THE EFFECT OF DISLOCATIONS ON THE MAGNETIZATION CURVES OF FERROMAGNETIC CRYSTALS

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THE EFFECT OF DISLOCATIONS ON THE

MAGNETIZATION CURVES OF FERROMAGNETIC

CRYSTALS

A. Seeger

To cite this version:

A. Seeger. THE EFFECT OF DISLOCATIONS ON THE MAGNETIZATION CURVES OF

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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 dislocations 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 information on dislocation density and arrangement.

1 . General Introduction. - The present paper deals with the interaction between dislocations and magnetization, or, speaking from a macroscopic viewpoint, with the effect of plastic deformation on the magnetization 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 understanding 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 magnetostriction, 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 interaction between magnetization J and external field H,

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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 magnetostatic 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 susceptibility

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 direction of the magnetic field and the motion of the ferromagnetic 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 distribution 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 l o p s cm. The magnetiza- tion process may occur either by the displacement of domain walls, so that the favourably orientated domains 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]),

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C 3 - 7 0 A. SEEGER

was well ahead of the interpretation of the stress- strain curves of metal single crystals in terms of dislocation 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 ferromagnetic 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 - x n 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 applications 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 magnetization 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 magnetostrictive 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 distribution 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 magnetocrystalline anisotropy, which tends to force the magnetization into the above mentioned crystallographically preferred directions, 3) the magnetostatic stray-fields associated with a divergence of the magnetization vector, 4) the magnetostatic energy of the magnetization in the applied field H, and, last but not least, 5) the magnetoelastic interactions between magnetization and internal stresses. Near saturation the magnetization is approximately parallel to the applied field ; local deviations from this direction may be treated as small quantities, so that the resulting equations may be linearized. These linearized equations may be solved by the method of Fourier transforms. The result is an integral for the susceptibility that may be written down for any dislocation arrangement. Since not only the applied field, but also the rather large magnetostatic energy associated with the stray-fields and the exchange interaction tend to align the magnetization, the approximations on which the linearization of the micromagnetic equations is based is quite good over the whole range of the approach to saturation.

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THE EFFECT O F DISLOCATIONS ON THE MAGNETIZATION CURVES C 3 - 7 1

axis. This can be done by two methods : In Stuttgart we are attempting to measure the approach to saturation in different crystallographic directions. A group at Jena will try to measure the magnetic anisotropy induced by the presence of dislocations.

The superiority of the present method over other non-destructive techniques for studying the disloca- tion arrangement is due to the following. The stresses and strains near the dislocation cores may become very high and thus usually dominate the effect of dislocations on macroscopic physical properties. These stresses and strains, however, are only slightly affected by the dislocation arrangement. In remarkable contrast to this, the various above mentioned interactions that tend to align the magnetization prevent the magnetization from following the rapid variations of the stresses near the dislocation core. This is illustrated in figure 2.

FIG. 2. -The distribution of the magnetization direction around an edge dislocation in an elastically isotropic medium in a uniform magnetic field H. The length scale is indicated by the bar denoting the exchange length K,;' (see sect. 4). The angles between the field direction (parallel to the Burgers vector) and the magnetization direction were calculated for H = 1000

mrsted and have been drawn five times enlarged. The sectors of positive and negative deviations from the field direction are indicated by f signs.

It may be said that the magnetization ignores the existence of the dislocation core. The main effect of a dislocation on the magnetization takes place of the order of several hundred Angstroms away from the dislocation line. In this region, however, the magnitude of the stresses is sensitive to the dislocation arrange-

ment. To give an example, the effect of a pair of dislocations of the same sign is quite different from that of a pair of opposite sign. Numerical examples have been discussed elsewhere [10, 121 and show that diffe- rent arrangements of a given dislocation density may give rise to order of magnitude differences in the magnetic susceptibility. Specifically, it has been shown that the measurements on stage I1 of plastically deformed f. c. c. single crystals cannot be explained without the existence of rather strong long-range stress fields of magnitude comparable with the macroscopic flow stress. For years this conclusion has been questioned by some of the people studying dislocation arrangements in cold-worked metals by transmission electron microscopy. However, it has recently received a striking confirmation by the transmission electron microscopy studies of Essmann on Cu single crystals [l31 and of Ramsteiner on Au single crystals 1141.

