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Note on Hartree models of the superfluid helium film
Volume 30A. n u m b e r 8
PHYSICS
LETTERS
A n i m p o r t a n t b y - p r o d u c t of t h i s c a l c u l a t i o n i s that exact transport coefficients, including their potential part, can be computed by solving the one-particle linearized generalized Boltzmann e q u a t i o n to s e c o n d o r d e r i n t h e u n i f o r m i t y p a r a m eter. Indeed, nowhere in this theory, is there a n y a p p e a l to t h e f u n c t i o n a l r e l a t i o n b e t w e e n t h e two p a r t i c l e d i s t r i b u t i o n f u n c t i o n a n d t h e o n e p a r t icle distribution function; this is in contrast with t h e t r a d i t i o n a l t h e o r y [e.g. 9]. T h e d e t a i l s of t h e s e c a l c u l a t i o n s w i l l b e p u b l i s h e d e l s e w h e r e [10].
References 1. H. Mori, Prog. Th. Phys. Jap. 28 (1962) 763; 33 (1965) 423. 2. L.Kadanoff and P. Martin, Ann. Physics 24 (1965) 419. 3. L. Kadanoff and J. Swift, Phys. Rev. 166 (1968) 89. 4. Y. Pomeau, Phys. L e t t e r s 27A (1968) 601 and Ph D Thesis (Universit~ de P a r i s , 1969, unpublished) 5. L. Van Hove, Phys. Rev. 95 (1959) 249. 6. G. Severnc, Physica 31 (1965) 877; E. G. D Cohen, Fundamental p r o b l e m s in s t a t i s t i c a l m e c h a n i c s II, ed. E. G. D. Cohen (North Holland, A m s t e r d a m , 1968). 7. L. Garcia-Colin, M. G r e e n and F. Chaos; Physica 20 (1966) 450. 8. D. Bohm; Quantum theory (Constable, London, 1954). 9. The proof is s i m i l a r to that given in P. R6sibois, J Chem. Phys. 41 (1964) 2979 and G. Nicolis and G. Severne, J. Chem. Phys. 44 (1966) 1477. 10. P. R6sibois, J. Star. P h y s i c s , to be published.
It i s g r a t e f u l l y a c k n o w l e d g e d t h a t t h e w o r k p r e s e n t e d h e r e f o u n d i t s g e n e s i s i n a s e r i e s of d i s c u s s i o n s w i t h P r o f . J . L e b o w i t z . We a l s o t h a n k P r o f . G. N i c o l i s f o r u s e f u l c o m m e n t s .
NOTE
ON
HARTREE
MODELS
OF
15 D e c e m b e r 1969
THE
SUPERFLUID
HELIUM
FILM
*
D. S. H Y M A N
Physics Department and Center for Materials Science and Engineering Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA Received 17 N o v e m b e r 1969
Phenomenological models of the superfluid helium film a r e discussed. It is shown that a r e c e n t H a r t r e e model due to G r o s s and Amit cannot lead to a completely r e a l i s t i c d e s c r i p t i o n of such films.
T o d a t e , t h e o r e t i c a l d i s c u s s i o n s of t h e s u p e r f l u i d h e l i u m f i l m h a v e b e e n b a s e d on p h e n o m e n o l o g i c a l [1,2] o r s e m i p h e n o m e n o l o g i c a l [3] m o d e l s in which the superfluid density is described by a complex wave function ~(r). However, a major p o i n t of u n c e r t a i n t y i n t h e s e a p p r o a c h e s i s t h e b e h a v i o r of ~P(r) a t t h e f r e e s u r f a c e of t h e f i l m . We shall show that one can use the Hartree model of G r o s s a n d A m i t [4] t o s t u d y a h e l i u m f i l m b o u n d t o a u n i f o r m s u b s t r a t e . In t h i s a p p r o a c h the boundary conditions on ~(r) are unambiguous; h o w e v e r , it w i l l b e s h o w n t h a t t h e e x p e c t e d s t r u c t u r e of m o d e r a t e l y t h i c k f i l m s c a n n o t b e o b t a i n e d with this model. T h e H a r t r e e m o d e l , g e n e r a l i z e d to i n c l u d e t h e e f f e c t of a w a l l w i t h a n a t t r a c t i v e p o t e n t i a l , l e a d s to t h e f o l l o w i n g e n e r g y f u n c t i o n a l f o r N i n t e r a c t i n g b o s o n s [4]: 466
E =
/Iv
12dr +
+ ½fv(r-r')[~(r)t2[~(r')]2drdr'
+
(1)
+ fVe(X)[~[2dr T h e d e n s i t y d i s t r i b u t i o n , g i v e n b y ] ~ ( r ) t 21 s a t isfies the normalization condition f[~ (r)l 2dr = = N. T h e p h e n o m e n o l o g i c a l i n t e r a c t i o n v ( r ) i s w r i t t e n a s v ( r ) = X6 ( r ) , w i t h X g i v e n i n t e r m s of t h e sGund s p e e d c a n d t h e n u m b e r d e n s i t y p / m of the bulk liquid by ~ = mc2/p. We have assumed t h a t t h e w a l l , s i t u a t e d a t x = 0, c a n b e r e p r e s e n t e d b y t h e o n e - d i m e n s i o n a l p o t e n t i a l re(X). A p p l i c a t i o n of t h e v a r i a t i o n a l p r i n c i p l e l e a d s to t h e f o l l o w i n g e q u a t i o n f o r ~h(x): * R e s e a r c h supported by Advanced R e s e a r c h P r o j e c t s Agency Contract SD-90, and in p a r t by the National Aeronautics and Space Administration.
