The Lindemann and Grüneisen Laws

Abstract

The Lindemann assumption of direct contact of neighboring atoms at fusion is replaced by the criterion that melting occurs when the root-mean-square amplitude of thermal vibration reaches a critical fraction $\ensuremath{\rho}$, presumed the same for all isotropic monatomic solids, of the distance of separation of nearest-neighbor atoms. The Debye-Waller theory of the temperature dependence of the intensity of Bragg reflection of x-rays is used, without further assumptions, to derive a generalized Lindemann law. In contrast to the Lindemann form, all physical quantities involved in this formulation are evaluated at the fusion point, and departure of the average energy of an atomic oscillation from the equipartition value is taken into account by the quantization factor of the Debye-Waller theory. If the Gr\"uneisen constant ${\ensuremath{\gamma}}_{m}$ of the solid at fusion is evaluated by its definition from the Debye frequency of the solid, use of the generalized Lindemann law and Clapeyron's equation permits one to express ${\ensuremath{\gamma}}_{m}$ in terms of the bulk modulus of the solid at melting and the latent heat and volume change of fusion. By means of Gr\"uneisen's law applied to the solid at fusion, ${\ensuremath{\gamma}}_{m}$ can be expressed likewise in terms of the corresponding bulk modulus, thermal expansion, volume, and heat capacity at constant volume; the two evaluations of ${\ensuremath{\gamma}}_{m}$ connect the Lindemann and Gr\"uneisen laws. These relations permit one to evaluate the slope and curvature of a fusion curve as functions of ${\ensuremath{\gamma}}_{m}$, and thus to express in terms of ${\ensuremath{\gamma}}_{m}$ the conditions that a fusion curve be normal in the sense of Bridgman. Experimental fusion data on 13 cubic metals are used to evaluate the constant of proportionality (inversely proportional to $\ensuremath{\rho}$) in the Lindemann frequency; the values are reasonably constant. The corresponding values of $\ensuremath{\rho}$ for Al and Cu show reasonable agreement with values deduced from x-ray intensity measurements. The average $\ensuremath{\rho}$ seems significantly below values estimated by Gr\"uneisen. Good agreement with the cubic metals is found for hexagonal close-packed elements, but not for elements with more complex lattices, in general. The two evaluations of the Gr\"uneisen constant ${\ensuremath{\gamma}}_{m}$ of the solid at fusion are shown to be in good agreement, experimentally, for 14 elements checked, but agreement fails for the elements Ga, Bi, and Sb, which show abnormal fusion curves.