T H E U N IV E R S I T Y O F M I C H I G A N COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report COLLISIONS IN IONIZED GASES Jordan D. Lewis ORA Project 07599 sponsored by: Advanced Research Projects Agency Project DEFENDER ARPA Order No. 675 uider contract with: U. S. ARMY RESEARCH OFFICE-DURHAM CONTRACT NO. DA-31-124-ARO(D)-403.DURHAM, NORTH CAROLINA administered through: OFFICE- OF RESEARCH ADMINISTRATION ANN ARBOR October 1966

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1966.

ACKNOWLEDGMENT This work would not have been possible without the guidance and advice of several teachers and friends. It is a pleasure to pay tribute to them at this time. I would like to acknowledge first and foremost the many long, fruitful hours spent in consultation with Professor Richard K. Osborn, Dr. Osborn has lent both his time and mind most freely and willingly over the past many months; he has been instrumental in my intellectual growth and has indeed set a high example as both teacher and scientist. The study of the Fokker-Planck collision operator contained herein was suggested by and performed largely under the guidance of Professors Fred Shure and Charles Dolph. Had I not been able to draw from their thinking on this problem, the study would have come to naught, I also wish to thank Professor N. D. Kazarinoff and Dr, R. K. Ritt for their careful reading of the manuscript and their most helpful suggestions pertaining thereto. Several interesting discussions with Professors Terry Kammash, George Summerfield, and Donat Wentzel, and with my fellow student, John Schaibly, have contributed considerably to my understanding and appreciation of the physical world, I would also like to thank the many teachers and friends not mentioned by name, who have provided the stimulating atmosphere within which I have worked and studied. Finally, ili-i

thanks are due to Chuck Martin for his assistance in computing some of the accompanying curves. This research was supported in part by the Advanced Research Projects Agency (Project DEFENDER, ARPA Order No. 675) and was monitored by the U. S. Army Research Office - Durham under Contract DA-31-124-AROD-403. iv

TABLE OF CONTENTS Page ACKNOWLEDGMENT......... d... iii LIST OF TABLES o a. o.......... Q 0 el 0 vii LIST OF FIGURES, o. o....... 0 vi.ii ABSTRACT,. o, o o o G O O O X, Chapter I1 INTRODUCTION............ 0, 0 1 1. The Many-Particle Problem 1 2. Outline of this Work 8 II, A BRIEF SURVEY OF PLASMA KINETIC THEORY 6. o. 16 1l Phenomenological Kinetic Equations 16 2, Kinetic Equations Derived from the Liouville Equation 24 A. The Absence of Correlations. The Boltzmann and Vlasov Equations 26 B, The Inclusion of Correlations, The Lenard-Guernsey-Balescu Equation 31 3. A Simple Collision Model 36 46 Relaxation in Velocity Space 40 5, Transport Phenomena; Electrical Conductivity 45 6. Collisional Effects on Small Amplitude Plasma Oscillations 53 III, AN EXPANSION THEOREM FOR THE LINEARIZED FOKKERPLANCK EQUATION.... C O... o... 60 1 Properties of the Equation 60 2, Expansion in Spherical Harmonics 62 3. Spectrum of the Radial Equation 66 4, Outline of the Expansion Theorem 70 5. The Electron Kinetic Equation 75 6, Discussion 76.V

TABLE OF CONTENTS (Concluded) Page IV. THE SCATTERING OF PHOTONS FROM A PARTIALLY IONIZED GAS,t,, I,0. I s. *. I... o 78 1, Some General Properties of the Scattering Function 78 2, The Electron Scattering Function for a Partially Ionized Gas 95 A, Reversible Theory 95 B, Irreversible Theory 104 3, Photon Scattering from a Partially Ionized Gas 114 4. Discussion 126 Appendix A, INTEGRALS.... * * * * * 0.., I *.0 4 130 B. THE SELF-ADJOINT PROPERTY...... e.,. oI. 132 C. PROOF OF THE EXPANSION THEOREM..... 134 BIBLIOGRAPHY., e, o o *...... *.., I * o 156 vi

LIST OF TABLES Table Page 4.1 Comparison of Scattering Functions for Different Collision Parameters..... a.. a 123 vii

LIST OF FIGURES Figure Page 2.1 Details of binary interaction,,........ o 30 2.2 Relations between kinetic theories discussed in text, d.. a.6.....* I s O.0.0 40 2.3 Resistivity vs. frequency for a fully ionized hydrogen gas... o le.. o. 50 4.1 Electron scattering functions for a singly ionized carbon plasma, showing dependence on the parameter.- (KA)-Ia 2- m, lb..e 120 4.2 Electron scattering functions for a singly ionized carbon plasma, showing the effect of neutrals for = Z 35.. le, ~ ~ ~ ^ ~ I.. e * * e.. 121 4.3 Electron scattering functions for a singly ionized carbon plasma, showing the effect of neutrals for! -1/,..........8.... eo. 122 viii

ABSTRACT This dissertation is a study of the description and effects of particle interactions in ionized gases. The principal results are: (i) An expansion theorem for the linearized Fokker-Planck collision operator for each component of a two-component fully ionized gas, and (ii) A description of photon scattering from a partially ionized gas, It is shown that the Fokker-Planck collision operator generates a complete, continuous set of velocity-space eigenfunctions, for which high-speed asymptotic forms are found, Since the set is continuous, the expansion formula has the form of a generalized Fourier integral. The effect of neutral atoms on the spectrum of photons scattered by electrons in a partially ionized gas is shown to be primarily a reduction in height and increase in width of the two electron-plasma "wings," The scattered photon spectrum is described for several characteristic cases, ix

I, INTRODUCTION The purpose of this dissertation is to investigate certain aspects of the kinetic behavior of ionized gases. The emphasis here is primarily on the description and effects of particle interactions. Two principal results are obtained. The first, developed in Chapter III, is an expansion theorem for the linearized Fokker-Planck collision operator for a two component fully ionized gas. The second, developed in Chapter IV, is a description of photon scattering from a partially ionized gas. The function of Chapter I is twofold: it presents a brief discussion of the present state of the art in the treatment of classical many-particle systems, thereby setting the stage for the rest of the work; in addition it contains an outline of this work designed to guide the reader. Chapter II contains a survey of some of the more pertinent results of plasma kinetic theory. The reader interested primarily in the results of this dissertation may choose to skip the first two chapters and proceed directly to Chapters III and IV, which are essentially self-contained. 1. The [lany-Particle Problem The theory of classical many-particle systems may be studied from three points of view. One may begin with the macroscopic, or fluid, equations with parameters such as density, mass velocity, and temperature as independent variables, and involving various transport -1 -

- 2coefficients, e.g.,, viscosity, heat conductivity, etc. Examples are the Euler and Navier-Stokes equations.l An altogether different approach involves the use of more fundamental and general microscopic formalisms. On the one hand, one may work with equations describing the evolution of one particle distribution functions, the well-known Boltzmann equation2 being a prime example. On the other hand, one may employ equations relating one-, two-, etc, particle distribution functions such as the hierarchy of equations derived from the Liouville equation,3 which latter treats the evolution of the distribution function for all N particles in the systems The fluid equations are generally considered adequate for treating wide classes of problems in gas dynamics. In this range the microscopic theory would not yield significantly different results, In fact, subject to certain conditions which define their range of applicability, the fluid equations, together with explicit formulas for the various transport coefficients entering into them, are derivable from the microscopic theory.2'4 There are, however, many important situations in which the macroscopic theory does not give a correct description. In general, this occurs when the length or time scales characterizing phenomena of interest are not long as compared with the scales on which the microscopic quantities fluctuate. Examples are the propagation of highfrequency or short wavelength waves, and behavior near boundaries. In these cases one must properly begin with the microscopic equations. The solution of the Boltzmann equation (or, more generally, any equation involving only the single particle distribution function) is, in general, a matter of considerable difficulty even in cases

- 3corresponding to the physically simplest situations. Significant progress has been confined almost entirely to the study of two limiting cases in which two different approximation procedures can be applied. A criterion for the range of validity of the approximate methods is provided by the comparison of a characteristic time r' or length L for the relevant process with the average time o or mean free path L~ between particle encounters. For high densities (t,>>, or L >> L) the Chapman-Enskog theory4'5 may be used. The first approximation of the theory consists in assuming collisions to be sufficiently frequent to maintain a local thermodynamic equilibrium. The next approximation corrects the distribution function by terms proportional to gradients in temperature T, flow velocity 8, and density n; this corresponds to the fluid equations with transport coefficients for heat conduction, viscosity, and diffusion. This high density region is in fact the range in which the fluid equations provide an adequate description. Higher approximations of the Chapman-Enskog theory lead to correction terms proportional to higher derivatives of 1Sl. The successive approximations of the 4 Chapman-Enskog theory correspond to an expansion of the distribution function in powers of the mean free path L.. For example if we consider sound waves with wavelength La>L,, the first and second approximations are already sufficient to give all significant features of the process. When L becomes comparable to L: however, it is necessary to go to the third and even higher approximations to obtain adequate results; the third approximation already involves formidable labor and has been used to solve only the simplest problems.3 Consideration of hiqher approximations is, in any case, of doubtful value since the

-4entire procedure breaks down in Just the range where the contributions from these higher-order terms becomes important. A different approach, using expansions in terms of Hermite polynomials in velocity space, has been given by Grad.6 He uses some moments of low order in addition to the usual ones representing n, ~t and -,. The procedure involves a gain in simplicity over the Chapman-Enskog theory but is still quite complicated, In any event, it is basically a high-density theory. The opposite limiting case of low densities ('e<<*: or L<<L) has been studied using iterative schemes beginning with the solution of the "collisionless" equation]7'8 In the case of ionized gases immersed in strong electromagnetic fields, such theories have been used extensively.7 As with the approximation schemes employed in the highdensity case, these iterative procedures become unwieldy if more than one iteration is necessary, It would naturally be very desirable to have a method capable of treating the microscopic equations over the whole range from low to high densities. Unfortunately the describing equations are generally non-linear, and even when linearized are extremely intractable, a prime source of difficulty being the term representing interparticle collisions. Relatively little work has been done in this intermediate density region, often referred to as the kinetic regime, Attention has generally focused on mathematical properties of the collision operator, on test particle treatments,5 or on a numerical solution of the kinetic equation in a few simplified situations,ll A notable exception is in the work of Chang and Uhlenbeck.12'13 These authors treated the propagation of small amplitude sound waves in a monatomic gas composed of atoms interacting via an inverse-fifth power force law, i~e Maxwell

- 5molecules. They were able to show that the linearized Boltzmann collision operator for this case generated a complete set of velocity-space eigenfunctions. Upon expanding the perturbed distribution function in terms of these eigenfunctions, they obtained from the linearized Boltzmann transport equation an infinite set of coupled algebraic equations which they solved by successive approximation. Their results were in quite good agreement with experimental observations on collections of neutral atoms. In view of the difficulties involved in solving the microscopic equations in the kinetic region, considerable interest has recently been focused on the mathematical properties of the terms representing collision effects. Motivation in this direction has been based in part on the feeling that a knowledge of the spectral properties of the (linearized) collision operator would lend insight into the kinetic behavior of the system.10'14 GradlO has considered the linearized Boltzmann collision operator for particles interacting via the general inverse power force law, where K is a constant and f is the interparticle separation. Grad found that in order to obtain mathematical results it was necessary to assume the interparticle force extended over a finite range; ie., angular integrations in the collision integral were truncated at small deflections. On the basis of this assumption he was able to show that the spectrum consists of two parts: a discrete spectrum and a continuous spectrum, The latter is bounded away from zero for "hard" potentials (S3 ), and approaches zero for "soft" potentials ( S < 5 ). For the special case of the Maxwell molecule (S 5') there is only a

-6discrete spectrum. For 5< 3 Grad was unable to find the spectrum, Recently Ferzigerl5 has used Grad's results for inverse power-law molecules to show that the linear Boltzmann collision operator generates a complete set of eigenfunctions, The form of the eigenfunctions was not, however, obtained, In any event, the spectral and completeness properties were not obtained for the Coulomb potential (s =2 ), A major problem in plasma physics is that of determining the properties of an isolated hot plasma; any material probe introduces impurities, while for a fully ionized plasma the emitted radiation is only moderately informative, any line structure arising from undesirable impurities. The interaction of an incident beam of radiation with a plasma has proved to be a useful method for determining the electron density; since the plasma acts roughly as a dielectric with coefficient E I Cue 6pe The /7Lh transmission is cut off below the plasma frequency. Radiation above the plasma frequency may also be used as a plasma probe, and several experiments have used the modificiation in phase velocity produced by the dielectric coefficient e as a measure of electron density.16 It has been known for some time that the scattering of photons or material particles from a system of interacting particles yields detailed information on the structure of the scattering system.l7 With the advent of intense light sources such as pulsed ruby lasers, considerable attention has been given to the scattering of photons by free electrons in ionized gases. Several authors have presented analyses of this phenomenon, usually basing their descriptions on

- 7semi-intuitive derivations, and employing collisionless kinetic theories, Notable among these are Salpeter,18 and Rosenbluth and Rostoker1l9 Shortly thereafter considerations of relativistic effects20 and nonlinear scattering21 appeared, The first experimental observation of photon scattering from an ionized gas was reported by Bowles,22 who observed the scattering of a radar beam from the ionosphere, More recently many workers have reported the measurement of optical photon spectra produced by scattering from ionized gases in the laboratory. The observations are generally in remarkable qualitative and quantitative agreement with theoretical predictions, which is a rarity in plasma physics. An illustration of this close agreement is given in the recently reported work of Anderson.23 Provided the photon wavelength A is of the order of the Debye length AD or larger, the scattered spectrum is characterized by a narrow central peak located at the incident frequency, and by two symmetrically placed satellite peaks separated from the central peak by 4wZcwpe, the electron plasma frequency. The central peak reflects the strong coupling of the electrons to the ions characteristic of long wavelength plasma phenomena, while the satellites are attributed to the resonant scattering of photons from longitudinal electron plasma oscillations. 1819 As the photon wavelength becomes large in comparison with the Debye length, the satellites become narrower and rapidly increase in height. This has been attributedl9 to a decrease in the effect of Landau damping on long wavelength plasma oscillations. Recently Ron, Dawson, and Oberman24 and Fante25 have computed

-8the scattered photon spectrum for a fully ionized gas including the effect of collisions. They find the difference between their results and the collisionless treatments to be very small, of order A', where Az'n'D is generally very large; in fact AZ is generally assumed large for the various theoretical models to be valid. In recent years considerable effort has been expended on the production and diagnosis of gases that are only partially ionized. Examples range from low temperature gas discharges to relatively high temperature ( 3oev) and high density plasmas generated by the laser bombardment of solids, Since photon scattering has been proven to be a most useful tool in the diagnostics of fully ionized gases, it is natural to expect this usefulness may be extended to include systems containing significant numbers of neutrals, In addition, since the Thomson scattering cross-section for electrons is several orders of magnitude larger than the ion cross-section or the Rayleigh scattering cross-section for neutrals,26 we would expect that photon scattering from electrons should be observable even when neutral densities exceed electron densities. The primary difference between fully- and partiallyionized gases in this respect would then be in the effects of neutral atoms on the scattering process, since charge-neutral collision frequencies may often be considerably larger than their Coulomb counterparts. 2. Outline of this Work The purpose of this dissertation is twofold, In the first part of this work we obtain the spectrum and prove an expansion theorem for the linearized Fokker-Planck collision operator27 for particles interacting via an inverse-square force law. In the second

- 9part of this work we employ a simple collision model to study the effects of collisions on the spectrum of photons scattered from a partially ionized gas. The plan of this paper is as follows. In Chapter II we present a brief survey of plasma kinetic theory, with particular emphasis on the description and effects of charged particle interactions. Due to the long range nature of the Coulomb interaction, when an ionized gas is in the kinetic regime, a great number of particles are "colliding" simultaneously, In this case the simple binary collision models employed to treat collections of neutral particles are often deemed to be inadequate, As a consequence many attempts have been made to develop suitable kinetic descriptions for plasmas, and a comparison of the various treatments has often led to confusion. The purpose of this chapter is to compare a few of the better known kinetic models in an attempt to cast some light on their similarities and differences. In Part 1 of Chapter II we give a very brief phenomenological derivation of the Boltzmann collision integral, and of the Fokker-Planck collision operator for inverse-square law forces, Emphasis is on the difference between short- and long-range interactions. In Part 2 of Chapter II we briefly discuss the hierarchy of equations generated by the Liouville equation. The effect of correlations between particles is studied by, on the one hand, neglecting correlations altogether and, on the other, retaining two particle correlations. In the first case we obtain the Boltzmann equation for short-range interactions, and the Vlasov28 equation for long-range interactions. In the second case we obtain a kinetic equation developed by Lenard,29 Guernsey,30 and Balescu.31 With proper assumptions this

- 10 - equation reduces to the Fokker-Planck equation. In view of the mathematical difficulties involved in the solution of the various kinetic descriptions, it is often advantageous to replace the more accurate and less manageable collision descriptions by a model that simplifies the solution of the kinetic equations, In Part 3 of Chapter II we consider such a simplified collision model, generally referred to as the Krook model.32 The ultimate test of any theory lies in a comparison of the predictions thereof with experimental observation, Due to the scarcity of relevant experimental and theoretical information in plasma physics, it is often instructive to compare the results of the various theories, One hopes, in so doing, to acquire physical insight into both the structure of the theories and the as yet unobserved properties of nature, In the last three parts of Chapter II we make such comparisons for a few illustrative cases, In Part 4 we review some recent numerical treatments of relaxation to equilibrium in velocity space, In some cases it is possible to compare the Fokker-Planck, Lenard-Guernsey-Balescu, and Krook descriptions. From the information currently available we observe a negligible difference between the predictions of the first two treatments. The limitations of a single parameter Krook-type model are discussed and compared with other results. In Part 5 of Chapter II we discuss the phenomenon of electron runaway and the contribution of collisions to plasma transport parameters. The failure of the Krook model in the description of runaway is noted, and the results of a computation using the Fokker-Planck description are discussed. The majority of this section is devoted to

- 11 - a comparison of the different collision descriptions in the computation of the plasma electrical conductivity. For field frequencies small compared to the collision frequency, the Krook, Boltzmann, and FokkerPlanck results are identical. For frequencies above the collision frequency but below the electron plasma frequency cape, the Boltzmann and Fokker-Planck results differ only slightly. For frequencies above Ope these collision descriptions break down; the reasons for this failure are discussed, For high frequencies conductivity computations based on the Vlasov equation and the first two members of the BBGKY hierarchy give similar results, and match the results of the collision description just below co Pe' In the final section of Chapter II we present a brief review of some recent work on collisional effects in plasma collective behavior. Since relatively little work has been done in this area, only a few comparisions of the different collision descriptions are possible. In general it is found that for wavelengths long compared to the Debye length damping is primarily collisional; for wavelengths of the order of the Debye length or less Landau damping33 predominates, Moreover, if a plasma is inherently stable, collisions increase the damping of small amplitude oscillations. In contrast if a plasma is unstable, collisions may increase the growth rate of the instability. In Chapter III we obtain the spectral properties, and develop an expansion theorem for the linear Fokker-Planck collision operator for a two-component fully ionized gas, In Part 1 of this chapter we take advantage of the small electron-ion mass ratio to decouple the equations for each species. For convenience we then concentrate our attention on the ion collision operator and later discuss the extension

- 12 - of our results to the electron case. To develop the expansion formula we follow the standard method of assuming solutions to the kinetic equation of the form fVt) = A(A) ee-( At) This reduces the equation to the form LK(&) IA(l -- A,, where L is a three-dimensional integrodifferential operator, In Part 1 we show that ReA>o and IAz- o as we would expect physically. In Part 2 we introduce a spherical harmonic expansion which replaces the three dimensional equation by an infinite set of uncoupled equations, where Lbj,,(v) is a singular integrodifferential operator. These are then cast into a self-adjoint form in Part 3 by introducing a suitable algebraic transformation on the functions a uA(V-AeX.). With boundary conditions obtained by combining the original kinetic equation with the conservation laws, we proceed to find the spectrum which is continuous and for,=o0I consists of all A4,, _, and for X-? consists of all A oQ, ->o -.Although the spectral resolution theorem34 implies the existence of an expansion theorem for self-adjoint operators, there always remains the task of constructing the expansion explicitly. We turn to this task in Part 4. Since La, is singular at -or and — oo, we temporarily replace the interval oe Vr-oo by the interval o<,<' v-z < and use a result of Tamarkin35 toshow that L~, generates a complete orthonormal set on this interval. To return to the original interval and

