Description of Crystallite Orientation in Polycrystalline Materials. III. General Solution to Pole Figure Inversion

Abstract

A method is presented here by which orientation distribution of crystallites in anisotropic polycrystalline samples can be derived from a set of plane-normal distributions obtained by x-ray diffraction measurements. It is the generalization of the similar procedure proposed previously for analysis of samples having fiber texture. It thus represents a completely general solution to the problem of pole figure inversion, applicable to samples having any arbitrary symmetry elements. The plane-normal distribution function is expanded in a series of spherical harmonics, the coefficients of which, Qlmi, can be determined by numerical integration of experimental diffraction data. The crystallite distribution function is expanded in a series of generalized spherical harmonics which appear as solutions to the Schrödinger wave equation of a symmetric top. The coefficients of the crystallite distribution function, Wlmn, are then obtained as linear combinations of Qlmi. Symmetry properties of Wlmn arising from crystallographic or statistical symmetry elements existing in the sample are examined. The methods of estimating the series truncation errors and of minimizing the experimental error by a least-squares method, previously proposed in connection with fiber texture analysis, are still applicable here with appropriate generalizations. In addition it is shown that the effect of diffraction line broadening due to finite size or imperfection of crystallites can also be allowed for at least approximately.

Effect of Chemical Composition and Thermomechanical Processing on Texture in Hot Bands of Ti and Ti+Nb Containing Ultra-low Carbon Steels

“…Texture measurements were conducted by X-ray diffraction using a Philips X'PERT system with an open cradle and copper K a radiation. Orientation distribution functions (ODF) of the global structure were calculated using the popLA algorithm based on the series expansion method proposed by Bunge 27) and Roe 28) from three incomplete pole figures of {200}, {110}, and {211}, which were measured up to an azimuth of 85 degrees. Experimental background and defocusing corrections were applied to the pole figures by measuring the texture of a pure random iron sample.…”

Use of the Velocity of Higher Order Lamb Modes in the Measurement of Texture

“…This fundamental consequence of the fact that the basis function having the coefficient w 400 varies only with the polar angle e, as can be seen from Eq. (1). When the polar axis is chosennormal to the plate, this contribution is independent of rotations of crystallites in the plane of the plate.…”

“…(0 e e (1) R.=O m=-R. n=-R. R.mn R.mn where 6, cp and ljl are Euler angles descri bi ng the ori entat ion of a part i cular crystallite with respect to the sample axes, ~=cos6, the ZR.~n are the Generalized Legendre functions, and WR. are the orientation d1stribution coefficients (ODC's).…”

“…The foundat i on for these procedures lies in the mathematical descdption of texture, in which the crystallite orientation distribution function (CODF) is expanded as a series of spherical harmon- ICS. In the notation of Roe [1,2], the expansion of the ODF takes the form R.…”

Explicit Formulae Showing the Effects of Texture on Acoustoelastic Coefficients

“…Following usual practice, we describe the crystallographic texture at X by an orientation distribution function (ODF) w. Using the convention initiated by Roe (3], we expand w as an infinite series of the generalized spherical functions:…”