Instability in Crystals and Thermal Expansion Coefficients

An analysis is made of the Gr\"uneisen parameter γᵢ=-dlnνᵢdlnV, where νᵢ is a normal mode frequency and V is the volume for certain ideal and real crystals. It is shown that negative thermal expansion or anomalous positive expansion in a solid is a possible result of an approach to instability of the modes of vibration.