1804 IEEE TRANSACTIONS ON MAGNETICS, VOL. 26, NO. 5, SEPTEMBER 1990

HYSTERESIS PHENOMENA IN ENSEMBLE OF DIPOLE-INTERACTING PARTICLES WITHOUT INTRINSIC ANISOTROPIE

D.V.Berkov, S.V.Meshkov

Institute of Chemical Physics USSR Academy of Sciences Chernogolovka, Moscow distr., USSR

Institute of Solid State Physics USSR Academy of Sciences, Chernogolovka, Moscow distr., USSR

ABSTRACT

In the present work we show, that hysteresis can be observed in an assembly of chaotic di- stributed particles without intrinsic magne- tic anisotropie due to interparticle magneto- static interaotion. To take this interaction into account, pair approximation and computer simulation of the remagnetization process is used.

I. INTRODUCTION

It is well known,that two kinds of mag- netic assemblies demonstrate hysteretic beha- viour: assembly of particles with intrinsic magnetic anisotropie (IMA) (which can be cry- stallographic or form anisotropie) and an as- sembly of isotropic (spherical) particles, which form anisotropic clusters (rings, cha- ins etc.) [1].In the present work we show, that hysteresis can be observed in an assemb- ly of chaotic distributed particles without IMA due to interparticle magnetostatic inte- raction. The simplest and most useful method' to study effects of such an interaction is the mean field approximation (MFA).

To describe the interparticle interacti- on in this approximation one must calculate the average local fieldElOc, acting on the given particle from all other particles of the assembly.Then the total field Et acting on the particle can be written as Et=Eo+Eloc, where Bo denotes the external field. For cha- otic distributed magnetic dipoles the distri- bution density of any projection of Hloc is the even function. This results in the zero average projections of Eloc, and the anhyste- retic remagnetization curve of the ideal (noninteracting) assembly of particles with- out IMA does not change qualitatively for the interacting assembly.

MFA neglects interparticle correlations. For dipole interacting particles such corre- lations must be of great significance. In pa- rticular, the dipole interaction of two dose particles aligns their moments along the pair axis and leads to the hysteretic remagnetiza- tion curve for such a pair with the coercivi- ty H c - p / r 3 , where p denotes the magnetic mo- ment of each particle and r is the interpar- ticle vector.

In this work we show, that the correct consideration of these correlations leads to the qualitatively new result - the hysteretic behaviour of the assembly of chaotically dis- tributed particles without IMA. We restrict ourselves with the case of identical spheri- cal particles with constant magnetic moment p . We consider the case of zero temperature

and large dissipation, which results in the quasistatic remagnetization process, so that the moment of the particle is always aligned in the field direction.

1I.PAIR APPROXIMATION

We start with the pair approximation, which takes into account only the nearest- neighbours interactions. The energy of the pair with the interparticle distance r and director e=r/r

remains the same under the substitution p l - p 2 , which is equivalent to permutation of particles. Therefore in the equilibrium state angles el and e2 between the moments and the external field are the same (e1=e2=e) and we can write

u = - 2 p E o - [ 3 ( e p ) " -p2 Ir-

If n denotes the particle concentratign,-%f@e average interparticle distance is r -n , the average e_",ergy of the interparticle inte- raction a - p ' r -p n and the- fjeld- from one particle at the distance r H - p r -pn . For this reason it is convenient to introduce di- mensionless energy u=U/p2n, f i.eld h=H/pn , mo- ment m = p / p and distance p 3 = 4 n r 3 / 3 . If the z-axis is aligned along the external field, we can rewrite (1) as

u = - ~ ~ ~ ~ c o s ( e ) t4n[l-3cos2 ( e-eo) ] / 3 p

where eo is the angle between the z-axis $nd the pair axis. The equilibrium angle e=m,b, can be determined from the condition au/ae=O:

hOzsi n ( e ) = -2nd n (2 ( e -eo) ) / p

To calculate the average magnetization VR >=<cos(e(h OZ ) )> one must find from ( 2 ) e(hoz,e0,p) and then take the average through the pair orientations eo and interparticle distances p. For chaotic assembly correspon- ding distribution densities are

f ( e O ) d e o = s i n ( e O ) d e O , f ( p ) d p = e - ' d p

and for unz> we obtain

m 4 2

0 0 < m z > = b - P d p I... ( e (hOz , p , eo) ) s i n ( eo) d eo ( 3 )

001 8-946419010900-1 804$01 .OO 0 1990 IEEE

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The substitution of variables ( p , e o ) + ( e , e o ) enables us to rewrite ( 3 ) as

unz>=]'si n( eo)deo@s( e ) expl - p ( e ,eo) 1%. ( 4) 4 2 X

enti n 0

where

p ( e ) =-2ns in(2( e-eo) ) /hOzs in ( e )

(see (2)). Thereby the problem is reduced to the evaluation of the integral and we needn't solve the transcendent equation (2) for all pairs ( p ,eo 1.

