Recasting Classical Expansion of Orientation Distribution Function as Tensorial Fourier Expansion

Tensorial Fourier expansion of orientation distribution function defined on the orthogonal group O(3)

“…For more detailed discussions on the preliminaries presented in §2a, see [6, Chap. 17], [16] and the references therein; for those in §2b, see [4].…”

“…20–21], which covers a broader context). Following this algorithm, we obtain and report in [16, tables 1 and 2] an orthonormal irreducible tensor basis falsefalse{bold-italicBml:−l≤m≤lfalsefalse} of scriptHl for 1≤l≤8.…”

“…For our present purpose, the following relation between the irreducible tensor basis falsefalse{bold-italicBmlfalsefalse} and the Wigner D-functions, namely (see [6, eqn (17.70)] and [16, eqn (5.3)]) right left right left right left right left right left right left3pt0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0emtrue⟨bold-italicBml,bold-italicR⊗lbold-italicBnl⟩ =Dmnl(bold-italicR)(1≤l<∞;−l≤m≤l,−l≤n≤l)for each bold-italicR∈SO(3) ,is crucial.…”

“…Standard equipment and software in texture measurements by XRD or electron backscatter diffraction report texture coefficients pertaining to some version of classical expansion of the ODF. There was no systematic study on the conversion between the classical and tensorial Fourier expansions until the publication of Man [6,16] in 2022. In [6, Chap.…”

“…In this paper, we consider ODFs defined on the orthogonal group O(3), each of which has its group of crystallite symmetry Gnormalcr and of texture symmetry Gnormaltex defined by a crystallographic point group and a subgroup of O(3), respectively. Our main objective is to extend the procedure given in [16] for writing down the tensorial Fourier expansion of an ODF w if its classical texture coefficients are known.…”