, Cold-beam-plasma interaction theory for finite transverse dimensions and finite magnetic fields, J. Appl. Phys. 37, 42014207 (1966).

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35. K. S. Karplyuk and S. M. Levitskii, Dispersion equation of the electron stream in plasma in the presence of a magnetic field, Soviet Phys.-Tech. Phys. (English Trans/.) 9, No. 8, 1063-1066 (1965). 36. V. E. Golant, A. P. Zhilinskii, I. F. Liventseva, and I. E. Sakhara, Electromagnetic radiation from a magnetized plasma traversed by an electron beam, Soviet Phys,Tech. Phys. 10, No. 11, 1559-1565 (1966). 37. H. L. Stover, Microwave and Electron Beam Interactions with a Finite Plasma, Tech. Doc. Rept. No. RADC-TDR-64-117, Techniques Branch, Rome Air Development Center, Griffiss Air Force Base, New York (1964). 38. J. E. Rowe, Harmonic currents in nonlinear beam-plasma interactions, in “Microwave Tubes,” Proc. 5/11 Intern. Congr. Paris, 1964, p p . 427431. Academic Press, New York, 1965. 39. J. E . Rowe, “Nonlinear Electron-Wave Interaction Phenomena,” Academic Press, New York, 1965. 40. J. E. Rowe, Nonlinear Beam-Plasma Interactions, Tech. Rept. No. 69, Contract No. AF33(657)-8050, Electron Physics Laboratory, Department of Electrical Engineering, The University of Michigan, Ann Arbor, Michigan (1964). 41. R. W. Could and M. A, Allen, Large signal theory of beam-plasma amplifiers, in “Microwave Tubes,” Proc. 5th Intern. Congr. Paris, 1964, pp. 4 4 5 4 9 . Academic Press, New York, 1965. 42. B. Geidne, Nonlinear Analysis of a Microwave Beam-Plasma Amplifier, Res. Rep. No. 52, Research Laboratory of Electronics, Chalmers University of Technology, Gothenburg, Sweden (1965). 43. C. A. Brackett, Harmonic Current Generation and Multi-Signal Effects in BeamPlasma Systems, Tech. Rept. No. 110. Contract No. DA-28-043-AMC-I315(E), Electron Physics Laboratory, Department of Electrical Engineering, The University of Michigan, Ann Arbor, Michigan (1968). 44. A. K . Berezin, G . P. Berezina, L. I. Bolotin, Yu. M. Lyapkalo, and Ya. B. Fainberg The interaction of modulated high-current pulsed electron beams with plasmas in a longitudinal magnetic field, J . Nucl. Energy: Pt. C 8, No. 3, 289-400 (1966). 45. J. Allison and G. S. Kino, A directly radiating beam-plasma oscillator, in “Microwave Tubes,” Proc. 5th Intern. Cong. Paris, 1964, p p . 378-382. Academic Press, New York, 1965. 46. A. Dattner, A study of plasma resonance, Proc. 5th Intern. Conf Ionization Phenomena Gases, Miinich, 1961, p p . 147771483, North-Holland Publ., Amsterdam, 1962. 47. J. C. Nickel, J. V. Parker, and R. W. Could, Resonance oscillations ina hot nonuniform plasma column, Phys. Rec. Letters 11, No. 5, 183-185 (1963). 48. S . Kronlund, Measurements on a Beam-Plasma Mici-owave Amplifier, Res. Rept. No. 68, Research Laboratory of Electronics, Chalmers University of Technology, Gothenburg, Sweden (1966). 49. C. S . Biechler and P. Chorney, The plasma fillcd cavity as a coupler for beam plasma amplifiers, Proc. 6th Intern. Conf Microwave Optical Generation Ainplifcation, Canibridge, 1966, p p . 515-519. IEE, Cambridge, 1967. 50. K. K . Chow, Applied Research on Methods of Coupling to Electromagnetic waves in Plasma Amplifiers, Tech. Doc. Rept. No. AL-TDR-64-150, Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio (1964). 51. J. Feinstein, The conversion of space-charge wave energy into electromagnetic radiation “ Electronic Waveguides,” Microwace Res. Inst. Sj’mp. Ser. Vol. 8, pp. 345-352. Polytechnic Press, Polytechnic Inst. Brooklyn, New York, 1958.

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52. J. Neufeld and P. H. Doy!e, Electromagnetic interaction of a beam of charged particles with plasma, Phys. Rev. 121,654-658 (1961). 53. M. A. Allen, G . S. Kino, J. Spalter, and H. L. Stover, Electron beam-plasma interaction, “Microwaves,” Proc. 4th Intern. Congr. Microwave Tubes, Seheveningen, 1963, pp. 327-333. Centrex Publ., Eindhoven, The Netherlands, 1963. 54. H. L. Stover, and G . S. Kino, A field theory for propagation along a non-uniform plasma, in “Microwave Tubes,” Proc. 5th Intern. Congr. Paris, 1964, pp. 374-378. Academic Press, New York, 1965. 55. M. A. Allen, C. S. Biechler, and H . S. Maddix, R F coupling to a! electron beam through a plasma, Appl. Phys. Letters 4, No. 6, pp. 107-108 (1964). 56. E. A. Pashitskii, Interaction of an electron beam with a plasma in a magnetic field. I. Longitudinal oscillations, Soviet Phys.-Tech. Phys. 8, No. 1, 34-38 (1963). 57. E. A. Pashitskii, Interaction of an electron beam with a plasma in a magnetic field. 11. Transverse oscillations, Soviet Phys.-Tech. Phys. 8, No. I , 3 9 4 2 (1963). 58. M. F. Gorbatenko, Interaction of an electron beam with a plasma existing in a magnetic field, Soviet Phys.-Tech. Phys. 8, No. 9, 798-804 (1964). 59. M. A. Heald and C. B. Wharton, “ Plasma Diagnostics with Microwaves,” Wiley, New York, 1965. 60. V. Ya. Kislov and E. V. Bogdanov, The interaction of slow plasma waves with an electron flux, Radio Eng. Electron ( U S S R ) (English Transl.) 5 , 145-162 (1960). 61. A. H. W. Beck, “Space-Charge Waves,” Pergamon Press, Oxford (1958). 62. V. Ya. Kislov, E. V. Bogdanov, and 2. S. Chernov, Physical foundations of plasma applications for generation and amplification of microwaves. Advan. Electron. Electron Phy.~.21, 287-332 (1965). 63. M . Chodorow and C. Susskind, “ Fundamentals of Microwave Electronics.” McGrawHill. New York, 1964. 64. G . T. Konrad, Harmonic Generation in Nonlinear Beam-Plasma Systems, Tech. Rept. No. 112, NASA Research Grant No. N G L 23-005-183, Electron Physics Laboratory, Department of Electrical Engineering, The University of Michigan, Ann Arbor, Michigan (1969). 65. G . M. Branch and T. G . Mihran, Plasma frequency reduction factors in electron beams, I R E Trans. Electron Devices 2, 3-11 (1955). 66. J . S. Cook, R. Kompfner, and C. F. Quate, Coupled helices, Bell System Tech. J. 35, 127- 178 (1956). 67. G . T. Konrad and S. K . Cho, Elliptic cavity couplers for traveling wave tubes, IEEE Trans. Electron Devices 10, 85-89 (1 963). 68. N . Marcuvitz, “Waveguide Handbook,” MIT Radiation Lab. Ser. Vol. 10. McGrawHill, New York, 1951. 69. S. Ramo and J . R . Whinnery, “Fields and Waves in Modern Radio,” 2nd ed. Wiley, New York, 1953. 70. R. Sasiela, Exploratory Development on Coupling Schemes for Plasma Amplifiers, Microwave Associates, Inc., Tech. Rept. AFAL-TR-68-109, Air Force Avionics Laboratory, Air Force Systems Command, Wright-Patterson Air Force Base, Ohio (1968). 71. D. A. Dunn, W. Nichparenko, J . E. Simpson, and K. I. Thomassen, Oscillations and noise generation in a beam-generated plasma, J . Appl. Phys. 36, 3273-3275 (1965). 72. S. M. Levitskii and I. P. Shashurin, Diffusion theory of a beam-generated plasma, Soviet Phys.-Tech. Phys. (English Transl.) 10, No. 7, 915-919 (1966). 73. J. Frey, Interaction components in beam-generated beam-plasma systems, J . Appl. P h r ~39, . 1887-1889 (1968).

The Formation of Cluster Ions in Laboratory Sources and in the Ionosphere ROCCO S. NARCISI Air Force Cambridge Research Laboratories, Bedford, Massachusetts AND

WALTER ROTH" The K M S Technology Center, San Diego, California LABORATORY SOURCES ............................ I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . ............................ 11. Experimental Techniques . . . . . . . . . . . . . . 111. Results ............................................................... A. Water Clusters ..................................................... .......................... B. Clusters Other Than Water.. . . . . .

19 80

82 82 94

THE IONOSPHERE 95 96 .......................... 96 B. Adsorption Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 C. Rocket Payload and Preflight Operations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 VI. Measurements. . . . . . . 99 A. Water Cluster Ion Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 B. Other D-Region Ion Species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 C. MeteoricIons ... .. . . .. . . . . . . . . .. .. .. .. .. .. .. . . . . . . . . . . . . . . .. .. .... . 102 D. Major E-Region Ion Species. . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. . . . . . 104 . . . . . . . . . . . . . . . 105 E. Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 VII. Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 A. Laboratory Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............... 108 B. Ionospheric Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............... 112 VIII. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 V. Experiment.

LABORATORY SOURCES I. INTRODUCTION The profound influence of water on the mechanisms and rates of gas phase chemical reactions has been recognized for many years. In flames, explosions,

* Present address : Diagnostic Instruments, I9

Inc., San Diego, California.

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ROCCO S. NARCISI A N D WALTER ROTH

and gas discharges, traces of water vapor often provide a source of H and OH free radicals, which enter into chain reactions and frequently are the initiators of reactions which, under extremely dry conditions, are much more difficult to promote. The reactions involving these radicals generally have relatively low activation energies and the regeneration of the H and OH accounts for the strong influence of H,O at very low concentration levels. Within the last decade, improvements in mass spectrometers and mass spectrometric techniques and the development of new techniques for the study of fast reactions in gas discharges and in ion beams, have resulted in much more detailed studies of ion-molecule reactions than previously possible. When reaction systems involve ions and polarizable molecules, ion-dipole forces may be sufficiently strong to form stable bonds and the new charged entities will have different mobilities, molecular weights, charge exchange cross-sections, recombination coefficients and reaction rate coefficients. These changes in properties may profoundly affect the relative ionic species concentration, the electron concentration, and the electron energy spectrum. The ramifications continue into areas of modern technology. The absorption, refraction, and transmission of radar waves interacting with a plasma depend upon the plasma frequency, which varies as the square root of the electron concentration. Thus, the presence of polarizable molecules in the ionosphere will affect the ability of a radar unit to “see” into the ionosphere and, if a sufficient concentration of polarizable molecules surrounds a reentry vehicle, its radar cross-section will be altered. The first observation, in the gas phase, of molecules composed of ions surrounded by polarizable molecules (cluster ions) was reported in 1959 by Beckey ( I ) who detected the hydrogen ion hydrates, H,O+, H 5 0 2 + H, , 03+, and H , 0 4 + . Subsequently, these observations were confirmed and extended (2) to H+(H20),, , and the hydrogen ion was discovered to attract clusters of CH,OH, C,H,OH, CH,OCH,, HCOOH, CH,CHO, and CH,COCH, (1-7). As may be expected, the hydrogen ion is not alone in its ability to attract and bind polarizable molecules. In other work, it has been shown (&If), that the solvated ions, NH4+(NH3),l,NO+(H,O), , NO,+(H,O), , O,+(H,O), , and Na+(H,O),l are formed. Negative ions have a similar ability and the ion clusters, 0-(H,O), , 0,-(H,O), , F-(H,O), , Cl-(H,O),, , Br-(H,O),, , I-(H,O),, NO,-(H,O),, , and OH-(H,O), have been observed under a variety of conditions (5,12-16). In this review, those cluster ions that may be formed in the ionosphere and corresponding laboratory observations will be considered.

11. EXPERIMENTAL TECHNIQUES A mass spectrometer of some type is required for analysis of the cluster ions, but their formation has been observed in a variety of sources.

THE FORMATION OF CLUSTER IONS

81

Inghram (17), Anway (7), and Beckey ( I ) used field-emission ion sources in conjunction with conventional magnetic mass spectrometers. Beckey (2) improved his system's detectability by the addition of electrostatic lenses to focus the ions on the entrance aperture of the mass spectrometer. Field (18) simply used the electron gun ion source of a magnetic mass spectrometer to create the ions. An electron multiplier was used as the detector. Knewstubb was the first to discover cluster ions in flames (19) and in direct current (dc) glow discharges (20). A magnetic mass spectrometer with an electron multiplier detector was used for mass analysis. In addition, a variable slit was located in the plane of the energy spectrum and allowed selection of any energy increment in the range of 2 to 25 eV. Similar work by Shahin ( I O ) involved a positive wire to negative cylinder corona discharge. Ions were extracted from a small hole in the side of the cylinder and focused by electrostatic lenses to a most probable energy of 45 eV into a quadrupole mass spectrometer. An electron multiplier was used in conjunction with an electrometer for ion detection. The combination of the quadrupole mass spectrometer and differential pumping allowed mass analysis of ions from corona discharges at pressures up to 1 atm. Kebarle (5, 13, 21, 22) irradiated He, Xe, and N,, in admixture with relatively small amounts of H,O, with wparticles from a 200-mCi Po source. Protons surrounded by clusters of water molecules were observed by focusing the ions formed, mass analyzing by means of a 90" magnetic mass spectrometer, and detecting with an electron multiplier. The ions were sampled by projecting a cone with a leak at its apex into the irradiated volume. Water clusters around the negative ions F-, C1-, Br-, I - , 0,-, and NO,- were observed in the same way. In later work, Kebarle (5, 6, 22) and Searles (23) used a 100-KeV proton beam penetrating lo-' in. of Ni foil as the ion source. The analysis and detection systems remained the same. The a-particle and proton beam ionization (8, 9 ) were used to produce ions consisting of NH, molecules clustered about N H 4+ ions. In his most recently reported work, Searles (23) used a proton-beam mass spectrometer that had the ability to be pulsed. A quadrupole mass spectrometer was used for mass analysis. Sieck ( 3 ) observed as many as three molecules of CH,OH or C,H,OH clustered about a single proton when using a time-of-flight mass spectrometer with a high pressure source. Observations of water clusters around negative ions are more recent, Moruzzi (12) used two different electron sources to produce negative ions in pure water vapor as well as H,O-0, and H,-0, mixtures. One source was a hot thoriated iridium filament and the other was a Au or Pd film irradiated by an ultraviolet lamp. The ions were admitted to a drift tube coupled to a radio frequency (rf) time-of-flight mass spectrometer with an electron multiplier detector. Electron energies were controlled by controlling E/p, the ratio of electric field strength to pressure. In this way, he was able to detect the

82

ROCCO S. NARCISI AND WALTER ROTH

cluster ions, (H,O),O,~, (H,O),O-, and (H,O),OH- with n 5 5 , with (H,O),O,- and (H,O),OH- predominating among their homologues. Golub (16) reported the observation of (H20)OH- in a double focusing mass spectrometer which sampled ions from a hot cathode arc discharge in a n NH,-H,O mixture. 111. RESULTS

A . Water Clusters I . Arourid Protons

In experiments with a field emission ion source ( I ) , gas pressures in the Torr and were reduced by differential pumping source were as high as to lop6 Torr in the mass spectrometer. When a mixture of H,O and D,O was investigated, mass peaks at 19, 20, 21, and 22 were observed. These corresponded to H 3 0 + , H,DO+, H D 2 0 + , and D 3 0 + . With H,O alone, as the pressure was increased, the intensity of the H 2 0 + peak increased almost linearly. At the same time, new mass peaks corresponding to H,O.H,O+ and (H,O),.H,O+ were observed. The intensities of these species also increased at an almost linear rate with pressure but the slopes of the intensity pressure curves for the latter two species were about half that for H 3 0 + .In addition, the intensity of H , 0 . H 3 0 + was always 10 times that of (H,0),.H30+. Later (2), with improved ion optics, and the ability to cool the tungsten tip of the field emission source, observations were made at - 195°C and clusters as high as (H,O),,H+ were seen.' The distribution of intensities as a function of 17 is shown in Fig. 1, where it may be seen that the predominant ion is (H,O),H+ and a subsidiary peak exists at n = 7. The relative concentrations were measured to teinperatures up to -78 C as well and the equilibrium constants were calculated for the reactions and assuming [H,O+] was a constant fraction of [HzO]. From the temperature dependencies of the equilibrium constants, the heats of the reactions were calculated as AHl = 4.0 & 0.8 kcal/mole and AH2 = 3.7 0.7 kcal/mole, about equivalent to the energy of a hydrogen bond. Anway (7) used a At this point, there is no intent to indicate structure by use of the notations (H,O),H', Instead, the notation used in the publication being dis(H,O),-lH,O+ or (HZn+lOn)+. cussed is generally adopted.

83

THE FORMATION OF CLUSTER IONS

?

P

FIG.1. Distribution of (H20)"Ht ions in a field emission source.

g i -

v)

t

0)

*

c

c 0 -

n

similar technique and observed H,O+-nH,O with IZ up to 3. He also detected H,Of. Studies of ionic reactions in pure water were made ( 4 ) at pressures up to 0.4 Torr in the ion source of a magnetic mass spectrometer. The source temperature was 200°C. The electron energy in the source was sufficiently low that H 2 0 + was the only ion formed by electron impact. This was followed by the rapid reaction. HzOf

+ HZO

+ H30f

+ OH.

(3)

At the pressure in the source and the residence time, essentially all of the H 2 0 + had reacted according to (3). Under these conditions, H,O+ was the predominant ion but smaller amounts of cluster ions up to H 9 0 4 + were observed. At about0.3 Torr,the relativeconcentrationsofH,O+, H,Oz+, H 7 0 3 + , andH,O,+ were0.73,0.23,0.03, and 1.O, respectively. Except for atrendtoward a peak at H904+,the results are quite different from those given in Fig. 1. It was found, in addition, that the ratio [H502']/[H30'] increased linearly

84

ROCCO S. NARCISI AND WALTER ROTH

with the square of the pressure and this suggested a third body stabilization collision, e.g., H30' + HzO HsOz+*, (4) H502+*

+ H2O

-+

H&+

+ H,O*,

(5)

where * indicates an excited state. Experiments in pure H,O and in Ar + H,O mixtures (5, 21, 22) using a-particle or proton excitation were in agreement with each other indicating that diffusion and flow were not playing a significant role. The proton beam work was used to calculate equilibrium constants, heats of reaction and entropies of reaction for the clustering reactions M

H3O+

+ H2O tH,O+(H,O)

'

(0, l),

(6)

where M represents a stabilizing collision with a third body. At equilibrium,

AFt-,,,

=

-RTln K n - l , n=

-

'H30'

RTln

(H20),

~

('H,O

(HLO), - I*'HIO

).

(9)

where P is the partial pressure of the subscript molecule. In a series of experiments PH20was varied from 0.1 to 6 Torr and the ratios of the partial pressures of the ionic species were measured. This allowed an evaluation of K n - l , and A F ; - , , n . In addition, at fixed pressure, the temperature was varied over the range of 250 to 1000°K and AHfl- and AS,were deterversus I/T mined from the slopes and intercepts, respectively, of log Kn-l,n plots. The results are shown in Fig. 2 from which it may be seen that AHnis a smoothly varying f u m b n of n - 1, n without any discontinuities to suggest a special stability for some of the cluster ions as indicated in Fig. I . The point at n - 1, n = - 1, 0 in Fig. 2 corresponds to the proton affinity of H,O. Studies were made in corona discharges in N, , 0 2 ,and air at pressures between 50 Torr and 1 atm containing H,O vapor at concentrations between 1.2 x lo-' and 0.65 mole % (10). Ions up to ( H 2 0 ) 8 H + were observed. The distribution of species (H,O),H' is shown in Fig. 3 for coronas in air at 1 atm pressure and various H,O concentrations. It is evident from the curves that, at low H,O concentrations, the n = 4 cluster ion is the favored specie but, as the H 2 0 concentration is increased, the preference shifts toward higher n, n = 6 being the most abundant ion at the highest H,O concentration studied. The same trend appears to hold for corona discharges i n N, and 0, at pressures of 1 atm. In air, at a constant H,O concentration of 0.65 mole %,

,,

85

THE FORMATION OF CLUSTER IONS

400

100

c

30 20

10 0.1

-1.0

1.2

2.3

(n

3.4 --

5.6

+

* H30f(H20),.

I ) ,n

FIG.2. Enthalpies for the reactions H 3 0 + ( H 2 0 ) n - I H 2 0 140 120

1

I

1

6,7

4,5

I

I

I

I

I

-

-

loo-

-

.t

60

-

0

0

1

2

3

4

5

6

7

8

9

n FIG.3. Distribution of (H,O),H+ in a corona discharge in air at 1 atm pressure and various H,O concentrations. (-) 65 x mole %, (- -) 12 x lo-, mole %, (-. -) 6.5 x lo-' mole %, (--) 4.1 x lo-' mole %, ( - . . -) 3 x lo-' mole ",).

86

ROCCO S. NARCISI AND WALTER ROTH

70

r

I

I

I

1

I

I

60 -

0

50

0'

I

2

.O

150 200 250 Discharge pressure, t w r

100

I

4

I

I

I

6

8

10

300

I

12

350

I

14

Accelerating region pressure, twr x lo4

FIG.4. Distribution of (H,O).H+ in a corona discharge in air, containing 0.65 mole % H,O, as a function of air pressure. The parameter is n.

as the pressure is increased from 50 Torr, the preferred n decreases until, at 350 Torr, / I = 1 is the most abundant ion. This is illustrated in Fig. 4. It seems clear that -collisions with H,O molecules lead to higher homologues and collisions with air molecules are destructive and lead to lower homologues. Cluster ions were produced (15) in the high pressure ion source of a quadrupole mass spectrometer and then introduced with controlled kinetic energy into a chamber for dissociating collisions with inert gas targets. Relative cross sections were measured for loss of one D 2 0 molecule from D,O,+, D , 0 3 + , and D , 0 2 + . I n addition, it was found that for the dissociative reactions,

%: !

H,O,+

"2H 5 0 2 +

H,03+ -

2 0

AH3 -H z 0

-+

H30+,

(10)

A H , was 0.25 eV (5.8 kcal mole). AH2 was 0.75 eV (17.3 kcalimole), and A H 3 was 0.95 eV (21.9 kcal/mole). Sampling of the gases from H,-O, flames has shown (19) the presence o f cluster ions up to H , 3 0 t ( H , 0 ) , with evidence to suggest that higher hydrates d o not exist in the flames. The mass spectrum including the isotopic distribution is shown i n Fig. 5 and the distribution of the three most prominent hydrates is shown as a function of distance along the flame axis in Fig. 6. I t may be seen in the latter figure that H 3 0 + reaches its maximum concentration in a very hot region of the flame while the higher hydrates reach their concentration maxima in the relatively cool part of the flame before reaction has

m~il THE FORMATION OF CLUSTER IONS I

p

-0

I

I

1

I

1

87

1

10%

7

+

e

10

_o

lo6 0

10

)

Moss number

FIG.5. Mass spectrum of H2-02 flame gases.

4

Reaction zone (opproxi

Distance along axis of flame gases Imm)

FIG.6. Relative concentrations of ions of masses 19,37, and 55 as a function of distance along the axis of a H2-02 flame.

begun, and then are rapidly destroyed in the reaction zone. The addition of a fine spray of an aqueous solution of a potassium salt resulted in a decay of the H 3 0 + concentration which was in accord with the increase of K + concentration. It appeared clear that the charge exchange reaction K+H,O++K+ +HzO+H

was responsible for this.

88

ROCCO S. NARCISI AND WALTER ROTH

Analysis of ions from the negative glow and Faraday dark space of dc discharges in water vapor (20) revealed the presence of cluster ions from H,O+ to H,O+(H,O), . It was speculated that the H 3 0 + was formed i n the rapid reaction (3) and that successive clustering follows in three-body reactions. The dominant cluster ion was found to be a function of the electric field between the cathode and the sampling probe. This was interpreted as an effect of the field on the translational energies of the ions in which the decomposition of the clusters by collision occurred more readily as the translational energies increased. The result was that the distribution of ion compositions was dependent upon sampling technique. It was also concluded that the stabilities of the clusters (bond strengths) decreased with successive hydration steps and fell sharply beyond H,O+(H,O), . In addition, up to the formation of H,O+(H,O),, the rates of successive hydrations appeared to increase in proportion to the increase of cluster ion size.

2. Around Other Positive Ions a. NH4+. Kebarle ( I I ) found that a-particle irradiation of water vapor at pressures of 1 to 5 Torr in admixture with several parts per million of NH, resulted in the reaction H3Ot'WHzO

+ NH3 -tNH,+.nHZO + (M. + 1 - r?)HzO

(12)

which is the equivalent of a proton transfer reaction. Higher concentrations of NH, lead to the complete disappearance of H,O+WH,O. These results are to be expected since the proton affinity of NH, is about 35 kcal higher than that of H,O. In addition, since H,O and NH, are isoelectronic, their polar-

FIG.7. Comparative solvation of (a)

NH,+ and (b) H 3 0 + by H 2 0 at I Torr H,O at 23°C.

3

4

5

6

7

n = Number of H20 molecules

THE FORMATION OF CLUSTER IONS

89

izabilities may be compared by comparing their molar refractions. The latter values are 2.75 cm3 for H,O and 5.67 cm3 for N H , . Thus, NH, is about 50 % more polarizable than H z O and the ion induced dipole forces would be correspondingly stronger. Data were obtained at pressures of 1 to 6 Torr of H,O. Figure 7 indicates the ion intensity distribution at 1 Torr. NH,+.nH,O were observed for / I as high as 6 with 12 = 4 the dominant ion. At the same time, w = 5 was dominant for H,O+.nH,O. At higher temperatures, / I for the dominant ions decreased by 1 for each cluster ion. It seems likely that, for NH,+.nH,O, n = 4 constitutes a first shell of water molecules around a symmetrical N H 4 + ion i n which the positive charge is shared equally by all five atoms. I t is not clear why, under the same conditions, n = 5 predominates for H 3 0 + . n H 2 0 . On the basis of an equally shared charge on the three hydrogens of a symmetrical H 3 0 + , 17 = 3 would be expected to dominate. In earlier work (f9),in flames of H, + 0, + N,, NH,+.nH,O was observed in the cooler parts of the flames with ?I up to 4. It was assumed that the parent ion was formed in the proton transfer reaction H30+

+ NH3

+ H20

+ NH4'.

(13)

In contrast to these results, corona discharges in N, at atmospheric pressure containing 7.5 x lo-, mole %, of water vapor, produced NH,'.3H20 as the dominant cluster. At the same time, H 3 0 + . 3 H , 0 dominated the H,O+.HH,O distribution but was produced i n lower concentration than the dominant NH4+ cluster (10).I n addition, the ions (NH3)z(H20),,-2H+and (NH3)3(H20)n-3H+were observed in lesser amounts with the dominant ions occurring at n = 4. In both these cases, there was no evidence to suggest the distribution in the molecules of the positive charge. b. Na'. In the course of his work on comparative solvation of NH,' and H,O+, Kebarle (If) observed ions at masses 59, 77, 95, . . . , 149 which he attributed to Na+.nH,O arising from a sodium impurity and resulting in water cluster ions up to n = 7 with a peak in the ion distribution spectrum at n = 5. The peak shifted to 12 = 4 at higher temperature, just as it did for NH,+.nH,O. It appeared from the results that the n = 4 and n = 5 clusters around NH,' and N a + had about equal heats and free energies of formation. By analogy with reaction (If), it is probable that the N a + was formed by NaW

+ H30+(g)-tNa+(g) + H20(g)+ H(g)

(14)

for which AH is about -24 kcal/mole. c. N O + , N O z + , and 0,'. In the studies involving corona discharges in moist air (fO),in addition to the ions (H,O),H+, the clusters NO+(H,O),, and NOz+(H,O), were observed. Figure 8 indicates ion distributions in air discharges at atmospheric pressure for several different water vapor pressures.

90

ROCCO S. NARCISI AND WALTER ROTH

n

FIG.8. Variation of relative abundance of (NO+)(H,O). (left) and (NO,+)(H20), (right) versus n, for the same conditions as in Fig. 3. The water concentrations for the different curves are: (----), 4.1 x mole %; (-. . -), 3.0 x lo-' mole %; (-. -), 2.0 A lo-' mole %; (-), 1.7 x lo-' mole %.

The peaks in the distributions occur at n = 2 for the high water concentrations and at n = 1 for the lower concentrations. Presumably, at still lower water concentrations, the parent ions NO+ and NO,+ would appear alone with no associated water. The binding energies of the water in the NO' and NO2+ clusters are unknown and accordingly, it is not known whether exchange reactions of the type

occur. However, work on photoionization of NO in a mixture of NO and H,O (24, 36) has shown that the following reactions occur. NO+

+ NO + NO -tNO+.NO + NO

(16)

+ HZO + M +NO+.(n + 1)HzO + M

(19)

NO+.nHZO

Reaction (20) suggests that water clusters around NO' in the presence of excess H,O would tend to form water clusters around protons. In the same work (lo), corona discharges in 0, containing 1.7 x lo-, mole "/, H 2 0 produced O,+(H,O), with n ranging from 1 to 3 and n = 2 dominating.

91

THE FORMATION OF CLUSTER IONS

3. Around Negative Ions

a. 0,- and 0-. The ions 0,- and 0- were produced (12) in an electron drift tube. In the presence of water, the following reactions occurred HZO.02or + H , O + M + or 0H2O.O0 2-

(21)

+M

(HzO)n- 1.01(Hz0)n.O~or +HZO+M+ or +M (H,O)n- 1.0(Hz0)n.O-

(22)

with n as high as 5. When the pressure of H,O was reduced to Torr, the only negative cluster ions observed were H,0.02- and (H,O),.O,-. While the authors suggest a third-body requirement in reactions (21) and (22), there was no experimental evidence to support this, and, especially for the higher hydrates, it is likely that the reactions are two body with the reaction energy being partitioned over the many degrees of freedom of the cluster ions. At E/p between 0.5 and 5.0 V cm-' Torr-', the predominant ion is (H,0),.02while at E/p between 6 and 10 V cm-I Torr-' the ions (H,O);Opredominate. b. OH-. In the same work (12), but with pure H,O, the negative ions observed were (H,O);OH- with n = 5 the highest member. The relative ion intensities were dependent upon E/p and the distributions are shown in Fig. 9. A suggested mechanism for the formation of the ions is N

HzO

+ e + H - + OH

H-+H,O+OH-+Hz OH(HZO).-I.OH-

+ HzO + HzO +HzO'OH- + HzO + HzO + HzO (H2O)"'OH- + HzO. +

(23) (24) (25) (26)

Here, reaction (24) is greater than or equal to 0.3 eV exothermic. Again, there is no experimental evidence to support the third-body requirements in reactions (25) and (26). When oxygen is added to the system, the OH- cluster ions disappear and the authors state that the 0- and 0,- cluster ions are formed by reactions (21) and (22). The possibility of exchange reactions of the type (H20):OH-

+

0-

or 0 2-

+ (H20);0H-

+

(HzO)n- ,.Oor (H20)n--m.02-

(27)

was not considered. In a hot cathode-arc discharge through an ammonia-water vapor mixture at a pressure of 0.05 Torr, the ion [OH(H,O)]- was observed (16). There

92

ROCCO S. N.4RCISI AND WALTER ROTH

I

5

200 E/P.Vcm-'iorr

FIG.9. The relative intensity of various ions as a function of E/p in water vapor. No negative ions were observed at low values of E / p . However, at high E/p values His the initial negative ion and is rapidly converted to OH-. Clustering of water molecules around this ion then occurs to form (H20):OH- ions. Ions with ti = 1-5 have been observed. ( 0 ) OH-, (0) H-, (0) (H20),.0H-, (m) (HZ0)5.0H-, (A) ( H 2 0 ) 3 . 0 H - , (0) (HZO)~.OH-, (A)(HzOWH-.

was no information on the structure of the ion or the distribution of the charge. The ion was observed to undergo photodetachment via a transition to a repulsive neutral state to form the products OH and H,O. c. Halogen negative ions. An a-particle ion source was used (13) to bombard various halogen compounds in an oxygen carrier gas to produce halogen atom negative ions. The compounds used to produce the respective negative ions were NF, , CCl,, CH,Br, , and CH,I. C1- could also be produced by addition of CHCI,, CH2Cl,, and CH,CI. The F- was produced in the exothermic reaction, 0 2 -

+ NF3 + + F - + NF2. 0 2

(28)

In an argon carrier gas, the NF, directly captured electrons and presumably dissociated to NF, + F-. When water vapor was admitted to the system at a pressure of 1 Torr, the ions, X-(H20), were observed with the distribution

THE FORMATION OF CLUSTER IONS

93

50

0 50

0 50

0 n ( n = number of H20 molecules 1

FIG.10. Experimentally measured equilibrium distribution of hydrates X-(HzO). at 1 Torr water pressure and 292"K,showing shift to lower hydration number with increase of ionic size. (a) F - , (b) CI-, (c) Br-, and (d) I-.

shown in Fig. 10 from which it is evident that the number of water molecules in the dominant clusters decrease as the ion size increases reflecting the stronger dipolar interactions at the smaller distances encountered with the smaller ions. When a comparative study was made (5) of the relative hydration of C1and H,Of, the interactions of H,O with the latter ion were found to be stronger, again because of the smaller effective radius. For example, at 1 Torr water pressure and 300"K, the dominant clusters were H30f.(H,0)5 and Cl-.(H20), . The same effect is demonstrated by a comparison of AF;-,,, values obtained from equilibrium constants at 290°K and listed in Table I. It is of interest that the difference in free energies between the two species decreases as the progression goes toward higher n - 1, n. To some extent this is due to a decreased importance of the central ion because of extensive shielding by the clustered water molecules.

94

ROCCO S. NARCISI AND WALTER ROTH

TABLE I AF;-_,,. AT 290°K a , b

n-l,ti

CI-

H30

2, 3 3,4 4, 5 5, 6

5 3.15 3 2.5

9.2 5.15 4.3 3.2

+

+

For the reactions H30+.(H20),-1 H 2 0 + H,O+.(H,O). and CI-.(H20),-1 HZO + Cl-.(HzO)n Data from Kebarle (5).

+

B. Clusters Other Than Water I . N H , around NH,'

Kebarle (8, 9, 1I, 23) analyzed the ions produced by a-particle bombardment of NH, and observed NH,+.nNH, with n as high as 7 and a function of temperature and pressure. Low temperature and high pressure favored the production of species with high n. This is illustrated in Fig. 11. The results were used in an equation similar to Eq. (9) to calculate equilibrium constants and AF for the clustering reactions NH,+.(n - 1)NHA +NH3 + N H . + + . H N H ~

(29)

and log K versus 1/T plots were used to determine A H and AS. The values are given in Table 11.

I

I

-

-

I

FIG.1 I . Schematic representation of ion intensities for clustered NH4+.rrNH3.(a) 10 Torr, 23°C; (b) 1 Torr, 23°C; (c) 1 Torr, IOO'C.

L 2

3

4

"

THE FORMATION OF CLUSTER IONS

95

TABLE I1 THERMODYNAMIC DATA AT 298°K FOR AMMONIA CLUSTERING REACTIONS OF THE TYPESHOWN IN EQ. (29) n

-

1, n

0, 1 1,2 2, 3 394 4,5

AF(kcal/mole)

AH(kcal/mole)

AS' (eV)

- 17.5

-21.0 - 17.0 - 17.8 -15.9

- 32.0 -26.8 -34.0 -36.0 -25.0

-8.0 - 6.4 -3.8 -0.2

-1.5

When water vapor was added to the system, the mixed clusters, NH,+-nNH,wH20 were observed in addition to NH,+.rNH, and NH,+.sH,O. The results clearly indicate that solvation of NH,' by NH, was preferred over H 2 0 . This is in accord with the results reported in Section A.2.a and relates to the higher polarizability of NH, . The change in the distribution of NH,+.nNH, ions with pressure suggested than an inner shell of four NH, molecules clusters around the NH,' ion and that, when n > 4,a second shell is formed.

2. N O around NO' A krypton resonance lamp was used to produce NO+ in an afterglow cavity containing NO and H,O (23). I n addition to the clusters NO+-nH,O, the ion NO'.NO was observed. The data suggested that a steady state exists in which the latter ion is formed by NO+ + N O + N O +NO+.NO + N O

(30)

and lost by ambipolar diffusion to the walls. The rate constant for reaction (30) was found, from loss of NO' and formation of the dimer, to be 5.0+ 0.8 x lo-,' cm6/sec. The ambipolar diffusion coefficient of NO+.NO was 84 & 13 cm2 Torr/sec at 294°K. T H E IONOSPHERE

IV. INTRODUCTION Water cluster ions H+(H20), were first measured in the earth's lower ionosphere in late 1963 with a rocket-borne mass spectrometer (25). Subsequent measurements over the past six years have essentially confirmed and extended the earlier results and showed water cluster ions to be of dominating

96

ROCCO S. NARCISI AND WALTER ROTH

importance from 85 km down to 60 km, the lower altitude limit of the measurements (26). This part of the review will be primarily restricted to a discussion of rocket-borne positive ion composition measurements in the D region (50-90 km), although some E region (90-140 km) measurements will be presented to show the distributions of atmospheric ion species at higher altitudes. It is the purpose of this review to describe the experimental techniques, to summarize the rocket measurements and their uncertainties and finally to discuss the ionospheric theories which have been proposed to explain the origin of the water cluster ions in the D region. Several problem areas are also described with the hope of stimulating further work in these areas. Negative ions are not considered in this part of the review, although these ions are of practical significance because they play an important role in determining the free electron density in the D region. Rocketed mass spectrometer measurements of negative ions in the D region have just begun in earnest over the past year or so and only a few measurements are available. However, it is worth noting here some recent evidence on the presence of negative ion clusters in the lower ionosphere. A rocket experiment was launched at 2100 hr local time on October 10, 1969 over Fort Churchill, Canada by the Air Force Cambridge Research Laboratories. The results indicate that the negative ions were predominately located below 90 km where ions with mass number close to 62-, 80- (most abundant), 98-, 116-, 134-, and 152- amu were measured. These ions were the dominant species in the D region and were tentatively identified as NO,-(H,O), with n = 0-5.’ Clearly, additional rocket measurements as well as future laboratory studies will be required before the negative ion situation in the D region is resolved. V. EXPERIMENT

A . Mass Spectrometer

A cryopumped quadrupole mass spectrometer is used for rocket-borne positive-ion composition measurements between 60 and 150 km (25, 27). A cross section of the instrument as it appears in a typical rocket-sampling configuration is shown in Fig. 12. Positive ions are drawn into the 0.030-in. inlet aperture by means of a - IOU placed on the forward plate. The ions are further accelerated into the field of the quadrupole rods where rf and dc voltages applied to the rods can be adjusted so that only those ions with a

* See E. E. Ferguson, F. C. Fehsenfeld, and D. K. Bohme, Planet. Space Sci. 17, 1759 (1969), concerning the origin and importance of NO3- in the D region.

THE FORMATION OF CLUSTER IONS

97

Surface heater Copper heat shield Liquid N p port Pumping cones Rocket skin Liquid nitrogen Electrical feedthrus

FIG.12. Cross section of the cryopumped quadrupole system in the sampling configuration. The quadrupole rods are 3 in. long and 0.150 in. in diameter.

specific mass-to-charge ratio will have the bounded oscillatory trajectories necessary to traverse the length of the rods to the electron multiplier. All other ions have amplitudes of oscillation that grow exponentially with time in the transverse direction and are removed from the ion beam. By removing only the dc voltage from the rods, the quadrupole can be operated in the highpass filter mode in which all ions above a preselected mass number can be focused by adjusting the rf voltage. The focused ion current is then amplified by an electron multiplier with a typical gain setting of lo4 and the output current from the multiplier is measured with a logarithmic electrometer which has a range from to lo-’ A. This permits a dynamic range of roughly 1 to lo5 ions/cm3 in the ionosphere. A mass scan is obtained by varying the rf and dc voltages with time such that the voltage ratio dc/rf is constant; this maintains a constant mass resolution over the mass range. With the quadrupole operating at a fixed frequency at about 5 MHz, a mass resolution between 16 and 30, defined at 10% peak height, is possible, and ample for most ionospheric measurements. The quadrupole is calibrated before flight using a standard electron impact ion source and known gases and gas mixtures. Flight mass spectra which show how the rocket measurements are programmed are shown and discussed later.

98

ROCCO S. NARCISI AND WALTER ROTH

B. Adsorption Pump Because of the relatively high ambient pressures in the mesosphere, a high speed, high gas capacity pump is required to maintain the spectrometer in its Torr. The atmospheric gas pressure operating pressure range, below 5 x Torr from 50 to 90 km. Since the rocket measuredecreases from 0.6 to ments are made at the stagnation point behind a detached bow shock, the gas pressure is further enhanced in front of the inlet orifice. For a Mach 3 vehicle, the pressure increases a factor of ten across the shock wave up to about 80 km, above which the shock is not well defined. The flyable pump is shown schematically in Fig. 12. Surrounding the conical support structure for the quadrupole rods is a liquid-nitrogen-cooled zeolite adsorption pump capable of pumping speeds exceeding 100 liters/sec at pressures near Torr. A pumping area of 2000 cm2 of zeolite is obtained by bonding the adsorptive zeolite onto seven concentric stainless steel cones. The cones are nested in a vacuum jacketed, stainless steel, liquid nitrogen dewar. Before flight, the system is pumped and baked at 160°C for over 10 hr to free the zeolite of most of its adsorbed gases. After cooling, the system is sealed off at a pressure in the IO-'Torr range. The system vacuum is monitored with a special thermocouple gauge both prior to and in flight. With the aid of the cryopump, measurements have been performed as low as 59 km, but the pump performs more reliably for measurements above 70 km. A vacuum cap which seals the aperture of the mass spectrometer is carried off by a split nose-cone which is ejected near 70 km on rocket ascent.

C. Rocket Payload and Preflight Operations

In addition to the quadrupole mass spectrometer, a typical rocket scientific package contains total charged particle sensors, such as cylindrical Langmuir probes, for simultaneous measurements of ion densities, electron densities and electron temperatures. The rocket payload also contains support instrumentation such as a vehicle aspect system, fmjfm telemetry capable of transmitting 16 clear analog channels of information, and a radar beacon for tracking. After the payload is mounted on the rocket, a dry nitrogen purge of the payload is begun 5 hr before cooling with liquid nitrogen and continued up to launch. The heater power for the resistive rod surface heater (Fig. 12), which maintains the forward surface of the mass spectrometer at 202C, and the liquid nitrogen are then turned on simultaneously. All this is done to reduce water vapor contamination in flight. About 3 hr of cooling are required to ensure proper pumping efficiency. Liquid nitrogen is supplied up to launch through a frangible nylon fitting, which breaks away upon first motion of the rocket.

99

THE FORMATION OF CLUSTER IONS

These experimental probes are carried aloft by two-stage solid propellant rockets ; such combinations as Nike-Cajun, Nike-Hydac, and Nike-Iroquois have been flown. With these types of boosters measurements are possible between 60 and 150 km. VI. MEASUREMENTS A . Water Cluster Ion Measurements

The rocket measurements of water cluster ions in the D region reported herein were performed exclusively by the Aerospace Composition Branch of the Air Force Cambridge Research Laboratories. There exists only one other rocket experiment reported recently by Goldberg and Blumle (28) in which water cluster ions were also detected in the D region. In a daytime flight designed to test the operation of a quadrupole system with a titanium sublimation pump, their spectrometer was not activated until 98 km on vehicle ascent, but on descent measurements in the vehicle wake showed masses 19' and 37+, H,Of and H 5 0 2 + ,as predominant ions below 80 and down to 64 km. The very first rocket measurements of ionospheric water cluster ions were made in October, 1963 (25) and are shown in Fig. 13. To date, successful measurements have now been performed on 23 passes through the D region from 13 rocket flights. In three cases, only either ascent or descent data were

37

601 I

' '

+

T+

'

I j " t l '

10

'

I I l I '

' '

lo2

I 1 1 1 1 1 '

10'

I

'

l I I L

I0 '

Ion density- lons/cc

FIG. 13. First measurements of water cluster ions in the D region. Mass 18+ has not been detected on subsequent flights. Mass 28+ is probably Si+, other mass number identities are given in the text. Meteoric ions which appear a t 83 km are not shown.

100

ROCCO S. NARCISI A N D WALTER ROTH

obtained. Results from four of these flights have been published (29, 30). In all cases, the results showed that water cluster ions dominated the ion composition of the D region below about 82 km. This was found to be true during the day and night, through sunrise and sunset, during an aurora, a meteor shower, a sporadic E event and a total solar eclipse. Generally the ion 3 7 + , H 5 0 2 + ,dominates the ion composition in the D region below 82 km with 19+, H 3 0 f , and 5 5 + , H 7 0 3 + ,also present but of lesser abundance. Heavier ions greater than 40 amu are typically less than 10% of the total ionization. The NO+ ion which was predicted to be the predominant ion in the D region is generally much less abundant than H,O,+ in the daytime midlatitude D region, but its relative abundance is found to be greater at higher latitudes. At midlatitudes in the night and at twilight, no NO+ ions are found below about 85 km. Other positive ion species are also present and are discussed below. To present a very rough picture of the measured variations of the water cluster ion density with time of day, consider the midlatitude, 10-km region between 75 and 85 km. Here, at midday, [H502+] is about lo3 ions/cm3; near midnight this decreases to less than or approximately equal to 100 ions/cm3. In this same altitude region, the H 5 0 2 +ions decay by roughly a factor of four through sunset from 400 to 100 ions/cm3, corresponding to a change in solar zenith angle from 88.5 to 98.6", and increase by about the same factor at sunrise from about 30 to 100 ions/cm3 for solar zenith angles of 102.5 and 90", respectively. There are considerable difficulties in obtaining accurate absolute ion densities below 85 km and, therefore, the values quoted above are subject to large errors, factors of two and more. The relative changes in density, which can be further checked by applying the calibrated instrument sensitivities in the laboratory to the measured ion currents in flight, are perhaps more accurate. B. Other D-Region Ion Species

To illustrate the types of positive ions found in the lower ionosphere, flight mass spectra obtained in the D region are shown in Figs. 14 and 15. These measurements were made near sunset over Fort Churchill, Canada by using a highly sensitive instrument to measure the minor constituents in the D region, in an attempt to obtain clues concerning the ion chemistry. It should also be pointed out that the ionization density over Fort Churchill is considerably greater, by a factor of 5 to 10, than that at midlatitudes for this same time of day. It is believed, although no unusual ionospheric activity was observed, that this enhanced ionization is due to precipitating energetic electrons, which are usually present at these auroral latitudes. Only two of several telemetry outputs are shown in Figs. 14 and 15 ; the dc

THE FORMATION OF CLUSTER IONS

101

FIG.14. The D region mass spectra obtained on rocket ascent in vehicle ram near Mach 4. [From Narcisi, in " Planetary Electrodynamics," Vol. 2, p. 69. Gordon and Breach, New York. Reproduced by permission of the publishers.]

sweep monitor which is one of the outputs used to determine mass number and the electrometer output showing the mass spectra. The mass spectra were scanned from 10 to 62 amu and the total ion current for masses larger than 41 amu was measured periodically, every 1.5 sec. The electrometer was calibrated in flight after every fourth spectrum sweep. The ions detected 24+-26+(Mg+). and our identifications are: 19'(H30+), 21 '(H3l8O'), 30+(NO+),32+(S+ below 86 km, O,+ above 86 km), 34+(S+),37+(H502'), 39+(H,'60180+),52+(Cr+), 54+-56+(Fef), 58+-60+(Ni+). For the following mass numbers, the identifications are less certain and possibilities are presented: 41 +[Na+.(H,O)], 46+(NO2+), 48+[(NO+.(Hz0) with possibly a smaller admixture of SO'], 50+[(02+.(HZO) also possibly with some

34so+]. As it is difficult to ascertain directly the relative abundances of the ions from the five-decade log scale in Figs. 14 and 15, this is shown in Fig. 16.

102

ROCCO S . NARCISI AND WALTER ROTH

FIG. 15. The D region mass spectra obtained on rocket descent. The current modulations reflect the motion of the vehicle which is backing into the atmosphere and severely coning with angles of attack varying between 180" and 90" down to 70 km. The absence of the larger mass ions below 75 km is probably due to cluster fragmentation and also collisional loss below 65 km. [From Narcisi in "Planetary Electrodynamics," Vol. 2. p. 69. Gordon and Breach, New York. Reproduced by permission of the publishers.]

Sulfur ions have been detected in the D region only since 1967, and whether the presence of the ions is of a transient nature has yet t o be determined. Sulfur ions were seen to be layered between 70 and 86 km at Fort Churchill, but five other rocket flights at midlatitudes detected these ions only in very thin layers near the mesopause a t 86 km.

C . Meteoric Ions One of the most striking and common features in the lower ionospheric measurements is seen in both the ascent and descent spectra in Figs. 14 and 15. This is the drastic transition in ion composition which occurs at the mesopause, about 86 km, where the water cluster ions (and sulfur ions) disappear completely and metal atomic ions appear abruptly, all within 2 km. Although

103

THE FORMATION OF CLUSTER IONS

lo2

1

.

I

1-

40

50

60

Mass number

FIG.16. The relative abundances of D region positive ions at 82 km over Ft. Churchill, Canada near sunset.

it is not the subject of discussion here, it is perhaps of interest to review briefly some of the measured characteristics of the metal ion distributions. Metal atomic ions and ions of some oxides of these metals have been measured on a total of 17 Air Force Cambridge Research Laboratories (AFCRL) rockets (31). These ions were always found'to be located predominantly between about 82 and 120 km and on two occasions, they were also detected at higher altitudes. Some typical and fairly consistent characteristics of the metal ion distributions are a layer composed mostly of magnesium and iron varying between 5 and 10 km in half-width with a peak density near 93 km and other narrower, higher altitude layers composed mainly of either silicon or magnesium, or iron. The lower altitude layer near 93 km is found to contain, in addition to magnesium and iron, smaller amounts of sodium, aluminum, calcium, nickel, and other metals. The relative abundances of these metal ions in the lower layer are in fair agreement with the relative

104

ROCCO S. NARCISI AND WALTER ROTH

abundances of these metals found in chondrite meteorites. Although differing in absolute densities, the different metallic species show remarkably similar altitude profiles. The magnitudes and altitudes of the narrower, higher altitude layers vary considerably, whereas the lower layer is more stable i n comparison. During conditions of high meteoric influx and sporadic E (which at midlatitudes was found to be composed entirely of metallic ions), the metal ion content in the lower ionosphere was seen to be significantly enhanced. Many of the physicochemical processes involving the metallic species are presently unknown, and a considerable amount of laboratory work and rocket experimentation will be required before the metal ion distributions can be explained [see Narcisi (31)].

D. Major E-Region Ion Distributions In general, the molecular ions of NO" and 0," dominate the ion composition in the E region with the meteoric ion species varying between 10 and 100 "/, of the total ionization over certain altitude intervals. The NO' is responsible for the ionospheric ledge near 82 km above which this ion rises steeply in density and becomes predominant. During the daytime, 0,' is significant, being about equal to the NO' density in the upper E region, while at night 02' is considerably less abundant than NO'. One example of the measured ion distributions is shown in Fig. 17. These are the results from a rocket fired at sunrise. The mass number assignments discussed previously apply to Fig. 17. 140 130 120

5

ll0

U m

-2

-

100

6

90 80

I

701 ' *'.""' ' ".'"'' Ioo I 0' I0'

'

""""

I

o3

'

'

"'"''!

I 0'

1J

"""': I

o5

Density- lons/cc

FIG.17. Concentrations of major positive ions in the D and E regions at sunrise, for a solar zenith angle of 90".

105

THE FORMATION OF CLUSTER IONS

E. Uncertainties

Errors due to contaminant gases, especially water vapor, have been argued to be small, and estimated errors of factors of two and more in the derived positive-ion densities below 85 km have been discussed (25, 29). The most serious problem, and therefore the one worthy of some discussion, arises from possible fragmentation of the cluster ions by the sampling method. This can occur either by collisional fragmentation with the energy for dissociation supplied by the draw-in electric field or by thermodynamic decomposition in the shock-heated gas. Preliminary laboratory experiments have been performed, simulating conditions close to the rocket experiment, in which gases with flame-produced ions, H,O+ and H 5 0 2 + ,were expanded with a Mach 2 nozzle and sampled with a quadrupole mass spectrometer (32). It was found that both the H,O,+ and H 3 0 + ions were essentially unaffected by either the electric field o r high shock-temperatures. However, fragmentation becomes more likely for the more weakly bound larger clusters. Electric field fragmentation of the larger clusters is presently under study in both laboratory and rocket experiments. 105

100

95

90 E

1

I

-=

Q 0)

.-

85

2

80

/

/--------

75

/ /*

/70 lo-'2

,I

lo-"

I

1 , 1 1 1 1 1

10-'O

I

I

I I I,,

Io-'

I

1

1

1

1

b

1

Ida

Current- amps

FIG.18. Results from a Mach 2 vehicle launched during a total solar eclipse. The ion currents were smoothed from a superposition of ascent and descent data. The 55' layer is not the result of the total solar eclipse as it was also measured on another similar rocket in sunlight prior to totality.

106

ROCCO S. NARCISI AND WALTER ROTH

Results of such studies should be available shortly. Evidence on the thermodynamic decomposition of the larger clusters is available from both sources. From laboratory measurements of Kebarle et al. (22), rate constants for M, can be reactions of the type H+(H20), M + H + ( H 2 0 ) n - 1 H,O calculated as a function of temperature. For example, the effective lifetimes near 7 5 km of H+(H20), for T = 370", 600", and 900°K corresponding to shock stagnation poini temperatures at Mach 2, 3, and 4, respectively, are sec as compared to a characteristic time of and roughly 0.03, about sec for an ion to traverse the distance from the rocket shock front to the sampling orifice. Clearly, H + ( H 2 0 ) , will start to decompose near Mach 3 from shock heating alone. This effect has also been observed in the rocket measurements. For rockets approaching speeds of Mach 4, relatively little H+(H,O), is measured, while at Mach 2 these ions are enhanced below 80 km, as shown in Figs. 15 and 18. Thus, although water cluster ions dominate the D region composition, there is some question as to their true relative abundances, especially below 80 km. Measurements at subsonic speeds with the present sampling geometry and at supersonic speeds with conical probes in which the sampling orifice at the cone apex penetrates the shock wave so that the undisturbed free stream is analyzed will be attempted in the future, in order to improve the lower altitude measurements.

+

+

+

VII. DISCUSSION A . Laboratory Results It has been amply demonstrated that many molecules cluster about a variety of ions to form new, higher molecular weight, ionic entities. Ion induced dipoles are clearly involved in forming the new bonds, and accordingly, the facility with which the bonds are formed is dependent upon the polarizability of the interacting molecule. It has been difficult to determine structures or charge distributions for the clusters. One reason for this difficulty is that, undoubtedly, during the process of polarization of the molecule, the ion itself becomes polarized, and this results in very small changes in the interaction forces as a function n. It is only after the reaction is complete that a redistribution of charge may occur. A statistical-mechanical treatment of ion cluster formation was developed (33) for ions, atoms, and molecules of hydrogen, but is equally applicable to other ion-molecule systems. The general reaction considered is A B+ A.B+ for which the equilibrium constant K may be calculated from partition functions F as follows :

+

K

= ( F A B + / F A. FB+)exp(-E,/kT).

+

THE FORMATION OF CLUSTER IONS

107

The orbiting model of Langevin (34) has been developed in detail (35) and considers the ion as a point charge whose interaction energy with a structureless polarizable molecule is given by: E,

=

-xe2/2r4,

(32)

where x is the polarizability of the molecule, e is the electronic charge, and r is the internuclear distance between molecule and ion. If we treat A.B+ as a diatomic ion and A and B’ as monatomics, then FA,+

=

[8n2Z,.,+ kT/h2][2 s i n h ( h V , . , + / k T ) ] [ ( 2 ~ ~ , . , + k T ) 3 ’ 2 / h 3(33) ],

F A= ( 2 ~ r n , k T ) ~ ’ ~ / h ~ ,

F,+

(34)

= (2nrn,+k~)3/2/h3,

(35)

where I,.,+ = pr2 with /I being the reduced mass of the A + B+ system. Substitution of (32)-(35) in Eq. (31) and assuming a temperature of 300°K and a vibrational frequency of 1000 cm-‘, gives

K

= [A.B+]/[A][B+] = 0.716(r2/p”2)exp(277a/r4)

moles-’ cm3,

(36)

where ct and r are expressed in units of cm3 and angstroms, respectively. Since in most of the experimental work absolute measurements of [A] are made, while only relative values of [B’] and [AvB’] are known (36) may be used to calculate [A.Bf]/[B+]. Alternatively, the equilibrium constants measured by Kebarle et al. (5) for the (H,O+)vzH,O system may be used to calculate internuclear distances in an attempt to elucidate structure. This procedure involves the assumption that the ion remains a point charge as the cluster ion grows in size. This assumption is valid for the thermal interactions reviewed here inasmuch as the slow relative approach of ion and molecule allows sufficient time for the ion to become polarized as well as the molecule. The internuclear distances rn-l, were calculated for the interactions: H30+.(n - 1)HzO

+ H20

+-H30C.fiHZ0.

(37) The results are illustrated in Fig. 19 and indicate an inflection point when H 3 0 + . 3 H 2 0is f ~ r m e dThis . ~ is more clearly demonstrated in the same figure where the second derivative of rn-l, n , determined by differencing, is plotted. Thus, while Fig. 2 indicates that AHn-l, varies smoothly withn, thevariation of r n - l , n indicates that a first completed shell is probably formed with H 3 0 + . 3 H 2 0and the interaction forces involved in forming the second shell vary in a different manner with n. The results are insensitive to the value of the vibrational frequency of the cluster ion.

108

ROCCO S . NARCISI AND WALTER ROTH

21 20

d2r dn2

008

FIG.19. Calculated internuclear distances and their second derivatives for the H,O'.(n - 1) H 2 0 - H 2 0 bonds.

0 06 004

0.02

OJ

1,2

2.3

3.4

4,5

5,6

0

(n-l),n

B. Ionospheric Theories

In the midlatitude D region, the three presently known major sources of ionization during quiet solar periods are solar Lyman CI ionization of nitric oxide (70-90 km), solar ionization of electronically excited 02('Ag) (75-90 km), and galactic cosmic rays (below 70 km). During solar disturbances, solar flare X-rays can considerably increase the ionization in the D region. At high polar latitudes, following certain solar flares, the D region ionization can also be substantially enhanced either by precipitating energetic electrons, or by energetic protons, observable at night by the accompanying aurora. During polar cap absorption events, precipitating protons can increase the free electron density so that there is a complete absorption of high frequency electromagnetic radiation in the D region. After the initial ionization, the resulting ion composition is determined by the subsequent ion chemistry. Only recently, vapor phase ion processes have been proposed to explain the origin of water cluster ions in the D region, but there are still problems to be resolved. From laboratory measurements, Fehsenfeld and Ferguson (36) have derived a qualitative reaction sequence for converting 02+to water cluster ions; this scheme was also independently derived in large part by Good et a/. (37) who were also able to measure the

THE FORMATION OF CLUSTER IONS

109

rate constants for most of the reactions. The reaction scheme is the following with the quoted rate constants for T = 300°K: kl =2.8 x

cni6/sec

k z = 1.2 x

-

(39)

3 x lo-''

cm3/sec

(40)

k3a 9 x lo-''

cm3/sec

(40a)

k3

kSb k,

-

cm3/sec

(38)

-

cm6/sec lo-'"

cm3/sec(est.)

(40b) (41)

k, - 3 x cm6/sec k - 5 : 1.4 x lo-', cm3/sec

(42)

k6 - 2 x k-6 =4 x

(43)

k,

=

k-,

=6

cm6/sec cm3/sec cm6/sec

(44)

cni3/sec.

x

Ferguson et al. (38) obtain k,,/k, 10 instead of the -3 of Good et al. The difference is attributed to reaction (40) being endothermic, and to the production of vibrationally excited 0 , + ( H 2 0 ) in reaction (39). The H,0/02 ratio in the experiment of Good et al. is higher, so that there is less vibrational relaxation between reactions (39) and (40). The H,O/Oz ratio in the experiment of Ferguson et al. is closer to D-region conditions. Other reactions for which Good et al. measured the rate constants at T = 300°K play a role:

+ HZ0 + Nz H+(HzO)z+ N L, H+(H,O)Z+ HzO + N Z * H+(Hz0)3+ N z ,

H30t

+

+

H+(HzO)~ H20

+ Nz 2 H+(H20)4+ N2

ks

- 3 x lo-''

k9 k-,

= : =

cm6/sec

(45)

2.3 x 1.9 K

cm6/sec cm3/sec

(46)

x

cni6/sec cni3/sec.

(47)

k,,

= 2.4

k-lo

=3

x

Another possible reaction route following (40a) and leading to H + . ( H 2 0 ) 3 determined by Fehsenfeld and Ferguson (36) is :

+ + M + H,O+.OH.OZ + M H3O+'OH'O2 + HzO + H3O+.OH'H,O + H,O+.OH.HzO + HzO H ~ O + . ( H Z O+ ) ZOH. H,O+.OH

(48)

0 2

(49) ( 50)

0 2

+

The ion H,O+.OH in (40a) is argued not to be the dimer ion (H20)2', because of the fast reaction H,O+ + H,O -+ H,O' + OH, k cm3/sec. Puckett (24) has isotopically identified this ion and also H,O, '.OH ceasing speculation that these ions might be NH,+.(H,O),, .

-

110

ROCCO S. NARCISI A N D WALTER ROTH

It is to be noted that, the rate constants may change considerably at D region temperatures of 160 to 240'K. The reverse reaction rate constants for reactions (42)-(44) and (46) decrease significantly with declining temperature, while the forward reactions increase with decreasing temperature, favoring larger cluster formation. It is now possible to derive the water cluster ion distribution in the D region in a semiquantitative manner. Several parameters are unknown and can only be estimated from theoretical approaches and laboratory experience. Some of the unknown parameters are: the neutral water vapor density distribution in the mesosphere, recombination coefficients at D region temperatures for water cluster ions both with electrons and negative ions, photodissociation cross-sections for the water cluster ions, temperature dependencies of the rate constants, and others. Ferguson et al. (38) performed a calculation estimating some of the unknown parameters. Starting with an 02+production rate as calculated by Hunten and McElroy (39) using the 02('Ag) distribution measured by Evans et a/. (40) and with a theoretical estimate of the water vapor concentration by Hesstvedt (41) of I t o 5 ppm, they were not only able to show that the 0,' ions produced can be quickly converted to the water cluster ions, but also that the derived ion densities, at least near 80 km, were very close to the observed densities. The calculations also suggest that the water vapor concentration decreases. rapidly above the mesopause to account for the sharp decrease in the water cluster ion density there, but this explanation may not be unique. The reaction scheme proposed by Fehsenfeld and Ferguson (36) was applied to the quiet midlatitude D region but it omits the NO+ problem. At midlatitudes, there is a need for a fast reaction sequence to convert the NO' ions to water cluster ions, since NO' from atmospheric measurements of both [NO] and Lyman x appears to be produced at a substantially faster rate than 02+ yet , NO' is not observed as the predominant ion below 80 km. Fehsenfeld and Fergiison found that following three, three-body cluster reactions the exothermic chain breaking reaction is NO+(HzO),

+ HzO

-+

H+.(H,O),

+ HNO,.

