Mott approximation

This implies for example that when a(t) has the Mott minimum value (e ifafit), the dc or large length scale (L — ) conductivity is zero. Thus, the conductivity goes continuously to zero at the Anderson transition, due to interference effects not envisaged in the Mott approximation. The exponent with which a vanishes is the same as that with which the localization length diverges. 

The quantum-mechanical ionization cross section is derived using one of several approximations—for example, the Born, Ochkur, two-state, or semi-classical approximations—and numerical computations (Mott and Massey, 1965). In some cases, a binary encounter approximation proves useful, which means that scattering between the incident particle and individual electrons is considered classically, followed by averaging over the quantum-mechanical velocity distribution of the electrons in the atom (Gryzinski, 1965a-c). However, Born s approximation is the most widely used one. This is discussed in the following paragraphs. 

The formula which approximately describes the R(T) dependence is the Mott s law [65] ... 

The modification of the three and four-particle system due to the medium can be considered in cluster-mean field approximation. Describing the medium in quasi-particle approximation, a medium-modified Faddeev equation can be derived which was already solved for the case of three-particle bound states in [9], as well as for the case of four-particle bound states in [10]. Similar to the two-particle case, due to the Pauli blocking the bound state disappears at a given temperature and total momentum at the corresponding Mott density. 

Or, as Nevil Mott put it, "If by making approximations and neglecting even large terms. . . one could account for something that had been observed, the thing to do was to go ahead and not to worry." 150... 

These classical formulas still do not account for the motion of the bound electrons in the atom or molecule. To be more appropriate to the interaction of an incident electron with the bound target electron, one must recognize that the velocity vector of the bound electron can be randomly oriented with respect to the incident electron providing a broadening of the energy of the secondary electron as calculated by the modified Mott cross section. If one integrates over the velocity distribution of the bound electron, the more familiar binary encounter approximation is derived that, in its simplest form, is given by Kim and Rudd [39] as... 

If a positive charge ze is immersed in a degenerate electron gas, the Coulomb field is screened by the electrons. The screening was first estimated by the present author using the Thomas-Fermi approximation (Mott 1936, Mott and Jones 1936, p. 86), and by this method one finds for the potential energy of an electron... 

This formula for the screened Coulomb field has been used for many applications it gives, for instance, a fairly satisfactory description of the residual resistance due to a small concentration of Zn, Ga, etc. in monovalent metals (Mott and Jones 1936, p. 293). Friedel (1956) points out, however, that the use of the Bom approximation gives too large a value of the scattering, and better values are obtained if one calculates the phase shifts exactly. 

For the metals Co, Ni and Pd and perhaps others it appears to be a good approximation to assume, in spite of the hybridization, that part of the Fermi surface is s-like with mrff me, and part d-hke with meff me. The current is then carried by the former, and the resistance is due to phonon-induced s-d transitions. This model was first put forward by Mott (1935) and developed by many other authors (e.g. Coles and Taylor 1962) for reviews see Mott (1964) and Dugdale and Guenault (1966). Applications of the model have also been made to ordered alloys of the type Al6Mn, Al7Cr by Griiner et al (1974), where the width A of the d-band is the same as it would be for an isolated transitional-metal atom in the matrix, but most of the Fermi surface is assumed to be (s-p)-like. The behaviour of the disordered Pd-Ag alloy series is particularly interesting. The 4d-bands of the two constituents are well separated, as shown particularly by... 

The intermediate range of concentrations between those at which resonant scattering and s-d transitions are appropriate has not been fully explored, except in the CPA approximation (Stocks et al 1973), which does not give the mean free path. For liquid transitional metals the present author (Mott 1972d) has suggested that one must introduce two mean paths, /s for the s-electrons and /d for the d-electrons, that / afor the latter (as in the alloy) and that s-d transitions are appropriate to describe the resistance. Other authors have described the resistance in terms of a single mean free path, determined by the resonant scattering of the s-electrons by the d-shells (Evans et al 1971). 

In this case the zero-order electronic wave functions are, in principle, referred to a Hamiltonian that contains the potential from the ions at their actual positions, i.e., the electrons follow the ionic motion adiabatically. Since both these approximations are sometimes referred to as the Born Oppenheimer approximation, this has led to confusion in terminology for example, Mott (1977) refers to the Born-Oppenheimer approximation, but gives wave functions of the adiabatic type, whereas Englman (1972) differentiates between the two forms, but specifically calls the static form the Bom Oppenheimer method. [We note that, historically, the adiabatic form was first suggested by Seitz (1940)—see, for example, Markham (1956) or Haug and Sauermann (1958)]. In this chapter, we shall preferentially use the terminology static and adiabatic. [Note that the term crude adiabatic is also sometimes used for the static approximation, mainly in the chemical literature—see, for example, Englman (1972, 1979).]... 

The conditions of the experiment discussed here are different than the restrictions imposed to obtain the Mott-Gurney equation. However, at least qualitatively, Equation 8.38 can describe the space charge-limited current effect. To test this hypothesis, Equation 8.38 was experimentally tested, and it was shown that it is approximately satisfied (see Figures 8.13 and 8.14) [112]. 

Thus, we have to conclude that, without knowing the physical nature of the frequency dependence of the differential capacitance of a semiconductor electrode, the donor (or acceptor) concentration in the electrode cannot be reliably determined on the basis of the Schottky theory, irrespective of the Mott-Schottky plot presentation format. Therefore, the reported in literature acceptor concentrations in diamond, determined by the Schottky theory disregarding the frequency effect under discussion, must be taken as an approximation only. However, we believe that the o 2 vs. E plot (the more so, when the exponent a approaches 1), or the Ccaic 2 vs. E plot, are more convenient for a qualitative comparison of electrodes made of the same semiconductor material. 

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