Fermi wave number

The volume of the system cancels out, so the Fermi wave number does not depend upon the size of the system, only upon the density of electrons. Wc may solve immediately for ky ... 

The simple metals, their prineipal crystal structures, and Fermi wave numbers kf-, in A Arrangement is by column and row of tlic Solid Stale Table. The values of fe].- shown here also appear in the Solid State Table. 

Calculate the Fermi wave number for C, Si, Ge, and Sn by minimizing the energy in Eq. (15-16) for both a face-centered cubic structure and a diamond structure. The values of /cp given for C, Si, and Ge in the Solid State Table are for an electron density corresponding to the diamond structure. That for Sn is for the metallic structure (not face-centered cubic but a structure of similar packing density). Compare the predicted ratio of kp values for the two structures in tin by noticing that the observed specific gravity is 5.75 in the diamond structure and 7.31 in the metallic white-tin structure. 

This leads to a local Fermi wave number, ky(r), given by / k. (r) /2m = L i.(r), and from Lq. (15-4), a local electron density N(r) =/<. (r) /(37c ). This is called the Fermi-Thomas approximation. The essential assumption that is required is that the potential does not vary greatly over the distance corresponding to the electron wavelength. 

It is most convenient to begin by constructing a (110) plane in wave number space, with the wave number lattice as shown in Fig. 16-12. The electron density is l6/because there are four atoms per cube and four electrons per atom, so the Fermi wave number can be obtained in units of Inja. Either a graphical solution or an approximate geometrical solution is adequate. These areas are directly measured in the de Haas van Alphen effect. The experimental values (Gold, 1958) are 1.00 and 0.11 times ftn/aY for the second band and third band, respectively. 

In most calculations done in this book, the plane waves ofimportance were those with the Fermi wave number, and the energy E of interest was the Fermi energy, so the kinetic energy cancels E. This can be made rigorous by suitable choice of the zero of energy for each matrix element, Then we have, from Eq. (D-5),... 

We may also compare the polar energies. This is particularly simple in the context of the empty-core model. Then, in gallium arsenide, for example, we may expect the unscreened pseudopotential (both Z and r ) for both gallium and arsenic to be the same as in each pure material. The screening depends only upon the Fermi wave number, and since the bond length does not change appreciably in an isoelectronic series, the entire denominator should be the same for... 

The effective interaction Kff is expected to depend not only on the wave vector but also on n, the number density of electrons in the medium. This, in turn, is related to the Fermi wave number fcp by the definition... 

TABLE 7.6 Free Electron Fermi Wave Numbers for Type II Liquids... 

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