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Phase transition in HeII films
Volume 26A, number 10
PHYSICS
LETTERS
8 A p r i l 1968
PHASE TRANSITION IN HeII FILMS D. J. AMIT Department o f Theoretical P h y s i c s , The Hebrew University, J e r u s a l e m , Israel Received 8 March 1968
Using a new functional for the thermodynamic potential the depression of the superfluid transition temperature is calculated and compared with experiment. It is shown that transition is of first order and that the range of validity of the theory extends to the transition temperature.
In a r e c e n t n o t e [1] a n e w t h e r m o d y n a m i c p o t e n t i a l f u n c t i o n f o r HeII j u s t b e l o w Tk w a s p r o p o s e d . T h i s f u n c t i o n r e p r o d u c e s the bulk p r o p e r t i e s of He a s t h e s u p e r f l u i d d e n s i t y and c r i t i c a l velocity, as far as the temperature behaviour is c o n c e r n e d . T h e t h r e e f r e e p a r a m e t e r s in t h e t h e o r y w e r e d e t e r m i n e d by t h e m e a s u r e m e n t s of the b e h a v i o u r of Ps - t h e s u p e r f l u i d d e n s i t y [2], v c - t h e c r i t i c a l v e l o c i t y [3] and Cp - t h e d i s c o n tinuity in t h e s p e c i f i c h e a t [4], a s one a p p r o a c h e s Tk f r o m below. T h e t h e r m o d y n a m i c p o t e n t i a l in the u n i f o r m c a s e i s of t h e f o r m ~ ( P , T) = ~I + + A ~ 2 + B ~ 4 + C~ 6, w i t h ~/2 = Ps, A = - a t 4/3 , B = - f i t 2/3, C = ~ , t = T~ - T a n d a , ~ , y >O[5]. W e now c o n s i d e r the c a s e of non u n i f o r m o r d e r p a r a m e t e r [6]. T h e v e l o c i t y i s t a k e n to be z e r o but s p a t i a l v a r i a t i o n s in @ o r in Ps a r e i n c l u d e d . T h i s i s done by c o n s i d e r i n g • to b e a f u n c t i o n a l of ~p(r) +(P,
T)
=
(1)
= ¢ I + f d 3 r { A ~ 2 +B~/4 + O~6 + ( / / 2 / 2 m ) I v ~ 12} . ~/(r) i s d e t e r m i n e d by t h e r e q u i r e m e n t t h a t 5 ~ , / 5 ~ v a n i s h . T h i s l e a d s to the f o l l o w i n g d i f ferential equation * - ( ] ~ 2 / 2 m ) V 2 ~ +AtP + 2B¢'3 + 3 C ~ 5 = 0 .
(2)
A s p e c i a l s o l u t i o n of t h i s e q u a t i o n i s ~/ = c o n s t a n t [1]. T h e e x p e r i m e n t s on the s u p e r f l u i d t r a n s i t i o n in f i l m s a r e p e r f o r m e d on f i l m s which a r e on t h e i n s i d e of c y l i n d e r s [7]. H o w e v e r , D / R ~-, 10 -6 w h e r e D and R a r e t h e t h i c k n e s s of t h e f i l m and the r a d i u s of the c y l i n d e r , r e s p e c t i v e l y . C o n s e * Prof. A. L. Fetter has pointed out that eq. (2) defines a coherence length ~ = ~(3y/4m132) 1/2 t -2/3 ~ 2.05 x x 10-8t-2/3 [9]. 448
