Spin waves, stoner modes and critical fluctuations in nickel

Solid State Communications, Vol. 6, pp. 1 8 9 - 1 9 2 , 1968.

Pergamon Press.

Printed in Great Britain

SPIN WAVES, STONER MODES AND CRITICAL FLUCTUATIONS IN NICKEL R.D. Lowde and C.G. Windsor Atomic Energy R e s e a r c h Establishment, Harwell, England (Received 20 December 1967)

Well-defined spin wave peaks are found by neutron scattering to exist in the generalized susceptibility ~((K, w) of nickel at t e m p e r a t u r e s up to and through the Curie temperature, while conversely an extended distribution having features of the h i g h - t e m p e r a t u r e limiting paramagnetic form p e r s i s t s with reduction of the t e m p e r a t u r e through T~ certainly down to r o o m temperature. Critical enhancement of the susceptibility in the region of energies above ~ ~ = 0.02 eV is not observed, the approach to criticality from either side being c h a r a c t e r i z e d instead by a pronounced anticipation of the features of the susceptibility on the other side of the transition point.

AS REPORTED in our earlier communications, ~-3 the neutron scattering from nickel at high t e m p e r a t u r e s is well explained by the elementary theory of Stoner p a r a m a g n e t i s m if an exchange enhancement calculated in random-phase approximation is introduced. In this note we wish to describe briefly the evolution of the scattering as the t e m p e r a t u r e is lowered into the f e r r o m a g netic region. The observed neutron scattering intensity is related to the K and w dependent spin c o r r e l a tion function,

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For a cubic paramagnet, or an ordered magnet without p r e f e r r e d domain directions, the differential c r o s s - s e c t i o n for unpolarized neutrons as a function of time-of-flight is (in conventional notation), ~2~ 2 ~ ~t = -3 (ro ~)~ ( d E / d t ) ( K ' / K ) I F(K_) 12 ~ ~ ( K , w ) ; o=~,Y," (2) 189

details of the spin-fluctuation spectrum ,~ and of the generalized susceptibility ~ ~ (K, ~ ) t o which it is related can be derived f r o m the m e a s u r e ments. Figure 1 presents a s e r i e s of t i m e - o f flight s p e c t r a for a single instrumental setting, showing the magnetic scattering observed at different t e m p e r a t u r e s when unpolarized incident neutrons of wavelength 3.0 ~ are deflected through 19 ° of an angle in such a way as to make the line of scattered neutron w a v e - v e c t o r s in r e c i p r o c a l space pass through the point 200. The magnetic scattering largely concentrates in the region of small K v e c t o r s close to 200 which, for this c r y s t a l orientation, r e l a t e s to energy t r a n s f e r s in the region of about 0. 05 eV. In the one particular case of T = 1.62 To, the solid line is drawn f r o m a theoretical calculation which we shall describe elsewhere; 4 it illustrates the good account that can now be given of the paramagnetic susceptibility of this metal. We have further shown in r e f e r e n c e 3 that the general features of the paramagnetic distribution for all t e m p e r a t u r e s above 1.05 To are explained by Stoner p a r a m a g n e t i s m with a constant RPA enhancement of 3 ± 1 .

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CRITICAL FLUCTUATIONS IN NICKEL

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FIG. 1 Time-of-flight neutron-intensity data (corrected for instrumental background) showing the s c a t t e r i n g of 3 . 0 A i n c i d e n t n e u t r o n s at 19 ° of a n g l e f r o m a c r y s t a l of Ni60 at v a r i o u s t e m p e r a t u r e s . The p e a k on the e x t r e m e r i g h t in a l l d i a g r a m s i s a phonon; the r e m a i n d e r of the s c a t t e r i n g i s m a g n e t i c in o r i g i n . In the one c a s e of 1.62 To the s o l i d line i n d i c a t e d shows the r e s u l t of a t h e o r e t i c a l c a l c u lation.

