of such a turbulence in the long-wavelength region of the s p e c t r u m in resonance with the b e a m i s s m a l l in the r a t i o of the corresponding phase volumes.
(T,, i s the t r a n s v e r s e t e m p e r a t u r e in the beam). Assuming that A < < E ~ , 16nW we shall in the p r e s e n t paper not take into account the effect of the trapped plasmons on the dynamics of the beam.
' ' ~ c c o r d i n ~ toll8' the development of the collapse leads t o the excitation of r a t h e r strong sound turbulence in the plasma.
However, the conversion of plasma noise i n resonance with '
the beam into density fluctuations which a r e produced by sound does not contribute greatly to the total energy balance a s the c h a r a c t e r i s t i c growth r a t e of such a p r o c e s s cannot exceed the growth r a t e of the modulational instability y M
-
w ~ ( ~ w / M ~ , , T ) ~ ' ~ , and hence r e m a i n s considerably s m a l l e r than v,,.'A. A. Ved:nov, Atomn. i n e r g . 13, 5 (1962).
' ~ a . B. Fainberg and V. D. Shapiro, Zh. Eksp. T e o r . Fiz.
47, 1389 (1964) [Sov. Phys. J E T P 20, 937 (1965)l; V. N.
Tsytovich and V. D. Shapiro, Zh. Tekh. Fiz. 35, 1925 (1965) [Sov. Phys. Tech. Phys. 10, 1485 (1966)).
3 ~ N. Tsytovich and V. D. Shapiro, Nucl. Fusion . 5, 228 (1965); V. D. Shapiro, Doctoral t h e s i s , Inst. Nucl. Phys.
Sib. Div. Acad. Sc. USSR, 1967.
4 ~ A. Vedenov and L. I. Rudakov, Dokl. Akad. Nauk SSSR . 159, 767 (1964) [Sov. Phys. Dokl. 9, 1073 (1965)l.
5 ~ . A. Galeev and R. Z. Sagdeev, Nucl. Fusion 13, 603 (1973).
%. E. Zakharov, Zh. Eksp. 'Teor. Fiz. 62, 1745 (1972) [Sov.
Phys. J E T P 35. 908 (1972)j.
'R. N. Sudan, Proc. 6th European Conf. Controlled Fusion and P l a s m a Phys. Vol. 11, Izd. OIYaI, Dubna, 1974, p. 126.
'B. N. Brerzman and D. D. Ryutov, Nucl. Fusion 14, 873 (1974).
'K. Papadopoulos, Phys. Fluids 18, 1769 (1975).
"A. A. Galeev, R. Z. Sagdeev, Yu. S. Sigov, V. D. Shapiro, and V. I. Shevchenko, Fiz. Plazmy 1, 10 (1975) [Sov. J.
P l a s m a Phys. 1, 5 (1975)l.
"I. P. Panchenko, T h e s i s , Inst. Phys. Acad. Sc. Georgian SSR, 1976.
"B. A. Al'terkop, A. S. Volokitin, and V. P . Tarakanov, Fiz.
Plazmy 3, 37 (1977) [Sov. J. P l a s m a Phys. 3, No. 1 (197711.
1 3 ~ . A. Galeev, Phys. Hot P l a s m a of the Magnetosphere (B.
Hultqvist and L. Stenflo Eds. ) Plenum P r e s s , New York- London, 1975, p. 251.
' * ~ a . B. ~ a l n b e r ~ , V. D. Shapiro, and V. I. Shevchenko, Zh.
Eksp. Teor. Fiz. 57, 966 (1969) [Sov. Phys. J E T P 30, 528 (1970)l.
1 5 ~ . I. Rudakov, Zh. Eksp. T e o r . Fiz. 59, 2091 (1970) [Sov.
Phys. J E T P 32, 1134 (1971)).
I6B. N. ~ r e i z m a n and D. D. Ryutov, Zh. Eksp. Teor. Fiz. 60, 408 (1971) LSov. Phys. J E T P 33, 220 (1971)l.
