Tutorial:Qualitative inspection

When analyzing data, the first thing one should do is get an overall sense of data. By applying a few simple rules-of-thumb, one can interpret a 2D x-ray scattering image, and infer quite a bit about the structure of the sample.

Amount of Order

TBD

Peak Position

Recall that reciprocal-space is inverted: i.e. peaks at large angle (large q) correspond to small-scale structures; whereas peaks at small angle (small q) correspond to larger (nanoscale) structures. Ultrasmall angle scattering (USAXS) probes yet larger (micron-scale) order. More specifically, when observing peaks at:

• Very large angle (0.5-4 Å⁻¹): Atomic packing distances (1-10 Å).
• Large angle (0.2-2 Å⁻¹): Molecular packing distances (0.3-3 nm). For instance, aromatic rings tend to pi-pi stack with a 0.3-0.4 nm repeat distance.
• Medium angle (0.03-0.3 Å⁻¹): Macromolecular distances (2-20 nm). For instance, polymers often crystallize into chain-folded lamellae with a period of 2-10 nm.
• Small angle (0.0002-0.04 Å⁻¹): Nanoscale distances (15-300 nm). For instance, block-copolymers and nanoparticle superlattices tend to organize in this size regime.
• Ultra-small angle (<0.0006 Å⁻¹): Micron sizes (>1 µm).

Peak Width

In scattering, sharp peaks correspond to large grain sizes, whereas broad peaks correspond to small grain sizes. This can be quantified through a Scherrer grain size analysis. Even qualitatively, however, it is usually easy to judge how well-ordered a material is based purely on peak widths. Consider a highly disordered system, such as an amorphous polymer. The polymer chains likely have some preferred chain-packing distance, but the 'lattice' only repeats once or twice before decorrelating; i.e. there isn't a well-defined crystal with well defined grain boundaries. In such a case one would see a very broad halo. In even more disordered systems, only diffuse scattering would be seen (this can be thought of as the ultimate limit of a broad peak).

On the other hand, extremely sharp peaks indicate that the lattice repeats in a well-correlated way over very large distances. Thus, sharp peaks arise when one has well-defined crystals.

Scattering Intensity

The scattering intensity (counts on the x-ray detector) scales with the amount of scattering material. Thus, a bigger sample yields a stronger scattering signal. (Of course if you try to scatter through a sample that is too thick, absorption will at some point instead reduce the signal.) For a given scattering volume, the intensity of the scattering can be thought of as a probe of the fraction of the material in the given state/phase/configuration. I.e. if a given peak is stronger in one sample vs. another, then this means that the phase (crystal form, etc.) corresponding to that peak appears more frequently in that sample. One must be careful, however, as many other things are implicated in peak heights (orientation, disorder, etc.).

Note that in general, well-ordered systems will appear to scatter more strongly than weakly-ordered systems. A broad scattering peak (small grains) will have lower maximum intensity than a sharp scattering peak (big grains). More generally, periodically ordered structures give rise to scattering events, whereas homogeneous systems do not scattering the incident radiation.

Higher Orders

Higher-order peaks in x-ray scattering generally imply a well-defined structure. Qualitatively, if you see lots of higher-orders, you can say you have a very well-organized structure. For molecular peaks, this usually means a high-quality crystal (with large grain sizes). For nanoscale peaks, this usually means a well-defined superlattice of some kind.

Conversely, disorder tends to broaden peaks, and also extinguish the intensity of higher-order peaks (c.f. Debye-Waller factor).

Observing only a single peak means a highly disordered system (e.g. amorphous packing); i.e. there is some preferred particle-particle distance, but no recognizable lattice order to longer distances.

Missing Orders

The intensities of higher-order peaks are affected by the exact shape of the structures sitting on the realspace lattice; i.e. the realspace lattice determines the peak positions, while the electron-density distribution within the unit cell controls the peak heights. This modulation of peak heights can be extreme: e.g. an appropriate electron-density distribution can entirely extinguish a particular peak.

This effect can be understood in terms of the form factor modulating the structure factor peak heights. Thus, if a given particle shape (or, more generally, density distribution in the unit cell) has a form-factor minimum at a particular q, then any structural peak at this q will not appear.

For instance, consider a lamellar structure with total layer spacing d; of course we expect reciprocal-space peaks at |q| = 2 π/d, and in fact we expect higher-orders at |q| = n 2 π/d for all integer n. However, for a well-defined two-layer structure where each sub-layer has the same thickness (square-wave density profile), the even-order reflections will be absent. If the two sub-layers have different thicknesses, then the even-order peaks will be present. More generally, the variation in intensity between the odd and even peaks can be used to deduce the duty-cycle.

On the other extreme, consider a 1D repeating structure where the density varies sinusoidally; the higher odd orders will be absent. In fact, a perfectly sinusoidal structure only needs the fundamental Fourier component to completely describe the density-profile. In other words, the higher-order peaks will be entirely absent.

Orientation Distribution

TBD

Crystal Order and Orientation

TBD (amorphous, 3D powder, in-plane powder, single-crystal)