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What is Rietveld
Refinement Method?
Thierry Roisnel
Laboratoire de Chimie du Solide et Inorganique
Moléculaire
UMR6511 CNRS– Université de Rennes 1 (France)
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
A powder diffraction pattern from the
data treatment point of view
A powder diffraction pattern can be recorded in numerical
form for a discrete set of scattering angles, times of flight or
energies. We will refer to this scattering variable as : T.
The experimental powder diffraction pattern is usually given as
two arrays :
{Ti , yi }i =1,...,n
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Powder diffraction profile:
scattering variable T: 2θ, TOF, Energy
Bragg position Th
yi
yi-yci
zero
Position “i”: Ti
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
1. The profile of powder diffraction
patterns
profile
The profile can be modelled using the calculated counts:
yci at the ith step by summing the contribution from
neighbouring Bragg reflections plus the background.
The model to calculate a powder diffraction pattern is
(for a single phase):
yci = bi +
h2
∑ I Ω(T − T )
h = h1
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
h
i
h
The profile of powder diffraction
patterns
yci = bi + ∑ I h Ω(Ti − Th )
h
∫
+∞
−∞
Ω( x)dx = 1
Profile function characterized by its
full width at half maximum (FWHM=H)
and shape parameters (η, m, ...)
Ω( x) = g ( x) ⊗ f ( x) = instrumental ⊗ intrinsic profile
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
The profile of powder diffraction
patterns
yc i = ∑ I h Ω(Ti − Th ) + bi
h
Ih = Ih ( β I )
Ω = Ω( xhi , β P )
bi = bi ( β B )
Contains structural information:
atom positions, magnetic moments, etc
Contains instrumental (spectral dispersion,
resolution parameters …) and sample
informations (defects,crystallite size, …)
Background: noise, diffuse scattering, ...
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
1.1 Integrated intensities
I h = { jL p O AC S
2
}
h
Integrated intensities are proportional to the
square of the structure factor S (cryst. and
magn. structure factors contributions).
The factors are:
Multiplicity (j), Lorentz-polarization (Lp),
preferred orientation (O), absorption (A),
other “corrections” (C) ...
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Crystallographic structure factor FH
F =
C
H
unit cell
∑
i =1
r r
N i . f i .exp  − Bi . ( sin θ λ )  .exp(2iπ H .ri )


2
H:
scattering vector
Ni:
% site occupation factor
fi:
X-rays:
Neutrons:
Bi:
Debye-Waller displacement factor
ri :
coordinates of the atom i in the unit cell
atomic scattering factor
coherent scattering length
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Magnetic structure factor FM
general formula of Halpern and Johnson:
r r 2 r r 2 r r r
F = F⊥ ( h ) = Fm ( h ) − e .Fm ( h )
(
2
r
h
r r
Fm (h )
r
e
magnetic structure factor
unit vector along scattering vector
r
h
scattering vector
r
k
propagation vector
r r r
h=H +k
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
)
2
Magnetic structure
Magnetic structure refined by FullProf must have a distribution of
magnetic moments than can be expanded as a Fourier series:
r r
µl , j = ∑ Skr , j .exp( −2iπ .k.Rl )
r
k
{}
In such a case, the magnetic structure factor is given by:
{ (
) }
nc
r r r
r r
r r r
Fm ( H + k ) = p.∑ f j ( H + k ).Skr , j .exp 2iπ H + k .rj
j =1
p
r r
fj ( H + k )
r
rj
Conversion constant
Magnetic form factor
Vector position of atom j
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
1.2 The profile function Ω(x) for cwl data
¾
analytical function to describe the observed diffraction
profile. Ω(x) is defined to model:
. each single Bragg line profile
¾ Full Width at Half Maximum: H (in deg. for cwl data)
¾ Integral breath β or β* (β* = β2θ.cosθ/λ)
¾ Profil parameter (form factor) φ = H/β
. angular dependence of the profile parameters
¾
The choice of Ω(x) should obey to some criteria for
structural and/or microstructural analysis:
. Fit the observed data
. Possible deconvolution (microstructural study)
. Refined parameters are physically meaningfull.
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Ω(x) modelisation: 1- the Gauss function
G( x ) = aG .exp( −bG .x 2 )
aG =
2 ln(2)
.
H
π
4. ln(2)
bG =
H2
βG =
φG =
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
1 H
π
= .
aG 2 ln(2)
HG
βG
= 2.
ln(2)
π
= 0.9394
Ω(x) modelisation: 2- the Lorentz function
aL
L( x ) =
1 + bL .x 2
2
aL =
πH
4
bL = 2
H
1 πH
=
2
aL
HL 2
φL =
= = 0.6366
βL π
βL =
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Ω(x) modelisation: combination of Gauss and Lorentz
¾ pseudo-Voigt function:
. Linear combination of Gauss and Lorentz
pV ( x ) =η.L( x ) + (1 −η ).G( x ) (0. ≤ η ≤ 1.0)
PV ( x ) = PV ( x , η, H )
Normalized pseudo-Voigt function:
β pV
H
2
=
η + (1 − η ). π .ln(2)
π.
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Ω(x) modelisation: combination of Gauss and Lorentz
¾ Voigt function:
. Convolution of Gauss and Lorentz
V ( x, H G , H L ) =V ( x, β G , β L ) = G ( x, H G ) ⊗ L( x, H L )
Complex function: approximation of Voigt by a pseudo-Voigt function
[Thompson, Cox, Hastings, J. Appl. Cryst. (1987), 20,79-83)]
( H ,η )  F ( H G , H L )
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Ω(x) angular dependence (for cwl data)
Caglioti, Paoletti, Ricci formula (1958, neutron case):
FWHM 2 = U tan 2 θ + V tan θ + W
U,V,W = f(αi, βM)
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Ω(x) angular dependence (for cwl data)
¾ Gauss function:
(
)
H 2 = U + Dst2 tg 2θ + Vtgθ + W +
¾ Lorentz function
(
)
H 2 = U + Dst2 tg 2θ + Vtgθ + W +
¾ pseudo-Voigt function
η = η0 + X 2θ
¾ Voigt function
IG
cos2 θ
(
IG
cos2 θ
)
H G2 = U + U st + (1 − ξ )Dst2 tg 2θ + Vtgθ + W +
H L = ( X + X st + ξ .Dst
Y + F (Sz )]
[
).tgθ +
IG
cos2 θ
cosθ
Instrumental parameters (beam divergency, spectral distribution, …)
Sample dependent parameters: strain, size
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
1.3 Background bi
¾
TDS (thermal diffuse scattering)
¾
incoherent scattering
¾
sample environment
¾
…
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
2. Rietveld method
The Rietveld Method consist of refining a crystal (and/or magnetic)
structure by minimising the weighted squared difference between
the observed and the calculated pattern against the parameter
vector: β
n
χ = ∑ wi { yi − yci ( β )}
2
i =1
σ
2
i:
n:
2
wi = σ12
i
is the variance of the "observation" yi
number of points in the pattern
The Rietveld method is a structure refinement method and not a structure
determination one. Structural parameters (space group, atomic positions,
magnetic configuration …) has to be known as well as possible.
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Some points of the L.S. refinement
χ
2
 ∂χ 2 
=0


