Temperature dependent stoner enhancement for nickel above the curie temperature

Volume 24A, n u m b e r 24

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ANGLE BETWEEN SOUNDAXIS [IOO] AND MAGNETICFIELD,DEGREES

Fig. 2. Absorption d e r i v a t i v e peak to peak amplitude for n u c l e a r magnetic acoustic r e s o n a n c e of 181Ta, A m= = ± 2 t r a n s i t i o n s , in the KTaO3 c r y s t a l as a function of angle 0 between magnetic field and sound axis [ 100]. of n o r m a l N M R on t h e s a m e c r y s t a l in i t s d e p e n d e n c e on a n g l e b e t w e e n t h e [100] a x i s a n d t h e m a g n e t i c f i e l d [2]; t h e s t r o n g c e n t r a l c o m p o n e n t w h i c h i s o b s e r v e d o n l y in N M R i s i n d e p e n d e n t of rotation angle. Thus, NMAR becomes potentially a n i m p o r t a n t t o o l in c r y s t a l c h a r a c t e r i z a t i o n . N M R t e s t s on t h e K T a O 3 s a m p l e (not s h o w n ) i n d i c a t e d l e s s t h a n 10 p e r c e n t s i g n a l a m p l i t u d e v a r i a t i o n w i t h r e s p e c t to r o t a t i o n a n g l e .

TEMPERATURE NICKEL

DEPENDENT ABOVE THE

LETTERS

5 June 1967

The deviation from sin40 behaviour cannot be a t t r i b u t e d p r i m a r i l y to a c r y s t a l d e f e c t s t r u c t u r e . G r e g o r y a n d B 0 m m e l [3] f o u n d s u c h d e v i a t i o n s in t h e i r r o t a t i o n a l p a t t e r n s on T a to b e u n a f f e c t e d by annealing or uniaxial straining. Moreover, where they employed solely longitudinal or shear mode transducers; they found evidence for both m o d e s in e a c h of t h e i r r o t a t i o n a l p a t t e r n s . T h i s m o d e m i x i n g m a y b e a t t r i b u t e d in p a r t [4] to t h e s m a l l r a t i o of t r a n s d u c e r a n d s a m p l e d i a m e t e r D * to s o u n d w a v e l e n g t h ~. D/~ w a s a b o u t 14 for their Ta samples and about 7 for the present t e s t s . In B o l e f a n d M e n e s t e s t s on Kr~[2] w h e r e D/~ w a s a b o u t 34, m o d e m i x i n g w a s m i n i m i z e d a n d c o n f o r m a n c e to t h e o r y w a s e v i d e n t . A l t h o u g h a m u c h h i g h e r D/X r a t i o i s n o t p o s s i b l e w i t h t h e present small size crystal, further tests should be directed toward a sufficiently high ratio and t o w a r d s t h e s t u d y of t h e r o t a t i o n p a t t e r n f o r t h e Am = ± 1 t r a n s i t i o n , w h i c h i s m o r e s u b j e c t to quadrupole broadening.

References 1. w. G. P r o c t o r , private communication. 2. D . I . B o l e f a n d M. Menes, Phys. Rev. 114(1959) 1441; M. Menes and D.I. Bolef, Phys. Rev. 109 {1958) 218. 3. E.It. Gregory and If. E. Bbmmel , Phys. Rev. Lett e r s 15 {1965) 404; E.H. Gregory, Ph. D. thesis, UCLA (1965) (unpublished). 4. H.J. McSkimin. J. Aeoust. Soc. Am. 28 (1956)484. * In the c a s e s cited, the t r a n s d u c e r d i a m e t e r D and that of the sample are approximately the same.

STONER ENHANCEMENT CURIE TEMPERATURE

FOR

E. P. WOHLFARTH

Departn~ent ,Of Mathematics, Imperial College, London, England Received 2 May 1967

The enhancement is estimated on the b a s i s of static susceptibility m e a s u r e m e n t s ; r e f e r e n c e is made to r e cent neutron s c a t t e r i n g m e a s u r e m e n t s which a r c also influenced by this enhaeement.

C a b l e e t al. [1] h a v e r e c e n t l y m e a s u r e d t h e n e u t r o n s c a t t e r i n g c r o s s - s e c t i o n in p a r a m a g ~ e t i c 666

nickel at temperatures the Curie temperature.

n e a r 1.6 Tc, w h e r e Tc i s This cross-section may,

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dence of ~T), si n ce it is this which h e r e d e t e r m i n e s the e n h a n c e m e n t of the c r o s s - s e c t i o n . This function m ay be d e t e r m i n e d , using (2), f r o m a knowledge of the m e a s u r e d static s u s c e p t i b i l i t y of n i c k e l above Tc. If Tc < T 4< TF, w h e r e T F is the d e g e n e r a c y t e m p e r a t u r e , i(T) has a l r e a d y been given [6]* as