"B. N. Breizman, D. D. Ryutov, and P. Z. Chebotaev, Zh.
Eksp. Teor. Fiz. 62, 1409 (1972) [Sov. Phys. J E T P 35, 741 (1972)J.
"A. A. Galeev, R. Z. Sagdeev, V. D. Shapiro, and V. I.
Shevchenko, P i s ' m a Zh. Eksp. Teor. Fiz. 24, 25 (1976) [ J E T P Lett. 24, 21 (1976)j.
T r a n s l a t e d by D. t e r Haar
The establishment of a stationary turbulence spectrum due to induced scattering of waves by particles
B. N. Breizman
Institute of Nuclear Physics, Siberian Division, USSR Academy of Sciences (Submitted August 18, 1976)
Zh. Eksp. Teor. Fiz. 72, 518-520 (February 1977)
We consider the problem of the evolution of the turbulence spectrum, taking into account the processes of creation, destruction, and induced scattering of waves. We show that any initial distribution of waves relaxes to a stationary distribution. We estimate the relaxation time.
PACS numbers: 52.35.Ra
In studying plasma turbulence one often has to deal particles. The scattering probability is characterized with the following equation for the occupation numbers by the kernel A&, kt). The actual form of the kernel is
n&, t ) : unimportant for what follows. Essential is only that the
number of quanta is conserved in the scattering. A for-
-n=2ym+n d
S A
( k ; k')n(ke; t)d3k'+er. ma1 expression of this fact is tine antisymmetry of theat kernel with respect t o the interchange of the arguments
k and k':
The problems of the spectra of the turbulence excited by beam o r parametric plasma heating (see, e. g., r1-41)
A ( k ; k') = - A (k'; k )
.
and also some problems in non-linear ~ ~ t i c s , ~ ~ ' ~ ' in par-
ticular, reduce to the solution of this equation. The The third t e r m on the right-hand side of (1) i s the inten- f i r s t term on the right-hand side of Eq. (1) describes sity of the thermal noise source.
the induced emission and a b s o r ~ t i o n Drocesses of waves
(depending on the sign of the growth r a t e y,), and the We assume that Eq. (1) has a stationary solution and second the processes of induced scattering of waves by we consider the problem of how this stationary solution 27 1 Sov. Phys. JETP 45(2), Feb. 1977 0030-5646/77/4502-0271$02.40 O 1978 American Institute of Physics 27 1
is established. It is possible in a number of cases follows:
to find the stationary solution (we denote it by
n,(k)) analytically. From the definition of n,(k) it fol- T = ( n - n . - n . 1 ~ - d3k I r k - (n-n,)'
"
) / nn. d8k.
lows that n , (7)
n. [ ? l k
+
~ ( k ; k') n. ( k ' ) d 3 k ]+
E.-0. A simple study of Eq. (7) a t the maximum shows that the quantity T is a maximum in the final stage of the relaxa- tion process when n and n, a r e close to one another.Hence we find the instability growth rate y, and we sub-
Therefore stitute y, into Eq. (1). As a result we get
n ' n ' 1 n ,
a n-n,
,...
= i n . ( - - 1 ) s k / 2 j e k ( - - 1 ) d 3 k < - m s x - . (8)-
+
nSA
( k ; k') [ n ( k ' ; t ) - n , ( k ' ) ] d 3 k ' . n . n , 2 r etat '=-"- n ,
It is clear from Eq. (8) that for a given spectral function We multiply both sides of this equation by (n -n,)/n and
n, the time T, is inversely proportional to the power integrate over k. Using the antisymmetry of the kernel
A&, k' ) we have of the thermal noise source. In fact, T, is equal to the
time during which the noise source "pumps" into each
where
I = (n-n,-n,
1.2)
n ,
- -
region of k-space just a s many quanta a s should be there (2) in the stationary state. Due t o the low intensity of the
noise source T, is usually considerably longer than the characteristic time f o r the growth of oscillations due to the instability, which equals y;'. We can thus distinguish two stages in the relaxation process. Initially the non- (3) linear stabilization of the instability is reached fast (af-
t e r a time of the o r d e r of 10 y;l) and a "rough" condition The right-hand side of Eq. (2) vanishes only when the
for balance between pumping and damping is ensured to spectrum n(k, t ) coincides with the stationary one. For
hold. In the second (longer) stage the details of the dis- any other distribution of waves it is l e s s than zero." In
tribution of the waves over the spectrum become estab- the non-stationary case the integral I therefore neces-
sarily decreases with time. We note now that this in- lished.