The three-body rate constant for the initial hydration of NO' NO+

+ HlO + M +NO+(H,O) + M

is 1.5 0.5 x cmh/sec at 293'K (24). This reaction chain is too slow compared to dissociative recombination of N O C with ionospheric electrons to be significant i n the D region. A possible fast exothermic reaction was recently suggested by Burke ( 4 2 ) , NO+(H,O) + H + N O + H,O+ which also applies to 0 , + ( H 2 0 ) but no rate constants have yet been measured nor is the concentration of hydrogen atoms in the mesosphere accurately known.

THE FORMATION OF CLUSTER IONS

111

Although not noted by Fehsenfeld and Ferguson, their reaction scheme is more directly applicable to disturbed conditions. It turns out that there is much less of a problem in explaining the origin of water cluster ions from energetic ionization sources which can ionize any atmospheric constituent more or less indiscriminately. Basically, any of the major constituent ions N,', Ar', etc., will, immediately after initial ionization, convert to 0,' by charge transfer, M + + O2 + 0,' M, k z lo-'' cm3/sec and [O,] > 1014/cm3below 85 km. The 0,' produced then initiates the above reaction sequence starting with Eq. (38). Furthermore, during moderately disturbed periods the ionization created by the usual solar radiation represents only a small portion of the total ionization and can be neglected. While the above reaction scheme represents a significant breakthrough regarding D region ion chemistry, other D region observations are not explainable by this scheme. For instance, a pair of rocket experiments were conducted around sunrise. The first rocket was fired at 102.5' solar zenith angle when the entire D region was in darkness. The second rocket was launched at sunrise when the entire D region was exposed to light but when all the hard radiation and even Lyman a were strongly attenuated or absorbed above 85 km and yet there was a factor of three increase in the H+.(H20), density in the second shot. There are considerable errors involved here, but if these measurements are believed, this suggests that water cluster ions are produced by fairly low energy solar radiation. Also the presence of sulfur ions in the D region is not understood. The ions at masses 32 and 34 amu were identified as sulfur ions, on the basis that the ratio of their concentrations was equivalent to the abundance ratio of the naturally occurring sulfur isotopes. The sulfur ions appear to be restricted to below 86 km, suggesting that they may be terrestrial in origin. Sulfur has been identified (43) in particles collected from noctilucent clouds; very thin clouds limited to the mesopause near 82 km and presumably composed of cometary dust particles coated with ice. How the sulfur (I.P. 10.36eV) is ionized is unknown. It is unlikely that atomic sulfur exists in the neutral state below 85 km because it can be readily oxidized. The location of the sulfur ion layer near the mesopause and the precipitous fall-off of the sulfur ion concentration along with a similar drastic decrease in density of the water cluster ions perhaps indicate a connection between the sulfur and water in the form of an aerosol or conglomerate. Prior to the knowledge of the vapor phase reactions, discussed earlier, H u n t (44) hypothesized that the simultaneous ionization and fragmentation of small water conglomerates, assumed to be present in the mesosphere and ranging in size up to cm could be the source of the ions of the type H + ( H 2 0 ) , . Furthermore, if the small water-conglomerates have very low ionization thresholds, the water cluster ions could be easily produced in the D

+

112

ROCCO S. NARCISI AND WALTER ROTH

region. Experiments designed to measure the ionization appearance potentials of small water-conglomerates created by a free jet expansion of water vapor are being carried out but no reliable results are presently available. Whether or not mesospheric aerosols or conglomerates play a significant role in the D region ionization remains to be determined. An approach to the sulfur problem would be to conduct laboratory studies of the ionization of aerosols containing sulfur. Such experiments may shed some light as to the origin of atomic sulfur ions in the D region. The sharp cut-off of metal atomic ions below about 85 km is another inexplicable observation. Causative processes are under investigation including the possibility of rapid clustering reactions of the metal-atomic ions with water vapor.

VIII. CONCLUSION The existence of cluster ions in the ionosphere and under a variety of laboratory conditions is indisputable. There remains, however, a considerable disparity among laboratory observations of the mass spectral distributions of these ions. It appears that sampling techniques may be at the root of these differences. Ionospheric observations may be similarly at fault, although the sparsity of experiments and experimental groups in this area does not permit a rigorous conclusion at this time. The preponderance of laboratory results and some cursory theoretical calculations suggest that there is a special stability for the cluster ion consisting of a singly charged arrangement of four water molecules and a hydrogen atom. At this time there are no conclusive experimental or theoretical results to suggest any definitive structure or charge distribution for these ions. The reader should be aware that the first observations of cluster ions were only a decade old at the time of preparation of this review and that this review of the published literature and private communications extends only through February, 1970. A considerable amount of work has been stimulated by these early observations and publications in this field have been burgeoning at a rapidly increasing rate. Accordingly, it is expected that this review will serve the purpose of sparing future workers in this field the tedium of reviewing the literature prior to early 1970.

REFERENCES 1. H. D. Beckey, Z . Naturforsch. 14a, 712 (1959). 2. H. D. Beckey, Z. Naturforsch. 15a, 822 (1960). 3. L. W. Sieck, F. P. Abramson, and J. H. Futrell, J . Chem. Phys. 45, 2859 (1966). 4 . M. S. B. Munson, J . Am. Chem. SOC.87, 5313 (1965). 5. P. Kebarle, S. K. Searles, A. Zolla, J. Scarborough, and M. Arshadi, Aduan. Muss Spectrometry 4, 621 (1967).

THE FORMATION OF CLUSTER IONS

113

P. Kebarle, R. M. Haynes, and J. G . Collins, J . Am. Chem. SOC.89, 5753 (1967). A. R. Anway, J. Chem. Phj.s. 50, 2012 (1969). A. M. Hogg and P. Kebarle, J . Chern. Phys. 43, 449 (1965). A. M. Hogg, R. M. Haynes, and P. Kebarle,J. A m . Chem. Soc. 88,28 (1966). M. M. Shahin, J . Chem. Phys. 45, 2600 (1966). P. Kebarle, Advan. Chern. 72, (1968). J. L. Moruzzi and A . V. Phelps, J . Chem. Ph.ys. 45, 4617 (1966). P. Kebarle, M. Arshadi, and J. Scarborough, J . Chcm. Phys. 49, 817 (1968). L. J. Puckett and W. C. Lineberger, 21st Guseoiis Electronics Conf., Boulder, Colorado, p. 261, October, 1968. 15. M. De Pas, J. J. Leventhal, and L. Friedman, J . Chem. Phys. 49, 5543 (1968). 16. S. Golub and B. Steiner, J . Chenz. Phys. 49, 5191 (1968). 17. M. G. lnghram and R. Comer, 2. Naturforsch. 10a, 863 (1955). 18. F. H. Field, J. L. Franklin, and M. S. B. Munson,J. Am. Chem. Soc. 85, 3575 (1963). 19. P. F. Knewstubb and T. M. Sugden, Proc. Roy. SOC.(London),A255, 520 (1960). 20. P. F. Knewstubb and A. W. Tickner, J . Chem. Phys. 38,464 (1963). 21. P. Kebarle and A. M. Hogg,J. Chem. Phys. 42, 798 (1965). 22. P. Kebarle, S. K. Searles, A. Zolla, J. Scarborough, and M. Arshadi, J . Am. Chem. SOC.89, 6393 (1967). 23. S. K. Searles and P. Kebarle, J . Phys. Chem. 72,742 (1968). 24. L. J. Puckett and W. C. Lineberger, private communication, Ballistic Research Laboratories, Aberdeen, Maryland; Bull. Am. Phj’s. SOC.14, 261 (1969). 25. R. S. Narcisi and A. D . Bailey, J . Geophys. Res. 70, 3687 (1965). 26. R. S. Narcisi, in “ Planetary Electrodynamics” (S. C. Coroniti and J. Hughes, eds.), Vol. 2, p. 69. Gordon and Breach, New York, 1969. 27. A. D. Bailey and R. S. Narcisi, Znstr. Paper 95, AFCRL-66-148 (1966). 28. R. A. Goldberg and L. J. Blumle, J . Geophyys. Res. 75, 133 (1970). 29. R. S. Narcisi, Ann. Geophys. 22, 224 (1966). 30. R. S. Narcisi, Space Res. 7 (R. L. Smith-Rose, ed.), p. 186. North-Holland, Anisterdam, 1967. 31. R. S. Narcisi, Space Res. 8 (A. P. Mitra, L. G. Jacchia, and W. S. Newman, eds.), p. 360. North-Holland, Amsterdam, 1968. 32. R . R. Burke and W. J. Miller, private communication, AeroChem Research Laboratores, Inc., Princeton, New Jersey (1969). 33. H. Eyring, J. 0. Hirschfelder, and H. S. Taylor, J . Chem. Phys. 4, 479 (1936). 34. M . P. Langevin, Ann. Chern. Phys. 5, 245 (1905). 35. G. Gioumousis and D. P. Stevenson, J . Chem. Phys. 29, 294 (1958). 36. F. C. Fehsenfeld and E. E. Ferguson, J. Geophys. Res. 74, 2217 (1969). 37. A. Good, D . A. Durden, and P. Kebarle, private communication, Univ. of Alberta, Edmonton, Canada (1969). 38. E. E. Ferguson, F. C. Fehsenfeld, and D. K . Bohme, J . Geophys. Res. 74, 5743 (1969). 39. D. M. Hunten and M. B. McElroy, J . Geophys. Res. 73, 2421 (1968). 40. W. F. J. Evans, D. M. Hunten, E. J. Llewellyn, and A. Vallance Jones, J. Geophys. Res. 73,2885 (1968). 41. E. Hesstvedt, Geophys. Pub(. 27, 1 (1968). 42. R. R. Burke, J. Geophys. Res. 75, 1345 (1970). 43. R. A. Skrivanek, in “Noctilucent Clouds” (I. A. Khvostikov and G. Witt, eds.), Intern. Symp. (Tallinn), p. 135 Moscow, 1967. 44. W. W. Hunt, Jr. in “Meteorological and Chemical Factors in D-Region Aeronomy” (C. F. Sechrist, Jr., ed.), Record 3rd Aeronomy Conf., Aeronomy Rept. No. 32, p. 311. Univ. of Illinois, Urbana, Illinois, 1969.

6. 7. 8. 9. 10. 11. 12. 13. 14.

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The Measurement of Lifetimes of Free Atoms, Molecules, and Ions A. CORNEY The Clarendon Laboratory, University of Oxford, England

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ 116 elationships for Electric Dipole Radiation. . . . . . . . . . . . . . . . . . 116 B. Justification for the Measurement of Lifetimes . . . . . . . . . C. Outline of Experimental Techniques for the Measuremen D. Techniques Not Discussed in Detail-the Direct Measurement off-Values. . 123 11. The Direct Observation of Exponential Decays Using Analog Methods. ....... 124

A. Experiments Using Electron Bomba

A . Details of the Technique.

.....

............................

159

. . . . . . . . . 169 B. Use of Optical Excitation ................................... 171 C. Use of Electron Bombardment Excitation.. . . . . . . . . . 178

B. Discussion of Experimental Results

A. Introduction

.........

...........................

C. Details of Experimental Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Experimental Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Discussion of Experimental Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Miscellaneous Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Methods Involving Las ............... B. Metastable Levels . . . . ...............

115

206 21 1 215 218 218 219

116

A . CORNEY

I. INTRODUCTION A . Theoretical Relationships for Electric Dipole Radiation

I . Spontaneous Emission It is well known that atoms, ions, and molecules in excited electronic levels tend to decay spontaneously to low energy levels of their respective systems by the emission of electric dipole radiation. (Often the atom or ion will possess one or more metastable levels from which decay by electric dipole radiation to the true ground level of the system is forbidden.) The time constants governing these decays lie typically in the range 10-9-10-6 sec, and the wavelengths of the emitted radiation lie mostly in the range 100 to 20,000 A. The reason for this spontaneous decay is most easily understood on the basis of the classical theory of electromagnetism. If we represent the excited electron in an atom by a harmonically bound point charge e which oscillates at the angular frequency wo , then the accelerated electron becomes a source of electromagnetic waves of angular frequency w o . The energy W stored in the oscillating system decreases at a rate given in CGS units by

- d Wjdt = e2wO4xo2/3c3,

(1)

where xo is the maximum amplitude of the motion, c is the velocity of light, and e is in esu ( I ) . Since the damping term introduced into the equation of motion of the oscillating electron by the energy loss through radiation is small, an approximate solution yields

Wt>= W(0)exp( - t b C J where

T~~

(2)

is the classical lifetime of the excited atom given by zC1= 3rnc3/2e2w02

(3)

For the 6 'P, level of mercury which emits radiation of wavelength 1850 8, Eq. (3)gives a value of zC1= 1.39 x lo-' sec which agrees well with the experimental value of z = (1.31 +_ 0.08) x sec obtained by Lurio (2). This agreement between the classical theory and the experiment is also found in other investigations involving the interaction of light and matter (3) and is useful because of the ease with which classical models may be visualized. However, it is unwise to attempt to apply the classical theory to transitions other than the resonance lines. To obtain results which are of general applicability, it is necessary to consider the interaction of atoms and radiation as described by the quantum theory. This may be done either by treating the atoms quantum mechanicaIly

117

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

while retaining a classical description of the electromagnetic field ( 4 ) or by a complete treatment in which the electromagnetic field is fully quantized (5). In each case the results are identical, namely that if an atom exists in an excited level k with energy E k ,which is described by a wavefunction $,, then there is probability per unit time A,, that the atom will decay spontaneously to a lower level n (characterized by energy E n ,wavefunction $,) which is given by

In this equation hw,, = E, - E n ,where h is Planck’s constant/2n and r i is the position vector of the ith electron in the atom with respect to its center of mass. The volume element dr indicates an integration over the coordinates of every electron in the atom. (In this and subsequent equations we use the convention that the first subscript indicates the initial level from which the atom makes a transition). Equation (4) applies to levels k and n only if they are nondegenerate, i.e., if they are Zeeman sublevels separated by the application of a large external field. For a system in which the levels k and n have total angular momenta Jkand J, and consist therefore of 2Jk 1 and 25, + 1 degenerate sublevels in zero external field, the transition probability per unit time from the excited level is given by an average over the initial state and a sum over the final states (6)

+

where gk = 23, + 1 is the statistical weight of the upper level; and the wavefunctions now refer to single Zeeman states with magnetic quantum numbers given by M k and M , . This equation assumes that the states of the excited level k are equally populated, as in most gas discharges. Thus if we have a system in which we excite Nk(0) atoms into the level k at time t = 0 and we suppose that there is no mechanism, other than electric dipole radiation, which depopulates the level k , then the rate of change of the population in level k is given by -dN,/dt = N ,

2 A,, , n

where the sum is over all levels n with energy less than that of k. If we let

Thus the mean lifetime of level k against spontaneous emission Eq. (7).

‘rk

is given by

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A. CORNEY

2. Stimulated Emission and Absorption If we consider again the classical model of an atom which contains an harmonically bound electron whose natural angular frequency is oo, then in the presence of an external oscillating electric field of angular frequency o the bound electron is set into forced oscillations at this frequency. Power may be either extracted from or fed into the external field by the electron depending on the relative phases of its motion compared with that of the external field. This process of stimulated absorption or emission of energy due to the interaction of an atom with an external electromagnetic field can also be treated quantum mechanically although the relations were originally derived by Einstein before the quantum theory was fully developed. If an atom is in a lower energy level n and is subjected to isotropic radiation of angular frequency w having an energy density per unit angular frequency range of p(w), then there is a probability P,, per unit time that the atom will be stimulated to make a transition to a higher level k with a consequent absorption of energy by the atom. Then Pnkis proportional to the energy density of radiation and is given by Pnk

(9)

= Bnk(P)P(w),

where B,k(P) is a coefficient that depends on the particular atom involved. Similarly, the radiation will stimulate an atom i n an excited level k to make a transition to a lower level n with the emission of energy with a probability Pkn given by pkn

(10)

= Bkn(P)P(w).

The coefficients for stimulated emission and absorption are related by gk

Bkn(P)

(1 1)

= gn B n k ( P ) ,

and are connected with the spontaneous transition probability through the relation Bn!,(P)

where

= (nZc3/fiw3)(g!,/gn)A!,n = 6.01L3(gk/gn)Akn

2

(12)

A. is the wavelength of the radiation measured in angstrom units.

3 . The f-values and Line Strengths It is found convenient to introduce 2 dimensionless quantity, known as the f-value, which is a measure of the strength of a given electric dipole transition. The emission f-value fkn is defined by

A,,

= (2e2w:njmc3)(

-Ikn).

(13)

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

119

Levels having allowed transitions to lower levels with f-values in the range 0.4-0.1 have short lifetimes in the range 10-9-10-8 sec. The negative sign associated with the emission f-value is really only of importance when sums over f-values occur, as in expressions for refractive indices. In a similar manner the absorption f-value f n k for a transition from a lower level to n higher level k may be defined by

where 1is measured in angstroms. From Eqs. (13) and (14) we see that -gkhn

(15)

=gnfnk,

and this product is often quoted in the literature as the gf-value. From Eqs. (13), (15), and (5) we obtain the relation

A quantity that is often used in theoretical calculations is the line strength S defined, for electric dipole transitions, by the relation

3hg,snk 2 2mwkn

fnk

=

'

From Eqs. (17) and (16) we see clearly that Snkis the quantum mechanical analog of the classical oscillating dipole moment. Since S k , = s n k the line strength is often denoted by S. When evaluated Eq. (17) gives 303.15

fnk

= -S &7n

with I in angstroms and S measured in atomic units, that is, in units of ao2e2= cm2 esu2, where a, is the radius of the first Bohr orbit. 6.459 x As an example of the use of these relations we consider the case of transitions from the ls2p 'P, level of helium. There are two allowed electric dipole transitions, 1 'So - 2 'P, at 584 A and 2 'So - 2 'PI at 20,581 A. (We consistently use the spectroscopic convention of writing the lower ofthe two energy levels first.) The line strengths for these transitions have been computed to be 0.5313 and 25.50 a.u., respectively (7). The application of Eq. (18) yields absorptionf-values off(584) = 0.2762andf(20581) = 0.3764. However, the use of Eq. (14) yields considerably different transition probabilites for the two lines due to the wavelength factor: A(584) = 1.799 x lo9 sec-' and A(20581) = 1.976 x lo6 sec-'. Thus, the lifetime of the 'PI level obtained from Eq. (7) is determined almost entirely by the vacuum ultraviolet transition and we have T('PJ = 5.552 x lO-''sec.

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A. CORNEY

4. Relations f o r Diatomic Molecules

We consider electric dipole transitions from an excited level of a diatomic molecule which has an electronic configuration labeled by k , vibrational quantum number u', and rotational quantum number J ' . For the lower level we have labels n, v", and J". In the approximation that the molecular wavefunctions may be written in the form $ = $ , $ , . $ J f M I / r and that the electronic part of the wavefunction $,(Ti), does not depend appreciably on the internuclear separation r, we find that Eq. (5) becomes

where M ' , M " are the magnetic quantum numbers associated with the rotational levels J ' , J " and where

R::"

=

j$~,M.$J..,..sin o

(14.

By summing over all transitions from a given vibrational level and using the following sum rule (8)

C

J" M'M"

lR:::"12 = 25'

+I

we have the transition probability for the (u', v") band given by

The lifetime of a given vibrational level is determined by T " , = l/cU.. Ahn,U.u.. (assuming that n is the only electronic level to which transitions are allowed). The emission ,fvalue for a band can now be defined by an equation identical to Eq. (13). If we make the further assumptions that all transitions between the electronic levels k and n have the same mean angular frequency Whn, then by using the sum rule (8) 1IR:Ifl2 = 1 U"

we have a total electronic transition probability A,,, given by

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

121

Because of the assumptions involved in the derivation of Eq. (21) it has very little meaning, and the use of a molecular emission fk, derived by the use of Eq. (13) is likely to be misleading. B. Just$cation f o r the Measurement of Lifetimes

The measurement of the lifetimes of excited levels of atoms, ions, and molecules provides information which is directly useful in the fields of spectroscopy, astrophysics, and plasma physics. For instance, the ability to maintain a population inversion in many gas laser systems depends on the total decay rate R , of the lower laser level obeying the inequality R , > A,,g,/g,, where A,, is the radiative transition probability of the laser line and g, , g1 are the statistical weights of the upper and lower levels, respectively. However, more generally, the fundamental data which are required are the f-values or transition probabilities for the electric dipole transitions of a given atom. (We frequently use “atom” to indicate that the discussion or experiment applies equally to atoms, ions, and molecules.) Thus the total power radiated by unit volume of a gas discharge in a transition from level k to where N k is the density of excited atoms and level n is given by Nk A,, hkn, stimulated emission is assumed to be negligible. Similarly a knowledge of the f-values allows one to calculate the absorption coefficient of a gas of atoms through which light is propagating. The f-values are also useful in problems which are not directly connected with radiation processes, thus expressions for the Van der Waals’ interaction between two neutral atoms involves a sum over f-values and the Stark shift of the energy levels of an atom may be calculated in a similar way. Because of the form of Eq. (7)fvalues for individual transitions may be obtained from lifetime measurements only if one of the following situations occurs: If an excited level k decays by a single electric dipole transition the (i) lifetime measurement gives the transition probability directly through the relation A,, = l / z k . This is the situation for the first excited P levels of the alkalis and a number of other atoms. It enables thefvalues for these transitions to be obtained unambiguously. (ii) I n a number of other atoms the decay of the first excited levels is dominated by one strong transition and thus a measurement of the lifetimes of these levels enables an upper limit to be placed on the transition probability of the strong line A,, < 1 / ~which ~ , is close to the actual value. This was the case for the 2 ‘PI level of helium discussed in Section I.A.3 and similar situations occur in the alkaline earths. (iii) In a situation where only the relative values of the transition probabilities contributing to the lifetime of a given level are known, a measurement

122

A. CORNEY

of the lifetime enables a set of absolute transition probabilities to be obtained. The techniques outlined in Section 1.D below frequently produce sets of relative data. (iv) In the case of molecules the form of Eqs. (7) and (20) means that thefvalues or transition probabilities for a given band can only be determined from the measured lifetime of a particular vibrational level if the FranckCondon factors IR",12 for the molecule are known. Finally it may be noted that in principle Eq. (16) would enable thefvalues of electric dipole transitions to be obtained if accurate wavefunctions of atomic energy levels could be calculated. In practice it has proved difficult to achieve, for although wavefunctions may be constructed which accurately reproduce the known energy levels of the atom, the expression for the fvalue effectively weights an overlap integral between different wavefunctions which may be considerably less accurate. It is therefore difficult to achieve an accuracy of better than 20 % in theoretical fvalues unless a great deal of care is taken with the calculation. Theoretical calculations are reviewed in three recent articles (9,9a, 10) and a Russian handbook has recently been translated (11). Only the Coulomb approximation of Bates and Damgaard is relatively simple to apply (12). Thus, it is of great value to obtain reliable experimental measurements to check theoretical estimates whenever possible. The measurement of lifetimes provides one means of doing this and the results have recently shown that wavefunctions calculated by neglecting configuration interaction may sometimes give results which are in error by 50% or more (13). For these reasons, the measurement of the lifetimes of excited levels of atoms is both necessary and useful. C. Outline of Experimental Techniques for the Measurement of Lifetimes

In the sections that follow, we discuss in detail the main techniques that are being used at present. Since they differ considerably in principle we outline them here. The methods described in Sections 11-IV depend on the direct observation of the exponential decay of intensity of light emitted from atoms in a given excited level which is populated by a brief pulse of excitation. The exciting pulse is removed in a time much shorter than a typical lifetime which it is desired to measure. In Section V use is made of the fact that light emitted from atoms which are subjected to a periodic excitation process will be modulated but with a difference in phase which depends on the lifetime of the levels involved. It is well known that energy levels which have a finite lifetime T due to spontaneous emission of radiation possess a finite energy width AE given by the uncertainty relation z * AE zh. The techniques described in Section VI

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

123

measure this width by the direct observation of the profile of spectral lines while the techniques of Section VII measure the width by the effect of magnetic fields on the polarization of resonance fluorescence. In Section VIII, we outline a number of more specialized techniques such as the use of the width of the Lamb dip in the curve of output power versus frequency of gas lasers and we discuss the problem of forbidden lines and metastable levels. The reader is referred to three recent review articles on the measurement of lifetimes (14-26) and to the well-known book by Mitchell and Zemansky (17) in which detailed references to experiments performed before 1933 are given. At present, new measurements of lifetimes are being reported with such regularity that any attempt at an exhaustive list of lifetimes or references would be soon out of date. For that reason we have concentrated on giving the details of the experimental methods and have used selected results for purpose of illustrating the accuracy of the different techniques. The National Bureau of Standards Data Center on Atomic Transition Probabilities is currently engaged in a program of compiling critically assessed transition probabilities for many elements. At present, two volumes of transition probabilities have been published, one containing data for the first ten elements, hydrogen t o neon (18), and the second for the elements sodium to calcium (19) together with a bibliography (20) containing references up to 1966. A further bibliography containing references up t o June 1969 has recently become available (2Oa).

D. Techniques Not Discussed in Detail-The f- Values

Direct Measurement of

It is easy to see from the relations given in Section 1.A that there are a number of ways of measuring thef-values of electric dipole transitions which do not involve the measurement of lifetimes. For example, the total intensity I,,, emitted spontaneously in a spectral line involving a transition from k to n is given by the luminous flux emitted into unit solid angle I,,,

= AknNk hWknv//47L,

(22)

where Vis the volume of discharge region which is filled with a uniform density of excited atoms Nk and which is optically thin to the radiation of angular frequency Wk,, . Thus a measurement of Iknwill allow the transition probability and thef-value for the line to be determined provided that the density of excited atoms is known. The techniques listed in this paragraph all depend on a knowledge of the density of atoms in a given level and since this is i n general difficult to determine, the absolute measurement of f-values by these methods is subject to error. This is avoided in the lifetime measurements provided the data are taken

124

A . CORNEY

at pressures low enough for radiation trapping and collision effects to be negligible. However, the techniques for the direct measurement of f-values have the advantage of investigating the ,f-value for a single transition while the measurement of lifetimes gives results which depend on a sum of transition probabilities. Thus the two approaches are complementary. The techniques which we shall not discuss in detail are as follows: (i) Total intensity measurements of light emitted from arcs and shock tubes. (ii) Measurement of the light absorbed by a vapor or atomic beam. (iii) Measurement of the anomalous dispersion close to an absorption line-the Hook method. The applications of these three techniques t o the measurement of atomic f-values are reviewed in references (14)and (21) and further discussion of the assumptions involved in the line intensity measurements may be found in (22) and (23). Nicholls and Stewart briefly discuss the application of the same methods t o molecules (9a). A detailed discussion of the Hook method has been given by Marlow (24) and many results obtained using this technique are given in two collections of translated articles by Russian workers (25). (iv) The Faraday rotation. This is a technique which is only infrequently used. Recent measurements of transition probabilities i n neon using this method are reported by Seka and Curzon (26). (v) The measurement of inelastic scattering cross sections for electrons on atoms. This is an unusual but useful technique. Recent work using this method has been reported by Hertel and Ross (27, 28) and Lassettre et al. (29, 30). 11. THEDIRECT OBSERVATION OF EXPONENTIAL DECAYS USING ANALOG METHODS

A . Principle of the Method

We showed in Section 1.A that, if we had a system that contained Nk(0) atoms in the excited level k at time t = 0 then, provided there was no subsequent repopulation of level k , the number of atoms in this level decreased exponentially as a function of time, Eq. (8). The intensity of light emitted by an optically thin source of volume V in a transition from k to a given lower level n is therefore given by Ikn(t)

= Nk(0)Ab.n h m k n v

exp(-

f/7k)/4z

(23)

decaying exponentially with a time-constant equal to the lifetime of the upper level, zk = l / C , A k , .

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

I25

The simplest experiment that could be imagined to measure lifetimes would consist of a sample chamber in which the atoms to be studied were excited to the desired upper level by electron bombardment. If the electron current is removed in a time much shorter than T ~ and , the intensity of a given transition from level k is measured by a photomultiplier followed by a fast oscilloscope, then a record of the exponential decay of level k may be obtained. B. Direct Observation of Oscilloscope Trace

This was the technique developed by Holzberlein and his collaborators (31-33) and applied by them to lifetime measurements in helium. Their apparatus is given schematically in Fig. 1. It consists of an excitation tube with a cylindrically symmetric three-electrode structure. The outer cylinder is a massive cathode heated by an external induction heater, the first and second grids serve to control the electron current and the mean energy of the electrons. The tube is connected to a vacuum and gas handling system. Light from the atoms excited in the central region enclosed by the second grid is observed

W+/"'y

I

Photomultiplier

0 Filter

I

0 I

/Platinum

'---Vacuum system r and gas SUPPIY

FIG.1. Schematic diagram of the apparatus used by Holzberlein (31) for direct observation of exponential decays in helium.

126

A. CORNEY

in a direction along the axis of the tube. The light is collected by a lens and passed through a multilayer dielectric interference filter. This defines the wavelength range of the light detected by the photomultiplier and enables selected excited levels to be studied. The photomultiplier supply is pulsed to a high level in synchronism with the electron excitation pulse in order to increase the gain over that obtainable under dc., operating conditions. The time-dependence of the photomultiplier output was displayed on a fast oscilloscope and the traces photographed. It was found possible to study the lifetimes of seven levels in helium over a range of densities from 1 x 1014 to 2 x 10l5 atoms/cm3. The lower limit was imposed by the decrease in the signal intensity and the upper limit by the difficulty of achieving a rapid cut-off of the excitation of atoms in the region between the two grids due to the formation of an ion sheath. The time resolution of approximately sec was limited mainly by the electron transit time across the diameter of the central observation region, d = 1.25 cm and the rise time in the cut-off voltage signal. Because of the wide-bandwidth characteristics necessary for this single shot operation it is essential to obtain as high an intensity as possible. For this reason Holzberlein used a cathode capable of supplying 5 A of electron bombardment current, and worked with electron energies in the range 35-100 eV. This is considerably higher than the threshold excitation energies of the levels being investigated. This means that at the instant that the electron excitation is removed there usually exist atoms in levels with energies greater than that of the level k whose lifetime is being studied. The atoms decay spontaneously and consequently level k will continue to be populated by cascade from the upper levels. The observed decay of intensity will consist of a sum of exponential components whose time constants are characteristic of the level k and of those levels above k. The treatment of multiple component exponential decays is discussed in detail in Section IIl.A.4 but the result is usually to increase considerably the uncertainty of the lifetimes being measured.

C. Use of Pulse-Sumpling Methods 1. Application to Atoms

The technique used by Holzberlein has been considerably improved by Pendleton and Hughes (34) in experiments on helium. While using the same basic idea they have increased the signal-to-noise ratio by feeding the amplified photomultiplier current into a pulse sampling oscilloscope. This enables the signal at a given point on a decay to be averaged over many repetitions of the pulse cycle. Repetition rates around 1 kHz were used and the output of the oscilloscope was recorded o n a chart recorder as the sampling window of the

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

127

oscilloscope was slowly moved through the decay curve. Pendelton and Hughes were able to confirm an observation made by Holzberlein that the decay curves of the 3 3D and 4 3 D levels of helium were composed of a rapid decay (z z 15 x sec) followed by a relatively long-lived component (z = 135 x sec). The intensity of the long-lived component in the decay curves for the 3D levels was found to increase strongly at higher pressures. This was interpreted as due to a collisional transfer from excited n 'P levels to n 'F and n 3F followed by radiative decay to the 3D level whose lifetime was being studied : He(n 'P) + He(l 'S) --f He(1 IS) + He(nF), (24) He(n' 3D) hv, n > n'. He(nF) --f

+

This collisional transfer has the effect of shortening the apparent lifetime of the n 'P levels and is clearly an undesirable effect if measurements of true radiative lifetimes are being made. However Kay and Hughes (35)have turned this pressure dependence of the decay curves into an advantage and by a detailed study have been able t o confirm that the collision process postulated in Eq. (24) is indeed responsible for the long-lived components in the 3D decay curves. They were also able to measure the total deactivation cross cm2 and sections for the 4 and 5 'P levels obtaining the values 2 x 6.4 x cm2, respectively. Unfortunately these cross sections are more than an order of magnitude larger than the values of (10.4 0.6) x cm2 cm2, respectively, obtained by Bennett et al. (47). and (10.4 0.3) x The two experiments differ in that Bennett et al. used electron energies close to threshold and helium pressures in the range 0.5-10 Torr while Kay and Hughes used energies well above threshold but a lower pressure range (1-20) x Torr. This discrepancy is discussed further in Section III.A.5c. In the pressure range used by Kay and Hughes the lifetimes of the n 'P levels are increasingly affected by the photon trapping or reabsorption within the sample of the strong vacuum ultraviolet radiation 1 'S - n 'P. Allowance has to be made for this effect in analyzing the pressure dependence of the n 'P lifetimes in order to obtain the collision cross sections, whereas at the pressures used by Bennett et al. the lifetimes of the n 'P levels are expected to be constant at a value corresponding to the resonance trapped limit. These collision processes have also been investigated by selective optical excitation and relative intensity measurements (3.5~).

2. Application to Molecules The pulse-sampling technique has also been used by Bennett and Dalby in an extensive series of lifetime measurements in the following molecules: N2+ (36), CO' (37), CH and NH (38), N 2 0 + (39), CN(40), and OH (41).

128

A . CORNEY

Master

Fixed

Clock

Delay

Gate 1

. D

-

Variable Delay

>

Gate 2

Photomultiplier

Integrating

.

-

Current Ampll f ie r

FIG.2. Schematic diagram of the pulse-sampling technique used for molecular lifetime measurements by Bennett and Dalby (36).

The exciting source consists of an indirectly heated cathode and a grounded accelerating grid which determines the electron energy. The current is controlled by an intervening grid and the excited molecules are observed in the space beyond the accelerating grid. A well-defined electron beam of 5 mA is obtained and electron energies in the range 80-200 eV are used. A block diagram of the electronic circuits is shown in Fig. 2. The master timing clock produces trigger pulses at 4 kHz. After a fixed delay this trigger activates gate 1 and a 70-V pulse of variable width is applied to the control grid of the electron gun. The pulse decay time of approximately 8 x sec contributes approximately half of the instrumental resolving time which is of the order of 15 x sec. The width of the pulse is chosen to be three or four times the lifetime which is to be measured. The master trigger pulse is also applied to a phantastron circuit which produces an output trigger pulse after a certain variable delay time. The trigger pulse activates gate 2 and a 1200-V pulse, also of variable width, is applied to the photomultiplier. Thus the photomultiplier is only sensitive for a short interval after the start of the excitation. The photomultiplier signal is integrated and recorded on a chart recorder as the photomultiplier gate delay is slowly increased. The

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

129

time delay is measured by observing the position of this gate pulse on an oscilloscope whose time-base is calibrated with a crystal controlled oscillator. A very similar apparatus has recently been used by Morack and Fairchild for lifetime measurements in argon ( 4 1 ~ ) . The excited molecules studied were produced from parent gases such as N, , CO, CH,, N,O, and H,O which were introduced into the electron gun at pressures in the range 0.2-5 x Torr. Over this pressure range the observed lifetimes were independent of pressure except for the A 'Z level of OH. This indicates the absence of quenching and photon trapping effects in most cases. Photon trapping was expected to be negligible since the parent gases are different from the molecules being studied. The possibility of cascade effects was investigated by measuring the lifetimes with varying electron energies and with varying excitation pulse widths as discussed in more detail in Section III.A.5. No variation of lifetimes was noted which could be attributed to cascades and the decay curves could be fitted with a single exponential. In some cases there are no upper levels from which cascades can occur while in others the transitions are observed to be considerably weaker than the lines actually being investigated. The emission bands of the excited molecules were isolated with interference filters which had full widths at half maximum transmission of approximately 100 A. This rather low resolution caused difficulties in the case of N,' due to unwanted radiation from excited N, molecules reaching the photomultiplier. Also, in the cases of NH, CN, and N,O+, the bandwidth of the interference filter included several vibrational bands having different values of v' so that only an average lifetime could be measured. Obviously the power of the method would be increased if greater resolution could be used. TABLE I THELIFETIMES AND SPONTANEOUS TRANSITION PROBABILITIES FOR THE VIBRATIONAL LEVELS OF THE A 21T STATE OF CO+ WHICH GIVE SYSTEM X 'X+-A 2n RISETO THE COMET-TAIL

a

Lifetime sec)

u*

2.78 j=0.2 2.61 * 0 . 2 2.36&0.15 2.22k0.13 2.11 k 0 . 1 3

0 1 2 3 4 5

0

1

2

3

9.47 17.5 11.3 1.71 0.92 6.51

7.63 3.90 7.84 13.30 6.84

-

u'

4.37" 14.2 2.37 26.6 22.4 14.8

Transition probabilities given in units of lo4 sec-'.

-

4.33 4.83 -

-

130

A. CORNEY

However the decrease in the signal which would result from such a change would probably necessitate the use of the multichannel delayed-coincidence methods described in Section 111. The lifetimes measured by this relatively simple technique range from (6.58 k 0.35) x sec for the u' = 0 level of the B 'C' state of N,+ to over sec in CO'. The results given in Table I for the A 'I3 state of CO' are typical of the accuracy that was attained. The variation of the measured lifetime with vibrational quantum number v' is due mainly to the factor o ~ , , ~ occurring .,,,, in Eq. (20). The absolute values of the measured lifetimes have been combined with relative intensity measurements by Nicholls (41b) to yield the values of the spontaneous transition probabilities Akn, which are given in Table I. 111. DELAYED COINCIDENCE PHOTON COUNTING

A . Experiments Using Electron Bombardment Excitation I . Principle of the Technique The technique of delayed coincidence is well known in nuclear physics for the measurement of lifetimes. It was first applied to atoms by Heron et al. (42, 43) but was not widely used. However, following the application of multichannel analyzers to this method by Bennett (44) in 1961, a large number of measurements on different elements have been made. The technique is possibly the most accurate and widely applicable of the methods currently available for the measurement of atomic lifetimes. The method again makes use of the fact that if we excite a sample of atoms to some level k by a pulse of electrons which is removed instantaneously at time t = 0, then the probability of detecting a photon on the decay transition from level k to level n in a time interval between t and t At after the removal of the excitation is given by

+

p ( t , At)

CT

A,, Nk(0) eXp( - t / T k ) At

(25)

The early experiments described in the next section differ from those described in Section I1 mainly in the application of instruments developed for use in nuclear physics. Thus each photon received by a photomultiplier is detected as a pulse of charge and the number of pulses received in a given time interval is counted by means of a scaler. Recent developments using multichannel analyzers have improved the detection efficiency by making the apparatus sensitive to the arrival of a photon at all times after the removal of the excitation pulse.

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

131

2. Single Channel Method The first application of the techniques of nuclear physics to the measurement of atomic lifetimes was made by Heron et al. ( 4 2 , 4 3 )in the measurement of the lifetimes of the 4 3S, 3 3P, 4 'P, and 3 3 D levels of helium. The apparatus used is shown in Fig. 3. The excited atoms were produced by electron bomto 5 x lo-' bardment of helium gas at pressures in the range from 5 x Torr. The electron beam pulse was obtained by applying a positive voltage pulse of 2 x lo-' sec duration to the grid of the electron gun which was normally held negative. The electron gun was pulsed a t a repetition rate of 10 kHz and produced a current of 100 pA during the pulse. The accelerating voltage, in the range 30-100 V, was adjusted to give the maximum intensity for a given line that was being studied. The light emitted by the decaying atoms was detected by a photomultiplier placed behind two collimating slits. The excited level k which was being To vacuum

_ -

0 Filter

Pulse Amplifier

Generator

Delay Cable

Coincidence Circuit

Pulse Shaper

FIG.3. Schematic diagram of the single channel delayed coincidence apparatus used by Heron ef al. (42) for lifetime measurements in helium.

132

A. CORNEY

studied was determined by the interference filter placed between the collimating slits. The arrivai of a photon at the photomultiplier produces a sharp negative voltage pulse at the cathode. This pulse was amplified and shaped before being applied to one side of a coincidence circuit. The pulses applied to the other input of the coincidence circuit were obtained from the electron gun pulse. These were delayed by a known amount by passing them down a fixed length of delay cable. The coincidence unit summed the two input voltages and provided an output pulse which was applied to a discriminator. The sec. The discriminator bias setting determines the resolving time of 3 x coincidence rate was measured as a function of the delay time by changing the length of the delay cable. The lifetimes of the different levels were obtained by plotting the log of the coincidence rate against the delay time and taking the slope of the linear portion of the decay curve. The resulting lifetimes were accurate to better than 5 %. The principle of the method is the same as that used by Bennett and Dalby (36) which was described in Section II.C.2. The difference being that Heron et al. were able to increase their signal-to-noise ratio by counting pulses over a considerable period and so obtaining an effective time constant which would be difficult to obtain using analog methods. The rather long limiting resolving sec achieved by Heron et al. could be considerably reduced time of 3 x using recent advances in pulsed electronic circuitry. However the method suffers from the disadvantage that it is sensitive to photons which arrive only within a specified short interval of time following the excitation pulse. Since the intensity of light produced by electron bombardment sources is usually low, a fact which prevented Heron et al. using the increased spectral resolution obtainable with a monochromator, only a small number of discrete delay times could be used in any given experiment. Obviously an instrument that is equally sensitive to photons at all times after the excitation pulse is removed would be a great advantage. The development of this technique is described in the next section. 3. Recent Improcements Using Multichannel Techniques

The extension of the technique used by Heron et al. (43) by the introduction of multichannel analyzers may be understood with reference to Fig. 4. Again a sample of excited atoms is prepared by a voltage pulse applied to an electron gun. The same voltage pulse is applied simultaneously to the start input of a time-to-pulse height converter. This initiates the charging of a capacitor by a constant current source. Some time after the excitation pulse an atom in the sample emits a photon on the decay transition of interest which is isolated by the spectrometer and detected by the photomultiplier. The resulting voltage pulse is amplified and applied to the stop input of the time-to-

133

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

_____.

Electron

I -

(p;{Fl

Gun

-- - - - -- - - - - - - -

-- - ---

-

,*

I

,-1

S

: ’\.

& \

I

Photomultiplier Start ,

Stop

Time-to-height Converter

i 256 Channel

Reset

t

300 MHz Amplifier

G Disc.

FIG.4. Schematic diagram of the multichannel delayed coincidence apparatus developed by Bennett (44). [From w. R. Bennett, Jr., in “Advances in Quantum Electronics” (J. Singer, ed.), p. 28. Columbia Univ. Press, New York, 1961.1

pulse height converter. At this instant the capacitor stops charging and a voltage pulse, whose amplitude is proportional to the time interval which has elapsed between the excitation and the detection of the first photon, is delivered to the multichannel pulse height analyzer. The nth channel of the analyzer corresponds to a voltage interval between V, and V, AV, , and if the incoming voltage pulse lies within this interval one count will be added in this particular channel. Thus each channel is made to correspond to a finite time interval after the removal of the excitation pulse. The system is therefore sensitive in all channels following the excitation pulse. If no photon is received before the capacitor reaches its peak charging voltage corresponding to the last channel of the analyzer, the time-to-height converter is reset ready for the start of a new excitation and decay cycle. After this sequence has been repeated many times, the number of counts in a given channel is proportional to the number of excited atoms present at that time. This application of the multichannel delayed coincidence method to the measurment of atomic lifetimes was developed by Bennett (44) and has been applied by Bennett and Kindlmann (45) to a study of helium relaxation rates and by the same authors to a detailed study of the lifetimes and collision induced relaxation in the 2p levels of neon (46).An excellent review of this technique is given by Bennett et al. (47).The same technique has also been

+

134

A. CORNEY

used by Klose in a study of atomic lifetimes in neon (48) and argon (49, 50). More recently Lawrence has extended the technique to the study of ions and atomic transitions lying in the vacuum ultraviolet region of the spectrum (51-53).

In the following sections we discuss various details of the experimental technique, methods used for the reduction of the data, and experimental difficulties. a. Electron gun design. The characteristics of the ideal electron gun for these experiments are as follows: The gun should produce a high current density beam of electrons with as small a spread of energies as possible. The mean energy of the electrons should be stable over a period of several hours and should be variable over a suitable range which will be determined by the system which it is desired to investigate. The electron beam should be capable of being cut-off instantaneously and the transit time for the electrons across the observation region must be as small as possible. Finally the background light from the cathode should be small. Different authors have used different approaches to optimize these competing requirements. Bennett et al. (47) describe a gun which consists of an oxide coated cathode 20 cm long by 1 cm wide with a grid 1.5 mm above the cathode and an anode 7 mm from the grid. The grid and anode are connected together so providing a field free drift region from which the light is taken in a direction parallel to the long axis of the cathode. By operating at low temperatures, both the background radiation and the spread of electron energies is reduced while the area of the emitting surface gives a relatively large total electron current of 100 to 200 mA. The electron energy spread expected was approximately equal to 0.1 eV and this was confirmed by observing the shape of electron excitation functions near threshold. The small spread in electron energies is important since it allows the selective excitation of a given excited level by working with the mean electron energy close to the excitation threshold for that level. This reduces greatly the possibility of the repopulation of the level under study by radiative cascade from higher lying excited levels and greatly increases the confidence in the analysis of the results. Since electron excitation cross sections close to threshold are generally small, 10-21-10-19 cm2, a large bombardment current is necessary in order to obtain signals of sufficient size. Klose (48) used an electron gun based on the designs of Simpson and Kuyatt (54) which provided a beam current of 100 pA in the energy range 20-100 eV. The energy spread of the beam was measured by the retarding voltage method and shown to be of the order of 0.5 eV for typical operating conditions. Because of the low electron current it was found necessary to operate at electron energies close to the maxima of the electron collision cross

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

135

sections rather than at threshold energies. Thus for the neon levels with thresholds in the region 18-19 eV, energies of 20 to 30 eV were required. Other authors (55) have used a dispenser cathode instead of the more usual oxide coated one. This has the advantages that it can withstand positive ion bombardment more readily, is less susceptible to poisoning by residual gases, and is capable of higher emission current densities. Lawrence (51)has developed a very compact source which uses a heated tungsten wire as the cathode. This has great advantages for a system which must handle molecular gases or be opened to the atmosphere at intervals. Most workers have chosen to bias the grid negative and to obtain the pulse of excitation by applying positive pulses to it. It is not necessary that the pulse should rise quickly, in fact it may be made to rise rather slowly in order to avoid overshooting the desired maximum voltage. For the last half of the pulse the level should be flat to within 30 to 50 mV. At this stage it is possible to sample the gun voltage and compare it with an accurate reference supply set to the required bombarding voltage. Any difference between the two is used to bias the gun to achieve the required positive peak voltage. Finally the gun pulse is cut off in a time which is typically 1-2 x l o p 9 sec. Repetition rates for the gun pulses are typically in the range 2- 100 kHz. In most multichannel systems the lifetime resolution is limited by the combined effects of the finite time taken for the gun pulse cut-off and the finite transit time of electrons across the source (a 20-eV electron takes 1.88 x l o e 9 sec to travel 0.5 cm). Resolution times of approximately 5 x sec are typical although the compact design of the source used by Lawrence (51) apparently gives resolving times of less than 1 x sec. b. Photon detection. In the original experiments of Heron et al. the number of excited levels that could be studied was limited by the low spectral resolution of the available filters. For experiments on neon and other inert gases the abundance of closely spaced lines throughout the visible spectrum makes the use of a monochromator with a dispersion of the order of 16 A/mm essential. It is also desirable to have a large aperture instrument to increase the light gathering power. If the monochromator is to be used at maximum efficiency several gratings blazed for different spectral regions are required. The photomultiplier used to detect the photons should have high sensitivity at the operating wavelength and low dark current. Since there is considerable variation of these properties even with tubes of the same type it is worthwhile obtaining a specially selected tube. This is particularly true for tubes operating in the near infrared which are usually cooled to reduce the dark current. Photomultiplier pulses fluctuate in amplitude due to slight

136

A . CORNEY

Trigger time

Time

I

Shlft

FIG.5 . Waveform illustrating the dependence of triggering time of a discriminator on pulse amplitude.

variations in gain. Two typical pulses are shown in Fig. 5. The pulse arrival time is usually determined by use of a discriminator circuit. However the exact arrival time obviously depends on the setting of the discriminator level. This variation in triggering time with the size of the output pulse is reduced if photomultipliers with fast rise times are used. In addition there are also fluctuations in the time taken for electrons to travel from the cathode to the anode. These are caused by variations in the electron path between dynodes, the point on the cathode from which the photoelectron originated, etc. Both the transit time spread and the discriminator jitter contribute to the finite resolving time of the system discussed in the previous section. For Bennett’s apparatus the resolving time was of the sec. order of 3 t o 4 x c . Various timing devices

(i) Differential Nonlinearity. In the ideal multichannel system the width of each channel is the same and remains constant over long periods of time. Thus each channel corresponds to the same fixed time interval. Any variation in channel width with channel number will lead to excessively high or low numbers of counts being recorded in a given channel and to the appearance of spurious peaks and valleys in an experimental decay curve. The term ‘‘ differential nonlinearity” is used to describe this defect. It is defined as the

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

137

percentage deviation from perfect linearity for a plot of pulse spacing against channel number. Differential nonlinearity arises in the time-to-pulse height converter system described in Section 111.3 because of departures from linearity in the saw-toothed waveform generated by the device. The multichannel system may be checked for this effect by providing the stop input of the time-to-height converter with pulses which are spaced at random intervals. A convenient way of doing this is to use the dark current pulses of the photomultiplier or to use a weak dc light source. Alternatively the start and stop inputs of the converter may be driven by two oscillators or pulse generators having incommensurate frequencies. Ideally the distribution of counts should be constant throughout all channels of the system. The deviation from the mean enables the relative time width of each channel to be measured, and used in the analysis of experimental runs. Thus, although in the system initially described by Bennett (44) the maximum variations of channel width were about lo%, the uncertainty in the lifetimes introduced by this effect was only 1 ”/,. (ii) Calibration. Having determined the relative width of each channel it is necessary to know the absolute time interval corresponding to the interval between the first and last channel. This may be obtained by introducing various lengths of delay cable before one input of the time-to-height converter. The time delays so introduced are obtained from the resonant frequencies of the given cables and the observed displacement of the peaks of the decay curves calibrates the time axis of the system. Lawrence (51) has combined the calibration and the measurement of the differential nonlinearity by using a crystal controlled time-base to provide pulses with repetition period T which alternately start and reset the time-to-height converter. A dc light source provides the random stop pulse and a fast scaler counts each of the delayed coincidences which is recorded in the pulse height analyzer. If N is the total number of these delayed coincidences and n , is the number of counts recorded in the ith channel of the analyzer, then the absolute width of that channel is Ati = n , T / N . If sufficient time is allowed for n i to reach lo4 a statistical accuracy of 1 % i s achieved. This scheme has the advantage that the calibration may be repeated regularly during a series of experiments. (iii) Alternatives to the Time-to-Pulse Height Converter. The time-topulse height converter has the advantages as a timing device in that it may be made to provide a total range from 100 x sec to 30 psec and is commercially available. On the other hand it is essentially an analog device with limited differential linearity. For decay curves consisting of a single exponential this is not too serious. However these decays tend to be the exception rather than the rule. Thus for the analysis of complex decay curves and greater attainable accuracy with single decays, Bennett and Kindlmann (46) used a digital method of time analysis with a phase-locked vernier chronotron

138

A. CORNEY

developed by Kindlmann and Sunderland (56). This and other methods for time analysis are discussed in (47).

4. Reduction of the Data a. Effect of jinite instrumental resoloing time. Methods of analyzing delayed coincidence curves have been discussed by Bay (57,58)and Newton (59), principally with reference to their use in nuclear physics. The effect of the finite instrumental resolving time is that, if the decay curve of a given excited level which is known to have a lifetime much shorter than the instrumental resolution is studied, a delay curve P(t) as a function of the delay time t , will be recorded. This prompt coincidence curve P ( t ) is determined entirely by the characteristics of the apparatus. A convenient definition of the instrumental resolving time 2, is then given by (60), 1

J 2plnax

"

7, = -

P(t)dt.

(26)

-a,

If we now study the time dependence of some photon arrival distribution which has the form f ( t ) , the delayed coincidence curve that is actually observed with the multichannel instrument is F( t ) given by the folding integral F(t) =

- 10

f ( x ) P ( t - x) dx. -m

0 Delay time

lo

-0

(units of 10

20 sec)

FIG.6. The coincidence counting rate as a function of delay time, where P ( t ) is the prompt coincidence curve and F ( t ) is the result of coincidences observed from a level with a finite lifetime.

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

139

Here it is assumed that the probability distribution functions have all been normalized to unit area. The form of the curves is illustrated in Fig. 6. The experimental problem is then to determinef(t) assuming that both F(r) and P(t) are accurately known. The most general way of doing this is to make use of a relation which exists between the rth moment of the delayed coincidence curve M,(F) and the moments M r - , ( P ) and M k ( f ) of the prompt curve and the required distribution respectively

where M,(F) =

j

m

T'F(T) dT -m

The set of equations (28) may be solved for the momentsf(t) and thus the form of the experimentally required function is obtained. Alternatively, if the functional form off(t) is known and contains n parameters which must be determined, then it is sufficient to solve n of Eqs. (28) for these parameters. If, in particular, the experimental distribution consists of a single exponential decay

f ( t ) = (l/t) exp( - t / z ) =O

for for

t 3 0, t < 0,

then by using r = 1 in Eqs. (28) we have, since all the distribution functions are normalized,

7 = M,(F)

-

M,(P).

(32)

Thus the mean lifetime of the exponential decay is equal to the distance on the time axis between the centroids of the delayed and prompt coincidence curves. By substitution of Eq. (30) into Eq. (27) and making the change of variables y = t - x we may obtain the result

m

and dF(t)/dt = (P(t) - F ( f ) ) / t .

(34)

This gives a second method of obtaining the lifetime T by using the slope of the delayed curve. By integrating Eq. (34) over the range t , to t , we obtain a

140

A. CORNEY

third expression for the lifetime z which depends on the measurement of the areas under the respective curves ["P(t) dt - r f z F ( t )dt

It is essential to use one of the three equations (32), (34), or (35) to determine the lifetime if it is of the same order as the width of the prompt coincidence curve P(t). However, for each of the three methods it is necessary to know the form of the prompt coincidence curve with reasonable accuracy. A search for fast transitions for this purpose has been made by several authors without much success. This is due principally to the fact that at the pressures normally used resonance trapping of photons effectively increases the observed lifetimes of the excited levels by one or two orders of magnitude. Recently, sec for his appaLawrence (51) has measured a resolving time of 0.7 x ratus using the Ne I1 transition at 461 A but in general lifetimes shorter than 5-8 x l o p 9 sec have not been measurable. Fortunately, in many cases, it is possible to determine the atomic lifetimes required by a fourth method, which does not require any detailed knowledge of the shape of P ( t ) . From Eq. (34) we have d

- [ln F(T)] = dt

Thus if the lifetime 5 is large enough so that at some point P ( t ) 4 F(t), then z may be determined from the slope of the curve of In F(t) at this point. In order to keep the error in z obtained by this method below 1 %, z should be at least three times the resolving time of the apparatus and only that part of the delayed coincidence curve beyond six times the resolving time should be used. This is the method of analysis which has been most widely used. Further details are given in the next section. b. Treatment of multiple component decays. If we consider the simplified energy level diagram of a hypothetical atom shown in Fig. 7, it frequently happens that we wish to study the lifetime of level 1, which has the same parity as the ground level. In general, this implies a rather small electron excitation cross section from the ground level. There often exist higher lying energy levels such as 2 with much larger direct electron excitation cross sections which can populate level 1 by radiative cascade. In order to obtain a sufficient signal on lines originating from level 1 it is usually necessary to increase the electron mean energy beyond the threshold, and, generally, level 1 is repopulated by cascade after the electron beam is cut off at time t = 0. If the atoms in level 2 decay freely with a total rate A , , then the population of level 2 at time t after the excitation pulse is removed is

Level 2

FIG.7. A simplified energylevel diagram showing the electron excitation and radiative cascade processes.

N

lo3

..

lo2

0

0.

10;'.

.

::'.

...

;*+a

. *

*I

.. 0

100 I

200

300

I

1

400 I

500

600

1

I

Time (in units of 10-9sec)

FIG.8. A plot of the log of the number of counts in a given channel of the multichannel analyzer as a function of the delay time represented by the channel address. The data were obtained on the ls5-2p, transition of Ne I by Klose (48) and illustrate the effect of cascade on the lifetime of the 2p6 level.

142

A. CORNEY

where n,(O) is the population of this level at time t = 0. The time dependence of the population of level 1 is determined by the total spontaneous decay rate A , plus the cascade contributions from level 2 at the rate A z l . Thus

If the: level 1 has population n,(O) at time t given by

= 0,

the solution of Eq. (38) is

The observation of radiative transitions from level 1 will therefore yield a decay curve consisting of a sum or difference of two exponentials with lifetime characteristic of the levels 1 and 2. This is a very common phenomenon and is illustrated by the decay of the 2p, level of neon shown in Fig. 8. In this plot the rapidly decaying component is due to the direct excitation of the 2p, level while the more slowly decaying component is probably due to cascade from the 2s and 3s group of excited levels. The appearance of multiple components in the observed decay curves is obviously an undesirable complication which should be avoided if at all possible. The only certain way of eliminating it is to use electron beams with very small energy spreads and to operate close to the excitation threshold of the level being studied. This requires careful attention to the design of the electron gun as was emphasized in Section III.A.3a. It is possible, however, to discriminate slightly against cascade effects between two excited levels provided that the lifetime of level 2 is much longer than that of 1. During the excitation pulse, the population of level 1 excited directly from the ground level will reach its equilibrium in a time of the order of 1/A, while equilibrium for level 2 will take considerably longer. Thus the ratio of direct excitation to cascade excitation of level 1 may be maximized by variation of the length of the excitation pulse. Provided that the lifetimes of the two levels involved differ by a factor of three or more a very simple method of analysis consists of finding the amplitude and lifetime of the longest lived component by using the slope of the tail of the graph in a semilogarithmic plot of coincidence counts against delay times. The contribution of the long-lived component may then be subtracted point by point from the earlier portion of the graph and the lifetime of the shorter lived component obtained from the new linear plot. The coincidence curve obtained by studying the decay of the population of level 1 given by Eq. (39) will have the form S(t) = C, exp(-A,t)

+ C, exp(-A,

t).

143

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

An alternative way of extracting the lifetime of the long-lived component is to perform a numerical integration of the data, indicated by 00

S’(t> =

S(t’) dt’ = ( C , / A , )exp(-A,t) f

+ ( C 2 / A 2 )exp(-A2

t).

(41)

Thus the contribution of the long-lived component to the new signal S 1 ( t ) is enhanced by a factor of A , / A , . This procedure is also useful for improving the statistical accuracy of a single component decay as is illustrated in Fig. 9. The large quantity of data obtained in many lifetime experiments means that the essentially graphical methods just described are too slow and inaccurate to do more than provide first estimates of the required lifetimes and signal amplitudes. For this reason most workers have extracted the data from the multichannel analyzer in digital form and have performed the analysis 2

8 7 6

c

5

C

0 ’ 4

0

j

3.k

lo2

~i f :*- I*

t (Delayed

coincidence

channel 1

+

FIG.9. Decay curves for the 2p6 level of neon obtained on the 2sz-2p6 transition by Bennett ( 4 4 ) . The lower curve gives the data obtained directly from the multichannel analyzer while the upper curve gives the integral of these data and illustrates the method used to discriminate against fast exponential decay components. [From W. R. Bennett, Jr., in “Advances in Quantum Electronics” (J. Singer, ed.), p. 28. Columbia Univ. Press, New York, 1961.1

I44

A. CORNEY

numerically with the aid of high speed computers. The basic program that has been used is one developed by Rogers (61) named FRANTIC. This program performs a least-squares fit of the data to a curve which is a generalization of that given by Eq. (40) by means of the Gauss iterative method. Since the number of counts in the ith channel m irepresents the integral of Eq. (40) over the interval t i to t i + A t i , the functional form used for the fit is mi =

N

1 - exp(-Aj Ati)

j = I

A j Ati

C C j exp( - A j ti>

where N is the number of exponential decay components which is thought to be present in the experimental curve. The program calculates, among other things, the best values of the exponential amplitudes C j and lifetimes l / A j together with their standard deviations. Further details concerning modifications of this method of analysis are given by Bennett et al. (47). With this method of analysis Bennett and Kindlmann (46) have clearly illustrated the dangers of using electron energies far above the threshold of the energy level being studied. They obtained measurements of the decay of the neon 2p, level both with the electron mean energy 0.1 eV above threshold, and also with a mean energy of 30 eV, that is 11 eV above threshold. The lifetimes of the exponential components resulting from a least squares analysis of the data are given in Table 11. TABLE I1 ANALYSIS OF DECAY CURVES FROM THE 2p2 LEVEL OF NEON Threshold data

Data taken at 11 eV above threshold

1 Exponential

1 Exponential

2 Exponentials

18.7 ik 0.3"

59.2 3 1.3

28.2 0.3 l 5 4 . 6 d ~1.8

+

3 Exponentials

*

25.2 0.5 90.3 j, 10 284 & 56

~

a

Lifetimes of exponential decay components given in units of

sec.

Analysis of the threshold data indicated the presence of only one exponential component. For the data taken above threshold, fits were made to curves containing one, two, and three exponential components. The resulting lifetimes illustrate the slow convergence of the lifetime of the fastest component toward the value obtained at threshold. Thus even with a three-component reduction the fast lifetime has a considerable systematic error. Other authors have observed much smaller increases in the cascade contribution as they

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

145

increased the mean electron energy due presumably to the difference in the electron energy distributions in their sources. The results given in Table I1 also show that errors may arise by forcing the data to fit a curve containing a smaller number of components than is actually present. This error may be detected by using different data record lengths for the fit. The presence of additional unsuspected components will be indicated by a variation of the lifetimes obtained as a function of the record length. This approach is discussed by Lawrence (51). Thus although the computer analysis is a powerful one the results obtained need to be treated with some care. 5. Experimental Dificulties a. Low signal intensity. The multichannel delayed coincidence system described above is obviously only able to count and measure the time delay of one photon per excitation pulse. Thus the system is saturable and if the photon arrival rate at the photomultiplier is too high the resulting delay curves will be distorted toward shorter lifetimes. The observed distribution is given by f ( t ) exp[ -n(t)], where f(t) is the true time-dependence of the input signal, and n(t) is the expected number of stop pulses which would arrive in the time interval t. Iff(t) = f ( O ) exp(-A,t), then the observed signal for A,t < 1 is

Thus the maximum fractional error introduced into the measured decay rate is 6 A , / A , =f(O)/A,. By keeping the photoelectron pulse rate at the start of the decay f ( O ) , sufficiently small, this effect may be easily avoided. An alternative way of overcoming this problem is to use a circuit which inhibits the multichannel analyzer from recording an event when two or more photons are detected following a single excitation pulse (61a, 61b). This method has the advantage that the time necessary for the collection of the data can be considerably reduced by increasing the photon arrival rate without simultaneously distorting the decay curve. In fact, the difficulty is usually to increase the photon arrival rate to a workable value. This is due to the combination of low electron bombardment current, low electron excitation cross sections, and poor detection efficiency of the spectrometer-photomultiplier combination. Thus although in favorable cases a decay curve may be built-up in a matter of minutes, in others several hours operation may be required. The speed with which the data may be gathered is determined by the excitation pulse repetition frequency which, for most atomic lifetimes, is fixed by the finite dead time of the apparatus. The maximum dead time for an N channel pulse-height analyzer with a

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A. CORNEY

time-to-height converter is given by t , = t , + N'/ where f is the internal oscillator frequency of the analyzer used for determining the channel address of a given event and t, is the memory storage time. Typical values for f and t , are 100 MHz and 1 p e c , respectively. At low signal counting rates, the dark current background becomes a problem. This may be reduced by a method which is essentially a lock-in detection method applied digitally. A slow square wave is used to modulate the dc reference supply for the excitation pulse amplitude. This alternatively raises the mean electron energy above and below the threshold of the energy level under study. When the pulses are above threshold the analyzer is run in the add mode, and when the pulses fall below threshold it is switched to the subtract mode. The result is that the background counts are kept as low as possible while the signal is accumulated during the add sweeps. This technique will also reduce the effect of signal from energy levels lower than that being investigated which might be present if the spectral resolution happens to be insufficient. b. The efect of cascade. This important experimental problem has been discussed in detail in Section 4.b. c. Efects of collisions. As discussed in Section II.C.l, at high enough pressures, the decay of excited singlet levels of helium is increased by inelastic collisions with helium atoms in the ground level : He(n'P)

+ He( 1 'S) -+ He( 1 'S) + He(n F).

