THE UNIVERS ITY OF MI CH I GA N COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report BREMSSTRAHLUNG OF SLOW ELECTRONS IN NEUTRAL GASES AND FREE-FREE ABSORPTION OF MICROWAVES A. Z. Akcasu L. H. Wald ORA Project 07599 sponsored by: Advanced Research Projects Agency Project DEFENDER ARPA Order No. 675 under contract with: U. S.'ARMY RESEARCH OFFICE-DURHAM CONTRACT NO. DA3-5-l24-ARO(D)-403 DURHAM, NORTH CAROLINA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR October 1966

This report was prepared as a paper for submission to The Physics of Fluids.

Bremsstrahlung of Slow Electrons in Neutral Gases and Free-Free Absorption of Microwaves A. Z. Akcasu and L. H. Wald Department of Nuclear Engineering, The University of Michigan Ann Arbor, Michigan ABSTRACT Bremsstrahlung of non-relativistic electrons in a neutral gas is investigated including the polarization and exchange effects. The intensity and spectrum of the bremsstrahlung and of the induced dipole radiation are obtained for a maxwellian distribution of electron energy in terms of the elastic scattering cross section of the atom for electrons and its polarizability. The interference of the induced dipole radiation with the bremsstrahlung is also considered. It is found that the exchange effects and the induced dipole radiation are negligible as far as the total radiated power is concerned. The latter, however, may be important at the short-wave end of the spectrum. Finally, the absorption coefficient is obtained from the bremsstrahlung cross section. The results are evaluated explicitly for a maxwellian distribution. 1

I. INTRODUCTION Bremsstrahlung of slow electrons in the field of neutral atoms has attracted interest in recent years in estimating the intensity of microwave radiation from slightly ionized gases and their free-free absorption coefficients. The radiation from a neutral gas containing free electrons is due to the deceleration of the free electrons in the field of a neutral atom and to the time-dependent dipole moment of neutral atoms induced by free electrons. In this paper, we shall refer to the former radiation mechanism as th.e "bremsstrahlunrg" and to the latter as the "induced dipole radiation."' The spectra of the radiation due to these mechanisms are entirely different. The observed spectrum will be a superposition of the bremsstrahlurlg and the induced dipole radiation. However, since these two emission mechanisms are not independent in so far as they are caused by the same collision event between a free electron and a neutral atom, the resulta'nt radiation can not be obtained by simply adding the two intensitieso The interference between these two radiation mechanisms should be taken into account. Furthermore, the exchange effect due to the indistinguishability of the incident electron and the bound electrons in the neutral atom may also play a role in estimating the intensity of the observed radiation. Th1e aim of this paper is to derive an expression for the ineltensity and spectrumn of the total radiation from a slightly ionized neutral gas taking into account the aforementioned effectso 2

Bremsstrahlung of slow electrons decelerated by neutral atoms was discussed previously by Firsov and Chibisov() who argued classically that the induced dipole radiation may account for the large portion of the radiation from a neutral gas. However, as will be apparent in the text, their quantum mechanical calculation includes neither the induced dipole radiation nor its interference with the bremsstrahlung. Their result gives the intensity of the bremsstrahlung only in terms of the elastic scattering cross-section of an electron on a neutral atom. However, the exchange effects and the polarization of the atom by the field of the incident electron are included implicitly through the scattering cross-section. In this paper, the induced dipole radiation and its interference with the bremsstrahlung are included in the quantum mechanical calculations, and the relative magnitudes of bremsstrahlung, induced dipole radiation and the interference effects are compared as a function of the gas temperature, assuming that the electrons and the neutral atoms are in thermal equilibrium. Moreover, the magnitude of the exchange effects on the radiation intensity is calculated explicitly, and its relative importance is discussed and shown to be negligible. It is hardly necessary to mention that the calculations are approximate in view of the complexity of the problem. Most of the approximations used in the derivations are standard in the study of the elastic scattering of slow electrons by a neutral atom. Some of the approximations are made only to derive a simple practical formula that contains all the qualitative features of the phenomenon under consideration. They can easily be relaxed if numerical precision is required. These approximations enable one to relate the intensity 5

of the observed radiation to some atomic parameters which are already known either experimentally or theoretically for many atoms, such as the elastic scattering cross section for slow electron scattering and coefficient of polarization, etco The exchange effects are discussed in the case of hydrogen atom for simplicitLy The calculation of the intensity of bremsstrahlung and induced dipole radiation is carried out in general for an arbitrary atom. 4