On s u r v e y i n g t h e c o m p l e t e s e t of c u r v e s it i s c l e a r that t h e r e i s in d e t a i l a s t e a d y and continuous e v o l u t i o n f r o m the w e l l - u n d e r s t o o d p a r a m a g n e t i c d i s t r i b u t i o n at high t e m p e r a t u r e s to the d i s t r i b u t i o n a t low t e m p e r a t u r e . T h e r e are, however, several striking features. Ignoring the phonon p e a k at the r i g h t - h a n d e x t r e m e of the d r a w n s o l i d l i n e s , the o b s e r v e d d i s t r i b u t i o n i s s e e n to b r e a k up, on p a s s i n g d o w n w a r d s t h r o u g h the C u r i e point, into a f o u r - m a x i m u m d i s t r i b u tion c e n t r e d a p p r o x i m a t e l y a r o u n d the p o s i t i o n K = 0. T h e i n t e n s i t y at the c e n t r e m a y b e r e garded as developing continuously, as a function of t e m p e r a t u r e , f r o m the p a r a m a g n e t i c S t o n e r s c a t t e r i n g ; r e m a r k a b l y , it a p p e a r s to p e r s i s t down to t h e l o w e s t t e m p e r a t u r e s , d i m i n i s h i n g in a m o u n t only s l i g h t l y o v e r a r a n g e in w h i c h the z c o m p o n e n t of s p i n r i s e s to 95% of i t s v a l u e at absolute zero. T h e two o u t e r m a x i m a , m o s t c l e a r l y s e e n

j u s t b e l o w the t r a n s i t i o n t e m p e r a t u r e , o c c u r at K ' s of the o r d e r of ~ K m a x. A m o r e c o m p l e t e investigation identifies them as spin wave peaks following a dispersion relationship indistinguisha b l e f r o m the p a r a b o l a ~w = D(T)K e m e a s u r e d o v e r the s a m e r a n g e of t e m p e r a t u r e b y the H a r w e l l B and B r o o k h a v e n 6 g r o u p s . A s a f u n c t i o n of r i s i n g t e m p e r a t u r e the s p i n w a v e m a x i m a a r e s e e n h e r e to p e r s i s t to t e m p e r a t u r e s c l o s e to T~ ; at s l i g h t l y d i f f e r e n t c r y s t a l s e t t i n g s - f o r i n s t a n c e , s a m p l i n g ~ ( K , w) at s m a l l e r % a s shown in F i g . 2 -- one can find w e l l - r e s o l v e d s p i n - w a v e - l i k e p e a k s u n a r g u a b l y p e r s i s t i n g a b o v e the C u r i e t e m p e r a t u r e . T h e s e p e a k s c l e a r l y c o r r e s p o n d to f a i r l y w e l l - d e f i n e d m o d e s of s p i n m o t i o n , and e v o l v e into the l i m i t i n g f o r m of p a r a m a g n e t i c s c a t t e r i n g only o v e r an e x t e n d e d r a n g e of t e m p e r a ture. In an e x t e n s i v e s u r v e y of the e f f e c t s at e n e r g y t r a n s f e r s g r e a t e r t h a n 0.02 eV, we find that t h e o n c e - e x p e c t e d c r i t i c a l e n l a r g e m e n t of the s c a t t e r i n g at t e m p e r a t u r e s n e a r the Cur~e point, p r e d i c t e d b y R P A , i s e s s e n t i a l l y a b s e n t . Of c o u r s e , with the t o t a l s c a t t e r i n g at any K b e h a v i n g r o u g h l y l i k e ( ~ e + K e)- 1, the i n t e n s i t y i n c r e a s e i s e x p e c t e d to be s l i g h t at l a r g e K ' s ; at s m a l l K ' s i t could p a s s u n o b s e r v e d if r e s t r i c t e d s e v e r e l y to s m a l l ~ ' s . Since, h o w e v e r , on a (nearest-neighbours-only) Heisenberg interpret a t i o n the l a r g e s t p r o b a b l e e n e r g y t r a n s f e r in n i c k e l i s a s g r e a t a s 4 Z J S ~ z3 eV, it i s r a t h e r remarkable that the critical enhancement should b e w h o l l y r e s t r i c t e d to a r a n g e of e n e r g i e s b e l o w 0.02 eV. The s t e a d i n e s s of the o b s e r v e d p a r t of

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the c o r r e l a t i o n function through the phase t r a n s i tion point, coupled with our e a r l i e r observation that the exchange enhancement a p p e a r s to be constant down certainly to 1.05 Tc, implies that the c r i t i c a l phenomena ~ in the susceptibility of nickel a r e confined v e r y s e v e r e l y to the neighbourhood of the origin in (K, ~) space.

of l o n g - r a n g e o r d e r , and develops peaks while still in the p a r a m a g n e t i c phase that a r e destined to take over the role of spin waves. Some p r e l i m i n a r y t h e o r e t i c a l considerations touching on this subject have been published by Kawasaki and Brout, 8 by Hubbard and Dalton 9 and by Resibois and de Leener. 10

