A Mathematical Algorithm for Estimating the Rotational Diffusion Coefficients from X-ray Photon Correlation Spectroscopy Data
The Brownian motion of particles can be decomposed into translational and rotational components, characterized by translational and rotational diffusion coefficients respectively. Measuring these two coefficients plays an important role in determining the structure and understanding the dynamic properties of materials, with benefit to research in such areas as molecular biology, material sciences. One of the promising tools for investigation of Brownian motion is the emerging X-ray Photon Correlation Spectroscopy (XPCS), which can capture dynamics of samples comprising large groups of particles in a broad range of time scales and length scales. Methods for estimating translational diffusion coefficients on the basis of the temporal auto-correlation analysis of XPCS images are used widely. However, to the best of our knowledge, there is no XPCS-based algorithm for assessing the rotational diffusion coefficients from such temporal auto-correlation data. In this thesis, we take a different route, and propose exploiting an angular-temporal cross-correlation function whose values are approximated by an estimator based on the collected experimental images. We prove the consistency of this estimator by deriving a tail bound. A numerical algorithm, MTECS, for estimating the rotational diffusion coefficients from the cross-correlation is designed and implemented. We demonstrate the capability of this algorithm by testing it on a range of simulated data.