- 13 - thus obtain the desired expansion theorem, we use the above completeness property together with an extension of the theory of singular differential equations.36'37 This finally yields a continuous, normalizable set of functions AdWv A/) that is complete with respect to functions square integrable in velocity space, Since the set is continuous, the expansion has the form of a generalized Fourier integral. For r ~>(2>/w)"'//we have found asymptotic forms of the expansion functions. Finally, in Parts 5 and 6 the extension to the electron kinetic equation is discussed, as well as certain implications of our results, In Chapter IV of this dissertation we develop a theory of photon scattering from partially ionized gases. The starting point for the present discussion is the description developed recently by Osborn,38 which treats photon scattering from a fully ionized gas. The primary concern in this work38 was with the establishment of a relationship between the observed distribution of scattered photons and the dynamical and statistical characteristics of the scattering plasma. It was assumed that (i) the dynamical variables of the plasma obeyed the classical equations of motion, and that (ii) the plasmas in question were sufficiently highly ionized that the presence of neutrals could be neglected, In Chapter IV we examine the second assumption described above, Our motivation in this direction is twofold. First, it is not clear a priori when the presence of neutrals will be truly negligible. Further, when their effect is significant, it must then be incorporated in a description of the scattering process. Vineyard,39 Salpeter,18 and Feyer40 have discussed the contribution of neutrals to the plasma

- 14 - scattering function, but each based his discussion on semi-intuitive arguments and in no case was a quantitative description developed. It was shown in reference 38 that, neglecting relativistic effects, the cross-section presented by an electron to a photon having frequency to and direction Q for the scattering of that photon into (c t ) and ( -L A' SL'+dL-' ) is given by Ad O ) Si 6r- (G, 1) where T-r is the Thomson cross-section. The so-called scattering function S(~,mo) is given by = w- - e dte- beee i) _o0 where Gee_<p - e The normalization is such that /Ne is the total number of electrons in the scattering volume, and the function is the Fourier-transformed configuration space electron density operator, with transform variable In Part 1 of Chapter IV we briefly review some of the more pertinent aspects of reference 38, Following Osborn,38 we employ a classical representation of the density operators and develop and discuss various aspects of the classical scattering function, Certain properties of a thermal equilibrium plasma relevant to the computation of the scattering function are also discussed, In general neutrals may influence the scattered photon spectrum in two ways, The first, and perhaps more obvious, contribution is significant when the number of photons scattered by neutrals into the

- 15 - frequency range of interest is not small compared with the number scattered by electrons. Secondly, neutrals may bias the scattered spectrum through their influence on the electron density operator. We investigate this latter effect in Part 2 of Chapter IV. We approach the problem along two somewhat different paths, The first is a simple extension of the treatment in refo 38, to include a neutral species in the description. We show that this approach does not lead to a noticable contribution from neutrals unless exceptionally high neutral densities are present. At this point we take a somewhat different tack, employing the simple Krook model to represent interactions with neutrals, and leading eventually to a modified scattering function displaying significant contributions from neutrals at neutral densities at least five or six orders of magnitude lower than in our earlier treatment. Part 3 of Chapter IV is devoted to a presentation and discussion of the classical scattering function, with and without the effects of neutrals, for several experimental configurations. Our results are in agreement with earlier predictions39'18'40 in that the effect of neutrals is to heighten and narrow the central spectral peak, while lowering and broadening the so-called electron plasma wings. In Part 4 we discuss the implications and limitations of the approximations and assumptions employed in the development of our results: a few recent experiments are cited as illustrations, Extensions of the present theory are suggested where such modifications may lead to significant differences.

II, A BRIEF SURVEY OF PLASMA KINETIC THEORY 1l Phenomenological Kinetic Equations One of the earliest and still most successful descriptions of a collection of free particles is the famous Boltzmann equationl'2 )FA a FA I a, which expresses how PA changes in time due to streaming and to encounters with other particles. Here A 1d % dtr is the expected number of type A particles in dfr'tcBr about V, 4r at time t, and A is the force on a particle of type A and (SFA/) represents the time rate of change of f produced by inter-particle encounters, For our purposes we assume that only elastic encounters are important. In the classical theory of non-uniform neutral gasesl'2 (F F/ /'t)) is taken to be the net number of particles of type A entering the phase space volume element cd1' d'ir/ per unit time due to instantaneous binary encounters. Thus if 0tr-_, is the relative velocity of a colliding pair, then the flux of particles of type l3 having velocity SJ incident on any particle of type A having velocity \r is dPrA f ( I tt) I Ad A,' If TAB(9 ) is the differential scattering cross-section, then the number of particles of type A scattered out of the phase space volume element j3 o, c)d into nJL during St by collisions with type B particles is - 16

- 17d3C,'J3W, FA (fl, ~,, + ) J 3Ol_ F 8(3S,, 1,n W) 9 Ne, (g.s~ e) J The number scattered into this same volume in dt is c13r, J3,, fA,, 8r E', FtB,, O))2 t5 ($d e) 6BSL dt As a consequence of momentum and energy conservation in the collision we have2 and hence the we 1 1 - knwon Boltzmann col l i s i on i nteg ral A binary encounter is often described in terms of the impact parameter b, the distance of closest approach if no interaction is present, This latter quantity is related to the differential scattering cross-section through the relation3 bdbcd= J G) JQ where dJ = e e dJ The derivation of the Boltzmann collision integral rests on three basic assumptions, and as yet there is considerable debate as to their significance: (i) The collection of particles is sufficiently rarefied so that only binary collisions need be considered; (ii) The probability of finding two particles in d3r about r simultaneously is proportional to the product of

- 18 - their individual distribution functions; (iii) the force %,IA is distinct from the interparticle forces, and it does not affect the collision process. Assumption (i) implies that the interparticle potential is so localized that, for a sufficiently dilute gas, the chances of finding more than two particles within "range" of each other simultaneously is negligible, While this assumption seems reasonable for dilute gases composed of neutral atoms or molecules, it causes considerable difficulty in the treatment of charged particles. The difficulty arises because electrostatic forces, being proportional only to the inverse square of the distance, permit many particles to be within range of each other at a given time. To see this, note that the effective interparticle potential for a fully ionized gas not far from thermal equilibrium is4 VA6(r) e e _ r/ D where the Debye screening length AD is given by4 CA 2. VIe where e is the temperature in energy units. It follows that the number of particles interacting simultaneously is roughly or (for a singly ionized gas) or (for a singly ionized gas) which is usually quite large.4 In contrast, charged particles will suffer large angle deflections only when3 the impact parameter is of the order of eZ/, the distance of closest approach in a head-on collision.

- 19 - Since interparticle spacings are ^-1/3v the fraction of particles making such encounters at any instant is Thus the binary collision assumption seems inadequate since it does not appear to account for the many overlapping long-range encounters. This was recognized by Chapman5'6 in the calculation of transport coefficients for an ionized gas. Chapman found that integrals over the impact parameter diverged for b-boo due to the long range nature of the Coulomb force, He overcame this difficulty by cutting off the integration at b ~ - $1/3, and assuming that the resultant force on a particle due to more distant encounters could be represented by an internal electrostatic force in the streaming term of the Boltzmann equation, Cohen, Spitzer, and Routly7 sought to overcome Chapman's difficulties by adopting a treatment developed by Chandrasekhar8'9 in a study of stellar dynamics, Chandrasekhar's work was based on Jeans'10 demonstration that when particles interact through inverse-square forces, the cumulative effect of the weak deflections resulting from the relatively distant encounters is more important than the effect of occasional large deflections, Chandrasekhar noted the strong similarity between the Brownian motion of a colloidal particle and the motions of particles interacting via inverse-square forces; his treatment is based on a description of Brownian motion due to A. D. Fokkerl and M. Plancki12 While Cohen, et al7 restricted themselves to slightly anisotropic velocity distributions, the general case of arbitrary distribution functions was considered by Rosenbluth, MacDonald, and Judd)13 We present a brief

- 20 - derivation, following Chandrasekhar9 and Rosenbluth, et al 13 Assume that there exist time intervals At long enough for a particle to suffer a large number of weak deflections but short enough for the net mean square change in velocity, <I'l>), to be small compared with the mean square velocity. Let PA() Ad) denote the probability that in the time interval At- a particle of type A having velocity v undergoes a displacement A. Assuming that pA does not depend explicitly on time, the distribution function for the At species is then given by9 FA(jAr S) (_/ tA-t) P At * aunt- atr) A( V;Do (2,1) Since /t and AV' are both assumed small, the integrand is expanded in a Taylor's series: jdlig)F4 (t 1 HL tFA |A(I J a, P As; P) - t,'" FA:,- F A PA Using the fact that equation (2.2) gives for the time rate of change of FAresulting from the cumulative effect of small deflections ~ (ll = - ~'-~' 2..,,, F l's (2.3)

- 21 - where A a} - J2 3(46V-) p~*P(; r) AS Afar F-d ar pA; V.bA trB d A (2,4) and so forth. Since A4vt represents the mean change in lr resulting from encounters during at, we write v- 4 52 { F,A.) jp u I at A iA It (2,5) and < 4 v A at (2.6) etc. for higher order terms, where a:.-r'/l is the magnitude of the relative velocity. Rosenbluth, et al 113 evaluated the Fokker-Planck coefficients (2.5) and (2.6) assuming a two-particle Coulomb crosssection, Since the integrals over QL diverge logarithmically at large impact parameters, they followed Cohen et al7 and cut off the integration at b..A,11 A the Debye screening length. After some algebraic manipulation Rosenbluth et al obtain whFe LoL.re/ 3r _ Af 2V t vH id+' r" M F O ) (2.* 7

- 22 - and /AA6 is the reduced mass. Assuming the logarithm to be a sufficiently slowly varying function of the relative velocity, the approximation is made MA e A, Aee/AD AI A e,, ~i(2.8) (It can be shown14 that this approximation introduces an error of less than one percent in the determination of transport coefficients based on (2.7).) In the derivation of (2,7) only those terms porportional to At phave been retained from the expansion (2.3) and in the evaluation of the Fokker-Planck coefficients (2,5) and (2.6). All other terms can be shown13 to be down by a factor A(g'A ). Thus the FokkerPlanck equation may be viewed as an expansion in powers of the ratio of mean kinetic energy to potential energy at a separation of /A, and is sound for.vI AAb4sufficiently large, or roughly 03(e ) ~ io-, (o- )v Equation (2.7) may be transformed into a more symmetric form. Since dVI Fe J- a 3v) 1 F.Y (2.7) can be rewritten in the form' t p oll A B~~~~

- 23 - This form of the Fokker-Planck equation was published by Landaul5 in 1936, While the above analysis treats the effects of a large number of overlapping small-angle deflections occurring in the time interval.at:, the form of the probability employed in (2.5) and (2.6) still assumes that these small angle deflections are themselves due to binary encounters. Moreover, we have retained the assumption (ii) of the joint probability being proportional to the product of the two singlet probabilities and have thereby ignored possible correlation effects. In addition, the present description explicitly ignores the effects of the relatively infrequent large angle deflections. Both Cohen7 and Rosenbluthl3 and their co-workers suggested the inclusion of a Boltzmann collision operator for impact parameters below an unspecified critical value, At the same time, they suggested the effect would usually be negligible. Finally, we note in passing that the Fokker-Planck equation as displayed above may be obtained4 by Taylor expanding the integrand of the Boltzmann collision integral with Coulomb cross-section, cutting off the integrals at b - A, and retaining only dominant terms. Several authors have presented descriptions which take into account the electrostatic properties of the plasma)16'17'18 The method is to consider a test particle as being subject to local fluctuating electric fields, and then calculating the Fokker-Planck coefficients on this basis. In all cases the results are quite similar, and are tantamount to including a dielectric constant in the functions r( in (2.7) An advantage of this work is that the Debye length enters the description as a natural cutoff distance, without having to be introduced in an ad hoc fashion as before, In contrast, the treatment yields a

- 24 - divergence at small impact parameters due to an improper handling of close encounters. To overcome this a cutoff is postulated at an impact parameter of the order of the distance of closest approachA e /e Hubbard19 surmounted this latter difficulty by retaining the entire infinite series in the expansion (2.2). 2. Kintetic Equations Derived From the Liouville Equation The construction of kinetic theories on the basis of phenomenological considerations naturally raises questions concerning the validity and range of applicability of the various descriptions. Perhaps the most satisfactory scheme for surmounting these difficulties is to begin with the most complete (and intractable) description available, the Liouvi 1e equation. For simplicity we consider a collection of N indistinguishable particles, occupying a volume V, with no external forces, The generalization to a multicomponent system is straightforward but tedious, and will be indicated by reference where appropriate. We define FN ((' /,. Ct)) L /, 2)...,A as the N/V-particle distribution function such that N F7.1 J','d-,~ is the probability of finding the t& particle in cI3r,5'r for each of the 11 particles, all at time t e Clearly T J,,,.d r,, 1. I I. According to Liouville's theorem,20 F is governed by the equation t L. + vr N _ _ (210)

- 25 - where -, I and \/ is the two-body potential. Thus we have a —- i C E, (2, 11 The information contained in the Liouville equation (2,10) or (2.11) is all inclusive but in general is inaccessible because of its complexity, A method for extracting useful information has been developed independently by Bogolyubov, Born, Green, Kirkwood, and Yvon, and is summarized in Montgomery and Tidman4 or in any good text on statistical mechanics such as de Boer and Uhlenbeck,21 To develop the so-called 8161KY hierarchy, we define reduced probab i 1 ity distributions F _ Vs JVci }, c<kN. (2.12) Multiplying (2,11) by VS and integrating as in (2.12) we obtain an expression for Fs which involves FH+I, In particular, for s =I and Sz we have -t + >-X iE (2.13) and F.N t 7F a br)' v s )(1 / V |O3s( Nc- + BTE zp)F; (2 14)

- 26 - While the Liouville equation uniquely determines the evolution of FN from the initial condition F{(t=o), the derived relations for Fs do not uniquely determine the evolution of Fs from Fs (t-o) Instead, the interactions of our S particles with the remaining 1\l-s particles are summarized in terms of a typical St- particle and we require a knowledge of Fstl to solve the problem. We shall now see how, for a given set of circumstances, the relations for F, and Fz can be simplified., We will make assumptions consistent with the physical conditions which we desire to treat and on the basis of these assumptions introduce approximations which are designed to retain the pertinent elements of information, Our first approximation is of quite general validity. We assume N1? to be a large number and expand in terms of \1-. At the same time, we allow the volume of the system to become arbitrarily large, but such that Nl/V=n remains finite. This removes interactions with the boundary, and reduces (2.13) and (2.14) to ~AL" \ r rT (2,15) and _ ( J3r(3 2 r( -a-.vr.tg,~ Y)gZ (2.16) A. The Absence of Correlations. The Boltzmann and Vlasov Equations. Writing F (L,,) IV,, )t in the form

- 27 - where g is a correlation function, and taking: o o, the system (2.15), (2.16) reduces to an equation for FI FT tV{ --- - ) dr*cJ\\ T7 F N (z) (2,17) It is interesting to consider (2,17) in two contrasting cases. In the first, we consider a dilute gas with short-range interparticle portentials VIT, such that the effective range of the potential is much smaller than the mean interparticle separation, Following Schonberg22 and Osborn23, we rewrite (2.17) in the form wicY [s which is equivalent to (2.17) since the additional terms are zero as may be verified by partial integration. Following Osborn23 we now introduce a set of assumptions designed to lend plausibility to the succeeding argument, (i) The potential VI is appreciable only over a sufficiently well defined region of radius cu- such that the r, integration in (2A18) is effectively confined to the volume I<,- ~5 I<. (ii) The length, t, is small compared to the mean interparticle spacing, ieo, << I This implies that Ko can be chosen such that the probability of finding more than two particles within a distance X& of each other at any time is negligible, ioe,, we choose oe so that the binary collision assumption is in some sense justified. (iii) The above choice of c. leads to a sufficiently restricted region for the spatial integration in (2.18) so

28 - that El,~,, VF ) t ) F % (ra. fia -) t) F-(,, g t) for all r defined by Iv,-$2l<c (iv) The product F.I) F, (z) does not vary appreciably over time intervals of order (v) Two particles within a distance cL of each other will be presumed to be interacting so strongly that they may be regarded as effectively decoupled from their environment, Assumption (v) permits us to interpret, aV': _V ____- ~ - L In >r ILwJ - 1 r ~. =L - ~ (2 19) where Ql and at are the accelerations experienced by particles 1 and 2 respectively throughout the duration of their close encounter. The approximation (2.19) plus assumptions (iii) and (iv) then enables us to approximate the integrand in (2,18) as follows: I y )' a (t) E\ ( FF rt)F,, t Fa,, 4>;2; F,r, ht) v F, (,,j:,t) -e ) t rV? (, z2t)F;(,,- jt) F F, ir,7 F. Evidently we may interpret the velocities ( r- ) c2 ) A) as the precollision velocities of a pair of particles entering into a strong binary interaction, whereas ( L, ) are the post-collision velocities of the same pair. Introducing the notation

- 29 - we see that we may now approximate (2,18) as ir, -i') <ck Note now that nothing in the integrand of (2.20) depends on C;, Thus the space integration is readily performed if we introduce the variable change R,- r whence fJ3 r-1, = 9(2021) If we take R to be the distance between the two particles during their close encounter and introduce a cylindrical coordinate system (see figure (2,1) ) with E -axis parallel to the pre-collision relative velocity, then (2E21) may be written as b idbd dc t' (2,22) 0.=O b=o - 0/h The quantity Abdk& is clearly the center of mass differential scattering cross-section introduced earlier, which we may write as JfL Thus (2,21) may be written as N/-1' = o J Lo (2.23) It is understood that the restriction on the range of impact parameters as displayed in (2,22) implies a corresponding limitation on the range of the angular integration in (2.23), In these terms then, equation (218) becomes, for the dilute short-range approximation,

- 30 - / / I CL + CL.2 Z Figure 2.1 Details of Binary Interaction

- 31 -'t) - F, t) (gti(2..24) Equation (2.24) is, apart from normalization differences, the Boltzmann equation in the absence of external forces. Finally, let us consider again equation (2.17) when VIZ is the Coulomb potential. In this case, as we have seen earlier, the notion of a binary interaction length ct is indistinct when the number of particles in a Debye sphere is large, We could pick a length & say of order a few times the distance of closest approach, but the choice is vague. In this case the potential term would be divided into a "binary" term and a "collective" term, this latter corresponding to the force term in the Boltzmann equation. For example if we consider only collective effects, then we may regard the potential term in (2.17) as producing an internal electric field which is a function of r alone, We define this field by gEt-) I j3 3 L rg and write (2i17) in the form WIV (2 25) which is known as the Vlasov equation. B, The Inclusion of Correlations, The Lenard-Guernsey-Balescu Equation, It is clear from the foregoing discussions that the neglect of correlations in the description of an ionized gas is a simplification

- 32 - which is difficult to justify, and which leads to problems such as the divergence at large impact parameters. Since the 863KY hierarchy (2.15), (2.16).,, includes correlations, we might hope that the kinetic equations we derive from it will be intrinsically free from this divergence, In the following we present the salient points of an important kinetic equation derived from the hierarchy which includes two-particle correlations, We outline the work of Lenard24 and Guernsey,25 which is based on earlier results of Bogolyubov,26 We begin with Mayer cluster expansions27 of the distribution functions, similar to that used in the previous section. We have, with F.(l,,t) - F,1-,t) F, z,t) t, h, ~,t) (2.26) and ~F;~ ~ i wjy) (,3&)e FJ 2,t) 3t) A ) (3,t) (6, t) +1 h Zl),3,t) (2.27) where h is a three-particle correlation function. We work in the so-called plasma limit, ) DA3>>1, and we further assume the threeparticle correlations to be negligible (b J). We consider only homogeneous plasmas, still with no external forces, so that the various FC() are independent of position and the correlation functions depend on ri and rj only in the combination r =IrW-~ Bogolyubov26 assumed that as a consequence of the assumption nYA>3> I, the correlation functions would vary much more rapidly in time than any of the distribution functions F,/l t). He thus

