Peaks spherical harmonics

For constant D, as the order parameter increases, a larger number of spherical harmonics are involved in producing a more peaked equilibrium distribution. This effect also means that more steps are needed for convngence. 

The quasispherical shape of the C6o molecule suggested that we should analyze the wave functions in terms of their expansion in spherical harmonics. We found that they were dominated by a single / component and we used it to label the lowest peal of the density of states in Fig. 1 is,p,d,... ). The highest peaks also have well-defined angular character, but they have significant splittings and overlaps that will be described in detail elsewhere. Features I and 2 in the spectrum, for example, correspond to tt states with / 5 and 4 angular character, respectively. 

The subsequent refinement included profile parameters X, Y, X , Ya, peak asymmetry, sample displacement and transparency shift. Preferred orientation was switched from the March-Dollase to the 8 -order spherical harmonics expansion (6 variables total) and 12 coefficients of the shifted-Chebyshev polynomial background approximation were employed. A reasonably good fit, shown in Figure 7.36, was achieved as a result. 

Fig. 7. Vertical resolution in the upper mantle of model S16B30 of Masters et al. (1996). The input velocity anomaly is a linear combination of three natural B-splines in radius and spherical harmonics laterally. The continuous curve shows the amplitude of the input anomaly as a function of depth, normalized to unit amplitude. The dashed curve is the peak recovered value. The data inversion tends to smear the anomaly c. 200 km radially. Adapted from Masters et al, (1996).

The following sections develop three subjects the classical approximations for the strain/stress in isotropic polycrystals, isotropic polycrystals under hydrostatic pressure and the spherical harmonic analysis to determine the average strain/stress tensors and the intergranular strain/stress in textured samples of any crystal and sample symmetry. Most of the expressions that are obtained for the peak shift have the potential to be implemented in the Rietveld routine, but only a few have been implemented already. 

Arguments for recent developments of the spherical harmonics approach for the analysis of the macroscopic strain and stress by diffraction were presented in Section 12.2.3. Resuming, the classical models describing the intergranular strains and stresses are too rough and in many cases cannot explain the strongly nonlinear dependence of the diffraction peak shift on sin even if the texture is accounted for. A possible solution to this problem is to renounce to any physical model to describe the crystallite interactions and to find the strain/ stress orientation distribution functions SODF by inverting the measured strain pole distributions ( h(y)). The SODF fully describe the strain and stress state of the sample. 

After the reconstruction, a cross-check should show that the reference reflection is degenerated to a 5-distribution, and there are no negative intensities in the desmeared image. If this is not the case, the found reference peak was broadened not only by imperfect orientation . In this case an iterative trial-and-error method is helpful the peak is proportionally narrowed, until over-desmearing can no longer be detected. The equations mentioned are directly applicable in case of fiber symmetry. If the symmetry of the scattering pattern is lower, the simplification must be reverted to a set of equations in the complete spherical harmonics instead of the Legendre polynomials. 

The first five even spherical harmonic terms describing the data of Fig. 20, derived using eqn (20), are shown in Fig. 21. The portion of the curve relating to the intense interchain peak at s — 1-4A is not drawn for some of the higher harmonic components since it is not... 

While major peaks are reproduced, the fine sfrucfures in fhe abundance spectra of the alkali clusters are not reproduced by any of the spherical models. To explain them we have to go beyond the spherical approximation. In order to accommodate nonspherical clusters, Clemenger [15] proposed a 3D harmonic oscillator model that can distort to spheroidal shapes. Spheroidal clusters have R = Ry, and a different R. Clemenger considered only those shape distortions that preserved the volume of the clusters. Therefore, if R was the radius of the spherical cluster before distortion, R ... 

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