THE UN:IV ERS-I'T: Y OF.M:I C HIGA N Qualified requestors may obtain copies of this report from the Armed Services Technical Information Agency, Arlington Hall Station, Arlington 12, Virginia. ASTIA Services for Department of Defense Contractors are available through the Field of Interest Register or a "need-to-know" certified by the cognizant military agency of their project or contract.

THE UNIVERSITY OF MICHIGAN 2764-5-T PHOTON TRANSPORT THEORY by R. K. Osborn and E. H. Klevans August 1960 Report No. 2764-5-T on Contract DA 36-039 SC-75041 The work described in this report was partially supported by the ADVANCED RESEARCH PROJECTS AGENCY,ARPA Order Nr 12Q-60, Proje.ct Code Nr. 7700. Prepared For The Advanced Research Prdjacts Agency and the U. S. Army Signal Researoh and Development Laboratory Ft. Monmouth, New Jersey

THE UNIVERSITY OF MICHIGAN 2764-5-T Qualified requestors may obtain copies of this report from the Armed Services Technical Information Agency, Arlington Hall Station, Arlington 12, Virginia. ASTIA Services for Department of Defense Contractors are available through the Field of Interest Register or a "need-to-know" certified by the cognizant military agency of their project or contract.

THE UNIVERSITY OF MICHIGAN 2764-5-T Table of Contents Page Abstract iv I. Introduction 1 II. Derivation of a Photon Balance Relation 3 III. Some Aspects of Equilibrium 24 IV. Some Applications of the Balance Relation 29 V. First Order Collective Effects on Photon Transport in the 49 Fully Ionized Plasma Appendix I 55 Appendix II 57 Appendix III 62 Appendix IV 65 Appendix V 69 111

THE UNIVERSITY OF MICHIGAN 2764-5-T Abstract A first order, momentum - configuration space transport equation for photons is derived for low energy (non-relativistic) systems. The derivation is first order in the sense that the transition probabilities characterizing photon scattering emission and absorption are computed only to the first non-vanishing order by conventional perturbation methods. The present approach provides an essentially axiom-deduction development of the theory of radiative transfer (albeit via several ill-evaluated approximations) within the context of which various processes and their interrelationships may be investigated. Most of these processes have hitherto been studied only phenomenologically and usually piecemeal. Specific application to photon scattering, cyclotron radiation, recombination radiation, de-excitation radiation, and bremsstrahlung is made in the text. The derivation of an H-theorem for photon-particle systems is sketched; and contact is made with the usual statistical mechanical treatment of the equilibrium states of such systems. It is also shown that some aspects of collective particle behavior can be introduced quite naturally into the description of photon transport in the fully ionized plasma.

THE UNIVERSITY OF MICHIGAN 2764-5-T I. Introduction It is the purpose of this paper to present in considerable detail some of the formal aspects of "first order" photon transport theory. By "first order" we imply that an explicit calculation of the effects of specific physical processes on photon balance shall be restricted to first (non-vanishing) order perturbation theory. This (as well as some other more subtle considerations to be discussed in detail later) seems to suggest that the validity of the subsequent analysis increases as the particle densities in the systems of interest decrease and as the importance of collective (coherent) particle behavior decreases. However, at this stage, it is perhaps unwise to attempt to formulate so simple a criterion of validity, as any such attempt is apt to be too stringent. For example, the equation whose derivation and implications are the concern of the present investigation has been employed extensively in the study of radiation transport in stellar systems(l) as well as in fission reactors. Thus we shall concern ourselves very little with such questions, but rather shall present in clearly stated operational terms a set of sufficient conditions in the context of which the equation of interest is expected to be useful. Some of the material to be presented herein was initially, but sketchily, developed in an earlier work(3) (hereafter referred to as I), particularily that of Section II in which we present the basic statement of our approach to the problem and a derivation of an equation of photon balance. This inclusion of repetitive detail

THE UNIVERSITY OF MICHIGAN 2764-5-T is for the purpose of completeness as well as to illuminate some subtleties that the earlier treatment glossed over. In section III we discuss briefly and in somewhat general terms some aspects of the thermodynamics of systems of interacting particles and photons. In section IV we obtain explicit formulae for the transition probabilities (or crosssections) germane to the description of photon balance in partially or completely ionized gases in the presence of externally applied, constant, uniform magnetic fields. In particular, it will be interesting to note that the results of Drummond (4) and Rosenbluth's calculations of cyclotron radiation losses from hot plasmas are contained nicely in the present "first order"'' treatment, as well as estimates of de-excitation and electron-ion recombination radiation losses. The emission and absorption of radiation by bremsstrahlung is also accounted for in the sense of the Born approximation, as well as photon scattering - which in the present nonrelativistic treatment reduces to Thompson scattering. Finally, in section V, it is shown that some aspects of the effect of collective particle behavior upon photon transport enters the theory quite naturally when dealing with fully ionized plasmas. Specifically, it is found that - in such instances - the photons of momentum Ok propagate between successive events with phase velocity c (l+wc / 2 c k ), where p = 4r n e2/m is the usual "plasma frequency. " It is to be emphasized that none of the results of the present treatment are original though the approach to radiation transport problems as developed herein is, so far as the authors know, new.

THE UNIVERSITY OF MICHIGAN 2764-5-T II. Derivation of a Photon Balance Relation The Hamiltonian to be employed in the present investigation is the same as the one presented in (I), i. e., H = T + TP +Hpe+V+Hp + HPe+HP2 (1) where d 2 d2 1 2 T =5d x [2rc P + ( x A)] (2a) c~r ~ 8r pe 53 + e c dXAe + 2 (2c) 2 (d x Ae, 2m c 5d V = d x (x (x') (2d) Hp lT- dxA. + (Ze) 2m c 3~~~~~~~~~[

THE UNIVERSITY OF MICHIGAN 2764-5-T 0- " 0-A 2 2m c 0 T T For some subsequent purposes it will be convenient to regroup some of the terms in the Hamiltonian as follows: p1 s3 Tl + +H +HU=_ dx (I II (3a) aT e T T (3b) where we have introduced the notation e - = - it v - Ae (4) c In (2) (or 3), / is a wave operator for particles of kind T; A is the divergenceless vector potential for an externally applied electromagnetic field, and A and P are the canonically conjugate wave operators for the photon field Note that d T A = He P = 0. =....x(I ~)' II ) 34

THE UNIVERSITY OF MICHIGAN 2764-5- T For calculational purposes it is convenient to transform to momentum space for the photons according to, "7 i e-ik- x = - _x (k), (sa) V kX k! ~ ~ -ik'x - P=i k e (k), (5b) 8ircV X kk where +(k)a (k) (k (-k) +a6 ((- (-k) (6) -X X -X X -x The a; (k) and acf (k) are destruction and creation operators for photons of momentum k and polarization X, and ~ (k) (X = 1, 2) are the unit polarization vectors of the photon field. The volume of quantization is designated by V and the sum over k is the usual sum over the integers permitted by the requirement that A and P be periodic on the boundaries of V. The c. and d. operators for photons obey the commutation rules ak, X' (k') =' Al (k - k'), (7) where as the wave operators for the particles will obey the rules

THE UNIVERSITY OF MICHIGAN 2764-5-T L[o- (x), ] (x ) = S (x-x) (8) depending on whether f represents a boson or fermion field. We employ the same notation for Dirac and Kronecker deltas, letting the context reveal which interpretation of the symbol is appropriate in a given case. Again as in I, we introduce a singlet photon density according to the definition 8 2 -ix-v,1 (x, k, t)= - e - (F, Pk (k, ) F), (9) where p, (k, a) =X (k + X (k-", (10) and F is the state vector for the system which satisfies the Schroedinger Equation aF HF =i - (11) 3t The sense in which L is to be interpreted as a density function is discussed in I, as well as the interesting question as to its statistical significance. It should be noted that the present introduction of a photon density in configuration and momentum space is somewhat in contrast to previous treatments of photon distri(6, 7) butions Analogous density functions for the particles were also introduced and discussed in some detail in (I), but we eeshall not be concerned with such densities in the present investigation.

THE UNIVERSITY OF MICHIGAN 2764-5-T To find an equation for the photon density, we introduce a method of temporal coarse graining which bears some formal resemblance to that employed (8) by Mori and Ross in their development of a transport equation for short-rangeforce gases. For convenience, we first rewrite the relation (9) as 8 _ -2ix q X;(t Tr D(t) e -(k, -), (1Z) where D(t) is the density matrix for the system in the photon interaction representation. Explicitly, if F is the state vector defined by equation (11), and if G is a new representation related to F by F =UG (13) where -iTrt/t U = e (14) then D (t) =b b (15) where the b's are given by b (t) =<n IG> (16) n =' (16) <n J UF> Thus the b's are simply the coefficients of the expansion of the state vector G

THE UNIVERSITY OF MICHIGAN 2764-5-T in terms of the set of base vectors n >. This set of base vectors is to be only partially specified at this point. The set is presumed complete and orthonormal and to diagonalize T with eigenvalues ~7 i. e., T ln> nj >= (17) The explicit determination of the particle-space dependence of these eigenvectors will be accomplished variously in the subsequent development, depending upon the specific quantity to be computed. The matrix elements of p t are given by p(k, i),=<n7 |Ut(t) p (k i) U c t)) n' >. (18) PXt~a7.1~,.,.-~.r lu~i~a,*e~uru'rul n., ~>. (18) Now consider X Xt +S X (t + S) =-He r ~ - 2iT Tr D (t + s) p (?, q) (19) The time (s) dependence of p t+s may be approximated by is pU+(s) p (s)p (S) -p + TP] (20) Xt+s Xt Xt i Pt A few of the succeeding steps in the derivation of an equation for X presented here in detail were erroneously summarized in I. X 8~~~~~

THE UNIVERSITY OF MICHIGAN 2764-5-T if only a linear dependence upon the time displacement is retained. If now we rewrite D (t + s) = D (t) + D (t, s), (21) we obtain the equation, (t +s) (t) +- s -2ixe TrD(t)- T P L L gT~~~~~~ ~~~(22) 8 -- +-Z) e z TrD(t, s)pt ) V Iv _ * ignoring the term containing the factor, s D (t, s). A straightforward calculation leads to an evaluation of the second term on the r. h. s. as -2sc; sin i n - Ik, (23) which, if we neglect terms of order t in the description of photon transport, becomes simply, -so n,, f (24) where D = k/| k * (25) See Appendix I.

