Chapter 1
INTRODUCTION

The maximum entropy method (MEM) [1,2] is very useful to modify structural models adopted in Rietveld analysis and determine distributions of electron and nuclear (coherent-scattering length, \(b_\mathrm {c}\)) densities from X-ray and neutron diffraction data, respectively [3]. The termination effect is less serious in MEM analysis than in Fourier/D synthesis because MEM is capable of estimating structure factors of high-\(Q\) reflections that have not been measured experimentally. In addition, after observed structure factors of overlapped reflections have been estimated on the basis of results of Rietveld analysis, subsequent MEM analysis more or less improves structure factors, which makes MEM very effective in the extraction of the maximum amount of structural information from powder diffraction data.

Three-dimensional (3D) visualization of electron/nuclear densities resulting from diffraction data achieves a better understanding of the highly disordered structure, chemical bonding (X-ray diffraction), or anharmonic thermal motion. For example, diffusion paths of mobile chemical species in ionic conductors have been visualized in three dimensions from X-ray and neutron powder diffraction data by a sophisticated structure-refinement technique called MEM-based pattern fitting (MPF) [3–6]. MPF may be able to reveal even anharmonic thermal vibrations, which are very difficult to analyze from powder diffraction data because of many parameters to be refined by a nonlinear least-squares method.

ERIS is a successor of MEM analysis programs Dysnomia [6–8], PRIMA, and ALBA [3,9]. PRIMA was a fortran program for MEM analysis of electron and nuclear density from X-ray and neutron diffraction data. ALBA was a fortran program for MEM analysis of the Patterson function from X-ray and neutron diffraction data to estimate intensities of unobserved and/or individual reflections of overlapped peaks in powder diffraction. PRIMA was first rewritten from scratch in the C++ language and released as Dysnomia in 2011.

Dysnomia provides the following features which were not supported in PRIMA:

1. linear combination of the generalized \(F\) and \(G\) constraints,
2. weightings based on lattice-plane spacings.
3. new implementation of optimization algorithms (a variant of the Cambridge algorithm [10] and the limited-memory BFGS algorithm [11]) that converge to solutions close to the true maximum-entropy conditions,
4. automatic selection of fast Fourier transform (FFT) and discrete Fourier transform (DFT),
5. parallel processing using an API (application programming interface), OpenMP, for multi-platform shared-memory parallel programming,

ERIS integrates the features of Dysnomia and ALBA, with a couple of new and unique features.

1. a new type of weighting function, proportional to exponential of \(d\)-spacing,
2. a feature to automatically determine the best weighting parameters,
3. implemented the maximum entropy Patterson method,
4. automatically impose constraints of extinction conditions in MEP analysis,
5. generate an input file of Superflip after convergence of MEP analysis to solve crystal structure by the charge flipping method.

ERIS is the name of the Greek goddess of strife and discord. She is mother of Dysnomia, who is also the goddess of discord. Since the lawlessness state stands for higher-entropy state compared to ordered state, it is suitable for the name of the MEM analysis program, in which the information entropy, \(S\), is maximized under some constraints. The program name ERIS also stands for Entropy-based Real-space Inference System.