Journal of Magnetism and Magnetic Materials 242–245 (2002) 116–124

Hysteresis models: non-linear magnetism on length scales from the atomistic to the macroscopic D.C. Jiles* Ames Laboratory, US Department of Energy and Department of Materials Science and Engineering and Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA

Abstract When materials are exposed to high levels of external magnetic field, the response of materials eventually extends beyond the simple linear regime. Under these conditions it is found that the change in response of the materials is no longer reversible. These phenomena are well documented in the experimental literature. However, the theoretical nonlinear description of the response of materials is only at a very early stage of development compared to with the low amplitude linear/reversible model theories. High performance computing is enabling researchers to model magnetic devices at smaller and smaller length scales, while at the same time accurate first principles calculations of magnetic properties now extend to systems involving thousands of atoms. The two different approaches, continuum versus discrete, are approaching each other at the mesoscopic length scales. The lecture will discuss some of the model theories that have been developed to describe the properties of magnetic materials. Beginning with first principles theories, the lecture will then trace the development of model theories at increasing length scales through micromagnetics, macroscopic Preisach, domain rotational models and phenomenological models of domain wall motion such as the stochastic process model. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Hysteresis

1. Development of model theories of hysteresis Hysteresis is a commonly occurring phenomenon in nature arising most often as a result of cooperative behaviour of a large number of identical interacting elements [1]. The most familiar examples occur in ferromagnetism, but similar behaviour occurs in ferroelectrics [2], ferroelastics [3] and fatigue. For a long time, theory and modelling of hysteresis in magnetic materials was a subject for the specialist investigator. However, in recent years the widespread, and increasing capability and accessibility of computers has made modelling available to a much wider range of investigators, so that this area of study has become of much wider interest. The magnetic modelling methods that are in use today are quite diverse and the choice of model depends crucially on the length scale of interest. Ultimately, these *Tel.: +1-515-294-9685.

models depend on the presence of magnetization (magnetic moment per unit volume) in certain materials. This magnetic moment per unit volume can be represented in terms of a net magnetic moment per atom, although in many cases, but not all, the magnetic moment is not actually localized on the atomic/ionic cores, but instead is caused by itinerant electrons. To understand how this net magnetic moment comes about theories were developed by Stoner [4] and Slater [5] concerning the electron band structure of magnetic materials, which explained why there is an imbalance of spin-up and spin-down electrons in ferromagnets. Band structure calculations provide a description of magnetism at a very fundamental level and much work is still being performed to refine and develop band structure calculations [6]. However, although these provide a basis for understanding magnetism on the scale of individual electrons and atoms, such theories are not central to this paper because they are not used for hysteresis modelling of materials.

0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 2 1 3 - 6

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2. Magnetism at the discrete level of individual atoms and beyond to the continuum level The principal idea of the Landau–Lifschitz–Gilbert (LLG) model is to describe the behaviour of individual discrete magnetic moments under the action of a magnetic field. Clearly from the classical times of Ampere, it was known that on the macroscopic scale a magnetic dipole moment of fixed magnitude would rotate under a magnetic field according to the equation t ¼ m0 m  H:

ð1Þ

This same concept, the general torque equation, was applied at the atomistic scale, or more correctly at the level of discrete magnetic moments of fixed magnitude but variable orientation, by Landau and Lifschitz [7]. This approach can be used to describe the behaviour of an individual magnetic moment. The behaviour of the entire material can then be determined by integrating the same process over the entire solid. The rate of change of magnetization with time then depends on the torque qM ¼ gr t: qt

ð2Þ

In the absence of damping a magnetic moment that is not initially aligned with the total field will precess around the field direction with a resonance frequency o0 ; which depends on the strength of the field and the magnitude of the moment o0 ¼ gr m0 H;

ð3Þ

where g is the gyromagnetic ratio of the magnetic moment. In the complete absence of damping, this precession will continue for infinite time. In practice, there is always some damping in solids and therefore for light damping (long time constant), there will be some precessional motion, and some rotational motion towards the field direction. The time taken to do this will depend on the damping coefficient. At high levels of damping (short time constant), the precessional component of the motion is suppressed because the moment reaches its final equilibrium orientation before precession has taken place. In a magnetic material there are interactions between the moments, so that magnetism in solids is a cooperative process. Therefore, the above equation must be modified when dealing with moments in a solid to take into account these interactions. Landau and Lifschitz suggested the following modification: qM 4pld ðM  M  HÞ; ¼ gt  M2 qt