THE UNIVERSITY OF MICHIGAN 2764-5- T Thus we now exhibit equation (22) as +cy(. +c * B jXI ) int (26) after identifying s = X (t),s) (27) and It 3 = - Ze e i ~TrD(t, s)p (k, q). (28) Our task now is to calculate to some approximation the effect of photonparticle interactions on the time rate of change of XX. To do this we choose our quantization volume V sufficiently small that we may assume that )X is essentially constant throughout it. This assumption enables us to reinterpret Xas a mean density which may still be usefully regarded as a continuously variable function of position such that dx y =v [+V ~ +2. (29) Since the volume of integration in (29) coincides with the volume of quantization, we find that 10

THE UNIVERSITY OF MICHIGAN 2764-5-T d3x Tr D (t, s) t (k) - Tr D (t, s) t (k, 0). Vs3 4. (30) Equation (29) implies an effective upper limit on the size of V if macroscopic spatial variation in a system of specified dimensions is to be meaningfully described. Conversely, equation (30) relates a lower limit on V to the maximum wave length of the photons to be considered. This latter condition obtains because the minimum relative uncertainity (A k/k) allowable in our specification of the momenta described by X Xis essentially given by the ratio of the uncertainity in the momenta assigned to the initial and final states of the emitting particles to that of the emitted photon, i. e., A/ k/k -AKf i/k But since the emitting particle is confined to the volume V (V = L ), it follows that A K > 1/L, and thus further that A k/k > l/kL = f, i = /L. Hence if X is the maximum wavelength of the radiation to be considered max. in a given case, it would seem that the quantization volume would have to be so chosen that X /L < < 1. max. This spatial coarse-graining is somewhat reminiscent of Ono's method of quantization in cells - though far less formally executed. The present treatment is admittedly cavalier with respect to these approximations - particularily so with regard to the possibility of reconciling the opposing assumptions leading to equations (29) and (30). But it was our stated intention here to merely make the assumptions and then explore the consequent implications.

THE UNIVERSITY OF MICHIGAN 2764-5-T We note that in the representation (17) the photon number operator is diagonal, and hence the matrix pXt (k, 0) has no off-diagonal elements, i.e., PXt (' n, r n' n' i Xk nn' &p' (31) where I>k is the eigenvalue of the number operator, oa (k) a (k). Our calculation of D is elementary but a little devious. Recalling equation (15) and observing that b (t+s) =b (t) +Q (s) b (t), (32) where Q is the matrix whose elements are Q" n'= ).. \ dsl... ds v (sl). n n d=l s=0 sd0 d n (33) nd-lv >2d-, (Sd) ) nd-_l d-l, _n' 7 ~i + v (s) - Ur (s) (H-T ) U (s), (34) n, n' Yl' 1

THE UNIVERSITY OF MICHIGAN 2764-5- T we find that + + D (t, s) =DQ +QD +QDQ. (35) Inserting (35) and (31) into (30) we obtain d x -.. Vt t Int s Xk n n'' (x) D + Q D,n'' n,' I' n I n'' I n + Q Q D s/ Xk n n'' n, n" " n' n' n" ". n n''n" (36) Finally, if we assume that the off-diagonal elements of the matrices D make a contribution to the desired balance relation which is small compared to that provided by the diagonal elements, we find for equation (26), + c n V'X - - Q +1Q D Vs X k'V n7, n 7 n,n n7,n n. g +~r Q D (37)

THE UNIVERSITY OF MICHIGAN 2764-5-T An explicit calculation of the Q's through terms second order in the interactions(3) leads to the equation, X +c f. V XX k (W(1) x + - n y 7> - L S ~k nU' ma n ma (38) + (2) ) (D - D n. ma, map ma n, nj where n(1) 2 w4 - (H- T nl, mar Vt2 n, ma s ma (2)T (H-T ) (H-T) |2 -(2) _nq,r r ro-, ma W = n, ma vt2 (w -w ) n? r 2i -0 sin2 n m s (x) 2 2 (39) s (t - 2 nvt ma As usual t w = E (40) n7 nv the energy of the system when in the state characterized by the occupation numbers {5ns<).~~~~~~~~1

THE UNIVERSITY OF MICHIGAN 2764-5-T The only second order process which will be considered here is bremsstrahlung, and is considered only because it first enters at this order. Furthermore, since only those transitions in which the photon number changes can contribute to the rate of change of 7 X, we see that W will be independent of TP + HPe. Hence, for subsequent purposes, we may explicitly exhibitW(1) and W(2) as Li -W W(1) = Hp 1+Hp e vt2 s(w -w )2 nj, ma n] ma 10 -10 21 4 ny ma 1+ (|H | I W(2) = sin 2 2 -t ~1nt ma V Hp1 2 (x) L r' ra ma (41) roa t (Lin10 -10 ) The cross terms that have been ignored in W(1) vanish since HP W1 + P e have non-vanishing elements only between states in which the photon number differs by one, whereas the matrix elements of Hp are zero for such pairs of statesbeing non-zero only if the photon numbers of the pair differ by two. The transitions described by HP F 2 (which is bilinear or quadratic in the photon c. and d. operators) are essentially those which represent the scattering of photons, while those accomplished by (HP ~ 1+ HP e) (which is linear in the photon c. and d. operators) are transitions in which either one or both of the

THE UNIVERSITY OF MICHIGAN 2764-5-T initial and final particle states are bound states or both are the magnetic states of free (spinless) charged particles. Because of the dependence of the relevant matrix elements upon the mass of the particle interacting with the photons, it is clear that we may largely ignore the ions and neutrals except insofar as they provide electron scatterers for one stage in the bremsstrahlung process and centers of force in the context of which atomic bound states can be defined. To proceed further it is necessary to specify in somewhat greater detail the nature of the particle-space dependence of the base vectors, I nj>. To do this we first note that, if we neglect terms in the Hamiltonian which describe particle-photon interactions, the wave operators satisfy the equation 2 > e e r, (x),(x')d3 (ii+ ) cr + - it X o-', (42) where t=ei H't/t - i H't/t (43) (x, t) =e Vi (x)e, (43) H' being the part of H (equation (2)) that survives after setting A = P = 0. Thus to find an appropriate set of base vectors in configuration space, we look for separable solutions of (42) with a time dependence of the form e (-iE t/th). De - fining an effective potential experienced by the wth particle at x by 16

THE UNIVERSITY OF MICHIGAN 2764-5-T e e +,) T r' 1,+, I x -3x v (x)=..... d x', (44) equation (42) then becomes, (]II)2 +v 0 =E. (45) _ 2 (r C For a quantization volume sufficiently large compared to the radius of the largest orbit of any bound state terminating transitions which are expected to contribute appreciably to our balance relation, and also large compared to the radii of gyration of the majority of the electrons in our system, we may anticipate that equation (45) defines a complete, nearly orthogonal set of states corresponding to both positive and negative eigenvalues. We are, of course, concerning ourselves only with electron eigenstates, and are working in the "binary collision" limit in which we assume that not more than one ion (or electron) is interacting with a given electron at any one time. Thus the potential v is to be regarded for the purpose of computing contributions from transitions involving bound states, as simply the coulomb potential of a single ion (which for the purpose of constructing our approximate representation may be taken to be infinitely massive and at rest) and consequently the states corresponding to negative eigenvalues will be bound coulomb states with - to first order in the external magnetic field - no azimuthal degeneracy. The states corresponding to positive eigenvalues are expected

THE UNIVERSITY OF MICHIGAN 2764-5-T to be quite well approximated by (for sufficiently large positive eigenvalues at least) the usual magnetic states for electrons in a spatially uniform, temporarily constant external magnetic field. Furthermore, for sufficiently high particle kinetic energies and sufficiently weak magnetic fields, the positive eigenvalue states should be further approximatable by plane waves. The states corresponding to positive and negative eigenvalues are not expected to be truly orthogonal for a finite volume of quantization. Furthermore the overlap between two such vectors will be the greater the smaller the absolute value of their respective eigenvalues. Nevertheless it will simply be asserted that equation (45)with periodic boundary conditions provides us with a sufficiently orthogonal set of base vectors to enable us to proceed to a calculation of the transition probabilities, equation (41). In accordance with these remarks, we designate the eigenstates of equation (45) by u (x) and their corresponding eigenvalues by EK, where here K is simply o-K -K a set of labels sufficient for complete specification of each state. We then expand (x_) = ( a (K)u (S), (46) K where now a (K) is a destruction operator for a o-th type particle in the Kth state. The factor 1/ j merely symbolizes that the eigenvectors have been normalized to unity.

THE UNIVERSITY OF MICHIGAN 2764-5-T In these terms it is a straightforward matter to compute the five "first' order transition probabilities contained in equation (41). The physical processes they represent and the further assumptions employed in their calculation are: 1) Photon scattering. Relevant matrix elements are those of HP For this calculation we approximate the initial and final particle states as plane waves. 2) Emission and absorption of cyclotron radiation comes from the matrix elements of lHp 1+ Hp +e Here we approximate the particle states as electron magnetic states - ignoring the perturbing influence of the coulomb potential. 3) The emission and absorption of photo-radiation produced by electrons undergoing free-bound and bound-free transitions respectively. In this case we approximate the free particle states by plane waves and the bound states as the usual coulomb states in the absence of external fields. 4) Emission and absorption of excitation radiation produced by electrons undergoing transitions between atomic bound states. Again we approximate these states by coulomb wave functions appropriate to the instance when no external fields are present. 5) Bremsstrahlungand inverse bremsstrahlung. Here, as in the scattering case, we approximate the initial and final electron states by plane waves.

THE UNIVERSITY OF MICHIGAN 2764-5-T Performing the calculations of the quantities in equation (41) as indicated and substituting the results into equation (38), and carrying out the indicated summations we finally obtain (after replacing averages of products of particle and photon occupation numbers by products of averages)(ll) 7 +c'V * 1 = s IK'k ) (k') (k)+lJ cr KK1X' k' (x)Vf (K) +V f (K - V \X 1V Q (k) + V f (K) 1 ~+Vff (- K 1 K1 K1 K1 ++ T] T (Xk) + TK (Kk) V + (K k (k ) + I Ico-K - rTK e a- K aKK K1 (x) Vf (K) + V f (K) -V X(k) Vf (K1) +Vf (KJ +TB (Xk)K, T' K1 ( Vf (K) V f, (Kl) r'- KK K2 K3 () l +V f (K2) l+ Vf (K3 V (k) V f (K2) Vf,(K3) (X)1 ~tvf (K)J V (47)f( ] See Appendix II.

