- Email: [email protected]
The effect of lattice vibration to the phase transition of the Ising spin systems
Volume26A. number 12
THE
PHYSICS LETTERS
EFFECT OF TRANSITION
LATTICE OF THE
6 May 1968
VIBRATION ISING SPIN
TO THE SYSTEMS
PHASE
N. MATSUDAIRA The Physical Science Laboratories, Nihon University at Narashino , Narashino , Chiba, Japan Received 4 April 1968 The effect of lattice vibration to the phase transition of the Ising models is calculated in the Weiss approximation.
It i s e x p e c t e d that the s e c o n d o r d e r p h a s e t r a n s i t i o n of spin s y s t e m s and the l a t t i c e v i b r a tion of t k e s e s y s t e m s affect each o t h e r . Many a u t h o r s [1] i n v e s t i g a t e d the a n o m a l o u s sound a t t e n u a t i o n and the h e a t c o n d u c t i v i t y n e a r the c r i t i c a l point. In t h i s note we w i s h to c a l c u l a t e d how the l a t t i c e v i b r a t i o n a f f e c t s the n a t u r e of the p h a s e t r a n sition. R o s s and t e r H a a r [2] s t u d i e d t h i s effect b y the E i n s t e i n model. We s h a l l adopt the Ising s p i n s with the n e a r e s t n e i g h b o r i n t e r a c t i o n s , i n t e r a c t i n g with D e b y e phonons. The effect of the v o l u m e change of the s y s t e m [2,3] w i l l be i n v e s t i g a t e d in f u t u r e , and it i s not c o n s i d e r e d h e r e . The t o t a l H a m i l t o n i a n [1] H = H~honon + H i s i n g + Hspin_phonon (1) can be t r a n s f o r m e d into the f o r m in which the phonon p a r t and t h e spin p a r t a r e c o m p l e t e l y separated, H = Hphonon + H i s i n g + H ' ,
(2)
H'---~ (J'/qS) 2 ~ q 2MNwW ~ (ran) (rs) ( q" R rnn)(q " R rs) × (3) × e x p { i q . ( R m - R r ) } {1 - e x p ( - i q . R m n ) } x
We s h a l l c a l c u l a t e eq. (4) by the W e i s s a p p r o x imation. T h e a v e r a g e of the p r o d u c t of spin o p e r a t o r s i s r e p l a c e d by the p r o d u c t of the a v e r a g e s . Since o u r H a m i l t o n i a n , eq. (2), i s t r a n s l a t i o n a l i n v a r i a n t , the a v e r a g e of a s p i n o p e r a t o r i s i n dependent of i t s s i t e and e q u a l s to the long r a n g e o r d e r of the s y s t e m , (Sm)o = L . It should be noted, h o w e v e r , that if the b o n d s (ran) and (rs) a r e connected, that i s , f o r e x a m p l e n = r, then SmSnSrS s = sins s. We thus obtain w h e r e A , B and C a r e p o s i t i v e c o n s t a n t s p r o p o r t i onal to the coupling (j,)2. We note that in o u r a p p r o x i m a t i o n the d i f f e r e n c e of the m o d e l s ( s q u a r e net, cubic l a t t i c e s ) a p p e a r s only in the e x p l i c i t e x p r e s s i o n of t h e s e c o n s t a n t s . T h e long r a n g e o r d e r L at t e m p e r a t u r e T = = 1/h~ i s d e t e r m i n e d b y m i n i m i z i n g the f r e e e n e r gy, eq. (4), with the r e s u l t L = tanh [ ~ L ( z J + 2 B - 4 C L 2 ) ] .
w h e r e J ' i s the d e r i v a t i v e of the exchange i n t e r a c t i o n with r e s p e c t to the a t o m i c d i s t a n c e , l~m n = = /~m - / ~ n , and 5 i s t h e l a t t i c e constant. To the l o w e s t o r d e r in the i n t e r a c t i o n the f r e e e n e r g y of o u r spin s y s t e m i s F = Fo + ( H ' ) o . (4) 610
(e)
F r o m eq. (6) we obtain v a r i o u s p r o p e r t i e s of the s y s t e m in the u s u a l m a n n e r . At the c r i t i c a l point ( L = 0) the exchange i n t e g r a l J i s r e p l a c e d b y the effective v a l u e J + 2 B / z . Thus the t r a n s i t i o n t e m p e r a t u r e i n c r e a s e s by the effect of the l a t t i c e vibration, Tc = T cO(1 + a )
× {1 - e x p ( i q . / ~ r s ) } SmSnSrSs,
(5)
(H') o = - N ( A + B L 2 - C L 4) ,
, a =2B/zJ.