II. GENERAL FORMULATION The physical system under consideration consists of an atom situated at the origin of the system of reference and an incident electron. We consider a radiative transition of this system from an initial state |i > to a final state If > with the emission of a photon. The energy intensity of the radiation emitted in all directions and in two polarization states per unit energy is 4 12 2 S i+ f ( ) 4e I < f Rli > (1) - f 3'5C3 where z+l R = j Lr (2) j=l and where rj denotes the position of the jth electron with respect to the nucleus of the atom which is assumed to be at resto There are z electrons in the atom. The frequency X is given by~w c =.-4f - where i andf are the energies of the initial and final states. The symbol I <fjRli > | in (1) is to be interpreted as <fiRli > 12 = I <f|Rvi > 1 (3) v=l where Rv are the cartesian coordinates of the vector R. The state vectors |i > and If > are the solutions of the Schroedinger equation for the atom + electron: 5

(H a + + V -i)i > =, (4) a e where Ha is the Hamiltonian of the atom, H is the kinetic energy of the incident electron, and V is the coulomb interaction between the atom and the electron. The final state If > satisfies a similar equation. We shall assume that the atom is in a ground state, and the energy of the incident electron is insufficient for the excitation of the atom. Thus, the atom will be found in a ground state after the collision in which the incident electron will be scattered from the initial momentum state Iki > to the final momentum state |kf >. Conservation of energy requires i= Eo + (2ki2/2m) (5) f = Eo + (i2k2/2m) (6) where m is the electron mass, and Eo energy of the atom in the ground state. The interaction potential V in (4) is the coulomb interaction between the electron and the atom: Ze 7 e2 V - Z --- (7) j=l I r-j where r is the position vector of the incident electron. The central problem is to compute the matrix elements < f|RIi > using the solution of the Schrodinger equation (4) for energies i and given by (5). Let the wave function associated with initial and final states be denoted by i(rll;l*..rz+lz7+l) and rf(rl, l;...,rz+rlCz+l). These functions must be antisymmetric with respect to the interchange of any pair (r_i,oi) and (rj,oj) 6

where ja denotes the spin of the jth electron. In the non-relativistic theory, the total spin of the system is a constant of motion, and the spin state of the system is not altered by the collision. However, the symmetry with respect to the interchange of the position coordinates of the wave function will depend upon the symmetry properties of the spin state. Therefore, the spin state of the system will affect indirectly the intensity of the radiation (exchange effects). In order to discuss the exchange effects in a most simple way, we shall focus our attention to the hydrogen atom. However, the other effects will be calculated for an arbitrary atom, In the case of a two-electron system, there are two possible spin states: a triplet and a singlet state~ The triplet state is symmetric whereas the singlet state is antisymmetric with respect the interchange of spins. Therefore, the coordinate wave function is antisymmetric in the triplet state while it is symmetric in the singlet stateo Let us denote the symmetrized initial + and final wave functions by i(rl,r2) and 4f (rl,r2) where the superscripts(+) and (-) indicate a symmetric and an antisymmetric wave function respectively. Using symmetrized wave functions one can modify the intensity formula (1) as follows + 42 + + 2 Sif = 4 <f < IRIi > I (8) 3c3 In the case of an unpolarized incident electron, the probabilities of find+ ing the system in a triplet and singlet state are 3/4 and 1/35 Hence, S i-f and S-i.f must be combined by the ratio 1 to 3 to yield the total intensity: 7

