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A numerical calculation of the excitation of helium II based on the hard-sphere model
Volume 30A. number 5
PHYSICS
A NUMERICAL
CALCULATION BASED ON THE
LETTERS
3 November 1969
OF THE EXCITATION HARD-SPHERE MODEL
OF
HELIUM
II
K. W. WONG
Department of Materials Science. University of Southern California, Los Angeles, California 90007, USA and Y. H. HUANG $$
Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas. USA Received 30 September 1969 In this letter we would like to report some numerical computations of a hard~sphere Boson system at liquid helium density. These computations are based on an exact iteration process from non-local field operators and with the core diameter taken as 2.1 ,~. The Hamiltonian includes only the repulsive hardcore taken into consideration via the algebra for these field operators developed in refs. 1 and 2. The numerical answer we obtained at liquid helium density shows apparently no zero-momentum condensate. However, the excitation spectrum is very close to the experimental one. It is interesting to point out also that such a conclusion implies that the three-body t e r m treated by Wu in the dilute case actually contributes zero here. O u r m e t h o d of n u m e r i c a l c o m p u t a t i o n is b a s e d on a p a r t i a l w a v e e x p r e s s i o n of t h e c o o r d i n a t e s p a c e r e s t r i c t i o n b e t w e e n p a r t i c l e s . T h e H a m i l t o n i a n i s t h e n f u r t h e r s e p a r a t e d into a p a i r - w i s e p a r t and a t h r e e - b o d y p a r t . T h e l a t t e r p a r t i s t r e a t e d to s e c o n d o r d e r p e r t u r b a t i o n . T h e p a i r - w i s e p a r t i n c l u d e s a t e r m of t h e BCS f o r m u s e d in s u p e r c o n d u c t o r t h e o r y . T h e z e r o - m o m e n t u m c o m p o n e n t is t r e a t e d e x a c t ly with K r o m i n g e r - B o l s t e r l i [3] t r a n s f o r m a t i o n , t h e p r o b l e m i s then s o l v e d by v a r i a t i o n a l m e t h o d . A l though in a s e n s e t h e B o g o l i u b o v p a i r - w i s e t e r m d o e s not c o n t r i b u t e n u m e r i c a l l y , but it i s e s s e n t i a l in t h e c o n v e r g e n c e of t h e c o m p u t a t i o n to i n c l u d e . T h e e x c i t a t i o n s p e c t r u m t h u s o b t a i n e d is
h ~ 2 {d _ S~I}½
(1)
Elx(k) = 2ma2 where
(1-? o)+ S1 withx
=
C) + 4 (19 + F ( c o s x - Jo(X)) +Bo(Jo(X) - 1) - ~
l=l
(\87ra3p - 4 c ) cos x + ~ ((cos X-jo(X) s o -
-
l=l
(4z + 1)E2I;2Z(x)
(4/+ 1)S21J2l(X)) (2)
+DJo(X))
-akand oO
C = j dxx 2
0
OO
/]2
1-U2'
1)(-1)lE21 ,
O = ~ (4/+ l=O
B2l = f dxx2j2l (x) 0
U2 1-?/2'
(3) oo
F : l~= o (4l+l)(-1)lB21,
2
E2l = fo d . x x J2l(X)
1 _U u2,
- :s2
- __
-s_
/sl .
