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Correlation of spins in ferromagnets in the immediate vicinity of the Curie point
Volume 36A, number 3
PHYSICS
CORRELATION IMMEDIATE
OF
LETTERS
SPINS
IN
VICINITY
30 August 1971
FERROMAGNETS
OF
THE
CURIE
IN THE POINT
J. KOCII~SKI Institute of Physics, Warsaw Technical University, Warsaw, Poland
L. W O J T C ZAK Institute of Physics, University of ~LOd~, I~6d~, Poland S.
MRYGO~
Institute of Physics, Polish Academy of Sciences, Warsaw, Poland Received 21 July 1971 The decisive role of the non-linear terms in the equation for the spin correlation function for t e m p e r a tures close to the Curie point has been demonstrated.
T h e c o r r e l a t i o n f u n c t i o n b e t w e e n s p i n s m a y be e x p r e s s e d in t e r m s of t h e m e a n d i s t r i b u t i o n in m a g n e t i c m o m e n t in a f l u c t u a t i o n t r e a t e d a s s u b s y s t e m in a r e s e r v o i r by the f o r m u l a
C
•
o.|
~ Z,( r ) (SZo(0) SrZ(0)) = (½A) 2 M
(1) Q03
v a l i d for s p i n s = ½, w h e r e A i s the a m p l i t u d e at the c e n t r e ( r = 0), in the u n i t s of B o h r ' s m a g n e t o n ~, a n d M Z ( r ) i s t h e m a g n e t i c m o m e n t [1]. T h e d i f f e r e n t i a l e q u a t i o n for M~(r) h a s b e e n d e r i v e d by the v a r i a t i o n a l m e t h o d [1] with a c c u r a c y to t h i r d o r d e r t e r m s . T h e s e c o r r e s p o n d to the f o u r t h o r d e r t e r m s in the t h e r m o d y n a m i c p o t e n t i a l by t h e v a r i a t i o n of which the e q u a t i o n h a s b e e n o b t a i n e d in the f o r m
(2) [V 2 - K21- K3(MZ)2 - K3v(grad MZ(r)) 2] MZ(r) = 0 , for t e m p e r a t u r e s T >/ T c a n d without e x t e r n a l m a g n e t i c field. H e r e K3 i s a f u n c t i o n of t e m p e r a t u r e , v ~ 1 . 4 1 a 2, a b e i n g the l a t t i c e c o n s t a n t in the b . c . c , l a t t i c e a n d K1 i s the O r n s t e i n Z e r n i k e p a r a m e t e r . T h e f i r s t two t e r m s in eq. (2) r e p r e s e n t the O r n s t e i n - Z e r n i k e e q u a t i o n for t h e c o r r e l a t i o n f u n c t i o n . T h e i m p o r t a n c e of the two r e m a i n i n g t e r m s d e p e n d s on t h e i r q u a l i t a t i v e l y d i f f e r e n t d e p e n d e g c e on t e m p e r a t u r e t h a n .~2.While c l o s e to Tc, ~ i s p r o p o r t i o n a l to T - T c zn o u r a p p r o x i m a t i o n , K3 i s a l m o s t t e m p e r a t u r e i n d e p e n d e n t a n d f o r s o m e t e n s of d e g r e e s above T c it is p r a c t i c a l l y c o n s t a n t . F o r a b . c . c , l a t t i c e
Fig. 1. The numerical solution ( ) of eq. (3) and the exponential approximation (4), (---), for A = 0.05, and C of (5) calculated numerically; C(Tc) = 0.7113. The variable ~ is in the units of the lattice constant a. and Mz in the units of g. 1 a n d s p i n s s = ~, one f i n d s K3(Tc) = 0.714 a - 2 p -2. T h e r e f o r e , when the c r i t i c a l point i s a p p r o a c h e d , the p a r a m e t e r K1 g r a d u a l l y l o o s e s i m p o r t a n c e in eq. (2) on b e h a l f of K3" We c o m e b a c k to the o r i g i n a l i d e a of S m o l u c h o w s k i [2] a l s o s u p p o r t e d by E i n s t e i n [3] that at the c r i t i c a l point the f o u r t h o r d e r t e r m s in the t h e r m o d y n a m i c p o t e n t i a l a r e i n d i s p e n s i b l e in the d e s c r i p t i o n of c r i t i c a l o p a l e scence. W h i l e l o o k i n g for s o l u t i o n of the n o n - l i n e a r e q u a t i o n we have b e e n g u i d e d by the a n a l o g y with o u r e a r l i e r a p p r o a c h of ref. [4], when the exponen. t i a l c o r r e l a t i o n of t h e a r g u m e n t ~ = Ix] + lYI+ I zl
171
Volume 36A, number 3
PHYSICS
h a s b e e n i n t r o d u c e d . In t e r m s of t h i s v a r i a b l e the e q u a t i o n (2) t a k e s the f o r m 3 d 2 M z_ 2
_
,~1Mz- ~3(MZ)3
/dMZ\ 2
0 (3)
It has b e e n s o l v e d n u m e r i c a l l y w i t h the b o u n d a r y c o n d i t i o n s Mz(O) = A, MZ(°°) = O. T h e a m p l i t u d e A r e m a i n s an u n d e t e r m i n e d p a r a m e t e r . T h e numerical solutions for 0 < A < 1 have been comp a r e d with t h e f u n c t i o n
MZ(x,y,z) = pdexp[-5(ixl+]y] + !zi)], 2 2~2 g +g p ~
1 -'w
3
j°
: c [-~
......