THE UNIVERSITY OF MICHIGAN 2764-5-T This is the balance relation sought. The quantities S,,, T, T, and T c r e B are transition probabilities per unit time for scattering, cyclotron emission, electron-ion recombination emission, de-excitation emission, and bremsstrahlung respectively. The scattering matrix is characterized by the symmetry property, S- K, Xk = K,' k'(48) a-K1, X'k' arK, Xk whereas the emission probabilities transform to the corresponding absorption probabilities under interchange of particle coordinates. As will be seen explicitly, all transition probabilities guarantee appropriate conservation of energy and momentum. The quantities f (K) are the particle analogues of )X, e.g., f~ (K) 3 is the expected number of particles per cm in the volume V having momentum t K at time t. The plus or minus signs arise because the a- type particles may be either bosons or fermions. (In most of the succeeding discussion we shall assume Boltzmann statistics for the particles, i. e., assume that states to which transitions go are sufficiently improbably occupied that we may neglect V f (K) compared to one. However, for the time being, and for most of the next section, we retain the quantum statistics as indicated.) The specific formulae for the transition probabilities occurring in equation (47), computed to the level of approximation discussed above, are:

THE UNIVERSITY OF MICHIGAN 2764-5-T so K, Xk = - (k) ~ (k') crK', X'k' 2 3, X X (x) (k'+ K' - k - K) (w Kk-wK') (49a) 2 2 K1 4xc / e o 4 74c e T K (k)c= co- VV m c k -ik-x a- 2 (x) k < K| e IIK (4K>9b) 4ir e2 tc K1 K ~K1 __T' (Xk)=T(k 2 2, rk K e aK mc m k a- a-,I -ik x 2 (x) <<K l e - (> (x) W (t - o 2(49c) K,k K 22

THE UNIVERSITY OF MICHIGAN 2764-5-T 2 - 2 2 2 4 oK2,oa'K 7r e e c T (Xk) 3 =G____ T ( K!= 4 2 B rK Ka'K 4V4 m k 1~2 E~ (k) ~ K2 (x U(i K1 -K31 W +L -10 -W K K1 K K2+ k + (K+K1-K2 K3-k) 6(w -W ), (49d) 1 ~ K2K k KK1 W +W -W -W_ 3 K2 K K1 K-k where the momentum S's are Kronecker a's, but 6(w) is a Dirac delta arising from the identification sin - 7r 2 (w)(50) SW 2 In (49c) we have lumped the formulae for T (recombination emission) and T r e (de-excitation emission) together, since they differ only in the selection of the states K > and jK1 > for the final step in the calculation of the emission and absorption coefficients conventionally employed in descriptions of radiative transfer. The fourier transform of the coulomb potential is defined by 3 dR iK- R u( IKI)= d e - 23

THE UNIVERSITY OF MICHIGAN 2764-5-T III. Some Aspects of Equilibrium Before proceeding to the final reduction of equation (47) to the form commonly employed in the description of problems in radiative transfer, it is convenient to digress briefly for a discussion of some of the anticipated implications of the present analysis for equilibrium systems. Though most of these implications are perhaps obvious from the form of the equation itself (and in fact are generally well known), it nevertheless seems to us of some interest to point them out in the present context which is not quite the usual one. Actually not all of these implications are completely obvious from the form of equation (47) itself. The one that is obvious is the fact that this equation admits steady state solutions appropriate to the description of the equilibrium state. But the further fact that the non-equilibrium system is in some sense driven irreversibly to that state in which the densities assume their conventional form is not obvious from equation (47). One needs an H-theorem deducible from (47), but this is not possible since it describes only the behavior of the photon density in terms of the particle densities. In order to deduce an H-theorem in terms of the densities directly, it would be necessary to have at hand the equations for the particle densities (I) completed to account for all processes to the same order of approximation as they are in the photon equation (47)(12) However, as we are not primarily concerned with the description of the particle densities in this paper, we shall base our discussion of an H-theorem in the present context upon an earlier phase of the analysis. 24

THE UNIVERSITY OF MICHIGAN 2764-5-T A cursory reappraisal of the argument leading to equation (38) reveals that it contains the more basic equation D = L W | Dm -D, (51) na, n m n ma l a nmV, mn where W =W (52) nj, ma ma, n Recalling that the diagonal elements of the density matrix, D, have the interpretation of the probability of finding the system in a state characterized by a particular set of occupation numbers, it is convenient to introduce a notation which emphasises this interpretation; so we define P(n t) =Dn7 (t) (53) and rewrite (51) as P (n rl, t) = W P (m a, t) - P (n t) (54) ma (13) This equation and its implications have long been well known, so we merely sketch the succeeding argument. We first define a function H by 25

THE UNIVERSITY OF MICHIGAN 2764-5-T H= P (n?) n P (n). (55) n It is then readily shown that dH -< 0, (56) dt the equality holding only when P (mc) = P (n), (57) all m, ac, n, and ). The monotonicity of the time derivative of H suggests that we may tentatively interpret it as closely related to the entropy of the system, hence we identify s=- K H, (58) where K is a constant of dimension ergs / OK. It then follows that we should interpret the state for which dS -= 0 S a maximum, (59) dt as the equilibrium state. A solution of (57) which immediately suggests itself by virtue of the energy conservation condition contained in W is * The interpretation as an entropy of a functional of the distribution functions whose time derivative is always zero seems somewhat inappropriate(14) 26

THE UNIVERSITY OF MICHIGAN 2764-5-T P (nv) =P(E n ). (60) A further condition on the solution if it is to describe the thermodynamic state of weakly interacting systems is; if E E +E n~ n then P (E +E ) =P(E ) P (E ). (61) A solution to the functional equation (61) is, of course, P (E) =Ce, (62) where C is to be determined by the requirement that P be a probability. Since P must be the same for both the photon and particle systems-and is the only macroscopic parameter they share - it follows that it must be related to the temperature, and is in fact 1/ K T One now readily establishes that the equilibrium photon and particle densities are: X (k)?= k=, a and f (K):=L nK P (n) E(63) BePEK +1 27

THE UNIVERSITY OF MICHIGAN 2764-5-T It is also now readily established that the function identified above as the system entropy becomes, in the equilibrium state, the sum of the photon and particle entropies respectively; and that the functional dependence of these partial entropies upon other thermodynamic variables is indeed that conventionally deduced by statistical mechanical arguments(15) Finally, employing the distribution functions (63), one easily shows that equation (47) is satisfied. One of the principal reasons for presenting the relatively familiar detail of this section is to emphasize the fact that this detail and these results obtain naturally in the context of a description of systems with many degrees of freedom which introduces no specifically statistical considerations other than those (16) inherent in the axioms of quantum mechanics themselves 28

THE UNIVERSITY OF MICHIGAN 2764-5-T IV. Some Applications of the Balance Relation Although the processes influencing photon transport are well known and, to the order of approximation characterizing the present analysis, have been more or less thoroughly investigated, equation (47) is not in a form that is easily recognized. Thus in this section, the photon balance equation will be reduced to a more familiar form. The processes contributing to the scattering, emission, and absorption of radiation, enumerated in Section II, will be discussed in somewhat greater detail; and the corresponding transition probabilities will be reduced, when feasible, to forms that have already found useful application. To initiate this reduction we now explicitly assume that the number of occupied particle states in any given energy range is small compared to the actual number of states in the same range. The effect of this assumption is to exclude from present consideration all systems characterized by particle degeneracy, and leads to a description of the particle densities in terms of Boltzmann statistics. We then go to the continuum in photon momentum space by defining L ~R = 7 dkdf (k), (64a) k d k and Pk dkRVk dkdn(k) 1 = Pkdkdfl(k) = (64b) k d3k (Zor) 29

THE UNIVERSITY OF MICHIGAN 2764-5-T Furthermore, for the treatment of photon scattering - for which free-particle states are employed to describe the particles before and after collision - it is also convenient to go to the continuum in particle momentum space, i. e., f f d K (64c) 3 K~d K Then equation (47) can be written as xcv ~ xxx - X P +c- p =(s +c +( +E+ k _ A X i c r e B A -(s + +a + A + ) +a (65) o c r e B A where the reaction rates for scattering (s), emission (E), and absorption (a) are now given by: X 3 3 3 K, Xk(66a) s = d Kd Kldk'dn' V X (k') fK1, (66a) i - Kl, X'k' X T' K, K,. k' AL 3 3 3 TK, Xk Pk = d Kd Kldk'da' V S k' + f (K)' Tj, X', k' V (66b) 30

THE UNIVERSITY OF MICHIGAN 2764-5-T = V T (Xk) f (K), (66c) c, r e V c, r,e K KK1 a =- L V T (Xk) f (K1), (66d) cre c, r, e K KK1 X V T (Xk) K f (K) f () (66e) B B T-K,cr'K1 a- - C ar' K K K2K3 x 3 a-K2, a'K3 3 (KL (K a = L V T (Lk), 3 f ( _K2) f I (K (66f) B B a-K,a-'K! a r c' K KK1 K2 K3 These reaction rates are, in general, complicated functions of the photon wave vector, photon polarization, position, and time - the space and time dependence arising through the dependence of )X and f on space and time. They represent total transition probabilities per unit time for transitions between all possible initial and final states such that a photon of momentum l k and polarization X is either gained or lost. The quantities s. are the "scattering in" and "scattering 1, 0 out" transition probabilities, whereas the E's and a's are the corresponding probabilities for emission and absorption respectively. The omission of the sum over the particle index for the cyclotron, recombination, and de-excitation radiation reaction rates is in accordance with the earlier discussion, in which it was indicated that essentially only electron transitions are important. 31

THE UNIVERSITY OF MICHIGAN 2764-5-T All of the emission and absorption processes enter the present analysis in the same fundamental way. In excitation and de-excitation for example, the photon field and the atom constitute two weakly coupled systems. The interaction between them causes an electronic transition from one atomic state to another accompanied by the emission or absorption of a photon. Electronic transitions leading to emission may proceed either spontaneously (at a rate independent of the presence or absence of photons) or at a rate proportional to the number of photons present (induced emission). The absorption rate is, of course, always proportional to the number of photons present. The other processes are described here in basically the same terms - cyclotron emission or absorption resulting from electronic transitions between unperturbed magnetic states, whereas bremsstrahlungand inverse bremsstrahlung are radiative free-free electronic transitions occurring in the field of another charged particle. Because of the present nonrelativistic treatment of the particles, equation (65) should probably be restricted to a description of systems in which themean particle energies are not expected to much exceed 50 kev. Under such circumstances, pair creation and annhilation should not contribute appreciably to the photon balance, and hence the necessary absence of a description of such processes in equation (65) should lead to neglegible error. The scattering process is not of particular interest when dealing with non-relativistic systems because the scattering rate (and corresponding crosssection) is small in comparison with that of some of the absorption and emission 32

