Empty lattice approximation

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered ... 

In various forms, lattice-gas models permeate statistical mechanics. Consider a lattice in which each site has two states. If we interpret the states as full or empty , we have a lattice-gas model, and an obvious model for an intercalation compound. If the states are spin up and spin down , we have an Ising model for a magnetic system if the states are Atom A and Atom B , we have a model for a binary alloy. Many different approximation techniques have been derived for such models, and many lattices and interactions have been considered. 

We can understand the behaviour of the binding energy curves of monovalent sodium and other polyvalent metals by considering the metallic bond as arising from the immersion of an ionic lattice of empty core pseudopotentials into a free-electron gas as illustrated schematically in Fig. 5.15. We have seen that the pseudopotentials will only perturb the free-electron gas weakly so that, as a first approximation, we may assume that the free-electron gas remains uniformly distributed throughout the metal. Thus, the total binding energy per atom may be written as... 

The pore space of a unimodal real material is represented by a three-dimensional cubic lattice, with unoccupied lattice sites considered "nodes", cubic site faces considered "bonds" and occupied sites considered solid matrix. The connectivity of a particular site is defined as the number of unoccupied site neighbours it has. For example, in the case of a completely unoccupied lattice all the sites (nodes) would have a connectivity of 6. For a cubic lattice, Elias-Kohav et al.[ ] have described a method for the determination of tortuosity. The tortuosity is approximated by the number of sideways diversions that a molecule needs to proceed in the void (unoccupied cubic sites). If M is the locally averaged number of blocked lattice sites adjacent to an empty site, then the probability of a one site diversion is M/6. M is obviously analogous to six minus the so-called connectivity of the lattice. After such a move there is a similar probability of a further diversion and when M does not vary with every diversion the local tortuosity after n steps is ... 

Table 5.4. Bragg angles calculated using the second approximation of the unit cell dimensions for LaNi4 85Sno,i5 obtained after a least squares refinement of lattice parameters employing the data from Table 5.3 a = 5.047, c = 4.017 A. Empty cells in the table correspond to the combinations of indices, which cannot be observed in the range 0 < 20 < 83°.

The distance from the nearest iron atom to the interstitial site is 1.4 A and 1.8 A respectively for body-centred and face-centred polymorphs. While the site in the body-centred cubic lattice is too small to accommodate the carbon without significant distortion, the site in the face-centred cubic lattice is suitable, and the carbon can be accommodated. Approximately one-third of the empty octahedral sites can be occupied, giving the formula FejC. 

The remaining empty cells on the lattice can now be filled by solvent molecules, but as there is only one distinguishable way in which this can be done, Qj = 1, there is no further contribution to Q. and the entropy of the system. The latter can now be calculated from the Boltzmatm equation. The factorials can again be approximated using Stirling s relation and, although this requires considerable manipulation, which will be omitted here, it can eventually be shown that... 

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