On a theory of multifield resonances

Volume 37A, number 5

PHYSICS LETTERS

ON A T H E O R Y

OF

MULTIFIELD

20 December 1971

RESONANCES

V. R. NAGIBAROV, N.K. SOLOVAROV and V. V. SAMARTSEV P h y s i c o - T e c h n i c a l I n s t i t u t e , K a z a n , USSR

Received 29 October 1971

The acousto-magnetic resonance has been considered, when it became possible to determine a sign of the spin-phonon interaction constant and to show the possibility of the strengthening of the exciting waves.

When speaking about a m u l t i f i e l d r e s o n a n c e we m e a n an a b s o r p t i o n of the e x t e r n a l g e n e r a t o r e n e r g y by a s y s t e m of quantum objects u n d e r going the influence of one, two, t h r e e or m o r e fields of the s a m e f r e q u e n c i e s s i m u l t a n e o u s l y . In this p a p e r we s t r e s s the fact that m u l t i f i e l d r e s o n a n c e s enable us to get the i n f o r m a t i o n , which was i m p o s s i b l e to get f r o m the d i r e c t o b s e r v a tion of e v e r y s e p a r a t e r e s o n a n c e . T h i s c a n be i l l u s t r a t e d by the c a s e of a difield a c o u s t o - m a g netic r e s o n a n c e . Let us c o n s i d e r a s a m p l e c o n t a i n i n g p a r a m a g netic p a r t i c l e s with effective spin S = ½ that is p l a c e d into a p e r m a n e n t magnetic field H ( H o / / Z ) . The s a m p l e u n d e r g o e s s i m u l t a n e o u s l y the i n fluence of the a l t e r n a t i n g magnetic field H I ( H I cos(c0t + F ) , H I sin(cot + F), 0) and of the longitudinal acoustic waves U(x,t)

= Rcos(cot

(1)

- kx + ~)

w h e r e H1, R a r e the a m p l i t u d e s of the a l t e r n a t i n g magnetic field and the acoustic wave; F , ~ a r e the i n i t i a l p h a s e s of e l e c t r o m a g n e t i c and acoustic fields r e s p e c t i v e l y ; k i s the modulus of the sound wave v e c t o r . F o r s i m p l i c i t y let us a s s u m e that the c r y s t a l has a cubic s y m m e t r y and the acoustic wave i n t e r a c t s only with the S x component of the effective spin. The H a m i l t o n i a n of the p r o b l e m in the s y s t e m of r e f e r e n c e r o t a t i n g with the w - f r e quency a s r e f e r e d to the z axis and r u n n i n g along with the acoustic wave, has the f o r m

(2)

= ~coogz + ¼ c n k ( g + - g _ ) +

+ ½~col(g+ exp(-i~ ) + g- exp(io,)), where G i s the s p i n - p h o n o n i n t e r a c t i o n constant; a = kx - • - F; S+=exp[;i(cot-kx+¢)}S+;

Sz=Sz,

the symbol ' ~ ' a t t r i b u t e s the o p e r a t o r s to the

r u n n i n g s y s t e m of coordinate; Wl = - y H 1 ; y i s a g y r o m a g n e t i c ratio. The Bloch equations which d e s c r i b e a spin s y s t e m behaviour in the c a s e u n d e r c o n s i d e r a t i o n a r e given for example in ref. [1]. The solution of these equations for a steady r e g i m e of the effect of exciting fields has the f o r m of M x = -MoTel(A2 + T~2) -1

× [ACOlT2Cosc~ + (A - colsino0], M y = -MoT~I(A 2 + T~2) -1

x [-A. T2(A -colsin

e~) + colCOS ot)],

M z : MoT22(A2 + T32)-1(1 + A2T~2)

,

(3)

where M o = - ½ g t a n h ( ~ w o / 2 k B T ) ; ~ is the m a g netic m o m e n t of a single p a r t i c l e ; k B is the B o l t z m a n n constant; T is the t e m p e r a t u r e ; A = = GRk/2~; A = w - COo; T 3 = T2[1 + T 1 T 2 ( A 2 - 2 A w 1 s i n s + ¢o2]-1/2. The r a d i o f r e q u e n c y power a b s o r b e d by the s p i n - s y s tem, is d e t e r m i n e d by the f o r m u l a [2] P = - M -d~H- = - w g l ( - , ~ x s i n ~ + MyCOSOt)

