THE UNIVERSITY OF MICHIGAN 2764-12-T 2764-12-T = RL-2080 A THEORY OF PHOTON TRANSPORT IN DISPERSIVE MEDIA by Edward H. Klevans May 1962 This report is identical to a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan, 1962 ARPA Order Nr. 120-61, Project Code Nr. 7400 Contract Nr. DA 36-039 SC-75041 Department of the Army Project Nr. 3A99-23-001-01 The work described in this report was partially supported by the ADVANCED RESEARCH PROJECTS AGENCY ARPA Order Nr. 120-61, Project Code Nr. 7400 Prepared For The Advanced Research Projects Agency and the U. S. Army Signal Research and Development Laboratory Ft. Monmouth, New Jersey

ACKNOWLEDGEMENTS The author wishes to express his appreciation to Professor R. K. Osborn for his invaluable guidance and assistance throughout the course of this investigation. He also wishes to acknowledge the stimulating discussions of the problem with his colleague, Ercument Ozizmir, and the criticism of the presentation given by Professor William Kerr. In addition, he wishes to acknowledge both the moral support and typing assistance given by his wife. Part of this work was supported by the Advanced Research Projects Agency and the U. S. Army Signal Research and Development Laboratories under ARPA Order No. 120-61, Project Code No. 7400, Contract DA 36-039 SC-75041 and carried out at the Radiation Laboratory, Electrical Engineering Department, The University of Michigan. Financial support for the completion of the investigation was provided by Conductron Corporation, Ann Arbor, Michigan. The author is especially grateful to both of these organizations. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS --- —----------------------------------- ii LIST OF ILLUSTRATIONS --- —--------------------------------- v I INTRODUCTION --- —------------------------------------ 1 II DEVELOPMENT OF THE PHOTON BALANCE RELATION --- — 12 III DISPERSION RELATIONS -------------------------------- 42 A. General Considerations --- —--------------------------- 42 B. Fully Ionized Gas with No External Fields --- —--------- 45 C. Fully Ionized Gas in a Constant, Uniform External Magnetic Field --------------------------------------- 46 D. The Neutral, Single Species Gas ------------------------ 49 IV MODIFIED DERIVATION OF THE PHOTON TRANSPORT EQUATION --- —--------------------------------------- 51 V THERMODYNAMICS 67 VI THE RADIATIVE TRANSFER EQUATION --- —--------------- 70 A. Reduction of the Photon Balance Equation --- —----------- 70 B. Radiation from a Plane Slab of Plasma --- —-------------- 75 VII EMISSION AND ABSORPTION COEFFICIENTS -------------- 82 A. General Considerations -------------------------------- 82 B. Bremsstrahlung -------------------------------------- 83 C. Cyclotron Radiation ---------------------------------- 85 D. Cerenkov Radiation ------------------------------------ 87 E. De-excitation Radiation -------------------------------- 90 F. Recombination Radiation ------------------------------- 91 VIII PHOTON TRANSPORT IN CRYSTALS ----------------------- 93 IX CONCLUSIONS ----------------------------------------- 102 APPENDIX A - CALCULATION OF THE TRANSITION MATRIX ELEMENT U, --- —------------------------ 104 APPENDIX B - CALCULATION OF THE PLASMA DISPERSION RELATION FOR PROPAGATION ALONG A CONSTANT EXTERNAL MAGNETIC FIELD ---------------- 108 iii

TABLE OF CONTENTS (Cont'd.) Page APPENDIX C - CALCULATION OF THE PLASMA DISPERSION RELATION FOR PROPAGATION PERPENDICULAR TO A CONSTANT, EXTERNAL MAGNETIC FIELD-.. APPENDIX D - THE HAMILTONIAN AFTER DIVISION OF THE SYSTEM INTO CELLS ------------------------- APPENDIX E - CALCULATION OF THE BRXEMSSTRAHLUNG EMISSION COEFFICIENT eB - -------------- APPENDIX F - CALCULATION OF THE CROSS SECTION FOR PHOTON SCATTERING IN A CRYSTAL --------- BIBLIOGRAPHY ----------------------------------------------- 111 116 120 125 129 iv

LIST OF ILLUSTRATIONS Figure Page 1. Two Cells of a One-Dimensional Model of the System 23 2a. Plane Two-Region Plasma Slab 76 2b. Plane One-Region Plasma Slab 76 3. Location of the jth Particle of the ath Atom 94 v

A THEORY OF PHOTON TRANSPORT IN DISPERSIVE MEDIA Edward Harris Kievans ABSTRACT A systematic, self-contained development of an equation describing photon transport in a dispersive, nonrelativistic medium is given. The postulate-deduction procedure has been developed along lines similar to those used by Ono in his quantum-statistical theory of neutral gas transport phenomena. The photon balance equation is exhibited as a time rate of change of a coarse-grained photon distribution function. The photons. A hcIkI described have an energy-momentum relation tw = where p is the index of refraction. Transport of these photons is characterized by a photon speed which is different from the speed of light. Transition probabilities for scattering, emission and absorption processes are computed to the lowest nonvanishing order. The development of the transport equation is self-contained in the sense that the transverse dispersion relations for different systems of interest are determined within the context of the analysis. With respect to these calculations, a method employed by Mead in his discussion of the neutral gas has been employed. Whenever comparison is possible the dispersion relations are found to be in agreement with those computed by other methods. Except for propagation parallel to the magnetic field, the method is not applicable to the plasma in an external magnetic field. Radiation from a plane plasma layer is studied and reasonably

ABSTRACT (Continued) good agreement with experiment is obtained, although the radiative transfer equation for the dispersive medium is not found to be in agreement with the more conventional equation which is developed by phenomenological balance considerations. Emission and absorption coefficients are modified as a result of the dispersive medium. For bremsstrahlung in a fully ionized gas, for instance, as the photon frequency approaches the plasma frequency the emission is reduced and absorption enhanced. The possibility of Cerenkov radiation for a plasma in an external magnetic field is also indicated. Photon transport through an Einstein model crystal is considered. The scattering cross section developed in x-ray scattering theory is obtained. In the limit of the infinite homogeneous crystal, the Bragg scattering condition emerges. The transport equation developed here is expected to be valid when (a) the medium is isotropic; (b) spatial variation of particle and photon distributions is slow over regions characterized by a length many times larger than the maximum wavelength of photons under consideration; (c) the frequency should be sufficiently above resonance regions (or the plasma frequency for the fully ionized gas) that radiation damping effects can be ignored. These conditions are only intended to be sufficient for the validity of the equation, but may not be necessary.

I. INTRODUCTION The problem of radiant energy transport is of fundamental importance in the study of astrophysics, in the shielding of a nuclear reactor, and in attempting to maintain thermonuclear reactions in a controlled nuclear fusion device. Gamma ray transport associated with fission reactors is discussed by Goldstein (1). The photons under consideration are very high energy, (>0. 5 Mev) and the most important interaction processes are pair production, Compton scattering and the photo-electric ionization. Furthermore, the photons are far from equilibrium. For radiation transport in stars and laboratory plasmas, we are usually concerned with low energy systems. Pair creation and annihilation are negligible and scattering is characterized by the Thomson cross section rather than the Compton cross section. For stellar systems the Thomson scattering may be important, but for laboratory plasmas it is usually negligible because of the small cross section. Depending upon the mean energy of the system, bremsstrahlung (and its inverse), ionization and recombination, and excitation and deexcitation can be important radiation processes. If a magnetic field is present, cyclotron radiation may also be important. We will be concerned in this thesis with a description of low energy photon systems. The equation of radiative transfer is usually developed by phenomenological consideration. In a medium in which the index refraction is unity the equation is written as (2, 3) 1

2 dIX - = S - aX I (I-1) ds X X where the intensity of radiation IX (x, O,, t! is defined so that I dSd 2dt is the energy of polarization X passing through a surface dS located at the position x going in the direction 2 in solid angle d 2 and with 2 oriented normal to the surface dS of frequency w in dw in the time interval dt. The absorption coefficient ar (x, w, ~2) is the probability per unit path traveled that energy of polarization X radiated with frequency w and going in direction 2 is absorbed, and the source term SI is defined so that S. (x, w, 2) d x dw dQ dt is the 3 energy of polarization X radiated in time dt in the volume d x about x with d frequency w in do going in direction Q in d2. The operation d means ds differentiation along the path, i. e. d dx. =_ - =Q (I-2) ds ds ax. ax. * J J When the me dium is characterized by a spatially varying index of refraction j(x), equationri'I-l) is modified. According to the derivation of Woolley (4), one obtains I \ I ads j- -2 (I-3) ds 2 w Hw ow As seen from equation (1-3), we do not have the simple conservation equation given by equation (I-1). The intensity is changing from point to point,

3 not only from creation and destruction, but from a convergence or divergence of the pencils of rays resulting from a change in the properties of the medium. Turning again to equation (I-l), we observe that in equilibrium, S = a B (T), where B (T) is the intensity of black body radiation of frequency w of temperature T. It will be shown later that B (T) is the Planck black body distribution function. The relation S =B (T) a w is known as Kirchoff's law. When /p 1, we obtain S w 2 (0 a = B (I-4) We can obtain greater insight into the emission and absorption terms in equation (I-1) by considering the microscopic processes. There are two types of source terms, spontaneous emission and induced emission, the latter being proportional to the intensity of the radiation present. Consider a simple two level system, with state 1 as the normal state and state 2 as an excited state (2, 3). The radiative transfer equation can be derived by use of the Einstein coefficients B12 A21 and B21 where B12 I is the probability per unit time that an atom exposed to isotropic radiation of intensity I dv will absorb the quantum hv making a transition to the state 2; A21 is the probability per unit time that an atom will decay by spontaneous emission from the state 2 to the state 1 and emit a photon of energy hv; and B21 I is the 21M

4 probability per unit time that an atom, exposed to isotropic radiation I will make a transition from the state 2 to the state 1. When ql and q2 are the statistical weights of the states 1 and 2, the Einstein coefficients are related by A21 _2hv3 q B21 q1 B 2 B = c q2; 2 (I-5) 12 c 2 122 where c is the speed of light. The coefficient B12 is related to the atomic absorption coefficient. Let n1 (v) dv be the number of atoms in the state n, capable of absorbing radiation of frequency v in dv. The number of such absorptions per second and per unit volume is n1 (v) dv B12 I. The incident isotropic radiation is given by 4 Xr I dv. Thus, the absorption coefficient per atom a (v) is nl (v) dv B12 I hv B12 hv nl(v) a (v) = = -, v nl 14n 4 r Irdv 1, 4 n Recalling that /n (v) d = n, and assuming n (v) is small except in the vicinity of v, B12 h v a (v)dv a (v )Av =. (I-6) 0 0 4 7r I Av dQdS We now consider h photons impinging upon a surface d S h v of a slab of thickness ds. Then, counting gains and losses d I hv En2 (A21 + I B21) n1 12 (I-7) ds 4r v - B 12

5 for isotropic emission and absorption. From the relations between the Einstein coefficients, we find dI 3 Clearly, S = L2 2 a accounts for spontaneous emission and c Y dl = - a I - B (I-9) where ql 2hy2 B = - -2 c2 --- (1-10) vql If the particles are in equilibrium, = e and equation (I-10) reduces to B (T) = 2 (e 1) (1-11) c which is the Planck distribution. We see then that in kinetic equilibrium, where the particles are in equilibrium although the radiation isn't,

6 S = a B (I-12) V V v and dI = - a I + a B (1-13) ds v v v v This is the equation for local thermodynamic equilibrium used in astrophysics. When we are no longer in local thermodynamic equilibrium it is seen from equation (I-10) that we need to know the distribution of n and n2. With this derivation we have progressed to a higher level of sophistication and insight than was achieved previously. Nonetheless, there are still some very important questions which this analysis does not answer. First, there is no obvious way to generalize this procedure to include an index of refraction, and second, the derivation does not exhibit in a clear cut manner, the approximations and limitations involved. To pursue the implications of the last comment, consider the analogous equation in neutral particle gas transport, the Boltzmann equation. This equation was obtained by Boltzmann by phenomenological balance arguments similar to those used to obtain equation (I-1). For instance, the Boltzmann equation does not reveal the relationship of the one-particle distribution to the two-particle or three-particle distribution functions. It does not indicate what, if any, modifications to the collision cross section might be necessary for different types of systems, e.g. the plasma. Also, we note that the Boltzmann equation is irreversible, i. e. it is not variant under the transformation t - - t, v - - v whereas the

7 Liouville equation, the fundamental equation of statistical mechanics, is invariant under this transformation. The relationship of these equations is not revealed by Boltzmann's phenomenological derivation. A considerable effort (5, 6, 7, 8) has been expended in the past twenty years in an attempt to achieve greater insight into the Boltzmann equation and to understand some of its limitations. A derivation of a radiation transport equation from fundamental considerations was made recently by Osborn and Klevans (9), who introduced a photon distribution function in analogy with the quantum mechanical Wigner distribution function for particles. A photon transport equation was developed for nonrelativistic systems and this equation was reduced to the conventional radiative transfer equation (I-1). A set of conditions was specified within which this equation would be expected to be valid. Further discussion of these points will be given in Chapter II. The extension of these fundamental considerations to dispersive media was begun in Section V of reference (9), where it was shown that certain aspects of collective particle behavior could be introduced quite naturally, into the description of photon transport in a homogeneous, fully ionized gas. We present in this thesis a further extension and reformulation of reference (9) so that greater detail of collective particle phenomena can be incorporated into the analysis from the beginning. A general framework is thereby provided within which we can discuss modifications of emission and absorption coefficients, as well as of transport of photons in dispersive media. In reference (9), the photon balance equation was developed for photons with an energy-momentum relation tiw = tic k. This is equivalent to saying that the

8 dispersion relation for transverse electromagnetic waves is given by w(k) = ck ck or that the index of refraction of the medium / is given by u - = 1. For w(k) ck the dispersive medium we can write =. In Chapter II we discuss te c k photons which have an energy-momentum relation t = where the (0 subscript X specifies the polarization. These photons are referred to (10, 11) as "dressed" photons, and they contain the effect of the collective particle phenomena. They propagate with a speed vI which is different from c. A systematic, selfcontained development of a transport equation for these photons is presented in Chapter II. The procedure is deductive, i.e. it is based upon clearly stated postulates. Nonetheless, it will be necessary to introduce many approximations in order to obtain the photon balance equation. These approximations will be made without exploring the corrections although it would be possible, in principle, to do so. Particular attention is given to the transport term. It is shown that, for a medium which is inhomogeneous, i. e. one which has spatial variation in the particle density or temperature, the procedure used in this chapter leads to an ambiguous result. The various interaction processes - emission, absorption and scattering - are computed to a convenient low order. In Chapter III, the dispersion relations for various systems are computed by the method suggested in Chapter II. For the fully ionized gas with no external fields, we find the transverse dispersion relation previously obtained by Bohm and Pines (12). For the fully ionized plasma in a constant, uniform magnetic field, we obtain, for propagation along the magnetic field, two circularly polarized waves, the ordinary and extraordinary waves. The zero temperature limit