THE UNIVERSITY OF MICHIGAN 2764-5-T processes. Since the cross-section for non-relativistic photon scattering is roughly proportional to the square of the radius of the electron, it is seen that, 18 3 even for free electron densities of order 10 per cm, the scattering mean-free6 path is of order 10 cms. Such a process can hardly be expected to significantly influence photon distributions in laboratory-scale systems. Thus, in investigations of radiative transfer in this energy range, scattering rates which would be characteristically dependent upon the photon densities in both the initial and final collision states are not usually given any consideration. Conversely, for the treatment of the problem of shielding high energy gamma rays from a nuclear reactor, the scattering process does become a significant competitor with other relevant photon reactions. This is due in large part to the fact that the interactions of such high energy photons with the electrons in the atoms in such systems may be satisfactorily treated as if the electrons were free, i. e., as if all such reactions are describable as Compton scattering. Consequently there is an enormous increase in the effective density of scatterers, leading to scattering mean-free-paths of the order of centimeters or less. However, in the description of scattering in these instances, the dependence of the scattering rates upon photon densities in post-collision states is always ignored(2) This is justified, of c ourse, because in these far-from-photon-equilibrium systems, the photon densities are always very small when compared to the densities of available states. 33

THE UNIVERSITY OF MICHIGAN 2764-5-T In the absence of an external magnetic field, information about photon distributions in particular polarization states is no longer significant. In fact, in such instances, it is reasonable to assume random polarization for the photons in which case, )X (k) =1 (k). Then equation (65) becomes 2XcB 2p 6f+ &[+ (s +a +a +a' + (i r e B 0 r e B V (67) where we have defined s = L i, 17 A 34 r e, B r, eB 1 r, e, B Zr, e, B The scattering rates may now be rewritten somewhat more explicitly as, 34

THE UNIVERSITY OF MICHIGAN 2764-5-T ~3 s 3 3 C r0=47r d Kd Kdk' df2' T ctc ~(k+K-k' -K1) kk' L J KKl k' i2 K 2 K2 K1 - (x) ( ick+ -lick' - ) (k) f (K) (68a) 2m Zm (8a) 3_ 3 3 C s =47r d Kd Kldk'dfl T tc (k kk' KKlk' ZK 22 K2 +K-k' - K1) (ck+ -ck' - ) 2m 2m (X) $k') + (k- r f (K), (68b) V I c where we have introduced the Thomson cross-section, 2 - ( ) 1+ (k' k') T T Equation (67) may now be readily reduced further to a form familiar in reactor (2) shielding studies. According to the above remarks about the ratio of photon densities to available state densities in these systems, we may drop ( (k) whenever it is compared with the corresponding density of states, Pk/ V. We then define a macroscopic linear absorption coefficient x by 35

THE UNIVERSITY OF MICHIGAN 2764-5-T where "scattering out" is considered as an effective absorption. In this sense, p represents a probability per unit path for small paths for the loss of a photon of momentum tk. Equation (67) may now be written as 1 _- Y~WX+FX= ( + + C VX c r e B + D dk d S n C((') o (k', f'; k,a), (69) where 3 3 f (K) tick -- - r e B c d Kd K1 k T (x) (k+K - k'-K1) (tick+ -ick' - ) Zm Zm When finally cognizance is taken of the fact that atomic de-excitation radiation is 36

THE UNIVERSITY OF MICHIGAN 2764-5-T considered to be of negligible importance in reactor shielding situations, and the high energy photon transport equation is reduced to non-relativistic form, it is seen that equation (69) is essentially the same as the one employed by Goldstein() For the remainder of this section we shall be primarily concerned with low energy plasma systems near kinetic equilibrium for which scattering can be neglected. Returning to equation (65) (without scattering), we accomplish a reduction to a form conventional in the discussion of low energy radiative transfer by defining a source function, J (x, k, t); an "effective" absorption coefficient, Xe ac (x, k, t); and a radiation intensity, I(x, k, t) such that: k m, (70a) V m =c, r, e, B m=c, r, e, B \e i X X Xe m m (70b) c m =c, r, e, B and I = itw c X (70c) (1) Then radiation transport is described by the familiar expression t- I +.O V I = J: -a I (71) c X - X 37

THE UNIVERSITY OF MICHIGAN 2764-5-T The effective absorption coefficient a is a probability per unit path for small paths for energy loss by "netr absorption. The qualifications "effective" and/ or "netr absorption imply a difference between the absorption and induced emission processes. The source function is essentially the rate of spontaneous emission 3 of energy per cm per unit k (frequency) per unit solid angle. When a strong magnetic field is present, cyclotron radiation can cause a significant energy loss from a plasma system. For fully ionized plasmas it may even be the dominant mechanism for radiant energy loss. When this situation obtains, equation (71) becomes 1. )Xe - I + ~s' I =M - X_ X I (72) c X -v, ~ c c A We note that the source function j is completely specified when L is specic C fied, and that 6 is known (at least formally) when the electron distribution c function is known and appropriate single-electron wave functions (previously discussed) are chosen. X (17) * To evaluate ~, we choose (following Parzen ) a coordinate system in which the external magnetic field is along the -axis, and the photon propagation Note that althougne of his approximations was not valid for the extreme relativistic case,Parzen's analysis is quite accurate for our problem. Also note that in his equation (26), k should be replaced by R. 38

THE UNIVERSITY OF MICHIGAN 2764-5-T vector k lies in the x - z plane. We specify the polarization vectors by the usual spherical base vectors in the polar and azimuthal directions. The calculation K1 of TK (Xk) then follows directly from the work of Parzen, after replacing cK his K and f by k and Y = vl /c respectively - except that we have allowed arbitrary electron momenta in the Z-direction, rather than restricting it to be zero. The results are 47r e 2 K, 2 V T (c, k) = 4 (mvL) &( k -K) ic khkl 2 (x) (K -k -K1) e - J' (n sin ) (73a) -K z z thw " n for the azimuthally polarized radiation, and 2 2 2 K1 4we V T (0, k) = (mvy) S(w - ) c K m2 ick K1k K i0 mv~ tK sine 0 J (n sin 0) Z tan 0 _ (73b) mc J nsin e Appendix III. 39

THE UNIVERSITY OF MICHIGAN 2764-5-T for the photons polarized in the direction of the polar unit vector. We have introduced the notation o to represent the electron gyromagnetic frequency. The symbol n occurring in equations (73a, b) is an integer equal to the difference between the radial energy quantum numbers characterizing the initial and final electron magnetic states appropriate to the transition under consideration. The quantity I kl / m vL occurring in the exponentials is the ratio of the perpendicular (to the magnetic field) component of the photon momentum to the corresponding component of the electron momentum. For frequencies of interest, this quantity is so small that for all practical purposes these exponentials may be replaced by unity. Energy and momentum conservation require that t k nw =ck-v k + (74) o z Z 2mx Direct substitution of equations (73a, b) into the definition (66c) of ~ gives, ~~n L~~~~~~ PL COS 0 ik cos 0 (x) i|nw ck (1 - + z), (75a) me 2mc 40

THE UNIVERSITY OF MICHIGAN 2764-5-T...=. d pf (p) P~ J - c m tick dpf()pm n 2 pL Coso ik cos cos (x) (pZ sin 0 - m c cot ) n c k (1 - + (75b) mc 2minc where we have converted to the continuum in momentum space for the description of the pre-transition particle distributions, f (p). Xe The evaluation of ac is also of great importance. This parameter (4, 19) has been calculated in various ways by various authors. It is seen from equations (66c, d) and (70b) that it can be written as Xe 1 2 K1 aXe =+ U V T (Xk) f (Ki) - f(K) (76) c cK - K, K1 This expression for the effective absorption coefficient is quite general. Kirchoff's law - consisting of a relation between a and ~ - can be developed at this c c point if we now assume local equilibrium for the particles, eg., take ( D)3 / 2 KZ / a m f (K) = n e K/ (77) 2mr where n is the (generally space and time dependent) particle density in configuration space, and f3 = 1/1 T is also permitted an arbitrary space-time dependence.

THE UNIVERSITY OF MICHIGAN 2764-5-T K1 Since T (Xk) conserves energy, f(K1) may be expressed as cK f(K1) =f(K) e with the consequences that X X Ptck Xe 1 tick X a = e, and a (e -1) (78) C C C c Observing that jX is proportional to E, equation (71) assumes the form, C C 1 *1Xe - I +fl I =-a (I I ), (79) c X c X 2 BB where [ hw/8r3 c2] dwo II do = Li. - - =ck BB es.... Thus again (see Section III) we have arrived at an expression which provides us with an equilibrium solution for the photons, namely IX independent of space and time and equal to IBB The condition for the thermodynamic solution can 2 BB also be rephrased as.X Xe 1 2 BB 42

THE UNIVERSITY OF MICHIGAN 2764-5-T which is a statement of Kirchoff's law for radiation of polarization X. When 3 tw << 1 (which is the situation discussed in references 4 and 19), the Rayleigh-Jeans approximation to the block body distribution is valid. Then equation (79), for the steady state, takes the form used in these analyses, i. e., ke L *s;V I =-CXe(I I ). (80) X c X 2 RJ Furthermore, in this instance, the effective absorption coefficients can be approximated from equations (75 a, b) and (78) as, 2 2 2 fe 47r e [3 7(' 3 ) 2 [ np, sinO a /. dp f() p' ( c 2L n m e me n p! cos 0 tk cos 0 (x) Xrnw -ck(l- + Z ) (81a) m me 2mc 0e 4r2 e21, 3 2r np. sin0 ac = f(p) Pt m J n 2 p cos 0 k cos0 (x) (p sin 0 - mccot0) nw -ck (1- +. (81b) mc 2mc Setting m = c = 1 and restricting attention to radiation proceeding nearly 43

