Thomas-Fermi screening function

Fig. 12. Thomas-Fermi screening function, t[)(R/a) (see Equation 4), for neutral atoms (-) and...

An especially useful approximation for the Thomas-Fermi potential has been developed by Lindhard and co-workers where the screening function is assumed to have the form ... 

Direct numerical integration is approached through a couple of transformations. The radius variable and the screening function are expressed in the traditional Fermi-Thomas scaled ones ... 

Here, q is the inverse of a screening length related to the valence electron density which contributes to the screening and /u. is a Lagrange multiplier controlling the total number of particles. The boundary conditions to be used with Equation (23) are that V(r) must match Vc r) at Rs and that rV(r) -> -1 as r -> 0. Once we have solved the Thomas-Fermi equation, we have calculated the screened function, defined as the bare impurity potential divided to the screened one, namely Vb/V. 

Figure 2J Screening function calculated for the S135H36 nanocrystal, using the Thomas-Fermi model (solid line), compared to a previous first principles calculation [127] (dashed line).

It is remarkable that the surface potential fall toward its bulk value is a similar exponential function, (4.148) or (4.152), in the semiclassical Thomas-Fermi theory of electronic screening in the Debye-Huckel/Gouy-Chapman theory of ionic screening and at semiconductor interfaces. Here we consider the following issue When two such phases come into contact as in Fig. 4.9, and a potential bias is set between their interiors, how is the potential drop distributed at the interface ... 

Simple mathematical expressions have been developed for the Thomas-Fermi, Bohr, Lenz-Jensen, and Moliere screening functions. Several forms exist for the Thomas-Fermi model. The earliest and best known of these is the Sommerfeld asymptotic form,... 

Lindhard and coworkers proposed two somewhat simpler and more approximative Thomas-Fermi screening functions given by... 

If %(r) is taken as the Thomas-Fermi screening function, Winterbon et al. (1970) have shown... 

Fig. 32-1.—The electron distribution function D for the normal rubidium atom, as calculated I, by Hartrec s method of the self-consistent field II, by the screening-constant method and 111, by the Thomas-Fermi statistical method.

Slater proposes an effective quantum number n = 3.7, the atomic factor can only be presented in the form of a sum with an infinite number of components. The series may be terminated if the effective quantum number for the N shell is taken as 3.5, 4.0, or 4.5. We calculated values of the atomic factor for the neutral Br atom with different values of n. The most satisfactory agreement with the theoretical form factors, calculated according to the Thomas—Fermi—Dirac model, was obtained at n — 4.5 screening coefficients proposed in [11] were used in the calculations. The equation of the atomic scattering function for the N shell in the case of a spherically symmetrical electron density distribution and n — 4.5 has the following form ... 

The most widely used screening function is that due to Moliere [206] using Thomas-Fermi statistics, which is approximated by... 

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