(44)

The initial effect of collisions is to increase the decay rate from the spontaneous emission rate EnA,, to

Here N is the density of atoms in the ground level, 6 is the mean relative velocity of the two colliding atoms, and o is a mean cross section which has been averaged over the velocity distribution of the two particles, 6 0 = (ua(u)),,

.

(46)

This increase in apparent decay rate due to atomic collisions is a very common phenomenon although the importance of the effect varies from level to level due to the wide variation of the inelastic collision cross sections. For near resonant collisions, where very little energy needs to be given up to or taken from the thermal kinetic energy, the cross sections tend to have very large values, of the order of cm2. If energy differences of the order of kT exist between the two sides of a reaction such as that shown in Eq. (44), the observed cross sections are of the same order as the gas kinetic cross sections, approximately cm', while for much larger energy differences, with AE

147

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

greater than a few tenths of an electron volt, the cross sections tend to be very small, about lo-'' cm2 and are negligible for most purposes. The numerical value for the collision induced relaxation rate at T"K is

+

NUa = 1.404Pa[(M1 M , ) / M , M , TI1/' x loz3 sec-I, (47) where P is the pressure in Torr and the cross section a is measured in units of centimeters squared, MI, M , are the atomic mass numbers of the two atoms involved in the collision. Thus, at 300°K the collision induced rate will be equal to a radiative rate of 10' sec-' for a pressure of 39.0 Torr, assuming MI = M , = 20 and a = cm'. The simple expression of Eq. (45) predicts that if measurements are made of the apparent relaxation rate as a function of pressure, a linear graph should be obtained. Results obtained by Bennett et al. (47) for the 4 'S level of helium are shown in Fig. 10, and it is obvious that at the highest pressures used the

FIG.10. The measured decay rate of the 4 'S level of helium obtained by Bennett et al. (47). The solid curve shows the result of a least squares fit of data to an equation of the form R = A BP CP2.At the highest pressures the nonlinear effect due to reciprocal transfer becomes noticeable.

+

+

0

10 Pressure In torr

decay rate is not a linear function of pressure. The reason for this is that at these pressures the effect of a reaction similar to the reverse of Eq. (44) becomes appreciable, a process which has been termed reciprocal transfer. The effect may be understood more clearly by reference to Fig. 11. For simplicity the transfer between only two levels is considered although experimentally more complex situations arise. Let the radiative decay rates for levels 1 and 2 be given, respectively, by A , and A , . Then a,, and a,, represent the effect of collision processes such as Eq. (44) which transfer atoms from level 1 to level 2 and vice versa, while a,, and a,, represent the effects of all other collision processes. Then after the electron excitation pulse has been removed,