k3(1 - K3 ~,p2A2)J
(4)
(5)
'
where C is a temperature dependent parameter n u m e r i c a l l y d e t e r m i n e d , fig. 1. A f r a c t i o n of a d e g r e e a w a y f r o m T c when C c h a n g e s f a s t and s t r o n g l y i n f l u e n c e s t h e b e h a v i o u r of the f u n c t i o n (4) a b e t t e r a p p r o x i m a t i o n to the n u m e r i c a l s o l u t i o n m a y b e found, h o w e v e r , f o r a q u a l i t a t i v e d i s c u s s i o n the e x p o n e n t i a l a p p r o x i m a t i o n i s preferable since its interesting consequences h a v e b e e n a l r e a d y d i s c u s s e d [4, 5]. T h e f o r m (5) of the i n v e r s e r a n g e of c o r r e l a t i o n e n a b l e s to e v a l u a t e the i n t e r v a l of t e m p e r a t u r e s w i t h i n w h i c h ~2 m a y be n e g l e c t e d . T h e l i m i t i n g t e m 1
172
LETTEttS
30 August 1971
p e r a t u r e s a r e T c +0.1 ° f o r A = 0.05 and 7"c + 0 . 3 ° f o r A = 0.1. C o r r e s p o n d i n g l y , b e l o w t h e s e t e m p e r a t u r e s K1 l o o s e s the m e a n i n g of the i n v e r s e r a n g e of c o r r e l a t i o n and is r e p l a c e d by 5, w h i c h is not e q u a l to z e r o at Tc. T h e c r o s s s e c t i o n c a l c u l a t e d w i t h (4) y i e l d s f i n i t e t r a n s m i s s i o n at Tc, a r e s u l t a n a l o g o u s to S m o l u c h o w s k i ' s in c r i t i c a l o p a l e s c e n c e t h e o r y and r e m o v i n g the u n p h y s i c a l c o n s e q u e n c e of i n f i n i t e t r a n s m i s s i o n in c o n v e n t i o n a l t h e o r y . W e n o t e that the t e m p e r a t u r e shift of the m e a n m a x i m u m of s c a t t e r i n g m a y be e x p l a i n e d by the c o r r e l a t i o n (4), h o w e v e r , the l a t e r a l m a x i m a [6], cannot a p p e a r in a t h e o r y o p e r a t i n g w i t h c o r r e l a t i o n s f o r m a l l y e x t e n d i n g to i n f i n i t y . F i n a l l y we n o t e t h a t the n o n - l i n e a r t e r m s a r e of l i t t l e i m p o r t a n c e in the e q u a t i o n f o r the IsinK2ri/r t y p e c o r r e l a t i o n s i n c e K2 i s p r a c tically temperature independent.
ReJere~tces [1] [2] [3] [4]
J. Koeit~ski, J Phys. Chem. Solids 26 (1965) 895. M. Smoluchowski, Ann. d. Phys. 25 (1908) 205. A. Einstein, Ann. d. Phys. 33 (1910) 1275. L. Wojtczak and J Kocirlski, Phys. Letters 32A (1970) 389. [5] L. Wojtczak and ,l Kocir]ski, Phys. Letters 34A (197t) 306. [6] R Ciszewski and K Blinowski. Phys. Letters 29A (1969) 513; 30A (1969) 68.
