Non-linear effects in the spin-wave conduction electron interaction

Volume 30A, number 4

PHYSICS

NON-LINEAR CONDUCTION

LETTERS

20 October 1969

EFFECTS IN THE SPIN-WAVE ELECTRON INTERACTION

K. M E N D E LSON * Department of Physics, Marquette University, Milwaukee, Wisconsin 53233, USA and H. N. S P E C T O R Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616, USA Received 19 September 1969

The effect of non-linear t e r m s on the interaction of spin-waves with conduction electrons has been studied. For small spin-wave amplitudes, the non-linear t e r m s introduce only small corrections to the linear approximation. We h a v e c o n s i d e r e d t h e e f f e c t of n o n - l i n e a r t e r m s on the d i s p e r s i o n r e l a t i o n f o r the i n t e r a c t i o n of spin w a v e s w i t h c o n d u c t i o n e l e c t r o n s . In our results the dispersion relation is only weakly d e p e n d e n t on t h e n o n - l i n e a r i t y . T h i s i s in c o n t r a s t to the c a l c u l a t i o n r e p o r t e d by L a l [1], w h i c h s h o w e d a v e r y s t r o n g d e p e n d e n c e on t h e non-linearity. T h e e q u a t i o n of m o t i o n f o r the m a g n e t i z a t i o n can be d i v i d e d into l i n e a r and n o n - l i n e a r t e r m s by w r i t i n g the m a g n e t i c f i e l d and the m a g n e t i z a t i o n in the s a m p l e a s H= H 0 + H 1

M= M 0 + M 1

w h e r e H o , M 0 a r e c o n s t a n t f i e l d s and H 1 , 3 4 1 a r e t i m e v a r y i n g f i e l d s . In a d d i t i o n to t h e e x t e r n a l d.c. m a g n e t i c f i e l d H o i n c l u d e s d e m a g n e t i z a t i o n and a n i s o t r o p y f i e l d s . In t e r m s of t h e s e f i e l d s , t h e e q u a t i o n of m o t i o n f o r t h e m a g n e t i z a t i o n i s [2]. 0t

- - 7 { 3 4 0 × (nV2 M I + H 1 ) + 341 × HO +

Z - a x i s , we l o o k f o r an a p p r o x i m a t e s o l u t i o n of the form + M 1 = M i x + i M l y = m e x p { + i ( k Z -co t)}

(3) : M l x - i M l y = (MI I* with s i m i l a r e x p r e s s i o n s f o r the e l e c t r o n i c and m a g n e t i c f i e l d s . H e r e k i s r e a l , but w m a y in g e n e r a l be c o m p l e x . T h e u p p e r s i g n in t h e e x p o n e n t i a l c o r r e s p o n d s to a l e f t hand c i r c u l a r l y p o l a r i z e d w a v e and the m i n u s sign to a r i g h t hand c i r c u l a r l y p o l a r i z e d w a v e . T h e c o m p o n e n t s of c u r r e n t d e n s i t y and e l e c t r o n i c f i e l d a r e r e l a t e d by J+,L= ~ E + ' w i t h t h e c o n d u c t i v i t y [3] 4 ~ = : i w ~ / ( + w +coc + i / z ) . H e r e COp = ( 4~ne2 / m ~ i s the e l e c t r o n p l a s m a f r e q u e n c y , coc = e B / m c i s t h e e l e c t r o n c y c l o t r o n f r e q u e n c y and ~" i s t h e e l e c . tron relaxation time. S i n c e M 1 << M 0 and t h e t o t a l m a g n e t i z a t i o n r e m a i n s c o n s t a n t , one can w r i t e 1

+ 3 4 1 × (nV2 M I + H 1 ) }

MIZ

=

(M~ - M ÷1

M0

o r , s u b s t i t u t i n g eq. (3). w h e r e g i s t h e g y r o m a g n e t i c r a t i o , e the m a g n i t u d e of t h e e l e c t r o n i c c h a r g e , m the e l e c t r o n m a s s and A the B l o c h w a l l c o e f f i c i e n t . In the f o l l o w i n g d i s c u s s i o n it i s a s s u m e d t h a t M 0 and H 0 a r e d i r e c t e d a l o n g the Z a x i s . It i s n e c e s s a r y to s o l v e eq. (2) p l u s M a x w e l l ' s e q u a t i o n to o b t a i n t h e d i s p e r s i o n r e l a t i o n s . F o r a circularly polarized wave traveling along the * Supported by the National Science Foundation P r o g r a m in Research Participation for College Teachers.

M I Z : - ( m 2 / 2 M 0) exp{~:i(co- co*) t} : =

-

(S)

(m2/2Mo)exp{+Imwt }

In o r d e r to o b s e r v e the w a v e s it i s n e c e s s a r y

that I Im co I<< [Re col- Then in the t i m e r e q u i r e d to observe several cycles of the wave [Im cotl <<[ 1 and one can m a k e the a p p r o x i m a t i o n M1Z = = - m 2 / 2 M O. U s i n g t h e s e r e s u l t s , one o b t a i n s d i r e c t l y f r o m eq. (2) and M a x w e U ' s e q u a t i o n s

213

Volume 30A, n u m b e r 4 the dispersion

PHYSICS

LETTERS

relation

p e n d o n t h e a m p l i t u d e of t h e w a v e . T h u s it d o e s not approach the linear dispersion relation for small amplitudes. Also for this configuration Lal found that the dispersion relation is independ e n t of ~. T h i s i s in c o n t r a d i c t i o n to o u r eq. (6) a n d to t h e e x p e r i m e n t a l r e s u l t s o b t a i n e d b y G r i m e s [4] in n i c k e l .

