An ideal powder diffraction experiment has as a sample a large number (typically millions) of crystallites1 that have random orientation. As will be discussed later (see 5.4), , the primary source of statistical error in measurement of diffraction intensities is due to the expected fluctuations in intensity values due to the nature of event counting, but in powder diffraction when the number of crystallites in the beam becomes smaller than ideal, either because the sample is small or because the crystallite size is large, intensities can also fluctuate due to the probability that a particular reflection is over- or under-represented becomes larger. These sort of fluctuations are normally referred to as particle counting statistics. They are not normally modeled, but the effect of them can be reduced in a few ways: if increasing the sample size is not possible, rotating or oscillating the sample can improve randomization of crystallite orientation, thus greatly improving the effective number of scattering crystallites. In addition, increasing the area of detection, by using an area detector, can also improve particle counting statistics significantly.
As we will discuss later, powder diffraction is commonly used on samples where the crystal orientation is not completely random. When crystallite orientation ordering is not random, the phenomenon is called preferred orientation and the description of how crystals are oriented is often referred to as texture. Powder diffraction is commonly used for the characterization of texture. Texture can also be seen as a defect in data that needs to be corrected in order to obtain an accurate crystal structure model or for accurate phase quantification. GSAS-II offers one mode of texture treatment one for correction of the effect and the other for texture characterization, noting that accurate characterization often requires measurement of diffraction either with detectors in different locations around the sample or with data collected while the sample is placed in multiple orientations.
I like to think of quantification of texture as a sphere-like shape where if you pick a crystallographic direction for a crystal system, every point on the surface of the spheroid represents the probability of finding a crystallite oriented in that direction. With a perfectly random powder, the probability surface is indeed a sphere, but as the probability for finding crystals oriented in a particular way increases, surface will expand in that direction (and in others related by symmetry) and will decrease in others. This probability surface could have very sharp maxima. (Think about the probability for orientation of long needles placed in a thin tube.) GSAS-II offers two ways to correct for texture. One is the March-Dollase model, where a particular direction is selected and the relative abundance of crystallites oriented in that direction is treated. The second uses a spherical harmonics representation to account for the distortion of the sphere into a spheroid. For texture characterization, only spherical harmonics representation is used. The allowed terms in the spherical harmonic series are restricted to match the Bravais lattice symmetry and sample treatment history. The number of terms in the expansion is determined by the user. The more terms that you allow, the more “spikey” the surface may become.
Powder diffraction is also used for partly- and non-crystalline materials. Noting that powder diffraction provides a structural fingerprint for a material, it will demonstrate different sorts of structural motifs, such as the differences in polymer ordering or other types of macromolecules as can be seen in different forms of cellulose.
It might be thought that a random orientation of crystallite would produce a random scattering of x-rays, electrons or neutrons with a featureless pattern, but this is not what happens. As we have seen, diffraction can only occur when the structure factor, \(F_{hkl}\), is non-zero and this only occurs for discrete values of \(h\), \(k\) and \(l\) and at specific Q values, where
\[F_{hkl}=\sum _j^N f_j \exp [-2\pi i(hx_j + ky_j + lz_j)]\] and \[Q = 2 \pi \sqrt {A_0 h^2 + A_1 k^2 + A_2 l^2 + A_3 hk + A_4 hl + A_5 kl}\] so powder diffraction only occurs at discrete Q values. Also, since \(Q = 4 \pi \sin \theta /\lambda \) and the maximum value for \(2\theta \) is \(180^\circ \) \(Q_{max} = 4 \pi /\lambda \), so the number of observable reflections is limited by \(\lambda \). Note that as the unit cell size increases, the A matrix terms (which are based on the reciprocal lattice tensor) decrease in size and an increasingly larger number of reflections can be observed.
With an ideal sample and a diffraction instrument with infinitesimal resolution, a powder diffraction pattern would be a series of delta functions. The intensity of each diffraction peak would be determined by \(I_{hkl}\), which is related to \(F^2_{hkl}\), but also by reflection multiplicity, as many values of \(hkl\) produce diffraction at exactly the same value of Q. However, the diffraction intensities will fall off in intensity with Q due to the Debye-Waller factor (see §3.8). In practice, peaks will be broadened both from the instrument and from sample effects, as will be discussed extensively in Chapter 14.
What if the crystallites are not randomly oriented? In the extreme case, all the crystallites will be oriented the same. This is a mosaic single-crystal. Diffraction will be seen in the powder diffractometer if the crystal is aligned so that the crystal is for diffraction, but if not, no diffraction occurs. This is commonly seen when a material is collected in a diamond-anvil pressure cell, where a small amount of material is placed between two diamonds. The observation of diffraction spots from the diamond will depend on how the gems are placed relative to the beam. Should we have a material that has an overabundance of some orientations for crystallites – think plates of mica dusted onto the surface of a flat plate sample holder. The habit of the mica sheets has the 001 axis perpendicular to the sheet and we can expect to have many more crystallites oriented with the sheets parallel to the sample holder than in other orientations. This would cause the intensities of the 00l reflections to increase as the intensities of all other reflections would fall, relative to a sample that has truly random orientation.
Note that if a material is composed of two different phases, the observed powder diffraction pattern will be the superposition of the component phases. The relative amount of the peaks from each phase will be determined by the relative amounts of each phase. The measurement of this is known as phase quantification. Powder diffraction is commonly used for this and can be employed with materials that contain as much as two dozen phases. As the relative abundance of a phase falls, it becomes harder to discern in the powder diffraction pattern. The exact sensitivity will depend on the pattern for the material (something with a few distinct peaks will likely be easier to measure than something that has a large number of relatively weak peaks) and the level of background in the sample. As a rule of thumb, with a laboratory diffractometer, the sensitivity is on the order of 1% and with a high-resolution (and low background) synchrotron measurement, 0.1%.