Ferro- and Antiferromagnetism in a Cubic Cluster of Spins

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Abstract
The Heisenberg exchange Hamiltonian has been solved exactly for a cubic array of eight spins (each with spin-½). Both energy eigenvalues and thermodynamic functions have been calculated. A Curie temperature can be defined and its value determined as a function of the strength of first-, second-, and third-neighbor interactions. For the antiferromagnetic state no simple definition is found for the Néel temperature. The temperature of the susceptibility maximum is determined and is plotted as a function of the strengths of the second- and third-neighbor interactions. For some values of the exchange constants, spiral-type antiferromagnetic states exist.