Algorithms of Two-Dimensional X-Ray Diffraction
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Algorithms of Two-Dimensional X-Ray Diffraction Bob B. He Bruker AXS Inc. 5465 East Cheryl Parkway, Madison, WI 53711, USA ABSTRACT X-ray diffraction pattern collected with two-dimensional detector contains the scattering intensity distribution as a function of two orthogonal angles. One is the Bragg angle 2T and the other is the azimuthal angle about the incident x-ray beam, denoted by J. A 2D diffraction pattern can be integrated to a conventional diffraction pattern and evaluated by most exiting software and algorithms for conventional applications, such as, phase identification, structure refinement and 2T-profile analysis. However, the materials structure information associated to the intensity distribution along J direction is lost through the integration. The diffraction vector approach has been approved to be the genuine theory in 2D data analysis. The unit diffraction vector used for 2D analysis is a function of both 2T and J. The unit diffraction vector for all the pixels in the 2D pattern can be expressed either in the laboratory coordinates or in the sample coordinates. The vector components can then be used to derive fundamental equations for many applications, including stress, texture, crystal orientation and crystal size evaluation.
(a) (b) Figure 1. A 2D diffraction pattern collected from a multilayer battery anode; (a) data frame as collected; (b) the 2D pattern displayed in 3D surface plot. INTRODUCTION Two-dimensional x-ray diffraction (XRD2) is the ideal, non-destructive, analytical method for examining samples of all kinds, such as metals, polymers, ceramics, semiconductors, thin films, biomaterials and composites for material science researches [1-2]. A 2D diffraction pattern
1921 Downloaded from https:/www.cambridge.org/core. Columbia University Libraries, on 15 Jun 2017 at 11:32:53, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1557/adv.2016.409
1922 Downloaded from https:/www.cambridge.org/core. Columbia University Libraries, on 15 Jun 2017 at 11:32:53, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1557/adv.2016.409
intensity distribution when the sample contains residual stress or applied stress loading. The dspacing variation at different orientation due to stress results in the 2T value variation measured at different J angles. Figure 2(d) shows a spotty intensity distribution along J due to only a limited number of large crystals satisfying the Bragg condition. Apparently, the spottiness is related to the crystal size and size distribution. A real material may contain texture, stress and/or large crystal size simultaneously so the 2D diffraction pattern can be a mix of the above four models. Table I. The comparison of Laue equation, Bragg law and diffraction vector containing 2T and J. Illustration Equation Description H The diffraction condition s s0 H hkl (1) for the incident beam O vector s0/O, diffracted beam vector s/O and the reciprocal latt
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