148

A. CORNEY

Level 2

FIG. 11. Simplified energy level diagram illustrating the collisional rates ( a i j )and the radiative decay rates (Ak.) involved in theeffect of reciprocal transfer.

~~~~l 1

the populations n, and n2 of the two levels are determined by the solution of the two coupled differential equations li, = - ( A , ri2 =

-(A,

+ a1>n, + a,1n,, + + U,,?I,,

(48) (49)

~ 2 ) n 2

+

where the total collisional destructive rates are given by a, = a,, a I 2and a, = a,, a,,. The solutions of Eqs. (48) and (49) for nl(t) and n,(t) both have the form of a sum of two decaying exponentials with decay rates R , and R , given by

+

A,+A,+a,+a, R1.2

=

2

1

k 2[(A,

+ a, - A,

-

+ 4u12 u,,11’2. (50)

Thus the observation of the decay of either level 1 or level 2 with a multichannel analyzer will yield in general a two-component decay, with the same decay rates R , and R, observed in each level. At the lowest pressures, the rates tend to the values

and are, therefore, linear with pressure. However, when the collisional deactivation rates become comparable with radiative decay rates, a nonlinear pressure dependence of R , and R , is observed. For this reason, it is usual to fit the decay rates observed for a given level as a function of pressure P , to a power series R = A BP CP2. (52)

+

+

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

149

For data obtained at low pressures, A gives the radiative decay, while B allows the total collision cross section to be determined from Eqs. (51) and (47). This analysis gives a powerful technique for the study of collision processes, both by a measurement of total cross sections and by allowing the main excitation transfer processes to be followed from level to level. The application of this technique to collision processes in helium is discussed by Kay and Hughes (35) and Bennett et al. (47). As noted in Section II.C.1, there is a considerable difference in the values obtained for deactivation cross sections by the two groups. A possible explanation for this is that when the reciprocal are much larger than a,, and p l 0 ,the decay rate R, transfer rates a12and measured at high pressures is approximately independent of density. It must be obvious that any detailed study of collision processes involves the analysis of multiple-component exponential decay curves. This can be done accurately only if the effects of cascade have been eliminated, and again emphasizes the importance of near threshold excitation. The effect of reciprocal transfer in the 2p level of neon (46) is complicated by the fact that the 10 levels lie within an energy range of 0.8 eV, and all have sec. The small energy spacing approximately the same lifetime of 17 x means that it is impossible to confine the excitation to a single level, while the nearly equal lifetimes prevent a meaningful analysis of the decay curves in terms of more than one exponential component. This forced reduction of the data i n terms of a single component decay leads to nonlinear pressure dependences of the decay rates which cannot be entirely attributed to reciprocal transfer. Only the linear term in the pressure-dependent decay curve can be associated directly with collisional effects. d. Photon trapping. When the lifetime of a level that has a strong allowed transition to the ground level is measured at pressures that are so low that collisional effects are negligible, a marked increase of the lifetime with density is observed. This is illustrated for the 3 2P level of sodium in Fig. 14, which is discussed in more detail in Section 1II.B. The effect is well known and, in this case, is due to the reabsorption within the source of photons in the 3 , S l I 2 3 ’PjI2, 1 1 2 transitions at 5890 and 5896 A, respectively. This is called “trapping,” ‘’ imprisonment,” or occasionally “ blockading ” of the resonance radiation and each reabsorption increases the observed lifetime by approximately the zero-pressure lifetime. The importance of the effect is due to the very large photon absorption coefficient k , at the centre of the Doppler broadened line 1,. Thus the probability that a photon will be reabsorbed while travelling a distance I , through a gas density No is given by x, where exp[ - y 2 k , L. exp( - y 2 ) ] d y ,

(53)

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A. CORNEY

and where

The transition probability for the resonance line is given by A,, and m is the mass of the atom. A theoretical treatment of the effect of trapping on the escape of radiation from a sample of excited atoms has been given by Holstein (62,63). The general solution of the problem is difficult but an approximate solution was obtained using the assumption, k, L 9 1. For the case when Doppler broadening is very much greater than natural or collision broadening of a given resonance line, the decay of excited atoms, which are initially distributed according to the normal diffusion mode of an infinite cylinder of radius R , is given by r

1

where g is an imprisonment factor given by g = l.60{k0 R[n ln(k, R)]'12}-'.

(56) For densities that give rise to optical depths k,L 6 1 Holstein's results are not applicable and the theories developed by Barrat (64) and by D'yakonov and Perel (65) are to be preferred. These theories predict that the observed decay rate will be given by

Barrat (64) uses the simplifying assumption that all the emitting atoms in the gas have the same velocity and so obtains an approximate expression for x given by x', where x' = 1 - e~p[-(n/6)'~~k,L]. (58) In helium, measurements of the decay curves of the 'P levels (47,48) at pressures to Torr have yielded decay rates corresponding to the completely resonance trapped limit, g z 0. Turner (66) has made similar measurements in krypton by studying the decay of resonance radiation in the afterglow of a pulsed discharge using the technique described in Section 11. C.2. However, in argon and neon (51,52), measurements at pressures below 2 x mm with a source in which the effective path length L was only 1 mm have enabled the transition probabilities of the resonance transitions to be estimated by extrapolation of the decay rates t o zero pressure. Although in discharges photon trapping can also occur on transitions which terminate on metastable and resonance levels with particularly large population density, this problem does not appear in the experiments described here

151

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

due to the very low density in these levels. However, for levels with transitions terminating in the ground level of the atom, accurate measurements over a wide pressure range are required in order to establish the true radiative lifetime of the excited level. A way of avoiding this problem is to use a molecular gas in a continuously pumped source (53)and to create the excited atoms by electron bombardment dissociation. The density of atoms in the parent gas may then be kept so low that photon trapping is negligible. Particular examples of resonance trapping are discussed in Sections 1II.B and 1II.C. e. S u m m a r y ofpressure-dependent eflects. The total effect of collisions and trapping on the observed decay rate of a given excited level may be illustrated with reference to Fig. 12. At very low pressures, the observed decay rate Atota,is the sum of the transition probabilities of all the allowed decay Akn. As the pressure is increased, trapping of modes, Atota,= AkO + the resonance transition occurs until the decay rate is reduced almost to ZnZO Aknat complete imprisonment. At still higher pressures, the observed rate increases linearly due to collisions until the reciprocal transfer region is reached. Figure 12 gives only a schematic impression, and the importance of

\

Low Dressure region

t

03

I

1 Pressure in torr

I 10

I

100

I

(log scale)

FIG. 12. A schematic diagram showing the pressure dependence of the experimental decay rate that would be observed on a given transition k +n from the excited level k which is connected with the ground level by a resonance transition.

152

A. CORNEY

the different effects varies widely from level to level even with one atom. In particular, the initial part of the curve will be frequently absent.

6. Comparison of Experimental Results We give in Table It1 the results of different workers for the lifetmes of the levels of the 2p5 3p configuration in neon. In each case, the method of delayed coincidences was used. Table 111 shows that the different authors are in close agreement for the lifetimes of these neon levels, although those of Bennett and Kindlmann are generally slightly shorter than those of other authors. A small cascade contribution to the decay curves might account for this discrepancy. There is good agreement between these experimental lifetimes and theoretical values recently computed by Feneuille et al. (66a). Some of the results obtained for the resonance lines of neon and argon (51, 52) are discussed in Section VI.B.2. TABLE I11 LIFETIMES OF THE LEVELS OF THE 2p5 3p CONFIGURATION OF NEON" Level (Paschen)

Klose (48)

Bennett and Kindlmann (46)

2Pl 2P2 2P3 2P4 2P5 2P6 2P7 2P8 2P9 2PlO

14.7 j,0.5 16.8 3Z 1.4 23.3 i4.8 22.4 j ,4.4 18.9 f 1.7 22.0 i 1.9 20.3 1.6 24.3 & 2.0 22.5 & 1.9 27.4 i2.9

14.4 0.3 18.8 i0.3 17.6 f 0.2 19.1 rt 0.3 19.9 5 0.4 19.7 f 0.2 19.9 & 0.4 19.8 i0.2 19.4 & 0.6 24.8 0.4

Bakos and Szigeti (67)

Osherovich and Verolainen (68) ~

+

Lifetimes given in units of

14.8

+ 0.15

28.2 & 0.85

22.0

+ 0.6

28.4 i 1.2 19.5 i 1.6

1411 20 & 1.6 18 f 1.3 24 j,1.5 23 i 1.5 21 & 1.4 22 & 1.3 25 i 1.4 24 i 1.5 2 6 k 1.5

sec.

B. Experiments Using Optical Excitation It is also possible to apply the method of delayed coincidences to the measurement of the lifetimes of atoms that have been optically excited. This method is restricted to those levels that can be excited from the ground level by the absorption of intense resonance radiation. Thus, only one or two levels may be studied in any one atom, and the technique is further limited to those atoms whose resonance lines lie in or near the visible spectrum. However, the technique does have the advantage that the very selective nature of

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

153

the excitation process eliminates the effect of cascading discussed in Section III.A.5. This has enabled Kibble et al. to make a detailed investigation of the effect of imprisonment of sodium resonance radiation on the measured lifetime of the 3 2P level of sodium (69) as well as investigating the effect of molecular (70) and atomic collisions (71) on this level. The apparatus used by Kibble et al. is described in detail by Habib et al. (72) and is indicated schematically in Fig. 13. Atoms of the sodium vapor Pulse shaping circuit

circuit Resomnce cell

I\

Gat ing pulse

I \

Pulse

:P%g

I

K e r r cell pulser

iI Sodium lamp

FIG.13. The multichannel delayed coincidence apparatus used by Kibble ef al. (69) for lifetime measurements of the 3 2Plevel of sodium using optical excitation.

contained in the resonance cell are excited to the 3 2P112,312 levels by the absorption of light of the sodium D lines 5890 and 5896 A, which is emitted by an Osram sodium lamp. The excitation is pulsed by focusing the exciting light through a Kerr cell shutter placed between crossed polarizers. The Kerr cell is filled with nitrobenzene and requires a voltage of 15 kV to open it fully. The resonance cell is designed (73) to give the incident and fluorescent radiation the shortest possible path within the absorbing vapor. This is necessary to overcome the effects of strong resonance trapping. The main body of the resonance cell is contained in an electrically heated oven. A side arm containing liquid sodium is maintaned at a slightly lower temperature and so determines the density of sodium vapor in the main cell. The cell is continuously pumped through a 3-mm bore tube in order to remove any gaseous impurities evolved during the experiments. The fluorescent radiation from the excited atoms is focused onto a Phillips 56 TVP photomultiplier operated at 2700 V across the dynode chain. The high voltage pulse on the Kerr cell produces light pulses of 10 to 20 x sec duration with a repetition rate of 60 Hz. Part of the voltage pulse is

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A. CORNEY

sec duration and fed to the shaped into a rectangular pulse of 100 x time-to-pulse height converter circuit. The delayed pulses from the photomultiplier are similarly shaped and supplied to the converter. This then supplies an output pulse whose amplitude is proportional to the time overlap of the two input pulses. Only 100 channels of a Victoreen P.I.P. 400-pulseheight analyzer were used during the experiments. The prompt spectrum of the apparatus could be obtained by replacing the sodium lamp with a tungsten lamp and arranging for the light pulses from the Kerr cell to pass directly through the side of the resonance cell to the photomultiplier. The calibration was achieved by inserting delay cables into one input t o the time-to-pulse height converter. Using this apparatus, Kibble et al. (69) found the lifetime of the 3 ’P level of sodium to be (16.6 & 0.4) x lou9sec. They also investigated the effect of resonance trapping on the lifetime of this level for a range of sodium vapor pressures from Torr. Their results are shown in Fig. 14. The to

V 0)

L

3

ln

0

f Id0

Id’

1o’*

No Vapor density (Atom cm”)

FIG.14. The variation of the measured lifetime of the 3 2Plevel of sodium obtained by Kibble et al. (69) as a function of sodium vapor pressure. The points are experimental, the solid curve is obtained from Milne’s theory, and the broken curve from Holstein’s theory. The encircled point and the zero density lifetime of T = 1.66 x lo-’ sec were used in the fit of the theoretical curves to the experimental data.

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

155

authors have compared their experimental results with the theories of Holstein (62,63) and Milne (74) by fitting the theoretical expressions to the data at the encircled point and at the low-pressure limit of the observed lifetimes. Some uncertainty in this fitting is due to the fact that the contributions of the two separate levels 3 ,Plj2 and 3 were not resolved in this experiment. This does not affect the measured lifetime of the levels at the lowest pressures, but at higher densities the fact that the absorption oscillator strength for the 3 ’S1,,-3 ,P3,, transition is twice as large as the 3 2S112-32Plj2transition will cause stronger resonance trapping of the former. Thus, in general, the observed decay curves will consist of two components with slightly varying decay rates. It was not possible to resolve these in the initial analysis of the data and it was, therefore, assumed that the observed lifetime was the mean of the lifetimes expected for the J = 4 and 3 levels. Figure 14 shows that Milne’s theory is in good agreement with the experimental values over the whole pressure range while Holstein’s expressions are in agreement only at pressures above 5 x lo-* Torr where the condition k , L >> 1 assumed by Holstein (62) is satisfied. This is rather surprising since Holstein has shown that Milne’s assumption of a characteristic mean free path for photon reabsorption is invalid in a region where the atomic absorption coefficient varies as strongly with frequency as it does near a resonance line. These experiments, however, represent one of the most accurate checks that has been made of resonance trapping over a large pressure range and illustrate clearly that Holstein’s results are not applicable at low density. The same apparatus has been used (70) to measure the decay rate of sodium atoms in the 3 ,P level as a function of the pressure of molecular gases N, , H, , HD, and D, . The collision processes that cause an increase of decay rate with increasing pressure in these cases, are usually referred to as quenching collisions since the energy of the sodium atom is converted into vibrational, rotational, and kinetic energy of the colliding molecules. These measurements yielded accurate quenching cross sections for these gases. Further measurements were made (71) on the effect of the inert gases on the lifetime of the 3 ,P level of sodium since Demtroder (75) had reported that the lifetime of this level was reduced by 20 % by 300 Torr pressure of helium. Demtroder’s results indicate a quenching cross section for sodium-helium collisions as large as 0.34 x cm2 and this was supported by measurements of Kondratiev (76) giving a quenching cross section of 0.3 x lo-‘’ cm2 for sodium-argon collisions at 860°K. The theoretical treatments of the problem (77, 78) were unable to decide whether cross sections as large as these were to be expected or not. Copley et al. (71), therefore, measured the lifetime of the 3 ’P level of sodium as a function of the pressure of He, Ne, Ar, Kr, and Xe over the range 100-800 Torr. The inert gases were carefully purified by passing the gas through a trap cooled in liquid air and storing the gas in flasks coated with metallic sodium or rubidium. The results showed no

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A . CORNEY

discernible change of lifetime with pressure. This negative result indicates that the quenching cross secti6ns for the inert gases are less than cm2. This has been confirmed by measurements of lifetimes of the 6 levels of cesium made by the phase-shift method as a function of helium pressure (78a). The quenching cross sections for this case are less than cm’. The results of previous workers (75, 76) must therefore be in error. It is possible that small quantities of impurities such as molecular gases evolved within the resonance cell or contained in the inert gas caused the observed quenching in these experiments. The work of Kibble et al. demonstrates the usefulness of the selective excitation by optical means for studying the lifetimes and collision processes involving atoms i n excited resonance levels. The technique deserves a wider application, especially since the range of applicability of different theories of resonance trapping seems to be poorly understood. C. Experiments Using Cascade Coincidences 1. Use of Cascade Photons

a. Principle of the method. A number of the experiments performed using electron excitation for delayed coincidence work have suffered from the effect of radiative cascade from higher levels, as discussed in Section III.A.5b. A technique by which this problem may be overcome was described by Brannen et al. (79). The cascade photon Bw,, is made to provide one pulse for a coincidence circuit. A decay photon hw,, from the intermediate level just created provides the second pulse to the coincidence circuit. By observing the coincidence rate as a function of a time delay inserted in the first photon channel they were able to measure the lifetime of the intermediate level which i n this case was the 7 3S1level of mercury. Suggestions for improving this technique using time-to-height converters and multichannel pulse height analyzers were made by Schatzman (80) and discussed by Camhy and Dumont (81). The method has actually been applied by Kaul (82) and Nussbaum and Pipkin (83)to measurements of the lifetime of the 6 3P1 level of mercury. The polarization correlation of cascade photons has also been studied by this technique (84). b. Experimental technique. The experimental technique used by Nussbaum and Pipkin (83) is indicated schematically in Fig. 15 together with the energy levels of mercury involved in their experiments. The atoms of mercury vapor within a sealed quartz tube were excited by a dc beam of 365 eV electrons from an electron gun enclosed within the same tube. Photons on the 6 3P,-7 3S, transition at 4358 A were selected by means of an interference filter and provided the start pulses for a time-to-amplitude converter. The

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

157

Light Source

/

Intmierence Filter

\

-

/

Mirrors

Interference Filter

Amplller

2537 A Disc

FIG. 15. A schematic diagram of the cascade delayed coincidence technique used by Nussbaum and Pipkin (83) in measurements of the lifetime of the 6 3P1 level of mercury.

photons on the 6 'S,-6 3P, resonance line at 2537 A are detected by a second photomultiplier and provide the stop pulses whose time delays are measured and stored in the pulse-height analyzer. The single channel counting rates N , + n, and N , + n, for the 4358 a and 2537 A channels, respectively, were monitored by scalers. Here N , , N , are the photomultiplier pulses due to the detection of photons while n,, n2 correspond to the dark current counting rates. In the experiments N , = N , = 1000 and n, z n2 = 100. A typical run, resulting in a multichannel decay curve containing a total of 3600 true coincidences, takes of the order of 21 hr. c. Experimental dificulties. The main difficulty in these experiments is the extremely low counting rate for true coincidences. It is of the order of to lop4 times that observed for most of the experiments described in Sections I1I.A and 1TI.B. This is mainly due to the low total efficiency for the detection of photons emitted by the source which enters twice in this experiment. Although the rates N , and N , could be increased to the value where the saturable nature of the detector becomes a problem, in fact, the upper limit for these single channel rates is set by the accidental coincidence rate N , ,

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A. CORNEY

which is determined by the resolving time T, of the system, N , = ( N , + n,) ( N 2 + n&,, and thus increases rapidly with the increase in single channel rates. Because of the inherent differential nonlinearity in this type of system, systematic fluctuation in the background could soon mask the required exponential decay of the true coincidence rate. For this reason the technique has not been widely applied. d. Experimental results f o r resonance trapping in mercury. Nussbaum and Pipkin (83) made measurements of the lifetime of the 6 3P, level of mercury over a range of pressures from 2.3 x lo-’ to 1.8 x Torr. The value obtained at the lowest density T = (1.14 f 0.14) x lo-’ sec agreed well with zero-pressure lifetimes measured by other workers. In Table LV,we give their results together with the optical thickness corresponding to their experimental geometry of L = 3.5 mm. In order to take into account the hyperfine structure and isotope shift of the 2537 A line, the value of k , used was 3 of that given by Eq. (54). The table also shows the effective lifetimes predicted by the theories of Barrat (64) and D’yakanov and Perel (65),using Eq. (57) with A,, = 0 in this case. Since Barrat’s expression for x , Eq. (58), is only an approximation while D’yakanov and Perel’s value, Eq. (53), takes into account the shape of the emission and absorption line profiles, it is not surprising that the latters’ theory fits the experimental results somewhat better than Barrat’s. The experiment provides a useful check on those theories of resonance trapping that are applicable in the region k , L 5 1.

En+,

TABLE I V EXPERIMENTAL AND THEORETICAL LIFETIMES FOR

Density Optical (atom ~ r n - ~ ) thickness, k,L 7.55 x 10” 2.08 x 1OI2 4.01 x 1OI2 1 . 7 0 ~10’3 5 . 7 7 ~1 0 1 3

THE

Experimental lifetime

0.033 0.092 0.176 0.749 2.54

1.14 i 0.14 1.26k0.11 1.24 i0.1 1 1.90 i 0.15 5.03 & 0.67

6 3P1 LEVEL OF M E R C U R Y ~

Theoretical lifetime D’yakanov and Barrat Pevel (65) (64) 1.16 1.22 1.29 I .89 4.95

1.16 1.22 I .30 1.96 7.18

Lifetimes given in units of lo-’ sec.

2. Use of Inelastically Scattered Electrons The true coincidence rate obtained by Nussbaum and Pipkin (83)has been increased by a factor of ten in a modification of their technique which was reported by Imhof and Read (85). In this new method a beam of helium atoms is crossed at right angles by a dc beam of electrons from a hemispheri-

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

159

cal electrostatic analyzer. The mean energy of the electrons was 75 eV with an energy spread approximately 70 meV. Electrons which are scattered by less than 5" enter a second energy analyzer and only those electrons with a predetermined energy are detected by a channel electron multiplier. By arranging to detect those electrons which have suffered an energy loss equal to the excitation energy of the level which it is desired to study, suitable start pulses for the time-to-pulse height converter are obtained. The stop pulses are obtained in the usual way from photons on the decay transition of interest, and the decay curve is displayed in a pulse height analyzer. In this initial experiment the lifetime of the 4 'S level of helium was measured to be (73 2) x sec. This method, besides eliminating the problem of cascade, is more widely applicable than the cascade experiments described in the previous section, since there, both photons must lie in regions where suitable photomultipliers and filters are available. The experimental system is, however, considerably more complex than the single-photon coincidence techniques described in Sections 1II.A and 1II.B.

IV. THE BEAM-FOILTECHNIQUE A . Details of the Technique 1. Introduction

The time-dependent exponential decay of excited atoms may be converted into a spatial decay of intensity by exciting a moving beam of atoms at a fixed position. This method was applied by several of the earlier workers (17). The accuracy of their results was limited by the low velocities of the thermal atomic beams and canal rays which were used. However, this idea has recently become the basis of an extremely important technique, following the discovery made independently by Kay (86,87) and Bashkin (88) that ions in the beam of a Van de Graaff accelerator were strongly excited by passing them through a thin carbon foil. Since then the technique has been considerably developed both by Bashkin's group at the University of Arizona (89-91) and by workers in other laboratories (92-94). A useful collection of papers presented at the Beam-Foil Spectroscopy Conference in November 1967 has been published recently (95). 2. Typical Experimental Arrangements

a. General outline. The general arrangement of a beam-foil experiment

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A. CORNEY

Van de Graaff

II Current lntegr Forodov

I

Drive Recorder

FIG.16. Schematic diagram of apparatus used for the measurement of the lifetimes of excited ions by the beam-foil technique.

is indicated in Fig. 16. A high energy beam of ions from the analyzing magnet of a Van de Graaff accelerator is passed through a thin carbon foil. During their passage through the foil, the ions are strongly perturbed and the beam that emerges on the downstream side of the foil consists of neutral atoms and ions. Usually, several stages of ionization are present (depending on the initial energy of the ions in the beam), and an appreciable proportion of the ions are in excited levels. Excited neutral atoms are more difficult to produce unless low beam energies are used. On the far side of the foil, the beam is seen to glow in a faint, usually blue, color as the excited atoms and ions decay by spontaneous emission. Due to the very high velocity of the beam, the excited particles are able to move a considerable distance downstream before they decay. The intensity of the light emitted by the beam decreases downstream and a measurement of the variation of the intensity of light from a given level of an ion as a function of distance from the position of the foil gives information about the lifetime of that level. In the apparatus shown in Fig. 16 the variation of intensity is obtained as a plot on an X- Y recorder. The excited beam has several important characteristics as a light source : (i) The de-excitation of the ions in the beam occurs in a high vacuum which enables the source to be coupled directly to avacuum spectrograph. Due to the very low effective density in the beam the ions decay freely and are not influenced by the effects of collisions or radiation trapping.

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

161

(ii) Ions of most of the light elements up to sodium have been excited. Since in many cases very highly ionized species are produced it is possible to study the lifetimes of levels which are not easily produced in other laboratory sources. The technique has also been applied to the study of lifetimes of molecular ions (96, 96a). (iii) The chemical purity of the excited beam is very high which considerably eases the classification of spectral lines not previously observed in other sources. b. Details of the Van de Graaff beam. Typically, beam energies of 2 MeV are used for ions H e f , C', and N + with an initial spread of beam energies, determined by the resolution of the analyzing magnet, of k0.02 MeV. This mean energy corresponds to a velocity of 5.2 x 10' cm/sec for N'. The beams are excited by passing through carbon, or occasionally aluminum, foils ranging in density from 5 to 40 pg/cm2. These correspond to foil thicknesses of 250 and 2000 A, respectively. Thus, the beam particles spend less than sec within the foil, and the ions are effectively excited instantaneously. Gaseous targets have also been used to provide a fast beam of excited atoms or molecules (93a, 96, 96a). In this method, the Van de Graaff beam passes through a chamber filled with gas at Torr before emerging into the observation chamber which is maintained at less than Torr. Beam currents are typically of the order of 1 pA which gives 6 x 10" ions/sec passing through the foil for a singly charged species. If we assume that the beam cross section is 0.25 cm', we find that the beam particle density is only 5 x lo4 ions/cm3. If all the ions were in excited levels and the radiative decay rate for the system were lo8 sec-', the total photon flux radiated into 4n solid angle would be of the order of 5 x 10" ~ r n - ~ ~ s e c - ' . The foils are prepared by evaporation from a carbon arc onto microscope slides that have first been coated with a thin film of soap or detergent. The carbon film can be floated off on water and pieces mounted on thin metal frames. The amount of light produced by the foils depends on their thickness (97) but due to surface effects, foils of equal thickness do not necessarily produce equally intense excitation of the beam. c. Spectrographs. For preliminary identification of the lines produced by a given ion beam it is useful to have available a fast photographic spectrometer. Bashkin (89) has described a special instrument designed by Meinel. Other aythors (98) have used a fast prism spectrograph to photograph new lines in the spectra of iron ions. However the majority of the authors use spectrographs that are commercially available. The lifetimes of excited levels can be obtained by the use of a stigmatic spectrograph, but this gives very inaccurate measurements due to the difficulty of the exposure calibration of photographic plates. Photoelectric techniques are therefore essential for accurate lifetime measurements. In the vacuum ultraviolet region, Heroux (99,100) has used a 2-m AFCRL rocket monochromator (101). Since this

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A. CORNEY

system has several unusual features we describe it in detail. Two interchangeable gratings enable the instrument to cover the wavelength ranges 60-320 and 300-1250 A. The entrance slit was 75-100 p wide. In order to scan through a spectrum, an exit slit, cut out of a thin Be-Cu belt, is moved along the Rowland circle. The instrument accepts light from a 1 mm length of the excited beam and has a numerical aperture of f/50. The instrumental line-width is determined by the exit slit and was approximately 0.5 8, between 60 to 320 8, and 2 8, in the range 300-1250 A. d. Photon detection and signal measurement. The detector used by Heroux was a Bendix M303 windowless multiplier with a tungsten cathode 8.9 cm long. Both this and the surfaces of the electron multiplication structure have high work functions, thus photons with wavelengths longer than 1300 A have a very low probability of producing a photoelectron. This is a very necessary characteristic since the detector is then insensitive to any light of longer wavelengths which is scattered by the grating. The Bendix multiplier also has a very low dark current. These characteristics are not shared by the combination, which has been used by some authors (102, 103), of a conventional photomultiplier sensitive in the visible or near ultraviolet together with a luminescent converter such as sodium salicylate. The signal level on transitions in the vacuum ultraviolet region of the spectrum is extremely small due to the low collection efficiency and poor reflection coefficients at these wavelengths. Consequently, it is necessary to use the most efficient detection method available and photon counting techniques are used. These are discussed in detail by Heroux (104, 105). The anode of the multiplier is operated at a high positive voltage and the pulse of charge is coupled capacitatively into the input of a pulse amplifier and then to a scaler. The pulses are counted for a preselected time interval, which in these experiments is usually 100 sec. Typically the number of counts received in this time interval would be 5 x lo3 with the monochromator selecting light from the beam close to the foil. Due to the X-ray radiation produced by the accelerator there is a source of noise which contributes to total background of between 1 and 4 counts/sec. For transitions in the visible region of the spectrum where the signal level is usually considerably greater, the advantages of photon counting techniques are not so obvious (106, 107) and a great deal of work has been done by recording the amplified photomultiplier current on a chart recorder as a function of the foil position. If the pressure in the target chamber is not sufficiently low, less than Torr, the beam may excite the background gas and a correction must be made to the observed signals. e. Calculations of the lifetimes from the data. After the observed transitions from a given ion beam have been identified the decay of a given excited level k can be investigated by adjusting the exit slit of the monochromator to

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

163

pass one of the radiative transitions from k to a lower level n. Let the number density of ions i n the level k at the downstream face of the foil, x = 0, be Nk(0). Then if the beam remains of uniform cross section S and the ions have a uniform velocity v , the number density of excited ions at position x = Z downstream is given by Nk(I) = Nk(0)exp( - t / T k ) ,

(59)

where t is the time taken for an ion to travel from the foil to the observation region, and Tk is the mean radiative lifetime of the level k . The number of ions which decay radiatively in the beam between I and I + A1 is Nk(Z, AZ)

= Nk(O) eXp(-Z/m,)[1

- eXp(-AZ/UTk)]S

AZ.

(60)

The observed signal is proportional to this quantity, the constant of proportionality being determined by the geometry of the ion beam withrespect to the monochromator, the detection efficiency of the monochromator, and the branching ratio for the decay transition observed. In the experiments reported by Heroux this factor is of the order of at 600 A. The quantity in brackets determines the initial counting rate. The signal will be weak if Al/uzk 4 1. In Heroux's experiments with A1 = 1 mm and u z5 x 10' cm/sec this restricts the apparatus to the measurement of levels with z < lO-'sec. Figure 17 shows a semilogarithmic plot of the counts observed in a 100-sec 3000

L

1000

n c C 3 0

u

r

's I

100 -

-

I

I

I

I

I

I

I

I

I

I

I

FIG.17. A semilogarithmic plot of the number of photons detected on the N I1 transition at 916 A as a function of foil position. The data were obtained by Heroux (99) using N + beam energy of 1 MeV and a foil thickness of 500 A. The transition is seen to be free of cascade.

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A. CORNEY

interval on the N It transition 2s' 2p' 3P-2s2p3 'Po at 916 A obtained by moving the foil upstream mm at a time with a stepping motor. The data have been corrected for a constant background count of 4 counts/sec. The accuracy with which the data fit the straight line leads to the conclusion that the lifetime for the level involved may be derived directly. The result in this case is T ( ~ P "=) (0.96 k 0.03) x sec.

+

3. Experimental DifJiculties a. The effect of cascades. Unfortunately, the data of Fig. 17 are not typical of many of the decay curves obtained by the beam-foil technique. The unselective nature of the excitation process makes it inevitable that many levels show decay curves influenced by the effect of radiative decay from higher excited levels. As discussed in Section ILI.A.4b, the data must then be analyzed as a sum of exponential decay curves,

IdZ, A 0 =

1 C j exp( - l / u r j ) . j

A typical experimental curve showing the effect of cascade is given in Fig. 18 for the He IL line at 304 8, (1 'S-2 'P). The curve can be interpreted as a

Foil position (mrn )

FIG. 18. A semilogarithmic plot of the number of photons detected on the He I1 transition at 304 8, as a function of foil position. The data were obtained by Heroux (100) using a He+ beam energy of 1 MeV and a foil thickness of 500 A. The transition is affected by cascade. 0,data corrected for background noise, T, = (1.61 i0.07) x lo-' sec. A,data corrected for noise and cascade, T~~ = (0.97 0.03) x lo-'' sec.

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

165

rapid decay due to the initial population in the 2p level followed by a slower decay determined by the repopulation of the 2p level by cascade. In this case, the data was analyzed by fitting a straight line to the tail of the curve and, by linear extrapolation, the data at earlier times has been corrected for the long-lived decay. The result is that the fast decay exhibits a lifetime

.r(2P),,,

= (0.97

& 0.03) x lo-''

sec.

Since the He I1 is a hydrogenic system an accurate theoretical value for the lifetime can be calculated :

z(2P)theor= 0.998 x lo-''

sec.

This is an important test that shows that it is possible to obtain accurate lifetimes for levels exhibiting fast decays with much slower decaying tails. It also indicates that the beam-foil method permits the measurement of very short lifetimes, which are, at present, beyond the range of most other techniques. The numerical methods discussed in Section III.A.4 also can be used to resolve the observed decay curves into a sum of exponential components. However, considerable care is required in analyzing and interpreting the results obtained from cascade decay curves, and the possibility exists of making quite serious errors. b. Calibration of the beam velocity. Two methods are available to determine the mean velocity of the particles in the excited beam. The first involves calibrating the accelerator voltage using nuclear reactions, which have sharp increases in cross section at accurately measured energies (108). For example, the reaction 7Li (p, n) 7Be has a threshold at 1880.60 f 0.07 keV. The heavy ions can be either accelerated at the calibrated voltage or the bending magnet of the Van de Graaff must be calibrated over a small energy range. The second procedure, applied by Bickel et al. (102) to a beam of 7Li+, uses an 18-cm radius, 90" electrostatic analyzer. The analyzer was connected directly to the target chamber, and the mean energy and the energy spread of the beam both before and after passing through the foil was measured. They found that the incident particles had an almost Gaussian distribution of energies with a mean energy of 56.2 keV and a full width at half maximum of 0.4 keV. After interaction with the foil the mean energy was reduced to 48.8 keV and the width of the distribution was increased to 3.0 keV. This appears to be the most satisfactory way of determining the energy loss of the beam after passage through the foil which is a general feature of these experiments. This loss can be large for ions of high atomic number moving at low velocities. An alternative way of correcting the velocity of the excited ions for the energy loss is to use the data obtained by Northcliffe (109) for the energy loss of different beams passing through carbon foils of varying thickness. The

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A. CORNEY

uncertainty of performing the correction in this way is estimated to be 1-2%. However, Chupp et al. (110) have noticed a systematic decrease in measured lifetime of 5 % when the same foil is used for measurements over a period of days. They interpret this as being due to an increase in the energy loss of the beam owing to the surface contamination of the foils. The effect of the velocity spread of the beam is usually negligible in comparison t o the other errors. c. The effect of Doppler shifts. The excited ions move with extremely high velocities and, although the beam is viewed at right angles to its direction of travel, the finite range of spectrometer acceptance angles can give rise to serious Doppler shifts. When the spectrograph slit is at right angles to the beam direction this results in a Doppler broadening of the lines, which may be of the order of 2 A. Although this does not affect the lifetime measurements directly, it does complicate the problem of identification of the observed lines. When the spectrograph slit is parallel to the beam direction, the problem is much more serious. d. Beamfluctuations and foil breakage. The Van de Graaff beam current cannot be held completely stable, and it is necessary to allow for this by normalizing the experimental data. This can be done by monitoring the beam current with a Faraday cup. The charge collected by the cup may be recorded continuously as a current or integrated for the same period as the photomultiplier signal. Difficulties arise in this method due to the fact that the Faraday cup measures the total charge collected. However, the excited beam usually consists of a distribution of charge that may change as the foil ages. Thus, the charge collected by the cup is not directly related to the number of ions in a given excited level. During the lifetime measurements the distance between the foil and the cup should remain constant to overcome the effects of beam divergence after passage through the foil. A better method of obtaining the data necessary for normalization is to monitor the intensity of light in the transition of interest at some fixed distance from the foil. This overcomes most of the difficulties associated with changes in the populations of charged states and excited levels. This technique also enables the instant when a foil breaks to be detected. Since foil breakage occurs frequently with 1pA beam currents of heavy elements (98), this information is necessary to avoid undue wastage of data. Since no two foils produce identical excitation of the beam, normalization of the data on a long run is essential. B. Comparison of Experimental and Theoretical Results

Lack of space and the rapid increase in the number of lifetimes measured by the beam-foil technique in the last few years makes it impossible to give a comprehensive list of references or results. It is also difficult to compare the

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

NY C E 0 0

IIIIII I

0.1

I

I

I

Bm I

I

0.2

Be1 I

167

LII I

0.3

I

1 0.4

FIG.19. Dependence of the sum of the absorptionf-values for the 2s 2Sl,2-2p 2Pi,2.3i2 transitions of the lithium isoelectronic sequence as a function of nuclear charge. 0 ,experimental values; 0 , theoretical values.

beam-foil results with those produced by other techniques since few other methods have been applied to the study o! highly ionized species. In order to illustrate the usefulness of the data obtained by the beam-foil technique, we give in Fig. 19 a plot of thef-values for the resonance lines, 2s 'S-2p 'Po, of the Li isoelectronic sequence as function of 1/Z, where Z is the atomic charge number. Wiese (13, IlOa) has shown that it is natural to expect a smooth progression of the f-values for a given transition as one moves up an isoelectronic sequence and that 1/Zis a convenient parameter to choose. Figure 19 shows t"e sum of the absorptionf-values for the transitions 2S1/2-2P1/2and 2S1/2-2P3/2calculated from the lifetime data of references (934 103, 111-111b, 112) using Eqs. (7) and (14). The theoretical data have been obtained using three different methods. Weiss (113) used variational calculations with 45 parameter wavefunctions for most of the ions, but for F VII and Ne VIII self-consistent field calculations were used. The data for

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A. CORNEY

very highly charged species were obtained by Cohen and Dalgarno (114) using the nuclear charge expansion method. Results for the lighter ions have also been obtained by this method ( 1 1 4 ~which ) are close to those obtained by Weiss (113). There is very close agreement between theory and experiment, which would give confidence in extrapolating or interpolating on other curves of this sort. Unfortunately, this simple situation does not hold generally, as may be seen by reference to Fig. 20 which shows the plot for thef-values of the tran-

03

$

1

/+

02-

>

/+

c

.

0.1-

c

.

c

P NeVl FV OIV I

0

0.05

01

1

I

NIII

C II

\ BI

I

I

I

0.15

0.2

1

'2

FIG.20. Dependence of the sum of the absorption f-values for the 2s22p 2P;,z, 3,z2s2p2 2D3,2,5,2 transitions of the boron isoelectronic sequence as a function of nuclear charge. Cohen and Dalgarno (charge expansion) ; 0 , Weiss (configuration interaction) ; A,Weiss (SCF calculation), 0, experimental.

+,

sition 2s' 2p 'P0-2s2p2 'D in the isoelectronic sequence of boron. Here, the transition involves levels due to equivalent electrons, and, since in these levels the electrons interact strongly among themselves, the independent particle approximation breaks down. The level wavefunction must then be described as a mixture of several configurations. This configuration interaction leads to experimentalf-values which rise to a maximum. The data in this case include results from beam-foil measurements (99, 115) and also the

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

169

phase-shift method (150). The theoretical fvalues obtained by Weiss (116) using the independent particle model and self-consistent field wavefunctions are seen to be in error by factors up to 2 or 3 and the same is true of the f-values obtained by the charge expansion method ( 1 1 6 ~although ) they lie on a misleadingly smooth curve. More elaborate calculatiors using configuration mixing gives much better agreement between theory and experiment. These two isoelectronic sequences illustrate clearly the usefulness of the lifetime measurement made by the beam-foil technique. Since the uncertainty in the beam velocity is usually less than 2 % the accuracy of the results, for those levels which are free of cascade, is usually limited by the fitting of an exponential to the experimental data. However, for those levels which show cascades, much larger uncertainties arise, especially if the decay rates of the different exponential components are of similar magnitude. It appears that these difficulties are not always realized by the authors. V. THE PHASE-SHIFT TECHNIQUE A . Theory of the Method 1. Use of a General Periodic Excitation Function

The phase-shift technique makes use of the fact that, since an excited level of an atom has a finite lifetime, the response of the population of this level to a periodic excitation process is shifted in phase with respect to the phase of the excitation. This phase shift is determined by the lifetime of the given level and the frequency of the excitation and so provides a means of measuring atomic lifetimes. Early applications of the technique are referred t o in Mitchell and Zemansky (17). We shall confine ourselves to recent applications of the method. Since it is difficult to arrange the excitation process to be a pure sinusoidal function of time, we consider the population Nk(t) of a level k when subjected to a general time-dependent excitation rate J ( t ) : dNk/dt =

-(l/Zk)Nk

+ J(t),

(62)

where Z~ = l/cn Akn is the mean lifetime of the level determined by spontaneous radiative decay to lower levels n. If the excitation rate is periodic with a fundamental angular frequency o,then J ( t ) can be expressed as a Fourier series a0

J(t) = -

2

+ 1 ( u pcos p o t + b, sin p o t ) . p=1

(63)

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A. CORNEY

Substituting (63) into Eq. (62) we obtain the steady-state solution

where the phase shift of the pth harmonic component is given by

2. Detection of a Single Frequency Component The problem of measuring the phase-shift 4,, and hence the lifetime, can be simplified by detecting only the intensity of light which is modulated at the fundamental frequency, p = 1. Then, for a transition from k to n, the signal observed, S(o) is given by S(O) = CA,,

[al cos(ot (1/Tk2

+ b , sin(& + m2)lI2

- 41)] 2

(66)

where Cis a constant that depends on the geometry, the efficiency of the detector, etc. By introducing the phase 4o of the excitation process at the first harmonic given by tan = b,/a, we may rewrite Eq. (66) in the form S(0) =

so cos(ot - 4s),

(67)

where SO

= CAk,[(alZ

+ b12)/(1/Tk2+ w2)]'/2

4 s = 41 + 4 0 Thus the measured phase-shift 4%is the sum of the phase shift

(68) (69)

introduced by the finite lifetime of the atoms, Eq. (65), and the phase of the excitation $, . The latter must be measured before Eq. (65) can be used to determine the lifetime. From Eq. (65), we find that the minimum uncertainty in the lifetime for a given phase error occurs when the phase shift 41 is 45".Thus for lifetimes of lo-', and lO-'sec, modulation frequencies of 159, 15.9, and 1.59 MHz are required. The lifetime can also be obtained from a measurement of the amplitude of a given harmonic in the emitted light but this method is seldom used because of the experimental difficulties involved. 3. Equivalence of the Phase-Shift and Delayed-Coincidence Methods The frequency dependence of the amplitude and phase-shift of S(o) are identical to those of the Fourier transform of the delayed coincidence signal S(t) = S(0) exp( - t/Tk) for t 2 0. Thus the phase-shift and delayed coincidence methods are completely equivalent. If the delayed coincidence signal is

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

171

observed at only a small number of fixed delay times, as in the method used by Heron et al. (43), equivalent information may be obtained by the phaseshift method by taking measurements at the same number of different frequencies. However the largest number of different frequencies used in a given experiment has so far not exceeded ten, and frequently the measurements are made at only one or two frequencies. Thus, the phase-shift method is definitely inferior t o the delayed coincidence method particularly when the latter is extended by the use of multichannel techniques. This is especially true when cascading makes an appreciable contribution to the observed signal. B. Use of Optical Excitation 1. Outline of the Technique

The use of modulated light to excite atoms and molecules for phase-shift measurements has been described by Ziock and his collaborators (117-119) working in Germany and a group in America under Brewer (120) whose apparatus has subsequently been used by a number of workers, for example (121-123). The principle of the method is illustrated in Fig. 21.

Q

Light source

Modulator

I I I

$->

Reference d Photomultiplier

Mixer

c

I

__f

Amplifier

I

Phase Shifter

I

I

I I

I I I I

Local Oscillator

Frequency Control

I I

I I

I I

Sample Mixer

Amplif ier

Phase Shifter

FIG.21. Schematic diagram of the apparatus used for the measurement of atomic lifetimes by the phase-shift technique with optical excitation [Cunningham (141)].

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A . CORNEY

Light from a suitable source is modulated at a convenient frequency f , typically 5 MHz, by one of the devices described in detail in the next section. This light excites atoms in the sample. Part of the exciting light is reflected off a plain glass sheet onto the reference photomultiplier while the light emitted by excited atoms is detected by the sample photomultiplier. The outputs from both phototubes are mixed with a quartz-crystal local oscillator signal of frequency (f- I ) kHz; the 1 kHz difference frequency is filtered out and amplified. The phase relationships between the reference and sample signals at the frequency f are still present in the 1-kHz signals following the heterodyne stage. The I-kHz signal in the reference channel is monitored, and the local oscillator frequency is servo controlled to within 0.1 Hz. This is necessary since circuits following this stage will cause changes of phase if the frequency drifts. Both reference and sample signal pass through calibrated phase-shifters to a phase null detector whose output is a minimum when the two signals are 90" out of phase. A phase difference of 0.2", corresponding to 0.1 x sec at 5 MHz, could be detected on clean signals with 15 V peak-to-peak amplitude. The experimental procedure consists of adjusting the sample photomultiplier supply to give a fixed 1-kHz signal amplitude and measuring the phase difference 4sbetween the reference and sample signals. Then the fluorescent cell is removed and an identical cell containing a dilute sol of MgO put in its place. Neutral density filters are inserted before the sample photomultiplier until the standard signal size is obtained when the phase difference 4o is measured. This procedure eliminates any spurious change of phase that might occur if the size of the sample signal varied. The phase difference due to atomic scattering (pl is then given by 41 = (ps - (po and can be used in Eq. (65) to calculate the measured lifetime. 2. Light Modulators

Methods of modulating light at high frequencies have been reviewed by Jones (124).Recently, a number of electro-optic modulators using crystals such as ADP and KDP have become commercially available. Although these are very suitable for the modulation of laser beams, the light sources used in phase-shift experiments require modulators with much larger angular apertures. A suitable modulator using KDP has recently been developed by Enemark and Gallagher (125). Ziock ( 1 1 7 ) used a Kerr cell operating at 1 MHz for measurements on the lifetime of the 5F, level of iron. Demtroder (126), Brehm et al. (127)and Hulpke et al. (128) have usedthediffraction of light by a standing ultrasonic wave to modulate light at I8 MHz in measurements of the lifetime of excited levels in Ga, Al, Mg, TI, and Na. The same technique has been widely used by other authors (120-123, 129) at slightly lower frequencies, and is described i n detail in the following paragraphs.

173

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

This technique makes use of an effect discovered by Debye and Sears (130), which is discussed in detail in textbooks (131, 132). If a standing ultrasonic wave is set up in a liquid, the alternate rarefactions and compressions in density produce similar spatial and temporal changes in the refractive index of the medium. A beam of light that is passed through the standing ultrasonic wave will suffer periodic phase changes across the wavefront, which will lead to interference effects if the light beam is focused on a screen. The spatial variations of refractive index act like a diffraction grating. Since these variations in refractive index are reduced to zero twice in every cycle of the ultrasonic wave, it is possible to obtain light modulated at twice the ultrasonic frequency by taking light from just one portion of the diffraction pattern.

Source

tl

Sample

Oscillator

FIG.22. The optical system of a light modulator using ultrasonic diffraction, after Link (121).

A typical experimental arrangement is shown in Fig. 22. Light from the source is concentrated onto the slit S, which is at the focus of lens L,. A collimated beam of light passes through the liquid and is focused by the lens onto a second slit S, which is adjusted to pass only the zeroth order of the diffraction pattern. The standing ultrasonic wave is generated by a 1%. diameter X-cut quartz crystal having a resonant frequency of 2.6 MHz. The crystal is mounted in a Lucite disk on the detachable bottom of the tank which is filled with a water-alcohol mixture in the proportions 81 : 19, respectively. This mixture has a zero temperature coefficient at 25°C and thus reduces the effects of thermal drifts. The water side of the crystal is held at ground potential by the use of metal contacts or conducting epoxy cement. Opposite the transducer is a steel reflector plate whose distance from the transducer is adjusted to be an exact multiple of the ultrasonic wavelength. To obtain good diffraction patterns, it is also necessary to adjust the plate to be parallel to the surface of the crystal.

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A. CORNEY

It is necessary to use an aperture stop in the optical system to confine the light beam t o that portion of the tank where the ultrasonic wave is uniform. The original 5.2-MHz modulator described by Brewer et al. (120) had a useful tank cross section of 1 in. x 2 in. and produced 80 % depth of modulation of the light in the zeroth diffraction order. However, the overall fractional This was greatly transmission for the optical system was only 4 x improved by Link (121) who introduced shorter focal length lenses and replaced the single slits S, and S, by multiple slit arrays. A similar type of highfrequency modulator, which uses the stress-induced double-refraction of a quartz block produced by standing ultrasonic waves has been described by Adrianova (133). An alternative approach to the problem of light modulation is to modulate the source directly. A phase-shift fluorometer using a hydrogen light source modulated at 10 MHz has been described by Birks and Dyson (134) and further details of modulated hydrogen sources are given by Hamberger (135). At lower frequencies, it becomes possible to construct sources other than hydrogen in which the light is directly modulated by radio-frequency excitation of the discharge (136). This would seem to be an easier solution than the rotating wheel reflection grating constructed by Brewer et al. (120) for modulation at 60 kHz. 3. Light Sources and Scattering Cells

For the measurements of the lifetimes of vibrational levels of the B 'rI& state of I, commercially available high-intensity mercury and tungsten lamps have been used as well as less powerful sodium and cadmium spectral lamps (137, 138). For the excitation of atomic fluorescence, however, Cunningham and Link (122)found it necessary to prepare microwave excited sources. Since the techniques for the construction of lamps and scattering cells are essentially the same as for the experiments described in Section VI1, the detailed discussion is given there. 4. Experimental DifJiculties

a. Determination of the zero-phase position. It is necessary to determine the phase of the exciting radiation 4o at the scattering chamber before the phase shift due to a finite atomic lifetime can be obtained from the measurements. Although, in principle, this can be performed as indicated in Section V.B.1, it is necessary to ensure that the optical geometry with the scattering MgO sol is identical to that used with the atomic sample. This is due to the fact that, after the light has been modulated by the ultrasonic grating, there is a phase variation of approximately 10" across the wavefront of the exciting beam due to variations of phase along the length of a given diffraction

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

175

order. Therefore, a difference in geometries during the zero-phase measurements could lead to a systematic error. Identical scattering geometries are particularly difficult to achieve when the atomic sample is a beam (122). Although, with care, the systematic errors due to the phase spread in the exciting light may be eliminated, it effectively limits the lifetime resolution of the system. This phase variation across the exciting light beam is much reduced in the modulator described by Enemark and Gallagher (125). Systematic phase-shift errors can also arise from changing geometrical arrangements due to the variation of the transit time of electrons through a photomultiplier, depending on the point of origin of the first electron on the photocathode. This effect has been investigated by Birks and Dyson (134) and Rees and Givens (139). Smaller changes of transit time are also caused by variations of the photomultiplier supply voltage or large changes in the wavelength of the light being detected, since these both affect the photoelectron velocity close to the photocathode (140). b. Calibration of the phase-shift apparatus. By using precision variable resistances in the RC phase-shift circuits the calibration may be made directly (120, 129). A small correction t o the usual analysis of the phase-shift circuit is discussed by Cunningham (141). To check the calibration, the apparatus may be used to measure the speed of light. A He-Ne laser gives a beam with a small divergence so that long optical paths may be used and the phase-shifter checked over a wide range. For modulation systems operating at lower frequencies, calibrated delay lines are used. Calibration accuracies better than 1 % are obtained. c. The effect of background scattering. The effect of the modulated exciting light which is scattered directly into the sample photomultiplier by the windows and walls of the sample cell is perhaps the most serious experimental difficulty encountered in this technique. The effect causes a phase shift 4, to be measured, which corresponds to a lifetime shorter than the true atomic lifetime. When the intensity of wall scattering is much larger than that of the atomic fluorescence, very large errors are caused. Fortunately, this effect often can be eliminated by measuring the amplitude of the total signal S , and that of the wall scattering S, which is assumed to have the phase 4o of the incident radiation. This assumption is correct if the cell is of small dimensions. Then if the signal due to atomic fluorescence has amplitude So and phase 4s as in Eq. (67) we have

s, cos(wt - +J

=

socos(wt - &) + s,cos(wt - &).

By making the substitution wt obtain

= wt’

+ 4o

(70) throughout Eq. (70) we may

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A. CORNEY

Since the measured phase is converted to a measured lifetime T, using the relation tan(+, - 4 0 )

=mzm,

(72)

and the true lifetime is given by tan(+, - 40)= wz,

(73)

we may obtain the following relation from Eq. (71) z, = T[ 1

+ &/So)( 1 +

W 2 T 2 ) ” 2 3 -1.

(74)

This result shows that when wall scattering is very much greater than the atomic fluorescence the measured lifetime is very small, while at the other extreme, the measured lifetime approaches a constant value equal t o the true atomic lifetime. This behavior is illustrated in Fig. 23 by data from reference (122) showing that measured lifetime of the 7s 2S1,2 level of thallium as a function of increasing atomic fluorescence to wall scattering signal So/S, obtained by increasing the scattering thallium beam density. At high values of the ratio So/S,, the measured lifetime approaches the true atomic lifetime of 7.65 x sec. The solid curve is a theoretical curve calculated using Eq. (74). Thus, in favorable cases, the effect of wall scattering may be easily overcome.

1

10

100

FIG.23. The measured lifetime of the 7s zS1,2level of thallium as a function of the atomic fluorescence to wall scattering ratio So/& obtained by Cunningham and Link (122). The circles represent the experimental data and the solid line was computed from Eq. (74), using the value of T = 7.65 x lO-’sec.

177

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

70

0

60

-

50 -

0

0 h

U

49 -

D

ltnl

0 0

C 0 0

I

1

I

I

I

I

I

I

I

I

10

I

I

I I I I I I

100

I

I

I

I

1 I l l

1000

FIG.24. The effect of resonance trapping on the lifetime of the 4p ‘P level of potassium measured by Link (121) using the phase-shift technique. The solid curve was calculatedfrom Eq. (74) using r = 27.8 x lO-’sec.

d. The effects of photon trapping and collisions. Although it is necessary to vary the density of atoms or molecules in the scattering sample in order to eliminate the errors due to wall scattering, too high an atom density may lead to photon trapping, as discussed in Section III.A.5. When this occurs, the lifetime measured by the phase-shift method will no longer tend t o a constant value as the density and atomic scattering increase, but will continue to rise to larger values. This behavior is illustrated in Fig. 24 for the measured lifetimes z, of the 4p ’P level of potassium as a function of increasing atomic t o wall scattering ratio S,/S,. The effect of radiation trapping increases the uncertainty associated with fitting Eq. (74) to the data in order t o obtain the atomic lifetime z. Since the atomic levels measured by this technique are necessarily connected to the ground level by strong transitions, the highest density in the sample being used is determined by the onset of resonance trapping. This occurs well before the region where atomic collisions begin to influence the lifetime appreciably. However, for the measurements of the lifetimes of vibrational levels in molecular iodine made by Chutjian et al. (138) the

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A. CORNEY

collisional quenching is appreciable at pressures of 1 x Torr and measurements over a range of densities must be made in order to obtain the radiative lifetime. 5 . Experimental Results The lifetime of the 3 2P level of sodium has been measured by a number of workers. Their results are given in Table V. There is obviously very satisfactory agreement between the different authors with the exception of the value obtained by Kartensen and Schramm (142, 143) and obviously a systematic error is indicated in their measurements. Unfortunately similar comparisons cannot be made for elements where the measurements are considerably more difficult than in sodium although results obtained by different lifetime techniques usually agree to within 10%. TABLE V LIFETIME OF THE 3 zP LEVEL OF SODIUM Lifetime"

Method used

15.9 i0.016 15.9 f 0.039 16.1 0.3 14.2 f 0.2 16.3 0.4 16.1 f 1.0 15.95 f 0.4 16.3 f 0.5 16.0 i0.5 16.1 i0.7

Phase-shift Phase-shift Phase-shift Phase-shift, electron excitation Delayed coincidence Delayed coincidence Hanle effect Level crossing Level crossing Optical double resonance

In units of

Reference

sec.

C . Use of Electron Bombardment Excitation 1. Outline of the Technique

The use of optical excitation restricts the application of the phase-shift method to a few levels in each atom which have strong electric dipole transitions to the ground level. The optical modulators used also limit the technique to a wavelength range 2000-8000 A. In order to extend the range of application of the phase-shift technique, Lawrence (149) developed an

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS r-----------7

179

D

Digital Phase

FIG.25. A block diagram of the electron excitation phase-shift apparatus used by Lawrence (149).

apparatus using a moauiatea eiecirun oomoarurncni 5 u u r ~ cW I U C I ~rias ~ I I I C C : been used by Lawrence and Savage (150) and other workers (151-153). Their apparatus is shown schematically in Fig. 25. The source consists of a directly heated tungsten filament. The electrons are accelerated to the required energy, 20-150 eV, by a voltage applied to the second and third grids. These grids are separated by 0.4 mm and form the entrance slit of a 0.5 m Seya-Namioka vacuum monochromator. The bandpass of the monochromator was 5-8 A. The fact that the source, monochromator, and photomultiplier are in one continuous vacuum system extends the wavelength range down to approximately 600 A. The sample gas was introduced into the source through a hollow anode, which also enabled pressure measurements to be made in this region. Typical source pressures were lo-' Torr. The electron current was modulated at the frequency f by a sinusoidal signal applied to the first grid of the source. By varying the distance between the tungsten filament and the first grid, anode currents of 2 mA with 85 to 95 % depth of modulation were obtainable. The modulated light emitted by the atoms in the source was detected by means of photomultipliers placed behind the exit slit of the monochromator.

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A. CORNEY

For the range 200-1300 A a Bendix magnetic multiplier with a cathode coated with CsT was used. Other photomultipliers, EMR 542F-08 (10503200 A) and EM1 6256A (1550-5000 A) extended the range into the visible region. To measure the phase of the emitted radiation a single sideband generator was used to provide a reference signal at (f+ 100) Hz. This was mixed with the photomultiplier signal and produced a 100-Hz beat at the output of the communications receiver that carried the phase-shift information of the original photomultiplier signal. The use of narrow band widths at the 100-Hz frequency enabled signals corresponding to photoelectron rates of 1000/sec to be measured. The phase-shift in the light signal is measured on a modified time-interval counter. The accuracy of the phase-shift measurement depends critically on the frequency stability of the side band generator. The elimination of harmonics from its output is also important since these could produce a beat at 100 Hz with suitable harmonics of the emitted radiation. Because of the unselective nature of electron bombardment excitation cascade transitions can contribute to the observed signal. In order to identify when this occurs, the phase-shift measurements must be made over a range of frequencies. In the experiments discussed here, 10 fixed frequencies from 0.54 to 54 MHz were available. For a transition free of the effects due to cascade, a smooth curve given by tan 41= ws can be drawn through the data points as

FIG.26. The absolute phaseshift versus modulation frequency curve for the u ’ = O level of the A ’ L I state of N O + observed on the X ’ C +- A IIT transition by Hesser (153).The lifetime obtained by the fit of the solid curve to the data points is (55.7i 6.0) i sec.

I

0.54

5.4 Frequency ( MHz )

, 54

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

181

illustrated in Fig. 26 for the phase-shift versus frequency curve for the u' = 0 level of the A 'Il state of NO+ (153). 2. Experimental DifJiculties

a. The efSect of radiative cascade. The presence of cascades is indicated when it is impossible to fit the data to the smooth curve as in Fig. 26. Since the lifetimes of higher levels giving rise to cascades are often longer than the lifetime of the level being studied, the effect is to cause anomalously large phase-shifts at low frequencies. The phase-shift due to the cascading contribution becomes much smaller as the modulation frequency is increased, and, in severe cases, will cause the plot of measured ph.ase shift against frequency to go through a maximum (149). The effect of cascades may be considered theoretically using a simple two-level system in which the upper level has lifetime T and is populated at a rate PF cos wt by modulated electron bombardment. The time-dependent equilibrium population of this upper level is Nit) =

BF COs(wt - 4 T ) (1 /T2 + w2)1'2 '

(75)

where tan = wT. The population n of the lower level with lifetime o which is populated at a rate Fcos wt by direct electronic excitation, and a rate aN(t)/T by radiative cascade, where ci is the branching ratio, is then given by solutions of the equation

P'F cos(wt - 4 T )

dn 1 _ ---n+ dt

z

(1

+ w2T2)1'2 + F cos wt

where tan

4c =

8' sin $T

(1

+ w2T2)1/2+ p' cos &

(77)

and P' = ED. Thus the phase 4s of the signal detected in the modulated light emitted by the lower level is given by

4 s = 41 + 4 c 2

(78)

where, as previously, tan = w r . By algebraic manipulation, this may be put in the form 4s= tan-I(wz) + tan-'(wT) - tan-'(wT/(l + p')). (79) Although more than one upper level may contribute by means of radiative cascade, the limited frequency range of the experimental data prevents a more

182

A. CORNEY

80

m

8 GOL

FIG.27. The absolute phase-shift versus frequency curve for the v ' = O levelof theD %+stateofNO.Thedata illustrate the effect of cascade and were obtained by Hesser (153) on the X - D ' C + transition using NOCl as the parent gas.

0

0

U v

-

40-

c

m

aJ

m

0

20-

0.54

5.4 Frequency

45 ( MHz )

detailed analysis being used. Figure 27 shows the effect of cascade on phaseshift measurements made by Hesser (153) on the X 2rI - D '2' transition in NO. A theoretical curve given by Eq. (79) has been fitted to the experisec. for the mental points resulting in a lifetime of z = (20.7 f 4.0) x u' = 0 level of the D 'C+ state, and cascade parameters of T = 125 x sec and p' = 0.54. The value of z determined in this way is only slightly influenced by the value of T chosen for the fit as long as T 3 lot. In other less favorable cases the correction for cascade can only be made if the important cascade transitions can be identified and if the lifetimes of their upper levels are known. Even then an approximate estimate of the lifetime 7 for the level being investigated must be known. Savage and Lawrence (154) report that in 30 % of the transitions investigated the cascade correction could not be made unambiguously and these transitions could not be used for lifetime measurements. The effects of the cascade obviously present severe experimental difficulties in the application of the electron excitation technique. b. Determination of the zero-phaseposition. The use of electron bombardment makes it impossible to measure the phase of the excitation process qbo by the technique used in the optical excitation method. It is therefore necessary to obtain 4oby measuring the phase of the light qb,(N) emitted by some standard optical transition, which has been shown to be free of cascade. Lawrence and Savage (150) chose the 1085-A line from the 2s2p3 3Dolevel

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

183

+

of N'. Then, since &(N) = $,(N) 40,we may obtain 4o if the phase-shift 4,(N) = tan-'(mN) due to the finite lifetime of the nitrogen ion can be calculated. Unfortunately this could only be obtained indirectly by a comparison of the measured phase-shifts in the 1085-A line and Lyman alpha radiation. This involves a cascade analysis for Lyman alpha radiation following Eq. (79) and the theoretical lifetimes of the 2p and 3d levels of hydrogen. This indirect method of obtaining the zero-reference phase introduces an uncertainty of the order of 5 % in the value of the measured lifetimes. This uncertainty is increased by the effects of phase meter error, changes in the photomultiplier delay with wavelength, and the effects of cascade, when present, to a total value of 10 to 15%. c. Spectral Resolution. It is necessary to ensure that light from only one excited level or multiplet lies within the band-pass of 4 to 8 A of the monochromator used in these experiments. This was checked by recording the spectrum with a 3-m Eagle monochromator with 0.4-A resolution. d. The effects of photon trapping and collisions. The effect of radiation trapping in these experiments was eliminated by exciting the spectra of atoms and ions from molecular parent gases. Thus Lawrence and Savage (150) have obtained excited levels of N I and N I1 using N, , levels of B I and B I1 from BF, , C I, C 11, and H I from CH, . Excited levels of molecules and molecular ions have been studied by the same technique (149,155-157) and are reviewed by Hesser (153). The source is continuously pumped and the low operating pressure ensures that the density of atoms and ions produced is never sufficient to cause photon trapping. The low source pressure also ensures that collisional deactivation is not significant in most of the experiments. However the small source geometry does mean that the lifetimes of atoms and ions which are longer than about 500 x sec will be affected by quenching on the electrodes and diffusion from the field of view of the monochromator. 3. Experimental Results

In Table VI, we compare a few of the results obtained for lifetimes of excited levels in N I and N I1 by the phase-shift method with values obtained by the beam-foil technique. Also shown are values for the lifetimes obtained using the NBS tables of transition probabilities (18). The agreement between the results obtained by the two techniques is seen to be satisfactory except for the final entry in the table, in spite of the fact that both techniques suffer from the effect of cascades. The NBS values for N I were taken from measurements with a wall-stabilized arc and seem in general to be slightly lower than the other experimental values. For N I1 theoretical values were used and, in comparison with the beam-foil measurements, are seen to be rather inaccurate.

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A. CORNEY

TABLE VI LIFETIMES OF LEVELS IN N 1

N I1

N 11"

Level

Exp. lifetime phase-shift

Exp. lifetime beam-foil

NBS lifetime'

2s22p23s'D 2s22p23s 2P 2s22p23s 4P 4~ 2s2p4 2s22p3s 3P' 2 ~ 2 ~ 33 ~

2.2 k 0.3' 1.9 i0.3" 2.5 3 0.3" 7.2 k 0.7b 0.9 i0.2d 0.4 i 0.2d

2.65 0.2' ,0.2' 2.28 j 2.35 C 0.2' 7.10 0.3' 1.01 0.15' 0.96 I0.05'

1.95 1.33 1.85 4.35 0.77 0.56

Atom N I

AND

'

Lifetimes given in units of

sec.

" Data from Lawrence and Savage (150). Data from Bashkin et a/.(158). Data from Hesser and Lutz (152). Data from Heroux (99). Data from Wiese et al. (18).

TABLE VII LIFETIMES OF MOLECULAR LEVELS Molecule Nz

Upper level

B

+

'xu+

0, = 0

Lifetime (lo-' sec)

Reference

6.58 0.35 6.60 i0.13 7.15 i0.50 4.50 i0.40 5.92 0.60

(36) (96) (163) (159) (153)

Pulse-sampling Beam-foil Pulse-sampling Phase-shift Phase-shift

(149) (96) (160) (153) (159~)

Phase-shift Phase-shift Beam-foil Pulse-sampling Phase-shift Delayed coincidence

11.80* 1.10 13.20 k 0.50 14.00 i0.40

(156) (161) (96)

Phase-shift Phase-shift Beam-foil

11.30f 1.10 13.90 i 1.00 14.90 i0.40

(156) (162) (96)

Phase-shift Pulse-sampling Beam-foil

+

co+

co,

+

B

'x+

v,

=0

%,+ 0, = 0

COr

+

2K 0, = 0

3.95 & 0.30 5.30 i 0.50 4.50 i 0.50 10.10 i 0.50 5.34 & 0.50 5.10 =k 0.07

(156)

Technique

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

185

A comparison of some molecular lifetimes obtained by different techniques is made in Table VII. The spread of experimental values indicates that the results are generally less reliable than those obtained for atoms and that systematic errors of 20 % or more are often present in the quoted results. In particular the spectral resolution used in many of the pulse-sampling experiments is rather low. Some of the experiments suffer from the effect of radiative cascade, while the results obtained by the beam-foil method depend on the correct elimination of the contribution of radiation from background gas in the viewing region. Fowler et al. ( 1 5 9 ~have ) recently applied the multichannel delayed coincidence technique to the measurement of molecular lifetimes in CO'. Their result for the D' = 0 level would seem to be the most accurate and is in good agreement with the phase-shift measurement (153).However this agreement is not shared by the values of (5.22k0.18) x lo-' sec and (5.79 f 0.08) x sec obtained by these authors (153,159a), respectively, for the v' = 1 level. WIDTH VI. THE RADIATION

A N D PRESSURE

BROADENING OF

SPECTRAL

LINES

A . Observation o j the Shape of Emission Lines

I . Introduction Both classical and quantum theories of radiation (5) predict that the distribution of intensity Y ( w o - w , r) as a function of angular frequency o, of a spectral line emitted by a stationary atom is given by

where oois the angular frequency corresponding to the difference in energy between the upper level k and the lower level n, ho, = Ek- E,, , and r is the sum of the natural or radiation widths, rk and I-,,, of the upper and lower levels, respectively. The total intensity emitted in the line is I , . The radiation width of an excited level is equal to the inverse of its radiative lifetime, thus

r = r k + r, = l/Tk + l / T n .

(81)

