Pseudopotential Theory for Interionic Interaction Potentials in Metals at Non-Zero Temperature. I. Elementary Derivation of Asymptotic Expressions on Basis of Hilbert Transforms

Abstract

An asymptotic expression V pair.as ( r ; T ) of a pair potential between ions in metals at non-zero temperature T ≠0 is given approximately by a form V pair.as ( r ; 0) D ( r ; T ), where V pair.as ( r ; 0)= V 3 cos (2 k F r )/(2 k F r ) 3 is the Friedel oscillatory potential at T =0, D ( r ; T )= x · cosech ( x ) with x =π( k B T / E F ) k F r is a damping factor, k B the Boltzmann constant, k F and E F the Fermi wave number and the Fermi level. Lighthill's method for estimating asymptotic form of Fourier transforms of f ( x ) that includes a singular point is proved to be applicable also to transformed functions \(F(x)= \int f(x-z)g(z; T) {\rm d}z\), where g ( z ; T )→δ( z ) as T →0. The proof is done by the use of Hilbert transforms, in which f ( x )=1/ x . This modified Lighthill's method derives V pair.as ( r ; T ) in the usual way that is used in pseudopotential treatment of V pair.as ( r ; 0). Earlier works by Kohn and Vosko, March and Murray, and Flynn and Odle are examined.