THE SURFACE DIELECTRIC FUNCTION AND ITS SUM-RULE FOR A SEMIINFINITE ELECTRON-SYSTEM

In this paper we define a surface dielectric function for a semi-infin ite electron system and a sum rule involving the imaginary part of the inverse of the surface dielectric function is derived. The sum rule h as almost exactly the same form as the well-known dielectric function f-sum rule in a bulk system, with the bulk quantities replaced by the corresponding surface quantities. However there is an important differ ence that the surface sum rule applies to all surfaces only in the Q - -> 0 limit, where Q is the two-dimensional wavevector parallel to the surface. For small Q, linear and higher order terms of Q will appear i n the sum rule whose coefficients may depend on the surface properties of the system considered. This is due to the fact that for Q = 0 both the surface dielectric function and the surface plasma frequency depe nd on the average electron density only, independent of the surface pr operties; however for Q not equal 0, the Q-dependence of the above two quantities both depend sensitively on the surface properties of the s ystem considered.