Rectangular d band model

Within the rectangular d band model the bond energy is proportional to the bandwidth, IF, through eqn (7.33). We may relate the bandwidth to... 

The heats of formation of equiatomic AB transition-metal alloys may be predicted by generalizing the rectangular d band model for the elements to the case of disordered binary systems, as illustrated in the lower panel of Fig. 7.13. Assuming that the A and transition elements are characterized by bands of width WA and WB, respectively, then they will mix together in the disordered AB alloy to create a common band with some new width, WAB. The alloy bandwidth, WAB may be related to the elemental bond integrals, hAA and , and the atomic energy level mismatch, AE — EB — EAt by evaluating the second moment of the total alloy density of states per atom ab( ), namely... 

Hence, within the rectangular d band model for the AB alloy density of states, from eqs (7.33) the bond energy becomes... 

We see that the simple rectangular d band model reproduces the behaviour found by experiment and predicted by Miedema s semi-empirical scheme. However, we must stress that the model does not give credence to any theory that bases the heat of formation of transition-metal alloys on ionic Madelung contributions that arise from electronegativity differences between the constituent atoms because in the metallic state the atoms are perfectly screened and, hence, locally charge neutral. Instead, the model supports... 

Fig. 8.12 The rectangular d band model of the (a) nonmagnetic, (b) ferromagnetic, and (c) antiferromagnetic states. (From Pettifor (1980).)...

This is the rectangular d-band model criterion equivalent to the exact second-order result, namely... 

The qualitative behavior predicted from the rectangular d band model can actually be seen in self-consistent DFT calculations, where the shape of the DOS is explicitly calculated. For example, we show this in the lower part of Figure 2.22 where the bulk and surface PDOS projected on to the valence d orbitals is plotted for bulk Zr and Ru and for Zr and Ru (0001) surfaces. For both Zr and Ru, the d PDOS associated with the surface atoms is narrower than that of the bulk. For Zr with <5 d electrons, the surface d PDOS moves down in energy ( 0.1 eV) relative to the... 

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