Osmotic pressure waves in superfluid 3He4He mixtures

Volume 30A, number 7

OSMOTIC

1~ H Y S I C S L E T T E R S

PRESSURE

WAVES

1 December 1969

IN S U P E R F L U I D

3He-4He

MIXTURES

A. C. GHOZLAN and E. J . - A . VAROQUAUX

Institut d'Electronique Fondarnentale, Laboratoire associ$ au C.N.R.S., Facult$ des Sciences, 91 ORSA Y, France Received 9 October 1969

The second sound velocity of superfluid 3He-4He mixtures is connected by a simple thermodynamical formula to the osmotic pressure. Within the framework of the BBP theory, the experimental data for these two quantities are found to be non-consistent. C o n s i d e r two v e s s e l s connected by a s u p e r leak, one filled with p u r e 4He at p r e s s u r e P0 and z e r o t e m p e r a t u r e , the other with a 3He-4He mixt u r e of 3He c o n c e n t r a t i o n by weight w, at t e m p e r a t u r e T. The osmotic p r e s s u r e ~(T,P, oo)and the fountain p r e s s u r e Pf(T, P) of the superfluid solution a r e defined by [1]: f i 4 ( T , P 0 + Pf + ~,w) = ~40(0, P0) ,

(1)

where ~40 and #4 a r e the c h e m i c a l p o t e n t i a l s p e r g r a m of 4He in the p u r e liquid and in the m i x t u r e . P = PO + P f + ~ is the r e a l h y d r o s t a t i c p r e s s u r e over the m i x t u r e and P 0 t u r n s out to be the fictitious p r e s s u r e such that d~4 = F40(0, P0 ) d ( P - P f - ~') ,

(2)

where v40 is the volume p e r g r a m of pure 4He at p r e s s u r e P 0 and z e r o t e m p e r a t u r e . With the help of eq. (2), we can solve the equations of h y d r o d y n a m i c s for h e l i u m m i x t u r e s u s i n g the s a m e p r o c e d u r e and notations a s London [2]; we find for the velocity of second sound uii: (3) 2 PsV40 wF

Ps(wa_p_~2-]-lFa(~+Pf)~

This r e s u l t can be shown to be equivalent to K h a l a t n i k o v ' s r e s u l t [3] but it r e v e a l s m o r e c l e a r l y the physical n a t u r e of second sound a s b e i n g a locally adiabatic motion (s/w constant) of the excitations in the superfluid "aether" [4]. At low t e m p e r a t u r e s and at c o n c e n t r a t i o n s which a r e not too s m a l l , P f v a n i s h e s and second sound b e c o m e s an osmotic p r e s s u r e wave; osmotic p r e s s u r e p r o p a g a t e s in a h y d r o d y n a m i c a l r e g i m e , except at v e r y low t e m p e r a t u r e s where diffusion finally b l u r s out h y d r o d y n a m i c s . This extends P o m e r a n c h u k ' s e a r l ~ r e s u l t [5] to the case where i n t e r a c t i o n between OHe q u a s i - p a r t i c l e s is taken into account.

426

Having thus e s t a b l i s h e d quite g e n e r a l l y a simple connection between osmotic p r e s s u r e and a second sound, we wish to point out the following consequence of f o r m u l a (3). F o r dilute solutions of 3He in Heii (x, m o l a r c o n c e n t r a t i o n ~ rn4w/rn3) in the s e m i - c l a s s i c a l region (T~ > T > 2TF), the o s m o t i c p r e s s u r e is given by [6]:

/T~3/2 Rv44T~{I+ ~ (1-Or)+ 0.133

I/_ ~ N \A2X" +~v~0-)

