Dispersion curves

Fig 1. (a) Phase (b) group velocity dispersion curves for aluminium. Circles show minimum dispersion points. Diamonds show excitation positions for transducer designed for X/d = 2.4 where d is the plate thickness. 

SASW-result (dispersion curve) showing Rayleigh Wave speed as a function of wavelength (depth)... 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Fig. 5.37 Comparison of the calculated phonon dispersion curve for Al with the experimental values measured using neutron diffraction. (Figure redrawn from Michin Y, D Farkas, M ] Mehl and D A Papaconstantopoulos 1999. Interatomic Potentials for Monomatomic Metals from Experimental Data and ab initio Calculations. Physical Review 359 3393-3407.)...

Fig. 19. The dispersion curve of a colorless soda-lime—silicate crown glass where the wt % of Na2 0 = 21.3 CaO = 5.2 and Si02 = 73.5.

Figure 5 (a) Dispersion curves for crystalline zwitterionic L-alanine at room temperature along... 

The optical rotatory dispersion curves of steroidal ketones permit a distinction to be made between the conformations, and assignment of configuration is possible without resorting to chemical methods (see, e.g. ref. 36) which are often tedious. The axial halo ketone rule and, in the more general form, the octant rule summarize this principle and have revealed examples inconsistent with the theory of invariable axial attack in ketone bromination. 2-Methyl-3-ketones have been subjected to a particularly detailed analysis. There are a considerable number of examples where the products isolated from kinetically controlled brominations have the equatorial orientation. These results have been interpreted in terms of direct equatorial attack rather than initial formation of the axial boat form. 

The P configuration at C-6 is based on mechanistic considerations and analysis of optical rotatory dispersion curves. 

Figure 8 shows the light dispersion curves for PVA (broken line) and mixed solvents of benzene and bromobenzene is (solid lines). Because the negative inclination of the dispersion curve for PVA is slower then those of... 

The effects of temperature on the color development of the porous film in chlorobenzene were shown in Table 6 [23]. The coloration was reversible thermochromism. The refractive index of the materials generally decreases as the temperature increases, and the temperature dependence of the liquid is greater than that of the solid. For example, the temperature dependence (A/id/°C) of PVA and chlorobenzene was found to be 3.0 x 10 and 4.5 x 10" at 589.3 nm. Consequently, it is interpreted that the wavelength of the crosspoint between the dispersion curves of PVA and chlorobenzene shifts from the long side to the short side with increasing tem-... 

Figure 3 Phonon dispersion curves obtained by inelastic neutron scattering revealing precursor behaviour prior to the 14M transformation in Ni-AI. The dip at q = 1/6 [110] (a) deepens upon cooling and (b) shifts under an external load .

In addition, theoretically calculated dispersion curves and Raman intensities have been reported as well as results of neutron scattering experiments [113, 115]. 

Figure 3 Frequency-dispersion curves of the longitudinal polarizability per unit cell of infinite periodic chains of hydrogen molecules according to the method used (RPA (bottom) and UCHF (top)). AH the values are in a.u.. The position of the first excitation energies which corresponds to the poles is indicated by vertical bars.

The generality of a simple power series ansatz and an open-ended formulation of the dispersion formulas facilitate an alternative approach to the calculation of dispersion curves for hyperpolarizabilities complementary to the point-wise calculation of the frequency-dependent property. In particular, if dispersion curves are needed over a wide range of frequencies and for several optical proccesses, the calculation of the dispersion coefficients can provide a cost-efficient alternative to repeated calculations for different optical proccesses and different frequencies. The open-ended formulation allows to investigate the convergence of the dispersion expansion and to reduce the truncation error to what is considered tolerable. 

From a basis set study at the CCSD level for the static hyperpolarizability we concluded in Ref. [45] that the d-aug-cc-pVQZ results for 7o is converged within 1 - 2% to the CCSD basis set limit. The small variations for the A, B and B coefficients between the two triple zeta basis sets and the d-aug-cc-pVQZ basis, listed in Table 4, indicate that also for the first dispersion coefficients the remaining basis set error in d-aug-cc-pVQZ basis is only of the order of 1 - 2%. This corroborates that the results for the frequency-dependent hyperpolarizabilities obtained in Ref. [45] by a combination of the static d-aug-cc-pVQZ hyperpolarizability with dispersion curves calculated using the smaller t-aug-cc-pVTZ basis set are close to the CCSD basis set limit. 

The CC2 model performes very different for static hyperpolarizabilities and for their dispersion. For methane, CC2 overestimates 70 by a similar amount as it is underestimated by CCS, thus giving no improvement in accuracy relative to the uncorrelated methods CCS and SCF. In contrast to this, the CC2 dispersion coefficients listed in Table 3 are by a factor of 3 - 8 closer to the CCSD values than the respective CCS results. The dispersion coefficients should be sensitive to the lowest dipole-allowed excitation energy, which determines the position of the first pole in the dispersion curve. The substantial improvements in accuracy for the dispersion coefficients are thus consistent with the good performance of CC2 for excitation energies [35,37,50]. 

Our results indicate that dispersion coefficients obtained from fits of pointwise given frequency-dependent hyperpolarizabilities to low order polynomials can be strongly affected by the inclusion of high-order terms. A and B coefficients derived from a least square fit of experimental frequency-dependent hyperpolarizibility data to a quadratic function in ijf are therefore not strictly comparable to dispersion coefficients calculated by analytical differentiation or from fits to higher-order polynomials. Ab initio calculated dispersion curves should therefore be compared with the original frequency-dependent experimental data. 

The present study demonstrates that the analytic calculation of hyperpolarizability dispersion coefficients provides an efficient alternative to the pointwise calculation of dispersion curves. The dispersion coefficients provide additional insight into non-linear optical properties and are transferable between the various optical processes, also to processes not investigated here as for example the ac-Kerr effect or coherent anti-Stokes Raman scattering (CARS), which depend on two independent laser frequencies and would be expensive to study with calculations ex-plictly frequency-dependent calculations. 

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