tegral i s bounded f r o m below a s for any values x>O the inequality x
-
1-
l n x > 0 holds. The minimum of the integral I is reached for the stationary distribution of waves.It follows from what we have said that for any initial condition the solution of Eq. (1) tends to a stationary state (provided, of course, that a stationary state exists).
The approach to a stationary state for small deviations from i t has been shown e a r l i e r in[".
Condition (2) enables us to obtain a useful estimate for the maximum number of quanta existing at any mo- ment of the change to the stationary state. To find this we write down the following relation:
In conclusion we note that all the results given here can easily be extended t o the case when scattering may lead to the transformation of the initial waves into waves of another kind. An example of such a process is, say, the scattering of Langmuir waves by ions in the plasma in which they a r e changed into electromagnetic waves.
The generalization of Eq. (2) to the c a s e of several kinds of waves ( n i b ; t), i = l , 2 ,
. . .
) has the following formwhere the quantities Zi a r e introduced by using Eq. (3), viz.
,
where the quantity I,, corresponds to the initial distribu- The author is grateful to D. D. Ryutov for his inter- tion of quanta while N and N, denote, respectively, the e s t in this work-
total number of quanta at time t and in the stationary state (we assume that the number of quanta in the sta- tionary state is finite). For the integral occurring in
Eq. (4) we can write down the following chain of inequal- ' ) T O avoid confusion we point out that ~ t , n ( k , t ) , and n,(k) are
itie s : positive definite functions. The introduction of negative occupa-
tion numbers has a sense only for waves with a negative N energy.
'"
In the present paper suchwaves are not considered.~n".ln~d3k<~n[m;x("-l;)] rz n , n n , d 3 k < L I n d 3 k = - (5)
Combining Eqs. (4) and (5) we find that ' B . N . Brerzman, D . D. Ryutov, and P. Z. Chebotaev, Zh.
Eksp. T e o r . Fiz. 6 2 , 1409 ( 1 9 7 2 ) [Sov. Phys. JETP 95, 7 4 1
N < ~ ( I , + N , ) . ( 1 9 7 2 ) l .
e - l (6) ' B . N . ~ r e r z r n a n , V. E . Zakharov, and S. L. Musher, Zh.
Eksp. ' r e o r . F f z . 64, 1297 (1973) [Sov. Phys. JETP 37,
We turn now to an estimate of the characteristic re- 658 ( 1 9 7 3 ) l .
laxation time T. It is natural to introduce this time a s ' E . Valeo, C . Oberman, and F . W . Perkins, Phys. Rev. Lett.
272 Sov. Phys. JETP 45(2), Feb. 1977 0. N. ~r&zrnan 272
28, 340 (1972). 1486 (1974) [Radiophys. Qu. Electron. 17, 1136 (1976)).
4 ~ L. . K r u e r and E. J. Valeo, Phys. Fluids 16, 675 (1973).
'v.
M. Dikasov, L. I. Rudakov, and D. D. Ryutov, Zh. Eksp.5 ~ . A. Akhmanov and Yu. E. D'yakov, Pis'ma Zh. Eksp. Teor. Teor. Fiz. 48, 913 (1965) [Sov. Phys. J E T P 21, 608 (1965)l.
Fiz. 18, 519 (1973) [ J E T P Lett. 18, 305 (1973)l.
%. A. Pasmanik and M. S. Sandler, Izv. vuzov Radiofiz. 17, Translated by D. t e r Haar
Charge movement in crystalline helium
K.
