Not only does the instrument create powder diffraction peak broadening, but all samples contribute broadening as well. While there are many types of defects in samples, such as stacking faults, screw dislocations or turbostratic disorder that can affect powder diffraction patterns, the two most significant effects are known as crystallite size broadening and microstrain broadening. Let us consider each effect. While instrumental broadening will depend only on the diffraction angle (or for TOF diffraction, both the angle and the wavelength), as we will see broadening can be different for different classes of reflections, where along some crystallographic directions, reflections will be broader than for other crystallographic directions. This is known as anisotropic peak broadening. While instrumental broadening is primarily Gaussian in nature, possibly with minor Lorentzian contributions, sample broadening is usually Lorentzian, though under unusual circumstances Gaussian contributions may be seen.
As was noted in chapter 2, diffraction arises from an infinite lattice of scatterers producing infinitely sharp diffraction maxima. Of course, all real crystals have finite size, so how does this affect diffraction? If we consider the diffraction from a single isolated crystal with a dimension d (we will also consider the crystal to be spherical) then all the diffraction maxima will be broadened in Q by a factor of 2\(\pi \)/d, though the actual shape of the broadening function can be complex, particularly where d is small, say with respect to 100*l. To relate this broadening to what is seen in measurements, we must consider that usually we measure diffraction patterns in experimental units such as an angle or for energy-dispersive measurements a unit (such as TOF) that we can relate to energy. Also, we have diffraction occurring from many crystalline domains, not one single crystalline particle, and further these domains may not be spherical. Let’s consider each of these in turn.
While crystallite size causes the same broadening for all peaks measured in Q, if we are looking at a plot of peak widths in degrees \(2\theta \), we will see a change in peak widths. Noting that \(Q = (4\pi /\lambda ) \sin \theta \), the derivative dQ/d\(\theta \) = (\(4\pi /\lambda \)) cos\(\theta \). Inverting this, we see that if broadening of peaks is a constant amount \(\Delta Q\), then \(\Delta \theta = \Delta Q \lambda / [4\pi \cos \theta ]\). Thus, crystallite size broadening, in units of \(2\theta \), \(\Delta 2\theta \) is proportional to \(1/\cos \theta \). This broadening formulation was first developed a century ago by Paul Scherrer and is commonly written as \[\tau =\ {\frac {K\lambda }{\beta \cos {\theta }}}\] where \(\tau \) is the mean crystallite size in the same units as the wavelength (l); \(\beta \) is the portion of full-width at half maximum of the peak that is due to crystallite broadening expressed in radians and \(\theta \) is one half the \(2\theta \) value for that peak. The unitless quantity K depends on the averaged morphology of the crystallites and is thus known as the shape factor. Winnie Wong-Ng in Volume H of the International Tables states that K will be in the approximate range of 0.6-2, depending on the assumption for this morphology, but is most commonly taken as 0.89. For GSAS-II we use K=1.
Note that crystallites need not have the same sizes in all directions. Anyone who has spent much time looking at powders, or for that matter minerals under a microscope knows that some materials will commonly grow with needle-like habits, where the size is much longer along one direction than in the perpendicular directions. Other materials, such as mica, may grow in a platy habit, where the length along one direction is much shorter than the perpendicular directions. With lower symmetry unit cells, many more complex unit cell anisotropic morphologies are possible. While crystallite broadening shows average properties for the entire sample that is illuminated, if the crystals have larger dimensions in some crystallographic directions than in others, we can expect to see broadening effects that differ for differing classes of reflections, since the class of the reflection (hk0 reflections vs 00l, for example) represent different orientations of the crystals. An easy analogy to think about the effect of orientation-dependent crystallite size on peak broadening is to imagine that the reflection “sees” the diffracting region in projection along the diffracting vector. Thus, if we consider the case where either needles or plates have their unique axis along the c direction (this is a common case for hexagonal and tetragonal materials), we can see that the (00l) reflections will have very different broadening from the (hk0) reflections and the (hkl) reflections will be intermediate between the two with the amount of broadening depending on the relative magnitudes of h, k and l. In the case of needles, (00l) reflections will be very sharp, since from that direction the crystal size is greatest. For plates, the (00l) reflections will be very broad, since that will project as the smallest crystallite size direction. Thus, crystallite size broadening can be anisotropic when the average sizes of the crystal vary significantly by crystallographic direction.
Since powder diffraction is a phenomenon from a large number of crystallites and these crystallites will never all have exactly the same size, we must also consider that the distribution of crystallite sizes will convolute our size broadening. In theory it might be possible to back out the size distribution from the broadening effects and some papers have explored this, but I have not seen any convincing examples where this has been done. In powder diffraction we usually use scalar value(s) for crystallite size that represent some sort of mean value.
