PHYSICAL REVIEW LETTERS

VOLUME 17 28 NOVEMBER 1966 NUMBER 22

EXTENSION OF THE PLASMA KINETIC EQUATION TO SMALL WAVE NUMBERS*

Allan N. Kaufman Physics Department and Lawrence Radiation Laboratory, University of California, Berkeley, California

(Received 19 October 1966)

The equation for the evolution of the electron velocity distribution, in a stable uniform electron-ion plasma, is extended to wave numbers so small that collisional damping ex­ceeds Landau damping for electron plasma waves.

It is generally realized1 that the standard plasma kinetic equation for a stable uniform Coulomb plasma, that of Lenard, Balescu, and Guernsey (LBG),2 has a serious defect in principle. Its derivation is based on the Bo-golyubov synchronization assumption, that the electron correlation function g(k) relaxes to an asymptotic state at a rate much faster than the rate of variation of the electron velocity distribution/(v). However, in the various der­ivations of the LBG equation, one finds that g-(k) relaxes at the rate y-r (k) (the Landau damp­ing rate) while /(v) characteristically relaxes at the rate v{v), the electron-ion collisional frequency. Comparing the two rates, one ob­serves that for

k<k (lnA)-1/2 (1)

(where kj) is the Debye wave number and A is the Debye parameter), yL(k) is exceeded by v [the mean collisional frequency, defined in Eq. (18)], so that the Bogolyubov assump­tion is violated.

It is also realized1 that g(k) actually relax­es not at the rate y ^ k ) predicted by the stan­dard derivation, but rather at the rate y(k), the true damping coefficient for an electron plasma wave k. The latter is known to be3

y(k)=yL(k)+£i7, (2)

for k «&£)• Thus, as k — 0, the rate y(k) =|i7 is appropriate for the relaxation of g(k).

The present paper utilizes this idea, here­tofore based on intuition, to derive a kinetic equation valid for arbitrarily small k. The Bogolyubov assumption is now valid at small k for fast electrons [such that v{v)«v}) this limitation is acceptable, since only fast elec­trons can resonate with fast plasma waves [k < kj}{lnA)~1/2], and only for the latter is the present modification in theory needed.

The validity of the Bogolyubov assumption implies that the momentum p of an electron is a Markov process. Neglecting large-angle scattering, we may thus utilize the standard Fokker-Planck equation4 for the momentum probability density /(p):

£ / & « - ^ 6 > . (3)

with J(p) = F(p)/(p)-(8/8p)-[D(p)/r(p)]. Here J is the flux density in momentum space, F is the dynamic friction, and D is the momen­tum diffusion tensor. As is shown by Thomp­son and Hubbard,5 these quantities may be ex­pressed as

J(p) = FP(p)/(p)-D(p)- (8/8p)/(p), (4)

fP{l)^AiJe2j^f2lmK^l{K^v), (5)

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/-\ i 2 r ^ kkC(k ,k -v ) . (6)

-zJ> • Here F is the contribution of polar izat ion to dynamic friction, 7T(k, co) i s the d ie lec t r ic func­tion, and C(k, co) is the micropotent ia l spec t ra l densi ty:

C(k, co) - JcPsJ^icpir, t)cp(r+s, t +r)>

x exp(-ek • s +ioor)dr. (7)

The LBG equation is obtained if one r ep laces K by the Vlasov function / fy , and C by the ex­press ion 5 obtained from the t e s t -pa r t i c l e the ­o rem. 6

For collisionally damped electron 7 p l a s m a waves , the damping is weak ( y « c o e ) . Hence, in (5), one may use the resonance approxima­tion for the range (1):

ImK-1<k,u) = -ii6[ReK(k,u)]sgn[ImK(k,(jo)] . (8)

Since the inclusion of coll is ions does not mod­ify ReK, we see that F ^ is insensi t ive to the damping mechanism, as one would expect on intuitive grounds.

