Lorentz weight

It is worth noting that, unlike the standard definition of the strength function with using 5 uj — Uv), we exploit here the Lorentz weight. It is very convenient to simulate various smoothing effects. 

Figure 1. Photoabsorption cross section for the dipole plasmon in axially deformed sodium clusters, normalized to the number of valence electrons N - The parameters of quadrupole and hexadecapole deformations are given in boxes. The experimental data [39] (triangles) are compared with SRPA results given as bars for RPA states and as the strength function (49) smoothed by the Lorentz weight with A = 0.25 eV. Contribntions to the strength function from p =0 and 1 dipole modes (the latter has twice larger strength) are exhibited by dashed curves. The bars are given in eVA. ...

The molar refraction, / m, is a measure of the size of a molecule. It is calculated with Eq. (8.5), the Lorenz-Lorentz equation, where , d, and M are the refractive index, the density, and the molecular weight, respectively. 

The weighting functions used to improve line shapes for such absolute-value-mode spectra are sine-bell, sine bell squared, phase-shifted sine-bell, phase-shifted sine-bell squared, and a Lorentz-Gauss transformation function. The effects of various window functions on COSY data (absolute-value mode) are presented in Fig. 3.10. One advantage of multiplying the time domain S(f ) or S(tf) by such functions is to enhance the intensities of the cross-peaks relative to the noncorrelation peaks lying on the diagonal. 

The Hamiltonian is insensitive to the direction of time, 7i(T) = T L(T ), since it is a quadratic function of the molecular velocities. (Since external Lorentz or Coriolis forces arise from currents or velocities, they automatically reverse direction under time reversal.) Hence both T and I1 have equal weight. From this it is easily shown that. (xl/s) = (exlL). 

Example 1-Propynyl acetate has nn = 1.4187 and density = 0.9982 at 20°C the molecular weight is 98.102. From the Lorentz and Lorenz equation,... 

Mean molecular polarizability can be calculated through the Lorenz-Lorentz- Equation from refractive index, n, molecular weight, MW, and density, d, of a compound, demonstrating that the parameters can be derived from these elementary molecular properties (Figure 3). 

The refractive index is an important quantity for characterizing the structure of polymers. This is because it depends sensitively on the chemical composition, on the tacticity, and - for oligomeric samples - also on the molecular weight of a macromolecular substance. The refractive indices (determined using the sodium D line) of many polymers are collected in the literature. In order to characterize a molecule s constitution one requires knowledge of the mole refraction, Rg. For isotropic samples, it can be calculated in good approximation by the Lorentz-Lorenz equation ... 

The velocity of light passing through a polymer is affected by the polarity of the bonds in the molecule. Polarizability P is related to the molecular weight per unit volume, M, and density p as follows (the Lorenz-Lorentz relationship) ... 

In appraising the average accuracy one must bear in mind that experimentally the mean polarizabilities are usually obtained from the refractive index n (at 5893 A, the sodium D-line) and the Lorenz-Lorentz equation (with M molecular weight, p macroscopic density, Vav Avogadro s number) ... 

Note.—A= atomic weight, nj=refraetivity according to Gladstone s formula, n, = refractivity according to Lorentz and Lorenz s formula.) The atomic refraction ArG of the element is 18-68 (solid), 18-89 (liquid),4 or 18-69 (mean of solid and liquid),5 while the value Arn was 9-10 (mean of solid and liquid).5... 

For X-ray diffraction, the effect of the absorption on the PID is normally negligible for the purpose of polytype identification, if a sufficient number (e.g., four or more) of periods along the same row are considered and the corresponding PIDs are weighted. The LP factors are critical, however, if the diffraction pattern is taken with a precession camera, because the Lorentz-polarization effect in the precession motion is severe. 

Tj corresponds to the tth interface distance, and to its corresponding standard deviation. Interface distances and their distributions were obtained by means of a nonlinear least-squares fitting of the /ideaiC ) 0 to Equation (19.A.2). A weight factor of l//ideai( ) 0 was used. The periodicity or long period L is the sum of + r2- The resulting parameters are shown in Tables 19.A.1 and 19.A.2. Finally, the ideal intensity, the Lorentz intensity, the interference function, and the interface distribution function were... 

The Lorentz-Lorenz molar refractivity, R, is obtained from Eq. (1), relating the geometric mean refractive index, n, the formula weight, M, and the density,(f. Where the unit cell dimensions and number of formula units per cell areknown, the second equation is useful, avoiding the unnecessary addition of the formula weight. 

The single-crystal study of the a-Ni(NCS) (4-ViPy), structure was performed by using a Siemens AED aut omated three-circle diffractometer (filtered MoKa). 5029 independent reflections were measured within 27° of 0 by using the 0) - 20 scan mode, but as little as 950 reflections having 2a(I) have been used for structure analysis. The intensities were corrected for Lorentz-polarization effects but not for absorption. The structure was solved by direct methods, SHELX was used (ref. 5). Full-matrix refinement was made but, in view of low data/parameters ratio, only Ni, thiocyanates and pyridine N atoms were given anisotropic temperature factors. H atoms were included in the refinement at calculated positions. The final R value is 0.056 the weighted = 0.046 (w = 1.7/(a (F) + 0.0002(F) ). 

---