Extended-zone representation

We turn now to the effects of a finite pseudopotential. Let us think specifically of silicon and begin with a free-elcctron gas of four electrons per ion, using the extended-zone representation, with energy equal to h k /2m for all k, rather than... 

FIGURE 3 The energy versus momentum curve of an electron traveling in a periodic potential, (a) Extended zone representation, (b) reduced zone representation, and (c) the endpoints of the wavevector for a three-dimensional cubic box of dimension d. 

Extended-zone and reduced-zone representations of band diagram for ID lattice with no external potential. 

Fig. 3.16 The efect of introducing a weak potential into the ID lattice is to lift the degeneracy of the energy levels mar to the edge of the Brillouin zone (shown in both extended-zone and reduced-zone representation).

Figure 6.9a-f illustrates a variety of the accepted band structure representations for nearly-free electron model. The Figure introduces the repeated-zone, extended-zone and reduced-zone images. The original free-electron parabola E = fi k Klme) is shown in Figure 6.9a. To leading order in the weak one-dimension periodic potential this curve remains correct except the value of k near the reciprocal lattice vector g. One can imagine that in this point the Bragg plane reflects the electron wave since the Bragg condition holds. Another free-electron parabola is centered at fe = g, and two parabolas are crossed each other at the...

It is difficult to visualize the Fermi surface of these metals because there re too many pieces. The bands are relativistic, so it is not possible to fold them out into the extended zone as was done for heavy rare earths. This makes the graphical representation of the Fermi surface rather difficult. In flg. 3.11 we show the cross sections of the various pieces of Fermi surface with the high symmetry planes of the Brillouin zone. A detailed description of the Fermi surface structure is given in the original article. 

The reaction interface can be defined as the nominal boundary surface between reactant and the solid product. This simple representation has provided a basic model that has been most valuable in the development of the theory of kinetics of reactions involving solids. In practice, it must be accepted that the interface is a zone of finite thickness extending for a small number of lattice units on either side of the nominal contact sur-... 

In the Universal Velocity Profile , the laminar sub-iayer extends to values of y+ = 5 and the turbulent zone starts at y+ = 30 and the range 5 < y+ < 30, the buffer layer, is covered by a second linear relation between and In, y+. What is the maximum difference between the values of u+, in the range 5 < y4 < 30, using the two methods of representation of the velocity profile ... 

Regrouping the intervals allowed as solutions of Eq. (3.99) fi-om the Figure 3.23 in terms of energy quantification (3.100), the representation of Figure 3.24 results (Putz, 2006) in its extended version (from the left side) it presents the discontinuities at the wave vectors values k = S = nK I a and respectively the energetic bands in the reduced version at the first Brillouin zone (from the right side). 

Figure 2 Schematic representation of the FEAt coupling scheme. The atomistic region (I) is embedded inside a continuum region (IV) and the two zones are coupled through an extended interface zone (shaded area). The transition region is divided into two zones (II and III), each providing boundary condition for either the continuum (zone II) or the atomistic region (zone III).

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