Statistical geometry. I. A self-consistent approach to the crystallographic inversion problem based on information theory

What did Erwin mean? The physics of information from the materials genomics of aperiodic crystals and water to molecular information catalysts and life

Erwin Schrödinger famously and presciently ascribed the vehicle transmitting the hereditary information underlying life to an 'aperiodic crystal'. We compare and contrast this, only later discovered to be stored in the linear biomolecule DNA, with the information-bearing, layered quasi-one-dimensional materials investigated by the emerging field of chaotic crystallography. Despite differences in functionality, the same information measures capture structure and novelty in both, suggesting an intimate coherence between the information character of biotic and abiotic matter-a broadly applicable physics of information. We review layered solids and consider three examples of how information- and computation-theoretic techniques are being applied to understand their structure. In particular, (i) we review recent efforts to apply new kinds of information measures to quantify disordered crystals; (ii) we discuss the structure of ice I in information-theoretic terms; and (iii) we recount recent investigations into the structure of tris(bicyclo[2.1.1]hexeno)benzene, showing how an information-theoretic analysis yields additional insight into its structure. We then illustrate a new Second Law of Thermodynamics that describes information processing in active low-dimensional materials, reviewing Maxwell's Demon and a new class of molecular devices that act as information catalysts. Lastly, we conclude by speculating on how these ideas from informational materials science may impact biology.

About the reliability of the Maximum Entropy Method in reconstructing electron density: the case of MgO

The reliability of the Maximum Entropy Method (MEM) to reconstruct finite temperature electron density (ED) is here discussed, investigating the case of periclase (MgO). A theoretical electron density has been generated by quantum mechanic calculations and folded with a function simulating atomic thermal motion, in order to produce a reference errorless ED [ρ(r)REF]. The Fourier coefficients of ρ(r)REF have been calculated, and used as “observed” diffraction intensities to reconstruct via MEM the original ED. The electron density attained by MEM [ρ(r)MEM] and ρ(r)REF have been compared with each other (pixel-by-pixel and critical points) to assess the ability of MEM to retrieve EDs, on the basis of a set of observed structure factors. We have carried out our study varying the number of observed structure factors [i.e. sin (θ)/λ cut-off], the nature of the prior-density [uniform density and procrystal-like model] and the way in which the prior-density is treated during MEM maximization [fixed or free to change]. We observe that (i) it is recommendable to use the prior-density as a start point only, and allow it to change during maximization; (ii) the closer is the prior-density to ρ(r)REF, the easier one attains by MEM a correct ED; (iii) if the prior-density is varied and a sufficient large number of observed structure factors used, then MEM tends to yield converging EDs, regardless of the prior-density chosen as a start point.

The MEM/Rietveld method with nano-applications – accurate charge-density studies of nano-structured materials by synchrotron-radiation powder diffraction

Structural studies of materials with nano-sized spaces, called nano-structured materials, have been carried out by high-resolution powder diffraction. Our developed analytical method, which is the combination of the maximum-entropy method (MEM) and Rietveld refinement, the so-called MEM/Rietveld method, has been successfully applied to the analysis of synchrotron-radiation (SR) powder diffraction data measured at SPring-8, a third-generation SR light source. In this article, structural studies of nano-porous coordination polymers and endohedral metallofullerenes are presented with the advanced technique of SR powder experiment. The structure of the adsorbed guest molecule in the coordination polymer and encapsulated atoms in the fullerene cage are clearly revealed by the MEM charge density. The methodology of MEM/Rietveld analysis is also presented.

Maximum entropy image reconstruction in X-ray and diffraction tomography

The authors propose a Bayesian approach with maximum-entropy (ME) priors to reconstruct an object from either the Fourier domain data (the Fourier transform of diffracted field measurements) in the case of diffraction tomography, or directly from the original projection data in the case of X-ray tomography. The objective function obtained is composed of a quadratic term resulting from chi(2) statistics and an entropy term that is minimized using variational techniques and a conjugate-gradient iterative method. The computational cost and practical implementation of the algorithm are discussed. Some simulated results in X-ray and diffraction tomography are given to compare this method to the classical ones.