The integration limits emin and emax as functions of eo must be determined separately for hOz>O and hOz<O. For positive hOz the low limit eminCO and the upper limit emax=eo. The reason is that for pairs with large p any fi- nite external field orients the moments along the z-axis ( e = e o ) and for close pairs ( p + O )

the interparticle interaction orients the mo- ments along the pair axis (€)=eo) although for any finite external field.

For negative hOz we must take into con- sideration leaps of the moments. If h O z + - m

both moments deviate from the z-axis so that the angle e>eo increases. At a certain nega- tive field they jump to the other equilibrium state between the negative direction of the z-axis and the pair axis. The stability of the initial equilibrium state can be lost in two ways (modes). In the first case random deviations S1 and S 2 from the equilibrium an- gle e have the same sign ( S l = S 2 = 6 , positive mode) and in the second case - opposite signs ( s l= - s2=6 , negative mode). The stability ana- lysis shows, that the positive mode takes place for pairs with

eozecr=arccos (3- " ) -arc& n ( 2 * 2' " / 3 ) / 2

and the negative one for eosecr . The transition from ( 3 ) to (4) requires

the evaluation of integration limits separa- tely for overturned and not overturned pairs. or the former, the lower integration limit is the angle e=eaft(eo) between the moments and the z-axis immediately after the leap and the upper limit emx=n. For the latter the integration must be performed from Omin=O to the angle efor(eO) between the moment and the z-axis before the leap.

To determine e for we have to solve sys- tems consisting of equilibrium equation !2) and the conditions of the loss of stability a2 u/aef =O , For positive and negative modes we find (bo= lhOz I , u 0 = p h d 2 z ) :

uosi n( e f) = s i n ( 2( ef -eo) )

uocos( ef) =2cos(2( ef -eo) ) I

uosin( er) =si n ( 2 ( of - e o ) )

u0cos ( er) =2/3

Excluding from this equations u o , wencan find ef and evaluate u o , The angle eaft=p,h can be found then from the equilibrium equation for overturned moments in a given field ( 2 ) :

-

- - s i n ( 2( eaf -eo)) =uosin( eaf t)

and the required angle e a f t = ' - E a f t = @,e,. The integral (4) can then be written as

o

efor x

The hysteresis loop in pair approximati- ons evaluated using (4)'and (4') is shown on fig.1 (curve 1). We have neglected the exclu- ded volume effe'ct, therefore the interaction field can be infinite (for pairs with p + O ) and the loop closes only in the infinity.

It is straightforward to show, that in this approximation the remanence jR=mZ(0)=0.5.for the pair axis are chaotic distributed. The coercivity calculated nume- rically is h,=5.04.

111. NUMERICAL CALCULATIONS

It is obvious, that the pair approxima- tion can not describe the whole loop correct- ly. This approximation is valid only for ex- ternal field much larger then the mean inte- raction field ( h O > > l ) . To describe the hyste- retic behaviour of the system for hO-1 the computer simulation of the remagnetization process is required.

The simulation was performed on the cu- bical assembly of chaotic distributed partic- les with periodic boundary conditions. We ha- ve started from the large external field and all moments aligned along this field.The to- tal field for each particle was calculated as the sum of external field and fields of all other particles of the assembly.The moments of all particles were then oriented along the total field direction at the point of the pa- rticle location. Then new total fields were calculated etc., until for all particles the angles between the moments and th? total fi- eld directions were less than -1 . For the next value of the external field we have used the orientations obtained for the previous external field as starting ones.

The hysteresis loop obtained with this method is shown on fig.1 (curve 2). An avera- ging was performed through 10 realizations of 400-particles assembly. The influence of ot- her particles narrows the loop calculated in the pair approximations (compare curves 1 and 2).The corresponding anhysteretic remagneti- zation curve evaluated in the mean field app- roximation with the field distribution densi- ty [ 2 1

p ( h l o c ) =T[- A/( A2 +h2 )'

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where A=4.47, is shown on the same fig.1 with the dashed line.

All these curves plotted as mz=mz(ho) are universal: the hysteresis loop in coordi- nates ( M Z , H O ) can be obtained from curves on

fig.1 using the definitions of h=H/pn and m=M/Ms, where Ms denotes the saturation mag- netization of the sample. The remanence jR=0.22 does not depend on the concentration and the coercivity increases with n linearly: hc=2.5 or Hc=2.5pn.

This universality results from the neg- lecting of the excluded volume effect (or the finite particle size). In this case there is only one length scale in the system - the me- an interparticle distance, which is used to construct all dimensionless quantities ( p , h and U). If we take the excluded volume effect into consideration, another characteristic length - the particle diameter - appears in the system, and the evaluation of such unive- rsal curves will be impossible. The excluded volume effect narrows the hysteresis loop in (m,h)-plane, for the reason that the introdu- cing of the minimum interparticle distance is equivalent to upper restrictign of the inte- raction field. The corresponding dependencies j R ( n ) and h c ( n ) are shown on fig.2.

REFERENCES

[l]. Wohlfarth E.P., Ferromagnetics Materials, v.2, Amsterdam, 1980. [2]. Berkov D.V., Meshkov S.V.,"Theory of remagnetizations curves of dilute random magnetics",Sov.Phys.JETP, v.94, pp.140-152, 1988.

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