q u e n t l y t h e b e h a v i o u r can b e t a k e n a s flat. W r i t i n g : x = u~, ~ = v~, D = u d a n d u = = (~ 2/2rnc~)l/2 t - 2/3, v 2 = (fl/31,)(1 +K)t ~/3Z PSBulk (T) w h e r e K = (1 + 3~y/f32) 1/2. W e o b t a i n dZT//d~ 2~+ 7/ + + Ml~?3 - M2~75 = 0 w h e r e M 1 ~ M 2 = 850, to t h e a c c u r a c y in w h i c h w e a r e w o r k i n g . T h e b o u n d a r y c o n d i t i o n s f o r t h e f i l m with one f r e e s u r f a c e a r e 77 = 0 on t h e s o l i d s u r f a c e and e i t h e r 77 = 0 o r dT//d~ = 0 a d i s t a n c e d away. T h e m a i n c o n c l u s i o n n a m e l y , that t h e p h a s e t r a n s i t i o n b e c o m e s of f i r s t o r d e r , i s i n d e p e n d e n t of w h i c h one the b o u n d a r y c o n d i t i o n s i s u s e d . In f a c t the s e c o n d one is s a t i s f i e d at half t h e d i s t a n c e on w h i c h t h e f i r s t one i s . T h e m a i n p r o p e r t y of the a b o v e t y p e of e q u a t i o n [5,6] i s t h a t the i n t e r v a l d o v e r w h i c h e i t h e r of t h e two b o u n d a r y v a l u e p r o b l e m s can be s a t i s f i e d , h a s a m i n i m u m g r e a t e r than z e r o . T h i s p r o p e r t y d e f i n e s t h e t r a n s i t i o n t e m p e r a t u r e in f i l m s T ~ F . W e find AT = T~ - T ~ F = = ~--J[~dc/(2m(~)~/2]3/2D , -~/2 w h e r e ~ = 0.48 x 10-18 erg(OK) -~/3 [1]. E v a l u a t i n g t h e m i n i m u m of d, dc, n u m e r i c a l l y , g i v e s 0.23, and 0.12 f o r t h e f i r s t and s e c o n d b o u n d a r y c o n d i t i o n r e s p e c t i v e l y . F i g . 1 s h o w s the c o m p a r i s o n w i t h e x p e r i m e n t [7] of A T x / T k f o r both b o u n d a r y c o n d i t i o n s . T h e a g r e e m e n t is b e t t e r f o r t h e f i r s t one. An i n c r e a s e of I0 - 15% in ~ w i l l i m p r o v e t h e a g r e e m e n t with e x p e r i m e n t . Such a c h a n g e i s not u n r e a s o n a b l e c o n s i d e r i n g t h e u n c e r t a i n t i e s in t h e m e a s u r e d v e l o c i t i e s [3]. C o n s i d e r , next, t h e f u n c t i o n d ( ~ ) w h e r e d i s t h e d i s t a n c e f r o m t h e i n i t i a l point, at w h i c h we t s t a r t i n t e g r a t i n g with ~ = 0, 77' = 7/i, to t h e c l o s e s t point a t w h i c h e i t h e r 7/ o r ~/' = 0. d c = = min[d(r/¼)] f o r ~/I > 0. If 7/~ > ~?c > 0 t h e b o u n d a r y c o n d i t i o n w i l l n e v e r be s a t i s f i e d [8]. F u r t h e r T m o r e , if d(7/~) a t t a i n s i t s m i n i m u m when ~I ~ 0 [6,5] t h e n t h e a m p l i t u d e of P s g o e s to z e r o a s T ~ TkF.
Volume 26A, number 10
•T•
PHYSICS
LETTERS tion r e a d s [11]
'
~2 kBT~/[81re ~ ~(T)] << P s ( T ) ,
1.0
D
02
(3)
w h e r e k B i s B o l t z m a n ' s c o n s t a n t and ~ i s t h e c o r r e l a t i o n l e n g t h * . T h i s r e l a t i o n is t e m p e r a t u r e i n d e p e n d e n t . I n s e r t i n g n u m b e r s in eq. (3) w e find t h a t t h e r i g h t hand s i d e i s about 24 t i m e s g r e a t e r than the l e f t hand s i d e . T h e t h e o r y , t h e r e f o r e , p r e d i c t s that t h e f l u c t u a t i o n s w i l l b e s m a l l up to t h e t r a n s i t i o n t e m p e r a t u r e .
