ClassicalClassical RietveldRietveldAnalysisAnalysis

D. D. ChateignerChateignerCRISMAT-ENSICAEN, Univ Caen, France

L. L. LutterottiLutterottiDIM, Univ. Trento, Italia

OutlineOutlineBrief historyMain computer softwaresWhat can this method do ?Parameters not-refinable by the Rietveld methodPrinciple of the Rietveld method

Intensity descriptionBackgroundPeak asymmetry - displacementsScale factorLorentz - PolarisationGeneralised structure factorThermal vibrations

atomic displacements in cartesian coordinatescrystal symmetry restrictionstemperature vibrations - special cases

Atomic scattering factor - Debye-Wallerx-raysneutrons

Volume - Absorption correctionMicroabsorption -roughness

Profile shape functionsProfile shape modifications

Origin of f(x)Instrumental geometryClassical texture correction

Cell parameters refinementMinimisation routines

Least-squaresDrivative difference minimisationGenetic algorithm

Reliability factorsExpressionsSome hints

Connecting parameters - constrainsStructure of a quite old input fileQuantitative phase analysisTips and tricks during the refinementTips and tricks to get good data !Grain size effect

As viewed from 2D detectorsIn the refinement don’t do !

example of corundum

Indexing of the diffraction pattern in unknown phasesQuality of the experiment

InstrumentHigh-res instrumentLow-res instrumentA good overall instrumentInstrument assessment

Data collectionStep sizeTotal collection timeRespecting statistics

SampleGrain statisticsAmbient conditions

Non Classical Rietveld applicationsQPA of crystalline/non crystalline mixturesRietveld Stress and TextureTextureStrain and stresses

Why Rietveld refinement widely used ?

Rietveld analysis Rietveld analysis –– brief historybrief history19641964--19661966 -- Need to refine crystal structures from powder. Peaks too much Need to refine crystal structures from powder. Peaks too much overlapped:overlapped:

Groups of overlapping peaks introduced. Not sufficient.Groups of overlapping peaks introduced. Not sufficient.Peak separation by least squares fitting (Peak separation by least squares fitting (gaussiangaussian profiles). Not for severe profiles). Not for severe overlapping.overlapping.

19671967 -- H.M. Rietveld H.M. Rietveld -- First refinement program of First refinement program of neutron data, fixed wavelength, neutron data, fixed wavelength, single single reflections + overlapped, no other parameters than the atomic pareflections + overlapped, no other parameters than the atomic parameters. rameters. ActaActa CrystCryst. . 22, 151, 1967.22, 151, 1967.19691969 -- First complete program with structures and profile parameters. First complete program with structures and profile parameters. Distributed 27 Distributed 27 copies (ALGOL). J. copies (ALGOL). J. ApplAppl. . CrystCryst. 1969.. 1969.19721972 -- Fortran version. Distributed worldwide. Wide acceptance in 1977Fortran version. Distributed worldwide. Wide acceptance in 1977..R.B. Von R.B. Von DreeleDreele -- neutron data, TOFneutron data, TOFD.B. Wiles & R.A. Young, then D.E. Cox D.B. Wiles & R.A. Young, then D.E. Cox -- XX--ray data, 2 wavelengths, more phasesray data, 2 wavelengths, more phasesHelsinki group Helsinki group -- spherical functions for preferred orientation but a single wavespherical functions for preferred orientation but a single wavelengthlengthJ. RodriguezJ. Rodriguez--CarvajalCarvajal, , FullprofFullprof, LHRL , LHRL -- surface absorptionsurface absorptionBGMN BGMN -- automatic calculation, crystallite size and automatic calculation, crystallite size and microstrainmicrostrain in form of ellipsoidsin form of ellipsoidsP. P. ScardiScardi, L. , L. LutterottiLutterotti -- size, strainsize, strainToday: the Today: the RietveldRietveld method is widely used for different kind of analyses, not only method is widely used for different kind of analyses, not only structural refinements.structural refinements.““If the fit of the assumed model is not adequate, the precision aIf the fit of the assumed model is not adequate, the precision and accuracy of the nd accuracy of the parameters cannot be validly assessed by statistical methodsparameters cannot be validly assessed by statistical methods””. Prince.. Prince.

