# An Alternative Formulation of the Harrison Model
## 2 A Brief Introduction to the Harrison Model
### An Alternative Elementary Unit (\"Hysteron\")  
As pointed out earlier, the reduced field strength \(h\) modeled with the original [PERSON] description rarely reaches values close to zero; for example, for Fe-Si electrical steels, the maximal value \(h_{\rm crit}\) is around 0.498 [8] since the reduced temperature \(t\cong 293/1043=0.281\)[10]. Our goal was to find out a functional dependence which would behave qualitatively like the one given with (3) yet able to reach \(h=1\) within the bounds \(m\in\langle-1;1\rangle\). For such a functional relationship, it might be easier to assign weights to individual units and interpret the results. Inspired by the work [29], which stressed the flexibility of logistic mapping and its usefulness in phenomenological descriptions of ferromagnetic hysteresis, we considered the following relationship for the description of the double-well energy landscape in an elementary hysteretic unit as:  
\[F(m)=am^{2}\Big{(}1-m^{2}\Big{)} \tag{5}\]  
where \(a\) is a constant. Following the monograph [3], we considered a connection between hysteresis phenomenon and metastability. The proper framework for examination of this relationship is thermodynamics of irreversible phenomena. If we consider a system with local memory, just three variables, \(H\), \(X\) ([PERSON]'s notation, cf. chapter 2.1.4, here identified as \(M\)) and \(T\), provide a complete characterization of states. If \(H\) and \(X\) are conjugate work variables, then \(G(H,T)=F\left(X,T\right)-H\,X\) is the corresponding Gibbs energy (\(F\) denotes the Helmholtz free energy of the system). To consider an energy profile with at least two minima symmetrically set with respect to the \(X\) axis, we need a polynomial of the order four. Imposing the condition \(\partial F(m)/\partial m=h\) for the equilibrium of competing thermodynamic forces [3], we can write:  
\[h=2 am\left(1-2m^{2}\right) \tag{6}\]  
Thus, the dependence of the applied field versus magnetization becomes a polynomial of one order lower, i.e., of the order three. Let us notice that, due to symmetry, this polynomial has just odd order terms. Of course, it is possible to consider higher-order polynomials, which may possess local global minima, yet, in that case, the polynomials will also possess only odd terms. Figure 1 depicts the dependencies \(F(m)\) and \(h(m)\) for an arbitrary value \(a=-2\). It is interesting to remark that the specific choice of (4) imposes several invariable values for some loop parameters such as remanence \(m_{\rm rem}=\mp 1/\sqrt{2}\) or \(m_{\rm crit}=\mp 1/\sqrt{6}\approx\mp 0.408\) regardless of the \(a\) value. It can be proven that the choice \(a=-0.75\sqrt{6}\approx-1.837\) allows one to obtain \(h_{\rm crit}=1\). A fixed remanence value might be considered a limitation since (4) is most appropriate for those materials for which the squareness of hysteresis loops (the ratio of remanence magnetization to saturation magnetization) [30] is close to 0.7. However, we would like to remark that alternative formulations for the dependence \(h=h(m)\) may also be applied. For example, a polynomial with odd powers is:  
\[h_{531}=8m^{5}-3m^{3}-2m \tag{7}\]  
where the subscript denotes that powers with non-zero values of coefficients also behave qualitatively in a similar way. The relationship (6) is depicted in Figure 2. The remanence values for this polynomial are \(m_{\rm rem}\approx\mp 0.849\), and the bifurcation magnetization \(m_{\rm crit}\approx\mp 0.602\), whereas the corresponding value of field strength (coercive field strength) \(h_{\rm crit}\approx\pm 1.22\). The value of magnetization when two branches of hysteresis loop coincide is approximately \(m_{\rm out}\approx\pm 0.9267\). Figure 1: The dependencies \(F(m)\)(broken line) and \(h(m)\) (solid line) for \(a=-2\). A thorough examination in which combinations of coefficients for polynomials with odd powers yield qualitatively similar loops is beyond the scope of the present paper and is postponed to another publication.