Volume 30A, number 8
PHYSICS LETTERS
-% Fig. 1. External wall potential, and density profile obtained from eq. (2).
r a n g e , and then f a l l to z e r o in a d i s t a n c e of p e r haps one or two a n g s t r o m s . The p h y s i c a l e f f e c t m i s s i n g f r o m the m o d e l is the influence of the a t t r a c t i v e i n t e r p a r t i c l e f o r c e s which would lead to a st ab l e, p r e d o m i n a n t l y flat f i l m . H o w e v e r , any a t t e m p t to i n c o r p o r a t e this e f f e c t through a m o d i f i c a t i o n of v ( r ) would f ai l , si n ce it can be shown [5] f r o m s t a b i l i t y c o n s i d e r a t i o n s that f v ( r - r')~2(r)tp2(r')drdr ' m u s t be p o s i t i v e . M o r e o v e r , with this condition, our concl u si o n c o n c e r n i n g the q u a l i t a t i v e nature of the s o l u t i o n s to (2) holds in g e n e r a l f o r all finiterange Ve(X). T h i s is e a s i l y s e e n by noting that f r o m (1) and (2) we can w r i t e the e n e r g y / a t o m as
E_ =-e= ~ /~2 d 2 ~+ h~3(x ) + 2m dx 2
Ve(X)~(x) = gt~(x)
(2)
~t is a L a g r a n g e m u l t i p l i e r which d e t e r m i n e s N, and ~ c l e a r l y can be c h o s e n to be a r e a l function of x alone. P h y s i c a l c o n s i d e r a t i o n s s u g g e s t that v e be of s h o r t r a n g e and infinitely r e p u l s i v e at x = 0. To i l l u s t r a t e our point, we s h a l l c o n s i d e r the e x a c t l y soluble m o d e l in which v e is taken to be a s q u a r e wel l ; our u l t i m a t e c o n c l u s i o n s , howe v e r , a r e independent of this s p e c i a l assumption. The po t en t i al is shown in fig. 1. v 0 and a a r e chosen in such a way that the well has a s i n g l e p a r t i c l e bound state. The boundary conditions a p p r o p r i a t e f o r a f i l m a r e : ~(0) = 0, and ~(x) ~ 0 as x ~ ~. With the aid of t h e s e boundary conditions, d i r e c t i n t e g r a t i o n of eq. (2) l e a d s to: tp(x) = ~f2 c o s e c h [x - a + a r e a c o s e c h (~(a)/~f2)] x>~a
~V(x) x= ao f d f / ~ ½ f 4 + 0
(l_a)f2 +a~2(a)]½
15 December 1969
-
N
~1 f
l)
( r - r')~2(r)~2(r')drdr '
(3)
f ~2(r)d r
Since ~(x) d e s c r i b e s a bound s y s t e m , g < 0. It then follows that e < 0. H o w e v e r , we can a l s o w r i t e E as
E = { (/r2/2m) f (V¢)2dr + f Ve(X)~2(r)dr + +½ f v ( r -
r')¢2(r)¢2(r')drdr'} { f
(4)
¢2(r)dr}-I
The only t e r m in eq. (4) that is negative i s the second. T h e r e f o r e , e v e n though we may i n c r e a s e N , ~2(x) m u s t s t i l l r e m a i n peaked in the r e g i o n w h e r e ve is a p p r e c i a b l y d i f f e r e n t f r o m z e r o , in o r d e r that the e n t i r e r i g h - h a n d side of eq. (4) be n eg at i v e. A s i m i l a r a n a l y s i s can be made in the context of the G i n z b u r g - P i t a e v s k i t h e o r y , but it i n v o l v e s f u r t h e r q u e s t i o n s beyond the scope of t h i s note. N e v e r t h e l e s s , if the f i l m s t r u c t u r e is to be d e s c r i b e d in the context of c o r r e c t bounda r y conditions, then a m o r e f u n d am en t al m i c r o s copic a p p r o a c h i s c a l l e d for.
