Rectangular band model

An interesting feature of this simple rectangular band model with ah = is that the cohesive energy in eqn (7.44) is independent of the coordination number,, so that the diamond, x — A, simple cubic, ( = 6), and close-packed, (x = 12), lattices, for example, would all be equally stable. The origin of this unexpected result may be traced back to the form of the binding energy, namely... 

The rectangular band model posits that the density of states is constant over some range of energies and zero otherwise. The constant value associated with the density of states is chosen such that... 

Figure 5. Dependence of rate of dissolution of 5pM Y-FeOOH in pH 4.0, 0.01M NaCl on concentration of a) tartaric acid, and b) salicylic acid. Fitted parameters obtained for rectangular hyperbolic model are given. Light source mercury arc lamp with 365nm band-pass filtering.

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... 

Thus, evolution of semiphenomenological molecular models mentioned in Section V.A (items 1-6) have led to the hat-curved model as a model with a rounded potential well. This model combines useful properties of the rectangular potential well and those peculiar to the field models based on application of the parabolic, cosine, or cosine-squared potentials. Namely, the hat-curved model retains the main advantage of the rectangular-well model—its possibility to describe both the librational and the Debye-relaxation bands. 

Fig. 4.16. Comparison of exact and rectangular band densities of states for osmium (adapted from Harrison (1980)) (a) density of states for osmium as computed using density functional theory and (b) density of states constructed by superposiug a rectaugular baud model with a free-electrou-bke deusity of states. Rising Unes in the two curves are a measure of the integrated density of states.

Rectangular Density of States Model for Electronic Entropy In this problem we imagine two competing structures, both characterized by electronic densities of states that are of the rectangular-band-type. In... 

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