Phonon-induced phase transition in a classical Heisenberg chain

The linear chain of classical Heisenberg spins coupled to phonons is studied. The coupling arises from the phonon-modulation of the nearest-neighbor exchange integral and the cases of both direct exchange and superexchange are treated. The phonons lead to an effective biquadratic interaction between spins, with important physical consequences. Thermodynamic functions at finite temperatures and zero field are evaluated exactly and discussed. It is shown that antiferromagnetically coupled spins in an external field at $T=0\mathrm{}°$K can undergo a first-order phase transition; as the field crosses a critical value, the magnetization and lattice spacing change discontinuously.