x
w h e r e a 2 = ~ 2 / 2 m [ ~ 1 , a n d s =Vo/l~ [ > 1. Continuity ot ~ ' / ~ at x = a l e a d s to an i m p l i c i t e q u a tion f o r ~(a) in t e r m s of an e l l i p t i c i n t e g r a l of the f i r s t kind. F o r all allowed v a l u e s of a ( and hence h0, the s o l u t i o n s f o r ~2(x) = n(x) have the f o r m shown above and s c h e m a t i c a l l y in fig. 1. In p a r t i c u l a r , they a r e s t r o n g l y peaked in the r e g i o n x < a. T h i s i s v e r y d i f f e r e n t f r o m the b e h a v i o r of the d en s i t y p r o f i l e of a r e a l p h y s i c a l film. We would e x p e c t the l a t t e r to have a peak n e a r the w a ll , r e m a i n e s s e n t i a l l y flat o v e r an a p p r e c i a b l e
I wish to thank P r o f e s s o r G. V. C h e s t e r f o r s u g g e s t i n g this i n v e s t i g a t i o n .
References 1. V L. Ginzburg and L. P. Pitaevskii, Soviet Phys. JETP 7 (1961) 858. 2. E. P. Gross, J. Math. Phys. 4 {1963) 195. 3. E. P. Gross: in Physics of many-particle systems, ed. E. Meeron {Gordon and Breach, Inc., 1966} p. 231. 4. E. P. Gross and D. Amit, Phys. Rev. 145 {1965) 130. 5. D. S. Hyman, Ph.D. thesis, Cornell University, 1969, unpublished.
467
PHYSICS
LETTERS
A n i m p o r t a n t b y - p r o d u c t of t h i s c a l c u l a t i o n i s that exact transport coefficients, including their potential part, can be computed by solving the one-particle linearized generalized Boltzmann e q u a t i o n to s e c o n d o r d e r i n t h e u n i f o r m i t y p a r a m eter. Indeed, nowhere in this theory, is there a n y a p p e a l to t h e f u n c t i o n a l r e l a t i o n b e t w e e n t h e two p a r t i c l e d i s t r i b u t i o n f u n c t i o n a n d t h e o n e p a r t icle distribution function; this is in contrast with t h e t r a d i t i o n a l t h e o r y [e.g. 9]. T h e d e t a i l s of t h e s e c a l c u l a t i o n s w i l l b e p u b l i s h e d e l s e w h e r e [10].
References 1. H. Mori, Prog. Th. Phys. Jap. 28 (1962) 763; 33 (1965) 423. 2. L.Kadanoff and P. Martin, Ann. Physics 24 (1965) 419. 3. L. Kadanoff and J. Swift, Phys. Rev. 166 (1968) 89. 4. Y. Pomeau, Phys. L e t t e r s 27A (1968) 601 and Ph D Thesis (Universit~ de P a r i s , 1969, unpublished) 5. L. Van Hove, Phys. Rev. 95 (1959) 249. 6. G. Severnc, Physica 31 (1965) 877; E. G. D Cohen, Fundamental p r o b l e m s in s t a t i s t i c a l m e c h a n i c s II, ed. E. G. D. Cohen (North Holland, A m s t e r d a m , 1968). 7. L. Garcia-Colin, M. G r e e n and F. Chaos; Physica 20 (1966) 450. 8. D. Bohm; Quantum theory (Constable, London, 1954). 9. The proof is s i m i l a r to that given in P. R6sibois, J Chem. Phys. 41 (1964) 2979 and G. Nicolis and G. Severne, J. Chem. Phys. 44 (1966) 1477. 10. P. R6sibois, J. Star. P h y s i c s , to be published.