 ∂α α =αopt
mini
Taylor expansion of
shifts
δα 0
yic (α )
around initial α 0
obtained by solving linear system of equations (normal equations):
Aδα 0 = b
A
b
p x p matrix (p free parameters)
Akl =
∑w .
i
i
vector (dimension p)
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
bk =
∂yic (α0 ) ∂yic (α0 )
.
∂α k
∂αl
 ∂yic (α 0 ) 
−
w
.
y
y
.
(
)
∑i i i ic  ∂α 
k


Some points of the L.S. refinement
Standard deviations of
αk :
σ (α k ) = ak
χν =
2
χ2
n− p
−1 2
Akk
χν
(reduced χ2)
ak coefficient of the codeword of the parameter αk
p parameters to refine
n points in the pattern having Bragg contributions
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Some points of the L.S. refinement
Cycle #1:
α1 = α 0 + δα 0
New α parameters are considered as starting ones in the next cycle and
the process is repeated until a convergence criterion is satisfied.
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Two sets of refinable parameters (1)
(1) Profile parameters
* peaks position: A, B, C, D, E, F, zeroshift
1/d*2 = A.h2 + B.k2 + C.l2 + D.kl + E.hl + F.hk
* profile: U, V, W, X, Y …
* asymetry P
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Two sets of refinable parameters (2)
(2) Structural parameters
* scale factor
* overal isotropic temperature parameter
* xi, yi, zi: atomic coordinates
* Bi: atomic temperature parameter
* Ki: components of the magnetic moment vector
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
3. Profile R-factors used in
Rietveld Refinements
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
R-factors and Rietveld Refinements (1)
∑ y −y
= 100
∑y
obs ,i
Rp
calc ,i
i
R-pattern
obs ,i
i
Rwp
 w y −y
i
obs ,i
calc ,i
∑
= 100  i
2
wi yobs ,i
∑

i

Rexp


(N − P + C) 
= 100 
 ∑ w y2 
 i i obs ,i 
2
1/ 2





R-weighted
pattern
1/ 2
Expected R-weighted
pattern
N: number of points in the pattern
P: number of refined parameters
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
C: number of constraints
R-factors and Rietveld Refinements
(2)
 Rwp 
χν = 

