# An Alternative Formulation of the Harrison Model
## 1 Introduction  
Hysteresis modeling is important for practitioners dealing with descriptions of magnetic properties of materials readily used in Nondestructive Testing and Evaluation. This topic is also in the spotlight of MDPI's _Applied Sciences_ journal, just to mention recent publications [1; 2]. As was pointed out in [1; 3], the hysteresis phenomenon is prevalent in different areas of science, yet its occurrence in ferromagnetic materials remains a paradigm [4].  
The present contribution is inspired by an excellent review work published recently [5]. It extends several concepts which have been mentioned in the previous publications [6; 7; 8; 9]. The fundamental concept of the paper is to modify the Harrison model of ferromagnetism to make it more useful for engineering purposes. The modification relies on the summation of responses of elementary bistable units operating on the quantum level using an approach like the stop models.  
The original description advanced twenty years ago [10] may be perceived as a \"toy model\" because of its spurious simplicity. However, it is based on a profound physical interpretation; moreover, it is consistent with the laws of irreversible thermodynamics. It distinguishes the reversible and irreversible magnetization processes and offers a simple interpretation of the hysteresis loop onset as related to quantum-mechanical effects.  
The irreversible magnetization processes are expressed in dimensionless units with the following relationship as:  
\[m=\tanh\frac{h+m}{t} \tag{1}\]in which \(m\) stands for reduced magnetization, and \(h\) is reduced applied field strength, whereas \(t\) denotes temperature, referring to the Curie temperature. This self-consistent relationship (since reduced magnetization \(m\) appears on both sides of Equation (1)) puts in the central perspective the concept of the \"effective\" field, which is the true internal field acting on magnetic moments within the ferromagnetic substance. It is different from the externally applied field. In physical units, this concept may be written as:  
\[H_{\text{eff}}=H+\alpha M \tag{2}\]  
where the dimensionless coefficient \(\alpha\) controlling the strength of interaction forces (referred to as the molecular field constant) is due to [PERSON] [11]. For a friendly introduction to the [PERSON] concept, the Readers are referred, e.g., to the textbooks [12; 13], or, e.g., to [14]. The theory of effective field has been used to explain spontaneous magnetic ordering and phase transitions in ferromagnets [3], and it is widely used in solid-state physics; some widely used hysteresis models such as the Jiles-Atherton formalism [15] heavily explore and rely on the concept. It is, however, interesting to mention that the idea of mutual interactions between individual entities has influenced seemingly different research studies, e.g., those devoted to collective behavior [16; 17] or focused on applied mathematics and computer science [18; 19; 20]. In the context of research on Nondestructive Testing and Evaluation, it should be stressed that the expression for the effective field may include additional terms related, e.g., to eddy currents, thermal viscosity, mechanical stresses or demagnetization effects [21], which opens an exciting perspective to develop new types of sensors of physical quantities based on monitoring of magnetic properties.