Bispiral Approach for Calculation of Electron Paramagnetic and Nuclear Magnetic Resonance Powder Spectra

The simulation of powder spectra involves summation of spectra calculated for N reference directions of the external magnetic field. Usually, the directions are given by regularly or randomly distributed points on a sphere. Due to an excessive number of points with the same polar angle θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document} but with different azimuthal angles φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}, axial distributions produce jagged spectra, especially for spin systems with a weak azimuthal anisotropy. To improve the quality of the obtained spectra, a triangulation and subsequent interpolation of resonance fields/frequencies for hundreds of additional directions between triangle vertices or an average over a range of magnetic fields/frequencies (tent) are applied. A single spiral method with graduate steps for both the θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document} and φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document} angles works better for systems with weak azimuthal anisotropy but allows for only a few interpolation points along the spiral. The proposed bispiral approach combines the best features of both spiral and triangular approaches: exact calculations for N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} reference spiral directions, joining of neighboring points of two spirals into a triangular net, and interpolation over hundreds of additional directions or the tent average. For systems with C1 symmetry, the angular space between primary and complementary spirals is exactly equal to the phase space of the magnetic fields (hemisphere). For systems with higher symmetry, the angular space can be significantly reduced by choosing the φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi$$\end{document}-shift for the second spiral, on par with the space reduction for axial distributions. Spectra simulated for axial, random, and bispiral distributions with two-dimensional interpolation over triangles and for the semispiral grid with one-dimensional interpolation are compared.