THE UNIVERSITY OF MICHIGAN 2764-5-T perpendicularily to the magnetic field (0 %i/2), it is seen that (81a) becomes the non-relativistic limit of the effective absorption coefficient obtained by Drummond (4) and Rosenbluth. For 0 = 0, the absorption coefficients for the different polarizations vanish for all transitions except those between the ground and first excited states - and are equal for these transitions. However, the mean free path for absorption when 0 = 0 is much greater than for absorption at 0 =. (Observe 2 that lim — 0 ). Thus it is reasonable to expect that, for systems for 0 —7r/2 c which all the linear dimensions are of the same order of magnitude, the radiation loss parallel to the magnetic field will be only a small percentage of the total radiation loss. This seems to be a reasonable inference to be drawn from the calculations of DR for the infinite slab, which also indicate that the bulk of the radiation is emitted into an angular interval for which ac < ac so that radiac c tion into the 0-polarization can probably be neglected entirely. (20) It is shown by Berman that for a hydrogen plasma with no magnetic field and a kinetic temperature from 3 ev to 200 ev, radiative recombination is the dominant energy emission process. The calculation of the emission coefficient for this case in the present context can proceed in a rather general way by choosing for the electron and atomic wave functions, iK x IK> e - - IK1>= n (82) 44

THE UNIVERSITY OF MICHIGAN 2764-5-T where n represents a sufficient set of labels to completely specify each atomic state. Since we have chosen a quantization volume such that only one ion is present, the ion density in the system is simply given by nI = 1/V. After converting to the continuum for the electron's initial momenta and summing over photon polarizations, we obtain for the recombination radiation emission coefficient 2 e ic 1 3 Cr I 2 me m k Xn 2 (x) ( K) (L -KiK (83) where ( ) = ~ (x) e d x n " n - and If = k - K. It is now a straightforward matter to obtain the results presented (21) by Heitler for example for transitions to the K-shell; or alternatively those presented by Bethe and Salpeter(2) for transitions into higher states.* The remarks about the effective absorption coefficient for cyclotron radiation are also applicable to radiative recombination. Calculations of See Appendix IV. 45

THE UNIVERSITY OF MICHIGAN 2764-5-T recombination radiation from a plasma have been performed by Berman() and (23) Kogan. For low energy hydrogen plasmas (less than 3 e v), or for higher energy (24) plasmas containing atoms of higher charge number, de-excitation radiation can be a serious energy loss mechanism. The transition probability T 1 (Xk) eoK can be put in a more convenient form for calculation in the dipole approximation by the elimination of the gradient operator from the matrix element. It is observed that (for the dipole approximation only), e < n HP j l ma > =_ < n7 | I H |ma > (84) and thus, instead of (49b), we may write 2 2 4r e m c T (X 3k) 2 (k ( s g ecrK ti V m c k 2 (x) | 6 < K,|x| K > (85) Equation (85) is related to the corresponding expression for the rate of spontaneous (21) de-excitation presented by Heitler (page 178, equation 10) by 46

THE UNIVERSITY OF MICHIGAN 2764-5-T 2p dKn e (ck)3 dfl 2 Kd Pk k 2 LwdAl=V T K1 (Xk k < K1 |x| K > cos 0, eo-K cV 2irtic3 (86) where 0 is the angle between the direction of polarization and the vector, x. If we now sum over polarization and integrate over angles, we obtain the transition probability A given by Berman; reference 20, equation (2-1). For further K, K calculations applied to a hydrogen plasma, see reference 20. The last radiation mechanism which we will consider is bremsstrahlung. The calculation of fB for electron-ion and electron-electron bremsstrahlung proceeds straightforwardly from equations (49d) and (66e). However, non-relativistic electron-electron bremsstrahlung contributes negligibly when compared with the electron-ion radiative collisions,(5) and hence shall be given no explicit consideration here. From the relations (49d), (66e), and (70a) we find that the source function in this instance may be written as Bdw I i vf (E, f) dEdE C d o dwdfl2, (87) E 2 after assuming that the scattering ions are at rest before collision. The crosssection TB dw d n2 is the one given by Heitler for non-relativistic electronion bremsstrahlung. Of course, equation (87) also effectively provides us with 47

THE UNIVERSITY OF MICHIGAN 2764-5-T the emission coefficientB (recall (70a) ). Hence if we again assume kinetic equilibrium for the particles, we may easily obtain the absorption coefficient ca according to equation (78). However, for most laboratory situations the B (26) large bremsstrahlung mean-free-paths imply that the photon densities will be exceedingly low (provided that bremsstrahlung is the principal emission mechanism). Hence,e a I/j <<1, (88) B B and consequently the rate of loss of radiant energy from such systems is essentially given by the rate of emission, jB. Extensive calculations of this emission rate have been carried out by Kvasnica, and an investigation of the range + of validity of the assumption in equation (88) is presented in reference (26). See Appendix V. Included among these calculations were rates of electron-electron bremsstrahlung as well. 48

THE UNIVERSITY OF MICHIGAN 2764-5-T V. First Order Collective Effects on Photon Transport in the Fully Ionized Plasma In the preceeding sections we have developed a description of photon transport which implicitly assumes that the photons travel with speed c between successive events. (This assumption is realized explicitly in the form of the transport term, crz*v fk). The assumption slipped into the analysis through the choice of the transformation operator (14) which defined the interaction representation (13). However, a brief reappraisal of the discussion in Section II reveals that a certain amount of the information available about the system in the Hamiltonian (2) has simply been discarded. It is our purpose in this section to show that this information can be exploited with but trivial modification of the preceeding analysis to enrich the treatment of radiative transfer in the fully ionized plasma. The point is that in the term Hp Y 2(equation (2g) ) in the Hamiltonian for the system there is a part that describes simply an energy level shift for the photons in the medium, as well as other parts describing interactions between photons and particles leading to changes in the states of the photons. Only the latter parts of this interaction term were employed in the calculation of the influence of transitions upon the rate of change of the photon distribution function. To incorporate the effect of the former part, one need only add it to the energy of the "free" photon in the definition of the unitary transformation taking us to an appropriate interaction representation. Since it is this unitary transformation that describes how the photons propagate between events, we will then find that the phase velocity 49

THE UNIVERSITY OF MICHIGAN 2764-5-T of photons of momentum tk is modified to be c (1 l+t2 / 2c2k ), where wo is the p p usual plasma frequency. The group velocity which enters into the transport term will be correspondingly modified to be c (1 - / 2 c k ) p For an explicit realization of the content of these remarks, we rewrite HP 2 in momentum space as HP( 2 =7_t L e a a (K) (K) (k) (-k) cV mT k XkKo- t7r r e2 a (K) a (K (k) I * (-k') (k+K - k' -K + 7 (90) cV m \Ikk' XkKT kX'k'K' where the prime on the second summation implies that the terms for which = X', k=k', and K=K' are to be deleted. These latter terms are just the ones that have been employed in the discussion of photon scattering and hence shall be largely ignored in the following. Recalling equation (6), it is seen that the terms in the first summation in (90) include some that are proportional to the photon number operator. These we single out for special consideration and designate them as 2 + + H 2irh t e a (K) a (K) aX (k)a (k) =T -,. (91) cV In k XkK o-50 50

THE UNIVERSITY OF MICHIGAN 2764-5-T Now define a transformation to an interaction representation by F =UG, (92) where now U =e-i (T ~+ HoP f 2) t/. (93) Proceeding as in section II, we find that equation (47) is reproduced with the following two modifications: (1) The transport term is altered from c z ~ v'X to a X a 2r e A k + (F, a (K) a (K) F(94) a x. ak. ckV m where we have employed the approximation + (F,a a a p F) (F, aa F) (F, p F). (95) (2) The energies of the "free" photons - defined as the eigenvalues of T~ + HP 2 in the representation that diagonalizes the number operators 0 for both particles and photons - are 2r 1 e + i2ck ( nK) Auk' (96) (ck) m Xk Kc- a

THE UNIVERSITY OF MICHIGAN 2764-5-T This shift in the photon energies requires a corresponding modification in the energy conserving delta functions contained in the various transition probabilities in equation (47). The transport term, equation (94), can be expressed in a more interpretable form. First note that 1 \ (F, a (K) a (K) F) =N (97) Na'- 0''(97) the expected density of particles of kind a- in the quantization volume V. Then, ignoring terms proportional to the ratio of the electron mass to the ion mass, one obtains aX a -n X _ (ck+ 2 / ck), (98) J ak P where we have introduced the notation o = 47r N e /m. This suggests the assignp e ment to photons of momentum tk a frequency o = ck + o2 / 2ck, a phase velocity o/k =v p p c (l+ L2 /2c2k2), and a group velocity awl/ak =v c (l w-2/cz2kZ) In these p g p terms, transport is described by -v - V (99) and we see that photons whose momenta are such that o / 2 c2k2/' 1 do not propagate through the plasma. This is substantially the same conclusion with regard to "first order" effects of collective particle behavior on electromagnetic wave 52

THE UNIVERSITY OF MICHIGAN 2764-5-T propagation in plasmas as is drawn from conventional macroscopic electrodyna(28) mics. The restriction of the remarks in this section to the fully ionized gas is a consequence of the fact that in systems in which electron bound states are important, considerable modification of, say, the energies (equation (96) ) is to be expected. Further investigation into the implications of the modifications (1 and 2) for transport processes will not be entered into here. A consequence of the present description of "non-interacting" photons for the equilibrium state is of some interest. Recall that in section III, the thermodynamic state was presumed characterized by the canonical distribution, -P (Tp +Tf ) P= C e (100) In the present instance, however, this distribution should be generalized to _~ (TP + T- + HP'2 P =Ce 0 (101) It is readily shown that the deinsity matrix (101) leads to particle densities the same as in equation (63). However, the photon density is altered to become X (k) = -(102) where = cck (1 + 7 2 n K). (103) C K 53

THE UNIVERSITY OF MICHIGAN 2764-5-T Since in this instance we are considering a large, spatially uniform system, the quantization volume V may comprise the whole system and In K =N (104) V K the density of particles of the a-t kind. Thus (102) may be written as'X=(k) P [ehc(l+WP/ 212k (105) Perhaps the most significant aspect of this modified thermal radiation spectrum is the prediction of the rapid decrease in the expected number of photons with momenta such that L2 / 2c2 k >>1. p 54

THE UNIVERSITY OF MICHIGAN 2756-3-T Appendix I A Theorem On Transport A general theorem concerned with the transport term of the rate equation can be stated: If = () a (k) a (k) Xk 8 Then - e - q (F, Ho,p (k, I j F) v 0 L X - -- (F, O (Xk) sin k P (k, F) e (I-) where PA( a) A (k+) A (k- q and O (Xk) represents an operator containing factors which commute with the a operators. Utilizing the commutation relation [ac(k), a (k')J = SxAA (k-k') we obtain o (X'k') [A' (k') aX (k ), X (ak + A'(k - k'X' Xk' 55