Main Computer Main Computer softwaressoftwares

H.M. RietveldH.M. RietveldDBW2.9, DBW3.2 (Wiles & Young)DBW2.9, DBW3.2 (Wiles & Young)FullprofFullprof (J. Rodriguez(J. Rodriguez--CarvajalCarvajal))GSAS (von GSAS (von DreeleDreele))BGMN (R. Bergmann)BGMN (R. Bergmann)LHRL (C.J. Howard & B.A. Hunter)LHRL (C.J. Howard & B.A. Hunter)RietquanRietquan (L. (L. LutterottiLutterotti, P. , P. ScardiScardi))MAUD (L. MAUD (L. LutterottiLutterotti))Jana (V. Jana (V. PetricekPetricek))RieticaRietica (B. Hunter)(B. Hunter)ESPOIR (A. Le Bail)ESPOIR (A. Le Bail)FOX (V. FOX (V. FavreFavre--NicolinNicolin, R. , R. CernyCerny))

L. M. D. L. M. D. CranswickCranswick, , CCP14 (Collaborative Computation Project No 14 for Single crystaCCP14 (Collaborative Computation Project No 14 for Single crystal and Powder l and Powder

diffraction). www.ccp14.ac.ukdiffraction). www.ccp14.ac.uk

What can this method do ?What can this method do ?

Analysis of the whole diffraction patternAnalysis of the whole diffraction patternProfile fitting is includedProfile fitting is includedNot only the integrated intensitiesNot only the integrated intensities

Refinement of the structure parameters from diffraction data Refinement of the structure parameters from diffraction data Quantitative phase analysis (crystalline and amorphous)Quantitative phase analysis (crystalline and amorphous)Lattice parametersLattice parametersAtomic positions and occupanciesAtomic positions and occupanciesTemperature vibrations (isotropic and anisotropic)Temperature vibrations (isotropic and anisotropic)Grain size and microGrain size and micro--strain (isotropic and anisotropic)strain (isotropic and anisotropic)Stacking and twin faultsStacking and twin faultsMagnetic moments (neutrons)Magnetic moments (neutrons)

Not intended for the structure solutionNot intended for the structure solutionThe structure model must be known before starting the Rietveld rThe structure model must be known before starting the Rietveld refinementefinement

Parameters not Parameters not refinablerefinable by the Rietveld methodby the Rietveld method

Space groupSpace groupChemical compositionChemical compositionAnalytical function describing the shape of the diffraction profAnalytical function describing the shape of the diffraction profilesilesWavelength of the radiation (except with the measure of a standaWavelength of the radiation (except with the measure of a standard)rd)Origin of the polynomial function describing the backgroundOrigin of the polynomial function describing the background

… in general no refinement of the models themselves… in general no refinement of the models themselves

Principle of the Principle of the RietveldRietveld methodmethodCalculated intensity at point i of the diagram:

G: normalised profile shape functionI: intensity of the k-th reflectionS: scale factor of phase ΦSummation performed over all phases Φ, and over all reflections k contributing tothe respective point.

Intensity of the Bragg reflections

∑ ∑Φ

ΦΦ −+=k

kkiibic IGSyy )22( θθ

mk: multiplicity of kLk: Lorentz-polarisation factor|Fk|2: structure factorPk: preferred orientation factor Ph(y)Ak: absorption factor

kkkkkk APFLmI 2=

•• The spectrum (at a 2The spectrum (at a 2θθ point i) is determined by:point i) is determined by:•• a background valuea background value•• some reflection peaks that can be described by different terms:some reflection peaks that can be described by different terms:

•• Diffraction intensity (determines the “height” of the peaks)Diffraction intensity (determines the “height” of the peaks)•• Line broadening (determines the shape of the peaks)Line broadening (determines the shape of the peaks)•• Number and positions of the peaksNumber and positions of the peaks

Iicalc = SF

f j

V j2 Lk Fk, j

2S j 2θ i − 2θk, j( )Pk, j A j + bkgi

k=1

Npeaks

∑j=1

Nphases

∑

BackgroundBackgroundPolynomial function in 2Polynomial function in 2θθ (the most commonly used)(the most commonly used)

