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JOURNAL DE PHYSIQUE Colloque C 3, Suppl&ment au no 7-8, Tome 37, juillet-aotit 1966, page C 3-68 THE EFFECT OF DISLOCATIONS ON THE MAGNETIZATION CURVES

OF FERROMAGNETIC CRYSTALS

Institut fur Physik am Max-Planck-Institut fiir Metallforschung, and Institut fiir theoretische und angewandte Physik der Technischen

Hochschule Stuttgart, Stuttgart, Germany

RCsumC. - La principale interaction entre les dislocations et la direction d'aimantation des rerromagnetiques est due aux effets magneto-8astiques (magnCtostriction). Apr&s un bref histo- rique, on considkre les sujets suivants : I. Voisinage de la saturation magne'tique. Les mesures de

susceptibilite dans cette region donnent un outil puissant pour etudier l'arrangement des disloca- tions dans les cristaux

dCformCs. - 11. Interaction entre les parois des domaines et les dislocations.

La thCorie a CtC rkcemment menCe assez loin pour permettre le calcul des courbes d'aimantation et des comparaisons

dCtaillees avec l'experience. - 111. Influence des dislocations sur les processus rotationnels dans les champs faibles. Elle a CtC etudiCe rCcemment en d6tail pour les monocristaux de nickel

< 100 > , et a donnC des renseignements complementaires sur la densite et l'arrangement des dislocations.

Abstract. - The principal interaction between dislocations and the direction of magnetization in ferromagnets is due to the magnetoelastic (magnetostrictive) effect. After a brief historical survey, the following subjects are considered

: I. The approach to ferromagnetic saturation. Susceptibility measurements in this range constitute a powerful tool for studying the dislocation arrangement in deformed crystals.

- 11. The interaction between domain walls anddislocations. The theory has recently been carried far enough to permit the calculation of magnetization curves and detailed comparisons with experiments.

-111. The efect of dislocations on rotationalprocesses in low applied fields. This

has recently been studied in detail for < 100 > -nickel single crystals and has given further infor- mation on dislocation density and arrangement.

1. General Introduction. - The present paper deals

with the interaction between dislocations and magne- tization, or, speaking from a macroscopic viewpoint, with the effect of plastic deformation on the magneti- zation curve of ferromagnets. It has been known for a long time that mechanically hard ferromagnetic metals are also (( magnetically hard )), i. e., that they have a lower initial permeability and a higher coercive force than well annealed, mechanically soft materials. This means that the presence of internal stresses impedes the magnetization processes in ferromagnets.

The breakthrough towards a quantitative under-

standing of the relation between mechanical stresses and the magnetization processes was achieved in the

1930's through the researches of

Becker and his colla-

borators, which culminated in the famous book by Becker and Dijring ct Ferromagnetismus )) [l]. The classical example for demonstrating the effect of mechanical stresses is polycrystalline nickel under

mechanical tension. The volume density of the free energy of the magnetoelastic interaction between a

tensile stress o and the spontaneous magnetization J, is given for a polycrystalline specimen by Here

8 is the angle between the tensile axis and the

direction of magnetization, il the constant of magneto- striction, which for nickel is negative and has the room temperature value

A = - 3 X 10-5. The negative sign

of the magnetostriction constant has the consequence that in a stretched nickel wire the magnetization vector

J tends to be perpendicular to the axis of the

wire.

A gradually increasing magnetic field H along

the axis of the wire has to rotate the direction of the magnetization

J against the torque exerted by the

magnetostriction. The volume density of the free energy of the inter- action between magnetization J and external field H, Article published online by THE EFFECT OF DZSLOCATIONS ON THE MAGNETIZATION CURVES C 3 - 69 the so called magnetostatic energy is given by

If only the magnetostrictive energy

@, and the magne- tostatic energy CD, are taken into account, the equation of the magnetization curve, i. e. the relation between the magnetization component

JH parallel to the field

and the field

H, can be found by minimizing

with respect to the angle

8. The result for the suscepti-

bility X is As shown in figure 1, for sufficiently large stresses the relation (4) is indeed fulfilled, apart from the hystere- RG. 1. - The effect of tensile stresses o of various magnitu- des on the room temperature magnetization curve of nickel wire (adapted from figure

62 of Becker and Doring [l]).

tic effects near the origin of the magnetization curve. The explanation of these effects requires to take into account phenomena which have been neglected in the preceding treatment.

