Instrumental broadening – smearing

Glatter, O.

doi:10.1107/97809553602060000581

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, p. 101

Section 2.6.1.6.2. Instrumental broadening – smearing

O. Glatter^a

2.6.1.6.2. Instrumental broadening – smearing

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These effects can be separated into three components: the two-dimensional geometrical effects and the wavelength effect. The geometrical effects can be separated into a slit-length (or slit-height) effect and a slit-width effect. The slit length is perpendicular to the direction of increasing scattering angle; the corresponding weighting function is usually called P(t). The slit width is measured in the direction of increasing scattering angles and the weighting function is called Q(x). If there is a wavelength distribution, we call the weighting function $[W(\lambda')]$ where $[\lambda'=\lambda/\lambda _0]$ and $[\lambda _0]$ is the reference wavelength used in equation (2.6.1.2). When a conventional X-ray source is used, it is sufficient in most cases to correct only for the Kβ contribution. Instead of the weighting function W(λ′) one only needs the ratio between Kβ and Kα radiation, which has to be determined experimentally (Zipper, 1969). One or more smearing effects may be negligible, depending on the experimental situation.

Each effect can be described separately by an integral equation (Glatter, 1982a). The combined formula reads $[\eqalignno{\bar I_{\rm exp}(h)={}&2 \int\limits^\infty _{-\infty} \int\limits^\infty_0 \int\limits^\infty _0 Q(x)P(t)W(\lambda ')\cr &\times I\bigg({\textstyle [(m-x)^2+t^2] ^{1/2}\over \lambda '}\bigg) {\,{\rm d}}\lambda '\,{\,{\rm d}}t\,{\,{\rm d}}x. & (2.6.1.56)}]$ This threefold integral equation cannot be solved analytically. Numerical methods must be used for its solution.

References

Glatter, O. (1982a). In Small angle X-ray scattering, edited by O. Glatter & O. Kratky, Chap. 4. London: Academic Press.Google Scholar

Zipper, P. (1969). Ein einfaches Verfahren zur Monochromatisierung von Streukurven. Acta Phys. Austriaca, 30, 143–151.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 2.6, p. 101