Thermodynamic Behavior of an Ideal Ferromagnet

TLDR

Spontaneous magnetization of an ideal Heisenberg ferromagnet follows a power‑series law in reduced temperature, with coefficients computed for cubic lattices. The study calculates the free energy of an ideal Heisenberg ferromagnet as a temperature power series using previously developed mathematical techniques. The authors apply the mathematical framework from an earlier paper to expand the free energy in powers of temperature. The expansion shows that the leading terms stem from noninteracting spin waves, while corrections from effective mass variation and spin‑wave interactions appear at higher orders, with the interaction term proportional to a second virial coefficient of order \(T^{5/2}\) and higher‑order interactions being negligible, contradicting earlier predictions.

Abstract

The free energy of an ideal Heisenberg-model ferromagnet is calculated as a power series in the temperature $T$, using the mathematical machinery developed in an earlier paper. The spontaneous magnetization in zero external field is given by $[\frac{M(T)}{M(0)}]=S\ensuremath{-}{a}_{0}{\ensuremath{\theta}}^{\frac{3}{2}}\ensuremath{-}{a}_{1}{\ensuremath{\theta}}^{\frac{5}{2}}\ensuremath{-}{a}_{2}{\ensuremath{\theta}}^{\frac{7}{2}}\ensuremath{-}{a}_{3}{S}^{\ensuremath{-}1}{\ensuremath{\theta}}^{4}+O({\ensuremath{\theta}}^{\frac{9}{2}}).$ Here $\ensuremath{\theta}$ is the temperature in dimensionless units, and ${a}_{0}$, ${a}_{1}$, ${a}_{2}$, ${a}_{3}$ are positive numerical coefficients which are computed for the three types of cubic crystal lattice. The first two terms are the result of the simple Bloch theory in which spin waves are treated as noninteracting Bose particles with constant effective mass. The ${a}_{1}$ and ${a}_{2}$ corrections come from the variation of effective mass with velocity. The ${a}_{3}$ term is the lowest order correction arising from interaction between spin waves. This result is in violent contradiction to earlier published calculations which gave interaction effects proportional to ${T}^{\frac{7}{4}}$ and ${T}^{2}$.The smallness of the thermodynamic effects of spin-wave interactions is discussed in physical terms, and partially explained, in the introduction of this paper. A general proof is given that the thermodynamic effects of the "exclusion principle," which forbids more than ($2S$) spin deviations to occupy the same atom, are of order $\mathrm{exp}(\ensuremath{-}a{\ensuremath{\theta}}^{\ensuremath{-}1})$ and give zero contribution to any finite power of $\ensuremath{\theta}$. The residual dynamical interaction between 2 spin waves gives rise to a second virial coefficient ${{b}_{2}}^{\ensuremath{'}}$ which is calculated and shown to be of order ${T}^{\frac{5}{2}}$. The ${a}_{3}$ term in the magnetization is proportional to ${{b}_{2}}^{\ensuremath{'}}$. Effects of interaction of 3 or more spin waves are estimated and found to be of order ${\ensuremath{\theta}}^{5}$ or higher.