Rietveld Analysis of X-ray and neutron diffraction patterns

1
Rietveld Analysis of X-ray and neutron
diffraction patterns

• Analysis of the whole diffraction pattern
• Profile fitting is included
• Not only the integrated intensities
• Refinement of the structure parameters from
  diffraction data
• Quantitative phase analysis
• Lattice parameters
• Atomic positions and occupancies
• Temperature vibrations
• Grain size and micro-strain (in the recent
  versions)
• Not intended for the structure solution
• The structure model must be known before starting
  the Rietveld refinement

2
Non-refinable parameters in the Rietveld method

• Space group
• Chemical composition
• Analytical function describing the shape of the
  diffraction profiles
• Wavelength of the radiation (can be refined in
  Fullprof or in LHRL suitable for the synchrotron
  data)
• Intensity ratio in Ka1, Ka2 doublet
• Origin of the polynomial function describing the
  background

3
Rietveld analysis

• History
• H.M. Rietveld - neutron data, fixed wavelength
• D.E. Cox - X-ray data
• R.B. Von Dreele - neutron data, TOF
• D.B. Wiles R.A. Young - X-ray data, 2
  wavelengths, more phases
• Helsinki group - spherical functions for
  preferred orientation but a single wavelength
• Fullprof, LHRL - surface absorption
• BGMN - automatic calculation, crystallite size
  and microstrain in form of ellipsoids
• P. Scardi et at - size, strain

• Computer programs
• H.M. Rietveld
• DBW2.9, DBW3.2 (Wiles Young)
• University of Helsinki
• Fullprof (J. Rodriguez-Carvajal)
• BGMN (R. Bergmann)
• LHRL (C.J. Howard B.A. Hunter)
• P. Scardi et al.
• Bärlocher
• GSAS

4
Integral intensity

• Calculated intensity
• G is the normalised profile function, I is
  the intensity of the k-th reflection. The
  summation is performed over all phases p, and
  over all reflections contributing to the
  respective point.
• The intensity of the Bragg reflections

5
Scattering by one elementary cell

• Structure factor
• Calculation is performed in the oblique axes (for
  the respective crystal system)

6
Temperature vibrations

• Atomic displacement (in Cartesian co-ordinates)

7
Crystal symmetry restrictions

• Six anisotropic temperature factors per atom in a
  general case (symmetrical matrix)
• For an atom in a site of special symmetry the
  B-matrix must be invariant to the symmetry
  operations (in the Cartesian axis system)
• An example - rotation axes parallel with z

8
Temperature vibrations - special cases

• Isotropic atomic vibrations
• Overall temperature factor

9
Scattering by one atom

• Atomic scattering factor
• a, b, c are from the International Tables for
  Crystallography
• Df, Df must be checked and changed for
  synchrotron radiation
• Another possibility
• Include our set of the atomic scattering factors

10
Preferred orientation of grains (texture)

• Gauss-like distribution
• March-Dollase correction
• Spherical functions

11
Absorption correction

• For flat samples - micro-absorption and surface
  absorption (Hermann Ermrich)
• Apparent decrease of the temperature factors or
  even negative temperature factors

12
Absorption correction

• For thin samples (powder on glass) in symmetrical
  arrangement
• thick sample, high absorption
• thin sample, low absorption

13
Extinction correction(for large crystallites)

• Extinction for the Bragg case (q 90)
• Extinction for the Laue case (q 0)

14
Profile functions

• Gauss
• Lorentz (Cauchy)
• Pearson VII
• Pseudo-Voigt

15
Background

• Subtraction of the background intensities
• Interpolation of the background intensities
• Polynomial function (six refinable parameters)
• Origin of the background - improves the pivoting
  of the normal matrix
• A special function for amorphous components

16
Minimisation routine

• Uses the Newton-Raphson algorithm to minimise the
  quantity
• Normal matrix

17
Reliability factors

• The profile R-factor
• The weighted Rp
• The Bragg R-factor
• The expected Rf
• The goodness of fit

18
Connecting parameters, constrains

• Young - parameter coupling
• Coding of variables number of the parameter in
  the normal matrix weight for the calculated
  increment
• Lattice parameters in the cubic system 41.00
  41.00 41.00
• Fractional co-ordinates at 12k in P63/mmc, (x 2x
  z) 20.50 21.00 31.00
• Fullprof - constrains
• Inter-atomic distances may be constrained
• BGMN - working with molecules
• Definition of the molecule (in Cartesian
  co-ordinates)
• Translation and rotation of the whole molecule