9 is in agreement with Spitzer (13), and our first order temperature correction has been given previously by Pradhan (14). For propagation perpendicular to the magnetic field, we have a wave polarized parallel to the magnetic field and one polarized perpendicular to the magnetic field. For the former wave, there is no coupling with the longitudinal mode and our results are in agreement with those obtained by the procedure employed by Bernstein (15). For the other wave, however, there is coupling between the longitudinal and transverse modes and our analysis is applicable only at frequencies sufficiently high that decoupling results. Lastly, we apply our technique to a neutral gas of one species and obtain the Kramers-Heisenberg dispersion formula (10). Because of the ambiguities which arose in the transport term in Chapter II, a new derivation of the photon balance equation is given in Chapter IV. The transport term is developed unambiguously, although it involves certain approximations. The photon speed is found to be different than the photon speed obtained in Chapter II. However, for the frequency region where the equation is expected to be valid, the two photon speeds are essentially the same. Additional terms which did not appear in the earlier derivation are present. They account for processes for which both emission and transport are involved. A condition is specified which enables us to ignore these terms. In Chapter V a brief discussion of the thermodynamics of particles and dressed photons is presented. An H-theorem is indicated, and a black body energy density spectrum in the dispersive medium is obtained. This spectrum,

10 which differs from the Planck distribution by a factor p. c/va' where v ak is in agreement with the result presented by Landau and Lifshitz (16). In Chapter VI, the photon balance relation is reduced to a radiative transfer equation which is a balance equation for the radiation intensity. This equation differs from equation (I-3), although it is difficult to evaluate the differences directly from the equations. In order to compare them, the radiation from a plane slab of plasma is computed from both equations. After the solution of equation (I-3) is modified to account for internal reflection, the solution of our equation is in qualitative agreement with that found from the phenomenological equation. Both solutions give reasonably good agreement with the experimental results of Bekefi and Brown (17). In Chapter VII, modifications to absorption and emission terms as a result of dispersion are indicated. In the fully ionized gas, for instance, emission of bremsstrahlung is strongly reduced as we approach the plasma frequency, whereas absorption is strongly enhanced. It is also shown in Chapter VII that Cerenkov radiation is possible for the plasma in a magnetic field. The calculation is severely limited, however, because the anisotropic nature of the medium is not considered. The problem of photon transport in an Einstein model crystal is discussed in Chapter VIII. The cross section for x-ray scattering is obtained and the Debye-Waller factor emerges naturally from the analysis. In the limit of the infinite, homogeneous crystal, the Bragg scattering condition will be found. An analogy with the low energy neutron scattering in a moderator is suggested.

11 Finally, in Chapter IX, we present the conclusions of our analysis. It should be emphasized that a primary goal of this thesis is the delineation of difficulties inherent in the development of a momentum-configuration space transport equation for photons in a nonrelativistic, dispersive medium. Questions will be raised which will not be answered and approximations will be invoked without investigation of corrections. But by pinpointing the difficult and obscure aspects of the development of such a theveory, we hope to focus attention upon areas in need of further investigation.

II. DEVELOPMENT OF THE PHOTON BALANCE RELATION The deductive development of the photon transport equation presented in this chapter is based upon two types of fundamental postulates, the dynamical postulate and the statistical postulate. The dynamical postulate characterizes the dynamics of the interactions between the particles and between particles and photons. It consists of specification of the hamiltonian of the system, and the Schroedinger equation il a = H { (II-1) for the wave function i which specifies the state of the system. For the nonrelativistic systems of interest this postulate can be considered to be on a reasonably firm foundation. The statistical postulate is introduced for the purpose of describing the behavior of a system with a large number of degrees of freeaorn It is presented in a formal manner and consists of specifying a distribution function for photons. This specification is not unique and we do not wish to claim that the particular distribution function chosen in this chapter provides one with the best method of development.o It does appear to be quite adequate for systems homogeneous in the particle distribution, although for inhomogeneous systems, an ambiguity arises in the study of the transport term. For this reason an alternative derivation of the transport equation based upon a different photon distribution function will be presented in Chapter IV. 12

13 Both derivations are being presented because we feel that the development of a photon balance equation for a dispersive medium from a postulate-deduction approach is quite complicated. At this stage of the development it seems desirable to look at it from various viewpoints. Each derivation presented here illuminates different aspects of the problem and each has different shortcomings. The Hamiltonian for a system of nonrelativistic charged particles interacting with a radiation field can be written as(18) H = T T H + T + He + V + HP1 + HP7e + HP2 (II-2) where Y d3 22 2 21 T = dx c + (Q x A )2 (II-3a) a2 2 P cP 82 T - 2- (II-3b) aC a a a a ea e, V (II-3c) a,a' x -x HPe Z [i ie Ae k + a Ae (II-3d) m c 2m c HP =l [I A (x_) 7] (II-3e) a a V2 a Ae e a HP^e = A (x) (II-3f) a 2m c a

14 2 H 2 S A2 (II-3g) a 2m c a In equations (11-3), A is the divergenceless vector potential for the external magnetic field and A and P are canonically conjugate transverse wave operators for the photon field. They satisfy the commutation relations A. (x), Pe (x) = A. (x) P (x') - P (x) A. (x) = ii 6, 6(x - xt - i(t ( I ) (I11-4) j- - ax. ax' 4?rJx-x \ The index a refers to the ath particle, and the sum over a and a' is to be interpreted in the following sense: If a and a' are the same kind of particle, take a' < a; if they are different kinds of particles, the sums are performed without restrictions. The various terms in equation (11-2) can be interpreted as follows: T7 represents the energy of the photon (or electromagnetic) field; T specifies Pe the kinetic energy of the particles; H is the interaction energy for particles with the external magnetic field; V is the Coulomb interaction between particles; H 7 and H 2 are particle-photon interactions, with H ' representing one photon processes - emission and absorption - and H ' being responsible for photon scattering from particles; and H Pe is the interaction energy for particles, photons and the external magnetic field.

15 The particles in the medium generate fields covering a wide spectrum (19) of possible frequencies. Following Osborn, we separate the vector potential into two parts A and A, where the former is associated with the "slowly varying" part of the field and the latter describes a "rapidly varying" part. The "slowly varying" field is of importance in developing a particle transport equation, and Maxwell's equations are used in the discussion of this part of the field (19). The "rapidly varying" field is employed in the study of photons. It is convenient for the development of a photon balance relation to regroup the terms in the Hamiltonian. This is accomplished by selecting from the system a cubical subvolume V (x), located about the point x with respect to some coordinate system, whose size will be unspecified for the moment. Then H = Hc + HR + H cR ( where H represents the energy of the particles and photons in the cell, HR represents the energy of the system outside the cell, and H is the interaction cR energy. The "rapidly varying" field in the cell constitutes the photons within the cell. The "slowly varying" transverse field within the cell is treated as an additional external field within which the particles move. Interactions between particles in different cells via long range Coulomb forces constitute a small effect upon photon-particle interactions within the cell and we may expect that neglect of this effect produces negligible error. Coulomb interactions within the cell are retained, however, and will be of importance in the discussion of bremsstrahlung.

16 Within the cell we expand the transverse wave operators into a fourier series with coefficients which are creation and destruction operators, i.e. F 2rtc2 -ikx (k) (l-a) AF e (k) (I-6a) Xk - Xk P (x) = i e - x (k) (II-6b) 87rc V Xk where 1 (k) = (k) a (k) + & (-k) a (-k) - -X- X - x k + The operators a (k) and a. (k) are creation and destruction operators for photons of momentum t k and polarization X (X = 1,2), and - (k) is a unit polarization vector of the photon field. The creation and destruction operators satisfy the commutation relations [c (k), a, (_ ) = 6 XX 6(k- k') *(I-7) (We will not be particularly careful to make a distinction between Dirac and Kronecker deltas, letting the context reveal which is appropriate.) The volume V in equations (11-6) is the quantization volume and is presumed the same as the cell volume.

17 The use of an arbitrary oscillation frequency in the expansions of A and P is somewhat unconventional. It was first used by Bohm and Pines (2) in their discussion of collective transverse oscillations in an electron gas. The condition for specifying wXk as a function of k will appear later and will yield the desired dispersion relations. The creation and destruction operators create and destroy photons of frequency aAk. They are different from the free space operators, although they become the same for w = c k. If af and aX are a k =1/2 2) (u c /2k) 2 ck ( ) a (k) = 1 1/2 Ek + /+k -+ ) a. I 2_ _ k ck 1 I X (-k) + - /c (k- (II-8b) xccx (-k) (-8c) 1/2 + 1/2 a(k) -X ek+ )l/2+(k) + (1k k )l/2 (a ( ) ) -8d) x 12(ock)l/2 L x x- k)+( x+ ~-J The Hamiltonian within the cell can be written as (1) (2) 0 I= H = H + H + H = H~ + H (I-9a) c where H = TY + H (I- 9b)

18 H M - (T + HPe) +V (III- 9c) = H y + H P~je (I- 9d) H (2) = (H Py2 + 3771) 0 + (H1P-y2 + Ty2 ) +H y 1 2 (ll-9e) 70 37 Xk tIw~, aX (k ax (k) (iI-10a) H M aE 2m - a a, a, y Ut S (xU, x ) x *-x (11-l0b) a, Xk e a m a 2I Vr e - - ( (k) IIf (fi-loc) HP2+ Tt 0 2 Xk a X c22 -2 c4 k W~k a 'Xk aX (k) (JIl-10d) HP7-2 H1 + Ty2_ 4 Xk L. a a X c2k2 -2 c k-( + + - + k [a (k) a (-k) + aX (kP aX (-k) 0Xk I - x' (II'-10e)

19 ' 2ie -i(k- k') x P7.2 a e - - HPy2= e - - H I mV (1 )l/2 XtAk tk, (x) I (k) +t (-kt) (II-lOf) e e where II = - ii v -- A - A - -a c c( yt (a a T Oyt and S (xO, x ) = 1 if both x and x are in the box V, and S (x, x ) = 0 if they are not. The sum over a is now restricted to particles within the box, but the conditions on the double sum over a and a' still apply. We introduce the statistical axiom by defining the singlet photon distribution function 8 2 G _, it) Vx (,kII-1) where Px (k ~) = a (k + ) k (k - %). The function Gk (, ks t), known as the fine-grained distribution function, can be interpreted, at least partially, through two of its properties. First, X d3x Gt (x', Ak, t) = (jP, at (k) at (k)) (II-12) V(x) - x

20 represents the expected number of photons of momentum ik and polarization X in the cell; and second, V(x) d3x' Xk G (x', k, t)t Xk Xk i E, w, a(k) a (k)1) Xk A., x _ (II-13) can be interpreted as the total photon energy in the cell, assuming ti wXk is the energy of a photon in the medium. The sense in which this assumption is valid is shown later in this section. A quantity of more direct physical interpretation is the coarsegrained distribution function 1 Fx (x k, t) - VVx) d x' G (x', k, t) (II-14) which is the expected number of photons per unit volume of momentum Ok and polarization X located in the volume V (x) about the point x. The equation of motion for GX (x, k, t) can be written xA, ( 8 Gx (x _, t) V e -2 (- ), () [H~, p (k, A)]) S 8 V.-2ix - (, [H' ( (,] ) ) SL_. (II-15)

21 The second term on the left hand side represents photon transport 'T and will be designated G. By straightforward manipulation, we find T 8 -2ix -q aWAk G- E e- - xak P (k, + {Terms of 9(i 2). (II-16) Assuming that terms of Q2i ) give negligible contribution to transport, and writing 1 aXk PX (k ) i ak k ) a [k ( w k _q)iE), o _s > = v (i' Px (k, q)), equation (II-16) becomes * T Gi - v' _ GA (11-17) aahk where v' = k From equation (II-16) it is seen that v' in equation (II-17) ak does not appear to the right of the divergence. However, from the expansions of A and P - equations (II-6) - it is clear that wXk cannot be a function of x within the cell. Since the development of this term was carried out within the cell, it is equally valid to write

22 ~T GX G = v'v G. A V Of course, this does not prohibit either w or v from varying from cell to cell. ' T Integrating GX over the cell, we find, for the coarse-grained transport term F = 1 d3x T (') x V _ V(x) = v Q - x G (xt) (1-18) where we have again observed that within the cell the photon speed is constant. The difficulty we face is the interpretation of equation (11-18) in terms of the quantity V F which is defined, in the one dimensional model shown in Fig. 1 as dF(Xl) (XX)-F(X) dx -. L (H-19) dx L Observe that for the inhomogeneous medium, i. e. a medium in which v7 and w change from cell to cell, the definition (11-19) clearly distinguishes between the dv7F y dF quantities d and v. This distinction is not possible from equadx dx tion (11-18), which is defined within the cell. It is because of this that the relation of equation (II-18) to a transport term expressed in terms of F is ambiguous in the inhomogeneous medium.

23 I I X (x2)K Figure 1. Two Cells of a One-Dimensional Model of The System

24 To pursue this point, let us again consider the one-dimensional model of of our system. (See Fig. 1). The quantity F represents the fine-grained distribution function for the whole system and G and G2 are the fine-grained distribution functions of the different cells. They are Fourier representations of F' within each cell and are expected to represent F quite well, except near the boundaries of the cells. The bars across the cells indicate the histogram approximation which characterizes the coarse-grained distribution function. From equation (II-18) we would write d dG T 1 X F = v dx' d L dx' X L Because of periodic boundary conditions, G (X + -) = G (X —) so that 2 2 evaluation of the integral at the end points would not give a meaningful result. However, as noted in Fig. 1, if we move a small distance e away from the boundary, we expect G to approximately represent F', so that 1 r( L) 1 2 - E F -) dG 2 dx' d G 2 2 L dx' L For IF slowly varying we might expect that this expression is a reasonably good approximation of equation (II-19). Thus, for the homogeneous medium, we are able to relate equation (II-18) in an approximate, but unambiguous manner to obtain -- V _ ' F

25 However, for the inhomogeneous medium it is simply not clear whether we should have a T 7 'T 7 FX = f ' v Fx or F_ = v - _ V F In Chapter IV we will present an alternative derivation of the transport equation in which the cell procedure will be treated more carefully. It is found in that derivation that the transport term should appear as F = * Vv7 F. (II-20) The photon speed v'J is slightly different in Chapter IV, but, as will be shown, it is approximately the same as the one presented here for systems for which our transport equation is expected to be valid. The right hand side of equation (11-15) represents the rate of change of G through interactions. Integrating over equation (11-15) we find, with the aid of equation (II-20, that FX,k, t) + Q. V vv F = V (i (^ OHI' PX (k, ' (II-21) We have not, as yet, discussed the limitations on the size of the cell. The left hand side of equation (II-21) provides an upper limit for the cell size, for if we wish to associate the behavior of the photons within the cell with the point x, then G x, k t) must be slowly varying in space within the cell.

