On the existence of a sharp transition in the coupled spin-¹/₂-phonon system

The existence of abrupt changes in the system containing a single spin-1/2 coupled with phonons is examined. A simple random-phase approximation analysis indicates that the transition between the weak-coupling and the self-trapping regime is of the second-order type. The consequences of this are the Curie-Weiss behaviour of the statistical susceptibility and the softening of the phonon mode when the critical point is approached. The abruptness of the transition depends mainly upon the ratio between the average phonon frequency and the free-spin precession frequency. In the self-trapping regime a theory, analogous to the small-polaron problem, is developed for the ground state and the dynamical susceptibilities. Theoretical results are shown to be in good agreement with exact computer data which are obtained using direct diagonalisation when phonon dispersion is not presented. The relevance of the theory presented here to the experiments on para-elastic defects is discussed.