It is a further advantage of the approach-to-saturation measurements that the scale at which the dislocation arrangement is probed can be changed by measu- ring the susceptibility at different magnetic field strengths. This permits sensitive tests of the dislocation models assumed. The unit of this scale is inversely proportional to the field strength (see sect. 4, eq. 6a). The smaller the applied field, the larger are the separa- tions over which the dislocation arrangement can be studied. Unfortunately, the method is generally applicable only down to field strengths at which domain walls begin to appear. In special cases it is nevertheless possible to perform analogous measurements in smaller effective fields. An example will be treated below (see sect. 4).

The deviations of the magnetization direction from the alignment parallel to the applied field that are caused by the presence of dislocations may be studied more directly by the diffraction of (polarized or unpola- rized) cold neutrons. The theory has been worked out in some detail by Kronmiiller, Seeger and Wilkens [l51 and verified by recent experiments by Stork [l61 on plastically deformed polycrystalline nickel. Without doubt this technique, when applied to single crystals, will become apowerful tool for studying the dislocation arrangements, too.

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C 3 - 7 2 A. SEEGER

structure is usually rather complex and depends on the mode of demagnetization, the size of the crystal, and the orientation of the crystal surfaces.

Particularly simple examples of domain structures as revealed by the transmission electron microscopy of thin single crystal foils of cobalt are shown in figure 3 and 4 [17, 181. At room temperature (to be precise, below 518 OK) the directions of easy magnetization of cobalt are parallel to the hexagonal axis. Cobalt belongs thus to the magnetically uniaxial materials. Adjacent domains are alternately magnetized in the two directions perpendicular to the basal plane. As shown in detail by Wilkens [19], the Lorentz force on the electrons due to the uniform magnetization within the domains acts to a rather good approximation like a uniform elastic curvature of the crystal, giving rise to the bright-dark contrast of the domains. (In figures 3 and 4, each of the bright or dark bands is the image of a domain). This contrast mechanism is distinct from the older methods [20, 211 for ima- ging ferromagnetic domains by electron-microscopy,

FIG. 4. -Transmission electron micrograph of a cobalt single crystal deformed on the basal plane to a resolved shear strain a = 0.03, showing ferromagnetic domains (faint back- ground line pattern) and dislocation lines (strong lines).

which required changes in the electron optics in the microscope. The Thieringer-Wilkens method is particularly advantageous if the interaction between the domain walls and the dislocations (black lines in figures 3 and 4) are to be studied, since the latter remain in focus.

In general, the magnetization process occurs by both the displacement of domain walls and by the rotation of the magnetization within the domain walls. The reliable treatment of the interaction between a domain wall and a dislocation is quite a formidable problem. Satisfactory solutions have been recently obtained by K.-H. Pfeffer [22], but we are unable to discuss them in any detail here. An important distinc- tion arises, depending on whether the domain walls belong to group I or group II (a nomenclature due to

_-

>

~ i e d e r [23]). &oup

fi

comprises those domain walls FIG. 3. - Transmission electron micrograph of an undefor-

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THE EFFECT OF DISLOCATIONS ON THE MAGNETIZATION CURVES C 3 - 7 3 tion of the domain-walls of group I extend also over the

adjacent domains, assuming a constant (finite) value for at least one component of the extra-stress tensor. The extra-stresses accompanying inhomogeneities in the spacial distribution of the magnetization are respon- sible for the magnetostrictive interaction between the domain-walls and the dislocations. (The force on the latter can be calculated by using the well-known Peach-Koehler formula. This has been done for a number of typical examples [24, 25, 261).

It is obvious that the displacement of the walls of group I should be much more impeded by the presence of dislocations than that of walls of group 11, since the walls of the first group interact with the dislocations in the neighbouring domains, whereas those of the second group have a strong interaction only with the much smaller number of dislocations within the wall. This prediction has been verified by Trauble in experiments on iron-silicon crystals (271. In the application of the preceding ideas on the interaction between dislocations and domain walls to the quantitative calculation of the effect of dislocation on the magnetization curves of ferromagnetic single crystals, one has to keep in mind the following. Even if the movement of a domain wall during the magnetization process may be considered to be independent of the other domain walls, it is nevertheless simultane- ously influenced by the interaction of the wall with a large number of dislocations. This requires a statistical treatment of the forces experienced by a domain wall during its passage through the crystal. Trauble [25, 26, 281 has developed the model of uncoupled displacements of domain walls to the extent that the magnetization curve, including magnetic hysteresis and Barkhausen jumps, may be treated theoretically. The comparison with experiment was mainly carried out on the initial susceptibility

X,

and the coercive field H,. Among the parameters characterizing the magnetization curve these two quantities have been investigated experimentally in most detail.

