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Theory of dilute antiferromagnetic alloys above the Néel temperature
PHYSICS LETTERS
Volume 31A, number 7 F o r / 9 >> ~, we have
ds2 =dT 2-dr 2-d 2-dZ 2+O(~/P). The m e t r i c is q u a s i - M i n k o w s k i a n , and its d e v i a t i o n f r o m f l a t n e s s is again a function of Z - u T , i.e. it p r o p a g a t e s with s p e e d u > 1. The e x a c t m e t r i c , h o w e v e r , i s s i n g u l a r on the s u r f a c e P = ½~. This is a h y p e r b o l o i d of two s h e e t s , which again can be i n t e p r e t e d as g r a v i t a t i o n a l ~ e r e n k o v radiation. A c o o r d i n a t e t r a n s f o r m a t i o n ( Z - u T ) / 4 u ~ = ~, ( T - u Z ) / u4u~-I = P sinh ~, x+iy = P c o s h ~ e x p i f l gives ds 2
2~t =-(l~)d~
2
d 2 -~+p.
2
2 . 2 (d~ - s l n h ~ d ~ 2 ) ,
It should be noted that the s i n g u l a r i t i e s at R = ~xm or /9 = ~g z a r e z e r o s of a s p a t i a l c o m p o n e n t of the m e t r i c . T h e i r m e a n i n g i s that s p a c e is c l o s e d (while the m e a n i n g of the o r i g i n a l S c h w a r z s c h i l d s i n g u l a r i t y was an infinite slowing down of a l l p r o c e s s e s ) . If follows that, in g e n e r a l r e l a t i v i t y
THEORY
OF
DILUTE
6 April 1970
t h eo r y , a tachyon p e r v a d e s throughout al l s p a c e , and cannot be l o c a l i z e d in the v i n i c i t y of a w o r l d line. I enjoyed many s t i m u l a t i n g d i s c u s s i o n s with c o l l e a g u e s at Haifa, London and C a m b r i d g e . I a m e s p e c i a l l y g r a t e f u l to P r o f . F . A . E . P i r a n i and to Dr. D.W. S c i a m a f o r t h e i r kind hospitality.
References 1. M. Sachs, Phys. Today, Dec. 1969, p.47; O M. Bilaniuk and E. C.G Sudarshan, Phys. Today, Dec. 1969, p. 50. 2. R. C. Tolman, Relativity thermodynamics and cosmology {Oxford University Press, 1934} p. 205. 3. N Rosen, Bull. Research Council Israel 3 (1954) 328. 4. J. Ehters and W. Kundt in Gravitation: an Introduction to Current Research, ed. L. Witten (Wiley, 1962). 5. I. Robinson and A. Trautman, Proc. Roy. Soc. A265 (1962) 463. I am indebted to Prof. I. Robinson for suggesting to compare this metric to his DS-spaee.
ANTIFERROMAGNETIC TEMPERATURE
ALLOYS
ABOVE
THE
NF, E L
M. J. ZUCKERMANN Eaton Electronics Research Laboratory, McGill University, Montreal, Canada Received 24 February 1970
The Ndel :cemerature and the criteria for the existence of localised magnetic states in alloys of type CrMe; C__rr-Feare calculated using a localised exchange model in the dilute limit, based on the Fedders and Martin model for pure Cr.
