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. 3.2, pp. 256-257
Section 3.2.2.3.3. Instrumental contributions^{a}Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA |
The instrument used to collect diffraction data affects the observed line shape in many ways. A complete discussion is beyond the scope of this article (but see Chapter 3.1 , and would have to consider many different diffraction geometries separately. One approach, generally known as fundamental parameters, is to model the effect of every optical element on the peak shape. The instrumental response function is then the convolution of all of these individual contributions.
The simplest (and most widely successful) application of fundamental parameters arises from the analysis of Caglioti et al. (1958, 1960) on the instrument response function of a step-scanning powder diffractometer at a reactor neutron source. Neutrons pass through a parallel-blade collimator to a monochromator crystal, through a second collimator to the sample, and then through a third collimator into the detector. It is a fair approximation to assume that the transmission functions of the collimators and the mosaicity of the monochromator are all Gaussians. In that case, the instrument response function produces diffraction peaks which are also Gaussians, with a width that depends on the diffraction angle aswhere the parameters U, V and W can be expressed in closed form as functions of the collimator acceptances and monochromator angle and mosaicity (Caglioti et al., 1958, 1960). Indeed, it was this simple form of the line-shape function that allowed Rietveld and co-workers to develop the important methodology of data analysis commonly known as the Rietveld or profile method (Loopstra & Rietveld, 1969; Rietveld, 1969).
The functional form used to describe the shape of the pulse that emerges from the moderator of a spallation neutron source, and the numerical values of the parameters which go into it, determine the peak shapes observed in the powder diffractometer at such an instrument. The treatment of such matters lies beyond this introduction; documentation of the widely used GSAS software includes a detailed description (Larson & Von Dreele, 2004; see also Chapter 3.3 ).
Analytical treatment of the resolution of X-ray diffractometers is generally complicated by the fact that the various contributions are more difficult to model, especially in the case of the para-focusing Bragg–Brentano geometry most commonly used in laboratory X-ray diffractometers. In the diffractometer, one generally considers factors that affect the line shape separately in the equatorial plane (the plane containing the source, sample and detector) and the axial direction (perpendicular to the equatorial plane, i.e., parallel to the diffractometer axis). In the equatorial plane the resolution is primarily governed by the divergence of the X-ray beam illuminating the sample and by the width of the receiving slit, but there are numerous contributions to the diffraction peak width and shape from such factors as non-conformance of the flat sample surface to the focusing circle, partial transparency of the sample and misalignment of the diffractometer (Cheary & Coelho, 1992). Even if a monochromator is used to select only the Kα_{1} line, the radiation spectrum from an X-ray tube consists of several Lorentzian functions owing to satellite transitions.
Axial divergence produces a pronounced asymmetry of low-angle diffraction peaks to the low-angle direction (and on the high-angle side of peaks with 2θ near 180°). If the incident or diffracted rays are out of the equatorial plane, they will be intercepted at a detector setting below (above) the actual diffraction angle if it is small (close to 180°), respectively. The effect can be minimized, but not completely eliminated, by narrowing the beam-defining apertures in the equatorial direction, or by introduction of parallel-blade Soller (1924) slits. The full treatment of the effect of axial divergence in a Bragg–Brentano diffractometer has been presented in a computationally convenient form (Cheary & Coelho, 1998).
It is generally easier to model the instrumental response function of a powder X-ray diffractometer based on a synchrotron-radiation source with an analyser crystal, because the transfer functions of the various optical elements are simpler to express. One approach is to approximate the wavelength- and angle-dependent reflectivity of monochromator and analyser crystals by Gaussians, and derive a closed-form expression for the width of the instrument response function (Sabine, 1987). This has been extended to include collimating and focusing (in the scattering plane) mirrors (Gozzo et al., 2006); see Chapter 3.1 for a full description of the relevant optical configurations. A shortcoming of this analytical approach is that the correct single-crystal reflectivity function is not a Gaussian, so it cannot account for the correct line shape, only provide an estimate of its width. Numerical convolutions to accurately model the line-shape function have been performed, and produce excellent agreement with measured profiles (Masson et al., 2003).
Instead of the analytical approach described in the previous few paragraphs, one frequently writes a parametrized function for the measured line shape without concern for the connection between the numerical values of the parameters and the microscopic properties of the sample or the geometry of the diffractometer. For example, one can use the Caglioti form for diffraction peak width [equation (3.2.13)] on any diffractometer, and adjust the parameters U, V and W to some measured standard sample. If size and strain contributions to sample-dependent broadening are both regarded as Lorentzians, they could be combined as Γ = X/cos θ + Y tan θ. (Here we make use of the fact that the convolution of two Lorentzians is a Lorentzian whose width is the sum of the individual widths. For Gaussians, the widths combine in quadrature, i.e., .) In the general case, where neither the instrumental response function nor the sample broadening is purely a Gaussian or Lorentzian function, one can write an empirical line shape as the approximate convolution of a Gaussian and a Lorentzian, with widths given byThis is one of the flexible line-shape models available in the widely used program GSAS (Larson & Von Dreele, 2004). It is still possible to make a semi-quantitative statement about sample properties by comparing refined parameters against a standard sample; for example, size broadening will increase the P and X parameters, and strain will increase U and Y. A similar model is used in the FullProf software (Rodríguez-Carvajal, 1993, 2001; Kaduk & Reid, 2011).
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