STRAY FIELDS AND BITTER PATTERN FORMATION ON FERROMAGNETS

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STRAY FIELDS AND BITTER PATTERN FORMATION ON FERROMAGNETS

K. Bray, P. Rhodes

To cite this version:

K. Bray, P. Rhodes. STRAY FIELDS AND BITTER PATTERN FORMATION ON FERROMAG- NETS. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-253-C1-255. �10.1051/jphyscol:1971183�.

�jpa-00214510�

JOURNAL DE PHYSIQUE

Colloque C 1, supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 253

STRAY FIELDS AND BITTER PATTERN FORMATION ON FERROMAGNETS

by K. BRAY

(*)

and P. RHODES Physics Department, University of Leeds, England

R&um6. - Des formules theoriques sont donnees pour les champs associBs B plusieurs structures des domaines magnbtiques. Elles sont employkes pour une investigation de la formation des figures de Bitter associees a ces structures, et des effets des champs appliquBs aux figures. Les resultats servent

a

donner une interpretation de certains traits singuliers des figures dkcouvertes dans le ferrite de baryum, et dans le fer-silicium et le cobalt en presence de champs appliquks

a

ces substances.

Abstract.

-

Theoretical expressions are given for the fields associated with

a

range of magnetic domain structures.

These are used

in

an investigation of the formation of Bitter patterns associated with such structures and of the effects of applied fields on the patterns. The results are used to provide interpretations of some unusual features of patterns repor- ted on barium ferrite, silicon-iron and cobalt

in

applied fields.

I. Introduction.

-

The stray fields associated with magnetic domains may be ascribed (i) to regions in which the normal component of magnetization, I,,, changes discontinuously, giving rise to surface

cr

magnetic charge

D,

and (ii) to regions in which div I

# 0,

giving rise to volume

⁽⁽

magnetic charge

⁾⁾

(see, for example, Rhodes et al. [I]). In the present work only surface magnetic charge is considered.

The stray fields derived from such ((charges

⁾⁾

play an important part in several of the methods used in the observation of domain structures (see, for exam- ple, Carey and Isaac

[Z]).

In general, it is difficult to calculate these stray fields, but a system of charges is considered berow, which corresponds in appropriate cases to systems of physical interest, and for which closed-form expressions for the fields may be derived.

The results of this treatment have made possible a more detailed analysis of the formation of Bitter patterns than has been given before (8 111).

11. Stray 'Fields. - The surface distribution of charge,

o,

to be considered is shown in figure 1. For

FIG. 1.

- Illustrating the surface

c(

magnetic charge

⁾⁾ distri-

bution.

In

the

cc

charged

>>

regions

the

surface density is

f 10,

and is assumed to extend over

the

range

- d y

< +

^a.

S

⁼

W i t corresponds to the

<<

simple plate

^)>

domain structure in a uniaxial material with magnetization normal to the surface. For W 4

S,

the magnetization is parallel to the surface, and the

⁽⁽

magnetic charge

⁾⁾

approximates to that due to the Bloch walls.

(*)

Now at

Trinity

and

All

Saints' Colleges, Horsforth, Leeds.

By expressing

o as a Fourier series and solving

Laplace's equation, subject to the boundary condi- tion -

(dV/dz) =

2 no at

z =

0, the potential V(x, z ) may also be obtained as a Fourier series. Then, using the relation H

=

- grad V, series are obtained for

H,

and Hz

;

fortunately, these latter can be summed explicitly, to give

1 sin a sin 8

- 2 10 H,(x,

ⁱ⁾ =

tan-' (-) ^sinh

^y

^+tan-' (-1, ^sinh

^y

^(2.2)

where

The variations of H,,

Hz and the total field

as functions of

= x/

W are shown for various values of

=

z/W in figures 2a and 26, for the particular case S

=

W. It can be seen from figure 2b that one of the main effects of an applied field Hap, in the

+ Oz direction is to shift the maxima in HT from

5

⁼ 0

to higher values of t.

