Teaching set #2 - Pattern size and shape

Teaching set #2 - Pattern size and shape effects

The overall size and shape of an ordered repetition of identical objects affects the size and shape of the peaks in its Fourier transform.

As a rule of thumb, a smaller size yields broader peaks, and vice versa. This is linked to how real space lengths and reciprocal space frequences are related. Indeed, as we observed in the previous teaching set, smaller shapes in real space cause longer oscillation periods in the Fourier space. While the form factor of the repeating object is transferred to the distribution of peaks intensities, here the form factor of the crystal shape is transferred to the very shape of each peak.

The following crystals are build from repeating a photograph of Dorothy Crawfoot as elementary unit cell. You don't know her? You can learn something interesting here:

https://www.nobelprize.org/prizes/chemistry/1964/hodgkin/

Patterns are shown with different intensity scales to facilitate comparison.

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1. Circular crystal (50 unit cells radius) and its Fourier transform.

2. Circular crystal (25 unit cells radius) and its Fourier transform.

Note: as the intensity remains the same, a smaller crystal causes it to distribute in broader regions around the peaks.

3. Circular crystal (10 unit cells radius) and its Fourier transform.

4. Circular crystal (3 unit cells radius) and its Fourier transform.

Note the ripples around every peak, particularly visible around the central one. These are a combination of truncation effects due to the very limited number of repetitions (6) and the very small size of the circular shape. The latter is arguably the dominant effect here.