Note that I say crystallite size and not particle size here. While these terms are commonly interchanged in discussions, they mean quite different things. As an example, consider that most metal objects that we encounter are polycrystalline. (An example of an exception to that would be single-crystal titanium jet engine turbine blades.) My wedding band, for example, is a single discrete object and thus is a single particle, but it is likely composed of hundreds of thousands of crystallites. To define our terms, a crystallite is an ordered region of a material lacking long-range defects, though point defects may be present. The term grain is sometimes used for crystallite. In a somewhat circular definition, a crystallite can be considered as largest region that is seen as undisrupted ordered region by a diffraction probe, but it should be understood that different diffraction probes and even different reflections will see this differently. In contrast, a particle is a discrete object that could be a single crystal, a group of twined crystallites or even an agglomerate of randomly oriented crystallites or even could be multiphasic. Even what we call a single crystal is likely composed of multiple crystallites, since the preference for single-crystal diffraction measurements are “ideally imperfect” mosaic crystals made up of small crystalline domains that are in good, but not perfect alignment. Large non-mosaic single crystals are likely to exhibit extinction for intense reflections, which complicates structural analysis.
There are a number of different methods that are used for measuring particle size, which include light scattering, small angle x-ray or neutron scattering, physical separations, electrical sensing and imaging. These measure the sizes of agglomerate particles in a powder or liquid slurry. In contrast, diffraction, by x-rays, neutrons or even electrons, measures the size of the regions where atoms are scattering coherently, e.g. from a periodic lattice or even quasicrystalline domain.
Note that for some types of long-range faulting, some reflections may not be sensitive to the lapse in coherence caused by the fault, since in projection from a certain direction that fault might be invisible. Thus, in faulted materials some classes of reflections may be more broad than others. The faulted area might even appear to represent a longer-range structural motif in projection from certain directions and thus create new reflections that would be forbidden according to the long-range structure. This is why stacking faults can cause such complex changes in a powder diffraction pattern relative to an unfaulted material. The program DIFFaX from Treacy, Deem and Newsam, provides an excellent mechanism to model the effects of stacking faults on diffraction patterns, where a material is modeled as a set of layers where rather than having a fixed order of layering, as would be found in a fully crystalline material, probabilities are supplied for the likelihood for each type layer following the previous. GSAS-II incorporates the DIFFaX code and provides an interface for computation of faulted diffraction patterns. GSAS-II also provides visualization for the layers are joined to help prepare this relatively complex, but sometimes necessary, approach to describing a structural model.
When a characterization goal is to better understand the average crystallite shape or distribution of sizes, small-angle scattering is likely to be a better measurement than [wide-angle] powder diffraction. GSAS-II includes a program, Shapes, that is intended to model the mean particle size from small-angle scattering data.
The second type of sample broadening we will consider was historically called residual stress, but a better name for this is microstrain. To consider what this is about, let us imagine that we take a single crystal particle and put it into a C-clamp so that along one direction there is a force being applied to the crystal. For simplicity, let’s label that direction as the c axis. This force, which is called an external stress, will cause the c axis to become slightly smaller since the lattice is being compressed by that force. If the same force were to be applied on the crystal uniformly, say by immersing the crystal into a container with a pressure-conducting liquid or gas and then squeezing on the container, we would then apply that force to all axes of our crystal. This would be hydrostatic stress. Thus, externally applied forces can change the lattice constants for a material. How much the lattice constants will change as a function of the force is described by a tensor populated by what are known as the elastic strain constants. The symmetry of the unit cell dictates the number of unique terms in this tensor, with 3 terms needed for a cubic system and 21 for triclinic. The name elastic stress implies that there can be a non-elastic condition. Elastic implies that if the force is released, the material will return to its original state. Indeed, if enough stress is applied, we can get outside the elastic regime. This will cause the crystal to undergo plastic deformation (think toothpaste) or catastrophic failure (cracking and breaking).
Now to get to back microstrain, imagine that I take a bunch of discrete unconnected crystals and put them into a container with random orientation and heat them to the point where crystals start to agglomerate, e.g. form connections to each other, but not hot enough so that the crystals start to recrystallize (where some crystals subsume others). When I remove the heat, I end up with a solid that has my crystals still randomly oriented as they started, but now the crystals are unable to move relative to each other since they are linked together. This heating process is known as sintering, but a similar process can occur when a molten material solidifies into a polycrystalline solid. Considering again our sintering material before we remove the heat, we can imagine that every crystal will have its lattice constants at equilibrium for that temperature. As the crystals start to cool, the lattice constants will get smaller, except the since there are now connections between the crystals, they are not free to shrink freely. If the contraction of the crystal lattices is at all anisotropic then some crystals will be trapped where they are unable to relax to their equilibrium lattice constants without breaking those connections. If they are unable to fully relax then they must be stretched (which we call being under strain). Since all forces must be matched, other crystals must be stress. This combination, which leads to an ensemble of crystals where there is a distribution of lattice constants, some larger than the equilibrium values and some smaller. This is the condition for residual stress. It commonly occurs in materials due to thermal and mechanical processing.