S f= S f+ (9) sif = 4 i f 4 if (9) The symmetrized wave function can be constructed from an unsymmetrized wave function as + +(r ) l 2 ( r,2) (10) where P12 is Ic c A-icLlaigge operator. The matrix element of R between two symmetrized wave functions can be expressed in terms of the unsymmetrized wave function as < *lRkI > = < TflRlHi > + < *flRP12l*I > (11) where we have used the fact that the exchange operator P12 commutes with R = Ll + t2,3 and that 2 (l~Pl2) = 2(1-P12)o (12) Substituting (il) into (8) and combining the resulting equation with (9), one obtains W ^(4o4e2 [I < *flRli > I + I < f2RP2li - Re < fl|R|ti >* < XflIRP121i (13) The last two terms accounts for the exchange effects. We shall now attempt to determine the wave function'f and *i in order to compute the matrix elements appearing in (13). For this purpose, one(2) (3) 8

expands 4(rl,r2) into the atomic wave functions On: f, i = Fn i(2 ) n,(rl) (14) n where Fn(2) are the state functions of the scattered electron when the atom is in the state On(rl), and satisfies V2 + (knfi)2 U U nn Fnlf, (15) In this equation, one defines Unn, m <TniVInt > (16) X2 and f i 2 2 2m a (kn 2 = kf i 2(En Eo) (17)' 2 n where En is the energy of atomic state |n >. In (14), the summation over n includes the integration over continuous spectrum also. One observes in (17) that (knf')2 < 0 for all n ~ o, and only (kn'i)2 > 0 since the incident electron energy is assumed to be insufficient for excitation of the atom. In other words the interaction of the electron with the atom is an elastic scattering collision. The solution of (15) with the asymptotic condition F e~ikYr6 e ikr18) Fn = e- +fno() (18) 9

is standard in the study of elastic scattering of electrons by neutral atoms(2)() and will not be repeated here. The relevant results are( Ff Vnf Fof (19) En-Eo ( +kf- Uff - Up) F =0 (20) where u(_2) - -,l_ (21a) nlf and Uff(_2) = 2m < ~OfIVlf > (21b) Similar equations are obtained for Fn (r2) by replacing the index f by i. The symbols |iE > refer to the initial and final ground states of the atom belonging to the energy Eo. They may differ from each other in their magnetic quantum numbers, viz., lf > = ITiJiMf > and t0i > = [JiM i > where Ji and Mi refer to the total orbital angular momentum and its projection respectively. The Tri denote the remaining quantum number describing the ground state of the atom. When the latter is an s-state, i.eo, when Ji=0 and thus Mi=Mf=0, there will be no distinction between the initial and final ground stateso Such atoms will be referred to as spherically symmetric. Although the hydrogen atom which is being used for the discussion of the exchange effect is spherically symmetric, we shall retain the distinction between the initial and final ground states, because most of the results here will be used for an arbitrary 10

atom. It is to be noted that the additional potential energy Upf(r2) in (20) (2) represents 2 the effect of the polarization of the atom by the field of the incident electron when the atom is in the state jf. >. The potential Uff(r2) represents the mean potential, or the "rigid" potential of the atom in the state 1|f> o Thus, (20) yields the wave function of the electron in the potential field Uff(r2) + Upf (2) We now return to the calculation of the matrix element < rf|R|jli > apf i pearing in (13). Substitution of the expansion of 4f and i in (14), and the use of the orthogonality of n's yield < flRti > = < FnfFnt > < nlrllDn? >+ n, n + <Fjni32|Fni >. (22) Note that the functions Fn1 and Fni are essentially the expansion coefficients in (14) and are not orthogonal. The double sum in (22) contains the matrix elements of the dipole operator er1 associated with the bound electron, and represents the induced dipole radiation. Similarly, the second term in (22) contains the matrix elements of the dipole moment of the incident electron between various states of the scattered electron, and represents the bremsstrahlung in the field of the neutral atom. The cross term which appears in the expression of I < JflRli >12 will account for the interference of the induced dipole radiation and the bremsstrahlung 11