•• NNbb

is the polynomial degreeis the polynomial degree•• BB

nnthe polynomial coefficientsthe polynomial coefficients

Gaussian components (direct beam or peculiar “bumps”)Gaussian components (direct beam or peculiar “bumps”)

A special function for amorphous componentsA special function for amorphous components

For more complex backgrounds specific formulas are available, e.For more complex backgrounds specific formulas are available, e.g. g. Fourier seriesFourier series

It is possible to incorporate also the TDS in the backgroundIt is possible to incorporate also the TDS in the backgroundSpecific physical models for incoherent scattering (Specific physical models for incoherent scattering (RielloRiello et al.)et al.)

( )∑= +

+++=n

m m

mmib QB

QBBQBBy1 12

12210

sin

ni

N

nn

G

ggi

b

BGbkg ]2[)2(01

θθ ∑∑==

+=

ni

N

nn

G

ggi

b

BGbkg ]2[)2(01

θθ ∑∑==

+=

)2*11cos(...2cos)( 1110 ii BKBKBKiB θθ +++=

Peak asymmetryPeak asymmetry--displacementsdisplacements

hhihihi signAA θθθθθθθ tan/)22).(22( 1)22( 2−−−=−Rietveld:

Finger, Cox, Jephcoat: most appropriate, uses instrument characteristics

Bérard, Baldinozzi: weak asymmetries

von Dreele: for TOF neutrons

Debye-Scherrer geometry: ∆2θ = a cosθ / R - b sin2θ / R

Bragg-Brentano: symmetric ∆2θ = -2 s cosθ / R (excentricity)asymmetric ∆2θ = b sin2θ / R sinω

transmission, flat plate: ∆2θ = −a sin(2θ) / R

sample transparency: ∆2θ = − sin(θ) / µR

sample planarity: ∆2θ = −6 cotan(θ) / α2

Scale factorScale factor

Iicalc = SF

f j

V j2 Lk Fk, j

2S j 2θi − 2θk, j( )Pk, j A j + bkgi

k=1

Npeaks

∑j=1

Nphases

∑

The scale factor (for each phase) is written in classical Rietveld programs as:

Sj = phase scale factor (the overall Rietveld generic scale factor)SF = beam intensity (it depends on the measurement)fj = phase volume fractionVj = phase cell volume (in some programs it goes in the F factor)

In Maud the last three terms are kept separated.

Sj = SFfj

Vj2

LorentzLorentz polarisationpolarisation

Iicalc = SF

f j

V j2 Lk Fk, j

2S j 2θi − 2θk, j( )Pk, j A j + bkgi

k=1

Npeaks

∑j=1

Nphases

∑

•• The LorentzThe Lorentz--Polarization factor:Polarization factor:•• it depends on the instrumentit depends on the instrument

•• geometrygeometry

•• monochromator (angle monochromator (angle θθ))•• detectordetector•• beam size/sample volumebeam size/sample volume•• sample positioning (angular)sample positioning (angular)

•• For a BraggFor a Bragg--Brentano instrument:Brentano instrument:

•• neutron TOF:neutron TOF: L = d4 sinθ

Lp =1+ Ph cos2 2θ( )

2 1+ Ph( )sin2 θ cosθ

Ph = cos2 2α( )

Generalised Structure FactorGeneralised Structure Factor

Iicalc = SF

f j

V j2 Lk Fk, j

2S j 2θi − 2θk, j( )Pk, j A j + bkgi

k=1

Npeaks

∑j=1

Nphases

∑

Under a generalized structure factor we include:Under a generalized structure factor we include:The multiplicity of the The multiplicity of the kk--thth reflection (with h, k, l Miller indices): mreflection (with h, k, l Miller indices): mkk

The structure factorThe structure factorThe temperature factor: The temperature factor: BBnn of atom n (eventually anisotropic)of atom n (eventually anisotropic)

N = number of atoms in the unitN = number of atoms in the unit--cellcellxxnn, y, ynn, z, znn coordinates of the coordinates of the nn--thth atomatomffnn, atomic scattering factor , atomic scattering factor or scattering length bj for neutrons