- 33 - suggested that (i) The distribution functions F}(1t) may be considered time independent in solving for - (I)j)t). (ii) The asymptotic value of the correlation function, (~'&j' o), represented as a functional of the F. (t't), may be used in solving for these latter quantities. (iii) As a consequence of (ii) the information contained in the initial condition would not appear in the F (lIt) and hence we may take) Employing these assumptions, Bogolyubov26 was able to reduce (2,15) and (2.16) to whereF =~ is a atvime id ais 1th- s lto(2.28) where 6 is a time independent function of F(' ), and is the solution of the integral equation &(- )t Y) - K (r-l V-) 1aF(- F(6) - - F(!l {X F- ) ( F(,) (2,29) whose kernel is 1<(n v 5 iL t ) The normalization is Id'JFr n and we take Vl'- e-/r Lenard24 and Guernsey25 reduced the system of equations (2A28), (2.29) to a single equation forF by following Bogolyubov's suggestion26

- 34 - of introducing spatial Fourier transforms, Their result was also obtained by Balescu28 using a diagram technique; at- -' _,. | ~ >,,)'L — Pax, ( —- -(2.30) where Q is a symmetric second rank tensor whose components are given by (~~=~JrZ I~ -Pybi~ ~OV) (2 31,.(\S V ) =- it IJ i N ( - IIt + ((a K Here k /ItIi, and iAm) - A (JcL~ce~ L wVt'z(rX;T;In. i d *v( IK FA ) F L e The integrand in (2,31), aside from the 6-function, becomes independent of * for *k-O (large impact parameters), so that the integral converges in this limit, Thus the incorporation of correlations in the kinetic description removes, in a natural way, the troublesome divergence at large impact parameters. For *k large, the integrand behaves as -z -and hence the integral diverges for k->aoo This is the same divergence encountered in the electrostatic treatments17'18 of the Fokker-Planck equation discussed above, and is due to the neglect of the three-particle correlation A when two of the three are of the order of e7/9 apart, In this case h is the same order of magnitude as the binary correlation 9, and cannot properly be ignored. To achieve convergence in (2,31) it is therefore necessary to make a short range cutoff, i e., integrate only within the sphere -- / o /e2

- 35 - Lenard24 simplified (2.31) somewhat by showing that, if one neglects contributions to Qj coming from speeds greater than a few times (2/,)/'Q reduces to Q& # ( As) e (2 32) The approximation introduced in going from (2,31) to (2,32) requires that F(I) must be small, for velocities greater than a few times the thermal speed, as compared with its values at lower speeds. Equation (2.30) with (2.32) is, aside from a missing factor of 2, just the Landau form (2o9) of the Fokker-Planck equationo We might consider the foregoinq as a rather convincing argument supporting the use of the Fokker-Planck equation for ionized gases with distribution functions satisfying the above condition, and which satisfy the Bogolyubov assumptions (i) through (iii). These assumptions are violated, for example, by systems in which F and 9 vary on the same time scale. Examples are the interaction of high-frequency waves with a plasma, or rapidly growing instabilities, The generalization of equation (2,30) for a multicomponent system may be found in Montgomery and Tidman,4 The generalization to a constant, uniform magnetic field was worked out by Rostoker,30 and is considerably more complex than (2.30), (2.31). Working with a multiple time scale theory developed by Bogolyubov for certain problems in nonlinear mechanics, Frieman and Book31 have developed a kinetic equation for homogeneous field-free systems that is free of divergences for all impact parameters, For small impact parameters their result resembles the Boltzmann collision integral; elsewhere

- 36 - it is similar to equation (2.30). The development of plasma kinetic equations is an active field, and it is beyond our purpose here to survey the topic in its entirety. In concluding this section we mention only briefly some of the problems of current interest in this area, and related work. In our derivation of equation (2,30) we assumed a homogeneous system with no external forces. We noted earlier that the phenomenological kinetic equations generally assume that inhomogeneities and local force fields do not significantly interfere with interparticle encounters. While these may often be valid assumptions, we would expect them to fail when length and time scales characteristic of inhomogeneities or of local forces are small compared with AID orAD/u-l (the time required for the establishment of a Debye screening cloud about each particle) where v- is the thermal speed of the particles. At this writing a kinetic equation that includes these generally neglected effects is not available. Many attempts have been made to overcome these difficulties. Perhans the most noteworthy are in the work of Bohm and Pines,32 Rostoker and Rosenbluth,33 Dupree,34 and Frieman)35 While none of these authors have presented a treatment satisfactory to all, their work lends considerable insight into the problem at hand, and often presents novel and significant mathematical tools. This work is certainly a good starting point for the interested reader. 3. A Simple Collision Model We have so far been concerned with the development of a kinetic equation for the description of ionized gases. While a satisfactory theory is yet unavailable, it is apparent from our earlier discussion

- 37 - that the more rigorous the treatment, the less tractable it is in terms of analytic solution, In point of fact, this problem is not unique to the study of ionized gases, The Boltzmann transport equation has so far been solved only for a few special cases, namely the inverse-fifthpower force law suggested by Maxwell, and for certain scattering kernels relevant to neutron transport theory.36 In view of these difficulties, it is often advantageous to replace the more accurate and less manageable collision descriptions by a model that simplifies the solution of the kinetic equations, With the current paucity of experimental information on fully ionized gases, such a model may often be a good starting point for the interpretation of the little information that is available, We thus consider briefly a collision model designed to satisfy the conservation laws and an H-theorem, and which considerably simplifies the mathematical analysis. One of the earliest of these models, generally referred to as the Krook model, was developed by Bhatnagar, Gross, and Krooki37 A similar but somewhat simpler model was suggested independently by Welanderi38 The first Krook model, for a single component system, is37 (w),i _ 4\(c~t)h |N()(YcQ) (2,33) with (ro~t) = (j3Crf t, (fjdrII 2ai) =~~~)' -7e

38 - and m30) _____= jits id F where t and ) define the flow velocity and kinetic temperature (in energy units) at (Et), and cr is the parameter of the model. The dimensions of INr- are inverse time, and 7- is generally chosen to yield an appropriate collision frequency on the basis of ~~39~ ~39 phenomenological considerations. For an ionized gas, O7 is generally chosen39 such that 8rwe \e \ A 3 (3 v, 0) 3 /7 ) A) k v While the Krook model is highly nonlinear, it is considerably simpler than, e.g., the Boltzmann collision operator, since the distribution function enters in (2.33) in a simple way: as FE id~F 3 3 For the particular case of small amplitude perturbations near equilibrium, the linearized form of the Krook model permits solution of the kinetic equation in closed form for several interesting cases37'40 To linearize (2.33) we write NF(&,t) = So t E ~ a) O (r, t) = 630 t f r,t)

- 39where E is a small parameter, and where n (,t, -) |j7 J (,t), No 9 = - Jf N0~l, -a,~ s~~rk ji~~~f~~~~~ r(2.34) We thus obtain, neglecting terms in 6, ( Noll 3 f No([t, t +o ) \ t) +I >(<t) F (; -i 3X"~) (2 35) with? (rA) =~? / V ( t) The extension of the Krook model to a two-component system was first given by Gross and Krook,40 together with an application of the linearized version to oscillations in a fully ionized gas. More recently Sirovich,41 Liboff,42 and Oppenheim43 have presented similar collision models for a general multicomponent system. While all of these models are similar in form, they are not identical.

- 40 - The relationship between the Krook model and the more sophisticated collision models rests primarily on intuitive grounds. In one special case, however, a more direct relationship has been demonstrated. For the case of a Maxwell molecule, Gross and Jackson44 showed that the linearized Boltzmann collision operator yields the linearized version of (2.33) if all the non-zero eigenvalues of the collision operator are approximated by a single constant which appears as c- in (2.33). In Figure (2.2) we have indicated for convenience the relations between the kinetic theories discussed above, Fig. 2.2 Relations Between Kinetic Theori-es Discussed in Text LI ouLVI LLE B B6KhY 4. Relaxation in Velocity Space The ultimate test of a theory lies in a comparison of the predictions thereof with experimental observation, At the present writing, the scarcity of relevant experimental and theoretical information is a major source of difficulty in kinetic physics.2 This is particularly so in the physics of fully ionized gases4 where on the one hand, the

- 41 - maintenance of a plasma and reliable observation techniques still present many unsolved problems and on the other hand, tractable theories allowing for inhomogeneities, rapid temporal variation, and boundary effects, are generally unavailable. In the presence of these difficulties it may be instructive to compare the results of the various theories, with the hope that this would lend physical insight into both the structure of the theories and the as yet unobserved properties of nature, In this and the following two sections we intend to make such comparisons for a few illustrative cases. Our division of these topics into three separate categories is not intended to imply that they are mutually exclusive; it has been effected for convenience alone. The problem of determining how a homogeneous expanse of gas behaves as it approaches equilibrium, ibe. "relaxes," is perhaps the simplest problem in the kinetic theory of gases, The problem is of interest here because it focuses attention on the collision operator, Perhaps the simplest description of the relaxation process is that obtained from the homogeneous isotropic Krook model (2.35). Thus if o is the value of the distribution function at t- O, then +w(t = o F g i 7nt- 0 (2.36) where F,0 is given in (2.34). The characteristic time (t4r)iappearing in (2.36) is generally referred to as a relaxation time. Due to the complexity of the more sophisticated models, a study of relaxation via analytic solution has generally not been achieved. While the simple Krook relaxation time may often be a sufficiently accurate estimate of

- 42 - relevant time scales, it is easily recognized that the description of a relaxation process by a single parameter could often be misleading. For example, the rate at which a given distribution becomes isotropic in velocity space if initially anisotropic, could be significantly different than the rate at which it relaxes to a Maxwellian, One method of estimating such rates without actually solving the kinetic equation is to single out and consider a single "test" particle in the gas. This procedure has been employed by Spitzer,39 who analyzed various aspects of the relaxation of the electron and ion components of an ionized gas, such as (i) removal of angular anisotropy, (ii) energy exchange, (iii) loss of energy of a particle by "dynamical friction," Bohm and Aller45 have similarly presented a detailed analysis on the relative importance of electron-electron collisions in establishing the velocity distribution of electrons in gaseous nublae and stellar atmospheres. Montgomery and Tidman4 perform a test particle analysis by assuming all particles except the test particle have a known (equilibrium) distribution, The kinetic equation (in this case FokkerPlanck) is then "linearized" about the test particle "distribution," and velocity moments of the linear equation are obtained, Relaxation times -, are then obtained by defining these as the ratio of the velocity moment M in question to its time derivative, ie,, I2M1 The relaxation times obtained from the test particle approach generally depend on the initial speed of the test particle, the relative temperatures of the species present, and the relative masses. For a

- 43 - two component ionized gas with equal electron and ion temperatures, it is found that electrons become isotropic primarily through collisions with ions; collisions with electrons play a small role in ion relaxation, and the relaxation to equilibrium of an isotropic electron distribution is primarily due to encounters with other electrons, Although the qualitative conclusions reached in a test particle treatment should generally be correct, they do not display all of the information available in the kinetic equation. Hence, in lieu of an analytic solution of the kinetic equations, several authors have presented numerical treatments for various situations, MacDonald, Rosenbluth, and Chuck46 have presented a numerical solution of the Fokker-Planck equation (247) for an isotropic electron gas imbedded in a positive neutralizing background. They assumed an initial Gaussianshaped distribution peaked in the vicinity of the speed (z0o/mThey found the time required for the distribution to come within a few percent of the final Maxwellian, throughout the range from zero energy to several times the average energy, is about ten times the self-collision time defined by Spitzer39 (the mean time required for a thermal particle to eventually suffer a 900 change in direction due to the cumulative effects of many small angle encounters with like particles), MacDonald, et al also found, as could be expected, that it takes considerably longer to fill out the high velocity "tail" of the Maxwell distribution. Recently Wu, Levans, and Primack47 have studied numerically the relaxation of a two-component plasma with initially anisotropic electron and ion temperatures, and with initially isotropic (but unequal) component temperatures, using the Lenard-Guernsey-Balescu equation (2 30), (2.31). They assumed that the distribution functions maintain a

- 44 - Maxwellian character throughout the relaxation process, having the form FA~(t)- (4...."'//'e) e, ) Z (t) Z (2 37) IT'9A N ( 2.37,,,,,,,~,~.3/where cN(t)Y msA i J3vFA) SAl VASi'J iF? and that _A — 1I << I/ ). | _ - 4 << I The results are in qualitative agreement with the predictions of the test particle theory, except when / > /C>. In this case, the anisotropic electron temperature relaxation is governed by collective phenomena, Since Wu, et al47 constrained their distribution functions to the form (2.37), a comparison of their isotropic relaxation results with those of MacDonald, et a146 would not be fruitful, We can only note that the collective effects manifest in (2.30), (2.31) are important in the relaxation process, under certain conditions of anisotropy. A direct comparison of the Fokker-Planck and Lenard-GuernseyBalescu equations has been achieved in another numerical relaxation study, performed by Dolinsky.48 Dolinsky solved both equations for several different initial conditions, for an isotropic electron gas in a neutralizing background. A comparison of the solutions showed a difference of less than two percent, for all speeds and for all time.

- 45 - 5, Transport Phenomena; Electrical Conductivity Many of the interesting phenomena in kinetic theory involve systems that are inhomogeneous or are subject to external fields. As we discussed in Chapter I, the role of particle interactions in such cases may or may not be important relative to other phenomena, depending on the nature of the system under consideration, A simple and yet interesting illustration is the phenomenon of electron runaway, which occurs when an ionized gas is subject to a sufficiently strong electric field. Kruskal and Bernstein49 have studied electron runaway using a transport equation with Fokker-Planck collision operator. For simplicity they neglected electron-electron collisions, and assumed the electronion mass ratio to be zero. Their analysis leads to a decomposition of velocity space into three regions, for electric fields greater than a critical value, In the first of these, the low velocity domain, the form of the electron distribution function is dominated by collisions and hence almost isotropic. The second region, one of intermediate velocity, is characterized by "quasi-steady" flow in velocity space, for which the low velocity region provides the source. Lastly there is a high velocity region, fed by the intermediate region, in which the electrons accelerate or "run away" almost freely under the action of the electric field, with only a very weak diffusion due to collisions. The phenomenon of'runaway, like the relaxation of high speed electrons discussed earlier, reflects the rapid decrease of the Coulomb cross-section with increasing relative velocity. It is apparent that a simplified collision model that does not take into proper account the

- 46 - nature of the interacting particles would here lead to erroneous resul ts. As we noted in Chapter I, there are many interesting circumstances wherein the macroscopic properties of a system exhibit only small variations in times of the order of the inverse collision frequency V, or in space over distances of the order of the mean free path L. -r/vc It is then possible to approximate the kinetic description by a fluid description treating macroscopic quantities such as density, mean velocity, pressure, etc,, where flows are linearly related to the generalized forces driving them, 2'4'50 For example the electric current is given by the product of the electric field and the conductivity. For an isotropic system the conductivity is a scalar; more generally it is a tensor. Since we are concerned in this work with the description and effects of particle interactions, it is instructive to consider the calculation of transport parameters briefly. While all transport coefficients are sensitive to particle interactions, the phenomena of interest here can be illustrated by a consideration of the electrical properties of a plasma. In the following discussion we will assume for simplicity that the system being considered is free from magnetic fields, temperature gradients, and inhomogeneities, We will further assume that the applied field is spatially uniform; iee,, that >>AAr where A is a length characterizing the field. Considerable attention has been turned in recent years to determining the conductivity of a fully ionized gas. The subject is not only of interest as a problem in kinetic theory, but is also of practical importance in that from a knowledge of the a.c. conductivity one can

47 - compute immediately the absorption coefficient for radiation in a plasma and hence, by Kirchoff's law, the emission properties.450951 The earliest calculations of electrical conductivity were based on phenomenological considerations. 239 Thus one simply assumes that the current carrying electrons suffer, on the average, equal accelerations by the electric field and decelerations due to collisions. Using such considerations, Spitzer39 calculates a conductivity assuming all current to be carried by the electrons, and neglecting interactions between electrons. He finds the conductivity -r to be Ne (2,38) where N0o and M are the electron number density and mass, respectively, and )_ is the electron-ion collision frequency. We can easily obtain similar results with a simplified Krook model. The dec. conductivity of a fully ionized gas has also been commuted using a Fokker-Planck collision operator. Spitzer, et al7'52 obtained a numerical result very close to that given by (2.38) with ~V the "self-collision" frequency defined earl i er. As we noted above, a significant contribution to the collisional processes in a plasma arises from long-range Coulomb encounters, and the duration of these encounters is quite sensitive to the relative speed of the particleso While a Krook-type collision model seems satisfactory for a fully ionized gas subject to low frequency or dec. fields, we would not expect such a simple representation of particle interactions to suffice for frequencies CO of order 2, or higher. The earliest treatments of the aoc. response of a plasma attempted to overcome these difficulties by employing velocity dependent

- 48 - collision frequencies54 or collision operators that described a diffusion in velocity space.2 These approaches were not, however, founded on a consideration of the nature of the interacting particles. Several authors have computed the impedance 2 = T-I of a plasma employing the various kinetic theories considered earlier in this chapter. Their results provide an effective means for comparing the various theories. In Fig. 2.3 we have displayed the quantity RAc/RDC where R is the real part of the plasma impedance?, as a function of W/cp The results are given for a fully ionized hydrogen plasma with A 2-'/1/o7, where A is given by N, A,, and are based on a similar display due to DeWolf,55 For frequencies co well below the collision frequency the purely resistive impedance is constant and the results of the simple Krook theory51 agree with the Fokker-Planck calculations.52 The low frequency resistivity has been computed51'52 both including and excluding (Lorentz gas) encounters between electrons. The effect of including these is to increase the low frequency resistivity by a factor ^'1l7 as is evident in the figure. Bernstein and Trehan,56 Robinson and Bernstein,57 Kauffmann,58 and Shkarofsky59 have obtained the aac. plasma impedance using a FokkerPlanck collision operator. Their results are summarized in Shkarofsky, Bernstein, and Robinson.60 Marshall61 performed a similar analysis using the linearized Boltzmann collision operator. None of these authors included the effects of internal "self-consistent" fields; i.e., they did not include the Maxwell equations in their analysis. The results for the Boltzmann and Fokker-Planck collision operators agree within a few percent, and this difference is likely due to different

- 49 - computational procedures. For frequencies below v: these results match the d.c. results, as is evident in the resistivity diagram- (For frequencies not small compared to the collision frequency the impedance has a reactive part2'54'60'61 reflecting inertial effects of the conducting charges.) As the frequency increases past ~~ the resistivity increases and eventually approaches a constant, independent of d, For frequencies well above c electron-electron collisions are seen to be insignificant as compared with electron-ion collisions, When the field frequency exceeds the plasma frequency Scheuer62 has argued that the resistivity should decrease, in contrast with the Fokker-Planck or Boltzmann results.60'61 When co>Az p Scheuer suggested that the maximum effective impact parameter should decrease from the Debye length urT/wP to the length ur/co. At distances larger than vrr/oo, encounters do not contribute to the resistivity since they are much longer in duration than the oscillations themselves, Dawson and Oberman63'64 computed the high frequency impedance of a plasma using the simple Vlasov equation including the internal electrostatic field, Their results (see Fig. 2.3)) agree with Scheuer's reasoning62 for co> op and join the Fokker-Planck and Boltzmann results60'61 for I < Ad, They observed a slight bump in the resistivity near c=o p which they attribute to the generation of longitudinal plasma oscillations. For very low frequencies their resistivity does not decrease, in contrast with thie collisional treatmenti6061 This latter difference in the predictions of the two treatments miqht be interpreted by reasoning as follows. For frequencies below the plasma frequency the dielectric response of the plasma is fast compared with the period of the imposed oscillation. For frequencies in the

Resistivity /.....- - Dawson and Oberman, << I Ope - - - Bernstein, et ol, Lorentz Gas Ra.c. 2 Rd.c. l/l 10A, 18 6 / I10I0 00 -8 10-2 I' Q 2p Figure 2.3 Resistivity vs. frequency for a fully ionized hydrogen gas. Taken from reference 54, A = 2-1/2 x 107.