The second and third terms in (15) represent the exchange effect as indicated by the presence of the exchange operator P12. It is in order to mention at this point the simplification introduced by considering the hydrogen atom for the discussion of the exchange effect. The crucial problem in the discussion of the exchange effect is the construction of the (Z+l)-electron function from the Z-electron and the incident electron wave functions. This problem has been discussed in detail for an arbitrary atom in reference 2. The coordinate wave function for the system of the electron plus the atom for a given total spin is rather complicated even in the case of the helium atom. It has a simple form only in the case of a 2-electron system already indicated by (10). Since the exchange effect is expected to be small as far as the radiation intensity at low electron energies is concerned, its inclusion for an arbitrary atom is considered as an unwarranted complication in the present analysis. However, the magnitude of the error due to the neglect of the exchange effect will be estimated quantitatively in the case of hydrogen atom as a guide by considering the last two terms in (13). 12

III. BREMSSTRAHLUNG The radiation due to the deceleration of the incident electron by the neutral atom is represented in (22) by Zn < FnfIrI|F > where we replaced r by r which we recall refers to the position of the incident electron. Since the energy of the latter is insufficient to excite the atom, the dominant contribution will come from the first term Ib - < Fof(r)lrlFoi(r) >, (23) where Foi(r) and Fo (r) are the wave functions of the scattered electron when the atom in the initial and final states, and satisfy (20). They can be expanded into spherical harmonics as F (r=) = 4 () (kfr) Ym(f) Ym *() (24) 2,m where Fy (kfr) are the solution of the radial Schroedenger equation. Substituting (24) and the similar expansion for Foi(r) into (23), performing the angular integration and retaining only the terms in the resulting equation containing the product FoFl, one obtains 00 Ib =i i 4k^ ki dr r3 *(kfr) Fl(kir) -b = of 1o A 00 7 - kf dr r3 F1*(kfr) Fo(kir) (25) 013 15

where k= ki/ki and kf = kf/kf. As pointed out by Firsov and Chibisov(l) the terms involving Fe for 2_2 correspond to electrons at a large distance from the atom at low energies, and do not interact appreciably with it. Therefore only the terms in (25) are of significance. Following this reference, we use Sin(kr+5 o) Fo(kr) = - (26a) Fl(kr) = jl(kr) (26b) where bo is the phase shift of the s-wave, and jl(kr) is the spherical Bessel function. We ignore the phase shift 6, associated with the p-wave which is justified at small incident-electron energies. The phase angle bo is a function of k, and represents the interaction of the electron with the atom. It is related to the elastic-scattering cross-section by a(k) = 4 sin 5o(k) (27) k2 Substituting (26) into (25) and performing the indicated integrals, one obtains lb = ij (a)2 I (kf) ki - k i J (28) This can be further simplified if the cross-section does not change appreciably in the region (0-ki) then a(kf);c(ki)_a(o), and Ib = i4/t ( ) 2m / _ ( 29) 14

where = k - kf (30) The intensity of the bremsstrahlung alone can be calculated substituting (29) into (13), multiplying the resulting equation by the density of final electron states per unit electron energy, viz., 2(m/ )3/ (2T)3 and integrating over the direction of kf, one obtains Sb (-iow) Na _T oca(o)c ( 2)-2E (31) 53i1 \ meC2 Ey I I Ei ii where a is the fine structure constant (e 2/fc), E. is the incident electron energy, and finally Na is the number of neutral atoms per unit volume. The spectral density for a maxwellian electron distribution is obtained from (31) by averaging it with 2r(Ot) 3/2Ne TEi exp[-Eili] as SbC =) 4c-2 / ao) m( )2 (x) K2(/2 ) e (32) where Ne is the number of electrons per unit volume, ~ is the temperature of the gas, and K2(x) is the modified Hankel function. Equation (32) gives the intensity of bremsstrahlung per unit energy from a unit volume of gas containing Na atoms and Ne electrons. The ratio of Na/Ne can be obtained from the Saha equation at the specified temperature. The total radiated power in all energies is obtained by integrating (32) over iw. The result is Sb 256~ ao )o (o20 /\ 2 N2a (5) 15 mein2 Krm N 15

