The difference in total energy of a crystal with and without a vacancy involves essentially three terms: (i) The change in the one-electron eigenvalues due to scattering of conduction electrons off the vacant site. (ii) The self-energy of the displaced charge. (iii) The change in exchange and correlation energies of the electron gas. We have investigated the contributions (i) to (iii) for Cu, Mg, Al and Pb. The change in the one-electron eigenvalues is shown to be insensitive to the Bloch wave character of the wave functions and also to the choice of the repulsive potential
V
(
r
) representing the effect of the vacancy on the conduction electrons. There is thus no difficulty in evaluating contribution (i) for metals of different valencies. In contrast, the self-energy of the displaced charge is shown to depend very sensitively on the choice of
V
(
r
), and it is, therefore, essential to make the calculation self-consistent. This we have done by properly screening the negative of the point ion fields for Cu
+
to Pb
4+
. The radial wave functions and phase shifts for the low order partial waves have been evaluated, and self-consistent displaced charges obtained. The exchange energy has been estimated from a Dirac–Slater type of approximation and is again not sensitive to the detailed form of the displaced charge, while the change in correlation energy is found to be unimportant in determining the vacancy formation energy. The formation energies for the polyvalent metals are in satisfactory agreement with experiment. Some results for excess resistivities due to vacancies in metals are also presented. Here, in contrast to the calculation of the formation energies, it is essential to account for the Bloch wave character of the electron waves scattered by the vacancy. It is also proposed that the displaced charge round a vacancy may be a useful building block (or pseudoatom) for forming the crystal density.
NONCOMMUTATIVE U(1) SUPER-YANG–MILLS THEORY: PERTURBATIVE SELF-ENERGY CORRECTIONS