ð4Þ

where the second term on the right hand side of the equation is a damping term which restrains the rotation of the moments under the action of a field. The existence of this damping term was, and remains, a hypothesis. There is no direct experimental verification, but the

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existence of such a term in bulk materials is reasonable from a theoretical standpoint. Twenty years after the original paper by Landau and Lifschitz, Gilbert wrote independently on the subject, from a continuum perspective, and developed a modified form of the equation [8]. The Gilbert form of the equation, as shown below, is used mostly for modelling today
 qM 4pld qM ¼ gm0 M  H þ ; ð5Þ M  qt qt gm0 M 2 In principle, the LLG model can be used either at the atomistic scale or at the ‘‘micromagnetic’’ continuum scale with the unit cells being much smaller than a single domain. The model is really directed towards volumes smaller than a single domain. Practical applications of the Landau–Lifschitz equation are made at length scales much greater than the atomistic. Typically, individual ‘‘moments’’ are defined at the level of 50 nm and each of these moments is treated as an indivisible unit. This can be justified if there is coherent rotation of the atomic moments occurring over the volume occupied by these moments. The model is suitable for time-dependent calculations of the reorganization of magnetic moments under the action of a field. It has the advantage over other models of not requiring the moments within a domain to be parallel, in fact it calculates the optimum orientations of moments, which in general are not necessarily parallel. The results of calculations based on the LLG model as shown in Fig. 1, can be averaged to provide hysteresis curves of materials. The model can be extended beyond the range of the single domain into simulations of multidomain specimens through the use of finite element

Fig. 1. Solution of the Landau–Lifschitz–Gilbert equation for an array of magnetic moments representing a thin film of permalloy of dimensions 2 by 0.5 mm. [After R.D. McMichael, http://www.ctcms.nist.gov/Brdm/mumag.org.html]. For the purposes of calculation each individual magnetic moment occupies an area of 0.05 mm  0.05 mm. The solution shows that the configuration of magnetic moments is not all parallel.

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methods. This approach is known as ‘‘computational micromagnetics’’ and enables both micro- and macroscale calculations to be incorporated into a single-model simulation [9,10]. This allows material microstructure to be included in the simulation and in this way it is possible to combine the advantage of exact theoretical equations developed for a small number of interacting magnetic moments with applications to practical materials.

3. Magnetism of domain rotation The Stoner–Wohlfarth model takes as its basis an array of single-domain particles, which can reorient their magnetization by coherent rotation of all moments within the domain. As such it takes no account of the details of the individual moments below the single domain in scale and unlike the LLG model it does not try to account for the orientations within the domain. Within domains, the model considers the competing effects of anisotropy and magnetic field on the orientation of moments. The domains themselves can have random alignments or they can be textured (meaning preferred orientation) or they can be even completely aligned in certain directions, although this would be a relatively trivial case requiring nothing more than a calculation of moment orientations within one singledomain particle. The model in its original formulation assumed that there were no magnetic interactions between the particles. In other words, the distribution of the particles was so dilute that each particle which was effectively isolated could not be influenced by the orientation of any other particles. This assumption can be changed, and has been changed by others, although including such interactions adds greatly to the computational complexity of the model. The model in its original condition also assumed axial anisotropy. This is the simplest type of anisotropic calculation to make. Other forms of anisotropy such as cubic can be included [11]. The basic idea of the model is to consider the reorientation of a magnetic moment within a singledomain particle in which the applied field is at some arbitrary angle to the anisotropic easy axis. In the case of the uniaxial easy axis along the field direction, this results in a bimodal switching behaviour with attendant coercive fields in the forward and reverse directions. In the other extreme case, where the magnetic field is applied along the anisotropic hard axis this results in a magnetization curve with no coercivity. In general, the domains will be oriented at an arbitrary angle relative to the field and such domains will have properties that lie between these two extremes (Fig. 2). A complete material may then consist of an assembly of domains, each at different angles to the field direction. The overall