IV. INDUCED DIPOLE RADIATION The dominant contribution to the double sum Znn'<Fn lIFnl> <rnlllIn1> in (22), representing the induced dipole radiation comes from terms for which either n = o or n' = o: Id = Z [ < FofIFni> < flDIn > + < FnfIFi > < nlD|lki >] (34) where z = rj (35) j=1 In (34) we replaced r1 by D such that the subsequent analysis will be valid for an arbitrary atom. The neglect of the terms for which both n $ o and n: J o can be justified by observing that they contain the product of two interaction potentials, viz., Vnf Vin /(En Eo)(En -Eo), whereas the terms in (34) are proportional to Vni/(En-Eo). Substituting Fn and Fn from (19) into (34) one obtains - F oV fnlFo > < nlDli > + < Fo VnilFi > < OfDIn > d En - Eo n o (36) where z v = ze2+ e2 r l|r-rj j = 1 In order to calculate the integration with respect to r appearing in <oflVfnl Fo > and < Fo fVniFo > we shall approximate Fo and Fo by plane waves. The result is 16

FJ 1JVfnIJi > 2 < e fl - z + e - > 2 - i 4te2 * < Kf Dl n >7) q where the last step is obtained by approximating exp(iq'rj) by l+iq.'r. This is justified when q'r < < 1 which is the case for low incident electron - -J energies. Substituting (37) into (36) yields 4t e2 = < eflg-Din > < jn Di > + < Onq D > i > < flD n > Id - - i n ~ n - o (58) Using Wigner-Echart theorem, one evaluates (58) as follows: A o(Mi)ez, for Mf = Mi Id =-i < + (-ex+iey), for Mf = Mi + 1 (59)'_ (ex+ieAy), for Mf = Mi - 1 A A where e ||^, and where o m 0 0 - En - Eo n o P+(M.) _ (-M ) bo 2M+ (J -M ) (J +M +) (41) ~ - 4 ii ii The expression ao and bo in terms of reduced matrix elements of D. In the case of a spherical atom for which Ji=Mi=O, 4xc I = - i q (42) q2 17

The quantity a is the polarizability of the atom, which is (9/2)r 3 for hydrogen(2). The intensity of the induced dipole radiation alone is obtained by substituting (39) into (13), summing over Mf, averaging over Mi, and integrating over k_: Sd IC) = Na (I24n )+ 4)l ]/[1- ] ix] (43) where 2 2 72 T2 ---- (ao +2P++2P_ ) 2J.+1 c2+2+2 ) Mi 2 b 2 2 ji(Ji+Z) = a + b [-2a + 2 2 (2J +2Ji +l) (44) and where x = (c/Ei). One observes in (43) that the spectrum vanishes when aioo in contrast to bremssurahlung of the incident electron given by (31). Certainly both spectra vanish at XA=Ei. The spectrum of the dipole radiation has a peak at approximately hc=0.97Ei. The major portion of the total induced dipole radiation for a given electron energy Ei is emitted in the frequency range under this peak. The spectrum for a maxwellian distribution of incident electron energy is found as Sd(1 ) = N-aNe 6 a4/ 2( e-/o (2 /2K8) (45) Sd(-)= NaNe 3 cz - 3 ec and the total radiated power 18

iSd = N, 215 ( 4 2 4 2(46) = NaNe K e Y a (46) It is interesting to compare the total radiated power in the case of bremsstrahlung and induced dipole radiation. From (33) and (46) one obtains their ratio as S d 27 2 (mc2)2 2 S - - ~~~(47) Sb 21 o(o) ( Xc)4 The magnitude of T and a(o) are of the order of 10 24cm and 10 16cm respectively, for most atoms. For example q=.62xlO 2cm3 and a(o)=4810 1cm for hydrogen (). For 0=0.5 ev which corresponds to a temperature 5000~K this ratio is less than 8% (for hydrogen.6%) indicating that the induced dipole radiation will be insignificant in most cases. This statement is particularly true in the microwave range of the spectrum because of the difference in the shape of the spectrum in the two cases as discussed above. This conclusion is at variance with that given in reference (1) by classical arguments. 19

