5.2 Bragg-Brentano Diffractometer

The concepts behind the Bragg-Brentano diffractometer were developed in the 1920’s by J. C. M. Brentano while working in the laboratory of Laurence Bragg, as an advance over ideas from H. Seemann and H. Bohlin but was first produced as a working instrument in the 1940s. A very successful commercial Bragg-Brentano instrument was developed in the early 1950’s by William Parrish for Norelco. The concept for this type of instrument is outlined in Fig. 5.1. To explain this concept, as x-rays leave the x-ray tube, they will spread in all directions, but slits and shielding limit the x-rays to diverge along a fixed angular range. The x-rays that impinge on the sample are diffracted back towards the detector, if the Bragg diffraction condition is satisfied, with slits used to prevent x-rays from other paths from reaching the detector. Thus, the loss of intensity from the spread of x-rays leaving the x-ray tube is partially offset by this parafocusing. It is called parafocusing, rather than focusing, because the focusing condition would only be satisfied if the sample were to be curved along the outlined focusing circle, as this would focus the point source of x-rays back onto the detector, but this is only approximated by a flat plate. As shown in the figure, this curvature would have to change with \(2\theta \) because the radius of the focusing circle changes as \(2\theta \) changes so creating a curved sample is never attempted. Note that one other effect of changing \(2\theta \) is that the length that the x-ray source projects onto the sample changes. This length is determined by slits that transmit a fixed divergence angle. The illuminated length becomes longer at lower \(2\theta \).

There are two variations on this diffractometer. In the example shown in Fig. 5.1, the detector rotates around the \(2\theta \) circle and the sample rotates at half the angle of the detector. This is called a in a \(\theta \)-\(2\theta \) diffractometer. In the less common and more expensive \(\theta \)-\(\theta \) diffractometer, the detector and x-ray source both rotate around the \(2\theta \) circle and the sample remains stationary.

An alternative to Bragg-Brentano geometry is the related Seemann-Bohlin diffractometer. Here, the x-ray tube and the detector not only rotate but also move towards and away from the sample to keep the focusing circle radius constant. This allows the sample to be curved along that fixed radius so that the ideal focusing condition is met, but this requires an exceedingly complex mechanical design with relatively minor gain. I have never seen a Seemann-Bohlin diffractometer in use.

Since the x-ray beam travels through the sample only as far as need be before it is diffracted, this geometry is called reflection mode diffraction. Note that the \(2\theta \) angle drops close to 0, the x-ray beam becomes close to parallel to the sample surface. This has two experimental implications. One is that if the sample is not perfectly smooth, the diffraction intensities become reduced as a result of surface roughness. The second is that the distance the x-ray beam penetrates into the bulk of the sample decreases with \(2\theta \). This produces a slight but perceptible shift in the effective sample position, in an effect known as sample transparency. The sample must be sufficiently thick so that at the highest measurement angle, no significant amount of x-rays will travel completely through the sample and will be lost. When this is true, diffraction intensities do not need to be corrected for sample absorption. There is no easy way to correct the intensity loss when samples are not sufficiently thick, but note that this would only be a problem for very low density materials or very thin samples. With highly absorbing samples, a different problem can arise. Only the very top of the sample is probed by the x-ray beam. The particles at the top may not be representative of the bulk with respect to composition, morphology or phase abundance.

The slits that define the angular range that determines the portion of the sample that is illuminated should be set so that at the lowest diffraction angle to be used, the x-ray beam does not extend past the sample. An alternative to this is to have the instrument adjust the slits defining the beam path to have the same length independent of \(2\theta \), this is known as \(2\theta \)-compensating slits. Traditionally, use of \(2\theta \)-compensating slits for Rietveld analysis is considered a bad idea, as the diffraction intensities must be scaled for the changing illumination range. With modern, highly computerized, diffractometers, I’m not so sure that it is not better to have the same region of sample used for all measurements, even though, as noted, the exact crystallites that will contribute to the pattern will still change as the amount that the x-ray beam penetrates into the sample decreases with angle.

It is common to use a sample holder for Bragg-Brentano diffraction consisting of a single crystal (typically sapphire) that is cut so that diffraction from this crystal will never occur when the crystal is mounted along the diffraction circle. It is then possible to measure a diffraction pattern from a small amount of sample placed on this so-called “zero background” sample mounting. While this is a good method for measuring a diffraction pattern with a small amount of sample, the diffraction intensities will have the same deviations from correct as a thin sample and should not be used for Rietveld analysis. That small amount of sample can be used for an accurate powder diffraction pattern if placed in a capillary tube and used in a transmission experiment, particularly if the capillary is spun. It is also common to spin samples in a Bragg-Brentano instrument. However, this type of spinning is done around the normal to the plane of the sample, which does not bring new crystallites into an orientation that allows diffraction. This does not improve the particle counting statistics; Bragg-Brentano sample-spinning is done to simplify possible texture, but forcing it to have cylindrical symmetry, if present. It is possible to oscillate a sample in a Bragg-Brentano diffractometer to improve particle counting statistics, but this must be done by increasing and decreasing the sample angle (properly this rotation of the \(\omega \) axis, but more commonly the angle is called \(\theta \)) or with an oscillation perpendicular to the \(2\theta \) circle. Neither is commonly done, as it interferes with parafocusing.

In Bragg-Brentano diffraction, the assumption is that the sample is placed exactly at the plane of the diffractometer circle with accuracy better than a fraction of a micron. This cannot be achieved routinely (a human hair is 50-100 microns) and must be corrected for in fitting, as it produces a slight but significant deviation between the measured \(2\theta \) angle and the actual diffraction angle. A second correction is important for samples that are of low density. As noted, the amount the x-ray beam penetrates into the sample changes with \(2\theta \) and for low density samples this affects the apparent location of the sample due to sample transparency and again this must be corrected. Both corrections will be discussed later in §12.2.