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Evidence for fischer's correlation functions in iron from critical neutron scattering
Volume 26A, number 9
EVIDENCE
PHYSICS LETTERS
FOR FISHER'S CORRELATION FUNCTION FROM CRITICAL NEUTRON SCATTERING
25 March 1968
IN IRON
D. BALLY, M. POPOVICI, M. TOTIA, B. GRABCEV ~and A. M. LUNGU Institute f o r Atomic Physics, Bucharest, Romania Received 13 February 1968
Temperature dependence of the critical small-angle scattering of 1.25 A neutrons in iron is found to indicate a small deviation of the asymptotic spin pair correlation function from the Ornstein-Zernike form, and to confirm Fisher's prediction concerning the character of this deviation.
The m e a s u r e m e n t s of the c r i t i c a l m a g n e t i c n e u t r o n s c a t t e r i n g in i r o n at t e m p e r a t u r e s above the C u r i e point TC [1-6] have been so far i n t e r p r e t e d m a i n l y in t e r m s of the O r n s t e i n - Z e r n i k e a s y m p t o t i c c o r r e l a t i o n function (in Van H o v e ' s notation [7]): ~(r) ~ r~ 2 exp ( - k l r ) / r .
(1)
S e v e r a l y e a r s ago, however, F i s h e r a r g u e d [8] that the p a i r c o r r e l a t i o n function at l a r g e d i s t a n c e s r in the v i c i n i t y of a second o r d e r phase t r a n s i t i o n point would r a t h e r have the a p p r o x i mate form: 7(r) ~ exp ( - k l r ) / r l + T l .
(2)
Since the expected value of 77 is l e s s than 0.1, the d i f f e r e n c e between (2) and (1) is difficult to put e x p e r i m e n t a l l y into evidence. A n a l y s i s of the c r i t i c a l n e u t r o n s c a t t e r i n g in t e r m s of the F i s h e r c o r r e l a t i o n function (2), p e r f o r m e d for i r o n [4] above the C u r i e t e m p e r a t u r e a s well as for E - b r a s s [9] n e a r the o r d e r i n g t e m p e r a t u r e showed that, b e c a u s e of the e x p e r i m e n t a l u n c e r t a i n t i e s , ~? cannot be d e t e r m i n e d f r o m the shape of the a n g u l a r d i s t r i b u t i o n s m e a s u r e d at fixed t e m p e r a t u r e s . However, f r o m the t e m p e r a t u r e dependence of the n e u t r o n s c a t t e r i n g it was p o s s i b l e to obtain i n d i c a t i o n s s a y ing that 7/is different f r o m zero in f l - b r a s s [10] and is v e r y likely to have a n o n - z e r o v a l u e in cobalt [11]. In o r d e r to s e a r c h for a p o s s i b l e n o n - z e r o v a l u e of 7/ in i r o n , the c r i t i c a l n e u t r o n s c a t t e r i n g at s m a l l e s t angles °(scattering v e c t o r s r a n g i n g f r o m 0.03 to 0.12 A- I ) was m e a s u r e d in a p r e v i o u s l y d e s c r i b e d s e t - u p [6] with an i m p r o v e d a n g u l a r r e s o l u t i o n . The t e m p e r a t u r e dependence of the i n v e r s e c o r r e l a t i o n r a n g e k 1 and of the 396
r e d u c e d s u s c e p t i b i l i t y Xo was deduced f r o m the c o r r e c t e d a n g u l a r d i s t r i b u t i o n s . The extraction of the data was p e r f o r m e d f i r s t through the u s u a l technique by u s i n g an O r n s t e i n Z e r n i k e function. The p a r a m e t e r s giving the b e s t l e a s t - s q u a r e fits of the data for 5° --< T - TC ~< 50 ° to the t h e o r e t i c a l l y expected power laws Xo ~ ( I - T C / T ) - T and k 1 = A ( T / T C - 1) v w e r e found to have the v a l u e s : ~ = 1.30±0.02; u = 0.67±0.015 a n d A = = (1.10+0.05)~, -1 ( O r n s t e i n Z e r n i k e c o r r e l a t i o n function). If the c o r r e l a t i o n function w e r e indeed of the O r n s t e i n Z e r n i k e f o r