Fig. 2. Rotation of the direction of magnetization Ms of a single-domain particle with uniaxial anisotropy, oriented with its easy axis HK at an arbitrary angle to the direction of the applied field H:

magnetization of the multi-domain sample is then the vector sum of all of the magnetic moments of the domains divided by the total volume. The turning force on a magnetic moment due to the applied field depends on the vector product of magnetic moment with magnetic field. The turning force on the magnetic moment due to anisotropy is the derivative of the energy with respect to angle. For a particle with uniaxial anisotropy, the equilibrium orientation is given by m0 HMs sin f  2K sin y cos y ¼ 0;

ð6Þ

where f is the angle of the magnetization relative to the field, K is the anisotropy coefficient and y is the angle of the magnetization relative to the magnetic easy axis. From the Stoner–Wohlfarth model, it is possible to calculate the saturation field Hs needed to rotate the magnetization of the most difficult domain oriented at 901 to the field direction 2K : ð7Þ Hs ¼ m0 Ms It is also possible to calculate the coercivity of the domains based on the switching field needed to reorient the domain aligned antiparallel to the field K : ð8Þ Hc ¼ m0 Ms The model has been widely used for describing the magnetic properties of materials on anisotropy and texture. In addition, many of the original ideas behind the model have been developed and extended and have found applications in fine particle systems. [12,13].

4. Magnetism of domain boundary motion Although in most cases, the boundaries of magnetic domains comprise only a small fraction of the total

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volume of a magnetic material, in multi-domain samples much of the magnetization change occurs in these regions and therefore it is of great interest to know and understand what changes are occurring at the boundary in order to predict and model the properties of such materials. The importance of domain boundary motion has been recognized by many investigators including Becker [14], Kersten [15,16], Neel [17,18], Globus [19–24] Escobar [25,26] and Bertotti [27,28]. The main idea in the treatment of domain boundary motion is to separate the motion into two components: reversible and irreversible. In most respects, this separation is somewhat artificial, since both processes take place together in multi-domain materials. However, the physics of the situation can be more easily analysed if these processes are separated. Irreversible processes necessarily cause energy dissipation and lead to coercivity and hysteresis while reversible processes do not. The motion of domain walls can further be divided into two types: flexible wall motion and rigid wall motion. The higher the surface energy of the domain walls the greater the tendency to move in planar fashion, particularly if the pinning sites that restrain the domain wall are weaker. On the other hand, domain walls with lower surface energy will tend to bend first, particularly if the pinning sites are stronger. In practice, components of both types of movement occur together as shown in Fig. 3. In the case of planar wall motion, much of the early modelling was developed by Becker, who was particularly concerned with the movement of domain walls interacting with regions of inhomogeneous strain, such as dislocations. Simplifying assumptions need to be made in order that an analytical solution can be obtained. Assuming a simplified periodically varying internal potential through which the domain walls move, an expression for the susceptibility for a material composed of 1801 domain walls can be derived in terms of the amplitude and wavelength of the periodic potential w¼

2Ac2 m0 Ms2 cos2 y ; 3p2 ls s0

ð9Þ

where y is the angle of the domain magnetization relative to the applied magnetic field, A is the cross sectional area of the wall, c is the periodicity of the internal potential with amplitude Emax ¼ 3ls s0 =2 due to periodically

Fig. 3. Flexible movement and translation of domain walls according to the Globus model.

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varying stress of amplitude s0 in a material of magnetostriction ls : The critical field, or local coercivity occurs when the domain wall moves over one of the local maxima in the slope of the potential and is therefore given by Hc ¼

