Debye-Waller factor, compressibility sum rule, and central peak at structural phase transitions

The temperature dependence of the Debye-Waller factor (DWF) near the critical point of ferroelectrics, "antiferroelectrics," $A15\mathrm{}$ structure compounds, etc., is investigated. Using the compressibility sum rule it is shown that the critical part of the DWF exponent quadratic in the momentum transfer is rigorously determined by the renormalized static phonon frequencies and thus will not be directly affected by the occurrence of a central peak. It is found that the mean-square particle displacement has a cusp at $T_c$, rather than the critical divergence predicted by various authors. The critical exponents associated with the cusp are estimated from scaling arguments and related to those of the specific heat. The extent to which this cuspshaped anomaly might be detected experimentally is briefly discussed. The results obtained here are also relevant for the electron-paramagnetic-resonance linewidth in the "slow-motion" regime.