However, D is modified by col l is ions . That the Rostoker express ion 5 ' 6 is no longer valid for C is c lea r , as one can check by tes t ing it, in the case of the rmal equi l ibr ium, with the f luctuat ion-dissipat ion theorem 5 :

C(k, co) = -(87r9A2co) ImK^HK w),

where , for k-+ 0,

K(k,co) = l-co 2/(<A>+iv)2. e

0)

(10)

To obtain a valid express ion for C in the range (1), we again make the resonance approxima­tion, now in the form

C(k,co)

= (4n2/k2)[S(®6(u-ur) + 8(-k)6(uj+u> £)], (11) K —"K

where cog is the posi t ive frequency of the wave k, and

5 ( k ) . 2 —Jo°°^coC(k,co) (12)

is the energy of the wave (k, cog). The con t r i ­bution of the range (1) to the diffusion tensor (6) is thus

D 1 ( p ) = 4 7 7 2 ^ 2 J ^ ^ 6 ( c o ^ k . f ) ^ ( k ) , (13)

where the integration is over the range (1). Not surpr is ingly , this i s identical to the e x p r e s ­sion obtained in the quasi l inear theory,8 although the conventional derivat ion and conditions of validity of the la t ter a r e quite different.

The wave energy <?(k), in turn, may be ob­tained from the s t eady-s ta te condition

5(k) = 2y(k)£(k), (14)

where 8 is the emiss ion r a t e . The par t of the la t te r due to longitudinal b remss t r ah lung may be calculated, by the method of Merc ie r , 9 to be

8B&)- (e2oo 2/3)(v~1)lnA,

where

(v ) = jdpv / ( p )

(15)

(16)

In addition, t he re is the col l is ionless emiss ion r a t e , &QQ&), the longitudinal Cherenkov r a d i a ­tion,10 which vanishes with J?. F r o m (14) and (2), we obtain

5R(k) + <?r(k) £ (k)

k in (1) (17)

The mean coll isional frequency v depends func­tionally11 o n / :

V = (e2u 2/3m)(v-3). (18)

As a check on (17), we may evaluate it for a Maxwel l i an / (p ) , and obtain the Rayle igh-Jeans formula <§(k) = 0. Noting that SB{k) is indepen­dent of k, we may define an effective electron t e m p e r a t u r e for non-Maxwel l i an / (p ) , by #eff = 5(k) in the range (1).

Substituting (17) into (13), we may easi ly evaluate ^ ( p ) . We find that, for v » (v) InA,

DX(P) 716 *~VV

' InA v (19)

valid approximately for any /.(p). In contras t , the s tandard theory, using <§(k) = <?c(k)/2yL(k), would yield a r e su l t highly sensi t ive t o / ( p ) . We note that, in magnitude, the contribution (19) i s sma l l e r , by a factor (InA)""1, than the contribution of all waves k <k D to D, and, by a factor (lnA)~2, than the "dominant" contr ibu­tion, from k >£]), r epresen t ing col l is ions with­in the Debye sphere .

Thus the p resen t cons idera t ions r ep re sen t

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a correction of the same order in A""1 as the terms in the LBG equation/2 but of higher or­der in (InA)"""1.

I wish to thank Dr. P. P. J. M. Schram and Mr. C. S. Liu for the benefit of helpful discus­sions.

*Work done under ausp ices of the U. S. Atomic E n e r ­gy Commiss ion .

1 E . A. F r i e m a n , Pr ince ton Univers i ty P l a s m a P h y s ­ics Labora to ry Repor t No. P P L - A F - 1 , 1966 (unpub­l ished) , pp. 19-20.

2D. C. Montgomery and D. A. Tidman, P l a s m a Kine t ­ic Theory (McGraw-Hill Book Company, Inc . , New York, 1964), Chap. 6; T . Y. Wu, Kinetic Equations of Gases and P l a s m a s (Addison-Wesley Publishing C o m ­pany, Reading, M a s s a c h u s e t t s , 1966), Chap. 6.

