Order-parameter saturation at low temperatures: Displacive phase transitions with coupled Einstein oscillators

Quantum saturation of displacive order parameters may be observed at temperatures higher than expected in a simple soft-mode model. The experimental results have been explained sometimes by coupling of the softphonon branch with hard mode branches. We analyse the effect of this coupling in the limit of coupled Einstein oscillators. It is shown that a significant change through this mechanism of the low-temperature quantum saturation of the order parameter requires abnormally large coupling terms. In any case, under favourable circumstances, the coupled hard-phonon spectrum can be derived from the second temperature derivative of the square of the measured order parameter.