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Submitted on 1 Jan 1981
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INTERACTION OF SOUND WITH VACANCIES IN
bcc 3He CRYSTALS
I. Iwasa, H. Suzuki
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C 6 , suppZiment au n o 12, Tome 42, dicembre 1981 page C6-896
INTERACTION OF S O U N D W I T H VACANCIES IN
bcc ' ~ eCRYSTALS
I. Xwasa and H . Suzuki
Department o f Physics, University o f Tokyo, Bunkyo-ku, Tokyo 113, Japan
A b s t r a c t . - L o n g i t u d i n a l sound v e l o c i t y was measured i n b c c 3 ~ e c r y s t a l s a t f r e q u e n c i e s of 1 0 , 30 and 50 MHz a s a f u n c t i o n o f t e m p e r a t u r e . Above 0.4 K , t h e v e l o c i t y always d e c r e a s e d w i t h i n c r e a s i n g t e m p e r a t u r e and no s y s t e m a t i c dependence on f r e q u e n c y was o b s e r v e d . The r e s u l t was i n t e r p r e t e d by t a k i n g i n t o a c c o u n t t h e c o n t r i b u t i o n of v a c a n c i e s t o t h e sound v e l o c i t y .
W e have measured t h e l o n g i t u d i n a l sound v e l o c i t i e s i n b c c 3 ~ e c r y s t a l s w i t h a r e s o l u t i o n o f 2 . 1 0 - ~ a t f r e q u e n c i e s o f 1 0 , 30 and 50 MHz a s a f u n c t i o n o f t e m p e r a t u r e . The r e l a t i v e change i n t h e v e l o c i t y Av/vo = ( v ( T ) - v o ) / v o is p l o t t e d i n F i g s . 1 and 2. The d a t a i n F i g . 1
3
a r e t a k e n f o r a c r y s t a l grown a t 33 atm (24.4 cm / m o l e ) . The v e l o c i t y change below 0 . 3
K
i s h y s t e r e t i c and a m p l i t u d e - d e p e n d e n t due t o d i s - l o c a t i o n s and i s d i s c u s s e d elsewhere.') Above 0 - 4 K t h e v e l o c i t y d e c r e a s e s w i t h t e m p e r a t u r e . The d a t a i n F i g . 2 t a k e n f o r a c r y s t a l3
grown a t 3 8 atm ( 2 3 . 8 c m /mole) show no s y s t e m a t i c dependence on t h e f r e q u e n c y
.
Our r e s u l t s a r e c o n s i s t e n t w i t h t h o s e o f Wanner, M u e l l e r and
airb bank.^)
They a n a l y z e d t h e d a t a by a 2-parameter e q u a t i o n2 2 4 6
p [ v ( T ) - v O ] = RT +ST
,
(1)where p i s t h e d e n s i t y and R and S a r e p a r a m e t e r s . The e q u a t i o n was d e r i v e d by u s i n g t h e Mie-Gruneisen e q u a t i o n o f s t a t e f o r a s o l i d
t o g e t h e r w i t h t h e s p e c i f i c h e a t . The f i r s t t e r m on t h e r i g h t - h a n d s i d e
4
was a d i r e c t consequence o f t h e l a t t i c e s p e c i f i c h e a t and t h i s T
-
dependence was o b s e r v e d i n h c p 4 ~ ecrystal^.^)
The s e c o n d term was r e l a t e d t o t h e e x c e s s s p e c i f i c h e a t ( h i g h - t e m p e r a t u r e anomaly) which2 was d e s c r i b e d by a T-dependent Debye t e m p e r a t u r e 8 (T) = O 0 (1-aT )
.
Although e q . ( l ) was w e l l f i t t e d t o t h e i r d a t a , t h e i r d e r i v a t i o n o f t h e s e c o n d term was u n j u s t i f i a b l e b e c a u s e t h e t e r m was comparable t o o r e v e n g r e a t e r t h a n t h e f i r s t t e r m .
The r e c e n t s p e c i f i c h e a t measurement o f fSreywal14) h a s shown t h a t t h e h i g h - t e m p e r a t u r e anomaly i s p r i m a r i l y c a u s e d by v a c a n c i e s . I n t h e f o l l o w i n g w e w i l l show t h a t t h e o b s e r v e d t e m p e r a t u r e dependence o f t h e
N O . 8 X 4 = 4 8 0 p p m
1
o r u n l 0 > \ -4-
Vo I 490 m l s > -a-
V"= 24 39 cm31rnole -6 - T, = 0 . 7 0 K 0 2 . 4 . 6 T E M P . ( K ) I I I I I l I I 1 NO. 16 X q = 200 ppm 0 10 MHz 30 MHz x 5 0 MHz V O s 5 2 1 m l s VM = 23.80 em3 I mole Tm=0.93 K-
0 .2 . 4 .6 .8 T E M P . ( K )F i g . 1 : Temperature dependence F i g . 2 : Temperature dependence o f t h e v e l o c i t y a t 10 MHz on a o f t h e v e l o c i t y a t 1 0 , 30 and c o o l i n g
r u n
( r u n 1) and a h e a t - 50 MHz. Phase s e p a r a t i o n d i d i n g run ( r u n 2 ).
