Rendering of Crystallographic Orientations, Orientation and Pole Probability Density Functions

Since the domain of crystallographic orientations is three-dimensional and spherical, their insightful rendering or rendering of related probability density functions requires (i) to exploit the effect of a given orientation on crystallographic axes, (ii) to consider spherical means of the orientation probability density function with respect to lower–dimensional manifolds, and (iii) to apply projections from the two–dimensional unit spheres S ⊂ IR onto the unit disk D ⊂ IR. The familiar crystallographic “pole figures” are actually the sum of two spherical X–ray transforms which associate with a real–valued function f defined on a sphere its mean values Xf along one–dimensional circles with centre O, the origin of the coordinate system, and spanned by two unit vectors. The family of views suggested here defines generalized cross–sections in terms of simultaneous orientational relationships of two different crystal axes with two different specimen directions such that their superposition yields a user–specified pole probability density function. Thus, the spherical averaging and the spherical projection onto the unit disk determine the distortion of the display. Commonly, the spherical projection preserving either volume or angle are favored. This rich family displays f completely, i.e. if f is given or can be determined unambiguously, then it is uniquely represented by several subsets of these views. A computer code enables the user to specify and control interactively the display of linked views which is comprehensible as the user is in control of the display. INTRODUCTION AND MOTIVATION A crystallographic orientation is basically the rotation g∗ in SO(3) which maps a right– handed coordinate system KS fixed to the specimen onto another right–handed coordinate system KC fixed to the crystal. It may be parameterized by three conventionally defined Euler angles, e.g. g∗ ∈ G ⊂ SO(3) : g∗ = g∗(α, β, γ) where the first rotation by α ∈ [0, 2π) is about the Z-axis, the second by β ∈ [0, π] about the (new) Y ′-axis, and the third by γ ∈ [0, 2π) about the (new) Z ′′-axis of the crystal coordinate system KC. Then (α, β) are the spherical coordinates of the direction Z ′′ = ZC = zKC with respect to KS . There exist 11 other ways to define Euler angles, and they are all in use, somewhere [3]. The coordinates of a unique direction denoted h with respect to the crystallographic coordinate system KC = g KS and r with respect to the specimen coordinate system KS are related to each other by h =M((g∗)−1) r =:M(g) r with M(g) =  cosα cosβ cos γ − sinα sin γ sinα cosβ cos γ + cosα sin γ − sinβ cos γ − cosα cosβ sin γ − sinα cos γ − sinα cosβ sin γ + cosα cos γ sinβ sin γ cosα sinβ sinα sinβ cosβ  . Copyright©JCPDS International Centre for Diffraction Data 2003, Advances in X-ray Analysis, Volume 46. 145