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Dynamics of adatoms interacting with metals and metal surfaces: The validity of the description given by Kramers' equation
Author(s) -
Mccaskill J. S.,
March N. H.
Publication year - 2009
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560180868
Subject(s) - constant (computer programming) , range (aeronautics) , perturbation (astronomy) , kinetic energy , perturbation theory (quantum mechanics) , physics , fick's laws of diffusion , classical mechanics , diffusion , chemistry , statistical physics , thermodynamics , quantum mechanics , materials science , computer science , composite material , programming language
The validity of the description of the dynamics of an adatom in a metal afforded by Kramers' equation is examined. The theory clearly involves the assumption that has been stressed previously by others that the kinetic energy of the moving adatom can be treated as a perturbation. This is equivalent to the use of an expansion parameter which is the ratio ( m/M ) 1/2 , M being the adatom mass and m the medium particle mass. By examining the derivation, it is pointed out that though this is a formally correct parameter, exploited for example by Lebowitz and Rubin, a screening length actually enters a more precise inequality for the range of validity. Second, the specific metal effects are shown to be subsumed into the phenomenological constants in Kramers' equation: in terms of an effective force, which in linear response involves the dielectric function & epsilon ( k ), and in terms of a modified diffusion constant (friction constant). This discussion is for the case of a single adatom. An expression is given for the averaged friction constant in a simple model of the surface, based on a barrier model of the metal and the Born approximation. In principle, the average force can be determined in this same model. Therefore, a local Fokker‐Planck equation is written in terms of local effective force and friction constant. Finally, a situation where adatoms interact is considered. It is emphasized that in an electron gas model the friction constant in the bulk has a long‐range contribution which varies asymptotically as cos 2 k f a/a 2 , arising from the scattering of Fermi electrons off a pair of adatoms at separation a, k f being the Fermi wavenumber. This is in contrast with the effective energy of interaction mediated by the electrons, which falls off as cos 2 k f a/a 3 . A brief discussion, though now restricted to the lowest‐order Born approximation, allows a formula to be written for the friction constant as a function of the position of the adatoms in the density gradient at a metal surface. This involves the two‐center potential screened by the dielectric constant, but now calculated in the presence of the surface. The evaluation of the friction constant is thereby reduced to quadratures on the screened potential.