Microstrain broadening is really measuring the range of lattice constants in a material and there can other chemical effects that change the lattice constants for a microscopic section of a sample other than residual stress. A material may not have uniform chemical composition. The bonding of atoms close to a surface differs from that of atoms deep in the bulk (due to the truncated bonds at the surface or modified chemical environment for those atoms, should the surface be modified by oxygen or water, etc.) so the outer layers of particles may not have the same lattice constants as the bulk.
To consider the effect of microstrain on a powder diffraction pattern, consider the () diffraction from two crystals, one where the lattice parameter is at the equilibrium value for c, but the other crystal has a slightly smaller lattice constant, c-d. Recalling that \(Q=2\pi d^\ast \) and for the 00l reflections \(d^\ast = l c^\ast \), we see that the reflection positions for 001, 002, 003,… reflections are, in units of Q, \(2\pi c^\ast \), \(2\pi c^\ast \), \(2\pi c^\ast \),… If we consider the crystal that has a lattice constant of \(c-\delta ^\prime \), the reciprocal cell length will be \(c^\ast +\delta \) and the reflection positions for those crystals will be \(2\pi (c^\ast +\delta )\), \(4\pi (c^\ast +\delta )\), \(6\pi (c^\ast +\delta )\),… Thus, the difference in the peak positions will be \(2\pi \delta \), \(4\pi \delta \), \(6\pi \delta \),… for Q values, \(2\pi c^\ast \), \(2\pi c^\ast \), \(2\pi c^\ast \),… respectively. So, the spacing between the peaks from these two crystals increases as Q increases, meaning that the spacing between the peaks for each crystal is proportional to \(\Delta Q/Q\). In a specimen with microstrain, we will have a distribution of lattice constants around the equilibrium value. This results in a distribution of peak positions which gives rise to peak broadening, but these peak positions also broaden proportional to \(\Delta Q/Q\).
When looking at a diffraction pattern plotted in \(2\theta \) units, we again must perform a coordinate transformation. Before we saw that \(Q = (4\pi /\lambda ) \sin \theta \) and \(dQ = (4\pi /\lambda ) \cos \theta d\theta \) so that \(d\theta = (dQ/Q) (\sin \theta / \cos \theta )\) or equivalently \(\Delta 2\theta \) is proportional to \(\tan \theta \).
As noted, the range of lattice constants seen in a material can arise from multiple effects, including elastic strain. The elastic strain constants for a material can be highly anisotropic, which in other words means that a material will be much more “squishy” in some directions than others. As an example, if we consider graphite, which consists of hexagonal sheets of carbon atoms, the effect of applying pressure in the plane of the sheets would be expected to be quite different than applying pressure in the perpendicular direction to reduce the space between the sheets. Likewise, if a material has a range of compositions, the change in chemistry may well affect some lattice directions more than others. Either effect results in anisotropic microstrain broadening where some classes of reflections are affected more than others.
In conclusion, please understand that crystallite size broadening will be present in all powder diffraction samples, but it is often negligible since a diffraction instrument must have very good resolution to discern the broadening due to the size range for crystallites typically used for powder diffraction, e.g. microns; microstrain broadening will usually be present as well and it is typical for this to produce observable peak broadening. The one case where the reverse is commonly true, where crystallite size is observable and microstrain is not is in samples where very small crystallites have been created intentionally (nanoscale materials). With lower resolution instruments, it may not be possible to detect either type of broadening. Likewise, anisotropic peak broadening usually must be fairly significant to be detectable.
The ability to differentiate crystallite size broadening from microstrain broadening depends on having discernible peaks over a sufficiently wide Q range. This is because the ability to distinguish the two sources of broadening depends on determining how that broadening changes with Q or \(2\theta \). Noting that in Q, crystallite size broadens all peaks (for anisotropic broadening, in a hkl class) by the same amount, while microstrain broadening increases with Q, so that \(\Delta Q/Q\) is constant. In \(2\theta \) units, broadening will be proportional to \(1/\cos \theta \) or \(\tan \theta \), for crystallite size and microstrain, respectively.