Emax p : cm0 Ms cos y

ð10Þ

Models have been developed by N!eel and Kersten in the case of domain wall bending. Kersten in particular was concerned with the movement of domain walls that were pinned by ‘‘inclusions’’, which meant simply inhomogeneous volumes within the material. In this case, equations can be derived for the initial susceptibility win for small deformation of the domain wall win ¼

m0 Ms2 l 3 h ; 3g

ð11Þ

where l is the spacing between the pinning sites and h is the length of the section of domain wall as shown. And for the local coercivity or critical field Hc above which the domain wall will break away from the pining site Hc ¼

g cos fcrit ; m0 Ms lðcos y1  cos y2 Þ

ð12Þ

where y1 and y2 are the angles of the magnetization relative to the magnetic field on either side of the domain wall. In the case of ferrites in which the defects are mostly confined to grain boundaries, the domain walls will be pinned principally at the grain boundaries and therefore the behaviour can be modelled assuming that the domain walls move simply by bending like an elastic membrane as described by Globus [19]. The Globus model depends on equations for deformation of the domain walls similar to those of Kersten. For the purposes of modelling, the grains are assumed to be spherical. Most applications of ferrites are for higher frequency magnetic field and therefore the domain walls can be considered to vibrate under the action of a timedependent field. Under the action of a ‘‘weak’’ applied magnetic field, the domain walls deform but remain fixed on the grain boundaries. From this it was predicted that the permeability depended linearly on the grain diameter. Comparison with experiment yielded good agreement [20]. Further studies showed that wall motion components dominate in ferrites, while rotational processes, which are dependent on anisotropy but not grain size, are of secondary importance in these materials [21]. The equation governing the deformation of the magnetic domain walls are described by Globus et al. [22]. Although the model was developed originally to describe the properties of ferrites, but was later shown to be valid for spinels and garnets. Dissipative processes resulting from translation of the domain walls at higher field

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strengths were added to the model subsequently by Guyot [23,24]. The most comprehensive treatment of the underlying theory of this model, including most of the equations, which were not given in the papers by Globus, has been given by Escobar et al. [25,26]. Accordingly the initial susceptibility due to domain wall bending is w¼

Ms2 D ; g

ð13Þ

where Ms is saturation magnetization, g is the domain wall surface energy and D is the grain diameter. The value of the critical field for the unpinning of domain walls is Hc ¼

2f ; Ms D

ð14Þ

where f is the pinning force per unit length on the domain wall along the grain boundary, which is assumed to be independent of the applied field. Bertotti et al. have extended some of the domain wall modelling concepts through the use of domain wall eddy current dissipation [27] and stochastic process models in which they considered the domain wall as moving through a randomly fluctuating potential [28,29]. The randomness of the potential seen by the domain walls can be quantified using two independent parameters, which represent mathematically the physical properties of the potential in terms of an average amplitude and an average fluctuation wavelength. If Hc is a randomly fluctuating function of position, then it will also be a randomly fluctuating function of the flux f: Local maxima in Hc represent physical pinning sites in the material, and for large displacements of domain walls there will be a correlation length x which represents the range of interaction of domain walls with pinning sites. These effects can be described if it is assumed that the local coercivity Hc obeys a Langevin equation of the form dHc Hc  /Hc S dW ¼ ; ¼ z dF dF

Hysteresis is found to be a direct result of discontinuous, dissipative processes occurring over small volumes which when summed together produces the familiar hysteresis over larger volumes. The Barkhausen emissions are fractal in nature, which means that the structure of the Barkhausen emissions is independent of the scale. The underlying physics of these complicated types of processes are only now being uncovered and discussed [32]. A recent review of the theory behind domain wall dynamics has been given by Jiles [33]. The domain boundary models are best applied to multi-domain materials in which the movement of magnetic domain boundaries is the principal magnetization mechanism. This means mostly bulk soft magnetic materials with low anisotropy and with large density of inhomogeneities either in the form of strains (dislocations) or inclusions (particles of a second phase).

5. Magnetism at the macroscopic scale: the integration of single-domain switching processes The Preisach model is a general mathematical model, which describes hysteresis on the macroscopic scale. The model was first developed to treat magnetic hysteresis but the mathematical structure is equally applicable to other physical systems exhibiting hysteresis, such as ferroelectric or mechanical hysteresis. The model treats magnetic hysteresis as simply a summation of a large number of microscopic switching events occurring in a magnetic material. It was actually a development of previous work on hysteresis by Weiss and Freudenreich and also derives some of its ideas from the earlier model of Ising. The Preisach model describes materials as an array of domains each with the same magnetic moment per unit volume (magnetization), but with different switching fields as shown in Fig. 4. The allowed microstate of the magnetic moment is either ‘‘up’’ or ‘‘down’’, as in the

