# An Alternative Formulation of the Harrison Model
## 2 A Brief Introduction to the Harrison Model  
In the original Harrison model, the [PERSON] mean field constant \(\alpha\) is assumed to be positive, just like in, e.g., the Jiles-Atherton approach--this corresponds to the positive feedback in the system. The argument of hyperbolic tangent in Equation (1) is written as the weighed sum \(h+m\). In that case, the curve given with Equation (1) is S shaped (if one considers \(m=m(h)\) dependence) or N shaped (if one considers \(h=h(m)\) dependence). It passes through the second and the fourth quadrants of the \(m\)-\(h\) plane; the extreme points are referred to as bifurcation points. Equation (1) may be easily inverted analytically; the description having magnetization as its input and field strength as its output is referred to as an inverse hysteresis model. The appropriate expression becomes:  
\[h=t\text{atanh}m-m \tag{3}\]  
where the reduced temperature \(t\) plays the role of a weighting parameter. The curve given with Equation (3) for a positive \(t\), \(0<t<1\), has two extreme points whose coordinates are computed from equating \(\text{d}h/\text{d}m\) to zero. The fragment of the curve between these extreme points is decreasing; in ferroelectric devices, this corresponds to the \"negative capacitance\" effect, which recently has been the subject of intensive research [22; 23] because of its potential applications, e.g., in low-power nanoscale devices. In ferromagnetic materials, in most cases, aside from in precise experiments carried out for single iron crystals [24], the intermediate region is hard to observe experimentally. As field strength reaches the threshold value equal to \(h_{\text{BIF}}\), an abrupt jump to another hysteresis loop branch occurs. The magnetization value \(m_{\text{EXIT}}\) at which both branches coincide cannot be given in an analytical form; however, it is straightforward to compute it numerically. Two extreme cases are:  
* \(t=0\) when the curvature of the \(h=h(m)\) function disappears. Then, the \(h=h(m)\) dependence is represented by a zigzag resembling the letter \"N\". The value of field strength at the bifurcation point (considered equivalent to coercivity) reaches unity;
* \(t=1\) when the hysteresis loop disappears. In the original model, [PERSON] used a phenomenological parameter \(\beta\), which was introduced to scale the irreversible hysteresis curves operating on the micromagnetic level up to the domain size (mesoscopic) level, on which the anhysteretic processes occurred. The afore-given statement is clear if one considers the expression (1) written not in relative units but physical ones. It reads [10]:  
\[\frac{M}{M_{\rm s}}=\tanh\frac{\mu_{0}\beta m_{\rm B}}{kT}(H+\alpha M) \tag{4}\]  
where \(\mu_{0}=4\pi\times 10^{-7}\) H/m is free space permeability, \(m_{\rm B}=9.27\times 10^{-24}\) Am\({}^{2}\) is the [PERSON] magneton, \(k=1.38\times 10^{-23}\) J/K is [PERSON]'s constant and dimensionless \(\alpha\) is [PERSON] mean field parameter (cf. (2)), whereas \(T\) measured in K is the absolute temperature. \(M_{\rm s}\) stands for saturation magnetization. From the inspection of the above-given relationship, all quantities have a clear physical interpretation, whereas \(\beta\) is merely a numerical means to scale the switching phenomenon, occurring at quantum scale up to the scale of magnetic domains. [PERSON] explained his approach in this way: \"... the new vector quantity \(\beta m_{\rm B}\) can be called a \"supermagneton\". In this model the \"supermagneton\" is a new quantum particle, that is acted upon by the effective field\" [6]. In other words, the original approach corresponds to the multiplication effect for a single hysteresis operator; thus, it means that the model is subject to averaging and homogenization, not just because of the presence of the averaged quantity \(\alpha\), but also because all operators are treated as the same.