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Advances in physics theories and applications

Thermodynamics Property of Magnetic Materials in a Ferromagnetic System using the Ising Model

Akaninyene Daniel Antia, Cyril C. Umoren

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

There is hardly any branch of physics one can approach successfully without resorting to quantum statistical mechanics.It has been proven that the quantum mechanical description of a system gives accurate results.To use quantum statistical mechanics, the system are treated at their microscopic level.Most systems consist of many particles and in order o deal with such large numbers, one has to resort to statistical method related to the partition function of statistical mechanics.In this study, we haves used the partition function which is a statistical parameter to explore the thermodynamic properties such as Helmoltz free energy, Entropy, internal energy and heat capacity of a magnetic materials in a ferromagnetic system using the first and second dimensional Ising model.The results show that in one-dimensional model, phase transition does not occur at finite temperature but susceptibility of a system can be measured from the fluctuations in the magnetization in the second dimensional Ising model.

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Investigation of the Finite Size Properties of the Ising Model Under Various Boundary Conditions

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Abstract The one-dimensional Ising model with various boundary conditions is considered. Exact expressions for the thermodynamic and magnetic properties of the model using different kinds of boundary conditions [Dirichlet (D), Neumann (N), and a combination of Neumann–Dirichlet (ND)] are presented in the absence (presence) of a magnetic field. The finite-size scaling functions for internal energy, heat capacity, entropy, magnetisation, and magnetic susceptibility are derived and analysed as function of the temperature and the field. We show that the properties of the one-dimensional Ising model is affected by the finite size of the system and the imposed boundary conditions. The thermodynamic limit in which the finite-size functions approach the bulk case is also discussed.

Zeitschrift für Naturforschung A•Dec 16, 2019•3 citations
There is hardly any branch of physics one can approach successfully without resorting to quantum statistical mechanics.It has been proven that the quantum mechanical description of a system gives accurate results.To use quantum statistical mechanics, the system are treated at their microscopic level.Most systems consist of many particles and in order o deal with such large numbers, one has to resort to statistical method related to the partition function of statistical mechanics.In this study, we haves used the partition function which is a statistical parameter to explore the thermodynamic properties such as Helmoltz free energy, Entropy, internal energy and heat capacity of a magnetic materials in a ferromagnetic system using the first and second dimensional Ising model.The results show that in one-dimensional model, phase transition does not occur at finite temperature but susceptibility of a system can be measured from the fluctuations in the magnetization in the second dimensional Ising model.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.
A magnetic property of the one dimensional spin 5/2 Ising model under the magnetic field has been investigated by means of transfer matrix method. Thermodynamic response functions are also obtained for varying values of scaling temperature and scaling magnetic field. Entropy and heat capacity of the system were calculated by benefiting from the temperature dependencies of Helmholtz free energy. Our simulation results demonstrate that as the strength of the magnetic field is increased, heat capacity tend to shift to the relatively higher temperature regions, and these findings are consistent with previous results for the low spin values in one dimensional Ising systems.
Many problems in statistical physics resist exact solution once the number of interacting particles becomes large, and the two-dimensional Ising model is a classic case where numerical simulation fills that gap. This paper reports a Monte Carlo study of the two-dimensional Ising model using the Metropolis algorithm, undertaken to examine how magnetization, internal energy, specific heat, and magnetic susceptibility change with temperature. Spin configurations were generated on a square lattice under periodic boundary conditions, with each proposed spin flip accepted or rejected according to the Boltzmann weight of the resulting energy change. The simulation reproduces the expected transition from an ordered ferromagnetic state at low temperature to a disordered paramagnetic state at high temperature, with magnetization falling sharply and both specific heat and susceptibility peaking near a critical temperature of Tc ≈ 2.3–2.4 (in units of J/kB), consistent with the exact value obtained analytically for this system. These results confirm that the Metropolis algorithm, despite its simplicity, remains a dependable tool for locating second-order phase transitions and extracting thermodynamic quantities in lattice spin systems, and the same computational framework can be adapted to more elaborate models used in condensed matter and materials research.
Using the Ising Model, the phase transition between ferromagnetic and paramagnetic phases of a 2-D square spin lattice are explored. A phase transition into the ferromagnetic phase occurs when the temperature of the system falls below some critical value. The ferromagnetic phase is characterized by a non-zero magnetization on the macroscopic level. Using the Mote Carlo method to obtain system's configurations (i.e., sets of microstates) in thermal equilibrium, we will explore system's macroscopic properties such as magnetization, magnetic susceptibility, and specific heat capacity as functions of temperature. We will evaluate the transition temperature, and discuss how the lattice size affects the results.
Parallel algorithm of partition function calculation of two-dimensional Ising model for systems with a finite number of spins was developed. Within a method of complete enumeration by using MPI technology with subsequent optimization of a parallel code time of calculations was reduced considerably. Partition function was calculated for systems of 16, 25, 36 Ising spins. Based on the obtained results, main thermodynamic and magnetic values dependences (such as heat capacity, magnetic susceptibility, mean square magnetization) for ferromagnetic and antiferromagnetic interactions was investigated. The analysis of a different configurations contribution showed, that states with the minimum energy have essential influence on dependences of thermodynamic values. Comparison with the results obtained by the Wang Landau algorithm was performed.
Abstract The one-dimensional Ising model with various boundary conditions is considered. Exact expressions for the thermodynamic and magnetic properties of the model using different kinds of boundary conditions [Dirichlet (D), Neumann (N), and a combination of Neumann–Dirichlet (ND)] are presented in the absence (presence) of a magnetic field. The finite-size scaling functions for internal energy, heat capacity, entropy, magnetisation, and magnetic susceptibility are derived and analysed as function of the temperature and the field. We show that the properties of the one-dimensional Ising model is affected by the finite size of the system and the imposed boundary conditions. The thermodynamic limit in which the finite-size functions approach the bulk case is also discussed.