ð15Þ

where now the flux F is the measure of displacement of domain walls, instead of the position, and W ðfÞ is a randomly fluctuation function (or ‘‘white noise’’ function), whose average value will be zero, and z is the interaction length, or correlation length, for domain walls with pinning sites. Eventually with some additional restricting assumptions the equation reduces to ’ 1 dF ’ ¼  1 dHc ’  m0 AMÞ ð16Þ þ ðF dt t sG dt The motion of domain walls through the internal randomly fluctuating potential leads to discontinuous changes in magnetization. These discontinuous processes are manifested as Barkhausen noise [30], which is closely connected with the existence of hysteresis [31].

Fig. 4. In the Preisach model, each domain is represented by its saturation magnetization and two characteristic switching field strength ha and hb which describe the field strengths needed to change the direction of magnetization.

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Ising model. This spin-up and spin-down restriction does limit the relevance of the model to actual physical reality in most magnetic materials. The domains in the Preisach model remain the same size, but there is no fundamental problem with this approach for empirical modelling since the volume fractions of the domains with particular combinations of switching fields can be varied within the model. The magnetic characteristics of a material are represented as the volume fraction of domains with particular combinations of switching field. This is described by probability distribution function Pðha ; hb Þ over the Preisach plane defined by all possible combinations of ha and hb ; which is the span of values of the two switching fields. The probability density function (also known as the Preisach function) P varies over the span of possible values of switching fields and this represents the distribution of different types of domain in the material. The magnetization M of the system as a function of applied field HðtÞ can be calculated by Z Z MðtÞ ¼ Ms Pðha ; hb Þdðha ; hb ÞHðtÞ dha dhb ; ð17Þ ha Xhb

where Ms is saturation magnetization. The value of dðha ; hb Þ is either +1 or 1 depending on the magnetic history. The Preisach function can also be expressed as a function of hc ¼ ðha  hb Þ=2 and hm ¼ ðha þ hb Þ=2; where hc is the coercive field of an elementary hysteresis loop and hm is the displacement or offset of the centre of the hysteresis loop from H ¼ 0: This representation of the Preisach function, Pðhc ; hm Þ; allows an integrated coercive field distribution Pðhc Þ to be computed by Z Pðhc Þ ¼ Pðhc ; hm Þ dhm ; ð18Þ where Pðhc Þ represents a probability distribution function for elementary hysteresis loops having a particular value of coercive field hc in the system. Calculation of the Preisach function in order to describe the magnetic properties of the material is therefore central to the use of the Preisach model since it determines how many domains, or what volume fraction of the material, will change orientation of its magnetization between two values of the magnetic field. The Preisach function itself is not an invariant. It can change with exposure to different field histories. The description of the model first appeared in a restricted form, which has come to be known as the classical Preisach model. The classical Preisach model has some characteristic properties, the most significant of which are (i) the memory (or wiping out property) whereby only the alternate series of dominant field exposure maxima are remembered and (ii) the congruency property, whereby all minor hysteresis loops

