The Rietveld Method Armel Le Bail Université du Maine, Laboratoire des oxydes et Fluorures, CNRS UMR 6010, Avenue O. Messiaen, 72085 Le Mans Cedex 9, France. Email : alb@cristal.org XX Conference on Applied Crystallography, Wisla, Poland, September 2006

OUTLINE « Rietveld method » according to the Internet - The Rietveld method in a few words/equations - Available/recommended software - Fullprof - Tutorials everywhere… - Demonstrations with FULLPROF XX Conference on Applied Crystallography, Wisla, Poland, September 2006

What Google says

http://sdpd.univ-lemans.fr/DU-SDPD/ A course about

http://home.wxs.nl/~rietv025/

Inside of that website : the complete original publication :

What Wikipedia says

What Google Scholar says

>1000 subscribers to the Rietveld Mailing List

>650 subscribers to the SDPD Mailing List

The Rietveld Method The Rietveld Method consists in refining a crystal (and/or magnetic) structure by minimizing the weighted squared difference between the observed and the calculated pattern against the parameter vector:  : is the variance of the "observation" yi

Least squares: Gauss-Newton (1) Minimum necessary condition: A Taylor expansion of around allows the application of an iterative process. The shifts to be applied to the parameters at each cycle for improving c2 are obtained by solving a linear system of equations (normal equations)

Least squares: Gauss-Newton (2) The shifts of the parameters obtained by solving the normal equations are added to the starting parameters giving rise to a new set The new parameters are considered as the starting ones in the next cycle and the process is repeated until a convergence criterion is satisfied. The variance of the adjusted parameters are calculated by the expression:

Several phases ( = 1,n) contributing to the diffraction pattern Several phases ( = 1,n) contributing to several (p=1,np) diffraction patterns

Least squares: a local optimization method The least squares procedure provides (when it converges) the value of the parameters constituting the local minimum closest to the starting point A set of good starting values for all parameters is needed If the initial model is bad for some reasons the LSQ procedure will not converge, it may diverge.

What to do when the information in the powder diffraction pattern is not enough ?

Constraints: reduce the number of free parameters (rigid body refinements) Restraints: same number of free parameters + additional observations (allowing some interatomic distances and/or angles to vary between some min and max limits)

What Rietveld software to use ? Recommendations inside of session 9 of the SDPD Internet Course :

List of Rietveld software used for final refinement in SDPD (list made in 1998)

Place of the Rietveld method in the SDPD maze : The ultimate step, helping as well at the structure completion stage (so, telling that a structure was « determined by the Rietveld method » is improper : it only can refine a structure)

Fullprof Note that Fullprof is able to perform so many more calculations than Rietveld refinements that only the main developer (Juan Rodriguez-Carvajal, now head of the Crystallography Group at the ILL, Grenoble, France) would be able to give a serious talk about it.

Rietveld and more…

What Fullprof can do is illustrated by the list of example files :

Rietveld refinements of crystal and magnetic structures from conventional X-ray data, or neutrons, TOF, multiphase, multipattern, quantitative analysis, solving in direct space by simulated annealing, intensity extraction (Le Bail method), use of restraints, constraints, rigid-body, size-strain estimations…

Tutorials are everywhere inside of the Internet Either at the ILL or CCP14 web sites for using WinPlotr, Fullprof or satellite programs for bond distance calculation of Fourier synthesis, applications to various kinds of datasets (T.O.F., constant wavelength) single or multiple, facing size/microstrain effects, using special features for solving (simulated annealing), etc.

Exercise 2 of the SDPD Internet Course, session 9

STRATEGY - CHECKLIST -3 - Make the best pattern you can. If 2-theta-max is small (<100°), you will have difficulties in refining the U and V profile width parameters. Have enough sample in order to intercept the whole beam at low 2-theta angles. Try to avoid preferred orientation, but have a plane surface in case of reflection geometry (quite hard sometimes...). Use zero background sample holder, or be prepared to have difficulties to locate the background. -2 - Make a preliminary approach : have good cell parameters and zeropoint (use ERACEL). -1 - Realize a Pawley or Le Bail fit, and use the profile parameters in the subsequent Rietveld fit. -0 - Go slowly with the Rietveld method : refine first some essential parameters (scale factor) 1- Depending of the percentage of atoms already known to be in the cell, you will be able to refine the cell and profile parameters or not (if not, keep fixed the values obtained at the Pawley or Le Bail methods stage, up to obtain sufficiently low Rp and Rwp values). 2- Add progressively new parameters to refine. 3- If the structure is incomplete, alternate Fourier synthesis and refinement with new atoms added after verifying interatomic distances. ...think to classical problems like preferred orientation, anisotropic broadening, etc. N- Some parameters are to be refined only at the end : separate thermal parameters for same atom-types and some instable parameters like the correction for sample transparency and so on...