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Critical exponent values at λ-transitions
Volume 26A, number 12
PHYSICS LETTERS
m a t e d f r o m an effective point charge model c a l culation [2]. Since p r e s s u r e dependent data for A a r e not a v a i l a b l e , the t h e r m a l expansion c o n t r i b u is unknown. A s s u m i n g this effect is s m a l l c o m p a r e d to the explicit effect, the value for C in eq. (1) is found to be (1.58 ± 0.05) × 10-12(OK) -4 with = 508°K [8], and -A(T=O) = (95.3 ± 0.2) × 10 -4 c m - 1 . The solid line in fig. 1 r e p r e s e n t s the t h e o r e t i c a l c u r v e u s i n g t h e s e values of C and 0. A n e a r e s t n e i g h b o r point c h a r g e e s t i m a t e of C can be obtained f r o m a modified f o r m of eq. (14) in ref. [2] m a k i n g u s e of known v a l u e s of the lattice p a r a m e t e r s , c r y s t a l d e n s i t y , v e l o c i t i e s of sound [8] and static h y p e r f i n e field [9]. The r e s u l t s gives Cth = 0.13 × 10-12(OK)-4, which is an o r d e r of magnitude s m a l l e r than Cexp. Covalency c o n t r i b u t i o n s of the I s , 2s, 3s and 4s s h e l l s of Mn 2+ with the f l u o r i n e 2p o r b i t a l s may play s o m e r o l e in e n h a n c i n g the hyperfine coupling t e m p e r a t u r e dependence as was shown to be the c a s e for Mn 2+ in MgO [3]. However, s i n c e the f l u o r i d e s a r e m o r e ionic than the oxides, one would expect c l o s e r a g r e e m e n t between Cth and Cexp for CaF2 c o m p a r e d to MgO, c o n t r a r y to the r e s u l t s . T h i s s u g g e s t s that p e r h a p s l a r g e a m p l i tude local mode v i b r a t i o n s which a r i s e due to the loose coupling of the Mn2+ ion at the l a r g e r Ca 2+ site m u s t be accounted for [10]. Such a c a l c u l a -
6 May 1968
tion would r e q u i r e a knowledge of the v a r i o u s coupling c o n s t a n t s for longitudinal and t r a n s v e r s e phonon v i b r a t i o n s for both the optical and a c o u s t i cal b r a n c h e s [5]. The author would like to thank P r o f e s s o r D. Kaup and P r o f e s s o r E. Simfinek for t h e i r helpful c o r r e s p o n d e n c e , to R. Zogby for a s s i s t i n g in the e x p e r i m e n t and to A. R o s s for c o n t r u c t i n g the v a r i a b l e t e m p e r a t u r e cavity.
References 1. w.M. Walsh J r . , J. Jeener and N. Bloembergen. Phys. Rev. 139 (1965) A1338. 2. E. ~im~[nek and R. Orbaeh, Phys. Rev. 145 (1966)
191.
3. E. Sim~tnek and Nai Li Huang, Phys. Rev. Letters 17 (19d6) 699. 4. R. Orbach and E. Sim~tnek, Phys. Rev. 158 (1967) 310. 5. R. Calvo and R. Orbach, Phys. Rev. 164 (1967) 284. 6. W. Low, Phys. Rev. 105 (1957) 793. 7. J.M. Baker. B. Bleaney and W. Hayes, Proc. Roy. Soc. (London) A247 (1958) 141. 8. D.R. Huffman and M. H. Norwoord. Phys. Rev. 117 (1960) 709. 9. J.S. Van Wierengen, Discussions Faraday Soc. 19 (1955) 118. 10. E.Sim~nek, private communication.
* * * * *
CRITICAL
EXPONENT
VALUES
AT k-TRANSITIONS
B. J. LIPA and M. J. BUCKINGHAM
Department of Physics, University of WesternAustralia, Nedlands, W.A. Received 11 April 1968 The values of the critical exponents of a cooperative transition approached along different thermodynamic paths are in general different. The relations between these values are discussed and illustrated by application to decorated Ising lattices.