Fk, j

2= mk fne

−Bnsin 2 θ

λ2 e2πi hxn +kyn + lzn( )( )n=1

N

∑2

Thermal vibrationsThermal vibrations

Atomic displacement (in Cartesian co-ordinates)

( )⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −==

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

==

cbbaaa

uuuuuuuuuuuuuuu

t

tjjj

00cossin10coscot1

;2

1 **

***

2

233231

322221

312121

αγβγ

πFβFFB

uuB

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

332313

232212

131211

ββββββββββ

β: symmetric rotation matrix describing the vibration ellipsoid

Crystal symmetry restrictionsCrystal symmetry restrictions

Six anisotropic temperature factors per atom in a general case Six anisotropic temperature factors per atom in a general case (symmetrical matrix)(symmetrical matrix)For an atom in a site of special symmetry the For an atom in a site of special symmetry the BB--matrix must be invariant matrix must be invariant to the symmetry operations (in the Cartesian axis system)to the symmetry operations (in the Cartesian axis system)

An example An example -- rotation axes parallel with rotation axes parallel with z

BBPP =t

z

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

1000cossin0sincos

αααα

P

Temperature vibrations Temperature vibrations -- special casesspecial cases

Isotropic atomic vibrationsIsotropic atomic vibrations

Overall temperature factorOverall temperature factor

( )[ ]

jj

n

jjjjjjjk

uB

BzkyhxifNF

22

12

2

8

sinexp2exp

π

λθπ

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−++= ∑

=

l

( )[ ]∑=

++×⎟⎟⎠

⎞⎜⎜⎝

⎛−=

n

jjjjjjk zkyhxifNuF

12

222 2expsin8exp lπ

λθπ

Atomic scattering factor and DebyeAtomic scattering factor and Debye--WallerWaller

an, bn, c are from the “International Tables for Crystallography”

f’, f”: anomalous scattering factors (dispersion and absorption resp.)

fifcbafni

nn ′′+′++⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑

=

4

2

2sinexpλ

θ

•• The atomic scattering factor for XThe atomic scattering factor for X--ray decreases with the diffraction angle and is ray decreases with the diffraction angle and is proportional to the number of electrons. For neutron proportional to the number of electrons. For neutron bbjj is not correlated to the is not correlated to the atomic number.atomic number.

•• The temperature factor (DebyeThe temperature factor (Debye--Waller) accelerates the decreases. Waller) accelerates the decreases.

Neutron scattering factorsNeutron scattering factors

•• For light atoms neutron scattering has some advantagesFor light atoms neutron scattering has some advantages•• For atoms very close in the periodic table, or isotopes, neutronFor atoms very close in the periodic table, or isotopes, neutron scattering may scattering may

help contrasting them.help contrasting them.

-0.5

0

0.5

1

1.5

H Li B N F Na Al P Cl K Sc V Mn Co Cu Ga As Br Rb Y Nb Tc Rh Ag In Sb I Cs La Pr Eu Tb Ho Tm Lu Ta Re Ir Au Tl Bi U Pu

VolumeVolume--Absorption correctionAbsorption correction

Iicalc = SF

f j

V j2 Lk Fk, j

2S j 2θi − 2θk, j( )Pk, j A j + bkgi

k=1

Npeaks

∑j=1

Nphases

∑

DebyeDebye--ScherrerScherrer geometry, absorption not constantgeometry, absorption not constant

There could be problems for There could be problems for microabsorptionmicroabsorption (absorption contrast)(absorption contrast)

MicroabsorptionMicroabsorption -- RoughnessRoughness

µµµµ

≤′′→

For flat samples For flat samples -- micromicro--absorption and absorption and surface absorptionsurface absorption

Apparent decrease of the temperature Apparent decrease of the temperature factors or even “negative” temperature factors or even “negative” temperature factorsfactors

0 20 40 60 800.980

0.982

0.984

0.986

0.988

0.990

0.992

0.994

0.996

0.998

1.000

1.002

1.004

1.006

1.008

1.010

SR

θ(°)