m , the equality ~ = 2u should be fulfilled. In fact, a s m a l l deviation f r o m this equality is observed. This deviation might be explained by a c o r r e l a t i o n function of the f o r m (2). Then, through the r e l a t i o n (2 - ~ ) v = ~ [8], the v a l u e ~? = 0.06 ±0.05 should be obtained. S t a r t i n g with this figure, we proceeded to the i n t e r p r e t a t i o n of the s a m e e x p e r i m e n t a l data by m e a n s of the F i s h e r and B u r f o r d f i r s t o r d e r a p p r o x i m a n t s [12]. This new a n a l y s i s of the data y i e l d the f i g u r e s : ~ = 1.344~:0.018; u = 0.697 ±0.014 and A = (1.18 ±0.05)A -1 ( F i s h e r c o r r e l a t i o n function). The v a l u e of 7/following f r o m t h e s e f i g u r e s i s ~? = 0.07±0.05. It is worth being pointed out that the n o n - z e r o v a l u e of 77 is able to explain, at l e a s t p a r t i a l l y , the e x p e r i m e n t a l l y o b s e r v e d too fast v a r i a t i o n with t e m p e r a t u r e of the i n t e r a c t i o n r a n g e r 1 in (1). More or l e s s m a r k e d deviations f r o m the t h e o r e t i c a l l y expected slow t e m p e r a t u r e dependence of r 1 (different t h e o r i e s give roughly the s a m e law r 2 ~ T - l ) , w e r e found in a l m o s t all m e a s u r e m e n t s and especially when s h o r t - w a v e length n e u t r o n s w e r e employed [1,2,5,6]. The open points in fig. 1, obtained by u s i n g an O r n s t e i n Z e r n i k e c o r r e l a t i o n function in the i n t e r -
Volume 26A, n u m b e r 9
PHYSICS
LETTERS
25 M a r c h 1968
ment with theoretical expectations, when the n o n - z e r o v a l u e of 77 i s t a k e n i n t o a c c o u n t .
References 1. H.A. Gersch, C. G. Shull and M.K. Wilkinson, Phys. Rev. 103 (1956) 525. 2. R.D. Lowde, Rev. Mod. Phys. 30 (1958) 69. 3. M. E r i c s o n and B. J a c r o t , J. Phys. Chem. Solids 13 (1960) 235. 4. L. P a s s e l , K, Blinowski, T. Brun and P. Nielsen, Phys. Rev. 139 (1965) A1866. 5. S. Spooner and B. L. Averbach, Phys. Rev. 142 (1966) 291. 6. D. Bally, B. Grabcev, A.M. Lungu, M. Popovici and M. Totia, J. Phys. Chem. Solids 28 (1967) 1947. 7. L . V a n Hove, Phys. Rev. 95 (1954) 1374. 8. M. E. F i s h e r , J. Math. P h y s . 6 (1964) 944. 9. J. A l s - N i e l s e n and O. W. Dietrich, P h y s . Rev. 153 (1967) 706. 1O. O.W. Dietrich and J. A l s - N i e l s e n , Phys. Rev. 153 (1967) 711. 11. D. Bally, M. Popovici, M. Totia, B. Grabcey and A. M. Lungu, Phys. L e t t e r s 25A (1967) 595. 12. M.E. F i s h e r and R. J. Burford, Phys. Rev. i[56 (1967) 583.
~a9 .
I
as d
1
/0
i
i
20 30 40 Temperature T-Tc in °C
I
_
so
Fig. 1. T e m p e r a t u r e variation of r~, obtained by assuming ~7 = 0 (open circles) and ~7= 0.07 (filled .~ircles~, r e s p e c t i v e l y . The solid line is the dependence r I ~ T -±. p r e t a t i o n of o u r e x p e r i m e n t a l d a t a , g i v e a p i c t u r e of t h e s e d e v i a t i o n s . T h e b l a c k p o i n t s a r e d e d u c e d f r o m t h e s a m e e x p e r i m e n t a l d a t a by a s suming ~ = 0.07. It turns out that the temperature d e p e n d e n c e of r 1 c o m e s i n t o a r a t h e r f a i r a g r e e * * * * *
R 4 INVARIANTS
FOR
DOUBLY
EXCITED
STATES
OF
HELIUM
C. E . W U L F M A N *
Center f o r Theoretical Studies, Coral Gables, Florida, USA and Mathematical Institute, Oxford, UK Received 13 F e b r u a r y 1968
The group R4 p r o v i d e s two new i n v a r i a n t s that usefully label those helium m e t a s t a b l e s t a t e s in which, to f i r s t approximation, both e l e c t r o n s a r e in the second quantum level.