I

-[(T) = 1 - a(T 2 - T2c) , 1"0

0.5 0.0

I 0.5

I 1.0

I 1-5

I 2-0

2.5

T/Tc

Fig. 1. on the b a s i s of the r a n d o m phase a p p r o x i m a t i o n , be e x p r e s s e d [2,3] in t e r m s of the i m a g i n a r y p a r t of the dynamic s u s c e p t i b i l i t y function X(q, co) in (1). T h i s function in turn is enhanced by the in= t e r a c t i o n s between the i t i n e r a n t e l e c t r o n s . If t h e s e i n t e r a c t i o n s a r e d e s c r i b e d by the w e l l known s h o r t - r a n g e m o d e l [4] and a single non=deg e n e r a t e band of e l e c t r o n s is a s s u m e d , then [3]

x(q,w) = x°(q,w)/[l-(I/2p2)x°(q,

w)],

(i)

w h e r e X° is the unenhanced s u s c e p t i b i l i t y and I the s h o r t - r a n g e i n t e r a c t i o n e n e r g y . The neutron c r o s s = s e c t i o n is thus a l s o enhanced by the i n t e r = a c t i o n s and this e n h a n c e m e n t , which depends on the t e m p e r a t u r e , i s t h e r e f o r e m e a s u r a b l e by t h e s e dynamic m e a n s . In the s t a t i c l i m i t , q=O, w= O, (1} r e d u c e s to the f o r m f i r s t given by Stoner [5] and which m a y be written, with X = X(0,0), X = X ° / [ 1 - I-(T)],

where

(2) i(T) = I f N(E)

5 June 1967

de.

H e r e N(E) is the density of s t a t e s and f = f(T) the F e r m i function. Doniach [3] has obtained the v a r i a t i o n of the i m a g i n a r y p a r t of X(q, co) for the e f f e c t i v e m a s s a p p r o x i m a t i o n , using I(0) as a p a r a m e t e r ; i n c r e a s i n g I-towards 1 r e d u c e s the f r e q u e n c y at which the c r o s s - s e c t i o n r e a c h e s a m a x i m u m and thus e n h a n c e s the s c a t t e r i n g at low f r e q u e n c i e s . F o r n i c k e l above the C u r i e point it is n e c e s s a r y to d i s c u s s the t e m p e r a t u r e d e p e n -

i .e., ] ( T c) = 1,

(3)

w h er e a - " ~ T F. T h i s r e l a t i o n is not, h o w e v e r , s t r i c t l y v al i d f o r n i c k e l and I(T) m u s t t h e r e f o r e be c a l c u l a t e d f r o m an a s s u m e d or c a l c u l a t e d N(E) function using that value of I which g i v e s the best a g r e e m e n t with the m e a s u r e d s u s c e p t i b i l i t y and at l e a s t s a t i s f i e s the condition [5] /(T c) = 1. Fig. 1 g i v e s s e v e r a l c u r v e s of / - c a l c u l a t e d in this way. C u r v e 1 c o r r e s p o n d s to a p a r a b o l i c band [7], 2 to an a s s u m e d r e c t a n g u l a r band [8], and 3 to an N(E) function obtained f r o m m e a s u r e d s p e c i f i c heats of nickel a l l o y s at low t e m p e r a t u r e s [9]. F o r p u r p o s e s of o r i e n t a t i o n the a p p r o p r i a t e c o r r e s p o n d i n g v a l u e s of A E , the exchange s p l i t ting at 0°K, p r o p o r t i o n a l to I, a r e also shown. The c u r v e s 1 and 2 a r e in c l o s e a g r e e m e n t with each other leading, for e x a m p l e at T = 1.6 Tc, to a value of T ~ 0.83. C u r v e 3 l i e s above the o t h e r s but h e r e the static s u s c e p t i b i l i t y does not, in fact, a g r e e with e x p e r i m e n t anything like a s well as f o r c u r v e s 1 and 2 and only the condition i ( T c ) = 1 is s a t i s f i e d in the l a s t c a s e . It would be of i n t e r e s t to c a l c u l a t e ~T) f o r m o r e r e a l i s t i c bands such as a r e now being obtained, to c o m p a r e t h ese r e s u l t s with those obtained f r o m neutron s c a t t e r i n g e x p e r i m e n t s and to i n v e s t i g a t e the influence on the c u r v e s of the ef f ect s of c o r r e l a t i o n c o n s i d e r e d by E d w a r d s [10]. Thanks a r e due to S. Doniach for helpful d i s c u s s i o n s and to R. D. Lowde and C. G. W i n d so r for allowing m e to s e e t h e i r data p r i o r to publication. 1. J.W. Cable, R.D. Lowde, C.G.Windsor andA.D.B. Woods, Paper M2, Washington Conference on Magnetism, 1966; J. Appl. Phys. 38 (1967), to be published. 2. T.Izuyama, D.J.Kim and R.Kubo, J. Phys. Soc. Japan 18 (1963) 1025. 3. S. Doniach. Proc. Phys. Soc., to be published. 4. C.Herring, Magnetism (Academic Press). Vol. 4 (1966). 5. E.C. Stoner, Proc. Roy. Soc. 154 (1936) 656. 6. E.P.Wohlfarth, Phys. Letters 20 (1966) 253. 7. E.P.Wohlfarth, Proc. Roy. Soc. 195 (1949) 434. 8. E.P.Wohlfarth, Phil. Mag. 42 (1951} 374. 9. M.Shimizu, T.Takahasi and A.Katsuki. J. Phys. Soc. Japan 18 (1963) 801. 10. D.M.Edwards, Phys. Letters 20 {1966} 362. * This formula was also given in ref.3 for the effective mass approximation. 667