Abstract
The vertices of regular four-dimensional polytopes are used to generate sets of uniformly distributed three-dimensional rotations, which are provided as tables of Euler angles. The spherical moments of these orientational sampling schemes are treated using group theory. The orientational sampling sets may be used in the numerical computation of solid-state nuclear magnetic resonance spectra, and in spherical tensor analysis procedures.
Highlights
Physical properties are anisotropic, meaning that they depend on the orientation of the object of interest in three-dimensional space, defined with respect to an external reference frame
If the physical system is macroscopically isotropic, all molecular orientations are encountered with equal probability
In order to facilitate exploitation of these results, we provide explicit tables of Euler angles derived from the vertices of the regular 4-polytopes
Summary
Physical properties are anisotropic, meaning that they depend on the orientation of the object of interest in three-dimensional space, defined with respect to an external reference frame. We derive by group theory the vanishing spherical moments for 3D rotation sets derived from each of the regular 3D and 4D solids.
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