máster en cristalografía y cristalización sevilla,...
Post on 20-Sep-2018
228 Views
Preview:
TRANSCRIPT
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Fundamentals of the Rietveld method
Máster en Cristalografía y cristalizaciónSevilla, 12 de Diciembre de 2012
Vicente Esteve Cano
Dpto. de Química Inorgánica y Orgánica
Universitat Jaume I de Castellón
1
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
1.- Rietveld method: principles
2.- Rietveld method: overall parameters
3.- Rietveld method: atomic parameters
hhhh Laboratory X-ray (synchrotron and neutrons)
2
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...3
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
1.- Rietveld method: principles
2.- Rietveld method: overall parameter
3.- Rietveld method: atomic parameters
4
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Experimental pattern
data: y(2ΘΘΘΘ) matrix
5
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...Hugo Rietveld’s great idea : Ihkl vs yi (2ΘΘΘΘ)
Experimental pattern: crossesCalculated pattern: continuous lineBottom: difference curve
6
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Rietveld method: refinement by least-square using
the function: Sy=ΣΣΣΣ i (yoi -y
ci)2
yoi are the experimental intensities
yci are the calculated intensities: (a) using approximated
structure factors (Fhkl ); or (b) estimating the structure factors
without prior knowledge (Le Bail`s method).7
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Meaning:
b(2θθθθi) background
Sα α α α scale factor of the pattern
mk multiplicity of k-reflection
Fk structure factor of k-reflection
h(2θθθθi-2θθθθk) function to distribute the intensities over 2ΘΘΘΘ range
Lp(2θθθθi) Lorentz-polarization correction
Pk other corrections: prefer orientation, absorption, extinction,
…
yic = b(2θθθθi) + Sα α α α ΣΣΣΣ k mk|Fk |
2 h(2θθθθi-2θθθθk) Lp(2θθθθi) Pk
yic = b(2θθθθi) + ΣΣΣΣnSn ΣΣΣΣ k mk|Fk |
2 h(2θθθθi-2θθθθk) Lp(2θθθθi) Pk8
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Essentially ΙΙΙΙhkl ∝∝∝∝ F2hkl
Summation extended over all atoms within the unit cellPositionsThermal/atomic displacement parameters (ADP)Occupation factors
(a) The diffraction intensities depend upon the structure.When we know a related structure, a set of (approximate)structure factors can be calculated
Fhkl = fºn e−Bsen2θ /λ2
e
(2π i (hxn + kyn +lzn)[ ]
n
∑
9
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Structure factors are extracted by an iterative approach:Fhkl are treated as variables (to be optimized) instead tobe calculated from an approximate structure.
Peak overlapping is a much severe problem in thisapproach
[ ] [ ]∑ ++−=n
)nl nyk n(h2(2/2sennhkl e e nfº F zxBoc iπλθ
(b) No prior knowledge of the diffraction intensities
10
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Le Bail’s method(“Whole profile matching” o “pattern matching”)
It does not use structural information
Ik(0) is (initially) a crude estimation
Ik= 0 (warning) is a fix point in the second equation
Recommendable for:
• To obtain a set of structure factors for ab-initio structure determination (also)
• Lack of precise information about the shape of the profile
• Large sample contribution to the profile
• Very crude initial structural model11
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Observation numbers
uuuu In the Rietveld method, the observations are the
set of measured intensities: yoi (2Θ)
uuuu However, useful information is only the set of
diffraction intensities : Ihkl , and it is needed at least 4-5 “good values” for each refined atomic
parameter (about 10 in SCD): not easy.
uuuu If there are not enough ‘good observations’ in order
to refine all atomic parameters; there are two choices:
(a) To implement (hard / soft) constraints; and/or
(b) To enlarge the number of observations.12
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Observation vs. number of refined parameters
uuuu The number of refined parameters can be decreased
using restrictions (Biso) or rigid body (phenyl groups)
uuuu The number of ‘good’ observations can be
increased by adding a new data set (like NPD, expensive)
uuuu The number of observations can be increased by
adding a new data set (soft constraints) which are
geometrical/chemical observations with a weight in the
refinement that can be optimized
(based on chemical knowledge and data base values)
SR= Sy + cwSGwhere SG=ΣΣΣΣ i w(Go -Gc)2
e.g. Go ≡≡≡≡ P-O 1.52(2)Å or Si-O 1.63(3)Å13
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Disagreement factors in the Rietveld methodThere are several indicators of the agreement between
observed and calculated patterns.