This intensity distribution is called the Lorentz line-shape and it has a full width at half its maximum intensity equal to the radiation width r. The finite width of excited energy levels may be regarded as due to the operation of the uncertainty principle AE.At z h , where At is t o be replaced by the finite lifetime of the level. Thus the measurement of the Lorentzian widths of

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A. CORNEY

a series of spectral lines would allow the lifetimes of the excited levels t o be determined. If z k and 2, in Eq. (81) are measured in seconds, then is given in angular frequency units, radians per second. Spectroscopists often quote the line cm-’) and the conversion factor is width in milliKayser (1 mK = F (rad/sec)= 67c x lo7 I’ (mK). Often units of megahertz are used for I’ and in this case I’ (MHz) = 30 I’ (mK). Unfortunately, the observed line-shape of spectral lines is usually dominated by the Doppler effect due to the thermal motion of the radiating atom. In the absence of natural broadening, this gives rise to a Gaussian intensity distribution 9(w - wo , A) given by

r

9(w

-

w,, , A) = (2/A&)

exp[ -4(w - oo>’/A2],

(82)

where A = 2wo(2kT/m)’/2/cand m is the mass of the radiating atom. For an atom with an excited level with lifetime lo-* sec which radiates at 5000 A the Lorentzian width, rk/27c = 15 MHz, while the Doppler width of this transition, assuming the atomic mass number = 100 and the gas temperature = 300”K, is given by Aw,/2n = A(ln 2)”’/27c = 720 MHz. The actual line-shape observed, I ( o - w o , r/A), is the result of folding the Lorentzian and Gaussian distributions together and is usually termed the Voigt profile (164). rn

I ( o - wo , r / A ) =

9(wo’ 0

- coo, A)Y(w,’

- w,

r)dw,’.

(83)

This integral cannot be evaluated analytically, but several tables of numerical values are available (165-167). The shape of the Voigt profile depends on the ratio r / A . For large values of this ratio, the profile approaches a Lorentzian function, while for small values the profile is nearly Gaussian. If r z A, accurate absolute values of both r and A may be obtained from a detailed analysis of the observed profile. This technique has been used to obtain the lifetimes and f-values of the resonance lines of the inert gases. A partial-level diagram of neon is shown in Fig. 28 and this is typical of all of the inert gases. The first four excited levels have been labeled using the nomenclature of LS coupling although the level spacings indicate intermediate coupling. The ‘PI levels of the inert gases are connected by strong resonance lines, with the ground level and they have radiative lifetimes in the range 5 x lo-’’ to 1 x sec. The intercombination line from the 3P1 level is weaker in neon and argon and the lifetimes of these levels are generally slightly longer. Because these transitions lie in the vacuum ultraviolet region, it is not possible to study the resonance lines

187

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

153 152-

1-

151h

7

2 p2 2p6

2 9 3P

150-

6

/

m

0 1497

k930

B 4 V)

._

C 3

Y

148-

> 136 <

2 2p53s

135W

134-

744

<' 0-

Ground level

736

1

2 P6

FIG.28. A partial energy level diagram of neon.

directly with the resolution necessary to determine their Lorentzian widths. However the widths of both 'PI and 3P, levels may be obtained by high resolution studies of spectral lines which terminate on these levels. Some of these transitions, which in neon are in the red, are shown in Fig. 28. They originate on levels which have radiation widths which are generally very small and which can be calculated from the measured lifetimes of these levels. The inert gases are very suitable for this type of high-resolution spectroscopy since isotopically pure samples of the even isotopes can easily be obtained. This eliminates the effects of isotope shift and hyperfine structure. They can also be used over a wide temperature range and the pressure of the gas may be determined accurately. Both these will be seen to be of importance in the discussion which follows. The techniques of high resolution spectroscopy are discussed in (168, 169).

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A. CORNEY

The techniques of optical double resonance, level crossings, and Hanle effect have also been used to measure the radiative width of excited levels. These methods are discussed in detail in Section VII.

2. The Experitnental Arrangenient A schematic diagram of the apparatus used by the Oxfordgroupin a series of detailed studies of spectral line profiles in helium, neon, argon, and krypton (170-174) is shown in Fig. 29. The spectral lines of interest are

D A R K ROOM

U

reservoirs

Double X - Y recorder

FIG.29. A schematic diagram of the apparatus used for high resolution studies of spectral line profiles in the inert gases. Hcre, P . M . denotes photoniultiplier.

excited in a low current dc discharge tube which is cooled i n a bath of liquid nitrogen or ice. The initial experiments (170) were performed using a microwave discharge cooled with liquid helium (175). The difficulty of measuring the gas temperature in this discharge caused it to be abandoned. The discharge is viewed in a constricted region to avoid the effects of self-absorption. Typical operating currents were 0.5 mA at pressures in the range 0.1-10 Torr. An f/16 Littrow spectrograph using Bausch and Lomb precision replica gratings is used to isolate the spectral line of interest. The high resolution is obtained with a Fabry-Perot etalon. The etalon plates used were 6-cm diameter and of very high quality. The imperfection profile due to surface irregularities had a half-width of 1/80 to ljl0Oth of an order at 5500 A. The

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

189

plates were coated with films of silver, aluminum, or multilayer dielectrics depending on the wavelengths of the lines being examined. The size of the etalon spacer is chosen so that overlapping of orders due to the width of the spectral line is avoided. The central spot of the Fabry-Perot fringe pattern is imaged onto a pinhole which replaces the entrance slit of the spectrograph. The Fabry-Perot fringe pattern is scanned across this aperture by slowly changing the pressure of carbon dioxide which fills a box surrounding the etalon. The intensity of light received by a photomultiplier placed behind a similar pinhole in the focal plane of the spectrograph is recorded as a function of pressure o n a form of X-Y recorder developed by Kuhn and Lucas-Tooth (176). The pressure scale of this instrument is linear and reproducible and can be converted into a wave number scale since the peaks of successive orders are separated by 1/2d cm-’, where d is the etalon spacing. This recorder has recently been improved to allow the simultaneous recording of the profiles of two separate spectral lines (177). Figure 30 shows a typical record for the profile of the 5852-w transition in neon, obtained at 1.09 Torr. The base line is recorded by interrupting the light from the discharge.

El

pressure

in

etalon box

( Arbitrary units

1

-

FIG.30. The spectral profile of 5852-A transition in *ONe obtained by Lewis using a pressure-scanned Fabry-Perot etalon. The discharge was cooled with liquid nitrogen. The measured width at half-intensity is 37.7 mK and the analysis, which is indicated in the diagram, leads to a total Lorentzian width I? = 20.1 mK and a Gaussian width A(ln 2)’’’ = 25.3 mK.

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A. CORNEY

3. Analysis of the Experimental Curves The analysis of the experimental traces is discussed in detail in (170, 173). The problem is that of unfolding the effect of the instrumental Fabry-Perot profile from the observed trace to obtain the true spectral line profile, and then finally of resolving the latter into a combination of Gaussian and Lorentzian functions. The method usually adopted is t o correct the base line for the small effect of overlapping orders and the peak height for the effect of the finite aperture of the pinhole. Then each order of the observed trace is treated as a Voigt profile and its component Lorentzian and Gaussian parameters, r and A, are found by measuring the widths at fractions of the peak intensity and using the tables of Davies and Vaughan (165). The analysis of the profile of the 5852-A line of neon shown in Fig. 30 is given above the experimental trace. This method assumes that the instrumental function can be represented as a combination of Lorentzian and Gaussian functions determined, respectively, by the reflectivity of the films and the imperfect flatness and parallelism of the etalon plates. It is a convenient property of the folding integral that the result of folding two Lorentzian functions having widths given by rl and T2 with two Gaussian distributions having parameters Al, A2 gives a Voigt profile. This profile may be considered to be the result of folding a single Lorentzian function, width r, with a Gaussian function, width A, where

r = rl + r2 A'

= A12 -k

(84)

AZ2.

(85) To obtain the breadth parameters for the spectral line (rl,A l ) from the results of analysis of the observed profile (r,A) it is necessary to know the instrumental contributions (r,, A2). These may be accurately determined by profile measurements on a given spectral line made with a range of different etalon spacers but with otherwise identical experimental conditions (170). 4. The Effect of Pressure Broadening

The Lorentzian widths rl of the spectral lines obtained by this techniquc were found to be dependent on the density of the inert gas in the discharge. This is mainly due to the fact that the lower levels for these transitions are strongly influenced by collisions with ground state atoms involving a resonant transfer of excitation

X('P,)

+ x(ls,)~x(ls,) + X('P1).

(86)

A similar transfer takes place for the triplet levels. This interaction leads to a broadening of the excited singlet and triplet energy levels and consequently to an increase in the Lorentzian widths of the

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

191

observed spectral lines. One of the purposes of the experiments described above was to investigate this in detail. In contrast with many other types of collision process the interatomic potential for resonance collisions is accurately known and has the leading term V K C/r3,where r is the interatomic distance and the constant C i s determined by thefvalue of the resonance line connecting the two levels involved. There have been many attempts to calculate theoretically the effect of this interaction on the width of spectral lines. Earlier methods are reviewed by Tsao and Curnutte (178) and Breene (179). More recently calculations based on the impact theory have been performed by Byron and Foley (180), Watanabe (181, 182), Ali and Griem (183, 183a), Omont (184, 187), and Stacey and Cooper (185). They predict that resonance broadening gives an extra contribution, rres to the Lorentzian width of a spectral line that is linearly proportional to the density N of the perturbing atoms,

(r,,,is given in radians per second). Heref,., and /? are the absorption oscillator strength and wavelength of the resonance line, and k,,., is a constant that depends on the total angular momenta J J ' of the upper and lower levels and the approximations made i n the calculation. Values of k,,, are given in Table VIlI for the 'P1-'S, resonance lines and should also apply to the 3P1-'S, intercombination lines. Table V l l I lists the theoretical results in the TABLE VIII THEORETICAL VALUESOF THE RESONANCE BROADENING CONSTANT Reference 1.33 1.44 1.41 1.45 1.53 1.532 1.54

Byron and Foley (180) Watanabe (181, 182) Ali and Griem (183, 183a) Omont (184) Omont (187) Stacey and Cooper (185) Berman and Lamb ( 1 8 5 ~ )

order of increasing accuracy and the last three values quoted should be correct to better than 2 %. The accuracy of these resonance broadening theories has been confirmed recently by independent measurements made by the Hanle ) recently published a comeffect (186-188). Berman and Lamb ( 1 8 5 ~ have prehensive account of similar calculations, and they discuss in detail the results of other authors.

192

A. CORNEY

Resonance broadening is characterized by the fact that the broadening constant k,,. is predicted to be independent of temperature and that, due to the symmetry of the interaction with respect to the spatial quantum number m,, there is no shift of the line center. Broadening due to higher order interactions, such as the Van der Waals' forces, does not share these properties and thus an important experimental check on the type of broadening observed can be made. It is therefore expected that the Lorentzian width rl of a spectral line that terminates on the 'PI level will be given by

The radiatio!i width P may then be determined by extrapolating a linear plot of rl against N to zero density. By subtracting from r the small contribution due to the upper level, Eq. (81), we may obtain the lifetime of the 'PI level and finally thef-value for the 'S,-'P, transition. Alternatively, thef-value of this transition may be obtained directly from the slope of the graph by using the theoretical value of k l o . As discussed in the Section VI.B, unresolved discrepancies have occurred i n results obtained by the extrapolation method but the results using the slope of the curve are considered reliable. The extrapolation method may also be used t o obtain thef-value of the S,-3P, transitions but when using the slope method a small correction must be made to the value of k,, in the case of neon and argon. This is discussed in more detail in Section 5.b. 5. Experinzeiital Dijficulties a. Tlie ejjiect of ser-absorption. The effects of self-absorption are frequently noticed, although the Lorentzian widths are obtained from the profiles of visible lines that terminate on excited levels which are not metastable. This is due to the relatively large population i n the resonance levels which is created by strong photon trapping of the vacuum ultraviolet resonance line. The effect occurs particularly at high current densities and, in extreme cases, causes the visible line profile to depart considerably from a Voigt profile. If the selfabsorption is less than 5 % at the peak, no departure from the Voigt profile can be seen and the effect can only be detected as a variation of the Lorentzian and Gaussian components as a function of discharge current. The effect is eliminated by working with weak visible lines and very low current densities. The use of low current densities ensures that Stark broadening by electrons and ions is always negligible. b. Tlie effect of nonresonant interactions. The theories of collision broadening predict that when both resonant and nonresonant interactions are

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

193

;?resent simultaneously the observed line broadening is heavily dominated by whichever is the stronger interaction. Thus for the lines which terminate on the 'P, levels the perturbation of both upper and lower levels by nonresonant interactions is negligible in comparison with the resonance broadening of the lower level. However since thef-values of the intercombination lines from the 3P, levels in neon and argon are much smaller than those of the true resonance lines, the observed broadening of lines terminating on the 3P, level is affected by nonresonant interactions. This is detected by a small shift in the line center with increasing pressure and a slight change in the observed broadening with temperature. Lewis (189) has devised a method for correcting the observed broadening for this effect which enables the f-value of the 1S,-3P, transition to be obtained from the slope of the curve. c. The measurement of the discharge temperature. It is necessary to know the temperature of the gas in the discharge in order to convert the measured pressure into a number density. The temperature of the discharge can be calculated from the measured Gaussian width provided that the instrumental contribution is known. The uncertainty in the measurement of the instrumental width usually leads to an uncertainty in the temperature of between 4 to 8 % which directly affects the accuracy of the measured broadening constant. d. Limit of validity of the impact theory. The assumptions contained in the impact theories of resonance broadening are valid only if the density of perturbing atoms N satisfies the inequality

where t; is the mean relative velocity of colliding atoms. It is therefore only possible to derive the lifetimes of atoms from the measured pressure broadening constant, Eq. (88), if the pressure range obeys this criterion. For argon the upper limit is 100 Torr while in krypton (173) the broadening becomes nonlinear at pressures greater than 15 Torr. The experimental results discussed below indicate that there is also a lower limit on the density for the impact theory to hold but no definite criterion can be given as yet. 6. Use of Atomic Beam Light Sources

Since one of the difficulties encountered in measuring the natural width of spectral lines is the large Doppler effect it is natural to ask whether this could not be greatly reduced by exciting a well-collimated beam of atoms. Such sources have been constructed by Odintsov (190) and Stanley (191). The excitation is produced by means of a crossed electron beam. These sources are considerably more difficult to construct than a dc discharge tube and have very low intensity. It was also discovered (192,193) that the observed Doppler

194

A. CORNEY

width was always considerably larger than the width estimated from the beam collimation and gas temperature of the beam source. This was attributed to the momentum imparted to the light atoms, helium and neon, by electron impacts. For these reasons the technique has not been widely applied.

B. Discussion of Experimental Results I . Use o j the Extrapolation Method In Table IX, the radiation widths of the 'P, levels of helium, neon, and krypton obtained by extrapolating the Lorentzian widths of spectral lines to zero density are given in mK = cm-'. For helium there exist very accurate theoretical calculations of the transition probabilities of the lines emitted by the 2 'P, level (7) which have been used to calculate the theoretical width of 9.56 mK given in the table. Other theoretical f-values in helium (195,196) give essentially the same result. Since the experimental results are very much larger than this value we conclude that there exists a definite anomaly in the experimental radiation widths obtained by extrapolation. Unfortunately, theoretical f-values of comparable accuracy do not exist for neon and krypton. However the table also shows the radiation widths calculated fromf-values of the resonance lines which have been obtained from the experimental pressure broadening constants by using Eq. (88) with the theoretical value of k , , = 1.53. For all three examples the width obtained by extrapolation is anomalously high. There is evidence of a similar but smaller discrepancy in argon. TABLE IX OBTAINED BY EXTRAPOLATION TO RADIATION WIDTHS OF 'Pi LEVELS ZERODENSITY

Element

He

Extrapolation width

Resonance broadening width

12.7 & 1.5" 13.5

9.04 & 0.55

Theoretical width

Reference

9.56

Vaughan ( I 71) Damaschini and Brochard (194) Schiff and Pekeris ( 7 )

-

Ne

5.67 4= 0.7

3.46 i 0.06

-

Kuhn and Lewis (172)

Kr

2.0 i0.5

1.50 i0.07

-

Vaughan (173)

a

Widths are given in mK-

cm-I.

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

195

Kuhn et al. (197) have suggested that this anomaly may be due to the resonance coupling of atoms by interaction terms which are neglected i n the usual theories of pressure broadening. Such effects have so far only been calculated for stationary two atom systems (198-200) and it is not possible to decide whether this explanation is correct. Other authors (201) consider that at low density the approximation of treating the Doppler broadening and collision broadening as two independent processes breaks down and that this may explain the anomaly. Whatever the true explanation, it is certain that at present it is not possible to obtain correct f-values by extrapolating the measured Lorentzian widths to zero density. We must also conclude that at some pressure below the range of the present experiments, i.e., <0.1 Torr, the impact theory of resonance broadening ceases to be valid.

2. Use of Resonance Broadening Results

In Table X are shown the absorptionf-values for the resonance transitions 'So-'P, and 1S0-3P, of the inert gases obtained by different techniques. For each element the results obtained from the observed resonance broadening of spectral lines is given first. These have been obtained using the value of k,, = 1.53 in Eq. (88), and the error quoted contains only the uncertainty of the measured broadening constants. The line-width measurements in helium and neon (192, 193) were made using an atomic beam light source at densities where resonance broadening is negligible. The results obtained by the lifetime measurements are systematically smaller than those obtained by resonance broadening. This may be due to the fact that the lifetime measurements for the 'P, levels of neon and argon (51, 52) barely satisfy the requirement, discussed in Section IIL.A.4a, that the lifetime being investigated should be longer than three times the resolving time of the apparatus. The effect of cascade also makes it difficult to use that part of the coincidence curve which is sufficiently long delayed from the end of the pulse. This is expected to lead to lifetimes which are slightly longer than the correct values and to consequently smaller f-values. However, the interaction V(r)K C/r3 used in the recent impact theories of resonance broadening is only an approximation to the full interatomic potential. Thus, although the broadening is found to be linear with density in the range of the present experiments, effects of the type discussed in Section VI.B.1. may reduce the effective broadening constant, k,,, and account for the small discrepancies apparent in Table X. For the 3P, levels of neon and argon, the experimental broadening constants are affected by nonresonant interactions as discussed in Section VI.A.5b. These have been corrected by the method described by Lewis (189) to obtain the values ~ f f ( ~ given P ~ ) in Table X.

TAHLt Atom He

Ne

Ar

Kr

Xe

x.

A R SOR P T~ ON f-VALUFS FOR THt.

Technique

f('P,)

-

Resonance broadening Line-width Line-width Electron scattering Hanle effect Theory

0 261 0.016 0.28 -0.02 0 26 I 0.07 0 31 I 0.04 0 270 0.01 1 0.2761 0.0001

Resonance broadening Line-width Lifetime Electron scattering Theory Theory Theory

0.159 & 0.003 0.16 & 0.014 0.130 ' 0.013 __ 0.120 0.140

Resonance broadening Lifetime Lifetime Electron scattering Theory Theory

0.256 f0.014 0.228 i0.021

Resonance broadening Absorption Lifetime Electron scattering Theory

0.173 0.008 0.135 iO.01 __

Hanle effect Absorption

0.238 0.01 5 0.26: 0.02

*

+

-

~

RFSONANCE LINMOF

fPPd -

Ref.

as for f('P1)

-

0.01 15 2 0.001

0.170

+ 0.004

-

-

0.0078 1 0.0004

0.138 1 0.013 0.14 0.01 0.132 0.175 0.163

~

0.012 0.035 -

0.063 $ 0.004 0.059 T 0.003 0.024 1 0.002 -

0.20

0.049

-

-

0.136

Zf

~

-

-

THE l N t R T G A S E S

+

+

0.319 0.018 0.287 1 0.024 -

0.233 0.02 0.249 0.330 0.366 t 0.01 1 0.293 I 0.02

0.193 0.003 0.158 0.01 0.166 __ 0. I38

0.346 t 0.06 0.274

0.256 $- 0.008

0.494

-

-

0.023 -

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

197

There is obviously a serious systematic error in the results fortheargon3P, level obtained by Morack and Fairchild (41a). Since their lifetime of (21 2) x sec is much closer to those of the next group of levels above 3P, (49) rather than the value of -8.5 x sec obtained by both (174) and (51) we conclude that their experiment suffered from the effect of cascade. The table also indicates that accurate f-values are very difficult to calculate for these transitions and that the electron-scattering measurements are rather inaccurate.

C. Observation of Absorption Line Shapes In principle, the shape of spectral lines can also be studied by absorption techniques and the results of a number of experiments on the resonance lines of the alkali metals are summarized by Moser and Schulz (214).More recent experiments are reported by Wilkinson (211)for the resonance lines of krypton and Chen and Phelps (215)for the resonance lines of cesium. There have been considerable discrepancies between the results of different workers indicating difficulties both of experimental techniques and interpretation of the observations. Most of these experiments have been performed with only moderate spectroscopic resolving power and the observations are confined t o the far wings of the lines. The widths of the absorption lines are almost entirely determined by the effect of collision broadening but unfortunately it is not possible to apply the results of the impact theories since these apply only to that part of the line profile satisfying Am @ Tci,'i,where d o is the displacement in angular frequency units from the center of the profile and Tcoilis the duration of a typical collision (22).When more accurate calculations are available for the wings of the resonance lines it will be possible to use the experimental data to calculatefvalues but at present the high-resolution studies of emission lines seems t o be the more useful method. VII. TECHNIQUES USINGRESONANCE FLUORESCENCE

A . In t roduc t ion The study of the intensity and polarization of resonance fluorescence has for a long time been a favorite method for the investigation of excited levels of atoms (17). I n this section, we discuss modern applications of the old technique known as the Hanle effect as well as the more recent methods of level crossings and optical double resonance. We emphasize the measurement of lifetimes although the level crossing and double resonance techniques have been used mainly for the measurement of the fine and hyperfine structure of energy levels. This is reflected in recent reviews of the field (216-219).

198

A. CORNEY

The techniques described in this section again make use of the finite width of the energy levels of excited atoms in order t o measure their lifetimes. However since in these experiments the signals are induced by radio frequency or dc magnetic fields, the Doppler effect is reduced in the ratio of the frequency of the radio frequency field to the frequency of the optical radiation. In all the experiments discussed here the Doppler width of the observed signals is at least a factor of lop3 smaller than the radiation width. This removes completely one of the factors which limited the accuracy of the lifetime measurements made by studying the width of spectral lines described in Section VI. The techniques described in this section also differ from those of the previous section in the fact that the width of the signal is determined by the width of only the upper excited level and the measurements may usually be made at densities which are so low that pressure broadening effects are negligible. We shall not refer explicitly to the early work on the Hanle effect since this was performed before the effects of resonance trapping and collision broadening were fully understood. Consequently some of the old numerical results are unreliable. The techniques of this section are most easily applied t o elements which are easily vaporizable and have resonance lines in the region 1800-8000 A, e.g., Hg, Cd, Zn, Na, K, Rb, and Cs. However, the methods have been recently extended and experiments have been reported on systems as diverse as Ca', Mg' ; He, Ne, Ar, Xe; Eu, Sm, Tm; NO and OH. Since the methods described here share many of the same experimental techniques and difficulties we discuss these details together after the principles of the different methods have been given. B. Principles and Theory of the Different Techniques I . Hanle Effect

a. Typical experimental arrangement. The principles involved in the use of the Hanle effect may be appreciated by reference to a classic series of lifetime measurements on the 'P, and 3P, levels in zinc, cadmium, and mercury (220-227). A typical experimental arrangement for the study of the Hanle effect in the even isotopes of mercury is indicated schematically in Fig. 31. Light from a quartz mercury lamp contains a strong line in the near ultraviolet at 2537 A which is the intercombination resonance line 6s' 'S0-6s6p 3P,. This light is filtered to remove other unwanted radiation and then is linearly polarized in a direction at right angles to a magnetic field H. This field is supplied by a set of Helmholtz coils which are not shown. The resonance radiation is focused onto a quartz vessel placed at the center of the Helmholtz coils. This resonance cell is prepared by evacuation at pressures below

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

-

C

199

A

-

To Helmholtz coils

t Magnetic Field Sweep

x- Y >Recorder

<

Amplifier

Torr and then sealed off after a small quantity of mercury has been distilled into it. The mercury atoms of the attenuated vapor in the resonance cell absorb the linearly polarized light and are excited from the ground level to the excited level 6 3P,. The excited atoms decay spontaneously to the ground level after a mean excited lifetime of T and re-emit the resonance radiation. The fluorescent light which is emitted in a direction at right angles to the direction of the incident light is collected by a lens, passed through a linear polarizer and detected by means of a photomultiplier. It is found that the intensity recorded by the photomultiplier changes as the magnetic field is slowly varied over a range of +_ 10 G about the zero field position. This constitutes the Hanle signal from which the lifetime may be obtained directly as is described below. It is due to the change i n polarization of the fluorescent light with magnetic field and so the Hanle effect is sometimes referred to as magnetic depolarization. b. Semiclassical theory. The detailed dependence of the intensity of the scattered light detected as a function of the magnetic field strength may be calculated by the use of a classical model in which the excited atom is treated as containing a single, harmonically bound electron. This approach is useful because it gives a clear picture of the process responsible for the Hanle effect and correctly predicts the shape of the signals observed in all experiments.

200

A. CORNEY

The excitation process is treated by assuming that the electron of one of the atoms receives an impulse at the moment of excitation to which causes it to start oscillating at the angular frequency w o in the direction of polarization of the exciting light. This electron, in the absence of a magnetic field, would produce an electric field E(t) at a point on the observation axis given by

E(t) = E(0) exp[ - r(t - to)/2]exp[ - iwo(t - t o ) ] .

(90)

The damping constant or radiation width is related to the quantum mechanical lifetime of the excited atom by r = l / r . The electron, however, also experiences a force due to the applied magnetic field which causes its plane of oscillation to process about the field direction with the Larmor angular frequency

where e , m are the charge and mass of an electron, g J is the Lande factor, and po the Bohr magneton. Thus at the time t the plane of oscillation makes an angle w,(t - t o ) - x with the plane of transmission of the polarizer. The transmitted intensity from one atom is therefore given by I ( H ) = Z, exp[ -r(t - t o ) ]cos2[w,(t

-

t o ) - 311.

(92)

The intensity that is observed at time t from a collection of atoms that were excited at a constant rate for all previous times is then

‘

I ( H ) = - J d t , exp[-r(t 2 -cc 10

+

- to>l{l cos 2[wL(t - to> - X I :

Thus the shape and sign of the observed signal depend on the orientation of the polarizer 31 in the detection beam. The dependence on the magnetic field, through the variable wL, will have either (i) a Lorentzian shape for x = 0 or n/2, (ii) a dispersion shape for 1= n/4 or 3x14. The experiment can i n fact be performed with a number of geometrical arrangements which differ from that shown in Fig. 3 1 but it is usually arranged so that the observed signal is Lorentzian. I n this case the importance of the Hanle effect lies in the fact that the field-dependent term falls to half its maximum value at magnetic fields H , given by gJP0 H k

2w L --2-=r

h

(94)

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

+

20 1

GAUSS

FIG.32. A typical Hanle signal observed on the ls5-2p, transition of neon with a wavelength equal to 6402 A. The full-width at half maximum, AH = 9.957 G, corresponds to an effective lifetime for 2p9 level of (8.11 f 0.16) x 10-gsec. Data taken with 1.37 Torr neon plus 0.68 Torr helium and discharge current of 10 mA.

This is illustrated in Fig. 32. The magnetic field width AH at half the maximum intensity is connected with the radiative lifetime through the relation gJ(P0 AH/h) =

= l/Z.

(95)

The data of Fig. 32 refer to the 2pg level of neon for which g J = 1.335 and A H = 9.957 G. Thus we obtain = 1.233 x lo8 rad/sec and the effective lifetime for the conditions of this particular run is z = (8.11 0.16) x lo-’ sec. It is important to notice that although optical radiation is being used both for excitation and detection the width of the signal is determined by the natural width of the level rather than by the Doppler width of the optical line. This is because the shape of the optical line is not resolved and the detection system is equally sensitive to all frequencies within the line profile. The signal appears effectively at zero (dc) frequency. c. Estimate of the size of the signal. The size of the signal is determined by the number of photoelectrons per second Q produced at the cathode of the photomultiplier by the scattered fluorescent light, where Q is given by Q

= FAN

d a,,,,, Ce

where F is the flux of photons incident on resonance cell = l O ”

(96)

sec-’

ern-', A is the cross-sectional area of cell w 3 cm‘, dis the length of resonance

202

A. CORNEY

cell z 2 cm, N is the density of atoms in the ground level z 10" ~ m - ~ , cscatt is the resonance scattering cross section zz cm2, C is the factor depending on the geometrical collection efficiency and e is the The resonance scattering quantum efficiency of the photocathode 5 x cross section is given by

where dv,, Avc are the Doppler widths for the atoms in the light source and scattering cell, respectively, ro is the classical electron radius, f is the absorption oscillator strength of the resonance line. For f = 1 , Av, = Avc z 1500 For these typical values we MHz we obtain the value given above for oscatt. find Q x 3 x lo7 sec-l. This is much larger than the dark current or the shot noise due to random fluctuations of the photocurrent so that accurate measurements of the shape of the Hanle curve can be obtained. This estimate illustrates the great sensitivity of resonance fluorescence techniques and the possibility of working at very low densities N . It also emphasizes the need for intense exciting light sources and good collection geometry. Equation (93) predicts that the maximum amplitude of the field-dependent Lorentzian term will be equal to that of the constant background signal given by the first term. This is only correct for excited levels which exhibit the classical or normal Zeeman effect. For other systems it is found that although the field-dependent term has the same Lorentzian shape as given by Eq. (93) it is generally a smaller fraction of the background term. This increases the experimental difficulty. The ratio of the signal to background in any given case is best calculated using the quantum mechanical expressions discussed below. 2. Level Crossings

a. Theoretical discussion. The Hanle effect is, in fact, only a special example of the general phenomenon of the spatial redistribution of fluorescent light which occurs when excited energy levels of an atom intersect. This was realized following the discovery of signals resulting from the intersection of fine structure levels at high magnetic fields in the 3 3P term of helium (228). The theoretical description of the effect was initially derived by Breit (229) and then rederived by Franken (230) and Rose and Carovillano (231). The rate R at which photons with polarization f a r e absorbed and photons with polarization g are emitted is given by

f

R(f, g) = k mm'

fm p r + icoUu,

fgp'm'gm'p

'

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

203

where k is a constant determined by the incident light intensity, density of atoms, etc. and n?,m' are magnetic states of the ground level while p , p' are are electric dipole magnetic states of the excited level. Then f ,, and g,,,,?,, matrix elements of the form f,,, = ( p If * rl n z ) . The energy separation of the states of the excited atom is given by up,,= ( E , - E,.)/h. If the states of the excited atom are all well separated so that o,,.$ r for p # p', then the different states effectively scatter independently giving an intensity which is field independent

The field-dependent signals are due to the interference of light scattered by two states which are close enough i n energy so that a single frequency of the incident light can excite both coherently. This occurs only over a small region z l-. determined by the radiation width of the excited levels olla, The level crossings which occur at nonzero field have been chiefly used for the investigation of the fine and hyperfine structure of atomic energy levels, although in principle they could also be used for lifetime measurements. Their

-1

0

+I m, 3

F:

0 -1

FIG.3 3 . Schematic diagram of the apparatus used for optical double resonance experiments on the 6 3PI level of mercury.

204

A. CORNEY

chief interest from our point of view is that they give a general expression from which the Hanle signals for special cases may be derived. Examples of the use of this formula are given in (223,232). More recently expressions using tensor operator methods have been derived (188, 233). Since the principle difference between these experiments and the Hanle experiments is the application of larger magnetic fields we d o not give any further details.

3. Optical Doirble Resonatice

a. Tjpical experimental arrangement. The optical double resonance method was first successfully applied by Brossel and Bitter (234)following a suggestion by Kastler and Brossel (235). The original experiments were again made on mercury but since then work on potassium (236),sodium (237),copper (238), thallium (232), and many more elements have been reported. The principles of the method may be understood with reference to Fig. 33. Light from a mercury lamp is passed through a linear polarizer so that the electric vector of the radiation is parallel to the constant magnetic field H which is applied to the atoms i n the resonance cell. This means that the incident 2537-A radiation can only stimulate electric dipole transitions with AmJ = 0. Thus, assuming the cell to contain only even isotopes of mercury, only the 171, = 0 state of the excited 6 3P, level of mercury is populated. In the absence of other perturbations these atoms would decay after a mean lifetime z emitting light which is also polarized parallel to the magnetic field (n polarized). This fluorescent light is detected by the photomultiplier after passing through a linear polarizer. If. however, the excited atoms are subjected to the influence of a strong radio-frequency magnetic field HI at right angles to the steady field H, then the conditions are right for magnetic resonance t o occur. When the Larmor frequency wL of the excited atoms i n the field H approaches the angular frequency w o of the applied rf magnetic field the atoms are transferred to the mJ = 1 states by stimulated magnetic dipole transitions. The rf field tends to equalize the populations of the excited states. Since the atoms i n the ni, = i- 1 levels decay by emitting light which is right and left circularly polarized ( a + and K )i n a plane perpendicular to H, the magnetic resonance may be detected as a change in the polarization of the fluorescent light. Since the analyzer only transmits n polarized light to the photomultiplier the intensity of this light is observed to decrease on resonance. Alternatively the resonance may be detected by an increase in the 0-polarized light emitted either parallel or perpendicular to the magnetic field H. The shape of the double resonance signal observed i n mercury as a function of the dc field H is shown i n Fig. 34 for various values of the rf field strength H,. h. Tlieoretical expressioti.sJor ohsewed signal. The variation of the double

*

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

205

rf

I

I

I

I

I

65

66

67

68

69

Magnetic Field

H

( Gauss

1

FIG.34. Optical double resonance signals observed on the 6 ’So-6 3P1 transition of mercury at 2537 8, as a function of the dc magnetic field H. The curves illustrate the effect of increasing the rf field strength H I .

resonance signal S as a function of the strength of the applied dc field H is given by

(1W where y is the gyromagnetic ratio = gjPo/fi and wL is the Larmor frequency given by Eq. (91); So is a constant which depends on the intensity of the

206

A. CORNEY

exciting light, etc. and is proportional to Q given in Eq. (96). The resonance curve is bell-shaped for yH, < but increases in amplitude and becomes double-peaked for yH, $ r. For small values of the rf field the full-width at half-maximum intensity AcoL is given by Aw,’ = 4r2[1

+ 5.8(yH1/r)’].

(101)

This increase in width is due to the induced magnetic dipole transitions which effectively reduce the lifetime of an atom in any given magnetic state. The lifetime z = I/r of the excited level can be obtained by plotting the measured Am,‘ against the square of the radio-frequency field strength H,’. Although we have considered here the magnetic resonance excited within the Zeeman states of the 3P, level of the even isotopes of mercury the technique is also applicable to resonance between hyperfine levels in zero magnetic field or between magnetic states of hyperfine levels. The only requirement is usually that the exciting radiation should create a nonequilibrium distribution within the excited states or levels. Equation (100) indicates that strong signals will only be observed when y H , E r. That is, the radio-frequency field must be strong enough to have an appreciable probability of inducing a transition between states within the lifesec and y = lo7 sec time of the atom. Thus for T = G-’, H , % 1 G is required. For shorter lifetimes, fields of sufficient magnitude are difficult to attain and they may induce a discharge in the resonance cell unless special care is taken. For this reason the level crossing technique is simpler although less flexible. The double resonance technique does have the advantage of yielding accurate g J values from the measured dc field at resonance and the known angular frequency wo of the rf field. Unless this information is already available the level crossing and Hanle effect techniques cannot be used. This difficulty is illustrated by recent work on level crossings in OH (239) and NO (240). In each case the g factors for the vibrational-rotational levels studied were unknown and the widths of the observed signals could not be converted into lifetimes. However in the case of NO, Crossley and Zare (241) were able to measure the g factor by a separate optical double resonance experiment and so obtain the value z = (2.09 & 0.16) x sec for the u‘ = 1 level of the A%+ state. The lifetime of the u’ = 2 level of the A ‘ll state of CO has recently been measured by the level crossing technique (241a). C. Details of Experimental Technique I . Light Sources

Since the size of the signal detected in all resonance fluorescence experiments is proportional to the intensity of the exciting light, Eq. (96), it is important that this should be as high as possible. In particular the total output

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207

in the resonance line being used should be in the range 10'5-10'8 photonslsec corresponding to output powers of 1 mW to 1 W for near ultraviolet lines. The spectral line must also be free from the effects of large self-reversal which is caused by light within the lamp passing through a region of cooler vapor or gas. Only light within the Doppler width of the line center can be absorbed by atoms in the cell. Other frequencies merely contribute to the undesirable background of scattered light. Commercial spectral lamps rarely satisfy these requirements of intensity and freedom from self-reversal without modification. The construction and design of light sources suitable for resonance fluorescence has been discussed in detail by Budick et al. (242) and for special cases by Chapman and Krause (243) and Krause (244). One of the simplest and most widely used sources is the microwave discharge. This consists of a small quartz tube, typically 2 cm long by 8 mm wide, which has been previously evacuated and baked down to Torr. The lamp is filled to by distilling in the metal whose resonance line is required and a few Torr of an inert starter gas is added before the tube is sealed off. The capsule is excited by placing it in a suitable microwave cavity using powers of up to 80 W at 2450 MHz. If the power dissipated is not sufficient to give a high vapor pressure of the metal, external heating may be required or alternatively the iodide of the metal may be used in the lamp (122). The latter technique is suitable for the following elements: Cu, Ag, TI, Bi, Pb, In, Ga. The microwave excited source is particularly convenient when it is necessary to use small quantities of rare isotopes. For elements which are difficult to vaporize or which react strongly with quartz, such as Mg, Ca, Sr, and Ba, the flow lamp illustrated in Fig. 35 has Raytheon Type 'X 2 4 kmHz antenna

Hollow molybdenum core with 4 0 holes 0 0 2 5 in dia in 0 0 2 0 1 n thick top-1;

'Quartz envelope

Lightly pac fibers t o c discharge Charge of Ca or Mg Ceramic cover encloslng heate wires a

Argon flow' (pressure 2 - 6 t o r r )

FIG.35. Diagram of the flow lamp used by Smith and Gallagher (245) for Hanle effect experiments on Ca+ and M g + .

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A . CORNEY

been used (245, 246). In this source the metal vapor is sprayed from an oven into a discharge produced i n a Rowing inert gas stream. The counter flow of gas prevents the metal condensing on the front window of the lamp. The excitation may be provided either by rf power at 15 to 30 M H z supplied t o an electrode wrapped around the outside of the lanip or by microwave power from an umbrella shaped antenna. For rare earth elements, e.g., ELI.Sm, Tni, etc., an intense hollow cathode light source has been developed (247, 248). Recently. Decomps and Dumont (249, 250) have used the intense radiation existing within a He-Ne laser cavity i n a series of Hanle effect measurements on excited levels in neon. Similar experiments are reported in xenon (251). In all cases the light source should be free from short term intensity fluctuations. If low signal intensity makes it necessary to use long time-constants i n the detection apparatus then the long-term drifts of the light source may also become a problem. 2. Scatteriiig Cells to For those elements whose vapor pressures are of the order of lo-’ Torr in the temperature range 300-600°K and which d o not react with quartz or Pyrex it is simple to make a sealed-off cell into which a small amount of the metal has been distilled. The vapor density is controlled by placing the cell i n an oven in which stray dc or ac magnetic fields have been reduced to lo-’ of the width of the signal. Light which is scattered off the walls of the cell into the photomultiplier should be reduced to a minimum by careful design of the cell and the optics of the incident light beam. It is also necessary to ensure that the cell is thoroughly outgassed before being sealed off otherwise the observed signals may be broadened by the effects of collisions. This type of cell again has the advantage that it can be filled with a very small quantity of separated isotopes. For refractory or reactive elements it is necessary to use a crude atomic beam produced by a heated oven in vacuum. This type of scattering chamber Iias been used by Lurio (225) for lifetime measurements in Mg and Ba, and by Steudel, Walther, and their co-workers in extensive measurements in the rare earths (247, 248, 252). It also has advantages for elements such as Cd (223) which can be used in sealed-off cells since the background of scattered light is reduced i n a beam and measurements can therefore be made at lower densities. The lifetimes of excited ions of Mg, Ca, Sr, and Ba have been measured using the Hanle effect (245, 246) by spraying a beam of these elements into a pulsed discharge run in argon. The use of ground state atoms i n the scattering beam usually litnits the application of the resonance fluorescence methods to the study of the first few excited levels in a given element. Brinkmann et al. (253) have overcome

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

209

this limitation by using an atomic beam source in which the beam passes through a discharge before entering the resonance scattering region. In this way, a beam of calcium and strontium atoms in their metastable ID, and 3P levels has been prepared, and optical excitation from these levels has allowed the lifetimes of several more excited levels to be measured. Similarly, a beam of metastable helium 2 IS atoms has been used in a measurement of the lifetime of the 2 'P, level (204, 205). The 2 IP, level would be difficult to excite from the ground level since the wavelength of the resonance line 584 A lies in the vacuum ultraviolet. However the 2 'S-2 'P transition at 2 p can be conveniently produced and isolated so that the Hanle effect may be induced by exciting from the 2 'S level. The technique of stepwise excitation from the resonance level has also been used as a method of studying higher levels (254-256).

2. Detection and Signal Measurement Since the Hanle effect and level crossing techniques depend on a spatial redistribution of light, it is usually possible to arrange two photomultipliers so that they observe equal but opposite field-dependent signals. Then by the use of a bridge circuit the difference of the outputs of the two photomultipliers can be obtained (234). This has the advantage that fluctuations in the light source are partly compensated for and the signal is doubled. This technique has continued to be used in France (257). The more common way of increasing the signal to noise is to modulate some experimental parameter at a low audio frequency and use narrow band phase-sensitive detection. Many experiments have been performed by modulating the steady magnetic field by the addition of a small component at 30 to 200 Hz. In order to avoid distortion of the signal (258) it is necessary to use an ac field amplitude which is a small fraction of the signal width. This naturally reduces the maximum amplitude of the signal. A further disadvantage is that the signal produced is the differential of that obtained for dc fields. A more satisfactory method of phase-sensitive detection is to use a rotating polarizer to modulate the scattered light (259). The use of long time-constants also makes it necessary to use slow field sweeps in order to avoid distortion of the signal. When sweep times reach the order of 30 to 60 min, long term drifts in the light sources and scattering cells become a problem. Recently multichannel signal averagers have been used (247, 260) and integration times of several hours have become possible. 4. Excitation by Electron Impact

a. Magnetic resonance experiments. The use of electron impact excitation greatly increases the number of excited levels which may be studied by the

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A. CORNEY

magnetic resonance and level crossing techniques. The success of these experiments relies on the fact that the magnetic states m, of a given electronic level are often unequally populated if excited by a beam of electrons whose energy is at or just above the excitation threshold. This inequality is the cause of the polarization of the light emitted by atoms excited by electron impact (261). The first application of this technique to the study of short-lived excited levels was made by Pebay-Peyroula et al. (262).The technique has since been widely used (263, 264) and is reviewed in two recent publications (265, 266). The experimental chamber consists of a small triode structure in a glass envelope. The electrons are produced by a heated oxide coated cathode and the excitation takes place in the grid-anode space. The emitted light is detected in the same manner as described previously, the only difference being that a monochromator or interference filters must be used to isolate a given spectral line. For magnetic resonance experiments the external dc field H is made parallel to the direction of the electron beam. Excitation then occurs preferentially to states with m, = 0. The rf field HI is applied externally to the tube and resonance is detected by an increase in the a-polarized light. If strong radio-frequency fields are used there is a tendency to excite electron cyclotron resonance in the beam. This gives rise to changes in the intensity and polarization of the emitted light. These are detected as very intense, broad resonances which mask the atomic magnetic resonance signals. This problem can be reduced by using a small anode-grid separation. Magnetic resonance experiments using electron impact excitation have been reported in mercury (263, 264) cadmium (267), zinc (268),sodium and cesium (269),and several inert gases. b. Hanle efSect and level crossing experiments. In order to use electron impact excitation to produce Hanle effect or level crossing signals it is necessary that the direction of the electron velocity should be perpendicular to the applied magnetic field H. This technique was first used for lifetime measurements in helium (270) by the Hanle effect and later for level crossing measurements of helium fine and hyperfine structures, (271) which also gave lifetimes. The technique has since been used in calcium (272) and neon (273). Since the electron velocity is initially perpendicular to the magnetic field the electron experiences a force which makes the trajectory into an arc of a circle. At high fields this causes the admixture of some dispersion shaped signal on the wings of the Lorentzian signal and a correction, which may affect the measured lifetime by 20% for T = lo-' sec must be applied. Thus the electron impact experiments are suitable only for levels whose lifetimes are longer than 3 x 10sec. Kaul(274) has shown that the use of fast beams of heavy ions in place of the electron impact excitation overcomes this particular limitation. c. The use of rf and dc discharges. It is difficult to use electron excitation in sealed triode structures with elements which poison the cathode or require

'

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

21 1

high temperatures for vaporization. In an attempt to overcome these difficulties Lombardi and Pebay-Peyroula (275) developed a method using excitation in an rf discharge. The experimental source consists of a cylindrical cell 1 cm thick which is placed between two flat condenser plates which terminate a 4 4 length of line. The principle of the technique is that at low enough pressures and with high rf electric field strengths the electrons in the discharge will oscillate in a direction parallel to the applied electric field. Typical operating parameters are 0.1 Torr with rf electric field strength of 250 V/cm at 250 MHz. The electron mean free path is of the order of 1 mm and the mean energy around 20 eV. This technique has been used for lifetime measurements in He, Ca, and Cd (275-277). In similar experiments May and Leung (265, p.167) reported the observation of magnetic resonances in the light emitted by a mercury dc discharge lamp while Carver (278) has observed nuclear polarization of 3He in a gas discharge. In these experiments the mechanism responsible for the nonequilibrium state populations cannot be clearly identified. However, in Hanle experiments using a dc discharge in neon, Carrington and Corney (259) have been able to show that at pressures in the range 1-10 Torr the signals are due to optical excitation from metastable levels by light produced within the discharge. The effect depends, as always, on the geometrical anisotropy of the excitation which in this case is due to the use of a long, narrow discharge tube. At lower pressures electron impact excitation becomes responsible for the observed Hanle signals as the mean free path of electrons within the discharge increases. Although the use of discharges extends the range of elements which may be studied the signals are often weak and their interpretation is made difficult by the large number of competing processes which occur. D. Experimental Dificulties 1. Resonance Trapping

Since in many of the experiments described in this section the signal is detected by a change in polarization of resonance radiation, effects due to the reabsorption of this resonance radiation by ground level atoms in the cell are to be expected. As each reabsorption increases the effective lifetime zerf resonance trapping results in experimental curves corresponding to a smaller effective radiation width reff. The effect has been investigated both theoretically and experimentally by Barrat (64). Improved calculations are given by D’yakanov and Perel (65) and are confirmed by measurements made by Omont (279). For a simple system such as the mercury 3P, resonance line at

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A. CORNEY

1, = 2537 A the measured width reff for the even isotopes depends on the density of the ground level atoms N as

where d’) is a constant less than 1, and x is the probability of a photon of the resonance line being reabsorbed within the resonance cell given by Eq. (53) or by Barrat’s approximation, Eq. (58). More detailed expressions are given by Lecler (227) for mercury and by Saloman and Happer (280) for lead. It is important to notice that trapping in resonance fluorescence experiments leads to a maximum increase of the measured lifetime to T/[ 1 - d k )rather ] than the indefinite increase predicted by Eq. (57) for measurements involving transient A,, = 0). Since in many cases d k )is much less than unity decays (for CnZO only small increases in lifetime are actually observed. Then Eqs. (102), (53), made over a range of densities to be and (58) enable measurements of reff extrapolated to obtain the true radiative lifetime z = I/r.However, in many experiments the sensitivity is such that the density can be reduced to the region where resonance trapping is negligible.

Density

(Atom ~ r n - ~

FIG.36. A plot of the peak-to-peak separation SH of the differentiated Hanle signal obtained from the 6s26p7s 3PIi’ level of lead by Saloman and Happer (280) as a function of density. The initial decrease in width is due to resonance trapping and the increase at high density to collision broadening. The magnetic field is giveii by H = 10.122 I and 6 H is by = 3’’’ yJp0 8 H / f i . related to the radiation width ref[

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

213

2. The Effect of Collisions

If measurements of the width of the signal ref, are made over a wide range of densities it is found that the data no longer obey Eq. (102) but the width increases again at higher densities due to the effect of collision broadening. This is illustrated in Fig. 36 for the 6s26p7s 3P10level of lead obtained by the Hanle effect method. This technique has been used to check the theories of resonance broadening (187, 188), but in general the effect only appears at densities much higher than those used for lifetime measurements. However in those experiments which use gas discharges as the scattering sample (245, 246, 259) it is necessary to make measurements over a range of gas pressures in order to determine the collision broadening cross section dk)and the atomic lifetime. It is found that the data can always be analyzed by assuming that the effects of collision broadening and resonance trapping are additive, thus

reff= r[i - d k ) x ]+

( 103)

where N is the density of the perturbing species. The value of the collision cross section d k measured ) in resonance fluorescence experiments depends on whether the signal is due to magnetic orientation ( k = 1) or to magnetic alignment ( k = 2) in the excited level (188). It generally differs slightly from the collision cross section of Eq. (45) which described the relaxation of the total excited level population k = 0. 3. Deviations from Perfect Geometry Large solid angle optics are generally used in both the exciting and detection beam in resonance fluorescence experiments in order to increase the signal. However the observed signal may deviate from those derived in Eqs. (93) and (100) which assumed well-defined geometries. The effect usually leads to a slight broadening of the curves and to a small asymmetry. These can usually be corrected for (222, 245). 4. Dificulties Encountered with Electron Excitation

In addition to the problems of cyclotron resonance and large distortion in Hanle effect signals the electron excitation method suffers from the usual problem of cascade. This was noticed in the early work of Pebay-Peyroula (263) when magnetic resonances were detected on the 6 3P2-6 3D3 line of mercury whose positions and widths corresponded to the g-values and lifetime of the 6 3F, level. It was postulated that the change of populations at resonance in the 6 3F, level is carried down to the 6 3D3level by cascade through

214

A. CORNEY

the infrared transition at 17,190 A. A more detailed study has revised the identification of the upper level to the 5d96s26p 3F, level (281). Similar cascade problems occur for Hanle effect signals. The situation has been treated theoretically by Nedelec (282) and the result indicates that the observed signal will be the product of the Hanle signals expected for the cascade and observed level. Thus if the lifetimes of the levels are sufficiently different the two contributions to the experimental curve may be resolved.

5. Eflect of Hyperfine Structure For the odd isotopes of an element with nuclear spin I the value of gJ used in the calculation of the Larmor frequency, Eq. (91), must be replaced by g F where, to a good approximation, gF =

gJ

F(F

+ 1) + J(J + 1) - I ( I + 1) 2F(F + 1)

1

and F is the total angular momentum quantum number for a given hyperfine level. Thus, except for those levels where J = I, the different hyperfine levels have different values of gF and will produce Hanle effect signals with differing widths. The relative contributions of these hyperfine levels to the observed signal depends on the form of the absorption and emission profiles of the resonance cell and lamp. It is therefore difficult to use the odd isotope Hanle signals for accurate lifetime measurements and experiments on pure samples of the even isotopes are to be preferred. When this is impossible, as in potassium and sodium, an even more complex situation arises for these elements have a zero-field hyperfine structure which is only slightly larger than the radiation width of the levels. Thus the magnetic fields required to depolarize the light to produce the Hanle effect are sufficient to cause appreciable decoupling of the total electronic and nuclear spin angular momenta J and I. This decoupling produces level crossing signals which overlap the zero-field Hanle signals. In cases such as these only a detailed comparison of the experimental and theoretical signals over a wide range of magnetic fields will enable the lifetimes and hyperfine structure constants to be measured. Work of this type in 39K has been reported by Schmieder et al. (260). 6. The Range of Lifetime Measurements

The atomic lifetimes measured by the techniques of resonance fluorescence range from (2.0 & 0.2) x sec for the 4s4p 3P, level of zinc (283) to (5.69 f 0.23) x lo-'' sec for the ls2p 'PI level of helium (204), although most of the measurements fall in the range 10-6-10-9 sec. The range is

LIFETIMES OF FREE ATOMS, MOLECULES, A N D IONS

215

limited at long lifetimes by the low photon scattering cross section oScatt, Eq. (97), of levels with smallf-values ;by the considerable contribution to the width made by wall collisions (283) and by the effects of field inhomogeneity. Thus for the 3P1level of zinc a field inhomogeneity of 1 m G over the scattering cell would broaden the signal by approximately 10% in the absence of wall collisions. Although it is possible to construct Helmholtz coils with the required homogeneity it is difficult to eliminate the effects of stray ac and dc magnetic fields at levels below 1 mG. At the short lifetime end of the range the double resonance method becomes impossible due to the requirement that the strength of the rf field HI should satisfy yHl x 1 / ~The . application of the Hanle effect becomes more difficult due to the fact that the fields required for appreciable depolarization cause Zeeman separations of the states of the absorbing atom which are of the same order as the Doppler width of the source. Consequently the intensity of the fluorescent radiation decreases. This effect is not predicted by Eqs. (93) or (98) since the derivations assume that the profile of the exciting radiation is essentially flat over the region of absorption of the cell. The effect of this Zeeman scanning of the lamp profile has been noted in helium (204) and xenon (213). It may be reduced by broadening the lamp profile by the application of a dc field of a few hundred gauss (226).

7. Experimental Accuracy The relatively simple apparatus, high sensitivity, and the selective nature of the optical excitation used in many of the resonance fluorescence experiments combine to make this method the most accurate of techniques for measuring atomic lifetimes. In many cases the experimental measurements may be made at densities which are so low that the effects of resonance trapping and collision broadening are completely absent. In these cases the experimental results are quoted with errors in the range of 3 to 5 %. The method is therefore suitable for precision lifetime measurements and enables sets of relative oscillator strength data obtained by other techniques to be placed on an absolute basis (232). The extension of the methods to nonresonance levels has usually meant increased experimental difficulties and a consequent increase in the uncertainty of the lifetime measurements although results may usually be quoted with errors not exceeding 10%. E. Discussion of Experimental Results Some results obtained by these methods for the lifetime of the 3 'P level of sodium have already been presented in Table V. They show good agreement between the results of different authors and techniques.

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A. CORNEY

Table XI gives the lifetimes of the np zP3,2levels of the alkalis and their first isoelectronic ions, together with the absorption f-value for the resonance line ns 2S1,2-np 2P3,2. The f-values of the isoelectronic atoms and ions exhibit a remarkable similarity, and do not change appreciably with changing values of n. The table shows the high accuracy attained by the resonance fluorescence techniques for measurements on true resonance lines. In Table XI1 we give results for the lifetimes of the lowest excited 'P1 and 3P, levels of group IIB elements together with the absorptionf-values for the transitions from the ground level. Again the true resonance lines exhibit remarkably constantjlvalues. This is also found for the resonance lines of the TABLE XI LEVEL CROSSING AND HANLE EFFECT RESULTS FOR THE ALKALI METALS AND THEIRISOELECTRONIC IONS" Atom

Lifetime

sec)

f-value

Reference

Na (n = 3) Mg+ (n = 3)

16.0 f 0.5 3.67 f 0.18

0.650 & 0.020 0.64 & 0.03

(245, 246)

K (n = 4) Ca+ (n = 4)

26.0 5 0 . 5 6.72 f0.20

0.678 0.013 0.66 & 0.02

(147, 260) (245, 246)

Rb (n = 5) Sr+ (n = 5)

25.5 f0.5 6.53 & 0.2

0.715 f 0.015 0.71 f 0.03

(147)

Cs (n = 6) Ba+ (n = 6)

32.7 1.5 6.27 f 0.25

0.666 f0.031 0.74 + 0.05

(147)

*

(147)

(246) (246)

a Lifetimes of the np 2P3,2levels and absorptionf-values for the transitions ns zSl,2-np2P3jz.

TABLE XI1 RESONANCE FLUORESCENCE RESULT^ Atom Zn (n

= 4)

ns2 'S0-nsnp 'P1 T = (1.41 & 0.04)

x

T = (2.0 f 0.2)

Hg (n = 6)

= (1.66 i- 0.05) f = 1.42 i0.04

X

T = (1.36 & 0.05)

x

f = 1.18 f0.07

x

Reference

(224, 283)

f = (2.13 f 0.20) x 10-4

f = 1.46 & 0.04 Cd (n = 5)

nsz 'So-nsnp 3P1

lo-'

7 = (2.39

& 0.04)

X

(222, 223)

f = (2.00 & 0.03) x l o u 3 T = (1.18

& 0.02) x lo-'

(226, 220)

f = (2.45 & 0.04) X lo-'

a The lifetimes of the nsnp 'P1 and 3P1 levels of group IIB elements and the absorption f-values of the corresponding resonance lines.

217

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

TABLE XI11 COMPAREON OF COLLISION CROSSSECTIONS IN NEON' Level (Paschen)

O&&e

'

3.3 f4.0 2.6 f2.6 8k3 5 f 4.5 3 f5.5 6.5 f 3.5 3.9 f 2.2 5.5f2.0 3.5 f 10.0 0.4 rt 1.7 a

1.5f0.5

9.10f0.6 5.25 & 0.12 2.51 & 0.16 5.07 f 0.28 8.36 f 0.48 10.51 f0.36 0.202 f 0.084

8.2 f0.4

10.3 7.75 2.48 6.24 8.17 10.1 0.765

Data given in units of emz. Measured at 500°K by Bennett and Kindlmann (46). Measured at 300°K by Decomps et al. ( 2 3 8 ~ ) . Measured at 315°K by Carrington and Corney (unpublished results). Theoretical at 315°K by Carrington and Corney (unpublished calculation).

group IIA elements. However, thef-value of the intercombination line to the 3P, level, which is forbidden in strict LS coupling, increases rapidly from zinc to mercury as LS coupling breaks down. The ratio of the lifetimes of the singlet and triplet levels can be used to determine the intermediate coupling coefficients of the wavefunctions describing these levels (226). Finally we present, in Table XIII, a comparison of collision cross-section measurements in the 2p levels of neon. The results in the second column were obtained from pressure fits to radiative decay rates obtained by the multichannel delayed coincidence technique (46).They represent the total population destructive collision cross section &2b-Ne. In column 3 results are given for &LNe which were obtained by resonance fluorescence measurements using laser excitation (233, 283a). Columns 4 and 5 present fluorescence measurements of the alignment destroying collision cross section &LNe which were obtained by the Hanle effect. In those cases where an accurate comparison is possible we see that the alignment destroying cross sections are of the order of three times the size of the population destroying cross sections. The cross sections for the destruction of the alignment of excited neon atoms by helium o&zL,, are found to be nearly equal to the values of CT~:L~~. Particularly noticeable are the small cross sections obtained for the 2p,, level which, by reference to Fig. 28, can be seen to be well separated from the other 2p levels. We have applied the theory of impact broadening in the asymptotic approximation (I85a, 2836) to obtain the theoretical cross sections that are

218

A. CORNEY

given in the last column of Table XIII. The magnitude of the Van der Waals' interaction was calculated from wavefunctions for the 2p levels supplied by Dr. S . Feneuille. We find that, although there is good agreement between the experimental and theoretical cross sections for broadening by neon at 315"K, we are unable to explain either the approximate equality of &LNe with &LHe or the temperature variation of the experimental cross sections. We conclude that it is necessary to improve the calculation by taking into account both the repulsive part of the interaction and also the deviation of the relative motion from straight line paths.

VIII. MISCELLANEOUS TOPICS A. Methods Using Laser Transitions It is well known that the power output of a single mode gas laser decreases as the oscillation frequency is tuned through the center frequency of the atomic resonance. The shape of this dip is given in a simple form by Lamb (284) and was observed experimentally by McFarlane et al. (285) and Szoke and Javan (286). In the case where the Doppler width of the laser transition is very much larger than the natural width, the shape of the Lamb dip is a Lorentzian of the form given by Eq. (80). Thus by careful measurements the sum of the natural widths of the upper and lower levels of the laser transition could be obtained. Unfortunately the shape and width of the Lamb dip are considerably affected by collisions. Generalization of Lamb's expression to higher pressures have been given by Fork and Pollack (287) and Szoke and Javan (288) and experimental measurements have been made over a range of pressures which have enabled collision broadening cross sections for the laser transition to be determined (286,288,289). However measurements of the lifetimes of individual levels obtained by this technique tend to be rather inaccurate. The Lamb dip is due to interaction of the standing wave of the radiation within the laser cavity with atoms which have nearly zero component of velocity parallel to the laser axis. These atoms are stimulated to emit and consequently the lower level of the laser transition will have an excessive population of atoms with zero axial velocity component. There should therefore be a peak at the center of any Doppler broadened optical line emitted by the lower laser level. This effect was predicted by White (290) and observed experimentally by Cordover et at. (292) and Schweitzer et at. (292). If the laser is tuned off center two peaks appear on either side of the Doppler broadened line. The widths of these peaks are determined by the widths of all three levels involved in the two cascade transitions. The correct expressions for the widths

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

219

were derived by Holt (293) who has shown experimentally (294)that by combining the measurement of the width of the Lamb dip with the widths of the peaks, that the effective lifetimes for each individual level can be obtained. A more general discussion of these effects is given by Feld and Javan (295). A more promising technique is that discovered by Schlossberg and Javan (296). They observed that resonances occurred in the output power of a gas laser when a magnetic field, coaxial with the laser tube, was slowly increased. The resonances occurred whenever the frequency difference v, between a pair of oscillating modes satisfied 2g, ,uoH = nhv, , where n is an integer, gJ is the Lande factor for the upper laser level, and H is the strength of the applied field. A theoretical treatment of this effect (297) shows that the width of the resonances is determined only by the width of the upper level and preliminary experimental results have been given for several levels in xenon (296). The interaction of the intense optical field within a laser cavity with the excited atoms of the discharge leads to considerable changes in the populations of the upper and lower levels of the laser transition. This can also lead to changes in the population of other levels which are connected by radiative or collisional transitions with the laser levels. Parks and Javan (298) have developed a means of studying the partial thermalization of two closely spaced levels as the pressure in the laser discharge is increased. Since at the highpressure limit the relative populations of the different levels are known it is possible to obtain relative transition probabilities for different lines by the measurement of relative intensities and the use of Eq. (22). This technique has been applied by Lilly and Holmes (299) to the measurement of relative transition probabilities in neon. In addition to the techniques discussed above, laser excitation has been used in experiments which are modifications of techniques described in earlier sections. Thus a pulsed argon ion laser at 5145 A has been used t o excite fluorescence in a beam of I, molecules and the lifetime measured by pulse-sampling techniques (300). The phase-shift technique described in Section V has been used by Fork et al. (301) to measure the lifetimes of the 3s, and 2p, neon levels by modulating the laser at 56.02 MHz with an internal piezo-elastic quartz block modulator. Finally the Hanle effect has been used with laser excitation as discussed in Section VII.C.l.

B. Metastable Levels

I . Collision Processes Involving Metastable Levels The inert gases, Ne, Ar, Kr, and Xe possess excited levels which in LS notation would be labeled 3P, and 3P,. These levels are metastable and they play an important part in the mechanism of gas discharges. Similar levels

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exist in the group IIB elements Zn, Cd, and Hg and in helium the corresponding metastable levels are 2 So and 2 3S,.In gas discharges, the lifetimes of these levels are determined by the combined effects of diffusion to the walls and collisional de-excitation involving ground state atoms and thermal electrons. These processes have been investigated in detail by Phelps (302-304) by absorption studies of the afterglow of pulsed discharges in helium and neon ) krypton. In these experiments light, whose and by Smith and Turner ( 3 0 4 ~in wavelength is such that it will excite metastables to higher levels, is passed through the sample and the transmitted intensity is recorded by a photomultiplier. In the early experiments the absorption was measured using the direct observation techniques described in Section I1.B but later considerable improvements in the sensitivity were obtained (305) by the use of a pulsesampling technique analogous to that described in Section 1I.C. Similar experiments have been performed in the metastable levels of Cd (306) and Hg (307) in conjunction with studies of the line-widths of magnetic resonances excited in these levels (308). The creation of the inequalities of population necessary for the study of magnetic resonance in metastable levels is usually achieved by optical pumping (309). A large number of studies of the line-widths of magnetic resonances in the optically pumped 2 3S, level of helium have been reported (310,311) and recently these techniques have been extended to include other inert gases (312-314). Unfortunately the relaxation of the nonequilibrium populations of the magnetic states of the metastable level usually occurs much more rapidly than the decay of the total metastable population. Thus the information obtained from a study of the line-widths cannot be used directly to give information about the lifetimes of the metastable levels.

'

2. Forbidden Single Photon Decay of Metastable Levels

A large number of atoms and ions possess low lying levels for which decay to the ground level by electric dipole radiation is strictly forbidden. Often, however, these levels can decay spontaneously by the emission of electric quadrupole or magnetic dipole radiation. Perhaps the best known examples are the IS and ID levels of 0 1 and the 'P and 2 D levels of NI which emit lines which are prominent features of the spectra of the aurora, nebulae, and the airglow. Expressions for the transition probabilities of these forbidden lines, similar to those presented in Section I for electric dipole radiation, may be found i n the NBS tables (18). The transition probabilities generally lie i n the range 10-1-10-8 sec-I. More general discussions of the subject are given in references (9, 315, 316). Although these forbidden lines have been produced in laboratory sources the first measurement of the very long lifetimes of the metastable levels

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221

