Analytical distance distributions in systems of spherical symmetry with applications to double electron-electron resonance.

Based on a simple geometrical approach, we derive analytical expression of the probability density functions (pdfs) of distance of probe molecules distributed homogeneously in spherical aggregates with shell structure. These distance distributions can be utilized in the investigation of double electron-electron resonance (DEER) data of disordered nanometer-sized spin clusters. Structural insights and geometrical parameters of the aggregates can be extracted by modeling the DEER time traces based on the analytical pdfs. This approach is efficient and avoids difficulties of the model-free solution of the inverse problem that are related to multi-spin effects, limited excitation bandwidth, bias introduced by the regularization scheme, or ambiguity resulting from broad distance distributions. The derived pdfs can serve as building blocks, from which the distance distributions in arbitrary spherically symmetric objects can be assembled. The scenario of the pumped species being chemically distinct from the observed species is covered as well as that of a single type of probe molecules. We demonstrate the merits of analytical distance distributions by studying the distribution of three different spin probes in SDS micelles. By simultaneously analyzing DEER data corresponding to different spin probe concentrations, the distribution of the spin probes over the micelle can be determined. Employing Bayesian inference it is found that for all probes studied, a spherical shell model is most appropriate among the studied models and by orders of magnitude more likely than a homogeneous distribution in a ball. This statement also applies to probes that are deemed nonpolar. We envisage that the spin probe distributions in disordered soft and hard matter systems can now be quantified using DEER spectroscopy with greater precision and reduced ambiguity.