Electronic Bands in Crystals

From Equation (3.61), we can define a density of energy levels or density of states N(s) in the crystal as As l. N(s) is a function giving the number or 

Electronic bands in linear polyene chain (double p) 

remembering from elementary analysis that  

T o introduce further details of the theory of solids in an elementary way, we can resort to the results given in Section 3.2 for the closed chain with N atoms. We have shown there that the general solution in complex form for the N-atom closed chain with N = odd is  

For solids, the quantum number k is replaced by the wave vector k  

Insulators can be distinguished from conductors or semiconductors in terms of their different conductivity at room temperature (T = 293 K) as shown in Table 3.3. 

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Fig. 2-4. Lattice potential energy and electron energy bands in crystals IB s inner band FB s frontier band.

Physical properties related to the electron motion in crystals fall essentially into two categories. Some, such as the electrical properties of crystals, arise from long-range interactions in the lattice here long-range forces from the electron--electron or the electron-core interactions play an important role. In these cases, the use of energy band theory is essential. On the other hand, in NLO effects the process of electronic excitation by the incident... 

Tables of compatibility relations for the simple cubic structure have been given by Jones (1962, 1975), and similar tables can be compiled for other structures, as shown by the examples in Tables 17.2 and 17.5. Compatibility relations are extremely useful in assigning the symmetry of electronic states in band structures. Their use in correlation diagrams in crystal-field theory was emphasized in Chapters 7 and 8, although there it is not so common to use B SW notation, which was invented to help describe the symmetry of electronic states in energy bands in crystals (Bouckaert el al. (1936)).

Another factor contributing to the asymmetry and breadth of absorption bands in crystal field spectra of transition metal ions is the dynamic Jahn-Teller effect, particularly for dissolved hexahydrated ions such as [Fe(H20)6]2+ and [Ti(H20)6]3+, which are not subjected to static distortions of a crystal structure. The degeneracies of the excited 5Eg and 2Eg crystal field states of Fe2+ and Ti3+, respectively, are resolved into two levels during the lifetime of the electronic transition. This is too short to induce static distortion of the ligand environment even when the cations occupy regular octahedral sites as in the periclase structure. A dual electronic transition to the resolved energy levels of the Eg excited states causes asymmetry and contributes to the broadened absorption bands in spectra of most Ti(m) and Fe(II) compounds and minerals (cf. figs 3.1,3.2 and 5.2). 

In Chapter 3, the Hiickel model of linear and closed polyene chains is used to explain the origin of band structure in the one-dimensional crystal, outlining the importance of the nature of the electronic bands in determining the different properties of insulators, conductors, semiconductors and superconductors. 

The first volume also includes papers on the correlation between the magnetic properties, the structure, and the electron distributions in crystals, and on that between the electron interaction, the distributions of the electron densities and potentials, and the band structure. New experimental results are also reported. 

One of the most striking properties of lanthanide metals and compounds is the relative insensitivity of electrons in the unfilled 4f shell to the local environment, compared to non-f electron shells with similar atomic binding energies. Whereas the 5d and 6s electrons form itinerant electron bands in the metallic solids, the 4f electrons remain localised with negligible overlap with neighbouring ions. For the maximum in the 4f radial charge distribution lies within those of the closed 5s and 5p shells, so the 4f shell is well shielded from external perturbations on the atomic potential, such as the crystal field. 

FIGURE 3.3 The generalization of the molecular orbitals to the electronic bands in a soli(h see (Further Readings on Quantum Crystal 1940-1978). 

An important consequence of quasi-free electrons treatment in solid consists in the possibility of classification of solids as metals, insulating, and semiconductors based on the width of the separation of energetic bands in crystal. 

Formulating the extended perturbation model through infinite ordered quantum well in crystal Kronig-Penney model) so generating the energetic multiple gaps in spectra of electrons in solids - the electronic stmcture of bands in crystals ... 

In many crystals there is sufficient overlap of atomic orbitals of adjacent atoms so that each group of a given quantum state can be treated as a crystal orbital or band. Such crystals will be electrically conducting if they have a partly filled band but if the bands are all either full or empty, the conductivity will be small. Metal oxides constitute an example of this type of crystal if exactly stoichiometric, all bands are either full or empty, and there is little electrical conductivity. If, however, some excess metal is present in an oxide, it will furnish electrons to an empty band formed of the 3s or 3p orbitals of the oxygen ions, thus giving electrical conductivity. An example is ZnO, which ordinarily has excess zinc in it. 

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