V. INTEHERENCE EFFECTS The interference of the dipole radiation and the bremsstrahlung is determined by the cross term in the expression of lIId+IbI l b i bl Idl +12Re( Idlt) _ -d+ J=d I+2ba.I) We have already discussed the first two terms. Using (29) and (39) we find that Iint = 2 Re Id'l = 32t ) () (48) Note that there is no contribution for MfMi. The interference correction to the radiation intensity is obtained by averaging I'nt over Mi, multiplying it by the density of final electron states, integrating over k and finally substituting the resulting expression into (13). The spectrum for a given electron Ei, the averaged spectrum and the total radiated power are calculated a s iSit(C) = - a Col ( 9) 3 Er c Sint.i) - -? (50) a.nd Sint =._ 1212\2c 87/2 (51) 105 d22 - m c \m( where 20

~My, ~ ~ U0 - 2Ji+1L cz0(lk) a=' - b,(52) 2Ji+l %(~) - o -3 Ji(J3+l) In performing the average over Id we have approximated the average of o (M ) v (o) by the product of the averages. When the atom is spherically symmetric in the ground state this approximation becomes unnecessary because there is no M. dependence in a (M;) = a and (o)'. 1i 1 o We shall now compare the combined effect of the induced dipole radiation and the interference term to the brems.strahlung intensity. Using (33), (46) and (51) we obtain d +Sint 2jr7 7o m2 ( 2 >2 2 2 Sd nt -2 i m e 8 3 - _ X Sb 21 a(o) \i2c2 e m c2 0J It is interesting to observe that this ratio depends on the type of the neutral atom only through the ratio (ao2/a(o)). To estimate the relative error we again use -'10 24cm3 and a(o)lO10 cm as typical values in (53), and obtain 0.3 ~ (8-1.7) where 8 is in ev. In the range of validity of the foregoing derivations, which require the incident electron energy to be small, and thus 8 < 1, one finds that the effect of the interference term on the radiation intensity is more important than the induced dipole radiation alone. It tends to decrease the total intensity as indicated by its negative sign. For 8=0.5 ev, the above ratio becomes 18% (for hydrogen it is less than 53) which is probably an upper estimate for many atoms. It can be concluded that the dipole radiation and its interference with the bremmstrahlung, which are associated with the polarization of the atom by the field of the incident electron, are insignificant as far as the total radiation intensity is concerned. 21

The relative error at a given frequency can be easily discussed with the foregoing formulas for the various spectra. At low photon energies the spectrum of the bremsstrahlung is flat whereas that of the induced dipole radiation and 22 the interference term decreases as i c 2. Hence, the emission due to the polarization of the atom can be ignored at low photon energies. It must be pointed out at this point that the polarization of the atom affects the intensity of the bremmstrahlung considerably. However this effect is taken into account through the elastic scattering cross section r(o), which is to be calculated from the asymptotic form of F (r). The latter is the solution of (20) which includes the effect of polarization of the atom through the additional interaction energy Up(r). The elastic scattering cross section decreases 2) where the effect of polarization of the atom is added to the rigid potential scattering. Hence, the intensity of the bremmstrahlung will also be smaller when the atom is polarizable, than when it is rigid, by a ratio which may be as high as 1/2 as is the case for hydrogen(2). 22