Sparks, t=10-4

Suortti, p=0.01, q=0.02 Pitschke, p=0.01, q=0.03 Sidey, s=0.005

⎥⎦⎤

⎢⎣⎡ −−=

21 πθtSRSparks

Suortti

Pitschke

Sidey

( ) ⎟⎠⎞

⎜⎝⎛+−−=

θsinq-exp p exp 1 qpSR

⎥⎦⎤

⎢⎣⎡ −−−−=

θθ sin1

sin)1(1 qpqqpqSR

⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛=

2/lnexp

2/ πθ

θπθ θ sS

s

R

Bragg-Brentano case

For thin samples (powder on glass) in For thin samples (powder on glass) in symmetrical arrangementsymmetrical arrangement

thick sample, high absorptionthick sample, high absorption

thin sample, low absorption, depends thin sample, low absorption, depends on 2on 2θ

0 .00 0 .05 0 .10 0 .15 0 .20 0 .25 0 .30 0 .35 0 .40

-0 .4

-0 .3

-0 .2

-0 .1

0 .0

e xp e rim e n ta l da ta

a bso rp tion fa c to r

a pp a re n t te m p e ra tu re

log

(Int

ensi

ty r

atio

)

(s in θ /λ )2

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−−=

θµ

µ sin2exp1

21

0tII

µ µt A→ ∞ =: ( )1 2

θ

µ θt A t→ =0: sin

Profile shape functionsProfile shape functions

Iicalc = SF

f j

V j2 Lk Fk, j

2S j 2θi − 2θk, j( )Pk, j A j + bkgi

k=1

Npeaks

∑j=1

Nphases

∑

( ) 2ln4;22exp 02

200 =⎥

⎦

⎤⎢⎣

⎡−

Γ−

Γ= CCC

G kikk

θθπ

( )4;

221

120

220

0 =−

Γ+Γ

= CCC

Lki

k

k θθπ

( )( ) ( )( )5.0

122;22124121

02

20

−−

=⎥⎦

⎤⎢⎣

⎡−

Γ−

+Γ

=−

mmCCP

mm

kik

m

kVII π

θθ

( )GLpV ηη −+= 1

Gauss (original Gauss (original RietveldRietveld neutron neutron function)function)

LorentzLorentz (Cauchy)(Cauchy)

Pearson VIIPearson VII

Voigt and PseudoVoigt and Pseudo--VoigtVoigt GLV ⊗=

Gaussianity : η = cn 2θ( )n

n= 0

Ng

∑

Profile shape modificationsProfile shape modifications

0 20 40 60 80 100 1200.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

FWH

M (2

θ°)

2θ(°)

D1B INEL CPS120

Half-width resolution function:

WVU kkk ++=Γ θθ tantan :al.et Caglioti 22

f(x): sampleg(x): instrument∫

+∞

∞−

−=⊗= dyyxgyfxgxfxh )()()()()(Profile contributions:

Origin of f(x)Origin of f(x)

l

l

hk

hk

dd cos 4 )(2

cos)2(

∆==∆

=∆

εθεθ

θλθ

TKLimited crystallite sizes Lorentzian broadening

(Scherrer size)crystallite microdistortions Gaussian broadening(microstrains)

Internal account of the broadening: e.g., parameterised PV (Thompson-Cox-Hasting):

)2()1()2()2( iPiPiP GLPV θηθηθ −+= 3L

2LLP /Y)(Y0.11116 + /Y)(Y0.47719 /Y)(Y1.36603 = −η

[ ] 5/1YY0.07842Y+YY4.47163+Y2.42843Y+YY2.69269+Y=Y 5

LG4

L2

G3

L3

G2

L4

GL5

G +

)()1(tantan)( 222 hklWVUY AkkkG Γ−+++= ξθθθ

)(cos/tan)( hklYXY AkkkL Γ++= ξθθθ

hklkhklA MMhkl /tan)()( 2 θσ=Γ

Gaussian partLorentzian partanisotropic param ….

