International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 2.5, pp. 143-144
Section 2.5.4.3.5. Data integration and peak evaluation^{a}Bruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA |
The purpose of data integration and peak evaluation is to generate a set of data points along distorted diffraction rings. Data integration for stress analysis is γ integration over several defined segments so as to generate diffraction profiles representing the corresponding segments. The peak position can be determined by fitting the diffraction profile to a given analytic function. Fig. 2.5.26 illustrates data integration over a diffraction frame. The total integration region is defined by , , and . The integration region is divided into segments given by . One data point on the distorted diffraction ring is generated from each segment. The γ value in the centre (denoted by the dot-dashed line) of each segment is taken as the γ value of the data point. γ integration of the segment produces a diffraction profile and the 2θ value is determined from the profile. The number of segments and the segment size () are selected based on the quality of the data frame. The larger the segment size is, the better the integrated diffraction profile as more counts are being integrated. γ integration also produces a smearing effect on the diffraction-ring distortion because the counts collected within the segment size are considered as a single γ value at the segment centre. The 2θ shift in the segment is averaged. The segment size should be sufficient to produce a smooth diffraction profile, but not so large as to introduce too much smearing. For data frames containing high pixel counts, the integration segment can be small, e.g. , and still have a smooth profile for each segment. For data frames having low pixel counts, for example the frames collected from a micron-sized area, from a sample with large grains or with a short data-collection time, it is critical to choose a sufficiently large segment size. The segment size can be determined by observing the smoothness of the integrated profile.
Peak evaluation in each segment can be done using the same algorithm used in the conventional method. The corrections to the integrated profiles are performed before or during the peak evaluation. Absorption correction eliminates the influence of the irradiated area and the diffraction geometry on the measured intensity distribution. The absorption for a given material and radiation level depends on the incident angle to the sample and the reflected angle from the sample. For 2D-XRD, the reflected angle is a function of γ for each frame. The polarization effect is also a function of γ. Therefore, the correction for polarization and absorption should be applied to the frame before integration. (Details of these corrections were discussed in Section 2.5.4.3.4.) The polarization and absorption correction is not always necessary if the error caused by absorption can be tolerated for the application, or if the data-collection strategy involves only ψ and ϕ scans.
In most cases, Kα radiation is used for stress measurement, in which case the weighted average wavelength of Kα_{1} and Kα_{2} radiation is used in the calculations. For samples with a broad peak width, diffraction of Kα_{1} and Kα_{2} radiation is merged together as a single peak profile, and the profile can be evaluated as if there is a single Kα line without introducing much error to the measured d-spacing. For samples with a relatively narrow peak width, the diffraction profile shows strong asymmetry or may even reveal two peaks corresponding to the Kα_{1} and Kα_{2} lines, especially at high 2θ angles. In this case the profile fitting should include contributions from both the Kα_{1} and Kα_{2} lines. It is common practice to use the peak position from the Kα_{1} line and the Kα_{1} wavelength to calculate the d-spacing after Kα_{2} stripping.
Background correction is necessary if there is a strong background or the peak-evaluation algorithms are sensitive to the background, such as in Kα_{2} stripping, peak fitting, and peak-intensity and integrated-intensity evaluations. Background correction is performed by subtracting a linear intensity distribution based on the background intensities at the lower 2θ side and the higher 2θ side of the diffraction peak. The background region should be sufficiently far from the 2θ peak so that the correction will not truncate the diffraction profile. The 2θ ranges of the low background and high background should be determined based on the width of the 2θ peak and available background in the profile. Based on a normal distribution, a 2θ range of 2 times the FWHM covers 98% of the peak intensity, and 3 times the FWHM covers more than 99.9%, so the background intensity should be determined at more than 1 to 1.5 times the FWHM away from the peak position. The background correction can be neglected for a profile with a low background or if the error caused by the background is tolerable for the application. The peak position can be evaluated by various methods, such as gravity, sliding gravity, and profile fitting by parabolic, pseudo-Voigt or Pearson-VII functions (Lu, 1996; Sprauel & Michaud, 2002).
References
Lu, J. (1996). Handbook of Measurement of Residual Stress. Society for Experimental Mechanics. Lilburn: The Fairmont Press.Google ScholarSprauel, J. M. & Michaud, H. (2002). Global X-ray method for the determination of stress profiles. Mater. Sci. Forum, 404–407, 19–24.Google Scholar