Thomas-Fermi screening

To put a more quantitative expression to the idea of electrons as screened quasiparticles, consider the following simple model. First, suppose that the electrons are maximally efiticient in screening each other s charge. This means that the Coulomb potential of an electron (—e)/r would be quickly suppressed as we move away from the electron s position (here taken to be the origin of the coordinate system). The most efficient way to do this is to multiply the Coulomb potential of the bare electron by a decaying exponential  

We can also define the induced charge through the Poisson equation  

For sufficiently weak fields, we take the response of the system to be linear in the total field  

As can be shown straightforwardly using perturbation theory (see Problem 8), for a system of single particles with energy = h k /2m e and Fermi occupation 

This is called the Lindhard dielectric response function. Notice that, at T = 0, in order to have a non-vanishing integrand, one of the occupation numbers must correspond to a state below the Fermi level, and the other to a state above the 

According to (4.157), the potential energy of two charges separated by a distance R is 

For two-point charges of magnitude Z, the Thomas-Fermi screening theory gives the result [4.69] 

Hint Evaluate the integral in spherical coordinates and use the result 

The theory of lattice vibrations which we discussed in the preceeding chapters has been based on the harmonic approximation which neglects all terms in the expansion of the potential energy (3.6) higher than the second-order terms. The most important consequences of the harmonic approximations are  

In real crystals, none of these consequences is satisfied accurately. 

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Z2 are the mass atomic number of the target atom, and is the Thomas-Fermi screening distance given by equation 4 ... 

It is also interesting that if some of the simple models for the bare metal surface are used to calculate the metal s contribution to the capacitance, a fit to experimental results would require unreasonable values for the solution contribution. Thus, the simple Thomas-Fermi result88 of C(dip) = 47r/ATF (Atf = Thomas-Fermi screening length) is greater than C(experiment)-1, and the same is true for the improved Thomas-Fermi results of Newns40 and the model of free electrons at an infinitely repulsive wall [see Eq. (12)]. These models are thus considered to be less realistic than the model of this work.30... 

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

Thus, the ionic coulomb potential is damped exponentially within a Thomas-Fermi screening length = 1 /ktf. It follows from eqs (2.41) and (6.10) that... 

Screening in metals is very effident even the low-density metal sodium with rs — 4 au has a Thomas-Fermi screening length as small as 1.3 au. 

In applying these ideas to metal solutions in ammonia, the restrictions on the Thomas-Fermi screening approach have generally been ignored. At metallic concentrations the concentration of free ammonia molecules is low, and the dielectric constant is near unity the pure ammonia value of Ke is only applicable when solvent molecules are not polarized by the ions. If we choose m /m less than unity, then fitting the theory to experimental data on conductivity becomes impossible. For the calculated conductivity to agree with the measured conductivity, we require... 

The exponential term is a Thomas-Fermi screening factor which accounts for the screening by the core electrons. Direct measurement of the ionic character of a bond is a complex operation. In principle, a number of techniques such as X-ray or neutron diffraction, nmr, photoelectron or Mossbauer spectroscopy provide information about electron distribution and charge density in practice the results are usually far from unambiguous. 

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

Examining (3.30) we see that e is a dimensionless energy unit. Physically, s gives a measure of how energetic the collision is and how close the ion gets to the nucleus of the target atom. For example, the value of the Thomas-Fermi screening distance, aTF, for He on Si is... 

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

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