Classical localization of an unbound particle in a two-dimensional periodic potential and surface diffusion

Abstract

In periodic, two-dimensional potentials a classical particle might be expected to escape from any finite region if it has enough energy to escape from a single cell. However, for a class of sinusoidal potentials in which the barriers between neighboring cells can be varied, numerical tridiagonalization of Liouville's equation for the evolution of functions on phase space reveals a transition from localized to delocalized motion at a total energy significantly above that needed to escape from a single cell. It is argued that this purely elastic phenomenon increases the effective barrier for diffusion of atoms on crystalline surfaces and changes its temperature dependence at low temperatures when inelastic events are rare.

Approximating densities of states with gaps

“…We formulate the problem of an extended quantum system as a three-term recurrence (TTR). The advantage of this approach is that any linear, Hermitian, wave-equation can be expressed in this form, not just Schrödinger's [2] and Heisenberg's [5] equations, but Maxwell's [6] equations and even Liouville's [7] equation for the classical evolution of distributions in phase space, can be transformed into a TTR. In this approach, the state of energy z is {ψ n (z)} which satisfies the TTR,…”

Densities of states, moments, and maximally broken time-reversal symmetry

“…19 can be evaluated for complex E and produces both the physical and second sheets of R 0 (E). As a numerical example this approximation is applied to the PDoS sech(πΕ) which arises in classical diffusion [6]. This is a case where the PDoS extends to infinite energy, but it satisfies the criterion for a 26 single band in Sec.…”

“…Another example is classical mechanics in the harmonic approximation where the coefficients of orbitals are replaced by displacements of atoms from equilibrium positions, and H is replaced by the dynamical matrix [5]. Finally, these ideas apply generally to classical mechanics in the Liouville formulation where functions of the dynamical variables, as well as the operators which act on those functions, obey linear equations of motion [6].…”