Preface

Knowledge of magnetism spans from Ancient Greece to present day. Current theories of magnetism bring to light new applications, new features, and new models.

Magnetism is the source of two phenomena in physics: electric current and spin magnetic moments of elementary particles. As such, much attention is paid to lowdimensional structures. As the space dimension of a physical system decreases, magnetic ordering tends to vanish as fluctuations become quite significant. As known, there is no spontaneous magnetization in systems in one dimension at any nonzero temperature; for instance, the isotropic spin-s Heisenberg model, hard-core, and any system with finite range interactions. Nevertheless, the mean-field approximation constitutes an example of the state of magnetic ordering of a chain of spins with long-range microscopic interactions giving anomalous ferromagnetism cases in one dimension. Therefore, an accurate description of magnetic ordering phases illustrates concepts about the critical behavior and phase change and possible applications of new magnetic devices.

Following the Introductory chapter, four chapters cover topics related to recent advances in the modeling and application of magnetometers.

M. Hsini, S. Zemni, In Chapter 2 "Modeling the Magnetocaloric Effect of Nd0.67Ba0.33Mn0.98Fe0.02O3 by the Mean Field Theory," M. Hsini and S. Zemni combine mean-field theory and the Bean–Rodbell model to justify the magnetocaloric effect (MCE) in a sample. They derive expressions to rating the magnetic entropy change under various magnetic fields and compare theoretical to experimental curves.

H. López Loera, In Chapter 3 "The Magnetometry—A Primary Tool in the Prospection of Underground Water," H. López Loera presents a geophysical methodology that shows the potential of combining natural and induced methods to locate confined aquifers in zones with a high probability of locating groundwater in the Mexican Mesa Central.

In Chapter 4 "Atomic Scale Magnetic Sensing and Imaging Based on Diamond NV Centers," M. Lee, J. Yoon, and D. Lee review the basic sensing mechanisms of the nitrogen-vacancy (NV) center and introduce imaging applications based on scanning magnetometry and wide field-of-view optics.

Finally, in Chapter 5 "SQUID Magnetometers, Josephson Junctions, Confinement and BCS Theory of Superconductivity," N. Khaneja discusses some theoretical aspects of a SQUID magnetometer, as its sensitivity makes it possible to apply to biomagnetism, materials science, metrology, astronomy, and geophysics.

> **Sergio Curilef** Universidad Católica del Norte, Antofagasta, Chile

**1**

Section 1

Introduction

Section 1 Introduction

**3**

**Chapter 1**

*Sergio Curilef*

context [1–5].

**1. Long-range interactions**

long-range interactions [5, 9].

Introductory Chapter: Statistical

and Theoretical Considerations on

Magnetism in Many-Body Systems

The description of systems with long-range interactions is relevant to statistical physics because we find appealing properties that deserve to be studied in detail. In this line, some variations of the Ising model that involves not only first neighbors, but also distant neighbors are employed; for instance, the Hamiltonian Mean Field (HMF), and the recently introduced dipole-type Hamiltonian Mean Field (d-HMF) model. We emphasize that the Ising model is a recurrent tool to study magnetic properties and the statistical behaviors of many-body systems in the broadest

In concern about long-range interactions, there are various challenges to face; these are related to the dynamics, size of the systems, and the theoretical framework to explain the behavior of systems, among others. In this type of system, we have typical consequences such as the loss of additivity and extensivity. The loss of additivity takes place for ensembles of interacting particles that cannot trivially separate into independent subsystems, which is explained by the presence of underlying interactions or correlation effects, whose characteristic lengths are comparable or more significant than the system linear size [6–8]. Additionally, the

If we have a system composed of N spins that interact one to each other as a power law of inter-particle distance, the total energy of the system is E. The system divided into two subsystems, 1 and 2, both composed of N/2 particles with energies *E*1 and *E*<sup>2</sup> , respectively. If the spins in subsystem 1 are up, and those in the subsystem 2 are down, the energies satisfy *E*<sup>1</sup> = *E*<sup>2</sup> . Since the sum of energies, *E E* 1 2 + is not equal to the total energy E; the system is nonadditive. In general, *EE E* < +1 2 . Besides, the extensivity ( *E N*/ →constant) is another fundamental property to consider in this kind of analysis. The recent literature shows a way to recover this property through Kac's prescription that has a standard thermodynamic structure because it preserves the Euler and Gibbs-Duhem relations. Therefore, this procedure allows us

to recover the extensivity, while the loss of additivity remains because of the

At equilibrium, an analytical procedure leads to solving the problem for obtaining the magnetization, the inverse temperature, the specific heat, in the canonical ensemble. Also, it is possible to get the microcanonical entropy. The caloric curve, ascertained using both, exactly coincides. At this stage, we emphasize that the solution to this system with long-range interactions becomes analytical, which solutions are not abundant in statistical physics. Nevertheless, out of equilibrium, we have data from simulations obtained from carrying out molecular dynamics.

loss of extensivity frequently accompanies the loss of additivity [5, 9].
