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The Journal of Chemical Physics

Statistical Mechanics of Nearest Neighbor Systems II. General Theory and Application to Two-Dimensional Ferromagnets

Elliott W. Montroll

Published Jan 1, 1942• 40 citations
Publisher

Abstract

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.

DOI
10.1063/1.1723622

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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.
In an effort to make statistical methods available for the treatment of cooperational phenomena, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics. A method is developed which yields the partition function as the largest eigenvalue of some finite matrix, as long as the manifold is only one dimensionally infinite. The method is carried out fully for the linear chain of spins which has no ferromagnetic properties. Then a sequence of finite matrices is found whose largest eigenvalue approaches the partition function of the two-dimensional square net as the matrix order gets large. It is shown that these matrices possess a symmetry property which permits location of the Curie temperature if it exists and is unique. It lies at JkTc=0.8814\frac{J}{k{T}_{c}}=0.8814kTc​J​=0.8814 if we denote by JJJ the coupling energy between neighboring spins. The symmetry relation also excludes certain forms of singularities at Tc{T}_{c}Tc​, as, e.g., a jump in the specific heat. However, the information thus gathered by rigorous analytic methods remains incomplete.
Using a method suggested by Montroll, we extend the well-known matrix formulation of the nearest-neighbor one-dimensional Ising problem to allow for interactions with an arbitrary finite range n, general spin l, and an applied magnetic field B. We exhibit the relevant matrix element explicitly and hence formally obtain the partition function via an eigenvalue problem of order (2l + 1)n. For the case B = 0, l = ½ we introduce a change of variable which simplifies the partition function while still allowing a matrix formulation. Using this approach we have computed specific-heat curves for infinite, ferromagnetic Ising chains with interactions of range n (n ≤ 7). We prove in an appendix that open and cyclic boundary conditions are equivalent for the system under consideration.
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.
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.
The static properties are investigated by computer simulation for the square Ising ferromagnet in which the nearest-neighbor interactions are ferromagnetic and the second interactions antiferromagnetic. Energy and magnetization at various temperatures are obtained by Monte Carlo calculations performed on the lattice with 20×20 spins. The Curie point determined as the inflection point of the energy-vs-temperature curve agrees with the result by Fan and Wu rather than other analytical approximations. The behavior of the spontaneous magnetization M 0 ( T ) for T → T c -0 as characterized by the critical exponent β is shown by comparing the data with an analytical equation for the magnetization. The exponent β remains probably unchanged from the original value 1/8.
The partition functions of the finite Ising models of spin S =1, 3/2, 2, 5/2 and 3 have been calculated by a high speed computer. The periodicity condition at the boundaries has been imposed, and only the nearest neighbor interaction has been assumed. The specific heats of these finite systems have been calculated as a function of temperature. The maxima of the specific heat increase proportionally to long N in the two-dimensional Ising model with spin S =1, N being the number of lattice points, which indicates that the specific heat for the above lattice has a singularity such as C v / k ∼ A log | T - T c |+ B ± : A ∼0.10 and B - - B + =0.24. The difference ( B + - B - ) has been estimated from the asymmetric distribution of the zeros of the partition function in the complex temperature plane.