Standard: e.g., LaB6, CeO2: WVU kkk ++=Γ θθ tantan :al.et Caglioti 22

+ deconvolution into Gaussian and Lorentzian parts+ anisotropy in spherical harmonics (Popa)

Instrumental geometryInstrumental geometry

Bragg-Brentano modified 2D detectors

Classical texture correctionClassical texture correction

Iicalc = SF

f j

V j2 Lk Fk, j

2S j 2θi − 2θk, j( )Pk, j A j + bkgi

k=1

Npeaks

∑j=1

Nphases

∑

( ) ( )( ) ( )( ) ( )kk

kk

kk

GGGP

GGGP

GGGP

α

α

α

3122

2122

2122

sinexp1

sinexp1

exp1

−−+=

−−+=

−−+=GaussGauss--like distributionlike distribution

MarchMarch--DollaseDollase correctioncorrection

Pk, j =1

mk

PMD2 cos2 αn +

sin2 αn

PMD

⎛

⎝ ⎜

⎞

⎠ ⎟

n=1

mk

∑−

32

PMD

: March-Dollase parametersummation over all equivalent hkl reflections (m

k)

αn: angle between the preferred orientation vector and the crystallographic plane (hkl) (in

the crystallographic cell coordinate system)

The formula is intended for a cylindrical texture symmetry (fibre) (observable in B-B geometry or by spinning the sample)

Cell parameters refinementCell parameters refinementIi

calc = SF

f j

V j2 Lk Fk, j

2S j 2θi − 2θk, j( )Pk, j A j + bkgi

k=1

Npeaks

∑j=1

Nphases

∑

The number of peaks is determined by the symmetry and space grouThe number of peaks is determined by the symmetry and space group of the p of the phase.phase.One peak is composed by all equivalent reflections mOne peak is composed by all equivalent reflections m

kkThe position is computed from the dThe position is computed from the d--spacing of the hkl reflection (using the spacing of the hkl reflection (using the reciprocal lattice matrix):reciprocal lattice matrix):

dhkl =VC

s11h2 + s22k

2 + s33l2 + 2s12hk + 2s13hl + 2s23kl

S=

2

2

sin21⎟⎠⎞

⎜⎝⎛=

λθ

hkld

min12

sin2

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛−∑

i hkli d

λθ

Minimisation routinesMinimisation routines

The cost function to minimise is the quantity (for most algorithThe cost function to minimise is the quantity (for most algorithms):ms):

Normal matrixNormal matrix

( ) iii

icioi ywyywWSS 1;2 =−= ∑

( )∑

∑

−∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=

∆=∆

iicio

m

icim

i n

ic

m

icimn

yyxywy

xy

xywM

0

yxM

( )

PN

yywM i

icioi

mmm −

−=

∑−

2

1σ

Least-squares

Derivative Difference Minimisation (Solovyov)

( ) ( ) ( ) yyw...yywyyw2

icioj

2

icio2

22

2

icio1∑

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡−

∂∂

++⎥⎦

⎤⎢⎣

⎡−

∂∂

+⎥⎦⎤

⎢⎣⎡ −∂∂

j

j

θθθ

Genetic (evolutionnary) algorithm

The problem depends on xThe problem depends on xii parameters, has a solution in the form of a parameters, has a solution in the form of a vector:vector:

The cost function to minimise is the quantity:The cost function to minimise is the quantity:

parameter sets are divided into father and mother sets, then sonparameter sets are divided into father and mother sets, then sons are s are generated:generated:

xison = (xi

father + ximother) / 2

at each step sons inherit mother or father properties:at each step sons inherit mother or father properties:

( )∑=

−=J

jicio yyiPF

1

2)()(v

)..,,..,( 21 ni xxxx=v

⎩⎨⎧

≥<

=1

motheri

1fatheri

p p if xp p if xson

ix

Quality of the refinements: Reliability factorsQuality of the refinements: Reliability factors