(4)

Let us assume that the inhomogeneous broadening is described by the Lorentzian lineshape

~1 "r~-I g(coo) = (coo - ~o) 2 + ~2

(~)

where f l l i s the i n h o m o g e n e o u s line width; ~2o i s the c e n t r a l frequency. The following e x p r e s s i o n can be obtained for the power of the r a d i o f r e quency field a b s o r p t i o n

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Volume 37A, number 5

PHYSICS LETTERS

P(¢o) = wH1MoT~l[(w - f~o)2 + (g21 + T~l)2] -1

(6)

x [(w 1 - Asino~)(1 + ~21T3) - (w - ~2o)AT2cos(~ ]. Choosing the v a l u e s of the i n i t i a l p h a s e s • , F and the s a m p l e d i m e n s i o n s one can always a c h i e v e the f u l f i l m e n t of the condition ~ = 2~n (n i s a whole n u m b e r ) . If t h e r e i s no a c o u s t i c f i e l d (A = 0) the c u r v e -P(w) has a m a x i m u m if u) = ~/~2 + (~1 + T31) 2" The i n c l u s i o n of the a c o u s t i c fi el d g i v e s r i s e to two e x t r e m e v a l u e s on the c u r v e of the e l e c t r o m a g n e t i c a b s o r p t i o n p o w e r if w = (AT2~2 o - B)-I{AT2g~ 2 + (~1 + T§ 1)2

(7)

± ~[A2T2(al + T31) 2 +B2l[ao2 + (f~l + T31)21}, w h e r e B = Wl(1 + ~21T3). One of t h e s e e x t r e m e v a l u e s is a m a x i m u m , the o t h e r is a m i n i m u m U n d e r this condition a shift of the m a x i m u m of the c u r v e P(¢o) will be o b s e r v e d e x p e r i m e n t a l l y . A d i r e c t i o n of the shift (to a l o w - f r e q u e n c y o r to a h i g h - f r e q u e n c y side) i s d e t e r m i n e d by the s i g n s of the spin-phonon i n t e r a c t i o n constant G and the g y r o m a g n e t i c r a t i o ~. If the signs a r e i d e n t i c a l the shift i s to the h i g h - f r e q u e n c y side and if they

456

20 December 1971

a r e d i f f e r e n t the shift is to the l o w - f r e q u e n c y side. Th u s, knowing the sing of the g y r o m a g n e t i c r a t i o , by the shift of the m a x i m u m of the P(¢0) c u r v e in:the e x p e r i m e n t on the a c o u s t o - m a g n e t i c difield r e s o n a n c e , we can o b t a i n both the v a l u e and the sign of the spin-phonon i n t e r a c t i o n constant. E s t i m a t e s show that f o r v a l u e s of the p a r a m e t e r s t y p i cal f o r e l e c t r o n p a r a m a g n e t i c r e s o nance (G ~ 1 0 - 1 7 e r g , ~ o ~ 1 0 1 0 s - l ; T2 ~ 1 0 - 7 s ; T I ~ 1 0 - 5 s ; h ~ 105cm;~21 ~ 1 0 9 s - l ; ~Wll 104 s -1) and f o r n u c l e a r m a g n e t i c r e s o n a n c e (G ~ 1 0 - 2 0 e r g ; ~2o ~ 1 0 7 s - l ; T2 ~ 104s; T 1 ~ l s ; k ~ 1 0 a m - l ; ~21 ~ 1 0 6 s - l ; 1~1t ~ 1 0 s - l ) a l r e a d y in the c a s e of R ~ 1 0 - g c m (the a b s e n c e of saturation) the shift of the m a x i m u m will be of o r d e r 1/10 of the i n h o m o g e n e o u s line width, i.e. it may be e a s i l y o b s e r v e d e x p e r i m e n t a l l y . It may be noted, that P(w) may be > 0 or <0. Hence in the d i f f e r e n t f r e q u e n c y r a n g e s one can o b s e r v e both a b s o r p t i o n and a s t r e n g t h e n i n g of the e l e c t r o m a g n e t i c f i e l d c a u s e d by the a c o u s t i c wave influence.

References [1] V. R. Nagibarov, V.V. Samartsev and N.K. Solovarov, Zh. Eksp. i Teor. Fiz. 61 {1971) 1636. [2] A. Abragam. The principles of nuclear magnetism (Clarendon Press, Oxford, 1961).