VI. EXCHANGE EFFECTS This section is devoted to the investigation of the exchange effects in the calculations of the bremmstrahlung intensity for the hydrogen atom. The matrix element associated with the exchange effect in (13) is < lf(rlr2)1 r +r2 li( r l) >. To evaluate this matrix element we substitute the expan-1 -2 i-2 - sion (14) for 4f and *i and retain the terms for which n=o and n'=o: -c = F(< r2) ( 2) > < o(r 1)r1Fi(r) + < Ff (r2)Ir2kI0(r2) > < %(r)lFoi(rl) > (54) f f In order to calculate 7- < Fo jlo > we use the expansion of Fof into spherical harmonics given by (24). Using the fact that 0o(r2) is a function of 121, one obtains 00 < FJlo > = 4nj FO*(kfr)o(r)r2dr. o Substituting Fo(kr)=sin(kr+bo)/kr as before one gets 4Tc co r ( cos 7 = - os 5o rdr% (r)sinkfr + sin5o rdr' (r) cos kfr. kf 0 Since we are dealing with slow electrons whose wave length is larger than the size of the atom, kfr < < 1. Then, using cos 8o1, sinkfrok r and cos r;], we find 7 = 4t / r drdo(r) + 4-t(o) rdrO(r) (55) o o 23

For the hydrogen atom, the wave function of the ground state is 0 (r)= (rO53)1/2 exp[-r/ro] where ro = 2/(me2) = 5.5 x lO9cm, the Bohr radius. Hence for the hydrogen atom Y = 2 r [4ro Jr + A7) ] (56) 2 In reference (2), the elastic scattering cross section is given as a(o) 60rr. With this value of a(o), we obtain y253.48ro hJro. Next we consider < %DrlFoI > Using again the expansion of Foi into spherical harmonics, and the fact that o0 is a function of Irl only, we find Co < oirlFoi > = i4rL Fl(k.r)$o(r)r5dr Putting Fl(kir) = jl(kir) and using the asumptotic form of jl(kir) for small arguments we get < olrlFo > = i S ki (57) where 00 = r4 O(r)dr (58) For hydrogen atom, 6 = 4! ro'Jr. Repeating similar calculations for the second term in (54), we obtain the desired matrix element as I = iqs (59) where 24

i = 4y = 64r 4 4r + (o(o)/)1l/2 (60) o o The corresponding radiation intensity is obtained as 2 /" 5/2 4 2 Sc(,o) = na e) 2 - (2Ei-O)) JEi T (61) 5a 3 T3;2 ), One observes again as in the case of dipole radiation, that the spectrum vanishes when -tcDo. The total power for a maxwellian distribution of incident electron energy follows from (61) as Sc = 4.54 o10 2 oa n2 NaNe (62) 2 - 2 8 3tr Tr c-i Using the value of the elastic scattering cross-section given in reference (2) i.e., ((o) = 60rro2, one finds that the ratio (Sc/ ) 10 8 where is in ev. Hence the contribution of this term is negligible for E < 1. As a final step, we consider the last term -2Re[ < VfLRIJi >* ~ < *f3R P12i > in (13), which is equal to -2 R [(Id + Ib) Ic*] Ignoring the induced dipole term Id one finds the spectrum associated with this term as Sc int(-)) = 3f2 t - tT2) (2E - C o)f / (63) The total average power is 21lf2oa mO 1 S =-. - Iz2 c ( m ) N Neo (64) c,int 6305O)5 c )S / a e The ratio of this term to the intensity of the bremsstrahlung is approximately equal to 82/10 where ( is in ev. Hence, for 0 < 1, this ratio is less than 10%, and decreases rapidly with the gas temperature. 25

The final conclusion is that the exchange effects can be ignored completely in estimating the intensity as well as the absorption of radiation in a neutral gas containing slow electrons. 26

