1 The Model
The term of hysteresis [1-3] from the etymological point of view is related to the lag that can be observed between the input, in magnetism, the applied magnetic field and the output, the magnetic moment of the sample. The most common lag-effect is observed when one applies a field that depends on time as sinusoidal function and the magnetic moment follows the same time dependence of the field with a certain phase-lag. When the applied field is zero, the magnetic moment has a remanence, and we have to apply a (coercive) field in the opposite direction to obtain the demagnetized state. This hysteresis type is dependent on the frequency and in many cases when the frequency is very small, the hysteresis disappears as well. This is a rate-dependent hysteresis. However, for many ferromagnetic materials, one observes even for measurements made at very low frequencies that the system still has a hysteresis loop which does not change much if the field rate in the measurement is modified in quite a wide range of values. This is a rate-independent hysteresis and will be discussed on details. We have to mention that this is however an approximation and that in many cases one can still evidence deviations from the rate-independent hysteresis behaviour. Magnetic relaxation which can include phenomena in a wide range of time scale is responsible in fact for these deviations.
As a first conclusion, one can introduce with a certain approximation the notion of rate-independent hysteresis, which will exclude both the phase-lag and magnetic relaxation. We can link this type of hysteresis with the existence in the system of entities with metastable states. To make our analysis simpler, one can concentrate our discussion on a system of ferromagnetic single-domain particles (a typical magnetic recording medium known as a particulate medium). Each isolated particle has its individual hysteresis loop that depends on the particle’s shape, anisotropy, volume, etc. This is also a rate-independent type hysteresis if the particles are not too small in which case they become superparamagnetic.
So, in a classical particulate magnetic medium each particle has its rate independent hysteresis loop. It is characterized by a free energy function that has for a definite domain of the applied field (between the negative and positive switching fields) two minima separated by a maximum. When the applied field is outside this domain the free energy has only one minimum and as a consequence the moment has only one equilibrium position. When the applied field is between the negative and positive switching fields the particle has two equilibrium positions. However, for a given value of the applied field, the particle will stay in one minimum and will not change that position (the relaxation is considered at the timescale of the experiment to be negligible). Only the applied field can produce an irreversible change between the two equilibrium states and this change is obtained when this field has a value equal with one of the two characteristic switching fields. As long as the particle maintains an equilibrium position and no switches occur, the changes in the moment position are reversible.
The behaviour of an ensemble of particles will also display hysteresis that will be controlled not only by the hysteretic properties of each particle seen as isolated but also by the interactions between the magnetic moments of the particles. We shall see that there are qualitative differences between the hysteresis of one particle and the hysteresis of a system.
Using the same example of a particulate medium one can explain in a simple manner how it was developed the well-known Preisach  hysteresis model, which will be referred to as the Classical Preisach Model (CPM).
For simplicity, let's suppose that all the particles have their anisotropy axes parallel to each other and with the applied field direction. In this case, if the field value is between the two switching fields, two equilibrium positions are possible; they are characterized by the angle between the anisotropy axis and the magnetic moment vector of the particle. In the case previously described, the equilibrium positions are characterized by the angles 0° and 180°. The hysteresis loop, in the absence of interactions, is symmetric with respect to the origin of the (H, m) co-ordinate system and has a rectangular shape. The negative and positive switching fields have the same absolute values.
For the beginning we consider that the interaction field is positive and parallel to the applied field direction. The effect of this interaction field is a displacement of the hysteresis loop to the left (towards smaller values of the applied field); the positive switch is assisted by the positive interaction field. In a similar way one may say that a negative interaction field will assist the negative switch, so a displacement of the hysteresis loop to the right (to higher fields) should be observed. In conclusion, the effect of the interaction field is in any case an asymmetry of the hysteresis loop of the constituent particles. The positive and negative switching fields will be unequal in absolute value (see Fig. 1). Even if this is a rather strong simplification for the effect of the interaction field on the particle’s hysteresis loops, we shall see that this will allow us to understand the main problems related to the effect of interactions in particulate systems.
In the Preisach model the switching fields of the particles are used as co-ordinates in a plane, named the Preisach plane (see Fig. 2), that is, the magnetic moment of each particle is associated to a point with the switching fields as co-ordinates (in the Preisach plane Hα is the positive switching field and Hβ is the negative switching field). Associating all the magnetic moments from a sample to this plane one obtains a distribution called the Preisach distribution which may be seen, taking into account the big number of particles in the system, as a continuous function of the switching fields of the particles (see Fig 2).
In any real system the interaction fields are not identical for all the particles. The particle anisotropy, and in consequence, the switching fields absolute values, in the absence of interactions, are not identical for all the particles, as well. For any dispersion, some of the particles are sufficiently far from the neighbours, so their hysteresis loops are symmetrical. The magnetic moments of these particles will be associated to the second bisector of the Preisach plane (Hα = -Hβ). The particles with the same coercivity, Hcc, but with different interaction fields will be associated to a line parallel to the first bisector:
Hc = (Hα - Hβ) / 2 = const. = Hcc → Hβ = Hα - 2Hcc (1)
and those with the same interaction field, Hii, are associated to a line parallel to the second bisector:
Hi = -(Hα + Hβ) / 2 = const. = Hii → Hβ = - Hα - 2Hii (2)
Usually, the positive and negative interaction fields, when the sample is demagnetized, are equally probable, so the Preisach distribution will be symmetrical with respect to the second bisector which corresponds to the zero interaction field. This property is confirmed by the symmetry of the major hysteresis loop of the sample with respect to the origin of the (H, m) system. Most of the real systems Preisach distributions have a maximum value corresponding to the most probable value of the anisotropy field of the particles from the system which is noted with Hc0 in Fig. 3.
