Theory of the Melting of Simple Metals: Application to Na

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Abstract
The melting curves of the simple metals may be calculated by a method which involves no adjustable parameters whatsoever if the electron-ion pseudopotential is known. This simplicity is made possible by the use of variational principles to determine the free energies of the solid and liquid states. The approach yields a volume- and temperature-dependent effective Debye temperature for the solid, and an effective volume- and temperature-dependent hard-sphere packing fraction for the liquid. As applied to Na, the method gives a melting curve in good agreement with experiment up to at least 40 kbar. The long-wavelength limit of the pseudopotential, which is not well known for most metals, is eliminated via the correct equilibrium density at 0 °K. (The melting curve is not, however, sensitive to the choice of this parameter.) Additional thermodynamic quantities are computed for both phases and along the melting curves in good agreement with available experiment. Lindemann's law is fairly well obeyed in the solid phase, although not perfectly, and its analog (i.e., constant hard-sphere packing fraction along the melting curve) holds in the liquid. The Lindemann ratio varies between 0.013 and 0.015 along the melting curve; the packing fraction is about 0.42. Above certain temperatures, there exist no Debye temperatures for which the free energy of the solid phase is stationary; following previous workers we interpret such temperatures as forming a line of mechanical instability which, however, lies far above the actual melting curve.

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