Band theory surface

Krasovskll E E and Schattke W 1997 Surface electronic structure with the linear methods of band theory Phys. Rev. B 56 12 874... 

The d-Band Theory of Surface Reactivity and Rationai Approach to Catalyst Design... 

Generally, all band theoretical calculations of momentum densities are based on the local-density approximation (LDA) [1] of density functional theory (DFT) [2], The LDA-based band theory can explain qualitatively the characteristics of overall shape and fine structures of the observed Compton profiles (CPs). However, the LDA calculation yields CPs which are higher than the experimental CPs at small momenta and lower at large momenta. Furthermore, the LDA computation always produces more pronounced fine structures which originate in the Fermi surface geometry and higher momentum components than those found in the experiments [3-5]. 

This book systematically summarizes the researches on electrochemistry of sulphide flotation in our group. The various electrochemical measurements, especially electrochemical corrosive method, electrochemical equilibrium calculations, surface analysis and semiconductor energy band theory, practically, molecular orbital theory, have been used in our studies and introduced in this book. The collectorless and collector-induced flotation behavior of sulphide minerals and the mechanism in various flotation systems have been discussed. The electrochemical corrosive mechanism, mechano-electrochemical behavior and the molecular orbital approach of flotation of sulphide minerals will provide much new information to the researchers in this area. The example of electrochemical flotation separation of sulphide ores listed in this book will demonstrate the good future of flotation electrochemistry of sulphide minerals in industrial applications. 

It is well known that the flotation of sulphides is an electrochemical process, and the adsorption of collectors on the surface of mineral results from the electrons transfer between the mineral surface and the oxidation-reduction composition in the pulp. According to the electrochemical principles and the semiconductor energy band theories, we know that this kind of electron transfer process is decided by electronic structure of the mineral surface and oxidation-reduction activity of the reagent. In this chapter, the flotation mechanism and electron transferring mechanism between a mineral and a reagent will be discussed in the light of the quantum chemistry calculation and the density fimction theory (DFT) as tools. 

The results of the electron theory as developed for semiconductors are fully applicable to dielectrics. They cannot, however, be automatically applied to metals. Contrary to the case of semiconductors, the application of the band theory of solids to metals cannot be considered as theoretically well justified as the present time. This is especially true for the transition metals and for chemical processes on metal surfaces. The theory of chemisorption and catalysis on metals (as well as the electron theory of metals in general) must be based essentially on the many-electron approach. However, these problems have not been treated in any detail as yet. 

An important aspect of semiconductor films in general with regard to electronic properties is the effect of intrabandgap states, and particularly surface states, on these properties. Surface states are electronic states in the forbidden gap that exist because the perfect periodicity of the semiconductor crystal, on which band theory is based, is broken at the surface. Change of chemistry due to bonding of various adsorbates at the surface is often an important factor in this respect. For CD semiconductor films, which are usually nanocrystalline, the surface-to-volume ratio may be very high (several tens of percent of all the atoms may be situated at the surface for 5 mn crystals), and the effects of such surface states are expected to be particularly high. Some aspects of surface states probed by photoluminescence studies are discussed in the previous section. 

The rigid band theory (RBT) of solids according to which there is a considerable transfer of electrons among the alloy components an alloy surface is then a structureless plane with atoms indistinguishable for gas molecules. 

The electronic structure of a solid metal or semiconductor is described by the band theory that considers the possible energy states of delocalized electrons in the crystal lattice. An apparent difficulty for the application of band theory to solid state catalysis is that the theory describes the situation in an infinitely extended lattice whereas the catalytic process is located on an external crystal surface where the lattice ends. In attempting to develop a correlation between catalytic surface processes and the bulk electronic properties of catalysts as described by the band theory, the approach taken in the following pages will be to assume a correlation between bulk and surface electronic properties. For example, it is assumed that lack of electrons in the bulk results in empty orbitals in the surface conversely, excess electrons in the bulk should result in occupied orbitals in the surface (7). This principle gains strong support from the consistency of the description thus achieved. In the following, the principle will be applied to supported catalysts. 

New experimental methods such as AES, ESCA, FEM, FIMS, LEED, SEM, and SIMS (2) allow for the first time to dispense with the above assumption since they permit direct study of surface processes. Stimulated by these developments, the theoretical treatment tends to turn more toward consideration of single surface atoms and their orbitals or bonds. However, it is clear that this direction of research is still in its beginning and therefore restricted to very special cases. The treatment of the more complex problems presented by supported catalysts by such methods—in experiment and in theory—is still ahead. Thus for the moment the use of band theory is much more promising as it can be based on a large amount of existing data. 

We finally arrive at a point where we can relate what we have learned to the principal theme of the text. Recall that small clusters essentially consist of surface whereas band theory applies to situations where the number of surface atoms is so small with respect to atoms in the bulk they can be ignored. In the Al77 cluster we have a shell-like structure of All A1i2 A144 (A1R)2o or a ratio of 20/57 = 0.35 surface to bulk. In addition, each of the 20 surface atoms is coordinated to... 

A metallic state predicted by one-electron band theory is not stable when its Fermi surface is nested, and becomes susceptible to a metal-to-insulator transition under a suitable perturbation. We now examine the nature of the nonmetallic states that are derived from a normal metallic state upon mixing its occupied and unoccupied band levels. For simplicity, consider the 2D representation of the nested Fermi surface shown in (100), where the vector q is one of many possible nesting vectors. The occupied and unoccupied wave vectors are denoted by k and k, respectively. Each unit cell will be assumed to contain one AO x Suppose we choose the k and k values to satisfy the relationship... 

The knowledge of the electronic structure of metal clusters is of key importance to understand the origin of their catalytic activity. The electronic structure of bulk solids and of solid surfaces is now well understood in terms of band theory. On... 

It is difficult to imagine any band theory for an amorphous chromia activated at 250°. In terms of semiconductor theory, it is difficult to imagine such large surface coverages as we observe at —78°. The close correlations which are developing between homogeneous catalysis and catalysis on chromia cannot be understood at all on the basis of semiconductor theory. 

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