14.2 Sources of instrumental broadening

So, what are the sources of instrumental broadening in powder diffraction? There are many, but let’s consider a few. First of all, how well is the wavelength determined for the probe? For conventional x-ray sources, there are usually several characteristic x-rays lines, typically labeled as Ka₁, Kb, etc. These lines are fairly sharp with respect to wavelength spread, but most instruments do not isolate a single one of these lines and thus the diffraction pattern that is observed is actually a superposition from several different wavelengths. With synchrotrons and reactor-source neutrons, at least one (with synchrotrons usually a series of two or more) monochromator crystals must be used to isolate the intended radiation. A key design aspect in these instruments is how broad the wavelength range that is allowed through the monochromator, where synchrotron instruments usually transmit a very narrow portion of the radiation, with a great loss of intensity, while for reactors, the monochromator crystals are usually deformed in some fashion to transmit a much wider range of the radiation in order to retain much more of the neutron flux.

With time-of-flight (TOF) neutrons we measure the wavelength by timing how long it takes for the neutrons to transverse the distance from the source to the detector. While a very sharp “flash” of neutrons is emitted when a pulse of protons hits a heavy metal target, these neutrons are very high energy and must be moderated by bouncing around in a material where they transfer some of their energy. Not all the neutrons will leave the moderator at the same time, and this results in a spread in the time of arrival for neutrons with the same energy. This delay effect is not symmetrical, so that TOF peaks are broadened more with neutrons arriving “late” than what would be seen if the probably for early arrival with respect to the mean time of flight were the same as late arrival. Further, since less energetic (slower) neutrons are likely to have bounced around in the moderator for longer than their higher-energy cousins, the time profile for when neutrons will arrive is narrower for higher energy (short wavelength) neutrons than for lower energy (long wavelength) neutrons; thus the resolution changes with wavelength.

For both lab and user facilities, the spectrum of the x-rays or neutrons will never be perfectly sharp, so there is always some range in energy, which results in some broadening in diffraction peaks.

A second contribution for aberration is the finite size of the radiation source. Remember that we need to measure the diffraction angle at infinite precision, if we want to see an infinitely sharp peak. If we do not know exactly where the photon or neutron was emitted, then we cannot know the angle of diffraction precisely. This uncertainty (equivalently, “blur”) in our peak position causes broadening. Of course, the smaller we make our source, the fewer quanta that we can measure, so the source size, like so many other aspects of instrument design, becomes a compromise between intensity and resolution. The instrument designer needs to select the intended resolution, often called the instrument response function, and then design so that each optical component in the instrument is matched to the intended resolution. If a single component degrades the resolution worse than intended, then that one factor will dominate the instrumental response. If one component alone would is set so that it would define the resolution much better than needed, it will reduce the intensity performance of the instrument with negligible gain in resolution. Thus, if we use a smaller source size, but do not improve our monochromator (or moderator) to match, we see negligible improvement in resolution but do see a significant drop in intensity.

The source is just one factor in the performance of an instrument. Every collimator, slit and optical component (monochromators, mirrors, etc.) will affect the overall performance of the instrument.

14.2.1 Low-angle peak asymmetry

Most instrumental effects (with the exception of that from TOF moderation) creates symmetrical broadening in peaks, but there is one common effect that broadens peaks at low values of \(2\theta \) and, when they can be observed, at \(2\theta \) values close to 180°. In nearly all laboratory diffractometers, as well as high-resolution synchrotron instruments and CW neutron instruments, the aperture or slit that defines what radiation can be observed by the detector is elongated in this direction as well. The effect of this elongation is axial divergence. This is shown schematically in Fig. 14.1 , where the x-ray or neutron beam diffracts from a small sample. Two Bragg diffraction cones are shown, one for a low angle reflection and one at significantly higher angle. Two slits are also shown schematically. The smaller one only admits the diffraction intensity at the correct \(2\theta \) angle, but for the elongated slit and the lower angle reflection, the slit starts two pass intensity well below the correct \(2\theta \) angle. This causes the diffraction intensity to be broadened towards lower angles, but not towards higher angles leading to asymmetry in the peak. This effect is much less severe for the higher angle Bragg cone which is why this effects lower angle peaks and falls off very quickly with angle. The same effect will occur at angles close to 180° as again the Bragg cones again have very small diameters, but the broadening is on the high-angle side. However, diffraction at such high angles is rarely possible other than with neutrons and most diffractometers already have very poor resolution at such high angles.

Likewise, in most instruments, the sample is also elongated in the direction of the \(2\theta \) rotation circle. One can think of the effect of this as if there are a series of diffraction cones stacked along the elongation direction, as shown in Fig. 14.2. This will cause the small slit to transmit radiation at angles lower than expected, thus causing approximately the same asymmetry as seen in Fig. 14.1 with the elongated slit. Most commonly, the sample length and the slit length are matched. When both are elongated, the sample length accentuates the effect seen from the elongated slit.

Note that axial divergence does not occur with area detection. In such instruments the radiation source and sample sizes are minimized. The radial integration process does not cause this peak asymmetry.