- 51range Vc-< X o< Ctup, collisions are too slow to affect the plasma response and a description incorporating only dielectric effects, i.ee, the Vlasov equation, yields results that are insensitive to frequency. For lower frequencies the collisions become important and the response is frequency sensitive. Evidently there is a range of frequencies below cop where the collisional and collective, or dielectric, descriptions produce similar results. Oberman, Ron, and Dawson65 have computed the high frequency conductivity of a fully ionized plasma by solving the first two members (2.13), (2.14) of the BBGKY hierarchy using a method due to Guernsey,66 We note that the Bogoliubov hypothesis was not employed; ie,, the two particle correlation function was allowed to vary on the same time scale as the one-particle distributions. The results of Oberman, et a165 are in complete agreement with the predictions of the much simpler Vlasov treatment.63'64 In concluding this section we note some other computations of plasma transport parameters for the interested reader, In references 57 through 60 the low frequency ( A< cop ) thermal diffusion and conductivity, and the viscosity have been computed as well as the electrical conductivity, for a plasma having small temperature and density gradients and immersed in a constant uniform magnetic field, Kivelson and Dubois67 have found the electrical conductivity for finite wavelengths using the kinetic equation (2.30) of Lenard, et al. Berk68 has obtained the conductivity for finite wavelengths, His approach was similar to that of Dawson and Oberman,63'64 Oberman and Shure69 used the first two BBGKY equations as in ref. 65 to compute the high frequency conductivity with a magnetic field. The high frequency

- 52 - electrical conductivity has been computed quantum mechanically by DuBois, Gilinsky, and Kivelson70 and by Ron and Tzoar.71 Oberman and Ron72 extended this work to include a magnetic field. The results are in agreement with the classical descriptions. We summarize briefly the effects of finite wavelengths, and magnetic fields. For finite wavelengths it is convenient to refer to the phase speed of the wave, Ir/ =cw/ where 5 27r/A, For p s vu or less, and co e-p, the principal contribution to the conductivity is electrostatic.68'70 For greater speeds collisional effects predominate. At high frequencies, C>eop, collisions are unimportant. In the presence of a magnetic field69 the conductivity in the direction of the field is unaffected. In contrast the transverse components decrease with increasing field strength until in the limit of infinite field strength no current flows across the field. Finally, Klevens, Primack, and Wu73 have computed the a.c. conductivity for wO>cup using the Lenard-Guernsey-Balescu equation (2.30). Two specific cases are considered in detail: in the first, the unperturbed plasma has different electron and ion temperatures; in the second, the unperturbed plasma is characterized by a relative drift between electrons and ions. For the first case they find that for,e/joc or 8'/de lo, the real part of the conductivity becomes negative. For the second case they find that if the electron drift speed exceeds 1.37 times the electron thermal speed, and if Ql >,O?76e the conductivity is again negative.

- 53 - 6. Collisional Effects on Small Amplitude Plasma Oscillations Until recently, most studies of plasma oscillations have been concerned with relatively high temperatures and low densities, as in thermonuclear devices, or with very weakly ionized systems such as the ionosphere. For systems in the first category the collision frequencies are generally very small compared with the oscillatory frequencies of interest, and this is used as a basis for disregarding collisional effects, For systems in the second category collisions with neutrals often predominate, and a simple Krook-type model is employed to account for these, In recent years considerable experimental attention has been given to fully ionized, relatively low temperature plasmas for which the foregoing collisionless or simple collision model assumptions are thought to be unsound. In addition, in the study of high temperature unstable plasmas it has been recognized that an inclusion of even very weak collisions can have a significant effect on the growth rate of the instability. For these reasons there has appeared an incentive to treat collective phenomena including the effects of particle interactions, In the following discussion we present a brief review of some recent work on collisional effects in plasma collective behavior, Since relatively few theoretical results are available, the work summarized here should be considered as a first step in the direction of understanding these phenomena, In keeping with the objective of the present treatise, our emphasis is on the nature and description of collisional effects. The multitude of possible collective modes that a plasma may

- 54 - support often makes a generalization of specific results very difficult, and few attempts to do so are made here. Since a considerable effort has gone into the analysis and classification of collisionless oscillations, the reader having more than a passing interest in collective phenomena would probably benefit by consulting the collisionless literature first.56'7483 In the analysis of small amplitude plasma collective phenomena it is frequently convenient to Fourier-Laplace transform the governing equations, together with the Maxwell equations for the electromagnetic field, and then to solve the transformed equations for the internal electric field, from which all other field quantities may be determined.83 The result is then displayed in the form83' )- - _ 1_ (2.39) where e is the plasma dielectric tensor (or constant for isotropic systems), the elements of 4 are the cofactors of their counterparts in e, and c 4() is a vector incorporating the initial conditions, The dependence of the electric field is given by the inverse Laplace transform of (2.39), and since gE and a are entire functions of S and * for many interesting cases,83 one is usually interested in the zeroes of the determinant |I(, S) I / Thus setting this quantity equal to zero yields a relation between the wave vector J and the Laplace variable $ ='co tf, and hence one estimates the growth or decay rates, etc. of various collective modes (it should be noted, however, that the dispersion relation 1e= o does not necessarily imply a one-to-one correspondence between frequency and wavelength84),

- 55 - In the following review the reader will note in some cases a remarkable similarity between the collisional effects on plasma collective behavior, and the collisional effects on transport phenomena discussed above. This is, of course, more than fortuitous, It can be shown70'85 that the longitudinal and transverse dielectric and conductivity tensors are related; for S= t' we have70 r (k, A) = T + Yrt ) with I the unit dyadic. In an early attempt to treat longitudinal plasma oscillations including collisions, Bhatnagar, Gross, and Krook37 employed the simple collision model discussed earlier. They treated a one component plasma consisting of electrons with fixed ions with no external fields, and neglected collisions between electrons. (The assumption of fixed ions implies the wave frequency is large compared to the ion plasma frequency,) Their results may be summarized as follows: (i) For wavelengths long compared to the Debye length or the mean free path a small change in the oscillation frequency was observed as the collision frequency varied from zero to infinity; the damping was slow (ioe, Y/o~<<I) and reached its maximum when the collision frequency equalled the plasma frequency. (ii) For wavelengths shorter than both the Debye length and the mean free path the damping was heavy and was primarily electrostatic, or Landau damping. Lenard and Bernstein36 treated the problem studied by Bhatnagar, et al37'38 using a pseudo Fokker-Planck collision operator designed to

- 56 - represent a diffusion in velocity space, and which conserved electron number density and yielded the Maxwell distribution for the equilibrium state. Their velocity dependent "diffusion coefficients" however, increased with velocity in contrast with the true Fokker-Planck coefficients. Their results are in general agreement with Bhatnagar, et al 37 Comisar,87 Gorman and Montgomery,88 Burgers,89 and Wu and Klevans90 have treated collisional damping of longitudinal electron oscillations in a one component plasma, including both electron-electron and electron-ion collisions. Comisar87 used the linearized Fokker-Planck collision onerator, Gorman and Montgomery88 used Guernseys reduction66 of the first BBGKY equations, Burgers89 solved a Boltzmann-like equation with the Debye potential replacing the Coulomb potential, and Wu and Klevans90 approximated the first two BBGKY equations and then employed a Guernsey-like reduction. All of these authors obtained similar results, which were restricted to weak collisions and long wavelengths. The results may be summarized as follows: (i) A wavelength-independent damping constant was found for electron-ion collisions, (ii) a damping constant porportional to z was found for both electron-electron and electron-ion collisions, (iii) collision damping dominated Landau damping, (iv) electron-ion collisions dominate the damping, and (v) a small, wavelength independent correction to the oscillation frequency was found, In each of the first two cases the damping constant Of was found proportional to the respective collision frequencies given by Spitzer.39 The work of Comisar87 has been extended by Buti and Jain91 to treat high frequency transverse plasma oscillations. Their results are essentially the same as Comisar's. The collisional damping of electron plasma oscillations is

- 5/ - easily described in terms of momentum transfer out of the collective modes, due primarily to electron-ion collisions, and the damping increases with increasing electron-ion collision frequency. In contrast, we might expect the effects of collisions on low frequency ion waves to be somewhat different, since momentum transfer to electrons is smallo Bhadra and Varma92 have investigated collisional damping of longitudinal ion waves using a simple Krook model and neglecting ion-electron collisions. For equal electron and ion temperatures, the damping decreased monotonically with increasing collision frequency. Their interpretation of this result is that since collisions do not transfer momentum out of the wave, their only affect is to enhance the propagation~ Kulsrud and Shen93 have investigated the propagation of ion waves using a Fokker-Planck collision operator in the limit of weak ionion collisions. They found the spatial damping to decrease with increasing collision frequency as with the time damping treated by Bhadra and Varma,92 and calculated the relation between wave speed and collision frequency for comparison with experiments on ion waves performed by Motley and Wong.94 Their results are in fair quantitative and qualitative agreement with the experimental results, but they suggest this may be only fortuitous since they attempted to extrapolate a time-damping theory to explain spatial damping lengths, We have so far been concerned with waves in isotropic plasmas, Liboff42 and Oppenheim95 have treated longitudinal electron plasma oscillations in the presence of a constant uniform magnetic field. Liboff used a Krook model to represent collisions, while Oppenheim employed a pseudo Fokker-Planck collision operator similar to that used by Lenard and Bernstein.86 The two treatments give similar results

- 58 - for long wavelengths and low temperatures, and for magnetohydrodynamic modes, in the absence of the magnetic field, The results differ, however, in the case of "microscopic Larmor resonance" modes, in the parameter range where wavelength is much longer than both the Larmor resonance and the collision length. Liboff's Krook model42 gives an infinite number of damped Larmor modes only at propagation precisely perpendicular to the applied magnetic field. Oppenheim's model, in contrast, gives an infinity of Larmor modes at arbitrary directions, except parallel to the field. Oppenheim suggests this difference reflects the velocity-space diffusion property of his collision operator. The damping constants found by Oppenheim and Liboff were quite similar, being proportional to the collision frequency in each case. It is well known83 that small amplitude disturbances of a homogeneous plasma near thermal equilibrium are stable; ie,, any such disturbances tend to decay in time. In addition, this inherent stability is not affected by the inclusion or exclusion of collisional effects in the describing equations, or by the imposition of a uniform magnetic field. In contrast the presence of currents or spatial gradients is known96 to be sufficient to induce unstable plasma behavior. The study of plasma instabilities is a relatively new field but nevertheless has received prominant attention in regard to both laboratory and extra-terrestrial phenomena, prime examples being the contain96 97,98 ment of hot plasmas and the growth mechanism of stellar flares. Due to the considerable complexity of the equations employed, the analysis of plasma instabilities has generally been restricted to collisionless treatments. Only within the past two years have attempts been made to include collisional effects. While these efforts have

- 59 - been few in number, the results indicate that these effects may have a profound influence on plasma behavior, Certainly more work is needed in this area. In an early treatment including collision effects, Kuckes99 analyzed the propagation of low frequency ion waves in a current carrying plasma without a magnetic field. Using a simple one-parameter collision model, he showed that "collisional effects of the electrons can lead to growth mechanisms for these oscillations," while "the thermal motions of the ions leads to a damping," More recently Bhadra,100 and Kulsrud and Shen93 have reported studies of ion acoustic waves, employing Fokker-Planck collision operators in an iterative weak collision analysis. Bhadra100 treated waves propagating parallel to a strong magnetic field with a perpendicular density gradient, and Kulsrud and Shen93 assumed a homogeneous plasma with a small external electric field. Bhadra found electron-electron collisions to have a destabilizing effect, while electron-ion collisions tended to stabilize. Kulsrud and Shen, in contrast, observed electron-ion collisions to decrease the critical current; electron-electron collisions had negligible effect. Bhadra100 also used a simple Krook model for purposes of comparison; he found only a slight difference in growth rates under some conditions.

III, AN EXPANSION THEOREM FOR THE LINEARIZED FOKKER-PLANCK EQUATION 1. Properties of the Equation In the first three sections of this chapter certain spectral properties of the collision operator are established, While these properties (apart from the reality of the spectrum) are not necessary for the later development of the expansion theorem, they are both useful by themselves, and enable certain conclusions to be drawn regarding the final form of the expansion. For our purposes it will prove convenient to write the FokkerPlanck equation in the Landau form (2.9); )-FA J,e i' FA A",, p', at ). SIr V2. (31) where Qt iB(s}\ ra) pint- uX-* rA is a positive constant, V = -v', and I is the unit dyadic. It is not difficult to showl that the Maxwell distributions FA, Fm satisfy (3.1) for DFJ t/t to,j= A,. In the vicinity of equilibrium we may write FJ: F I t+4I+ (', ~)J. Neglecting terms quadratic in 4. we then obtain from (3.1) the linear equation M a~~~~~t aJM M J 1 ( - 60 -

- 61 - We will refer to the quantity F~fJ as the perturbation from equilibri um. Equation (3.2) as it stands is in fact a pair of coupled equations for fA and ~B. Due to the quite small value of the electronion mass ratio the equations are however only very weakly coupled, Thus for example the effect of the ion perturbation on the electron perturbation is small when compared with the effect of the ions and electrons in the unperturbed equilibrium distributions. In addition, it can be shown2 that in the approximation r o>a~ me, the ions act like a single component gas. In the following we will consider the equation for the ions. The treatment of the electron equation is quite similar, and the modifications necessary for this case will be indicated later. We have then, Cl- (13I-C'r F>1 4 7 (3,3) We will for convenience drop the subscript "i" from,L and it, If satisfies the conditions ZVI tvf=,F. (3.4) it is possible to show that (3.4) conserves number, momentum, and kinetic energy densities. Introducing fY{t) = >AQ)e/p(-At) in (3.3), we find -AR 2 f, g~a ir ~ E l' Gab']7 (3.5) Multiplying (3.5) by, and integrating over r we have, after a parts integration,

- 62 - v- It z-' fJ3r _sV F.-' - CPT C c (3 6) The second term on the right in (3.6) vanishes provided 9, satisfies the second of conditions (3.4) and fI/z S,, w 44) = 0. (3.7) Assuming these conditions hold we exchange r- and v' in (3.6), noting that Qfv {' (t}j ). We add the result to (3.6), obtaining iJ-F,,,/~1 - |F Ftt( Q ~ p a~gh (3.8) Since G is a real positive quadratic form it follows that the right side of (3.8) is real and positive or zero. Hence,\,A=Oand A 0 O Employing standard methods3 we can find from (3.8) the most general form of 9, when -O; (s =+ al - t-3 j(3.9) with ol ) and c3 arbitrary, but necessarily independent, constants. 2. Expansion in Spherical Harmonics From (3.5) we have

- 63 - — F d=r13 Q' (3,5) To perform a parts integration on the second term, consider the quantity With F.,- =o (/)3 eX t' -i= - /, /2 we have rF;;//~at - - zot 4' F,' Also v/. Q, so (3.10) gives G,:~&,,-'v-' F,,Q ='' 1 The first term on the right in (3.11) vanishes if satisfies (3.4), Using the relations _Q r l - zr and introducing a dimensionless time r and dimensionless velocity c in (3.3), Th(- firs= teronthe ri t (3-11)vi f satisfies/34 (3.3),

- 64 - the kinetic equation takes the form - SrrcI e-c' c') =. i'<v ]c_cZ- - -2 I~~ eE A~ - ~' o~ (.3.12) The time r is measured in units of the "Spitzer self-collision time,"4 and c _I / is in units of the rms thermal speed. Equation (3,12), in three dimensions, may be replaced by a set of uncoupled equations in one dimension by introducing the spherical harmonic expansion -o - s~A 2 S c) (, A,{, ) Y W(_, _) We find (see appendix A) $JC 1 e 9l e,/~/- =.7 fI (3.13) IJ3 C/ ICC/ vt h 3. 1 3) -C I C I = -T ( Jc)l-'e "jte~ ZI 1 "s (3.14) idle-~ where RQ = J,3 (CC ~- /c 3'},

- 65 - wih Se d/)/- (~ / [ZL- () U-jeC' 9( i'C ( T I eCC (C 9 );cerf(c), with efCc) C 2crr f- c/A ep(-x ). The expressions for RP\ and, were found by Rosenbluth et al.5 using a different method, for the axially symmetric case m - 0. Combining (3.12) through (3,15) we find Z2 ~y " = Z ~i'7l) a - 2- M R -- C Y). Le (3.16) _ X I I );C * Ic e - Performing the indicated angular differentiations in (3,16) and then employing the orthogonality property of the spherical harmonics, we obtain the uncoupled equation - ~-T _t [T' + (? c)T - c ~3'~ 3eL 17I (3.17) ~ [cs: +(3~2ca)S - cSA~ S~~j~j

- 66 - The index m is clearly superfluous and will be deleted in the following. Performing the primed differentiations in (3.17) we find + [(#2c3 _ (~( + [Zeq~g 4(X+1), +(-I 5) ( -I ) d e -)' I C / /e ( )/, ( - _____,d (I cI/. (3 18) For boundary conditions we will use conditions (3.4), which were obtained from the conservation laws. Although (3.7) is stronger than the first of conditions (3.4) we will see below that the solutions of (3.18) which satisfy (3.4) also satisfy (3.7). 3. Spectrum of the Radial Equation If we introduce the transformation q (Wd (' 2,. ) -c' e=/ c/2 g (~' )( C (3. -

- 67 - we obtain from (3,18) the formally self-adjoint equation dc /CP C+7 Iji dG!~ ti ~A dc7R C (re=7 + with P(<) r/ti_ e3 t (l C e- + I -- / ) A. Transforming (3.4) via (3.19) we have C, -: 0 ( 3 d c, C iv+thc$ =Q. (3.22) We will later show (Appendix B) that the problem (3.20)-(3.22) is selfadjoint, The spectrum of (3.20) is that set of numbers ~,/1 such that (3.20) has non-trivial solutions which satisfy (3.21) and (3.22). We have already seen that the A/~ must be real, and must be positive for >2a and positive or zero for3 <2.