It i s e a s y to s e e f r o m t h e f o r m u l a s t h a t the p r o b l e m we h a v e h e r e is a highly n o n - l i n e a r one. E x c e p t f o r t h e t e r m i n v o l v i n g t h e d e p l e t i o n f a c t o r C and t h e d e l t a f u n c t i o n a l s c a t t e r i n g t e r m g i v e n by E o t h e a s s y m p t o t i c p a r t in U a p p r o a c h e s z e r o l e a v i n g r e a l l y no d i v e r g e n c e s at l a r g e m o m e n t a . T h e t e r m i n v o l v ing E o s h o u l d be z e r o , s i n c e it is e s s e n t i a l l y L i e b ' s s u b s i d i a r y c o n d i t i o n [5]. W e r e a d i l y n o t i c e that D $ On leave of absence from the University of Kansas. $$ Present address: R . C . A . Research Laboratory, Pennsylvania. 292
Volume 30A. n u m b e r 5
PHYSICS
LETT~.RS
3 November 1969
20
15
EXPERIMENTALI
v o
.S IO ,,K w
,
l
K inA 2 Fig. 1. A plot of the excitation s p e c t r u m using these n u m b e r s . The c l o s e n e s s to the experimental curve can be adjusted by changing "a" slightly. d i v e r g e s p r o v i d e d t h a t o n e r e m e m b e r s t h a t D c o m e s f r o m t h e k i n e t i c e n e r g y o p e r a t o r on a w a v e f u n c t i o n t h a t h a s a d i s c o n t i n u o u s d e r i v a t i v e on t h e s u r f a c e of t h e c o r e . S i n c e U(x) c o m e s f r o m t h e g r o u n d state vector as follows I@)G = 1-[ exp {- U(x)b*b*} x - x I 0) , x>0
(4)
w h e r e bx a r e K r o m i n g e r - B o l s t e r i operators which has no zero-momentum component, we must treat t h e s u r f a c e d e r i v a t i v e f r o m o u t s i d e t h e c o r e . W h i c h i m p l i e s E2l s h o u l d r e a l l y b e t r e a t e d m o r e c a r e fully with the following limiting process
E21 = l i m f dxx2j21(x) E'--' 0
U(x(1 +¢)) •, , '-,~ - r r 2 'v,, ~'l+e' i •
(5)
E>O S u c h l i m i t i n g p r o c e s s e s l e a d to t h e p r e s e n t n u m e r i c a l n u m b e r s f o r l i q u i d h e l i u m d e n s i t y . W e h a v e c o m p u t e d t h e s e c o n s t a n t s b y i n c l u d i n g u p to Jl8(X) t e r m s , a n d a r e g i v e n b e l o w : C = 27r2a3p , B0
=0.82
where p = ,
B2
=
(3_~6)3 ~-3 ,
0.18
,
B4
a = 2.1 A ; =0.096,
B6
F = -0.17 =0.0401
;
,
B8
D = 8.99 ;
B10 = 0.006 ,
B12 =
0.003 ,
B14 = 0.0
,
B16 : 0.0
,
B18 = 0.0
E0
E2
0.306,
E4
= 1.01
,
E6
=-0.37
,
g 8
E l 4 = 0.05
,
E16 =-0.02
,
E18 =0.003
= 0.0
El0 =0.085,
,
=
El2 = -0.06
,
(6)
=0.016, ;
=0.119, •
293
Volume 30A, number 5
PHYSICS L E T T E R S
3 November 1969
Our conclusion, t h e r e f o r e , is that superfluidity in helium is a purely q u a s i - p a r t i c l e pair c o n d e n s a tion and probably has nothing to do with the visual f r e e - p a r t i c l e z e r o - m o m e n t u m condensation.
References 1. K.W.Wong, J. Math. Phys. 5 {1964} 637. 2. A.J.F.Siegert, Phys. Rev. 116 (1959) 1057. 3. A.J.Kromminger and M.Bolsterli, Phys. Rev. 128 (1962) 2887. 4. T.T.Wu. J. Math. Phys. 2 (1961) 105. 5. E.Lieb. Proc. Nat'l. Acad. Sci. 46 {1960) 1000.
GENESIS
OF
THE
SPIN-FLIP
RESISTIVITY
PHENOMENON
IN CHROMIUM
G. T. MEADEN and N. H. SZE Deparlrnenl of Physics. Dalhousie University, Halifax. Nova Scotia. Canada Received 29 September 1969
We report the discovery of a small wave-like variation of the electrical resistivity near the spin-flip temperature, and demonstrate that with annealing and grain-gro~h it develops into a prominent step-like change of 1.7~ magnitude.
