On the paramagnetic resonance and the longitudinal relaxation of ferromagnets in the critical region above Tc

~Solid

State Communications,Vol.39, pp.17-21.

0038-1098/81/010017-05502.00/0

Pergamon Press Ltd. 1981. Printed in Great Britain.

ON THE PARANAGNETTC RESONANCE AND THE LONGITUDINAL RELAXATION OF FERRO~AGNETS IN THE CRITICAL REGION ABOVE T c A.V.Lazuta, S.V.Maleyev a,~ B.P.Toperverg L e n i n g r a d Nuclear Physics ~n-titute

Gatchina,

Leningrad

188350, USSR

(Received 1 March 1981 by E.A.Kaner) Uniform d~namical susceptibility of cubic ferromagnets above T~ in magnetic field is investigated. Resonance frequences and dampin~ constants are determined in the limiting cases of low and strong magnetic field. It is shown that in the exchange temperature region when 4 ~ F ~ ~'- I owing to spin diffusion the ,m~form susceptibility is depended nontrivially on the frequency and the magnetic field. From this dependence the temperature behaviour of spin diffusion coefficient can be determined.

The paramagnetlc resonance in ferromagnets above T c and the longitudinal relaxation in the magnetic field have been intensively investi~ted experimentally (see, e.g.,Ref./1-3/). Recent-

where Nj.~ i s the t e n s o r o f demagnetize,t i o n . The t r a n s v e r s e p a r t of the s u s ceptibility X~/~ ~))- describes the ,,.4I

ly it has been demonstrated in Ref./4,D/" -" that the scattering of the polarized neutrons from magnetized ferromagnets above T c may provide valuable information on the dynamics of critical fluctuations. At the same time there no exists detailed theoretical description of the critical dynamics of ferromagnets in the magnetic field above T c. In this paper we consider uniform dynamical susceptibility of the cubic ferromagnets, taking into account the exchange and dipolar interactions only. Our analysis is purely phenomenologlcal and based on the dynamical scaling hypothesis./6/

f o m spin precession and the longitudinal one d e s c r i b e s t h e r e l a x a t i o n of

m a g n e t i z a t i o n to i t s e q u i l i b r i u m v a l u e . For t h e s e two p a r t s we assume t h e f o l lowing expressions : (2)

X,(H.)):X(H) ~ i C (H,a)) +

'

C.(H,k) (3>

where H is the intrlnslo time i n d e p ~ ent magnetic field; X o ~ H ) = aM/~H and X ( H ) , MH -I , M is the magnetizatlon

It is well-known that the dynamical susceptibility of the sample X describes the llneae response to the external maL~netic field. The respose to the internal field is described by the susceptibility of the substance ~ . These two susceptibilities in the uniform case are connected by the well-known relation

i n t h e f i e l d H. F i r s t , l e t us c o n s i d e r t h e exchange t e m p e r a t u r e r e g i o n when 4~rXo/a~ 1, ,~obe:£ng the sero-fleld static susceptibility. In this region the e n e r ~ of critical fluctuations is

~ ven /),

by ~ e ( a £ ) - T c ( ~ ) 5 / 2 (see Ref. where ~ . a ' l T ~ i s t h e i n v e r s e c o r r e l a t i o n l e n g t h , ~ ' l (T-Tc)Tcl a n d ~

17

FERROMAGNETS IN THE CRITICAL REGION ABOVE TC

18

is a q u a n t i t y of the order of the l a t t i c e c o n s t a n t and we n e g l e c t t h e s m a l l F i s h e r exponent ~ . In accordance with static scaling theory the magnetic f i e l d is small i f g/~H Z.~.~(~). In this case all static physical quantities can be expanded in the power series of (g~H/~'~(~)) 2. In the dynamical theory the above condition means that the Larmore frequency is small as compared to the energy of the critical fluctuations. We shall discuss here the behaviour of the quantities ~ N '~ in the case of small ~0 and H. To analyse the quantity ~ in eq. (2),(3), we use following Ref./7'8/the Kubo expression

where A tioms

static

~

Here the lines represent the Green functions which are the generalizations of Eqs. (2), (3) to nonzero momentum q ~4. ~ . If the H and C~ are small, we have i ~ 0 + i D q2 Gal (q'uJ)=Go(q) ~O+ i Co+ i D q2 G+_(q,

