ACOUSTIC SHEAR IMPEDANCE OF SUPERFLUID 3He-B

(1)

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ACOUSTIC SHEAR IMPEDANCE OF SUPERFLUID

3He-B

Hiromichi Ebisawa

To cite this version:

(2)

JOURNAL DE PHYSIQUE Colloque

C6,

supplPment au no

8,

Tome

39,

aolit

1978,

page

_{~ 6 - 4}

ACOUSTIC SHEAR IMPEDANCE OF SUPERFLUID

3He-B Hiromichi Ebisawa

Department

of

Applied Science

Tohoku University, Sendai, Japan

R&sum&.- Une e x p r e s s i o n de lfimp&dance a c o u s t i q u e du mode t r a n s v e r s e e s t obtenue dans l e kCgime

Hw

<< 28 e t w.r = pour l a phase B de 3 ~ e s u p e r f l u i d e .

Abstract.- An e x p r e s s i o n f o r t h e a c o u s t i c s h e a r impedance of t h e s u p e r f l u i d B phase of 3 ~ e i s o b t a i n e d f o r

yw

<< 2A and w r = m

.

Recently evidences of e x i s t e n c e of t r a n s - v e r s e zero sound a r e r e p o r t e d

111

i n t h e normal and i n t h e s u p e r f l u i d phases of 3 ~ e . I n t h e s u p e r f l u i d phase t h e o b s e r v a t i o n i s based on t h e measurement o f t h e t r a n s v e r s e a c o u s t i c impedance.

T h e o r e t i c a l l y , o n l y t h e d i s p e r s i o n r e l a - t i o n i s s t u d i e d i n t h e s u p e r f l u i d phases by Combes- cot-Combescot 121, Bolton 131, Maki-Ebisawa 141 and W'6lfle 151. A s Landau p r e d i c t e d , t h e t r a n s v e r s e zero sound can propagate i n t h e Fermi l i q u i d i f t h e Landau parameter s a t i s f i e s F , > 6 . The v e l o c i t y de- c r e a s e s from t h e normal s t a t e v a l u e a s temperature goes down through t h e c r i t i c a l temperature, e x c e p t n e a r t h e r e g i o n of resonance of t h e o r d e r parameter c o l l e c t i v e mode where t h e v e l o c i t y h a s a s t r u c t u r e and t h e a t t e n u a t i o n has a peak, u n t i l t h e sound c a n no longer propagate because F1 i s e f f e c t i v e l y reduced 12-51.

We e x p e c t t h a t , i n t h e r e g i o n where t h e sound can propagate, t h e a c o u s t i c impedance behaves j u s t i n t h e same f a s h i o n a s t h e v e l o c i t y does on t h e analogy o f c l a s s i c a l formula Z = p c where p

i s t h e d e n s i t y and c i s t h e v e l o c i t y . To s e e t h i s d i r e c t l y i t i s i n t e r e s t i n g t o g e t an e x p r e s s i o n of t h e impedance. However t h e impedance f o r t h e strong- l y damped mode i s determined by t h e q u a s i p a r t i c l e s n e a r t h e w a l l of t h e c o n t a i n e r through the Landau damping. So t h e s t u d y of t h e impedance g i v e s some i n f o r m a t i o n s about t h e n a t u r e of o r d e r i n g n e a r t h e w a l l i f i t i s d i f f e r e n t from t h a t i n t h e bulk a t a l l . We n o t i c e t h a t t h e p e n e t r a t i o n depth (may be o r d e r of vF/w%l-lO~, u s u a l l y ) i s of n e a r l y t h e same o r d e r a s t h e d i p o l e h e a l i n g l e n g t h .

For t h e f i r s t s t e p of works f o r t h a t pur- pose, I p r e s e n t h e r e an e x p r e s s i o n of t h e a c o u s t i c impedance i n t h e B phase f o r t h e frequency range

2and wr>>l under a simple assumption on boundary c o n d i t i o n s .These r e s t r i c t i o n s on t h e frequency c o n f i n e our c o n s i d e r a t i o n t o t h e temperature range below t h e r e g i o n of i n t e r e s t i n g p o s s i b l e s t r u c t u r e o f t h e impedance and only t o t h e r e a l p a r t . Fur- theremore t h e Landau parameter i s n e g l e c t e d except F and F 1 f o r s i m p l i c i t y .