An interesting point of principle is t h e decomposition of the s c a t t e r i n g into contributions f r o m p a r a l l e l (zz) and perpendicular (xx and yy) spin c o r r e l a t i o n s . T h e s e must be equal above the Curie t e m p e r a t u r e but b e c o m e different below it as the z axis of spin ordering b e c o m e s defined. The continuity of our data through the Curie t e m p e r a t u r e implies that with rising t e m p e r a t u r e the different components of s u s c e p t i bility anticipate their convergence together, a growing contribution f r o m Xzz rendering the spin waves in c u r r e n t terminology "sloppy", and p r o ducing an extra, hitherto l i t t l e - c o n s i d e r e d contribution to the fluctuation s p e c t r u m that evolves into the p a r a m a g n e t i c distribution. C o r r e s p o n dingly, with lowering of the t e m p e r a t u r e f r o m infinity the susceptibility anticipates the onset

We should mention that in 1961, Riste ~ implied that spin waves had been o b s e r v e d above the N4el t e m p e r a t u r e in a n t i f e r r o m a g n e t i c Fe30~, although the u n p r o c e s s e d data have n e v e r been published. The continuous nature of the evolution through T N of ~ at large K in MnF~ was f i r s t noted by T u r b e r f i e l d and Okazaki. ~ Simultaneously with the p r e s e n t work, Cowley et al. ~ and Nathans et al. ~ have demons t r a t e d the existence of s p i n - w a v e - l i k e peaks above the N4el t e m p e r a t u r e in CoF~ and RbMnF3 respectively. Acknowledgements - We a r e grateful to Drs. W. Brinkman, S. Doniach, J. Hubbard and P . C . Martin for illuminating discussions.

References 1.

CABLE J . W . , (1967).

2.

LOWDE R.D.

3.

LOWDE R.D. and WINDSOR C . G . , Magnetism, Boston 1967).

4.

ALLAN G.,

5.

STRINGFELLOW M . W . ,

6.

PICKART S . J . et al.

7.

JACROT B . , KONSTANTINOVIC J . , PARETTE G.G. and CRIBIER D . , of Neutrons in Solids & Liquids II_._, 317 IAEA, Vienna, (1963).

8.

BROUT R . , Phys. Lett. Japan. 2.99, 156 (1963).

9.

DALTON N.W. and HUBBARD J . ,

LOWDE R . D . ,

WINDSOR C.G. and WOODS A . D . B . , J. appl. Phys. 3__88, 1247

and WINDSOR C . G . ,

LOMERW. M . ,

J. appl. Phys.

LOWDE R . D .

P r o c . phys. Soc.

Phys. Rev.

156,

1_8.8, 1136 (1967).

in p r e s s (1968) (International Congress on

and WINDSOR C.G. ( t o b e p u b l i s h e d l . (1968) (in p r e s s ) .

623 (1967). Inelastic Scattering

24.__A, 117 (1967)J KAWASAKI K . and BROUT R . ,

10. RESIBOIS P. and de LEENER M., 11. RISTE T . ,

Phys. Rev. Lett.

AERE Report P R / T P Phys. Lett.

25A, 65,

1__33, 20,

P r o g r . th~or. Phys.

H.M. Stationery Office, (1967).

(1967).

J. phys. Soc. Japan 1_.77, Suppl. BIII, p. 60 (1962).

12. TURBERFIELD K . C . and OKAZAKI A . , P r o c . Phys. Soc. 13. COWLEY R . A . and MARTEL P . , Magnetism, Boston 1967).

85__, 743 (1965).

J. appl. Phys. in p r e s s (1968) (International Congress on

192

CRITICAL FLUCTUATIONS IN NICKEL

14. NATHANS R . , MENZINGER F. and PICKART S . J . , Congress on Magnetism, Boston 1967).

Vol. 6, No. 4

J. appl. Phys. in p r e s s (1968) (International

Die E r g e b n i s s e von Neutronen-Streuexperimenten an Nickel zeigen, class auch bei T e m p e r a t u r e n bis zur und oberhalb der C u r i e - T e m p e r a t u r wohldefinierte Spinwellen-Maxima in der v e r a l l g e m e i n e r t e n Suszeptibilitat ~((K, w) bestehen bleiben, wahrend a n d e r e r s e i t s ein diffuser Anteil~nit Merkmalen, die dem p a r a m a g n e t i s c h e r Zustand bei hohen T e m p e r a t u r e n entsprechen, bei Verminderung der T e m p e r a t u r yon To mindestens bis zur Raumtemperatur welter vorhanden ist. Ein kritisches Ansteigen der Suszeptibilitat im Energiebereich oberhalb ~w = 0. 02 eV tritt nicht auf; start dessert werden bei Armaherung an den kritischen Punkt schon die Eigenschaften der anderen Seite deutlich vorweggenommen.