T h i s l e t t e r p r e s e n t s the a p p l i c a t i o n of the t h e o r y of Doniach et al. [1] f o r s t r o n g l y enhanc e d p a r a m a g n e t i c a l l o y s to dilute a n t i f e r r o m a g ne t i c a l l o y s above the N~el t e m p e r a t u r e T N. The i m p u r i t i e s a r e taken to be t r a n s i t i o n m e t a l a t o m s i s o e l e c t r o n i c with the host a t o m s . The a n t i f e r r o m a g n e t i c s t a t e of the p u r e m e t a l i s d e s c r i b e d by the two band m o d e l of F e d d e r s and M a r t i n [2] f o r Cr . The H a m i l t o n i a n f o r the a l l o y s m a y be d e s c r i b e d as follows : (i) a n o n - i n t e r a c t i n g H a m i l tonian d e s c r i b i n g e l e c t r o n s in the two bands. T h e s e bands c o r r e s p o n d to i d e n t i c a l e l e c t r o n and hole s h e e t s of the F e r m i s u r f a c e of p u r e C r s e 362
p a r a t e d by a wave v e c t o r O in r e c i p r o c a l space. (if) an i n t e r b a n d exchange i n t e r a c t i o n which binds e l e c t r o n s in one band (band a) to h o l es (with the s a m e spin a s the e l e c t r o n s ) in the s e c o n d band (band b). F e d d e r s and M a r t i n have shown that the f o r m a t i o n of t r i p l e t bound e l e c t r o n - h o l e p a i r s l e a d s to the f o r m a t i o n of a t r a n s v e r s e l i n e a r spin density w a v e ground st at e which d e s c r i b e s the a n t i f e r r o m a g n e t i s m of the p u r e m e t a l below the N~el t e m p e r a t u r e TN. The i n t e r b a n d i n t e r a c t i o n h as coupling c o n s t a n t I at e a c h host l a t t i c e site a n d / ' = / + A / a t e a c h i m p u r i t y si t e. A / m a y be p o s i t i v e o r n e g a t i v e depending on the type of i m -
Volume 31A, number 7
PHYSICS LETTERS
purity. The t h e o r y of Doniach et al. [1] is now a p p l i e d d i r e c t l y to this m o d e l in the enhanced p a r a m a g netic r e g i m e f o r t e m p e r a t u r e s T > T N. The d y n a m i c s u s c e p t i b i l i t y ×alloy (q, oJ) of the a l l o y f o r m o m e n t u m q and f r e q u e n c y o~ i s : × a l l o y ( q , co)
×cr(q,
=
¢o)/[1-ni~Ieff(T , W)Xcr(q , ¢o)] (1)
Then ×Cr(q, ~o) i s the d y n a m i c s u s c e p t i b i l i t y of p u r e C r and is given by : XCr = xQr(q, w ) / [ 1 - I x Q r ( q , 60)]
(2)
where
1D xQr(q , w)
=
-
i(E~+Q)-i(c b) a
b
(3)
V k W-Ek+Q+E k
n I i s the i m p u r i t y c o n c e n t r a t i o n ; f(e) is the F e r m i function and e~ and e b a r e the k i n e t i c e n e r g i e s of e l e c t r o n s f r o m band a, and h o l e s f r o m band b, r e s p e c t i v e l y , and V i s the v o l u m e of the s y s t e m . Aleff(T, oJ) iS the effective l o c a l exchange c o r r e c t i o n and is given by :
A/eff(T , w) = AI[1 - A I X c r ( 0 , w)] -1
(4)
w h e r e X c r ( r , ~) i s the d y n a m i c s u s c e p t i b i l i t y of p u r e C r in r e a l s p a c e a s a function of f r e q u e n c y , i.e. the F o u r i e r t r a n s f o r m a t i o n of × c r ( q , °J)×Cr i s a function of t e m p e r a t u r e T . Equation (4) i m m e d i a t e l y g i v e s the c r i t e r i o n f o r l o c a l i s e d m o m e n t f o r m a t i o n , i.e. when 1 = A/×Cr(0 , 0), a l o c a l i s e d m a g n e t i c s t a t e is f o r m e d . The p r o p e r t i e s of the l o c a l i s e d m a g n e t i c s t a t e a r e dependent anly on t h e e x c e s s l o c a l i n t e r band exchange A / a t the i m p u r i t y s i t e , and not i n t r a - b a n d i n t e r a c t i o n s a s p r o p o s e d by B e h e r a and V i s v a n a t h a n [3]. H o w e v e r , the p r e s e n t t h e o r y cannot account for the d e t a i l e d p r o p e r t i e s of l o c a l i s e d m a g n e t i c m o m e n t s in dilute C r - F e , C r - N i , C r - C o a l l o y s ( s i n c e the i m p u r i t i e s a r e not i s o e l e c t r o n i c with C r a t o m s ) thoi~gh it a c counts f o r t h e i r o r i g i n . A d e t a i l e d a n a l y s i s i n v o l v e s the p r e s e n c e of v i r t u a l bound s t a t e s at the i m p u r i t y s i t e and will b e p r e s e n t e d in a n o t h e r publication.
