Quantum field theory lattice

With the absorption of a quantum with an energy of more than 3.05 eV resp. 3.29 eV, an electron is lifted out of the valence band and into the conduction band, thereby forming an exciton (Fig. 5). This interpretation is also supported by the molecular orbital theory and the crystal field theory regarding the bonding conditions in the TiC lattice. 

Several interesting topics have been excluded, perhaps somewhat arbitrarily, from the scope of this book. Specifically, we do not discuss analytical theories, mostly based on the integral equation formalism, even though they have contributed importantly to the field. In addition, we do not discuss coarse-grained, and, in particular, lattice and off-lattice approaches. At the opposite end of the wide spectrum of methods, we do not deal with purely quantum mechanical systems consisting of a small number of atoms. 

The classical theory for electronic conduction in solids was developed by Drude in 1900. This theory has since been reinterpreted to explain why all contributions to the conductivity are made by electrons which can be excited into unoccupied states (Pauli principle) and why electrons moving through a perfectly periodic lattice are not scattered (wave-particle duality in quantum mechanics). Because of the wavelike character of an electron in quantum mechanics, the electron is subject to diffraction by the periodic array, yielding diffraction maxima in certain crystalline directions and diffraction minima in other directions. Although the periodic lattice does not scattei the elections, it nevertheless modifies the mobility of the electrons. The cyclotron resonance technique is used in making detailed investigations in this field. 

In the first part of this introductory section, we summarize the main collective phenomena acquired by the dipolar exciton from the lattice-symmetry collectivization of molecular properties. The crystal is considered as an assembly of electrically neutral systems, the molecules, physically separated from each other and in electromagnetic interaction. This /V-body problem will be treated quantum-mechanically in the limit of low exciton densities. We redemonstrate the complete equivalence of this treatment with the theories of Lorentz and Ewald, as well as with the semiclassical approximation. In Section I.A, in a more compact but still gradual way, we establish the model of the rigid lattice of dipoles and the general theory of low-exciton-density systems in interaction with the radiation field. Coulombic excitons, photons,... 

Before the quantum theory of solids (see description in Chapter 21), microscopic descriptions of metals were based on the Drude model, named for the German physicist Paul Drude. The solid was viewed as a fixed array of positively charged metal ions, each localized to a site on the solid lattice. These fixed ions were surrounded by a sea of mobile electrons, one contributed by each of the atoms in the solid. The number density of the electrons, is then equal to the number density of atoms in the solid. As the electrons move through the ions in response to an applied electric field, they can be scattered away from their straight-line motions by collisions with the fixed ions this influences the mobility of the electrons. As temperature increases, the electrons move more rapidly and the number of their collisions with the ions increases therefore, the mobility of the electrons decreases as temperature increases. Equation 22.7 applied to the electrons in the Drude model gives... 

Th.e refinements of the theory, which have been worked out in particular by Houston, Bloch, Peierls, Nordheim, Fowler and Brillouin, have two main objects. In the first place, the picture of perfectly free electrons at a constant potential is certainly far too rough. There will be binding forces between the residual ions and the conduction electrons we must elaborate the theory sufficiently to make it possible to deduce the number of electrons taking part in the process of conduction, and the change in this number with temperature, from the properties of the atoms of the substance. In principle this involves a very complicated problem in quantum mechanics, since an electron is not in this case bound to a definite atom, but to the totality of the atomic residues, which form a regular crystal lattice. The potential of these residues is a space-periodic function (fig. 10), and the problem comes to this— to solve Schrodinger s wave equation for a periodic poten-tial field of this kind. That can be done by various approximate methods. One thing is clear if an electron... 

First of all we have to mention that the above described situation of resonance is not related to any quantum effects. Moreover, the role of the transverse electromagnetic field in crystal oscillations in the infrared part of the spectrum was discussed by means of the classical dynamics of crystal lattices a long time ago by Born and Ewald (2) (see also (3) and (4)), and later by a semiphenomenological approach in (5), (6). It is evident, however, that a quantum theory of polaritons in the region of electronic transitions can also be important particularly for the discussion of quantum effects. 

There are also applications of quantum theory for instance in the onset of a failure in a material. The failure starts on the atomic scale when an interatomic bonding is stressed beyond its yield-stress threshold and breaks. The initiation and diffusion of point defects in crystal lattice turn out to be a starting point of many failures. These events occur in a stress field at certain temperatures. The phenomena of strain, fatigue crack initiation and propagation, wear, and high-temperature creep are of particular interest The processes of nucleation and diffusion of vacancies in the crystal lattice determines the material behavior at many operation conditions. 

---