KINETIC

EQUATION

PARTICLES I.

i.

FOR

HIGH-ENERGY

INTERACTING Ol'khovskii

WITH

and

M.

LOW-MASS

A CRYSTAL

B.

Ependiev

UDC 531.19:539.2.01:539.188

F r o m a previously obtained kinetic equation describing the motion of an attenuated gas of fast particles in a c r y s t a l , an equation f o r l o w - m a s s p a r t i c l e s is obtained, taking into a c count the r e c o i l of p a r t i c l e s of the c r y s t a l in the interaction; in p a r t i c u l a r , an equation is obtained for the case of the potential of absolutely solid s p h e r e s of sufficiently s m a l l radius. 1. In [1], a kinetic equation was obtained for an attenuated gas of fast g particles moving in a medium of c p a r t i c l e s , in the f o r m OF,,n

,

OF1 o

Ot

=

(1)

Oqg

7-:-T-y_ {FI,0(Q~' ,

, l)

0,I,~ . . . . .

t)) dq~dvC;

Fi, 0 and F0, i a r e the distribution functions of g and c p a r t i c l e s , r e s p e c t i v e l y ; ~ = ~(Iclg -- qC I) is the i n t e r a c tion energy of t h e s e p a r t i c l e s ; and ~rtl,l)

,,

V~'I,I)

(2)

where the dynamic o p e r a t o r sT(i'i) may be written in the f o r m [2, 3] A

S~TM = exp {~LOa)}; ^L (1:) = v # - -0 + ~oc - 0 1 80 0 Oqg Oq~ rng Oqg Ovg

I O0 0 m~ Oqc Ov C "

To t r a n s f o r m Eq. (2), the following changes of variable and notation a r e introduced q = q g - - qr

u = v~ - - v c,

q g == r ,

(3)

v~ = v .

In t h e s e variables

^

^ 0 LO,U = v ar 0

^

I aO 0 ^ ~ L; r% aq c)v 1 00 0 (~: m~mc

L = L ( q ' u ) = n Oq

~

Oq

c)g

]

r n g - ~ rrl c

,

Note also that, for integer n >_ 1, the following relations a r e identically t r u e ^ (U:.u)

" vg

=-

--

^ (.L) "-I

~

1 00

mg A

^

- -

^

;

(LO:))"

Vc =

--

(LO,l)Yq ~ = ( t o , D p -~ v;

A (L) "-1

~

1 0r

~ -

m c Oq"

Oq A

A

(LOJ})nq ~ = ( L ( U y -~ (~ - - u ) ,

and it will be expedient to use the integral r e p r e s e n t a t i o n

M. V. L o m o n o s o v Moscow State University. T r a n s l a t e d f r o m I z v e s t i y a u Uchebnykh Zavedenii, F i z i k a , No. 10, pp. 7-12, October, 1978. Original a r t i c l e submitted F e b r u a r y 14, 1977.

0038-5697/78/2110-1255507.50 9 1979 Plenum Publishing Corporation

1255

[~-~(__ ,.,;)n(LO,')f' A A ) ,-,.oo= ~

(S~;') A)-...~ = A + ~

~L(1,1)~(1,I)Ad'~..

A - - j ,.,_= 0

which is valid f o r any function A ( r , q, v, u).

Then Eq. (2) takes the f o r m

q ~ : ) = r + s (q, u);

q~"~ = r - q - -ts (q, u);

u); -v(',', , =v--u--Tw(q,u)

v~'" = v + w ( q ,

( ~ = m_~) ,

(4)

where

,=L

,s_:~

m, Jo.

Oq

a:;

= • f s _ ) * n:; ra, J~

Oq

Substituting Eqs. (3) and (4) into Eq. (1) and taking into account that

{r,.o (0(,',', v(~'>, t) eo., (0:'"'. v<:'', 0} au

= o,

the following r e s u l t is obtained

OF,.o. + v OF,,o = -n-e ~ c)(D~ {F,,o (r + s, v + r Ov %J Ot dr 2.

t) Fo., (r -- q -- 7s, v -- u -- 7r

/)} aq clu.

(5)

The distribution function F0,1 f o r the component is d e t e r m i n e d f r o m a chain of equations [1] 0Fo~ , = {He:, Fo~} ' +

Ot

X -nc OFo.s+, c - S *0 i..~*, 0 ~ d q ,~+ , d v ,~+ , ,~i~s rac Oq~ v~

(s=

1, 2 . . . ) .