PHYSICS
CORRELATION IMMEDIATE
OF
LETTERS
SPINS
IN
VICINITY
30 August 1971
FERROMAGNETS
OF
THE
CURIE
IN THE POINT
J. KOCII~SKI Institute of Physics, Warsaw Technical University, Warsaw, Poland
L. W O J T C ZAK Institute of Physics, University of ~LOd~, I~6d~, Poland S.
MRYGO~
Institute of Physics, Polish Academy of Sciences, Warsaw, Poland Received 21 July 1971 The decisive role of the non-linear terms in the equation for the spin correlation function for t e m p e r a tures close to the Curie point has been demonstrated.
T h e c o r r e l a t i o n f u n c t i o n b e t w e e n s p i n s m a y be e x p r e s s e d in t e r m s of t h e m e a n d i s t r i b u t i o n in m a g n e t i c m o m e n t in a f l u c t u a t i o n t r e a t e d a s s u b s y s t e m in a r e s e r v o i r by the f o r m u l a
C
•
o.|
~ Z,( r ) (SZo(0) SrZ(0)) = (½A) 2 M
(1) Q03
v a l i d for s p i n s = ½, w h e r e A i s the a m p l i t u d e at the c e n t r e ( r = 0), in the u n i t s of B o h r ' s m a g n e t o n ~, a n d M Z ( r ) i s t h e m a g n e t i c m o m e n t [1]. T h e d i f f e r e n t i a l e q u a t i o n for M~(r) h a s b e e n d e r i v e d by the v a r i a t i o n a l m e t h o d [1] with a c c u r a c y to t h i r d o r d e r t e r m s . T h e s e c o r r e s p o n d to the f o u r t h o r d e r t e r m s in the t h e r m o d y n a m i c p o t e n t i a l by t h e v a r i a t i o n of which the e q u a t i o n h a s b e e n o b t a i n e d in the f o r m
(2) [V 2 - K21- K3(MZ)2 - K3v(grad MZ(r)) 2] MZ(r) = 0 , for t e m p e r a t u r e s T >/ T c a n d without e x t e r n a l m a g n e t i c field. H e r e K3 i s a f u n c t i o n of t e m p e r a t u r e , v ~ 1 . 4 1 a 2, a b e i n g the l a t t i c e c o n s t a n t in the b . c . c , l a t t i c e a n d K1 i s the O r n s t e i n Z e r n i k e p a r a m e t e r . T h e f i r s t two t e r m s in eq. (2) r e p r e s e n t the O r n s t e i n - Z e r n i k e e q u a t i o n for t h e c o r r e l a t i o n f u n c t i o n . T h e i m p o r t a n c e of the two r e m a i n i n g t e r m s d e p e n d s on t h e i r q u a l i t a t i v e l y d i f f e r e n t d e p e n d e g c e on t e m p e r a t u r e t h a n .~2.While c l o s e to Tc, ~ i s p r o p o r t i o n a l to T - T c zn o u r a p p r o x i m a t i o n , K3 i s a l m o s t t e m p e r a t u r e i n d e p e n d e n t a n d f o r s o m e t e n s of d e g r e e s above T c it is p r a c t i c a l l y c o n s t a n t . F o r a b . c . c , l a t t i c e
Fig. 1. The numerical solution ( ) of eq. (3) and the exponential approximation (4), (---), for A = 0.05, and C of (5) calculated numerically; C(Tc) = 0.7113. The variable ~ is in the units of the lattice constant a. and Mz in the units of g. 1 a n d s p i n s s = ~, one f i n d s K3(Tc) = 0.714 a - 2 p -2. T h e r e f o r e , when the c r i t i c a l point i s a p p r o a c h e d , the p a r a m e t e r K1 g r a d u a l l y l o o s e s i m p o r t a n c e in eq. (2) on b e h a l f of K3" We c o m e b a c k to the o r i g i n a l i d e a of S m o l u c h o w s k i [2] a l s o s u p p o r t e d by E i n s t e i n [3] that at the c r i t i c a l point the f o u r t h o r d e r t e r m s in the t h e r m o d y n a m i c p o t e n t i a l a r e i n d i s p e n s i b l e in the d e s c r i p t i o n of c r i t i c a l o p a l e scence. W h i l e l o o k i n g for s o l u t i o n of the n o n - l i n e a r e q u a t i o n we have b e e n g u i d e d by the a n a l o g y with o u r e a r l i e r a p p r o a c h of ref. [4], when the exponen. t i a l c o r r e l a t i o n of t h e a r g u m e n t ~ = Ix] + lYI+ I zl
171
Volume 36A, number 3
PHYSICS
h a s b e e n i n t r o d u c e d . In t e r m s of t h i s v a r i a b l e the e q u a t i o n (2) t a k e s the f o r m 3 d 2 M z_ 2
_
,~1Mz- ~3(MZ)3
/dMZ\ 2
0 (3)
It has b e e n s o l v e d n u m e r i c a l l y w i t h the b o u n d a r y c o n d i t i o n s Mz(O) = A, MZ(°°) = O. T h e a m p l i t u d e A r e m a i n s an u n d e t e r m i n e d p a r a m e t e r . T h e numerical solutions for 0 < A < 1 have been comp a r e d with t h e f u n c t i o n
MZ(x,y,z) = pdexp[-5(ixl+]y] + !zi)], 2 2~2 g +g p ~
1 -'w
3
j°
: c [-~
......