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corresponding to cycling of magnetic field between the same two extrema are congruent. The first of these characteristic properties seems to be universally applicable to hysteresis phenomena. The second is more limited in its applicability, and there are certainly many instances where it is not valid. The switching fields, or local coercivities, are assumed to be different from one domain to the next. Furthermore, within this model the switching fields (coercivities) can be different in the up and down directions for the same domain, although the model offers no explanation for this. Originally, there were no interactions allowed between the domains. Again this restriction is rather unrealistic since there is unquestionably an exchange interaction within magnetic materials, although the strength of this interaction between domains is crucially dependent on domain size. A comparison of the classical Preisach model to measured magnetic hysteresis loops shows that the model does broadly describe hysteresis but systematic discrepancies occur. Generalizations of the original concept have been developed to address a wider range of magnetic hysteresis phenomena [34]. The generalized Preisach model does not have the congruency restriction. These later developments of the model also included interactions between domains, but once this is included the actual locations within the material of domains with particular orientations becomes significant. One way to circumvent this latter problem is to use a mean field approach to model the interactions. This is equivalent to assuming that each domain in the material interacts equally with all other domains within the material. The quasi-static Preisach model describes the hysteresis losses in materials. When the frequency of the exciting field is changed, however, there are additional power losses, which are traditionally separated into ‘‘classical losses’’ arising from the solution of the classical Maxwell’s equations in a magnetically permeable and electrically conducting medium, and ‘‘excess losses’’. Dupre et al. [35] investigated these frequencydependent power losses using the dynamic Preisach model. It was known from experimental observations that the excess losses, due to domain wall processes, depend on the frequency of excitation according to a f 3=2 power law and it was found that this can be described under restricted conditions by the Preisach model. Subsequently, Dupre et al. investigated the description of power losses using the generalized moving dynamic Preisach model under a unidirectional applied magnetic field [36]. Agreement was generally good over a range of frequencies up to 1 kHz. Some systematic discrepancies were noted at the higher end of this frequency range, which is probably attributable to the influence of partial penetration of the magnetic field, due to the skin effect, on the dynamic parameter obtained

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from the statistical theory of domain processes. Furthermore, it was shown that the classical losses cannot be derived through the statistical theory of domain processes. The relationship between microstructure and magnetic properties is important for understanding the behaviour of magnetic materials. This can be achieved through computational micromagnetics as described above, and as shown by Dupre et al. through the generalized dynamic Preisach model [37]. The effects of grain size and texture were separated within the model and the Preisach function was rewritten containing explicit terms representing grain size and texture. These two terms were found to be the same for all magnetization processes, including quasi-static reversible, quasistatic irreversible and excess losses. As shown by Pasquale and Jiles, the Preisach and Jiles–Atherton models are equivalent under certain conditions [38]. In particular, if the Preisach distribution function decreases exponentially with hc according to the above equations, then the rate of change of magnetization with field in the two models can be shown to be equivalent. In conclusion, the Preisach model provides a mathematical basis for describing hysteresis. It reconstructs the behaviour in terms of a set of more elementary components and has some similarities to the use of Fourier analysis for periodic functions [39] in as much as the total response (hysteresis) is represented as the sum of a large number of elements (hysterons, each with different switching fields). The model is widely applicable on the macroscopic scale for describing hysteresis in magnetic materials but the mathematics of the process gives switching field distributions that may not be physical. One severe limitation of the Preisach model as formulated is that it really only allows irreversible processes to be modelled. Reversible processes, if they are to be included, must be taken care of in an arbitrary manner. In addition, the basic Preisach model is scalar in nature (due to the spin-up/spin-down restriction) whereas magnetization processes are in practice vector. The generalized Preisach model includes an extension to describe the vector nature of these processes.

6. Magnetism at the multi-domain level In dealing with the behaviour of materials, in particular their bulk magnetic properties such as coercivity, remanence, permeability and hysteresis loss, other problems arise that make it difficult if not impossible to simply scale up the predictions of models that are based on consideration of one or two domains. Therefore, a more general approach is needed in order to develop equations that represent the average behaviour of the materials. These models necessarily use