0

The a c o u s t i c impedance i s t h e r a t i o of t h e momentum flow t o t h e v e l o c i t y of t h e l i q u i d ans i s

c a l c u l a t e d a t t h e boundary between t h e l i q u i d and t h e t r a n s d u c e r . The theory of t h e impedance i n t h e normal Fermi l i q u i d i s developed by Bekarevich- Khalatnikov / 6 / and Flowers-Richardson-Williamson 171. They solved t h e l i n e a r i z e d Landau t r a n s p o r t e q u a t i o n i n a h a l f space w i t h a p p r o p r i a t e boundary c o n d i t i o n s . Using t h e Wiener-Hopf f a c t o r i z a t i o n , they o b t a i n e d e x a c t s o l u t i o n s . I n t h e s u p e r f l u i d phase, t h e k i n e t i c e q u a t i o n i s matrix-formed ( s e e I 8 1 where the o s c i l l a t i o n of t r a n s v e r s e c u r r e n t i s

coupled with t h e f l u c t u a t i o n of o r d e r parameter. I n t h e p r e s e n t c a s e , Mw<<2A, t h e e q u a t i o n i s much s i m p l i f i e d a s i s d e r i v e d i n r e f e r e n c e s / 2 / and 151. I n t h e e q u a t i o n , t h e v e l o c i t y of q u a s i p a r t i c l e s vF i n ~ a l a r i s reduced t o vF Sk/Ek where E 2 = 5 k 2 + ~ 2 , and d e v i a t i o n s of t h e d i s t r i b u - k t i o n f u n c t i o n s a r e p r o p o r t i o n a l t o - f ' ( ~ - a f ( E )/aE ) k k

where f(Ek) i s t h e Fermi d i s t r i b u t i o n f u n c t i o n . I use t h e boundary c o n d i t i o n t h a t Bogolons w i t h mo-

mentum d i r e c t e d i n s i d e t o t h e l i q u i d a t t h e boun- d a r y have t h e e q u i l i b r i u m d i s t r i b u t i o n on t h e moving frame w i t h a v e l o c i t y u p a r a l l e l t o t h e w a l l , i . e .

-

m:vFft, and t h e t o t a l mass flow i s mupY, where Y i s Yoshida f u n c t i o n given below e q u a t i o n

( 6 )

-

The impedance i s given by 3+FlY

Z = pVF

7

(SO-1

+

@ )

1 (1)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978602

(3)

dx tan-' ( A ~ / A ~ ) / X ~

X X+I

w (X) =

7

l o g

-

_x-

₁

-

I

(5)

where

2

s i s t h e s o l u t i o n of A(s)=O. I f t h e r e i s no s o l u t i o n t h e n so-1 should be d i s r e g a r d e d i n e q u a t i o n ( I ) . These e x p r e s s i o n s a r e simple exten- s i o n of those given i n r e f e r e n c e s / 6 / and 171. We should n o t i c e t h a t o t h e r assumptions on t h e boundary c o n d i t i o n might give more complicated o r even no e x a c t s o l u t i o n . Numerical e v a l u a t i o n o f the impedance h a s n o t been f i n i s h e d .

F i n a l l y I give some remarks on t h e next s e v e r a l s t e p s . ( i ) For t h e A phase, t h e impedance depends s t r o n g l y on the d i r e c t i o n of & r e l a t i v e t o t h e w a l l . I t is r a t h e r e a s y t o perform some a n g u l a r i n t e g r a l s i n t h e case p e r p e n d i c u l a r t o t h e w a l l . ( i i ) A t temperatures where c o l l e c t i v e modes a r e e x c i t e d , i t seems n o t e a s y t o g e t any a n a l y t i c so- l u t i o n of t h e k i n e t i c e q u a t i o n f o r two, a t l e a s t , unknown f u n c t i o n s . ( i i i ) I n c l u s i o n of the e f f e c t of q u a s i p a r t i c l e c o l l i s i o n s i s e a s y i f we adopt a r e l a x a t i o n time approximation and i n f a c t i t h a s been done although we remember t h a t r e s u l t s t h u s c a l c u l a t e d i n t h e normal Fermi l i q u i d a r e u n s a t i s - f a c t o r y i n e x p l a i n i n g e x p e r i m e n t a l r e s u l t s on t h e imaginary p a r t of t h e impedance.

References

/ I / Roach, P a t R., K e t t e r s o n , J.B., J. Low Temp. Phys.

5

(1976) 637, Phys. Rev. L e t t .

36

(1976) 736.

/ 2 / Combescot, M . , Combescot, R . , Phys. L e t t .

58A

(1976) 181.

/ 3 / Bolton, J.P.R., J . Phys. C

9

(1976) L 565. / 4 / Maki, K., Ebisawa, H., J. Low Temp. Phys.

2

( 1 977) 627.

151 Walfle, P . , Sound Propagation and K i n e t i c Coef- f i c i e n t s i n S u p e r f l u i d 3 ~ e , p r o g r e s s i n Low Temperature Physics Vol. V I I .

161 Bekarevich, I . L . , Khalatnikov, I . M . , Soviet Phys.- JETP

12

(1961) 1187.

/ 7 / Flowers, E.G., Richardson, R.W., Williamson, S . J . , Phys. Rev. L e t t . 2_5 (1976) 309.