Ideal second sound resonance in He - II

Volume 36A, number 3

P HYSI C S L E T T E R S

IDEAL

SECOND

SOUND

30 August 1971

RESONANCE

IN

He

-II

F. VIDAL and Y. SIMON

Groupe de Physique des Solides de l'Ecole Normale S~perieure, Laboratoire associ$ au C . N . R . S . , Paris 5, France Received 19 July 1971 The second sound resonance is studied with a very sensitive technique• Excellent agreement with the linear theory is found. Absence of time delays (to 5 × 10-6s) in the a.c. heat flow across a metal-HeII interface is shown.

In analogy with t r a n s m i s s i o n line p r o b l e m s , the a m p l i t u d e of l i n e a r standing w a v e s of second sound n e a r the r e s o n a n c e iS [1]

' --.

~ ~ ~ "e~"4" -.

C->:, . . . . .

,,

TL (co) = T L ( % ) / {

1 + i(co - con)Q/w n } ,

~11'~

T=151OK

* 2- - - ,:,: ".

] | 1

I)1 1 2 3

302.1 300.1 301.1 303.1

i

4

304.1

s-1 s1 S-1 s4

(1)

w h e r e T L (con) = (- 1)n zooqo (con)/aL is the a m plitude of the r e s o n a t i n g second sound at the c l o s e d end (y = L) of the c a v it y , co - con is the deviation f r o m the r e s o n a n c e - f r e q u e n c y con f o r the mode n, Zoo = (pCu2)-I is the c a r a c t e r i s t i c i m p e d a n c e of H e - I I and Q =- n u / a L , w h e r e a is the e f f e c t i v e attentuation c o e f f i c i e n t (Q = ¢On/Aco w h e r e Aco is the width of the r e s o n a n c e at 1/~f2 of the m a x i m u m amplitude). T h u s , the c o m p l e x a m p l i t u d e of the t e m p e r a t u r e o s c i l l a t i o n T L (co) d e s c r i b e s a c i r c l e with a d i a m e t e r T L (COn) in the c o m p l e x plane. T h e c e n t e r of the c i r c l e {½ TL(con) , 0} i s on the r e a l axis whose p h a s e i s that of the incident heat c u r r e n t density in h e l i u m qo exp(icot). In t h i s l e t t e r we r e p o r t d i r e c t m e a s u r e m e n t s of the c o m p l e x a m p li tu d e r e s o n a n c e of second sound in r e c t a n g u l a r c a v i t i e s . T h i s m e a s u r e m e n t s give a v e r y s e n s i t i v e method of checking the f i r s t o r d e r h y d r o d y n a m i c a l e q u a tions. Th e e x p e r i m e n t a l a r r a n g e m e n t has been a l r e a d y d e s c r i b e d [1]. Special caution has been taken h e r e in o r d e r to avoid d e f o r m a t i o n s of the r e s o n a n c e c u r v e due to the coupling between second sound and e x t r a n e o u s a.c. heat flows in the bath (vibrations, a l t e r n a t i v e pumping, etc.). We have u s e d r e c t a n g u l a r c a v i t i e s with a c h o i c e of d i m e n s i o n s avoiding coupled wave m o d e s [2]. T h e Q r a n g e s f r o m 150 to 500 at 1.6°K, in the fundamental mode. T h e t r a n s m i t t e r i s a 5 to 20 ~z Cu f i l m . A J o u l e effect at co is obtained by

I "-]----;

., 1 0 0 (~v~l))

•

Exp.

C_,

.q.(2)

•

/t

y

B

•

•

qo~0)~6.~,,,2

Fig. 1. Typical resonance-curves v(co) in the complex plane• In (B), a carbon bolometer (dV/dT < 0) is used. The lower curves represent non-linear behaviors. c r o s s i n g a.c. and d.c. c u r r e n t s . The r e c e i v e r is a g r a n u l a r A1 f i l m (1000 ,~) on Epoxy. Th e A1 i s m a d e g r a n u l a r by the method of Cohen and A b e l e s [3] and T c in the r a n g e 1.4 - 2°K a r e obtained. T h i s b o l o m e t e r has high s e n s i b i l i t y ( d V / d T ~ 1 V / ° K ) f o r low n o i se l e v e l (0.2 ~V) and t h e r m a l i n e r t i a is not o b s e r v e d at the f r e q u e n c i e s u s e d (100C Hz). The b o l o m e t e r signal i s v (co) = A ( d V / d T ) " T L (co) {2} w h e r e A ( c m / V ) i s the c o e f f i c i e n t of the a m p l i f i c a t i o n chain. Th e c u r v e (A) in fig. 1 is a t y p i cal r e s o n a n c e c u r v e in the c o m p l e x plane f o r the fundamental mode. Th e r e a l a x i s has the phase 0 {I~ of the a.c. c u r r e n t in the t r a n s m i t t e r . 0 {I~ i s m e a 165