111. Formation of Bitter Patterns.

-

Following the Boltzmann distribution argument of Kittel [3], earlier discussions of the formation of Bitter patterns have been largely qualitative, and assumed that colloid particles are attracted to positions of maxima in the field. However, it seems important to take into account the fact that particles reaching the surface of the material will have any further motion impeded, and hence the Boltzmann distribution may not be attained. In order to give a more quantitative treat- ment the following model has been used. An initially spatially uniform distribution of spherical single- domain particles suspended in a viscous medium has been assumed, and the motion of the particles under

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971183

K. BRAY AND P. RHODES

(a) Field components with zero applied field. (b) Magnitudes of the total field ,HT. Full Full curves : (Hx/Zo). Broken curves : (H,/Zo). curves : zero applied field. Broken curves :

applied field nlo in the 4 Oz direction.

FIG, 2.

-

Magnetic fields as functions of = x/W at various heights above the charge plane shown in @ure 1, for the case S =

w.

The numbers on the curves show the values of 5 = z/ W.

(a) Zero applied field. (b) Applied field xZo in the

+

Oz direction.

FIG. 3.

-

Histograms showing the calculated relative density of colloid on the basal plane for a ^cc simple plate >> domain structure (S = W).

the action of the stray and applied fields calculated numerically from the following equation of motion ..

mr

=

vI, grad I H I - ⁶

^narl;!

(3.1) where m,

^Q,

v and 1, are the mass, radius, volume and magnetization of a particle, and q is the viscosity of the medium. It was found that for typical values of the constants involved the term on the left-hand side of (3.1) could be neglected. Figures 3 and 4 show the relative numbers of particles calculated as reaching the surface.

Figure 3a shows colloid collecting most densely at the boundary for zero applied field ; while figure 3b, corresponding to a field of

n&

applied normally, indicates that although there is still a local maximum at the boundary there is also a proncrunceil rnaxifnum at the domain centre. This result is in good agreement

FIG. 4.

-

Histogram showing the calculated relative density of colloid on the axial plane for a <<simple plate >> domain structure (S

-

W) with an applied, field 4 In in the

-

Ox direc- tion, where In is the component of magnetization normal to

the surface in the absence of an applied field.

STRAY FIELDS AND BITTER PATTERN FORMATION ON FERROMAGNETS C1-255

with some experimental observations of Craik

141

on patterns on the basal plane of barium ferrite.

It may be noted that use of the Boltzmann distribu- tion in conjunction with the fields shown in figure 2b would not lead to an interpretation of this unusual feature of the patterns.

Figure 4 indicates that a field applied in the - Ox direction causes the colloid density to be drastically reduced at alternate domain bounda~ies (e. g. at

t

^{= 1.0)}

and enhanced at the intervening boundaries

(5

=

0.0 and 2.0). The

<<

disappearance >> of alternate

boundaries on the application of such a field has been observed, for example, in Fe-Si and Co by Bates and Neale

[5]

and by Nee [6]. Inspection of the stray fields arising from the

^(<

magnetic charges

^>)

associated with boundaries, calculated from (2.1) and (2.2) with W + S, shows that this effect cannot be inter- preted in terms of these fields. Hence it must be con- cluded that there are non-zero magnetization compo- nents normal to the so-called

⁽⁽

axial planes >>, and that these are producing fields similar in form to those considered above for basal planes.

References

[I]

RHODES

(P.),

ROWLANDS

(G.)

and BIRCHALL

@.

R.),

^[4] CRAIK @. J.), Brit. J . Appl. Phys., 1966, 17, 873.

J. Phys. Soc. Japan, 1962, 17,

B-1,

543. [5]

BATES

(L. F.)

and NEALE

(I?. E.), Proc. Phys. SOC., [2]

CAREY

(R.)

and

ISAAC (E. D.), Magnetlc

Domains

1950, A 63, 374.

(E.

U.

P.,

London),

1966. [6]

MEE

(C. D.), Proc. Phys. Soc., 1950, A 63, 922.

[3]

KIT~EL (C.),

Phys. Rev., 1949, 76, 1527.