In pure c h r o m i u m at a r o u n d 121 to 123°K (known as T F) the d i r e c t i o n of the spin p o l a r i z a tion r e l a t i v e to the s p i n - d e n s i t y - w a v e w a v e - v e c 1 tor flips through an angle of ~ . This m o d i f i c a tion of the m a g n e t i c s y m m e t r y is known to affect s e v e r a l physical p r o p e r t i e s but for a long time it appeared that any effect on t r a n s p o r t p r o p e r t i e s , including the e l e c t r i c a l r e s i s t i v i t y , was too s m a l l to be o b s e r v e d [1-7]. Recently, however, t h e r e have been r e p o r t s of changes of slope of the e l e c t r i c a l r e s i s t i v i t y n e a r T F [8,9]. We r e p o r t h e r e not m e r e l y a change of slope of the e l e c t r i c a l r e s i s t i v i t y but the o c c u r r e n c e and development of a step-type anomaly at T F in a 99.999% pure p o l y c r y s t a l l i n e s p e c i m e n for t h r e e different states of anneal. The r e s u l t s a r e p r e s e n t e d in figs. 1 and 2. C r - 0 r e f e r s to the u n a n n e a l e d s a m p l e of r e s i d u a l r e s i s t i v i t y r a t i o 178, C r - 1 to the s a m p l e after one hour a n n e a l i n g at 1200°C, and C r - 5 0 ( r e s i dual r e s i s t i v i t y ratio 295) to the s a m e s a m p l e after 75 hours annealing above 1000°C of which 50 h o u r s were p a s s e d at 1200°C. In C r - 0 there o c c u r s an undulation of ~ 1% amplitude, c e n t e r e d n e a r 121.2°K and s p r e a d over 1.8 deg. (AT). The t e m p e r a t u r e exponent is 2.45 below T F and 2.25 above T F. The o n e - h o u r a n n e a l enhanced the anomaly to 1,3%, and reduced the a p p a r e n t t r a n 294
sition width (T F ~ 120.3°K, AT ~ 0.6 deg.). The difference in the t e m p e r a t u r e exponents was also r a i s e d , becoming 2.55 below and 2.31 above T F. F o r both C r - 0 and C r - 1 the c r o s s - o v e r points, obtained by extrapolating the upper s e c t i o n s of the c u r v e s of fig. 2 (> T F) toward the lower p a r t s (
PHYSICS
A NUMERICAL
CALCULATION BASED ON THE
LETTERS
3 November 1969
OF THE EXCITATION HARD-SPHERE MODEL
OF
HELIUM
II
K. W. WONG
Department of Materials Science. University of Southern California, Los Angeles, California 90007, USA and Y. H. HUANG $$
Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas. USA Received 30 September 1969 In this letter we would like to report some numerical computations of a hard~sphere Boson system at liquid helium density. These computations are based on an exact iteration process from non-local field operators and with the core diameter taken as 2.1 ,~. The Hamiltonian includes only the repulsive hardcore taken into consideration via the algebra for these field operators developed in refs. 1 and 2. The numerical answer we obtained at liquid helium density shows apparently no zero-momentum condensate. However, the excitation spectrum is very close to the experimental one. It is interesting to point out also that such a conclusion implies that the three-body t e r m treated by Wu in the dilute case actually contributes zero here. O u r m e t h o d of n u m e r i c a l c o m p u t a t i o n is b a s e d on a p a r t i a l w a v e e x p r e s s i o n of t h e c o o r d i n a t e s p a c e r e s t r i c t i o n b e t w e e n p a r t i c l e s . T h e H a m i l t o n i a n i s t h e n f u r t h e r s e p a r a t e d into a p a i r - w i s e p a r t and a t h r e e - b o d y p a r t . T h e l a t t e r p a r t i s t r e a t e d to s e c o n d o r d e r p e r t u r b a t i o n . T h e p a i r - w i s e p a r t i n c l u d e s a t e r m of t h e BCS f o r m u s e d in s u p e r c o n d u c t o r t h e o r y . T h e z e r o - m o m e n t u m c o m p o n e n t is t r e a t e d e x a c t ly with K r o m i n g e r - B o l s t e r l i [3] t r a n s f o r m a t i o n , t h e p r o b l e m i s then s o l v e d by v a r i a t i o n a l m e t h o d . A l though in a s e n s e t h e B o g o l i u b o v p a i r - w i s e t e r m d o e s not c o n t r i b u t e n u m e r i c a l l y , but it i s e s s e n t i a l in t h e c o n v e r g e n c e of t h e c o m p u t a t i o n to i n c l u d e . T h e e x c i t a t i o n s p e c t r u m t h u s o b t a i n e d is
h ~ 2 {d _ S~I}½
(1)
Elx(k) = 2ma2 where
(1-? o)+ S1 withx
=
C) + 4 (19 + F ( c o s x - Jo(X)) +Bo(Jo(X) - 1) - ~
l=l
(\87ra3p - 4 c ) cos x + ~ ((cos X-jo(X) s o -
-
l=l
(4z + 1)E2I;2Z(x)
(4/+ 1)S21J2l(X)) (2)
+DJo(X))
-akand oO
C = j dxx 2
0
OO
/]2
1-U2'
1)(-1)lE21 ,
O = ~ (4/+ l=O
B2l = f dxx2j2l (x) 0
U2 1-?/2'
(3) oo
F : l~= o (4l+l)(-1)lB21,
2
E2l = fo d . x x J2l(X)
1 _U u2,
- :s2
- __
-s_
/sl .
