Extended zone

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).

What are the mechanisms by which slow, laminar combustion can be transformed into an intense, blast-generating process This transformation is most strongly influenced by turbulence, and secondarily by combustion instabilities. A laminar-flame front propagating into a turbulent mixture is strongly affected by the turbulence. Low-intensity turbulence will only wrinkle the flame front and enlarge its surface area. With increasing turbulence intensity, the flame front loses its more-or-less smooth, laminar character and breaks up into a combustion zone. In an intensely turbulent mixture, combustion takes place in an extended zone in which... 

The Boreal and Sub-Boreal Forest ecosystems represent the forests of cold and temperate climate. These ecosystems occupy an extended zone in the northern part of the Northern Hemisphere. The total area is 16.8 x 106 km2, or 11.2% from the whole World s territory. 

In summary, junctions are more or less extended zones in crystals in which the disorder type changes and transport occurs along with simultaneous (local) reactions of the SE s. Junctions exhibit complex kinetic behavior due to the coupling of fluxes and reactions. The (p-n) junction is an interesting limiting case but has served to introduce the fundamental concepts of junctions. 

Equation (4) holds generally at the face center but is valid over the whole face if the crystal point group contains a reflection plane through the zone center that is parallel to the face. It also holds for all k vectors that terminate on a line in the BZ face that is parallel to a binary axis. The E(k) may be described either by a singlevalued function of k (with k > 0), which is called the extended zone scheme, or by a multivalued function of k within the first BZ, the reduced zone scheme (see Figure 17.2). 

Figure 17.2. E(k) for the 1 -D free electron model in (a) the extended zone scheme and (b) the reduced zone scheme.

The values of k larger than fcedge are usually "folded" back into the first Brillouin zone, since no new physics arises from the extended-zone scheme. 

Figure 7.7a shows the extended-zone electronic band structure for a one-dimensional crystal - an atom chain with a real-space unit cell parameter a and reciprocal lattice vector Tr/n - containing a half-filled (metallic) band. In this diagram, both values of the wave vector, +k, are shown. The wave vector is the reciprocal unit cell dimension. The Fermi surface is a pair of points in the first BZ (Fig. 7.7c). When areas on the Fermi surface can be made to coincide by mere translation of a wave vector, q, the Fermi surface is said to be nested. The instability of the material towards the Peierls distortion is due to this nesting. In one dimension, nesting is complete and a one-dimensional metal is converted to an insulator because of a Peierls distortion. This is shown in Figure 7.7b, where the real-space unit cell parameter of the distorted lattice is 2a and a band gap opens at values of the wave vector equal to half the original values, 7r/2a.

A (110) section of the Fermi surface of aluminum in the extended zone scheme, showing the distortion at the Bragg planes. [After Harrison, 1966a.]... 

The rounding off of the Fermi surface in the periodic-zone scheme also corresponds to a distortion of the free-electron sphere in the extended-zone scheme. That distortion, calculated for aluminum, is shown in F ig. 16-9. It has the effect of reducing the total area of free surface and, as we shall sec, in the case of covalent solids the same effect eliminates the entire surface. 

A central cross-scction of the Fermi sphere for aluminum in the extended-zone scheme was shown in Fig. 16-9. By a study such as that illustrated in Fig. 16-5, you can identify two orbit types, which correspond to cross-sections of the surfaces for the second and third bands, shown in Fig. 16-8. The topology of ihe F ermi surface is the same for lead but, with four electrons per atom in lead, the sphere is larger in comparison to the zone than that for aluminum. Estimate the area of cross-sections of the surfaces for the second and third bands for lead, in units of (Inlay, with a the cube edge. 

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... 

A treatment of transport properties in terms of this surface is no more complicated in principle than that in the polyvalent metals, but there is not the simple free-clectron extended-zone scheme that made that case tractable. Friedel oscillations arise from the discontinuity in state occupation at each of these surfaces, just as they did from the Fermi sphere. When in fact there arc rather flat surfaces, as on the octahedra in Fig. 20-6, these oscillations become quite strong and directional. A related effect can occur when two rather flat surfaces are parallel, as in the electron and hole octahedra, in which the system spontaneously develops an oscillatory spin density with a wave number determined by the difference in wave number between the two surfaces, the vector q indicated in Fig. 20-5. This generally accepted explanation of the antiferromagnetism of chromium, based upon nesting of the Fermi surfaces, was first proposed by Lomer (1962). 

Figure 27. Extended zone plot for MgO(001) showing the data obtained from a number of TOF spectra. The solid curve is the calculated Rayleigh wave dispersion, while the dashed curve in the <110> direction is the S7 shear horizontal mode which lies below the sagittal plane modes for this crystal in this direction. The dot-dashed line is a scan curve at the angles indicated. (Reproduced from Fig. 3 of Ref. 82, with permission.)...

In Phase C (Rutqvist et al. 2003), the research teams performed calculations for the case where one or several water-bearing discrete fractures intersect the repository. As in Phase B, the temperature field shows nearly no difference between THM, TH, TM calculations and is very similar to the one in Phase B. The fracture(s) accelerates the resaturation of the buffer/backfill and prevents the desaturation of the rock mass. From a mechanical point of view, the fracture(s) constitutes a zone of weakness and results in a more extended zone of damage as compared to the homogeneous scenario. TH or THM calculations produce very similar pore pressure fields. With respect to the stresses in the buffer, the conclusions are similar to the homogeneous case, with a predominant effect of pore pressure on total stresses, compared to the thermal stresses. 

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