More on the Rietveld Method,

Selected Examples

John Faber

Scientist Emeritus, ICDD

faber@icdd.com

(610)325-9814

Objectives (Positive)

Introduction to practice Problem solving Real examples Obtainable information Pitfalls Structural parameters Global parameters

Objectives (Negative)

Basic crystallography Least-squares theory Diffraction theory Program tutorial (GSAS, Fullprof)

Estimates of Quality of Fit

Rw y obs y calc

w y obswp

i i i

i i

2

2

1 2/

R

y obs y calc

y obsp

i i

i

1/ 2

exp / i oiR N P w y

2 2Goodness of fit= / ( ) /( )wp e m o cR R w y y M P

The Rietveld Method is a refinement technique, not a structure solution method. A good starting model is required!

How good is good?

All atoms within 0.5-0.6 Å of their true positions, but�

x% of the total scattering power

How do you get the model?

Primary literature Inorganic Crystal Structure Database (ICSD) Metals Data File (CRYSTMET) Linus Pauling File (LPF) Cambridge Structural Database (CSD) (Protein Data Bank) (PDB) Crystal Data Identification File (CDIF) Powder Diffraction File (PDF) ab initio structure determination

What determines the intensities?(1) Structure Factors

Atomic positions Occupancies Atomic scattering factors Thermal/disorder parameters Lattice parameters Symmetry

What determines the intensities?(2) Global Parameters

Concentration Incident intensity Background Extinction Absorption

Preferred orientation Multiplicity Lp factor Profile function Diffractometer

parameters

To get accurate results, we must model all these quantities correctly (correlations)! The advantage of the Rietveld method is that it uses all the information in the powder pattern, and yields the most information.

WE NEED TO GET THE RIGHT MODEL

� Examining the nature of the errors in the difference plot can give insight into the source of the errors.

You can also see what worked well in the model.

GRAPHICS TOOLS ARE: INDISPENSIBLEINDISPENSIBLE!

Can use a wide range of data

The least squares �response surface� can be flat, and prone to false minima, so care is needed. It is also possible to ask too much from the data. We need to remember that we are fitting a model to data, and that our answers will only be as good as the model is appropriate.

Two steps

Preparation Refinement

In some programs (like GSAS) these two steps are separate,and in others, e.g. DBWS, FULLPROF, these are combinedinto a single operation.

The Sample

Need a powder, but� Random is best, but� Resolution � more is better, but can generate

size/strain by trying to get powder Phase purity An advantage of the Rietveld method is that ideal

samples are rare, and the method provides a way of dealing with real samples

The Instrument

Alignment/systematic errors (zero, shift, trns) Wavelength(s) Profile function

Data Collection

Compromises! Fixed step sizes (but new GSAS format) Wide 2 range 5-10 steps across FWHM of sharpest peaks Constant or variable counting time Step or continuous scan 5-10,000 counts for strongest peaks Programs assume fixed slits!

Background

Crucial to get right � affects integrated intensities (and thus the structure) � especially the displacement coefficients

Interacts with the profile function Peak tails Use a few parameters as possible Crystalline sample � slowly varying Background parameters are highly-correlated

PANalytical

Variable incident and scattered beam slit apertures

2 2 2

Other Sources

2 2 2

I(counts)

And if: 0.025 ( )

= ( I(counts)) (0.025 ( ))

Total Statistics OtherSources

Statistics

Total

I counts

I counts

Background Functions

Polynomials Chebyschev polynomials Cosine Fourier series Real space pair correlation function

(linear with �bumps�)

More-complex functions Linear interpolation between selected points Read from file

Ni powder

Panalytical Alpha-1

Accelerator Detector

Ni sponge

Scintag PAD-V

Energy Dispersive Detector

Profile Coefficients

Crucial to getting the right answers Structure/intensities/overlap/tails Valuable information in profile

coefficients

X-ray Profile Functions

Gaussian Lorentzian (Cauchy) Modified Lorentzians Split Pearson VIII (pseudo)Voigt Empirical �learned�

Stacking fault model � DIFFax, � Fundamental parameters

Instrument Profile Function

Some programs require one Helps interpret refined values Use a sample free of size and strain

broadening � SRM 660a (LaB6) orSRM1976 (corundum plate)

Hands-On to Define The Instrument Calibration (Sample Instrument)

Al2O3� Zero=-1.76(06) (centidegrees)� Lx=3.08(02)� Ly=2.14(04)� trans=-1.28(05)� Asym=-.92(08)� Shift=-3.43(01)� GP=1.78(03)

LaB6� Zero=-0.77(04) (centidegrees)� Lx=4.19(02)� Ly=0.19(04)� trans=0.0� Asym=2.12(02)� Shift=-0.48(05)� GP=1.15(03)

Instrument Zero (avg)=-1.26 (centidegrees)

Synchrotron Data from

Peter Stephens

Ellipical Mirror � PANalyticalX�Pert PRO MPD, 240 mm

Cylindrical Geometry -Capillaries

How do you know when you�re finished?

�A Rietveld refinement is never finished, only abandoned�

P. W. Stephens

Quality of Refinement (1)

R

I obs I calc

I obsF

K K

K

' '

' '

/ /

/

1 2 1 2

1 2

R

I obs I calc

I obsB

K K

K

' '

' '

R

y obs y calc

y obsp

i i

i

Rw y obs y calc

w y obswp

i i i

i i

2

2

1 2/

S S N P R Ry wp e / //1 2

R N P w ye i oi //1 2

Quality of Refinement (2)

Convergence Chemical reasonableness Distances and angles Displacement coefficients Bulk composition Graphical fit / and magnitude of

Mullite

There is no one measure of the quality of a Rietveld refinement!

Decreasing the Size of the Problem

Constraints thermal parameters by atom type occupancies Restraints = soft constraints bond distances, angles, torsions, chemistry,

chiral volume, planar group, / start strong, then relax Rigid Bodies

Damping � an aid to convergence

Apply only part of the calculated shift Scale factors and lattice parameters generally

well-behaved Background terms are highly-correlated (50%) Profile coefficients (50%) Atom coordinates (OK to 60%) Sometimes need to fix parameters (occupancy,

U, profiles) at physically-reasonable values

Rietveld analysis lets your extract the maximum information from your data, and thus make the maximum impact. It needs to be part of any X-ray analyst�s �arsenal� of tools.

AlphaAlpha--D GlucoseD Glucose((KadukKaduk, Faber and Needham), Faber and Needham)

Ground specimen mortar and pestle

Rietveldrefinement: preferred orientation seen:

Rwp=23%

X2=16

Glucose Monohydrate, CGlucose Monohydrate, C66HH1414OO77

Rietveld refinement:

preferred orientation seen:

Rwp=23%

X2=16

MicronizingMill employed to reduce PO, fine particles

TGA & DSCTGA & DSC

XRD unequivocal: Upon grinding, we see a complete transition of anhydrate to

monohydrate

LeBail

Like Rietveld without the constraints of the structure in determing the integrated intensities of the Bragg peaks; however, peak positions are constrained by lattice parametersand space group � Great for helping to define the profile

parameters� Separate structure and profiles� Indepensible for structure solution (Pawley)