inverse problems · configuration space what the projection keeps · underdetermined experiments, collapsed configuration spaces, and the tools brought from outside · started October 2026

Rietveld refinement and mode crystallography

Rietveld refinement and mode crystallography

The case Iñigo and Josu know best (T06, T07): the one in which the external tool exists, is used daily, and is exact mathematics on top of one hypothesis.

The projection

A powder pattern is the intensity scattered by a crystalline powder against the scattering angle. Three projections happen before the detector. The crystal's electron or nuclear density is projected onto its Fourier amplitudes, the structure factors \( F_{hkl} \); the measurement keeps \( |F_{hkl}|^2 \) and discards the phases. The powder average projects the three-dimensional reciprocal lattice onto one axis, the modulus \( |\mathbf{Q}| \) or equivalently \( 2\theta \); every reflection with the same \( d \)-spacing lands on the same peak, and orientation is gone. And the instrument convolves each peak with a profile of finite width, so that neighbouring reflections overlap and only their sum is seen. What arrives is thousands of intensities, of which a few hundred resolved peak areas, positions and widths are independent.

The Rietveld model (Rietveld 1969) computes that pattern from the configuration: atomic coordinates, occupancies, displacement parameters, cell, profile parameters, zero shift, background and scale, and refines them all by least squares against the measured points. It is the known theory of T01 in its purest form: nothing in the forward map is in doubt. What is in doubt is which of the many configurations that fit the pattern is the structure.

Where it folds

  • Exactly. The phase problem. The powder average. Homometric structures (Patterson 1944): different arrangements with the same set of interatomic vectors and hence the same intensities. With X-rays, a light atom next to a heavy one contributes almost nothing: its coordinates are in the null space for practical purposes, and exactly so when its scattering vanishes.
  • At the noise. The sloppy valleys every refiner knows and names by their symptoms: an occupancy and a displacement parameter of the same site that trade against each other; two atoms related by a pseudo-symmetry whose displacements the pattern sees only as a sum; a background polynomial that eats the tails of the peaks; size and strain broadening with the same angular dependence over a short range. These are the off-diagonal elements of the correlation matrix that programs print and refiners glance at.
  • Off the manifold. A space group too high, so that the refinement is forced to put in the displacement parameters what belongs in a distortion; a profile function that cannot make the anisotropic broadening of a stacking-faulted sample; a second phase not in the model. The fit looks acceptable and the parameters are biased.

The external tool

Mode crystallography (Perez-Mato, Orobengoa and Aroyo 2010) describes a distorted structure as a parent of higher symmetry plus a displacement field, and writes that field in the basis of the symmetry-adapted modes of the parent: displacement patterns that transform as irreducible representations of the parent space group, each with an amplitude. AMPLIMODES (Orobengoa et al. 2009) performs the decomposition on the Bilbao Crystallographic Server; ISODISPLACE, now ISODISTORT (Campbell et al. 2006), does the same and writes the amplitudes as refinable parameters for Rietveld programs. The configuration space is unchanged; the coordinates are new.

What the new coordinates buy is written on the Discourse page and summarised here. The physics of a displacive transition says that one or two primary modes carry the distortion and the rest are small: in mode coordinates that is a prior a refinement can use, in atomic coordinates it cannot be stated. Displacements that the pattern sees only jointly are one amplitude, not two correlated coordinates. And the refined amplitudes come ranked, which is the dimension reduction Perez-Mato et al. name among the method's purposes: fundamental against marginal degrees of freedom, hidden correlations detected, fewer effective parameters. Iñigo's T07 says the same from the practitioner's side: the exploration is done in mode space, and when the structure is written back in coordinates, many of the values the coordinates could have taken have been discarded by the symmetry.

The one thing the tool does not supply is the parent. It is a hypothesis from outside the pattern: a related compound, a high-temperature phase, a group–supergroup chain. The decomposition is exact given it; the choice of it is judgement. A script can do the first and must record the second.

What is already scripted, and what is not

The decomposition is scripted: AMPLIMODES and ISODISTORT take a parent and a distorted structure and return the modes. The release strategy is not: which amplitudes to refine first, when to free a secondary one, when to stop, is still the refiner's hand. The published machine-learning work on powder patterns goes around the refinement altogether and learns classifications from simulated patterns — crystal system, extinction group and space group from about 150 000 patterns (Park et al. 2017; 95, 84 and 81 % respectively), phases in mixtures from 1.8 million synthetic patterns (Lee et al. 2020), with physics-informed perturbations and probabilistic outputs (Szymanski et al. 2021). What none of these does is what T07 and T09 together ask for: the refinement itself, in mode coordinates, done over a corpus, with the whole tube of each refinement kept so that the identifiability of each amplitude can be learned as a function of the data.

What this case gives the programme

The toy of P2 — a synthetic perovskite distortion, patterns simulated under an instrument model, refined from many starting points in atomic coordinates and in mode amplitudes, the valley drawn in both — is a few days' work with any scriptable Rietveld engine, and it is the figure that makes the whole thesis visible. The corpus of P3 starts from the same family, because the parent is unambiguous there and the mode decomposition is textbook. The diff-ref-struct core, which already carries mode-described phases and the eigen-spectrum of a refinement (its Explore module, L114–L115 of that project), is one candidate engine; GSAS-II's scripting interface is another.