One s i g n i f i c a n t a s p e c t of the p r o b l e m of coope r a t i v e t r a n s i t i o n s is the identification of those p r o p e r t i e s of a s y s t e m on which the a s y m p t o t i c f o r m of the c r i t i c a l s i n g u l a r i t y depends. It has b e c o m e c u s t o m a r y to c h a r a c t e r i z e d the n a t u r e of the s i n g u l a r i t y by the v a l u e s of the c r i t i c a l exponents, a, fl, r, 5, for example, d e s c r i b i n g r e s p e c t i v e l y the a s y m p t o t i c f o r m of the specific heat, c o e x i s t e n c e c u r v e , g e n e r a l i z e d s u s c e p t i b i l i t y and the c r i t i c a l i s o t h e r m . In addition to the o r d e r
p a r a m e t e r and its conjugate i n t e n s i v e v a r i a b l e involved in the cooperative t r a n s i t i o n , a s y s t e m in g e n e r a l p o s s e s s e s other d y n a m i c a l v a r i a b l e s on which the t h e r m o d y n a m i c behaviour depends. The t r a n s i t i o n t e m p e r a t u r e Tc for e x a m p l e depends on t h e s e other v a r i a b l e s , the l o c u s of Tc often being r e f e r r e d to a s the k - l i n e , when one other v a r i able only is c o n s i d e r e d . It is i m p o r t a n t to note that a s y s t e m whose p a r t i t i o n function p o s s e s s e s a p a r t i c u l a r c r i t i c a l s i n g u l a r i t y may display dif643
Volume 26A. number 12
PHYSICS LETTERS
f e r e n t c r i t i c a l exponents depending on the t h e r m o d y n a m i c path by which the k - l i n e i s approached. It is our p u r p o s e h e r e to set down the r e l a t i o n s between the v a l u e s of the c r i t i c a l exponents in the p a r t i c u l a r c a s e s when the path is specified by the constancy of an i n t e n s i v e or e x t e n s i v e v a r i a b l e r e s p e c t i v e l y , t h e s e b e i n g the paths of g r e a t e s t practical importance. As an example we have the r e s u l t that if, as n u m e r i c a l e s t i m a t e s [1] of the incompressible 1 c a s e suggest, ~ = ~ and a = i for a t h r e e d i m e n sional c o m p r e s s i b l e I s i n g model at c o n s t a n t p r e s s u r e , for the s a m e s y s t e m at c o n s t a n t d e n s i t y the v a l u e of ¥ would be ~ and the specific heat would approach a finite m a x i m u m at a cusp, with a = - ~ . We c o n s i d e r a s y s t e m p o s s e s s i n g an e x t e n s i v e t h e r m o d y n a m i c v a r i a b l e , A, and conjugate i n t e n sive v a r i a b l e a = ( a F / a A ) T , where the f r e e e n e r g y F = F ( A , T ) has a A-line of s i n g u l a r i t i e s at t e m p e r a t u r e TA(A). Except in s p e c i a l c a s e s ( a A / a T ) k = = AA' and (aa/aT)A = aA' a r e finite and non z e r o . To d e r i v e the n e c e s s a r y t h e r m o d y n a m i c r e l a t i o n s we employ a method u s e d by Buckingham and F a i r b a n k [2] for a n a l y s i n g the k - t r a n s i t i o n of l i quid helium. We introduce two functions of state t = l ( T , a ) = T - TA(a) and 0 = O(T,A) = T - TA(A), both of which v a n i s h on the A-line. It is easy to show that ( a A / a T ) a d i v e r g e s at the A l i n e in the s a m e way a s Ca, t h e specific heat at c o n s t a n t a, (Ca ~ t - a ) . T h i s can be u s e d to d e t e r m i n e the a s y m p t o t i c r e l a t i o n * between 0 and t: O~t l-a, a>0
.
