Tables for
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.5, pp. 138-139

Section Texture-data processing

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: Texture-data processing

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For a specific diffraction ring, 2θ is a constant or at least assumed to be constant for texture analysis, and the sample-orientation angles (ω, ψ, ϕ) for a frame are also constants. Therefore, the pole-density information is given by the diffraction-intensity distribution as a function of γ only, or I = I(γ). Integration of the diffraction intensities in the 2θ direction converts 2D information into the function I(γ).

Fig. 2.5.21[link](a) shows a 2D diffraction ring for texture analysis. The low and high background and diffraction-ring 2θ–γ range are defined by three boxes, noted as BL, BH and I(γ), respectively. All three boxes have the same γ range from γ1 to γ2. The 2θ ranges for the diffraction ring, low background and high background should be determined based on the width of the 2θ peak and available background between adjacent peaks. Assuming a normal distribution, a 2θ range of 2 times the FWHM covers 98% of the intensity peak, and 3 times the FWHM covers more than 99.9%. The 2θ range should also be broad enough to cover the possible 2θ shifts caused by residual stresses in the sample. Fig. 2.5.21[link](b) is the 2θ profile integrated over the section Δγ in Fig. 2.5.21[link](a). The background ranges on the low and high 2θ sides are given by 2θL1–2θL2 and 2θH1–2θH2, respectively. The 2θ-integrated diffraction intensities as a function of γ are plotted in Fig. 2.5.21[link](c). The background can be calculated and removed from the intensity values of the low and high backgrounds or ignored if the contribution of the background is very small.

[Figure 2.5.21]

Figure 2.5.21 | top | pdf |

Pole-figure data processing: (a) a frame with the 2θ integration ranges for the (220) ring; (b) 2θ profile showing the background and peak; (c) integrated intensity distribution as a function of γ.

2θ integration without a background correction can be expressed as[I(\gamma) = \textstyle\int_{2{\theta _1}}^{2{\theta _2}} {J(2\theta, \gamma)\, {\rm d}(2\theta)}, \quad {\gamma _1} \le \gamma \le {\gamma _2}.\eqno(2.5.61)]A similar equation can be used for 2θ integration of the low and high backgrounds BL(γ) and BH(γ). Assuming a linear background change in the vicinity of the 2θ peak, the background under the peak, B(γ), is then given by[\eqalignno{B(\gamma) &= {B_L}(\gamma){{(2{\theta _2} - 2{\theta _1})(2\theta_{H2}+2\theta_{H1}-2\theta_2 - 2\theta_1)} \over {(2{\theta _{L2}} - 2{\theta _{L1}})(2\theta_{H2}+2\theta_{H1}-2\theta_{L2}-2\theta_{L1})}} &\cr&\quad+ {B_H}(\gamma){{(2{\theta _2} - 2{\theta _1})(2{\theta _2} + 2{\theta _1} - 2{\theta _{L2}} - 2{\theta _{L1}})} \over {(2{\theta _{H2}} - 2{\theta _{H1}})(2{\theta _{H2}} + 2{\theta _{H1}} - 2{\theta _{L2}} - 2{\theta _{L1}})}}.&\cr&&(2.5.62)}]Then the background B(γ) can be subtracted from the integrated intensity distribution I(γ).

The algorithms of γ integration given in Section[link] can be easily modified for 2θ integration by exchanging γ and 2θ in the equations. Algorithms with solid-angle normalization should be used to get consistent integrated intensity over all areas of the detector. The 2θ-integrated intensity distribution can then be mapped onto a pole figure based on the fundamental equations (2.5.53)[link] and (2.5.54)[link]. When a pole-figure pixel is overlapped by more than one data point from different scans, as shown in the region covered by two scans in Fig. 2.5.20[link](b), the average value should be mapped to that pole-figure pixel. Fig. 2.5.22[link](a) shows pole-density mappings on the pole figure. There are big gaps between the measured pole-density data points due to the large ϕ-scan steps of 5°.

[Figure 2.5.22]

Figure 2.5.22 | top | pdf |

Pole-figure processing: (a) I(γ) mapped to the pole figure; (b) Pole figure after interpolation and symmetry processing.

All factors affecting relative intensities, such as Lorentz, polarization, air scattering, and Be-window and sample absorption, will have an effect on the measured pole densities for the pole figures. Some or all these corrections may be applied to the diffraction frames before 2θ integration if the texture study demands high accuracy in the relative pole densities. Among these factors, the most important factor is sample absorption, since data sets for pole figures are typically collected at several different incident angles. A ridge between the pole-density regions covered by two different incident angles may be observed if sample absorption is not properly corrected.

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