Temperature dependence of the elastic shear moduli of the cubic metals

A theoretical model for the temperature dependence of the elastic shear moduli of the cubic metals is presented which accounts for the quantitative relationship of these elastic properties to the thermal expansion. The particular phenomenon which the model is concerned with is the extremely rapid temperature variation of the shear moduli, which typically is about 25 times as great as the rate of thermal expansion. The model has the following salient features: (i) The interatomic pair pseudopotential is approximated over the important thermal range by means of a Morse potential; (ii) the thermal expansion is assumed to arise from anharmonic Morse oscillators which are further assumed not to be significantly coupled to each other; (iii) the exact quantum-mechanical solutions for the Morse oscillators are then combined with the Debye model to give an analytic formula for the thermal expansion; (iv) the elastic shear moduli are calculated from this, assuming nearest-neighbor interactions only. The calculated thermal expansions are found to agree quite well with the experimental values, particularly at low temperatures. The rapid temperature variation of the shear moduli is explained by the model in terms of the thermal movement of the nearest neighbors with respect to the interatomic-potential minima. Complete agreement between calculated and experimental shear moduli has been limited, however, by the contributions beyond the nearest neighbors which are not zero, and which have not been included in the formulas.