(1)
It is not h a r d to p r o v e the following identity for CA, the specific heat at c o n s t a n t A : CA - Ct = C f { 1 + C f / ( C a - C t ) } -1
(2)
where C t = T(OS/aT) t and C f = T a A ' ( ~ A / O T ) t a r e both finite. Thus if Ca d i v e r g e s , (2) shows that CA a p p r o a c h e s a finite m a x i m u m , C m a x = = TA(aS/OT)A + TAaA'A A at a cusp n e a r which C m ax
- CA ~ t a ~ O a / 1 - A
(3)
T h i s r e s u l t was d e r i v e d in ref. 2 and has b e e n d i s c u s s e d r e c e n t l y by Griffiths [3 I. The " s p o n t a n e o u s o r d e r p a r a m e t e r " is the s a m e quantity r e g a r d l e s s of the t h e r m o d y n a m i c path so that eq. (1) gives i m m e d i a t e l y flA = = / 3 a l ( l - a').
Using the fact that, in language appropriate to * If Ca ~ In tt I- 1 the corresponding transformation will be 19 ~ t l n l t l - 1 , which must be used to derive the thermodynamic functions at constant A. For example the nsuontaneous order parameter" is given by
(lellnlel-1)~.
644
6 May 1968
the f e r r o m a g n e t i c s y s t e m , (aM/~H)T, A c a n be shown to b e c o m e equal to ( a M / ~ H ) s , a on the k line, e n a b l e s one to obtain the r e s u l t s : 5 T,A = 5S,a;
rT, A = ~ S , a / ( 1 - or)
(4)
T h e s e adiabatic exponents a r e l e s s than or equal to the c o r r e s p o n d i n g i s o t h e r m a l ones, the equality p r e v a i l i n g if, for example, the homogeneity conditions [4] a r e s a t i s f i e d , and in any case TS,a = ~ T , a for T > T k. (On the low t e m p e r a t u r e side a and y m u s t , in a u s u a l notation, be r e placed by a' and ~' in all r e l a t i o n s above). As an e x a m p l e of the g e n e r a l c a s e d i s c u s s e d c o n s i d e r a d e c o r a t e d Ising l a t t i c e a s d e s c r i b e d by F i s h e r [5]. His a n a y l i s can e a s i l y be u s e d to show that such a lattice, with d e c o r a t i n g spin v a l u e s a given by a p r o b a b i l i t y d i s t r i b u t i o n function W(a, (r) (say exp af(~2)), s y m m e t r i c a l about a = 0, has a c r i t i c a l s i n g u l a r i t y with the s a m e a s y m p t o t i c p r o p e r t i e s as the u n d e c o r a t e d I s i n g l a t t i c e ; the c r i t i c a l t e m p e r a t u r e Tc(a) defines a A-line if a is r e g a r e d as an i n t e n s i v e t h e r m o d y n a m i c p a r a m e t e r . The a s y m p t o t i c p r o p e r t i e s for the s y s t e m in which the e x t e n s i v e v a r i a b l e conjugate to a (in the example, A = ~ i f ( ~ i 2 ) ) , is c o n s t r a i n e d to be constant then follow f r o m the r e s u l t s above. Other a u t h o r s [6] have obtained the s a m e r e s u l t s in v a r i o u s s p e c i a l c a s e s by exp l i c i t c a l c u l a t i o n s . We now see that the s a m e p r o p e r t i e s will r e s u l t for any extensive constraint. One of us (B. J. L.) wishes to thank m e Corn monwealth Scientific and I n d u s t r i a l R e s e a r c h O r g a n i s a t i o n for the a w a r d of a Senior P o s t g r a d u a t e Studentship.
Refe~'ences 1. See for example D. S. Gaunt, Proc. Phys. Soc. 92 (1967) 151. 2. M.J. Buckingham and W. M. Fairbank, Progress in low temperature physics Vol.HI ed. C. J. Gorter, (North Holland, Amsterdam, 1961} p.80. 3, R. B. Griffiths (preprint). 4 . B. Widom, J. Chem. Phys,43 (1965) 3898; L.P.Kadanoff, Physics 2 (1966) 263. 5. M.E. Fisher, Phys. Rev. 113 (1959) 969. 6. I. Syozi, Progr. Theoret. Phys. (Kyoto) 34 (1965) 189; C. J. Thomson, J. Math. Phys. 7 (1966) 531; J. W. Essam and H. Garelick, Proc. Phys. Soc. 92 (1967) 136.