26 A lower limit of the cell size is partially indicated by the interaction term. An emitting particle localized within the box has an uncertainty in its 1 change of momentum, dictated by A K > i. This uncertainty appears as f, i L a similar uncertainty in the photon momentum, i. e. AK ^Ak > If we k f,i L X fL -k max wish k < 1 we must certainly require that << 1 where X is k L max the maximum wavelength of the radiation to be considered. We now assume the existence of a complete and orthonormal set of eigenfunctions{| n rR >} such that HR |nT7R> = ERInr7R> (II-22a) HM + T0 nr> = (E +) nrR >. (II-22b) The eigenlabel n specifies the eigenstate for particles within the cell, while the label r7 refers to the set of numbers needed for specification of the occupancy of photon momentum states within the cell. The label R is attached to the eigenstate used to complete the specification of a state of the whole system. Expanding b [n?7R > = r bR In7 R and defining * D nRn =b b nriRnt v'Rt ntI'RI n77R the interaction term can be written

27 1. n ~R pjk (okj 0)?X k DnnRnnR t) (II-23) where we have noted that X nrqR nR (k O) = UXk 8 n'r'R' nqR and L J nrR, n,:R = 0. Substituting equation (11-23) into (II-21), we obtain Fx _ ~ 0 7 FX F V A -. Xk nrnRnnR ~ nrl - (nI-24) The succeeding steps in the study of D RR are similar to those niRnrR presented by Van Hove in deriving the Pauli equation. We begin by applying a emporal oarse-graing(21)heeby temporal coarse-graining whereby D nRnR (t+s) D nnRnnR (t) nRnnR niRnnR D (t) -_ n -Rn:. R (nII-25) s in which s is a time interval short compared to the lifetime r of a state, but long compared to the period of oscillation of the emitted, absorbed, or scattered photon. (These conditions are to be considered as sufficient for our purposes, although they may not be necessary. )

28 From the Schroedinger equation I (t+s) e 1H /T1(t) i(HR + Hc)s/ti e (t) - e-H lt U(s) (t). (II-26) where U(s) = e iH / and where we have assumed the commutivity of H and H. Noting that U (s) is diagonal with respect to the R part of the representation DRnnR (t+s) = < n7R J I(t+s) > 2 D= n Rn'r'R'n""R" (t) (e-l s/ U(s) )nrRn"r"R" n'trl"R" (x) (e-R(s))nrlRn''R' - H n'?' n"t " DnRnrR (t) Un tRn"r" )UnRn'R (s) (n1-27) For the remainder of the discussion, we will suppress the label R. If an average is performed over the phases of the expansion coefficients b (t) at the time t, and we invoke the postulate of a random a nr priori phases, we find from equation (nI-27)

29 D (tts) - D (t) = [D ntnt, (t) I U ()12 nri nr nrinr' ' n'rT' - Dnn (t) | Un.,n (s) 2] Wfay]7 nt 77n-0ni (11-28) The method of radiation damping theor22 2324 is employed to calculate the matrix elements of U (s). Restricting ourselves to those transition and scattering processes which are accounted for by the lowest power of HI, we find U(1) (s) = (-i) HIn n' n'nnrl 7+ ioo 1 1 (x) 27ri a- iOD ezs/l (z -- -----— 5. --- —---------- ^ --- —(z+iE + (I ))(z +i~, + I.(.)) nrl 2 nn nz z 2 nr'n'rl (z) (II-29) where, to second order in the interaction strength e, 1 ir (z) = i HI 2 nrlnr nrlnr - (i)2 E n'ri' nr HI HI nnn'n' r nl'nn z + i n n r' (1-30) The integral in equation (11-29) can be performed when the poles of the integrand are known. We wish to show that all of these poles are on the imaginary axis. Suppressing the index r7, we define 1. See Appendix A.

30 K (z) = ni 1 z.+iE +y it, (Z) W (Z) = iF (Z) n 2 nri - iRH'. nr7 The poles of K (z) occur at the zeros of K1 (z, and n n KC () z + iE + iHI IIn nfl + w n(Z) Let z = x + iy. Then w (x + i y) nt ~n H'HI nnt fin x + i (y + End =xz Wii7n H'HI fn?, n, n X2 +(Y+E )2 - i zt~ I (y + -E ) x2 + y+E)2 xn H IHI nnt nl Then K-1 + Y = + nti ~f + y+E +H I) - zt~ y +En 2+ (y + 2, HI HI Hn H].

31 -1 The first term in brackets is positive definite, so that K cannot vanish unless n x = O. We see then that -1 lim K ( + i y) = 0 if and only if x-0 n y = - H + (1-31) n nn n'n Y+Ent Assuming the last two terms are small compared to en, we can write HI HI I_ _C nnt n'n y - - H + n. (II-32) n nn fn - E + E nt'n n n The choice of y given by equation (11-32) is equivalent to evaluating the integral in equation (1-29) using (z) = lim r (x-i ). nrwn xO n nrl (24) Corrections to this are discussed elsewhere, but are not of importance for the processes considered here. Define B and S by n7l n?7 B = Re lim i r (x- i ) (II-33a) n? x-n4O nrln nr and S = Im lim (x - i (-33b) n n x-4 On 2 n(nn3 nr

32 where Re and Im mean the real and imaginary parts. From equation (II-30) we find B = -2 E | 8(W -,,) (II-34a) nr1 t nt'nrtnn nt71nT nn n 71 S H + 2 p nnn'7?' n'ttn' ] (II-34b) n [ ninnr nE En - en where P indicates that a principle value should be taken. The quantity B is referred to as the level broadening, while S is called the level shift for bound state problems, although it also exists for continuum states. A physical interpretation of these quantities is obtained from equation (II-29) by assuming B nr Bn,. Then writing En = E +Sn n] nt nwriting = e +S, equation (II-29) becomes u (s)| 2 = | H1 | 2 e Bnns 35 U (s): I (S) 2 H e -..... (II-35) nr(- n'7 ' /t Recall that Un(1) (s) 2 is the probability that the system will make a transition from the state \nr > to the state n'r' > during the time interval s. Of course, the probability of being in the state | nr > is decreasing with time -B s as the result of transitions out of the state. The factor e nr? accounts for the depopulation of the state | nn >, and suggests that Bn is interpretable as the total transition probability per unit time for transitions out of the state n| >, and that B Z- where r is the lifetime of the state. If we limit s such that n7 or

33 -B s s/r <<1, we see that e nr 1. This is equivalent to taking lim B - 0 n?7 in the integrand of equation (11-29) before performing the integral. The quantity S is associated with a shift of the energy level E. Note from equation (II-34b) that when no photons are present, Sn0 is nonvanishing and represents the self energy of the medium. It will be shown in Chapter III that S - S = z Xk: kg:kg k nr nO k where g (UWk) is a function of bUik and includes sums over particle momenta. Recalling equations (II-9b), (II-lOa), and (II-22b), we write = S = E + E nr nrw nq n q where = Zn k [l+g()k]- (II-36) Xk k Xk Xk We have discarded the infinite self energy which could have been removed at an earlier stage by a mass renormalization. It is not of any particular interest to us. In a representation in which o = c k, the additional term would give a finite energy level shift from the unperturbed eigenvalues. This additional energy results from the virtual interactions of the photons with the medium. Tidman refers to this energy as the polarization energy of the medium, while (25) Van Hove, in discussing systems with continuous eigenvalues, refers to this as

34 the persistent effect of the medium. It is now possible to appreciate the advantage of the arbitrary wXk By choosing Xk so that g (t k) = 0, we will be working in a representation in which the polarization energy of the medium is included in the photons from the 2 beginning. These photons, which have a dispersion relation different from t = ck, are the "dressed" photons mentioned in Chapter I. The unperturbed photon eigenvalues are given by Xk with wk chosen so that g (Wk) = 0 or, equivalently, S - S: 0. For cAhoe sokth nr7 nO simplicity we drop the bar in the eigenvalue notation. Equation (1-25) can now be expressed in the form of the Pauli equation nrnrl nr(mo) mama n:nrl D (- D (11-37) where W, the transition probability per unit time for a transition from the nrma state | ma > to the state |nr >, is related to equation (II-35) by W(1 lim nnm 21 HI I 2 ) (11-38) nrm -- ~ - 2 lI nm ( -7m)(n-3 mCr 2. The idea of incorporating the level shift into the photons was borrowed from MeadO0)Bymodifying the radiation damping theory of Heitlet23) and other422 24) we have adapted Mead's calculation to the photon transport problem,

35 with w = e /. The limit s-aoo reflects the condition s w > > 1 nr) nri discussed earlier. The transition probability Wn has the symmetry property W = W nr ma manr To proceed further, we note that 2 HI | H(1) |2 + |Py 2 222 H' 2 IH 2 + (H 2) i2T nm n- inr7ma 2 nirma No cross terms appear because H(1) is linear in creation or destruction Pzy2 operators and HP2 is bilinear; thus, a matrix element of both terms cannot be simultaneously nonvanishing. The transitions described by HPY2 are photon-particle scattering, while those accomplished by H(1) include: (a) electron transitions between n rima two magnetic states, resulting in emission or absorption of cyclotron radiation; (b) free-bound and bound-free electron transitions, producing recombination radiation or photoelectric absorption, respectively; (c) transitions giving excitation absorption or de-excitation emission; and (d) free-free electron transitions, where the electron is under the influence of a Coulomb potential. These latter transitions account for bremsstrahlung and inverse bremsstrahlung. It will also be shown in Chapter VII, in connection with cyclotron radiation calculations, that Cerenkov radiation is possible for a plasma in a magnetic field. Because of the inverse mass dependence of the matrix elements, only electrons need to be considered for both scattering and radiative transitions. We will specify the particle state according to the process and medium under consideration. The subject of x-ray scattering in a crystal will receive special attention in Chapter VIII.

36 For other cases discussed here, we will treat the system as a collection of spinless particles. It will be assumed that the Hamiltonian of the M medium H can be written as a sum of terms, each representing either an electron (for the fully ionized gas) or a molecule (for the neutral gas). This allows us to employ appropriate product wave functions for the system under consideration. For example, for the fully ionized gas, product wave functions for the electrons will be chosen. Relevant aspects of completeness of the particle states have been discussed in reference (9). For specific calculations, the following approximations will be employed: (a) Emission and absorption of cyclotron radiation - Ignore interactions between particles and take the particle states as electron magnetic states. Cerenkov radiation will appear as transitions in which the component of the particle momenta along the magnetic field changes. (b) Recombination radiation and photoelectric absorption - The bound state is chosen as the state of an appropriate central potential, e. g. the hydrogen atom state of a Coulomb potential. Plane waves are used for the free particle state. (c) De-excitation radiation and excitation absorption - Use Coulomb wave functions without any external fields for both bound states, (d) Bremsstrahlung and inverse bremsstrahlung - Use positive energy Coulomb wave functions for both initial and final particle state. Expand the wave functions in a perturbation series(26) and keep only the first nonvanishing matrix element. From equations (II-25) and (II-37) we write (with a little manipulation)

37 F'I n1fntflT w (1)k] V Dnnrl /Xk - /k ' (II-39) In order to simplify the remaining calculations in this section, we will neglect photon scattering. Then, from equation (II-38) 1 w(1) V nrln':' - 2 7r H (1) 2 6 (a V2 I nt rlI nwr a- wn I d n 1 The expression for H (1) is found from equation (II-lOc) to be nr n'rIt (1) nrnr' acr, X -cr V Xk [ (x) d ' -ik - x T' " e - -, — (k) ~ O^., n + < 7 ia (k) | Tt I A ik- x () a > n X- - - (n1-40) N where d T = T1 dx. i=l 1 The relation between the decomposition of Vin (x1 x2... XN) into appropriately symmetrized product wave and the second quantized formalism (9) (27) used by Osborn and Klevans is given by Landau and Lifshitz for both bosons and fermions. We obtain

38 * ikk x -ik.x -- (k) )II K) (x-41) (1+nK) nK d x I e - (k) II (-41) where the positive sign indicates bosons and the negative sign fermions. The labels K and K1 appended to the one particle wave functions represent a sufficient set of labels to specify the one particle state. Then we have 1 (1) K1 - w = T (Xk) nK (1 + ) 7V nrn ' 13, K, K1 (x) < | c (k) r > 2 (II-42a) aK 2 or = T ik) nK (1+:n.) < a (k) > (11-42b) 3, K,K PK K I-2 depending upon whether we have emission or absorption. The quantities K1 aK1 To and T are given by TK OK

39 K1 T (X k) = OK 422) ( 4r c \Iv2 2 / e 2 1 Imc WXk ( Kk - 'K) 1 (x) < K1 e - - (k) * IK > (II-43a) aKi T O(k) = K - ( 422 \Ie 2 4r2 c / e tc 2 / mc 1 8 (K1k- wK) ~Xk K K (x) <K1 e (k) - II K> a. (II-43b) The index 13 refers to the various emission and absorption processes. For the emission term k = 'Xk - 1 and for the absorption term r1k = 1xk + 1. Equation (II-37) can therefore be written "'kkrk~ ' 3nyKK1 D T (Xkk) nK (1 - nK ) (rXk + 1) mii i "K - n-k - T (Xk) nK(l+ ) (In-44) It will be seen in Chapter III that U2k is dependent upon the particle occupation numbers so that T is dependent upon particle occupation numbers, Replacing the averages of products of particle and photon occupation numbers by products of averages, and employing the symmetry property

40 K (Xk) aK 1 (X k) we obtain F f3KK 1 -K1 T (X k) P3K L{ VFX k) +1jK f (K) I v f(K,)j - V FX_ k) Vf (K1) ~1+ V f(K)T (11-45) where K1 If; (A.k) nrl 2 2 ^. 47r c t v2 (x < K11 KI D T1 m7O 2 -ik- x (A.k) -1 WXk ~ (~ l - 'K) E 2 _X (k) - II (II-46) We have used the approximation in equation (II-46) (and it will be used throughout) that z? D n rly A (to) = A (W) where A (w,) is some function of w, and where w n rl D nr to.

41 Finally, proceeding along entirely similar lines, the contribution of photon scattering can be computed. We obtain for the photon balance relation F + Q2 V v7 - F KK PKKI K )f(K) - _ (x) V f (KI) - V F (k) V f (K) { V f (K)] KK X'k' -KXk SK X'k' V FX, (k') VFx (k)+l V f(Kl1) 1+Vf(K)} +1 Vf (K){1+ V f (K1)] (II-47) V FX (k) {V FX (k') where -KXk K1, X'k' I e2 3 8r3 \mc V3 4 c WXk 'Xk x(k) ~ ' 2 (x) 6 (k' + K' -- K (wKk - Kk') ( ' (nI-48)

II. DISPERSION RELATIONS A. General Considerations In Chapter II, it was pointed out that the unperturbed photon eigen values are given by r =k Ak rXkk L k where wXk is chosen such that Snr - Sno = 0. In this chapter, we will investigate this condition. The quantity Snr was given in Chapter II as Snr= Hn + n~o =! H ra HI HI nrlra rnrj enr - er ( nr} ruc (II-34b) It is readily established that HI = (pY2 + T1) nr7nrj o nfnrn 1 2 Xk 7Xk tWXk 4, 2 c2k2+ 4r e 2 WXk nK -wX k - (III-la) and HI HI p nrnra ranr? Enr -E ra?kKK1 e 2 2ir tWXM k( ) - m V ( 1.-K) nKj 2 ( 1Xk k I.k(K1 I K)12 (x) k -EK -EK +kk + (yk-+ 1) I I~k (K K1) 21 - 1) — k — — ) (III-lb) EK-EK Xk EK1 k 42

43 where -ik,x IAk (Kj1 IK)= < Kj e 6A (k) II | K> (III-2) and where we now employ a prime to indicate the exclusion from the sum terms for which a denominator vanishes. By straightforward manipulations we find (except for the self energy of the medium) Snr = k i CA k g (wkk) - kk where 1i i!2 I/4tr4 e2 2 ((WXk) = 4 U nK + W2k2 - 2 +Vm A2 K *- (III-4) m KK1 EK-K - E K X k k As discussed in Chapter II, we choose wk so that g(wok) vanishes. The obvious non-trivial choice is 4vr e2 I W =2 c2k2 + 7 nK C~~ mV L K 1 E I (K|K)|2 -1 -- -- k K- K (nK - n ). (III-5) Km m KK1 EK- EK1 WXk In accordance with remarks in the last chapter, averages of functions of wo will be replaced by functions of the average of ow.

