A qualitative investigation of the wave functions of metallic uranium

Abstract

The potential field and wave functions in metallic uranium have been calculated approximately by determining Hartree self-consistent fields for the four configurations (6 d ) 6 , (5 f ) 2 (6 d ) 2 (7 s ) 2 , (5 f ) 4 (6 d ) 2 , and (5 f ) 6 using the Wigner–Seitz boundary condition at the surface of the equivalent atomic sphere. In the self-consistent field obtained for each configuration, wave functions falling to zero at the surface of the sphere were evaluated for the outer electrons. The differences between the Hartree ∊-parameters for the two boundary conditions were used to give an indication of the relative band widths. The (5 f ) wave function has its principal maximum inside the (6 s ) (6 p ) shell, but is appreciable at the surface of the sphere and must participate in bonding. The binding energy of an electron in the (5 f ) wave function is 0.3934 rydbergs in the configuration (5 f ) 2 (6 d ) 2 (7 s ) 2 as given by the Hartree ∊-parameter. The (6 d ) function has its maximum charge density at the boundary and, with a binding energy of 0.1749 rydbergs in the same configuration, is likely to form a good metallic band. The (7 s ) function has a large negative binding energy and is not likely to occur. In the configuration (5 f ) 2 (6 d ) 2 (7 s ) 2 the band widths of the (5 f ), (6 d ) and (7 s ) functions are in the ratio 1:9∙8:22∙7. As the number of (5 f ) electrons is increased all the binding energies decrease and the band widths increase. At the same time the (6 s ) and (6 p ) functions are markedly perturbed, the functions (5 s ), (5 p ) and (5 d ) to a lesser extent.