PHYSICS LETTERS
m a t e d f r o m an effective point charge model c a l culation [2]. Since p r e s s u r e dependent data for A a r e not a v a i l a b l e , the t h e r m a l expansion c o n t r i b u is unknown. A s s u m i n g this effect is s m a l l c o m p a r e d to the explicit effect, the value for C in eq. (1) is found to be (1.58 ± 0.05) × 10-12(OK) -4 with = 508°K [8], and -A(T=O) = (95.3 ± 0.2) × 10 -4 c m - 1 . The solid line in fig. 1 r e p r e s e n t s the t h e o r e t i c a l c u r v e u s i n g t h e s e values of C and 0. A n e a r e s t n e i g h b o r point c h a r g e e s t i m a t e of C can be obtained f r o m a modified f o r m of eq. (14) in ref. [2] m a k i n g u s e of known v a l u e s of the lattice p a r a m e t e r s , c r y s t a l d e n s i t y , v e l o c i t i e s of sound [8] and static h y p e r f i n e field [9]. The r e s u l t s gives Cth = 0.13 × 10-12(OK)-4, which is an o r d e r of magnitude s m a l l e r than Cexp. Covalency c o n t r i b u t i o n s of the I s , 2s, 3s and 4s s h e l l s of Mn 2+ with the f l u o r i n e 2p o r b i t a l s may play s o m e r o l e in e n h a n c i n g the hyperfine coupling t e m p e r a t u r e dependence as was shown to be the c a s e for Mn 2+ in MgO [3]. However, s i n c e the f l u o r i d e s a r e m o r e ionic than the oxides, one would expect c l o s e r a g r e e m e n t between Cth and Cexp for CaF2 c o m p a r e d to MgO, c o n t r a r y to the r e s u l t s . T h i s s u g g e s t s that p e r h a p s l a r g e a m p l i tude local mode v i b r a t i o n s which a r i s e due to the loose coupling of the Mn2+ ion at the l a r g e r Ca 2+ site m u s t be accounted for [10]. Such a c a l c u l a -
6 May 1968
tion would r e q u i r e a knowledge of the v a r i o u s coupling c o n s t a n t s for longitudinal and t r a n s v e r s e phonon v i b r a t i o n s for both the optical and a c o u s t i cal b r a n c h e s [5]. The author would like to thank P r o f e s s o r D. Kaup and P r o f e s s o r E. Simfinek for t h e i r helpful c o r r e s p o n d e n c e , to R. Zogby for a s s i s t i n g in the e x p e r i m e n t and to A. R o s s for c o n t r u c t i n g the v a r i a b l e t e m p e r a t u r e cavity.
References 1. w.M. Walsh J r . , J. Jeener and N. Bloembergen. Phys. Rev. 139 (1965) A1338. 2. E. ~im~[nek and R. Orbaeh, Phys. Rev. 145 (1966)
191.
3. E. Sim~tnek and Nai Li Huang, Phys. Rev. Letters 17 (19d6) 699. 4. R. Orbach and E. Sim~tnek, Phys. Rev. 158 (1967) 310. 5. R. Calvo and R. Orbach, Phys. Rev. 164 (1967) 284. 6. W. Low, Phys. Rev. 105 (1957) 793. 7. J.M. Baker. B. Bleaney and W. Hayes, Proc. Roy. Soc. (London) A247 (1958) 141. 8. D.R. Huffman and M. H. Norwoord. Phys. Rev. 117 (1960) 709. 9. J.S. Van Wierengen, Discussions Faraday Soc. 19 (1955) 118. 10. E.Sim~nek, private communication.
* * * * *
CRITICAL
EXPONENT
VALUES
AT k-TRANSITIONS
B. J. LIPA and M. J. BUCKINGHAM
Department of Physics, University of WesternAustralia, Nedlands, W.A. Received 11 April 1968 The values of the critical exponents of a cooperative transition approached along different thermodynamic paths are in general different. The relations between these values are discussed and illustrated by application to decorated Ising lattices.