44 Throughout our discussion we will find it convenient to apply various approximations with respect to the averaging of WXk. In particular, we will use 2Xk - (WXk) 1 ) 1 (Ak WXk ) ) W~k t'k' WXk WX' k' Although these approximations are employed in a rather cavalier manner, it should be remembered that, at least in principle, it would be possible to investigate the errors involved. Applying an average to equation (III-5), we now find 2 2 k2 + 2 (III-6) Xk P L k 2 2 where w = 47rne /m p 2 1 Xk (Ki E - /( m KK1 EK-EK wk 1 EK- EKk EKl Xk 1 KK I IK 2 and f (K) -(K) K). From equation (III-6), the index of refraction /u can be i ten written

45 -2 k 2 XkJ I2 = ( )2 - -= 1 - Pk (in-8) WXk -Xk Because we will only be interested in J and Jn throughout the remainder of the thesis, we will drop the bar and use w andy as the averaged quantities. In order to compute Ik(K1I K), it is necessary to specify the particle eigenfunctions. Three special systems will be considered, and the eigenfunctions will be chosen accordingly. They include: the fully ionized plasma with no external magnetic field; the fully ionized plasma in a constant homogeneous magnetic field; and the neutral gas of one species, B. Fully Ionized Gas With No External Fields For the fully ionized gas without external fields, we use plane wave states for the particles. After going to the continuum in velocity space, we obtain 2 ck2 +2 + P d3vf(v) 2 p 3 2 (WA.-V. k_ 2 -(m (III-9) A quantum correction to the classical result is in evidence in the integrand. If the particle distribution function is isotropic in velocity space, the frequency wXk is independent of which polarization is chosen if linear polarization is used. (12) This is the same result obtained by Bohm and Pines. In the zero temperature limit, equation (III-9) reduces to the Langmuir dispersion relation c= 2 k2 + 2 P

46 Dropping the quantum corrections and assuming a symmetric velocity distribution, equation (III-9) can be written, by straightforward manipulation, in the form 2 c2 k2+ c2 PSd f (v) o = k + w P d v -. —. (II-10) (3 - k - v) This transverse dispersion relation for the fully ionized gas was obtained by Bernstein5) by solving the linearized set of equations consisting of Vlassov's equation and Maxwell's equations. Our dispersion relation (with no external fields) is obtained from his equation (32) when his Laplace variable s is set equal to - iw. C. Fully Ionized Gas in A Constant, Uniform External Magnetic Field When an external magnetic field H is present, the medium becomes anisotropic. In general, this will introduce coupling between the transverse and longitudinal modes. Radiation then propagates in a direction different from the normal to the wave front. The concept of the photon does not enter naturally (or, perhaps, even legitimately) in this context. Nonetheless, under certain circumstances the different modes decouple and ET = 2 where ET is the transverse dielectric constant for photons of polarization X. One such case is propagation parallel to the magnetic field. For convenience the magnetic field is chosen along the z axis. For particle eigenstates we choose the magnetic states of Johnson and Lippmann 8. When circular polarization is employed, the waves propagate independently and

47 we obtain Vz kz 2 22 -— ) =c2k2+ P dvzf(vz) - w( - - (III-ll) + P zVz kz (ua- bjd ---CeH where WC =. The positive sign is associated with a wave whose electric mc (13) vector rotates in the same direction as the electrons gyrate. This is called( the "extraordinary" wave. The negative sign refers to the "ordinary" wave, for which the electric vector rotates in the opposite sense. In the zero temperature limit equation (II-11) reduces to 2 2 22 P w+ =c + (III-12) 1+ - ( The apparent difference between this result and that given by Spitzer(13) is accounted for by our associating a sign with the charge e, while Spitzer uses e( = - e for electrons. A first order temperature correction can be obtained from equation vz kz -1 (III-11) by expansion of (1-, assuming we are not near a resonance. i We find 2 c2 1.-1 2 - (III-13) e2 (c _ +)2 3. Appendix B

48 / k T \1/2 where xD = ( kOT 1 is the Debye shielding distance. This is the same result as obtained by Pradhan(14) for the ordinary wave. He does not give a result for the extraordinary wave. The inadequacies of the present method for anisotropic media have been mentioned. Nevertheless, it is of some interest to consider propagation of radiation at some angle e with respect to the magnetic field and compare the results achieved through our perturbation procedure with those obtained by the more conventional approach(15). For simplicity we consider propagation perpendicular to the magnetic field along the y axis in a left-handed coordinate system. The polarization vectors are chosen as E and E. For polarization parallel to the magnetic field, there is no longitudinal-transverse coupling and we find4 92 92 2 12 Lw2 =w c2k - - (III-14) z n= -oo nc -B where n — c v k 2 n j0v ( e. (III-15) n Jo I- L)n c ~c For polarization perpendicular to the magnetic field, we obtain oo x Wi ck -= W (- ) - (III-16) where k = Boltzmann's constant o 4. Appendix C

49 3X -^ dv, fV(Y)v2 Jn (J IC) (III-17) c For comparison, consider the solution of the linearized set of equa(15) tions containing the Vlassov equation and Maxwell's equation Bernstein does not exhibit the transverse dispersion relations in a form which is easily compared with our results. However, by performing the 0 and v integraz tions in his equation (21), we can, from his equations (21), (23), and (27) obtain these dispersion relations. We find first that equation (III-14),which represents a wave whose polarization vector is parallel to the magnetic field, exhibits no coupling and is in agreement with Bernstein without further approximation; and second that, because of longitudinal-transverse coupling, equation (III-16) is in agreement only for ck >> wt and ck > >. At these frec p quencies the coupling is negligible. D. The Neutral, Single Species Gas As a final application of our method of determining dispersion relations, we consider the neutral gas of one species. The particle eigenstates are the eigenfunctions of H where H is the unperturbed mol mol Hamiltonian for the ath molecule, and where we have written H - H() j mol a For the dipole approximation we find C22 2 2 K K, L 1-2 2 ck k L 1 2 (111-18) KK1 Xk KK 1 where bKK1, the oscillator strength is given by KK1'^

50 2m, 1 2 KK1 = f K1 1 KK1 (m-19) with K1 -( () xDKK1 n = E (k)- <K xi K >. If f(K) = S 1 K10 c2 k2 2 (all molecules in the ground state), we obtain - K 0 (IIIP K W0- to x K 0 -20) which is the Sellmeyer-Drude formula for the dielectric constant of a medium of undamped oscillators with frequency wK 0' When equation (III-20) is written in the form 8a- e2 N 2 -1 V K WK0 o ((k) X)K02 ~t"lo-"A (II-21) we have the Kramers-Heisenberg dispersion relation. With respect to this result our procedure is equivalent to the calculation of Mead(1).

IV. A MODIFIED DERIVATION OF THE PHOTON TRANSPORT EQUATION In the derivation of the transport equation presented in Chapter II, a certain ambiguity arose in connection with specifying the coarse-grained transport term in an inhomogeneous medium, i. e. a medium in which the photon speed vYand the frequency w are functions of position. In this chapter we will present an alternative derivation of the photon balance equation in an effort to shed more light on the question of photon transport. A coarse-grained distribution function will be introduced at the outset, and a formal cell procedure developed(29) This will be in contrast to Chapter II, where the decomposition of the Hamiltonian was carried out on a somewhat intuitive basis. Other aspects of this derivation will also be of intersv est. For instance, it will be shown that the condition - < < 1, where s is the time employed in connection with coarse-graining and L3 is the cell volume, is necessary to eliminate terms describing combinations of interaction and transport. The statistical postulate for the present development is introduced by defining 1 F (X, k, t) - Tr D(t) pX (X, k) (IV-1) x L3 as the expected number of photons per cm3 of momentum i k in the cell of volume L located atX. The density matrix D is again defined as Dnrnrl = n b* nr, bnr where b is an expansion coefficient of the expansion = bnj I nnr > nr? 51

52 and I nr1 > } is a set of orthonormal base vectors presumed to be complete. The meaning of the labels n and rn will be given later. The singlet density cell operator is given by p(, k)= + (-=, k)a (X k) (IV-2) where the cell creation and destruction operators obey the commutation relations pA (X., k), c (-', k' = s.g A, (k-k ') The equation of motion for F is _ = V (T) Trp D, = (-)(-) TrD F Hp A. v i- v (IV-3) (IV-4) where the quantized Hamiltonian is the same as equations (11-3) except that the second (9) formalism will be used for the particles o In order to accomplish the averaging of [H, p] we expand A(x) = E - c2 E(X,x) + (X, k),L -,k, A. wX(1, k) ' - t -ik x -(- 8 c2L3 k W_) e^STC-L0 k )x (IV-5a) E(_; x) L_ (:, k) where OC Wi = I a (3, K) UaK() E(, x ) (A-_ k) = (., k) ~,(k) c.(M, -k) (-k); h A. - - -?L - ~ A~ A. - (IV-5b) (IV-5c)

53 X (Ig7X, k) is the oscillation frequency associated with the cell located atiXa; u vK (x) are a complete set of particle eigenstates that will be chosen later L^ -J in accordance with the radiation mechanism under discussion; and aa (X., K) and a. (T~X, K) are creation and destruction operators for particles of type a and momenta i K in the cell located at X'. They have commutation relations a (I, LK), a, ('g:, KI+ - S,,,, s(K-K') (IV-6) where the plus and minus signs refer to fermions and bosons respectively. The step function E (ZX, x) can be written E(_2, x) = E(X1, x1) E(X2, x2) E~(X3, x3 (IV-7) where E( X., x = 1 for X. - L < x. < X + L J J J 2 J J 2 = for x.= 2 (IV-8) 2 (Jv 2 = 0 otherwise. Two properties of E which are of special interest are 1) E (X xj) E(Xj', xj) E(j., xj) $, (IV-9a) dE( X*j, x) L L 2) ( xj= (xIj - + ) - (xj -j - ) (IV-9b) dx. 2 J 2 J The cell procedurej as introduced here, has some unconventional aspects. It is seen, for instance, that

54 -ik - x V A= A 2 C2 e - rx- ( 7VE( x), L3 X,kX ^( X,k) - which vanishes within the cell, but not on the boundary. In a similar manner, the (Vx A)2 term in the Hamiltonian will result in the appearance of new terms which we will call H. It must be kept in mind that the cell procedure was introduced to perform the operation Tr D [, p, so that the effect of these additional terms should be judged with respect to the equation for IA rather than on some intermediate stage where it may appear unconventional. In fact, the terms in T H will lead to photon transport, as well as other meaningful effects which will be discussed later. After a considerable amount of straightforward calculation we can write the Hamiltonian as 0 (1) (2) 'YT PT IT cc H = + H+H + H + H + H + H + H V (IV-la) where Ho = Tyo + HM HM = T + HP + V (IV-lOc) H ) =H + HP ye (IV-lOd) (2) P -y2 -y l Py2 y2 P y 2 H 2 (H +T1)+(H 72+T72 )+H (IV-Oe) 0 1 2 HT "YT 'yT HT =HT + H7T (IV-lOf) 1 2 The H and H = H1 + H(2) are the same as those discussed in Chapter II. They refer to photon and particle energies and photon-particle

55 interactions within cells, and can be written H +HM+ HI= HO(~X)+HM( x)+HI (x. YT PT The terms HT have already been mentioned. The term HPT is associated with particle transport and HIT will describe an interaction in which a particle moves from one cell to another during an emission or absorption of a photon. Lastly, the term V is the Coulomb interaction for particles in different cells. All of the terms are shown in Appendix D. The Schroedinger equation can now be written ii a. (H+ HI+ HT + HPT + HIT + Vcc). (IV-11) at The set of eigenfunctions {!n r >j mentioned earlier are chosen such that H~ | nr> = > (E () + (X) In>. -X - The eigenlabels nr7 specify the number of particles and photons of all possible momenta (and of different kinds of particles and different photon polarization) within every cell; i. e. | nr> = n....nK n. n Xnr> I i'I K 1:lgj Kj.2 jol Kj M2 a Kj n Xl"j Kj' K' X1 j k" X211 kl;.> Expanding _= ]bnr(t) nn> >, we observe that the diagonal elements of nrl 2 the density matrix Dnrnn = | bnn give the probability of finding the

56 entire system in the state I nr? >. It will be convenient for further calculations to transform to an interaction representation. Let i= U (IV-12) -i(H~ + Hi^^t/~f where U = e satisfies the equation it au = (H+ HT) U(t). (IV-13) at Because U is unitary, we can write F - Tr p D, (IV-14) V where D=U DU andp =U+ U. (IV-15) Then - Tr p D+p D. (IV-16) X V A It follows from equation (IV-13) that P =(i) H+H7, p (X, k) (IV-17) A* T The term Tr p D leads to photon transport and will be designated F. From equation (IV-17), and noting that H0, p( X, k = 0, we find

57 FT = (I ) TrD X v [HYT. p1( k i 1 h v 1 nrl n I -o (IV-18) Dnn,, H, p _, kn'' n,,' D H IT k( By straightforward calculation, we find H1T, px (X, k7 ii c2k. a ax:1 F (t, k, t) wx(IX-, k) 2 + it c2 2 Lt'k' (kl +kl ) E(k') -X (k) j w ( X, k) S(k2-kk) (kg-k0) (x) a (, k) a a-.Ii + t, (.., k'),(:, k') AX --- + a a) + (Xp, k) X c -i ati + cyclic permutation, )X, ( z, k') w;,(-X, k') - (IV-19) where the a operation -- operating on a function h is defined by ax^ h(XC1 + L, X2 -3) -h((Z) L (IV-20) am: Equation (IV-19) is not exact. Letting h( X j) be each of the above functions which follow a differentiation, we have approximated

58 ah (.-L) ah(X.) a2h(X: ) ____J L J L 3- L J - + *.. axm ax. aX J J J ah(J.) by -, neglecting all higher derivatives. This implies, for instance, F that - must be slowly varying, and partially specifies the upper bound on the IL cell size. Assuming the off-diagonal elements of the density matrix in equaT tion (IV-18) give negligible contribution, we find that only H1 is of importance, and then only the first term. Thus FT Q ~ v ) (I., k) Fk (X:, k, t) (IV-21) where 2 v7= c k =cM (X, k). wo (X, k) Equation (IV-21) indicates that v7 should be operated on by the divergence. Note that for the nondispersive medium we obtain the expected result v7= c. a w The photon speed obtained in Chapter II was given by v = vg =whereas in equation (IV-21) we find v7 = cp;(XT, k). In order to determine the difference in these results, let us consider two isotropic dispersive media, the fully ionized gas and the neutral, one species gas. The dispersion relation for the fully ionized gas was given by equation (III-10) as 2 = c2k - dv..f(v) (III-10) p (w-k - v)