If A i and L i a r e t h e R u n g e - L e n t z a n d a n g u l a r mom_~,ntum v e c t o r s of e l e c t r o n i i n a n a t o m a n d A = Z-~Ai, L = ~ L i , t h e n t h e C a r t e s i a n c o m p o n e n t s of A a n d L o b e y t h e c o m m u t a t i o n r e l a t i o n s of t h e g e n e r a t o r s of R 4 [1]. O n e , t w o . . . m a n y electron functions can therefore be labeled by t h e e i g e n v a l u e s N 2 - 1 = 2 [ J a ( J a + 1) + J b ( J b + 1)] of t h e R 4 C a s i m i r o p e r a t o r A 2 + L 2, a n d b y t h e e i g e n v a l u e s [ J a ( J + 1) - "b(Jb + 1)] 2 of t h e R
q
rtic invari
nt
[2]. If we denote (ga -
irreducible
representations
of R 4 w i t h N = n,
Q : o [3]. B i e d e n h a r n ' s s t u d y of t h e r e d u c t i o n of r e p r e s e n t a t i o n s of R 4 × R 4 i n t o i r r e d u c i b l e r e p r e s e n t a t i o n s of R 4 s h o w s t h a t w e c a n w r i t e t w o - p a r t i c l e eigenstatesofA ~+L 2,A'L, L 2, L z a s [2]
LM
:
IVlP'OpO;POLM)
x ~ n , l , m , ( 1 ) ~Onlm(2 ) ;
×
( p = n - 1)
b y Q, t h e n h y d r o g e n a t o m s t a t e s ~ n l m b e l o n g t o * NSF Science Faculty Fellow f r o m U n i v e r s i t y of the Pacific, Stockton, California.
S t a t e s w i t h Q * 0 o c c u r i n p a i r s w i t h Q = ~:1QI. T h e s e m a y b e c o m b i n e d t o f o r m e i g e n s t a t e s of ( A . L) 2 w h i c h h a v e d e f i n i t e p a r i t y ~. W h e n n = n ' ,
397
EVIDENCE
PHYSICS LETTERS
FOR FISHER'S CORRELATION FUNCTION FROM CRITICAL NEUTRON SCATTERING
25 March 1968
IN IRON
D. BALLY, M. POPOVICI, M. TOTIA, B. GRABCEV ~and A. M. LUNGU Institute f o r Atomic Physics, Bucharest, Romania Received 13 February 1968
Temperature dependence of the critical small-angle scattering of 1.25 A neutrons in iron is found to indicate a small deviation of the asymptotic spin pair correlation function from the Ornstein-Zernike form, and to confirm Fisher's prediction concerning the character of this deviation.
The m e a s u r e m e n t s of the c r i t i c a l m a g n e t i c n e u t r o n s c a t t e r i n g in i r o n at t e m p e r a t u r e s above the C u r i e point TC [1-6] have been so far i n t e r p r e t e d m a i n l y in t e r m s of the O r n s t e i n - Z e r n i k e a s y m p t o t i c c o r r e l a t i o n function (in Van H o v e ' s notation [7]): ~(r) ~ r~ 2 exp ( - k l r ) / r .
(1)
S e v e r a l y e a r s ago, however, F i s h e r a r g u e d [8] that the p a i r c o r r e l a t i o n function at l a r g e d i s t a n c e s r in the v i c i n i t y of a second o r d e r phase t r a n s i t i o n point would r a t h e r have the a p p r o x i mate form: 7(r) ~ exp ( - k l r ) / r l + T l .