(6)

Introducing the variables t = t / A t , ~c = q C / r c , vC = vC/v0c, and ~c = r c ' where r e and ~c a r e the c h a r a c t e r i s t i c radius and e n e r g y o f c - - c interaction, v0C is the mean c - p a r t i c l e velocity, and At is the t i m e n e c e s s a r y to establish the kinetic s t a t e of the g p a r t i c l e , Eq. (6) may be written in d i m e n s i o n l e s s f o r m

0 0, no(re) 3

v

+ ,__

~ 0 ,.s+t

"--Z

051, (oL, qs+ldvs+,}

(s

0T0 / 1,2. .),

~C

where % = rc/voC; e = me (voC)2/~c. Now note that the a r g u m e n t of the shock function F0, i in Eq. (5) depends on the g - - c interaction, whereas the f o r m of F0,1 is d e t e r m i n e d by the chain in Eq. (6), in which no quantities a s s o c i a t e d with the g - - c p r o c e s s a p p e a r . T h e r e f o r e , in the r i g h t - h a n d side of Eq. (7), all the dimensionless variables and t h e i r derivatives a r e of o r d e r unity, and hence a F 0 , s / ~ " ~ A t / r c. If the behavior of g p a r t i c l e s in a steady infinite c r y s t a l is to be investigated using Eq. (5), the following conditions must be imposed .%t (( % , L )> vog At,

(8)

lO-t3 - i0-12 sec f o r the c r y s t a l and where L is the c h a r a c t e r i s t i c l i n e a r dimension of the c r y s t a l . Since r e in e x p e r i m e n t s , usually, L ~ 10 -2 c m , it is r e a s o n a b l e to take At ~ 10 -~5 sec f o r a g p a r t i c l e -- a proton of e n e r g y ~1 MeV. In fact, in this t i m e a proton m a y t r a v e l a distance ~v0g&t ~ 10 -5 c m , which c o r r e s p o n d s to t h e t h i c k n e s s o f ~ 10 s a t o m i c l a y e r s of the c r y s t a l , and a f t e r a time At a kinetic state of the g p a r t i c l e s will cert,~_n]y have b e e n r e a c h e d . T h u s , the steady equilibrium function will be used as the distribution function of the c r y s t a l

Fo.s ----Fs {q~... qe) I-I B (v~), l
1256

(9)

where B (v) = (mj2r.O~) ~'- exp

m~ v "~20~}.

{--

The equation f o r the configuration function F s is obtained by multiplying Eq. (6) t e r m by t e r m by vlC and i n t e grating o v e r all v e l o c i t i e s . Then, taking account of Eq. (9), the following s t e a d y chain is obtained [2] ,~~ ~arqlT n ~, O~ ~a ~ . -7 I
~ar aqf""

~ F ~ + Id q~§

(10)

( s = l, 2...),

and this is used to i n v e s t i g a t e the e q u i l i b r i u m steady s t a t e of the c r y s t a l [4]. 3.

With a view to the substitution of Eq. (9) into Eq. (5), B(v) is written in the f o r m of an expansion B(v) = ~ ( v ) + gc

d~ ~ ( v ) + . . .

(11)

2rnc avov

This is p o s s i b l e if the m e a n - s q u a r e deviation of the g - p a r t i c l e velocity ((v

--

~)2), 2 >> (0~ me)' ~.

(12)

F o r c r y s t a l s in which O c ~ 0 . 0 1 - 0 . 1 eV, and for f a s t p r o t o n s , f o r which the k i n e t i c - e n e r g y s c a t t e r ~1 keV, Eq. (12) is s a t i s f i e d . In this e a s e , only the f i r s t t e r m need be retained in the expansion in Eq. (11), and it is a s s u m e d that Fo,, = F, (go) ~ ( v g .

(13)

Substituting Eq. (13) into Eq. (5) and introducing the notation f = F1,0, F = neF0,1, the r e s u l t obtained is

Of , v O f _ at

' ~

1 ~o~ m~

aq

O_O_K(r, q, v, t)dq,

(14)

a~

where

K= ~f(r +s, v + w, t) F(r--q--Ts)~,(e--tt--.;w)du.

(15)

Now c o n s i d e r the c a s e when T = m g / m c << 1 (this condition is obviously s a t i s f i e d f o r the s c a t t e r i n g of protons and neutrons on single c r y s t a l s ) . Substitution of the expansion

q)+... FL(r -- q -- Ts) = F (r -- q) -- Ts O---F(r-Or ~(,v- u-

-,'w) = ~ ( v -- u) -~- ..... , ~ aau. o ( v - u )

!-.

. .

into Eq. (15) and integration with r e s p e c t to u g i v e s , ignoring t e r m s of o r d e r l a r g e r than T, the following result

K= f(r§

v§

t) F ( r - - q ) - - ; , s ~ F ( r - - q )

[wf(r=s,

v--w.

(16)

t) F ( r - - q ) ] 9

u = ~

~

It is now expedient to introduce the functions c~

s:~ =

=S_= (q, v) -~q d:,

W .

9

1

=

,~1

- -

-~q d:, i'S_=(q, v)aq~ 0

where A S:(q, v) =exp,.L,,,..

?' L L, = ] .....

1257

Then sl.~.=s~(l

--

-~,) L ,

wl==.=w~(1

*''1

.

.

;).

