Convergence of the hyperspherical harmonic expansion for crystallographic texture

Advances in instrumentation allow a material texture to be measured as a collection of spatially resolved crystallite orientations rather than as a collection of pole figures. However, the hyperspherical harmonic expansion of a collection of spatially resolved crystallite orientations is subject to significant truncation error, resulting in ringing artifacts (spurious oscillations around sharp transitions) and false peaks in the orientation distribution function. This article finds that the ringing artifacts and the accompanying regions of negative probability density may be mitigated or removed entirely by modifying the coefficients of the hyperspherical harmonic expansion by a simple multiplicative factor. An addition theorem for the hyperspherical harmonics is derived as an intermediate result.

CALCULATION OF ENERGY EIGENVALUES AND EIGENVECTORS FOR THREE-BODY MOLECULES IN JACOBI AND HYPERSPHERICAL COORDINATES

The Jacobi coordinates is used to eliminate center of mass motion of three-body systems. We write the results in hyperspherical coordinates and expand eigenfunction in a series of orthonormal complete set of Ykαi(Ωi) in partition i of Jacobi coordinates. The matrix elements of two-body interaction potential in hyperspherical harmonic approach are determined exactly using computed analytical form of Raynal–Revai coefficients to change the base set of Ykαi(Ωi) to other set such as Ykαj(Ωj). The generalized Laguerre functions are used to change the second order coupled differential equations to a non-differential matrix equation. This is solved to find energy eigenvalues and eigenfunctions of three-body molecules. The obtained results are in agreement with computational methods.