Past exam of the condensed matter physics course of the University of Oxford / 2019 / 2 / f

Square the stated Lindemann melting criterion and insert the high-temperature displacement:Thus, with $m_{\mathrm{Ar}}=(0.040\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1})/N_A$,Here $T_m/T_E\simeq 1.66$, so the use of the high-temperature approximation is only moderately justified. It is an estimate, not a controlled $T/T_E\to \infty$ limit. Retaining the exact Einstein mean-square displacement in the same heuristic criterion gives about $97\mathrm{K}$, indicating the size of that particular correction. The empirical threshold $0.05d$ and the independent-oscillator model introduce additional uncertainty.