To be able to calculate the magnetic moment of the sample in the CPM one has to know its Preisach distribution. The determination of the Preisach distribution of a sample, also called identification, requires a number of experimental data. However, we wish to present first how one calculate a magnetization curve if we already know the distribution. This will help us to understand also how an identification process can be designed.
Fig.1 - A representative rectangular hysteresis loop associated to the point of co-ordinates (Hα , Hβ) in the Preisach plane. The coercive field of particle is Hc = (Hα - Hβ) / 2 and the characteristic interaction field is Hi = -Hs = - (Hα + Hβ) / 2 (Hs is the shift of the rectangular hysteresis loop in the H direction). We noted the magnetic moment of the particle with m0 and the saturation magnetic moment with m0s
Fig.2 - The Preisach plane. The saturation magnetic moment m0 of the particle whose hysteresis loop is represented below is associated to the point M. The T0 triangle is containing all the particles in the system. The Everett integral in the point M is calculated as the integral of the Preisach distribution over the T triangular region (hashed region). The P0Q0 line is the first bisector of the Preisach plane.
Fig.3 - The Preisach plane and a continuous Preisach distribution
Fig.4 - The Preisach plane. One applies the field H to the system. The limit triangle is partitioned into three zones. The "non-local memory" appears due to the existence of the region III, marked with "memory"
Let's suppose that we know the Preisach distribution of the system and let's observe what is happening in the Preisach plane when a (positive) field H is applied. As we may see on Fig. 4, two lines, L1 and L2, could be plotted in the Preisach plane; they are obtained from the condition that either the positive switching field, Hα, is equal to the applied field or the inferior switching field, Hβ, equals the same value. One mention that all the particle magnetic moments are associated to a triangular zone, which will be referred to as the limit triangle, bounded by three lines: the first bisector, L1s corresponding to a field sufficient to saturate the sample in the positive direction (Hα,max) and L2s for a field sufficient to saturate the sample in the negative direction (Hβ,min). Due to the symmetry of the major hysteresis loop, these positive and negative saturation fields are equal in absolute value. Outside this triangular limit there is no magnetic moment associated to the Preisach plane since i) the left-up zone of the Preisach plane is characterized by hysteresis loops with the superior switching field less than the inferior switching field, which is non-physical, and ii) the absolute value of all the switching fields in the system is less than the saturation field.
As one sees on Fig. 4, the L1 and L2 lines are generating from their intersection with the limit triangle three zones, marked with Roman numbers. The particles associated to the zone “I” have positive switching fields less than the applied field, so it will be all on the positive branch of their hysteresis loops, we shall say that they are on the "+" position. The particles associated to the zone “II” have their negative switching field greater than the applied field, so they will be on the negative branch, named "-" position. For the particles from the zone “III” the applied field is between the inferior and superior switching fields so their state could not be changed by the applied field. The existence of this zone makes the memory effect possible in ferromagnetic particulate medium. This memory is qualitatively different from the memory exhibited by one isolated particle which is also characterized by a hysteresis loop. The single-domain particle has a local memory while the system has a non-local memory. That is, if the actual value of the magnetic moment projection in one single particle case is completely determined by the values of the field and magnetic moment at an infinitely near previous moment, the actual value of the magnetic moment projection in the particle system case is also determined by the history of applied fields. However, when the field value is changing it is not necessary to see what is happening with all the three zones discussed previously. It is easy to observe that if the field value is increasing only the particles which are entering in the “I” zone change their state to "+" state. The others remain unchanged. In the same way, for the decreasing fields only the particles entering in the “II” zone are changing their state. So, a simple rule may be formulated (see Fig. 5):
- for the increasing fields the L1 line is going to the right (higher fields) and all the moments at the left of this line are switched to the “+” position and
- for the decreasing fields, the line L2 is going down (lower fields) and all the moments over that line are switched to the “-” position.
After a sequence of fields a staircase line, L, separates the “+” and “-” zones in the limit triangle (see Fig. 6).
Fig.5 - Left: increasing field applied to the sample; Right: decreasing field applied to the sample.
Fig.6 - The Preisach plane after a field sequence
The memory of the system is "concentrated", in fact, in this staircase line. One observes that the co-ordinates of the segments which are forming the L staircase line are given by the local extreme values of the applied fields (local minima and maxima). Finally, the total magnetic moment of the system is the difference between the magnetic moment on the “+” and “-” zones, that is:
One also observes that the calculus of these two integrals over the S+ and S- regions can be decomposed in a sum of integrals over triangular zones such us the one represented in Fig. 2. These integrals are named Everett integrals. The Everett integral for the triangle T(Hα1 , Hβ1) is given by:
which may be written in simplified manner for any point in the Preisach plane as:
If one performs a normalization at the saturation value of the magnetic moment which is
ms = E(Hm , -Hm) (6)
where Hm = Hα,max = -Hβ,min is the maximum absolute value of the critical field of the particles in the system and one uses the notation
then the normalized Everett integral will be:
 G. Bertotti, Hysteresis in magnetism: for physicists, materials scientists, and engineers. San Diego: Academic Press, 1998.
 I. D. Mayergoyz, Mathematical models of hysteresis and their applications, 1st ed. Amsterdam; Boston: Elsevier, 2003.
 E. Della Torre, Magnetic hysteresis, New York: IEEE Press, 1999.
 F. Preisach, "Über die magnetische Nachwirkung," Z.Phys., vol. 94, pp. 277-302, 1935