6 April 1970
The N~el t e m p e r a t u r e T N of the a l l o y i s o b t a i n e d f r o m eq. (1) by equating the d e n o m i n a t o r on the r i g h t hand side of eq. (1) to z e r o f o r q = w = 0, i.e. 1 = nlAIeff( T, 0)×Cr(0 , 0). F r o m eqs. (2), (3), (4) and the e x p r e s s i o n f o r TN0 of F e d d e r s and M a r t i n [2], the equation f o r T N i s given by : TNO In TN _
_
i
ni[/~leff (TN, 0)/I]
N(O)I[l+nl,xleff(TN, 0)/I]
(5)
It is difficult to apply eq. (5) d i r e c t l y to e x p e r i m e n t b e c a u s e of the a p p r o x i m a t e n a t u r e of the F e d d e r s M a r t i n model. However, a rough e s t i m a t e of AI~ff can be m a d e f o r -C- r - M o • The s u g g e s t e d v a u e s f o r I and N(0) in C r in r e f e r e n c e [3] a r e : I = 5eV, N(0) = 0.025eV -1 and TN0 = 311OK. F o r C_xr-Mo, AT N = TI~ - TN0 = - 14oK at 1 at.% Mo [4]. The s u b s t i t u t i o n of t h e s e v a l u e s into (5) g i v e s ,xleff(0) ~ - 2.9eV f o r Mo in Cir. F o r C_xr-W, ,XTN = - 2 9 ° K a t 1 a t . % W . Both Mo and W a r e i s o e l e c t r o n i c with C r and t h e r e f o r e it i s e x p e c t e d that ,x TN would be n e a r l y the s a m e in both C r - M o and C__rr-W. A r a j s [4] p o i n t s out that, a s the d - e l e c t r o n s wave functions f o r W a r e much l e s s l o c a l i s e d than for m o l y b d e n u m , the d i s c r e p a n c y can be e x p l a i n e d d i r e c t l y on the b a s i s of the F e d d e r s and M a r t i n m o d e l [2]. In t h i s l e t t e r we have p o i n t e d out that the e x i s t e n c e of l o c a l i z e d m o m e n t s and the change in N~el t e m p e r a t u r e with i m p u r i t y c o n c e n t r a t i o n can be q u a l i t a t i v e l y e x p l a i n e d by c h a n g e s in i n t e r band r a t h e r than i n t r a b a n d i n t e r a c t i o n s f o r c e r tain a l l o y s f o r C r with t r a n s i t i o n m e t a l s . No useful quantitative p r e d i c t i o n s can be m a d e s i n c e a c c u r a t e v a l u e s f o r I, XCr(0 , 0) and A I a r e not a s y e t a v a i l a b l e . F u r t h e r work i s in p r o g r e s s to i n v e s t i g a t e the c o l l e c t i v e spin m o d e s of such a l l o y s both above and below T N.
References 1. s. Engelsberg, W . F . Brinkman and S. Doniach, Phys. Rev. Letters 20 (1968) 1040. 2. P . A . F e d d e r s and P.C.Martin, Phys. Rev. 143 (1966) 245. 3. S.N.Behera and K.S.Visvanathan, Can. J. Phys. 47 (1969) 477. 4. S.Arajs, J. Appl. Phys. 39 (1968) 673.