involved was made by Omholt and Harang (317) by observations on the green line ('D-'S, 5577 A) of atomic oxygen emitted by aurorae. They assumed that the same process which excited the allowed first negative bands of N,' also excited the metastable oxygen atoms. Thus, in the simplest case, the time dependence of an exciting pulse may be observed on the prompt emission of the N,+ bands and the delayed emission from 01 enables the lifetime to be measured. Unfortunately the excitation rarely occurs as a pulse but fluctuates in a random way and a sophisticated correlation of the data recorded on the two transitions is necessary in order to obtain the lifetime (318). The values obtained for the lifetime of the 'S level range from 0.45 f 0.1 to 0.80 0.2 sec and illustrate the difficulty of. the technique. It is uncertain as to whether this spread of values should be regarded as real and accounted for by the occurrence of quenching in some of the observations. A similar spread of results was obtained in measurements on quiet aurorae (319). The same method has been applied to the red oxygen line 3P-'D (320). The data of this experiment have been reanalyzed by Omholt (321) and yield lifetimes in the range 150-200 sec for the 'D level. Recently advances in the design of photomultipliers and in signal averaging techniques have made it possible to study the time-dependence of the population of these metastable levels in the laboratory. In most cases (322,323) the main interest has been to obtain the collisional deactivation cross sections for these metastable levels caused by atoms and molecules present in the upper atmosphere. Laboratory measurements have also been made of the transition probabilities of the forbidden lines of oxygen by the measurement of the total intensity emitted by atoms in a discharge (324, 325) However these results are rather inaccurate due to the difficulty in determining accurately the density of atoms in the metastable level. Several molecules also possess metastable states which can decay by the emission of electric quadrupole or magnetic dipole radiation. The lifetime of the A3C,+ state of N,, which decays by the emission of the VegaardKaplan bands (X 'C,+ - A 3Eu'), has been the subject of a number of investigations. An early attempt (326) using an excited thermal beam of molecules failed due to the long lifetime of this state. More recently flowing afterglows (327, 328) and time-resolved studies of pulsed discharges (329,330) have been used in direct measurements. The lifetime has also been obtained indirectly by combined studies of the intensities of absorption and emission spectra (331).This experiment has recently been reanalyzed and a new value for the lifetime of the A 3Cu+ state proposed (332).The results of these experiments are collected in Table XIV. The wide variation between the results of different workers illustrates the difficulty of estimating the effects of diffusion and quenching in the direct lifetime measurements. Measurements on other metastable states of N, having considerably shorter lifetimes (326, 333) are

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TABLE XIV LIFETIME OF THE A 3Cu+,u' = 0 LEVEL OF N2 MEASURED IN SECONDS Technique Flowing afterglow Pulsed discharge (indirect) Absorption and emission (C = 0) ( C = *I)

Lifetime -1 1.1 -0.9 >10 12.6 f4.3 2.0 f0.9

1.36 2.70

Reference (327) (328) (329) (330) (332~) (331) (332)

likely to be more accurate. Important metastable states also exist in molecular oxygen. However, their very long lifetimes have prevented any direct measurements, although a number of estimates of the transition probabilities have been made by absorption techniques. 3. Two Photon Decays of Metastable Levels

The metastable 2 2Sl,z levels of hydrogenic systems and the metastable levels 2 'So and 2 3S0 of helium and its isoelectronic ions form exceptional cases. For the 'So levels decay by any single photon emission of either electric or magnetic type is forbidden while for the 3S1 and 2S1,2 levels, although decay by magnetic dipole radiation is allowed, the transition probability is extremely small. All of these levels decay radiatively by the simultaneous emission of two photons. This two photon decay has been detected in the case of the 2 2S1,2 of Hef (334, 3342) and the 2 'S level of NeXI (335). However, while the transition probabilities for both hydrogenic (336)and helium-like (337) systems have been calculated, the only successful lifetime measurement reported so far has been of the 2 'S level of helium ( 3 3 7 ~and ) ~ this remains an interesting, yet extremely difficult problem. C . Autoionizing Levels

Up to this point we have been concerned with the lifetimes of levels which lie below the lowest ionization energy of their respective systems. However if an electron other than the most loosely bound one is excited, or if two electrons are excited simultaneously, the resulting energy level often lies above the ionization energy. In this case the phenomenon of autoionization is observed; the excited atom can undergo a radiationless decay by ejecting an

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electron and forming an ion in its ground level. The transition probability for this process is often much larger than that of the competing radiative decay, and the lifetimes of the autoionizing levels are consequently very short, 10-12-10-15 sec. The lifetimes of these levels can thus be obtained by a measurement of the width of the observed lines r. However, as illustrated by the theory of Fano and Cooper (338) and the absorption measurements of Codling et al. (339), the profile of the absorption line is generally not a simple Lorentzian function due to the interaction of the autoionizing levels with the adjacent continuum states. In exceptional cases, however, there are no adjacent continuum states with which the excited level can connect directly. In these cases the lifetimes of to the levels are of the order of sec, and it becomes possible to measure them directly. This has been done for a number of metastable autoionizing levels in the alkalis (340, 341) and their isoelectronic ions (342, 343).These measurements have been performed using thermal beams of atoms excited by electron impact and by the use of beams from ion accelerators.

ACKNOWLEDGMENTS I wish to thank the authors who have given me permission to use their figures and data together with many others who have contributed by sending reprints of their results. I also wish to thank my colleagues in the Clarendon Laboratory (particularly C. G. Carrington) for many useful discussions. My wife gave invaluable help during the preparation of this article.

REFERENCES 1. W. Panofsky and M. Phillips, “ Classical ElectricityandMagnetisni.” Addison-Wesley, Reading, Massachusetts, 1962.

2. A. Lurio, Phys. Rev. 140,A 1505 (1 965). 3. B. P. Kibble and G. W. Series, Proc. Roy. SOC.(London) A274,213 (1963). 4. L. I. Schiff, “Quantum Mechanics.” McGraw-Hill, New York, 1968.

5. W. Heitler, “The Quantum Theory of Radiation.” Oxford Univ. Press, London and New York, 1957. 6. E. U. Condon and G. H. Shortley, “The Theory of Atomic Spectra.” Cambridge Univ. Press, London and New York, 1951. 7. B. Schiff and C. L. Pekeris, Phys. Rev. 134,A638 (1964). 8. G. Herzberg, “ Molecular Spectra and Molecular Structure-Diatomic Molecules.’’ Van Nostrand, Princeton, New Jersey, 1950. 9. R.H.Garstang, in “Atomic and Molecular Processes,” p.1 (D. R. Bates, ed.) Academic Press, New York, 1962. 9u.R. W. Nicholls and A. L. Stewart, in “Atomic and Molecular Processes,” p.47 (D. R. Bates, ed.), Academic Press, New York, 1962. 10. D. L a y e r and R. H. Garstang, Ann. Rev. Asfron. Asfrophys.6 , 4 4 9 (1968).

224

A. CORNEY

11. I. B. Levinson and A. A. Nikitin “Handbook for Theoretical Computation of Line

Intensities in Atomic Spectra.” Israel Program for Scientific Translations, Jerusalem, 1965. 12. D. R. Bates and A. Danigaard, Phil. Trans. Roy. SOC.A242, 101 (1949). 13. W. L. Wiese, Appl. Opt. 7, 2361 (1968). 14. W. L. Wiese, in “Methods of Experimental Physics” (B. Bederson and W. L. Fite, eds.), Vol. 7A, p.117. Academic Press, New York, 1968. 15. K. Ziock, in “Methods of Experimental Physics” (V. W. Hughes and H. L. Schultz, eds.), Vol. 4B, p.214. Academic Press, New York, 1967. 16. W. R. Bennett, P. J. Kindlniann, and G . N. Mercer, Appl. Opt. Suppl. 2, 34 (1965). 17. A. C. G . Mitchell and M. W. Zemansky, “Resonance Radiation and Excited Atoms.” Cambridge Univ. Press, London and New York, 1961. 18. W. L. Wiese, M. W. Smith, and B. M. Glennon, “Atomic Transition Probabilities,” Vol. 1 , NSRDS-NBS4. U.S. Gov’t. Printing Office, Washington D.C., 1966. 19. W. L. Wiese, M. W. Smith, and B. M. Miles, “Atomic Transition Probabilities,” Vol. 2, NSRDS-NBS2I. U.S. Gov’t Printing Office, Washington, D.C., 1969. 20. B. M. Glennon and W. L. Wiese, Natl. Bur. Standards ( U . S . ) Misc. Publ. 278 (1966). 20a.B. M. Miles and W. L. Wiese, Bibliography on atomic transition probabilities, Narl. Bur. Stands. ( U S . ) Special Publication 320 (1970). 21. E. W. Foster, Rept. Progr. Phys. 27, 469 (1964). 22. J. Cooper, Rept. Progr. Phys. 29, 35 (1966). 23. H. Grjen~,“Plasma Spectroscopy.” McGraw-Hill, New York, 1964. 24. W. C. Marlow, Appl. Opt. 6, 1715 (1967). 25. I. Meroz, ed., “Optical Transition Probabilities,” Vols. I and 11. Israel Program for Scientific Translations, S. Monson, Jerusalem, 1962. 26. W. Seka and F. L. Curzon, J. Quanr. Specty. Radiative Transfer 8, 1147 (1968). 27. I. V. Hertel and K. J. Ross, Proc. Phys. SOC.(London) (At. Mol. Phys.) 2, 285 (1969). 28. I. V. Hertel and K. J. Ross, Proc. Phys. SOC.(London) ( A t . Mol. Phys.) 2,484 (1969). 29. E. N. Lassettre, M. E. Krasnow, and S. Silverman, J. Chem. Phys. 40, 1242 (1964). 30. E. N. Lassettre, S. M. Silverman, and M. E. Krasnow, J. Chem. Phys. 40,1261 (1964). 31. T. M. Holzberlein, Rev. Sci. Instr. 35, 1041 (1964). 32. R. G. Fowler, T. M. Holzberlein, C. H. Jacobson, and S. J. B. Corrigan, Proc. Phys. SOC.84, 539 (1964). 33. R. G. Fowler, T. M. Holzberlein, and C. H. Jacobson, Phys. Rev. 140, A1050 (1965). 34. W. R. Pendleton and R. H. Hughes, Phys. Rev. 138, A683 (1965). 35. R. B. Kay and R. H. Hughes, Phys. Rev. 154, 61 (1967). 35a. J . Bakos and J. Szigeti, Proc. Phys. SOC.(London)(At. Mol. Phys.) 1, 1 1 15 (1968). 36. R. G. Bennett and F. W. Dalby, J. Chem. Phys. 31,434 (1959). 37. R. G. Bennett and F. W. Dalby, J . Chem. Phys. 32, 1 1 1 1 (1960). 38. R. G. Bennett and F. W. Dalby, J. Chem. Phys. 32, 1716 (1960). 39. I. E. Dayton, F. W. Dalby, and R. G. Bennett, J. Chem. Phys. 33, 179 (1960). 40. R. G. Bennett and F. W. Dalby, J. Chem. Phys. 36, 399 (1962). 41. R. G. Bennett and F. W. Dalby, J. Chem. Phys. 40, 1414 (1964). 41a.J. L. Morack and C. E. Fairchild, Phys. Rev. 163, 125 (1967). 41b. R. W. Nicholls, Can. J. Phys. 40, 1772 (1962). 42. S. Heron, R. W. P. McWhirter, and E. H. Rhoderick, Nature 174, 564 (1954). 43. S. Heron, R. W. P. McWhirter, and E. H. Rhoderick Proc. Roy. SOC.(London) M34, 565 (1956). 44. W. R. Bennett, Jr., in “Advances in Quantum Electronics” (J. Singer, ed.), p.28. Columbia Univ. Press, New York, 1961.

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

225

45. W. R. Bennett and P. J. Kindlmann, Bull. Am. Phys. SOC. 8, 87 (1963). 46. W. R. Bennett and P. J. Kindlmann, Phys. Rev. 149, 38 (1966). 47. W. R. Bennett, P. J. Kindlmann, and G. N. Mercer, Appl. Opt. SuppI. 2, 34 (1965). 48. J. Z. Klose, Phys. Rev. 141, 181 (1966). 49. J. Z . Klose, J. Opt. Soc. Am. 57, 1242 (1967). 50. J. Z. Klose, J. Opt. Soc. Am. 58, 1509 (1968). 51. G. M. Lawrence, Phys. Rev. 175, 40 (1968). 52. G. M. Lawrence and H. S. Liszt, Phys. Rev. 178, 122 (1969). 53. G. M. Lawrence, Phys. Rev. 179,134 (1969). 54. J. A. Simpson and C. E. Kuyatt, Rev. Sci. Znstr. 34, 265 (1963). 55. K. A. Bridgett and T. A. King,Proc. Phys. SOC. 92, 75 (1967). 56. P. J. Kindlmann and J. Sunderland, Rev. Sci. Znstr. 37, 445 (1966). 57. Z. Bay, Phys. Rev. 77, 419 (1950). 58. Z. Bay, IRE (Znst. Radio Engrs.) Trans. Nucl. Sci. 3, No. 4 (1956). 59. T. D. Newton, Phys. Rev. 78,490 (1950). 60. Z. Bay, V. P. Henri, and H. Kanner, Phys. Rev. 100, 1197 (1955). 61. P. C. Rogers, M.I.T. Lab. Nucl. Sci. Tech. Rept. No. 76, NYO-2303 (1962) (unpublished). 61a.C. C. Davis and T. A. King, Proc. Phys. SOC. (London)(Gen.Phys.) 3,101 (1970). 61b.0. M. Williams and W. J. Sandle, Proc. Phys. SOC.(London)(Sci.Znstr.), to be published. 62. T. Holstein, Phys. Rev. 72, 1212 (1947). 63. T. Holstein, Phys. Rev. 83, 1159 (1951). 64. J, P. Barrat, J. Phys. Radium, 20, 541, 633, 657 (1959). 65. M. I. D’yakanov and V. 1. Perel, Soviet Phys. JETP 20, 997 (1965). 66. R. Turner, Phys, Rev. 140, A426 (1965). 66a.S. Feneuille, M. Klapish, E. Koenig, and S. Liberman, Physica, (1970). To appear. 67. J. Bakos and J. Szigeti. Opt. Specty. 23, 255 (1967). 68. A. L. Osherovich and Y. F. Verolainen, Opt. Specty. 22, 181 (1967). 69. B. P. Kibble, G. Copley, and L. Krause, Phys. Rev. 153, 9 (1967). 70. B. P. Kibble, G. Copley, and L. Krause, Phys. Rev. 159, 11 (1967). 71. G. Copley, B. P. Kibble, and L. Krause, Phys. Rev. 163, 34 (1967). 72. E. E. Habib, B. P. Kibble, and G. Copley, Appl. Opt. 7, 673 (1968). 73. L. Krause, Appl. Opt. 5 , 1375 (1966). 74. E. A. Milne, J. London Math. Soc. 1 , 1 (1926). 75. W. Demtroder, 2.Phys. 166,42 (1962). 76. V. N. Kondratiev, Dokl. Akad. Nauk. SSSR 153, 1108 (1963). 77. E. E. Nikitin and V. K. Bykhowski, Opt. Specty. 17, 444 (1965). 78. J. H. Stamper, J. Chem. Phys. 43, 759 (1965). 78a.J.N. Dodd, E. Enemark, and A. Gallagher, J. Chem. Phys. 50,4838 (1969). 79. E. Brannen, F. R. Hunt, R. H. Adlington, and R. W. Nicholls, Nature 175,810 (1955). 80. E. Schatzman, Ann. Astrophys. 28, 111 (1965). 81. C. Carnhy and A. M. Dumont, Ann. Astrophys. 28, 113 (1965). 82. R. D. Kaul, J . Opt. SOC.Am. 56, 1262 (1966). 83. G. H. Nussbaum and F. M. Pipkin, Phys, Rev. Letters 19, 1089 (1967). 84. C. A. Kocher and E. D. Commins, Phys. Rev. Letters 18, 575 (1967). 85. R. E. Imhof and F. H. Read, Chem Phys. Letters 3, 652 (1969). 86. L. Kay, Phys. Letters 5 , 36 (1963). 87. L. Kay, Proc. Phys. SOC.85, 163 (1965). 88. S. Bashkin, Nucl. Znstr. Methods 28, 88 (1964).

226

A. CORNEY

89. S. Bashkin, Appl. Opt. 7, 2341 (1968). 90. W. S. Bickel, Appl. Opt. 7, 2367 (1968). 91. U. Fink, Appl. Opt. 7, 2373 (1968). 92. M. R. Lewis, F. S. Zimnoch, and G. W. Wares, Phys. Rev. 178,49 (1969). 93. J. P. Buchet, A. Denis, J. Desesquelles, and M. Dufay, Phys. Letters Azs,529 (1969). 93a.J. P, Buchet, A. Denis, J. Desesquelles, and M. Dufay, Compt. Rend.265,471 (1967). 94. L. Heroux, Phys. Rev. 180, 1 (1969). 95. S. Bashkin, ed., “Beam Foil Spectroscopy.” Gordon and Breach, New York, 1968. 96. J. Desesquelles, M. Dufay, and M. C. Poulizac, Phys. Letters A27, 96 (1968).

96a.M. Hollstein, D. C. Lorents, J. R. Peterson, and J. R. Sheridan, Can. J. Chem. 47, 1858 (1969). 97. W. S. Bickel, J. Opt. SOC.Am. 58,213 (1968). 98. W. Whaling, R. B. King, and M. Martinez-Garcia, Astrophys. J. 158, 389 (1969). 99. L. Heroux, Phys. Rev. 153, 156 (1967).

100. L. Heroux, Phys. Rev. 161, 165 (1967). 101. H. E. Hinteregger, in “Space Astrophysics” (W. Liller, ed.) McGraw-Hill, New York, 1961. 102. W. S. Bickel, I. Bergstrom, L. Lundin, and I. Martinson, Phys. Rev. 178, 118 (1969). 103. K. Berkner, W. S. Cooper, S. N. Kaplan, and R. V. F‘yle, Phys. Letters 16, 35 (1965). 104. L. Heroux, Appl. Opt. 7, 2351 (1968). 105. L. Heroux and H. E. Hinteregger, Rev. Sci. Znstr. 31, 280 (1960). 106. R. R. Alfano and N. Ockman, J. Opt. SOC.Am. 58,90 (1968). 107. J. K . Nakamura and S. E. Schwartz, Appl. Opt. 7 , 1073 (1968). 108. J. B. Marion, Rev. Mod. Phys. 38, 660 (1966). 109. L. C. Northcliffe, Ann. Rev. Nucl. Sci. 13, 89 (1963). 110. E. L. Chupp, L. W. Dotchin, and D. J. Pegg, Phys. Rev. 175,44 (1968). 1lOa.W. L. Wiese and A. W. Weiss, Phys. Rev. 175, 50 (1968). 111. W. S. Bickel, I. Martinson, L. Lundin, R. Buchta, J. Bromander, and I. Bergstrom, J . Opt. SOC.Am. 59,830 (1969). 1IIa.T. Andersen, K. A. Jessen, and G. Sorensen, Phys. Letters A29, 384 (1969). I I l b . 1 . Bergstrom, J. Bromander, R. Buchta, L. Lundin, and I. Martinson, Phys. Letters A28, 721 (1969). 112. W. S . Bickel, H. G. Berry, J. Desesquelles, and S. Bashkin, J . Quant. Specty. Radiative Transfer, 9, 1145 (1969). 113. A. W. Weiss, Astrophys. J. 138, 1262 (1963). 114. M. Cohen and A. Dalgarno, Proc. Roy. SOC.(London) A275,492 (1963). II4a.A. Dalgarno and E. M. Parkinson, Phys. Rev. 176,73 (1968). 115. W. S. Bickel, Phys. Rev. 162, 7 (1967). 116. A. W. Weiss, Phys. Rev. 162, 71 (1967). 116a.M. Cohen and A. Dalgarno, Proc. Roy. SOC.(London) A280,258 (1964). 117. K. Ziock, Z. Physik 147, 99 (1957). 118. 0. Osberghaus and K. Ziock, Z. Naturforsch. l l a , 762 (1956). 119. C. Ottinger and K. Ziock, 2. Naturforsch. 16a, 720 (1961). 120. L. Brewer, C. G. James, R. G. Brewer, F. E. Stafford, R. A. Berg, and G. M. Rosenblatt, Rev. Sci. Znstr. 33, 1450 (1962). 121. J. K. Link, J. Opt. SOC.Am. 56, 1195 (1966). 122. P. T. Cunningham and J. K. Link, J. Opt. SOC.Am. 57, lo00 (1967). 123. P. T.Cunningham, J. Opt. SOC.Am. 58, 1507 (1968). 124. 0. C. Jones, J. Sci. Znstr. 41, 653 (1964). 125. E. A. Enemark and A. Gallagher Rev. Sci. Znstr. 40,40 (1969).

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

227

W. Demtroder, 2. Physik, 166,42 (1962). B. Brehm, W. Demtroder, and 0. Osberghaus, 2. Nuturforsch. 16a, 843 (1961). E. Hulpke, E. Paul, and W. Paul, 2. Physik 177, 257 (1964). E. A. Bailey and G. K. Rollefson, J. Chem. Phys. 21, 1315 (1953). P. Debye and F. W. Sears, Proc. Natl. Acad. Sci. US.18, 409 (1932). M. Born and E. Wolf, “Principles of Optics,” Chapter XII. Pergamon Press, Oxford, 1959. 132. M. V. Berry, “The Diffraction of Light by Ultrasound.” Academic Press, New York, 1966. 133. I. I. Adrianova, Opt. Specty. 14, 70 (1963). 134. J. B. Birks and D. J. Dyson, J. Sci. Znstr. 38, 282 (1961). 135. S. M. Hamberger, J. Nucl. Energy, Part C 5, 73 (1963). 136. A. Corney and G. W. Series, Proc. Phys. SOC.83, 213 (1964). 137. L. Brewer, R. A. Berg, and G. M. Rosenblatt, J. Chem. Phys. 38,1381 (1963). 138. A Chutjian, J. K. Link, and L. Brewer, J. Chem. Phys. 46,2666 (1967). 139. 3. D. Rees and M. P. Givens, J. Opt. SOC.Am. 56,93 (1966). 140. A. Muller, R. Luniry, and H. Kokubun, Reu. Sci. Znstr. 36, 1214 (1965). 141. P. T. Cunningham, Ph.D. Thesis, Univ. of California,Berkeley, U.C.R.L. 18419 (Nov. 1968). 142. F. Karstensen and J. Schramm, 2.Physik 195,370 (1966). 143. F. Karstensen, 2.Physik 187, 165 (1965). 144. R. J. Wolff and S. P. Davis, J . Opt. SOC.Am. 58, 490 (1968). 145. I. Agarbiceanu, I. Kukurezianu, I. Popesku, and V. Vasiliu, Opt. Spectry 14,8 (1963). 146. G. V. Markova and M. P. Chaika, Opt. Spectry 17, 170 (1964). 147. R. Schmieder, Ph.D. Thesis, Columbia Univ. New York, New York (1969). 148. H. Ackermann, 2. Physik 194, 253 (1966). 149. G. M. Lawrence, J. Quant. Spectry. Radiative Transfer 5 , 359 (1965). 150. G . M. Lawrence and B. D. Savage, Phys. Rev. 141,67 (1966). 151. J. E. Hesser, Phys. Rev. 174, 68 (1968). 152. J. E. Hesser and B. L. Lutz, J. Opt. SOC.Am. 58, 1513 (1968). 153. J. E. Hesser, J . Chem. Phys. 48, 2518 (1968). 154. B. D. Savage and G. M. Lawrence, Astrophys J . 146, 940 (1966). 155. J. E. Hesser and K. Dressler, Astrophys J. 142, 389 (1965). 156. J. E. Hesser and K. Dressler, J. Chem. Phys. 45, 3149 (1966). 157. J. E. Hesser and K. Dressler, J. Chem. Phys. 47, 3443 (1967). 158. S. Bashkin, H. G. Berry, W. S. Bickel, and J. Desesquelles, Compt. Rend. 268, 234 (1969). 159. E. Fink and K. H. Welge, Z. Naturforsch. 19a, 1193 (1964). 159a.R. G. Fowler, P. R. Skwerski, R. A. Anderson, G. E. Copeland, and T. M. Holzberlein, J. Chem. Phys. 50, 4133 (1969). 160. R. P. Schwenker, J. Chem. Phys. 42, 1895 (1965). 161. T. Wentink, to be published. 162. R. P. Schwenker, J. Chem. Phys. 42, 2618 (1965). 163. M. Jeunehomme, J . Chem. Phys. 44,2672 (1966). 164. J. Stone, “Radiation and Optics.” McGraw-Hill, New York, 1963. 165. J. T. Davies, and J. M. Vaughan, Astrophys J. 137, 1302 (1963). 166. B. D. Fried and S. D. Conte, “The Plasma Dispersion Function.” Academic Press, New York, 1961. 167. D. G . Hummer, Mem. Roy. Astron. SOC.70, 1 (1965). 168. P. Jacquinot, Rept. Progr. Phys. 23, 267 (1960). 126. 127. 128. 129. 130. 131.

228

A . CORNEY

169. W. H. Steel, “Interferometry.” Cambridge Univ. Press, London and New York, 1967. 170. H. G. Kuhn and J. M. Vaughan, Proc. Roy. Soc. (London) A277,297 (1963). 171. J. M. Vaughan, Proc. Roy. Soc. (London) A295, 164 (1966). 172. H. G. Kuhn and E. L. Lewis, Proc. Roy. Soc. (London) A299,423 (1967). 173. J. M. Vaughan, Phys, Rev. 166, 13 (1968). 174. E. L. Lewis and D. N. Stacey, to be published. 175. J. M. Vaughan, J . Opt. SOC.Am. 54, 318 (1964). 176. H. G . Kuhn and H. J. Lucas-Tooth, J . Sci. Znstr. 35,413 (1958). 177. H. G. Kuhn, E. L. Lewis, D. N. Stacey, and J. M. Vaughan, Rev. Sci. Znstr. 39, 86 (1968). 178. C. J. Tsao and B. Curnutte, J . Quant. Specfry. Radiative Transfer 2, 41 (1962). 179. R. G. Breene, Handb. Physik. 27, 1 (1964). 180. F. W. Byron and H. M. Foley, Phys. Rev. 134, A625 (1964). 181. T. Watanabe, Phys. Rev. 138, A1573 (1965). 182. T. Watanabe, Phys. Rev. 140, AB5 (1965). 183. A. W. Ali and H. R. Griem, Phys. Rev. 140, A1044 (1965). 183a.A. W. Ali and H. R. Griem, Phys. Rev. 144,366 (1966). 184. A. Omont, J . Phys. Radium 26, 26 (1965). 185. D. N. Stacey and J. Cooper, Phys. Letters A30,49 (1969). 185a.P. R. Berman and W. E. Lamb, Phys. Rev. 187, 221 (1969). 186. A . Omont, Doctoral Thesis, Ecole Normale Superieure, Paris (1967). 187. A. Omont and J. Meunier, Phys. Rev. 169,92 (1968). 188. W. Happer and E. B. Saloman, Phys. Rev. 160,23 (1967). 189. E. L. Lewis, Proc. Phys. Soc. 92, 817 (1967). 190. V. I. Odintsov, Opt. Specfy. 10, 202 (1961). 191. R . W. Stanley, J . Opt. SOC.Am. 56, 350 (1966). 192. F. A. Korolev, V. I. Odintsov, and E. V. Fursova, Opr. Specty. 16, 304 (1964). 193. F. A. Korolev and V. I. Odintsov, Opt. Specfy. 18, 547 (1965). 194. R. Damaschini and J. Brochard, Compt. Rend. 264, 1350 (1967). 195. A . Dalgarno and A. L. Stewart, Proc. Phys. Soc. 76,49 (1960). 196. A . W. Weiss, J . Res. Natl. Bur. Std. (CJ.S.A.)71A, 163 (1967). 197. H . G . Kuhn, E. L. Lewis, and J. M. Vaughan, Phys. Rev. Letters 15,687 (1965). 198. D. A . Hutchinson and H. F. Hameka, J . Chem. Phys. 41, 2006 (1964). 199. P. R . Fontana and D. D. Hearn, Phys. Rev. Letters 19, 481 (1967). 200. M. R. Philpott, Proc. Phys. Soc. 89, 217 (1966). 201. V. A. Alekseev, A. V. Vinogradov, and I. I. Sobelman, J. Quant. Specfy. Radiative Transfer 10, 55 (1970). 202. R . Lincke and H. R . Griem, Phys. Rev. 143, 66 (1966). 203. J. Geiger, Z. Physik 175, 530 (1963). 204. E. S. Fry and W. L. Williams, Phys. Rev. Leffers20, 1335 (1968). 205. E. S. Fry and W. L. Williams, Phys. Rev. 183, 83 (1969). 206. J. Geiger, Z. Physik 177, 138 (1964). 207. A . Gold and R . S. Knox, Phys. Rev. 113, 834 (1959). 208. H. Statz, C. L. Tang, and G . F. Koster, J . Appl. Phys. 34,2625 (1963). 209. J. W. Cooper, Phys. Rev. 128, 681 (1962). 210. R . S. Knox, Phys. Rev. 110, 375 (1958). 211. P. G . Wilkinson, J . Quant. Specfy. Radiative Transfer 5 , 503 (1965). 212. J. D. Dow and R . S. Knox, Phys. Rev. 152, A51 (1966). 213. D. K . Anderson, Phys. Rev. 137, A21 (1965).

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

229

213a.P. G. Wilkinson, J. Quant. Specty. Radiative Transfer 6, 823 (1966). 214. H. Moser and H. Schultz, Ann. Physik 4, 243 (1959). 215. C. L. Chen and A. V. Phelps, Phys. Rev. 173, 62 (1968). 216. G. W. Series, Rept. Progr. Phys. 22, 280 (1959). 217. G. zu Putlitz, Ergeb. Exakt. Naturw. 37, 105 (1964). 218. H.E.Radford, in “Methods of Experimental Physics ” Vol. 4B, p.105. Academic Press, New York, 1967. 219. B. Budick, Aduan. At. Mol. Phys. 3, 73 (1967). 220. J. P. Barrat, J. Phys. Radium 20, 657 (1959). 221. A. Landman and R. Novick, Phys. Rev. 134, A56 (1964). 222. A. Lurio and R. Novick, Phys. Rev. 134, A608 (1964). 223. F. W. Byron, M. N. McDermott, and R. Novick, Phys. Rev. 134, A615 (1964). 224. A. Lurio, R. L. de Zafra, and R. J. Goschen, Phys. Rev. 134, A1198 (1964). 225. A. Lurio, Phys Rev. 136, A376 (1964). 226. A. Lurio, Phys. Rev. 140, A1505 (1965). 227. D. Lecler, J. Phys. Radium 29, 611 (1968). 228. F. D. Colegrove, P. A. Franken, R. R. Lewis, and R. H. Sands, Phys. Rev. Letters 3, 512 (1959).

229. G. Breit, Rev. Mod. Phys. 5 , 91 (1933). 230. P. A . Franken, Phys. Rev. 121, 508 (1958). 231. M. E. Rose and R. L. Carovillano, Phys. Rev. 122, 1185 (1961). 232. A. Gallagher and A. Lurio, Phys. Rev. 136, A87 (1964). 233. M. Dumont and B. Decomps, J. Phys. Radium, 29, 181 (1968). 234. J. Brossel and F. Bitter, Phys. Rev. 86, 308 (1952). 235. A. Kastler and J. Brossel, Compt. Rend. 229, 1213 (1949). 236. G. J. Ritter and G. W. Series, Proc. Roy. SOC.(London) A238,473 (1957). 237. H. Ackermann, Z . Physik 194, 253 (1966). 238. 5. Korvalski and G. zu Putitz, Z. Physik 208, 459 (1968). 239. A . Marshall, R. L. de Zafra, and H. Metcalf, Phys. Rev. Letters 22,445 (1969). 240. D. R. Crosley and R. N. Zare, Phys. Rev. Letters 18,942 (1967). 241. D. R. Crosley and R. N. Zare, J. Chem. Phys. 49,4231 (1968). 241a.W. C. Wells and R. C. Isler, Phys. Rev. Letters 24, 705 (1970). 242. B. Budick, R. Novick, and A. Lurio, Appl. Opt. 4,229 (1965). 243. G. D. Chapman and L. Krause, J. Opt. SOC.Am. 55,1269 (1965). 244. L. Krause, J. Opt. SOC.Am. 57, 1162 (1967). 245. W. W. Smith and A. Gallagher, Phys. Rev. 145,26 (1966). 246. A. Gallagher, Phys. Rev. 157, 24 (1967). 247. E. Handrich, A. Steudel, R. Wallenstein, and H. Walther, Coll. C1. Supp. J. Phys. Radium 30, C1-18 (1969). 248. H. Kretzen, W. Lange, A. Steudel, and H. Walther, Z. Physik (to be published). 249. B. Decomps and M. Dumont, ZEEE J. Quant. Elec. QE-4,916 (1968). 250. B. Decomps and M. Dumont, J. Phys. Radium 29,443 (1968). 251. M. Tsukakoshi and K. Shimoda, J. Phys. SOC.Japan 26,758 (1969). 252. H. Kretzen, W. Lange, J. Luther, A. Steudel, and H. Walther, Proc. At. Specty. Congr., Paris, July 1969, p.31. 253. U. Brinkmann, A. Steudel, and H. Walther, Z. Angew. Phys. 22,223 (1967). 254. B. P. Kibble and S. Pancharatnam, Proc. Phys. SOC.86, 1351 (1965). 255. J. P. Barrat, J. L. Cojan, and Y. Lecluse, Compt. Rend. 262, 609 (1966). 256. B. Laniepce, J. Phys. Radium 29,427 (1968). 257. B. Decomps and M. Dumont, Compt. Rend. 262, 1004 (1966).

230 258. 259. 260. 261. 262. 263. 264. 265. 266. 267. 268. 269. 270. 271. 272. 273. 274. 275. 276. 277. 278. 279. 280. 281. 282. 283.

A. CORNEY

H. Wahlquist, J . Chem. Phys. 35, 1708 (1961). C. G. Carrington and A. Corney, Opt. Commun. 1, 115 (1969). R. W. Schmieder, A. Lurio, and W. Happer, Phys. Rev. 173,76 (1968). H. S. W. Massey, E. H. S. Burhop, and H. B. Gilbody, “Electronic and Ionic Impact Phenomena,” Vol. 1. Oxford Univ. Press, London and New York, 1969. J. C. Pebay-Peyroula, J. Brossel, and A. Kastler, Compt. Rend. 244, 57 (1969). J. C. Pebay-Peyroula, J. Phys. Radium 20, 669 (1959). J. C. Pebay-Peyroula, J. Phys. Radium 20, 721 (1959). “Excitation Electronique d’une vapeur atomique.” CNRS, Paris, 1967. J. C. Pebay-Peyroula, Proc. Arnold Sommerfeld Centennial Meeting, Munich, 1968. M. Barrat, Ph.D. Thesis, Univ. of Caen, France, 1965. A. D. May, Compt. Rend. 250, 3616 (1960). Y . Archambault, J. P. Descoubes, M. F’riou, A. Omont, and J. C. Pebay-Peyroula, J. Phys. Radium 21, 677 (1960). A. Faure, 0 .Nedelec, and J. C. Pebay-Peyroula, Compt. Rend. 256, 5088 (1963). C. Galleron-Julienne and J. P. Descoubes, Compt. Rend. 261, 916 (1965). M. Chenevier, These 3eme Cycle, Universite de Grenoble (1966). A. Faure, These 3eme Cycle, Universite de Grenoble (1966). R. D. Kaul, J. Opt. SOC.Am. 57. 1 1 56 (1967). M. Lombardi and J. C. Pebay-Peyroula, Compt. Rend. 261, 1485 (1965). M. Chenevier, J. Dufayard, and J. C. Pebay-Peyroula, Phys. Letters A25,283 (1967). M. Giroud, These 3eme Cycle, Universite de Grenoble (1968). T. Carver, Phys. Rev. Letters 19, 485 (1967). A. Omont, Compt. Rend. 260, 3331 (1965). E. B. Saloman and W. Happer, Phys. Rev. 144,7 (1966). G. Goullet and J. C. Pebay-Peyroula, Compt. Rend. 259, 93 (1964). 0. Nedelec, J, Phys. Radium, 27, 660 (1966). F. W. Byron, M. N. McDermott, R. Novick, B. W. Perry, and E. B. Saloman, Phys.

Rev. 134, A47 (1964). 283a.B. Decomps, Doctoral Thesis, Univ. of Paris (1969). 283b.C. H. Wang and W. J. Tomlinson, Phys. Rev. 181, 115 (1969). 284. W. E. Lamb, Phys. Rev. 134, A1429 (1964). 285. R. A. McFarlane, W. R. Bennett, and W. E. Lamb, Appl. Phys. Letters 2, 189 (1963). 286. A. Szoke and A. Javan, Phys. Rev. Letters 10, 521 (1963). 287. R. L. Fork and M. A. Pollack, Phys. Rev. 139, A1408 (1965). 288. A. Szoke and A. Javan, Phys. Rev. 145,137 (1966). 289. P. W. Smith, J. Appl. Phys. 37, 2089 (1966). 290. J. A. White, J. Opt. SOC.Am. 55, 1436 (1965). 291. R. H. Cordover, P. A. Bonczyk, and A. Javan, Phys. Rev. Letters 18,730, 1104 (1967). 292. W. G. Schweitzer, M. M. Birky, and J. A. White, J. Opt. SOC.Am. 57, 1226 (1967). 293. H. K. Holt, Phys. Rev. Letters 19, 1275 (1967). 294. H. K. Holt, Phys. Rev. Letters 20, 410 (1968). 295. M. S. Feld and A. Javan, Phys. Rev. 177, 540 (1969). 296. H. R. Schlossberg and A. Javan, Phys, Reu. Letters 17, 1242 (1966). 297. H. R. Schlossberg and A. Javan, Phys. Rev. 150,267 (1966). 298. J. H. Parks and A. Javan, Phys. Rev. 139, A1351 (1965). 299. R. A. Lilly and J. R. Holmes, J. Opt. SOC.Am. 58, 1406 (1968). 300. S. Ezekiel and R. Weiss, Phys. Rev. Letters 20,91 (1968). 301. R. L. Fork, L. E. Hargrove, and M. A. Pollack, Appl. Phys. Letters 5, 5 (1964). 302. A. V. Phelps and J. P. Molnar, Phys. Rev. 89, 1202 (1953).

LIFETIMES OF FREE ATOMS, MOLECULES, AND IONS

303. A. V. Phelps, Phys. Rev. 99,1307 (1955). 304. A. V. Phelps, Phys. Reo. 114,1011 (1958). 304a.D. S. Smith and R. Turner, Can. J. Phys. 41, 1949 (1963). 305. A. V. Phelps and J. L. Pack, Rev. Sci. Znstr. 26,45 (1955). 306. M. Barrat, Compt. Rend. 259, 1063 (1964). 307. M. Baumann, Z . Physik 173, 519 (1963). 308. K. Tittel, Z . Physik 187, 421 (1965). 309. C. Cohen-Tannoudji and A. Kastler, Prog. Opt. 5 , 3 (1966). 310. L. D. Schearer, Phys. Rev. 166, 30 (1968). 311. L. D. Shearer, Phys. Rev. 171, 81 (1968). 312. L. D. Schearer, Phys. Rev. Letters 21, 660 (1968). 313. L. D. Schearer, Phys. Letters A28, 660 (1969). 314. L. D. Schearer, Phys. Rev. 180, 83 (1969). 315. S. Mrozowski, Rev. Mod. Phys. 16, 153 (1944). 316. A. Rubinowicz, Rept. Progr. Phys. 12, 233 (1949). 317. A. Omholt, and L. Harang, J . Atm. Terr. Phys. 7, 247 (1955). 318. A. Omholt, Geofys. Publ. 21, 1 (1959). 319. K. V. Paulson and G. G. Shepherd, J. Atmos. Terrs. Phys, 27, 831 (1965). 320. W. Stoffregen and H. Derblom, Nature 185, 28 (1960). 321. A. Omholt, Planet Space Sci. 2,246 (1960). 322. E. C. Zipf, Can. J. Chem. 47, 1863 (1969). 323. F. Stuhl and K. H. Welge, Can. J . Chem. 47, 1870 (1969). 324. F. J. Blanc, 0. Oldenberg, and N. P. Carleton, J. Chem. Phys. 45,2200 (1966). 325. J. W. McConkey and J. A. Kernahan, Planet Space Sci. 17, 1297 (1969). 326. W. Lichten, J . Chem. Phys. 26, 306 (1957). 327. J. F. Noxon, J. Chem. Phys. 36,926 (1962). 328. L. F. Phillips, Can. J. Chem. 43, 369 (1965). 329. E. C. Zipf, J . Chem. Phys. 38,2034 (1963). 330. H. H. Bromer and F. Spieweck, Planet Space Sci. 15,689 (1967). 331. N. P. Carleton and 0. Oldenberg, J. Chem. Phys. 36, 3460 (1962). 332. D. E. Shemansky and N .P. Carleton, J. Chem. Phys. 51, 682 (1969). 332a.T. Wentink and L. Isaacson, J. Chem. Phys. 46, 822 (1967). 333. R. S. Freud, J . Chem. Phys. 50, 3734 (1969). 334. M. Lipeles, R. Novick, and N. Tolk, Phys. Rev. Letters 15, 690 (1965). 334a.R. Novick, Comments on Atomic and Molecular Physics 1, 91 (1969). 335. R. C. Elton, L. J. Palumbo, and H. R. Griem, Phys. Rev. Letters 20,783 (1968). 336. J. Shapiro and G. Breit, Phys. Rev. 113, 179 (1959). 337. G. W. F. Drake, G. A. Victor, and A. Dalgarno, Phys, Rev. 180, 25 (1969). 337a.A. S. Pearl, Phys. Rev. Letters 24, 703 (1970). 338. U. Fano and J. W. Cooper, Phys. Rev. 137, A1364 (1965). 339. K. Codling, R. P. Madden, and D. L. Ederer, Phys. Rev. 155,26 (1967). 340. P. Feldman and R. Novick, Phys. Rev. 160, 143 (1967). 341. P. Feldman, M. Levitt, and R. Novick, Phys. Rev. Letters 21, 331 (1968). 342. D. J. Nicholas, C. W. Trowbridge, and W. D. Allan, Phys. Rev. 167, 38 (1968). 343. V. V. Balashov, V. S. Senashenko, and B. Tekou, Phys. Letters A25,487 (1967).

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Recent Advances in Particle Accelerators JOHN P. BLEWETT Brookhaven National Laboratory. Upton. New York

I . Introduction . . . . . . . ............................................... 233 I1. Electron Ring Accele s .............................................. 234 I11. Heavy Ion Accelerators ................................................ 236 IV. Superconducting Accelerators ........................................... 238 A . Accelerators Using Superconducting Magnets .......................... 238 B . Accelerators Using Superconducting rf Cavities ........................ 240 V . Proton Synchrotons .................................................... 240 A . Serpukhov ......................................................... 240 B The National Accelerator Laboratory .................................. 241 C . The European 300-GeV Project ...... .............................. 243 D. Design Studies for Proton Synchroton ..................... E . Proton Synchroton Improvement Programs . . . . . . . . . . . . . . . . . . . VI . Electron Accelerators .................. ............................ 247 A . SLAC............................................................. 247 B. Cornell ............................................................ 247 C. Other Electron Synchrotrons ......................................... 248 VII . Storage Rings and Colliding Beams ...................................... 248 A . The CERN Intersecting Storage Rings ................................. 248 B . The Novosibirsk Proton-Antiproton Ring ............................. 249 C . Colliding Beam Studies at the National Accelerator Laboratory .......... 250 D . Electron Storage Rings .............................................. 250 V I E Cyclotrons ............................................................ 252 IX . Meson Factories ....................................................... 253 X . R.1.P ................................................................. 254 XI . Conclusion ........................................................... 255 References .......................................................... 256

.

I . INTRODUCTION At national and international accelerator conferences held in 1967. 1968. and 1969 three new topics seemed to stand out in the interest and excitement that surrounded their discussion. First was the concept of the electron ring accelerator announced by the group of the late Veksler in Dubna and quickly adopted in several centers. This appears to be a successful culmination of many years of attempts to accelerate positive ions by the high electric fields that can be generated in bunches of electrons. Second. the idea that there 233

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may be islands of stability at masses much higher than those in the standard periodic table began to be accepted, and many centers have proposed a variety of heavy-ion accelerators to explore this possibility. Finally, there has been notable progress in the design and construction of superconducting magnets for pulsed operation at high fields. Speculation is now rife regarding the possible design of synchrotrons using these magnets. Parallel progress on superconducting cavities for use in linear accelerators gives promise of important improvement in the duty cycle of electron linacs. In the meantime, a number of new machines of more conventional design are under construction or have been completed. In this article, we shall attempt a review both of the new ideas and of progress on major accelerators throughout the world. rr. ELECTRON RINGACCELERATORS The minimum overall length of linear accelerators for high energies has, in the past, been set by the maximum electric field that could be maintained without breakdown between parts of the accelerating structure. It has long been evident that, in principle, much higher fields could be maintained within a dense bunch of electrons or ions; the unsolved problem was that of preventing the bunch from blowing up in its own fields. It now appears that Veksler and his associates have solved this problem. The solution was announced at the Sixth International Accelerator Conference in Cambridge, Massachusetts in 1967 ( I ) . The procedure proposed by the Dubna group is illustrated in Fig. 1. In a rather weak magnetic field, falling off with radius at a rate slower than I/r, a ring of electrons is injected at an energy of a few million electron volts. This ring is stable for the same reasons that the beam is stable in a betatron. The field is now increased by a factor of 20 or so and the ring shrinks both in major and minor radius. Due to the electric field induced by the changing

0%FTED

0

0

0 FIG.1 . Electron ring accelerator.

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magnetic flux, the electrons are accelerated at the same time. Provided only that enough electrons have been injected, the fields in the ring can now be much higher than can be maintained between electrodes. Typical parameters for this process might be: Initially--10'4 electrons at 4.5 MeV injected into a field of 750 G to produce a ring of 20-cm major radius and 5-mm minor radius. After compression by raising the field to 20 kG-the ring now has a major radius of 3.7 cm and a minor radius of 1 mm.

In the compressed ring the maximum electric field will be 1300 MVjm. This is to be compared with the maximum field of about I5 MV/m contemplated in the Stanford Linear Accelerator Center (SLAC) two-mile linac. With the increase in electron energy, the space-charge forces tending to blow the ring apart are decreased by the usual relativistic factor of (1 - p'). A burst of gas is now admitted into the chamber. The gas is ionized by the electrons and some ions are trapped inside the ring. The number of ions trapped should be small (perhaps 1 %) compared with the number of electrons in the ring. These ions will now make the ring completely stable even without the focusing action of the shaped magnetic field. As indicated in Fig. I , the magnetic field can now be varied in such a fashion as to allow the ring to drift along its axis into a region of smaller field. The circumferential momentum of the electrons will be partially converted into axial momentum; as the ring moves, it will carry the trapped ions with it and they will be accelerated to the same axial velocity as that achieved by the electrons. By this procedure it appears that proton energies of the order of 1 GeV should easily be attainable in a drift distance of a few meters. If still higher ion energies are desired, the electron ring can be accelerated in a radio frequency (rf) system like that in an electron linear accelerator. This procedure has its problems since only one large bunch is accelerated at a time. An alternative procedure has been proposed at Berkeley; acceleration here is accomplished by the fields, when a radial transmission line is discharged to provide a roughly square wave of accelerating field lasting a few nanoseconds. The Dubna proposal has been accepted with some enthusiasm at the Lawrence Radiation Laboratory, and an experimental program there is now well advanced. A symposium on electron ring accelerators was held at Berkeley in February of 1968. Its proceedings (2) are a useful reference on the subject. Work on electron ring accelerators is now in progress at several other centers including the University of Maryland, the Rutherford Laboratory, CERN, Munich, and Karlsruhe. At the 1969 International Accelerator Conference held in Erevan in the USSR (3), successful production and compression of electron rings was

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announced by the Berkeley and Dubna groups. Also, at Dubna, a ring containing triply ionized nitrogen ions has been allowed to escape and expand; the nitrogen ions were accelerated to about 4 MeV per nucleon. Nothing has yet been done about acceleration of electron rings by applied rf or pulsed electric fields. A theoretical controversy has arisen about how radiation losses of energy of electron rings passing through acceleration systems vary with electron energy. Possibly, these losses will set an upper limit to acceleration. However, these losses may be reduced by a system of acceleration using a smooth, dielectric-loaded waveguide, as proposed by Schopper of Karlsruhe (3). Whether or not the electron ring accelerator will be a cheap multigiga electron-volt machine for proton acceleration is not yet clear. Its duty cycle will be very small, but for many applications this does not matter. Probably, it will be useful for the acceleration of heavy ions. Work carried out during 1970 and 1971 probably will lead to a better evaluation. In the meantime, the ideas of Veksler and his associates seem to be the most exciting contribution in the field of particle acceleration of the past decade. Review articles describing the present (late 1969) status of the electron ring accelerator will appear early in 1970 in a new journal Particle Accelerators. 111. HEAVYIONACCELERATORS

I n the past, research on transuranium elements has been concentrated in the laboratories of Seaborg and Ghiorso in Berkeley and of Flerov at Dubna. Recently, however, world-wide interest has flared up in the possible production of very heavy elements, which theoretical studies during the past decade ( 4 ) have indicated might possibly be stable. The goal of most of the enthusiasts is production of ions of all masses up to uranium with energies of 6 to 10 MeV:nucleon. Heavy ion accelerators are, of course, useful for many other purposes than production of transuranic elements. Experiments on nuclear reactions and Coulomb excitation of nuclei have been i n progress since about 1940, using the lighter heavy ions of such elements as carbon and nitrogen. These ions were accelerated in cylotrons, linear accelerators, or electrostatic accelerators. The catalog of heavy ion machines now in operation includes four linear accelerators capable of accelerating ions of mass up to argon (:gAr) up to 10 M.=V/nucleon,three classical cyclotrons of which the largest, at Dubna, can accelerate ions up to :GZn to over 6 MeVinucleon, four isochronous cyclotrons, and a large number of tandem Van de Graaff accelerators. These machines all have different mechanisms setting upper limits to the mass of the ion that can be accelerated to any desired energy. These limitations depend on the ratio of Q to A , the number of electronic charges borne by

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the ion and the weight of the nucleus in nucleons, respectively. The linear accelerator is designed to accelerate ions to a given velocity and, hence, to a constant energy per nucleon. As Q / A decreases from unity (for protons) to the attainable values in the neighborhood of 0.15 for very heavy ions, the accelerating field must be increased until limited by electrical breakdown. Cyclotrons are characterized by the relation : energy per nucleon

=

Tp( Q / A ) ’ ,

where T, is the peak energy to which protons can be accelerated. Thus, a proton cyclotron capable of proton acceleration to 300 MeV can accelerate heavy ions having Q / A of 0.15 to about 7.MeV/nucleon. Electrostatic accelerators have terminals at constant potential; if the ion source is in the terminal, ions will be accelerated to energies per nucleon of @ / A ) times the terminal voltage. In a tandem machine, negative ions with a single charge can be accelerated from ground to the terminal and there stripped in a gas jet or a foil to a high positive state of ionization; acceleration back to ground will yield ions of energy [(I + Q ) / A ] times the terminal voltage. The advantage of this method is that higher values of Q can be achieved by stripping at high energy than can be reached at low energy in an ion source. In most of the numerous proposals now under consideration in the United States, a cyclotron is proposed as the final stage. To provide a high state of ionization, a tandem Van de Graaff is proposed as the injector. At Oak Ridge, Argonne, the University of Rochester, and other centers it is felt that the Van de Graaff machine should have an energy as high as possible; it should be either the High Voltage Engineering Corporation’s MP tandem which has a guaranteed terminal voltage of 10 MV (16 MV has been reached) or its new TU tandem with a guaranteed terminal voltage of 16 MV (and a design voltage of 20 MV). The high injection energy is chosen because the degree of ionization by stripping increases with energy. The cyclotron for the final acceleration could then be of the order of 300(Q/A)2MeV/nucleon. A dissenting voice is heard from Michigan State University where it is felt that a smaller tandem and a cyclotron for 600(Q/A)2MeV/nucleon would be cheaper for the same final result. Elsewhere linear accelerators are favored. At the Lawrence Radiation Laboratory where the heavy ion linear accelerator (Hilac) has been in use in studies of transuranium elements since 1957 a “ Hilac improvement program ” is proposed to extend the capabilities of the machine (now capable of accelerating ions of mass up to 40) to make it possible to accelerate all ions up to uranium. Ions will be injected from a high-voltage machine of 2.7 MV with a Q / A of about 0.04, accelerated to 1 MeV/nucleon, and then stripped to a Q / A of 0.15. In a final linac section, the ions will be accelerated to an energy of 8.6 MeV/nucleon.

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In Germany, a project has been under study for some years under the direction of Schmelzer in Heidelberg. This project has resulted in initiation of construction in Darmstadt of a heavy ion linear accelerator. Several local institutions are collaborating in this effort. This machine will include three different types of linear accelerator, each peculiarly suited to the velocity range where it is used. The first stage will be a modified Sloan-Lawrence machine having its drift tubes connected alternately to the inner and outer conductor of a coaxial line. The second stage will be a conventional drift-tube accelerator. The final stage includes a series of independently powered accelerating cavities. An interesting parallel study was made at Frankfurt using a helix for the accelerating unit-this is the inverse of the conventional travelingwave tube. The Darmstadt machine will bring all ions to between 6 and 10 MeV/nucleon-it is known as the UNILAC because of its universal capability. Another scheme with intriguing possibilities has been proposed in Heidelberg by Hortig. This idea depends on the fact that a heavy ion passing through a foil stripper will emerge with a higher-charge-state than will be reached after passage through a gas stripper. Ions entering a tandem Van de Graaff are stripped in a foil and accelerated to the terminal of the tandem. Here, they pass through a gas stripper. This reduces the number of charges on the ion and as it proceeds to the other end of the tandem it loses less energy than it gained on its way to the terminal. The ion is now deflected in a magnet through an angle of 180" and returned to the tandem where the same process is repeated. After a rather large number of passes through the tandem, this process should result in the attainment of very high energies. There is some question as to how many ions would survive this multiple acceleration and no one has yet made an experimental attempt to verify Hortig's predictions. At Orsay, France, a cyclotron using a Sloan-Lawrence linear accelerator was in operation at the end of 1969. This machine will not reach the very heavy ion range; ions up to about mass 80 (krypton) are accelerated to 5 to 10 MeV/nucleon. In all, over 30 laboratories are studying, actively proposing, or constructing heavy ion machines. If there are islands of stability beyond the end of the periodic table, they should be discovered during the next decade. IV. SUPERCONDUCTING ACCELERATORS A . Accelerators Using Superconducting Magnets

For more than half a century, Nature has alternately held out promises of major applications of superconductivity, and then hastily retracted them. At first, superconducting magnets for high fields could not be built because superconductivity is destroyed in a high magnetic field. With the advent,

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a decade ago, of type I1 superconductors, high-field magnets again appeared possible. However, these magnets were plagued with instabilities associated with losses when the field was turned on or changed. Parts of the magnet would “go normal” and the stored energy in the field would be dissipated in the normal region, often with destructive results, This was cured by dilution of the superconductor with such large quantities of copper that the conductor could still carry the current without instability when the superconductor became unstable. This worked nicely for large magnets for large bubble chambers, but for magnets with small apertures suitable for use in synchrotrons, the current density had been so drastically reduced in the process of stabilization that small magnets could not be designed for high fields. The probable solution of this dilemma was presented at a 6-week “ Summer Study on Superconducting Devices and Accelerators ” (5) held at Brookhaven during 1968. Smith, of the Rutherford Laboratory, building on the work of many earlier workers, proposed that cables including many strands of superconductor each having a diameter of 5 to 10 p , and with the strands transposed as in a Litz cable, should be inherently stable and should show quite tolerable losses when used under pulsed or ac conditions. The latter half of 1968 and all of 1969 have been spent in verifying these predictionsthey appear to be correct both qualitatively and quantitatively. No unpleasant new surprises have been experienced, and work is now turning toward the design of synchrotrons for energies of several hundred giga electron volts using magnets with dipole fields of 60 kG or higher. This work is in progress at Brookhaven, Berkeley, the Rutherford Laboratory, and many other centers, but it will be several years before the electrical and mechanical design problems are solved. At the present writing, however, all of these problems appear to be soluble, and it is to be expected that the next decade will see the construction of synchrotrons having superconducting bending magnets operating at fields of 60-80 kG. In a parallel, low-temperature program, studies have been made of the properties and behavior of very pure metals. Although these are not superconducting metals, this program is mentioned here since it parallels, in many ways, the program on superconducting magnets. Of the various metals suitable for use in magnets, aluminum seems to be the most promising. Aluminum, pure to about one part in lo6, has a resistivity at 4°K that is lower by a factor of about 15,000 than its resistivity at room temperature. In a magnetic field, a magnetoresistivity component appears, but saturates at reasonable fields to give an increase by a factor of about three in coil resistance. Taking refrigerator efficiency into account, an optimum operating temperature for aluminum coils appears to be between 15 and 20”K, where cooling with liquid hydrogen is possible (6). Losses in aluminum coils are, of course, purely resistive, whereas losses in

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superconducting coils are hysteretic in character (loss per cycle is constant, relatively independent of frequency). Hence, for very rapid cycling, aluminum may have advantages. For the slower cycles appropriate for very high energy machines, probably superconducting coils will be preferable. B. Accelerators Using Superconducting rf Cavities

Linear accelerators operated at room temperature require such enormous quantities of rf power, simply to make up for losses in the walls of the accelerating cavity, that they can be operated only in short pulses. The low dutycycle is very undesirable for many applications, and attempts have been made, by cooling the walls, to reduce losses. Superconducting walls now appear very promising. Two superconductors are outstanding because of their high critical magnetic fields-the rf magnetic field at the wall is proportional to the accelerating electric field. In lead, the critical field is about 800 G ; in niobium, the field is over 1600 G. In a TMolo cavity these correspond to accelerating fields of 50 and 100 MV/m. Losses are very small, but are still large enough at 4°K that it is not possible to provide adequate cooling by boiling helium outside of the cavity. Hence, it appears necessary to drop the temperature to less than 2"K, at which temperatures, it is possible to make use of the desirable properties of superfluid helium as a coolant. The leading role in development of superconducting cavities has been taken by the group under the direction of Fairbank and Schwettman at the High Energy Physics Laboratory of Stanford University. In the laboratory, this group has achieved Q of 10" factors at low levels. Electric fields as high as 70 MV/m have been observed in niobium cavities with attendant Q's of 8 x l o 9 . These figures are more than adequate to encourage the construction of an accelerator-an electron machine 150 m in length with a design energy of 2 GeV is now well advanced. Less ambitious projects using superconducting cavities will be found at Brookhaven, Karlsruhe, CERN, the University of Illinois, and a number of other centers.

SYNCHROTRONS V. PROTON A . Serpukhov In 1967, two new records were set at Serpukhov in the USSR. In June of that year, the world's largest proton linear accelerator was brought into satisfactory operation. This is the 100-MeV linac injector for the "70-GeV" proton synchrotron at Serpukhov. Four months later, on October 13th, protons were accelerated in the synchrotron to a peak energy of 76 GeVmore than twice the previously highest energy in the alternating-gradient synchrotron (AGS) at Brookhaven.

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The Serpukhov machine is conservatively designed, well built, and can be expected to continue to give a very impressive performance. It is almost 1.5 km in circumference. Its magnet system, which includes 20,000 tons of steel and 700 tons of aluminum coils, requires a peak input power of 100 MW. Its experimental areas are enormous ; they are distributed around a central hall 90 x 150 m in dimensions. By 1969, the Serpukhov accelerator was running with a peak intensity of over 10” protons/pulse with 5-10 pulses/min. A full-blown experimental program was in full swing, including several collaborative efforts with outside groups. Perhaps the most interesting cooperative arrangement is between Serpukhov and CERN. CERN has undertaken to provide a fast beam-extraction system for the 70-GeV accelerator. Also, CERN will provide rf particle separators to be used at Serpukhov for at least 10 yr. In return, CERN scientists will be allowed to take an active part in the 70-GeV experimental program, generally as members of joint CERN-Soviet teams. France also has its own collaborative arrangement to allow French scientists to join the new experimental program. In return she is providing a 6000-liter liquid hydrogen bubble chamber known as “ Mirabelle.” Mirabelle is under construction at Saclay, and received her first test in July 1969, when photographs of cosmicray showers were taken. B. The National Accelerator Laboratory The rather flamboyant search for a site for the US national 200-GeV accelerator finally was terminated in December of 1966, when the Atomic Energy Commission announced that it had chosen an area to the west of Chicago, rather close both to the Argonne Laboratory and to O’Hare Airport. The Universities Research Association, which is to manage the project, was already in being, and early in 1967, it chose Robert R. Wilson to be director of the new National Accelerator Laboratory (NAL). A design group was established near the site, and set to work in June of 1967. It soon was announced that the accelerator would be designed in such a fashion that it could later be extended to 400 GeV and, possibly, to 500 GeV. Its completion is planned for July of 1972. Ground was broken for the first building, the injector housing, in December of 1967, and construction of conventional buildings began immediately. During 1968, design and construction of prototypes proceeded apace and, by the end of 1969, many of the machine components were on order. A landmark in 1969 was acceleration of a proton beam to 10 MeV in the prototype first section of the injector. In contrast to the Brookhaven and CERN synchrotrons, where the same magnet system serves both to bend and focus the beam during acceleration, “

”

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the NAL synchrotron will have a " separated function " magnet ring. Bending of the beam into its circular orbit will be accomplished by dipole magnets with uniform fields, whereas focusing will be by separate quadrupole magnets. This idea was considered as early as 1952 for use in the Brookhaven AGS, but it has not previously been used in a major accelerator. Injection will be accomplished in three stages. A 750-keV CockcroftWalton preinjector will provide a beam for a 200-MeV linear accelerator very similar to the one under construction at Brookhaven. The linac, in turn, will be the injector for a " booster " synchrotron which will raise the proton energy to about 8 GeV before transfer into the main ring. The arrangement of the various components is shown in the plan of the National Accelerator Laboratory shown in Fig. 2. WEST CHICAGO

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It is too early to predict whether this major project can be completed within its rather ambitious schedule. However, by the end of 1969, its progress is very impressive.

C . The European 300-Ge V Project The efforts preparatory to initiation of the American 200-GeV project have been paralleled in the European CERN community of nations. A design study for a 300-GeV accelerator was organized at CERN ; this group had presented a comprehensive report and cost estimate at the end of 1964. The machine described was, in general, similar to that resulting from the American studies, except that its magnet system combined the functions of bending and focusing in the conventional fashion. In 1963, a “ European Committee for Future Accelerators (ECFA) was organized under the chairmanship of Professor Amaldi of Rome. By 1967, ECFA has surveyed the whole European program in high energy physics and presented a set of recommendations, both for international programs and for development of regional facilities. It recommended the construction of a 300-GeV accelerator “ with the least possible delay.” With the assistance of the accelerator experts at CERN, an extensive study was made of possible sites for the 300-GeV machine. Over 100 sites were investigated in 1 I European countries. Then, by the simple procedure of allowing each country to submit only one site, the number of sites was cut down to nine (some CERN countries did not submit site recommendations). A committee of “three wise men” from three of the CERN countries that did not submit sites made a careful study of the nine sites and gave them graded ratings under several related headings; the final choice was not made by this committee, however, and this was still the status at the end of 1969. Two important events occurred during 1968. In June, for financial reasons, the United Kingdom withdrew from the 300-GeV project. The decision of the British government was by no means popular with Britain’s scientists, who have made great efforts to have this decision reversed. In December of that year, the CERN Council chose a director-general for the 300-GeV program; their choice was John B. Adams, a British scientist who had directed the construction of CERN’s 28-GeV synchrotron and who, for a short time, had been Director-General of CERN. A few months later, Adams moved back to CERN and took up direction of the 300-GeV program which, by now, was somewhat attenuated as it waited for European approval. In April of 1970, the future of this project was still not clear. The members of the CERN council still had not agreed on a site for the machine, and the international debate was continuing at the ministerial level. ”

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D. Design Studies f o r Proton Synchrotrons By 1967, three design studies had been completed for proton synchrotrons in the 40- to 60-GeV range. One, in Tokyo, had receivedinterim approval by the Japanese government. Since then, it has fallen into political difficulties and, in 1969, its future was still doubtful. A second project, in France, was in the hands of a capable group at Saclay and a third, German project was centered in the Nuclear Physics Intitute at Karlsruhe. The French and German projects were uncomfortably close to each other and questions were raised in the European community about the wisdom of such parallel expenditures. In 1969, the situation was resolved by France’s fiscal troubles; the French program has been set aside indefinitely. At Karlsruhe, much enthusiasm still remains. What will become of the German machine probably depends, t o a considerable extent, on the clarification of the 300-GeV situation. Two more speculative study programs are aimed at very high energies. At the Radio Technical Institute in Moscow a study was inspired in 1960 by an abortive attempt to organize a program for an “ intercontinental accelerator ” to have an energy in the 300- to 1000-GeV range. Discussions between Soviet and American study groups were to have been held in 1961. The Radio Technical Institute study resulted in proposals for a “cybernetic accelerator ” (7) in which information derived from the proton beam was to be used automatically to apply magnetic corrections in order to restore the beam to its correct orbit. It was claimed that this could result in important reduction of machine aperture and consequently in its cost. A I-GeV model of this machine was constructed and is now in operation, performing essentially as predicted. Since the idea of more or less automatic correction of beam position from pickup electrode information is now generally accepted and since most other design groups have used this procedure to make reductions in aperture, this scheme does not, by now, appear particularly novel. The Radio Technical Institute will terminate its model study in 1970; Mints, Director of the Institute, continues to be enthusiastic regarding the practicability of a 1000-GeV accelerator and, in 1969 ( 3 ) proposed again that consideration be given to collaboration in its construction between the Soviet Union and the United States. The American effort in the abortive 1960 and 1961 intercontinental studies was centered at Brookhaven, where also it was concluded that 1000-GeV synchrotrons are practical. Low keyed efforts have continued at Brookhaven and, by 1969, became concentrated on the use of superconducting or cryogenic magnets. As has been discussed i n Section IV, problems connected with pulsing superconducting magnets have largely been solved. Bending magnets for peak fields of 60 kG or higher are under design and should be in test operation during 1970.

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Because of the possibility that the NAL 200-GeV accelerator may be extended to 400 GeV or higher, the Brookhaven study has been extended upward toward 2000 GeV. With 60-kG magnets, a 2000-GeV synchrotron would have a radius of about 1.5 km, only 50% larger than the 200-GeV accelerator. It is hoped that this reduction in size will result in a corresponding reduction in cost. To what extent this is true will be shown by the Brookhaven studies during 1970 and 1971.

E. Proton Synchrotron Improvement Programs All proton synchrotrons seem either to be in the process of being improved or preparation and submitting of such proposals is under way. Here we mention only the two major programs now in progress. Both CERN and Brookhaven are in the midst of programs aimed at increasing the intensity of their proton synchrotrons (PS). Both are raising intensity per pulse by increasing the energy of their injectors. At Brookhaven, the present 50-MeV linac injector will be replaced by a new 200-MeV linac. At CERN, injection energy will be increased by inclusion of a booster synchrotron that raises proton energy from the present 50-MeV linac t o 800 MeV. For reasons connected with the fact that the PS beam is to be used to inject into a colliding beam system (see Section VII), the booster consists of four synchrotrons stacked one above the other and each having a radius exactly one quarter of that of the PS. This rather complicated system will be located adjacent to the PS and will be half in France and half in Switzerland (see Fig. 3). Average intensity at the two machines is to be increased by increasing the cycling rate. This calls for larger power supplies for the ring magnets and for increased power for the rf accelerating stations. In both cases, the cycling rate can be approximately doubled or, alternatively, the field can be held at its peak value for long periods while the beam is slowly extracted. Both machines now have both fast and slow extracted beams, the former referring to the process of extraction during one revolution of the proton beam, the latter to extraction during periods of tens or hundreds of milliseconds. Slow extraction is the more difficult and has advanced rather slowly; however, both machines now have available slow beams with extraction efficiencies between 80 and 90 %. At both machines and at most other high energy accelerators, computers play increasingly important roles in the control room. Beam operations of great complexity can now be monitored and controlled. The difficult process of analyzing and correcting beam-position errors is now carried on at high speed with computer assistance. More and more, the computer is in " on-line " use both in machine operation and in the experimental programs.

FIG.3. CERN geography. Legend: ( I ) SC, (2) PS, (3) North Hall, (4) South Hall, (5) East Hall, ( 6 ) neutrino area, (7) ISR ring, ( 8 ) transfer tunnels, (9) interaction regions, (10) switchyard, (1 1) West Hall, (12) BEBC buildings, (13) booster, (14) central computing.

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VI. ELECTRON ACCELERATORS

A . SLAC In January of 1967, the latest in a series of historic electron linear accelerators was brought into full operation at the Stanford Linear Accelerator Center (SLAC). This is the " two-mile accelerator " designed for initial operation at 20 GeV, later to be pushed upward toward 40 GeV by increases in rf power levels. The two-mile machine was completed on schedule and within its cost estimate. This performance was capped by the preparation of a monograph of more than 1000 pages (8) detailing the history and organization of the project and describing all important details of construction. The accelerator, like previous Stanford linacs, uses iris-loaded waveguide excited at 2856 MHz. Power is supplied by 245 klystron amplifiers each capable of a pulsed output of 24 MW. The SLAC accelerator supplies electrons of the highest energy available in any laboratory in the world. Accelerated beams pass through a beam switchyard where they can be directed to an experimental area housing three spectrometers rated at 1.6, 8, and 20 GeV, to a second experimental area designed for study of secondary particles, or to a large bubble chamber. SLAC is not resting on its laurels. Close collaboration is maintained with the High Energy Physics Laboratory at Stanford where work on superconducting linacs is far advanced (see Section 1V.B). Plans are already in preparation for converting the two-mile machine to a superconducting linac. It is expected that its energy could thus be raised from 20 to 100 GeV and its duty cycle from 0.07 to 100% at 25 GeV or 6 % at 100 GeV. A further flight of fancy involves deflecting the accelerated beam, returning it to the injection point and reaccelerating to 200 GeV. Possibly, this process could be repeated more than once to give 300 or even 400 GeV. B. Cornell

Before leaving Cornell to become the Director of the NAL, R. R. Wilson presided over the completion of a 10-GeV electron synchrotron. Preliminary tests were in progress at that time. Finally, in March of 1968, the machine operated at its full energy. To keep energy loss by synchrotron radiation within controllable limits, this machine has a very large radius of 100 m. For this reason, the peak magnetic field is only 3.3 kG; this field can be provided with rather meager magnets. The machine has no vacuum chamber. The whole magnet structure including its coils is enclosed in a stainless-steel skin, which serves as the

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vacuum chamber. The accelerator is cycled at 60 Hz as is the smaller Cambridge Electron Accelerator (CEA). It should be possible, by adding to the rf accelerating system, to raise the energy of this machine t o about 15 GeV. C. Other Electron Synchrotrons

Two machines inspired by the CEA and operating in the 6- to 7-GeV range are now in full operation. The Deutsches Electronen Synchrotron (DESY) in Hamburg was first operated in 1964. Its construction was directed by W. K. Jentschke. The other machine is in the laboratory of Alikhanian i n Erevan in Soviet Armenia. Its first operation was in the summer of 1967. Both machines are, in general, similar in concept and construction to the Cambridge accelerator. A British electron synchrotron, nicknamed “ NINA,” was brought into operation at its design energy of 4.5 GeV in December of 1966. Consideration has been given to using NINA as an injector for a much larger ring to increase energy to 15 to 20 GeV. At Bonn, an electron synchrotron for 500 MeV was started in 1953 and completed in 1958 to be the first alternating-gradient synchrotron in Europe. This was replaced in 1967 by a new electron synchrotron for 2.5 GeV. The later accelerator is distinguished by the fact that its magnet was entirely designed by a computer.

VII. STORAGE RINGSAND COLLIDING BEAMS There are many serious problems associated with the construction, operation, and use of colliding beam systems. To avoid beam loss, pressures must be below lo-’ Torr. To achieve usable collision rates, circulating currents must be of the order of amperes, yet such high circulating currents are plagued by a host of instabilities. In electron or positron storage rings, the stacking process is aided materially by radiation damping. Several electron-electron and electron-positron systems have been built and operated. As yet, however, there have been no experiments using protons. Two systems using protons are under construction, one at CERN for protonproton collisions and one at Novosibirsk for proton-antiproton collisions. In the following paragraphs we review the proton projects and then turn to electron rings in operation, under construction, and proposed.

A . The CERN Intersecting Storage Rings It has been evident for many years that colliding beam experiments are, in principle, possible and that probably they should be undertaken with beams either from the 33-GeV AGS at Brookhaven or from 28-GeV CERN PS.

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Two colliding beams at energies of the order of 30 GeV will make available for particle production as much energy as could be yielded from 2000-GeV protons impinging on protons at rest. After much soul searching at Brookhaven and CERN, the decision was made to proceed with construction at CERN. At Brookhaven, it was felt that duplication of the CERN effort was not advisable. Construction was approved at CERN in 1965 and the project is now well advanced. Completion is expected at the end of 1971. The intersecting storage ring (ISR) system shown in Fig. 3 consists of two concentric rings distorted in such a fashion that they intersect at eight points. They are 300 m in diameter, 100 m larger than the PS. At the intersection points, the two counter-rotating protons beams cross each other at an angle of 15”. It is expected that circulating beams of 20 A of protons can be built up by multiturn injection from the PS. Experimentation with the ISR is complicated by the fact that the primary interactions cannot be observed directly ; information must be derived from the secondary particles. If intermediate bosons or quarks are produced, however, they should readily be detectable. Intense preparations are in progress so that experimentation can begin as soon as the machine is in operation. B. The Novosibirsk Proton-Antiproton Ring

In many ways, collisions between protons and antiprotons are even more interesting than proton-proton collisions. Antiprotons are produced in some quantity by present accelerators in the 30-GeV range, but their angular distributions are sufficiently broad that it has not appeared possible to produce a sufficiently intense beam for experimentation in a colliding beam system. However, Professor Budker and his associates at Novosibirsk have pointed out the fact that, if coaxial beams of antiprotons and electrons of the same velocity are maintained, the radial deviations of the antiproton beam will gradually be transferred to the electrons and a concentrated antiproton beam can result. This process is now referred to as “electron cooling.” It will probably be tested in one of the smaller machines at Novosibirsk. There seems to be no theoretical reason to doubt that it will be successful. Construction has been started at Novosibirsk on a ring for studies of 25-GeV colliding beams of protons and antiprotons. Protons will initially be accelerated in an injector synchrotron to about 3 GeV. Several pulses from the injector will build up the beam in the main 25-GeV ring to lOI4 protons. This beam is accelerated in the main ring to 25 GeV and allowed to strike a target. The antiprotons produced are transferred to a third ring about the same size as the injector synchrotron. In the straight sections of this ring they will be electron cooled, transferred back to the main ring and there accelerated

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simultaneously with a new charge of protons. It is hoped that collisions can be observed between circulating beams of l O I 4 protons and 10’ antiprotons. It is hoped that construction of this very ingenious system can be completed during 1971. It is known as VAPP-4.

C. Colliding Beam Studies at the National Accelerator Laboratory At the NAL it was felt that a study should be made of possible colliding beam storage rings before the 200-GeV accelerator was too far advanced, since future colliding-beam programs might call for modifications in the machine itself. This study took place during the summer and fall of 1968 and is described in a report published at the end of 1968 (9). It was concluded that storage rings for the full energy of the machine would be prohibitively expensive and the design study concentrated on 100-100-GeV colliding beams. The storage rings could be located in a convenient location adjacent t o the main ring, as indicated in Fig. 2. It is hoped that, at some later date, developments of superconducting magnets can lead to replacement of the storage ring magnets by superconducting magnets yielding much higher fields and that, by this means, the colliding beam energies can be raised to 200 GeV or higher. D . Electron Storage Rings

Almost all colliding-beam systems involving electrons are designed for studies of collisions between electrons and positrons. At Orsay, France, a 500-MeV ring (known as “ACO”) has been operating since 1966. It is supplied with electrons and positrons by the Orsay linear accelerator. At the Frascati laboratory in Italy, a larger ring for energies up to 1.5 GeV (called “ADONE”) came into operation at the end of 1967. It is capable of maintaining circulating beams of the order of I A. At Novosibirsk, VEPP-2, a 700-MeV ring, has been in use since 1966. It is about to be superseded by VEPP-3, a 3.5-GeV ring. It is this ring whose straight sections will be used for a test of electron cooling of a beam of protons. To a visitor to the Soviet Union, the laboratory in Novosibirsk is one of the most interesting. Some of the most ingenious and resourceful accelerator physicists in the Soviet Union have gathered there and the visitor is continually amazed at the unconventional techniques that have been invented and made to work. The laboratory has one deficiency, however. When confronted with the success of many of their projects, the laboratory managers have realized that they do not have enough high energy physicists to utilize the working machines. To rectify this situation, they have made an agreement with Japan t o allow Japanese physicists to come to Novosibirsk and to help to organize an experimental program, and several Japanese physicists already are in residence.

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At the DESY electron synchrotron in Hamburg, colliding beam rings for electrons and positrons were authorized in 1969 and should be operating by 1973. In this case, the electrons and positrons will not be stored in a single ring; an electron ring and a separate positron ring will be located one above the other. The primary reason for this is the observation that interactions between the two beams can often lead to instabilities that blow up the minor diameter of one or both beams. In the Hamburg system, the beams will collide at only one point. The scheme has the further advantage that it can also be used to study electron-electron collisions. It is expected that 1 A of circulating current can be stored at 3 GeV in each ring. At 1.5 GeV it should be possible to store 10 A in each ring. There are no colliding beam systems operating in the US. In 1964 proposals for electron-positron colliding beams of about 3-GeV energy were submitted by the CEA staff and by the staff of the SLAC. A panel set up by the Atomic Energy Commission recommended that the SLAC proposal be implemented, but since then, neither proposal has received support. In the meantime, the CEA group has invented a clever procedure for modifying the electron accelerator so that it can be used as a storage ring for electrons and positrons. To increase the probability of collisions, a bypass has been introduced at one point in the ring, including focusing lenses that reduce the beam cross-section at the point where collisions will be observed. The vacuum chambers in the remainder of the ring have been replaced with ceramic sections in which the necessary high vacuum can be attained. It is expected that the system can be tested during 1970. Circulating beams of electrons and positrons, provided by a new combination of injector linacs, should reach intensities of about 100 mA in each beam. The SLAC proposal was modified and resubmitted in 1969. It calls for two, separate, somewhat distorted rings intersecting at two points. The maximum energy is now 2 GeV at which it is expected that circulating beams of 500 mA can be maintained. The project now bears the name “SPEAR” (Stanford positron-electron asymmetric rings). An unusual storage ring project is to be found at the Physical Sciences Laboratory of the University of Wisconsin (formerly the MURA laboratory). Here, a 240-MeV electron storage ring has been in operation since 1967. Its injector is a 50-MeV FFAG accelerator built by the MURA group to study the FFAG principle. Colliding beam experiments have never been planned for this storage ring. Originally it was built to study instabilities and other phenomena in storage rings. Recently, however, much interest has arisen in possible applications of the spectrum of synchrotron radiation emitted by the circulating electrons. The ultraviolet region of the spectrum has found a number of applications in research programs on solid state physics and atomic physics. The machine now operates continuously as a research tool in these programs.

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VIII. CYCLOTRONS The old-fashioned cyclotron of the type built by Lawrence and Livingston has now been twice superseded. The postwar invention of the synchrocyclotron led to the construction of a number of machines, many of which have had very honorable histories. Notable among these are the 180-in. machine at Berkeley, the 600-MeV SC at CERN, and the 680-MeV synchrocyclotron at Dubna. In November of 1967, the largest synchrocyclotron ever built came into operation at Gatchina, near Leningrad. The design energy of this machine is 1 GeV; its magnet weighs 8500 tons. Probably the Gatchina synchrocyclotron will be the last major synchrocyclotron ever to be built, since the present trend is toward isochronous cyclotrons that can operate continuously, rather than in the pulsed fashion of the synchrocyclotron. These cyclotrons depend for focusing on azimuthal variation of the magnetic field, which yields focusing in the same fashion as is used in alternating gradient systems. With this focusing, the average field need not fall off with radius, as is necessary in conventional cyclotrons, and it is now possible to increase field with radius until the period of an orbit becomes independent of radius. This can be extended into the relativistic range of energies and appears practical for energies up to about 1 GeV and possibly to still higher energies.