THE UNIVERSITY OF MICHIGAN 2756-3-T = [o(X k + )O(X, 1pk, ) (1-2) The 1.h. s. of equation (I-1) thus becomes 8 i - E e" 41i -(F, (xIk o, k+q. P) (k-(, ~k)Q,)F) v m - (1-3) y?k' V 7k' 8 e- 2ix _) (F, O (kk) P (k ) F)e _h v 2f 2 8 (4 - )r _(F, O (kk) sink F e (1-4) where the exponential operators are defined by their series expansion. Two cases of interest are H =T ~ and H = T +H. For H =T o o o o O (Xk) = ick and equation (I-4) reduces to equation (23). When we use H = T 0 + Hp we obtain, in the classical limit, equation (94). 56

THE UNIVERSITY OF MICHIGAN 2756-3-T Appendix II Derivation of the Cyclotron Radiation Term in Equation (47) from Equation (38). As pointed out in section 3, the cyclotron radiation results from that part of equation (38) which contains H + HP. Ignoring scattering and bremsstrahlung, equation (38) becomes c X.V- k cWi mc (D -D ) (II-1) X -:QXJk nv ma ma, ma n7, nl rn ma where (1) )(H + H e) 2 (II-2) ny, ma V 2 nr ma ne, mca sin2( no ma s) and we have replaced 2 by (w It has been s (W - ) 2 nv2 ma nv ma ob s erv e d in section 2 that a diagonal element of the density matrix D is interpretable as the probability of finding the system in the state characterized by the occupation numbers In a > at time t. Consequently, we introduce P (n, t) (t) = D and note that n7,n? V'~)t(k) = X rTk P (n9, t) n7 Thus, (=) P (n Y, t) n? 57

THE UNIVERSITY OF MICHIGAN 2764-5-T with P(ny, t) = w, ma(P (ma, t) -P (ny, t). (II-3)'n n-~, ma' m a Instead of performing this sum algebraically, we appeal to the physical process, and recognize that the probability of the system being in the state characterized by occupation numbers InKn K1 k > is affected in the following way: 1) P (nTK n K Ok) is increased by: a) A photon absorption when the initial state has occupation numbers n -K, 7#n +1 >. b) A photon emission when the initial state has occupation numbers n|K -1 nTK +1 - 1Ak > 2) P (n,K ncK VBEK) is decreased by the reverse processes. Now H() = HP + can easily be put in the form ~~(1) ie 2 hrc + H a (K1) a_ (K) (11-4) a- XkKK1 mc Vk - x )[a+()X 1k+(k ((k) s 3 -ikx +aW (x) + (k) At (k)W + a! (-k) G d x e uaK IIT u (11-4) K magnetic states discussed in Appendix III. The sum over k is over both positive and negative values, so that when (-k) appears it can be replaced everywhere by (k). 58

THE UNIVERSITY OF MICHIGAN 2756-3-T Equation (II-2) can now be written W (1) = T 1 (Xk') <n a(k) a + (K1) a (K) ma >l (II-) nj, ma c T K + or T caK (X'k') <n (k a k)a (Kj) a (K) Ima> where the first matrix element represents photon emission and the second matrix element represents absorption, and where 2 2 - K14 T (X'k') = (w cTK V 2 (mc K k (x) - e -*l II' K> (II-a) 0= K, L t Eqyuation (11-1) can now be reduced to int nr ma T 1Xk T ()Xk') (i=nK) n K 1 cKK1~X'k' Ink (x)P (n aK -1, )' 59

THE UNIVERSITY OF MICHIGAN 2756-3-T F+ T;k T rKT ('k') (1 tn )n +)P(n -1 +1) f ~Xk ~c aoK crK a-K1 X(k' cTK TK X nn Tr KK1X'k' k c aT K (X'k') nK (1+n) P (K "-K1? X'k' n cTKK1X'k' - TK (X'k) n (l+nK a (, ( 7) II kk E TcaT-K - rK -K ( k 1) (n K K )'?k nr cTKK1X'k' where the positive sign is appropriate for bosons and the negative sign is for fermions. When the sums are broken into two parts, (X'k' = Xk and X'k' f Xk) and the indices are appropriately shifted in the first two terms, we get 21 ca-K a-K a t T (K k) n (n +1) P(n n K k) ca TK - K arK1'k K,rK' c K, K1 nq TK (kk) n (n +1) ( +k )(n K +1) - + 1)n X t ca-K! Lkk a-K a-1 c KKln7 (x) P (naK n K1 kk) (11-8) 60

THE UNIVERSITY OF MICHIGAN 2756-3-T If we approximate the average of the products by the product of averages, we obtain atX t = Z TcK1 (Xk) [(k) +j V f (K)+V f (13 at int TKK1 - V X (k) V f K f (3 (11-9) The minus sign where (+) appears would result from a rederivation with anticommutation rules.

THE UNIVERSITY OF MICHIGAN 2764-5 -T Appendix III Reduction of Equation (49 b) to Equations (73 a), (73 b). We first rewrite equation (49 b) in the form VTcrK(X_) = Kmc ) k 8(wKlk-WK) ('me ik-E z1 (k X II+ + Ek xII_+ Ek;IIz (III-1) where (10) (x) <' miKz e |K>= IjRKZ> =uj0(Pkl <CY p2 -1) - 1 (oap2) (j ei(j -) 0+ iKz Li (a2) with a =-, L 2 being the box normalization length, and where Lq 1 (a 2) is the associated Laguerre polynomial. We have defined II_ = IIx +iIIy and - = EkX i kX The operators IIT are creation and annihilation operators with the property IIujj= imTo0 b 2(j +1) uj+l1..I_ (III-2) IIT ujj= -im0ow b 2J2j uj-1, R where b i e When the k vector is oriented in the x -z plane with the

THE UNIVERSITY OF MICHIGAN 2764-5-T magnetic field parallel to the z axis, V Tc -K (kk) m ) k - (W~ -Kk K) -x | 11+ g b(Kz -kz -Klz) I (ijl j1 ) + 2 m~w E bkZ(j+l) S(Kz -k -Klz) I(j''j +1, ) + Ek Tin S(Kz-kz -Klz) I(j''i j) -k,X Zwhere I(jR' j) = <j'?' e ipcos 0| j > (III-3) and S(Kz-kz-Kiz) results from the z integration. Taking = 0 (the "wellcentered orbit" approximation(l7 {which is valid to first order, (39) it is seen from Parzen that for cases of interest in plasmas only transitions between different j states are important. Thus, letting n = j' -j, we obtain(l7) I (j', ol jo) = ine - tomv Jn(n T s in 0) I(ji o Ij-1, o) - in+le.htjml' Jin- (11 1-) [ ikk-mv (nYsin (III-4) -n-l tckik 1 We specify the polarization vectors by the usual spherical base 63

THE UNIVERSITY OF MICHIGAN 2764-5-T vectors in the polar and azimuthal directions, so = cos = + k,cs k+ - k =-sin 0 and k = 0 If we assume that in the argument of the Bessel functions, n-l _ nn+l _n (this is moderately good since the bulk of the radiation is expected for n>4) we obtain the desired result V ZT KK( ) k)= 42 (mvo-v)2 WKLk -K m- tckk K) (x) S(Kz-kz-Klz) e [1 toWmvL v J, (na'sino)]2 2z K1 4i,4r2e2 V;TK1 (0,k)= m2ick (mv ) ()Kk- K) co-K m - m.K (x) g(Kz-kz-Klz)e [-th m v cos 0 1 mo-c tan L 6' sin 0 1 (I1-5) where we have made use of Bessel function recursion formulae and the relations 1 R_, b (n) -, where R is the radius of the orbit. 64

THE UNIVERSITY OF MICHIGAN 2764-5-T Appendix IV Reduction of ar to a Cross Section. The quantity 5r is related to the cross section by j-r =Ca c (K1, k) f(K1 )dK (IV-i) XK1 where x, r p' (K1, K k) = zli 3 c KEd K Then X _dE d (m - V T ( k) up (K1, E, k) dE dQ = V 3 - k (IV-2) (27r) where we have assumed t2 K2 _ tick. 2m 1 ik.x We choose the plane wave state for the free electron as I K> e - -i the factor - indicating the number density of ions. Then 65

THE UNIVERSITY OF MICHIGAN 2764-5-T X VdEd2 e2 2mck ap (K, E, 2 k) dEd2 - mc2k mck mc k (x) s(E k-EK)I & K1 e - - VIK_1 We can arrive at the result of Bethe and Salpeter(2) by defining k X _ mLKV-ik. * x =k1 -h m K1 le vi k Q V K 7 (IV-4) where DQK' represents a matrix element with normalization different from that used above. The cross section can now be written a, (K1, Q, k) d2 = jCX (Ki, E,, k) dE 2 222 2 2 2 e DKI dQ2 (IV-5) 2 2 D f2K1 m ck which is just the result given in reference (22), equation (69. 5) when obvious notation changes are made. A further reduction can be achieved by use of the hydrogen-like atom ground state wave function 66

THE UNIVERSITY OF MICHIGAN 2764-5-T IK1 >- /ao e-a t2 where a0 = 2 The integration over x yields me x 32Z 2mck e Cyp(K1, E, 2k) dE dQ = dE dQ2 a~k mc 2 6, K (x) (E E-E) (IV-6) K,_k K 1_ + (kK)2]2 Assuming 2 (a) hi K tick 2m -2 2 2 (b) ao < k + K (IV-7) (c) tck << 1 2C) m c we obtain c (Kl,2, k) dQ2 = dE(K1, E, E,K) 7/2 2 4 (~hk 0 ( (h (x) sin 0 d~2 (IV-8) 67

THE UNIVERSITY OF MICHIGAN 2764-5-T k*K where /a cos0 k-. Lastly kK a (K1, k)1 djOa(KI, Q2. k) p p Z5 m /2 = 0 Z0 2 22(c (IV-9) 2 2 2 where =o, ro = 2 and a = If we multiply 3 m e2 ac 137 by a factor 2 to account for two electrons in the K shell, we obtain the result found in Heitler!21) equation (14), pg. 207. 68

THE UNIVERSITY OF MICHIGAN 2 764 -5 -T Appendix V Reduction of e B In equation (66 e) we let c denote electrons and T' ions, then convert the sums to integrals to obtain X 1 2 2 B = ( ) r Za d Kd K d Kd K 2m m tIck+ tk2 _ t2k. K 2 2 xa-) IU( K - ) - e3 Km 2 t2 k h2hk2 2 2k o K 22m2 k*K m m In a non-relativistic approximation hck >> and tck >> - 6 2m m m 69

