Thermal Expansion and Structure of Anisotropic Monatomic Solids

An explanation is given for the anisotropy of both the structure and thermal expansion of certain monatomic crystals. Only those crystals are considered which have a face-centered-tetragonal structure with four atoms per unit cell. The anisotropic properties are explained by the introduction of an effective atomic pair interaction which is itself anisotropic. The interaction energy φ between any two atoms of the crystal is assumed to be of the form φ=-A^'Rᵐ(1+ε^'P₂)+B^'Rⁿ, where R is the distance between the atoms, A^', B^', m, n, and ε^' are constants, and P₂=-1/2(1-3cos²θ), where θ is the angle between the line joining the two atoms and one of the principal axes of the crystal. This interaction is summed over all pairs in order to obtain the total energy U of the crystal at the absolute zero of temperature. The contribution to the energy of the 0^∘{}K lattice vibration is neglected. The structural anisotropy is then seen to result from the condition that U be an absolute minimum at zero pressure. This structural anisotropy is found to depend only on the value of ε^' and disappears completely in the case ε^'=0. With the structure given, the linear thermal expansion is calculated along the principal axes of the crystal. Computation is made possible by expressing the three isobaric linear thermal-expansion coefficients in terms of the interaction energy. The anisotropy of the thermal expansion is likewise found to depend only on ε^' and to disappear in the case ε^'=0. Applications of the theory are made to indium and γ-manganese.