The field round an impurity, screening and virtual bound states

5 The field round an impurity screening and virtual bound states 

If a positive charge ze is immersed in a degenerate electron gas, the Coulomb field is screened by the electrons. The screening was first estimated by the present author using the Thomas-Fermi approximation (Mott 1936, Mott and Jones 1936, p. 86), and by this method one finds for the potential energy of an electron 

This formula for the screened Coulomb field has been used for many applications it gives, for instance, a fairly satisfactory description of the residual resistance due to a small concentration of Zn, Ga, etc. in monovalent metals (Mott and Jones 1936, p. 293). Friedel (1956) points out, however, that the use of the Bom approximation gives too large a value of the scattering, and better values are obtained if one calculates the phase shifts exactly. 

More recent work has shown, however, that an exponential decay of the screening potential as in (21) is not correct, and that round any scattering centre the charge density falls off as r 3 cos 2kFr. This we shall now show, by introducing the phase shifts t/, defined as follows (cf. (13)). Consider the wave functions Fx of a free electron in the field of an impurity. These behave at large distances from the impurity as (Mott and Massey 1965) 

We can evaluate the sum by multiplying by R dk/n and integrating, which leads to the expression 

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