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Structure of the input file(Fullprof for
anglesite)
COMM PbSO4 D1A(ILL),Rietveld Round Robin, R.J.
Hill,JApC 25,589(1992) !Job Npr Nph Nba Nex
Nsc Nor Dum Iwg Ilo Ias Res Ste Nre Cry Uni Cor
1 7 1 0 2 0 0 0 0 0 0 0
0 0 0 0 0 ! !Ipr Ppl Ioc Mat Pcr Ls1 Ls2
Ls3 Syo Prf Ins Rpa Sym Hkl Fou Sho Ana 0 0
1 0 1 0 0 0 0 1 6 1 1 0
0 1 1 ! ! lambda1 Lambda2 Ratio Bkpos
Wdt Cthm muR AsyLim Rpolarz 1.54056
1.54430 0.5000 70.0000 6.0000 1.0000 0.0000
160.00 0.0000 !NCY Eps R_at R_an R_pr R_gl
Thmin Step Thmax PSD Sent0 5
0.10 1.00 1.00 1.00 1.00 10.0000 0.0500
155.4500 0.000 0.000 ! ! Excluded regions
(LowT HighT) 0.00 10.00 154.00
180.00 ! 34 !Number of refined
parameters ! ! Zero Code Sycos Code
Sysin Code Lambda Code MORE -0.0805
81.00 0.0000 0.00 0.0000 0.00 0.000000
0.00 0 ! Background coefficients/codes
207.37 39.798 65.624 -31.638
-90.077 47.978 21.000 31.000
41.000 51.000 61.000 71.000
! Data for PHASE number 1 gt Current
R_Bragg 4.16 PbSO4
!Nat Dis
Mom Pr1 Pr2 Pr3 Jbt Irf Isy Str Furth ATZ
Nvk Npr More 5 0 0 0.0 0.0 0.0 0 0 0
0 0 0.00 0 7 0 P n m a
lt-- Space group symbol !Atom Typ X
Y Z Biso Occ /Line
belowCodes Pb PB 0.18748 0.25000
0.16721 1.40433 0.50000 0 0 0
171.00 0.00 181.00 281.00 0.00 S
S 0.06544 0.25000 0.68326 0.41383
0.50000 0 0 0 191.00 0.00
201.00 291.00 0.00 O1 O 0.90775
0.25000 0.59527 1.97333 0.50000 0 0 0
211.00 0.00 221.00 301.00
0.00 O2 O 0.19377 0.25000 0.54326
1.48108 0.50000 0 0 0 231.00
0.00 241.00 311.00 0.00 O3 O
0.08102 0.02713 0.80900 1.31875 1.00000 0
0 0 251.00 261.00 271.00
321.00 0.00 ! Scale Shape1 Bov
Str1 Str2 Str3 Strain-Model 1.4748
0.0000 0.0000 0.0000 0.0000 0.0000
0 11.00000 0.00 0.00 0.00
0.00 0.00 ! U V W
X Y GauSiz LorSiz Size-Model
0.15485 -0.46285 0.42391 0.00000 0.08979
0.00000 0.00000 0 121.00 131.00
141.00 0.00 151.00 0.00 0.00 !
a b c alpha beta
gamma 8.480125 5.397597 6.959482
90.000000 90.000000 90.000000 91.00000
101.00000 111.00000 0.00000 0.00000
0.00000 ! Pref1 Pref2 Asy1 Asy2 Asy3
Asy4 0.00000 0.00000 0.28133 0.03679-0.09981
0.00000 0.00 0.00 161.00 331.00 341.00
0.00
20
Quantitative phase analysis

• Volume fraction
• Weight fraction
• Use the correct occupancies N occupancy / max
  of Wyckoff positions

21
Tips and tricks (on the course of the refinement)

• Instrumental parameters
• Scale factor (always)
• Background (1)
• Line broadening and shape (3)
• Zero shift (4)
• Sample displacement or transparency (5)
• Preferred orientation (7)
• Surface absorption (7)
• Extinction (7)

• Structure parameters
• Scale factor (always)
• Lattice parameter (2)
• Atomic co-ordinates (6)
• Temperature factors (8)
• Occupancies (8), N occ/max(N) important for
  quantitative phase analysis

22
Tips and tricks (how to obtain reliable data)

• Use only good adjusted diffractometer
• Bad adjustment causes the line shift and
  broadening the latter cannot be corrected in the
  Rietveld programs
• Use only fine powders
• Coarse powder randomises the integral
  intensities
• Coarse powder causes problems with rough surface
• Use sufficient counting time
• The error in intensity is proportional to sqrt(N)
  as for the Poisson distribution
• Apply dead-time correction
• For strong diffraction lines, the use of the
  dead-time correction is strongly recommended

23
Effect of the grain size

• Variations in observed intensities (bad
  statistics)
• Figure Effect of specimen rotation and
  particle size on Si powder intensity using
  conventional diffractometer and CuKa radiation.
• International Tables for
• Crystallography, Vol. C,
• ed. A.J.C. Wilson,
• Kluwer Academic Publishers, 1992.

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In the Rietveld refinement dont

• refine parameters which are fixed by the
  structure relations (fractional co-ordinates,
  lattice parameters)
• refine all three parameters describing the line
  broadening concurrently
• refine the anisotropic temperature factors from
  X-ray powder diffraction data
• use diffraction patterns measured in a narrow
  range
• forget that the number of structure parameters
  being refined cannot be larger than the number of
  lines

25
Corundum
26
Auxiliary methods and computer programs

• The most critical parameters for the convergence
  of the Rietveld refinement - lattice parameters
• FIRESTAR
• Only the crystal system must be known (not the
  space group)
• The diffraction pattern must be indexed

27
Problems with positions of diffraction lines

• Residual stresses in bulk materials
• Anisotropic deformation of crystallites
  (anisotropy of mechanical properties)
• Presence of errors in the structure (stacking
  faults, )
• Use of the programs working with net integral
  intensities (POWOW, POWLS) is recommended
• How to get the net intensities?
• Numerical integration (not for the overlapped
  lines)
• Profile fitting using analytical functions (for
  overlapped lines) - DIFPATAN

28
Indexing of the diffraction patternin unknown
phases

• Computer program TREOR (Trials and Errors)
• Requirements
• A single phase in the specimen
• High-quality data (particularly, the error in the
  positions of diffraction lines must not exceed
  0.02 in 2q)
• Very good alignment of the diffractometer or the
  use of an internal standard (mixed to the
  specimen)