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, p. 130

Section Frame integration

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: Frame integration

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2D frame integration is a data-reduction process which converts a two-dimensional frame into a one-dimensional intensity profile. Two forms of integration are generally of interest in the analysis of a 2D diffraction frame from polycrystalline materials: γ integration and 2θ integration. γ integration sums the counts in 2θ steps (Δ2θ) along constant 2θ conic lines and between two constant γ values. γ integration produces a data set with intensity as a function of 2θ. 2θ integration sums the counts in γ steps (Δγ) along constant γ lines and between two constant 2θ conic lines. 2θ integration produces a data set with intensity as a function of γ. γ integration may also be carried out with the integration range in the vertical direction as a constant number of pixels. This type of γ integration may also be referred to as slice integration. A diffraction profile analogous to the conventional diffraction result can be obtained by either γ integration or slice integration over a selected 2θ range. Phase ID can then be done with conventional search/match methods. 2θ integration is of interest for evaluating the intensity variation along γ angles, such as for texture analysis, and is discussed in more depth in Chapter 5.3[link] .

The γ integration can be expressed as[I(2\theta) = \textstyle\int\limits_{{\gamma _1}}^{{\gamma _2}} {J(2\theta, \gamma)\,{\rm d}\gamma }, \quad 2{\theta _1} \le 2\theta \le 2{\theta _2},\eqno(2.5.23)]where J(2θ, γ) represents the two-dimensional intensity distribution in the 2D frame and I(2θ) is the integration result as a function of intensity versus 2θ. γ1 and γ2 are the lower limit and upper limit of integration, respectively, which are constants for γ integration. Fig. 2.5.13[link] shows a 2D diffraction frame collected from corundum (α-Al2O3) powder. The 2θ range is from 20 to 60° and the 2θ integration step size is 0.05°. The γ-integration range is from 60 to 120°. In order to reduce or eliminate the dependence of the integrated intensity on the integration interval, the integrated value at each 2θ step is normalized by the number of pixels, the arc length or the solid angle. γ integration with normalization by the solid angle can be expressed as[I(2\theta) = {{\textstyle\int_{{\gamma _1}}^{{\gamma _2}} {J(2\theta, \gamma)(\Delta 2\theta)\,{\rm d}\gamma } } \over {\textstyle\int_{{\gamma _1}}^{{\gamma _2}} {(\Delta 2\theta)\,{\rm d}\gamma } }},\quad 2{\theta _1} \le 2\theta \le 2{\theta _2}.\eqno(2.5.24)]Since the Δ2θ step is a constant, the above equation becomes[I(2\theta) = {{\textstyle\int_{{\gamma _1}}^{{\gamma _2}} {J(2\theta, \gamma)\,{\rm d}\gamma } } \over {{\gamma _2} - {\gamma _1}}},\quad 2{\theta _1} \le 2\theta \le 2{\theta _2}.\eqno(2.5.25)]

[Figure 2.5.13]

Figure 2.5.13 | top | pdf |

A 2D frame showing γ integration.

There are many integration software packages and algorithms available for reducing 2D frames into 1D diffraction patterns for polycrystalline materials (Cervellino et al., 2006[link]; Rodriguez-Navarro, 2006[link]; Boesecke, 2007[link]). With the availability of tremendous computer power today, a relatively new method is the bin method, which treats pixels as having a continuous distribution in the detector. It demands more computer power than older methods, but delivers much more accurate and smoother results even with Δ2θ integration steps significantly smaller than the pixel size. Depending on the relative size of Δ2θ to the pixel size, each contributing pixel is divided into several 2θ `bins'. The intensity counts of all pixels within the Δ2θ step are summarized. All the normalization methods in the above integration, either by pixel, arc or solid angle, result in an intensity level of one pixel or unit solid angle. Since a pixel is much smaller than the active area of a typical point detector, the normalized integration tends to result in a diffraction pattern with fictitiously low intensity counts, even though the true counts in the corresponding Δ2θ range are significantly higher. In order to avoid this misleading outcome, it is reasonable to introduce a scaling factor. However, there is no accurate formula for making the integrated profile from a 2D frame comparable to that from a conventional point-detector scan. The best practice is to be aware of the differences and to try not to make direct comparisons purely based on misleading intensity levels. Generally speaking, for the same exposure time, the total counting statistics from a 2D detector are significantly better than from a 0D or 1D detector.


Boesecke, P. (2007). Reduction of two-dimensional small- and wide-angle X-ray scattering data. J. Appl. Cryst. 40, s423–s427.Google Scholar
Campbell, J. W., Harding, M. M. & Kariuki, B. (1995). Spatial-distortion corrections, for Laue diffraction patterns recorded on image plates, modelled using polynomial functions. J. Appl. Cryst. 28, 43–48.Google Scholar
Rodriguez-Navarro, A. B. (2006). XRD2DScan: new software for polycrystalline materials characterization using two-dimensional X-ray diffraction. J. Appl. Cryst. 39, 905–909.Google Scholar

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