The d r o p i n n o t o c c u ri n
t h i s c r y s t a l . t h e v e l o c i t y a t 0.08 K and t h e The c u r v e i s c a l c u l a t e d from h y s t e r e s i s below 0 . 3 K a r e due e q . ( 4 ) w i t h k = -1.10 and t o d i s l o c a t i o n s m u l t i p l i e d byL =
-0.55.phase s e p a r a t i o n . The c u r v e i s c a l c u l a t e d f r o m eq. ( 4 ) w i t h k = -1.53 and
L
= -0.51.sound v e l o c i t y i s c o n s i s t e n t w i t h t h e s p e c i f i c h e a t which
i s
composed o f t h e l a t t i c e and vacancy t e r m s .According t o Greywall t h e s p e c i f i c h e a t o f a b c c 3 ~ e c r y s t a l
is
a sum o f (1) t h e l a t t i c e t e r m C8 w i t h t h e Debye t e m p e r a t u r e 0 , ( 2 ) t h e vacancy t e r m C w i t h t h e vacancy f o r m a t i o n energy Q , ( 3 ) t h e exchange
4
t e r m CJ w i t h t h e exchange e n e r g y J , and ( 4 ) t h e r e s i d u a l e x c e s s s p e c i f i c h e a t Cx whose o r i g i n i s n o t g i v e n . Each o f t h e f i r s t t h r e e major t e r m s h a s a f u n c t i o n a l form f (a/T) where
a =
8 , @ o r J. Then t h e g e n e r a l i z e d Mie-Griineisen e q u a t i o n of s t a t e r e a d sC6-898 JOURNAL DE PHYSIQUE
where YE, = d l n Y,/d l n V a n d w e s e t YE, = 0 f o r n u m e r i c a l e s t i m a t i o n s . A s t h e b u l k modulus i s a l i n e a r c o m b i n a t i o n o f t h e e l a s t i c c o n s t a n t s which i n t u r n a r e r e l a t e d t o t h e s o u n d v e l o c i t y , t h e r e l a - t i v e c h a n g e i n t h e v e l o c i t y c a n b e w r i t t e n a s
where k and
R
a r e a d j u s t a b l e p a r a m e t e r s . Bo, B1 and B 2 a r e c a l c u l a t e d from t h e d a t a o f t h e s p e c i f i c h e a t 4 ) and t h e e l a s t i c c o n s t a n t s . 5 ) E q u a t i o n ( 4 ) i s t h e n f i t t e d t o o u r v e l o c i t y d a t a . The c u r v e s i n F i g s . 1 and 2 a r e c a l c u l a t e d f o r L = - 0 . 5 1 w i t h k = 3 R a n d R = - 0 . 5 5 w i t h k = 2L, r e s p e c t i v e l y . E q u a t i o n ( 4 ) i s p h y s i c a l l y more r e a s o n a b l e t h a n e q . (1) : A s t h e b u l k modulus i n c r e a s e s a n d t h e l o n g i t u d i n a l sound v e l o c i t y d e c r e a s e s w i t h t e m p e r a t u r e , t h e s h e a r modulus i n t h e i s o t r o p i c a p p r o x i m a t i o n s h o u l d d e c r e a s e . A s i m p l e i n t e r p r e t a t i o n o f t h e s e c o n d termi n
e q . ( 4 ) i s t h e n t h a t t h e s h e a r modulusi s
d e c r e a s e d by v a c a n c i e s b e c a u s e t h e r e a r e no b o n d i n g s a t t h e v a c a n t s i t e s . A s f o r t h e f i r s t t e r m , t h e p r e - s e n t v a l u e s o f k = - 1 - 5 3 and -1.10 a r e c o m p a r a b l e t o t h o s e o b t a i n e d f o r h c p 4 ~ e ( - 0 . 8 - - 1 . 8 ) . I n c o n c l u s i o n , t h e c o n t r i b u t i o n o f v a c a n c i e s t o t h e v e l o c i t y which was n o t c o n s i d e r e d by Wanner e t a l . s h o u l d b e t a k e n i n t o a c c o u n t . We t h a n k R. Wanner f o r u s e f u l d i s c u s s i o n s .R e f e r e n c e s
1) I . I w a s a , N . S a i t o and H. S u z u k i : t o b e p u b l i s h e d .
2 ) R. Wanner, K . H . M u e l l e r , J r . a n d H.A. F a i r b a n k : J . Low Temp. Phys. 1 3 , 1 5 3 ( 1 9 7 3 )
-
3 ) I . I w a s a and H . S u z u k i : J . Phys. Soc. J p n .
