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. 130-131
Section 2.5.3.3.4. Lorentz, polarization and absorption corrections^{a}Bruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA |
Lorentz and polarization corrections may be applied to the diffraction frame to remove their effect on the relative intensities of Bragg peaks and background. The 2θ angular dependence of the relative intensity is commonly given as a Lorentz–polarization factor, which is a combination of Lorentz and polarization factors. In 2D diffraction, the polarization factor is a function of both 2θ and γ, therefore it should be treated in the 2D frames, while the Lorentz factor is a function of 2θ only. The Lorentz correction can be done either on the 2D frames or on the integrated profile. In order to obtain relative intensities equivalent to a conventional diffractometer with a point detector, reverse Lorentz and polarization corrections may be applied to the frame or integrated profile.
The Lorentz factor is the same as for a conventional diffractometer. For a sample with a completely random orientation distribution of crystallites, the Lorentz factor is given as
The Lorentz factor may be given by a different equation for a different diffraction geometry (Klug & Alexander, 1974). The forward and reverse Lorentz corrections are exactly reciprocal and effectively cancel each other. Therefore, it is not necessary to perform the Lorentz correction to the frame before integration if relative intensities equivalent to a conventional Bragg–Brentano diffractometer are expected. The Lorentz correction can be done on the integrated diffraction profiles in the same way as on the diffraction profiles collected with conventional diffractometers.
When a non-polarized X-ray beam is scattered by matter, the scattered X-rays are polarized. The intensity of the diffracted beam is affected by the polarization; this effect is expressed by the polarization factor. In two-dimensional X-ray diffraction the diffraction vectors of the monochromator diffraction and sample crystal diffraction are not necessarily in the same plane or perpendicular planes. Therefore, the overall polarization factor is a function of both 2θ and γ. Fig. 2.5.14 illustrates the geometric relationship between the monochromator and detector in the laboratory coordinates, X_{L}, Y_{L}, Z_{L}. The monochromator is located at the coordinates , which is a translation of the laboratory coordinates along the X_{L} axis in the negative direction. The monochromator crystal is rotated about the axis and 2θ_{M} is the Bragg angle of the monochromator crystal. The diffracted beam from the monochromator propagates along the X_{L} direction. This is the incident beam to the sample located at the instrument centre O. The 2D detector location is given by the sample-to-detector distance D and swing angle α. The pixel P(x, y) represents an arbitrary pixel on the detector. 2θ and γ are the corresponding diffraction-space parameters for the pixel. Since a monochromator or other beam-conditioning optics can only be used on the incident beam, the polarization factor for 2D-XRD can then be given as a function of both θ and γ:
If the crystal monochromator rotates about the axis, i.e. the incident plane is perpendicular to the diffractometer plane, the polarization factor for two-dimensional X-ray diffraction can be given as
In the above equations, the term cos^{2} 2θ_{M} can be replaced by for different monochromator crystals. For a mosaic crystal, such as a graphite crystal, n = 2. For most real monochromator crystals, the exponent n takes a value between 1 and 2. For near perfect monochromator crystals, n approaches 1 (Kerr & Ashmore, 1974). All the above equations for polarization factors may apply to multilayer optics. However, since multilayer optics have very low Bragg angles, approximates to unity. The γ dependence of the polarization factor diminishes in this case. The polarization factor approaches
References
Kerr, K. A. & Ashmore, J. P. (1974). Systematic errors in polarization corrections for crystal-monochromatized radiation. Acta Cryst. A30, 176–179.Google ScholarKlug, H. P. & Alexander, L. E. (1974). X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials. New York: John Wiley & Sons.Google Scholar