(2)
Since the expected value of 77 is l e s s than 0.1, the d i f f e r e n c e between (2) and (1) is difficult to put e x p e r i m e n t a l l y into evidence. A n a l y s i s of the c r i t i c a l n e u t r o n s c a t t e r i n g in t e r m s of the F i s h e r c o r r e l a t i o n function (2), p e r f o r m e d for i r o n [4] above the C u r i e t e m p e r a t u r e a s well as for E - b r a s s [9] n e a r the o r d e r i n g t e m p e r a t u r e showed that, b e c a u s e of the e x p e r i m e n t a l u n c e r t a i n t i e s , ~? cannot be d e t e r m i n e d f r o m the shape of the a n g u l a r d i s t r i b u t i o n s m e a s u r e d at fixed t e m p e r a t u r e s . However, f r o m the t e m p e r a t u r e dependence of the n e u t r o n s c a t t e r i n g it was p o s s i b l e to obtain i n d i c a t i o n s s a y ing that 7/is different f r o m zero in f l - b r a s s [10] and is v e r y likely to have a n o n - z e r o v a l u e in cobalt [11]. In o r d e r to s e a r c h for a p o s s i b l e n o n - z e r o v a l u e of 7/ in i r o n , the c r i t i c a l n e u t r o n s c a t t e r i n g at s m a l l e s t angles °(scattering v e c t o r s r a n g i n g f r o m 0.03 to 0.12 A- I ) was m e a s u r e d in a p r e v i o u s l y d e s c r i b e d s e t - u p [6] with an i m p r o v e d a n g u l a r r e s o l u t i o n . The t e m p e r a t u r e dependence of the i n v e r s e c o r r e l a t i o n r a n g e k 1 and of the 396
r e d u c e d s u s c e p t i b i l i t y Xo was deduced f r o m the c o r r e c t e d a n g u l a r d i s t r i b u t i o n s . The extraction of the data was p e r f o r m e d f i r s t through the u s u a l technique by u s i n g an O r n s t e i n Z e r n i k e function. The p a r a m e t e r s giving the b e s t l e a s t - s q u a r e fits of the data for 5° --< T - TC ~< 50 ° to the t h e o r e t i c a l l y expected power laws Xo ~ ( I - T C / T ) - T and k 1 = A ( T / T C - 1) v w e r e found to have the v a l u e s : ~ = 1.30±0.02; u = 0.67±0.015 a n d A = = (1.10+0.05)~, -1 ( O r n s t e i n Z e r n i k e c o r r e l a t i o n function). If the c o r r e l a t i o n function w e r e indeed of the O r n s t e i n Z e r n i k e f o r m , the equality ~ = 2u should be fulfilled. In fact, a s m a l l deviation f r o m this equality is observed. This deviation might be explained by a c o r r e l a t i o n function of the f o r m (2). Then, through the r e l a t i o n (2 - ~ ) v = ~ [8], the v a l u e ~? = 0.06 ±0.05 should be obtained. S t a r t i n g with this figure, we proceeded to the i n t e r p r e t a t i o n of the s a m e e x p e r i m e n t a l data by m e a n s of the F i s h e r and B u r f o r d f i r s t o r d e r a p p r o x i m a n t s [12]. This new a n a l y s i s of the data y i e l d the f i g u r e s : ~ = 1.344~:0.018; u = 0.697 ±0.014 and A = (1.18 ±0.05)A -1 ( F i s h e r c o r r e l a t i o n function). The v a l u e of 7/following f r o m t h e s e f i g u r e s i s ~? = 0.07±0.05. It is worth being pointed out that the n o n - z e r o v a l u e of 77 is able to explain, at l e a s t p a r t i a l l y , the e x p e r i m e n t a l l y o b s e r v e d too fast v a r i a t i o n with t e m p e r a t u r e of the i n t e r a c t i o n r a n g e r 1 in (1). More or l e s s m a r k e d deviations f r o m the t h e o r e t i c a l l y expected slow t e m p e r a t u r e dependence of r 1 (different t h e o r i e s give roughly the s a m e law r 2 ~ T - l ) , w e r e found in a l m o s t all m e a s u r e m e n t s and especially when s h o r t - w a v e length n e u t r o n s w e r e employed [1,2,5,6]. The open points in fig. 1, obtained by u s i n g an O r n s t e i n Z e r n i k e c o r r e l a t i o n function in the i n t e r -
Volume 26A, n u m b e r 9
PHYSICS
LETTERS
25 M a r c h 1968
ment with theoretical expectations, when the n o n - z e r o v a l u e of 77 i s t a k e n i n t o a c c o u n t .
