Reduced wave vector

Here uf = u exp(277ig r) is, like w, periodic with the period of the lattice, and k = k - 27rg is a reduced wave vector. Repeating this as necessary, one may reduce k to a vector in the first Brillouin zone. In this reduced zone scheme, each wave function is written as a periodic function multiplied by elkr with k a vector in the first zone the periodic function has to be indexed, say ujk(r), to distinguish different families of wave functions as well as the k value. The index j could correspond to the atomic orbital if a tight-binding scheme is used to describe the crystal wave functions. 

Fig. 6. 27 Reduced diffusion coefficient versus reduced wave vector by scaling it to the length scale ls=(a Qy. Data were obtained near the 0-temperature T=39.1 °C. Different symbols diamonds, circles and squares correspond to concentrations c=0.057,0.081 and 0.177 g/cm respectively. (Reprinted with permission from [327]. Copyright 1996 The American Physical Society)...

Examples of these two behaviours are provided by (TMTSF)2Re04 and (TMTSF)2C104. In (TMTSF)2Re04 the anions order at c. 180 K with a reduced wave vector (, and undergo a sharp metal-insulator transition with Ea = 0.16 eV (Jacobsen et al, 1982). The anion ordering transition of (TMTSF)2C104 occurs at 24 K with = (0, 0). This qr value implies that the... 

Therefore k is generally restricted (hence referred to as the reduced wave vector) to a region of k space such that no two points in this region are separated by any vector K. This is a unit cell in reciprocal space, and is referred to as the first Brillouin zone. 

Wilson ((>85) has pointed out that if a Brillouin zone is full, the electrons occupying the states of this zone can make no contribution to the electric current. This fact follows from the definition of the zone as a region enclosing all reduced wave vectors. Imagine all electrons of Figure 6 shifted Akx by an external field. The electrons in states within Akx of the zone boundary are reflected to the opposite zone boundary, so that the zone remains filled and there is no transfer of charge. This observation permits a sharp distinction between metallic conductors, semiconductors, and insulators. Because of the high density of states in a band, a crystal with partially filled bands is a metallic conductor. If all occupied zones (or bands) are filled, the crystal is a semiconductor if Ef kT, is an insulator if kT. 

The solution of the Schroedinger equation for a Morse potential is well known (see e.g. [5]), so that the suppression-enhancement amplimde can be calculated using the available information on the solution of the respective equation (see, e.g. [17]). Let us introduce two dimensionless quantities, the reduced wave vector k, associated with the incoming wave, and the wave-vector-like quantity ko, associated with the potential well depth ... 

Thus these points in a small but well-defined region of k space include all possible irreducible representations of the translation group the vectors of the reciprocal lattice transform points in the Brillouin zone into equivalent points. The Brillouin zone therefore contains the whole symmetry of the lattice, each point corresponding to one irreducible representation, and no two points being related by a primitive translation. The smallest value of k ki, k2, kz) belonging to the rep is called the reduced wave-vector. The set oi reduced wavevectors is called the first Brillouin zone. 

Let us now compare the experimental results on the AOTsystem to the theoretical prediction deduced from the droplet model. In the hydrodynamic regime, significant deviation from the vr behavior of the scattered intensity is deduced from the theory. The expected upward curvature in the intensity vs. temperature plot is in qualitative agreement with the theoretical predictions. Fits of the experimental data to the extended droplet model theory lead to results in quantitative agreement with theory in both the critical and hydrodynamics regimes. The reduced value of T, F, is plotted as a function of the inverse reduced wave vector, for four... 

A particular presentation of experimental data is used. In the theory of lattice vibration it is generally accepted to denote the phonon wave vector by q. The reduced wave vector 5 is plotted along abscissa. This dimensionless quantity equals to the ratio of the magnitude of the phonon wave vector to its maximal magnitude in the given crystallographic direction. By definition. 

The equation (12.22) allows us to calculate the dependence of group velocity of lattice waves on the wave vector (on the wavelength). It is possible if the dispersion curve, that is the dependence frequency v on reduced wave vector has been measured experimentally. 

Surface phonon bands along symmetry lines of the SBZ are given for fee metals in Figs. 5.2-49-5.2-55 and in Table 5.2-20. In all figures the horizontal axis is the reduced wave vector, expressed as the ratio to its value at the zone boundary. Table 5.2-21 gives the surface Debye temperatures for some fee and bcc metals, as well as the amplitudes of thermal vibrations of atoms in the first layer p as compared with those of the bulk pb-In the harmonic approximation, the root mean square displacement of the atoms is proportional to the inverse of the Debye temperature. 

More precise numerical calculations of the reduced threshold voltage (Uth/Uo) and the reduced wave vector d/2w (at the threshold) agree well with experimental data, Fig. 6.13, for the first four Grandjean zones. For thicker cells (or higher zones) formulas (6.17)-(6.21) work quite well. In [25] theoretical and experimental data have also been obtained for the planar texture with the directors at opposite boundaries at an angle 7t/2. 

FIGURE 6.13. Thickness dependence of the reduced wave vector (d/2w) and threshold voltage (Uth/Uo) for the instabiUty shown in Fig. 6.12 [25] (m is number of the Grandjean zone). Crosses (experiment) and solid lines (theory) are threshold voltages circles (experiment) and dash Unes (theory) are wave vectors. 

Figure 6.23 shows the experimental data (obtained on 4-butyl-4 -methoxy-azoxybenzene, BMAOB, doped with 1% of cholesteryl caprinate) and the numerical calculations of the threshold voltage and the reduced wave vector... 

FIGURE 6.23. Threshold voltage (a) and the reduced wave vector (b) in the lowest three Grandjean zones for various dielectric anisotropy. Ae = —0.1 (rectangles), 0 (circles) and +0.25 (triangles). Solid lines are computer calculations [87]. 

A theoretical approach of such Boson peaks is developed, for example, in Refs. 55 and 56 (see Fig. 37). One important characteristic of the Boson peaks is that their spectral profile is universal in the sense that it does not depend on the composition of the glass, the polarization of the light, or the temperature [57], as shown in Fig. 38 in reduced intensity and reduced wave-vector scales. However, these low-lying peaks may not be due purely to acousticlike states localized optical states, medium-range-order effects, and so forth may coexist in the same frequency range and be revealed by Raman scattering. 

Fig.3.18. Observed (points) and calculated (full lines) phonon dispersion of aluminium in the (100) and (llO)-directions at 300 K. 03 in 1013 radians per sec against reduced wave vector. The corresponding Bril-louin zone is shown in Fig. 3.6b. Calculation nearest-neighbour interactiohs with 3 force constants derived from the 3 elastic constants. Experimental values from [3.25]...

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