H) = -Q0(q) -gyH - i [ 0 -

~-

(7)

iDq2

E~H + i V o, iDq 2

'

where D is t h e spin diffusion c o n s t a n t . It has been shown in Ref./8/that if0J = - O, H = O, the first diagram as well as the entire series depicted in Fig. I have the same order of magnitude, and the sum gives the Eq.(6). We shall see that the H and cO dependence of ~A is determined by the first diagrams only. This diagram can be easily calculated using the method of Ref./8
labels three different func-

(8.a)

/"÷_ , ~ + . There Go, ~ i s t h e Green f u n c t i o n and f o r ~ we have ,

(5)

w?

Vol. 39, No. I

6 f f +us

~dt e '~'~1 ,~-.

2

o

+

)+l( +2 gp,/-t)} ~. ~)

-,{

~lI(w-y/,.~ + Ic~o11} (s.c)

~. (,k+, where UJo= 42~(g~) 2 .~-1 is t h e dipolar enerEy and ~fo is the volume of the unit ceel. One has as usually ~)-GA. = 4/T;~o A . I t has been shown in aef. /7,8/ t h a t (6) a

r~o,o)=C=¥ ~ (~i, + =~ L~c,.~ 1+~, x j where ~ and ~ t q u a n t i t i e s of o r d e r o f u n i t y . Now l e t us c o n s i d e r t h e d i a g r a m s e r i e s f o r F . r e p r e s e n t e d i n F i g . 1.

+

~

+

.

'

Z• + ; D h ~

(8.a)

GJ~)=@k)~+;r .z D¢ 0

Using (8c) and (8d) we find:

z(~) = z(o) +az(~)

a ~ ++

(9)

~Dk~C oo+Z:(D~'+9-

2

-

G.%>~or. ~', Fig. I

rdx,dxz~,6"p,~X,.~.~%~

~

FERROMAGNETS IN THE CRITICAL REIGON ABOVE T

Vol. 39, No. I

19

C

~(x).. 1 -1/V2"" [1 +(1,x2) 1/2 ] I/2 (10) %Oz(x).. x/V2"[(1+x2)1/2* 1 ] - 1 / 2 • Prom these expressions we get the following formulae for the shift of the resonant frequency an~ the correction for the line width:

,._A 2

(11)

~2

tar irregularity occurs in the case of more c o m p l i c a t e d d i a ~ a ~ • l e o . How-. ever, it can be shown in this case that when k ~ 0 the renorsmli=ed zverrices are proportional t o k k _ / a e . As p a c o n s e q u e n c e , t h e tO d e p e n d e n t corr e c t i o n s s h o u l d be s m a l l e r by t h e factor ~/~e (x) , than considered • b o r e . I t s h o u l d be n o t e d t h a t i f u) or g/uH are of order of 5"~e(~).~ ,all corrections become o f t h e o r d e r ~ o , and we h a v e no s i m p l e e x p r e s s i o n f o r d

/-'A ( ~ ) ) . In the case of strong magn e t i c f i e l d , when 8~H > ~ c ( K ) , the

w h e r e b~C= ~91I'I. A s i m i l a r e x p r e s s i o n c a n be o b t a i n e d f o r ~ , (~0). I f H . 0 t h e f r e q u e n c y d e p e n d e n t p a r t o f ~ (03~ has the

form

(12)

~'~ ~ - ( 0 - - ~ z

F(o) =

-

D~

correlation l e n g t h R c = ~ - 1 d e p e n d s on the magnetic field and has the followi n s foz~: Rc . a-I(Tc/Sy=H)2/5-aRH "1. As a result all ph~sioal quantities depends on the magnetic field onl~. In

particular

we have 4 ~ ' X n ~p=(Tc/g/~):~ v l q - ~

-

-

<'-1 and Xo,(H) - 5 X,,(H). In this case we get: (13)

(T#)'" li

.~

,o,-

-~.

