Basics of Rietveld Refinement
Scott A Speakman
13-4009A
x3-6887
speakman@mit.edu
Uses of the Rietveld Method
• The Rietveld method refines user-selected parameters to
minimize the difference between an experimental pattern
(observed data) and a model based on the hypothesized crystal
structure and instrumental parameters (calculated pattern)
• can refine information about a single crystal structure
– confirm/disprove a hypothetical crystal structure
– refine lattice parameters
– refine atomic positions, fractional occupancy, and thermal parameter
• refine information about a single sample
– preferred orientation
• refine information about a multiphase sample
– determine the relative amounts of each phase
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Requirements of Rietveld Method
• High quality experimental diffraction pattern
• a structure model that makes physical and chemical sense
• suitable peak and background functions
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Obtaining High Quality Data
• issues to consider
– aligned and calibrated instrument
– beam overflow problems
– thin specimen error
– good counting statistics
– appropriate step size
– sample transparency
– surface roughness
– preferred orientation
– particle size
• go to XRD Basics pg 102
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Describing the Crystal Structure
• space group
• lattice parameters
• atomic positions
• atomic site occupancies
• atomic thermal parameters
– isotropic or anisotropic
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The Crystal Structure of LaB6
• LaB6
• Space Group Pm-3m (221)
• Lattice Parameter a=4.1527 A
Atom
Wyckoff x
Site
y
z
B
occ.
La
1a
0
0
0
0.00157
1
B
6f
0.1993
0.5
0.5
0.0027
1
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Where to get crystal structure information
• check if the structure is already solved
– websites
• Inorganic Crystal Structure Database (ICSD) http://icsd.ill.fr/icsd/index.html
4% is available for free online as a demo
• Crystallography Open Database http://www.crystallography.net/
• Mincryst http://database.iem.ac.ru/mincryst/index.php
• American Mineralogist
http://www.minsocam.org/MSA/Crystal_Database.html
• WebMineral http://www.webmineral.com/
– databases
• PDF4 from the ICDD
• Linus Pauling File from ASM International
• Cambridge Structure Database
– literature
• use the PDF to search ICSD listings and follow the references
• look for similar, hopefully isostructural, materials
• index the cell, and then try direct methods or ab-initio solutions
– beyond the scope of today’s class
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Instrumental Parameters
• background
• peak profile parameters
– cagliotti parameters u, v, w
– pseudo-voigt or other profile parameters
– asymmetry correction
– anisotropic broadening
• error correcting parameters
– zero shift
– specimen displacement
– absorption
– extinction
– roughness
– porosity
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How many parameters can we refine?
• Each diffraction peak acts as an observation
– theoretically, refine n-1 parameters
• refining a tetragonal LaNi4.85Sn0.15 crystal structure, there might be:
– scale factor
– 2nd order polynomial background: 3 parameters
– 2 lattice parameters
– no atomic positions (all atoms are fixed)
– 3 or 5 thermal parameters
– 2 or 4 occupancy factors
– zero shift and specimen displacement
– 5 profile shape parameters
• 22 parameters maximum with 43 peaks (20 to 120 deg 2theta)
– does this mean we can refine all parameters?
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background functions
• manually fit background
• polynomial
• chebyshev
• shifte chebyshev
• amorphous sinc function
• many others for different programs
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profile functions
• vary significantly with programs
• almost all programs use Cagglioti U, V, and W
H 2  W  V tan   U tan 2 
• HSP uses pseudo-voigt, Pearson VII, Voigt, or pseudo-voigt 3 (FJC
asymmetry)
• GSAS uses functions derived more from neutron and synchrotron
beamlines
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• go to parameters_calc_pattern.pdf
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How do you know if a fit is good?