In the multiaxial ferromagnetic crystals of nickel and iron, the variation of

X,

and H, with temperature and amount of plastic deformation is found to be well accounted for by the model of uncoupled wall displacement up to temperatures at which the magnetic anisotropy due to magnetoelastic effects becomes comparable with the magnetocrystalline anisotropy 124, 25, 271. At such temperatures one has to take into account the simultaneous occurrence of magnetization processes by rotation of the magnetization direction within the domains and by domain wall displacements.

In the magnetically uniaxial CO crystals the combi- ned magnetization processes c( rotations plus wall displacements occur at all temperatures. Taking advantage of the fact that the domain structure of such crystals is comparatively simple, a rather complete experimental and theoretical study of the effect of tensile plastic deformation on the magnetization processes in CO single crystals has been carried out. The agreement between experiment and theory is quite good. The reader is referred to the literature for a detailed account [29 to 341.

4. The Effect of Dislocations on Magnetization by Rotation at Small Fields. - In sect. 3 it has been said that the initial susceptibility

X,

of a multiaxial ferromagnetic single crystal can be explained theoretically by the displacement of domain walls, provided the anisotropy due to magnetoelastic effects is small compared with the magnetocrystalline anisotropy. Let us consider what happens if at a given temperature the dislocation density N is increased, e. g. by plastic deformation. In the absence of dislocations or other crystal imperfections, the ferromagnetic domains (outside the domain walls) are uniformly magnetized in one of the (( easy directions I), e. g., in nickel in one of the eight

<

111

>

directions. Near dislocations, however, the magnetization points locally in directions that are determined by a compromise between the magnetoelastic interaction with the dislocations and the crystalline anisotropy. The situation is analogous to that discussed in sect. 2 (compare Fig. 2) with the modification that the magnetization is approximately parallel to one of the easy directions, and not to the applied field. If a small magnetic field is applied, it is relatively easy to rotate the magnetization in the environment of the dislocations, giving rise to a contribution to the magnetic susceptibility. The mathe- matical techniques mentioned in sect. 2 are applicable and lead for a cubic crystal to an effective constant of anisotropy due to the magnetoelastic interaction of the form

In eq. (5) G denotes the shear modulus, b the disloca- tion strength, K, the first constant of crystalline anisotropy

*,

and c a function of the magnetostrictive cons- (*) Definitions : The volume density of the free magnetocrystalline energy of a cubic ferromagnet is given by

@I

= K,(Y: Y:

+

Y: Y:

+

Y: Y:)

+

K, Y: Y: Y:

+

...

@a,

where y, are the direction cosines of the magnetization vector J.

The volume density of the free exchange energy is

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C 3 - 7 4 A. SEEGER tants A,,, and A,

,,,

of the dislocation character and

arrangement, and of the ratios of the elasticconstants of the crystal. If the dislocations are sufficiently far apart from each other, for nickel at room temperature c is typically of the order of magnitude 1 0 - ~ . For the reasons discussed in sect. 2 in connection with the approach to saturation, the influence of the disloca- tion arrangement on the magnitude of c may be rather pronounced. In the case of the approach to saturation the scale for probing the dislocation arrangement is given by the socalled exchange length of the applied field H

in the present case its place is taken by the socalled exchange length of the magnetocrystalline anisotropy = (3 A/2

I

K t

1)

'l2. _(6b)

In nickel at room temperature the experimental values are for the anisotropy constant

K, =

-

4 . 5 X 104 erg/cm3,

the exchange constant A = 8.6 X 10-7 erglcm, and

the spontaneous magnetization J, = 485 Gauss. Inser- ting these values into eq. 6 gives us for the exchange length of the magnetocrystalline energy at room temperature

x i 1 = 535A, (7a)

and for the equivalent field strength (i. e., the field leading to = K,)

H, = 124 Oersteds . _(7b)

In practice, the measurements of the susceptibility in the range of the approach to saturation cannot be 'extended to field strengths as low as H,. At such low field strengths the basic assumption of the theory outlined in sect. 2 (that the crystal consists of one ferromagnetic domain only) breaks down. The deter- mination of K, through measurements of the contribution of the rotation processes to the reversible susceptibility at small applied fields (e. g., the initial susceptibility

X,)

provides thus in principle a method of employing larger exchange lengths than can be obtained by approach-to-saturation measurements. Raising the temperature above room temperature decreases

1

K,

1

and permits K,' to be extended to higher values.