(7)
It can be shown that (i) the e n t r o p y of o u r spin s y s t e m i s continuous at the c r i t i c a l point: the t r a n s i t i o n r e m a i n s to b e of s e c o n d o r d e r (ii) T h e h e a t c a p a c i t y j u s t below the c r i t i c a l point i s r e duced, C/N=3k/2(l+a')
, a' =4C/zJ
.
(iii) T h e e x p r e s s i o n f o r the spin s u s c e p t i b i l i t y
(8)
Volume 26A. number 12
P H Y SI C S L E T T E R S
above and just below the c r i t i c a l point i s of the u s u a l f o r m , with T ° r e p l a c e d by Tc given by eg. (7). F a r below the c r i t i c a l point the t e r m 4CLZ in eq. (6) p l a y s the role and the e x p r e s s i o n b e c o m e s different f r o m the u s u a l one. More e l a b o r a t e c a l c u l a t i o n s f o r the a v e r a g e in eq. (4) will be r e p o r t e d in n e a r future.
6 May 1968
References 1. K. Tani and H.Mori, Phys. Letters 19 (1966) 627; V. N. Kashcheev, Phys. Letters 24A (1967) 627; 25A (1967) 71; K. Tani and N. Tsuda, Phys. Letters 25A (1967) 529. 2. A.W.Ross and D.ter Haar, Physiea 25 (1959) 343. 3. C.P. Bean and D. S. Rodbell, Phys. Rev. 126 (1962) 104; R. H. Donaldson, Phys. Rev. 157 (1967) 366.
CORRECTION TO THE PAPER "ATOMIC ENVIRONMENT EFFECT ON T H E M A G N E T I C PROPERTIES O F V A u 4 ', ![ 1] H. CLAUS, A.K. SINHA and P. A. BECK
University of Illinois, Urbana, Illinois 61801, USA Received 8 April 1968
In o u r r e c e n t p a p e r [1] the p a r a m a g n e t i c m o m e n t in o r d e r e d and i n d i s o r d e r e d VAu 4 was c o m p a r e d with that in a dilute alloy of V in Au. The l a t t e r value (Peff = 0.25~B p e r V atom) was c o m puted f r o m low t e m p e r a t u r e s u s c e p t i b i l i t y data by Lutes and Schmit [2]. Since the p u b l i c a t i o n of o u r p a p e r , we b e c a m e a w a r e of the r e c e n t detailed work by Kume [3], which r e l i a b l y e s t a b l i s h e d the value of Peff 3.0~tB p e r V atom in dilute a l l o y s with Au by m e a n s of s u s c e p t i b i l i t y m e a s u r e m e n t s o v e r a wide t e m p e r a t u r e range. (Kume a l s o found c o m p e n s a t i o n b e low the Kondo t e m p e r a t u r e of 290°K. K u m e ' s susceptibility measurements are, furthermore, in r e a s o n a b l e a g r e e m e n t with the e a r l i e r r e s u l t s of Vogt and G e r s t e n b e r g [4]. i n view of the l a r g e r m o m e n t p e r V atom in dilute alloys with Au, the c o n c l u s i o n to be d r a w n f r o m c o m p a r i s o n with o u r data for VAu4 m u s t be r e v i s e d . The p r e s e n c e of V a t o m s in the s e c o n d and t h i r d - n e a r e s t shell a r o u n d a V atom in o r d e r e d VAu4 now a p p e a r s to decrease the m o m e n t
a s s o c i a t e d with this V atom, i n s t e a d of i n c r e a s ing it, as p r e v i o u s l y i n f e r r e d . The c o n c l u s i o n d r a w n in the p a p e r [1] f r o m c o m p a r i s o n of the m o m e n t in o r d e r e d VAu 4 with that in the d i s o r d e r e d alloy r e m a i n s unchanged: the m o m e n t d i s a p p e a r s on V a t o m s having one or m o r e V n e a r e s t neighbors. E r r a t u m : On p. 39, right column, line 11 f r o m the bottom, "Au + 2% V" should be c o r r e c ted to "Au + 20% V". In fig. 2 the alloy c o m p o s i tions should read: V0.sNb0.2Au 4 and V0.9Ta0.1Au 4.
References 1. H. Claus, A.K. Sinha and P.A. Beck, Phys. Letters 26A (1967) 38. 2. O.S. Lutes and J. L. Schmit, Phys. Rev. 134 (1964) A676. 3. K. Kume, J.Phys. Soc.Japan 23 (1967) 1224. 4. E. Vogt and D. Gerstenberg, Ann. Phys. 4 (1959) 145.