The profile RThe profile R--factor ………factor ………

The weighted The weighted RpRp ………………………………………………………………………………

The Bragg RThe Bragg R--factor ………factor ………

The expected The expected RfRf ………………………………………………………………………………

The goodness of fitThe goodness of fit

∑∑ −

=

iio

iicio

p y

yyR

( ) 21

2

2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

∑∑

iioi

iicioi

wp yw

yywR

∑∑ −

=

iko

ikcko

B I

IIR

21

2exp

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

=∑

iioi yw

PNR

( ) 2

exp

2

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−

−=

∑RR

PN

yywGoF wpi

icioi

N=number of points, P=number of parameters

The R indicesThe R indices

•• RRwpwp

more valuable than R: its absolute value does not depend on themore valuable than R: its absolute value does not depend on theabsolute value of the intensities. absolute value of the intensities. ButBut it depends on the background. it depends on the background. With a high background it is more easy to reach very low values.With a high background it is more easy to reach very low values.Needs Needs to be calculated on net intensities. Increasing the number of to be calculated on net intensities. Increasing the number of peaks (sharp peaks) it is more difficult to get a good value.peaks (sharp peaks) it is more difficult to get a good value.

•• RRwpwp

< 0.1 correspond to an acceptable refinement with a medium < 0.1 correspond to an acceptable refinement with a medium complex phasecomplex phase

•• For a complex phase (monoclinic to triclinic) a value < 0.15 is For a complex phase (monoclinic to triclinic) a value < 0.15 is goodgood•• For a highly symmetric compound (cubic) with few peaks a value <For a highly symmetric compound (cubic) with few peaks a value < 0.08 0.08

start to be acceptablestart to be acceptable

•• The RThe Rexpexp

is the minimum Ris the minimum Rwpwp

value reachable using a certain number value reachable using a certain number of of refinablerefinable parameters in given experimental conditions. It needs a parameters in given experimental conditions. It needs a valid weighting scheme to be reliable.valid weighting scheme to be reliable.

WSS and WSS and GoFGoF

•• The weighted sum of squares is only used for the minimization roThe weighted sum of squares is only used for the minimization routines. Its utines. Its absolute value depends on the intensities and number of points. absolute value depends on the intensities and number of points. Helps Helps following the convergence during refinement.following the convergence during refinement.

•• The goodness of fit is the ratio between the RThe goodness of fit is the ratio between the Rwpwp

and Rand Rexpexp

and cannot be and cannot be lower then 1 (unless the weighting scheme is not correctly valualower then 1 (unless the weighting scheme is not correctly valuable: for ble: for example in the case of detectors not recording exactly the numbeexample in the case of detectors not recording exactly the number of r of photons or neutrons).photons or neutrons).

•• A good refinement gives A good refinement gives GoFGoF values lower than 2.values lower than 2.

•• The goodness of fit is not always a very good index to look at aThe goodness of fit is not always a very good index to look at as, with a s, with a noisy pattern, it is quite easy to reach a value near 1.noisy pattern, it is quite easy to reach a value near 1.

•• With very high intensities and low noise patterns it is difficulWith very high intensities and low noise patterns it is difficult to reach a value t to reach a value of 2.of 2.

•• The The GoFGoF is sensitive to model inaccuracies.is sensitive to model inaccuracies.

Connecting parameters Connecting parameters -- constrainsconstrains

Young Young -- parameter couplingparameter couplingCoding of variables: number of the parameter in the normal Coding of variables: number of the parameter in the normal matrix + weight for the calculated incrementmatrix + weight for the calculated incrementLattice parameters in the cubic system: 4.00 Lattice parameters in the cubic system: 4.00 4.004.00 4.004.00Fractional coFractional co--ordinates at 12k in P6ordinates at 12k in P633/mmc: (x 2x z): 0.205 0.210 /mmc: (x 2x z): 0.205 0.210 0.3100.310

FullprofFullprof -- constrainsconstrainsInterInter--atomic distances may be constrainedatomic distances may be constrained

BGMN BGMN -- working with moleculesworking with moleculesDefinition of the molecule (in Cartesian coDefinition of the molecule (in Cartesian co--ordinates)ordinates)Translation and rotation of the whole moleculeTranslation and rotation of the whole molecule

Modern versions (Modern versions (FullprofFullprof, MAUD, GSAS …), MAUD, GSAS …)Most constrains types can be enteredMost constrains types can be entered

Structure of a quite old input fileStructure of a quite old input file((FullprofFullprof for anglesite)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 Rpolarz1.54056 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 Sent05 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 10.00154.00 180.00