• difference pattern
• Residuals R
– R is the quantity that is minimized
during least-squares or other
fitting procedures
Rwp   wi Yio  Yic 
2
i
– Rwp is weighted to emphasize
intense peaks over background
– Rexp estimates the best value R
for a data set
• an evaluation of how good the
data are
– RBragg tries to modify the R for a
specific phase
• GOF (aka X2)
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Refinement Strategy
• Rietveld methods fit a multivarialbe structure-backgroundprofile model to experimental data
– lots of potential for false minima, diverging solutions, etc
• need to refine the most important variables first, then add more
until an adequate solution is realized
– a correct solution may not result …
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Ray Young’s Refinement Strategy
•
•
•
•
•
•
•
•
•
•
•
scale factor
zero shift or specimen displacement (not both)
linear background
lattice parameters
more background
peak width, w
atom positions
preferred orientation
isotropic temperature factor B
u, v, and other profile parameters
anisotropic temperature factors
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HSP Automatic Refinement Strategy
• Very similar to Prof Young’s recommendations
• a good choice for beginners
• you can set limits on any of these parameters
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Additional Files
• XRD_Basics_HSP_2006.pdf
– large collection of information about X-ray diffraction, instrumentation,
and different techniques
• X’Pert HighScore Plus Tutorial.pdf
– overview of the different functionality available in HighScore Plus
• Introduction.pdf
– overview of Rietveld
• parameters_calc_patterns.pdf
– overview of parameters involved in calculating a diffraction pattern
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further reading
• “Rietveld refinement guidelines”, J. Appl.Cryst. 32 (1999) 36-50
• R.A. Young (ed), The Rietveld Method, IUCr 1993
• V.K. Pecharsky and P.Y. Zavalij, Fundamentals of Powder
Diffraction and Structural Characterization of Materials,
Kluwer Academic 2003.
• DL Bish and JE Post (eds), Modern Powder Diffraction, Reviews
in Mineralogy vol 20, Min. Soc. Amer. 1989.
• CCP14 website http://www.ccp14.ac.uk/tutorial/tutorial.htm
• prism.mit.edu/xray/resources.htm
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Rietveld Programs
• Free
–
–
–
–
–
–
GSAS + ExpGUI
Fullprof
Rietica
PSSP (polymers)
Maud (not very good)
PowderCell (mostly for calculating patterns and transforming crystal
structures, limited refinement)
• Commercial
– PANalytical HighScore Plus
– Bruker TOPAS (also an academic)
– MDI Jade or Ruby
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Examples
• Silicon
• LaB6
• intermetallic LaNi4.85Sn0.15
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Silicon
• Open the datafile in HSP
• Add the structure model
– insert the structure manually
– import (insert) a struture file
• usually use the CIF format– the ubiquitous standard for crystal
structures
• HSP can also import ICSD *.cry files and structures from other
refinement programs
• GSAS can import CIF or PowderCell files
• try the automatic refinement
• manually improve the fit
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Silicon Crystal Structure
• Fd3m
– which setting? (2)
• a=5.43 A
• Si at 0.125, 0.125, 0.125
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Lanthanum hexaboride LaB6
• Open the datafile
• insert the crystal
structure CIF file
• Note that boron
(z=5) makes little
difference in the
XRD pattern
compared to the
lanthanum (z=57)
• what can we do to
improve the fit
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LaNi4.85Sn0.15
• The data was taken from Chapter 6 of Fundamentals of Powder
Diffraction and Structural Characterization of Materials, by
Pecharsky and Zavalij
• The structure is a bit more complex that our earlier example,
which allows us to explore more features of HighScore Plus
• The data (Ch6_1.raw) is in GSAS format, which can be read into
HighScore Plus
• I have also included a CIF file from the ICSD (#104685) with all
the main features of the structure described
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Issue to Consider
• How can I work without knowledge of the structure?
– Use LeBail or Pawley method to determine lattice parameters
– Try indexing and solving the structure using the HighScore Plus tools
– You will find that there are 16 possible space groups for this material, but
picking the most common (and simplest) choice, P6/mmm, is the right way to go
• Where do I put the atoms?
– You can use a Fourier map to find out wherein the structure the electron densities
are greatest. Put the heaviest atoms (La) at these sites, then work your way
through the chemistry
• What variables do I refine and in what sequence?
– Take a look at the “automatic” option in HSP - this is not a bad strategy to use.
We will go through these in detail…
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