- 68 - We can find the spectrum of (3.20) by first considering the related problem ci j dCS ] /\[e "4 - (3.23) with conditions on the functions ~(L) identical to (3.21) and (3.22). Clearly P) Je/cC, and (~ are bounded and continuous for all finite c except possibly near c=o. For c<<I we have ()-= - - c *+ O(So), (t ( o + () -~,,?tl) [3.,r * -(. In general we can write pn =t - fJxr Ihe e- and thus P(6)> O for all c~oo. It follows that for.1o 1 (3,23) has a regular singular point at c~=. For c small, (3.23) has the asymptotic solutions La > (c c/+l (c<<1) (3.24) The first of these satisfies (3.22) for all (, It also satisfies the stronger condition obtained from (3.7),,(c o)c-//z;(C)>o The second solution satisfies neither condition. This is clear for / o. For J= 0, the second solution is a constant which cannot be zero since the solutions (3,24) are linearly independent. For c sufficiently large and for Ah4 #o, (3.23) takes the asymptotic form

- 69 - l4. -3 2? t And qAc5 13 ouo) We find (A,>o rn, (<) A~, C 3/Y < (24\,- ) (3.25) with -Y Ah /~ r //-/ Given. and Ad, equation (3.23) has only one solution which satisfies the condition at c=O. This solution thus contains only one arbitrary constant and it follows that A,, and /o in (3.25) are not independent. Whatever the relation between Aq and q1t is, (3,25) satisfies (3.21) for all positive Aq,. Hence the spectrum of (3.20) contains all positive' A,, for each / To determine the spectrum of the integrodifferential equation (3.20) with (3.21) and (3.22) we note that It (~~,c') is a HilbertSchmidt kernel and the symmetric integral operator in (3,20) is consequently completely continuous.6 According to Weyl's perturbation theorem,6 the addition of a completely continuous symmetric operator cannot alter the continuous spectrum of any symmetric operator to which it is added, Since (3.23) is self-adjoint (see Appendix B) it is symmetric, and it follows that the spectrum of each / -component of the linearized Fokker-Planck equation contains all positive real Al For L-o and/=l we found A,==o belongs to the spectrum; this corresponds to a shift to an equilibrium different from that originally postulated. For l> / the spectrum is (1,41 )-fold degenerate, corresponding to the (z2/-l ) different spherical harmonics of order /

- 70 - 4. Outline of the Expansion Theorem We now proceed with the development of an expansion theorem based on (3.20). Our method is essentially an extension of the theory due to Weyl and Levinson,7 to include singular integrodifferential equations with Hilbert-Schmidt kernels. We give a brief outline in this section, The details are left to the appendix. Since the eigenfunctions of the kinetic equation are bounded and continuous on every finite interval, it is natural to pursue an expansion formula for functions u(c) square integrable on the interval A: oc< C oo. As with the earlier theory we first establish an expansion formula on a finite subinterval S of aL,: c' b, < bc-< so that the singularities of the linear operator are external to S. The expansion formula we seek is then obtained by taking dt a in a suitable manner. In the following we will mean by La the integrodifferential operator in (3.20) and by LS the operator obtained when the lower and upper limits of the integral in (3.20) are replaced by e and 6, respectively. In the following the index g will be retained only where it is necessary to avoid confusion. We have already seen that P) P! and 9 are continuous on g and that K (C)c') is bounded and integrable on the square r- b-, ~zc-''/. Tamarkin8 has shown that subject to these conditions, the solutions of LS =-A- which satisfy homogeneous bondary conditions atc-= and c-b form a complete orthogonal and normalizable set of eigenfunctions Ih,;, on i, with an associated denumerable sequence of real eigenvalues i/~l. Assuming the sh to be normalized, the expansion formula on 6 is thus

- 71 - (~)= hit ~ eJcc )() (3.26) where A(c) is any function square integrable on 5 We now use the Weyl-Levinson theory to take -'A. Since the subsequent development of the expansion theorem is in every respect a duplication of the earlier theory, we will display only the salient features, Given A, the most general solution of Li 0#=-A 0 is a linear combination of the two linearly independent solutions, say l, Thus we can write, (ce, A, ) = fri q, (c)\ A St) k r(, ) (3.27) where rat, and rgz are complex constants. With (3.27), (3.26) becomes Following Levinson7 we define an Hermitian, positive semidefinite matrix /s, called the spectral matrix, with elements w/SJ'k which consist of step functions with jumps at the eigenvalues /AS, given by faxk b C o ~) -gS0' (A~t -~) - \j )T Let A )( +o)-: (A), and let 1r(o) be the zero matrix. We use the spectral matrix to replace the infinite series in (3.28) by a LebesgueStieltjes integral Z (e ) = r) (A) Jest t) (3.29).- ~k

- 72where b As S->a (that is, Xv-o, b -oo), /o approaches a limit matrix t To find gob let -, +t', w oo, and let -.= q+,mk(A) z be a solution of L&S=-A- satisfying the homogeneous boundary condition and similarly let Xb, I, tMb(- ), be a solution of the same equation satisfying cg (j 3d) f g / P(b) X/(b) -c, Clearly ra' [i.C)/f'T (%) and similarly for Mb As zoo and - v oo, Ac and vzb approach limiting values in the complex m plane denoted respectively by mo(A) and,(A). These limiting values are clearly determined by the behavior of ~1 and 04 for small and large c, for A complex. For 1 and 0, to be linearly independent it is necessary and sufficient that their Wronskian equal a nonzero constant, say one; eP U _ I= I (330) This last will be satisfied if f and K satisfy the conditions +1 (s ll) = K rjt(S v ) - w, T P) - Cc. z rts)~l SA )= Kor g where S is an interior point of ~ and o ~<O lr. These conditions are also sufficient to ensure that,I ~ are entire functions of A

- 73 - for each fixed c on & (this follows from Tamarkin8). With these properties secured we can find7 the limit values.ao and u and hence the limit matrix go, whose elements are given by,PA;Jk ("'A) -k( (1) t+0+ 5'I' JkM t (3.31) Aj -k 0+ Tk where M,, (A): 1 ~qeo.- VV~ M12(A) mH A22( () - A - ATo find the tjk we need asymptotic forms of ~! and <, for large and small c. These are given by (3,24) and (3,25), as may be verified by direct substitution. Taking t, cz to be asymptotic respectively to c-, C for c small, we apply the homogeneous boun — dary condition to ) and then take Po+o to find ~o: oo (X ~ O) wo =- - Shot ( &-=o)o (3.32) Thus for; /0 only MvZ can have a nonzero imaginary part and consequently only q) will contribute to the expansion formula (3,29), When 1=o both solutions are regular atc -zo and the limit matrix is not determined until we specify o(. The boundary condition (3,22) dictates the choice =-O.e For c large we take

- 74 X - C, 3/A 4. (C.I/q _ ) (* C- 5/Z/ f1 Ah) > ~ and find QI by integrating (3.30): c E/Y ByLV (c 6/- _ cI)7 (5r/T/'? AH v c (1 Co.- M. ) where co is a constant of integration. Applying the homogeneous boundary condition to 2B'-EtkbM )~z and then taking 1-ow with e\cA >o we find'9e1p rid (l cf"/ - <97 Combining(3.31) through (3.33) we have finally Combining(3.31) through (3.33) we have finally.d, 5 (4,) z Jdszz) = t) 7~> ~),//,,1>0o(3.34) Since the spectrum is empty for Ac <o, o(a) is constant on this range. The expansion formula (3.29) becomes with The expansion converges in the mean for all funct) square The expansion converges in the mean for all functions a(p) square integrable on (o,oo). If the spectral function g is not continuous at -- =o, this point will contribute to the integral in (3.35). We return to the description of perturbations from equilibrium. If exi(-~~)t('[,o) is square integrable in velocity space, then from

- 76 - As we indicated earlier we decouple the electron kinetic equation from the ion equation by dropping the term (m,e/'i) /'//' in (3.2). This amounts to neglecting the effect of the ion perturbation on the electron perturbation, but retains the effects of encounters with ions in the thermal distribution. The uncoupled equation conserves electron number density provided (3,4) holds, but does not conserve momentum or kinetic energy in the electron gas. This is as it should be, since a substantial portion of the electron momentum, and a small amount of the energy, is lost to the ions. Applying the methods of section 1 we find as before w Ao, A>, and for A=o we findg o const., corresponding to (369), The remainder of the development proceeds as before. A spherical harmonic expansion yields a set of singular integrodifferential equations, and the transformation (3.19) brings these into self-adjoint form, As before, the expansion formula has the form of a generalized Fourier integral If, for example, the ions are protons, then we can take ree= rei If we use o(e in place of o(t in the definition of'? and 6, then the electron equations may be obtained from the ion equations by replacing T(C) in (3.15) by T(c)tdc/ZT(f/-~e) where Ea o(e/ Id e/ — CL 6. Discussion We have used boundary conditions obtained by requiring the solutions of the kinetic equation (3.3) to be consistent with the conservation laws. The Hilbert space then emerged as a natural function space for the framework of the mathematical development. The question

- 75 - (3.19) and (3,35) we have e ( r l f(ed)d (326) with and The functions ~1 correspond to q in (3,35) and are the solutions of (3,20) satisfying (3,22). We have defined the density, mean velocity, and kinetic temperature of the ion gas as being proportional respectively to the first three moments of theaquilibrium distribution F, If / is continuous at Ao — o this point will not contribute to the expansion formula (3.36) and the eigenfunctions (3.9) for At -o will not be contained in the expansion, By virtue of the conservation laws the functions (3.9) will then be orthogonal to (3,36). Thus (3,36) is complete only if o has a jump at Al= 0 for 1-D}o I. It follows that the exclusion of (3.9) from (3,36) yields an expansion which is complete with respect to all square integrable perturbations conserving m,*<w%>, and 0 5. The Electron Kinetic Equation We have developed an expansion theorem based on the uncoupled kinetic equation (3,3) for the ions, The extension to the electron kinetic equation is straightforward and requires only a little algebra.

- 77 - persists (see e,g., the discussion of Uhlenbeck and Ford9) as to whether square integrability should be a requirement on the distribution functions from the beginning, In the light of the present work this condition does not appear to be necessary, and for our purposes it would not have been sufficient, To see this we note the condition JIdu I F I1 < o leads to cI/' l0f = o- 0 (3.37) which is weaker than the corresponding condition (3.4). Since both solutions of (3.20) satisfy (3.37) for,(= 0, it would be possible to have an expansion theorem for solutions of the kinetic equation which are square integrable but do not satisfy the conservations lawso

IV. THE SCATTERING OF PHOTONS FROM A PARTIALLY IONIZED GAS 1. Some General Properties of the Scattering Function In the first part of this chapter we present a brief review of a classical derivation of the scattering function, and discuss certain properties of an equilibrium gas relevant to the computation of the scattering function, The rest of the chapter treats photon scattering from a partially ionized gas. The photon scattering can be characterizedl by a cross-section describing the effective area that a particle in the sample presents to an incident photon, having directions-4 and energy f7t, for the scattering of that photon into. a small solid angle about the direction V and into a small energy increment about ed'. It can be shown that, neglecting relativistic and dispersion effects, the electron cross-section is given byl T M'' )S) C) A(8) S*41) 4(41) where o-. () is the Thomson cross-section and e, e and Ax are given by K A=W =: eA' =, with c sk, w= ck, and - 78 -

- 79 - The so-called scattering function S is given by -o S A) L) ~ tinge i do e it'" U 6kt/ (402) where of e^((g ) =( <etrlC~)F (-))>r (4d3e The normalization is such thatl'/e is the total number of electrons in the scattering volume, It can be shownl that the scattering crosssection for ions with mass mv is of order (me/mi)D smaller than the electron cross-section (4,1), where Fme is the electron masse We will thus neglect the photon scattering from ions, assuming local charge neutrality in the scattering system. Scattering from neutrals will be considered later, The function e in (4.3) is the Fourier-transformed electron density operator. Since the derivation of (4,1) was necessarily quantum mechanical, it follows that the density operators should be described quantum-mechanically, It was argued in reference 1 that the difference between the quantum and classical descriptions of the density operators will often have negligible quantitative significance. On this basis the somewhat simpler classical description was employed, We will continue to assume the validity of this approximation here, The reader interested in a quantum mechanical description of the scattering function would do well to consult the work of Rosenbaum, Zweifel, et al 293 We are clearly concerned with the electron density operators p' (it),where

- 80 - and %J% rrb Si (4.5) We seek ultimately the thermal average of the product of the Fouriertransformed electron density operators as displayed in (4.3); i.e., with (4.4) we wish to obtain &ee ( Oei) = cI5 r 3cP( S ('2c'e ) ha To compute the thermal average above we follow Osborn1 and generate a set of equations for the phase-space density operators, which we then solve subject to certain well-defined approximations. Since the procedure for generating these equations has been delineated elsewhere,1 we present only a brief summary here. Assuming the dynamical variables of the system obey the classical equations of motion, we have = - - x If where H= T7V is the plasma Hamiltonian. The symbol i 3 means Poisson bracket, and for any function A of the system dynamical variables, A el) 2_ IE> r_ p - W aW —I ~~A)I( J`~~J~,~

- 81 - It thus follows that _- LA and hence In particular, the phase-space density operator cA for particles of kind A is given by A t e 1 (')) _L e-(o)) (4,6) If the plasma Hamiltonian is taken to be H4Z {2Z ~;7.. Vi No) Ng AA then it is a straightforward matter to show that gA(,v)t) satisfies the equation -W Eqato ( isq ( 9 in r 1, - d'J~v L a J t'~)8 ) _ ~/'*'(4 -8) Equation (4,8) is similar to eqns (II,19) in reference 1, but is now generalized to include any number of species in the scattering system Now as in reference 1 we let the average of 4 be FA, ie,,

- 82 - and further define the fluctuation operator SA, MA ~~4) v-,r )_(4 1 0) Still proceeding as in reference 1 we combine (4.8), (4.9) and (4,10) to obtain an equation for the fluctuation operators a A, and then approximate this equation by neglecting terms quadratic in the fluctuation operators, We obtain aS4A~t- Ir ~' ~aS-A - KA, ^ T ~x (X) [F( ~, t) SUB (.li) t SxA xA) FO( U 0f(t4. =11) Equation (4.11) is now further simplified by assuming that the target plasma is in the thermodynamic state, and further that the singlet densities F FB are independent of space and time, and are Maxwellian functions of the velocity. Euqation (4.11) now reduces to as^A i a A _ _ _ FRA (x) daq'at to \r WIA V r abx ( (X) (4.12) It follows from (4.2) and (4.3) that we must solve the system of equations (4.12) for the Fourier-transformed electron fluctuation operators as functions of time for all t, - < t<-'. To this end we introduce respective Laplace transformations4 for t>o and t< o -

- 83 - Si e,)=jt e~-dgPt A{XSt 0 p = tte'9 tto r_> ( (Xt (jPpePt p) = Jj-6'~)(4j13) -= (t<Po) i: A I j p = e VptI At / {st) o p'" (4,14) where _ A(X-rj )' S s (4.15) and further introduce the Fourier transformation We thereby obtain from (4.12) -I/L \r )- 1~ _(" x1 We now divide (4.17) by ( p - ), integrate over, and define

- 84 - h= (,p - WVsa (,, ) (4.18) and )A )I5. V (-) Togo, X(4,19) obtaining (tAk) t-;& S2 hA ) ~ (4.20) Now let L represent the determinant implicit in the system of equations (4.20) and let te be the cofactor of h. Then solving for we find he ) ~ (, p) t3J (e dO (4221 ) Finally, we multiply this last expression by s0em( ), thermal average the product, and integrate over v-'. These operations yield I3tr j5S/<,gae0#g ) 48ebd ~ -e o) (4.22) where G6e~o _ < %C'e (t"I)s4o);:B(}o) (4.23) We now perform the inverse Laplace transformations on (4,22), as per (4.13) and (4.14), and employ the Laplace convolution relations,4 obtaining

- 85 - d- ePt' p) GAO > > e __ (c->o)t1>o) _n~| D ((,p) _( t) _,0 ( (0f- ~t<o) (4 24) where we have defined Df~~~a (pr, )~ -,l (Jpe<;c (4,25) ltiri ting we can easily show (provided the interparticle potentials depend only on the magnitude of the separation), since /B and A are functions of the -AA6 only, that DB obeys the symmetry relations D(% ) v8(- D(-Y-2~) (4 26) and further Combin (.) = D(41 0), (4,27) Combining (4.3), (4.4) (4.10), and (4d15) we have, with dfr3,Jr'c3xd3p~'et F!)Se(yt)F6c') >(2n)e Ne~ (~)>

where IVe e x scattering volume, ee(it) = (z~r)3 hle Ne i() + + dip td>,d3,,,V Y(,S. 4, 2 8) Fourier transforming with respect to the time variable as per (4.2) we find S ~,2 n ) _ (7r)3xe VI() V I vo) + He X & (x)6 (PeEcZ) s1/2e - D {El - (4.29) To complete the description, i e., to portray a given experiment, we must specify the interparticle potentials and the quantity, e(o) defined in (4,23). The complete specification of this latter quantity requires a fairly detailed knowledge of the scattering system, and is generally a formidable computational task. Nevertheless certain general properties of G6e() are readily established and are germane to a description of the structure of S6, a/ ). Consider the thermal average of the product of the time-independent density operators for the species ( A, 8 ). With (4.5) or (4J10) this quantity may be written either as Ni ^ NB o 2(a)9 {(9 )r a 2s 6(Q-Q@ )(@QC @)r(4,30) or as + < M (@) 64 B (Q,, (431 )

- 87 - where @_ v As before we assume the functions FA, F6 are space-and-time independent Maxwellian functions of velocity frG F,e For later convenience we denote 46(9,9) Q - <al (Q) iM 6.Q0) (4,32) Since 5 J Q C i' KA(Q) 8B(")> A/AVNB (4,33) by definition of the density operators, it follows that Q a 4' ). (4,34) Our interest here is in the functions G A, the Fourier transform (G tb 6t) being needed for (4.29). We begin our analysis by separating (4.30) into two terms; tJA NG +< Z E-a (4'-9;'>T (435) We take the system Hamiltonian H TtV to be as given in (4,7), and define - Jr - l ~ e BT/9'I f' Je-v/ where the respective integrations run over the coordinates of all N particles in the system. We now write the second term in (4.35) as

- 88 - J ~6 _I-Ie-(Tt; (lQ-e) s (' ~) ~r~ t+*'~ A- A V/G)~iNA S r M A[) M BN') JJx $-e-v/~ ~ (~-XV);[C,) MAvi)M A (fr/) )8/\ B{) V) ~A, k(4,36) where nA M (Fm FMA(y) and hA is the number density of the A th species. To determine the functions nYf defined in (4.36) we take the gradient of (4,36) with respect to X Y" - A J3xe e(6 IVA N8V/6 _ ) I-i 4* xi)6f/-x- -- e a, (4,37) From (4.7) we have V = z 2 V C(, 7-r 1) +Z 7 Vclr-_ ( I C C7,) 47 c Q

- 89 - and hence N\IA v A"(I)d-AAD at 6w A 2 X BV *D (1-D)N A _vAA + 2I axZ' (4.38) In the following we will neglect the second term on the right in (4,38). This term is proportional to the force exerted on a particle by that same particle, and we expect that the effects of neglecting this term will not be manifest in any observable results. The relation (4.37) now becomes A A ax = -~ cJ N3x~ E l(x-x );(- N)e?D v/) P) AWe now separate this last into two terms; one for o-= ( and hence 8 =D ) and one for ca ( A N A A D)iAV,.

- 90 - NANaIJ~) S' (4.39) eD P(,k,~~' ~~? e, Recalling the definition (4.36) of n j6 (".,/) and introducing pdf~,:' 9-= <~ D D= A the relation (4,39) becomes,,A~B (X I.').n 4 + + - X|J3XI/ A D( /y (4.40) To compute the two-particle correlation functions AB, we write n D as n3 ( ))= n An D t A s 6 AO {X EN)+ - v A9 ('&,. + h tD (4.41) where h is a three particle correlation. Inserting (4,41) in (4.40) and neglecting k3 then yields the system of equations _~X ~ a~" (4.