0.5 0.4 0.3
8 A p r i l 1968
m
0.1
I l l u m i n a t i n g d i s c u s s i o n s of t h e n a t u r e of t h e s o l u t i o n of eq. (2) with D r . A c h i B r a n d a r e g r a t e fully acknowledged. 00~
I
3
I
10
I
I
2O 30
~,"
References
Fig. 1. Depression of transition temperature. (9 experiment of ref. 7. Curve A: Theory first boundary condition. B: Second boundary condition. In t h e p r e s e n t c a s e d i s m i n i m i z e d by 771 ~ 10 ( d - " ~ a s nI ~ 0 and d - ~ oo a s n l ~ 12, d(10) = = 0.23. T h u s 7/ ¢ 0 a s T ~ TAF and r e a c h e s a v a l u e of 0.8. A b o v e T k F 7/ = Ps = 0. T h i s i s a f i r s t o r d e r p h a s e t r a n s i t i o n . A s D ~ oo, Tk F T~ and t h e t r a n s i t i o n b e c o m e s of t h e s e c o n d order again. T h e l a t e n t h e a t of t r a n s i t i o n i s l = = - TF(a@/~T) P. U s i n g eq. (1) f~r @ with t h e c r i t i c a l ~p we find l = Q I T k F D - - ~ w h e r e Q = 0.025,
,/c
I = dg
d~ [2~/2 +½MI~/4] ,
O
and D i s m e a s u r e d in A to g i v e l in e r g / c m 2. F i n a l l y , we c o n s i d e r the r a n g e of v a l i d i t y of t h e t h e o r y [10,11]. S i n c e any t h e o r y of t h e L a n dau t y p e h a s an o r d e r p a r a m e t e r - o r d e r p a r a m e t e r c o r r e l a t i o n f u n c t i o n of a Y u k a w a f o r m t h e c o n d i -
1. D . J . A m i t , to be published. 2. J . R . C l o w and J . D . R e p p y , Phys. Ray. Letters 16 (1966) 887. 3. J . R . C l o w and J . D . R e p p y , Phys. Rev. Letters 19 (1967) 289. 4. W.M. Fairbank, Liquid helium ed. G. C a r e r i (Academic P r e s s , N.Y., 1963). 5. Yu.G.Mamaladze, J. Exp. i Teor. Fiz. 52 (1966) 729, Soviet Phys. J E T P 25 (1967) 479. 6. V.L. Ginzburg and L. P. Pitaevskii, J. Exp. i Teor. Fiz. 34 (1958) 1240, Soviet Phys. J E T P 7 (1958) 858. 7. R. Bowers, D. F. Brewer and K. Mendelson, Phil. Mag. 42 (1951) 1445; D. F. Brewer, A . J . Simonds and A. L. Thomson, Phys. Rev. Letters 15 (1965) 182. 8. D . J . A m i t , to be published. 9. J . A . Tyson and D, H. Douglass, Phys. Rev. Letters 17 (1966) 472; B.D.Josephson, Phys. L e t t e r s 21 (1966) 608. 10. V.L.Ginzburg, Fiz. Tverd. Tela 2 (1960) 2031, Sov. Phys. Solid State 2 (1960) 1824. 11. L . P . Kadanoff et al. Rev. Mod. Phys. 39 (1967) 395 Sec. F. * See footnote on previous page.
*****
449
PHYSICS
LETTERS
8 A p r i l 1968
PHASE TRANSITION IN HeII FILMS D. J. AMIT Department o f Theoretical P h y s i c s , The Hebrew University, J e r u s a l e m , Israel Received 8 March 1968
Using a new functional for the thermodynamic potential the depression of the superfluid transition temperature is calculated and compared with experiment. It is shown that transition is of first order and that the range of validity of the theory extends to the transition temperature.
In a r e c e n t n o t e [1] a n e w t h e r m o d y n a m i c p o t e n t i a l f u n c t i o n f o r HeII j u s t b e l o w Tk w a s p r o p o s e d . T h i s f u n c t i o n r e p r o d u c e s the bulk p r o p e r t i e s of He a s t h e s u p e r f l u i d d e n s i t y and c r i t i c a l velocity, as far as the temperature behaviour is c o n c e r n e d . T h e t h r e e f r e e p a r a m e t e r s in t h e t h e o r y w e r e d e t e r m i n e d by t h e m e a s u r e m e n t s of the b e h a v i o u r of Ps - t h e s u p e r f l u i d d e n s i t y [2], v c - t h e c r i t i c a l v e l o c i t y [3] and Cp - t h e d i s c o n tinuity in t h e s p e c i f i c h e a t [4], a s one a p p r o a c h e s Tk f r o m below. T h e t h e r m o d y n a m i c p o t e n t i a l in the u n i f o r m c a s e i s of t h e f o r m ~ ( P , T) = ~I + + A ~ 2 + B ~ 4 + C~ 6, w i t h ~/2 = Ps, A = - a t 4/3 , B = - f i t 2/3, C = ~ , t = T~ - T a n d a , ~ , y >O[5]. W e now c o n s i d e r the c a s e of non u n i f o r m o r d e r p a r a m e t e r [6]. T h e v e l o c i t y i s t a k e n to be z e r o but s p a t i a l v a r i a t i o n s in @ o r in Ps a r e i n c l u d e d . T h i s i s done by c o n s i d e r i n g • to b e a f u n c t i o n a l of ~p(r) +(P,
T)
=
(1)
= ¢ I + f d 3 r { A ~ 2 +B~/4 + O~6 + ( / / 2 / 2 m ) I v ~ 12} . ~/(r) i s d e t e r m i n e d by t h e r e q u i r e m e n t t h a t 5 ~ , / 5 ~ v a n i s h . T h i s l e a d s to the f o l l o w i n g d i f ferential equation * - ( ] ~ 2 / 2 m ) V 2 ~ +AtP + 2B¢'3 + 3 C ~ 5 = 0 .
