Three-band model

Although the extended Cauchy equation fits experimental data well [7], its physical origin is not clear. A better physical meaning can be obtained by the three-band model which takes fliree major electronic transition bands into consideration. 

The major absorption of an LC compound occurs in two spectral regions ultraviolet (UV) and infrared (IR). The electronic transitions take place in the vacuum UV (100-180 nm) 

The three-band model takes one — 7 transition (the Aq band) and two transitions 

The three-band model clearly describes the origins of refractive indices of LC compounds. However, a commercial mixmre usually consists of several compounds with different molecular 

To model the refractive indices of an LC mixture, we could expand Equation (6.12) into power series because in the visible and IR spectral regions, A By keeping up to the terms, the above extended Cauchy equation (6.10) is again derived. 

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We have shown the least complicated one which turns out to be the simple cubic lattice. Such bands are called "Brilluoin" zones and, as we have said, are the allowed energy bands of electrons in any given crystalline latttice. A number of metals and simple compounds have heen studied and their Brilluoin structure determined. However, when one gives a representation of the energy bands in a solid, a "band-model is usually presented. The following diagram shows three band models ... 

Figure 6.2 Wavelength-dependent refractive indices of 5CB at T= 25.1 C. Open and closed circles are experimental data for and n , respectively. Solid line represents the three-band model and dashed lines are for the extended Cauchy model. The fitting parameters are listed in Table 6.1. Li and Wu 2004. Reproduced with permission from the American Institute of Physics.

Table 6.1 Fitting parameters for the three-band model and the extended Cauchy equations. LC 5CB at...

Gallium Antimonide (GaSb). Transport in n-t)fpe GaSb is cort5)licated by the contribution of three sets of conduction bands with minima located at F, L, and X. The data on transport coefficients can be consistently explained by a three-band model, the X bands contributing to transport above 180 °C. Intrinsic carrier concentrations i of the order of 10 cm have been estimated for a tert5)erature T = 365 K. 

The three-band model has several parameters and is still rather complicated. Therefore, an even simpler model, the one-band Hubbard model, is often used as a crude model for high Tq materials. The Hubbard model arranges the tight-binding states on a square lattice. Each electron can hop to a nearest-neighbor site and two electrons (of different spin) on the same site have a repulsive interaction. No other electron-electron interactions are included therefore the Hubbard model assumes the repulsions are heavily screened by the ions in the crystal. There have been attempts to justify the reduction of the three-band model to the one-band Hubbard model, although the justifications must be considered very weak. Therefore, the one-band Hubbard model would not describe the details of high Tq materials correctly, but at best qualitatively. 

The spectrum shown in Fig. 7.5 shows the appropriate portion of the spectrum for a copolymer prepared from a feedstock for which fj = 0.153 It turns out that each polyene produces a set of three bands The dyad is identified with the peaks at X = 298, 312, and 327 nm the triad, with X = 347 367, and 388 nm and the tetrad with X = 412 and 437 nm. Apparently one of the tetrad bands overlaps that of the triad and is not resolved. Likewise only one band (at 473 nm) is observed for the pentad. The identification ol these features can be confirmed with model compounds and the location and relative intensities of the peaks has been shown to be independent of copolymer composition. 

Fig. 2. Electrical resistance as a function of the temperature at the indicated magnetic fields for a bundle of CNTs. The dashed lines separate three temperature ranges, while the continuous curve is a fit using the two-band model for graphite (see inset) with an overlap of 3.7 meV and a Fermi levei right in the middie of the overlap [9].

They found that the order for the M—0 stretching modes (M—0 force constants) was (02)Pd(02) > Pd(Oj) and (02)Ni(02) > Ni(02), and since the 0-0 force constant increased as the M-0 force constant decreased, reversibility could not be equated with shorter 0—0 bond length. To test the proposed isosceles model they predicted the various absorptions that would be expected on the basis of the symmetrical structure (a) and unsymmetrical structure (c), Fig. 8. If the isotopic ratio of 0 0 is 1 1, then three bands would be expected for structure (a), with relative intensities 1 2 1, and four bands would be expected for structure (c), with relative intensities 1 1 1 1, the two oxygen atoms now being in different environments. These workers 189) then obtained the IR spectrum for the cocondensation product from the reaction between nickel and O2 (4.2—10°K), in isotopic ratio 0 =1 1, and found three bands with relative inten-... 

Fig. 4. Energy level diagrams showing possible electronic configurations for positively-charged polaron (a) and bipolaron (b) defects and (c) a schematic bipolaron band model. The negatively-charged polaron would carry three electrons and the bipolaron four. Also shown is the neutral polaron-exciton (d) which would decay to restore the chain structure.

Ultrafast ESPT from the neutral form readily explains why excitation into the A and B bands of AvGFP leads to a similar green anionic fluorescence emission [84], Simplistic thermodynamic analysis, by way of the Forster cycle, indicates that the excited state protonation pK.J of the chromophore is lowered by about 9 units as compared to its ground state. However, because the green anionic emission is slightly different when it arises from excitation into band A or band B (Fig. 5) and because these differences are even more pronounced at low temperatures [81, 118], fluorescence after excitation of the neutral A state must occur from an intermediate anionic form I not exactly equivalent to B. State I is usually viewed as an excited anionic chromophore surrounded by an unrelaxed, neutral-like protein conformation. The kinetic and thermodynamic system formed by the respective ground and excited states of A, B, and I is sometimes called the three state model (Fig. 7). 

The complexes with NCO-, HS- and sulfonamides all give molar absorbancies from approximately 400 to 600 M-1 cm-1 which is in the low range of pseudotetrahedral models. Moreover, the bands are broader and show more structure than in the case of CN- (cf. Figures 6—8). These properties suggest substantial contributions from fields of lower symmetry. Coleman (51) has fitted Gaussian curves to the observed spectra, and his analysis indicates the presence of five closely spaced bands, while the cyanide complex was resolved into three bands. 

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