59 We find, with a few manipulations, that 2 A 1+ _ p d-vfy- k v v - Op2 v -a- = 2c -- (IV- 2 _22) g 3k i 2 f() 1 Y_ 2c f( (w-k-v)2 where k is the unit vector in the direction k. It is seen, then, that if the index of refraction p is not appreciably different from one, (t >> wp),then v gv =c(. p g In the zero temperature limit, Vg = cp is exact. As we will see in Chapter VII, when p -# 0 the absorption coefficient gets very large and the present analysis becomes of questionable validity, so that the frequency region for which p does not differ greatly from one is the region where our development is expected to have greatest validity. For the monomolecular gas, we found in Chapter III that 2 8r Ne2 2 (~k) Kol) 1 ). (III-21) V K (to2 -2t) KO X If X. >> KO, this becomes c2k2 1 87rNe2 - 1 - W2 W2 V A X K K - - I (x(k) x) Ko so that, multiplying by o2, we obtain a Wc = ---- = V = ell. ak g X The frequencies for which wo >> wK0 are above the resonance region. For o A< u KO we are below the resonance region where, from equation (III-21), it is found that Vg / v- =cu. However, we haveno assurance

60 from either derivation of the transport term that it is valid to pass through the resonance region where, because of strong absorption, the transport equation is not expected to be appropriate. We see then that in the frequency region where the present analysis is expected to be valid, the photon speed given in Chapter II and the one obtained in this chapter are approximately the same and it is not obvious, from the point of view of measurement, which is more appropriate. The second term in equation (IV-16) accounts for interactions, and we will write F I - Tr p D. (IV-23) X V X The time coarse-graining will be employed in a manner similar to that already used in Chapter II. Thus D- 5D (t+ s) - D(t) D....-. (IV-24) s We will again require that su >> 1 and s/r< < 1, where X is the frequency of the photon which was emitted, absorbed or scattered, and ' is the lifetime of a state. Equation (IV-23) now becomes. Vs Trp LD(t+s)- D(t) = - rTr p D(t+s) - Tr pX D(t). (IV-25) Recall that D(t+s) = U (t+s) D(t+s) U(t+s)

61 so that Tr pxD(t+s) = Tr U(t+s) U (t) pLU(t) U (t+s) D(t+s) But = Tr U(s) X U +(s) D(t+s).D (t +s) = iJ (s) D (t) fj+ (s) (IV-26) (IV-27) where -i Hs/h, U(s) = e Utilizing the Feynmann calculus(30) J, we write (Iv.-28) U0(s) = U(s) (I +QI) (IV.-29) where QI= (- i 5j d 1 j =1 0 S.~ 3 1.. J (IV9-30) and (HIi+ HIT), = e IiWH +HiT)sj/t (~I+HIT )e- ('H)S l (IV-31) The terms H PT and Vcc~ have been dropped from the discussion because they do not account for any photon processes. With the aid of equation (IV-29), equation (IV-26) becomes I I+ Tr p.D(t+s) =Tr pX(I+Q')D(I+Q1 ) (IV-32) so that

62 Tr p D(t+s) p D(t) =Tr [Q D+DQI Q D QI+ MY-k nrVn jQn rn'trl 1 [ - D | Q + terms proportional to the offnrnU| niQnI 2 L diagonal elements of the density matrix (IV-33) These latter terms will be assumed to give negligible contribution. I We turn now to a calculation of Qnrn't ~ There are two contributions to this matrix element, one from Hi and the other from HIT. In order to indicate the relative importance of these terms, we calculate HI and H using plane wave states for the particles and find, HI KL This quantity is much smaller than one for all cell sizes of interest. Thus, we may expect that HIT will give a negligible contribution and it will be dropped henceforth. The interaction processes of interest are contained in QI ) where QI(l) = (l) ds (e(H i H1 e) si ) / n0nnf? o nr nr~V (IV-34) Writing -i(H~+H)i -iH s/h e e (I+QT(s1) (1IV -35)

63 with S 1 rf-1 QT( Z = (-L)J... S j= 0 j=O i H s/t T -iHs s, ds'.... ds'. (e HT e ) 1 i H1 (x) ~(x) (e s! /HT1 -iE~ s ) e J) (IV-36) the expression (IV-34) can be written I(1) (2i i) Qnrin r' 7 - )L nrintrl e in nrfn' r' sin 2 I - - +J "nnirn'rt Inr rntr (IV-37) where s(sl) i + T inTn (- ' ds QT+(sl) Uo HIUO+ UO+HIUQ ( nrl n' ri 0 T+ + + Q (sl) U0O H U QT(s) 1 nrin'trt (IV-38) The first term is the usual interaction term for photons within a cell. The remaining terms describe processes in which both transport and interactions take place, e. g. creation in the celli- L and transport to the cellX Let us examine the first of these "correction" terms. To lowest order T(1)+ i T lnrf n' 7Tl1 Jnn' (- ) S ds nrntl) H n (IV-39) where T (1)+ T()* T i(nr7-Wn" r ) s2 Q (s) = Q n ' (sn) = (- Hi) H n n ds2 e nrnn"rl' n"r7 nr 1 - ln"' nr 0

64 The exponent of the exponential contains the factor s(wnt-wnt nr), where Wnq0 is the frequency of the photon in the cell _K and in,, rIt is the frequency of this photon in an adjacent cell. This change of frequency reflects the condition that in an inhomogeneous medium, a photon of a given k will have a different energy in each cell. Further discussion of this point will be given in Chapter VI. If we restrict our systems of interest to be those for which L w ~ -- << 1 ) 8ax we may expect that nil ^ nT -t and, consequently s(w - ) <<1. wn" n' Using this condition we obtain -i5l T QT (1) (S H (IV-40) n'n() ( ) H ln"fnr The magnitude of this term (as seen from Appendix D) is indis1c2 k sj vy cated by the quantity = where vais the photon speed. If now 0wL L vT S we require the cell to be of such size that L < < 1 then Q will be very small and the terms J can be neglected. The implication of this condition is that s should be small compared to the time for a photon to cross a cell. This restriction was also used by Ono(29) in a similar context. sv' s) 2 Note that sL = 2, so that for p not appreciably different from one, s < < kL. But s >>~ 1, so that kL < 1. This implies that the kL wavelength of the photon must be much smaller than the cell size.

65 *1 Flig F -, From equation (IV-25), (IV-33) and (IV-37), we obtain, finally w(1),k-)= 2'n1xk 7 r nl n^nrnl V j -41) where nfl n, r) I..m 2? ~^ 4 I 2 - H A2 nrmn IrI. 9 Wnr? BY r sin2 n --- n —' s S(0 - )n2 s(Cwn,7 - W~n rT') In 2 nln'f [ (Wnor - Wn' r?). (IV-42) ~ I The quant:ity F A indicates the rate of change of the number of photons in the cell IX by interactions. It would appear from equation (IV-33), however, that the sum over nWIr implies we are summing over the interactions in all cells. Except for the cell:X, the sum over nrl can be taken to the right of reXk and all of the terms will cancel. Thus the states I nr > and I nr1' > can be considered the same except for photon and particle numbers in the cell Lastly. it should be noted that the procedure used here does not provide us with a condition for determining Ak (I, k). This was not because of any fundamental difficulty in developing such a procedure, but rather because of expediency. The present procedure took us quickly to a discussion of QT and of the relative importance of the interaction terms. The alternative procedure, in which a level shift analysis is introduced, is quite lengthy for

66 considering the additional terms arising from a combination of interactions and transport. It leads to the same final result, however, and the condition for determining A. (Ixu, k) is the same one discussed in Chapter II. The final reduction to a photon balance equation is the same as we have already shown in Chapter II, and will not be repeated here. Our final result is the same as equation (II-47).

V. THERMODYNAMICS The deduction of equilibrium properties for our system is conveniently achieved by first obtaining an H-theorem to describe the irreversible nature of the approach to equilibrium. The development of such a theorem for the system of particles and photons is not possible from equation (II-47) since this equation does not describe the rate of change of particles. An additional equation would be required. This procedure has been employed by Osborn (19) tp obtain an H-theorem for the complete system. Our deduction will follow the argument presented in reference (9) which is based upon the use of the Pauli equation, D = W D -D r. (II-37) nnrW ma mama nrnrn Defining H = D In D (V-l) nrlnr? nrjnr1 nr7 it is easily shown by use of equation (II-37) that < 0, (V-2) dt the equality holding when D = D nrnrl7 mama Since D (t) = P(n7, t) is the probability of finding the system in the state nr > at time t, the H-theorem suggests the identity 67

68 S= -k H where kgis the Boltzmann constant and where S, the entropy, obeys dS > 0 dt with S a maximum for the equilibrium state. By standard thermodynamic argument (44), it is readily established that F (k) = r P(r1) = - (V-3a) r e -1 f(K) = nK P(n) = (V-3b) n K Be + 1 where,3 = 1/1, and where the (+) in equation (V-3b) indicates fermions and bosons, respectively. The equilibrium radiation law is obtained by converting to the continuum in photon momentum space, i. e. o FX Pk FX (k) = Fdkd(k) dkdQ2 kcd3k X V where Vk2 k (2 )3 is the density of states. Defining the energy density UX (k) by U (k) w=i. F. and transforming to v space (vx = - ), we find, after integrating over A 2 7r

69 2, that 2 43rh ay Yx y o 2r (16) for black body radiation in a transparent medium. Letting v v and summing over polarization, we obtain the usual black body radiation spectrum U (v ) 4rh (V-5) 0 3 7 1/ ck C where z = of more direct interest. It is seen from equation (V-4) that iv) hv=k= d8 v (V-6) c 2 (e c (e - l)c c ck 2 where = -- is the index of refraction. Note that IL /Ax is independent of the medium.

VI. THE RADIATIVE TRANSFER EQUATION A. Reduction of the Photon Balance Equation The photon balance equation (II-47) describes the rate of change of the expected number of photons in the cell located at the positionX, with wave number k in dk, going into direction 2 in d 0. As pointed out in Chapter I, the quantity usually studied in astrophysics is the radiation intensity I(x, tW, 0. t). In this chapter, equation (11-47) will be reduced to an equation for the intensity I. The resultant equation will differ in appearance from the phenomenological equation (1-3). In order to compare them, we will calculate the radiation from a plane slab of plasma and compare the results. It will be assumed that the number of occupied particle states in a given energy range is small compared to the number of states available, so the degenerate particle systems are excluded and Boltzmann statistics apply. Recalling equations (11-46), (II-47) and (II-48), we define X 2 K1 E = V TK O2T k)f(K) (VI-la) e KK K -K /3K X-2 K1 Oy V TPK 0mk ) f(K (VI-lb) KK1 = V 2 K1 a - V T K ) [f(K) - f(K (VI-lc) KK! si d K d Kddk' d 2' V3 SK, F ) (VI-ld) A'K, (V K, Klk' 70

71 x1I f /d3K d3KdkT d~ V3 L~? ~iK~jf Xk o ( EK1 dk' dQ' V3 S (k A)+ f (K) (VI-le) K,_K1 k' where F (k) p and f (K) are defined by A- K I F(k)=F dk dQ(k) (VI-2a) 3 X X ked k 7 V k2dk dQ(k) ~ (1(k) d = 3 (VI-2b) 3 k (2 7r)3 ke d k f(K) = fO(K) dK. (VI-2c) KE d3K Then equation (11-47), the photon balance relation, can be written F~(k) + Q v7 F~(k) A - - a ZI) F - a(k) Fi(kj (VI-3) SiPk o + -(s -s) F (k) V 0 1 X - For nonrelativistic plasma systems, the scattering rate is usually small in comparison with at least one of the absorption and emission processes. Even with free electron densities of the order 10 electrons per cm, the mean free path is of order 10 cm. For large stellar systems, scattering can still become important and a considerable body of literature exists treating this case(2 3) become important and a considerable body of literature exists treating this case

72 For laboratory-size systems the effect of scattering is expected to be negligible, and for the remainder of our discussion, the scattering process will not be given further consideration. The radiation transport equation is usually exhibited as a function of frequency rather than k. If the medium is spatially homogeneous with respect to particle distributions, w, v y and Ad are independent of position. When the medium is inhomogeneous, however, w, for a given k, will be different at different places in the medium. It was seen in Chapter IV that k rather than W, was the natural variable for our description, for the assumption was made that only terms in transport for which k remains unchanged as we move across the boundary are important. Modifications to this are possible in principle, but rather impracticable. It was also pointed out in Chapter IV, in connection with terms for which we have photon creation in cell X-L and transport to X (or the inverse) that w, for a given k, could change only by a small amount from one cell to the next. The physical origin of the change of w from one cell to the next was given in Chapter II where the method for choosing w was developed. The radiation present induces collective particle behavior which can be associated with the polarization of the medium. The "dressed" photons being considered have been defined so that the collective effects are included in the photons. In the inhomogeneous medium, the "dressing" is changing from cell to cell. Consider the low temperature plasma, for instance. where, to a good approximation, the dispersion relations for transverse oscillations is given by

73 2 2 2 2 aw (x)= c k + w (x). pPhotons which in free space have energy to = ck are now modified as above and w varies slowly in space as a result of the space dependence of to. Thus, although we have total energy conservation for the system, the energy of the photons changes from cell to cell. Consider the transformation from the variables x, k, Q, t to x', t, Q', t' where XT -X ww = J(x, k) t' =t. Thus a a (a a ax. ax )&o J where (&o/ ax.) is independent of x'. Defining 3 3 x, w, Q', t') d x' dw ' = F (x, k,x,t) dx dk dQ (VI-4) we obtain a___ a -v +' 7'v7(X', W) L+V (V oW) - v7^ 2 ((VI-5)

74 The intensity of radiation is given by I(x,, B, t) —= h Xv'Y k(X, t, Q, t) (VI-6) so that (dropping the prime) +1 a+- V +wa ' \) Y &t 0 W / V (VI-7) = ( W S - - (x,, n) where Y=(V w) X 2 3 X (x W, 7r)3 aw (VI-8a) a'3(x, ) -- _ (VI-8b) xv For the homogeneous, stationary state system, y. = 0 and L IV aX (VI-9) which is of the same form as equation (1-3) when the index of refraction p, is not a function of position. In thermodynamic equilibrium, with a Maxwellian particle distribution, we have, after integrating over _2 X13 2 B (u) (VI-lOa) where BX(w) daX = x/ko - - c (VI-lOb) /koe 47r 3c e~- 1

