The thermal expansion of axial metals

The thermal expansion of axial metals is surveyed with particular reference to recent very low temperature data for zinc, cadmium, magnesium, β-tin, indium, bismuth and antimony. For a given solid, the independent thermal expansion coefficients α_⊥ and α_‖ are conveniently compared at low temperatures by plotting α_⊥/T ³ and α_‖/T ³ against T. The plots show clearly how contraction in one principal symmetry direction ensues from rapid expansion in the other direction. The Grüneisen functions γ_⊥ and γ_‖ are discussed for each solid using data from lattice-dynamical models and neutron-diffraction measurements. They show how features of the individual thermal expansions arise from a combination of vibrational, elastic and electronic effects. For these solids and for zirconium, titanium, yttrium and thallium the larger of γ_⊥ and γ_‖ at high temperatures is that referring to strains in the direction of the stronger forces (judged from the axial ratio). At lower temperatures the inequality reverses, except for tin, implying that the highest normal-mode frequencies depend most strongly on strain in the direction of the stronger forces. More experimental studies of thermal expansion at the lowest temperatures are desirable, as well as more second and third-order elastic constant data. Theoretical work is needed on realistic lattice models applicable to non-cubic metals.