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IMPURITY INTERACTION ANISOTROPY AND
QUANTUM DIFFUSION OF 3He IN SOLID 4He
V. Slusarev, M. Strzhemechny, I.A. Burakhovich
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C6, supplkment au
no
8 , Tome 39, aoiit 1978, page
C6-117
IMPURITY INTERACTION ANISOTROPY AND QUANTUM DIFFUSION OF
3HeI N
SOLID
4HeV.A. S l u s a r e v , M.A. Strzhemechny and 1.A.Burakhovich
PhisicaZ-Technical I n s t i t u t e for Low Temperatures o f t h e Ukrainian Academy of Sciences,
Kharkou,
USSRResum@.- On determine l e p o t e n t i e l d ' i n t e r a c t i o n e n t r e impuretgs 3 ~ e dans 'He s o l i d e 1 grande d i s t a n c e 1 p a r t i r des c o n s t a n t e s B l a s t i q u e s de l a m a t r i c e . On u t i l i s e e n s u i t e l e p o t e n t i e l a i n s i determine pour c a l c u l e r l a s e c t i o n e f f i c a c e de c o l l i s i o n impuriton-impuriton e t l e coef- f i c i e n t d e d i f f u s i o n . On montre que l a s e c t i o n e f f i c a c e d e c o l l i s i o n e s t t r S s s e n s i b l e
1
l ' a n i s o t r o p i e du p o t e n t i e l d ' i n t e r a c t i o n .
A b s t r a c t . - The lon-range i n t e r a c t i o n p o t e n t i a l f o r 3 ~ e i m p u r i t i e s i n s o l i d 'He i s c a l c u l a t e d based on t h e r e a l v a l u e s of t h e dynamic m a t r i x c o n s t a n t s . The p o t e n t i a l found i s used t o c a l - c u l a t e t h e impuriton-impuriton c r o s s - s e c t i o n and t h e d i f f u s i o n c o e f f i c i e n t . It i s shown t h a t t h e a n i s o t r o p y changes e s s e n t i a l l y t h e c r o s s - s e c t i o n . As known, i n t e r a c t i o n of d e f e c t s i n a c r y s - t a l a t l a r g e s e p a r a t i o n s r i s h i g h l y a n i s o t r o p i c :
E .
i n t= u o
13 (:E-P(;)]
+
where a i s t h e n e a r e s t neighbor d i s t a n c e ; m, t h e u n i t i n t e r i m p u r i t y r a d i u s v e c t o r ;P(;),
t h e c h a r a c t e r i s t i c polynomial which depends on t h e c r y s t a l s t r u c t u r e and, g e n e r a l l y , o n t h e e l a s t i c c o n s t a n t s . I n terms o f t h e e l a s t i c i t y t h e o r y and assuming d e v i a t i o n s fromi s o t r o p i c medium p r o p e r t i e s s m a l l , we have c a l c u l a t e d Uo and P(;) f o r bcc and hcp s o l i d helium f o u r . Thus, f o r t h e e x p e r i m e n t a l l y most i n t e r e s t i n g hcp phase
Q:
Uo = (1+
-
(3C3)-Cll-2c13-4C,+*) C11Here Qo i s t h e volume change caused by s u b s t i t u t i o n of one r e g u l a r
article
by an impurity; C i j , t h e e l a s t i c c o n s t a n t s of t h e %e m a t r i x ; m Z , t h e com- ponent o f t h e d i r e c t i o n u n i t v e c o t r p a r a l l e l t o t h e hexagonal a x i s ;B = (C33+ C11- 2C13- 4C44)/(3C33- C11- 2C13- 4C44 ) E s t i m a t e s f o r a molar volume of 20.5 cm3 y i e l d
Uo = 24
mK,