It i s e a s y to s e e f r o m t h e f o r m u l a s t h a t the p r o b l e m we h a v e h e r e is a highly n o n - l i n e a r one. E x c e p t f o r t h e t e r m i n v o l v i n g t h e d e p l e t i o n f a c t o r C and t h e d e l t a f u n c t i o n a l s c a t t e r i n g t e r m g i v e n by E o t h e a s s y m p t o t i c p a r t in U a p p r o a c h e s z e r o l e a v i n g r e a l l y no d i v e r g e n c e s at l a r g e m o m e n t a . T h e t e r m i n v o l v ing E o s h o u l d be z e r o , s i n c e it is e s s e n t i a l l y L i e b ' s s u b s i d i a r y c o n d i t i o n [5]. W e r e a d i l y n o t i c e that D $ On leave of absence from the University of Kansas. $$ Present address: R . C . A . Research Laboratory, Pennsylvania. 292
Volume 30A. n u m b e r 5
PHYSICS
LETT~.RS
3 November 1969
20
15
EXPERIMENTALI
v o
.S IO ,,K w
,
l
K inA 2 Fig. 1. A plot of the excitation s p e c t r u m using these n u m b e r s . The c l o s e n e s s to the experimental curve can be adjusted by changing "a" slightly. d i v e r g e s p r o v i d e d t h a t o n e r e m e m b e r s t h a t D c o m e s f r o m t h e k i n e t i c e n e r g y o p e r a t o r on a w a v e f u n c t i o n t h a t h a s a d i s c o n t i n u o u s d e r i v a t i v e on t h e s u r f a c e of t h e c o r e . S i n c e U(x) c o m e s f r o m t h e g r o u n d state vector as follows I@)G = 1-[ exp {- U(x)b*b*} x - x I 0) , x>0
(4)
w h e r e bx a r e K r o m i n g e r - B o l s t e r i operators which has no zero-momentum component, we must treat t h e s u r f a c e d e r i v a t i v e f r o m o u t s i d e t h e c o r e . W h i c h i m p l i e s E2l s h o u l d r e a l l y b e t r e a t e d m o r e c a r e fully with the following limiting process
E21 = l i m f dxx2j21(x) E'--' 0
U(x(1 +¢)) •, , '-,~ - r r 2 'v,, ~'l+e' i •
(5)
E>O S u c h l i m i t i n g p r o c e s s e s l e a d to t h e p r e s e n t n u m e r i c a l n u m b e r s f o r l i q u i d h e l i u m d e n s i t y . W e h a v e c o m p u t e d t h e s e c o n s t a n t s b y i n c l u d i n g u p to Jl8(X) t e r m s , a n d a r e g i v e n b e l o w : C = 27r2a3p , B0
=0.82
where p = ,
B2
=
(3_~6)3 ~-3 ,
0.18
,
B4
a = 2.1 A ; =0.096,
B6
F = -0.17 =0.0401
;
,
B8
D = 8.99 ;
B10 = 0.006 ,
B12 =
0.003 ,
B14 = 0.0
,
B16 : 0.0
,
B18 = 0.0
E0
E2
0.306,
E4
= 1.01
,
E6
=-0.37
,
g 8
E l 4 = 0.05
,
E16 =-0.02
,
E18 =0.003
= 0.0
El0 =0.085,
,
=
El2 = -0.06
,
(6)
=0.016, ;
=0.119, •
293
Volume 30A, number 5
PHYSICS L E T T E R S
3 November 1969
Our conclusion, t h e r e f o r e , is that superfluidity in helium is a purely q u a s i - p a r t i c l e pair c o n d e n s a tion and probably has nothing to do with the visual f r e e - p a r t i c l e z e r o - m o m e n t u m condensation.