(2)
A s p e c i a l s o l u t i o n of t h i s e q u a t i o n i s ~/ = c o n s t a n t [1]. T h e e x p e r i m e n t s on the s u p e r f l u i d t r a n s i t i o n in f i l m s a r e p e r f o r m e d on f i l m s which a r e on t h e i n s i d e of c y l i n d e r s [7]. H o w e v e r , D / R ~-, 10 -6 w h e r e D and R a r e t h e t h i c k n e s s of t h e f i l m and the r a d i u s of the c y l i n d e r , r e s p e c t i v e l y . C o n s e * Prof. A. L. Fetter has pointed out that eq. (2) defines a coherence length ~ = ~(3y/4m132) 1/2 t -2/3 ~ 2.05 x x 10-8t-2/3 [9]. 448
q u e n t l y t h e b e h a v i o u r can b e t a k e n a s flat. W r i t i n g : x = u~, ~ = v~, D = u d a n d u = = (~ 2/2rnc~)l/2 t - 2/3, v 2 = (fl/31,)(1 +K)t ~/3Z PSBulk (T) w h e r e K = (1 + 3~y/f32) 1/2. W e o b t a i n dZT//d~ 2~+ 7/ + + Ml~?3 - M2~75 = 0 w h e r e M 1 ~ M 2 = 850, to t h e a c c u r a c y in w h i c h w e a r e w o r k i n g . T h e b o u n d a r y c o n d i t i o n s f o r t h e f i l m with one f r e e s u r f a c e a r e 77 = 0 on t h e s o l i d s u r f a c e and e i t h e r 77 = 0 o r dT//d~ = 0 a d i s t a n c e d away. T h e m a i n c o n c l u s i o n n a m e l y , that t h e p h a s e t r a n s i t i o n b e c o m e s of f i r s t o r d e r , i s i n d e p e n d e n t of w h i c h one the b o u n d a r y c o n d i t i o n s i s u s e d . In f a c t the s e c o n d one is s a t i s f i e d at half t h e d i s t a n c e on w h i c h t h e f i r s t one i s . T h e m a i n p r o p e r t y of the a b o v e t y p e of e q u a t i o n [5,6] i s t h a t the i n t e r v a l d o v e r w h i c h e i t h e r of t h e two b o u n d a r y v a l u e p r o b l e m s can be s a t i s f i e d , h a s a m i n i m u m g r e a t e r than z e r o . T h i s p r o p e r t y d e f i n e s t h e t r a n s i t i o n t e m p e r a t u r e in f i l m s T ~ F . W e find AT = T~ - T ~ F = = ~--J[~dc/(2m(~)~/2]3/2D , -~/2 w h e r e ~ = 0.48 x 10-18 erg(OK) -~/3 [1]. E v a l u a t i n g t h e m i n i m u m of d, dc, n u m e r i c a l l y , g i v e s 0.23, and 0.12 f o r t h e f i r s t and s e c o n d b o u n d a r y c o n d i t i o n r e s p e c t i v e l y . F i g . 1 s h o w s the c o m p a r i s o n w i t h e x p e r i m e n t [7] of A T x / T k f o r both b o u n d a r y c o n d i t i o n s . T h e a g r e e m e n t is b e t t e r f o r t h e f i r s t one. An i n c r e a s e of I0 - 15% in ~ w i l l i m p r o v e t h e a g r e e m e n t with e x p e r i m e n t . Such a c h a n g e i s not u n r e a s o n a b l e c o n s i d e r i n g t h e u n c e r t a i n t i e s in t h e m e a s u r e d v e l o c i t i e s [3]. C o n s i d e r , next, t h e f u n c t i o n d ( ~ ) w h e r e d i s t h e d i s t a n c e f r o m t h e i n i t i a l point, at w h i c h we t s t a r t i n t e g r a t i n g with ~ = 0, 77' = 7/i, to t h e c l o s e s t point a t w h i c h e i t h e r 7/ o r ~/' = 0. d c = = min[d(r/¼)] f o r ~/I > 0. If 7/~ > ~?c > 0 t h e b o u n d a r y c o n d i t i o n w i l l n e v e r be s a t i s f i e d [8]. F u r t h e r T m o r e , if d(7/~) a t t a i n s i t s m i n i m u m when ~I ~ 0 [6,5] t h e n t h e a m p l i t u d e of P s g o e s to z e r o a s T ~ TkF.
