Peierls and spin-Peierls phase transitions in systems with coupled softening modes

Peierls and spin-Peierls-type phase transitions are studied in quasi-one-dimensional systems where the distortion of the lattice is described by more than one normal mode. A model is chosen to describe this situation in the case of two coupled normal modes. A possibility of intrinsic structural instability is also included. A mean-field treatment is effected for a three-dimensional lattice and the following results have been obtained as a function of $\Gamma$, the strength of the coupling between the two normal modes: (1) The distortions corresponding to each normal mode occur at the same temperature, i.e., the transition temperature is unique; (2) the transition temperature increases when $\Gamma$ decreases; finally, (3) for negative coupling energies ($\Gamma <0\mathrm{}$) there exists a value ${\Gamma }_c$, of $\Gamma$, such that for $\Gamma <{\Gamma }_c$ the transition is of first order and for $\Gamma >{\Gamma }_c$ the transition is of second order. This model is applied to the alkali-metal tetracyanoquinodimethanide salts. It is found that, qualitatively, the dependence of the transition temperature on cation size and the order of the phase transition can be adequately understood in terms of the present model.