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, pp. 134-136

Section Sampling statistics

B. B. Hea*

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

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In powder X-ray diffraction, the number of crystallites contributing to each reflection must be sufficiently large to generate reproducible integrated peak intensities (see Chapter 2.10[link] ). A larger number of contributing crystallites gives better precision or sampling statistics (also referred to as particle statistics). Sampling statistics are determined by both the structure of the sample and the instrumentation. For a powder sample in which the crystallites are perfectly randomly oriented, the number of contributing crystallites for a diffraction peak can be given as[N_s = p_{hkl} {Vf_i \over v_i} {\Omega \over 4\pi },\eqno(2.5.47)]where phkl is the multiplicity of the diffracting planes, V is the effective sampling volume, fi is the volume fraction of the measuring crystallites (fi = 1 for single-phase materials), vi is the volume of individual crystallites and Ω is the angular window of the instrument (given as a solid angle). The multiplicity term, phkl, effectively increases the number of crystallites contributing to the integrated intensity from a particular set of (hkl) planes. The volume of individual crystallites, vi, is an average of various crystallite sizes. The combination of the effective sampling volume and the angular window makes up the instrumental window, which determines the total volume of polycrystalline material making a contribution to a Bragg reflection. For 2D-XRD, the instrumental window is not only determined by the incident beam size and divergence, but also by the detective area and the sample-to-detector distance (γ angular coverage).

In B-B geometry, the effective irradiated volume is a constant,[V_{\rm BB} = A_oA_{\rm BB} = {A_o/2\mu },\eqno(2.5.48)]where Ao is the cross-section area of the incident beam measured on the sample surface, [A_{\rm BB} = 1/(2\mu)] is the transmission coefficient for B-B geometry, and μ is the linear absorption coefficient. For 2D-XRD, the effective volume is given as[V = A_oA = A_oT/2\mu,\eqno(2.5.49)]where A is the transmission coefficient and T is the transmission coefficient with B-B normalization for either transmission or reflection as given previously.

The angular window is given as a solid angle. The incident beam has a divergence angle of β1 within the diffraction plane and β2 in the perpendicular direction. The angular window corresponding to the incident-beam divergence is given by[\Omega = {\beta _1}{\beta _2}/\sin \theta \ {\rm or}\ \Omega = {\beta ^2}/\sin \theta \ {\rm if}\ \beta = {\beta _1} = {\beta _2}.\eqno(2.5.50)]

For 2D-XRD, the angular window is not only determined by the incident-beam divergence, but also significantly increased by γ integration. When γ integration is used to generate the diffraction profile, it actually integrates the data collected over a range of various diffraction vectors. Since the effect of γ integration on sampling statistics is equivalent to the angular oscillation on the ψ axis in a conventional diffractometer, the effect is referred to as virtual oscillation and Δψ is the virtual oscillation angle. In conventional oscillation, mechanical movement may result in some sample-position error. Since there is no actual physical movement of the sample stage during data collection, virtual oscillation can avoid this error. This is crucial for microdiffraction. The angular window with the contributions of both the incident-beam divergence and the virtual oscillation is[\Omega = \beta \Delta \psi = 2\beta \arcsin [\cos \theta \sin (\Delta \gamma /2)],\eqno(2.5.51)]where β is the divergence of the incident beam. While increasing the divergence angle β may introduce instrumental broadening which deteriorates the 2θ resolution, virtual oscillation improves sampling statistics without introducing instrumental broadening.

In the cases of materials with a large grain size or preferred orientation, or of microdiffraction with a small X-ray beam size, it can be difficult to determine the 2θ position because of poor counting statistics. In these cases, some kind of sample oscillation, either by translation or rotation, can bring more crystallites into the diffraction condition. Angular oscillation is an enhancement to the angular window of the instrument. The effect is that the angular window scans over the oscillation angle. Any of the three rotation angles (ω, ψ, ϕ) or their combinations can be used as oscillation angles. Angular oscillation can effectively improve the sampling statistics for both large grain size and preferred orientation. As an extreme example, a powder-diffraction pattern can be generated from single-crystal sample if a sufficient angular window can be achieved by sample rotation in such as way as to simulate a Gandolfi camera (Guggenheim, 2005[link]). Sample oscillation is not always necessary if virtual oscillation can achieve sufficient sampling statistics.


Guggenheim, S. (2005). Simulations of Debye–Scherrer and Gandolfi patterns using a Bruker Smart Apex diffractometer system. Bruker AXS Application Note 373.Google Scholar

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