While the presence of dislocations allows rotational processes to contribute to the reversible susceptibility at small fields, it impedes the motion of domain walls. As Kronmiiller [24] has shown in detail, the existence of these two effects due to the dislocations leads'to a

non-zero contribution of the rotational processes even under conditions under which in an ideal crystal the magnetization process would occur by domain wall displacements only. A particularly interesting case, which has been treated theoretically by Kronmul- ler [24] and investigated experimentally by Koster and Kronmuller [35] is that of a rod-shaped nickel single crystal with crystal axis parallel to the [l001 direction. In the demagnetized state of such a [loo]-crystal the eight easy directions

<

111

>

occur with equal probability [36]. The fractions of the total volume occupied by each of these eight c( phases are called (( phase volumina )) and are denoted by v,,

...,

v,. If a finite field H is applied parallel to the [l001 direction, the magneto-static energies of the four phases with positive [l001 components of the magnetization vector are all equal, and so are those of the four phases with negative [l001 components. It is therefore expedient to introduce cc super-phases )) representing these two groups of magnetization directions, and to characte- rize them by ct super-phase volumina )) v + and v- with

The increase in the average magnetization parallel to the applied field, JH, takes place by both displacements of 1800 walls, described by AV, and by rotations of the magnetization vectors inside the domains. These rotations are best characterized by the angle a between the average direction of the magnetization vector in each phase and the < 11 1

>

direction of the magnetization vector of the phase in zero field. This angle must be the same for all phases in order to avoid the high magnetostatic energy associated with strayfields at the domain walls. The energy (per unit volume) necessary for the displacement of the domain walls against the interaction with the dislocations is written for small displacements as

R is to be calculated by the statistical treatment outlined in sect. 3.

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THE EFFECT O F DISLOCATIONS ON THE MAGNETIZATION CURVES C 3 - 7 5 walls and the dislocations, and a term contributing to

the effective anisotropy constant KO (eq. 5), taking account of the local rotations of the magnetization vector in the stress field of the dislocations inside the domains. It is characteristic of this phase theory that the energy of and the volume occupied by the domain walls do not appear in the total energy expres- sion ; both are neglected compared with the other terms.

By minimizing the total energy with respect to variations of a and AV, one obtains [24, 351 for the angle of rotation

1 J, H sin a =

-

--- ,

J6

Keff

for the difference of the super-phase volumina

and for the initial susceptibility

Here

2 K,,, = K ,

+

-

I

K1

I

3 (13)

is an effective constant of anisotropy taking into account both the magnetoelastic and the magnetocrystalline anisotropy. Eq. (10) to (12) are valid for sufficiently small fields H.

Eq. (12) is similar to eq. (4). It is more general than that equation in that a) it applies to single crystals, b) it takes into account inhomogeneous (internal) stresses, c) it allows for the crystalline anisotropy and is therefore applicable at lower stresses than eq. (5). From the point of view of utiiizing the magnetic measurements for the investigation of the dislocation arrangement, one wishes to derive from experiments the parameters K,, which according to theory should be proportional to the dislocation density N, and R, for which theory predicts a proportionality to N1I2 [24,25]. According to eq. (10) and (1 l), the determina- tion of K, and R is equivalent to obtaining sina and AV. These quantities are related to the reIative length change I due to the magnetization (i. e. the magnetostriction) and the average magnetization JH by the equations [24, 351

I = A,,,

(k

sin a

+

A v

$

cos a sin

1

a (14)

If measurements of 2. and JH are available, eqs (14) and (15) may be solved for sin a and AV. The physical basis for this analysis is the fact that displacements of 1800 degree walls do not contribute to the magnetostriction, but only to the average magnetization, whereas the rotation of the magnetization vector contri- butes to both quantities.

Figure 5 shows the results of measurements [35] at 60 OC on a [l001 Ni single crystal deformed at room

internal magnet& field H

FIG. 5. -Difference between super-phase volumina Au (full lines) and sin a (scale 10 times enlarged, dashed lines) for a nickel [l001 single crystal as a function of magnetic field H for various flow-stresses z (room temperature deformation ;

T O = resolved critical shear stress). Measurements at 60 OC.