The best is the (shape) of the difference curve (must be flat)
Must be as low as possible!
RWP = 100 ×××× √√√√ (ΣΣΣΣ i w|yoi -y
ci |2 / ΣΣΣΣ i w|y
oi |2)
RP = 100 ×××× ΣΣΣΣ i |yoi -y
ci | / ΣΣΣΣ i |y
oi |
RB (≡≡≡≡ RI) = 100 ×××× ΣΣΣΣ k |Iok -I
ck | / ΣΣΣΣ k |I
ok |
RF = 100 ×××× ΣΣΣΣ k |Fok -F
ck | / ΣΣΣΣ k |F
ok |
REXP = 100 ×××× √√√√ ((N-P+C) / ΣΣΣΣ i w|yoi |2)
χχχχ2 = RWP / REXP
14
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Minimization and correlation matrix
uuuu Standard analysis by least-squares with a non-linearresidue function. The equations are solved by invertingthe normal matrix.
uuuu The results are not found in just one step but using an iterative method which calculates the shifts of each parameter to be refined/optimized:
∑ ∂∂=∆ −
k
1jkk
x
Sy x M
uuuu A damping factor may be applied
uuuu The correlation matrix allows to see the parameter relationships15
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Precision, accuracy and standard deviations
uuuu The calculation of the standard deviation of the n-variable
CPN
SyM1nn
+−=
−
nσ
uuuu This calculation assumes that the ‘unique’ source of errorsis the data statistics (there are several other sources: improperfitting of the peak shape, non-random particle distribution, …)
uuuu To evaluate the accuracy of the result we must know the finalvalue of the refined parameter (in our case the ‘truth’ is the valueobtained in single crystal studies where is available).
To be in agreement the ‘Rietveld derived’ σσσσ’s must be ×××× ~3.16
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
1.- Rietveld method: principles
2.- Rietveld method: overall parameters
3.- Rietveld method: atomic parameters
yic = b(2θθθθi) + Sαααα ΣΣΣΣ k mk|Fk |
2 h(2θθθθi-2θθθθk) Lp(2θθθθi) Pk
17
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Overall parameters
They affect the whole pattern
- Scale factor (several phases ⇒⇒⇒⇒ several scale factors)
- Coefficients (or functions) to describe the background
- Unit cell parameters
- Goniometer-zero (and sample height/shift)
- Peak shape functions and parameters
- Polarization factor
- Preferred orientation correction
18
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Scale factor(s)
uuuu The optimization of the scale factor does not
usually show correlation unless the occupation factor of
a heavy cation site. Bi2-xFexO3
u For mixtures of crystalline phases, it can be obtained the
amounts of each crystalline phase and even the overall
amorphous phase (indirectly, by addition of a suitable
crystalline standard).
m(ZMV)
WK
mV
WKS e
2e µµρ α
α
αα
αα = =
∑=
= n
1i
ii (ZMV)S
(ZMV) S W
ααα
19
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Main requisite to carry out a quantitative phase analysis
with the Rietveld method (and XRPD data):
* Crystal structures must be well known (this is the
calibration). It is not needed for special analysis
methodologies.
The composition of a rock may be determined;
(there is no need of internal standard nor calibration curve)
Phase analysis
20
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Al2O3
CaF2
ZnO
Three phases sample: ~ 33 wt% ZnO, 33 wt% CaF2 & 33 wt% Al2O321
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Overall parameters
They affect the whole pattern
- Scale factor (several phases ⇒⇒⇒⇒ several scale factors)
- Coefficients (and functions) to describe the background
- Unit cell parameters
- Goniometer-zero (and sample height/shift)
- Peak shape functions and parameters
- Polarization factor
- Preferred orientation correction22
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
To correctly describe the background is key to attain
a good Rietveld fit.
