The calculations that you performed in the previous task allowed you to compute the equilibrium volume of aluminium at zero temperature, and without including quantum effects (see below). We now want to compute the equilibrium volume as function of temperature, so that we will be able to extract the coefficient of thermal expansion. In order to do that we have to compute the Helmholtz free energy of the system, and find its minimum with respect to volume for the chosen temperature. We will start to compute free energies using the harmonic approximation that we discussed in the previous chapter. The crucial point here is that the harmonic frequencies depend on volume (and for this reason often people like to refer to it as the quasi-harmonic approximation) and so calculations performed at different temperatures will produce different values of free energy. In general, vibrational frequencies are lower at higher volumes, and since the free energy decreases monotonically with decreasing phonon frequencies and increasing temperature, the net effect of temperature is that of increasing the equilibrium volume and therefore cause expansion upon heating.
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