Three-Dimensional Reciprocal Form Factors and Momentum Densities of Electrons from Compton Experiments I. Symmetry-Adapted Series Expansion of the Electron Momentum Density

Abstract

All methods of reconstruction of the electron momentum density from Compton scattering data are based on a series expansion of this density in terms of symmetry-adapted surface spherical harmonics (polyhedral harmonics). Owing to the improvements of y-ray and X-ray Compton spectrometers during the last few years, measurements of Compton spectra of single crystals in a shorter time and with higher angular resolution have become feasible. Therefore, in addition to the current studies, an increased number of investigations of a greater variety of crystals with large sets of directional data will soon be performed. This work is at first concerned with the accurate and efficient computation of associated Legendre functions of the first kind, Pf, with high / and m and for the whole range of their real argument, which are needed for the description of data with high angular resolution. We further show that electron momentum densities obey Laue-class symmetries in the case of crystals and point-group symmetries with an inversion centre in general, in absence of external magnetic fields. All necessary information about the polyhedral harmonics belonging to the totally symmetric representations of these point groups is given, with particular emphasis on the group O h for which the hexoctahedral harmonics with I ^ 20 are displayed graphically. In addition, the locations of the extrema on the unit sphere are tabulated.

Explicit Orthonormal Fixed Bases for Spaces of Functions that are Totally Symmetric Under the Rotational Symmetries of a Platonic Solid

“…Work on the construction of basis functions for various representations of the icosahedral group, including the T~ (O, ~o) functions for the identity representation, has been done previously by Altmann (1957), Cohan (1958), Elcoro, Perez-Mato & Madariaga (1994), Heuser-Hofmann & Weyrich (1985), Jack & Harrison (1975), Kara & Kurki-Suonio (1981), Laporte (1948), Liu, Ping &Chen (1990), McLellan (1961) and Meyer (1954), among others. The approaches used include group-theoretical approaches and approaches through combining primitive invariant polynomials in the Cartesian coordinates.…”