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 reflectionUse grouped intensitiesCurve fit the overlapping peaksFit the whole powder pattern

Curve fitting

The 112 / 200 reflections of tetragonal Y doped ZrO2

Parameters are often highly correlated Constraints can help

The birth of the Rietveld methodThe 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, Uiso

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(-ax2)

Angular variation of peak width

G2 = U tan2q + V tanq + WCaglioti, Paoletti and Ricci, NIM 3, 223 (1958)

Lorentzian peak shapes Peaks from “high resolution” instruments

often have a strong Lorentzian contribution to their shape

20

22/

2/1)(

xxxL

0

5

10

15

20

25

30

35

10 10.2 10.4 10.6 10.8 11

2Theta

Angular variation of Lorentzian FWHM often described by:

tancos

YX

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

ZrP2O7 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, Rwp, Rp, RF, RI, RB

– Goodness of fit, c2

– Serial correlation indicators, DWd

Normal probability

Very valuable indication is visual quality of fit

Profile R factors can be misleading

Rwp = 7.8%c2 = 16.5DWd = 0.19

Rwp = 2.1%c2 = 1.17DWd = 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 BaBiO3

Ultrahigh resolution

FWHM ~0.02o in some cases at ~ Cu Ka

Ultrahigh peak to backgroundPeak 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 analysisLearn about crystallite size and strainDetermine texture in a material

Achievements Major contribution to almost every hot area of “hard”

materials in the last 15 years– High temperature superconductors– Buckyballs (C60)– Colossal Magnetoresistance– Thermoelectrics– Hydrogen storage– Batteries

Now making an inroad in biological science and organic materials– Drugs, polymers, proteins?

High Tc superconductors

Much of the solid state chemistry of these materials was worked using neutron diffraction and Rietveld refinement

C60 - Buckminsterfullerene

The structure of C60 and its metal doped variants have all been examined using the Rietveld method

Orientational ordering in C60

At high T, C60 is rotationally disordered, but at low temperatures the molecules order

Polymer electrolytesPowerful 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 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.