Fermi pseudopotential

Taking into account the quantum mechanical probability of the transition of a neutron from a state characterized by wavevector k and energy E to another state characterized by wavevector k and energy E, caused by scattering from a target-system via potential V (taken as the extremely short ranged Fermi-pseudopotential), one arrives at the following expression for the partial differential cross section (see, e.g.. Squires 1996) ... 

Neutron scattering from a monatomic system can have variations in the bound scattering length, b, from atom to atom. These differences can occur due to a system being comprised of several spinless isotopes, a single isotope with non-zero spin, or a combination of the two. It is important to note that b is complex and can be negative, which is equivalent to an attractive Fermi pseudopotential [48] and therefore a k phase change in the wave. If we assume that there is no correlation between atomic position and b then the cross section can be split into two contributions ... 

The scattering of a neutron (A 10 cm) by a nucleus (radius 10" to 10 cm) may be regarded as an S-wave process, so that there is no angular dependence, and the interaction may be described by the so-called Fermi pseudopotential. In general, the scattering of a neutron by a single bound nucleus is described within the Bom approximation by the Fermi pseudopotential. 

The EPM required some measured data to determine the Fourier coefficients of the pseudopotential. However, the most modem approaches follow the Fermi [5] concept of developing a pseudopotential to yield a wave function without nodes that coincides with the all-electron atomic wave function outside the core and is still normalized. Several methods were developed [16-19] in the 1970s and 1980s, and new methods for constructing useful pseudopotentials continue to appear in the literature. The applications discussed here are mostly based on the pseudopotentials developed using the approach described in Ref. [19]. The important point to empha-... 

Fig. 5.12 The Heine-Abarenkov (1964) pseudopotential for aluminium which has been normalized by the Fermi energy. The term q0 gives the position of the first node. The two large dots mark the zone boundary Fourier components, vDS(111) and (200).

Thus, the screened pseudopotential form factor of aluminium normalized by the Fermi energy will approach q = 0 at —2/3 as observed in Fig. 5.12. 

Fig. 1 Asymptotic structure coefficients as(j8), c ks,xG ) crw03), a CS), and aKS gOS) as fimction of barrier height parameter )S =VW/eF, where W is the barrier height and eF the Fermi energy. Corresponding values of the Wigner-Seitz radius rs for jellium and structureless-pseudopotential models over the metallic range of densities are also given. The relationship between rs and ff is via self-consistent calculations in the local density approximation for exchange-correlation.

FIGURE 6.8 Band structure of graphite calculated by a mixed-basis pseudopotential method. The solid and dotted lines denote the O- and re-bands, respectively. The Fermi level is indicated by EF. (From Holzwarth, N.A.W., et al., Phys. Rev. B, 26, 5382, 1982. With permission.)... 

We know that when the pseudopotential is at full strength, all of the Fermi surface must disappear, since none is present in the semiconductor. We can, in fact, see from the figure that what must happen is that it disappears into the slanted and vertical planes of Fig. 18-3,a the horizontal planes as well as the omitted (111) planes are noncssential. Indeed, the vertical and slanted planes are among the twelve (220) Bragg planes that make up the Jones Zone (Mott and... 

The ratio wJEf) of pseudopotential form factor to frec-clcctron Fermi energy for silicon, showing that the direct Joncs-Zone diffraction [220J should be weak. 

The identification of the covalent energy with a pseudopotential has relevance to the f/ -dependence that has recurred throughout our studies. The pseudopotential form factors, when divided by the Fermi energy and plotted against, as in Fig. 18-3, are almost a universal eurve, approaehing —2/3 at small q and crossing the axis near q/k = 1.6 or 1.7 in most systems. To the extent that this curve is universal, the value at q ky= 1.108, and therefore, IT, would be a universal constant times as we have found to be true. The case for such a... 

It was mentioned earlier that through the resonance region, the phase shift increased by 7t, corresponding to the insertion of an extra state. If the Fermi energy is well above. the resonance is completely occupied in the sense that the probability density for the atomic state r/>, which wc used to describe the resonance in the formulation of transition-metal pseudopotentials, is unity. Similarly, it is empty if ,j is much less than Ey. It is not difficult to show that in fact the probability density for occupation is just di/n at intermediate energies also. This is a special case of the Friedel sum rule, which states that the number of excess electrons located at a scattering site is... 

M. L. Cohen, Applications of the Fermi Atomic Pseudopotentials to the Electronic Structure of Nonmetals, in Highlights of Condensed matter Theory, Corso, Soc. Italiana di Fisica, Bologna 89, 16 (1985). 

Fig. 5.2. a) Static structure factor of an amorphous or liquid metal b) band-structure characteristic c) pseudopotential d) perturbation characteristic for two different Fermi-sphere diameters, a-c are qualitatively drawn in arbitrary units... 

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