Einstein mean-square displacement ($⟨x^2⟩=\hslash \coth (T_E/(2T))/(2m{\omega }_E)$)

For one Cartesian direction in an Einstein solid, $⟨x^2⟩=\hslash [2⟨n⟩+1]/(2m{\omega }_E)$. Substituting the thermal oscillator occupation proves the displayed result. At $T\gg T_E$, equipartition gives $⟨x^2⟩=k_{B}T/(m{\omega }_E^2)={\hslash }^{2}T/(m{k}_B{T}_E^2)$. The total displacement variance of an isotropic atom is three times the one-coordinate result, a distinction that matters in applying a melting threshold.