- 91 - Thus given the system potentials, the correlations may in principle be computed from (4.42). Two pertinent properties of the r can be demonstrated without specifying the potentials. To obtain the first of these we introduce a change of variables in (4.42) according to Vr _ -('', x"-x/ This gives ay A SB v (ll) A AD A? j@i~f,aV By + a0 (46 43i Equation (4.43) is clearly invariant under the transformation ~ a- If we now add a constant vector O, to f' such that A6 we find that (4,43) is unchanged. It follows that mL is a function of I~1 alone; i.e., cE ASB nof (Ae 1) - t (ffi-X (4,44);q~ (,,',) =' In the sequel it will be necessary to have on hand information regarding the normalization of Aw Since the normalization is already specified by (4,33), we merely combine (4,30), (4,31), (4.35), and (4,36) to find G( ((9~) as a function of'v2L. v- v-,) n(4,45) Integrating now over ( X, x', B, im) we find, with (4.34)

- 92 - j]X op x d(,x, N A (N B )SA (4.46) or i j3'r A8B () n A (N& B _,e) We return now to our discussion of the scattering function. Fourier transforming G&Bey (x,, we have & ^e (a, ~ )) -= I3xd3x'e'X) ~ /, where G%~(o) is given in (4,23), With (4.44) it follows that G ie ( } - ) (4,47) and since'As is real we also have G-e V,,V, Sh) = te( t V- j. ) (4.48) Combining (4.29 and (4.47) gives s (a) ^$5 = -S(-5 AC$n). (4.49) It follows easily with (4,27), (4,29) and (4.48) that S(', 4a) is real, as we would expect,5 With (4.28) we also have G7ee, t ) _ E ee(vtO) = G&eew *t) ~ (4.50) Combining (4,2) and (4,50) we find K (K, d a >) - )S l.e 6itO Je(t ast), (4.51) Hence when G eesatisfies (4.50) we can compute the scattering function

- 93 - having only Geefor t1>0 At this point, instead of inverting the Laplace transform as in (4.24) and (4.25), an operation straight-forward in principle but generally Herculean in practice, we adopt a procedure due to Rostokero Thus we consider first the identity = dur Jl jt jte J dt e' s') (4.52) where S t 2 ~ t 27r Pi |- vv (4.53) since4 () dt e' 9 ~9) _ -) tl P yiV) ZoI dt Jr -i PyC23 ~(4 54) with P indicating principal value. Now let S+e P)), =0r-t+ be the Laplace transform of some function, and consider further the following inverse transformation P-f —to oJp~r-ip IJJ~Ls s- teo

941t-7 r 00 00 /0 S d.e C t F) ) i)dte l(te )' oo = O Q3 Jv S t, v) I t/ "o ) = )to.(4.55) 000 In arriving at (4.55) we have used the definition (4.53) of the function St(+<l{') a Substituting (4,2) into (4.55) we have a r-i oo.o i dpe P St7z 2e fet)) t >o -o tOo (4.56) It follows that. and the Laplace-transformed correlation function for t> O are related via No- Stt' lt" ) = — ~t 2re- O',I (4.57) From (4,53) and the knowledge that the scattering function is real we have, with (4.51)) 75Ne C-Yot t We now combine this last with (4.3), (4.4), (4.10), and (4,22). After Laplace transforming the forward scattering term (2rr)36ngne S(0-) and noting that

- 95 - GI - h-,pl otY) I/2nr3 e fig) tejro)t o PAW (2eCOt)(r)3eX()) _( ) we obtain (S A(w)c ) = (2)3 e,~('Ac) # tran 6 / tl' - -.l (4.58) 2. The Electron Scattering Function For A Partially Ionized Gas In this section we derive the electron scattering function for a system composed of electrons and one species, respectively, of positive ions and neutral atoms, The extension to a more general multicomponent system is straightforward but adds considerably to the algebraic complexity. We will continue to assume that the equilibrium plasma is characterized by a single temperature common to each species, and is free from spatial gradients or external fields. We present two different treatments, with somewhat different results. A, Reversible Theory From (4,58) we have, neglecting forward scattering, S (faw) = I e~ Q _'(x) TrNe arsc+ (x)[g/ 4iA(l+ACtht,. n,~ G.ee;+ -(A~/B~e i.4Gote I (4'59)

- 96 whe re -h I -Aee (ie I+Am < A11-A1') tA1,(ItA,,,vLt +,, -i; and B se I/ p _ l J v/ < ~ In writing (4.59) we have made use of the identities e; -c = ee AL'L' We have displayed the scattering function in (4.59) in a form that will facilitate an estimate of the significance of the terms involving the neutrals, To this end it will prove useful to write A.lBin the form ( —6- w (4.60) where and the sumbol "/7" next to the integral means the path of integration is deformed above the singularity. To estimate VAeO[() when either or both of the pair ( A, B ) is a neutral atom, we assume the potential may be approximated by a Yukawa potential, VA/(r) CA6 f-I exp (-r/a)

- 97 - where &. is the effective range of the potential, and cA, is a constant to be determined, We have easily VA%)) 7 I CAB; With K/-I >> io-, k a ~<<l and thus VAB(UK) B # 77LCAB (46)61 To estimate the constant cas we recall that the center-of-mass differential elastic scattering cross-section for the pair ( A, 6 ) having relative momentum ~ is given by7 () ^ (@ ) = A X /tAV A ()| where /,A. is the relative mass. From (4.61) it follows that, in the energy range of interest,.0'6(6) is approximately isotropic. Writing A6 - _ C 6 (e) i= 10- (a where8 0- is of order 1- coc gi, we have CA no J) At To estimate the various terms in (4,59) we assume ~-4,, and take mV to be the mass of the C Z atom, y,2, 5'/o-2 t.

- 98 - Assuming for simplicity equal electron and ion densities, we have, for A 6 —- 6.3 A Atl' he l' (/KL) e'I 2ezxi&q 6, ~ he Y A Xo t) AL e HE ee. (4A62) Here e is the laboratory scattering angle, and appears only in the argument of the sine, in contrast with the kinetic temperature, also denoted by e. Noting that9 the quantity in square brackets in (4.60) is of order one or less for all values of A6o//, we find (for A= 691f3f ) [^~y, Tn, rv 1./o6x o lL~-Ao 6 6 I) +/ E:;7 L23( r) [.//.- L i (4.63) and It is clear that the quantities in (4.62) and (4.63) are smaller for larger Me(or n, ) and are largest (about a factor of twelve larger than the abofe values) for a hydrogeneous scattering system.

- 99To determine the significance of electron-neutral correlations in the scattering function, we must estimate the magnitude of the quantity (see (4.45) and (4.59)) al'~~~e ~~ ld2+~~~~~e"oi 0 r) (4,64) where we have assumed m.v = l, To facilitate computation of the pair correlation functions De appearing in (4.64) we introduce the assumption that the average distance between any pair of particles is large compared with the effective range of the charge-neutral or neutral-neutral potentials. This range being typically of the order of 1o -8ly the assumption implies particle densities small compared with lo -3.V\ Under this assumption it follows that the contribution of neutrals to the correlation between charged particles may be neglected, We thereby obtain from (4,43) a pair of equations for ylee(r),.e(r 6 Taking advantage of (4.44) we can write these as ee- _ q d1 see L I ee l eeI ee'L bt Ie es r l H) Jrr V where gle is the ionic charge and ve= e, The equations above are similar to a pair of equations treated by Lambl0 for a singly ionized

- 100 - gas; with' — I Lamb's equations coincide with ours. Following Lamb, we first perform the angular integrations and then differentiate with respect to r, The result is a pair of differential equations I d (ttL/n~ee _ e;/ Eke,q/rT e(8! = ~~~ -~ +.__ +f l;J2 a t~ Je P/ q7rie;(ee Fnve)=o.(4.65) Ctdr & The system (4,65) permits a non-trivial constant solution. Taking we find Al - p; AX. LambiO has shown that for distances r large compared with e /6 the first derivatives in (4.65) contribute negligibly to the solutions. Neglecting these terms, it is then a simple matter to show by direct substitution that the functions Pee e D r ) satisfy (4.65) provided B. =-';L', and where We thus take the solutions of (4 65) to be e)1fl-A C/ K. e-/ - ~ A

- 101 - The constants A,., - are determined easily with the normalization conditions (4.46), so that finally we have te (r) = re C-+ I - ""rN Ye =age t e /+l~ v1 (4.66) where N/ = A e t N Nl To estimate the significance of electron-neutral correlations as per (4.64), we write lfe(r') in the form Ynke t\ e Nv We~r E where (1 is a function that we expect will differ appreciably from zero only for f'' /lo " e, the range of the electron-neutral potential. The normalization condition (4,46) gives Substituting the above form of Y e together with e from (4,66) into (4,64) now gives, neglecting the contribution from terms in I/N (which corresponds to the neglect of forward scattering) ~ -- (, ) () t ) Sdtret'e ~ve(c) With Kg A-I' (7s lo-ac- I where A is the photon wavelength, we approximate the integral above;

- 102 - Y (M~0t~e) I Thus finally we have, with A - Io-5X', G,.e - (It U-J U (e4D r) e1 4e -4|/ oclg olE5m We are now in a position to estimate the magnitudes of the terms involving neutrals in (4.59). First, we note from (4.62) that j.~, /~A/~ is negligible unless / eh o, or larger. In such cases the total light intensity scattered by the electrons is greatly exceeded by that scattered by the neutrals. Since our interest here is in the influence of neutral atoms on the electron scattering function, and since we do not anticipate an experiment in which scattering from electrons could be observed at such extreme density ratios, we will not consider these extreme cases here, It follows from (4.63) that (4.59) may be further simplified provided ~ to co o 3for graphite; Yt Ieo l8, rn), C- at atmospheric pressure),

- 103 - we would expect (4,66) to hold for almost all plasmas of interest here. Thus when (4.62) and (4.63) are satisfied (4.59) reduces to -S (-, -eo ) + ZgAce r t ] A eot A Glti e 4.68) where now zA' I + Aee + i (I-Ae e) The dependence of the scattering function displayed in (4,68) upon electron-neutral correlations (the term containing je" ) disappears when ~ ~< Io 2 (4.69) If in addition the condition..C....l......o<<o (4.70) holds, then Ae,,.,Ae < I and (4.68) reduces to the result1 for a fully ionized gas, For v ~" a -'}, (4.70) becomes. c 1o22 (4,71) It is thus apparent that, unless very high neutral and electron densities are present, together with relatively low temperatures, the present theory does not predict an observable effect of neutral atoms on the electron scattering function,

- 104 - Before abandoning our quest for an observable effect of neutrals in the spectrum of electron-scattered photons, we turn to a somewhat different and more realistic formulation of the scattering function, B. Irreversible Theory It was noted in Chapter II above that, since there are generally many charged particles within range of each other simultaneously for most plasmas, the effect of close binary encounters may often be neglected. In this case one may represent the effects of particle interactions by an appropriate electric field term in the kinetic equation. In contrast, we would not expect such a "field representation" to be suitable for the representation of encounters between particles having ranges of interaction that are small compared with the mean interparticle distanceo In the following discussion we adopt a scheme outlined in Chapter II designed to give a more realistic treatment of the interactions between charges and neutrals than that employed above, We begin with equation (4 11) for the fluctuation operators of the A th species-; it- + * X - It 2 | 2 3x " 3I' a v () 1 (4.72) At this point in ref, 1 and in Part 1 above, the assumption of thermal equilibrium was introduced for the target plasma, Before doing so here it is convenient to exploit the difference between the relatively

- 105 - long-range Coulomb forces between charged particles and the relatively short-range charged-neutral and neutral-neutral forces, To this end, and with an eye on our ultimate goal, we multiply (4.72) by Ie( /o), average the result, and introduce rP Ae(2(> X.jdT. P tB - (X/ Fry i) 9 fl/x \rt) (4 73) We thus obtain from (4,72) Ae I/Ae We now introduce a change of variables according to so (4.74) takes the form BAte Ae Bt Ma ) ___Flil (I,'(' t) <S8(,f,| trrat9 e, t (4275) In writing (4,75) we have suppressed the dependence on the variable K' Our identification of the integrand in p e(Q\, =do7' 9 finso) imle tnto n (4 76) as the equilibrium phase-space correlation function for the pair ( As e ) implies that rA (~) is a function of f and f only. It

- 106 - follows from (4.75) that re(t) is a function of r,, and t alone, The identification of Ae(, t), -(_re ~, Vit) (4,77) as a time dependent correlation function, and the relationship between the scattering function and the space-time Fourier transform of Gee has been exploited above, The description of r1e by a linear transport equation, as in (4.75), has been suggested on the basis of a semiintuitive argument by Nelkin and Ghatak,ll and has been employed by them and by Yip and Nelkin12 in a study of slow neutron scattering from liquids and dense gases. Recently Van Leeuwen and Yipl3 have derived a similar kinetic equation for [A, for short range potentials, from the cluster expansion of a one-particle distribution function, At this point we introduce an approximation into the treatment of the charged-neutral and neutral-neutral interactions, This leads us ultimately to the binary collision description attained by Van Leeuwen and Yip,13 and employed by Nelkin, et al 11,12 We thus adopt the treatment outlined in Chapter II in going from equation (2217) to the Boltzmann collision integral (2.24), Thus the terms in (4.75) involving the relatively short-range neutral interaction potentials VA(r-Irl) are approximated as in Chapter II by a linear Boltzmann collision integral, and (4.75) becomes ee FeVeA( K-~) at wdeAe, - eQI

- 107 - j- ljd3 n l, < le\r ar,, )t)rhei ) t t F,rw~t~)P&,.t,) - Fe(r,lt)tVI,i T _ FN (rl st) ree(rl t)7 (4.78) FN o, (4.78) and similarly for r. At this point we introduce the assumption that the target plasma is in the thermodynamic state, and hence that the singlet densities F, Fq are independent of space and time and are Maxwellian functions of velocity. We have argued in the preceding section that electron-neutral correlations should, under most conditions, contribute negligibly to the electron scattering function. It is worth noting that Salpeterl4 has argued semi-intuitively that pair separations which are small compared with both the photon wavelength and the Debye length contribute negligibly to the spectrum of photons scattered by electron density fluctuations, This is in good agreement with the experimental observations of Ramsden and Davies.15 This suggests that, since the electron-neutral correlation is significant only for separations of order )O' eA or less, it would be reasonable to ignore electron-neutral correlations in our computation of the scattering function. A somewhat different argument in support of this assumption is suggested by the observation that fluctuations induced in the neutral distribution by recoiling electrons should be insignificant. Referring to (4073) we see that neglecting neutral fluctuations implies neglecting electronneutral correlations. With these assumptions (4.78) becomes

- 108 - aPee re Ae M ) — S ar 1 1El: [^(S8)ree( If)- MAtAd2 Ie (SI and similarly for. It is apparent that the neutral species collision operator in (4379) conserves both electron and ion number densities, but does not conserve total momentum and kinetic energy in the system, in contrast with the collision operator in (4,78) which satisfies all three conservation laws. This, of course, results from our neglect of fluctuations in the neutral distribution. An important consequence of the collision approximation manifest in (4.78) or (4.79) is that while (4,75) is invariant under the time reversal transformation t —t, r>-a, these last two equations are not invariant. We will return to consider the consequences of this i rreversi bi 1 i ty shortly. We now introduce a further approximation into the collision description in order to avoid the complexities of the collision operators in (4,79) in their present form. We thus replace these operators by the linearized version of the simple single parameter collision model first proposed by Bhatnagar, Gross, and Krook,16 and discussed in Chapter II above, The model is constructed to satisfy, in this case, the requirement of number conservation for each species, preserves the irreversible nature of the above description, and provides a considerable simplification for the subsequent analysis. Finally then, the kinetic equations that we use are

-109 - pet t3 ) --. ) =f 2 LM( QV) $J3~r pfe( t3) _ reepsr f)J (4180 d r te t r-' | rI/c gv, ((4.80) The parameters > are clearly electron-neutral and ion-neutral collision frequencies. Fourier-Laplace transforming (4.80) and (4,81) as before, we eventually find, with (4.77) -pA -&- ( ) V_(4 83) ~A (} t~~Aee ff>')(.L Ae L -'LOP) -- ie )'A c(4 84) and ll.AS I

- 110 - The functions A4eK o,0) are obtained from (4.76) and the Fouriertransformed equilibrium phase-space correlations. Thus with (4.45) and (4.66) we have, neglecting forward scattering, p ee(t,, o) N e Me4it) (-l },. )I -It (l 9) (4,86) and r "e(K( qo Ne Mt( )(l?') [1+ (~A)~ (4.87), M Y) (4,87) where ie is the average ionic charge. As a result of our having employed an irreversible theory in the description of the correlation function, it is readily shown that this quantity is not symmetric under the interchange t -- t, in contrast with (4,28). In addition Gee(r l t) diverges exponentially for t-.- ao so that the integral in (4,2) does not exist, To overcome this difficulty we follow Nelkin, et al.,11'12 and prescribe a behavior for negative times different from that we would obtain by solving the system (4.78), ( Fe pte) for t< o, The prescription ensures the convergence of (4.2), yields a real scattering function, as it must, and the result is symmetric in M and zBoas required for classical systems 5 The prescription is G6e ( - ) - C-ee (t A) (4 88) Fourier transforming (4,78) for both species we can show Geet,th) _yee(y pof) (4.89) With the symmetry property (4.88) we can now compute the

- 111 - scattering function as before using only the Fourier-Laplace transformed electron correlation function for t > o. Thus with (4,59) and (4.82) we eventually obtain S (,~O) —' + z a-' ( t X- k,, i 1- (p t'-v)Q 0 (4 90) where -_ _< L(/i')(g (Pt ) (X):| t%;)(~p3; i- (~;)Q - and x +~,) [ KAD) It will prove convenient for computational purposes to write (4.90) in terms of dimensionless variables. We thus introduce the new vari abl es i- (2K D3~ - /cpe ( 5 /~ (491 ) where supe = ) D-e( L~~~fee ~ 1:'~ -—'

- 112 - We obtain from (4.83) /Ct,< _XZA l )3A; CT g 11 (Jue M 7h'~ -ot-' -'"' e> h —-'f~ 1-.t.'J. cope We We) j (i 4 - )Pe'AY( ) t' A (4,92) where the complex function' (xw'yj) is the plasma dispersion function tabulated by Fried and Conte9 and,/A (/ 2 )/Z, (4.93) The scattering function now takes the form s (/, X ):= (7rK)-1 (t)1/ ( 2 )- Q' () (X) ((2tl et qie -6 4 \ (me) }.Z.~, 2;.1("5)/ (4,94) with 9- - B' and 2/~"')' - z FI/' t' t~ f(t)7,,/ ~"

- 113 - The scattering function displayed in (4,94) will differ negligibly from the result for a fully ionized gasl when the imaginary parts of the arguments of the functions:(t),(t) are small compared to unity; <h^s /Z7 llA ar~ It, (4.95) To secure an estimate of when this condition holds, we write the collision frequencies'~" as VA < C A"",,) h (a A(CAM /O- (4 96) where (4.95)is of order one to ten cm2 as previously. With (4.91, (4.95) becomes Y\A x /o.:.- << or e ~ 3,6~ w' i/o i (4~97) For photons emitted by a ruby laser ( A-6 983A ), this can be written as ~-<< t/ 6 ~X'toz t (4 98) Comparing (4.98) with (4.66) and (4.70) we find that the scattering function (4.94) obtained with the irreversible theory will

- 114 - display neutral atom effects at neutral densities at least five or six orders of magnitude lower than those necessary to see these effects in the reversible result (4.59). In the following section we describe and discuss the photon scattering function as displayed in (4,94). 3, Photon Scattering From a Partially Ionized Gas In this section we discuss the results of the previous section for various plasma configurations. For convenience we take Aw'= Mw To obtain quantitative information regarding the scattering function, it is necessary to claculate the real part in (4.94). To this end we write PAA RA l IS ) RAKIF (4.99) and define m ~Fv' I/e, and t; s ( I., /(R' RIK - 7691' A~l ) qk -e + * Re),. %, (p-,'L R- -rlI) ('te'+T;) (' -'R,) b (4.100) with ~( /1t,') as per (4,93). After some algebraic

- 115 - manipulation (4.94) becomes g (K /\ )= _ g )-1Y( e )r < Ng/R -N. r -V I --.. (4,101) where 1, - (zell tl)(Re4 1 + Ie 22) - 2PmA/Yl3 N= (Zti'2 +I) (Re, ~Ieq, ) -t w~ jql'Y 3 /\ R - b, bl lL 3~b -1 Al b /\r~ llb, t bwb, t pWllY 6 (4,102) Using tabulated values9of RA, A, IA and rA we have computed S as a function of 9=4 &A/Wpe for different values of ( and re, rt, I Some typical results are plotted in Figures 4.1 through 4.3. To enhance our understanding of these results, we now develop approximations to (4,103) in various limits, To this end we employ both the power series and asymptotic expansions of the plasma dispersion function. These are, respectively,9 with t=X l- y, g Z2(t) =XL1/2 e-t-tr-n tl-c~1 2 -t(4,103) and (t) - lf 7T'/ett-l 1(1/2) t -i t(4. 104) where r = I ~ - d (4e105)