THE UNIVERSITY OF MICHIGAN 2764-5-T If now we take the mass of the ions as infinite, then perform the K3 and K1 integrations, we find that =X 8wro Zak3 B 0 3 2 ~(V-2) (x)32 Sd K 8(E2 - E+hck) 4 (V-2) I K2 -K+k Note that d K2 =' 3/2 E d EdQ2. Now average over polarization, placing k along the z axis, and perform integrations over K1 and E2 to get 2 1/2 2 2 C m (fo: =2 7rn r Z a f (E) E(E-ick) dEdfdnZ2 B' r -k E sin 0 + (E -tck) sin2 02 - 2 E(E-hck) sin sin2z cos (Z -02) (x) 2 (V-3) [E+(E-tick)-c -2o \E (E -hc k)]2 where we have neglected k in comparison K2 -K, and written 4 (2m)2 [E IK2 - K| ( T E ) E+(E-Cck)2, \|E(E - h ck) -2p with/po =/A/I2 + sin 0 sin 02 cos ( - 02) andyu = cos 0. We arrive at a more familiar form by observing that j dw = dWn n tiwvf dEd~ s dnD (V-4) B e IeB 70

THE UNIVERSITY OF MICHIGAN 2 764 -5 -T where f f = e n But e 2k - d= 6B i wd(ck) B B (2 7r)3 so that 1 r m c E -tw c- dnr2 _ B 2 w~2 137 w V E E sin 0+ (E -ick) sin2 -02 2E(E-hck) sinsin sin02 cos (-2) (x) 2(V-5) (E + (E -tck) -2 \I/ (E-tIck)] When appropriate variable change is accomplished and one 0 integration (21) is performed, this result is equivalent to that found in Heitler, equation (17), pg. 245.

THE UNIVERSITY OF MICHIGAN 2764-5-T References (1) S. Chandrasekhaar, Stellar Structure, Dover Publ., 1957. (2) H. Goldstein, The Attenuation of Gamma Rays in Reactor Shields, AddisonWesley, 1959. (3) R. K. Osborn, Univ. of Mich. Radiation Laboratory report no. 2756-1-T. (4) W. E. Drummond and M. N. Rosenbluth, Phys. of Fluids, 3, 45 (1960). (5) L. I. Schiff, Quantum Mechanics, McGraw-Hill Book Co., Inc., 1949. (6) W. E. Brittin, Phys, Rev., 106 843, (1957). (7) A. Simon and E. G. Harris, Phys. of Fluids, 3, 245 and 255, (1960). (8) H. Mori and J. Ross, Phys. Rev., 109, 1877, (1958). (9) S. Ono, Prog. Theor. Phys. (Japan), 12, 113, (1954). (10) M. H. Johnson and B. A. Lippmann, Phys, Rev., 76, 828 (1949). (11) S. Ono, Proceedings of the International Symposium on Transport Processes in Statistical Mechanics, Interscience Publ., Inc., N. Y. (1959), P. 229. In this reference, as in (9), Ono develops a transport equation in which the influence of scattering is described in terms formally similar to those presented herein. (12) S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. Press, 1958. (13) C. Kittel, Elementary Statistical Physics, John Wiley and Sons, Inc. 1958. (14) T. L. Hill, Statistical Mechanics, McGraw-Hill Book Co., Inc. 1956. (15) J. E. Mayer and M. G. Mayer, Statistical Mechanics, John Wiley and Sons, Inc., 1940. (16) N. G. van Kampen, p. 239 of reference (11). 72

THE UNIVERSITY OF MICHIGAN 2764-5-T (17) G. Parzen, Phys. Rev., 84, 235, (1951). (18) D. C. Judd et al, Phys. Rev., 86, 123, (1952). (19) B. Trubnikov and V. S. Kudryavtsev, Proceedings of the Second International Conference on Peaceful uses of Atomic Energy, Vol. 31. p. 93, (1958). (20) S. M. Berman, Electromagnetic Radiation from an Ionized Hydrogen Plasma, PRL 9-27, Space Technology Laboratories. (21) V. Heitler, The Quantum Theory of Radiation, Oxford Press, 3rd Edition, 1954. (22) Bethe and Salpeter, Quantum Mechanics of One - and Two - Electron Atoms, Academic Press Inc., 1957. (23) V. I. Kogan, Plasma Physics and Controlled Thermonuclear Fusion, Vol. III, Pergamon Press, 1959. (24) F. H. Clauser, Plasma Dynamics. Addison-Wesley, 1960. (25) R. F. Post, Annual Review of Nuclear Science, Vol. 9, 1959. (26) V. V. Babikov and V. I. Kogan, Plasma Physics and Controlled Thermonuclear Reactions, Vol III, Pergamon Press, 1959. (27) J. Kvasnica, Czech. J. Phys,, B 10, 14 (1960). (28) L. Spitzer, Jr., Physics of Fully Ionized Gases, Interscience Publ., Inc. N.Y., 1956. (29) U. Fano, Phys, Rev. 103, 1202 (1956). (30) H. Olsen and H. Wergeland, Phys. Rev. 86, 123 (1952). 73

Title of Report Photon Transport Theory Contract No. DA 36-039 SC-75041 Period Report No. 2764-5-T Covered or Date August 1960 Contractor The University of Michigan DISTRIBUTION LIST Address Copy Nr. Commanding Officer 1-4 U. S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey ATTN: Logistics Division Marked for: Mr. O. C. Woodyard, Project Engineer SIGRA/SL-SR Commanding Officer 5 U. S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey ATTN: SIGRA/SL-SR(T), Thence: MFandRU No. 3 (for file) Commanding Officer 6 U. S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey ATTN: Technical Documents Center, SIGRA/SL-ADT Commanding Officer 7-11 U. S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey ATTN: Technical Information Division (For retransmittal to accredited British and Canadian Government representatives and to Department of Commerce) Commanding Officer 12 U. S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey ATTN: Director of Research, SIGRA/SL-DR Commanding Officer 13 U. S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey ATTN: SIGRA/SL-XS-Mr. A. Harris Commanding Officer 14 U.S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey ATTN: SIGRA/SL-SA (t)- Mr. M. Miller 74

Distribution List, The University of Michigan, Report No. 2764-5-T (Cont.) Address Copy Nr. Commanding Officer 15 U. S. Army Signal Material Support Agency Fort Monmouth, New Jersey ATTN: SIGMS-ADJ Corps of Engineers Liaison Office 16 U. S. Army Signal Research and Development Laboratory Fort Monmouth, N. J. U. S. Navy Electronics Liaison Office 17 U. S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey Marine Corps Liaison Office 18 U. S. Army Signal Research and Development Laboratory Fort Monmouth, New Jersey OASD (Rand E), Rm 3E1065 19 The Pentagon Washington 25, D.C. ATTN: Technical Library Chief of Research and Development 20 OCS, Department of the Army Washington 25, D.C. Chief Signal Officer 21 Department of the Army Washington 25, D.C. ATTN: SIGRD Director, U.S. Naval Research Laboratory 22 Washington 25, D.C. ATTN: Code 2027 Director 23-28 Advanced Research Project Agency Office of the Secretary of Defense The Pentagon Washington 25, D. C. Chief, U.S. Army Security Agency 29-30 Arlington Hall Station Arlington 12, Va. 75

Distribution List, The University of Michigan, Report No. 2764-5-T (Cont) Address Copy Nr. Deputy President 31 U. S. Army Security Agency Board Arlington Hall Station Arlington 12, Va. Commanding Officer and Director 32 U. S. Navy Electronics Laboratory San Diego 52, California Commander 33-34 Wright Air Development Division Wright-Patterson Air Force Base, Ohio ATTN: WCOSI-3 Commander, Air Force Cambridge Research Center 35 L. G. Hanscom Field Bedford, Massachusetts ATTN: CROTR Commander, Air Force Cambridge Research Center 36 L. G. Hanscom Field Bedford, Massachusetts ATTN: CRZC - Dr. M. R. Nagel Commander, Rome Air Development Center 37 Air Research and Development Command Griffiss Air Force Base, New York ATTN: RCSSLD Commanding General 38 U.S. Army Electronic Proving Ground Fort Huachuca, Arizona Commander 39-48 Armed Services Technical Information Agency Arlington Hall Station Arlington 12, Virginia ATTN: TIPDR Commanding Officer, 9560th TSU 49 U. S. Army Signal Electronics Research Unit P.O. Box 205 Mountain View, California 76

Distribution List, The University of Michigan, Report No. 2764-5-T (Cont) Address Copy Nr. Commanding General 50 Army Rocket and Guided Missile Agency Redstone Arsenal Huntsville, Alabama ATTN: ORDXR-RED This contract is supervised by Radar Division, Surveillance Department, U. S. Army Signal Research and Development Laboratory, Belmar, N.J. For further information contact Mr. Orville C. Woodyard, Senior Scientist, Radar Division, Surveillance Department, USASRDL, Belmar, N. J. Telephone PRospect 5-3000, Extension 61393. Contracting Officer's Technical Representatives: Messrs. V. L. Friedrich, O. C. Woodyard and J. Maresca, Radar Division, Surveillance Department, USASRDL, Belmar, N. J. 77

