Axial thermal expansion from free-energy minimization with temperature-dependent force constants in phonopy: a technical report, with $α$- and $ω$-Ti as the worked example

In a hexagonal crystal the $a$ and $c$ axes change their lengths at different rates with increasing temperature, and the thermal expansion is described by two coefficients, one for each axis. The equilibrium lattice parameters at each temperature are found by minimizing the Helmholtz free energy ove