F w = ½wH( m / M 0 )2 +

COS

{

1

I{ 1 -

+1 com+ ws w h e r e cos = 4nTM0, WH = ~IIo and corn

20 October 1969

TM

= w s n K 2 / 4 n + w H. F o r r n / M 0 <'- 1, t h e n o n - l i n e a r t e r m s i n eq.(6)

Refe fence s

a r e s m a l l c o m p a r e d to t h e l i n e a r t e r m s a n d t h e y a p p r o a c h z e r o a s m / M 0 ~ 0. I n t h i s l i m i t eq. (6) g o e s o v e r t o t h e d i s p e r s i o n r e l a t i o n o b t a i n e d in the usual linear approximation. As noted above, t h i s i s in d i r e c t c o n t r a d i c t i o n to t h e r e s u l t s q u o t e d b y L a l . He o b t a i n e d a d i s p e r s i o n r e l a t i o n w h i c h w a s v e r y d i f f e r e n t f r o m t h a t o b t a i n e d in the linear approximation and which does not de-

1. P. Lal, Phys. L e t t e r s 29A (1969) 168. 2. C. H e r r i n g and C.Kittel, Phys. Rev. 81 (1951) 869. 3. L. Solymar and C. N. L a s h m o r e - D a v i e s , Int. J. Electron 22 (1967) 549. 4. C.C. G r i m e s in P l a s m a effects in solids, 7th Intern. Conf. on the P h y s i c s of semiconductors. P a r i s 1964 (Academic P r e s s , New York, Dunod, P a r i s . 1965) p.87. ~

HCP-BCC

PHASE

TRANSITION

IN

*

*

*

THE

*

GROUND

STATE

OF

SOLID

3He

J.-P. HANSEN Laboratoire de Physique Th~orique et Hautes Energies, 91-Orsay, France * Received 18 S e p t e m b e r 1969

The well-known HCP-BCC t r a n s i t i o n of solid 3He is exhibited at 0OK by the variational method using a J a s t r o w x Gaussian t r i a l wave function and the Monte Carlo method to compute the g r o u n d - s t a t e energy as a function of density. Contrarily to all other rare gases, helium 3 does not crystallize in a close-packed structure a t low t e m p e r a t u r e s , b u t i t s low p r e s s u r e p h a s e i s b o d y - c e n t e r e d c u b i c w i t h a t r a n s i t i o n to a hexagonal close-packed structure at higher press u r e s [1,2]. T h e e n e r g y d i f f e r e n c e b e t w e e n b o t h p h a s e s i s e x p e c t e d t o b e v e r y s m a l l , of t h e o r d e r of a f e w t e n t h s of a d e g r e e p e r a t o m w h i c h r e n d e r s a t h e o r e t i c a l d e s c r i p t i o n of t h e t r a n s i t i o n d i f f i c u l t . W e h a v e n e v e r t h e l e s s a t t e m p t e d to e x h i b i t t h i s s o l i d - s o l i d t r a n s i t i o n in t h e g r o u n d s t a t e b y t h e variational method, using a Jastrow × Gaussian trial wave function: ~=/<]~jexpt-½(~)5

} ×~i

e x p { - ½ A ( r i - R i ) 2}

where A and B are variational parameters and the Ri are the lattice sites. This trial wave func* L a b o r a t o i r e associ~ au Centre National de la Rec h e r c h e Scientifique.

214

t i o n t o g e t h e r w i t h t h e M o n t e - C a r l o m e t h o d to compute the energy expectation value have proved s u c c e s s f u l to d e s c r i b e t h e g r o u n d - s t a t e p r o p e r t i e s a n d m e l t i n g of s o l i d 3He a n d 4He [3]. In r e f . 3 , t h e L e n n a r d - J o n e s i n t e r a t o m i c p o t e n t i a l v(r) = = 4 e [ ( c r / r ) 12 - o(cr/r)6], w a s u s e d w i t h E = 10.22OK a n d ~ = 2.556 A, b u t t h e c a l c u l a t i o n s w e r e d o n e only for a face-centered cubic lattice which is stable only under very high pressures. Keeping the same interatomic potential and the same type of t r i a l w a v e - f u n c t i o n we t h o u g h t it w o r t h w h i l e to repeat the calculations for both HCP and BCC lattices. The Monte Carlo procedure is the same a s t h a t d e s c r i b e d in r e f . 3; 800 a t o m s w e r e p l a c e d in a b o x w i t h p e r i o d i c b o u n d a r y c o n d i t i o n s . A l l a t o m s w e r e a l l o w e d to i n t e r a c t w i t h t h e i r f i r s t 5 s h e l l s of n e i g h b o u r s (i.e. 58 a t o m s in t h e BCC c a s e a n d 56 a t o m s i n t h e H C P c a s e ) a n d t h e c o r reaction terms from all other shells were calc u l a t e d by p u t t i n g t h e J a s t r o w f a c t o r s f ( r i j ) =