- 116Consider first S as a function of A. For,<< / collective effects play a negligible role in the scattering process. With (4,103), (4.101) reduces under this condition to (4.106) This is identical in form with the result of Ghatak and Nelkin.11 For ~=c -, (4,106) reduces to the well-known ideal gas form M 1 Ye/ e (4.107) For e>O, the scattering function given in (4.106) becomes narrower and increases in height as Be increases in magnitude.11 For <> I, A<< 1, v=(,/ll and teril or less, (4.101) becomes TShis lat is sm) lar t (4106) but now the) sc n ect, e This last is similar to (4,106) but now the scattering electrons are strongly coupled to the ions, and the ion mass replaces the electron mass in the scattering function, resulting in a narrower scattered photon spectrum, Next consider S with the condition we (4.109)

- l7 - When (4.109) holds we find (x) [((4.10t );' e my'(7Sl f t) In this case the center of the scattered photon spectrum is dominated by a relatively narrow "ion peak" provided wl'.1' s> (2,,,t)or A/ > 1 0(utt')/-17 * (4,111) Under the appropriate conditions the scattering function will have a resonance at the electron and/or ion plasma frequencies. Thus for(g 3, lec< and Ad I or larger, S takes the approximate * forms (x) e_'e (4,112) II-/ cZI 7 +F(2Fl/) r-2(2 le] (lefo)'.ao 2t- o..l..... * The reader will note that, because of the two different asymptotic forms (4.105), the form (4.112) cannot be obtained from (4.113) with

- 118The resonance at Ac=c/e is apparent in (4.112) and (4.113), It is clear from (4,113) that the effect of collisions is to inhibit the resonance effect, as we might expect. The height of the resonance is inversely proportional to ~e while the width increases as Be increases, For.1> /, vg'l, and 41=0, we find S (K-, 6X) - zf; X' K- ( ), ( (~....e.p.-....., (4,114) 1- (i(I Jr ii (2 te (1 c3 ) E -, ( Cl ) where p, and hv ev is the electron number density as before, For to we find a form similar to (4,113). Finally, consider the scattering function in the limit of very strong collisionso With (e-, (m" >/, and mv I, S takes the approximate form S (~, ~~ 1C, Hence in the strong collision limit collective phenomena are unimportant (except possibly for Aft I; see (4.110) ), as we would expect. The effects of electron-neutral collisions dominate, and the approximate form (4,115) of the scattering function is similar

- 119 - to (4,106), In Figure 4.1 we have plotted the normalized scattering function, for a single ionized ( -. =I ) gas of carbon atoms, vs. ~ as a function of the parameter for l e='= (zero effective neutral density), The general qualitative dependence of S on the value of is clearly apparent. For g< /, E has the Gaussian form (4 107), ForI /I the resonance at - I is present, becoming narrower and higher as, increases, as per (4,112), When f is small the effect of ions dominates in S as suggested by (4.110) while an incipient ionplasma resonance is evident in the vicinity of = /,IYSxlo / For >> /, S approaches the "strong coupling" form (4.108). In Figure 462 we display the normalized scattering function, as in Figure 4.1 for = },18 as a function of the dimensionless collision parameters M1e, Ad. For convenience we have assumed n e~ MV from (4.96) this implies equal electron-neutral and ion-neutral collision cross-sections, i.e To. e= o- a As ye increases relative to -! collective effects are seen to disappear; the resonance near Ao isle becomes lower and broader as per (4.113). When Ae-.>/, S approaches the form (4.115). Figure 4.3 is similar to Figure 4.2, only for 1= 2,36 To understand the behavior of the scattering function when yef.al; we have computed S for a few values of P with &/,/t when (i)je= o,3,A1 -t0'.,1 and when (ii), 7e=o,/, [-OA3 The results are displayed below in tabular form, It is evident that changes in the ion collision parameter have only a very small effect on S in the vicinity of ~- /, while changes in the electron parameter similarly do not significantly disturb S for ~ small.

- 120-: =5.25 =2.5 a 1.66 I~~~~~~~~~~~~~ = ~i 11=1.18 I:0.744-' fB=0.235 f~_ =0.100 | 10i ~~=0.235 =2.3 f~:~~~~~~~~~~~~~~~~0.25 F.100 4.10 I. 118 /= 2.3_5 10-4 10-3 10-2 lo-, 10 Figure ~4.1 Electron scattering functions for a singly ionized carbon plasma, showing dependence on the parameter I = (2KAD3-1. The results have been normalized to the ideal electron gas scattering function at 5 = dh/acpe = 0. Broken line represents ideal ion gas scattering function for I = 5.25. Effective neutral density is zero for all cases.

10 10 I \ \ t- \ \ I eO ~f\\\\\ al a,r/O N.0 to-2 10~ 4 0- aro 101 jnizedcb tt fetions for a (singD) The eUt' shovie been the effzed to thie ideal- electron gas SC te ~mcandltis habeeed.n ta T e pare-Mtel) mTe bro1en line re sents ~~~tion at bw 1 that V' ~lee = I 0. the approxmate form (eqn (4.115)) for e.3 -

- 122\0X I~,=0.' I0 10 \ Figure 43 ElectrOn 5eatteriflg functions for a singly ionize6 carbon plasma, showing the effect of neutrals for I (2 funl) = 1.18. Te results have been normalized to the ideal electron gas scattering functionr ates/ e 0. The parameter rje is defined by rie Vfepe The broken line represents the approximate form (eqn. (4.115)) for I Cue = 10.

- 123 - Table 4.1 Comparison of Scattering Functions for Different Collision Parameters (o.lo-3 =5 x 102 1 3(0. 1,)0.3) 1.21 1.14 0.986 (o.Io, I) 36(.1)0.V$ 0,992 0,85 1,23 8(o,3, o,3) S(o~. ~,o, 3 o 0.998 1 09 0.81 So 13 O. I) 0.83 1.12 0,989 S(o, 3 ) O.3) This is as we would expect; when u'-LrP, = o where J =(2W./Yk/2the function Z1 is well approximated by the leading terms in its asymptotic form (4,105) and electron dynamics dominate ion dynamics in the scattering function. Similarly when lie06 << oe ion dynamics play the dominant role, These observations are reflected in the approximate forms (4.110), 4,112), and (4,113) of the scattering function, In most experimental situations the scattered spectrum is observed as a function of the shift AA in wavelength from the wavelength of the incident photons, Consider for example the differential

- 124 - photon scattering cross-section as a function of ZA for ruby laser photons (/ l bqC3 A ), for a singly ionized carbon plasma with h elo 108-s and =(10>) - + Ok 4.116) From (4,1) we have, converting to wavelengths, (22 ^XE/) = 2mC ({tat)3~-S _ M)o(2) (41 7) where AA/=/-,\ 6 It is clear from (4,116) and (4,117) that retaining the terms in AA/A yields a cross-section that is asymmetric about A =O, for constant 1< Comparing (40116) with the scattering function as displayed in Figure 4.1, it is evident that, for constant K, retaining AA/A in (4,116) has the effect of increasing the height of the peak at'=+1 and lowering the peak at.= -I In addition, the upper peak is shifted nearer LA,=Owhile the lower peak is shifted further away. Since it is not the scattering function but the cross-section that is measured, however, we must also account for the factor ( ItA)/A )-3 in (4,117). Clearly this factor will contribute to a lowering of the upper peak and an increase in height of the lower peak. The two effects are thus competitive, For the example at hand we have found the net effect to be a 5,9 per cent decrease in amplitude of the upper resonance, and an equal increase in amplitude of the lower resonance, In addition, the location of the upper resonance is shifted about 1.3 percent closer to a,=-O while the lower resonance is shifted the same amount further

- 125 - away. We have so far neglected contributions to the scattered photon intensity due to elastic (Rayleigh) scattering of photons from the neutral atoms. For simplicity we characterize the spectrum of neutralscattered photons by the ideal gas, or Doppler scattering function, The ratio of light intensity scattered by neutrals to that scattered by electrons is then (7 ( le))I, E) <<E Ore -- 1 t (4.118) where S (LA, co) is the electron scattering function discussed above, and (tR' lo-r8 ivl~g-~fen~c- 1 is the Rayleigh cross-section.17 It is evident that scattering from neutrals is unimportant when ~~he i~ Ic4C / (4,119) Moreover when scattering from neutrals is significant, it is clear that the effect will be manifest in the observed spectrum only in the vicinity of the central ion peak; i.e,, when Acw<<epe From the foregoing discussion it is evident that the scattered photon spectrum will contain electron plasma wings whenever

- 126 - In addition, collisions with neutrals eliminate the wings when /) e SA y o —e > I e e It thus follows, with (4,96), that the wings should be observable whenever 4, Discussion In the preceding analysis we have employed a treatment based on a temporally irreversible kinetic theory to describe the effect of neutral atoms on the spectrum of photons scattered from electron density fluctuations in a partially ionized gas, Our analysis was quite similar to that suggested by Yip et al. 12,13 for a description of neutral particle (i.e. photons or neutrons) scattering from moderately dense neutral gases, We remarked in Section 2 that an analysis based on a reversible kinetic theory predicted neutral atom effects only at unusually high neutral densities, A recent experiment reported by Greytak and Benedekl8 provides striking quantitative support for the irreversible treatment. These authors observed the spectrum of 6328A photons scattered from thermal fluctuations in neutral gases near standard temperature and pressure.

- 127 - Their observation of a symmetric pair of spectral lines located at AA ~f lo- A is in excellent agreement with the theoretical prediction of Yip and Nelkin12'19 for scattering from neutral gases, These results clearly contradict the predictions of the reversible theory, i.e., a Gaussian-shaped spectrum, at the relatively low densities ( Z X Z 1o j-3) involved, We have assumed in our computation of the classical scattering function that each of the particle species could be characterized by a Maxwell velocity distribution with a common temperature for all, For many experiments this assumption is invalid, and could lead to erroneous conclusions. The extension of the present work to allow for different component temperatures is straightforward but adds considerably to the algebraic complexity. While such considerations are beyond 14,20,21 the scope of our purpose here, we note that several authors have investigated the effects of unequal temperatures for a fully ionized two-component plasma. They showed that the scattering function for such a system can be qualitatively different from that computed with a single temperature model, In a recent experiment Kronast, et al22 have employed Salpeter's resultsl4 in a measurement of electron and ion temperatures in a theta pinch. For their particular experiment they found 0e / Po.. In our development of the electron correlation function in Part 2, we tacitly assumed that the inclusion of close encounters (collisions) between charged particles could be neglected. To lend support to this assumption we employ a simple Krook model to estimate the significance of Coulomb collisions. From the analysis of Part 2 it is evident that these effects should be negligible provided

- 128 - ~,'< I (4.95) where now =,tA _':J(4.120) t the Spitzer collision frequency for charged particles, with VC the Spitzer collision frequency23 for charged particles, ok<,.TF 8VAe > AA A- (4.121) Here,/A is the reduced mass for the pair in question and we assumed for simplicity a singly ionized gas, Combining (4.95), (4,120), and (4,121), the condition for neglecting Coulomb collisions becomes _<< (4.122) 23 where Ar\-A. Since x is generally a very large number, we would generally not expect Coulomb encounters to be significant here. (Ron, Dawson and Oberman24 and Fante25 have recently estimated the effects of Coulomb encounters on the electron scattering function using somewhat different analyses than the simple model employed here, They found the inclusion of these effects produced a change in, of the order of _A7_ ). The principal result of this chapter is the electron scattering function for a partially ionized gas, as discussed and displayed in Part 3. It is apparent that for given values of Ie, -t, A, 1 and A, the scattering function as a function of $A is uniquely determined. Even so, we would not expect that a single experiment

- 129 - could serve to measure all of these quantities for a given plasma. Our results apply for instruments with infinitely sharp spectral resolving power and uniform average plasma density over the scattering volume, Average density nonuniformities and the finite resolving power of instruments together with the natural width of the incident photon beam will add to the width of the observed spectral structure and it may be necessary to take these into account in a given experiment. In addition to the extension of this work to allow for different componenent temperatures, it would be most interesting to consider the effects of magnetic fields and small spatial gradients, for fully or partially ionized gases.

APPENDIX A. INTEGRALS Lets be the angle between ~ and _'. From the generating function relation for Legendre polynomials we have I ~ ~'1-' [c +rX- 2rs'367 / and similarly for C'-. Writing A(') - Z,MYM and employing the addition theorem for Legendre polynomials, we obtain (3,13). To find (3.14) we use the relation Li + x 2- 2ZJ - Jxx x IEx — 2x J1 JxId IrX + x~- ~ 1/,i (A2) Combining (Al) and (A2) we have e-ll ='C L - Zcc c ] 1 CI = IC _I2 [ /)/)(c' c)./=o and similarly for c - c'. From the pure recurrence relation for Legendre polynomials we have c(..~() g ((>) = _L ~- i ( 2e) +,! l (,4() (A4) - 130 -

- 131 - Combining (A3) and (A4) and then using the addition theorem as before we find (3.14). Finally (3.15) is obtained from (3,14) with Nm = goo with s,[ the Kronecker delta.

APPENDIX B. THE SELF-ADJOINT PROPERTY Let LS be either (i) the differential operator in (3.20) or (ii) the integrodifferential operator in (3.20) defined as in Chapter III, Part 4 on the closed interval I: Sac S b 6, o L, 1 < oo. Let La be similarly defined on (,oo), Then in either case (i) or (ii) there exists a complete orthonormal set of functions iCv;} on 6, generated by ~ -— /d with homogeneous boundary conditions at o. and 6b 1,2 Let e and, be any nonzero functions square integrable on ( oo), and consider the inhomogeneous problems Ls A +- A \f) Ls Ar+ A*, (B1) with the same homogeneous boundary conditions at X and b as in the above homogeneous problem. Let Q/A1 o so that A will not belong to the spectrum of the set (J,~t ~ Then the problems (B1) have nontrivial solutions -oo A- with b / )\, ag k I )J g )dj m (c) ~(c)A) JHe (B2) Now let - 132 -

- 133 - L/A~d Z xk*- A -/ jk= - with -k and 0 as in (B2). Multiplying by 9* and * by { and then integrating over o we have (I M(>'k Ur)(A) (B3) and similarly _/ AV k-A Since (3.24) and (3.25) are asymptotic solutions for both the differential and integrodifferential equations, the limit matrix P, is the same in either case. Taking -r_> in (B3) and (B4) we have Sj,= /mz and thus | 9*6)Z, (C) Jc'A | e t J4; =;e~) PS )c (B _t d- p. - (B5) After taking A -oo and employing (B1), (B5) becomes w h t d re C4d resultredo which is the desired result.

APPENDIX C. PROOF OF THE EXPANSION THEOREM This proof is based largely on the so-called Weyl-StoneTitchmarsh-Kodaira-Levinson theorem, as outlined by Yosida,l and by Coddington and Levinson,2 Our goal is an expansion theorem for real valued continuous functions K(c) in (ooo) with eC Ic/(c)l <oo The expansion functions are to be the solutions of the linear integrodifferential equation Jd (p 4-6.. + (.J K(' C') oK e Here?P P(') P, and /<: Kg are real and continuous and P>o, (o, ao), and,,c'll<('c~ l<. The function q(Q(,c) is real and continuous, is regular at c~O for,=:o, and has a regular singularity at c =0 for, An expansion theorem based on the solutions of Jo Pid^/\q + J,' I<(CI,' ) +Ac~') O (Cl) which satisfy homogeneous boundary conditions at o<., J<oo has been established by J,. D. Tamarkin,3 Our task here is to extend the interval (b, ) to the interval (o)fo ). Preliminaries, Tamarkin s Results Let L; represent the linear integrodifferential operator in (C1). The following properties have been demonstrated by Tamarkin3 (i) For a fixed complex / let ~l, K represent a pair of - 134 -

- 135 - linearly independent solutions of L _-A4, real for real A, satisfying the conditions ( P(s) $' (s,n l=o P(s). r(s,) = I (C2) Then 4n,,, IQ/ are entire in A for every fixed c on (ii) For the self-adjoint boundary value problem eo - 4 A*) t S P(a ) d/( ) - O,(ad + A'~~~~ k - (C3) there exists a sequence of real eigenvalues IAt and a complete orthonormal set of eigenfunctions Asr. In terms of these functions the expansion formula for any t E tl(t) is del ( 6), (C4) To extend the interval 6:&(a)) to f(,o,oo) we proceed as follows, Since 4, z, form a basis for the solutions of LSf=-AQ we can write hkn (r) - f (/31 (<s,)t Thuz (C) Ash (C5) where'[nl,' are complex constants. With (C5), (C4) becomes ~2 ~~b Now define an Hermitian, positive semidefinite matrix ~, called the

- 136 - spectral matrix, with elements /Gk consisting of step functions with jumps at the eigenvalues JR given by kWe let ) be the zero matrix and define We let p (O) be the zero matrix and define pl away from the eigenvalues byg r(A.o) = g(A). We employ the spectral matrix to replace the infinite series in (C6) by a Lebesgue-Stieltjes integral; 00t 0 (A() 5 I X tE f()A ) *K('l)?PGk (A) (C7) _, J~k~l where (C8) As A (that is, o, 1- ), {I approaches a limit matrix, Our task to to find the matrix p, and to prove the convergence of the expansion (C7), (C8) in the limit, Weyl's Limit Point and Limit Circle Theory For any number Mb, the expression %b=-t-mbsatisfies equation (Cl). We now choose mb so that $~ satisfies the boundary condition at the p nt p Tbn m us =0 (C9) at the point b Then Ub must satisfy w 1{1t /) ( - %/() +?()n()

- 137 - Note that the one-point boundary condition (C9) has not restricted A to real values. Since qI5,,, and OIj are all entire functions of A,Mb (A) is a meromorphic function of /. It is readily shown that every zero of the entire function Xb is real, and hence all the poles of ^6 A) lie on the real axis of the A -plane. Consider Mb as a function of A,, and ~. If we let e= ~j5 and maintain A and / at fixed values, we can write Aa as 16:.. (C(o) C~rftD Since AD - 1C = A,{ )'(b ) - A)(bA)),'(,) P);rub ( A ) $o where Wb(/,k,) is the Wronskian2 of q1,'. evaluated at b, the transformation (C10) is a one-to-one conformal mapping which transforms the real axis of the e -plane into a circle Cb in the complex v (A) plane. Therefore if LvA /A cX o, then vIbf/Ay,) varies on the circle CL(A) with a finite radius, as e varies over the real axis of the t plane. The equation of the image of the real axis, -x =o, is found from (C10); * C o r) I * (A Cll D ) or (Cll) V\b (b MX )=

138 - which is the equation for Cb (^)A It follows easily that the center of C, is M, Wb (#-)' ) and the radius is b (A| W (C12) For the moment let QI,. satisfy LS.- A,It Li =-/z with AI/t /A1. Then the symmetry of Lg permits demonstration of the Greens formula, (n-z, S 01 ~ld = ts (oW- W (%}i)6 Now with 1 - /L = A and by virtue of (C2), VVs (8, ) - /4 (4, )ft) = 1. (C13) Further, with cI (c) AiK)= 0n'(cA), A9 (')AL)-= t/(A A), and making use of Greens formula, we have 2 ) Combing (C, (C1 / j A) A) a) (C,) AJ tc - A)- t~ S L (reA)l ckCP Cobiin () (1 A, a 0,4,)o t a,,, (C14) Combining (C12), (C13), and (C14) we obtain

- 139 - (A) L2 (Dl 2 1 (eA OijA7j A eQ o (C 15) Lemma 1, If Wo=,A/-1o, then the interior of the circle b A) is mapped onto the lower half plane of the e -plane by the transformation (Cl0). Proof, Since the real axis of the g -plane is the image of the circle C6 (A) by the transformation (C10), the interior of Cb {A) is mapped onto either the upper half plane or the lower half plane of the e -plane and further, the point at infinity of the M -plane is mapped onto the point- Y()lA{ibA)/&(,/A)of the e -plane. On the other hand, we can write R [ pIa,;v,,7 L dllb) i) } $ (br), - > 0. This means that - ()qA,/ /) belongs to the upper half plane of the — plane. Hence the point at infinity, which is not contained in the interior of Ce (A), is mapped into the upper half plane. This proves the lemmao Since WS (~ ~>l, the transformation (C10) has an unique inverse which is given by Inb vw of L a, ifA) Yw bltthe a --- {A)+, i L L_ In view of Lemma 1, if B A: ~o> o, wat belongs to the inter ior of