It i s g r a t e f u l l y a c k n o w l e d g e d t h a t t h e w o r k p r e s e n t e d h e r e f o u n d i t s g e n e s i s i n a s e r i e s of d i s c u s s i o n s w i t h P r o f . J . L e b o w i t z . We a l s o t h a n k P r o f . G. N i c o l i s f o r u s e f u l c o m m e n t s .
NOTE
ON
HARTREE
MODELS
OF
15 D e c e m b e r 1969
THE
SUPERFLUID
HELIUM
FILM
*
D. S. H Y M A N
Physics Department and Center for Materials Science and Engineering Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA Received 17 N o v e m b e r 1969
Phenomenological models of the superfluid helium film a r e discussed. It is shown that a r e c e n t H a r t r e e model due to G r o s s and Amit cannot lead to a completely r e a l i s t i c d e s c r i p t i o n of such films.
T o d a t e , t h e o r e t i c a l d i s c u s s i o n s of t h e s u p e r f l u i d h e l i u m f i l m h a v e b e e n b a s e d on p h e n o m e n o l o g i c a l [1,2] o r s e m i p h e n o m e n o l o g i c a l [3] m o d e l s in which the superfluid density is described by a complex wave function ~(r). However, a major p o i n t of u n c e r t a i n t y i n t h e s e a p p r o a c h e s i s t h e b e h a v i o r of ~P(r) a t t h e f r e e s u r f a c e of t h e f i l m . We shall show that one can use the Hartree model of G r o s s a n d A m i t [4] t o s t u d y a h e l i u m f i l m b o u n d t o a u n i f o r m s u b s t r a t e . In t h i s a p p r o a c h the boundary conditions on ~(r) are unambiguous; h o w e v e r , it w i l l b e s h o w n t h a t t h e e x p e c t e d s t r u c t u r e of m o d e r a t e l y t h i c k f i l m s c a n n o t b e o b t a i n e d with this model. T h e H a r t r e e m o d e l , g e n e r a l i z e d to i n c l u d e t h e e f f e c t of a w a l l w i t h a n a t t r a c t i v e p o t e n t i a l , l e a d s to t h e f o l l o w i n g e n e r g y f u n c t i o n a l f o r N i n t e r a c t i n g b o s o n s [4]: 466
E =
/Iv
12dr +
+ ½fv(r-r')[~(r)t2[~(r')]2drdr'
+
(1)
+ fVe(X)[~[2dr T h e d e n s i t y d i s t r i b u t i o n , g i v e n b y ] ~ ( r ) t 21 s a t isfies the normalization condition f[~ (r)l 2dr = = N. T h e p h e n o m e n o l o g i c a l i n t e r a c t i o n v ( r ) i s w r i t t e n a s v ( r ) = X6 ( r ) , w i t h X g i v e n i n t e r m s of t h e sGund s p e e d c a n d t h e n u m b e r d e n s i t y p / m of the bulk liquid by ~ = mc2/p. We have assumed t h a t t h e w a l l , s i t u a t e d a t x = 0, c a n b e r e p r e s e n t e d b y t h e o n e - d i m e n s i o n a l p o t e n t i a l re(X). A p p l i c a t i o n of t h e v a r i a t i o n a l p r i n c i p l e l e a d s to t h e f o l l o w i n g e q u a t i o n f o r ~h(x): * R e s e a r c h supported by Advanced R e s e a r c h P r o j e c t s Agency Contract SD-90, and in p a r t by the National Aeronautics and Space Administration.
Volume 30A, number 8
PHYSICS LETTERS
-% Fig. 1. External wall potential, and density profile obtained from eq. (2).
