Thomas Fermi radius

Within a jellium atom, the electron frequency is of order 1017/sec. compared with the plasmon frequency for jellium (1.1 x 1016/sec.) so an isolated jellium atom behaves as a dielectric. However, the valence electron screens any electric field caused by polarization. The screening length (Thomas-Fermi) is 0.47Ang., or 0.36 of the radius of the jellium atom. Thus the field of the positive ion is reduced by about 30% at R. 

Suppose we take an electron out of one of the hydrogen atoms, leaving behind a H ion. The electron will be re-attracted to the ion by a potential - e /R. It will be bound in the ground state for this potential, which has a radius Uh = h /me. If, however, there are enough ionized electrons in the lattice, they will screen the electron from the positive ion core, according to a potential - (e /R) exp(- kR), where X is a screening parameter in the Thomas-Fermi modeP X... 

The Thomas-Fermi approximation is, unfortunately, a poor approximation for the sp-valent metals. It is based on the assumption that the potential varies much more slowly than the screening length of the electrons themselves, so that the local approximation for the kinetic energy, eqn (6.6), is valid. In practice, however, the variation in the ionic potential is measured by the core radius, Rc (cf Fig. 5.11), which is not large but of the same size as the screening length, XTF. Thus, we do not satisfy the criterion for the validity... 

As already remarked, the idea underlying the Thomas-Fermi (TF) statistical theory is to treat the electrons around a point r in the electron cloud as though they were a completely degenerate electron gas. Then the lowest states in momentum space are all doubly occupied by electrons with opposed spins, out to the Fermi sphere radius corresponding to a maximum or Fermi momentum pt(r) at this position r. Therefore if we consider a volume dr of configuration space around r, the volume of occupied phase space is simply the product dr 47ipf(r)/3. However, we know that two electrons can occupy each cell of phase space of volume h3 and hence we may write for the number of electrons per unit volume at r,... 

Fig. 1 Exact and Thomas-Fermi electron density n as a function of position z for the Airy gas model with force F = 0.10. The scaling length is 1 = 1.71. The edge region is —/ < z < / and the Thomas-Fermi density is reasonably accurate for z > I- The infinite barrier is at z = 201 = 34.2. The magnitudes of the densities in this figure are valence-electron-like the density parameter Ts (the radius of a sphere containing on average one electron) is about 3.3 at z = I and about 1.3 at z = 10 (atomic units)...

It is also a more subtle effect, which is why Goppert-Mayer (1941) had difficulty computing it by using the Thomas-Fermi model. Griffin et al. (1971) in their numerical study foimd that the mean radius of the 4s electron in the 3d sequence lies very close to the knee between the two wells. Thus adding a 4s electron has a profound effect on the 3d effective potential, which accoimts for the otherwise mysterious effect known as competition between the filling of the d and s subshells. 

A semiquantitative explanation of these facts is afforded by a simple model [34] involving the statistical Thomas-Fermi theory [35]. In this model, the nuclei of the cage are replaced by an uniform positive charge distributed on a sphere with a radius equal to the radius of the cluster. The charge distribution generates a hollow sphere potential given by... 

In this equation, n is the conduction electron density, Ep the Fermi energy and kp the radius of the Fermi sphere. e(Q) given by (4.159) is also known as the Lindhard dielectric constant [4.69]. For Q 0, the quantity in the square brackets is equal to 1 and e(Q) then reduces to the Thomas-Fermi dielectric function [4.69],... 

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 ... 

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