* * * * *
363
Volume 31A, number 7 F o r / 9 >> ~, we have
ds2 =dT 2-dr 2-d 2-dZ 2+O(~/P). The m e t r i c is q u a s i - M i n k o w s k i a n , and its d e v i a t i o n f r o m f l a t n e s s is again a function of Z - u T , i.e. it p r o p a g a t e s with s p e e d u > 1. The e x a c t m e t r i c , h o w e v e r , i s s i n g u l a r on the s u r f a c e P = ½~. This is a h y p e r b o l o i d of two s h e e t s , which again can be i n t e p r e t e d as g r a v i t a t i o n a l ~ e r e n k o v radiation. A c o o r d i n a t e t r a n s f o r m a t i o n ( Z - u T ) / 4 u ~ = ~, ( T - u Z ) / u4u~-I = P sinh ~, x+iy = P c o s h ~ e x p i f l gives ds 2
2~t =-(l~)d~
2
d 2 -~+p.
2
2 . 2 (d~ - s l n h ~ d ~ 2 ) ,
It should be noted that the s i n g u l a r i t i e s at R = ~xm or /9 = ~g z a r e z e r o s of a s p a t i a l c o m p o n e n t of the m e t r i c . T h e i r m e a n i n g i s that s p a c e is c l o s e d (while the m e a n i n g of the o r i g i n a l S c h w a r z s c h i l d s i n g u l a r i t y was an infinite slowing down of a l l p r o c e s s e s ) . If follows that, in g e n e r a l r e l a t i v i t y
THEORY
OF
DILUTE
6 April 1970
t h eo r y , a tachyon p e r v a d e s throughout al l s p a c e , and cannot be l o c a l i z e d in the v i n i c i t y of a w o r l d line. I enjoyed many s t i m u l a t i n g d i s c u s s i o n s with c o l l e a g u e s at Haifa, London and C a m b r i d g e . I a m e s p e c i a l l y g r a t e f u l to P r o f . F . A . E . P i r a n i and to Dr. D.W. S c i a m a f o r t h e i r kind hospitality.
References 1. M. Sachs, Phys. Today, Dec. 1969, p.47; O M. Bilaniuk and E. C.G Sudarshan, Phys. Today, Dec. 1969, p. 50. 2. R. C. Tolman, Relativity thermodynamics and cosmology {Oxford University Press, 1934} p. 205. 3. N Rosen, Bull. Research Council Israel 3 (1954) 328. 4. J. Ehters and W. Kundt in Gravitation: an Introduction to Current Research, ed. L. Witten (Wiley, 1962). 5. I. Robinson and A. Trautman, Proc. Roy. Soc. A265 (1962) 463. I am indebted to Prof. I. Robinson for suggesting to compare this metric to his DS-spaee.
ANTIFERROMAGNETIC TEMPERATURE
ALLOYS
ABOVE
THE
NF, E L
M. J. ZUCKERMANN Eaton Electronics Research Laboratory, McGill University, Montreal, Canada Received 24 February 1970
The Ndel :cemerature and the criteria for the existence of localised magnetic states in alloys of type CrMe; C__rr-Feare calculated using a localised exchange model in the dilute limit, based on the Fedders and Martin model for pure Cr.