One s i g n i f i c a n t a s p e c t of the p r o b l e m of coope r a t i v e t r a n s i t i o n s is the identification of those p r o p e r t i e s of a s y s t e m on which the a s y m p t o t i c f o r m of the c r i t i c a l s i n g u l a r i t y depends. It has b e c o m e c u s t o m a r y to c h a r a c t e r i z e d the n a t u r e of the s i n g u l a r i t y by the v a l u e s of the c r i t i c a l exponents, a, fl, r, 5, for example, d e s c r i b i n g r e s p e c t i v e l y the a s y m p t o t i c f o r m of the specific heat, c o e x i s t e n c e c u r v e , g e n e r a l i z e d s u s c e p t i b i l i t y and the c r i t i c a l i s o t h e r m . In addition to the o r d e r
p a r a m e t e r and its conjugate i n t e n s i v e v a r i a b l e involved in the cooperative t r a n s i t i o n , a s y s t e m in g e n e r a l p o s s e s s e s other d y n a m i c a l v a r i a b l e s on which the t h e r m o d y n a m i c behaviour depends. The t r a n s i t i o n t e m p e r a t u r e Tc for e x a m p l e depends on t h e s e other v a r i a b l e s , the l o c u s of Tc often being r e f e r r e d to a s the k - l i n e , when one other v a r i able only is c o n s i d e r e d . It is i m p o r t a n t to note that a s y s t e m whose p a r t i t i o n function p o s s e s s e s a p a r t i c u l a r c r i t i c a l s i n g u l a r i t y may display dif643
Volume 26A. number 12
PHYSICS LETTERS
f e r e n t c r i t i c a l exponents depending on the t h e r m o d y n a m i c path by which the k - l i n e i s approached. It is our p u r p o s e h e r e to set down the r e l a t i o n s between the v a l u e s of the c r i t i c a l exponents in the p a r t i c u l a r c a s e s when the path is specified by the constancy of an i n t e n s i v e or e x t e n s i v e v a r i a b l e r e s p e c t i v e l y , t h e s e b e i n g the paths of g r e a t e s t practical importance. As an example we have the r e s u l t that if, as n u m e r i c a l e s t i m a t e s [1] of the incompressible 1 c a s e suggest, ~ = ~ and a = i for a t h r e e d i m e n sional c o m p r e s s i b l e I s i n g model at c o n s t a n t p r e s s u r e , for the s a m e s y s t e m at c o n s t a n t d e n s i t y the v a l u e of ¥ would be ~ and the specific heat would approach a finite m a x i m u m at a cusp, with a = - ~ . We c o n s i d e r a s y s t e m p o s s e s s i n g an e x t e n s i v e t h e r m o d y n a m i c v a r i a b l e , A, and conjugate i n t e n sive v a r i a b l e a = ( a F / a A ) T , where the f r e e e n e r g y F = F ( A , T ) has a A-line of s i n g u l a r i t i e s at t e m p e r a t u r e TA(A). Except in s p e c i a l c a s e s ( a A / a T ) k = = AA' and (aa/aT)A = aA' a r e finite and non z e r o . To d e r i v e the n e c e s s a r y t h e r m o d y n a m i c r e l a t i o n s we employ a method u s e d by Buckingham and F a i r b a n k [2] for a n a l y s i n g the k - t r a n s i t i o n of l i quid helium. We introduce two functions of state t = l ( T , a ) = T - TA(a) and 0 = O(T,A) = T - TA(A), both of which v a n i s h on the A-line. It is easy to show that ( a A / a T ) a d i v e r g e s at the A l i n e in the s a m e way a s Ca, t h e specific heat at c o n s t a n t a, (Ca ~ t - a ) . T h i s can be u s e d to d e t e r m i n e the a s y m p t o t i c r e l a t i o n * between 0 and t: O~t l-a, a>0
.
