An empirical finding is that the ratio of glass transition temperature $T_g$ to the crystal melting temperature $T_m$ is near $\frac{2}{3}$ for many glass-forming materials. The key to understand the origin of the $\frac{2}{3}$ rule is the availability of a theoretical method to determine $T_m$ and its relation to $T_g$. In this paper, we compare the mean-square displacements of caged molecules in the glass $\langle u_a^2\left(T\right)\rangle$ with that in the crystal $\langle u_c^2\left(T\right)\rangle$, and surmise whether $\langle u_a^2\left(T\right)\rangle$ at $T_g$ and $\langle u_c^2\left(T\right)\rangle$ at $T_m$ have comparable values. If the surmise $\langle u_a^2\left(T_g\right)\rangle \approx \langle u_c^2\left(T_m\right)\rangle$ is realizable, it offers a method to determine $T_m$ and $T_g/T_m$. Justification of the surmise is made possible by the recent temperature-modulated differential scanning calorimetry experiments [M. Pyda et al., J. Phys. Chem. Lett. 17, 61 (2026)] in finding a reversible heat capacity signature in melting similar in shape and magnitude to that observed at glass transition in molecular and polymeric glass formers. Hence, present at $T_m$ is a structural relaxation involving the same molecules as that at $T_g$. The result motivated us to verify the surmise from neutron scattering data of four materials, ortho-terphenyl, cis-decalin, selenium, and glycerol, and by Brillouin light scattering data of ortho-terphenyl. Thus, the relation $\langle u_a^2\left(T_g\right)\rangle \approx \langle u_c^2\left(T_m\right)\rangle$ can be used as a criterion of melting. Our melting parameter ${\mathrm{\Delta }}_m$ ≡$\sqrt{\langle u_c^2\left(T_m\right)\rangle }/d_{\text{NN}}\approx \sqrt{\langle u_a^2\left(T_g\right)\rangle }/d_{\text{NN}}$ with $d_{\text{NN}}$ as the nearest-neighbor distance obtained for several materials varies over the range from 0.05 to 0.32. It seems that molecules of larger size and with stronger interaction have larger ${\mathrm{\Delta }}_m$ values.

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• Crystal melting
• Glass transition
• Molecular solids
• Brillouin scattering & spectroscopy
• Neutron scattering