r a n g e , and then f a l l to z e r o in a d i s t a n c e of p e r haps one or two a n g s t r o m s . The p h y s i c a l e f f e c t m i s s i n g f r o m the m o d e l is the influence of the a t t r a c t i v e i n t e r p a r t i c l e f o r c e s which would lead to a st ab l e, p r e d o m i n a n t l y flat f i l m . H o w e v e r , any a t t e m p t to i n c o r p o r a t e this e f f e c t through a m o d i f i c a t i o n of v ( r ) would f ai l , si n ce it can be shown [5] f r o m s t a b i l i t y c o n s i d e r a t i o n s that f v ( r - r')~2(r)tp2(r')drdr ' m u s t be p o s i t i v e . M o r e o v e r , with this condition, our concl u si o n c o n c e r n i n g the q u a l i t a t i v e nature of the s o l u t i o n s to (2) holds in g e n e r a l f o r all finiterange Ve(X). T h i s is e a s i l y s e e n by noting that f r o m (1) and (2) we can w r i t e the e n e r g y / a t o m as
E_ =-e= ~ /~2 d 2 ~+ h~3(x ) + 2m dx 2
Ve(X)~(x) = gt~(x)
(2)
~t is a L a g r a n g e m u l t i p l i e r which d e t e r m i n e s N, and ~ c l e a r l y can be c h o s e n to be a r e a l function of x alone. P h y s i c a l c o n s i d e r a t i o n s s u g g e s t that v e be of s h o r t r a n g e and infinitely r e p u l s i v e at x = 0. To i l l u s t r a t e our point, we s h a l l c o n s i d e r the e x a c t l y soluble m o d e l in which v e is taken to be a s q u a r e wel l ; our u l t i m a t e c o n c l u s i o n s , howe v e r , a r e independent of this s p e c i a l assumption. The po t en t i al is shown in fig. 1. v 0 and a a r e chosen in such a way that the well has a s i n g l e p a r t i c l e bound state. The boundary conditions a p p r o p r i a t e f o r a f i l m a r e : ~(0) = 0, and ~(x) ~ 0 as x ~ ~. With the aid of t h e s e boundary conditions, d i r e c t i n t e g r a t i o n of eq. (2) l e a d s to: tp(x) = ~f2 c o s e c h [x - a + a r e a c o s e c h (~(a)/~f2)] x>~a
~V(x) x= ao f d f / ~ ½ f 4 + 0
(l_a)f2 +a~2(a)]½
15 December 1969
-
N
~1 f
l)
( r - r')~2(r)~2(r')drdr '
(3)
f ~2(r)d r
Since ~(x) d e s c r i b e s a bound s y s t e m , g < 0. It then follows that e < 0. H o w e v e r , we can a l s o w r i t e E as
E = { (/r2/2m) f (V¢)2dr + f Ve(X)~2(r)dr + +½ f v ( r -
r')¢2(r)¢2(r')drdr'} { f
(4)
¢2(r)dr}-I
The only t e r m in eq. (4) that is negative i s the second. T h e r e f o r e , e v e n though we may i n c r e a s e N , ~2(x) m u s t s t i l l r e m a i n peaked in the r e g i o n w h e r e ve is a p p r e c i a b l y d i f f e r e n t f r o m z e r o , in o r d e r that the e n t i r e r i g h - h a n d side of eq. (4) be n eg at i v e. A s i m i l a r a n a l y s i s can be made in the context of the G i n z b u r g - P i t a e v s k i t h e o r y , but it i n v o l v e s f u r t h e r q u e s t i o n s beyond the scope of t h i s note. N e v e r t h e l e s s , if the f i l m s t r u c t u r e is to be d e s c r i b e d in the context of c o r r e c t bounda r y conditions, then a m o r e f u n d am en t al m i c r o s copic a p p r o a c h i s c a l l e d for.
x
w h e r e a 2 = ~ 2 / 2 m [ ~ 1 , a n d s =Vo/l~ [ > 1. Continuity ot ~ ' / ~ at x = a l e a d s to an i m p l i c i t e q u a tion f o r ~(a) in t e r m s of an e l l i p t i c i n t e g r a l of the f i r s t kind. F o r all allowed v a l u e s of a ( and hence h0, the s o l u t i o n s f o r ~2(x) = n(x) have the f o r m shown above and s c h e m a t i c a l l y in fig. 1. In p a r t i c u l a r , they a r e s t r o n g l y peaked in the r e g i o n x < a. T h i s i s v e r y d i f f e r e n t f r o m the b e h a v i o r of the d en s i t y p r o f i l e of a r e a l p h y s i c a l film. We would e x p e c t the l a t t e r to have a peak n e a r the w a ll , r e m a i n e s s e n t i a l l y flat o v e r an a p p r e c i a b l e
I wish to thank P r o f e s s o r G. V. C h e s t e r f o r s u g g e s t i n g this i n v e s t i g a t i o n .
References 1. V L. Ginzburg and L. P. Pitaevskii, Soviet Phys. JETP 7 (1961) 858. 2. E. P. Gross, J. Math. Phys. 4 {1963) 195. 3. E. P. Gross: in Physics of many-particle systems, ed. E. Meeron {Gordon and Breach, Inc., 1966} p. 231. 4. E. P. Gross and D. Amit, Phys. Rev. 145 {1965) 130. 5. D. S. Hyman, Ph.D. thesis, Cornell University, 1969, unpublished.
467