 Rexp 
2
S=
Rwp
Rexp
2
Reduced Chi-square
Goodness of Fit indicator
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
R-factors and Rietveld Refinements (3)
Two important things:
• The sums over “i” may be extended only to the regions
where Bragg reflections contribute
• The denominators in RP and RWP may or not contain the
background contribution
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Crystallographic R-factors used
in Rietveld Refinements
∑ ' I '− I
= 100
∑ 'I '
obs , h
RB
calc , h
h
Bragg R-factor
obs , h
h
∑ ' F '− F
= 100
∑ 'F '
obs , h
RF
calc , h
h
obs , h
h
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Crystallographic RF-factor.
Calculation of the “observed”
integrated intensities
 Ω(Ti − Th )( yobs ,i − Bi )  Provides ‘observed’
' I obs ,h ' = I calc ,h ∑ 
 integrates intensities for
( ycalc ,i − Bi )
i 
 calculating Bragg R-factor

' Fobs ,h ' =
' I obs ,h '
jLp
In some programs the crystallographic RFfactor is calculated using just the square
root of ‘Iobs,h’
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Important points
(1) The absolute value of the profile R-factors has little
significance because their values depend on the quality of the
data as well as on the goodness of the structural model. The Rfactors obtained by a refinement of the whole pattern without
structural model provide the “expected” values for the best
structural model.
(2) The most important assessments of the quality of the final
refinement are:
• the absence of important disagreement in the plot of
the observed versus calculated patterns (detailed
examination of the “Rietveld plot”: Yobs, Ycalc, Yobs-Ycalc)
• the chemical significance of the final structural model
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
(3) All these reliability factors are not satisfactory from a statistical
point of view. Number of more significant parameters are calculated
in FullProf:
o deviance (A. Antoniadis et al. Acta Cryst A46, 692-1990)


 y 
D = 2∑  yi ln  i  − ( yi − yci ) 
i 
 yc ,i 


o from D, one can derive two others measures of discrepancy which are
useful as model selection criteria. They take account of both the g.o.f.
of a model and the number of parameters used to achieve that fit.
Q = D + a. p
p number of parameters to refine
a represents the coast of fitting an additionnal parameter
(Akaike: a=2; Schwartz: a=ln(p))
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Q can be compared to Hamilton criteria used in single crystal
crystallography, to use the opportunity to add a refinable
parameter.
p= p +1 Î χ2 ↓
.
If QA ↓ and QS ↓:
. If QA ↑ and QS ↑ :
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
valid parameter
not valid parameter
o Durbin-Watson statistic parameters: d and QD
Measure of the correlation between adjacent residuals
(serial correlation)
n
d=
∑
i =2
{
(
w y − y c
i
 i i
)
2
(
∑{ (
wi yi − yic
i
)
}
2
− wi −1 yi −1 − yic−1 

)}
2
 n − 1 3.0901 
QD = 2 
−

n+2 
n − p
¾ d < QD :
positive serial correlation: sussessive value of the residuals
tends to have the same sign (most common situation in profile
refinement)
¾ QD < d < 4-QD :
no correlation
¾d > 4-QD :
negative serial correlation: sussessive value of the residuals
tends to have the opposite sign
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
4. Some References on R.M.
¾ Rietveld H.M., A Profile Refinement Method for Nuclear
and Magnetic Structures, J. Appl. Cryst 2, 65 (1969)
¾ The Rietveld Method, edited by R.A. Young, IUCr, Oxford
Science Publications (1996)
¾ A.W. Hewatt, High Resolution Neutron and Synchrotron
Powder Diffraction, Chemica Scripta 26A, 119 (1985)
¾ A. Albinatti and B.T.M. Willis, The Rietveld Method in
Neutron and X-Ray Powder Diffraction, J. Appl. Cryst. 15,
361 (1982)
¾ A.K. Cheetham and J.C. Taylor, Profile Analysis of Powder
Diffraction Data: Its Scope, Limitations and Applications in
Solid State Chemistry, J. Solid State Chem. 21, 253 (1977)
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
Some References on R.M.
¾ L.B. McCusler, R.B. Von Dreele, D.E. Cox, D. Louër and P. Scardi
Rietveld refinement guidelines
J. Appl. Cryst. (1999) 32, 36-50
¾E. Jansen, W. Schäfer and G. Will
R values in analysis of powder diffraction data using Rietveld refinement
J. Appl. Cryst. (1994) 27, 492-496
¾A. Antoniadis, J. Bernoyer and A. Filhol
·
Maximum-likelihood methods in powder diffraction
refinements
Acta Cryst A46, 692 (1990)
¾R.J. Hill and H.D. Flack
·
The use of the Durbin-Watson d statistic in Rietveld analysis
J. Appl. Cryst. 20, 356 (1987)
The Rietveld Method, T.R. (Mexico 21-25 June 2004)
The R.M. on the Web
¾ http://home.wxs.nl/~rietv025/
* mailing list
* R.M. FAQ’S
* seminal papers
*…
¾ http://www.ccp14.ac.uk
Collaborative Computationnal Project in Single Crystal and Powder Diffraction
* Tutorials
* Software download
*…
¾ http://www-llb.cea.fr/fullweb/fp2k/fp2k.htm
FullProf web site:
* FullProf manual
* Tutorials
* examples
* ...
The Rietveld Method, T.R. (Mexico 21-25 June 2004)