The University of Michigan, Ann Arbor, Michigan Unclassified The University of Michigan, Ann Arbor, Michigan Uncasfe PHOTON TRANSPORT THEORY PHOTON TRANSPORT THEORY E. H. Klevans and R. K. Osborn 1. Aspects of first-order E. H.- Klevans and R. K. Osborn 1. Aspect ffrs-re Radiation Laboratory Report No. 2764-5-T, November 1960, 73 pp. U. S. photon transport theory Radiation Laboratory Report No. 2764-5-T, November 1960, 73 pp. U. S. photontasother Army Signal Research and Development Laboratory Contract DA 36-039 2. Advanced Research Projects Army Signal Research and Development Laboratory Contract DA 36-039 2. AdvancdRiaebPoet SC-75041, ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report Agency, ARPA Order Nr. SC-75041, ARPA Order Nc. 120-60, Project Code 7700, Unclassified Report AgencAPOreN. A first order, momentu~m-configuration space transport equation for 120-90, Project Code 7700 A first order, momentum-configuration space transport equation for 120-60 rjc oe70 photons Is derived for low-energy (non-relativistic) systems. The de-3.US.AmSinlRsac photons is derived for tow-energy (non-relativisticj systems. The de- 3.S rivation is first order in the sense that the transition probabilities char- 3. n.dS ArvlomyeSi Resabrhor rivation is first order in the sense that the transition probabilities char- 3. nU S.AmI inlRsac acterizing photon scattering emission and absorption are computed oniy to andtrDevelopment3Laboratory acterizing photon scattertng emission and absorption are computed oniy to aondteeomntLbrtr the first non-vanishing order by conventional perturbation methods. CotatD 609S 54 the first non-vanishing order by conventional perturbation methods.CotatD3609C74 The present approach provides an essentially axiom-deduction develop- The present approach provides an essentially axiom-deduction development of the theory of radiative transfer (albeit via several ill-evaluated ment of the theory of radiative transfer (albeit via several ill-evaluated approximations) within the context of which various processes and their approximations) within the context of which variou's processes and their interrelationships may be investigated. Most of these process have hither- Interrelationships may be Investigated. Most of these process have hitherto been studied only phenomenologically and usually piecemeal. Specific to been studied only phenomenologically and usually piecemeal. Specific application to photon scattering, cyclotron radiation, recombination radia- application to photon scattering, cyclotron radiation, recombination radiation, de-excitation radiation, and bremsstrahlusig is made in the text. tion, de-excitation radiation, and bremsstrahiu-ng is made in the text. The derivation of an H-theorem for photon-particle systems is sketched; The derivation of an H-theorem for photon-particle systems is sketched; and contact is made with the usual statistical mechanical treatment of the and contact is made with the usual statistical mechanical treatment of the equilibrium states of such systems. equilibrium states of such systems. It is also shown that some aspects of collective particle behavior can It is also shown that some aspects of cotiective particle behavior can be Introduced quite naturally Into the description of photon transport in the be introduced quite naturally into the description of photon transport In the fully ionized plasma. fully ionized plasma. The University of Michigan, Ann Arbor, Michigan Unclassified The University of Michigan, Ann Arbor, Michigan Unlasfe PHOTON TRANSPORT THEORY PHOTON TRANSPORT THEORY E. H.- Klevans and N.- K. Osborn 1. Aspects of first-order E. H. Klevans and R. K. Osborn 1. Aspecsofit-re Radiation Laboratory Report No. 2764-5-T, November 1960, 73 pp. U. S. photon tmrasport theory Radiation Laboratory Report No. 2764-5-T, November 1960, 71 pp. U. S. photontasotter Army Signal Research and Development Laboratory Contract DA 36-039 2. Advanced Research Projects Army Signal Research and Development Laboratory Contract DA 36-039 2. AdvaneRsarhPoct SC-75041, ARPA Order Nr. 120-60, Project Code 7700, UnelassIfied Report Agency, ARPA Order Nr. SC-75041, ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report AgencAPrre c A first order, momentum-configuration space transport equation for 120-60, Project Code 7700 A first order, momentum-configuration space transport equation for 120-60 rjc oe70 photons is derived for low-energy (non-relativistic) systems. The de-3.US.AmSin Rsac photons Is derived for low-energy (non-relativistic) systems. The de- 3.S rivation is first order in the sense that the transition probabilities char_-.U S.d ArmyloSignal Resabrhor rivation is first order in the sense that the transition probabilities char- 3. nU S.AmI inlRsac acterizing photon scattering emission and absorption are computed only to andtrDevelopment3Laboratory acterizing photon scattering emission and absorption are computed only to aondeeomntLbrtr the first non-vanishing order by conventional perturbation methods. CnrcDA3-9SC701the first non-vanishing order by conventional perturbation methods. Cnrc A3-3 C54 The present approach provides an essentially axiom-deduction develop- The present approach provides an essentially axiom-deduction development of the theory of radiative transfer (albeit via several ill-evaluatmentnofothehtheoryroforadiativevetransferr(albeittviaaseverallill-evaluated approximations) within the context of which various processes and their approximations) within the context of which various processes and their Interrelationships may be investigate. Most of these process have hitInterrelationshipshimaymbe Investigated.d.Mosttofftheseeprocessshaveehitherto been studied only phenomenologically and usually piecemeal. Specific to been studied only phenomenologically and usually piecemeal. Specific application to photon scattering, cyclotron radiation, recombination radia- application to photon scattering, cyclotron radiation, recombination radiation, de-excitation radiation, and bremsstrahiung Is made in the text. tion, de-excitation radiation, and bremsstrablung is made in the text. The derivation of an H-theorem for photon-particle systems is sketched; The derivation of an H-theorem for photon-particle systems is sketched; and contact is made with the usual statistical mechanical treatment of the and contact is made with the usual statistical mechanical treatment of the equilibrium states of such systems. equilibrium states of such systems. It is also shown that some aspects of collective particle behavior can It is also shown that some aspects of collective particle behavior can be introduced quite naturally into the description of photon transport in the be introduced quite naturally into the description of photon transport in the fully ionized plasma. fully ionized plasma.

The University of Michigan, Ann Arbor, Michigan Unclassified The University of Michigan, Ann Arbor, Michigan Uncasfe PHOTON TRANSPORT THEORY PHOTON TRANSPORT THEORY E. H. Kievans and R. K. Osborn 1.- Aspects of first-order E. H. Klevans and R. K. Osborn 1. Aspectsofit-re Radiation Laboratory Report No. 2764-5-T, November 1960, 73 pp. U. S. photon transport theory Radiation Laboratory Report No. 2764-5-T, November 1960, 73 pp. U. S. photon tasotter Army Signal Research and Development Laboratory Contract DA 36-039 2. Advanced Research Projects Army Signal Research and Development Laboratory Contract DA 36-039 2. Advance4 eerhPoet SC-75041, ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report Agency, ARPA Order Nr. SC-75041, ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report AgencyAP re r A first order, momentum-configuration space transport equation for 120-60, Project Code 7700 A first order, momentum-configuration space transport equation for 120-60,Poet oe70 photons Is derived for low-energy (non-relativistic) systems. The de-3.US.AmSinlRsac photons is derived for tow-energy (non-relativistic) systems. The de- 3.S rivation is first order in the sense that the transition probabilities char- 3. n.dS ArmyloSignal Resabrhor rivation Is first order in the sense that the transition probabilities char- 3. nU S.AmD Sga Rsac acterizing photon scattering emission and absorption are computed oniy to anCDvlomntrctD Laboratory504 acterizing photon scattering emission and absorption are computed only to aondteelpetaortr the first non-vanishing order by conventional perturbation methods. CnrcDA3-9S-541the first non-vsnishing order by conventional perturbation methods.CotatD3609C-54 The present approach provides an essentially axiom-deduction develop- The present approach provides an essentially axiom-deduction development of the theory of radiative transfer (albeit via several ill-evaluated ment of the theory of radiative transfer (albeit via several ill-evaluated approximations) within the context of which various processes and their approximations) within the context of which various processes and their interrelationships may be investigated. Most of these process have hither- Interrelationships may be investigated. Most of these process have hitherto been studied only phenomenologically and usually piecemeal. specific to been studied only phenomenologicaiy and usually piecemeal. Specific application to photon scattering, cyclotron radiation, recombination radia- application to photon scattering, cyclotron radiation, recombination radiation, de-excitation radiation, and bremsstrahiung is made in the text. tion, de-excitation radiation., and bremsstrshlung is made in the text. The derivation of an H-theorem for photon-particle systems is sketched; The derivation of an H-theorem for photon-particle systema is sketched; and contact is made with the usual statistical mechanical treatment of the and contact is made with the usual statistical mechanical treatment of the equilibrium states of such systems. equtilibriumn states of such systems. It is also abown that some aspects of collective particle behavior can It is also shown that some aspects of cotiective particle behavior can be introduced quite naturally into the description of photon transport in the be introduced quite naturslly into the description of photon transport in the fully ionized plasma. fully ionized plasma. The University of Michigan, An Arbor, Michigan Unclassified The University of Michigan, Ann Arbor, Michigan Uncasife PHOTON- TRANSPORT THEORY PHOTON TRANSPORT THEORY E. H.- Klevans and R. K. Osborn 1. Aspects of first-order E. H. Klevans and R. K. Osborn 1.- Aspectso is-re Radiation Laboratory Report No. 2764-5-T, November 1960, 73 pp. U. S. photon transport theory Radiation Laboratory Report No. 2764-5-T, November 1960, 73 pp. U. S. photon ranprter Army Signal Research and Development Laboratory Contract DA 36-039 2. Advanced Research Projects Army Signal Research and Development Laboratory Contract DA 36-039 2. Advance eerhPoet SC-75041, ARPA Order Nr..120-60, Project Code 7700, Unelassified Report Agency, ARPA Order Nr. SC-75041, ARPA Order Nr. 120-60, Project Code 7700, Unclassified Report Agency RAOre r A first order, momentum-configuration space transport equation for 120-60, Project Code 7700 A first order, momentum-configuration space transport equation for 120-60PrjcCoe70 photons is derived for low-energy (non-relativistic) systems. The de- 3. U. S. Army Signal Research rivations is feirst fordr Inohe-enserg thatnthelatransitionytes prbblTies cha- 3. U. S. ArySgalRsac rivation is first order in the sense that the transition probabilities char- and Development Laboratory vtell i phostonrsaderin temisesio tandth tabsortion prompuiltied onlyrt and Devlpetlbrtr acterizing photon scattering emission and absorption are computed only to Contract DA 36-039 SC-75041 actrizng phtnsatrn msinadasrto r optdol oContratD3609C-54 the first non-vanishing order by conventional perturbation methods. the first non-vanishing order by conventional perturbation methods. The present approach provides an essentially axiom-deduction develop- The present approach provides an essentially axiom-deduction development of the theory of radiative transfer (albeit via several ill-evaluated ment of the theory of radiative transfer (albeit via several ill-evaluated approximations) within the context of which various processes and their approximations) within the context of which various processes and their Interrelationships may be investigated. Most of these process have hither- interrelationships may be Investigated. Most of these process have hitherto been studied only phenomenologically and usually piecemeal. Specific to been studied only phenomenologically and usually piecemeal. specific application to photon scattering, cyclotron radiation, recombination radia- application to photon scattering, cyclotron radiation, recombination radiation, de-excitation radiation, and bremastrahlung Is made in the text. tion, de-excitation radiation, and bremsstrahlung is made in the text. The derivation of an H-theorem for photon-particle systems is sketched; The derivation of an H-theorem for photon-particle systems is sketched; and contact is made with the usual statistical mechanical treatment of the and contact is made with the usual statistical mechanical treatment of the equilibrium states of such systems. equilibrium states of such systems. It is also shown that some aspects of collective particle behavior can It is also shown that some aspects of collective particle behavior can be introduced quite naturally into the description of photon transport in the be Introduced quite naturally into the description of photon transport In the fully ionized plasma. fully ionized plasma.

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