Lindemann melting criterion in two dimensions

Abstract

It is demonstrated that the Lindemann's criterion of melting can be formulated for twodimensional classical solids using statistical mechanics arguments. With this formulation the expressions for the melting temperature are equivalent in three and two dimensions. Moreover, in two dimensions the Lindemann's melting criterion essentially coincides with the Berezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young melting condition of dislocation unbinding.

Sound Velocities of Lennard-Jones Systems Near the Liquid-Solid Phase Transition

“…The Lindemann’s criterion of melting can be expressed as [ 5 , 6 ] where is the Debye frequency, a is the characteristic interparticle separation (the Wigner–Seitz radius is used here), and C is expected to be a quasi-universal constant. In this form the Lindemann expression for the melting temperature applies to both 3D and 2D solids (the constants C can be different for the 2D and 3D cases) [ 6 ].…”

“…The Lindemann’s criterion of melting can be expressed as [ 5 , 6 ] where is the Debye frequency, a is the characteristic interparticle separation (the Wigner–Seitz radius is used here), and C is expected to be a quasi-universal constant. In this form the Lindemann expression for the melting temperature applies to both 3D and 2D solids (the constants C can be different for the 2D and 3D cases) [ 6 ]. The Debye frequency in 3D can be expressed via the longitudinal and transverse sound velocity as For soft repulsive interactions the strong inequality usually holds and this can be used to further simplify the melting condition [ 6 ].…”

“…An important consequence of this formulation is that the ratio of the transverse sound velocity to the thermal velocity is predicted to have a quasi-universal value along the melting curve (which can be different in 3D and 2D). This condition has been verified on soft repulsive interactions (in particular, using the Yukawa or exponentially screened Coulomb potential) in both 2D and 3D, where it works reasonably well [ 6 , 16 ]. A natural question arises whether this condition remains valid and useful also for potentials with a long-ranged attraction.…”

“…The high frequency (instantaneous) elastic moduli and the corresponding sound velocities are important characteristics of condensed fluid and solid phases, which affect and regulate wave propagation [ 1 ], the instantaneous Poisson’s ratio [ 2 ], the coefficient in the Stokes–Einstein relation [ 3 , 4 ], the Lindemann melting rule [ 5 , 6 ], as well as some other simple melting rules [ 7 ], the relaxation time in the shoving model [ 8 , 9 ], just to give few examples.…”

“…It has been recently demonstrated that the Lindemann’s criterion of melting can be re-formulated for two-dimensional (2D) classical solids using statistical mechanics arguments [ 6 ]. With this formulation the expressions for the melting temperature are equivalent in three dimensions (3D) and 2D.…”