extension of the plasma kinetic equation to small wave numbers
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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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VOLUME 17, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 28 NOVEMBER 1966
/-\ 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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V O L U M E 17, N U M B E R 22 PHYSICAL REVIEW LETTERS 28 NOVEMBER 1966
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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