Tutorial: Thermal expansion of aluminium

The main aim of this tutorial is to introduce the basic tools that allow the calculation of energies and free energies of crystals. We will do this with an example application: the calculation of the coefficient of thermal expansion of aluminium at zero pressure, as function of temperature, $\alpha_0 = 1/V (\partial V/\partial T)_{p=0}$. What will be learnt here will be easily applicable to the calculation of other thermodynamic properties of materials, which can all be derived from the free energy.

The pressure is defined by $p = -(\partial F / \partial V)_T$, and therefore the equilibrium volume at $p = 0$ is the one that minimises the Helmholtz free energy $F$. The main objective of this tutorial, therefore, will be to compute $F(V,T)$, and for each $T$ find the volume for which it assumes its minimum value.