Furthermore, it must be properly fitted to be
strongly correlated to other interesting/important
parameters such as Biso, occupation factors, ...
There are different equations and algorithms that allow
a good ‘automatic’ fitting. There is also the possibility of
a hand fitting. (Or a combination of both approaches).
All these procedures have advantages and disadvantages.
23
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Overall parameters
They affect the whole pattern
- Scale factor (several phases ⇒⇒⇒⇒ several scale factors)
- Coefficients (and functions) to describe the background
- Unit cell parameters
- Goniometer-zero (and sample height/shift)
- Peak shape functions and parameters
- Polarization factor
- Preferred orientation correction
24
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
uUnit cell parameters. Their optimization is not
problematic if there is not pseudo-symmetry (then,
damping factor has to be increased).
They may be correlated with the peak shape parameters.
A suitable ‘small’ step size (e.g. 0.02 °/2θ for CuKα1) is
needed for a good refinement of these parameters.
uuuu The centering of the goniometer (zero-shift) is
correlated with the parameters that describe the axial
divergence S/L and H/L. Furthermore, the zero-shift
may include the sample height error (not perfect
alignment) as the correlation is very high.
Metric and symmetry 25
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Sample height error
26
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Overall parameters
They affect the whole pattern
- Scale factor (several phases ⇒⇒⇒⇒ several scale factors)
- Coefficients (and functions) to describe the background
- Unit cell parameters
- Goniometer-zero (and sample height/shift)
- Peak shape functions and parameters
- Polarization factor
- Preferred orientation correction
27
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Shape of an isolated peak
Lorentzian:
Gaussian:
Pseudo-Voigt:
πβο2
2wFWHM; 22
2
==+
Φ = )(Φxw
wx
Φ(x) = Φo [ηL + (1-η)G]
πββπο
Ln22wFWHM ;)( 22 ==Φ=Φ − xex 2
28
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Peak shape: (isotropic) variation with 2θθθθ
hhhh Gaussian component variation (with 2θ )
wG2 = U tan2θ + V tanθ + W + P/cos2θ
wL = X tanθ + Y/cosθ
hhhh Lorentzian component variation (with 2θ )
29
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Peak shape: Lorentzian variation with 2θθθθ
XY
30
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Peak shape: 2θ θ θ θ dependence: axial divergence
Mainly for low-angle peaks, 2θ θ θ θ < 30 °°°°It can be modelled with the S/L and H/L parameters
These parameters may be strongly correlated with zero-shift
Good starting values: S/L=H/L=0.02 (but depends on the Soller slits)
31
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Peak shape: anisotropic variation (hkl dependent)
Broad peak Sharp peak
Difference curve
Ellipsoidal correction:
in this case along [001]
32
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Peak shape of Al2O3: variations for two diffractometers
CuKα1,2-D5000_overall FWHM (yellow); Loren. (pink); Gaussian (light-blue)
CuKα1-X’Pert_overall FWHM (blue); Loren. (green); Gaussian (red)
FWHM=0.10-0.20º (overall)
FWHM=0.04-0.12º (overall)
33
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
XRPD pattern of the
same alumina:
CuKα1 (top)
and
CuKα1,2 (bottom)
110-120º range (2θ)
34
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
CuKαααα1CuKαααα1,2
Synchrotron X-R
La8.65Sr1.35Si6O26.325
Three patterns:
35
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
4
-0.5
0.
0
0.5
1.
0
2-Theta, deg
Coun
ts
5.0 10.0 15.0 20.0 25.0 30.0
X10E
4
0.0
2.
0
4.0
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
4
-0.5
0.
0
0.5
1.
0
2-Theta, deg
Coun
ts
5.0 10.0 15.0 20.0 25.0 30.0
X10E
4
0.0
2.
0
4.0
2-Theta, deg
Coun
ts
29.0 30.0 31.0 32.0 33.0 34.0 35.0
X10E
4
-0.5
0.