k3(1 - K3 ~,p2A2)J
(4)
(5)
'
where C is a temperature dependent parameter n u m e r i c a l l y d e t e r m i n e d , fig. 1. A f r a c t i o n of a d e g r e e a w a y f r o m T c when C c h a n g e s f a s t and s t r o n g l y i n f l u e n c e s t h e b e h a v i o u r of the f u n c t i o n (4) a b e t t e r a p p r o x i m a t i o n to the n u m e r i c a l s o l u t i o n m a y b e found, h o w e v e r , f o r a q u a l i t a t i v e d i s c u s s i o n the e x p o n e n t i a l a p p r o x i m a t i o n i s preferable since its interesting consequences h a v e b e e n a l r e a d y d i s c u s s e d [4, 5]. T h e f o r m (5) of the i n v e r s e r a n g e of c o r r e l a t i o n e n a b l e s to e v a l u a t e the i n t e r v a l of t e m p e r a t u r e s w i t h i n w h i c h ~2 m a y be n e g l e c t e d . T h e l i m i t i n g t e m 1
172
LETTEttS
30 August 1971
p e r a t u r e s a r e T c +0.1 ° f o r A = 0.05 and 7"c + 0 . 3 ° f o r A = 0.1. C o r r e s p o n d i n g l y , b e l o w t h e s e t e m p e r a t u r e s K1 l o o s e s the m e a n i n g of the i n v e r s e r a n g e of c o r r e l a t i o n and is r e p l a c e d by 5, w h i c h is not e q u a l to z e r o at Tc. T h e c r o s s s e c t i o n c a l c u l a t e d w i t h (4) y i e l d s f i n i t e t r a n s m i s s i o n at Tc, a r e s u l t a n a l o g o u s to S m o l u c h o w s k i ' s in c r i t i c a l o p a l e s c e n c e t h e o r y and r e m o v i n g the u n p h y s i c a l c o n s e q u e n c e of i n f i n i t e t r a n s m i s s i o n in c o n v e n t i o n a l t h e o r y . W e n o t e that the t e m p e r a t u r e shift of the m e a n m a x i m u m of s c a t t e r i n g m a y be e x p l a i n e d by the c o r r e l a t i o n (4), h o w e v e r , the l a t e r a l m a x i m a [6], cannot a p p e a r in a t h e o r y o p e r a t i n g w i t h c o r r e l a t i o n s f o r m a l l y e x t e n d i n g to i n f i n i t y . F i n a l l y we n o t e t h a t the n o n - l i n e a r t e r m s a r e of l i t t l e i m p o r t a n c e in the e q u a t i o n f o r the IsinK2ri/r t y p e c o r r e l a t i o n s i n c e K2 i s p r a c tically temperature independent.
ReJere~tces [1] [2] [3] [4]
J. Koeit~ski, J Phys. Chem. Solids 26 (1965) 895. M. Smoluchowski, Ann. d. Phys. 25 (1908) 205. A. Einstein, Ann. d. Phys. 33 (1910) 1275. L. Wojtczak and J Kocirlski, Phys. Letters 32A (1970) 389. [5] L. Wojtczak and ,l Kocir]ski, Phys. Letters 34A (197t) 306. [6] R Ciszewski and K Blinowski. Phys. Letters 29A (1969) 513; 30A (1969) 68.