3 J . Dawson and C. Oberman , P h y s . Fluids 5^ 517 (1962). In addition to damping caused by e lec t ron- ion col l i s ions , t h e r e i s a contribution to wave damping from the e las t i c sca t t e r ing of the wave by ions . The l a t t e r is s m a l l e r than the fo rmer by a factor of the o r -

Although KC1 with color centers has been investigated many times, little is known about the mechanism by which they are formed dur­ing irradiation.1 It is also unclear what type of disorder they represent in the crystal lat­tice. The model in which the F center is a negative-ion vacancy having captured an elec­tron is well established. There are some in­dications that during x irradiation of alkali halides negative ion vacancies are formed to­gether with halogen interstitials as complement defects.2"7

The purpose of this paper is to report about the disorder in KC1 crystals containing color centers. Combined measurements of the mac­roscopic volume change and the change of the lattice parameter of KC1 were performed, in­troducing color centers. The defect concen­tration was obtained from optical absorption measurements. In additively colored KC1 Schott-ky-type disorder in the halogen sublattice was found. In x-irradiated KC1 the disorder was of Frenkel type.

The volume change was studied by measur­ing the mass density change using a hydrostat-

der (fc/ferjMhiA)"""1, which i s l e s s than (lnA)~3 /2 in the range (1). Hence, the l a t t e r contr ibutions will be n e ­glected h e r e .

4 S. Gasiorowicz , M. Neuman, and R. Riddel l , Phys . Rev. jjOl, 922 (1956).

5W. B. Thompson and J . Hubbard, Rev. Mod. Phys . 32, 714 (1960); J . Hubbard, P r o c . Roy. Soc. (London) A260, 114 (1961).

6N. Ros toker , P h y s . Fluids 7_, 491 (1964). 7We ignore ion waves in the p re sen t d iscuss ion .

The i r p rope r t i e s a r e quite sensi t ive to col l is ions [see R. Kulsrud and C. Shen, Phys . Fluids £, 177 (1966)], and mus t be t r e a t e d with grea t c a r e .

8 For a c r i t i ca l s u m m a r y , see I. Berns te in and F . Engelmann, P h y s . Fluids 9., 937 (1966).

9R. P . M e r c i e r , P r o c . Phys . Soc. (London) 8.3, 819 (1964).

10N. Ros toker , Nucl . Fusion 1, 101 (1961). ^In evaluating (18) with (16), one m u s t impose a low­

e r l imit of A"""1^) on the speed v. 12We note that the divergence of the LBG equation at

l a rge k has been t r ea t ed by a number of a u t h o r s , l e a d ­ing to a co r rec t ion o rde r ( InA) - 1 .

ic flotation method. The lattice parameter change was obtained from the shift of the (800) Cu Ka x-ray diffraction peaks.6 The concen­tration of color centers was calculated from the optical absorption spectrum. For large concentrations the /f-band absorption was used to obtain the F-center concentration.8

Color centers were produced by two differ­ent methods. Additive coloration was performed at 720°C in a potassium atmosphere with sub­sequent quenching according to the technique described by van Doom.9 X irradiation of the crystals was performed at room temperature by a tungsten anode tube; it was operated at 120 kV, 12 mA. The x rays were filtered through the tube window and 0.02-mm copper foil, and irradiation was given from both sides to obtain maximum uniformity of coloration. Optical absorption measurements showed that the max­imum inhomogeneity was less than 1%. KC1 single crystals were obtained from the Harshaw Chemical Company.

An answer on the type of defects present in a crystal lattice is derived from the compari­son of relative lattice-parameter change Aa/a

SCHOTTKY AND FRENKEL DISORDER IN KC1 WITH COLOR CENTERS

H. Peisl, R. Balzer, and W. Waidelich I. Phys ika l i sches Insti tut der Technischen Hochschule, Darms tad t , Germany

(Received 9 August 1966)

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