Thermal expansion of germanium and silicon at low temperatures

The linear expansion coefficients (α) of silicon and germanium are reported at temperatures down to ca. 0/50. At temperatures below 16°K, the expansion coefficients are very small but are positive and hence the Grüneisen parameter γ=3α. VK/C (K is the bulk modulus and C/V is heat capacity per unit volume) is positive. At T∽0/15, γ has a minimum value of —0·1 for germanium and —0·42 for silicon. At higher temperatures α and γ become positive and agree with data of Gibbons. Individual γ-values for the transverse acoustic modes must be reduced considerably by dispersion to account for the deep minimum in γ(T).