The Einstein oscillator model of the Wigner electron crystal takes a specified force constant, which is an explicit function of the electronic separationrs. Here a method is devised for its self-consistent calculation. For the gaussian density of the Wigner oscillator, with width λ in terms ofrs, experimental data on the melting temperatureTmversus Landau filling factorvis employed to plot the critical overlap parameter, λcsay, versusTm. This shows that λcincreases asTmdecreases, which is in contrast to Lindemann's Law.