References 1. H.A. Gersch, C. G. Shull and M.K. Wilkinson, Phys. Rev. 103 (1956) 525. 2. R.D. Lowde, Rev. Mod. Phys. 30 (1958) 69. 3. M. E r i c s o n and B. J a c r o t , J. Phys. Chem. Solids 13 (1960) 235. 4. L. P a s s e l , K, Blinowski, T. Brun and P. Nielsen, Phys. Rev. 139 (1965) A1866. 5. S. Spooner and B. L. Averbach, Phys. Rev. 142 (1966) 291. 6. D. Bally, B. Grabcev, A.M. Lungu, M. Popovici and M. Totia, J. Phys. Chem. Solids 28 (1967) 1947. 7. L . V a n Hove, Phys. Rev. 95 (1954) 1374. 8. M. E. F i s h e r , J. Math. P h y s . 6 (1964) 944. 9. J. A l s - N i e l s e n and O. W. Dietrich, P h y s . Rev. 153 (1967) 706. 1O. O.W. Dietrich and J. A l s - N i e l s e n , Phys. Rev. 153 (1967) 711. 11. D. Bally, M. Popovici, M. Totia, B. Grabcey and A. M. Lungu, Phys. L e t t e r s 25A (1967) 595. 12. M.E. F i s h e r and R. J. Burford, Phys. Rev. i[56 (1967) 583.
~a9 .
I
as d
1
/0
i
i
20 30 40 Temperature T-Tc in °C
I
_
so
Fig. 1. T e m p e r a t u r e variation of r~, obtained by assuming ~7 = 0 (open circles) and ~7= 0.07 (filled .~ircles~, r e s p e c t i v e l y . The solid line is the dependence r I ~ T -±. p r e t a t i o n of o u r e x p e r i m e n t a l d a t a , g i v e a p i c t u r e of t h e s e d e v i a t i o n s . T h e b l a c k p o i n t s a r e d e d u c e d f r o m t h e s a m e e x p e r i m e n t a l d a t a by a s suming ~ = 0.07. It turns out that the temperature d e p e n d e n c e of r 1 c o m e s i n t o a r a t h e r f a i r a g r e e * * * * *
R 4 INVARIANTS
FOR
DOUBLY
EXCITED
STATES
OF
HELIUM
C. E . W U L F M A N *
Center f o r Theoretical Studies, Coral Gables, Florida, USA and Mathematical Institute, Oxford, UK Received 13 F e b r u a r y 1968
The group R4 p r o v i d e s two new i n v a r i a n t s that usefully label those helium m e t a s t a b l e s t a t e s in which, to f i r s t approximation, both e l e c t r o n s a r e in the second quantum level.
If A i and L i a r e t h e R u n g e - L e n t z a n d a n g u l a r mom_~,ntum v e c t o r s of e l e c t r o n i i n a n a t o m a n d A = Z-~Ai, L = ~ L i , t h e n t h e C a r t e s i a n c o m p o n e n t s of A a n d L o b e y t h e c o m m u t a t i o n r e l a t i o n s of t h e g e n e r a t o r s of R 4 [1]. O n e , t w o . . . m a n y electron functions can therefore be labeled by t h e e i g e n v a l u e s N 2 - 1 = 2 [ J a ( J a + 1) + J b ( J b + 1)] of t h e R 4 C a s i m i r o p e r a t o r A 2 + L 2, a n d b y t h e e i g e n v a l u e s [ J a ( J + 1) - "b(Jb + 1)] 2 of t h e R
q
rtic invari
nt
[2]. If we denote (ga -
irreducible
representations
of R 4 w i t h N = n,
Q : o [3]. B i e d e n h a r n ' s s t u d y of t h e r e d u c t i o n of r e p r e s e n t a t i o n s of R 4 × R 4 i n t o i r r e d u c i b l e r e p r e s e n t a t i o n s of R 4 s h o w s t h a t w e c a n w r i t e t w o - p a r t i c l e eigenstatesofA ~+L 2,A'L, L 2, L z a s [2]
LM
:
IVlP'OpO;POLM)
x ~ n , l , m , ( 1 ) ~Onlm(2 ) ;
×
( p = n - 1)
b y Q, t h e n h y d r o g e n a t o m s t a t e s ~ n l m b e l o n g t o * NSF Science Faculty Fellow f r o m U n i v e r s i t y of the Pacific, Stockton, California.
S t a t e s w i t h Q * 0 o c c u r i n p a i r s w i t h Q = ~:1QI. T h e s e m a y b e c o m b i n e d t o f o r m e i g e n s t a t e s of ( A . L) 2 w h i c h h a v e d e f i n i t e p a r i t y ~. W h e n n = n ' ,
397