Fermi energy Free-electron value

It should be mentioned that a free-electron value for the Fermi energy may be... 

The electrochemical potential of single ionic species cannot be determined. In systems with charged components, all energy effects and all thermodynamic properties are associated not with ions of a single type but with combinations of different ions. Hence, the electrochemical potential of an individual ionic species is an experimentally undefined parameter, in contrast to the chemical potential of uncharged species. From the experimental data, only the combined values for electroneutral ensembles of ions can be found. Equally inaccessible to measurements is the electrochemical potential, of free electrons in metals, whereas the chemical potential, p, of the electrons coincides with the Fermi energy and can be calculated very approximately. 

Adsorption related charging of surface naturally affects the value of the thermoelectron work function of semiconductor [4, 92]. According to definition the thermoelectron work function is equal to the difference in energy of a free (on the vacuum level) electron and electron in the volume of the solid state having the Fermi energy (see Fig. 1.5). In this case the calculation of adsorption change in the work function Aiqp) in... 

The NFE behaviour has been observed experimentally in studies of the Fermi surface, the surface of constant energy, F, in space which separates filled states from empty states at the absolute zero of temperature. It is found that the Fermi surface of aluminium is indeed very close to that of a spherical free-electron Fermi surface that has been folded back into the Brillouin zone in a manner not too dissimilar to that discussed earlier for the simple cubic lattice. Moreover, just as illustrated in Fig. 5.7 for the latter case, aluminium is found to have a large second-zone pocket of holes but smaller third- and fourth-zone pockets of electrons. This accounts very beautifully for the fact that aluminium has a positive Hall coefficient rather than the negative value expected for a gas of negatively charged free carriers (see, for example, Kittel (1986)). 

Morrison [232,233] finds that the free energy of electrons in the bulk phase (Fermi energy) is about the same for different selective and active catalysts. He notes that this value is very near (or just above) the electron exchange level of oxygen and hence makes reduction of oxygen possible. 

A numerical evaluation of the Fermi energy lor a simple metal having one or two conduction electrons per atom yields a value of approximately ID-11 erg. or a few electron volts. The equivalent temperature. E,/b. is several lens of thousands of degrees Kelvin. Thus, except in extraordinary circumstances, when dealing with metals. bT -SC ( i.e.. the energy range or partially filled states is small, and the Fermi surface is well defined by the foregoing statement. It must be noted, however, that this is not necessarily true for semiconductors where the number of free electrons per unit volume may be very much smaller. 

Band Structure for the Free-Electron Case. If the electron is free, then the Bloch functions are simple plane waves, because the wavef unctions nk(r) used for the expansion Eq. (8.4.2) are themselves plane waves. For an electron gas with no lattice and no imposed symmetry, Fermi-Dirac statistics apply At 0 K all electrons pair up (spin-up and spin-down), with an occupancy of 2 for every k value from k 0 to the Fermi wavevector kF =1.92/rs = 3.63 a0/rs, and from zero energy up to the Fermi energy eF = h2kF2 /rn 50.1 eV rs/a0) 2, where rs is the radius per conduction electron and a0 is the Bohr radius, and the energy levels are spherically symmetric in k-space. The Fermi surface is a sphere of radius kF. Note that the ratio (rs/a0) varies from 0.2 to 1.0 nm for metals (Table 8.3). 

The Seebeck coefficients (thermoelectric powers) of Na WOs have been measured over a wide range of x values at room temperature (300° K.). At this temperature, the residual resistance, p0, and thermal resistance, pt, are comparable, the value of p0 being between pt and 2pt. Nevertheless, one would expect to a first approximation (10) that S = (1/3) (ir2k2T/e ), where S is the Seebeck coefficient, k is Boltzmann s constant, e is the electronic charge, and f is the Fermi energy. For free electrons, the Fermi energy f = (h2/2m ) (3n/87r)2/3 where h is Planck s constant, m is the effective mass, and n is the density of free electrons. Since n is proportional to x, f varies as x2/3 and S varies as xr2/3. 

Table 6. Fermi energies Ef observed for Rb and Cs and their suboxides in the Hel spectra. (40,66) The numbers of free electrons no are calculated from these values and compared with the values Uc, which are expected from the bond model...

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