+-

-~

@

These

corrections

have the order

of

ma~t~e 4~X. ~Af. == •re~__ ~eseent i a l in t h e f r e q u e n c y range~e~9~a)>/l o where they are proportional to

From t h e e x p e r i m e n t a l i n v e s t i g a t i o n of the considered phenomena the diff u s i o n c o n s t a n t D c a n be d e t e r m i n e d . I t should be noted, that the corrections (8)-(12) exist n o t on]¢ i n fer~omagaet~ n e a r Tc , b u t a l s o i n a l l m a g n e t i c e y e toms where spin diffusion is p r e d o m i nant over uniform relaxation. The s i m i l a r phenomena i n t w o - d i m e n s i o n a l m a ~ -

netics

are more pz~ainent

and have been

investigated earlier (see R e f . / 9 / ) .

I t i s e a ~ t o show t h a t i r r e g ~ a r oomrecttons oonsidered •bore sze dee to the integration over the z~nge of sm~U

memant• x2~, 12 C - i ' ° l D'I- ~ " s~-~-

From t h e l a t t e r e x p r e s s i o n s we s e e t h a t with the increeusing field the par•magnetic resonance line width narrows. It c a n be eholm t h a t temperature dependent corrections to the considered quantities have the relative order of magnitude r (~/' . / % ) - 3 / 5 .

the

H e r o we m•ko • l e o • s h o r t r e m a r k on %" d e p e n d e n c e o f dampimg c o m e t • r i t e

b r . _ ( H r ) , V,CH?) at the f~.xod f i n d ~p~ >> r~ ( a , r ) . It is maturaUl to expect that ff~(H,r) will either ino r e a o e m o n o t o n o u s l y w i t h /" d e e r s • s i n g o r h a v e t h e mmxim~a a t %"= oonst(E~/To)3/5. ~ h e r e m L l t s o f Ref. 121 m o s t p r o b a b l y ~ L i o s t e • n o n m o n o tone b e h a v i o u ~ o f ~. ( H , ~ ) w i t h • meatinn. It is olelrly ~t i n % h ~ c•,so

FERROMAGNETS IN THE CRITICAL REGION ABOVE Tc

2o

In the dipolar temperature region when 4/~)( >> 1 or the characteristic cal fluctuations

~'2d ( ~ )

[

~.

~ ~.~. q0 ( q0 =(::~-l(~)o/~J~ energy of the critihas the form:

(14)

Vol. 39, No. I

large S = 7/2 (EuO, h~S) has been investigated in Ref./3'11/ and unconventional behaviour was observed. Nevertheless, it is not excluded that in Ref. /11/ the beginning of the cross over to the unconventional behaviour has been seen, since the observed dynamical

-:

critical exponent was smaller than the conventional one. At the same time the experimental results obtained by the where ~ and

~ are quantities of order

of unity. This expression is the generalization of the results of Ref./8/and~" can be easily obtained by the method used there. In that paper it is pointed out that the unconventional behaviour takes place due to rescattering of the critical fluctuations and therefore the quantity

~

smallness.

must have some numerical Therefore,

if c4 > > ~ ,

we

have wide temperature region, where the dipolar dynamics has the conventional form and its temperature dependence is determined by factor GO1

Nevertheless

at temperature very near Tc, when ~ c

<< ~ T c G(qo(1)2"~' 4 W X of ~d(~)

, the b e h a v l o u r

is unconventional.

In this

case one has

S~- d

(~) ~- To(q0(1)3/2(~Ta) ~51/2,

(15)

a n d one d e a l s w i t h h a r d d i p o l a r dyr,--,,,~cs I n R e f . / 1 0 / i t has b e e n p o i n t e d o u t t h a t b e c a u s e o f t h e odd d y n a m i c a l s p i n correlations the critical dynamics depends strongly on t h e m a g n i t u d e o f t h e atomic spin ~ . In particular, the rescattering processes which take place due t o t h e t h r e e - s p i n d y n a m i c a l c o r relations f o r l a r g e ~ a r e damped. T h e r e fo~e, the quantity ~ is strongly ~ dep e n d e n t and d e c r e a s e w i t h t h e i n c r e a s i n g o f S . As a r e s u l t , one has wide temperature range of conventional crit i c a l dy-mm~cs f o r f e r r o m a g n s t s w i t h l a r g e ~ . On t h e c o n t r a r y , i n t h e c a s e o f s m a l l ~ t h i s r e g i o n c a n be more n a r r o w , o r c o m p l e t e l y a b s e n t . The d i polar dynamics of ferromagnets with

polarized neutron scattering in Fe (S = = 1) are in a qualitative agreement with the unconventional d.vnamics (see Her. "'/4'5/). It is necessary to point out here that in the unconventional case the W-dependent part of the dynamical susceptibility has not the Lorentzian

shape (see Ref./12/).

