Band theory Brillouin zones

The theory of band structures belongs to the world of solid state physicists, who like to think in terms of collective properties, band dispersions, Brillouin zones and reciprocal space [9,10]. This is not the favorite language of a chemist, who prefers to think in terms of molecular orbitals and bonds. Hoffmann gives an excellent and highly instructive comparison of the physical and chemical pictures of bonding [6], In this appendix we try to use as much as possible the chemical language of molecular orbitals. Before talking about metals we recall a few concepts from molecular orbital theory. 

This expression will be used later on when we are calculating the contribution of the free electrons to the molar specific heat capacity of the metal at constant volume (see section 1.8.2.1). However, in order to do that, we need to know the term , which is the number of free electrons per atom of the metal. Sommerfeld s model does not provide us with this number, but Brillouin s band theory, or zone theory, can be used to evaluate it. 

Simply doing electronic structure computations at the M, K, X, and T points in the Brillouin zone is not necessarily sufficient to yield a band gap. This is because the minimum and maximum energies reached by any given energy band sometimes fall between these points. Such limited calculations are sometimes done when the computational method is very CPU-intensive. For example, this type of spot check might be done at a high level of theory to determine whether complete calculations are necessary at that level. 

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

The reciprocal lattice is useful in defining some of the electronic properties of solids. That is, when we have a semi-conductor (or even a conductor like a metal), we find that the electrons are confined in a band, defined by the reciprocal lattice. This has important effects upon the conductivity of any solid and is known as the "band theory" of solids. It turns out that the reciprocal lattice is also the site of the Brillouin zones, i.e.- the "allowed" electron energy bands in the solid. How this originates is explciined as follows. 

Discuss the origin of the Hume-Rothery electron phases within the framework of Jones original rigid-band analysis. How does second-order perturbation theory help quantify Mott and Jones earlier supposition on the importance of the free electron sphere touching a Brillouin zone boundary ... 

By physicists in terms of their conductivity, magnetic property and work functions. And they came up with a theory that deals with Fermi electrons, Brillouin zone and band structure. 

Fig. 5.12. (a) Variation of electron energy with wave number for a two-dimensional metal on the basis of the Bloch theory. ka, kb and kc are values of the first forbidden wave number in different directions in the first Brillouin zone. The band of forbidden energy in (b) indicates that the energy levels of the first and second zones do not overlap, (c) As (a) but for the case in which there is overlap between the energy levels of the first and second zones, as shown in ([Pg.99]

The next advance came from the application of Fermi-Dirac statistics to the electrons in metals, which led to the band theory of a quasi-continu-ous series of energy levels, and to the concept of Brillouin zones, which is of special value for alloys. This sets the stage for a detailed study of the electronic factor in catalysis on metals. 

It is evident that many of the older ideas about the poisoning of nickel catalysts must be re-examined in the light of data now available. The two nickel sulfides, NisSa and NiS, which can exist under the conditions prevailing in some catalytic systems, are catalysts for some reactions also catalyzed by metallic nickel and for other reactions not yet found to be catalyzed by the metal. Explanations of this behaAuor in terms of band theory (or Brillouin zones) can be expected in the near future. 

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