T h i s l e t t e r p r e s e n t s the a p p l i c a t i o n of the t h e o r y of Doniach et al. [1] f o r s t r o n g l y enhanc e d p a r a m a g n e t i c a l l o y s to dilute a n t i f e r r o m a g ne t i c a l l o y s above the N~el t e m p e r a t u r e T N. The i m p u r i t i e s a r e taken to be t r a n s i t i o n m e t a l a t o m s i s o e l e c t r o n i c with the host a t o m s . The a n t i f e r r o m a g n e t i c s t a t e of the p u r e m e t a l i s d e s c r i b e d by the two band m o d e l of F e d d e r s and M a r t i n [2] f o r Cr . The H a m i l t o n i a n f o r the a l l o y s m a y be d e s c r i b e d as follows : (i) a n o n - i n t e r a c t i n g H a m i l tonian d e s c r i b i n g e l e c t r o n s in the two bands. T h e s e bands c o r r e s p o n d to i d e n t i c a l e l e c t r o n and hole s h e e t s of the F e r m i s u r f a c e of p u r e C r s e 362
p a r a t e d by a wave v e c t o r O in r e c i p r o c a l space. (if) an i n t e r b a n d exchange i n t e r a c t i o n which binds e l e c t r o n s in one band (band a) to h o l es (with the s a m e spin a s the e l e c t r o n s ) in the s e c o n d band (band b). F e d d e r s and M a r t i n have shown that the f o r m a t i o n of t r i p l e t bound e l e c t r o n - h o l e p a i r s l e a d s to the f o r m a t i o n of a t r a n s v e r s e l i n e a r spin density w a v e ground st at e which d e s c r i b e s the a n t i f e r r o m a g n e t i s m of the p u r e m e t a l below the N~el t e m p e r a t u r e TN. The i n t e r b a n d i n t e r a c t i o n h as coupling c o n s t a n t I at e a c h host l a t t i c e site a n d / ' = / + A / a t e a c h i m p u r i t y si t e. A / m a y be p o s i t i v e o r n e g a t i v e depending on the type of i m -
Volume 31A, number 7
PHYSICS LETTERS
purity. The t h e o r y of Doniach et al. [1] is now a p p l i e d d i r e c t l y to this m o d e l in the enhanced p a r a m a g netic r e g i m e f o r t e m p e r a t u r e s T > T N. The d y n a m i c s u s c e p t i b i l i t y ×alloy (q, oJ) of the a l l o y f o r m o m e n t u m q and f r e q u e n c y o~ i s : × a l l o y ( q , co)
×cr(q,
=
¢o)/[1-ni~Ieff(T , W)Xcr(q , ¢o)] (1)
Then ×Cr(q, ~o) i s the d y n a m i c s u s c e p t i b i l i t y of p u r e C r and is given by : XCr = xQr(q, w ) / [ 1 - I x Q r ( q , 60)]
(2)
where
1D xQr(q , w)
=
-
i(E~+Q)-i(c b) a
b
(3)
V k W-Ek+Q+E k
n I i s the i m p u r i t y c o n c e n t r a t i o n ; f(e) is the F e r m i function and e~ and e b a r e the k i n e t i c e n e r g i e s of e l e c t r o n s f r o m band a, and h o l e s f r o m band b, r e s p e c t i v e l y , and V i s the v o l u m e of the s y s t e m . Aleff(T, oJ) iS the effective l o c a l exchange c o r r e c t i o n and is given by :
A/eff(T , w) = AI[1 - A I X c r ( 0 , w)] -1
(4)
w h e r e X c r ( r , ~) i s the d y n a m i c s u s c e p t i b i l i t y of p u r e C r in r e a l s p a c e a s a function of f r e q u e n c y , i.e. the F o u r i e r t r a n s f o r m a t i o n of × c r ( q , °J)×Cr i s a function of t e m p e r a t u r e T . Equation (4) i m m e d i a t e l y g i v e s the c r i t e r i o n f o r l o c a l i s e d m o m e n t f o r m a t i o n , i.e. when 1 = A/×Cr(0 , 0), a l o c a l i s e d m a g n e t i c s t a t e is f o r m e d . The p r o p e r t i e s of the l o c a l i s e d m a g n e t i c s t a t e a r e dependent anly on t h e e x c e s s l o c a l i n t e r band exchange A / a t the i m p u r i t y s i t e , and not i n t r a - b a n d i n t e r a c t i o n s a s p r o p o s e d by B e h e r a and V i s v a n a t h a n [3]. H o w e v e r , the p r e s e n t t h e o r y cannot account for the d e t a i l e d p r o p e r t i e s of l o c a l i s e d m a g n e t i c m o m e n t s in dilute C r - F e , C r - N i , C r - C o a l l o y s ( s i n c e the i m p u r i t i e s a r e not i s o e l e c t r o n i c with C r a t o m s ) thoi~gh it a c counts f o r t h e i r o r i g i n . A d e t a i l e d a n a l y s i s i n v o l v e s the p r e s e n c e of v i r t u a l bound s t a t e s at the i m p u r i t y s i t e and will b e p r e s e n t e d in a n o t h e r publication.
