Theory of Dipole Interaction in Crystals

TLDR

Dipole arrays can be represented as vectors in a high‑dimensional space, and their classical interaction energy is a quadratic form in the dipole components. The study proposes extending the analysis to more general dipole configurations in a subsequent paper. The method reduces energy calculation to diagonalizing the quadratic form, using basic arrays as eigenvectors, and expressing any array as a linear combination of these to obtain additive energies from the eigenvalues. The approach successfully solves the characteristic value problem for highly symmetric cubic lattices, yielding minimum‑energy arrays with and without external fields that qualitatively match de Haas and Wiersma’s experiments, though some differences are ascribed to quantum effects and incomplete saturation.

Abstract

It is shown that dipole arrays may be represented as vectors in a many-dimensional vector space. The classical dipole interaction energy is a quadratic form in the components of the dipole moments. Its calculation is reduced to the diagonalization of this form. The characteristic vectors are so called basic arrays. An arbitrary array may be decomposed into a linear combination of basic arrays, the energies are additive and may be obtained from the characteristic values of the quadratic form. The method is demonstrated by the complete solution of the characteristic value problem of a highly symmetric class of cubic arrays. The minimum energy arrays are obtained without and with an external magnetic field for the simple cubic, body-centered cubic, and face-centered cubic lattices. The results are in good qualitative agreement with the experiments of de Haas and Wiersma on Cs Ti alum. Some discrepancies are attributed to quantum effects and to incomplete saturation (entropy $S>0$). The extension to these more general cases will be considered in a following paper.