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

International Tables for Crystallography (2006). Vol. C. ch. 2.3, p. 50

Section 2.3.1.1.7. Axial divergence

W. Parrisha and J. I. Langfordb

a IBM Almaden Research Center, San Jose, CA, USA, and bSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England

2.3.1.1.7. Axial divergence

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Divergence in the axial direction (formerly also called `vertical divergence') causes asymmetric broadening and shifts the reflections. The aberration is illustrated in Fig. 2.3.1.11[link] for a low-2θ reflection in the transmission-specimen mode (Subsection 2.3.1.2[link]). The narrow profile was obtained with δ = 4.4° parallel slits placed between the monochromator and detector, and the broad profile with the slits removed. The slits caused a 33% reduction in peak intensity. This problem was recognized in the first design of the diffractometer using the X-ray tube line focus when parallel slits were used in the incident and diffracted beams to limit the effect (Parrish, 1949[link]). Increasing the radius reduces the effect if the slit length is kept constant. The intensity is also reduced because the chord length intercepted is a smaller fraction of the longer radius diffraction cone. The construction of parallel (Soller) slits (Soller, 1924[link]) is shown in Fig. 2.3.1.5(d)[link].

[Figure 2.3.1.11]

Figure 2.3.1.11| top | pdf |

Effect of axial divergence on profile shape. Narrow profile recorded with parallel slits (PS), δ = 4.4° between monochromator and detector (Fig. 2.3.1.12[link]), and broad profile with these parallel slits removed. Faujasite, Cu Kα, αES 2°.

The calculation of the aberration and the present status is summarized by Wilson (1963[link], pp. 40–45). The results depend on the aperture of the parallel slits, the length of the entrance and receiving slits, and 2θ. In the limit of small s, the shift of the centroid is [\Delta(2 \theta, {\rm rad}) = (s/l)^2 \cot 2 \theta /6, \eqno (2.3.1.15)]where s is the spacing and l the length of the foils. The shift becomes very large at small 2θ's but not infinite as equation (2.3.1.15[link]) implies. The shift is to smaller 2θ's in the forward-reflection region and to larger 2θ's in back-reflection. However, the mathematical formulation is difficult to quantify because in the forward-reflection region the axial divergence convolves with the flat-specimen aberration to increase the asymmetry. In the back-reflection region, the effect is not so obvious because the distortion is smaller and the Lorentz and dispersion factors also stretch the profiles to higher angles.

References

First citation Parrish, W. (1949). X-ray powder diffraction analysis: film and Geiger counter techniques. Science, 110, 368–371.Google Scholar
First citation Soller, W. (1924). A new precision X-ray spectrometer. Phys. Rev. 24, 158–167.Google Scholar
First citation Wilson, A. J. C. (1963). Mathematical theory of X-ray powder diffractometry. Eindhoven: Philips Technical Library.Google Scholar








































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