Double spiral

Double spiral

Simulations of EPR/NMR powder spectra are based on the summation of spectra calculated for large array of orientations {q, j} of the applied magnetic field. Conventional partition schemes (simple square grid, "igloo", equilateral triangular partitions, SOPHE partitions with equal solid angle triangles) give rather jagged initial spectra, which demand additional smoothing. The spiral choice of {q, j} nodes essentially reduces the "banding" of powder spectra, however, until now the nodes were determined at each angular step using a minimizing routine. This allowed implementing only one-dimensional interpolation along the spiral line, which is less efficient than two-dimensional interpolation for triangular partitions.

We have developed a new approach combining advantages of the spiral partition with the powerful averaging procedure. We found a way to determine the nodes without minimization. Resonance spectra are calculated for the each node of the spiral with the help of exact diagonalization of spin-Hamiltonian matrixes or by perturbation theory. A second spiral shifted with respect to the first one on Dj=180/N degrees is used to build basic triangles for the interpolation: N is equal to 1, 2, 3, 4 or 6, depending on the point group symmetry of the considered site. Since magnetic field is axial vector, there is no necessity to calculate spectra for the second spiral; this halves required computer time. Apexes of smaller triangles (sub-partition) are obtained by recurrent dichotomy of angles of the basic triangles. The nodes between two spirals fill up uniformly the {q, j} space, if these spirals circle from pole to pole. Therefore, a correction of intensity contributions, which is unavoidable for other partition schemes, falls off (for instance, the uncorrected absorption calculated for square grid diverges, when q approaches zero). The choice of the minimal integration domain as a function of the site symmetry is also simplified, since double spiral domains are automatically optimal. Two methods of spectrum smoothing are used: a nonlinear interpolation of resonance fields or frequencies for apexes of small triangles and/or a covering of the spectral area between calculated intensity values ("tent").

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