0
0.5
1.
0
2-Theta, deg
Coun
ts
13.0 14.0 15.0 16.0
X10E
4
0.0
1.
0
2.0
3.
0
4.0
5.
0
CuKαααα1
1.5406Å
MoKαααα1,2
0.7093Å0.7136Å
36
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Overall parameters
They affect the whole pattern
- Scale factor (several phases ⇒⇒⇒⇒ several scale factors)
- Coefficients (and functions) to describe the background
- Unit cell parameters
- Goniometer-zero (and sample height/shift)
- Peak shape functions and parameters
- Polarization factor
- Preferred orientation correction37
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Polarization
uuuu The Lorentz-polarization correction is implemented in all common
programs. Its value (in the equation given below) depends upon de
monochromator used.
θθθ
cossin 2
)2cos P(1 P Lp
2
2hh −+=
uPh may be optimized with a standard with known - Biso.
u Ph=0.80 (IPOLA=1)38
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Overall parameters
They affect the whole pattern
- Scale factor (several phases ⇒⇒⇒⇒ several scale factors)
- Coefficients (and functions) to describe the background
- Unit cell parameters
- Goniometer-zero (and sample height/shift)
- Peak shape functions and parameters
- Polarization factor
- Preferred orientation correction39
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Preferred orientation correction
uThere are several functions/corrections but the
March-Dollase methodology is the most used.
3/2-oj
2j
22o
n
1j
hp, ))/RA (sin A cos (R O +=∑=
where Aj is the angle between the prefer orientation vector
(for instance [001]) and the given reflection
Ro is a optimisable parameter:
<1.0 for plaques >1.0 for needles
uuuu More than one axis can be defined for each phase40
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Preferred orientation for layered - Pb(HO3PC6H5)2
Loss of information
41
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Lambda 1.5406 A, L-S cycle 462 Obsd. and Diff. Profiles
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
4
0.0
1.
0
2.0
3.
0
4.0
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
3
0.0
1.
0
2.0
Lambda 1.5406 A, L-S cycle 462 Obsd. and Diff. Profiles
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
4
0.0
1.
0
2.0
3.
0
4.0
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
3
0.0
1.
0
2.0
2-Theta, deg
Coun
ts
10.0 15.0 20.0 25.0 30.0 35.0
X10E
4
0.0
0.
5
1.0
1.
5
2.0
2-Theta, deg
Coun
ts
10.0 15.0 20.0 25.0 30.0 35.0
X10E
3
0.0
1.
0
2.0
CuK αααα1Reflection
CuK αααα1,2Transmission
CS
H2
CS
H2
Lambda 1.5406 A, L-S cycle 462 Obsd. and Diff. Profiles
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
4
0.0
1.
0
2.0
3.
0
4.0
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
3
0.0
1.
0
2.0
Lambda 1.5406 A, L-S cycle 462 Obsd. and Diff. Profiles
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
4
0.0
1.
0
2.0
3.
0
4.0
2-Theta, deg
Coun
ts
10.0 20.0 30.0 40.0 50.0 60.0 70.0
X10E
3
0.0
1.
0
2.0
2-Theta, deg
Coun
ts
10.0 15.0 20.0 25.0 30.0 35.0
X10E
4
0.0
0.
5
1.0
1.
5
2.0
2-Theta, deg
Coun
ts
10.0 15.0 20.0 25.0 30.0 35.0
X10E
3
0.0
1.