- 140 - the circle Cb {A) if and only if o z<o, namely, t'( -t) >o. From (C17) it follows that + PF() (6 A)X qA' 7,A) By Green's formula we have Combining (C18, and (C20) we have ByLemma 2Greens fteormula w e havecircle (A)if;and only ifrw (+ t v +, Q t*;)27 (Cl9) and from (C2), V\hs (WSt + hnpLK) = WS (}s) t W5 (07 t+)F t tW/Js (4-)~ )+ (+ _M w\ K (C20) Combining (C18, (Cl9), and (C20) we have and only if

- 141 j t+ (,A t- i(K ), |~ de < Z and M lies on Cb (Aif and only if (Note: it is easily shown that Lemma 2 also holds when co: = / A<O) It follows that, if vw is inside 6,, and i'< i, then jL9 }t wf 2 < i }+ twv ~<t;Jc < b Hence va is also in Cb,, even though the centers of C and Cb, may not coincide, We thus have Lemma 3, The circle C;, contains Cb for 1'<l and A /1 0o. It follows that, as b6-'o, the circlesCb converge either to a limit-circle or to a limit-point. In the limit circle case we have from (C15) that Si is Ly(soo); the same property is readily demonstrated for I e Next consider the boundary point X, o,<< b. For an arbitrary real number oi, the boundary condition at the point x with h -, t-v,, determines Caot( c)i )+ Pd4)r y q(A) and also the circle Cg(A), described by the equation WOK (t, A< YK )= -G,(C21 )

- 142 - Similarly as in Lemma 2 we can prove Lemma 4. If o Y- A / o, then vY lies on the circle C /(A) or in its interior depending on whether or Similarly as in Lemma 3 we can prove Lemma 5. The circle CO, contains C. for,<t<o' and $o Afo. As before, as.*o the circles C, converge either to a limitcircle or to a limit-point. The Limit Matrix p Recall that LSd -d I(pci ) + + 4 Sc'/K4e')de(c') Let 6,, be solutions of Lg=: —A9 satisfying the conditions Pgs) 4'h,/ - PAs) Y(As,A) I2) For the self-adjoint boundary value problem ( o<, b1 o< ) CL4o d Qu) t Ad4o( P(a)c c4)-A

- 143 - there exists a sequence of real eigenvalues lJ~.n and a complete orthonormal set of eigenfunctionsm f)~. The expansion formula for any function A ~ (r) is M/k)-t = 2 h(C) | Js'2(Cr) hobe (r((C4) Multiplying (C4) by &'(') and integrating over $ gives the Parseval relation 6 6 u (l rcl) ~ J I c ~ (c/) k~. c~, lc. |(C22) Similarly if tAr()) Ctr(') 4 1(s() then )( (C23) Employing the representation (C5) and the spectral matrix p, (C22) may be rewritten as 0a6 _ o J' k where Mk = c) d eI). Applying the Parseval relation (C22) to any continuous function A on (o) oo ) which vanishes outside some interval 51, contained in one obtains / 00 - I A )hr 1(e)I1 d = C )( (C25) where

- 144 - Let -aqRtR wit be a solution of Lg =A-AA, v/1Mo satisfying the boundary condition and similarly let &b,1-yvb z be a solution of the same equation satisfying Then w and w\ lie on circles c, and c, in the complex vi -plane whose equations are, respectively, V\^ (X,) 0 ) =0 ) /b (3Gb/ ~f ) ~ (C26) It is easily shown1 that Green's function for the boundary value problem (C3) is ( 0AiF o ) ( a. A) A (A) )A) - l w, (A where (AI - nA, (A)= We (A) b) The Parseval relation in the form (C23) is now applied to the functions (A., )J' k c yielding

- 145 - i w tk JC= |aJ,) h JckGIr5 h6-Sv -; c(C27),~c' 91~k - c =-c*)J k c-N From the definition of G; it follows that G (, S. A) c S (c, l) - b(A) (C28) /JCA)- c>A) and ael p<> Lc~1M, I- ] b( c-" S (C29) p ELv) ()b wt)'A Using (C28) and (C29) and Green's formula, the integrals in (C27) can be evaluated. For example Z;~.~jb tcs(CS/h~l~d L 2' ^ 1v (A ) - VO (A)K (where we have made use of (C26) to arrive at the second step). Therefore

- 146 - x {ris2a) =~) /A (C30) Simi 1 arly and hence -* ((,. ) -)* ( - (* (C31) I~, A/n- -' "h A In arriving at (C31) we have used the fact that VOwh (?i js)=Owhich follows since both 3 and h satisfy the same boundary condition at We; similarly (A) tbha Recalling the definition of the spectral function we combine (C27), (C30), and (C31); ( JcolI- (A) _=TAD, (L ) where We can similarly show that

- 147 - J; - =..... -= (C33) where M6l, is given by (C32) and 0 {t /) =- I /)~ 52 >0 {)... Mb A. vv taA) (b A) From Lemma 2 and Lemma 4 we have eA9 2 sIb t - M b is) Thus for a fixed A, \.:A $0; v1/1^) and wm (A) are in oppostie half planes. Suppose / —l in (C33). Then points kA~') lie on a circle Ca which is in Cs/1, for az whereas points wn(~') lie on C4 which is in C3S/tfor Lb3s/Z. Thus there is a constant k 0> o such that I~lt')-a().')/> kl for a 5, b 3. Since w,') and " /l') are uniformly bounded for E < 4., >s, it follows from (C33), and the definition of the /Vqk that _~ I' - ~ < k' Thus for v>o This last together with Pjq,(o) _O gives

- 148 - lloug F~)I < K (l+A~) -an</\ ~~O (C34) We are now able to prove the existence of the limit matrix g., For this we need the Helly Selection Theorem.2 Let V(lu ), h 1, 2, be a sequence of real nondecreasing functions - <o < O, and let /(A) be a continuous nonnegative function on the same interval. If then there exists a subsequence X,~,j and a non-decreasing function h such that 1h >)' ) _H(c) and It follows from (C34) and the Helly Selection Theorem that there exists a sequence of intervals = (c I), S- (oDo) and corresponding boundary conditions prescribed by o(,,, such that,4jk (A) tends to a limit /oj>(kA), 1 --. It is easily seen that the limit matrix An, like At, is Hermitian, is positive semi-definite, and is of bounded total variation on every finite A interval. It remains to establish an explicit formulation for From (C33) we have, with /Wt, and d ~ a, K MA A~ I d /

- 149 - Let Al, As be points of continuity of Ao. Then integrating the above with cu0o held fixed and finally taking co-o, we have:k - -I JA4-(A' AcOk A) S W k (p ) where is given by M - A l I\/^/z MobA/A eMn (( ) Itpo e, )o LMo (A) -M o (A) If both points ok,Q b-oo are in the limit point case, M. and Ml. are unique and it follows easily that is unique. If either point is on a limit circle, the spectral matrix is not unique without the specification of a boundary condition at the point in question, Whether a particular case is limit point or limit circle is readily determined from the asymptotic solution and the expression (C15) for the radius of C, or its analog for C The Parseval Relation in the Limit S3'A Consider a function t(c) having a continuous second derivative on o~c<oo 5 and which vanishes outside some interval So, contained in 6. Then applying (C25) to Lg4 we have

- 150 - LSoIL 1 i (Li)& t / c ) L 4g) J( f Jk Applying Green's formula, oo oO 0 =_A X;.,JA) and hence (C35) becomes 0)011|J A 7 g k sJ pdJk (). (C36) o - J} - I Now for A large, ~-A~~~ AZ AA k-1 A-1 i 1 d A'"-z0 J ~' k iq q'k (C37) G L L'LAJ cle this last following from (C36). It is convenient to rewrite (C25) in the form ( + I k (C25) A-t, A Combining (C25) and (C37) i~ 9 i,) Sk1 d('Nj4 Af j- IL AcJc (C38)

- 151 - To take the limit SH A (that is, o, bb -oo ) in (C38) we need the following Integration Theorem. 2 Suppose 5iiP)gis a real, uniformly bounded, sequence of nondecreasing functions on a finite interval JdAl~e, and as s ume Lg Kis)- hiA) JolAer Voo If C is any continuous function on (d-A-e ), then We established earlier the properties of p'ik required by the integration theorem, Thus letting -~' through the sequence of intervals A, found above, it follows, using (C38) and the integration theorem, that {i t(C rcncL; - iA 2 XdkM'JfIk I ~ A; i }LAFdc -A 2 Now allowing A4-c, there results the Parseval equality 1) (C39) for any k(ec) restricted as above. We now show that the Parseval equality holds for any (se) in YZ(o),) First suppose A(c)f E('o) and vanishes for cr sufficiently large and sufficiently small, Then there exists a sequence of functions y,, E z//ooo) possessing continuous second derivatives and vanishing near c=O and for all large c such that

- 152 - Applying (C39) to He-, Mvk- O M \ M" - V)Ad k (C40) k=1 Since the left side of (C40) tends to zero as Ij - o, it follows that the sequence of vectors ti, 4A 4, ash where Cam = )(JC 1 (s), (C41) converges in the mean in j~(/ ), and since the latter space is 2 complete there exists a vector M which is the limit in the mean of this sequence, It is clear from (C41) that the components of iX are the continuous functions 00 Returning to (C39) (000A fI|A6C)Iltc = c_( | (|' otk d/%k oi ~._ oo }'l k -I a J3k=I which proves the Parseval relation for any E (c)L ~f(&cO) vanishing for all c sufficiently large and sufficiently small. Suppose now that t(e) is any function of class;I(6,ao) and define CU) ycc

- 153 - and - )( ) (',A) Jc and similarly for Mrr(c) Since (X<f) C cS) |fs)xXk tr)Ck ( ) it follows that the set of vectors'xZ converges (as? o, > in the mean o I ) to a vector function It By letting X-0o,0 o0 in | t Y"Ak OX4' dJ4ek(A) 2 L4c-2cc Qoo Jk=l k there now follows the Parseval equality IV' 4 4 (o LA (el (C42) k_ Jlk=l for any t (c) e (o1 The Expansion Theorem for the Singular Interval With the Parseval relation established, the proof of the expansion theorem may now be giveno Let D- (-7,7) and define M ) ( ) - i' l), ) 2), (a (C43) If ct1(c) and'i(L) are in,g{Q) then the relation (C42) implies

- 154 - |I (L)3 J(C)6JC = k dc J 6 k=1 (C44) which follows since we can write qLIT r =;t + =f I -| -t Ct:1 IBI~~+ILl Now consider some function rt (,q) which vanishes for ccc, cL ~, rand represent the transform of T by the vector T Multiplying (C43) by T~ and integrating we have SS J t1( jlk=i cI 2 _-v |T~J(J k. (C45) -i,)ki From (C44) for r #= M and cl =T, t, T c, T d7Dgk (C46) Subtracting (C45) from (C46), and using the Schwarz inequality, I S2(w-"TJoI < (Lf1 )3 i r:s.T( 2)~Ik l lk= C (o- O) k >*JAA % -= 1')JJk ~o' J

- 155 - Applying this inequality to the function T(c) given by T 6r ) f Z ( ) — DC ~ ) O c c r t (ccz CI C X we obtain 2-~ W (c, M D Ct ) { S' ( _ -1 Jk =1 or finally CI L %k5I j- ( k ~-k tl ) =k Since the right side does not depend on cl, c., the above holds with c-, C~ ~-~ Letting — 0 yields the expansion which clearly converges in the mean in which clearly converges in the mean in Jf olao)o

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- 162 - Chapter II, contd, 60. I. P. Shkarofsky, I, B, Bernstein, and B. B. Robinson, Phys. Fluids 6, 40 (1963). 61. W. Marshall, The Kinetic Theory of an Ionized Gas, in three parts, Harwell AERE T/R 2247 (1957), 2352 (1958), 2419 (1960). 62. P. A. G. Scheuer, M. N, Roy, Astron, Soc. 120, 231 (1960). 63. J. Dawson and C. Oberman, Phys. Fluids 5, 517 (1962). 64. J. Dawson and C. Oberman, Phys. Fluids 6, 394 (1963). 65, C. Oberman, A. Ron, and J, Dawson, Phys. Fluids 5, 1514 (1962). 66. R. L. Guernsey, Phys. Fluids 5, 322 (1962). 67. M, G, Kivelson and D. F, DuBois, Phys, Fluids 7, 1578 (1964). 68. H. Berk, Phys. Fluids 7, 257 (1964). 69, C. Oberman and F. Shure, Phys. Fluids 6, 834 (1963). 706 D. F, DuBois, V. Gilinsky, and M. G. Kivelson, Phys, Rev. 129, 2376 (1962), 71. A. Ron and N, Tzoar, Physo Revo 131, 12 (1963). 72. CO Oberman and A. Ron, Phys. Rev. 130, 1291 (1963). 73. E. H. Klevans, J. R. Primack, and C,-S. Wu, Phys. Rev. 149, 1 (1966), 74. T. H. Stix, The Theory of Plasma Waves, McGraw-Hill, New York, 1962, 75, W. P. Allis, S. J. Buchsbaum, and A. Bers, Waves in Anisotropic Plasmas, MI.T. Press, Cambridge, Mass,, 1963. 76. J. J3 Brandstatter, An Introduction to Waves, Rays, and Radiation in Plasma Media, McGraw-Hill, New York, 1963. 77. F. F. Denisse and J. L, Delcrois, Theorie des ondes dans les plasmas, Dunod, Paris, 1961, 786 T. E. Stringer, J. Nucl. Energy C5, 89 (1963). 79. I. B. Bernstein, S. K. Trehan, and M, P. H, Weenink, Nuclear Fusion 4, 61 (1964). 80. L. Oster, Revs, Mod, Phys. 32, 141 (1960), 81. D, Bohm and E, P. Gross, Phys, Rev, 75, 1851 (1949).

- 163 - Chapter II, contd, 82, E. P. Gross, Phys. Rev. 82, 232 (1951), 83. I. B. Bernstein, Phys, Rev. 109, 10 (1958). 84. K. M. Case, Ann. Phys, (N.Y.) 7, 349 (1959). 85, P. C. Martin and J, Schwinger, Phys. Rev. 115, 1342 (1959). 86. A. Lenard and I., B, Bernstein, Phys, Rev. 112, 1456 (1958). 87. G. G. Comisar, Phys, Fluids 6, 76 (1963); errata: Phys. Fluids 6, 1660 (1963). 88. D. Gorman and D, Montgomery, Phys, Rev. 131, 7 (1963), 896 J. M. Burgers, Phys. Fluids 6, 889 (1963). 90. C.-S. Wu and E, H. Klevans, in Proceedings of the Seventh International Conference on Ionization Phenomena in Gases (to be pub.). 91. B. Buti and R. K. Jain, Phys. Fluids 8, 2080 (1965), 92. D. K. Bhadra and R. K, Varma, Phys, Fluids 7, 1091 (1964). 93. R, M, Kulsrud and C. S. Shen, Phys. Fluids 9, 177 (1966), 94. R. Wo Motley and A. Y, Wong, Proc. VI Internat'l, Conf. on Ionization Phenomena in Gases, P. Hubert and E, Cremien, ed,, S,E.R.M,,A, Paris, 1964. 956 A. Oppenheim, Phys. Fluids 8, 900 (1965). 96. See, for example, almost any recent issue of The Physics of Fluids, 97, H. J. Smith and E, v. P. Smith, Solar Flares, The MacMillan Company, New York, 1963, Chapter VI, 986 The Physics of Solar Flares, W. N. Hess, ed., National Aeronautics and Space Administration, Scientific and Technical Information Division, Washington, D. C., 1964, Section II, 996 A. F. Kuckes, Phys. Fluids 7, 511 (1964). 100. D. K. Bhadra, Dissertation, The University of California, La Jolla, 1965 (unpublished).

BIBLIOGRAPHY Chapter III 1. D. C. Montgomery and D, A. Tidman, Plasma Kinetic Theory, Mc-GrawHill Book Company, New York, 1964, Chapters 2 and 3. 2, B. B. Robinson and I. B. Bernstein, Ann, Phys6 (N,Y.) 18, 110 (1962). 3. Ref. 1, pi 85. 4. L. Spitzer, Jr., Physics of Fully Ionized Gases, Interscience Publishers, New York, 1962, 2nd ed., pp. 132-136. 5. M. Rosenbluth, W. MQ MacDonald, and D. L, Judd, Phys. Rev. 107, 1 (1957). 6. F, Riesz and B. Sz-Nagy, Functional Analysis, Frederick Ungar Publishing Co,, New York, 1955. 7. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York, 1955, Chapter 9. 8, J. D. Tamarkin, Trans, Amo Math, Soc, 29, 755 (1927). 9. G. E. Uhlenbeck and G. WO Ford, Lectures in Statistical Mechanics, Amer. Math. Soc., Providence, R I., 1963, po 88. - 164 -

BIBLIOGRAPHY Chapter IV 1, R. K. Osborn, An Elementary Review of the Scattering of Photons by Highly Ionized Plasmas, The University of Michigan, Office of Research Administration, Ann Arbor, February, 1966. 2. R, Aamodt, K. M. Case, M, Rosenbaum, and P. F. Zweifel, Phys, Rev, 126, 1165 (1962). 3, M, Rosenbaum and PO F. Zweifel, Phys. Rev. 137, B 271 (1965). 4. P. M. Morse and H, Feshbach, Methods of Theoretical Physics, McGrawHill, New York, 1953, 5, L. Van Hove, Phys, Rev. 95, 249 (1954), 6, N. Rostoker, Nuclear Fusion 1, 101 (1961). 74 L, I, Schiff, Quantum Mechanics, McGraw-Hill, New York, 1955, 8. N.F, Mott, The Theory of Atomic Collisions, 3rd ed,, Clarendon Press, Oxford, 1965.. 9. B. D. Fried and S. D. Conte, The Plasma Dispersion Function, Academic Press, New York, 1961. 10. G. L. Lamb, Jr., Phys. Rev, 140, A 1529 (1965), 11. M. Nelkin and A. Ghatak, Phys, Rev, 135, A 4 (1964)6 12. S. Yip and M, Nelkin, Phys. Rev, 135, A 1241 (1964). 13. J. M, J. Van Leeuwen and S. Yip, Phys, Rev, 139, A 1138 (1965), 146 E. E, Salpeter, Phys. Rev. 120, 1528 (1960), 15. S. A. Ramsden and W. E. R. Davies, Phys. Rev, Letters 16, 303 (1966). 16, P. L. Bhatnagar, E. P. Gross, and M, Krook, Phys, Rev. 94, 511 (1954). 17, H. C. Van de Hulst, Light Scattering by Small Particles, Wiley, New York, 1957. 18, To J. Greytak and G. B, Benedek, Phys. Rev. Letters 17, 179 (1966). - 165 -

- 166 - Chapter IV, contd. 19. M. Nelkin and S. Yip, Phys, Fluids 9, 380 (1966), 20. I. B. Bernstein, S. K. Trehan, and M. P. A. Weenink, Nuclear Fusion 4, 61 (1964). 21. M. N. Rosenbluth and N. Rostoker, Phys. Fluids 5, 776 (1962). 22. B. Kronast, H. Rohr, E. Glock, H. Zwicker, and E. Funfer, Phys. Rev. Letters 16, 1082 (1966). 23, L. Spitzer, Jr., Physics of Fully Ionized Gases, Interscience, New York, 1962. 24. Ao Ron, J. Dawson, and C, Oberman, Phys, Rev. 132, 497 (1963). 25. A, Fante, Phys. Fluids 8, 149 (1965).

BIBLIOGRAPHY Appendix B 1. J. D. Tamarkin, Trans. Am. Math. Soc. 29, 755 (1927). 2, E. A, Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. - 167 -

BIBLIOGRAPHY Appendix C 1. K, Yosida, Lectures on Differential and Integral Equations, Interscience Publishers, New York, 1960. 2. E. A. Coddington and N, Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. 3. J, D. Tamarkin, Trans. Am. Math, Soc. 29, 755 (1927). ~ 168

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