Volume 26A, number 10
•T•
PHYSICS
LETTERS tion r e a d s [11]
'
~2 kBT~/[81re ~ ~(T)] << P s ( T ) ,
1.0
D
02
(3)
w h e r e k B i s B o l t z m a n ' s c o n s t a n t and ~ i s t h e c o r r e l a t i o n l e n g t h * . T h i s r e l a t i o n is t e m p e r a t u r e i n d e p e n d e n t . I n s e r t i n g n u m b e r s in eq. (3) w e find t h a t t h e r i g h t hand s i d e i s about 24 t i m e s g r e a t e r than the l e f t hand s i d e . T h e t h e o r y , t h e r e f o r e , p r e d i c t s that t h e f l u c t u a t i o n s w i l l b e s m a l l up to t h e t r a n s i t i o n t e m p e r a t u r e .
0.5 0.4 0.3
8 A p r i l 1968
m
0.1
I l l u m i n a t i n g d i s c u s s i o n s of t h e n a t u r e of t h e s o l u t i o n of eq. (2) with D r . A c h i B r a n d a r e g r a t e fully acknowledged. 00~
I
3
I
10
I
I
2O 30
~,"
References
Fig. 1. Depression of transition temperature. (9 experiment of ref. 7. Curve A: Theory first boundary condition. B: Second boundary condition. In t h e p r e s e n t c a s e d i s m i n i m i z e d by 771 ~ 10 ( d - " ~ a s nI ~ 0 and d - ~ oo a s n l ~ 12, d(10) = = 0.23. T h u s 7/ ¢ 0 a s T ~ TAF and r e a c h e s a v a l u e of 0.8. A b o v e T k F 7/ = Ps = 0. T h i s i s a f i r s t o r d e r p h a s e t r a n s i t i o n . A s D ~ oo, Tk F T~ and t h e t r a n s i t i o n b e c o m e s of t h e s e c o n d order again. T h e l a t e n t h e a t of t r a n s i t i o n i s l = = - TF(a@/~T) P. U s i n g eq. (1) f~r @ with t h e c r i t i c a l ~p we find l = Q I T k F D - - ~ w h e r e Q = 0.025,
,/c
I = dg
d~ [2~/2 +½MI~/4] ,
O
and D i s m e a s u r e d in A to g i v e l in e r g / c m 2. F i n a l l y , we c o n s i d e r the r a n g e of v a l i d i t y of t h e t h e o r y [10,11]. S i n c e any t h e o r y of t h e L a n dau t y p e h a s an o r d e r p a r a m e t e r - o r d e r p a r a m e t e r c o r r e l a t i o n f u n c t i o n of a Y u k a w a f o r m t h e c o n d i -
1. D . J . A m i t , to be published. 2. J . R . C l o w and J . D . R e p p y , Phys. Ray. Letters 16 (1966) 887. 3. J . R . C l o w and J . D . R e p p y , Phys. Rev. Letters 19 (1967) 289. 4. W.M. Fairbank, Liquid helium ed. G. C a r e r i (Academic P r e s s , N.Y., 1963). 5. Yu.G.Mamaladze, J. Exp. i Teor. Fiz. 52 (1966) 729, Soviet Phys. J E T P 25 (1967) 479. 6. V.L. Ginzburg and L. P. Pitaevskii, J. Exp. i Teor. Fiz. 34 (1958) 1240, Soviet Phys. J E T P 7 (1958) 858. 7. R. Bowers, D. F. Brewer and K. Mendelson, Phil. Mag. 42 (1951) 1445; D. F. Brewer, A . J . Simonds and A. L. Thomson, Phys. Rev. Letters 15 (1965) 182. 8. D . J . A m i t , to be published. 9. J . A . Tyson and D, H. Douglass, Phys. Rev. Letters 17 (1966) 472; B.D.Josephson, Phys. L e t t e r s 21 (1966) 608. 10. V.L.Ginzburg, Fiz. Tverd. Tela 2 (1960) 2031, Sov. Phys. Solid State 2 (1960) 1824. 11. L . P . Kadanoff et al. Rev. Mod. Phys. 39 (1967) 395 Sec. F. * See footnote on previous page.
*****
449