611
THE
PHYSICS LETTERS
EFFECT OF TRANSITION
LATTICE OF THE
6 May 1968
VIBRATION ISING SPIN
TO THE SYSTEMS
PHASE
N. MATSUDAIRA The Physical Science Laboratories, Nihon University at Narashino , Narashino , Chiba, Japan Received 4 April 1968 The effect of lattice vibration to the phase transition of the Ising models is calculated in the Weiss approximation.
It i s e x p e c t e d that the s e c o n d o r d e r p h a s e t r a n s i t i o n of spin s y s t e m s and the l a t t i c e v i b r a tion of t k e s e s y s t e m s affect each o t h e r . Many a u t h o r s [1] i n v e s t i g a t e d the a n o m a l o u s sound a t t e n u a t i o n and the h e a t c o n d u c t i v i t y n e a r the c r i t i c a l point. In t h i s note we w i s h to c a l c u l a t e d how the l a t t i c e v i b r a t i o n a f f e c t s the n a t u r e of the p h a s e t r a n sition. R o s s and t e r H a a r [2] s t u d i e d t h i s effect b y the E i n s t e i n model. We s h a l l adopt the Ising s p i n s with the n e a r e s t n e i g h b o r i n t e r a c t i o n s , i n t e r a c t i n g with D e b y e phonons. The effect of the v o l u m e change of the s y s t e m [2,3] w i l l be i n v e s t i g a t e d in f u t u r e , and it i s not c o n s i d e r e d h e r e . The t o t a l H a m i l t o n i a n [1] H = H~honon + H i s i n g + Hspin_phonon (1) can be t r a n s f o r m e d into the f o r m in which the phonon p a r t and t h e spin p a r t a r e c o m p l e t e l y separated, H = Hphonon + H i s i n g + H ' ,
(2)
H'---~ (J'/qS) 2 ~ q 2MNwW ~ (ran) (rs) ( q" R rnn)(q " R rs) × (3) × e x p { i q . ( R m - R r ) } {1 - e x p ( - i q . R m n ) } x
We s h a l l c a l c u l a t e eq. (4) by the W e i s s a p p r o x imation. T h e a v e r a g e of the p r o d u c t of spin o p e r a t o r s i s r e p l a c e d by the p r o d u c t of the a v e r a g e s . Since o u r H a m i l t o n i a n , eq. (2), i s t r a n s l a t i o n a l i n v a r i a n t , the a v e r a g e of a s p i n o p e r a t o r i s i n dependent of i t s s i t e and e q u a l s to the long r a n g e o r d e r of the s y s t e m , (Sm)o = L . It should be noted, h o w e v e r , that if the b o n d s (ran) and (rs) a r e connected, that i s , f o r e x a m p l e n = r, then SmSnSrS s = sins s. We thus obtain w h e r e A , B and C a r e p o s i t i v e c o n s t a n t s p r o p o r t i onal to the coupling (j,)2. We note that in o u r a p p r o x i m a t i o n the d i f f e r e n c e of the m o d e l s ( s q u a r e net, cubic l a t t i c e s ) a p p e a r s only in the e x p l i c i t e x p r e s s i o n of t h e s e c o n s t a n t s . T h e long r a n g e o r d e r L at t e m p e r a t u r e T = = 1/h~ i s d e t e r m i n e d b y m i n i m i z i n g the f r e e e n e r gy, eq. (4), with the r e s u l t L = tanh [ ~ L ( z J + 2 B - 4 C L 2 ) ] .
w h e r e J ' i s the d e r i v a t i v e of the exchange i n t e r a c t i o n with r e s p e c t to the a t o m i c d i s t a n c e , l~m n = = /~m - / ~ n , and 5 i s t h e l a t t i c e constant. To the l o w e s t o r d e r in the i n t e r a c t i o n the f r e e e n e r g y of o u r spin s y s t e m i s F = Fo + ( H ' ) o . (4) 610
(e)
F r o m eq. (6) we obtain v a r i o u s p r o p e r t i e s of the s y s t e m in the u s u a l m a n n e r . At the c r i t i c a l point ( L = 0) the exchange i n t e g r a l J i s r e p l a c e d b y the effective v a l u e J + 2 B / z . Thus the t r a n s i t i o n t e m p e r a t u r e i n c r e a s e s by the effect of the l a t t i c e vibration, Tc = T cO(1 + a )
× {1 - e x p ( i q . / ~ r s ) } SmSnSrSs,
(5)
(H') o = - N ( A + B L 2 - C L 4) ,
, a =2B/zJ.