.

and

f (r + s, v + w, t).=o = f (r ~, v*, t) -- 7 (r* -- r) ~r f (r* , v*, l) -- 7 (v* -- v) ~ f (r*, v*, t ) + . . . ,

(17)

w h e r e r * =r +sg, v* =v+w/~. Note that the o p e r a t o r I~0 has the p r o p e r t y A

-Lot ~ -

9 . =.

._ ~),

A

Lov ~ =

O.

(18)

Substituting Eq. (17) into Eq. (16) gives

K = f ( r * , v "~, t ) F ( r - - q ) - - ~ ( r * - - r )

;

o

[f(r*,v*, t)F(r--q)]--7~v[(V*--v)f(r*,

v*, t ) F ( r - - q ) I.

(19)

Finally, taking into account Eq. (19), Eq. (14) yields the equation

•

at

~, ~

~ - - - a lf(r*, v*, t)F(r--q)]d~ m~ J aq Ov

ar

(r*--r)

If(r*,v*,t)F(r--q)l

+~vl(v*--v)f(r*,v*,t)F(r--q)]

(20)

dq,

where

r.=r+l

~:S_.d~I)d:;

j

oq

O.=V+

I ~'S

a(Dd

TJ -:Tq

0

:"

0

Equation (20) is a kinetic equation f o r fast l o w - m a s s g p a r t i c l e s , a c c u r a t e to t e r m s of o r d e r T, taking into account the r e c o i l of atoms of the equilibrium c r y s t a l . The functions r* and v* a r e d e t e r m i n e d by the dynamic motion of a g p a r t i c l e of m a s s ~ in the field of a fixed s c a t t e r i n g c c e n t e r with potential r 4.

In the approximation as T -~ 0, Eq. (20) may be written in the f o r m 1

IOq~a

~+V~=I-~-~q~-~p{f(r~,

v*, t) F(r--q)}dq

(re=me),

(21)

where it has been taken into account that p = mg. This equation d e s c r i b e s the behavior of a p a r t i c l e without taking into account the r e c o i l of atoms of the c r y s t a l . Suppose that q _~ r 0 is the significant region of interaction; this means that r < r 0. The spatial inhomogeneity of the function f is d e t e r m i n e d by the inhomogeneity a@ of the function F. T h e r e f o r e , if r 0 << a o , it may be a s s u m e d that r* = r and F ( r - q) = F ( r ) i n t h e region q _~ r 0. Then the integral on the right-hand side of Eq. (21) is

l=l

C ar -~of(r, c) (r) 3~q v*, t)dq. q~ro

Hence, taking into account Eq. (18) when p = m, the following r e s u l t is obtained

/ : F ( r ) f V ~ q f ( r ' v*' t ) d q : r ~ q~r~

where ds

,,i v q f ( r ' v% q =

ro

is an element of solid angle.

On the s p h e r e q = r0, v* = v ff qv < 0 and v* = v ' ff qv > 0; v ' is the velocity which the p a r t i c l e must have before s c a t t e r i n g If, a f t e r s c a t t e r i n g , It moves with velocity v. Hence, it follows f r o m Eq. (22) that

1258

(22)

1 =r~(ro)~-vF(r)

qvf(r, v', t)d-~

v, t) .

(23)

qv>O

Since the i n t e r a c t i o n is c e n t r a l , the v e c t o r s v ' and q may be written in the f o r m of an expansion in the unit vectors v' = v (n~,cos 0 + n : sin 0); q = q (n~,cos ,~ --' n_ sin,~), where n v = v / v ; n •

O.

In the c a s e of the potential of an absolutely solid s p h e r e

(I)(q)=

{~, O,

the angles 0 and ~ a r e r e l a t e d as follows: 0 = u -- 2r m a y be t r a n s f o r m e d to give 2g

O

q < ro q~.~-ro

Then, taking into account that the i n t e g r a l in Eq. (23)

r.'2

~r.

fl

0

-

O

Equation (21) m a y be r e p l a c e d by the following r e l a t i o n

Of, vOf r.(ro)~_vF(r){l ~ f ( r , vn, t)dO~--f(r, v, t)} Ot r Or =

(24)

This equation d e s c r i b e s the motion of f a s t p a r t i c l e s , without e n e r g y l o s s , in a c r y s t a l c o n s i s t i n g , f r o m the point of view of g - - c i n t e r a c t i o n s , of m a s s i v e absolutely elastic s p h e r e s of v e r y s m a l l r a d i u s . LITERATURE

CITED

J

1.

2. 3. 4.

I. I. O l ' k h o v s k i i , M. B. Ependiev, and N. M. Sadykov, Izv. V y s s h . Uchebn. Z a v e d . , Fiz. ~ No. 4, 79

(1977). N. N. Bogolyubov~ P r o b l e m s of Dynamic T h e o r y in Statistical P h y s i c s [in Russian], M o s c o w - - L e n i n g r a d (1946). P. Libov, Introduction to K i n e t i c - E q u a t i o n T h e o r y [in Russian], Moscow (1974). I. I . O l ' k h o v s k i i , Izv. V y s s h . Uchebn. Z a v e d . , F i z . , No. 1 (1976).

1259