Volume 36A, number 3

PHYSICS

s u r e d by i n d u c t i o n in a c o i l to 0.5 °. S i n c e the e l e c t r i c a l and t h e r m a l skin e f f e c t s of t h i s Cu f i l m s a r e n e g l i g i b l e at t h e f r e q u e n c i e s u s e d , the r e a l a x i s i s at the p h a s e of qo (co). N e a r t h e r e s o n a n c e , a l l p o i n t s c o r r e s p o n d i n g to e q u a l i n c r e m e n t s in co a r e e q u a l l y s p a c e d . T h e c u r v e v (co) b e i n g a c i r c l e , the e r r o r in the d e t e r m i n a tion of the p h a s e (in r a d i a n s ) and a m p l i t u d e of the r e s o n a n c e s i g n a l 4, i s due to the r e l a t i v e width of the n o i s e - s p o t (4 ~ A~, (co)//v (wn)) and the r e l a t i v e e r r o r in t h e d e t e r m i n a t i o n of the r e s o n a n c e f r e q u e n c y i s ~/Q. With A l - f i l m s (Av ~ 2 ram) and qc (w) = 0.5 m W , / c m 2 (v (COn) ~ 2 0 c m ) . ~ 10 -2. W i t h t h i s a c c u r a c y , no p h a s e d i f f e r e n c e b e t w e e n qo (co) and v (Wn) h a s b e e n o b s e r v e d . T h u s , e x t r a n e o u s t i m e lag (to 5 × 10 .6 s) is a b s e n t in t h e h e a t f l o w a c r o s s a m e t a l - H e - I I i n t e r f a c e . T h e fig. 1 (B) is the r e s o n a n c e c u r v e o b t a i n e d with a c a r b o n b o l o m e t e r . A t i m e d e l a y , due a t h e r m a l skin e f f e c t of b o l o m e t e r , is p r e s ent, but the r e s o n a n c e c u r v e is s t i l l a c i r c l e . A d i r e c t c o m p a r i s o n in a w i d e r a n g e of heat p o w e r d e n s i t y w a s m a d e b e t w e e n the m e a s u r e d s i g n a l c (w), and t h e v a l u e c a l c u l a t e d f r o m eq. (2) (fig. 1 (D)). A m p l i t u d e d e p e n d a n t e x t r a - a t t e n u a t i o n s h a v e b e e n o b s e r v e d with a . c . h e a t c u r r e n t d e n s i t i e s qoc (co) m u c h s m a l l e r than the c r i t i c a l d.c. h e a t c u r r e n t d e n s i t y qoc (0). A s t r a n g e dep e n d a n c e of qoc (w) w i t h the r e s o n a t o r g e o m e t r y and the n a t u r e of the c o u p l i n g with t h e bath h a v e b e e n o b s e r v e d . In fig. 1 (D) t h i s n o n - l i n e a r e f f e c t

166

LETTERS

30 August 1971

i s shown f o r a r e c t a n g u l a r c a v i t y (L = 33 m m . s 6 x 12 m m 2) c o m m u n i c a t e d w i t h the bath t h r o u g h 1 two h o l e s (1 x 6 m m 2) p r o v i d e d at 7L. Nonl i n e a r e f f e c t s in t h e r a n g e qo (¢o) << qoc (0), p r o b a b l y due to the t u r b u l e n c e c r e a t e d by high a . c . r e s o n a n t h e a t c u r r e n t , h a v e b e e n o b s e r v e d by B r u c e [4] in high Q r e s o n a t o r . In o u r c a s e the r e s o n a n c e c i r c l e i s not d e f o r m e d f o r qo(W) > qoc (co) (fig. 1 (C)), w h i c h a l l o w s u s to s u p p o s e that m e c h a n i s m of s e l f a t t e n u a t i o n i s not p r e s e n t . In the l i n e a r r a n g e an e x c e l l e n t a g r e e m e n t , w i t h in the e s t i m a t e d e x p e r i m e n t a l e r r o r (5%), is found b e t w e e n t h e v a l u e s of qo (co) c a l c u l a t e d r e s p e c t i v e l y f r o m the J o u l e e f f e c t and f r o m the b o l o m e t e r s i g n a l (eq. (2)). T h e s u g g e s t i o n s of D r . M. L e Ray a r e g r a t e fully a c k n o w l e d g e d ; we h a v e b e n e f i t e d f r o m d i s c u s s i o n s with P r o f e s s o r J. Bok and D r . M. Frangois.

R ejerences [1] F. Vidal, Y. Simon, N. Le Ray and P. Thorel, Rev, Phys. Appl. 4 (1969) 50. [2] P . J . Bendt, Phys. Rev. 136 (1964) A918. [3] R. W. Cohen and B. Abeles, Phys. Rev. 168 (1968) 44. [4] R. El. Bruce, in: Proe. Ninth Int. Conf. on Low. temp. phys.. Colombus, Ohio. 1964 eds. J. G. Daunt and a l (Plenum P r e s s , New York. 1965).