International
Tables for
Crystallography
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. 130-131

Section 2.5.3.3.4. Lorentz, polarization and absorption corrections

B. B. Hea*

aBruker AXS Inc., 5465 E. Cheryl Parkway, Madison, WI 53711, USA
Correspondence e-mail: bob.he@bruker.com

2.5.3.3.4. Lorentz, polarization and absorption corrections

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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[L = {{\cos \theta } \over {\sin ^22\theta }} = {1 \over {4\sin ^2\theta \cos \theta }}.\eqno(2.5.26)]

The Lorentz factor may be given by a different equation for a different diffraction geometry (Klug & Alexander, 1974[link]). 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[link] illustrates the geometric relationship between the monochromator and detector in the laboratory coordinates, XL, YL, ZL. The monochromator is located at the coordinates [X_L, Y^\prime_L, Z^\prime_L], which is a translation of the laboratory coordinates along the XL axis in the negative direction. The monochromator crystal is rotated about the [Z^\prime_L] axis and 2θM is the Bragg angle of the monochromator crystal. The diffracted beam from the monochromator propagates along the XL 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:[\eqalignno{&P(\theta, \gamma) =\cr&\quad {{(1 + \cos ^22\theta _M\cos ^22\theta)\sin ^2\gamma + (\cos ^22\theta _M + \cos ^22\theta)\cos ^2\gamma } \over {1 + \cos ^22\theta _M}}.&\cr&&(2.5.27)}]

[Figure 2.5.14]

Figure 2.5.14 | top | pdf |

Geometric relationship between the monochromator and detector in the laboratory coordinates.

If the crystal monochromator rotates about the [Y^\prime_L] 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[\eqalignno{&P(\theta, \gamma) =\cr&\quad {{(1 + \cos ^22\theta _M\cos ^22\theta)\cos ^2\gamma + (\cos ^22\theta _M + \cos ^22\theta)\sin ^2\gamma } \over {1 + \cos ^22\theta _M}}.\cr&&(2.5.28)}]

In the above equations, the term cos2 2θM can be replaced by [\left| \cos ^n2\theta _M \right|] 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[link]). All the above equations for polarization factors may apply to multilayer optics. However, since multilayer optics have very low Bragg angles, [\left| \cos ^n2\theta _M \right|] approximates to unity. The γ dependence of the polarization factor diminishes in this case. The polarization factor approaches[P(\theta, \gamma) \simeq {{1 + \cos ^22\theta } \over 2}.\eqno(2.5.29)]

References

Kerr, K. A. & Ashmore, J. P. (1974). Systematic errors in polarization corrections for crystal-monochromatized radiation. Acta Cryst. A30, 176–179.Google Scholar
Klug, H. P. & Alexander, L. E. (1974). X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials. New York: John Wiley & Sons.Google Scholar








































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