VII. THE EFFECTIVE ABSORPTION COEFFICIENT DUE TO FREE-FREE TRANSITIONS We have seen that for the low electron energy range, the only processes responsible for transitions between states of the whole system (neutral atom + electron + radiation field) are bremsstrahlung and inverse bremsstrahlung. It is possible to use the energy intensity of emission due to bremsstrahlung of equation (32) to calculate the effective absorption coefficient for radiation. This calculation will now be done. If it is assumed that the medium is isotropic, then the effective absorp(4) tion coefficient for unpolarized photons is aeff f= 1 X d (Qk e(k) - O(k)) (65) elf 8itc - - - A where X is the index of the polarization state e% is the transition probability per unit time for emission of a photon of polarization X into direction QK -K A is the transition probability per unit time for absorption of a photon of polarization X traveling in direction QK It is possible to show that for a Maxwellian electron distribution the absorption and emission transition probabilities are related by the equation c(k) = e(/@ e(k) (66) also, the total radiation intensity S(lMc) per neutral atom per unit energy interval is related to E (k) by 27

S(AU) f= 2 3S dQ x (k) (67) The use of equations (66) and (67) in equation (65) yields for the effective absorption coefficient aeff = - 2 S(Al)(1 - et/) eff =a Substitution of S( w) which was obtained previously in equation (32) by ignoring polarizability and exchange effects and approximation of K2 (&hwD/2@) and exp [Xc6/$] in the result yields for aeff 2 1 32 (o) (68) Jeff a E (o) Na Ne(68 eff - m m3/2c 2 It is interesting to note that if one defines an "effective collision frequency" in the standard way (cf. Ref 5 where veff = 8/3 - a (o) Na -), Tt ~m the absorption coefficient may be seen to agree with the power absorption coefficient aeff of the Maxwell-Lorentz theory(5) for non-dispersive media and small 2 2 collision frequencies (vff << ),e ff weff = c 0C where p2 = plasma frequency = 4te2Ne/m 28

VIII. CONCLUSION The present analysis indicates that the dominant contribution to the intensity of radiation from a neutral gas containing slow electrons comes from the deceleration of the free electrons by the field of the neutral atoms. The contribution of the induced dipole radiation is always negligible. However, the interference of the induced dipole radiation with the bremsstrahlung may decrease the total intensity as much as 18%, depending on the ratio of the polarizability of the atom and its elastic scattering cross section for slow electrons with energies less than 1 ev. The induced dipole radiation may become appreciable in the short-wave limit of the emitted spectrum. It is also found that the exchange effect in the calculation of radiation intensity for the hydrogen atom is much less than 10%, and can be ignored entirely. The same conclusion is expected to be also true for a multi-electron atom. The intensity of the bremsstrahlung of the free electrons in the field of the neutral atoms which is the dominant emission mechanism is shown to be proportional to the elastic scattering cross section of the atom for slow electrons in the limit of zero incident energy. This cross section includes the effect of the polarization of the atom by the field of the incident free electron. Thus, the polarizability of the atom affects, and decreases, the intensity of the bremsstrahlung, although the induced dipole radiation which is due to the polarization of the atom is negligible. In view of these results, it is concluded that the free-free absorption of the microwaves in a slightly ionized neutral gas will be predominantly due 29

to the inverse bremsstrahlung. The absorption coefficient has been found to be in good agreement, above the plasma frequency, with that given by the classical Lorentz formula of electro-magnetic theory for nondispersive media. 50

ACKNOWLEDGMENT The writers wish to acknowledge the aid and encouragement obtained from Dr. R. K. Osborn of The University of Michigan and Dr. M. L. Barasch of the Radiation Laboratory of The University of Michigan. 31

REFERENCES 1. 0. B. Firsov and M. I. Chibisov, Soviet Phys.-JETP, 12, 1235 (1961). 2. G. G. Drukarev, The Theory of Electron-Atom Collisions, Academic Press, New York (1965). 3. N. F. Mott and H. W. W. Massey, The Theory of Atomic Collisions, 2nd. ed., Clarendon Press, Oxford (1950). 4. E. H. Klevans, "The Theory of Photon Transport in Dispersive Media," The University of Michigan Radiation Lab. Report No. 2764-12-T (1967). 5. J. M. Anderson and L. Goldstein, Phys Rev. 100, 1037 (1955). 32