(7)
It can be shown that (i) the e n t r o p y of o u r spin s y s t e m i s continuous at the c r i t i c a l point: the t r a n s i t i o n r e m a i n s to b e of s e c o n d o r d e r (ii) T h e h e a t c a p a c i t y j u s t below the c r i t i c a l point i s r e duced, C/N=3k/2(l+a')
, a' =4C/zJ
.
(iii) T h e e x p r e s s i o n f o r the spin s u s c e p t i b i l i t y
(8)
Volume 26A. number 12
P H Y SI C S L E T T E R S
above and just below the c r i t i c a l point i s of the u s u a l f o r m , with T ° r e p l a c e d by Tc given by eg. (7). F a r below the c r i t i c a l point the t e r m 4CLZ in eq. (6) p l a y s the role and the e x p r e s s i o n b e c o m e s different f r o m the u s u a l one. More e l a b o r a t e c a l c u l a t i o n s f o r the a v e r a g e in eq. (4) will be r e p o r t e d in n e a r future.
6 May 1968
References 1. K. Tani and H.Mori, Phys. Letters 19 (1966) 627; V. N. Kashcheev, Phys. Letters 24A (1967) 627; 25A (1967) 71; K. Tani and N. Tsuda, Phys. Letters 25A (1967) 529. 2. A.W.Ross and D.ter Haar, Physiea 25 (1959) 343. 3. C.P. Bean and D. S. Rodbell, Phys. Rev. 126 (1962) 104; R. H. Donaldson, Phys. Rev. 157 (1967) 366.
CORRECTION TO THE PAPER "ATOMIC ENVIRONMENT EFFECT ON T H E M A G N E T I C PROPERTIES O F V A u 4 ', ![ 1] H. CLAUS, A.K. SINHA and P. A. BECK
University of Illinois, Urbana, Illinois 61801, USA Received 8 April 1968
In o u r r e c e n t p a p e r [1] the p a r a m a g n e t i c m o m e n t in o r d e r e d and i n d i s o r d e r e d VAu 4 was c o m p a r e d with that in a dilute alloy of V in Au. The l a t t e r value (Peff = 0.25~B p e r V atom) was c o m puted f r o m low t e m p e r a t u r e s u s c e p t i b i l i t y data by Lutes and Schmit [2]. Since the p u b l i c a t i o n of o u r p a p e r , we b e c a m e a w a r e of the r e c e n t detailed work by Kume [3], which r e l i a b l y e s t a b l i s h e d the value of Peff 3.0~tB p e r V atom in dilute a l l o y s with Au by m e a n s of s u s c e p t i b i l i t y m e a s u r e m e n t s o v e r a wide t e m p e r a t u r e range. (Kume a l s o found c o m p e n s a t i o n b e low the Kondo t e m p e r a t u r e of 290°K. K u m e ' s susceptibility measurements are, furthermore, in r e a s o n a b l e a g r e e m e n t with the e a r l i e r r e s u l t s of Vogt and G e r s t e n b e r g [4]. i n view of the l a r g e r m o m e n t p e r V atom in dilute alloys with Au, the c o n c l u s i o n to be d r a w n f r o m c o m p a r i s o n with o u r data for VAu4 m u s t be r e v i s e d . The p r e s e n c e of V a t o m s in the s e c o n d and t h i r d - n e a r e s t shell a r o u n d a V atom in o r d e r e d VAu4 now a p p e a r s to decrease the m o m e n t
a s s o c i a t e d with this V atom, i n s t e a d of i n c r e a s ing it, as p r e v i o u s l y i n f e r r e d . The c o n c l u s i o n d r a w n in the p a p e r [1] f r o m c o m p a r i s o n of the m o m e n t in o r d e r e d VAu 4 with that in the d i s o r d e r e d alloy r e m a i n s unchanged: the m o m e n t d i s a p p e a r s on V a t o m s having one or m o r e V n e a r e s t neighbors. E r r a t u m : On p. 39, right column, line 11 f r o m the bottom, "Au + 2% V" should be c o r r e c ted to "Au + 20% V". In fig. 2 the alloy c o m p o s i tions should read: V0.sNb0.2Au 4 and V0.9Ta0.1Au 4.
References 1. H. Claus, A.K. Sinha and P.A. Beck, Phys. Letters 26A (1967) 38. 2. O.S. Lutes and J. L. Schmit, Phys. Rev. 134 (1964) A676. 3. K. Kume, J.Phys. Soc.Japan 23 (1967) 1224. 4. E. Vogt and D. Gerstenberg, Ann. Phys. 4 (1959) 145.
611