6 April 1970
The N~el t e m p e r a t u r e T N of the a l l o y i s o b t a i n e d f r o m eq. (1) by equating the d e n o m i n a t o r on the r i g h t hand side of eq. (1) to z e r o f o r q = w = 0, i.e. 1 = nlAIeff( T, 0)×Cr(0 , 0). F r o m eqs. (2), (3), (4) and the e x p r e s s i o n f o r TN0 of F e d d e r s and M a r t i n [2], the equation f o r T N i s given by : TNO In TN _
_
i
ni[/~leff (TN, 0)/I]
N(O)I[l+nl,xleff(TN, 0)/I]
(5)
It is difficult to apply eq. (5) d i r e c t l y to e x p e r i m e n t b e c a u s e of the a p p r o x i m a t e n a t u r e of the F e d d e r s M a r t i n model. However, a rough e s t i m a t e of AI~ff can be m a d e f o r -C- r - M o • The s u g g e s t e d v a u e s f o r I and N(0) in C r in r e f e r e n c e [3] a r e : I = 5eV, N(0) = 0.025eV -1 and TN0 = 311OK. F o r C_xr-Mo, AT N = TI~ - TN0 = - 14oK at 1 at.% Mo [4]. The s u b s t i t u t i o n of t h e s e v a l u e s into (5) g i v e s ,xleff(0) ~ - 2.9eV f o r Mo in Cir. F o r C_xr-W, ,XTN = - 2 9 ° K a t 1 a t . % W . Both Mo and W a r e i s o e l e c t r o n i c with C r and t h e r e f o r e it i s e x p e c t e d that ,x TN would be n e a r l y the s a m e in both C r - M o and C__rr-W. A r a j s [4] p o i n t s out that, a s the d - e l e c t r o n s wave functions f o r W a r e much l e s s l o c a l i s e d than for m o l y b d e n u m , the d i s c r e p a n c y can be e x p l a i n e d d i r e c t l y on the b a s i s of the F e d d e r s and M a r t i n m o d e l [2]. In t h i s l e t t e r we have p o i n t e d out that the e x i s t e n c e of l o c a l i z e d m o m e n t s and the change in N~el t e m p e r a t u r e with i m p u r i t y c o n c e n t r a t i o n can be q u a l i t a t i v e l y e x p l a i n e d by c h a n g e s in i n t e r band r a t h e r than i n t r a b a n d i n t e r a c t i o n s f o r c e r tain a l l o y s f o r C r with t r a n s i t i o n m e t a l s . No useful quantitative p r e d i c t i o n s can be m a d e s i n c e a c c u r a t e v a l u e s f o r I, XCr(0 , 0) and A I a r e not a s y e t a v a i l a b l e . F u r t h e r work i s in p r o g r e s s to i n v e s t i g a t e the c o l l e c t i v e spin m o d e s of such a l l o y s both above and below T N.
References 1. s. Engelsberg, W . F . Brinkman and S. Doniach, Phys. Rev. Letters 20 (1968) 1040. 2. P . A . F e d d e r s and P.C.Martin, Phys. Rev. 143 (1966) 245. 3. S.N.Behera and K.S.Visvanathan, Can. J. Phys. 47 (1969) 477. 4. S.Arajs, J. Appl. Phys. 39 (1968) 673.
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