Bloch bandwidth

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

Bainbridge, 292 Bandwidth, 282 Bar spectrum, 290 Baseline correction (UV), 212 Bathochromic effect, 196 Beer-Lambert s law, 205, 220 Biosensor, 365 Bloch s theory, 130 Blue-shift, 195 Boltzmann distribution, 131 Bonded silica, 53 Bragg s law, 245 BSA, 336 Burner (AAS), 261... 

Consider a J-electron system, such as a transition metal compound. The valence d atomic orbitals do not range far from the nucleus, so COs comprised of Bloch sums of d orbitals and, say, O 2p orbitals, tend to be narrow. As the interatomic distance increases, the bandwidth of the CO decreases because of poorer overlap between the d and p Bloch SUMS. In general, when the interatomic distance is greater than a critical value, the bandwidth is so small that the electron transfer energy becomes prohibitively large. Thus, the condition for metallic behavior is not met insulating behavior is observed. 

The selectivity of the excitation is characterized by the bandwidth of the magnetization response. The response spectrum is determined by the Fourier transform of the selective pulse only in first order. Generally, the NMR response is nonlinear, and nonlinear system theory can be applied for its analysis (cf. Section 4.2.2). A model suitable for describing the NMR response in many situations applicable to NMR imaging is given by the Bloch equations (cf. Section 2.2.1). They are often relied upon when designing and analysing selective excitation (Frel). 

Fig. 9.20. Predictmg the band structure ot (PtH )co- (a) The Bloch functions tor 1 = 0 and k = corresponding to the atomic orbitals 6p-(a type orbitals). Stl.ry (S type orbit s). 5J x- in type orbitals, similar for 5

Bloch diagram for the first Brillouin zone resulting from the overlap of Is AO basis functions for a ring of H atoms. The bandwidth is given by 4. ... 

Bloch functions and Bloch diagrams for the p AO basis sets, where the p-orbitals lie (a) perpendicular to and (b) parallel with the linear chain connecting the atoms in the crystal. The greater overlap in (b) implies a larger bandwidth for (b) than for (a). 

Bloch, and Hamermash (1947), the radar cross sec- FIGURE 17.64 tion off of resonance is expressed in reciprocal terms chaff, of ln(rod length/rod radius). The effective halfpower bandwidth of dipole scatters is, thus, usually referenced to the ratio of dipole length to dipole diameter as shown in Fig. 17.64. As it is desirable to have some bandwidth in television antennas, it is also desirable to have some bandwidth in the radar scattering from chaff, which leads to a measurable diameter of the rods or width of the ribbon strips used for chaff. 

If a periodic polymer chain has a very narrow bandwidth [like the so-called narrow-band (widths of order 10" eV) periodic nucleotide base stacks or base pair stacks in Table 9.11) or a polymer consists of a nonperiodic sequence of different types of units, then the electronic states become localized molecular states and coherent Bloch-type conduction is no longer possible. 

It is useful to describe the form of localization that occurs in a homogeneously disordered material in contrast to the model described in the previous section. The three-dimensional models described below assume that the materials are isotropic i.e., the materials should be electrically the same in all directions. In a perfect crystal with periodic potentials, the wave functions form Bloch waves that are delocalized over the whole solid [35]. In systems with disorder, impurities and defects introduce substantial scattering of the electron wave function, which may lead to localization. Anderson demonstrated [64] that electronic wave functions can be localized if the random component of the disorder potential is large enough compared with the electronic bandwidth (Fig. 3.6). In this case, the localized wave functions have the form... 

Fig. 6.2. The Bloch functions at special points in -space for the lattice in Fig. 6.1, and the qualitative shape of the corresponding energy bands, for (left) the -orbital and (right) the p-orbital. The horizontal lines are the energies of the isolated atomic orbitals. If there is one electron in each orbital, the bands are half-fiUed and the Fermi level is at half the bandwidth with a zero band gap.

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