I

{V 0 - ~(V) } = ~kin + ~int '

where v40 is the m o l a r volume of pure 4He, ot is the BBP p a r a m e t e r [7], and V 0 - I ( V ) =f(T) contains the t e m p e r a t u r e dependence-of ~int the p a r t of the osmotic p r e s s u r e due to 3He-3-I~e i n t e r a c t i o n s in the H a r t r e e - F o c k a p p r o x i m a t i o n . As V is independent of x [ B B P a p p r o x i m a t i o n ] , we have from (4): a~int P , s / w

m4(NA'2xlf

+Tf'(S40 +Rx)l

where C is the m o l a r heat capacity of solutions and $40 the m o l a r entropy of p u r e 4He. The left hand side of (5) is evaluated f r o m the e x p e r i m e n t a l v a l u e s of Sandiford and F a i r h a n k for u 2 [8] by s u b s t r a c t i n g f r o m the total u ~ a s givenn by (3) the kinetic p a r t which c o m e s i r o m Ykin" A c o m p a r i s o n can be c a r r i e d through with E d w a r d ' s aTrint/Ox]p ' T [9] at T = 0.32 and 0.65 OK, using the value at 0.32°K a s b o u n d a r y value in the i n t e g r a t i o n of (5) and checking with the other. The a g r e e m e n t is r a t h e r poor; we conclude that the two sets of data do not yield cons i s t e n t v a l u e s f o r f ( T ) . This c o n c l u s i o n l i e s a) on the fact t h a t f ( T ) does not depend on x, a s is indeed shown by the x 2 dependence of ~int found

Volume30A, number 7

PHYSICS

e x p e r i m e n t a l l y b ) o n t h e v a l i d i t y of H a r t r e e - F o c k a p p r o x i m a t i o n f o r T > 2 T F [10]. We gratefully acknowledge very helpful conv e r s a t i o n s w i t h J . S e l d e n , a n d w i t h A. L a n d e s mann and J. Winter.

References 1. Work p a r t i a l l y supported by D.R.M.E. on c o n t r a c t 750/68. 2. F. London, Superfluids (Vol. H) (Dover N.Y. 1962) .~• p . 194.

THRESHOLD

EXCITATION

OF

LETTERS

1 D e c e m b e r 1969

3. I. M. Khalatnikov, An introduction to the theory of superfluidity, (W.A. Benjamin N.Y., 1965)p. 163. 4. J. Wheatley, Am. J. Phys. 36 (1968) 181. 5. I. P o m e r a n c h u c k , Zh. Eksp. Theor. Fiz. 19 (1949) 42. 6. C . E b n e r , Thesis, University of Illinois (1967) p. 46, unpublished. 7. J. Bardeen, G. Baym and D. P i n e s , Phys. Rev. 156, (1967) 207. 8. F. Wilson, D. O. Edwards and J. T. Tough, Phys. Rev. L e t t e r s 19 (1967) 1368. 9. D. J. Sandiford and H. A. Fairbank, Phys. Rev. 162 (1967) 192. 10. J. Seiden, J. Phys. 30 (1969) 267.

H 2 AND

D 2 BY

ELECTRON

IMPACT

R. I. H A L L , J . M A Z E A U a n d J . R E I I ~ . A R D T Laboratoire de Physique et Optique Corp~culaires, Facult~ des Sciences de Paris, France Received 18 September 1969

Using an improved " r e t a r d i n g potential difference • technique and a p r e c i s e energy calibration, the s t r u c t u r e in the t h r e s h o l d excitation s p e c t r a of H 2 and D 2 between 11.7 and 14 eV has been identified with certitude as belonging to the vibrational s e r i e s of the c3II u state. We have built a "retarding potential differe n c e " e l e c t r o n g u n [1] a s s o c i a t e d w i t h a p a r a l l e l plates collision chamber in which electrons having lost all their energy in an inelastic collision are trapped and measured. T y p i c a l d a t a of t h r e s h o l d e x c i t a t i o n i n H 2 a n d

pw',|.

c'n'u[

t

i i

e.H2

i

11.87

s'X"

!l

B'x;! ! i

.%

i s

"~. 11

12

13 Electron

14 energy

15 (eV)

Fig. 1. T h r e s h o l d excitation s p e c t r u m of H 2 (see the text for energy scale calibration). Dashed liffes r e p r e sent spectroscopic values of the indicated states.

12

Electron energy

13

(eV)

Fig. 2. T h r e s h o l d excitation s p e c t r u m of D 2. In the i n s e t is shown the c a l i b r a t i o n of the s p e c t r u m against the nitrogen v = 0 ~E, state which a p p e a r s between the v = O and the v = 1 l e v e l s of the c 3 H 4 state of D 2. 427