, ,

FIG.4. Indiana cyclotron

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This principle has been applied in a rather extreme design illustrated in Fig. 4. This design has been evolved at the University of Indiana by Rickey, Sampson, and their associates. The magnet has been separated into four quite separate sectors, narrow enough so that only about half of the orbit lies between magnet poles. The space between poles is now available for injection and extraction of beams and, if desired, for localized rf accelerating systems. Injection is from another cyclotron which is a scaled-down version of the main machine. Construction of this machine was well advanced at the University of Indiana by the end of 1969. The Indiana cyclotron has aroused great enthusiasm among cyclotron designers. In particular, many of the proposals for heavy ion accelerators include either four-sector or six-sector versions of the Indiana design. Probably most cyclotrons to be built in the near future will be of this type. IX. MESONFACTORIES

In 1965, four groups in the US were actively studying machines of high intensity to produce proton beams in the range between 500 MeV and 1 GeV for the purpose of generating intense beams of pions, muons, and other secondary particles. These were inspired by intense interest in the many interesting experiments in meson physics and nuclear structure that could be made possible, provided that beams of sufficient intensity could be produced. At Yale University, a linear accelerator was proposed; later this became a joint Yale-Brookhaven project. A similar linac was under study at Los Alamos. At Oak Ridge, two possibilities excited interest, a conventional proton cyclotron and a “ separated-orbit cyclotron,” a cyclotron with separated magnet sectors and so high a rate of acceleration that the machine became a sort of spiral linear accelerator. At the University of California at Los Angeles, negative hydrogen ions were to be accelerated in a cyclotron; their charge would be reversed in a stripping foil and they then would be extracted easily. An “Ad Hoc Panel on Meson Factories ” chaired by Professor Bethe had recommended to the Office of Science and Technology in 1964 that one, but only one, meson factory be built. Finally, in July of 1967, the “Los Alamos Meson Physics Facility ” (LAMPF) was chosen and authorized. Construction began almost immediately and, in 1969, was about half completed. LAMPF will be a drift tube linac from its preinjector energy of 750 keV to 100 MeV. At 100 MeV, the beam will pass from the 200-MHz drift tube section into an 800-MHz iris-loaded waveguide-system in which it will be accelerated to 800 MeV. Average current at 800 MeV is to be 1 mA. The duty cycle will be 6%, possibly to be increased later to 12%. Both sections of the accelerator have new and ingenious modifications from previous linear

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JOHN P. BLEWETT

accelerators; unfortunately these features are somewhat too esoteric for description in this brief review. Completion is scheduled for 1972. While the complex situation in the US was being resolved, two meson factories were started elsewhere in the world. At the Eidgenossische Technische Hochschule in Zurich, studies on possible meson factories led to the choice of a complex of two isochronous cyclotrons to yield a beam of 100 FA of 500-MeV protons. The injector cyclotron is a four-sector machine which yields 70-MeV protons. With so high an injection energy, it is possible to leave out the central section of the main cyclotron since the protons are injected on an orbit of radius 2.1 m. Acceleration brings the protons to a final energy of slightly more than 500 MeV on an orbit of radius 4.5 m. The cyclotron magnet consists of eight completely separate sectors between which there is room for four rf accelerating cavities each capable of adding 500 keV to the proton energy. This makes for a clear separation between orbits and allows the extraction efficiency to approach 100%. It is expected that other ions (deuterons, alpha particles, etc.) will be used in this machine. Heavy ions can be accelerated to 10 MeV/nucleon. A site for this machine has been chosen at Villigen, near Zurich. Here will arise the “ Schweitzerisches Institut fur Nuklearforschung ” which has been abbreviated to “SIN.” Construction of the accelerator will begin in 1971 and it is expected that the machine will be finished by 1974. The second meson factory outside of the US is a joint project of four western-Canadian universities. Initially, three universities were involved, the University of British Columbia, the University of Victoria, and Simon Fraser University, and the project was named the “ TRI-University Meson Facility ” or “ T R I U M F ” for short. Later, the University of Alberta joined the three British Columbia universities, but the original name has been retained. TRIUMF will be an isochronous, sector-focused cyclotron with six sectors. In it, negative hydrogen ions will be accelerated; they will be stripped at extraction. As protons, they will be deflected outward by the magnetic field. This is the same scheme as was proposed by Richardson for the nonapproved cyclotron meson factory at Los Angeles. Indeed, Richardson is one of the more important consultants on the TRIUMF program. Since negative hydrogen ions tend to lose their electrons when deflected by too strong a magnetic field, it has been necessary to reduce the field to less than 6 kG. For 500-MeV protons the orbit diameter in this field is about 16 m; the total magnet weight is about 3500 tons. The cyclotron will be built on the campus of the University of British Columbia in Vancouver. Completion is expected to be in 1974. X. R.I.P. During recent years, a number of older machines have been turned off and several proposed projects have been canceled. Four of these are particu-

RECENT ADVANCES IN PARTICLE ACCELERATORS

255

larly notable. On December 30, 1966, the Brookhaven Cosmotron, the first machine to accelerate charged particles to energies over 1 GeV, was shut down after a productive lifetime of 14 yr. The machine has now been completely dismantled. In January of 1969, the 68-MeV proton linear accelerator at the University of Minnesota was closed down. its lifetime also was 14 yr. Until the 100-MeV linac injector at Serpukhov began operating in 1967, the Minnesota linac provided protons of the highest energy attained in a linear accelerator. The other two machines died before they were born. Both were bold extrapolations of accelerator technology, and it seems sad that they failed to be supported. First is the “ Omnitron of the University of California. This was a combination of a synchrotron and a storage ring to be used in acceleration of heavy ions. By transferring ions back and forth between the synchrotron and the storage ring passing through appropriate strippers, heavy ions could have been accelerated to several hundred million electron volts per nucleon. The Omnitron proposal was withdrawn in 1968. The second unsuccessful project was the “ Intense Neutron Generator ” (ING) proposed by the Chalk River Laboratory in Canada. It was to be a long linear accelerator to be used in continuous operation accelerating a current of 65 mA of protons to l GeV. Its target was to be a stream of molten lead and bismuth. The neutron flux derived from this process would have been higher than that yet achieved at any reactor. Unfortunately, it would have been very expensive (over $150 million) and, although it was approved by the Canadian Science Council, the Canadian government canceled it in 1968. These few sad demises fade into insignificance, however, when viewed against the many new accelerator projects described in the preceding sections of this summary. It would appear that acceleration of charged particles is still a vigorous and exciting endeavor. Much remains to be done in the study of the nucleus and its components. ”

XI. CONCLUSION

No claim for completeness can be made for the preceding presentation of recent highlights in accelerator development. In the last five years, thousands of pages of proceedings of accelerator conferences have appeared describing interesting developments far too numerous for inclusion in so brief a summary as this one. For example, no mention has been made of the conversion of Columbia’s Nevis synchrocyclotron to higher energy and intensity operation. Also neglected or barely mentioned are the new linac injector and pressurized preinjector for the 3-GeV proton synchrotron at Saclay, the high intensity electron linac at MIT, the splendid new tandem accelerators developed by High Voltage Engineering, the Dynamitron accelerators developed

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by Radiation Dynamics, various high current electron accelerators for use as flash X-ray tubes, and many proposals for modification and improvement of existing accelerators. The reader who wishes more detailed information is referred to the proceedings of the International Conferences on High Energy Accelerators held in 1965 ( I O ) , 1967 ( I ) , and 1969 (3), and to the proceedings of the American Accelerator Conferences held in 1965 ( I I ) , 1967 (12), and 1969 (13). A less technical but very readable source of information about current accelerator developments is the CERN Courier published monthly in French and English by CERN’s Public Information Office.

REFERENCES 1. V. I. Veksler er al., in “Proceedings of the Sixth International Conference on High Energy Accelerators, 1967” (R. A. Mack, ed.), p. 289, CEAL-2000, Cambridge Electron Accelerator, Cambridge, Massachusetts, 1967. (Available from the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia.) 2. “ Symposium on Electron Ring Accelerators,” UCRL-18103, Lawrence Radiation Laboratory, Berkeley, California, 1968. (Available from Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia.) 3. “ Proceedings of the Seventh International Conference on High Energy Accelerators, 1969,” Erevan, USSR (to be published). 4 . G. T. Seaborg, Ann. Rev.Nucl. Sci. 18, 53 (1968). 5. A. G. Prodell and H. Hahn, eds., “Proceedings of the 1968 Summer Study on Superconducting Devices and Accelerators,” BNL 5015 5 , Brookhaven National Laboratory, Upton, New York. (Available from Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia.) 6 . G. T. Danby, in “Proceedings of the 1968 Summer Study on Superconducting Devices and Accelerators ” (A. G. Prodell and H. Hahn, eds.), p. 1 115. BNL 501 5 5 , Brookhaven National Laboratory, Upton, New York. (Available from Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia.) 7. A. A. Vasil’ev, ed., “ 1000 GeV Cybernetic Proton Accelerator.” English translation issued by the US Atomic Energy Commission, AEC-tr-6949, 1968, and available from Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia. 8. R. B. Neal, ed., “The Stanford Two-Mile Accelerator.” Benjamin, New York, 1968. 9. “ Design Study for Proton-Proton Colliding-Beam Storage Rings for the National Accelerator Laboratory,” National Accelerator Laboratory, Batavia, Illinois, 1968. 10. M. Grilli, ed., “Proceedings of the Fifth International Conference on High Energy Accelerators, Frascati, 1965,” Comitato Nazionale per I’Energia Nucleare, Rome, 1966. 11. IEEE Trans. Nucl. Sci. NS-12 No. 3 (1965). 12. IEEE Trans. Nucl. Sci. NS-14 No. 3 (1967). 13. IEEE Trans. Nitcl. Sci. NS-16 No. 3 (1969).

Broadened Energy Distributions in Electron Beams BOD0 ZIMMERMANN Krupp Research and Development Center, Essen, Germany

................ 11. Energy Spread and Internal Energy . . . . . . . . . . . . . 111. Estimating the Energy Spread. ......................... ............................... IV. Broadened Energy Distributions. . . . A. General Considerations ............. .................... B. Broadened Normal Energy Distribution. . . . . . . . . . . . . . . ........

261 261 211

VI. Absolute Calculation of the Energy Spread . .

217

......................

260

. . . . . . . . . . 211

B. Divergent Beams. . . . . . . . . . . . . . D. Crossover.. A. B. C. D. E.

....

............................

Energy Shift. ..................... Internal Energy at Equilibrium. . . . . . . . . . . . . . . Energy Spread and Entropy. . . . . . . . . . . . . . . . . . Boltzmann Equation. . . . . . . . . Relativistic Beam Energies. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . 297 . . . . . . . . . . . . . . . 291

C. Electron Beams in Magnetic Fields ........................ Appendix. ..................... .................................. References. ........ .................................

308 311

I. INTRODUCTION The calculation of velocity o r energy distributions in electron beams is a very complicated problem. First of all there is a great variety of geometrical configurations and electrical fields which influence the velocity distributions. I n principle, the electron paths and hence the velocity distribution can be calculated if the electric field is known. But apart from the tedious calculation work, there are still further difficulties. Even if we restrict ourselves t o thermionically emitted electron beams, the velocity distribution of the emitted electrons is still not quite clear. 251

258

BOD0 ZIMMERMANN

Another difficulty which enters the calculation is the electric field which is produced by the beam itself-the space-charge field. The space-charge field can be divided into two parts, a smooth field which is produced by the equivalent smeared-out charge distribution and the fluctuating part which comes from the discrete nature of the charges. In almost all theoretical investigations only the smooth part of the space-charge field was taken into consideration. However, in 1953 Mott-Smith (1) calculated a change of electron temperature in electron beams as a result of the Coulomb interaction between individual electrons, and thus considered the fluctuating part. The change of temperature, however, was so small that he did not ascribe any significance to it; so did Lindsay (2) and Simpson and Kuyatt (3). The Coulomb interaction of the electrons at a mean distance of about cm in dense electron beams is indeed small. But Ash and Gabor ( 4 ) in 1955 proved experimentally the presence of Coulomb interaction between electrons and positive ions in a plasma. They found it to be consistent with a great number of calculations made by Wilson (5), Thomas (6), Langmuir and Jones (7), Landau (8), Druyvestein (9), Chapman and Cowling (lo),and Chandrasekhar (11). The interaction of one charged particle with another particle of an ensemble of particles with charge + e can be characterized by the mean free path A. According to the above-mentioned calculations, A is given by

where q is the particle density. The constant a varies between 1 and 3, depending on author. The expression for 2 is derived by considering only binary collisions and cutting off the Coulomb force for distances greater than the mean distance of the electrons for noncompensated ensembles and at the Debye radius for charge-compensated ensembles. Since this distance appears only in the argument of the logarithm, the result is not particularly affected by this procedure. In 1968, Zimmermann (12) proved, proceeding on ideas of Hartwig and Ulmer (13), the Coulomb interaction between individual electrons to be responsible for the experimentally found great energy spreads in electron beams of high current densities. In spite of the small change in electron temperature, energy spread in the laboratory system suffers a considerable change owing to a certain magnification which depends on the mean energy. Rejecting the relaxation concept of Hartwig and Ulmer but considering collisions between individual electrons, as they do, Loeffler (14) derived the energy spread generated in crossovers. Nevertheless his result is in agreement with the calculations of Zimmermann. The first systematic investigation on anomalously broad energy spreads

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

259

was made by Boersch (15). In beams of 30-keV mean energy he found that, as the beam current was increased, the energy distribution changed from Maxwellian to a more symmetrical distribution and that the energy spread increased from about 0.2 to about 2eV. This effect has been confirmed by Haberstroh (26) and Dietrich (27). Hartwig and Ulmer (23), and Ulmer and Zimmermann (28) observed the effect at 200 to 1OOOeV beam energy and Simpson and Kuyatt (3) at 2 to 10 eV. In the meantime, many other workers have also found the energy distribution to broaden in beams of high currentdensity. We refer to the experimental results in Section VIII. The experimental results obtained by Boersch induced other workers to offer explanatory statements. Fack (19) considered interaction of the electrons with statistical longitudinal space-charge oscillations and found the broadening of the energy spread to be proportional to the square root of the current density. Veith (20) explained the effect, which he observed in magnetically confined electron beams, by statistical fluctuations of the current density; Epsztein (21) by transverse instabilities in the electron beam. Lenz (22) proposed that a weak bond of the electrons be assumed at a point of high space-charge density. Schiske (23) considered the fluctuations of the electric microfields, which are distributed according to the Holtsmark distribution, and also took into account the beam boundary. He found the broadening of the energy spread to be proportional to j 5 I 9 , where j is the current density. Ulmer held a relaxation process to be responsible for the broadening effect. Although the derivation of his formula, according to which the energy broadening is proportional to j 1 ’ 3 ,is theoretically incorrect, it proves to be in very good agreement with experiment. In agreement with Mott-Smith, Zimmermann, and Loeffler, the relation (AE)’

= (A&)’

+ C‘j

(2)

for the energy spread AE can be derived theoretically, taking into consideration binary encounters of the electrons. The magnitude C’ depends on the beam parameters, and to a small extent on j and is tightly related to the mean free path 1.The observed energy-spreads extending up to about 5 eV are in quantitative agreement with the calculations of Loeffler and Zimmermann. In this article, the theory leading to Eq. (2) and to the absolute value of C’ is presented. We begin with the relation between energy spread and internal energy, which is a key to understanding why the weak Coulomb interaction can cause such drastic energy broadenings as were observed (Section TI). In Section I11 relation (2) is derived, except for the factor C’, by very general considerations. In Section IV general considerations are made as to the shape of the broadened distributions and special distributions are calculated. On the basis of these results, the relations between several energy widths of the same distributions can be stated (Section V). In Section VI the

260

B O D 0 ZIMMERMANN

absolute values of the energy spread for several beam geometries are calculated. The comparison with experimental results is made in Section VIII.

11. ENERGY SPREAD AND INTERNAL ENERGY When we consider an ensemble of particles as a whole, we are mainly interested in mean values. In this article we consider only mean values, which are derived from the velocities of the particles. In general, the velocities are found t o be scattered about the mean velocity (v). For every direction we define for a given ensemble the velocity spreads according to (Au~)’= (

(3)

( ~ i- ( ~ i ) ) ’ ) .

The velocity component ui is the projection of the particle velocity v on the direction i. From Eq. (3), we derive the corresponding internal kinetic energies, which are also known as peculiar energies, ui = fm(Aui)’.

(4)

We further define to each direction i the energy

E 1. = ‘mu2 2 1

(5)

and the energy spreads AEi according to (AEi)’ = ( ( E i - (Ei))’).

(6)

The energy spread and the internal energy of a single direction can differ by an enormous factor. To demonstrate this, we at first prove the relation ( A E ~ z) ~~ ( E , ) u ~ , (7) which is valid for ( E i ) % u i . Putting 6ui = ui - ( u i ) gives us the desired relation as follows: (AEJ’ = $m2((u,”> - (ui2>’)

+

= $m2{(((ui) - (((ui> z m2(ui)’ . ( ( 6 ~ ~ ) ‘ )

+SU~)~)’}

% 4(E,)u,. (8) The physical meaning of (7) becomes evident if we derive the differential from (5) : d E i = mui d v i . (9) This may be interpreted as the relation between energy and velocity difference of two particles with mean velocity u i. Squaring (9) we obtain Eq. (7). In the following, we use a Cartesian coordinate system, in which

(u,>

= (0,) = 0

(10)

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

26 1

and omit the subscript z, when we think of magnitudes corresponding to the z-direction :

Then we have’

In agreement with our suppression of the subscript z, we mean by the simple term “internal energy” the longitudinal internal energy u, = u and by “energy spread” the longitudinal energy spread AE, = AE. From Eq. (7) we see that AE is larger than u by the factor 4 ( E ) / A E , which is large when compared with one in those cases that we are interested in. Furthermore, we see that for constant ( E ) even a small change du of the internal energy produces a considerable change d(AE) in the energy spread. This follows immediately from (7) : d(AE) = 2 ( ( E ) / A E ) du S du.

(13)

We consider now the process which leads to a change of u in the drift region of the electron beam. We assume that the electron beam has been accelerated relaxation-free by a homogeneous electric field and is then drifting in the subsequent field-free space. We call an acceleration “ relaxation-free,” if interaction between the electrons can be neglected during acceleration. The ideal case, therefore, is acceleration by a potential step. For every point in the beam an energy spread can be defined. Depending on the exact definition of AE it remains exactly constant during acceleration or tends to a finite, nonzero limit for large mean kinetic energies. We will first consider the consequences of this fact and prove it at the end of this section. Since AE is limited, the internal energy, as a consequence of (7), tends toward zero as the mean energy increases. On the contrary to u the transverse internal energy u, + uy is not affected by the acceleration process. Thus, the equipartition of the internal energies among the transverse and longitudinal degrees of freedom is destroyed. This is illustrated in Fig. 1. At the cathode (a) there is only a small inhomogeneity, whereas at the anode (b) only a small part of the whole internal energy is contained in the longitudinal degree of freedom. Therefore, in the subsequent drift region (c), the Coulomb interaction between the electrons introduces the restoration of the equipartition by transferring energy from the transverse degrees of freedom into the If the transverse velocities ux and u, are Maxwell-distributed, we further have AE,

AE,

= d 2 u X= d 2 u Y .

=

262

BOD0 ZIMMERMANN

(aI

(b)

(C

1

(d)

FIG.1 . Destruction of the approximate equipartition of the internal energy at the cathode in the acceleration region and restoration of equipartition by relaxation in the subsequent field-free space, thus increasing the energy spread. The “components ” of the internal energy, corresponding to different velocity directions, are plotted in apolar diagram. (a) at the cathode, (b) at the anode, (c) in the drift region, and (d) in the drift region after relaxation.

longitudinal degree of freedom. Thus, u increases and because of (7) AE also increases, but by a much greater amount. This great increase of AE can be determined easily by experiment, because it is approximately proportional to the width of the energy distribution in the laboratory system. We examine now the behavior of AE during the relaxation-free acceleration of the electrons. As mentioned above, this behavior depends on the exact definition of AE. If we attach AE to an ensemble of given particles, then AE will be the same before and after the acceleration, because each electron will gain the same energy. Also, if we define AE at a point in a continuous beam as the energy spread of those electrons that pass through the plane perpendicular to the beam axis during a given time, AE will be exactly constant along the beam. The proof of this statement requires a short calculation, which we begin by denoting the number of electrons within the velocity interval u ... u + du and the volume element F dz ( F being the cross section of the beam) at the cathode by dEo = f o ( E ) du dz.

(14)

Denoting the potential energy of the electrons at z by -E,(z), the distribution, according to Liouville’s theorem, is given by dn, = f o ( E - E,) du dz = ( 2 ~ E ) - ” ~ j b (E E,) dE dz

(15)

for E > E , . During the time dt

dN,, = (2/m)fO(E- E,) dE dt

(16)

263

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

+

electrons of the energy E * * E dE are passing the cross section at z . The distribution dNo/dE dt is called the “ current distribution.” The energy spread AEj, related to this current distribution, is then given by

(AE’)’

= Jm

( E - ( E ) j ) y o ( E - E,) d E . (JE:fb(E - E,) d E )

-1

Eo

= ((E

- El)’):

with

El

= (E)’ - Eo

The symbol (. . .)f means the average with respect to the current distribution at the cathode. Observing the relation

=

loW Ef,(E) dE . (Jomro(E)d E ) - l = ( E ) f

(19)

we obtain

(AE’)2 = ( ( E - (E)f)’)f

= (AEf)’.

(20)

Despite this beautiful feature of AEj, we shall not use it any further. Instead, we define the local energy spread AE and the local internal energy u, which are attached to the electrons being simultaneously present in a volume element. These magnitudes are more adequate to our problem, because thermodynamical equilibrium means equipartition of the local internal energies. The local energy spread is not constant along the acceleration region. It tends, however, towards the current-related energy spread AEj for large values of E , , as a consequence of the relation

(g(E))’ = (g(E)E”’)/( El”).

(21)

denotes hereafter the average with respect to particles The symbol simultaneously present in a volume element. For sufficiently narrow functions g(E) the right-hand side of (21) tends toward ( g ( E ) ) for a large value of E , . Thus we have AE-+AE’=AEf for E o + 0 3 ( . . a )

u+--

1 (AE~)’ 4 (E)

for E ,

-,03.

264

BOD0 ZIMMERMANN

111. ESTIMATING THE ENERGY SPREAD In this section, we shall determine to what extent energy spread depends on the beam parameters current density, current, mean energy, cathode temperature, beam length, and geometry, excluding the factor that determines the absolute value of the energy spread (24), which will be determined in Section VI. The fundamental experimental device is shown in Fig. 2. Between cathode K and anode A, an electric field accelerates the emitted electrons, which are then drifting up to the electrode A, in a field-free space before being decelerated to the cathode potential of cathode K,. For sufficiently high currentdensities, the energy spread AE' at K1 exceeds the energy spread AEc at the cathode. To give an example, we assume a cathode temperature of T = 2300°K equivalent to AEc z kT z 0.2 eV, an acceleration energy of E , z ( E ) = 1 keV, and a current density of j z 5 A/cm2. For these values, there has been measured an energy spread of AE' z 2.5 eV, which corresponds t o u - u, z 0.005 eV.

FIG.2. Sketch of the experimental setups at which energy broadenings were observed. K, cathode; A-A1, drift region; A1-KI, deceleration region. AE, =: AEo < A E =: AE'. u, ug < u << u'.

*

Let us first assume that the electrons are moving in a fictitious, ideally reflecting tube, so that the current density remains constant along the beam. In the drift region, transverse internal energy is transferred into the longitudinal degree of freedom. The exponential function describing this relaxation process can be replaced for short times by its linear approximation. Thus at a stationary point in the reference system that moves with the mean beamvelocity we have for the internal energy u

z uo

+ u,

t/z,

(23)

where u, is the (longitudinal) internal energy at the end of the relaxation process. Its value is given by the simple fact that 324, is equal to the whole internal energy u,, + uyo+ uo z ux0+ uyo at the anode:

Although the real process is not quite so simple, relation (24) is confirmed by more detailed consideration in Section VII. Approximation (23) is sufficient

BROADENED ENERGY DISTRIBUTIONS I N ELECTRON BEAMS

265

to describe all previous experiments, because z is found to be large when compared with the mean flying time t of the electrons. This means that the investigated beams are too short for the relaxation to be attained. We formulate the estimate of the relaxation time in terms of field fluctuations. These may be caused by collective and irregular motions of the electrons. We d o not consider the collective motions that give rise to spacecharge waves. Therefore the fields are, in the absence of space-charge waves, distributed according to the Holtsmark distribution. The correlation time zK and the mean amplitude F of the fluctuations can be expressed in terms of the mean distance d and thermal velocity uT of the electrons: zK = d/uT, F z e/d. (25) The correlation length is about equal to the mean distance d. Let us designate the momentum transferred to an electron during the ith fluctuation as 6 p i . Then, after several fluctuations, the momentum change scatters within the margins - Ap * * . Ap, where

+

Averaging is performed over several electrons and, therefore, does not depend on a special electron. In the expression on the right, we suppress the mixed terms (6pi6 p j ) because 6pi have alternating signs and are not correlated. Thus we obtain We simplify this expression further by introducing the average number of fluctuations n, to which an electron is submitted during the time t and have ( A P )z ~ n ((6Pi>'>,

(28)

where n = t/sK. Since, in most cases, 6piare small when compared with the transverse momentum components of the electrons, we have

where the ci and ( c i 2 > are numbers of magnitude 1. Substituting (29) in (28) we have finally ( A P )= ~(~,~)e~tu;'d-~.

(30)

Thus ( A ~ ) ~ / 2can r n be interpreted as the increase of the (longitudinal) internal energy. Comparing (30) with (23) we can, therefore, estimate the relaxation time as m2 z=--' uT3d3. 3 (ci2>e4

266

BOD0 ZIMMERMANN

Note that the relaxation time z is not identical with the correlation time zK .2 Introducing the current densityj, the length z of the drifting region, a n d the cathode temperature T by the relations 21rm j z _t - -.d3-

3e

(E)’

1 2

- mvT2 = k T

we obtain from (23) and (7) 7c(ciz)e4

u=uo+-.-,

(AE)’ = (AE,)’

jz

(33)

3e

+ Cjz(kT)-’/’,

(34)

where C = 47ce4(ci’)m1/’(18e2)-’/’ is calculated in Section VI and found to be not constant but depending to a small extent on the beam parameters. This arises from the divergent behavior of the integrals including the Coulomb interaction. The value of ( c i ’ ) can be estimated from the commonly known mean free path 1,which is given in (I), by the relation A = zvT and (31). In agreement with the exact result of Mott-Smith (I), we thus obtain

( c i 2 ) = 3.78 ln(kTd/2.22e2).

(35)

Unfortunately, the application of Eq. (35) is very restricted because beams of constant current density are not often used. More often found are diverging beams a n d crossovers. A n unfocused beam diverges in a field-free space in consequence of the transverse thermal velocities with a semiangle CI of about CI = (kT)’/’(E)-’/’. Therefore, after an effective length zeff the particle density is so small that further encounters take place only rarely. If 2R is the diameter of the beam at the anode, z e f fis proportional to Rcc. For reference we give here the result derived later : Z,fr z 0.1 32R(E)’/Z(kT)-1i2. (36) Thus, we have with (34) for the divergent beam

(AE)’

= (AE,)’

+ cdJ’R(E)’’2(kT)-’

(37)

with c d % 0.132C, valid for z > Z , f f . The effective length of a crossover of diameter 2r0 and semiangle ct is zelf= 2 . ro/cr. The transverse internal energy is ( E ) cr2 and must be inserted in Eq. (34) instead of k T , in order to derive the energy spread at a distance greater than r,/a from the smallest cross section. It is then given by

-

(AE)’ = (AE,)’

+ 2C,jro(E)-’iZ~-2,

This difference had not been recognized in references (13),( I @ , and (25).

(38)

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

267

where AE, is the energy spread at a distance of at least ro/abefore the smallest cross section. The exact calculation of C , in this case has been carried out by Loeffler. We review his paper in Section V1.D. The relations (34), (37), and (38) can be derived from the unique equation

+

( A Q 2 = (AE0)2 j C ( u , , ) - ’ / 2 j ( z )dz,

(39)

where ut, is the transverse internal energy. Although the validity of this equation is not proven for general cases, it may help to estimate the energy spread to be expected in complex beam geometries. We conclude this section by indicating how Hartwig and Ulmer (13) derived their relation, according to which AE - AE, should be proportional t ~ j ’ ’ First ~ . a linear relation corresponding to (23) was assumed to apply to the energy spread AE instead for u. Secondly z was confused with zK. We mention this here, because experiments are, as we will see, in very good agreement with the relation of Hartwig and Ulmer. The following section gives, however, further theoretical support for the relation (23).

IV. BROADENED ENERGY DISTRIBUTIONS A . General Considerations

We begin with the definition of the energy distribution and the velocity distribution. The energy distribution is that function of v or E , the product of which with dE gives the number of particles in the energy interval E * . . E dE. The velocity distribution is, however, that function, the product of which with dv gives the number of particles in the velocity interval v * * v dv. Nevertheless, these two functions may be the same except for a multiplicative constant. This occurs because we failed to define the particles we are considering. The velocity distribution of the particles which are simultaneously present in a volume element may be denoted by

+

+

dn c c f ( E ) du. (40) It does not matter whether E or v is the argument off. Then the velocity distribution of those particles which during a given lapse of time pass a plane perpendicular to the beam axis is given by the function f ( E ) v : dn‘ f~ { f ( E ) u } du .;f(E) dE. (41) Because v dv = dEjm the function f ( E ) is the energy distribution of the just defined particles. Thus, we have seen that the local velocity distribution is identical with the “ current-related ” energy distribution. According to the just assumed point of view we denote, therefore, the function f ( E ) either by velocity distribution or energy distribution.

268

BOD0 ZIMMERMANN

Calculating the energy distribution is important for two reasons. First, it is a result which can be compared with the experimentally obtained curves, and secondly, the energy spread AE of the distribution can be related to the usually communicated energy widths AE,,, and AE,,, which can be obtained more easily from the measured distributions than AE. These relations prove to depend on the shape of the distributions. Independent of the initial distribution at the anode we expect at complete relaxation a drifting Maxwellian distribution. In addition we will show in this section that in a beam of constant current density the broadened distribution is a convolution of the initial distribution at the anode and a drifting Maxwellian distribution. This relation is valid approximately also for diverging beams. We denote a point in the beam by its distance z from the anode plane and its distance a from the beam axis. To every point ( z , a) we attach three local velocity distributions, namely: (1)

the really existing distribution f(v; z, a) dv in the volume element

W ,4 at ( z , a), (2) the velocity distribution fo’(v; z, a) dv, which the electrons in V(z,a) had when they passed the anode plane z = 0, (3) the velocity distribution fo(v; z, a) dv,which the electrons would have, if they would not interact. From definition,f,‘ andf, are identical in the noninteracting case. In addition to these definitions we introduce the probabilityf,(w - v ; v , z, a) dw that an electron in V ( z ,a), which started at z = 0 with v , has in V(z, a) the velocity w. Then the relation betweenf, fo’, and f l is given by

(42) In the noninteracting case,fl is the Dirac functionf, = 6(w - u) so that we obtain the above relationf,’ =fo again. For the first stage of the relaxation process we now make the approximation to replace f,’ by f o . As a second approximation we considerf, to be only dependent on ( w - v) and not on u explicitly. This is certainly allowed for beams of constant current density which can be seen as follows: An electron alters its velocity in consequence of several encounters. The mean momentum transfer m(w - v) during one encounter is the same for all electrons with the same transverse velocity because it is determined by the relative velocity of the colliding electrons. This difference is, however, approximately equal to the difference of the transverse velocities, because these are much greater than the difference of the longitudinal velocities (ux + u,, & u ! ) .

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

269

The above arguments are approximately true for divergent beams, too. Electrons with different u but equal transverse velocities have approximately the same path, because the differences of u of the electrons are very small so that they meet with the same densities and velocity distributions. With these approximations we show that the internal energy of the electrons in V(z,a) is the sum of the internal energy uo , which the electrons at ( z , a) would have without interaction, and a second term u , ; u , is that longitudinal energy (referred to the moving reference system) which an electron receives from the transverse velocity components during its way from the anode t o the point (z, 4 : u = uo + u1. (43) We see the similarity of the last equation with (23). We will now prove the relation (43). Denote by (v) the longitudinal mean velocity of the electrons in V(z,a) if there is no interaction and by ( w ) the longitudinal mean velocity if the interaction is present:

Then the internal energy of the electrons in V(z,a) is given by rn 2

u = - j j ( w ) ( w - ( w ) ) ~dw

The integral before the last is uo . The last integral can be evaluated with the aid of the relation s f i ( w - u)(w - v) dw = jJ’fo(v)fl(w - u)(w - v) du dw

=

(w) - (u)

(46)

and is found to be identical with the internal energy ul(u) of those electrons in V(z,a) which started at z = 0 with the velocity v. Since ul(u) is independent of v, (43) is proven. There is another important consequence of the approximation, thatfis a convolution of the two functionsf, and fi. In the case of a beam of constant current density f l is found t o be a Maxwellian distribution with the two

270

BOD0 ZIMMERMANN

parameters u1 = k T J 2 and the velocity shift (w) - ( u ) . In the Appendix we show

From this it follows immediately that for u1 % u, the distribution f tends toward a drifting Maxwellian distribution. Although (47) was derived only for beams of constant current density, it is found experimentally to be valid for divergent beams, too. In the following we consider only points in the beam, where

fo(u; z ) =fo(u; z = 0). (48) For a homogeneous beam of constant current density this is true for all points in the beam. For a not focused and therefore divergent beam this holds on the beam axis only for z 9 R(eUo/u,r)lizand eUo/u % 1, where U , is the acceleration voltage and 2R is the beam diameter at z = 0. To show this we introduce the distribution g(utr)utrdutrof the transverse velocities. Then the longitudinal velocities on the beam axis are, if interaction is neglected, distributed according to

=

Jo

R

=j,(u;

f0(u; z = 0)g 2 = O)g(O)-.

uo R 3

(49)

32

The last line is valid, as asserted, only if rn(u)ZR2(2u,rz2)-1< 1 and if ( u ) is great compared with the velocity spread offo(u; z = 0), What remains to be determined is the distribution at the anode. We will assume here the simplest case, that is a homogeneous acceleration field between the plane cathode and the plane anode. According to Liouville’s theorem we can derive from the velocity distribution yo(u) at the cathode the velocity distribution at the anode where U , is the acceleration voltage. The problem, related to the distribution at the cathode, will be discussed in Section VIII. For the present we will consider only two special distributions, the normal velocity distribution K

(

Z)

foK(u; T ) d u = e x p --

do,

u 2 0

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

27 1

and the total velocity distribution B

The distributions (51) and (52) are not normalized, because the following calculations can be carried out easier with the given expressions. The normal velocity distribution is that expected under theory (2), whereas the total velocity distribution is often found in experimental devices (13, 15). In calculating the broadened distributions, we neglect the velocity shift (w) - ( u ) to make things less complex. It is not difficult to take this shift into account in the final result.

B. Broadened Normal Energy Distribution Having collected all fundamental relationships we can obtain the broadened normal energy distributions by a straightforward calculation. From (51) we obtain with the aid of Liouville’s theorem at the anode

After partial integration this distribution, inserted in (42), yields (54) where

[“k“70

F,(w)=exp --

1

2k(T+ mw2Tl)

For incipient relaxation (TJT 4 1) and great acceleration voltages (eUo/kT % l), Eq. (54) is approximated by fK(w; y ) w *ex+”[l

- erf(y

+ x/2y)l

(56)

with

x = (eUo - mw2/2)kT, y = (eUokT1)’/’/kT. (57) For y 2 10 the distribution can be further approximated with a maximum error of 1 % in the interval 1x1 5 yz/10 by the even simpler expression MW;

Y 2 10) x (1/2Jky) exPC-

(X

+ 1>2/4y21.

(58)

272

BOD0 ZIMMERMANN

This follows from (26)

1 z - exp[ - y 2 - x] exp 27

(59)

Substituting the variables w, T , , T, uo in Eq. (58) we obtain

kT -_ ( e U , kTl)-1'2

*

2

exp

-m(w - vo - kT/mu,)' 2kT1

'

(60)

We see already that for a relaxation temperature T, with kT, (lOkT)'/eU0 the broadened normal energy distribution agrees fairly well with a drifting Maxwellian distribution with the mean velocity (w)K

= (u)K

z~o(+ l kT/2eUo).

The maximum OffK(W) is shifted to smaller energies for increasing Tl in consequence of the change of shape of the distribution. It is easy to see from the above equations that the maximum is accepted at the reduced energies - y( -2 l n ( 4 ~ y ) ) ' ~ ~for

xKz

( -1

y2 6 1 for y 2 1.

+ty-2

+

In experiments which analyze the distribution with a retarding field analyzer the integrated energy distribution iK(w) is measured. For a given velocity w the value of iK(w) gives that part of the beam current, to which only these electrons contribute, the velocities of which are greater than w. We have, therefore,

1

m " kT w

~K(w) = -

w~K(w)

dw

=

s'

~K(w(x))

dx.

-m

(62)

Inserting (54) in (62) we have by integration iK(w; T ) = +{(I

+ Ti/T)'/'Fi(w) + F ~ ( w ) }

with

[

F2(w) = 1 - erf (w - uo) (2kmT,)1i2]'

(63a)

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

273

For Tl/T Q 1 and eUo/kT b 1 we obtain with (56)

+

+ x/2y)} + 1 + erf(x/2y)],

i d w ; y) % $Cexp(x y2){1 - erf(y which can be approximated by

+

(64)

+

iK(w;y >, 10) FZ +[I erf{(x 1)/2y}] (65) for y 2 10. The last equation can also be obtained by direct integration or(58). C . Broadened Total Energy Distribution The broadened total energy distributions can be easily obtained from the corresponding broadened normal energy distributions. Observing that &B(U; T ) = (Ta/aT)fo,(U; T), (66) we see that the operator TdIdTtransforms a normal energy distribution in the corresponding total energy distribution. Thus, we have

(mv2/2 - eUo) kT

I

(eu, - mu2/2) kT

fOB(u; T) do = -

fB(w; T) dw

=-

(67)

(+ TI

2 T

2(T

+ Ti) F , ( w ) + G(w)] dw,

(68)

with

- w)2 + (2kT(T + Tl))1'2(uo + T +T TI w ) exp - m(uo 2kT1 mT1

~

(69) Further we obtain

274

BOD0 ZIMMERMANN

From the last equation we see that even at relaxation temperatures TI with kT, 2 (lOkT)Z/eUo, the broadened total energy distribution agrees fairly well with a drifting Maxwellian distribution with the mean velocity ( w ) ~= ( u ) B z ~ , ( l kT/eU,). The distribution (70) has its maximum value at the reduced energies

+

XB %

- 1 - y2(2 +y2)/(1

+ 37' + y4)

Here the first line is an interpolation formula which for small and large y behaves in the manner as derived from the distributions. The integrated total energy distributions can be deduced from the corresponding normal energy distributions using the relations iB(w; T)

= (1

+ Td/aT)iK(w; T)

(74)

and iB(w;

Y) = ( I

- yd/dy)iK(w(x,y);

y) - xfK(W; 71,

(75)

respectively. We obtain iB(w; T ) = 2l [ ( l +$)(l

+$)-1'2F1(1(.)

+ (1 + 2)1'2G(w) + F 2 ( w ) ] ,

+ 1 + e r f -X + 2 ~ c - ' ' ~ y e x p 2Y

(77)

V. DIFFERENT ENERGY WIDTHSOF THE SAME DISTRIBUTION A broadened energy distribution is defined by three parameters: y, kT, and the mean velocity (w). From these, only y and kTdetermine the energy width. We first derive the dependence of AE on the parameters y and kT. B Then we define the other energy widths and write down their relations to AE.

BROADENED ENERGY DISTRIBUTlONS IN ELECTRON BEAMS

275

According to Eqs. (7) and (43) we have AE = (4eU0 u,

+ 4eU0 u , ) ” ~ .

(79)

Since at the anode u1 = 0, the energy spread BE, at the anode is AE, = 2 ( e U , ~ , ) ” ~ According . to Eqs. (22), (20), (51), and (52) we obtain, after relaxation-free acceleration, for normal (superscript K) and total (superscript B) energy distributions AEoK= k T ; UoK

kT = -k T ; 4eU,

AEOB= 2‘I2kT,

(80)

kT ugB=kT. 2eU,

Observing the definition (57) of y , Eq. (79) can be written AEK = kT(1

+2

~ ~ ) “ ~AEB ; = kT(2 + 2 ~ ’ ) ~ ’ ~ .

(82)

Determining AE from measured energy distributions is a tedious procedure because it requires numerical integration. Therefore other energy widths are usually communicated. For integrated energy distributions i(w) we consider the two energy widths (Fig. 3)

FIG.3. Integratedenergydistribution in anelectron beam, as obtained with retarding-fieldenergy analyzers. The interval 1 - l/e z 0.632 is taken in such a manner that the corresponding energy interval is minimal. This minimum valueis denoted byAE63%.

AE,,%: the electrons within that energy interval supply the part (1 - l/e) of the beam current and the interval is the smallest with that feature, and AE,,%: the currents at the boundaries of this interval are 5 and 95% of the beam current, respectively. The energy width AE63y0was introduced by Boersch and is usually communicated in papers. In order to determine AE of a measured distribution without numerical integration the knowledge of AE,,, alone is not sufficient. Either kT or AE,,, is necessary in addition.

276

BOD0 ZIMMERMANN

For " differential " energy distributions di(w(x))/dxwe consider only the energy width A E , , , , which is the full width at half maximum of the distribution. In order to be able to extract easily AE and kT from a measured integrated distribution we need the functions

AE = f 2 ( - )A&,, kT

(84)

Ego%

for broadened normal and total energy distributions. If two energy spreads are known, AElk Tcan be determined and from this AE,,,lk T, for example. Then k T is known and from this BE. With the aid of Tables I and 11 the required functions (83) and (84) can be found by fitting simple functions to TABLE I DIFFERENT ENERGY WIDTHS FOR BROADENED TOTAL DISTRIBUTIONS AS A FUNCTION OF y ENERGY y

AEB/kT

0 1/2 4112 1

1.414 1.581 1.732 2.000

492

2.646 3.162 1.414~

2 >10

AE,B,oh/kT

AE,B,yJkT

2.00 2.48 2.85 3.41 4.65 5.65 2.547~

4.40 5.09 5.60 6.44 8.74 10.31 4.652~

AEyI,/kT 2.445 2.90 3.54 4.30 5.88 7.20 3.330y

TABLE I1 DIFFERENT ENERGY WIDTHS FOR BROADENED NORMAL ENERGY DISTRIBUTIONS AS A FUNCTION OF y

0 112 4112 1 110

1.000 1.225 1.414 1.732 1.414~

1.oo 1.91 2.34 2.97 2.547~

2.94 3.65 4.55 5.63 4.652~

0.693 2.33 2.83 3.86 3.330~

277

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS 2

I

2

7

2

4

3

71/

/

B 112

B 63%

/

/

- 2

/

/

/

15

/

/

/

q5

/

/’

1.5

/

KK 63% 63%

/

I

/

/

/

//

I

A 63%

05

I

2

FIG.4. (a) Relation between the energy spread A E and the full width at half minimum A E l l z of broadened total energy distributions (Sg)and broadened normal energy distributions (KB). The dashed line represents the asymptotic relation between AE and A E l l z . (b) Relation between energy spread AE and 63%-width AE,,, of broadened total energy distributions (B63 %) and broadened normal energy distributions (K63 %). The dashed line represents the asymptotic relation between A E and AE,,% .

the selected values. Some functions found from Tables I and I1 are shown in Fig. 4 and are given by the following formulas:

+ 0.3344~+ 0.2208( ~+~ 1.434)”2 = -2.035 + 0.7652 + 0.13 exp(- lO(2 - 3)2) =

K

1.202

(85) (86)

with J = A E B / k T , 52 = AE:,,/kT, K

= AE&%/kT - 2.165.

(87)

(88)

VI. ABSOLUTE CALCULATION OF THE ENERGY SPREAD A . General Considerations The energy distribution and, consequently, the energy spread are characterized, as we have seen, by the relaxation parameter y , which is related to u1 by the equation y = (2eU0 u,)’’2/kT. (89)

278

BOD0 ZIMMERMANN

The starting-point of the following calculation of u1 and 2y’ is the remark that (2rn~,)’/~ is the spread Ap, of the momentum components the electrons gain in the z-direction on their way from z = 0 to the point (z, a) by impacts with other electrons. Here, as in the foregoing considerations, we neglect the macroscopic space-charge field and the z-components of the relative velocities in comparison with the transverse relative velocities. Let 6p,(qi,x i ) denote the z-component of the momentum an electron obtains at its ith impact with the impact parameter x i and the (transverse) relative velocity qi = (q,, q J i ; qi is assumed to be perpendicular to the beam axis. After several impacts, the mean square spread of the momentum transfer is

Here N(q, x , t ) dq, dq, 2nx dx dt is the number of impacts an electron suffers with impact parameters between x . . . x + dx and relative velocities between q . . . q dq in the time interval t . . . t + dt. Thus,

+

d j = W q , x, t ) 4 , 4,

(91)

is the current density of those electrons which, in the reference system applying to the electron under consideration, have velocities within the interval q . . . q + dq. On the other hand, S j is given by d j = lqln 4, 4

2 ,

(92)

where n = n(q + Q, r(t)) is the electron density in phase space (more exactly: the phase space density integrated over the z-components of the velocities) at the point r(t) of the electron with the transverse velocity Q. Thus, we have

The bracket ( ), denotes the average with respect to x. The upper limit of the impact parameter, which is necessary to avoid divergence, is given precisely later. We introduce the component 6p,(q, x ) of the momentum transfer, which is perpendicular to the impact direction. Since every angle between 6p,

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

279

and the z-direction is equally probable and since d p , is the projection of dp, onto the z-axis, we have

( ( h ( q ,XN2>, = N d P * ( q , X N 2 > , . (94) The momentum transfer 6 p , ( q , x ) can be taken from every textbook on mechanics : (95) Partial integration yields

(96) We replace the density n(q + Q, r) which is still lacking in calculating the internal energy

by the density no(q + Q, r) of the noninteracting case. This is justified since it is our intention to consider incipient relaxation only. The mean internal energy of an electron within a volume element is given by

where the average is taken with respect to all transverse velocities which are present in the volume element. We shall now consider divergent beams, homogeneous beams of constant current density, and crossovers. We follow up the calculation in the case of divergent beams in greater detail. For homogeneous beams we give only the result of the calculations, because Mott-Smith has given a detailed calculation in ( I ) . In the section dealing with crossovers the paper (14) of Loeffler is reported. B. Divergent Beams We mean by divergent beam a beam which diverges in a field-free space in consequence of the transverse thermal velocities. We assume the beam to have rotational symmetry. Then the phase space density no on the beam axis can easily be found. At the point (z = (v>t, a = 0) it assumes the value

280

BOD0 ZIMMERMANN

( q is spatial particle density at z = 0.) In the following we restrict ourselves to the calculation of u1 at a point ( z , 0) on the beam axis with z % R ( e U 0 / k T ) ' l 2 . For such z we have u1 = ( U ~ ( Q ) ) z~ u1 (Q = 0) because electrons which reach such points have only small transverse velocities compared with the velocities of those electrons with which they impact. The energy u1 can now be calculated using Eqs. (97), (93), (99), (94), and (96). With the abbreviations

we obtain

After partial integration of the integrals containing the term (1 we obtain

We calculate the integrals on the premises the approximation

In(1

t1<$

+ a4t4) cz (4a4c4 ln(a5)

+ a4t4)-'

and a > 4. With the aid of

for a5 < 1, for a t > 1

they can be reduced to the integrals JOm

d t . 5e-c2 In

d5 . te-5'(ln

5=

d z e - z In z =

1 "

<)2

=8

J0

C

- - = -0.14430 4

(104) d z e-'(In z ) = ~

where C is Euler's constant C % 0.5772.

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

28 1

Further calculation must be divided into two cases, namely, into the case of the short beam, that is ?tl > 1, and into the case of the long beam, that is a t l < 1. 1. Short Beam

For a t 1 > 1 we obtain from (102) and (103) for the internal energy

1

+tl

(1

d5(5 - 252)e-r2ln(a5)

1lLz

+2

I

m

2

dt(? - t)e-5’ ln(a5).

(105)

51

Partial integration of the last integral yields

j5yd t -25 e-<’ In(at) = -e-5”[ln(a~l)]2 + 2

a2

d t . <e-~2[ln(a<)]’.

(106)

51

We split up the last integral into two integrals, extending from 0 to co and from 0 to tl.We put all this in Eq. (105) and calculate all integrals of Eq. (105) with finite integration interval with the approximation exp( - 5’) z 1 - 5’ and obtain

1 +(21 4a5

$1).

With the aid of (104) we finally obtain u1 =

2nqRe4 ~

kT

[In

2.04

~.

ln(0.167a25,)

51

Replacing a and t1by the original magnitudes according to (100) we obtain with x13= 3(4nq)-l and j = eq(zj), where j is the current density at the

282

BOD0 ZIMMERMANN

beginning of the drift region, and with regard to Eq. (57) and the relation kT, = 2u, the result3

*)

(ln(4.16 R2eWo - 3.54

. ln(3.52 . 106(kT)312

+zj116

+ 0.521 ( e ~ , ) ' ~ ' '( I

0.648 * 10-2j"6 - (kT)"2(eUo)1i'2

From this we see that the factor C, in Eq. (37), given by C,

= 2y 2(kT )

(,jR) '( e Uo)~

(1 10)

depends only logarithmically on the beam parameters. As mentioned above, Eq. [I091 holds only for a t , > 1, that is

Tables I11 and IV contain numerical examples for [lo91 and the corresponding A E in columns 3 and 4.

2. Long Beam In the case a t , < 1 the energy u1 is to be found independent of z. According to (102) and (103) we obtain

The first part of the third integral we integrate partially. Decomposing the integrals with the intervals from l/u to co in intervals from 0 t o co In this and all following formulas that are numbered by a number in square brackets, the magnitudes R and z have to be inserted in centimeters, i in amperes per square centimeter and eU, and k T i n electron volts. Thus, the formulas can be immediately applied to experiment.

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

TABLE I11 ENERGYSPREADS OF BROADENED NORMAL ENERGY DISTRIBUTIONS' Divergent beam 1

2

kT (eV) 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

'

3

4

2Y2

AEK kT

Homogeneous beam 5 2Y2

__ 19 7 .10-4 1.18. 5.84.10-3 2.44.10-2 8.98 . 2.92. lo-' 8.54. lo-' 2.28 5.68 13.12

0.026 0.119 0.45 1.48 4.24 10.9 25.4 54.8 110.0 206.0

1.013 1.058 1.204 1.575 2.29 3.45 5.14 7.47 10.54 14.4

0.099 0.472 1.88 6.43 19.4 52.6 129.0 287.0 613.0 1215.0

6

AEK kT 1.05 1.21 1.70 2.73 4.54 7.32 11.4 17.0 24.8 34.9

a Assuming eU0 = 1000 eV, R = 0.25 mm, z = 10 cm, and j = BT2 exp(-eUA/kT)with B = 120 A c m - 2 deg-2 and eUA = 4.54 eV.

TABLE IV ENERGY SPREADS OF BROADENED TOTAL ENERGYDISTRIBUTIONS" Divergent beam 1

2

kT (eV) 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

9.84.10-5 5.90.10-4 2.92.10-3 1.22 ' 1 0 - 2 4.49.10-2 1.46 . lo-' 4.27. lo-' 1.14 2.84 6.56

Homogeneous beam

3

4

5

2Y2

AEB kT

2Y2

A EB kT

1.419 1.436 1.496 1.667 2.062 2.79 3.95 5.6 7.8 10.7

0.051 0.243 0.98 3.13 10.1 27.4 67.4 153.0 322.0 641.o

1.432 1.498 1.73 2.27 3.48 5.42 8.33 12.45 18.0 25.4

0.014 0.062 0.238 0.78 2.25 5.77 13.60 29.4 59.3 112.0

6

* Assuming eUo = 1000 eV, R = 0.25 mm, z = 10 cm, and j = BT2 eXp(-CfJA/kT) with B = 60 A c m - 2 degg2 and eUA= 4.54 eV.

283

284

BOD0 ZIMMERMANN

and 0 t o l/a and approximating the exponential function according to exp[ - 5’3 % 1 - t2 we obtain

7 nqRe4 u 1 = -* (1 4kT

m

+ 16 j0

d< . teCr’(ln at)’

or with the aid of (104)

-

With (57) and kTl

E4 ((In 0 . 5 8 4 ~ (+ ) ~0.47 kT

= 224,

we obtain

2y2 = 1.523 . 10-3(eUo)1i2R(kT)-3j{[ln(3 . 107(kT)3

. (eUo)1i2/j)]2+ 16.9 - 0.035 .j’i3(kT)-1(eU0)-116}.

~1151

The condition of validity, at1 < 1, can be written

For a t 1 = 1 the formulas [lo91 and [115] are identical.

3. Limits of Validity The formulas [lo91 and [115] for 2y2 are only valid to an accuracy of about 1%, if a > 4 and tl < 4,that is j

< 2 . 105(kT)3(eU)1’2

z <4~(e~/kT)*”.

Moreover, the results in this section are only then obligatory if the longitudinal internal energy is small compared with the transverse internal energy ut,, that is, if AE(u,,/u)”~& AE. (1 18) The energy u,, diminishes with increasing z mainly because of the spacial beam divergence. From the expression (99) for the phase space density we obtain

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

285

FIG.5. Curves a and b represent the energy spread AEB‘ofdivergent electron beams in unitsofkTasfunctionofthebeamlengthz, according to Eqs. (82) and [109]; eU, = 1000eV, k T = 0.24 eV, R = 0.25 mm. The curves a and b are only valid, if A E B G A E B ( ~ , , / ~ ) ”The 2. function AEB(u,,/u)l’Z/kTis represented by curve C. In curve a, j = 5 A cm-2. In curve b, j = 1 A crn-’.

In Fig. 5 the energy spread AEB, calculated from (82) and [109], and the function AEB(utr/u)1’2, calculated from (1 19), are shown in dependence on the beam length z for a special example. C . Beams of Constant Current Density

In beams of constant current density the energy distribution in the drift region is space-independent, if interaction is neglected. The same is valid for the phase space density

, is averaged with respect The internal energy u1 = ( 1 / 2 m ) ( ( A p l ( Q ) ) 2 )which to Q, is obtained by inserting

286

BOD0 ZIMMERMANN

in Eq. (93). The calculation, which we suppress here,4 leads t o the result u = 1.312e4yz(eUok T ) - "'[ln((

kT/e2)3/q) 6.741.

(122)

Equation (122) is identical with the result of Mott-Smith ( I ) , who obtains it using Boltzmann's equation. From (122) follows 2y2 = 0 . 0 1 1 4 7 ~ j ( k T ) - ~ln(3.78 '~

. 106(kT)3(eU)''2/j).

w31

From this C in Eq. (34) can be calculated:

c = 2y2(k~)5/2(jz)-1.

(124) Its dependence on the beam parameters is only logarithmic. In Tables I11 and IV in columns 5 and 6 numerical values for 2yz and AE are given.

D. Crossover Contrary t o the foregoing sections, we will consider here the process of energy broadening in the laboratory system. This is a point of view of Loeffler (14) whose ideas and calculations we will follow in this section. The theoretical investigation of a crossover is important because in many cases beam geometries can be considered as a sequence of crossovers. We assume that in the smallest cross section of the crossover the current density and brightness is constant within the beam boundaries. We consider only the energy spread of those electrons which are moving along the beam axis. By impacts with other electrons the axis electrons gain or lose energy in beam direction. Since the longitudinal velocity differences of the electrons are very small, the longitudinal component of the distance vector of two interacting electrons can be assumed to be constant during an impact. That is, we restrict ourselves to the case that the distance of the impacting electrons is changed faster by the transverse velocities than by the longitudinal relative velocities. This condition is expressed by the formula A E 4 muo, (1 25) where c i is the beam semiangle. Thus an axis electron remains over a long path in the nearly constant field of another electron, so that the product of force and length is the greater, the greater is the mean longitudinal velocity. This is the magnification effect mentioned earlier in Eq. (13) considered in the laboratory system. The interaction time is limited by the diverging trajectories of the electrons. We will emphasize that the interaction length and not the interaction time is increased by increasing the mean velocity. During the calculation the integral

f," In 6exp(-&z" appears.

=

tfz

- ~ In ' ~ze-" dz = -(n-"*/4)(ln 4

+ C ) = -0.871

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

287

If ri is the distance vector of an axis electron j to an electron i the axis electron suffers an energy shift

in z-direction, where 2 is the unity vector in z-direction. The ri depends on z, whereas the (Ti * 2) are independent of z. After having passed the crossover the axis electron has experienced the energy change 6Ej = C 6E,’.

(127)

I

The average energy spread 6 E of an axis electron, which is caused by collisions is then given by ( C ~ E=) ((6Ej)’)j ~

=

(1(SE,’)’>j i

where (dN/d(GE))d(6E) denotes the number of impacts at which an axis electron gains or loses energy between 6E . . . 6E d(6E) on its way through the crossover. Loeffler has calculated this to be

+

W

N(6E

(dN/d(GE)d ( 6 E ) = 32nro3qO.

= e’/aro) = 2 e2/aro

The total number N(6E = 0) of impacts is infinite. But this singularity does not lead to difficulties in calculating the integral (128) with the expression (129). An upper limit has t o be introduced, however, for the energy shift variable 6E in order to enforce convergence. This can be done by various methods. Loeffler has met this difficulty by digitizing the energy shift variable in such a manner that N(6E,) = n - 4, (130) and by replacing the integral (128) by a summation. He obtains

(32e>ro (3 + In 2 + 2 ln(32nro3q) + 41 ln2(32nrO3q) -

(GE)’

=

if 32nrO3qp 1, if

32nrO3q6 1,

288

BOD0 ZIMMERMANN

In order to see that it is this special method which yields the linear dependence of 6E on y for small rO3y,we integrate Eq. (128) with the expression (129), introducing an upper limit 6E,. We obtain

6Em

if

e2 6Em<--, UrO

The maximum energy shift 6 E , should be about 3aeUo, because the maximum gain in longitudinal internal energy during one impact is about equal to the maximum transverse energy a2eUo. Loeffler has not justified Eq. (130) analytically and claims in the discussion of his result (131) that the largest contribution to 6 E (that is 6E,) causes about 90% of the total energy spread in the region 47rrO3y= 0.01 and about 57% of the total in the region of 47rro3y = 100. This means that his result depends critically on the relation (130). The analytical treatment by Loeffler is confirmed by computer calculations, made by him and Hudgin, concerning the quadratic dependence of ( c ~ Eon ) ~ y for small y and the linear dependence of (6E)’ on y for great y. The computer calculations are based on Monte Carlo methods and on numerical integration and convolution of the initial probability distributions. The closed form approximations (1 31) yield consistently higher results than the computer calculations, ranging from about +40 %. deviation at 47rrO3q= 0.01 t o about 20% higher values at 47rrO3y= 100. The computer calculations are reported to have an estimated accuracy of about +2%, if the length 2L of the crossover is sufficiently large: 2L > 6ro/cr. (133) This means that at larger distances from the smallest cross section the impacts are negligible. Electrons, which are moving under the maximum angle a, will have smaller energy spreads at first because the mean distance to the other electrons is greater and secondly because the interacting length with the next electron is smaller. This is also mentioned in Ulmer and Zimmermann (25). We will emphasize that the results obtained by Loeffler are in agreement with the relaxation concept as adopted by Mott-Smith, Hartwig, Ulmer, and Zimmermann, as already shown in Section 111. It should be noted that Loeffler does indeed reject the relaxation concept. This, however, seems to be only a play with words. The actual calculations, in order to obtain quantitative results, are essentially the same. Contrary to Zimmermann, Loeffler has

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

289

considered, however, not only collisions between neighboring electrons but the collisions of the axis electrons with all other electrons in the beam. Loeffler claims to have proven that there is a broadening in beams, which do not have any initial energy spread (including the transverse velocity components). It is, however, a matter of fact that the electrons in a crossover with a # 0 have a nonvanishing transverse internal energy. In addition, Leoffler’s calculations are based on the supposition that the transverse velocities are large compared with the longitudinal relative velocities. Thus, Loeffler did not give any quantitative proof that there is energy spread generation in dense monochromatic electron beams. It is evident, however, from the extended relaxation concept that there must be an energy spread generation in those beams. Since there is some nonvanishing internal potential energy, the principle of increasing entropy requests a generation of kinetic internal energy by a relaxation process. So far, the energy spread generation in monochromatic beams has not been quantitatively investigated. To conclude this section we will consider a crossover which the electrons enter with an energy spread A E , . If there occur several impacts along the crossover, the energy spread AE after the crossover is given by (AE)’

= (AE,)’

+( c ~ E ) ~ .

(134)

This is in agreement with Eqs. (39) and (43). Equation (134) can be proven directly as follows: The energy of an electron j may be EOi before and E’ after the crossover. Then Ej

= E,’

+ 6E’.

(135)

The energy spread after the crossover is then given by (AE)’

=

.

((E’ - (E))’)’

= ((Eoj

-(E)

= ((Eoj

-

= (AEo)2

+ 6Ei)2)j

(IT))’)

+ (6E)’.

+

((SEj)2)’

(136)

The term ((E,’ - ( E ) ) . ( d E j ) ) is 0 because E,’ and 6Ej are not caused by the same impact and are thus not Correlated. If collisions occur only scarcely by reason of a small particle density, 6E,’ and 6E’ may be correlated. But in this case, too, the above average vanishes because we can expect as many 6E,’ 6Ej as 6E,’ 6E’ = -6E,’ 6Ej. From this it follows that Eq. (134) may be applied to this case, too. The energy spread occurring after several successive crossovers of a beam with initial energy spread can be calculated with the aid of Eq. (136). In contrast to this, Loeffler proposes the simple addition of the energy spreads if

290

BOD0 ZIMMERMANN

the geometry of the subsequent crossovers is identical and if the energy spread is small enough not to significantly affect the axial separations throughout the system. Against the handling of AE according t o Eq. (134), Loeffler raises the objection that the existence of the variances is not proven and calls attention to the strong interactions, which may cause distributions with rather extended tails. Comparison of calculated and measured distributions, as made in Section VIII, show that this apprehension seems not t o be necessary. INVESTIGATION OF BEAMS OF CONSTANT CURRENT DENSITY VII. FURTHER

The geometry of a homogeneous beam of constant current density is so simple that further quantitative results can be obtained. In connection with the energy broadening there occurs a shift of the mean energy. This is seen to be very small, however, and will be masked in experiment by other effects, such as the current-dependent macroscopic space charge at the cathode. In a later subsection we derive the longitudinal internal energy of the equilibrium distribution, assuming the electron ensemble t o be an ideal gas. A very interesting question is whether a restoration or even a narrowing of the initial energy distribution can occur by any process, for example by deceleration of a beam. We will give an answer to this question by entropy considerations. In a following subsection we derive the collision term of the Boltzmann equation from the above results. It is a pure differential expression and the Boltzmann equation resembles a diffusion equation. It is, however, not confirmed by theoretical considerations or experimental evidence whether this expression is valid in the acceleration region, too. In the last subsection of this section the energy spread of beams with relativistic mean energy is derived.

A . Energy Shift A consequence of the increasing longitudinal internal energy is an increase in the longitudinal pressure. Thus, every volume element of the beam runs against a decelerating field, which causes a shift of the mean energy t o smaller values. We will now derive the magnitude of this shift. The difference d ( w ) of the mean velocity is related t o the pressure difference d P by Euler’s equation.

m j d ( w ) = - e U‘P. Together with the equation of state for ideal gases [ O = 2u/k = T +

(137)

kB = P/q = eP(w>/j (138) TI], which we assume to be valid, Euler’s equation yields

m d(w)

=

-d(kO/(w)).

(139)

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

29 1

From this we obtain by integration

( w > = (u> - k T l ( z ) ( m ( ~ > ) - ’ ,

(140)

where ( v ) is the initial velocity. In deriving (140) we have assumed ktl 4 m(v>’/2. The energy shift is then given by

+m(w)’ - +m(v)’

= -kT,.

(141) It is small compared with the energy spread. Therefore, experimental proof cannot be expected. B. Internal Energy at Equilibrium

The macroscopic state of the electron beam is described by the integrated Euler equation

(w) - (0)

= e(Po -

P)(mj)-’

(142)

and by the law of energy conservation

+ m ( ~ > ’+ u

+ Utr + ( P / v ~=) *m(u>’ + uo + u t r , 0 + ( P / v >> ~

(143)

where h

=u

+ ut, + (Plr])

( 144)

is the heat function end utr the transverse internal energy. These equations yield with r] = j ( e ( w ) ) - ’ , go = j ( e ( u ) ) - ’ , and elimination of P u

+ utr, 0

=z ~0

+ u t r , 0 + +m((w>

-

(0))’

- ( e / j ) ( ( w >- (u>)Po *

(145)

Observing (141) we see that the sum of longitudinal and transverse internal energy is approximately constant : u

+ ut, = uo + utr,

(146)

0.

For the isotropic equilibrium distribution we have ut,(z + co) = 2 4 2 + 00). With uo 6 u,,, = kT we have u(z + 00)

FZ

kT/3,

8(z -+

00)

= +T.

(147)

We can now estimate the beam length z,, at which relaxation is nearly finished. From (57) and (147) follows 2yz(z -+ co) = 4eU0(3kT)-’. Then according to [I231 a beam of length

z , = 11 6 ( k T ) 3 i 2 e U o [ln(3.78 j . 106(kT)3(eU,)’i2/j)]-

[148]

has approximately its maximum energy spread A E , = (+eU, kT)’l’.

( 149)

292

BOD0 ZIMMERMANN

All energy spreads measured so far are small compared with this upper limit, and this is the justification for considering throughout this review only the linearized relaxation law. The value of the maximum energy spread depends strongly on the assumption that there is no external macroscopic electric field along the beam. The maximum energy spread is AEm = ($eUo kT)''' ( 150) .. if a constant mean velocity is enforced by an external acceleration field of

This follows from the above equations replacing (137) by m j d ( w > = - e dP

+ eyFdz.

(152)

In (18) constant mean velocity was assumed and the value (150) was given as an upper limit of the energy spread. We see that the small, nearly undetectable energy shift has a considerable effect on the maximum energy spread. C . Energy Spread and Entropy One may wonder whether for an appropriate external field a decrease of the energy spread may occur. Especially, it may be interesting to know whether an energy broadening can be restored to the original condition by deceleration. To answer these questions let us consider the electron beam as hydrodynamic flow and see what happens with entropy. A hydrodynamic flow is described by Euler's equation, energy conservation and an equation for the entropy. In an isentropic and therefore reversible flow the local entropy per particle remains constant along the beam. The entropy per electron is given by (153) where (Ax Ay AZ)-' is the particle density and Apx Apy Apz is the product of the mean momentum spreads. Putting utr= Apx Apy/2m,

AE

=pz

ApJm,

y = (Ax Ay Az)-'

(154)

we obtain S = kln

observing that p , y

= p , , ylo c c j .

u t r AE u t r , o AEo

+ so,

(155)

Since entropy cannot decrease, we have

AE 2 (utr, olutr) A E o .

(156)

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

293

From this it follows that AE can only be smaller than AE, if the transverse internal energy has increased. As further consequence we derive that AE can never be smaller than AE, if ul,, = 2u, = AE, . For this purpose we first derive the energy width of an adiabatically accelerated beam. In this case the relations u = +ut, and S = const. hold everywhere in the beam. From Eq. (155) we obtain

,

AE

= (2(E)(AEo)2)”3.

(157)

This energy spread is smaller than the equilibrium energy spread after relaxation-free acceleration. We now consider an accelerated electron beam, for which AE, > ut,, o . In this case we can bring about a reduction of AE, by deceleration. It is most efficient, if we first decelerate relaxation-free to the mean energy ( E l ) , at which u = +ut,,o . The value of ( E l ) is given by

AEo

= ( 2 ( E , )Utr, 0)”’.

(158)

Successive adiabatic deceleration to the mean energy AEa yields an energy spread AE, , which is given by

AE,

= (2(El)(AEa)2)1/3.

(159)

Elimination of ( E l ) from the last two equations yields

AEa = (AEo . ut,, ,)l/’.

( 160)

From this we see that an energy spread AE, can be diminished if the transverse energy ut,, is smaller than the energy spread. If this is not the case as in usual thermionic electron guns, the energy spread of beams of constant current density can only be increased. The narrowing effect as just described has not been observed so far. It would be necessary to use magnetic fields in order to confine the electron beam over the necessary length.

,

D. Boltzmann Equation In principle, all the results on energy broadening presented in this article might be obtained from the Boltzmann equation with a collision term containing binary collisions of neighboring electrons. Such calculations would be n o less involved than the method presented here. As a matter of fact, only the case of homogeneous beams is investigated with the aid of the Boltzmann equation with collision term ( I ) . Another approach to the explanation of energy broadening was performed by Fischer (27). He starts with the collisionless one-dimensional Boltzmann equation including a time and space-dependent electric field, which is related

294

BOD0 ZIMMERMANN

to space-charge fluctuations by the one-dimensional Poisson equation. Thus, no transfer of transverse energy into longitudinal energy is taken into account. Fischer’s energy broadening is due to noise fed into the beam from the electron gun and depends periodically on the beam length. For small beam lengths the energy spread depends linearly on the length. The process considered by Fischer does not generate entropy. Since there are some points which are not quite clear, we can regard the results published in his paper as tentative and will, therefore, refrain from going into further detail. Zimmermann thinks that this effect is too small to be observed because it is exceeded by the relaxation process. We will now derive a Boltzmann equation, which describes the relaxation process. From this equation all previous results can be reobtained. With the aid of Eqs. (47) and (141) the distribution function (42) can be written in the form

where, according to (57) and [123],

kT,

= pjz

P=

eUo( 0.00574 kT)’’2

i

In 3.78 . 106(kT)3

J

It can easily be verified that f ( w ; z) is a solution of

T o obtain the differential equation for the phase space density g = 9f = (j/e<w>)fweuse

which follows from (141) and (162). We obtain

In the case u 4 kT eUo the average velocities (v) and (w) can be replaced by the variable w. Since (166) is equal to the collision term in the Boltzmann equation describing the relaxation process, it reduces in field-free space to the differential equation

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

295

We see that a force ( - p e j ) acts on the electrons. This was already shown in Section V1I.A. The last term on the right-hand side is by first approximation negligible. Thus in a reference system moving at mean local beam velocity, g is a solution of

This is a diffusion equation in the velocity space. The relaxation process in electron beams is, therefore, not described by the usual relaxation time concept introduced in investigations where the collision term is a small perturbation. This difference comes from the fact that the distribution actually existing in the beam is far from equilibrium distribution. If at greater times it approaches equilibrium distribution, dynamical friction as investigated by Chandrasekhar (28) in order to calculate lifetimes of galaxies must be considered.

E. Relativistic Beam Energies For relativistic beam energies Eq. ( 7 ) does not hold between energy spread and internal energy. To derive the appropriate relation in the relativistic case we consider, for the sake of simplicity, a " beam " consisting of only two identical particles, the transverse velocities of which are zero. The energy of the particle i ( i = 1, 2 ) is given by

Ei

= (Eo2

+pi2~2)1'2,

(169)

where E , = m, c2 is the rest energy and p i the momentum of the particle. As usual, we denote four-vectors by ( E i , cpi). Then

(4CP) = (El>C P i ) + (4C P 2 ) 7

( 2 AE7 2c AP) = ( E l , cpi) - ( E 2

9

~ 2 ) .

( 170)

(171)

Addition of the squares of (170) and (171) yields

( E , C P )+ ~ ( 2 AE, 2c A P ) = ~ 2(E1, c p J 2

where

+ 2(E2, c p J 2 .

(172)

E", is the rest energy of the two-particle system, one further obtains ( 2 AE)2 - c2(2 A P ) ~= 4E02 - Eo2.

(175)

296

BOD0 ZIMMERMANN

On the other hand, we have

E AE

(176)

AP

=

because of E l 2 - E,2

= c’(pl’

-p2’).

(1 77)

Elimination of Ap from (175) and (176) yields 4(AE)’ = ( E 2 - E“02)(E”02 - 4EoZ)/Eo2.

(178)

With

E

= E“,

+ 2eU0,

E“,

- 2E, = 2u,

(179)

one finally obtains the relation (AE)’

= 2(2 + ( e u o / e o ) ) e u ,u

( 180)

between energy spread AE, acceleration voltage eU, , and longitudinal internal energy u ; here we have restricted ourselves to the case u < E, which is the only one of practical importance. For nonrelativistic beam energies, e U , < E , , Eq. (180) reduces to our earlier result (7). For ultrarelativistic energies, eU0 % E , , we have (AE)’ = ( 2 ~ / E , ) ( e u , ) ~ .

(181)