References 1. K.W.Wong, J. Math. Phys. 5 {1964} 637. 2. A.J.F.Siegert, Phys. Rev. 116 (1959) 1057. 3. A.J.Kromminger and M.Bolsterli, Phys. Rev. 128 (1962) 2887. 4. T.T.Wu. J. Math. Phys. 2 (1961) 105. 5. E.Lieb. Proc. Nat'l. Acad. Sci. 46 {1960) 1000.
GENESIS
OF
THE
SPIN-FLIP
RESISTIVITY
PHENOMENON
IN CHROMIUM
G. T. MEADEN and N. H. SZE Deparlrnenl of Physics. Dalhousie University, Halifax. Nova Scotia. Canada Received 29 September 1969
We report the discovery of a small wave-like variation of the electrical resistivity near the spin-flip temperature, and demonstrate that with annealing and grain-gro~h it develops into a prominent step-like change of 1.7~ magnitude.
In pure c h r o m i u m at a r o u n d 121 to 123°K (known as T F) the d i r e c t i o n of the spin p o l a r i z a tion r e l a t i v e to the s p i n - d e n s i t y - w a v e w a v e - v e c 1 tor flips through an angle of ~ . This m o d i f i c a tion of the m a g n e t i c s y m m e t r y is known to affect s e v e r a l physical p r o p e r t i e s but for a long time it appeared that any effect on t r a n s p o r t p r o p e r t i e s , including the e l e c t r i c a l r e s i s t i v i t y , was too s m a l l to be o b s e r v e d [1-7]. Recently, however, t h e r e have been r e p o r t s of changes of slope of the e l e c t r i c a l r e s i s t i v i t y n e a r T F [8,9]. We r e p o r t h e r e not m e r e l y a change of slope of the e l e c t r i c a l r e s i s t i v i t y but the o c c u r r e n c e and development of a step-type anomaly at T F in a 99.999% pure p o l y c r y s t a l l i n e s p e c i m e n for t h r e e different states of anneal. The r e s u l t s a r e p r e s e n t e d in figs. 1 and 2. C r - 0 r e f e r s to the u n a n n e a l e d s a m p l e of r e s i d u a l r e s i s t i v i t y r a t i o 178, C r - 1 to the s a m p l e after one hour a n n e a l i n g at 1200°C, and C r - 5 0 ( r e s i dual r e s i s t i v i t y ratio 295) to the s a m e s a m p l e after 75 hours annealing above 1000°C of which 50 h o u r s were p a s s e d at 1200°C. In C r - 0 there o c c u r s an undulation of ~ 1% amplitude, c e n t e r e d n e a r 121.2°K and s p r e a d over 1.8 deg. (AT). The t e m p e r a t u r e exponent is 2.45 below T F and 2.25 above T F. The o n e - h o u r a n n e a l enhanced the anomaly to 1,3%, and reduced the a p p a r e n t t r a n 294
sition width (T F ~ 120.3°K, AT ~ 0.6 deg.). The difference in the t e m p e r a t u r e exponents was also r a i s e d , becoming 2.55 below and 2.31 above T F. F o r both C r - 0 and C r - 1 the c r o s s - o v e r points, obtained by extrapolating the upper s e c t i o n s of the c u r v e s of fig. 2 (> T F) toward the lower p a r t s (