statistical thermodynamic principles to describe the resulting magnetization behaviour of a very large number of magnetic domains. The earliest thermodynamic approaches were developed for the simplest systems, specifically paramagnets. Paramagnets have the simplicity of being magnetically homogeneous, unlike ferromagnets. Later models were developed for the more technically important class of ferromagnets without including hysteresis, and finally, hysteresis models were developed. The model of hysteresis developed by Jiles and Atherton depends on statistical mechanics and is most relevant on the mesoscopic scale. It works well for materials with low anisotropy for which the main mechanism is domain boundary movement. It can be used for simple anisotropies such as axial and planar anisotropies with minor modifications. For highly anisotropic materials, it can still be used with the understanding that a simple analytic anhysteretic equation cannot in general be developed for anisotropic materials and therefore the mathematical approximations become less realistic the greater the anisotropy and the larger the number of magnetic easy directions. The classical model for magnetism is the Langevin– Weiss model, which considers an array of magnetic moments in thermal equilibrium at a particular temperature. This was used by Jiles and Atherton as the basis for developing a model of hysteresis. The orientations of the magnetic moments are distributed statistically and integrating the distribution of moments over all possible orientations leads to an equation for the bulk magnetization. The details of this depend on the restrictions imposed by anisotropy, so that for example different solutions are obtained depending on whether the magnetic moments experience axial anisotropy, planar anisotropy or are in a completely isotropic environment [40]. The extension of the Langevin–Weiss theory used to describe ferromagnetic materials, incorporates a coupling among magnetic moments acting as a strong magnetic field to align the magnetic moments in a domain parallel to each other. To quantify this coupling a mean field, which is proportional to the bulk magnetization, He ¼ H þ aM is used. This mean field approach describing the interactions needs to be applied with some caution, but recent work by Chamberlin [41] has shown that the mean field approach is viable for clusters of spins on the nanoscopic scale. By replacing the classical magnetic field H with the effective magnetic field H þ aM; which includes coupling to the magnetization, an equation for the anhysteretic magnetization of a ferromagnetic material can be obtained as follows.
 m mðH þ aMÞ M ¼ Ms L 0 : ð19Þ kB T

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Instead of considering coupling between each individual magnetic moment the mean field is used to represent the inter-domain coupling. For isotropic materials, the anhysteretic function is

texture coefficient ftext ; which is a statistical evaluation of the volume fraction of the textured portion of a material, was also introduced. The anhysteretic magnetization can then be given as

M ¼ Ms fcothðxÞ  1=xg;

Man ¼ ftext Maniso þ ð1  ftext ÞMiso ;

ð20Þ

ð28Þ

where x¼

m0 mH* and kB T

H* ¼ H þ aM:

Alternatively for materials exhibiting axial anisotropy, M ¼ Ms tanhðxÞ

ð21Þ

and for materials exhibiting planar anisotropy M¼

kB T d I 0 ðxÞ log Z ¼ Ms 0 ; m0 dH* I0 ðxÞ

ð22Þ

where I0 ðxÞ ¼

N X 1 x2s 2 2 s¼0 ðs!Þ

ð23Þ

N X s x2s1 : 2 2 s¼1 ðs!Þ

ð24Þ

and I00 ðxÞ ¼

A more generalized extension to cover other more complicated anisotropies was made [42], in which the energy of a magnetic moment with anisotropic perturbation was calculated in three dimensions and therefore different kinds of anisotropic materials could be described. An increasing range of magnetic materials in which anisotropy and texture play a significant role, for example hard magnetic materials, as it can be seen in Section 3. Following the development of the generalized anhysteretic function [43] P eE=kB T cos y Pmoments Maniso ¼ Ms all ; ð25Þ E=kB T all moments e where y is the angle between the direction of the magnetic moment and the direction of the applied field, and E ¼ m0 /mSðH þ aMÞ þ Eaniso ;

ð26Þ

Eaniso is the anisotropy energy which depends on the anisotropic structure of material. For example, in the case of cubic anisotropy Eaniso ¼ K1

3 X

cos2 yi cos2 yj

ð27Þ

iaj

with the normal convention on symbols. In this description, only the first anisotropy coefficient K1 was used since this approximation is in most cases sufficient to provide an accurate description of the different magnetization curves along different directions. A

where Maniso is the anisotropic anhysteretic magnetization contribution, and Man is the isotropic anhysteretic magnetization contribution. For more complicated textured materials, there may be several different texture orientations such that each particular direction has a proportion of the grains oriented along it. In these cases, the anisotropic contribution of each part must be calculated separately and the net anisotropic portion of the anhysteretic magnetization is the weighted sum of the components of magnetization of these orientations along the direction of the applied field. From this description of the thermodynamic anhysteretic magnetization it is possible to develop a description of hysteresis through the consideration of energy dissipation mechanisms. The irreversible and reversible components of magnetization can be described separately in the mathematics, although they are linked physically. The two components of magnetization can then be combined to give an equation for the total magnetization.

Acknowledgements This research was supported by the US Department of Energy, Office of Basic Energy Sciences, Materials Sciences Division, under contract W-7405-Eng-82 and by the Computational Materials Science Network http:// cmpweb.ameslab.gov/cmsn/.

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