i n agreement with Ref. / I / , and B ;:0.51.The p o t e n t i a l o b t a i n e d was used t o c a l c u l a t e t h e impuriton-impuriton c r o s s - s e c t i o n , t h e mutual s c a t t e r i n g p r o c e s s being t r e a t e d a s s c a t t e r i n g by an immovable c e n t r e . I n t h i s approximation, t h e small- a n g l e s c a t t e r i n g proves t o be most e f f i c i e n t , and t h e problem i s solved a n a l i t i c a l l y . The t o t a l c r o s s -
+ +
s e c t i o n U i s
o
= cr0<lQ(m, n)1
>where
= d
2Lh-2
fivp o t e n t i a l (v being t h e band-averaged impuriton
+ -f v e l o c i t y ) ; Q(m, n)
,
t h e c h a r a c t e r i s t i c polynomial depending on t h e d i r e c t i o n of t h e s e p a r a t i o n vec- -f t o r w i t h r e s p e c t t o t h e c r y s t a l (m) and t h e mutual+
v e l o c i t y ( n ) . C a l c u l a t i o n s show t h a t t h e c r o s s - s e c t i o n w i t h t h e a n i s o t r o p i c i n t e r a c t i o n d i f f e r s e s s e n t i a l l y from t h a t o b t a i n e d u s i n g s t a n d a r d l y t h e+
i s o t r o p i c p o t e n t i a l ( i . e . t h e p o t e n t i a l w i t h P(m) =+
+
0 o r Q(m,n) = 1 ) . Thus, f o r hcp 'He w i t h a molar
3
+
volume of 20.5 cm3, we e s t i m a t e <IQ(rn,n)
I>
5 0.21.Using t h e o b t a i n e d d i f f e r e n t i a l c r o s s - s e c t i o n we have c a l c u l a t e d i n terms of t h e k i n e t i c e q u a t i o n t h e impurity d i f f u s i o n c o e f f i c i e n t f o r t h e s i t u a t i o n where t h e most s i g n i f i c a n t f a c t o r i s t h e mutual im- p u r i t o n s c a t t e r i n g . The f a v o r i n g circumstance, which s i m p l i f i e s t h e t r e a t m e n t c o n s i d e r a b l y , i s t h a t t h e temperatures of i n t e r e s t exceed e s s e n t i a l l y t h e im-
p u r i t o n width, t h e r e b y r e s u l t i n g i n uniform impuri- t y d i s t r i b u t i o n w i t h i n t h e band. F i n a l l y , f o r t h e d i f f u s i o n c o e f f c c i e n t averaged over d i r e c t i o n s i n t h e c r y s t a l we o b t a i n I 3 where Z i s t h e c o o r d i n a t i o n number; V, t h e u n i t v e c t o r s of t h e n e a r e s t neighbors s u b j e c t t o summa- t i o n ; Do, t h e d i f f u s i o n c o e f f i c i e n t w i t h o u t account f o r t h e i n t e r a c t i o n t o t e n t i a l a n i s o t r o p y : 5 Do =
--
Z=I
~ a ~ 2 1 3 3 v r ( 1 / 3 )$7)
-
-
fm (L) UO X being t h e i m p u r i t y c o n c e n t r a t i o n ; J , t h e 3 ~ e - 4 ~ e exchange i n t e g r a l . The q u a l i t a t i v e form of t h e l a s t r e l a t i o n h a s been f i r s t r e p o r t e d by Kagan 121.i s t h e c r o s s - s e c t i o n f o r t h e f i c t i t i o u s i s o t r o p i c
Using t h e e x p e r i m e n t a l d a t a of R e f e r e n c e / 3 / on d i f f u s i o n o f 3 ~ e i n s o l i d 4 ~ e , we e s t i m a t e
J = 2 . 6 x 1 0 - ~ K. Note t h a t t h e " r e n o r m a l i z a t i o n " f a c t o r i s q u i t e a p p r e c i a b l e : D I D o = 3.5.
R e f e r e n c e s
/ I / W. Huang, H.A. Goldberg and R.A. Guyer., Phys. Rev.
11
(1975) 3374.1 2 1 Yu., M., Kagan., Sov., J.Low Temp.Phys.