75 2 The quantity u2 B is the same equilibrium distribution obtained in Chapter V from a somewhat different approach. B. Radiation From a Plane Slab of Plasma For the stationary state, equation (VI-7) can be written dIA +(d ) a (\k =/ (x,, Q)- a(x,c, i) (VI-il) ds ' + di - - where, employing the summation convention, dx. d _ d a a ds ds ax. j ax. J J The comparison of equation (VI-11) with the phenomenological equation dl 1 ds j I (I-3) ds - ds is not easily accomplished without a knowledge of the intensity. Rather than attempting to compare these equations as they stand, we will calculate the radiation from a plane slab of plasma and compare the results. Instead of equation (VI-11), however, we will solve equation (VI-3) (without scattering) and then transform to frequency space after obtaining the solution. In order that this be done unambiguously, we will subdivide the slab into two regions A and B. (See Figure 2a). In region A, of thickness L1, the absorption and emission coefficients are assumed constant. In region B, of thickness L2, the density of the particles decreases monatonically from the density in region A to zero. Thus, at the point P, where the radiation emerges going in direction _2 in dQ, we are in free space. In the end we will take the limit L2 - 0. 2L

76 S A S2 - 'n s "M L1 L2 L Figure 2a. Plane Two-Region Plasma Slab n S' L Figure 2b. Plane One-Region Plasma Slab

77 In the plasma with no external fields, the frequency and the emission and absorption coefficients are independent of the choice of polarization (for linear polarization). Assuming random polarization of the photons, we take F =- F and obtain, for the stationary state, X 2 d YFO= Pk -avY FO (VI-12) ds V where we have dropped the label X from the emission and absorption coefficients. We have also assumed that bremsstrahlung was the only radiation mechanism. Calculation of these coefficients for bremsstrahlung will be given in Chapter VII, but will not be necessary for our purposes here. The path of integration for the solution of equation (VI-12) is shown in Figure 2a. The length S is related to L by S = L/cos 0 where cos 0 = - f2 and n is the outward unit normal to the slab surface at P. The boundary condition F (0, k, Q) = 0 seems appropriate for the present problem. Assuming a uniform temperature, the solution of equation (VI-12) can now be written as p P -2 ds'a (s') c F0(P, k, 0) = ds Vd e 0 P -I dsa (s) +./ ds s (s) eb V M b (VI-13)

78 To perform the remaining integrations, we assume the space dependence a (s) %(s) = S2 (P - s) CCb E (S) a (P- S). The result is given as a 1- 2 a c F(P k 0) 1 aSl aS2 (VI-14) a Finally, taking S2- 0 and S1 —S we obtain o k dk E a - aaSl F (P, k, 0) dk = 3 - (VI-15) a At the point P. we still use free space conditions, so we can write v02 ea - F (P., k, 0) dk =(P,w, 0) dw = dw 3 a 47r c a Assuming kinetic equilibrium for the particles, it can be shown that e vq. a a a ew/ko T ~a e -1 so that I(P w, 0) = ci w 0(P, w, 0) =B(w(-) - e a (VI-16) For vT' = cu we have a I(P, w,0) =pB(w) 1 - e * (VI-17)

79 Assuming that the dispersion relation for the plasma is given with sufficient accuracy by 2 2 2 2 w =c k + w P it is seen that 2 2 Thus, as we approach the plasma frequency (where, in fact, Ta gets large) the a radiation appearing outside the plasma is strongly reduced and approaches zero. Before giving further discussion to our results, let us consider the solution of equation (1-3) for the same problem. This time, however, it will not be necessary to consider two regions. (See Figure 2b). The boundary (2) condition at P is I(P, c, 0') I(P, A 0) = 2 a where the plus and minus signs indicate that we are approaching the boundary from the right (free space) and left (plasma) respectively. The angles 0 and 0' are related by Snell's law sin 0' = sin 0. Ha Note that the path length S' = L/cos 0' is different from S. The boundary condition at the other boundary will be taken as I(0, w, 0') = 0. For the same conditions discussed for the previous problem, it is a straightforward calculation to show that I(P+,, 0) = B(w) =B() [ - ea. (VI-18) ~1 -c~IS' )

80 Equations (VI-17) and (VI-18) are quite different and would predict entirely different results as we approached the plasma frequency. However, (17) as pointed out by Bekefi and Brown, equation (VI-18) must be modified to account for internal reflection at the boundaries. When this is done equation (VI-18) becomes, to a good approximation, - a SII(P, o, 0) =(1 - r) B(w) 1 - e (VI-19) where F is the reflection coefficient. As a reasonable low order approximation, we take 31) 2 r.p-. + 1 With this expression for rF equations (VI-19) and (VI-17) are in qualitative agreement, although they can differ by as much as a factor of four. Nonetheless, both of them give reasonably good agreement with the (i7) experimental results of Bekefi and Brown, and over the range where our equation would be expected to be valid, they are essentially in agreement. The experimental results do not suggest a clear preference for either equation (VI-17) or (VI-19). As a consequence of the analysis presented here, we feel that the validity of equation (1-3) is still open to question. The limitations of the phenomenological equation are not obvious and we have not been able to obtain this equation from the postulate-deduction method described in Chapters II and IV. On the other hand, we have obtained an equation by this method which is to represent the transport of photons in a dispersive medium. Many approximations

81 and assumptions have been invoked, but we have attempted at each stage of the development, to point these out and to suggest the limitations within which we expect this equation to be valid. The experimental work carried out to this time does not demonstrate which result, the one obtained from the phenomenological equation or the one we have developed, is better.

VII. EMISSION AND ABSORPTION A. General Considerations In this chapter we will discuss the modifications to the emission and absorption coefficients resulting from collective particle behavior. Recall that, for the 3th type of process X ( ) 2 X 2 3 c2 (2r)3 o (VI-8a) a- (w, 2) = v KK1 V2 TUK (Xk) (f (EK) - f (K OKK (VI-8b) (VI-la) X E - z KK 2 K1 V T (X k) f (K) 3BK 2 K1 22 2 2 V2 T (Xk) ( OK - mc wXk Xkk 6 (w Y -) (x) < EK e xE (k) II K > 2,, - _ (11-46) X Aside from changes which may appear in e/ as a result of the medium behavior, \ tc it is seen that j differs from the "free space" result by a factor, which for v =v gives for v7 = c gives 82

83 ic -r -( VII-1) c\ (27r) In the fully ionized gas with no external fields, the index of refraction p is always less than one. Consequently, as we approach the plasma frequency, where p-+ 0, we may expect the emission inside the body to be -X) strongly reduced. Furthermore, since v -4 0 as wU-+ and a 0 is inversely proportional to v, we anticipate that the absorption coefficient would get very large. However, this analysis is notappropriate to investigate these coefficients in the vicinity of the plasma frequency. B. Bremsstrahlung The content of these remarks can be illustrated by consideration of the bremsstrahlung process. Taking positive energy Coulomb states for the particles and going to the continuum for the particle momenta distribution, it can be shown5 that 8 rc r e Z nI -I d3 Kf (K) 3 - (x) d K1 6 (EK - EK) - X (A) K1-I (k)-2) 2 where r = - and n is the number density of ions. Averaging over mec polarization and integrating over EK, we find. Appendix E

84 1 - X 2 2 a2 5m /2 EB 2 2 EB 2= 2f r nI r Z 3 X L) (x) /d EdQd f (E) - E- (E, 0, 0 0 0 1) (VII-3) where e (E, 0, 0, 01, 01) E sin 0 + (E - ) sin 0 2 E(E-) sin 0 sin 0 cos (0 - 0) 2 [E + (E - t) - 2 p E (E -ti)2 e2 1 Po = cos 0 cos 0 sin 0 sin 01 cos (0- 01); a= c, 37 - Define the quantity aB d d 1 as the cross section for emission of a photon of frequency w in dw) in which an electron with initial energy E and direction Q has a final state in which the electron is going into -1 in d 12 The source term 2 2 Bd - 2 d3 = e2 3 2 'wdw (VII-4a) B 2 X B 32 ' B (2r3 c2 is related to the cross section by J dw = dwn niw vf dE dQ2 d (VII-4b) B e e B 1 E,f 1 where n is the number density of electrons and f is the electron e e tribution normalized to unity. From equations (VII-3), (VII-4z)and (VII- 4b) find that

85 2 2 dudh r2 2 32,, — aB d ri d 2 = I / (VII-5) 2 137 V E This differs from the corresponding result obtained in reference (9) by the 2 factor C which is just the factor suggested by equation (VI-8a) and vY discussed earlier. Note that as a function of frequency, eB is not altered by collective particle behavior of the medium. C. Cyclotron Radiation * It was pointed out in Chapter III that a plasma in a constant, uniform magnetic field is highly anisotropic and that our procedure is not sufficiently general to treat this case properly. Nonetheless it is both qualitatively and quantitatively interesting to consider this system. For instance, we will observe the possible appearance of Cerenkov radiation. Also, we will indicate how the results of this analysis differ from those presented previously (9). 2 K1 For magnetic transitions, V T rK (X k) can be written 22 2 v mK (Xk) = 4 c ( mK - M mk K k K (x) I (j' ' K' | j K ) (VII-6) where (j K' | jK ) = < j' K' e k - (k)o-I j I K K Z K Z Z A -' Z

86 and the eigenfunctions j Q K > are given in Appendix B. Transitions for which ' z j ' j are associated with the emission and absorption of cyclotron radiation. Relevent matrix elements for the transitions are given in Appendix C. From equations (C-7), (C-8), (VI-8b) and (VII-6) we obtain 2 aC (W, kQ v() i X v n 9 P / ^v eos 0 (x) UJ (v YO sin 0) [w0- (n cw Po tV W COS 0 2 0 0 (VII-7a) 2 m 2 22 3 r (nw ) c -~,. \ P m_\ c vn co n 0 C C (x d vf(v) cotont- v % ) ] [ ] 2 F ( vzCos 0 s tiw Cos 0 ~ (x) n - n w - c 2 2 (VII-7b) L'r 2 m c where = and where the initial and final particle distributions c w0 have been taken as Maxwellian. The 0 and 0 polarizations are associated with the polarization vectors which have been chosen as the spherical base vectors in the polar and azimuthal directions.

87 In the limit v -- c and M -e 1 we find the results given in reference (9), The absorption coefficient for this case has received a great deal of attention (32, 33, 34) in the past few years because of the importance of cyclotron radiation as an energy loss mechanism for laboratory thermonuclear systems. Numeri-0 cal calculations of the absorption coefficients (34) have indicated that only a c is of importance for systems where cyclotron radiation is important and then only for propagation nearly perpendicular to the magnetic field. For 0 = - d ( — vf(v) v1 (x) c (Lcio ) 0 c (VII-8) It is seen from this equation that the deviation from I = 1 is reflected c in the factor -- and in the argument of the Bessel function. Do Cerenkov Radiation Cerenkov radiation results from the matrix element < jK | e -ik (k)-II j 0 K >. Z< K - - Z This element was not considered in the calculations of Osborn and Klevans (9) because when p = 1, it is not possible to simultaneously conserve energy and momentum and the term must vanish. For a plasma in a magnetic field, however, it is possible for p > 1 and this implies the possibility of simultaneously conserving energy and momentum. Thus, the radiation associated with this matrix element should now be included in the discussion.

88 Since our analysis can at best give qualitative results we will consider the simple case of a single particle moving along the magnetic field. Then, taking k propagating in the y - z plane (using a left handed coordinate system), we need only calculate I (0 e K' | 0 0 K ). The index IH refers to the position of the z z orbit.6 Because the photon is emitted at some angle 0 with respect to the z axis, it is necessary for momentum conservation that the particle make some movement in the x - y plane. Such motion is accomplished by a transition to a different I state. We then find e 2 /2 a I (O K' OK) = (K + k -K ) - ( ) e 23t ) /t 1/2 (VII-9) where a = kK H sin 0. Substituting equation (VII-9)into equation (II-46), we obtain K2 K47 c22 e 2 1 2 -a2 Tce K Mic2 mc wi t -K (Kz + ( k - z + k - ) (VII-10) Xz Kk K Z z z Combining energy and momentum conservation, we fin. 6 The position coordinates of the center of the orbit do not commute and cannot be simultaneously specified (28). The center of the orbit is quantized with eigenvalues — H (2i + 1).

89 / A v cos 6 c \ / AL v cos 0 \ ( -) cs O (VII-li) k z where 0 is the angle between k and the z axis. Since w / 0, equation (VII-11) is simply the Cerenkov condition = cos 0. (VII-12) MAV AXz From equations (VI-la), (VII-10) and (VII-11), we obtain, after carrying out the sum over., 3 2 2 /22\ 2 \2 __\ X (47r c) ( _e_ -- --- 2., = e 2 e ce \ mec / v Cos 0 \ () zc (VII- 13) 2 \,a2 / Noting that e 2 1j substitution of equation (VII-13) into (VI-8a) gives 2 2 x ^ ^ (<3 )2 ( v cos 0 1 je ( ) -2 2. M -1 (VII-14) c 27r v / A Using the spherical base vectors 0 and 0 as polarization vectors, we have = sin 0 and = 0. Thus

90 jd d dw = d d j (, a) = (1 22 ) d. (VII-15) - 2 2 v c wovz With v = c A, this gives the classical result for Cerenkov radiation (18). It must again be emphasized, however, that the applicability of this result is limited since we cannot obtain [0 within the context of this analysis, except for frequencies sufficiently high that the index of refraction is nearly equal to one. E. De-excitation Radiation For de-excitation radiation from an impurity in a highly ionized gas, frequencies are too high and densities too low for the present discussion to be of much interest. The results presented by Osborn and Klevans (9) are sufficiently accurate, and for completeness will be included here. For a dipole approximation 22 2 2 Ki 47Tr c V Td (X k) = mcc K k K (X) i<K, | (k).j |K>. (VII-16) From the commutation relations Exj, Pk ] i= i kPk and [x., V (x) = 0 jS Pk jk] we obtain

91 (k) < K p K >2 = I jk)( )<K [x,, H K>2 2 2\~ C l 2 m2 <K1 K (k) x K > (E -Ei) so that 2 K, 47r c V T (X ) <K1 x K> cos2 0 S( k-K dK 2J K k K C (VII- 17) where 0 is the angle between the direction of polarization and the vector x. From equation (VII-17) we can obtain the source term j or the absorption coefficient a. Or we can relate it to the rate of spontaneous de-excitation ydS2 presented by Heitler (35) by 2 3 ydQ = V V TK (X k) - d < K K, x K> 2 os2. V dk 3 K1 2 7rtCi (VII-18) Further calculations with this quantity are presented by Berman (36). F. Recombination Radiation Lastly, we consider recombination radiation, which is important for hydrogen plasma systems with kinetic temperature from 3 ev, to 200 ev. We choose the electron and atomic wave functions as K > 1 eiKx K > = (VII-19) -1 n 4v

92 where n represents a sufficient set of labels to completely specify an atomic state. After converting to the continuum for the electron's initial momenta, the emission coefficient for recombination radiation can be written 2 2 3 X X 2 e tic 1 V 3 3 ( E = -n - /Qj dKf(K) r 7*Y 2 M m cox n v 2me (x) (< * K) S ( Kk- ) | ((VII-20) where _ _A) = Jn (x) e - d x and K = k - K. As a result of the electron normalization, the ion density is given by nI = V The frequency of such radiation is very high, and 2 2 w ~ >>. Dispersive effects would be expected to be negligible and deviation P2C cpX from = 1 should be small. Additional calculations concerning recombination radiation are given in reference (36).

