Bloch wave functions

Coulson and Baldock (14) have applied Bloch wave functions to an examination of surface states in graphite. They find that one edge of the basal plane has surface states and the other does not—that is, along n and along m, respectively, as shown below. 

For a crystal, our one-electron s take the form of Bloch wave functions ... 

Some supplementary remarks to the theory of Penn might be appropriate here. There are additional effects which are of relevance if a more quantitative theory of the photoemission process from an adsorbate-covered surface is envisaged. The first point is that the Anderson model as applied to chemisorption is a clearly oversimplified model to describe real metal-adsorbate systems. Besides overlap effects due to the nonorthogonality of the states k) and a), there are several interaction effects which are neglected in the Hamiltonian, Eq.(5). The adsorbed atom, for instance, may act as a scattering centre for the metal electrons and thus modify the Bloch wave functions characteristic of the free substrate. This can be accounted for by adding a term... 

The one-electron wave function in an extended solid can be represented with different basis sets. Discussed here are only two types, representing opposite extremes the plane-wave basis set (free-electron and nearly-free-electron models) and the Bloch sum of atomic orbitals basis set (LCAO method). A periodic solid may be considered constmcted by the coalescence of these isolated atoms into extended Bloch-wave functions. On the other hand, within the free-electron framework, in the limit of an infinitesimal periodic potential (V = 0), a Bloch-wave function becomes a simple... 

For solids with more localized electrons, the LCAO approach is perhaps more suitable. Here, the starting point is the isolated atoms (for which it is assumed that the electron-wave functions are already known). In this respect, the approach is the extreme opposite of the free-electron picture. A periodic solid is constructed by bringing together a large number of isolated atoms in a maimer entirely analogous to the way one builds molecules with the LCAO approximation to MO (LCAO-MO) theory. The basic assumption is that overlap between atomic orbitals is small enough that the extra potential experienced by an electron in a solid can be treated as a perturbation to the potential in an atom. The extended- (Bloch) wave function is treated as a superposition of localized orbitals, centered at each atom ... 

The most important and difficult quantity to calculate in the above equation is the electronic coupling matrix element. The initial state wave function, TC, of a conducting electron in the metal electrode is not taken as a Bloch wave function because the periodicity of the metal is no longer effective at the interface (normal to the surface). The wave function of a free particle is taken as that given in [35], but allowance is made for the interaction with the metal using an internal effective mass of the conducting electron [39] ... 

Bloch wave function - A solution of the Schrodinger equation for an electron moving in a spatially periodic potential used in the band theory of solids. 

The explicit periodicity of the Bloch wave function poses a major challenge for the calculation of aforementioned chemical shielding tensor in the solid state. Under these circumstances, the calculation of NMR parameters requires the inclusion of a macroscopic magnetic field described by a nonperiodic vector potential, yielding a new Hamiltonian ... 

This basic quantity may be related to the true electron momentum distribution when the positron wave function is known through calculations. The many-body factor y f) is known to play some role but it has been established that the correlations have no effect on the position of the Fermi surface breaks (Majumdar 1965). Equation (1) shows that positron annihilation is well suited for Fermi surface studies as only ni(k) has breaks. This can be further seen if one expresses the positron and electron wave functions as Bloch wave functions. One obtains... 

The band theory applies to the perfectly regular organization of a crystal, leading to delocalized Bloch wave functions, Eq. (6). In a now classical paper, Anderson [7] has shown that disorder may result in a localization of the states. In that case, the one-electron wave function takes an exponential form... 

In the quantum theory the key elements are the Hamiltonian and the Bloch wave functions. Nanosolid densification modifies the wave functions slightly as, in this case, no chemical reaction occurs. 

Similar to the present introduction, in Sections 5.2 and 5.3 mainly (but not exclusively) one-dimensional models are used to describe metal surfaces and their properties. However, in contrast to the present section, we (mostly) do not rely on an expansion of the Bloch wave function in terms of atomic orbitals, but rather use plane waves. For metals with s,p-derived valence/conduction electrons, the latter approach turns out to be particularly advantageous as often a few plane waves are suflident for a semiquantitative description. 

The dispersion relation (Eq. (5.34)) yields two bands, which for an infinite ( bulk ) crystal are displayed by the thick lines in the left part of Eigure 5.12. As can be seen, an energy gap of width 2Vg opens up at the Brillouin zone boundary k = f Note that for an infinite crystal the wave vector k has to be real, since otherwise the Bloch wave function F(z) = e" Ut(z) would diverge exponentially for either z -r- +00 or z —00. 

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