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Journal of Engineering Research (Libya)

Ground State Properties of Frustrated Ferromagnetic Ising Chains in a Uniform-Random-External Magnetic Field.

Dr.MANSOOR Dr.MANSOOR, A-Z. HABEEB

Published Jun 30, 1995• 0 citations

Abstract

The methods of Williams,Doman and Williams,and Doman and Habeeb are used to describe the ground state properties of infinite ferromagnetic Ising chains in a random field.Analytic results for the ground state energy,the magnetisation per spin in the direction of the local field and the entropy as a function of magnetic field are obtained. When the probability factor of the zero local field is zero,our results are in agreement with thut obtained previously by Williams. terminators.A terminator could be defined as the simplest group of spins whose direction is indepen- dent of that of its neighbours,but depends on the value of the magnetic field.The behaviour of the system is now dominated by these subchains and so we take into account all possible configurations for such subchains.The idea of superspins and superb- onds introduced by Williams(1981)and applied later to examine the properties of many Ising

DOI: 10.66411/jer.v4i4.384

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The methods of Williams,Doman and Williams,and Doman and Habeeb are used to describe the ground state properties of infinite ferromagnetic Ising chains in a random field.Analytic results for the ground state energy,the magnetisation per spin in the direction of the local field and the entropy as a function of magnetic field are obtained. When the probability factor of the zero local field is zero,our results are in agreement with thut obtained previously by Williams. terminators.A terminator could be defined as the simplest group of spins whose direction is indepen- dent of that of its neighbours,but depends on the value of the magnetic field.The behaviour of the system is now dominated by these subchains and so we take into account all possible configurations for such subchains.The idea of superspins and superb- onds introduced by Williams(1981)and applied later to examine the properties of many Ising

For pt.II see ibid., vol.15, p.1693 (1982). The methods developed by Williams (1981) to describe the ground-state properties of a random-bond infinite Ising chain are extended to apply to a system of finite-length random-bond Ising chains. It is shown how to evaluate the magnetisation, ground-state energy, and entropy as a function of magnetic field. It is shown that the steps in the magnetisation and entropy when B=2J/r behave differently for r even and r odd. The magnetisation and entropy remain finite as B becomes very small.

The transfer matrix method is used to describe the properties of the random bond of a simple periodic frustrated Ising chain in a uniform magnetic field. Doman and Williams(1982) are found expressions for the thermodynamic functions of the system at low temperature. Their methods is extended to apply to a system of five-antiferromagnetic bond Ising chain terminated by a ferromagnetic bond at low temperatures. New expressions for the magnetization, energy, and entropy as a functions of magnetic field is found. All results are in agreement with the previous results.

Recently published methods used to examine the ground-state properties of the random-bond Ising chain in a uniform magnetic field are extended to describe the system at low temperatures. The entropy spikes broaden asymmetrically, and the excitation energy and specific heat become asymmetric double-peaked functions of the magnetic field with zero values between the peaks at B=2J/r for integer r. The magnetisation steps at B=2J/r are smoothed out asymmetrically. Both the ground-state and finite-temperature properties of a simple periodic frustrated Ising chain are investigated using the transfer-matrix method. The ground-state properties are similar to those of the random bond chain; the ground-state entropy, for instance, is finite for J<or=B<or=2J, with spikes at J and 2J.

The properties of the ground state of one of the simplest models of frustrated magnetic systems, a dilute Ising chain in a magnetic field, are considered for all values of the concentration of charged non-magnetic impurities. An analytical method is proposed for calculating the residual entropy of frustrated states, including states at the boundaries between the phases of the ground state, which is based on the Markov property of the system under consideration and allows direct generalization to other one-dimensional spin models with Ising-type interactions. The properties of local distributions and concentration dependences of the composition, correlation functions, magnetization and entropy of the phases of the ground state of the model are investigated. It is shown that the field-induced transition from the antiferromagnetic ground state to the frustrated one is accompanied by charge ordering and the absence of pseudo-transitions in the dilute Ising chain is proved.

The features of the response of frustrated states to the external field are considered on the example of a diluted Ising chain. In the ferromagnetic case, partial ordering occurs, which leads to a decrease in entropy. In the antiferromagnetic case, the switching on of the field leads to the appearance of a long-range order in the system, although the state remains frustrated. A mapping of a one-dimensional spin model to a Markov chain is proposed, which makes it possible to study in detail the properties of correlation functions and local distributions of the states in the spin chain.