An introduction to the Rietveld method Angus P. Wilkinson School of Chemistry and Biochemistry Georgia Institute of Technology

Outline  History and fundamentals –The birth of the Rietveld method »what is Rietveld refinement? –Adolescence »new instrumentation expands scope –Maturity »new software enhances scope  Achievements –the technique is invaluable to materials scientists –and is becoming a serious tool for examining organics

Historical background  Powder diffraction is viewed primarily as a tool for phase identification and quantitative analysis.  With few exceptions, most of the crystal structures refined using powder data prior to Rietveld refinement were simple.  The compression of a 3D diffraction pattern into 1D can lead to overlapping peaks and information loss

Solutions to the problem of overlap  Do not use overlapped reflection  Use grouped intensities  Curve fit the overlapping peaks  Fit the whole powder pattern

Curve fitting The 112 / 200 reflections of tetragonal Y doped ZrO 2  Parameters are often highly correlated  Constraints can help

The birth of the Rietveld method  The Rietveld method was developed (1967, 1969) to extract the maximum amount of information from a pattern –initially only applied to neutron data due to simple peak shape

Experiments limited by low resolution

Parameters  Structural Variables –X, Y, Z, fractional occupancies, U iso  Correction terms –Absorption, extinction »These really do belong in the model!  Profile parameters –Unit cell constants, wavelength –Peak shape, including width, asymmetry and anisotropy

The peak shape model  Peak shape is determined by: –Sample characteristics –Instrument characteristics  Medium resolution CW neutron diffractometers give Gaussian peak shape. exp(-ax 2 )

Angular variation of peak width  2 = U tan 2  + V tan  + W Caglioti, Paoletti and Ricci, NIM 3, 223 (1958)

Lorentzian peak shapes u Peaks from “high resolution” instruments often have a strong Lorentzian contribution to their shape Angular variation of Lorentzian FWHM often described by:

The limits of Rietveld refinement?  We have to consider structural complexity, data quality and what we already know  Structural complexity is determined by: –unit cell size –symmetry  Data quality includes factors such as: –How many resolved peaks do we have? –Is both neutron and X-ray data available  Existing information –Bond lengths –Chemical composition

ZrP 2 O 7  Material is pseudo cubic (actually Pbca) with 136 unique atoms in the unit cell (402 coordinates!) –Synchrotron X-ray plus neutron data combined with simulated annealing to get away from a pseudosymmetric starting point gave a good refinement. Restraints used. Stinton, Hampson and Evans, Inorg. Chem. 45, 4352, (2006). This is an extreme example!

How good is your model ?  Many ways of judging the quality of a refinement: –Agreement indices, R wp, R p, R F, R I, R B –Goodness of fit,  2 –Serial correlation indicators, DWd  Normal probability  Very valuable indication is visual quality of fit

Profile R factors can be misleading R wp = 7.8%  2 = 16.5 DWd = 0.19 R wp = 2.1%  2 = 1.17 DWd = 1.79

Profile plots can be very helpful  Zero point error / sample height problems

Profile plots can be very helpful  Peak shape model wrong

Better instruments  Instrument developments have enhanced the information content of powder patterns –high resolution time of flight and reactor based instruments developed in the 80s –very high resolution x-ray diffractometers developed at synchrotron sources in the 80s  However, the extra information comes at a price –The peak shape is often determined by the sample –TOF diffractometers have highly asymmetric peak shapes  Modeling high resolution data is more demanding

Synchrotron data for BaBiO 3

Ultrahigh resolution  FWHM ~0.02 o in some cases at ~ Cu K 

Ultrahigh peak to background  Peak to background > 1000:1 possible

TOF diffraction patterns  Asymmetric peak shapes

Advances in data analysis  Use multiple data sets to get extra information  Use constraints and restraints to handle very complex structures  Perform phase analysis  Learn about crystallite size and strain  Determine texture in a material

Achievements  Major contribution to almost every hot area of “hard” materials in the last 15 years –High temperature superconductors –Buckyballs (C 60 ) –Colossal Magnetoresistance –Thermoelectrics –Hydrogen storage –Batteries  Now making an inroad in biological science and organic materials –Drugs, polymers, proteins?

High T c superconductors  Much of the solid state chemistry of these materials was worked using neutron diffraction and Rietveld refinement

C 60 - Buckminsterfullerene  The structure of C 60 and its metal doped variants have all been examined using the Rietveld method

Orientational ordering in C 60  At high T, C 60 is rotationally disordered, but at low temperatures the molecules order

Polymer electrolytes  Powerful solution procedures combined with constrained Rietveld refinements reveal details of electrolyte structure

Drug structures can be determined  Powerful structure solution methods (often simulated annealing) combined with Rietveld refinement and constraints have been used to examine drugs

Battery electrodes u Powder diffraction and Rietveld analysis are widely used to characterize electrode materials and follow structural changes in-situ Followed phase composition as a function of discharge. Over 300 citations as of 2012

Conclusions  Rietveld refinement has become a very powerful and widely used tool. It makes the most of the available information –Quite large structures can be refined ~ 200 structural parameters –The complexity of the problem is limited by instrument resolution and sample quality  Rietveld analysis is limited by the requirement that you have a reasonable structural model before you start  When performing a refinement consider all possible indicators of model quality and make sure the visual fit is OK.