VIII. TRANSPORT IN CRYSTALS The last subject to be discussed is radiation transport through crystals. It is of interest, not only because we obtain the anticipated x-ray scattering cross sections as a natural consequence of the analysis, but also because of the analogy that can be made with the problem of low energy neutron transport through moderators or filters. Since a quantum mechanical transport equation for cold neutrons has yet to be given, such an analogy is perhaps worth pursuing. As mentioned in Chapter II, we will employ the relatively simple Einstein model of the crystal. In this model, the atoms (nuclei surrounded by electrons) execute independent isotropic oscillations about fixed equilibrium positions. The force constant of the Hooke's Law restoring force is identical for all nuclei. In accordance with these remarks, and with the appropriate quantities shown on Figure 3, the Hamiltonian for the systemcan be written.N. N 2 H = + HU] = + [ M w u 1- 0 e UJ 2M 2 a o 2 N n p + + 2m. * 1n (VIII-) n a=l j jwhere H is the Hamiltonian for the oth nucleus, and H is the Hamiltonian o e 93

94 a u -j Figure 3. Location of the j Particle of the th Atom

95 for the electrons surrounding this nucleus. The natural oscillation frequency is given by w. The nucleus has an equilibrium position R, and an instantaneous displacement u. The instantaneous position of the jth electron of the oth nucleus is x, displaced by. from the position of the nucleus. We have assumed a complete, orthonormal set of medium eigenstates I m >3 such that Im > =7Tm a m U (u )0 (0U) (u n - 0' (VIII-2) where H Im > = E Im > m (VIII-3a) H (u ) = E r (u) o m -a m m - Uy Uy Uy (VIII-3b) (VIII-3c) H 0 ( ) = e 0 (0 ) e n - n - U U U and a = CT..., e. From equations (VIII-1) and (VIII-2), it

96 is seen that m (u ) is a harmonic oscillator wave function. For notational m "a convenience, we denote (u ) - |m > (VIII-4a) m -a (> 0 (I ) In >. (VIII-4b) n -a I a We will only be concerned her e with scattering of radiation from electrons, and absorption will be neglected. For the high frequencies under consideration, it is also anticipated that we may assume an index of refraction of unity with little error. In accordance with these remarks we take M py2 H = H + H 2 where /e2 + + a-iK HPy2 Ak V mcJk k'1112 _ (k)-~ (-k') e - x2 - Xk \mc2[k k ' -a Xk X k' X'k' (VIII-5) with k. = k - k'. Charge and mass are not subscripted with the index a since only photon-electroh interactions are considered. The derivation of the transport equation will be assumed valid. For some scattering problems, the spatial coarse-graining might become controversial, but we will not concern ourselves with this possibility. Writing ) it (e -e +E -E )/ ( - ) = dt e a n nr1 ma 27r J v -c

97 and defining 3 2 2 Ak V3kk(mc2) ' k) (k') CT — i (R +) -i i ( Heor+He) t/~ - i ( H +H) t/ 0 e e - - k Xk,, -X tk' A; ''V F.k ( V F 'k ' + 1 ) -Xk Xk ( +k, I) (VIII- 6) where XIk /Xk 2r d t Ak P( < m (t) m > ~ m (VIII-7a)

98 I -A - dt e Z P(m) < m | (t) I m >. Xk Xk 2 7r au'I m (VIII-7b) The quantityP(m) = D is the probability of finding the particle system in h sTt kX tt'k' + the state I m >, Note that Ik I' since Qa (t) and 2+ do not commute. Ak 7-Xk Assuming random polarization for the radiation, we take F = F X F Then, summing equation (VIII-6) over polarization and converting to the continuum in photon momentum space, we find 1 aF_ ) + Q F(k) c T - - - = N 1 + v Fk)) /dk' dQ' F(k') c (k', Q'; k, Q) V Fr k')\ - N Fk) /dk' d' ( + - ( k, k k', Q) (VIII-8) 2 p kt \ 2p~- - - where ~+ c k'1 ';t k, _ T " a(k' Q';k, L) = ack( ) dt e1"^ L P(m) < m I Q Q'(t) I m > ca' m (VIII-9a) a(k, k' 2') = Tk dt e - Z Z P(m)<m f2'(t) J m > rcac7' m (VIII-9b) 1 2 r +(k') PT - e2 r o mnc2 N = number of nuclei

99 The quantity o(k',Q2;kQ) dk' d~' is a cross section. For neutron transport, even near thermal equilibrium, the number of occupied states is small compared to the number available for occupation. Thus, scattering terms dependent upon the final state particle density could be dropped. For a crystal at room temperature, this approximation is also valid for x-ray scattering, since we are on the high frequency tail of the Planck distribution where the above remarks on occupation of available states again applies. 7 We can obtain, by a rather tedious calculation, the result ok, 0k,~) ~T o\ 6/-\ 2 -DK2 1 V1 -K. Rco a(k',';k,Q) = aT c6() f e12 K - - c+aT c 6() eD [f|2 I (B22) - f12 +- D I 20 +aT T- c e If I (Bl) (3+no)o [n=1 -Mt Ho + e ( (-n (VIII-10) where D = coth (VIII-11 a) 2Mwlo 2 koT B = csch 0 (VIII-llb) 2Mw 2 kjT O 7Appendix F *Appendix F

100 f = P(n)f- (VIII-llc) f-1 nn n f- = < n |1e1 'j I| > (VIII-lld) nfl j and where I is the modified Bessel function of the first kind of order n. We n have made the approximation that there are no transitions between electronic (37) states. The quantity f — is called the atomic scattering factor. It has the meaning that if the electronic system is in the state I|n >, then | f — 2 T nn T gives the scattering from the electrons surrounding the nuclei. The cross section aT is the Thomson cross section for scattering from a free electron. The first term of equation (VIII-10) represents interference scattering. (Note that such scattering is elastic.) The remainder of the terms comprise what is called direct scattering. In the second term no oscillator states are changed during the scattering, wfiile in the last term the oscillator state is changed. The last term accounts for inelastic scattering. It should be noted also that e is the Debye-Waller factor and 1I\7- -ilC* RU I (37). | e E 2 will give rise to the Bragg scattering condition in an N infinite homogeneous crystal. Equation (VIII-10) reduces to the cross section for neutron scattering from a monoisotropic crystal of spinless nuclei() if f -- 1 and o- a, the nuclear scattering length. A thorough discussion of this formula is given (38) in reference38 It is perhaps worth a final observation that when inelastic scattering can be neglected, we find that

101 1 aQ ~i) + ' v Fqk) C at = N/d d Fck, Q') (k, Q, Q' ) - N F~(k, n) fd f ', (k, ) (VIII-12) where a(k,, _ ') = TTe 2?12 ( )[ —'1 )R 2- I2 +f2I (B ) (VIII-13) Thus, within the context of the Einstein model (and the additional approximations we have made), elastic scattering terms which are dependent upon the final state photon density cancel.

IX. CONCLUSIONS In the foregoing, a transport description of electromagnetic phenomena in terms of creation, destruction, scattering and flow of photons has been developed. The usefulness of characterizing electromagnetic phenomena in such terms is limited by several factors. First, it is limited in a practical sense to physical problems for which one-photon processes give the only significant contribution to the rate of change of the photon density. It is also restricted to those systems for which spatial variation of particle and photon distributions is slow over regions characterized by a length many times larger than the longest photon wavelength under consideration. This geometric optics condition suggests that our description is applicable for investigating the energy balance of short wavelength radiation, whereas the full set of Maxwell's equations is needed to describe electromagnetic phenomena for which longer wavelengths are of interest. There are additional conditions which tend to restrict the expected range of validity of our transport equation. These conditions are intended only as sufficiency conditions, however, and it is not claimed that they are necessary. First, the medium should be isotropic. In the anisotropic medium, there is in general a coupling between the various transverse and longitudinal modes and our procedure is not appropriate to treat these systems. In a few special cases, e. g. photon propagation in a plasma, parallel to a uniform external magnetic field, the different modes decouple and our procedure can still be applied. A second restriction is that the photon frequencies should be sufficiently far above the resonance regions that level broadening effects can be neglected. 102

103 It may also be possible to apply the equation between resonance regions or below them, but it is not obvious from the derivation that it is legitimate to pass through the resonance region, where the equation is not expected to be valid. The radiation transport equation which we have obtained is not in agreement with the equation obtained phenomenologically (equation (I-3)). However, when the solution of equation (I-3) is modified to account for internal reflection, both equations yielded solutions which were in reasonably good agreement with the experimental results of Bekefi and Brown (17). Thus it does not yet appear possible to reject either of the equations. Because there have been many approximations employed without investigation, and because there is an unresolved descrepancy for the inhomogeneous medium between our transport equation and the more conventional phenomenological equation, we suggest that further study in the development of a radiation transport equation for the dispersive medium is in order. This thesis offers a beginning, and it is hoped that further work can shed light n some of the difficulties encountered in our derivation.

APPENDIX A CALCULATION OF THE TRANSITION MATRIX ELEMENT Unn We wish to apply radiation damping theory to the study of U(s) = exp [-iH s/li]. Define K(z) z+i(H~- ) K(z) = - n~ n b (A-l) Then we can write 7+ioo Un'r7nIS) = 2T i f ^'n I) 2Ti 7-ioo From equation (A-l) we obtain I (z+i ) K +i H K +i n nn nn nn zs/' f dz ez / K (z) n'r7'nr) (suppressing the index r7 for simplicity) H K = 1 nn' n'n n' in I I (z+i ) K +i H K +iHI K n' n n n'n nn nn' n'n Define Q (z) by K =K Q K Kn'n Kn~n'Qn nKnn n n'n n'n nn Q = 0 n'n' + i HI K n'n"' n"n' n" ~ n, n' n' n all n' (A-2) (A-3) O. (A-4) (A-5a) (A-5b) 104

105 Define z) by r HI -;::L it, z) = nfl H'I nfl? K n~tQn (A-6a) 1 2 itiF,(Z) =0 (A-6b) From equations (A-3) and (A-5) we see that 1 ihF(z) K = 1- (Z +iE ) K 2 n nnn n nn. Thus 1 K = - nn Z+iEn + 1 itilnn n2 (A-7) From equations (A-4), (A-.5), and (A-7) we Obtain = -iH' n'n i2 I i n' - iH )K Q ntnt n~n? n'n n- t in nnn Kn "n n In From equation (A-6) we see that equlation (AL-8) can be written (A-8) Q -i H I+1i n'n n'n ZH'I K? TQ K Q n'ntt nlnt n??n? nln? n'n -i H H Kn??ntt? n?n (A9 (A-9)

106 We note that Q first appears to first order in H, which is equivalent to first order in the interaction strength. We wish to obtain Qnn in an expansion in powers of e in terms of H and K. To first order, we would have Q =-iH + R n'n n'n where R is the remainder and is assumed small compared with the first term. R is still a function of Q. Substitution for Q,, into equation (A-9) n? n would give I 2 T I I Q n -i +(i) Hn n K1in + R. (A-10) 'n n'n #, nn n'n nly i n, nt Thus, we have exhibited Q to second order in powers of H, plus a n'n remainder. By similar substitution into equation (A-9) whenever a Q appears on the right hand side, we can express Qnn in terms of H and K up to any desired order, plus a remainder. By a similar procedure, we find, from equations (A-6a) and (A-9), 'iii (z) i ( H + i(-i) H K H 2 nn nn nn' nn' n'n ' 'n +(i)(-i)2 K H K H n' nn' n'n' n'n" nWn" n"n n" n, n' + R". (A-1) Except for the remainders R' and R", equations (A-10) and (A-ll) are the same as the first few terms of equations (16) and (15), respectively, of

107 reference (24), except for differences in notation. For the purposes of this paper we will need only the first term of equation (A-10) and the first two terms of equation (A-ll). As a consequence of the above, equation (A-2) now becomes y+ioo U (s) d= e K dz es K n Q K (A-12) n'r1'nn 2 7ri J n'r'n n 'n nrn,. y-ioo Writing the first term in the expansion of U (s) as U(1 (s) nTr'nrj n 'nr) and noting that all first order processes and scattering are contained in this term, we have U() (s) = (-i)HI n'r)'nr n'r'nrj y+ito z s/f 1e ( dz i () 27ri 7-t +'n (Z))k+i E n 2 7-ioo D nrl 2 nrlnr7 n'J 2 n (A-13)

APPENDIX B CALCULATION OF THE PLASMA DISPERSION RELATION FOR PROPAGATION ALONG A CONSTANT EXTERNAL MAGNETIC FIELD From Johnson and Lippmann28, we take the result iK z K > =| jKz > = uj (p 0) e -l {L (B-l) where uj ( = ) /!; "' -e 1/2 i I - 1/2 +io (j -1) (- i) b- e (j-2) 2, -., (j ) J ( ) (x) (B-2) and where b2 tc eH; 2 -. 1/2 is the box normalization; 0 is the a:zi muthal angle with respect to the y axis in a left-handed coordinate system; and v(-) = J iW j e a -e j+v e ~ (B-3) (39) is the associated Laguerre polynomial as defined in Magnus and Oberhettinger It will be useful to employ the operators + 1 II-= - II + i IIx] which have the properties: II | jjKz > = mcb {j + j+l Kz > II | JKz> =mcb |-j-l Kz>. (B-4) (B-5) 108

109 We can now write, for kll H, i.e. k = k z, Ii I- w I (j'l' KfjK) = < j'Q' K |je -i kz z ~ (k)- II jKz> -A = (K' + k -Kz) < j'' (k)' II j >. Z Z -A (B-6) Taking 1 1 =W (e2+iel) -2 =- 2 el) where e and e2 are the unit vectors for the x and y axes, respectively, we have 1 ' II = II 2 a II = II. Thus I i(j'' K~ jtKz) =mwcb jT jI,+ 1 ' (K +kz - Kz) I~~~~~ j (B-7a) I2(Jt K Z jKz) =m wcbf j S j(Kz+ kz- Kz). (B-7b) We will complete the calculation of equation (III-11) by use of equation (III-7). In a spatially uniform, nondegenerate system, we find 1 I SJ- - i12 (Kz k- Kz) P =mz LKz)f( 2 Kt 2K 2 2 K2 'i(-j)tw-tiw 2 2m Ix (j' lj)2 g(Kz +kz -Kz) + I (B-8) (i2 Ki2 (j-j) w +Titohw +, c A 2m.tr K 2 2m 2m

110 From the symmetry properties I (j' | j) = 2 (i | ') I2 (i'lI) = I (i lj') and the assumption 4t kz i Z, < < 1, it is straightforward to obtain the result Kz w2 c2k2 + 2 P dv f(vz) _ P Vz kz t. ( 1 - W+ ) Vz kz (a+ C) (1- - ) _- c (B-9) where ow = w+ and w2 = ~-. 1L +a ^