The tensile stresses required to modify the

room- temperature magnetization curve of nickel thoroughly are of the order of

10-' to 10-4 of Young's modulus,

i. e. small compared with the stresses in the immediate environment of dislocations, and comparable with the flow-stress of moderably deformed nickel single crystals. This means that we may expect a strong influence of plastic deformation on the magnetization processes in nickel single crystals. By the presence of dislocations the magnetization is locally forced into

magnetoelastically preferred directions. This impedes both the rotation of the magnetization into the direc-

tion of the magnetic field and the motion of the ferro- magnetic domain walls. Before discussing the influence of the dislocations on magnetization processes in more detail, we should like to make some historical remarks. The develop- ments of the understanding of the magnetization of ferromagnets and of the plastic deformation of metal single crystals show some interesting parallels. Ferromagnets show the rather striking property that comparatively small applied magnetic fields may increase the magnetization enormously, compared with paramagnetic or diamagnetic substances. This was explained by Pierre Weiss [2] as being due to the existence of spontaneously magnetized domains, whose directions of magnetization may be rotated under the influence of the torques exerted by small applied fields. Pierre Weiss conceived the non-magnetic state of a ferromagnet of macroscopic dimensions as consisting of (c grains with a statistical distri- bution of magnetization directions. We know now that in the non-magnetic state of large ferromagnetic single crystals the magnetization vector is aligned almost everywhere in one of a few crystallographic directions of (( easy magnetization D, the socalled (c crystallographically preferred )) directions. The regions, in which the magnetization lies in one of these crystallographically preferred directions are called (( domains D. The gradual transition between the magnetization direction in one domain and that in a neighbouring domain takes place in a c( domain- wall )) or Bloch wall. The width of the domain walls varies from material to material and is temperature dependent, usually in the sense that it decreases with decreasing temperature.

A typical order of magnitude

for the domain wall width is lops cm. The magnetiza- tion process may occur either by the displacement of domain walls, so that the favourably orientated do- mains grow at the expense of less favourably orienta- ted domains, or by the rotation of the magnetization vector inside the domains (or, of course, by a combi- nation of both processes).

The preceding concepts where arrived at mainly by

theoretical studies. The Bitter technique (see, e. g., [3]), which was extended and brought to perfection after

World War

11, enabled the domain walls to be seen

and be studied directly. The theoretical ideas on domain configurations and domain movements res- ponsible for the various features of the magnetization curve could be checked, and further theoretical studies were suggested. During this period the understanding of the magnetization curves in terms of domain walls

C3-70 A. SEEGER

was well ahead of the interpretation of the stress- strain curves of metal single crystals in terms of dislo- cation arrangements and movements. As is well known, dislocations had entered solid state physics as a theoretical concept designed to account for the strikingly low critical shear stress of metal crystals. It took quite some time until dislocation theories, e. g. of work-hardening, could be related to observations on a microscopic scale. The first really powerful technique in this respect were electron- microscopic slip line studies. More recently, transrnis- sion electron microscopy and X-ray techniques are making contributions of growing importance and are influencing the development of dislocation theory. Related techniques have also been applied to domain studies, but they have so far proved less powerful. It appears that at the present time the development in the dislocation field has overtaken the field of ferro- magnetic domains, thus reversing the situation as of a few years ago. Nevertheless, with regard to the basic questions and the methods of research, in particular the interrelation between experiment and theory, the two fields are very similar, and experiments in one field may help in the other.

A particularly interesting

domain of study is of course the interaction between the two fields, i. e. the effect of dislocation processes on the magnetization processes. This field has been studied quite intensively in recent years at Stuttgart, and the present paper intends to outline some of this research. The quantitative studies of the effect of dislocations on the magnetization have sofar been based on the formulation of the magnetoelastic elastic interactions given by

Becker and Doring [l], sometimes called

the Voigt approximation [4]. This approximation consists of assuming that the elastic stress field of the dislocation remains unaffected by changes in the direction of magnetization.

W. F. Brown, jr., working

as a visiting scientist at Stuttgart, has recently ~i-xn an exact formulation of the problem, avoiding the

Voigt approximation

[4, 5, 6,7]. The ensuing theory is rather involved, and applications have not yet been made. It can be shown, however, that for the applica- tions touched upon in this report, the additional terms are indeed negligible.

2. The Approach to Ferromagnetic Saturation. -

W. F. Brown, jr., was the first who studied quantitati- vely the effect of crystal dislocations on magnetization processes. He was concerned with the region of the magnetization curve near saturation, where the magne- tization has almost the direction of the magnetic

field, and where a further increase of the magnetization component parallel to the applied field occurs by the

rotation of the magnetization against the magneto- strictive action of internal stresses. With remarkable foresight, he attacked the problem in a way that still, twenty-five years later, forms the basis of the present work in this field [8]. A number of applications to more recent problems have been made. Using the theory of continuous distributions of dislocations,

Seeger and Kronmiiller

[9, 101 were able to generalize Brown's theory in such a way that for a given distri- bution of dislocations the magnetic susceptibility in the approach to saturation may be calculated in a straightforward way.

The theory may be briefly

outli,ned as follows : The spatial variation of the direction of magnetization is described by a set of non-linear partial differential equations, the socalled micromagnetic equations or (( Brown's equations D. These equations allow for 1) exchange interaction between neighbouring spins, which tends to align these spins parallel,

2) the magne-

tocrystalline anisotropy, which tends to force the magnetization into the above mentioned crystallogra- phically preferred directions, 3) the magnetostatic stray-fields associated with a divergence of the magne- tization vector,

4) the magnetostatic energy of the

magnetization in the applied fieldquotesdbs_dbs19.pdfusesText_25

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