Section 2.3.1.1.4. Instrument broadening and aberrations

The asymmetric form, broadening and angular shifts of the recorded profiles arise from the Kα doublet and geometrical aberrations inherent in the imperfect focusing of the particular diffractometer method used. There are additional causes of distortions such as the time constant and scanning speed in rate-meter strip-chart recording, small crystallite size, strain, disorder stacking and similar properties of the specimen, and very small effects due to refractive index and related physical aberrations.

Perfect focusing in the sense of reflection from a mirror is never realized in powder diffractometry. The focusing is approximate (sometimes called `parafocusing') and the practical selection of the instrument geometry and slit sizes is a compromise between intensity, resolution, and profile shape. Increasing the resolution causes a loss of intensity. When setting up a diffractometer, the effects of the various instrument and specimen factors should be taken into account as well as the required precision of the results so that they can be matched. There is no advantage in using high resolution, which increases the recording time (because of the lower intensity and smaller step increments), if the analysis does not require it. A set of runs to determine the best experimental conditions using the following descriptions as a guide should be helpful in obtaining the most useful results.

In the symmetrical geometries where the incident and reflected beams make the same angle with the specimen surface, the effect of absorption on the intensity is independent of the θ angle. This is an important advantage since the relative intensities can be compared directly without corrections. The actual intensities depend on the type of specimen. For a solid block of the material, or a compacted powder specimen, the intensity is proportional to , where μ is the linear absorption coefficient of the material. The transparency aberration [equation (2.3.1.13)], however, depends on the effective absorption coefficient of the composite specimen.

The need to correct the experimental data for the various aberrations depends on the nature of the required analysis. For example, simple phase identifications can often be made using data in which the uncertainty of the lattice spacing Δd/d is of the order of 1/1000, corresponding to about 0.025 to 0.05° precision in the useful identification range. This is readily attainable in routine practice if care is taken to minimize specimen displacement and the zero-angle calibration is properly carried out. The experimental data can then be used directly for peak search (Subsection 2.3.3.7) to determine the peak angles and intensities (Subsection 2.3.3.5) and the data entered in the search/match program for phase identification.

However, in many of the more advanced aspects of powder diffraction, as in crystal-structure determination and the characterization of materials for solid-state studies, much more detailed and more precise data are required, and this involves attention to the profile shapes. The following sections describe the origin of the instrumental factors that contribute to the shapes and shift the peaks from their correct positions. Many of these factors can be handled individually. With the use of computer programs, they can be determined collectively by using a standard sample without profile broadening and profile-fitting methods to determine the shapes (Subsection 2.3.3.6). The resulting instrument function can then be stored and used to determine the contribution of the specimen to the observed profiles.

A series of papers describing the geometrical and physical aberrations occurring in powder diffractometry has been published by Wilson (1963, 1974). His work provides the mathematical foundation for understanding the origin and treatment of the various sources of errors. The major aberrations are described in the following and are illustrated with experimental profiles and plots of computed data for better visualization and interpretation of the effects. The information can be used to correct the experimental data, interpret the profile broadening and shifts, and evaluate the precision of the analysis. Chapter 5.2 contains tables listing the centroid displacements and variances of the various aberrations.

The magnitudes of the aberrations and their effects are illustrated in Figs. 2.3.1.8 (a) and (b), which show the Cu spectrum inside the experimental profile. At high 2θ's, the shape of the experimental profile is largely determined by the spectral distribution, but at low 2θ's the aberrations are the principal contributors. The basic experimental high-resolution profile shapes from specimens without appreciable broadening effects (NIST silicon powder standard) are shown in Figs. 2.3.1.8(c)–(f). The solid-line profiles were obtained with a reflection specimen (Fig. 2.3.1.3). The differences in the doublet separations are explained in Subsection 2.3.1.2. These profiles are the basic instrument functions which show the profile shapes contained in all reflections recorded with these methods. The shapes are modified by changing slit sizes.

Figure 2.3.1.8| top | pdf | Diffractometer profiles. (a) and (b) Spectral profiles λ of Cu Kα doublet (dashed-line profiles) inside experimental profiles R (solid line). (c)–(f) Experimental profiles with reflection specimen (R) geometry (Fig. 2.3.1.3) with αES 1° and αRS 0.046° (solid line profiles), and transmission specimen (T) (Fig. 2.3.1.12) with αES 2° and receiving axial divergence parallel slits (dotted profiles). Cu Kα radiation. (a) Si(531), (b) quartz(100), (c) Si(111), (d) Si(220), (e) Si(311), and (f) Si(422).