Before proceeding any further, we shall show, with the aid of this equation, that the relaxation process is unimportant for ultrarelativistic electrons which have been accelerated by a high-frequency accelerator. In such an accelerator not all electrons gain the same energy. Thus the accelerator itself generates an energy spread. In order to show the relaxation effect, this spread would have to be small compared with the enhancement of the energy spread caused by the relaxation process. A numerical example illustrates the circumstances. If kT = 0.14 eV, then we have with E , = 0.5 . 106 eV at complete relaxation ( A E J e U , ) = 0.05%. The relative energy spread in high-frequency accelerators, however, amounts to about 1 % according to (29), which is 20 times more. Finally we determine the internal energy u as function of the beam length z for such z at which the longitudinal internal energy is small compared with the transverse internal energy (or, what is the same, for such z at which the exponential function describing the relaxation process can be replaced by its linear approximation). For this purpose we write down Eq. [123] using (57) and obtain MI =

0.01147eq, to ln(3.78 . 106(kT)3((m,/2)”2 2rnO(kT)1”

erl0

(182)

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

297

where qo is the rest density and to is the mean flying time measured in the rest system. With C Z t , 2 = C 2 t 2 - z 2 = 2 2 ( c 2 ( u ) - 2 - l), (183) c2(eu]o)2= c2(eq)2- , j 2 = j z ( c 2 ( u ) - 2 and (E,

+ eUo)2

FZ

Eo2(1 -

11,

(1 84)

(V)~C-~)-',

(1 85)

-

we obtain from [182] Ul =

0.00574E0 zj eUo(eUo 2E0)

+ . ( ~ u , ( I+ z))1'2jj). Equation (180) and u = uo + u1 finally yield the equation (AE)2 = (AE0)2 + 0.01 1 4 7 ~ j ( k T ) - "In~

for the energy spread.

VIIL EXPERIMENTAL SUPPORT FOR THE THEORETICAL RESULTS In this section we describe the experimental devices with which energy broadenings were found. The experimental results will be compared with the theoretical expectations. One cannot expect exact agreement because the geometry of electron beams is in general not simple and in n o device identical with the one theoretically assumed. The following comparisons will help the reader to apply the formulas in this article t o his own device. I n order to be able to make this comparison, it is necessary to begin with the shape of the initial and broadened distributions. Then we will compare the measured energy spreads with the calculated ones. A . Shape of the Distributions

[. Distributions at a Tlievmionic Cathode It is known from textbooks and from Lindsay ( 2 ) , for instance, that the distribution of the velocity components perpendicular to the emission surface of a plane cathode should be a half-Maxwellian distribution as given by (51). This is a consequence of the Fermi distribution existing within the cathode. In

298

BOD0 ZIMMERMANN

precision experiments Shelton (30) and Kisliuk (31) measured such distributions at tantalum single crystals. There are, however, several circumstances which can disturb this distribution, especially at low energies. Quantum mechanics predict an energy-dependent reflection coefficient at the cathode surface (32), which causes a deficiency of low-energy electrons. There is no experiment which proves this effect without any doubt, because it depends strongly on the microscopic surface conditions. Moreover, another effect also causes a deficiency of low-energy electrons. Nonsingle crystals d o not have a constant work function ail over the surface, because every crystal direction has a different work function. Therefore, several different local energy distributions exist on the surface, the sum of which differs from a half-Maxwellian distribution in the low-energy region. The extreme values of the work functions of tungsten are 4.39 and 4.69 eV depending on crystal direction. The difference of 0.3 eV is even greater than the thermal kT = 0.2 eV of the emitted electrons at T = 2300°K. Nonconstant work function can also result from an inhomogeneous adsorption of gas at the cathode. At oxide cathodes, ohmic resistance at the surface may explain the measured distributions which look like total-energy Maxwell distributions belonging to temperatures which are larger than the actual cathode temperatures by a factor 1.5 t o 2.2 (33-36). From these arguments it is evident that prediction of the energy distribution at a cathode is, in general, not easy. In Fig. 6 several measured energy distributions are shown.

FIG.6. Energy distributions at a cathode with temperature T = 0.229 eV/k. Curve I is the energy distribution of electrons emitted by an ideal plane cathode (normal energy distribution), proportional to e-€IkT. It is the high energy tail of the Fermi distribution.The normal distribution was measured by Shelton (30)at a tantalum single crystal cathode. Curve I1 is the nornialdistribution,multiplied by the "transmission function" T ( E )= 1 - exp(-€/0.191 eV), proportional to TEe-''". The transmission function accounts for the quantum-mechanical reflection of the electrons at the cathode (37). Curve 111 is the total energy distribution, proportional to € e - t ' A r . It was measured by several authors (3, 13, I S ) at different cathode geometries.

2 E/hT

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

299

2. Distributions at the Anode It is only in the presence of homogeneous acceleration fields and in the absence of spectrometer effects by geometrical apertures that one can apply Liouville’s theorem in the form (50). In the case of space-charge-limited operation of the electron gun, an inhomogeneous space charge may lead to deviations from the half-Maxwellian distribution. Tracing the electron paths in the whole variety of usual electron guns is a tedious process. Recently Hanszen and Lauer (38) began to study the influence of the different electron paths on the energy distributions and broadenings. We will avoid all these difficulties and point only to the consequence of a spectrometer effect, according to which only those electrons are passing the anode aperture for which u,,/c < a, where the angle a may depend on anode and bias voltage. The distribution of the longitudinal velocities is then given by

d n , cc (exp(-

s) g) - 1) exp( -

do.

For small LY this is the Maxwellian total-energy distribution; for great a this represents a half-Maxwellian distribution. At low electron currents usual electron gun geometries generate beams with energy distributions differing not significantly from the total energy distribution. Moreover, Hartwig and Ulmer (13) found indications that a increases with the acceleration voltage as might be expected from geometrical considerations. This is supported by a simple calculation of Andersen and Mol (39),who stated that the distributions to be expected are superpositions of normal and total energy Maxwellian distributions.

3. Broadened Distributions Distributions, which we call broadened, are not caused by the above mentioned effects, because they cannot explain the great energy spreads which have been observed. The appearance of such “ anomalous ” energy spreads is sometimes called “ Boersch-effect,” because Boersch (15) made the first systematic investigation of this effect. The following comparison shows that the effect may be quantitatively explained by Coulomb interaction between individual electrons. To begin the proof of this statement we compare the theoretically expected distributions with the measured ones. In Fig. 7 measured distributions are shown by solid curves, calculated distributions by dotted curves. The distribution 4 was measured by Hartwig and Ulmer (13) in a beam generated by a Ba-0-cathode. The distributions M6 and M1 were measured by Ulmer and Zimmermann (18) in a beam generated by an indirectly heated plane tungsten

300

BOD0 ZIMMERMANN

(-

mw2/2 + const )/eV

FIG.7. Comparison between measured(-)and calculated (. . .)energy distributions. The measured distributions M1 and M6 are taken from reference (18), the measured distribution 4 from Fig. 5 of reference (13).

cathode. For comparison with theory, we must decide whether the measured distributions emerged from Maxwellian total or normal energy distributions. Both comparisons were made. Using the results of Section V the " calculated " cathode temperatures were determined. The table in Fig. 7 shows that only those temperatures are compatible with the measured temperatures Tgwhich apply to broadened total energy distributions. A decision based on comparison of the two types of broadened distributions with the measured distributions is not possible, because the agreement is of the same quality. Thus, in general, it is not possible t o decide whether a measured distribution is a broadened normal or a broadened total distribution. The above comparison shows that the symmetry of the distributions is obtained at a faster rate than is energy spread. B. Energy Spread of the Distributions

In this section we describe the experimental devices and report the opinions of several investigators and the conclusions they have drawn from their measurements, and compare them with the theoretical results. In Fig. 8 the energy spread AEB(j) as a function of the current density is shown by solid lines. The corresponding beam parameters are given in the table included in Fig. 8 and are in agreement with the experimental setup of

BROADENED EKERGY DISTRIBUTIONS IN ELECTRON BEAMS

30 1

Ulnier and Zimmerniann (f8).Observing the results of the foregoing section, we can expect broadened total energy distributions. In addition to the calculated curves, measured values of the energy spread are entered in this figure. ,jg)of values corresponding to each measured distribuThe couple (AEb3,>,, tion, where j , is the current density measured at the Faraday cage, has been transformed into the couple (AE,;), where j is the current density at the anode. Converting AE63uointo AE, the measured distributions were assumed t o be broadened total energy distributions. This can be expected from the foregoing section. Keeping in mind that there is no parameter which could be adapted, the agreement is very good. The remaining discrepancy which occurs especially for intermediatej may be caused by the nonidentity between the theoretical assumptions on beam geometry and the really existing beam geometry. There is, for example, the interaction between the electrons in the acceleration region which may be described by another relationship than [109]. So far, consideration of interaction between individual electrons in the acceleration region has not yielded any sound result. If we assume Eq. (39) to be valid, however, we obtain the same dependence of (AE)* on j(0) as in the case of a homogeneous beam of constant current density, ifj(z) is proportional to j(0). Therefore, the dependence o n j should be the same whether interaction in the acceleration region is negligible o r not. 7

- 0

1

10'

1G2 j

I

1

'0 10

m~/cT'

FK;. 8. Energy spread A.EB of divergent electron beams as a function of the current density j . The solid lines (-) are calculated according to Eqs. (82) and [109]. The points repi-csent results of the measurements M, R, S taken from reference (18); the current densities attached to the points are identical with the ratio of the anode current/emission area.

302

BOD0 ZlMMERMANN

There remains a serious cause for the discrepancy shown in Fig. 8. As i n other experiments the beam current was altered by variation of the voltage of a control grid. Without considering the variations of the electron paths in consequence of the variations of the bias voltage, we merely find that the beam divergence, the beam diameter, and the current density at the control grid are changed (15). The influence o f t h e electron paths on the energy spread was recently investigated and will further be investigated by Hanszen and Lauer (38). In the above interpretation we assumed that only the current density changed. In order to estimate the possible error, it was now assumed that the variation of the control grid may change the measured beam current I = nR2j only by variation of the beam diameter 2R, leaving j constant, in which case the measured energy spreads proved to be slightly smaller than the calculated ones. Thus, observing the dependence of the beam diameter caused by the grid voltage, the discrepancy between theory and experiment, as seen in Fig. 8, can be understood. The first systematic and very extensive experimental investigation into the broadening of energy distributions in electron beams was carried out by Boersch (15). He found that the anomalies are caused mainly by the current density and that they depend to a smaller extent on the whole beam current and average beam energy. Boersch used for his investigation triode systems, each of which consisted of a directly heated hairpin cathode, a control grid, and an anode. The acceleration voltage was about 30 keV, the measuring system was a Faraday cage. In order to avoid the voltage drop along the cathode wire, he operated the electron guns with those bias voltages, at which only those electrons were analyzed which were emitted from an area of about 0.01 mm 4. The influence of voltage drop was avoided by intermittent heating in the experimental setup used by Hartwig and Ulmer (40). No appreciable broadening was found. On heating with alternating current of 50 Hz Boersch found only a broadening smaller than 0.1 eV which is compatible with the temperature oscillations of about 0.06 eV/k. By many experiments he determined the influence of cathode temperature, grid voltage, distance between electron gun and Faraday cage, macroscopic space-charge, and acceleration voltage on the energy broadening and excluded them as agents responsible for the anomalous broadening. He stated that no spectrometer effect in the gun was responsible for the broadening. He found that the energy distributions at the outer regions of the beam are narrower than those on the beam axis and that they have a lower mean energy than the axis electrons. The first effect we can understand, because the outer electrons experience a smaller number of impacts than the axis electrons, the latter effect is caused by the macroscopic space-charge in the gun, which depends on the distance

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

303

from the beam axis. Boersch proved, however, that the broadening of the distributions is not caused by a superposition of the different distributions of the axis and boundary electrons. After he found that the current density is responsible for the broadening, Boersch investigated the consequences of focusing and defocusing of the beam. As expected, broadening occurs during focusing and diminution of the broadening upon defocusing. Boersch found that broadening occurs in the drifting region. This does not mean, however, that there is no broadening in the acceleration region. At distances of more than 18 cm between cathode and Faraday cage Boersch found no further broadening. This is in agreement with theory (see Fig. 5). Boersch found that AE63c,oincreases with increasing average beam energy ( E ) , which is in agreement with Eq. (37). TABLE V COMPARISON BETWEEN THEORY AND EXPERIMENTS PERFORMED BY BOERSCH~

A E Z ,lk T

i,

i

(Aicni’)

(Aicm’)

Theory

Experiment

0.0000

0.0000

2.00

0.0005 0.0010 0.0020 0.0040 0.0070

0.0125 0.0250 0.0500 0.1000 0.1750

3.10 3.88 5.00 6.68 8.50

2.00 2.53 3.07 4.14 6.28 9.45

a

According to Eqs. (85) and [109].

In Table V a quantitative comparison is made as far as possible between theory and Boersch’s experiments. Boersch measured the current density j g at a distance of 10 cm from the electron gun. His results and Eq. [lo91 agree fairly well if we take the value j = 25j, as current density at the anode. Boersch found that the observed shifts of the mean energy are not in a causal relationship with the broadenings. The energy shifts are caused by the current-dependent space-charge cloud at the cathode and by the currentdependent covering of the cathode with gas, which influence the work function. Both effects cannot cause an appreciable broadening. In his paper Boersch proposes together with Fack (19) that longitudinal space-charge waves are responsible for the energy broadenings. It seems that this attempt to explain the phenomena is not further pursued in electron beams without magnetic fields.

304

BOD0 ZIMMERMANN

After Boersch and Dietrich (17), Hartwig and Ulmer (13) continued the experimental investigation of the energy broadening. They used electron beams generated by tungsten hairpin and Ba-0-cathodes with mean energies between 200 and 1000 eV. The experimental results of Boersch could be confirmed. In addition, Hartwig and Ulmer propagated the idea that broadening is caused by the relaxation process tending to restore thermal equilibrium by collisions between individual electrons. Hartwig and Ulmer obtained the equation

AE,,,

- AE63%(o)= D AE63%(0)j1/3

(189)

on the ground of some plausibility considerations. Although so far there exists no serious derivation of Eq. (189), it proved to describe their and Dietrich’s experiments with surprising precision. It is, however, fair to say that the two parameters AE,,%(O) and D were adapted by regression analysis. By this, AE63%(0)is found to assume values between k T and 2kT and t o decrease with increasing mean energy. This dependence of AE6300(0)was explained by a spectrometer effect similar to the one described by Eq. ( 1 88), assuming that ci increases with the acceleration voltage. As expected, the extrapolated AEs3%(0) of the hairpin cathode was approximately 2kT, whereas that of the plane Ba-0-cathode was approximately kT. Thus, D was found t o increase with increasing acceleration voltage. This was explained by the assumption that according t o the above-mentioned spectrometer effect the transverse internal energy increases with increasing accelerating voltage. Thus, the deviation from thermal equilibrium within the beam is greater for higher beam energies, which was assumed to cause a faster relaxation. The behavior of D just described must be expected from Eq. (37). This shows that the assumption of a spectrometer effect is not necessary in this case. Equation [lo91 describes the experimental results of Hartwig and Ulmer with the same precision as the measurements shown in Fig. 8. Hartwig and Ulmer performed the calculation according to Eq. (39) using, however, the relationship (189) with the result that the relaxation takes place mainly in the acceleration region. This may be valid especially for diverging beams. We will remember, however, that the validity of Eq. (39) is not proven until now. Hartl (41) found that the energy spread of electrons with a mean energy of 20 to 40 keV, which passed a monochromator, increases with current density. Even at small current densities, he obtained energy widths which are twice the initial width of 12 meV. A quantitative comparison with theory is not possible because the relevant values of R , j , and the transverse internal energy u,, cannot be taken from his communicated data. One may assume that the transverse internal energy is only about 12 meV. Then, according to Eq. (37), the considerable broadening can be understood. Hartl found agreement with Eq. (189), but Eq. (37) is likewise in agreement with Hartl’s measurements.

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

305

Simpson and Kuyatt ( 3 ) found energy widths u p to 0.9 eV in beams of 2 to 10 eV mean energy, the electrons being emitted by an indirectly heated dispenser cathode. They condensed their experimental results in the relation

This relation fits very well t o the experiments with small broadenings, in which a linearization of Eq. (34) is justified. AE71Zbeing proportional to at small values, we expect from Eq. (82) w , 2

= AG/Z(O) ‘ (1 + (Y2/2)).

(191)

For the device “Version A, 10 e V ” this formula is in quantitative agreement with Eq. (190), if the beam is assumed to be homogeneous and of constant current density, The dependence of AE,,, on the beam energy, as found by Simpson and Kuyatt, is not in agreement with Eq. (191). This discrepancy may be caused by the special geometry of the experimental set-up. In order to decide whether energy broadening is necessarily accompanied by an energy shift, the energy spread was altered by focusing and defocusing the beam. Within the precision of the experiment (- 5 of the energy width), n o shift was found. This is in agreement with Eq. (141). The energy spread was found to be independent of gas pressure between and Torr. Simpson and Kuyatt emphasized that energy broadening in low-energy dense electron beams is a real effect and not an analyzer artifact. Also it does not arise in the space-charge cloud before the cathode. In contrast t o the results of Simpson and Kuyatt, Beck and Maloney (42) claimed that the measured broadenings in diverging electron beams are an effect of the energy analyzers. They measured the energy distributions in electron beams of 20 eV mean energy and failed in finding any broadening. Even if we assume a homogeneous beam of constant current density we can expect from their beam parameters (cathode temperature = 1100”K, beam lengths z 5 cm, and j < 5 . lo-’ A cm-’) and Eq. [I231 only2y2 < 2.2 . That is to say that, according to our insight into the process ofenergy broadening, Beck and Maloney could not find broadening. Thus, we cannot support their conclusion. Collin and Magnee (43) performed several experiments on electron beams of 100 u p t o 600 eV mean energy. In their paper many numerical values of measured energy widths u p t o 3.26 eV are given. They confirmed the relationship (1 89) and, in addition proposed a slightly different relationship, containing also the variable bias voltage, which fits their measured values somewhat more closely. Their experimental results are, with the same accuracy, also in agreement with Eq. (37) concerning the dependence on current density. A

306

BOD0 ZIMMERMANN

quantitative comparison is not possible because the current density at the anode is not communicated. The energy distributions shown in the paper do not show a symmetrization with increasing energy width. We suppose a spectrometer effect or an incomplete measurement of the distribution to be responsible for this. Speidel and Gaukler (44) measured energy widths in electron beams originating from hairpin cathodes (emission radius R, = 75 pm) and point cathodes (emission radius R, = 5 pm) as function of the beam currents I , and I,, respectively. The great difference in the beam diameter enables the dependence of AE on the beam diameter to be checked. From Fig. 5 of Spiedel and Gaukler (44) the relations (1hRL2 - l,Ri2)(Rh -

(I,

- I,>(Ri, - R,)-’

< 0,

(192)

> 0,

(193)

can be found, assuming constant A E , , , . Observing I yield

= nR2j,these

relations

I5

IC

FIG.9. Calculated energy spread A,EB of divergent electron beams as a function of the initial beam diameter 2R according to Eqs. (82) and [109]; eU,, = 1000 eV, k T - 0.24 eV, z = 10 cm. Curve a shows j = 5 A ~ m - curve ~ ; b shows j = 1 A cm-2.

5 \

W

a

I I

0 2

I 04 R/rnm

I

I

06

08

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

307

a A E ( R , Z ) / a Z > 0 was also measured,

the rela-

Since a A E ( R , j ) / a j > 0 and tions (194) yield

3 AE(R,j ) > 0, aR

a AE(R, I ) aR

< 0.

(195)

This is in agreement with [lo91 and (82) or (131). A quantitative comparison with theory is not possible because the different beam geometries cannot be taken into consideration quantitatively. In Fig. 9 a numerical example for the dependence of AE on the beam diameter is given. C . Electron Beams in Magnetic Fields

Anomalous broad energy distributions were also found in electron beams moving in magnetic fields. Energy spreads were observed which are greater than those in electron beams without magnetic fields. This may be expected because the mean current density and the mean flying time of the electrons can be increased so that any exchange of energy is rendered more efficient. It is probable, however, that interaction between individual electrons is not the only reason for these broadenings, but that macroscopic space-charge and spectrometer or other effects also play an important role. For completeness we report some experiments and possible explanations of the observed effects. Veith (45) measured integrated energy distributions of electron beams moving in parallel electric and magnetic fields. The magnetic field strengths were about 600-1100 G, the acceleration voltage was about 500 V. Veith varied the length of the deceleration field and of the drift region. The measured distributions can best be described if we divide them into the high-energy part covering the electrons with excess energy and the low-energy part covering the electrons with a deficiency of energy. The high-energy parts can be approximated by Maxwellian distributions with temperatures equivalent to 1 eV up to 15 eV (and in one case 50 eV), the low-energy parts by Maxwellian distributions with temperatures equivalent to 30 eV and greater. I t is inadequate to consider the energy spread of these distributions because this would put too much value on the low-energy part, where energy deficiencies of up to 70 eV (and in one case 500 eV) were observed. These deficiencies can be caused by inhomogeneous fields and are not the problem in which we are interested. Veith was mainly interested in the electrons with excess energies. He found that their number and their temperature increase if the length of the deceleration field is increased. Excess energies of 50 eV were measured. This can neither be caused by stationary fields nor by differences of work functions. Veith found space-charge waves in the drift region to be very unlikely to be responsible for the excess energy. He proposed to seek the reason in an

308

B O D 0 ZIMMERMANN

energy exchange between electrons with participation of high-frequency space-charge fields of a wide-spread frequency spectrum. These would be generated by fluctuations of current density and electron velocities, as present at the entrance of every flow. The role of direct electron-electron interaction is not quite clear. Suffice it t o say that the observed temperatures of the electrons with excess energy are, in principle, not too great in order to exclude explanation by direct electronelectron interaction. In electron beams with axial magnetic fields of about 400 G Ulmer and Zimmermann (18) observed energy broadening of the same magnitude ( 55 eV) as in beams without magnetic field, although the analyzed current was greater. The distributions did not symmetrize but approached curves which look like the lower half of a distribution shown in Fig. 7. A spectrometer effect was assumed to be responsible for this. Mihran and Andal (46) reported on excess energies of 100 eV in 0-type electron beams and tested possible explanations without any great success. Later Mihran (47) published that the plasma frequency is a key parameter for the energy spread. In electron beams in crossed electric and magnetic fields, Miller and Dow (48) found electrons with excess energies up to 50 eV, which are distributed according to Maxwellian distributions with temperatures equivalent to about 10 eV. They proposed an interaction which is characterized by a combination of the cyclotron and plasma frequency and the rotational component of the average electron motion. If these are in resonance, a fairly effective energy exchange can occur. The energy exchange should occur in the neighborhood of the cathode, where a coherent rf signal of about 2 t o 3 kHz has been observed. We conclude this experimental review by pointing to a paper of Sutherland (49) who observed plasma effects in very dense beams in poor vacua which led to excess energy spreads. Even from this short discussion, the conclusion can be drawn that in addition to collisions of individual electrons, other effects play a role in the excess energy spread of electron beams in magnetic fields. Although the relaxation process as promoted by Ulmer and Zimmermann may be able to give energy spreads of the reported magnitudes, the relaxation time seems to be too long to be entirely responsible for the observed energy spreads. Thus, these problems well merit further theoretical and experimental research effort.

APPENDIX We show that the function fi(w - u), defined in Section IV, is given by Eq. (47) in the case of a homogeneous beam of constant current density. We

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

309

remember that according t o Eq. (42) the broadened distributionfis given by a convolution offo andf‘,. In the following we denote a convolution by the symbol ‘‘ x.” If the local internal energy at a plane z = z1 > 0 is greater than the internal energy at z = 0 by the amount u1 and if the mean velocity is changed by the amount v l , then the distribution at z = z1 is given by f ( u ; Zl) =fo(t.>xft(v; u1,u1>.

(A.1)

Equation (A.l) is the abbreviated form of Eq. (42). We now assume the plane z = z1 to be a new emission source. In a plane z = zl’ > z1 the internal energy and the mean velocity may have changed by the amounts ul’ and u I ’ , respectively, with respect t o the plane z = zl. Thus

f ( u ; z1‘)

=

Cfo(u)Xfi(u;

u1,

c1)13f1(v;

=fo(u)xCfi(z:;111, vl)xfl(u;

Ul’,

I.,’)

Ul’, r1’)l.

(A4

+ z.1’)

(A.3)

On the other hand, we have f ( u ; Zl’) =fo(v)xf1(u;

u1

+ Ul’,

01

if we consider the development of the beam from z = 0 up to z = zl’without interruption. The last two equations yield u1

+

u1’,

1 ’~ , 1 ’ )= f i ( ~ p U; I

+

u I ’ , 1‘1

f 1 ( u ; I l l , U])xf](u;u1’, u1’) = f l ( u ;

c1

+ c1’).

(A.4)

+ 1’1‘).

(A.6)

Thus, the Fourier transform

.+

1

satisfies the relationship f i ( C p ; 111, 2‘1)

.fi(q; ~

From this, we have for u1

= el‘ = 0

f,(cp; u , , 211

= 0) = ec(q)’ul,

(A.7)

where t ( q )is a function of cp. From (.A.6) and (A.7) it follows that

f,(cp ; I / v ,) . ec(q)Ol’-f,((P;

u1

+ u1 ‘>0 1 ) .

(‘4.8)

Differentiating this equation with respect t o ul’ and putting u1 = 0 we obtain the differential equation

310

BOD0 ZIMMERMANN

Here C(v,, cp) is an as yet unknown function of v, and cp. Since (A.4) is symmetrical with respect t o the interchange u 1 t f u , , u l ' tf u l ' the following relation holds likewise fl(cp; u l , u l )

=

c"(ul, cp)e4(q)CI.

(A. 11)

This yields f,(cp;

u1, u1

= 0) = C"(u1, 40).

(A.12)

Comparison with (A.10) results in f1(cp;

4) = exp(t(cp)u1) . exp(S(cph).

u1,

(A.13)

By reasons of dimension we have (((PI =

-w2,

Rep) =

--acp,

(A. 14)

where am and p are dimensionless constants. Fourier transformation of (A.13) yields

(A.15) From this

=

ipu,.

(A.16)

Comparing this with(46) we obtain with v1 = < w ) - ( 0 ) the equation B = - i. Since u, is the internal energy off,, we obtain CI = ljm. With 2u, = kT, we finally obtain from (A.15)

This is identical with Eq. (47).

ACKNOWLEDGMENTS This work could never have been completed without generous support from the Krupp Research and Development Center, who made it possible for me to attend the Symposium in Gaithersburg in 1969, and thus to have a discussion with several workers on this field and who placed all the necessary facilities at my disposal.

BROADENED ENERGY DISTRIBUTIONS IN ELECTRON BEAMS

31 1

I am also grateful to Professor P. A. Lindsay who was kind enough to read and correct the first draft of the manuscript. He is, however, not responsible for the remaining insufficiencies. I am very indebted to Professor K. Ulmer, University of Karlsruhe, for his support and encouragement of my efforts. I will not forget to thank Mrs. Muller for writing the manuscripts conscientiously in a foreign language.

REFERENCES H. M. Mott-Smith, J . Appl. Phys. 24,249 (1953). P. A. Lindsay, Advan. Electron. Electron Phys. 13, 181 (1960). J. A. Simpson and C. E. Kuyatt, J . Appl. Phys. 37, 3805 (1960). A. E . Ash and D . Gabor, Proc. Roy. Soc. (London) A228,477 (1955). H. A. Wilson, Proc. Roy. SOC.(London)A102,9 (1923). L. H. Thomas, Proc. Roy. SOC.(London) A121, 464 (1928). I. Langmuir and H . A. Jones, Phys. Rev. 31, 390 (1928). L. D. Landau, Soviet Phys. JETP 7, 203 (1937). M. J. Druyvestein, Physica 5, 561 (1938). 10. S. Chapman and T. G . Cowling, “ T h e Mathematical Theory of Nonuniform Gases.” Cambridge Univ. Press, London and New York, 1952. 11. S. Chandrasekhar, Astrophys. J . 94, 511 (1941). 12. B. Zimmermann, Ph.D. Thesis, University Karlsruhe, 1965. 13. D. Hartwig and K . Ulmer, 2.Physik 173, 294 (1963). 14. K. H. Loeffler, Z. Angew. Phys. 27, 145 (1969). 15. H. Boersch, Z. Physik 139, 115 (1954). 16. G . Haberstroh, Z. Physik 145, 20 (1956). 17. W. Dietrich, Z. Physik 152, 306 (1958). 18. K. Ulmer and B. Zimmermann, 2.Physik 182, 194 (1964). 19. H . Fack, Physik. Verhandl. 6, 6 (1955). 20. W. Veith. 2. Angew. Phys. 152, 306 (1955). 21. B. Epsztein, Compt. Rend. 246, 586 (1958). 22. F. Lenz, Proc. 4th Intern. Conf. Electron Microscopy, Berlin, 1958 1, 39. Springer, Berlin, 1960. 23. P. Schiske, Proc. 5th Intern. Conf. Electron Microscopy, Philadelphia KK-9. Academic Press, New York; Physik Verhandl. 12, 143 (1961). 24. B. Zimmermann, Record 10th Symp. Electron, Ion, Laser Beam Technol. Natl. Bur. Stand. Gaithersburg, 1969. 25. K. Ulmer and B. Zimmermann. in Proc. 8th Ann. Electron Laser Beam Symp., (G.I. Haddad, ed.), p. 449. Univ. of Michigan, Ann Arbor, Michigan, 1966. 26. Jahnke-Emde-Losch, “Tables of Higher Functions,” p. 26. Teubner, Stuttgart, 1960. 27. M. Fischer, Verhandl. Deut. Phys. Ges. [6] 4, 276 (1969); J . Appl. Phys. July 1970; in “ Report K 69-03 of Bundesministerium fur wissenschaftliche Forschung,” Germany, Symposium on One-Particle Distribution Functions in Plasmas (M. Fischer and K. Wieseniann, eds.), pp. 21-1 to 21-13. Braun, Karlsruhe 1969. 28. S. Chandrasekhar, Ann. N . Y . Acad. Sci. 45, 131 (1943). 29. M. Livingston and J. P. Blewett, “Particle Accelerators,” p. 316. McGraw-Hill, New York, 1962. 30. H. Shelton, Phys. Rev. 107, 1553 (1957). 31. P. Kisliuk, Phys. Rev. 122, 405 (1961). 32. C. Herring and M. H. Nicols, Rev. Mod. Phys. 21, 185 (1949). 1. 2. 3. 4. 5. 6. 7. 8. 9.

312 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

BOD0 ZIMMERMANN

W. B. Nottingham, Phys. Rev. 41, 793 (1932). L. R. Koller, Phys. Rev. 25, 671 (1925). H. Rothe, 2.Physik 37, 414 (1926). G. Koslov, W. G. Terpigorev, A. S. Shapovalov, and L. V. Vinenko, Zh. Tekh. Fiz. 38, 2061 (1968). W. B. Nottingham, in “Handbuck der Physik” (S. Fliigge, ed.), Vol. XXI, p. 1. Springer, 1956. K.-J. Hanzsen and R. Lauer, Z. Naturforsch. Ma, 214 (1969). W. H. J. Andersen and A. Mol, Proc. 4th European Regional Conf. Electron Microscopy, Rome, p. 349 (1968). D. Hartwig and K. Ulmer, 2. Angew. Physik 15, 309 (1963). W. A. M. Hart1,Z. Physik 191,487 (1966). A. H. W. Beck and C. E. Maloney, Brit. J. Appl. Phys. 18, 845 (1967). J. E. Collin and A. MagnBe, Bull. SOC.Roy. Lieges 9-10, 522 (1967). R. Speidel and K. H. Gaukler, Z. Physik 208,419 (1968). W. Veith, Z. Angew. Phys. 7, 437 (1955). T. G. Mihran and B. K. Andal, ZEEE Trans. Electron Deuices ED-12, 208 (1965). T. G. Mihran, J. Appl. Phys. 38, 159 (1967). M. H. Miller and W. G. Dow, J. Appl. Phys. 32, 274 (1961). A. D. Sutherland, IRE Trans. Electron Devices ED-7, 268 (1960).

Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed.

A Abranison, F. P . , 81(3), 83(3), 112 Ackermann, H., 178(148), 204(237), 227, 229 Adlington, R. H., 156(79), 225 Adrianova, 1. I., 174,227 Agarbiceanu, I., 178(145), 227 Alekseev, V. A,, 195(201), 228 Alfano, R . R., 162(106), 226 Ali, A. W., 191,228 Allan, W.D., 223(342), 231 Allen, M.A,, 6(20), 9,ll(20, 53), 12(55), 60 (55), 76, 77, 78 Allis, W. P., 5(16), 76 Allison, J., 10, 77 Andal, B. K . , 308,312 Andersen, W. H. J., 299,312 Anderson, D. K., 196(213), 215(213), 228 Anderson, R. A., 184(159a), 185(159a), 227 Anderson, T., 167(11 la), 226 Anway, A. R., 81, 82,113 Archanibault, Y . ,210(269), 230 Arshadi, M., 80(5, 13), 81(5, 13, 22), 84(5,

22), 91(22), 92(13), 93(5), 106(22), 107(5), 112, 113 Ash, A. E., 258, 311

Bates, D. R., 122(12), 224 Baumann, M., 220(307), 231 Bay, Z., 138(60), 225 Beck, A. H. W., 25(61), 78, 305,312 Beckey, H.D., 80, 81, 82(1, 2), I12 Bennett, R . G . , 127(36, 37,38,39,40,41), 128,132,184(36), 224 Bennett, W. R., 123(16), 127,130,133(46), 134, 137, 138(47), 144, 147, 149(46), 150(47), 152,217(46), 218(285), 224, 225, 230 Bennett, W. R., Jr., 130, 133, 137,224 Berezin, A. K., 7,10, 76, 77 Berezina, G . P., 7(25,26), 10(44), 76, 77 Berg, R . A,, 170(120), 174(120, 137), 175 (1 20). 226, 227 Bergstroni, I., 162(102), 165(102), 167(11 I , I l l b ) , 226 Berkner, K., 162(103), 167(103), 226 Bernian, P.R . , 191,217(185a), 228 Berry, H.G . , 167(112), 184(158), 226, 227 Berry, M. U., 173(132), 227 Bers, A,, 5(16), 76 Bevc, v., 5, 75 Bickel, W. S . , 159(90), 161(97), 162(102),

165, 167(111, 112), 168(115), 184(158),

B Bailey, A . D., 91(25),96(25,27), 99(25), 105

(25),113

226, 227 Biechler, C . S., 6(20), Il(20, 49), 12(55), 60 (59, 76, 77, 78 Birks, J. B . , 174,175,227 Birky, M.M., 218(292),230 Bitter, F., 204,209(234), 229 Blanc, F. J., 221(324), 231 Blewett, J . P., 296(29), 311 Blumle, L. J., 99,113 Boersch, H., 258,271(15), 298(15), 299,302,

Bailey, E. A., 172(129), 175(129), 227 Bakos, J . , 127(35a), 152,224, 225 Ralashov, V. V., 223(343), 231 Barrat, J. P., 150, 158, 198(220), 209(225), 311 21 I , 216(220), 225, 229 Bodt, H., 7(30), 76 Barrat, M . , 210(267), 220(306), 230, 231 Bogdanov, E. V., 5(14), 6(14), 7,8,23(60), Bashkin, S., 159(89, 95), 161,167(112), 184, 26(62), 75, 76, 78 _776, 227 Bohni, D., 3, 10, 75 31 3

314

AUTHOR INDEX

Bohme, D. K., 109(38), IiO(38), 113 Bolotin, L. I., 7(25), 10(44), 76, 77 Bonczyk, P. A., 218(291), 230 Born, M., 173(131), 227 Boyd, G. D . , 7, 76 Brackett, C. A., 10, 12, 77 Branch, G. M . , 36(65), 78 Brannen, E., 156, 225 Breene, R. G . , 191, 228 Brehm, B., 172, 178(127), 227 Breit, G., 202, 222(336), 229, 231 Brewer, L., 171, 172(120), 174(137, 138), 175 (120), 177(138), 226, 227 Brewer, R. G., 172(120), 174(120), 175(120), 226 Bridgett, K. A., 135(55), 225 Brinkniann, U., 208, 229 Brochard, J., 194, 228 Brornander, J., 167(1 1 I , 11 1b), 226 Bromer, H. H., 221(330), 222(330), 231 Brossel, J., 204,209(234), 210(262), 229, 230 Brown, S . C., 3, 75 Buchet, J. P., 159(93, 93a), 161(93a), 167 (93a), 226 Buchsbauni, S. J., 5(16), 76 Buchta, R., 167(111, l l l b ) , 226 Budick, B., 197(219), 207, 229 Burhop, E. H. S . , 210(261), 230 Burke, R. R . , 105(32), 110(42), 113 Bykhowski, V. K., 155(77), 225 Byron, F. W., 191, 198(223), 204(223), 208 (223), 214(283), 215(283), 216(223, 283), 228, 229, 230

C Camhy, C., 156, 225 Carleton, N. P., 221(324, 331, 332), 222(331, 332), 231 Carovillano, R. L., 202, 229 Carrington, C. G . , 209(259), 21 I , 213(259), 230 Carver, T., 21 I , 230 Chaika, M . P., 178(146), 227 Chandrasekhar, S., 258, 295, 3 / 1 Chapman, G. D., 207, 229 Chapman, S . , 258, 311 Chen, C. L., 197, 229 Chenevier, M., 210(272), 21 1(276), 230

Chernov, Z . S . , 26(62), 78 Cho, S. K., 54(67), 78 Chodorow, M . , 32(63), 78 Chorney, P., 3 , 6(20, 22), 11(49), 75, 76, 77 Chow, K. K., 11(50), 77 Chupp, E. L., 166, 226 Chutjian, A,, 174(138), 177, 226 Codling, K . , 223, 231 Cohen, M., 168, 169(116a), 226 Cohen-Tannoudji, C., 220(309), 231 Cojan, J. L., 209(255), 229 Colegrove, F. D . , 202(228), 229 Collin, J. E., 305, 312 Collins, J . G., 81(6), 113 Comniins, E. D., 156, 225 Condon, E. U . , 117(6), 223 Conte, S. D., 186(166), 227 Cook, J. S., 54(66), 78 Cooper, J., 124(22), 191, 197(22), 224, 228 Cooper, J. W., 196(209), 223, 228, 231 Cooper, W. S., 162(103), 167(103), 226 Copeland, G. E., 184(159a), 185(159a), 227 Copley, G., 153(69, 70, 71, 72), 154(69), 155 (70, 71), 178(69), 225 Cordover, R. H., 218, 230 Corney, A,, 174(136), 209(259), 211, 213 (259), 227, 230 Corrigan, S. J. B., 125(32), 224 Cowling, T. G., 258, 311 Crawford, F. W., 8(32), 76 Crepeau, P. J., 5( I5), 76 Crosley, D . R . , 206, 229 Cunningham, P. T., 171, 172(122, 123), 174 175(122), 176(122), 207(122), 226, 227 Curnutte, B., 191, 228 Curzon, F. L., 124, 224

D Dalby, F. W., 127(36, 37, 38, 39, 40, 41), 128, 132, 184(36), 224 Dalgarno,A., 168(114, 114a),169(116a), 194 (195), 222(337), 226, 228, 231 Damaschini, R., 194, 228 Danigaard, A., 122(12), 224 Danby, G. T., 239(6), 256 Dattner, A,, l0(46), 77 Davies, J. T., 186(165), 190, 227 Davis, C. C., 145(61a), 225

315

AUTHOR INDEX

Davis, S. P., 178(144), 227 Dawson, J., 4(10), 75 Dayton, 1. E., 127(39), 224 Debye, P., 173, 227 Decomps, B., 204(233), 208, 209(257), 229, 230 Demtroder, W., 155, 156(75), 172(127), 178 (126, 127), 225, 227 Denis, A . , 159(93, 93a), 161(93a), 167(93a), 226 D e Pas, M., 80(15), 85(15), I13 Derblom, H., 221(320), 231 Descoubes, J. P., 210(269, 271), 230 Desesquelles, J., 159(93, 93a), 161(93a, 96), 167(93a, 112), 184(96, 158), 226, 227 D e Zafra, R. L., 198(224), 206(239), 216 (224), 229 Dietrich, W., 258, 304, 311 Dodd, J. N., 156(78a), 225 Dotchin, L. W., 166(110), 226 Dow, J. D., 196(212), 228 Dow, W. G., 308,312 Doyle, P. H., 11, 78 Drake, G. F. W., 222(337), 231 Dressler, K., 183(155, 156, 157), 184(156), 227 Druyvestein, M. J., 258, 311 Dufay, M . , 159(93, 93a), 161(93a, 96), 167 (93a), 184(96), 226 Dufayard, J., 21 1(276), 230 Dumont, A. M . , 156, 225 Dumont, M., 204(233), 208, 209(257), 217 (233), 129 Dunn, D. A,, 60, 78 Durden, D. A., 108(37), 113 D’yakanov, M. I., 150, 158, 21 1, 225 Dyson, D. J., 174, 175, 227

F Fack, H., 258, 303, 311 Fainberg, Ya. B., 7(25, 26), 10(44), 76, 77 Fairchild, C . E., 129(41a), l95,196(41a), 224 Fano, U., 223, 231 Faure, A,, 210(270, 273), 230 Fehsenfeld, F. C., 90(36), 107(36), 108, 109, 110(38), 113 Feinstein, J., 11, 77 Feld, M. S., 219, 230 Feldman,, P., 223(340, 341), 231 Feneuille, S., 152, 225 Ferguson, E. E., 90(36), 107(36), 108, 109 (38), 110, 113 Ferrari, R. L., 6(21), 76 Field, F. H., 81, 113 Field, L. M . , 7(23), 76 Fink, E., 184(159), 227 Fink, U., 159(91), 226 Fischer, M., 293, 311 Foley, H. M . , 191, 228 Fontana, P., 195(199), 228 Fork, R. L., 219, 287, 230 Foster, E. W., 124(21), 224 Fohler, R. G . , 125(32, 33), 184(159a), 185, 224, 227 Franken, P. A,, 202(228), 229 Franklin, J. L., 81(18), 113 Freud, R. S., 221(333), 231 Frey, J., 61(73), 78 Fried, B. D . , 186(166), 227 Friedman, L., 80(15), 85(15), 113 Fry, E. S., 196(204, 205), 209(204, 205), 214 (204), 215(204), 228 Fursova, E. V., 193(192), 195(192), 196 (192), 228 Futrell, J. H., 81(3), 83(3), 112

E G Ederer, D . L., 223(339), 231 Elton, R . C., 222(335), 231 Eneniark, E. A,, 156(78a), 172, 175, 225, 226 Epsztein, B., 258, 311 Evans, W. F. J., 110, 113 Everhart, T. E., 5, 75 Eyring, H,. 106(33), 113 Ezekiel, S . , 219(300), 230

Gabor, D., 258, 311 Gallagher, A,. 156(78a), 172, 175, 204(232), 207, 208(245, 246), 213(245, 246), 215 (232), 216(245, 246), 225, 226, 229 Galleron-Julienne, C., 210(271), 230 Garstang, R. H., 122(9, lo), 220(9), 223 Gaukler, K. H., 306, 312 Geidne, B., 9, 12, 77

316

AUTHOR INDEX

Geiger, J., 196(203, 206), 228 Getty, W. D., 6(17), 76 Gilbody, H. B., 210(261), 230 Gioumousis, G., 107(35), 113 Giroud, M., 21 1(277), 230 Givens, M. P., 175, 227 Glennon, B. M., 123(18, 20), 183(18), 220 (18), 224 Golant, V. E., 8, 77 Gold, A , , 196(207), 228 Goldberg, R. A , , 99, 113 Golub, S., 80(16), 91(16), 113 Comer, R., 81(17), 113 Good, A,, 108(37), 113 Gorbatenko, M. F., 12(58), 78 Goschen, R. J., 198(224), 216(224), 229 Could, R. W., 4, 5(11), 7(23), 9, 10(47), 75, 76, 77

Goullet, G., 214(281), 230 Green, H. R., 196(202), 228 Grieni, H. R., 124(23), 191, 222(335), 224, 23 I Grilli, M., 256(10), 256 Groendijk, H., 7(29), 76 Gross, E. P., 3, 10, 75

Heron, S . , 130, 131, 132, 224 Heroux, L., 159(94), 161, 162, 163, 164, 168 (99), 184, 226 Herring, C., 298(32), 311 Hertel, I. V., 124, 224 Herzberg, G., 120(8),223 Hesser, J. E., 179(151, 152, 153), 181, 182, 183(155, 156, 157), 184, 185(153), 227 Hesstvedt, E., 110(41), 113 Hinteregger, H. E., 161, 226 Hirschfelder, J . O., 106(33), 113 Hogg, A. H., 81(8, 9, 21), 84(21), 91(21), 94 (8, 91, 113 Hollstein, D. C., 161(96a), 226 Holmes, J . R., 219, 230 Hulpke, E., 178(128), 227 Holstein, T., 150, 155, 225 Holt, H. K., 219(293, 294), 230 Holzberlein, T. M., 125, 184(159a), 185 (159a), 224, 227 Hughes, R. H., 126, 127, 149, 224 Hummer, D. G . , 186(167), 227 Hunt, F. R., 156(79), 225 Hunt, W. W., Jr., 111(44), 113 Hunter, D. M., I10(40), 113 Hutchinson, D. A , , 195(198), 228

H I Haberstroh, G., 258, 311 Habib, E. E., 153, 225 Hahn, H., 239(5), 256 Hahn, W. C., 8, 76 Haniberger, S. M., 174, 227 Hanieka, H . F., 195(198), 228 Handrich, E., 208(247), 209(247), 229 Hanzsen, K.-J., 299, 302, 312 Happer, W., 191(188), 204(188), 209(260), 212, 213(188), 214(260), 228, 230 Harang, L., 221, 231 Hargrove, L. E., 219(301), 230 Hartl, W. A. M., 304, 312 Hartwig, D., 258,267,271(13), 298(13), 299, 300(13), 302, 304, 311, 312 Haynes, R. M., 81(6, 9), 113 Heald, M. A,, 15(59), 78 Hearn, D. D., 195(199), 228 Hedvall, P., 6, 61(19), 76 Heitler, W., 117(5), 185(5), 223 Henri, V. P., 138(60), 225

Imhof, R. E., 158, 225 Inghram, M. G.. 81, 113 Isler, R. C . , 206(241a), 22Y Issacson, I ., 222(332a), 231

J

Jacobson, C. H.. 125(32, 33), 224 Jacquinot, P., 187(168), 227 Jahnke-Enide-Losch, 272(26), 311 James, C . G., 172(120), 174(120), 175(120). 226

Javan, A,, 218(291), 219(295, 296, 297, 298), 230

Jessen, K. A., 167(11la), 226 Jeunehomme, M., 184, 227 Jones, H . A ,, 258, 311 Jones, 0. C., 172, 226

317

AUTHOR INDEX

K

Kanner, H., 138(60), 225 Kaplan, S. N., 162(103), 167(103), 226 Karplyuk, K. S . , 8, 77 Karstcnsen, F., 178(142, 143), 227 Kastler, A., 204, 210(262), 220(309), 229, 230, 231 Kaul, R. D., 156,219, 22.5, 230 Kay, L., 159, 225 Kay, R . €3.. 127, 149, 224 Kebarle, P., 80(5, 13), 81(8, 9, 23) 84(5. 21, 22), 88, 89(11), 91(21, 22), 92(13), 93(5), 94, 95(23), 106, 107, 108(31, 37), 112, 113 Keegan, T., 5(15), 76 Kernahan, J. A,, 221(325), 231 Kibble, B. P., 116(3), 153(69, 70, 71, 72), 154, 155(70, 71), 178(69), 209(254), 223, 229 Kindlniann, P. J . , 123(16), 133(46, 47), 137 (47), 137, 138, 144(46), 147(47), 149(46, 47), 150(47), 152, 217, 224, 225 King, R. B., 161(98), 166(98), 226 King, T. A . , 135(55), 145(61a), 225 Kino. G. S.. 10, 11(53), 12, 77, 78 Kisliuk, P., 298, 311 Kislov, V. Ya., 5(14), 6(14), 7(24), 8(24), 23 (60), 26(62), 75, 78 Klapish, M., 152(66a), 225 Klose, J . Z., 134(48, 49, 50), 134, 141, 150 (48), 152, 195(49), 225 Knewstubb, P. F., 81(19,20), 86(19), 88(20), 89(19), 113 Kiiox, K. S . , 196(207, 210, 212), 228 Kocher, C . A., 156, 225 Koenig, E., 152(66a), 225 Kokubun, H., 175, 227 Koller, L . R., 298(34), 312 Kompfner, R., 54(66), 78 Kondratiev, V. N., 155, 156(76), 225 Konrad, G. T., 35(64), 78 Korolev, F. A. , 193(192, 193), 195(192, 193), 196(192, 193), 228 Korvalski, J., 204(238), 229 Koslov, G., 298(36), 312 Koster, G. F., 196(208), 228 Krasnow, M. E., 124(29, 30), 224 Krause, L., 153(69, 70, 71, 73), 154(69), 155 (70, 71), 178(69), 207, 225, 229

Kretzen, H., 208(248, 252), 229 Kronlund, S . , 11(48), 77 Kuhn, H. G., 188(170, 172), 189(176, 177), 190(170), 194, 195, 196(172), 228 Kukurezianu, I . , 178(145), 227 Kuyatt, C. E., 134, 225, 258, 305, 311

1,

Lamb, W. E., 191, 217(185a), 218(285), 228, 230 Landau, L. D., 258, 311 Landman, A , , 198(221), 229 Lange, W., 208(248, 252), 229 Langevin, M. P., 107, 113 Langmuir, I., 3, 75, 258, 311 Laniepce, B., 209(256), 229 Lassettre, E. N., 124, 224 Laucr, R., 299, 302, 312 Lawrence, G . M., 134(51, 52, 53), 135, 137, 140, 145, 150(51, 52), 151(53), 152(51, 52), 169(150), 178, 179, 181(149), 182, 183 (149), 185, 195(51, 52), 196(51, 52), 225, 227 Layzer, D., 122(10), 223 Lecler, D., 198(227), 212, 229 Lecluse, Y . , 209(255), 229 Lenz, F., 258, 311 Leventhal, J. J., 80(15), 85(15), 113 Levin, L. P., 6, 76 Levinson, I. B., 122(1 I), 224 Levitskii, S. M., 8, 61(72), 77, 78 Levitt, M., 223(341), 231 Lewis, E. L., 188(172, 174), 189(177), 193, 194, 195(174, 197), 196(172, 174), 197, 228 Lewis, M . R., 159(92), 226 Lewis, R. R . , 202(228), 229 Libernian, S., 152(66a), 225 Lichten, W., 221(326), 231 Llewellyn, E. J., 110(40), 113 Lilly, R . A., 219, 230 Lincke, R . , 196(202), 228 Lindsay, P. A , , 258, 271(2), 297, 311 Lineberger, W. C . , 80(14), 90(24), 91(24), 109(24), 110(14), 113 Link, J . K., 171, 172(121, 122), 173, 174 (138), 175(122), 176(122), 177(138), 178 (121), 207(122), 226

318

AUTHOR INDEX

Lipeles, M., 222(334), 231 Liszt, H. S., 134(52), 150(52), 152(52), 195 (52), 196(52), 225 Liventseva, I. F., 8(36), 77 Livingston, M., 296(29), 311 Loeffler, K. H., 258, 279, 286, 311 Lombardi, M., 21 I , 230 Looney, D. H., 3, 75 Lorents, D. C . , 161(96a), 226 Lucas-Tooth, H. J., 189(176), 228 Lumry, R., 175, 227 Lundin, L., 162(102), 165(102), 167(111, 11 1b), 226 Lurio, A,, 116, 198(222, 224, 225, 226), 204 (232), 207(242), 208, 209(260), 213(222), 214(260), 215(226, 232), 216(222, 224, 226), 217(226), 223, 229, 230 Luther, J., 208(252), 229 Lutz, B. L., 179(152), 184, 227 Lyapklo, Yu. M., 10(44), 77

Meroz, I., 124(25), 224 Merrill, J . H., 3(6), 75 Metcalf, H., 206(239), 229 Meunier, J . , 191(187), 213(187), 228 Mihran, T. G., 36(65), 78, 308, 312 Miles, B. M., 123(19, 20a), 224 Miller, M. H., 308, 312 Miller, W. J., 105(32), 113 Milne, E. A,, 155, 225 Mitchell, A. C. G., 123, 159(17), 169, 197 (17), 224 Mol, A., 299, 312 Molnar, J. P., 220(302), 230 Morack, J. L., 129(41a), 195, 196(41a), 224 Moruzzi, J. L., 80(12), 81, 91(12), 113 Moser, H., 197, 229 Mott-Smith, H. M., 258, 266, 279, 286, 293 ( I ) , 311 Mrozowski, S., 220(315), 231 Muller, A,, 175, 227 Munson, M. S. B., 81(18), 83(4), 112, 113

M McConkey, J. W., 221(325), 231 McDerniott, M. N., 198(223), 204(223), 208 (223), 214(283), 215(283), 216(223, 283), 229, 230 McElroy, M. B., 110, 113 McFarlane, R. A,, 218, 230 McWhirter, R. W. P., 130(42, 43), 131(42, 43), 132(43), 224 Madden, R. P., 223(339), 231 Maddix, H. S., 6(22), 12(55), 60(55), 76, 78 Madore, R. J., 6(22), 76 Magnee, A,, 305, 312 Maloney, C. E., 305, 312 Marcuvitz, N., 54(68), 56(68), 78 Marion, J. B., 165(108), 226 Markova, G. V., 178(146), 227 Marlow, W. C., 124, 224 Marshall, A,, 206(239), 229 Martinson, I., 162(102), 165(102), 167(11I , 1 I 1 b), 226 Martnez-Garcia, M., 161(98), 166(98), 226 Massey, H. S. W., 210(261), 230 May, A. D., 210(268), 230 Mercer, G. N., 123(16), 127(47), 133(47), 134(47), 138(47), 144(47), 147(47), 149 (47), 150(47), 224, 225

N Nakamura, J . K., 162(107), 226 Narcisi, R. S., 91(25, 26), 95(30), 96(25, 26, 27), 99(25), lOO(29, 30), 103(31), 104(31), lOS(25, 29), 113 Neal, R. B., 247(8), 256 Nedelec, O., 210(270), 214, 230 Neufeld, J . , 11, 78 Newton, T. D., 138, 225 Nicholas, D. J., 223(342), 23Z Nicols, M. H., 298(32), 311 Nicholls, R. W . , 122(9a), 124(9a), 130, 156 (79), 223, 224, 225 Nichparenko, W., 60(71), 78 Nickel, J. C . , 10, 77 Nikitin, A. A,, 122(11), 155(77), 224, 225 Northcliffe, L. C., 165, 226 Nottingham, W. B., 298(3?, 37), 312 Novick, R . , 198(221, 222, 223), 204(223), 207(242), 208(223), 213(222), 214(283), 215(283), 216(222, 223, 283), 222(334, 334a), 223(340, 341), 229, 230, 231 Noxon, J. F., 221(327), 222(327), 231 Nunnink, H. J. C . A., 7(27), 8(27), 76 Nussbaum, G. H., 156, 157, 158, 225

319

AUTHOR INDEX

0 Ockman, N., 162(106), 226 Odintsov, V. I., 193(192,193), 195(192,193), 196(192, 193), 228 Oldenberg, O., 221(324, 331), 231 Omholt, A., 221(317, 318,321), 231 Omont, A., 191(186), 210(269), 211, 213 (187),228, 230 Osberghaus, O . , 171, 172(127), 178(127), 226, 227 Osherovich, A. L., 152,225 Ottinger, C . , 171,226

P Pack, J. L., 220(305), 231 Palumbo, L. J., 222(335), 231 Pancharatnam, S., 209(254), 229 Panofsky, W., 116(1), 223 Parker, J. V., 10(47), 77 Parkinson, E. M., 168(114a), 226 Parks, J. H., 219,230 Pashitskii, E. A,, 12(56, 57), 78 Paul, E., 178(128), 227 Paul, W., 178(128), 227 Paulson, K. V., 221(319), 231 Pearl, A. S., 222(337a), 231 Pebay-Peyroula, J. C . , 210(263, 264, 266, 269,270), 211,213(263), 214(281), 230 Pegg, D. J., 166(110), 226 Pekeris, C . L., 119(7), 194,196(7), 223 Pendleton, W. R., 126,224 Perel, V. I., 150,158, 211,225 Perry, B.W., 214(283), 215(283), 216(283), 230 Peterson, J. R., 161(96a), 226 Phelps, A. V., 80(12), 81(12), 91(12j113, 220(305), 229, 230, 231 Phillips, L. F., 221(328), 222(328), 231 Phillips, M., 116(1), 223 Philpott, M.R., 195(200), 228 Pierce, J. R., 3, 75 Pipkin, F. M . , 156,157,158,225 Pollack, M.A., 218,219(301), 230 Popesku, I., 178(145), 227 Poulizac, M. C., 161(96), 184(96), 226 Priou, M.. 210(269), 230 Prodell, A. G., 239(5), 256

Puckett, L. J., 80(14), 90(24), 91(24), 109 (24), 110(14), 113 Pyle, R. V., 162(103), 167(103), 226

Q Quate, C . F., 54(66), 78

R Radford, H. E., 197(218), 229 Ramo, S., 56,78 Read, F. H., 158,225 Rees, J. D., 175,227 Rhoderick, E. H., 130(42, 43), 131(42, 43), 132(43), 224 Ritter, G. J., 204(236), 229 Rogers, P. C . , 144,225 Rollefson, G . K., 172(129), 175(129), 227 Rose, M. E., 202,229 Rosenblatt, G. M., 172(120), 174(120, 137), 175(120), 226, 227 Ross, K . J., 124,224 Rothe, H., 298(35), 312 Rowe, J. E., 9,lO(38, 39), 24,29,31, 77 Rubinowicz, A,, 220(316), 231 S Sakhara, I. E., 8(36), 77 Saloman, E. B., 191(188), 204(188), 212,213 (188), 214(283), 215(283), 216(283), 228, 230 Sandle, W. J., 145(61b), 225 Sands, R.I*., 202(228), 229 Sasiela, R.,60(70), 78 Savage, B.D., 169(150), 179,182,183,184, 227 Scarborough, J., 80(5, 13), 81(5, 13,22), 84 ( 5 , 22), 91(22), 92(13), 93(5), 106(22), 107 (9,112, 113 Schatzman, E., 156,225 Schearer,L.D.,220(310,311,312,313,314), 231 Schiff, B., 119(7), 194,196(7), 223 Schiff, L. I., 117(4), 223 Schiske, P., 258,311 Schlossberg, H. R., 219(296, 297), 230

320

AUTHOR INDEX

Schrnieder, R. W., 178(147), 209(260), 214, 216(147), 227, 230 Schramm, J., 178(142), 227 Schultz, H., 197, 229 Schwartz, S. E., 162(107), 226 Schweitzer, W. G., 218, 230 Schwenker, R. P., 184(160, 162), 227 Seaborg, G. T., 236(4), 256 Searles, S. K., 80(5), 81(5, 22), 84(5, 22), 91 (22), 93(5), 94(23), 95(23), 106(22), 107(5), 112, I13 Sears, F. W., 173, 227 Seka, W., 124, 224 Senashenko, V. S., 223(343), 231 Series, G. W., 116(3), 174(136), 197(216), 204(236), 223, 229 Shahin, M. M., 81, 84(10), 89(10), 90(10), 113 Shapiro, J., 222(336), 231 Shapovalov, A. S., 298(36), 312 Shashurin, I. P., 61(72), 78 Shelton, H., 298, 311 Shemansky, D. E., 221(332), 222(332), 231 Shepherd, G. G., 221(319), 231 Sheridan, J. R., 161(96a), 226 Shimoda, K., 208(251), 229 Shortley, G. H., 117(6), 223 SiecK, L. W., 81, 83(3), 112 Silverman, S. M., 124(29, 30), 224 Simpson, J. A., 134, 225, 258, 298(3), 305, 311 Simpson, J. E., 8, 9, 60(71), 76, 78 Skwerski, P. R., 184(159a), 185(159a), 227 Smith, D. S., 220, 231 Smith, M. W., 123(18. 19), 183(18),220(18), 224 Smith, P. W., 218(289), 230 Smith, W. W., 207, 208(245), 213(245), 216 (245) Smullin, L. D., 3, 6(17), 75, 76 Sobelman, I. I., 195(201), 228 Sorensen, G., 167(111a), 226 Spalter, J., 11(53), 78 Speidel, R., 306, 312 Spieweck, F., 221(330), 222(330), 231 Stacey, D. N., 188(174), 189(177), 191, 195 (174), 196(174), 228 Stafford, E. F., 172(120), 174(120), 175(120), 226 Stamper, J. H., 155(78), 225

Stanley, R. W., 193, 228 Statz, H., 196(208), 228 Steel, W. H., 187(169), 228 Steiner, B., 80(16), 91(16), 113 Steudel, A., 208(247, 248, 252), 209(247), 229 Stevenson, D. P., 107(35), 113 Stewart, A. L., 122(9a), 124(9a), 194(195), 223, 228 Stoffregen, W., 221(320), 231 Stone, J., 186, 227 Stover, H. L., 8, 11(53), 12, 77, 78 Strivanek, R. A., 111(43), 113 Stuhl, F., 221(323), 231 Stupak, V. G., 7(25), 76 Sturrock, P. A., 3, 75 Sugden, T. M., 81(19), 86(19), 89(19), 113 Sunderland, J., 138, 225 Susskind, L., 32(63), 78 Sutherland, A. D., 308, 312 Szigeti, J., 127(35a), 152, 224, 225 Szoke, A., 218, 230

T Tang, C . L., 196(208), 228 Targ, R., 6, 76 Taylor, H. S., 106(33), 113 Tchernov, Z. S., 7(24), 8(24), 76 Tekou, B., 323(343), 231 Terpigorev, W. G . , 289(36), 312 Thomas, L. H., 258, 311 Thomassen, K . I., 60(71), 78 Tickner, A. W., 81(20), 88(20), I13 Tittel, K . , 220(308), 231 Tolk, N., 222(334), 231 Tomlinson, W. J., 217(283b), 230 Tonks, L., 3, 75 Trivelpiece, A. V., 4, 5(11), 75 Trowbridge, C . W., 223(342), 231 Tsao, C . J., 191, 228 Tsukakoshi, M., 208(251), 229 Turner, R., 150, 196(66), 220, 225, 231

U Ulmer, K., 258, 267, 271(13), 288, 292(18), 298(13), 299, 300(18), 301, 302, 304, 308, 311, 312

32 1

AUTHOR INDEX V

Vallance Jones, A., 110(40), 113 Vasil’ev, A. A., 244(7), 256 Vasiliu, V., 178(145), 227 Vaughan, J. M., 186(165), 188(170,171, 173, 175), 189(177), 190(170, 173), 193(173), 194, 195(197), 196(171, 173), 227, 228 Veith, W., 258, 307, 311, 312 Veksler, V. I., 234(1), 256(1), 256 Verolainen, Y . F., 152, 225 Victor, G. A., 222(337), 231 Vinenko, L. V., 298(36), 312 Vinogradov, A. V., 195(201), 228 Vlaardingerbroek, M. T., 7(30, 31), 8, 76

Wells, W. C., 206(241a), 229 Wentink, T., 184(161), 222(332a), 227, 231 Whaling, W., 161(98), 166(98), 226 Wharton, C. B., 15(59), 78 Whinnery, J. R., 56, 78 White, J. A., 218(292), 230 Wiese, W. L., 122(13), 123(14, 18, 19, 20, 20a), 167(13, 110a) 183(18), 184, 220(18), 224 Williams, 0. M., 145(61b), 225 Williams, W. L., 196(204, 205), 209(204, 205), 214(204), 21 5(204), 228 Wilkinson, P. G., 196(211, 213a), 197, 228, 229 Wilson, H. A,, 258, 311 Wolf, E., 173(131), 227 Wolff, R. J., 178(144), 227

W

Wahlquist, H., 209(258), 230 Wallenstein, R., 208(247), 209(247), 229 Walther, H., 208(247, 248, 252, 253), 209 (247), 229 Wang, C . H., 217(283b), 230 Wares, 0. W., 159(92), 226 Watanabe, T., 191, 228 Webb, H. W., 3(6), 75 Weimer, K . R. U., 7(27, 28, 29, 30, 31), 8(27), 76 Weiss, A . L., 194(196), 228 Weiss, A. W., 167(110a), 168, 169, 226 Weiss, R., 219(300), 230 Wdlge, K. H., 184(159), 221(323), 227, 231

Z Zare, R. N., 206, 229 Zernansky, M. W., 123, 159(17), 169, 197 (1 7), 224 Zhilinskii, A. P., 8(36), 77 Zirnmermann, B., 258, 264, 288, 292(18), 299, 300(18), 301, 308, 311 Zimnoch, F. S., 159(92), 226 Ziock, K., 123(15), 171, 172, 224, 226 Zipf, E. C., 221(322, 329), 222(329), 231 Zolla, A., 80(5), 81(5, 22), 84(5, 22), 91(22), 93(5) 106(22), 107(5), 112 zu, Putlitz, G . , 197(217), 204(238), 229

Subject Index A absorption, 118-119, 121 absorption f-values, 195-197, 216-217 absorption lines, 197, 223 absorption oscillator strength, 202 absorption spectra, 221 accelerators, 233-256, 296 Ad Hoc Panel on Meson Factories, 253 ADP crystal, 172 adsorption pump, mass spectrometer and, 98 AFCRL rocket monochromator, 161-1 62 Air Force Cambridge Research Laboratories, 96-97, 103 airglow, 220 Alberta, University of, 254 alkalai metals, 197 alkaline earths, lifetime measurements of, 121 alkalis lifetime measurement of, 121, 216 metastable autoionizing levels in, 223 aluminum foil, in Van de Graaf accelerator, 161 aluminum ions ambipolar diffusion, 95 American Accelerator Conferences, 256 ammonia clusters, 94-95 analog methods, exponential decay nieasurement and, 124-1 32 argon, 61, 152, 186, 188, 192, 194-195, 197 lifetime measurements of, 129, 134 Argonne Laboratory, 237, 241 Arizona, University of, 159 astrophysics, lifetimes measurement and, 121 atomic beam light sources, 193-194 Atomic Energy Commision, 241, 251 atoms electron bombardment excitation of, 130-1 52 ground state of, 208-209 322

lifetime of, 116-223 pulse sampling of, 126-127 aurora ion composition during, 100 spectra of, 220 autoionizat ion, 222-223

B ballistic theory, 9-10 beam collimation, 194 beam-confined mode, 60-64, 67, 69 beam density, 61 beam electron velocity, 14 beam energies nonrelativjstic, 296 relativistic, 295-297 ultrarelativistic, 296 beam fluctuations, 166 Beam-Foil Spectroscopy Conference, I59 beam-foil techniques, 159-169, 183, 185 beam-generated plasma, beam-confined, 62 beam loss, 248 beam-plasma amplifiers, 7, 67, 70 beam-plasma collisions, 35-36, 4 1 4 3 beam-plasma interactions experimental study of, 51-72 Lagrangian equations for, 23-39 study of, 3 theory of, 7-12 beam-plasma systems, coupling to, 10-12 beam velocity calibration, 165-166 beam-wave synchronism, 50 Bendix M303 windowless multiplier, 162 Bessel functions, 10, 24, 57 Boersch effect, 299 Bohm-Gross theory, 3 Bohr magneton, 200 Bohr orbit, 119 Boltzmann equation, 290, 293-295 boron, 168 Brillouin field. 26

323

SUBJECT INDEX

Brillouin flow, shielded, 33 Brillouin value, 33 British Columbia, University of, 254 broadened energy distributions, 267-274, 299-300 in electron beams, 257-310 broadened normal energy distribution, 271-273 broadened total energy distribution, 273-274 Brookhaven Cosniotron, 255 Brookhaven laboratory and accelerator, 240-249, 255 Busch’s theorem, 32

C cadmium, 198, 210 cadmium lamp, 174 calcium, 209-210 calcium ions, 103 California, University of (Los Angeles), 253, 255 Cambridge Electron Accelerator (CEA), 248 Canadian Science Council, 255 carbon, 236 carbon foil, in Van de Graaff accelerator, 161 cascade, radiative, 181-182, 185 cascade analysis, 183 cascade coincidences, 156-1 59 cascade decay curves, 1 6 4 165 cascade effects, 129, 146, 152-153, 164-165. 180-183, 195, 213-214 cascade photons, 156-1 58 cascade transitions, 180, 182 cathode poisoning, 60 CERN (European Committee for NuclearResearch), 235, 240-241, 243, 245, 248, 251-252,256 CERN accelerator, 243 CERN Courier,256 CERN intersecting storage rings, 248-249 cesium, 197, 210 Chalk River Laboratory, Canada, 255 chondrite meteorites, 104 circuit equation

derivation of, 19-22 in Lagrangian coordinates, 29-30 for plasma-wave propagation, 12 cluster ions experimental studies on, 80-95 formation of, 79-1 12 Cockcroft-Walton preinjector, 242 colliding beams, 248-253 collision, resonant, 146-147 collisional deactivation, 183 collision broadening, 192-193, 195, 197198, 213, 215, 218 collision effkcts, 7-9, 35-36, 4 1 4 4 , 66, 72-77, 124, 146-149, 177-178, 183, 197, 213, 218, 248, 289, 308 on beam-plasma interactions, 7-8 metastable levels and, 219-220 on nonlinear beam-plasma interactions, 9 collision frequency, 4, 15, 36, 44, 67 collision term, 290, 293-295 convection current density, 18 copper, 204 Cornell University, accelerator at, 247-248 corona discharges cluster ions in, 84-86 in moist air, 89-90 in nitrogen, 89 Coulomb excitation, 236 Coulomb force, 258 Coulomb interaction, 258-259, 261, 266, 299 coupling and coupling parameters, 10-12, 27, 54-60, 62-72, 74-75, 186, 195 crossover, 286-290 cryopumped quadrupole mass spectrometer, 96-97 cybernetic accelerator, 244 cyclotron, 236-238, 252-253 cyclotron frequency, 31 cyclotron frequency resonance, 5, 21 3

D damping, 4, 8, 200 dc beam current, 10, 27 dc charge density, I8 dc field, 30 dc sweep monitor, 101

324

SUBJECT INDEX

Debye length, 2, 8 Debye radius, 258 Debye sphere, 2 decay curves, beam-foil technique and, 159-1 69 decay photon, 156 delayed coincidence photon counting, 130-159, 170-171 DESY (Deutsches Elektron Synchrotron), 25 1 diatomic molecules, 120-121 dielectric interference filter, 126 dielectric loading factor (DLF), 53-54 differential nonlinearity, 136-137 diffusion equation for, 295 lifetime measurements and, 221 Dirac function, 268 discharge temperature, measurement of, 193 dispersion equation, 8, 13-16 dispersion shape, 200 divergent beams, 279-285 Doppler broadening, 149-1 50, 166, I95 Doppler effect, 166, 186, 193-194, 198 Doppler width, 186, 193-194, 198,201-202, 207, 215, 218-219 double-focusing mass spectrometer, 82 double resonance, 197, 204-206 D region, of ionosphere, 100-102, 106, 108-112 Dynaniitron accelerator, 255-256

E

Eagle monochromator, 183 Easitron amplifier, 25 Eidgenossische Technische Hochschule, 254 electric dipole radiation, 116-121 electric dipole transition, 121 f-values of, 123-124 electric field, variation across plasma column in, 24-30 electric quadrupole radiation, 220-222 electromagnetism, spontaneous decay and, 116-117 electrometer output, 101 electron accelerators, 247-248

electron beam broadened energy distributions in, 257310 of constant current density, 285-286, 290-297, 30 1, 308-3 10 crossed, 193-194 divergent, 279-285 drifting, 12 energy loss by, 7 finite, 8 infinite, 8 linear, 7-8, 74 long, 282-284 in magnetic fields, 307-308 nonlinear, 8-1 2 plasma and, 2 short, 281-282 thermionically emitted, 257 electron-beam excitation, 6 electron beam-plasma systems, 5-6 electron bombardment excitation, 130-1 52, 178-185 multichannel techniques in, 132-1 38 single-channel method in, 131-1 32 electron bunching, 50-51 electron coding, 249 electron crossing, 10 electron cyclotron frequency, 69 plasma frequency and, 23 electron density, in plasma, 23 electron4ectron interaction, 308 electron-electron systems, 248 electron excitation, difficulties in, 213-214 electron gain, 66-67, 70 electron gun, 52, 75, 81, 299 design of, 134-1 35 electron impact, 223 electron impact excitation, 209-21 1 electron linacs, 234 electron multiplier, 81, 97 electron multiplier detector, 81 electron-positron systems, 248 electron ring accelerator, 233-236 electron storage rings, 250-251 electron stream in beam-plasma interactions, 23-36 bunched, 34 electro-optic modulators, 172-174 electrostatic accelerators, 236- 237 electrostatic analyzer, 165-1 66

325

SUBJECT INDEX

electrostatic lens, mass spectrometer and, 81 emission lines, shape of, 185-194 emission spectra, 221 energy distribution broadened, 267-274 broadened normal, 271-273 broadened total, 273-274 current-related, 267 defined, 267 differential, 276 integrated, 275 energy shift, 287, 290-291, 303, 305 energy spread, 259, 268, 300-307 absolute calculation of, 277-290 entropy and, 292-293 estimation of, 264-267 internal energy and, 260-263 magnetic fields and, 307-308 maximum, 291-292 energy widths, 274-277 entropy, energy spread and, 292 E region, of ionosphere, 104 etalon, Fabry-Perot, 188-189 Eulerian analysis, 27 Euler’s equation, 290, 292 European Committee for Future Accelerators, 243 European Committee for Nuclear Research, Aee CERN exponential decays, observation of by analog methods, 124-1 30, 132

F

Fabry-Perot etalon, 188--189 Fabry-Perot fringe pattern, 189 Fabry-Perot profile, 190 Faraday cage, 301-302 Faraday cup, 166 Faraday dark space, 88 Faraday rotation, 124 Fermi distribution, 297 field emission ion source, 82 flames cluster ions in, 79, 81, 89 hydrates in, 86 forbidden lines, 123 forbidden single photon decay, 220-222

FORTRAN lV, 36 Fourier expansion, 30 Fourier series, 169 Fourier transform, 170, 309-310 Franck-Condon factors, 122 FRANTIC, 144 free-electron density, 96 free radicals, 80 $value, 120-122, 167-169, 186, 191-197 direct measurement of, 123-124 line strengths and, 118-119

G gain parameter, 27 gas inert, 186-187, 190, 196-197, 207-208, 210 ionized, 1-3 gas discharges, 80 gas lasers, 123 Gatchina synchrocyclotron, 252 Gaussian function, 190, I92 Gaussian intensity distribution, 186 Gaussian width, 194 Gauss iterative method, 144 gyroresonance, 5

H Hahn plasma model, 8 half-Maxwellian distribution, 297-299 halogen negative ions, 92 Hanle cure, 202 Hanle effect, 188, 197-202, 206, 208-210, 213-214, 219 Hanle signal, 199, 204, 214 harmonic generation in beam-plasma system, 10 in nonlinear beam-plasma systems, 1-75 harmonics, computer solution of, 36-51 harmonic signals, 62-77 heavy-ion accelerators, 234, 236-238 helium, 119, 150, 155-156, 158-159, 188, 194-195, 202, 209-210, 220, 222 collision process in, 149 lifetime measurements of, 121, 125-126, 131

326

SUBJECT INDEX

Helium (cont.) pulse-sampling of, 126-127 Helmholtz coils, 195 High Energy Physics Laboratory, 240, 247 High Voltage Engineering Company, 255 Hilac accelerator, 237 Holtsmark distribution, 259, 265 Hook method, of f-value measurement, 124 hot-cathode discharge, 60 hydrates, in flames, 86 hydrodynamic flow, 292 hydrogen, plasma-density studies of, 61 hydrogen-ion hydrates, 80 I Illinois, University of, 240 impact broadening, 21 7-21 8 impact theory, validity of, 193 Indiana, University of, 253 inelastically scattered electrons, 158-1 59 inelastic collisions, 146 inelastic scattering, 124 injector cyclotron, 254 Intense Neutron Generator (ING), 255 internal energy, energy spread and, 260263 International Accelerator Conference (1969), 235-236 International Conferences on High Energy Accelerators, 256 iodine, lifetime measurements of, 177-1 78 ion iron, 103 lifetime of, 103 ion beams, 80 ion cluster bond strength, 88 ion cluster decomposition, 88 ion clusters see also water cluster ion (s) electric field fragmentation of, 105 thermodynamic decomposition of, 106 ion-dipole forces, 80 ionic reactions, in pure water, 83-84 ion-molecule reactions. 80 ionosphere, 95-1 12 cluster ion formation in, 79-112 D region of, 100-102, 106, 108-112 E region of, 104

theories of, 108-1 12 isochronous cyclotrons, 254

K Karlsruhe Nuclear Physics Institute, 240, 244 K D P crystal, 172 Kerr cell, 172 klystron, multicavity, 25 krypton decay-curve measurements of, 150 &values for, 194 resonance lines for, 197 spectral profiles of, 188 krypton resonance lamp, 95

L Lagrangian analysis one-dimensional, 27-29 two-dimensional, 29-30 Lagrangian coordinates circuit equation in, 29-30 force equations in, 30-33 Lagrangian formulation, of beam-plasma interaction equations, 23-36 Lagrangian theory, 9 Lamb dip, 123, 218-219 Landau damping, 4 Lande factor, 200, 219 Langevin orbiting model, 107 Langmuir probes, 98 Larmor frequency, 200, 205, 21 4 laser beams, 173 laser systems, lifetimes measurement and, 121 laser transitions, 218-219 Lawrence Radiation Laboratory, 235, 237 lead, resonance trapping in, 212 level crossings, 188, 197, 202-204, 210 lifetimes measurement, 116-223 analog methods of, 124-130 beam-foil technique of, 159-185 delayed-coincidence photon counting method in, 130-1 59 experimental techniques for, 122-123 justfication for, 121-122

327

SUBJECT INDEX

radiation width and pressure broadening of spectral lines in, 185-197 resonance fluorescence techniques in, 197-218 light absorption measurements, 124 light intensity, lifetimes measurement and, 122-125 light modulators, 172-174 linear accelerators, 234-236, 240 linear beam-plasma interaction, 74 linear beam-plasma theory, 7-8 line intensity measurements, 123-124 Liouville’s theorem, 262, 270-271, 299 Littrow spectrograph, 188 Litz cable, 239 longitudinal internal energy, 290 longitudianl relative velocities, 286, 289 Lorentz force equation, 9, 30-33, 35 Lorentz gauge condition, 14 Lorentzian function, 186, 190, 192 Lorentzian line-shape, 185-186, 200, 202 Lorentzian signal, 210 Lorentzian width, 185-187, 190-192, 194 Los Alamos Meson Physics Facility, 253 Lyman alpha radiation, 183

M magnesium ions, 103 magnetic alignment, 213 magnetic depolarization, 199 magnetic dipole radiation, 220-222 magnetic dipole transitions, 206 magnetic field beam-plasma interactions and, 23-36 electron beams in, 307-308 lifetimes measurements and, 123 plasma density and, 61 magnetic orientation, 213 magnetic resonance, 206, 209-210 Maryland, University of, 235 mass number, 101 mass spectra, 101 mass spectrometer cluster-ion experiments and, 80-81 double-focusing, 82 magnetic, 81 quadrupole cryopumped, 96-98 rocket-borne, 95

Maxwell equations, 4, 14-1 5 , 18 Maxwellian distribution, 268-272, 274, 298-300, 307-308 Maxwellian total-energy distribution, 299 mercury cascade photons and, 156-158 ground level atoms of, 212 Hanle effect and, 198-199 optical double resonance in, 204, 210 mercury lamp, 174, 198, 204 meson factories, 253-254 mesopause, 102 metal atomic ions, 102-104 metastable levels, problem of, 123, 219-222 meteoric ions, 102-104 Michigan State University, 237 Microwave Associates, 60 Minnesota, University of, 255 Mirabelle accelerator, 241 mode coupling, 10 molecular emission, 121 molecular lifetimes, 184-185, 220-223 molecules diatomic, 120 pulse-sampling of, 127-1 30 momentum transfer, 278-279 multichannel analyzers, 130, 132-1 38, 145 multichannel delayed-coincidence methods, 130, 140-145, 185 multichannel signal averagers, 201 multiple-component decays, 140-145 multisignal effects, in nonlinear beamplasma systems, 1-75

N National Accelerator Laboratory, 241-243, 250 National Bureau of Standards, 123 narrow-band phase-sensit ive detect ion, 209 nebulae, spectral lines of, 220 negative cluster ions, 91-93, 96 negative glow, 88 neon, 186-188, 190, 192-195, 197, 208, 210-211,217, 219-220 atomic lifetime of, 134 lifetimes measurement for, 152 reciprocal transfer in, 149 transition probabilities in, 124

328

SUBJECT INDEX

Nevis synchrocyclotron, 255 nickel ions, 103 Nike-Cajun rocket, 99 Nike-Hydac rocket, 99 Nike-Iroquois rocket, 99 NINA synchrotron, 248 nitrogen excited levels of, 183 plasma density measurement in, 61 triple ions of, 236 nonlinear beam-plasma systems harmonic generation and multisignal effects in, 1-75 one-dimensional equation results in, 3645 two-dimensional equation results in, 46-5 1 nonlinear beam-plasma theory, 8-12 nonresonant interactions, 192-193, 197 Novisibrisk laboratory, 250 Novisibirsk proton-antiproton ring, 249250 nuclear expansion, 168 Nuclear Physics Institute, Karlsruhe, 244 nuclear reactions, heavy-ion accelerators in, 236

0 Oak Ridge (Tenn.) Laboratory, 237, 253 Omnitron, 255 optical double resonance, 188, 197, 204206 optical excitation, 171-178, 182 limits of, 178 optical radiation, 198 oscillatory motion, 4 oscilloscope trace, in helium lifetime measurements, 125-1 26 oxygen cluster ions, 91

P particle accelerators, recent advances in, 233-256 Particle Accelerators (pub.), 236 particle distribution, in beam-plasma interaction, 9

Penning discharge, 22, 54 periodic excitation function, 169-185 permittivity, 4 phase-meter error, 183 phase-shift technique, 169-1 85 photocathode, 175 photoionization, of nitrous oxide ions, 90 photomultiplier, 126-136, 162, 172, 175, 179, 189, 199, 204, 221 photon blockading, 149-151 photon counting, 162 photon decay, forbidden, 220-222 photon detection, 135-136, 162 photon imprisonment, 149-151 photon reabsorption, 149-151 photon trapping, 129, 149-151, 177-178, 183, 192 Physical Sciences Laboratory, 25 1 Pierce-type gun, 52 plasma arc discharge, 7 beam-generated, 6, 22, 60 cold, 8, 10, 13-16 defined, 1-2 dielectric, 4-5 electric and magnetic fields of, 2 infinite, uniform, and warm, 8-9 ions and, 80 surface charge effects in, 8 warm, 9 wave propagation through, 4-5 plasma angular variations, 51 plasma collisions, 30, 4 3 4 4 plasma column, 12, 75 characteristics of, 60-62 derivation of circuit equation for, 19-22 dispersion equation for, 13-1 6 harmonics of, 36-51 one-dimensional, 12-22 quasi-two-dimensional, 23-36 variations of electric field across, 24-30 plasma column displacement current, 18 plasma density, 6-7, 12, 61, 74 plasma dispersion, 30 plasma effects, summary of, 308 plasma frequency, 11, 23, 30, 39, 64-66 74, 80, 308 amplification and, 7 oscillation growth rate and, 7 plasma frequency reduction factor, 36

329

SUBJECT INDEX

plasma model, zero-temperature, 4 plasma oscillations, 3 plasma particle collisions, 30, 43-44 plasma physics, lifetimes measurement and, 121 plasma radial variations, 51 plasma surface wave, 26 plasma wave propagation, circuit equation for, 12-22, 72-75 Poisson’s equation, 30, 294 polarization, 200 positive-ion composition measurements, 95-1 12 potassium, lifetimes measurements of, 177, 204 potassium salt, in cluster-ion tests, 87 pressure broadening effects of, 190-192, 198 of spectral lines, 185-197 pressure-dependent effects, summary of, 151-1 52 proton-antiproton collisions, 248 proton-beam ionization, 81 proton-beam mass spectrometer, 81 proton cyclotron, 253 proton excitation, 84 proton synchrotron, 240-245 proton transfer reaction, 88-89 pulsed electronic circuitry, 132 pulse-sampling, 185

Q quadrupole mass spectrometer, 81, 86 quantum-mechanical lifetime, 200 quantum mechanics, 298 quantum theory, 1 1 8 quenching, 129, 183, 221

R radar waves, plasma interaction with, 80 radiation, theories of, 185 Radiation Dynamics (Company), 256 radiation trapping, 124, 177-1 78, 183 radiation width, of spectral lines, 185-197 radiative lifetime, 185-197 radio-frequency excitation, 174 Jee a h o rf

Radio Technical Institute, Moscow, 244 regression analysis, 304 relativistic beam energies, 295-297 relaxation-free acceleration, 261-262, 275 relaxation methods, 264-267, 289, 294-296, 304, 308 relaxation parameters, 277 resonance broadening, 191-192, 195-197, 213 resonance cell, 198-199, 210-202 resonance coupling, 195 resonance curve, 206 resonance fluorescence, 123, 197-21 8 resonance-fluorescence light sources, 206208 resonance frequence, of plasma surface wave, 26 resonance radiation, 149-151 resonance scattering cross section, 202 resonance techniques, 198-199 resonance transitions, 195-196 resonance trapping, 149-151, 158, 177-178, 198, 211-212, 215 resonant interactions, 192-193 rf (radio-frequency) accelerating cavities, 2 54 rf beam current, 10 rf cavities, superconducting, 240 rf charge density, 18 rf coupling, 54-60, 62 rf energy, 6, 12-1 3, 75 rf excitation, 174 rf field, 4, 6, 8, 12, 25, 30-33, 46-51, 62-72 rf magnetic field, 204-206 rf plasma wave, 28 rf potential, 23 rf space-charge effects, 46 rf velocity, 16 rf wave, 8, 10, 36-40 Rochester, University of, 237 rocket-borne positive-ion composition measurements, 95-1 12 rotating-wheel reflection grating, 174 Rutherford Laboratory, 235, 239

s saturation, in beam-plasma interactions, 26, 4 0 4 1 , 43, 45-50, 64-66, 69-72 saturation lengths, 37

330

SUBJECT INDEX

scattering, 176 scattering cells, 174-175, 208-209 Schweitzerisches Institut fur Nuklearforschung (SIN), 254 self-adsorption, 192 separated-orbit cyclotron, 253 Serpukhov accelerator, 240-241 Seya-Namioka vacuum monochromator, 179-1 80 sheath formations, 12 signal measurement, 209 signal-to-noise ratio, I32 silicon ions, 103 Simon Fraser University, 254 Sixth International Accelerator Conference, 234 Sloan-Lawrence linear accelerator, 238 sodium in double resonance experiments, 204 in lifetimes measurements, 178 in magnetic resonance studies, 210 in optical excitation measurements, 153155 sodium ions, in lower-atmosphere studies, 103 sodium lamp, 174 solar disturbances, 108 solar eclipse, ion composition during, 100 solar ionization, 108 spacexharge effects, 2, 304-307 spacexharge field, 258-259 spaceecharge oscillations, 10 spacexharge waves, 303 SPEAR (Stanford positron-electron asymmetric rings), 251 spectral lines radiation width and pressure broadening Of, 185-197 width of, 198 spectral resolution, 183, 185 spectrograph in beam-foil technique, 161-162 stigmatic, 161 spectrometer effect, 307 spectroscopy high-resolution, 186-187 lifetimes measurement and, 121 mass, see mass spectrometer spectrum analysis, 62-64, 69-71 spontaneous decay or emission, 116-1 17,

124-1 30, 199, 220-222 Stanford Linear Accelerator Center(SLAC), 23.5, 247, 251 Stanford positron-electron asymmetric rings (SPEAR), 251 Stanford University, 240 Stark broadening, 192 Stark shift, 121 stepwise excitation, 209 stigmatic spectrograph, 161 stimulated absorption, 118 stimulated emission, 118 Stokes’ theorem, 18 storage rings, 248-253 strontium, lifetimes measurements of, 209 sulfur ions, in atmosphere rocket mesurements, 102, 111-112 Summer Study on Superconducting Devices and Accelerators, Brookhaven Laboratories, 239 superconducting accelerators, 234, 238-240 with superconducting cavities, 240 with superconducting magnets, 238-240 superconducting cavities, 234, 240 synchrocyclotrons, 252 synchrotrons, 234

T tensor operator methods, 204 thallium, in optical double resonance experiments, 204 thermionic cathode distributions, 297-298 time-of-flight mass spectrometer, 81 time-to-pulse-height converter, 137-1 38 titanium sublimation pump, 99 Tonks-Dattner resonances, 10-1 1 total glow mode, 60-61, 67 transition probabilities, in lifetimes nieasurements, 116-124, 130, 183 transverse velocities, in electron-beam energy distributions, 286, 289 traveling-wave amplifier, 70, 74 traveling-wave tube-loss parameter, 30 traveling-wave tube theory, 12, 22-23, 52 TRI-University Meson Facility(TRIUMF), 254 tungsten lamp, in optical excitation, 174

33 1

SUBJECT INDEX

two-mile accelerator, 247 two-photon decay, 222 U

UNlLAC accelerator, 238 Universities Research Association, 241 uranium, heavy-ion accelerators and, 236

around positive ions, 88-90 around protons, 82-85 temperature and, 109-1 10 theories of, 108-1 12 water vapor free radicals and, 80 ionization of, 108-1 12 wave propagation, through plasma, 4-5 Wisconsin, University of, 251

V

vacuum ultraviolet resonance line, 192 vacuum ultraviolet transition, 119 Van de Graaff accelerator, 159-160, 165, 236-238 Van de Graaff beam, 161, 166 Van der Waals’ forces, 121, 192, 218 vapor-phase ion processes, 108-1 12 Vegaard-Kaplan bands, 221 velocity distribution, 267-274 velocity shift, 271 Victoria, University of, 254 Voigt profile, 186, 190, 192 VSWR coupler, 67

w water, effect of on gas-phase chemical reactions, 79-SO water cluster ion(& 81-93 density of, 100 in ionosphere, 95-96, 102 measurements of, 99-100 around negative ions, 91-93

X xenon in beam-plasma experiments, 60-62 in lifetimes measurements, 208

v Yale University, 253

z Zeeman effect, 202 Zeeman scanning, 21 5 Zeeman separations, 21 5 Zeeman states, I 17, 206 zero- phase position, determination of, 182-1 83 zinc H a n k effect and, 198 in lifetimes measurements, 215 magnetic resonance experiments and, 210

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