Hybrid NFE-TB bands

Transition metals are characterized by a fairly tightly bound d band of width W that overlaps and hybridizes with a broader nearly-free-electron sp band as illustrated in Fig. 7.4. This difference in behaviour between the valence sp and d electrons arises from the d shell lying inside the outer valence s shell, thereby leading to small overlap between the d orbitals in the bulk. 

4 A schematic representation of transition metal sp (dashed curve) and d (solid curve) densities of states when sp-d hybridization is neglected. 

For example, from eqn (2.55), the average radial distance of the hydrogenic 3d and 4s wave functions are in the ratio 0.44 1. Thus we expect the band structure of transition metals to be represented accurately by a hybrid NFE-TB secular equation of the form 

The band structure of nonmagnetic fee and bcc iron is shown in Fig. 7.5, being computed from the hybrid NFE-TB secular equation with resonant parameters Ed = 0.540 Ry and = 0.088 Ry. The NFE pseudopotential matrix elements were chosen by fitting the first principles band structure derived by Wood (1962) at the pure p states Nv (tiuo = 0.040 Ry), L2 ( U1 = 0.039 Ry), and X (t 200 = 0.034 Ry). Comparing the band structure of iron in the 100 and 111 directions with the canonical d bands in Fig. 

we see that there is only the one level with symmetry At and Ab respectively, which hybridizes or mixes with the lowest NFE band, with the other four d levels of different symmetry unperturbed. Because of the canonical nature of the pure d bands reflected in eqn (7.24), the band structure of aU the nonmagnetic fee and bcc transition metals will be very similar to that shown in Fig. 7.5 for iron. 

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Fig. 7.5 The hybrid NFE-TB band structure of fee and bee iron in the nonmagnetic state. The solid circles represent the first principles energy levels of Wood (1962). (From Pettifor (1970e).)...

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