other words, the dispersion of

In

~(o~)

must take place when ~ ~ 5 q d ( e ~ .This dispersion has been observed in R e f P 1/. In the dipolar temperature region the condition for the smallness of the field is the same as in the exchange region: g/~H<<~e(~) = T c ( ~ a ) 5 / 2 , where H is the intrinsic magnetic field.At the same time in the dipolar region if q~, t h e momentum d e p e n d e n c e o f t h e

dynamical susceptibility is nonessent i a l , and we h a v e no n o n r e g u l a r cO d e pendent corrections to the static damping c o n s t a n t a t ~ << ~d(~). Therefore,

for the small fields we get

~('.~,O,Hl = F('~,(~J["

(1G)

z'gfH

(('')

If ~ <~ ~J (~) the second form in the expression for I+_ may be c o n s i d ered as the temperature dependent renormalization of the g-factor. In this case the rsnormalisation is very strong The susceptibility ~ +_ has the maxl-

Vol. 39. No. I

FERROMAGNETS IN THE CRITICAL REGION ABOVE TC

In the strong magnetic field if 4 ~ - ~ ( ( H ) >>I anddfH~-'..qO, t h e e x p r e s sions obtained above for dipolar dynamics hold true with substitution of

to~.

However, if 4 ~ ( H ) ~ 1

and~H>

> q o ' we a c t u a l l y h a v e t h e e x c h a n g e case in the strong magnetic field dis-

21

one He in the same limiting cases. In the exchange region He--~ H , in the dipolar one in the case of small fields H --~-He (4/'2~oN)-I~~ He, and the condition of the smallness of the external field takes the form (17)

cussed above.

The expressions for the susceptibility of the sample can be easily evaluated using Eq.(1). For example, the expressions for the resonant frequencies are obvious generalizations of the results of Ref. /13/. For practical purposes it is useful to discuss shortly the relation between the internal f i e l d H and t h e e x t e r n a l

In the strong field in dipolar region if 4/TX(r,H) >>I we have

H

~'He

(.,~T~--) 4- ~-%.

•

(181

References I. 2. 3. 4.

J.K~tzler, G.Kamleiter, G.Weber. J.Phys.Cg, L361, 1976. J.Kgtzler , H.V.Philipeborn. Ph~e.Rev.Lett.L40, 790, 1978. J.Kotzler, W.Scheithe. Ph~e.Rev.,B18, 1306, 1978. A.I.Okorokov, A.G.Gukasov, V.V.Runo~, M.Roth. Solid State Comm. (to be published). 5. A.V.Lazuta, S.M.Maleyev, B.P.Toperverg. Solid State Co---.(to be published). 6. P.C.Hohenberg, B.I.Halperln. Rev.Mod.Ph~s.,49, 435, 1978. 7. D.L.Huber. J.Ph.Ts.Chem.Solids, 32. 2145, 1971. 8. S.V.Maleyev. Zh.Eksper.Teor.Fiz.,66, 1809, 1974.(Sov.Phys.JETP, 39, 889, ---1974) • 9. H.Benner. Pb~e.Letters, 70A, 225, 1979. 10. A.V.La~uta , S.V.galeyev, B.P.Toperverg. Zh.Eksp.Theor.Phys.,75, 764, 1978. (Sov.Ph~s.JETP, 48, 386, 1978). 11. M.Shlno, T.Husb/moto. J.Ph~s.Soc.Japan, 45, 22, 1978. 13. D.L.Huber, LS.Seehrm.

Ph~s.Stat.Sol.(b),

?4,

145, 1976.