Continuous Diffraction of Molecules and Disordered Molecular Crystals

Abstract

The diffraction pattern of a single non-periodic compact object, such as a molecule, is continuous and is proportional to the square modulus of the Fourier transform of that object. When arrayed in a crystal, the coherent sum of the continuous diffracted wave-fields from all objects gives rise to strong Bragg peaks that modulate the single-object transform. Wilson statistics describe the distribution of continuous diffraction intensities to the same extent that they apply to Bragg diffraction. The continuous diffraction obtained from translationally-disordered molecular crystals consists of the incoherent sum of the wave-fields from the individual rigid units (such as molecules) in the crystal, which is proportional to the incoherent sum of the diffraction from the rigid units in each of their crystallographic orientations. This sum over orientations modifies the statistics in a similar way that crystal twinning modifies the distribution of Bragg intensities. These statistics are applied to determine parameters of continuous diffraction such as its scaling, the beam coherence, and the number of independent wave-fields or object orientations contributing. Continuous diffraction is generally much weaker than Bragg diffraction and may be accompanied by a background that far exceeds the strength of the signal. Instead of just relying upon the smallest measured intensities to guide the subtraction of the background it is shown how all measured values can be utilised to estimate the background, noise, and signal, by employing a modified "noisy Wilson" distribution that explicitly includes the background. Parameters relating to the background and signal quantities can be estimated from the moments of the measured intensities. The analysis method is demonstrated on previously-published continuous diffraction data measured from imperfect crystals of photosystem II.