Teaching set #3 - Crystallization in reciprocal space

When illuminated with suitable radiation (typically X-ray, electron or neutron beams) crystals produce diffraction patterns. Why are these patterns the way they are? Why do they have pointy Bragg peaks, or broad clouds, or sharp streaks of intensity?

The answer is both complex and simple: the complex part is that diffraction patterns are produced primarily by the Fourier transform of the distribution of atoms* in space. In simple terms, the pattern we see depends on how atoms are distributed in the structure of what is producing the diffraction pattern. Every diffraction pattern is a continuous distribution of intensities, since strictly periodic and perfect crystals (which would produce only infinitely sharp Bragg peaks and no other intensities) do not exist.

The continuous nature of diffraction patterns and the importance of atom-atom spatial relationships (a.k.a. correlations) are exemplified in this collection by considering the step-wise growth of a naphtalene crystal.

Below, molecular structures are paired with their simulated X-ray diffraction pattern. The former are shown on the left-hand side, viewed along the b axis of the considered unit cell (structure with refcode NAPHTA49 in the Cambridge Structural Database). Patterns show the h0l plane of the diffraction space, which contains information on the projection of the structure along the b crystallographic direction. This makes the figures on the left-hand side related to the pattern displayed to their right.

* to be precise we should rather say "the distribution of scattering centres". Their positions depend on where the atoms are located, but their identity depends on which radiation we use for a diffraction experiment: X-rays are scattered mainly by electrons, neutron beams are scattered by atomic nuclei, and electron beams are scattered by the electrostatic potential surrounding the atomic positions.

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01. Single molecule of naphtalene and its diffraction pattern.

02. Complete unit cell of a naphtalene crystal and its diffraction pattern.

03. Naphtalene 1D "crystal" made by 2x1 unit cells and its diffraction pattern.

04. Naphtalene 1D "crystal" made by 5x1 unit cells and its diffraction pattern.

N.B.: fringes can be observed along the h direction of the diffraction space (horizontal). These oscillations are 5 from peak to peak (peaks included), due to the fact that in the a direction we have 5 unit cells.

05. Naphtalene 2D crystal made by 5x5 unit cells and its diffraction pattern.

N.B.: fringes now also extend to the l direction of diffraction space, since we have 5 unit cells in the c direction. However, peak-to-peak oscillations are 10 and not 5. This is due to the presence of 10 repeating units instead of 5 in this specific projection of the structure, even though the number of unit cells in the 3D structure is 5.

06. Naphtalene 1D "crystal" made by 10x1 unit cells and its diffraction pattern.

07. Naphtalene 2D crystal made by 10x10 unit cells and its diffraction pattern.

08. Molecular form factor effects.

It is possible to recognize how the form factor of the repeating unit (in the images, that corresponds to a single naphtalene molecule) modulates the intensity of the features in diffraction space. While their distribution is a result of the overall atomic structure and the presence and extent of repetition, the intensities always follow the shape of the form factor of the repeating object. Below, the form factor is overlayed as semitransparent white mask onto each diffraction pattern.