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C 3 - 7 6 A. SEEGER , erg ' U ' z 3 12 -

.

0.6 to

-

Q = R/K#, 8 - 0.4 U 2 6 - d 4 0.2 2 o

i

3 1

i

7 k 8 0 flow stress r - zo 1

A2

FIG. 6. - Same crystal as in figure 5 ; K,ff and R as deduced

from figure 5 for H - + 0 as function of (r - 70). Q gives the ratio of the contributions of rotational processes and domain wall displacements to the initial susceptibility at 60 O C . deformation is proportional to (2-2,)'12. I t should be

noted that the ratio

of the rotational processes to the magnetization processes by domain wall displacements passes through a maximum as a function of the amount of plastic deformation. This result had been predicted by Kron- miiller [24].

We have seen that the theoretical predictions on the field and flow-stress dependence of K, and R are borne out well by the experiments. On account of the sensitivity of the function c to the dislocation arrangement, the absolute magnitude of K, may be used to derive information on the dislocation arrangement in an analogous way to the differential susceptibility in the range of the approach to saturation (see sect. 2). Since K, depends strongly on temperature, the exchange length may be varied by varying the temperature and can be made to overlap with the exchange lengths available in the approach to saturation measurements. Nevertheless, the information obtainable from the two types of experiments is different in detail, since in the measurements a t small fields small deviations of the magnetization vector from the < 11 1 >

-

directions are used to probe the dislocation arrangement, whereas in the high field measurements small deviations from the [l001

-

direction are employed. A quantitative comparison of the results of the two techniques is therefore not possible without a detailed theoretical analysis. This is in the process of being carried out by A. Holz a t Stuttgart. The results sofar available show that the

absolute magnitude of K, cannot be accounted for without the existence of long-range stress fields in stage I1 of the work-hardening curve, in agreement with the conclusions from the approach to saturation measurements.

Acknowledgment.

-

The author is grateful to the large number of coworkers that have contributed to the results reviewed in this paper, above all Dr. Hel- mut Kronmiiller. The support of the Deutsche Fors- chungsgemeinschaft of some of this work is grate- fully acknowledged.

[l] BECKER (R.) and DORING (W.), Ferromagnetismus, Julius Springer, Berlin 1939.

[2] WEISS (P.), J. Phys. Rad., 1907, 6, 661. For the history of the subject, see Colloyue National de Magnt- tisme commtmoratif de l'euvre de Pierre Weiss (Centre National de la Recherche Scientifique), Paris 1958, in particular the contributions by

L. NCel (p. l), and Ch. Guillaud and R. Vaudier (P. 25).

[3] CRAIK (D. J.) and TEBBLE (R. S.), Magnetic Domains, in : Report on Progress in Physics, 1961, 24, 116. [4] BROWN, Jr (W. F.), Micromagnetics (Interscience Tracts on Physics and Astronomy), Interscience, New York 1963.

[S] BROWN, Jr (W. F.), Proc. Intern. Conf. on Magnetism, Nottingham 1964, p. 681.

[6] BROWN, Jr (W. F.), J. appl. Physics, 1965, 36, 994. [7] BROWN, Jr (W. F.), Magnetoelastic Interactions (Springer Tracts in Natural Philosophy, Vol. g), Berlin, Heidelberg, N. York 1966.

[8] BROWN, Jr (W. F.), Phys. Rev., 1941, 60, 139. [9] SEEGER (A.) and KRONMWLLER (H.), J. Phys. Chem.

Solids, 1960, 12, 298.

[l01 KRONMULLER (H.) and SEEGER (A.), J. Phys. Chem. Solids, 1961, 18, 93.

[l11 KRONMULLER (H.), Z . Physik, 1959, 154, 574. [l21 DE WIT (H. J.), Recovery of Magnetic, Electrical

and Mechanical Properties of Nickel and Iron after Plastic Deformation, Thesis, University of Amsterdam 1965.

[l31 ESSMANN (U.), Phys. stat. sol., 1965, 12, 723. [l41 RAMSTEINER (F.), Thesis, T. H. Stuttgart 1966,

Mater. Sci. Engineering, in press.

[l51 KRONMULLER (H.), SEEGER-(A.), and WILKENS (M.),

2. Phvsik. 1963. 171, 291.

[l61 STORK (A:), Thesis, T.

HI

Miinchen 1965.

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