Energy-wave number characteristic

The energy-wave-number characteristic, ( )> depends only on the density of the free-electron gas and the nature of the pseudopotential core but not on the structural arrangement of the atoms. Its behaviour as a function of the wave vector, q, is illustrated in Fig. 6.6, where we see that it vanishes at q0 as expected. It also has a weak logarithmic singularity in its slope at q = 2kF. 

The coordinate system used for the calculation of the energy wave number characteristic q remains fixed as we integrate over k. 

Let us first look a little further into how higher-order terms in the pseudopotential affect properties. The counterpart of the energy wave-number characteristic given in Eq. 17-5, for third-order terms, has been evaluated (Lloyd and Sholl, IflOS Brovman, Kagan, and Holas, 1971). It contains three powers of the form factor, and a double sum over a q and a q is required in the evaluation. An interesting test for the importance of such terms was noted several years ago by... 

As already mentioned in the previous section, this condition is indeed fulfilled by the energy-wave number characteristic. In essence, this relation (Eq. 37) establishes the internally consistent treatment of the potential between the ions and the electrons, as compared to the ion-ion interaction. 

The zero order contribution to the dielectric matrix imposes that Gi = G2. One then immediately recognizes the energy-wave number characteristic in the first term of this expression for the dynamical matrix. Furthermore, in the framework of the local ionic pseudopotential, the induced electron density is given by ... 

The appearance of the energy-wave number characteristic is not essential for the derivation, but is quite natural from perturbation theory with pseudopotentials. In the more elaborate treatments, as e.g. the dielectric formulation, the electronic contribution is more complicated, as is outlined in the related section. This contribution has then to be treated in the appropriate way. The main purpose of the Ewald-Fuchs method is to handle the ionic contribution, which is written as in the previous appendix, where q is an arbitrary constant, chosen to obtain optimum convergence for the direct and the reciprocal lattice sum. Denoting the equilibrium positions of the ions by... 

Concentrating now on the ionic contribution, which is obtained by neglecting the energy-wave number characteristic in H q), one finds ... 

Table 4 collects all characteristic results of analysis of the JT interaction in a number of crystals doped with several 3d -ions like V +, Cr , Mn" +. As seen from the table, the JT stabilization energy is always of the order of several hundred wave numbers, varying from 257 cm for Cs2NaYF6 Cr + to 584 cm K2NaScF6 Cr . ... 

The h rpotheses of Kolmogorov allow a number of additional deductions to be formulated on the statistical characteristics of the small-scale components of turbulence. The most important of them is the two-third-law deduced by Kolmogorov [84]. This law states that the mean square of the difference between the velocities at two points of a turbulent flow, being a distance x apart, equals C(ex) / when x lies in the inertial subrange. (7 is a universal model constant. Another form of this assertion (apparently first put forward by Obukhov [116] [117]) is the five-third law. This law states that the spectral density of the kinetic energy of turbulence over the spectrum of wave numbers, k, has the form Cke / k / in the inertial subrange. Cj, is a new model constant (see e.g., [8], sect. 6.4). 

Equation (10.2) can be used to determine the concentration of a compound in a solution if the value of K is known for that compound. Chemical bonds, such as C-C, C-F, etc., absorb different amounts of infrared energy over various wavelengths. Absorption patterns vary from sharp to broad for different bonds. Peak IR absorption wavelength (wave number) is a characteristic of chemical bonds. Absorption over a range of wavelengths called the infrared spectrum is a fingerprint characteristic of an organic material. Qualitative identification can be achieved... 

The vibrational wave-numbers , and characteristic temperatures of ethane are given (Hansen and Dennison, J. Chem. Phys. 1952, 20, 317) in table 2. There are six non-degenerate modes including the internal rotation (mode 4) and six pairs of doubly-degenerate modes. The potential energy u restricting the internal rotation has three equal minima and three equal maxima. It may be represented approximately by the form... 

Each of these vibrations has a characteristic frequency and can occur at quantized frequencies only. When IR light of the same frequency is incident on the molecule, the energy is absorbed by the molecule and the amplitude of the particular mode increases. However, this absorption occurs only if this vibrational mode can cause a change in the molecular dipole. Consequently, not all vibrational modes are IR active and the molecular symmetry plays a key role in the reduction of IR spectrum patterns. In addition to these fundamental vibrations, overtone peaks may also be observed with much reduced intensity at two, three times, and so on, the wave numbers, the sum of two or three times the wave numbers, or the difference between two wave numbers. Detailed IR spectroscopic theory and group theory can be found elsewhere [60-62]. 

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