APPENDIX C CALCULATION OF THE PLASMA DISPERSION RELATION FOR PROPAGATION PERPENDICULAR TO A CONSTANT, EXTERNAL MAGNETIC FIELD In order to obtain equation (III-14), we must calculate I (j 'K' j K ). Take I = 0 (the well-centered-orbit approximation (40, 41)) and write +. - - e +3 II _E-X (k).= II = If + XXII + X I z where - k 1 * 2 [E I + i~ E Then (K' + k - K ) I (j'it' j 0) Z Z z I K3K ) = I(jI'' K' [j0K )= E tiKz f Z Z z + S(K' +k -K)mw b j 1 f- I (j' 'tj +l,0) Z Z Z C X +j+l + I (j't | j,0)] Elx (J' I L J-,) (C-l) where I (j '' Ij) = <j'' l]eiklcos0 j0> It was shown by Judd, et. al. (42) that I(j', t j 0) = Ri' ]/ L1 J \ 2 \ a 2 I (' 0lj 0) where a = k b sin 0, 0 being the angle between the z axis and the propagation vector k, the latter being restricted to the y-z plane. Since + 8. For the magnetic state vector and for properties of ITr, see Appendix B. We also use IIZ | jK >=li Kz |jjQK>. 111

112 a2 1 — 0-12 p2H 2 e1 a.#10 Hj-j'1)~1 we can takeII ~=O0. Writing ~ 2.1j:Qj, 0)=[.jj? / it is a straightforward calculation to obtain 1200 d9 e~ 9 -n na(2~)1~2 (C-2) where n = j -j '. Substituting JnH2~) 1'j2 = OD m2m+n M+ n/2 (-1), a m+ nY 22 mI into equation (C-2) and integrating, we find 1/2 OD (1 m a2m+n (+) I(j? O jjO)= =K j2 I m=0 m m n/2) Using Stirling's approximation, and assuming j ~ m, we have (j+ M)V -j *,.m (C-3) so that ]-1/2 i(j'I 0 Ii )~ j- ) 1 * n/ 2 Jn H(2j"2 (C-4) 0 Again applying Stirling's approximation, and assuming j;>n, equation (C-4) becomes Similarly I(j I Oji 0) — J fa(2j )1/2] J(j'I 0 1 i + 1, 0)z= Jl/(j~~2jJ (C-5) (C-6)

113 Specifying the polarization vectors as the spherical base vectors ~1 = C and - we find =2 -si0 &.+ c0Cos 0; ^- 4T 3 el = - sin 0 &~ Then, noting that R., the radius of orbit of quantum number j, and v., the j J speed of an electron in that orbit are given by R. = ( (iL)' - v = ( 2 j )/2 ( c) Vi 2 m we obtain, after using Bessel recursion formulas, I(j' 0 K' | j 0 K ) = m wj b2j (K' +k -K ) J(v.,y0 sin 0) I z z c z z z n j (C-7) I (jK' O j 0K ) = - K sin 0 6 (K'+k -K ) J (v.jy sin 0) 0 z z z z z z n j 0 J (v.-Y sin 0) + mno b 2j cos 0 $(K'+k -K ) nj — e c z z z v. jp sin e (C-8) where yx cw c Writing equation (III-7) in the form dK dK', f(K ) f(j) - f(K ) f(j) zJ z Z Z 1 PXk m jj' (C-9) (C-9) IXk(jK IJ K ) 2 (x) K -EK -wXk 1 k

114 we find, by substitution of equation (C-7) into (C-9) 00 Pe= j=0 L Jd K f(K ) f(j) e n Cf(K ) Z n=-ooD 1t wn/kT / -1 (C-10) (x) cot 0 m (c)sin cot - szC c 2 mc 2n k~ 2 - 2 k K 2 k2 z z z nt w -'i w- + - c 0 m 2m To obtain this, we have written f(j') f(j) w n/kT = e C o a result which follows by taking an equilibrium distribution - k wcAT ^ OT f(j)= (1- e /oT) e /o This distribution is obtained by convetinal statistical mechanical methods assuming the energy levels are the magnetic state levels and that f(j, K ) ' f(j) f(K ). Observing that z m 2 ci wcJ 2v. it is seen that m v. j)dj =f(j)=f - - d v hi w c m vm e (mvj/2)koT - dvL ko T = f(v ) dvL

115 The upper limit of the sum over n can be extended to infinity since the terms for j large give negligible contribution. Finally, for propagation along the y axis, it is readily established that n wc P / '- (C-ll) n where n d v f(vl ) [Jn(V' y7)e2 0 We can now write 2 2 2 2 ) (C-12) -C k L) +. (C-12) epn nw-w n p n n n c -o n But eo 2 (x) = 1 n n=-oo so that Yn n and 2 22 2 W W=- n' n (C-13) p np n nc -we the result shown as equation (111-14). Equation (11-16) can be obtained by similar considerations with the use of equation (C-7).

APPENDIX D THE HAMILTONIAN AFTER DIVIDING THE SYSTEM INTO CELLS We present here the various terms in the Hamiltonian. Recall equations (IV-lOa) to (IV-1Of). H = H + H(1) + H(2) + Hr + HPT + HIT + V (IV-lOa) o 0O HM H = T +HM (IV-lOb) HM = T + HPe + Vc (IV-lOc) H(1) = HP^1 + HPYe (IV-lOd) H(2)= (HP72 + T71) + (HP72 + T72) + H^ (IV-lOe) H +T 1 +T 2e H T = H1 + H2 (IV-lOf) 2 Explicit calculation gives: 70 T = liw (X, k) a (X, k) a (X, k) (D-l) X, Xk P +HPe 1 + f_ aTx +H = 2 a (K)a (K')- I u K( (D-2) IIuK,(_x) E(X, x) 116

117 c + + vc = ~ a (X,K)a,(X, K') a (X,K") a,(X, K"') a — (t a - a Z, - a ca a' Z, K,, KT, KII (D-3) (x) 12 fd x d Ix' E(X, x)E(X, x') ()v/d + + (x' UaK(X) K'(") K, a'K I x - x' I M -k XkK1 2 1/2 /27r tc e. \~V / m cJ i k (X, k);.(, k) (D-4) (x) a (X,K)a (X,K')V 1 d a a'- - V — 3 + 3x E(X,x)u(X) e -- aK -ik-x U aK(X) A Py2 Tyl= _ H0 +T 2 _X4k 2 47r e -- - - a (X, K) a (X, K) mru (X, k) ar- a (D-5) c2k2 2 k) - + c (X k) (2 k)a (X, k) +X(X, k) Hy2 + T72 = 7 47r e 4 - m (X_, k) X,Xk La ( - - - 2k2 2 (X +..~t(;, k) I (D-6) (x) a[x (X, k) lA (x, -k) + (x, k) (x, -k)

118 2 P-y2 _ ' arr H 2 =m Xfkf$Xk KK?,vK#~Kf 1 (X,ik4,(x3, k') 'I(. k) Q u~fjXp kt) (D-7) N> a (.,K) a (X, K') d x E(X,a) e ~(X) u K(-X PT_ 1 + 1/d3[+_x a H = a (X, K)a(X,K6) (IIW E(X, x)). E(:X1x)ll u 2m a — C- v K _ -.UcK' uKK' XX' (D-8) + E(X x~)(lla u+ + a* ) UaK, IIUE (1', x) +uku ELX, - (IIE (LV, x) - qKl(ll 1/2 HIT= aX Xr~k K K' ) (2e 1 Ik) (X k) o I + (x + (X, K) a ( s K')+ (X., K) a (X',K) (X)a a,- a- a (D~-9) () 1 d3 E(:.E+ We-ik- x u WUE(,x cx) dx(X~xu (~e - U (x)~lE(~x) V UK-c~

119 cc I V = a+( K) a+,(X K?) a -(Xilit) a (XI, K t 1) K KT, K'TKTTT (D-1O) (X d3 xd 3xTIE (X~j )E(XT I, xT) + +I u (u (x ') u (x u,,(x) uK(3 &,KT( cK"2 o~I LIX - XT I yr _____ H1 4L k1lj(X. ii) \L, k) 5(k +kI) S(k +k') (D-11) IjXX+L.,X2"XT -.k') u~,X1+, 2'X 3'kt + _ I _LJ w(XI-L.,X2~X3. ' I + cyclic permutation itic2 2 4L 2I XXkXT kT + L ~(k +kT)5(k + k') C\X., k) 2 2 +33 (D-12) ___ X T (X,-L., X2, X3, -~t) cyclic kI pemutation 33, kt caXI(X_-L_, X2 _X3 _I pe (XI +LI X2p X I uO?(X,+L.,X2X 0-0

APPENDIX E CALCULATION OF THE BREMSSTRAHLUNG EMISSION COEFFICIENT Ex B In order to calculate e, it is first necessary to consider V2TK1 k)- 4r c2 (e2 1 BK - me wXk (x) I (K1 JK)| (II-34) where 3 * -ik-x x I(K1 JK)= d3 x u (x) e - - (k) - (E-1) Writing 3/2 iK' x K (x) = (-)3/ d3 K' e ~ /(K') (E-2) equation (E-1) can be written I(Ki K)=-it d3K' ~I/ (K')(KI + k)( K). (E-3) The eigenstates uK (x) are to satisfy the equation (E - Ho - V) u = 0, (E-4) J2 K2 Z e2 where H and V=- e. (E-5) o 2m r Let uo be a plane wave eigenstate of Ho with eigenvalue E. Then we can write9 u = uo +; (E- H) u = 0 (E-6) 9. The procedure used here for obtaining uK(x) and ~ (K) was originally developed by Lippmann and Schwinger(43T. However, we will follow the presentation of this method of Bethe and Salpeter(26). 120

121 where X; satisfies the equation (E - H -V) ) V uo = 0. (E-7) 0 It can be shown (26) that YL has a cut on the real axis and therefore represents two functions X +, which satisfy the relation 1 7+ = V(u + +) (E-8) E -Ho +ie where e is an infinitesimally small quantity. (This integral equation representation of +K is not unique.) The eigenstates u can now be written usu+ = uo. (E-9) It can be shown(26) that there is one unique state u+ (and u_) corresponding to each eigenstate of u0 of Ho. We will choose the eigenstates u+ for the remainder of the discussion. Define + (K) as the Fourier transform of u+(x) i. e.,+ (K) = )3/2 d3x e — + x) (E-10) From equation (E-9) 03+(K) $ (K Kv) 1 = S(K-K + -i2K'2 -i2K2 2m 2m 1 3/2 d3 -iK - x \ d xe - V(x)u (x) (E-11) where u was chosen as o

122 = " u 1)3/2 -iK- x 27r (E-12) Substituting v(x) ( )/2 2-r d3 K" \J iK" ' x e v (K") into equation (E-11), we obtain 0 (K)= - + - TV 1 - K' + - 2 2 (K' + ie)_-K (2m) '2 3 d K" (K)V (K-K"). (E-13) In the first Born approximation we write (K") = (K" -K') d+ - Vo - band equation (E-13) becomes (K)= - (K- KI) + ( 2 t 2 V(K - K') K'2 _-K2 (E-14) For the potential V = Ze2 r V(IK|) = - Ze2 U(IKI) (E-15) where u(lKI) = 1 )3/2 27r -iK r 3 e - - d r I r 1 3/2 22 (2r) 2ff2 K2 (E-16)

123 Substituting (E-14) and (E-15) into equation (E-3), we find -it I(Kll K) = (m- V /2mZ e2 "- l~ y, d3 K' (~E K') S(K' -K) - - g(K' + k - K) S(K -Kl) U(IK' +k-KI) K2- K'+kl2 S(K' +k-K) U(IK'-Kl) K2 - K'2 (E-17) (E-17) The first term in brackets must vanish since K1 + k- K = 0 is not compatible with energy conservation. Thus 2im Z e2 I(K1 IK) = U(q) -1 i UV (x K1) K2- IKl+k 12 (~- K) K2-IK-k!2 J (E-18) where q = K + k - K. Assuming a nonrelativistic limit, we take k <K1 < -- K and, substituting equation (E-18) into (II-46), obtain V2 T 1 (k) = BK (x) 8rnIc4 r2 e2 Z2 S(EK k- EK) V 3 A. C~h. K1`E. LE~ q4 e2 1 and one factor of has been me2 V (E-19) associated with the ion where r = density n density nI. Converting to the continuum in particle momenta and observing that K (1) = d3K1 P(K)=V Id3K1 E can now be written B

124 87-c4r2e2 Z2 n A. ~ 0I EB= B W3 A.X S dKf (K) K d3 K ~ (EKlk -EK) (x 4, (E-20)

APPENDIX F CALCULATION OF THE CROSS SECTION FOR PHOTON SCATTERING IN A CRYSTAL We wish to reduce equation (VIII-9a) ck 1(___1 a(k, k, Q)= a k( ) dt e to equation (VIII-10). Define 1 ft + f - 2 /1 P (m) m| ' (t) m) a 'l m (F-l) Y =YolYY 1 YY =- C P(m) rav' m + 1 ~72 P(m) on m + T (ml Qf (t) m) (m | QQ nt) | m) (F-2) and consider YO. Expanding [m) into the product wave function and utilizing orthogonality, we obtain Y~= 2 P(m )P(n ) - 0 -mn n f a(c CT U ik *.ucr \ai jUi a ( i I (X) - p( P,)p(n,) (m, I e -i~ ' m,, m,n a' (X) e P(m ) (m ale oTr m Uy (7, e-i-g -U'Iu IU > () 1 NX i&- (R -R') e (F-3) 125

126 where f = n 0y -i&. P(T ) ( |Z4 e J a a |I > (F-4) By use of the Corallory to Bloch's theorem, _l 7 P(m)(m IQ21m) 2 P(m) <m eQ m) =e m Z —\M (F-5) m where Q is a harmonic oscillator variable. We take the thermal distribution for P(m ) so that a0 i ( 1 k0 ei wo V 2 e 2kC P(m ) = 2 sinh a (F-6) From equations (F-3), (F-5) and (F-6) it is easily shown that 7 P(m )(m le m -iK- ui 2 -Dk -D~ m I) =e i- coth - d c 2kT O (F-7) where D= - 21 Finally, then, we have -D02 Y =e rf12 { 1 e itR I 2- 1 (F-8) 1 Turning nowto Y, we first rewrite it in the form 1 1 N (X) FP m iii - -0" n n o' (:r! e (t e E- )t P(ff) 0" J I 2 (F-9) ( m a I A AI (t)I m)

127 where -i&' u A(~__e After a somewhat tedious calculation, we find sin o t -iC- [u cos w t + P0' /\ (t) = e o Hence, sin w t -i/' [u cos tot + P /( l A + i\(t) | )= \ i e e (in AA(t)|ii =Knle e By use of the formula A B A+B+ [A,B] e e =e which holds as long as A and B comrnute with LA, B], we find fP(m) m /\+ in itiK2sint t/2Mo A(t) m) =e sin tot i- (l1-cos to0t)u - M P] (x) p(m) (m e m) (F-10) Again utilizing equaation (F-10), after laborious computation, e [ o(B 2 )+ n = 1 In(BK.2) n (x) in(wot - -2 ) e +( B2Mc h w -in(w t - 2kT) i oo csch -- O 2T (F-ll) where

128 Writing n,n' J nnnn /! in e, 2 --- eP() | + If 12 (F-12) For scattering in which electronic states do not make any transitions, we obtain J f 1 (F-13) Substituting equations (F-ll) and (F-13) into (F-9), equation (VIII-10) follows directly from equations (F-1), (F-2), (F-8) and (F-9). If we had also included electronic transitions, we would have approximated W +(e, E /h = w - and employed 2 nn so that J c 1. The modification of equation (VII-10) is to replace (fj by unity.

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