0.
KeshishevInstitute of Physical Problems, USSR Academy of Sciences (Submitted March 5, 1976)
Zh. Eksp. Teor. Fiz. 72, 521-544 (February 1977)
The mobilities of the carriers in crystalline helium are determined by investigating the diode volt-ampere characteristics and the temperature dependence of the currents induced by a tritium source. The diode current is found to be proportional to the square of the voltage in a comparatively narrow range of voltages and temperatures. In the range of high voltages, the form of the volt-ampere characteristics is determined by the dependence of the drift velocity of the charges on the electric field strength. For direct measurement of the camer drift velocity, a three-electrode time-of flight method is employed. The temperature dependences of the positive and negative carrier mobilities are measured at different molar volumes. The dependence of the carrier drift velocity on the electric field strength is also measured. The data are compared with the results of existing theoretical studies.
PACS numbers: 67.80.Mg
The f i r s t experiments on the movement of c h a r g e s in 2 ) measurement of the charge velocity by a t h r e e - solid helium w e r e made by ~hal'nikov"] and w e r e con- electrode time-of-flight method.
tinued in Refs. 2 and 3. In these studies specially grown
helium c r y s t a l s s e r v e d a s the f i r s t object of the investi- CONSTRUCT~ON OF THE APPARATUS gation. (An attempt a t c u r r e n t measurement in poly-
crystalline s a m p l e s obtained by the blocked capillary method showed that the c u r r e n t s in t h i s c a s e w e r e not amenable t o m e a ~ u r e r n e n t . ~ ~ ' ) In Refs. 1-3, the helium c r y s t a l s w e r e grown f r o m the liquid phase a t constant p r e s s u r e in a cylindrical g l a s s container. The growth r a t e was determined by the t e m p e r a t u r e gradient along the container axis. The growth p r o c e s s could be moni- tored visually. The d e c r e a s e in the induced diode cur- rent during the solidification p r o c e s s was the c r i t e r i o n of the quality of t h e c r y s t a l sample. Thus, in the solid phase, the c u r r e n t amounted a t most t o 2-3% of the c u r r e n t in liquid helium f o r the w o r s t s a m p l e s and reached 60% f o r the better samples. Usually, the bet- t e r samples w e r e obtained a t a growth r a t e of
-
5 p m /sec.
The described method of obtaining the c r y s t a l s was used by Mezhov-Deglin in the study of the thermal con- ductivity of solid The r e c o r d values of t h e r m a l conductivity measured by him, up to 50 W/cm-deg, and the absence of an "annealing" effect, indicate that the samples obtained under these conditions w e r e very near- ly perfect single crystals. A d i r e c t verification of this fact was obtained in Refs. 6 and 7 by x-ray analysis.
The purpose of the present work was the study of the p r o c e s s of charge t r a n s f e r in the He4 c r y s t a l s obtained
The construction of the apparatus used in the present r e s e a r c h and the method of growing t h e c r y s t a l s were described in detail in Refs. 2 and 5. The c r y s t a l s were grown in containers of two types: g l a s s (diameter 10 m m ) and metal (diameter 40 mm). Inside each of them, depending on the goal of the experiment, was placed a diode o r triode electrode system. The construction of the g l a s s container with a s m a l l triode inside is shown in Fig. l a . Rings of f e r r o c h r o m e (alloy N47KhB) w e r e
fSS Copper Molybdenum
Ferrochrome Glass FZZB Stainless steel Mica -
by the described method. The r e s e a r c h followed two
basic directions: FIG. 1. Construction of containers and triodes. l,6-housing;
2,7-bottom with cold conductor; 3,8-P-active electrode; 4, 1 ) measurement of the c u r r e n t s induced in the diode; 10-collector; 5,9-grid; 11-guard ring; T-thermometers.
273 Sov. Phys. JETP 45(2), Feb. 1977 0038-564617714502-0273$02.40 O 1978 American Institute of Physics 273