(1)
It is not h a r d to p r o v e the following identity for CA, the specific heat at c o n s t a n t A : CA - Ct = C f { 1 + C f / ( C a - C t ) } -1
(2)
where C t = T(OS/aT) t and C f = T a A ' ( ~ A / O T ) t a r e both finite. Thus if Ca d i v e r g e s , (2) shows that CA a p p r o a c h e s a finite m a x i m u m , C m a x = = TA(aS/OT)A + TAaA'A A at a cusp n e a r which C m ax
- CA ~ t a ~ O a / 1 - A
(3)
T h i s r e s u l t was d e r i v e d in ref. 2 and has b e e n d i s c u s s e d r e c e n t l y by Griffiths [3 I. The " s p o n t a n e o u s o r d e r p a r a m e t e r " is the s a m e quantity r e g a r d l e s s of the t h e r m o d y n a m i c path so that eq. (1) gives i m m e d i a t e l y flA = = / 3 a l ( l - a').
Using the fact that, in language appropriate to * If Ca ~ In tt I- 1 the corresponding transformation will be 19 ~ t l n l t l - 1 , which must be used to derive the thermodynamic functions at constant A. For example the nsuontaneous order parameter" is given by
(lellnlel-1)~.
644
6 May 1968
the f e r r o m a g n e t i c s y s t e m , (aM/~H)T, A c a n be shown to b e c o m e equal to ( a M / ~ H ) s , a on the k line, e n a b l e s one to obtain the r e s u l t s : 5 T,A = 5S,a;
rT, A = ~ S , a / ( 1 - or)
(4)
T h e s e adiabatic exponents a r e l e s s than or equal to the c o r r e s p o n d i n g i s o t h e r m a l ones, the equality p r e v a i l i n g if, for example, the homogeneity conditions [4] a r e s a t i s f i e d , and in any case TS,a = ~ T , a for T > T k. (On the low t e m p e r a t u r e side a and y m u s t , in a u s u a l notation, be r e placed by a' and ~' in all r e l a t i o n s above). As an e x a m p l e of the g e n e r a l c a s e d i s c u s s e d c o n s i d e r a d e c o r a t e d Ising l a t t i c e a s d e s c r i b e d by F i s h e r [5]. His a n a y l i s can e a s i l y be u s e d to show that such a lattice, with d e c o r a t i n g spin v a l u e s a given by a p r o b a b i l i t y d i s t r i b u t i o n function W(a, (r) (say exp af(~2)), s y m m e t r i c a l about a = 0, has a c r i t i c a l s i n g u l a r i t y with the s a m e a s y m p t o t i c p r o p e r t i e s as the u n d e c o r a t e d I s i n g l a t t i c e ; the c r i t i c a l t e m p e r a t u r e Tc(a) defines a A-line if a is r e g a r e d as an i n t e n s i v e t h e r m o d y n a m i c p a r a m e t e r . The a s y m p t o t i c p r o p e r t i e s for the s y s t e m in which the e x t e n s i v e v a r i a b l e conjugate to a (in the example, A = ~ i f ( ~ i 2 ) ) , is c o n s t r a i n e d to be constant then follow f r o m the r e s u l t s above. Other a u t h o r s [6] have obtained the s a m e r e s u l t s in v a r i o u s s p e c i a l c a s e s by exp l i c i t c a l c u l a t i o n s . We now see that the s a m e p r o p e r t i e s will r e s u l t for any extensive constraint. One of us (B. J. L.) wishes to thank m e Corn monwealth Scientific and I n d u s t r i a l R e s e a r c h O r g a n i s a t i o n for the a w a r d of a Senior P o s t g r a d u a t e Studentship.
Refe~'ences 1. See for example D. S. Gaunt, Proc. Phys. Soc. 92 (1967) 151. 2. M.J. Buckingham and W. M. Fairbank, Progress in low temperature physics Vol.HI ed. C. J. Gorter, (North Holland, Amsterdam, 1961} p.80. 3, R. B. Griffiths (preprint). 4 . B. Widom, J. Chem. Phys,43 (1965) 3898; L.P.Kadanoff, Physics 2 (1966) 263. 5. M.E. Fisher, Phys. Rev. 113 (1959) 969. 6. I. Syozi, Progr. Theoret. Phys. (Kyoto) 34 (1965) 189; C. J. Thomson, J. Math. Phys. 7 (1966) 531; J. W. Essam and H. Garelick, Proc. Phys. Soc. 92 (1967) 136.