0
2.0
CuK αααα1Reflection
CuK αααα1,2Transmission
CS
H2
CS
H2
RQPA results:
c-Ca4Al6O12SO4, 23.6(2) wt%
o-Ca4Al6O12SO4, 15.9(2) wt%
Ca5(SiO4)2SO4, 16.7(3) wt%
gypsum, 13.5(1) wt%
β-belite, 9.9(1) wt%
anhydrite-II, 8.3(1) wt%
alite-M3, 6.1(1) wt%
CaTiO3, 4.7(1) wt%
(Mg,Ca)CO3, 1.2(1) wt%
March-Dollase PO correctionfor gypsum [010]:
PO-coeff.=0.499(8) reflection
PO-coeff.=1.37(2) transmission
42
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
1.- Introduction to powder diffraction
2.- Rietveld method: principles
3.- Rietveld method: overall parameter
4.- Atomic parameters
[ ] [ ]∑ ++−=n
)nl nyk n(h2(2/2sennhkl e e nfº F zxBoc iπλθ
Occupation
factors ADP’s Positional parameters43
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Ion conductor series (four compositions): Na1+x(Zr2-xInx)(PO4)3
44
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Positional parameters
+ soft constraints
+ rigid body
+ increase the number of useful data (e.g. NPD)
Correlations of the positional atomic parameters (xyz)
- Between atoms in complex structures
- Atoms with a large number of electrons dominate
the scattering of the sample. So the errors in the positions
of the light atoms are much higher.
45
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Simulation of the effect of the ADPs for standard sample Al2O3:Left: normal-Uiso Right: Large - Uiso
0.003 Å2 0.03 Å2
Thermal parametersADP’s
Correlations
+ Polarization (properly inserted)+ Good background description+ Disorder (positional/compositional)+ Presence of heavy atoms+ Complex structures
46
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Occupation factors
Correlations
+ Scale factors (and heavy atoms)+ ADP’s+ Disorder+ Background+ Between them (if more than one)
Warning! Take care of the input format
It is always very useful to have the chemical analysis. However,we have to distinguish between the overall elemental chemical composition and that from the crystalline phases.
Final thought: Not all that can be refined makes sense!
Just refine what's needed and if there is information for that!47
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Thanks so much for your attention!
48
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...49
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
This presentation is entirely based on the presentation
prepared by
Aknowlegments and References
50
Miguel Ángel García Aranda
Departamento de Química
Inorgánica
Universidad de Málaga
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Ca3-xMgxSiO5, main cement phase, x=0.00, 0.02, 0.04, 0.06, 0.08 and 0.10.
The figure shows the evolution in the peak positions, consequence of the evolution of
the unit cell parameters. Important for reactivity (water hydration / hardening).
# 2 Determination of the unit cell parameters.
28 30 32 34 36 38 40
2θθθθ
I (u.
a)0Mg
02Mg
04Mg
06Mg
08Mg
1Mg
51
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
Sources of peak broadening
Instrumental BroadeningInstrumental Broadening
A antiphase domainB interstitial atomG, K grain boundaryL vacancyS substitutional impurity/dopingS’ interstitial impurityP, Z stacking faults┴ dislocations
Microstructural featuresMicrostructural features
2 θ (º)2 θ (º)Finite Crystallite Size
FWHM α cos -1 θ (if isotropic)size < 0.2 µm
Lattice Strain (microstrain)
FWHM α tan θ (if isotropic)fluctuations in cell parameters
Extended Defects
Anisotropic broadeningAntiphase Boundaries, Stacking Faults
# 5 Determination of the microstructure of the phas e
52
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
# 5 Determination of the microstructure of the phas e: student’s patterns
53
Máster en Cristalografía y cristalización Sevilla, 2012
The Rietveld method...
XRPD Applications
# 1 Identification of crystalline compounds (using the PDF database). (Based on Ihkl and dhkl)
# 2 Determination of the unit cell parameters. (Based on dhkl)
# 3 Determination of the crystal structure (atomic parameters). (Based on Ihkl and dhkl)
# 4 Quantitative phase analysis (sample purity). (Based on Ihkl)
# 5 Determination of the microstructure of the phas e. (Based on the shape-‘FWHM’ of the Ihkl)(average microparticle size and shape, microstrains, residual stress, etc.)
# 6 XRPD can be coupled to thermal variation (therm odiffractometry):Uses for: phase transitions, chemical reactions, melting/crystallization, thermal expansion, …
# 7 XRPD can be coupled to pressure variation:Uses for: phase transitions, equation of state determination, …
# 8,9, … XRPD coupled to chemical gradient , magnetic fields, …; and combinations54
top related