!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/codes207.37 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 ==> Current R_Bragg: 4.16PbSO4

!Nat Dis Mom Pr1 Pr2 Pr3 Jbt Irf Isy Str Furth ATZ Nvk Npr More5 0 0 0.0 0.0 0.0 0 0 0 0 0 0.00 0 7 0

P n m a <-- Space group symbol!Atom Typ X Y Z Biso Occ /Line below:CodesPb PB 0.18748 0.25000 0.16721 1.40433 0.50000 0 0 0

171.00 0.00 181.00 281.00 0.00S S 0.06544 0.25000 0.68326 0.41383 0.50000 0 0 0

191.00 0.00 201.00 291.00 0.00O1 O 0.90775 0.25000 0.59527 1.97333 0.50000 0 0 0

211.00 0.00 221.00 301.00 0.00O2 O 0.19377 0.25000 0.54326 1.48108 0.50000 0 0 0

231.00 0.00 241.00 311.00 0.00O3 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 011.00000 0.00 0.00 0.00 0.00 0.00

! U V W X Y GauSiz LorSiz Size-Model0.15485 -0.46285 0.42391 0.00000 0.08979 0.00000 0.00000 0121.00 131.00 141.00 0.00 151.00 0.00 0.00

! a b c alpha beta gamma8.480125 5.397597 6.959482 90.000000 90.000000 90.00000091.00000 101.00000 111.00000 0.00000 0.00000 0.00000

! Pref1 Pref2 Asy1 Asy2 Asy3 Asy40.00000 0.00000 0.28133 0.03679-0.09981 0.00000

0.00 0.00 161.00 331.00 341.00 0.00

Quantitative phase analysisQuantitative phase analysis

Volume fractionVolume fraction

Weight fraction Weight fraction

( )∑Φ

ΦΦΦ

ΦΦΦ = 2

2

uc

uc

VSVSV

( )∑Φ

ΦΦΦΦΦ

ΦΦΦΦΦ = 2

2

uc

uc

VMZSVMZSm

Z = number of formula unitsM = mass of the formula unitV = cell volume

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

Instrumental parametersInstrumental parametersScale factor (always)Scale factor (always)Background (1)Background (1)Line broadening and shape (3)Line broadening and shape (3)Zero and other shifts (4)Zero and other shifts (4)Sample displacementSample displacement or or transparency (5)transparency (5)Preferred orientation (7)Preferred orientation (7)Surface absorption (7)Surface absorption (7)Extinction (7)Extinction (7)

Structure parametersStructure parameters

Scale factor (always)Scale factor (always)Lattice parameter (2)Lattice parameter (2)Line broadening and shape (3)Line broadening and shape (3)Atomic coAtomic co--ordinates (6)ordinates (6)Temperature factors (8)Temperature factors (8)Occupancies (8), N = Occupancies (8), N = occ/max(Nocc/max(N) ) important for quantitative phase important for quantitative phase analysisanalysis

Standard if possible (LaB6)

Tips and tricks (how to obtain reliable data)Tips and tricks (how to obtain reliable data)First get a good experiment/spectrum/First get a good experiment/spectrum/diffractometerdiffractometerKnow your sample as much as possibleKnow your sample as much as possibleCare about grain statisticsCare about grain statisticsUse sufficient counting timeUse sufficient counting time

The error in intensity is proportional to The error in intensity is proportional to √√N as for the Poisson N as for the Poisson distributiondistributionFor strong diffraction lines, the use of the deadFor strong diffraction lines, the use of the dead--time correction is time correction is strongly recommended (but rarely operated !)strongly recommended (but rarely operated !)

Do not refine too many parametersDo not refine too many parametersAlways try first to manually fit the spectrum as much as possiblAlways try first to manually fit the spectrum as much as possibleeNever stop at the first resultNever stop at the first resultLook carefully and constantly to the visual fit/plot and residuaLook carefully and constantly to the visual fit/plot and residuals during ls during refinement process (no “blind” refinement)refinement process (no “blind” refinement)Zoom in the plot and look at the residuals. Try to understand whZoom in the plot and look at the residuals. Try to understand what is causing at is causing a bad fit.a bad fit.Do not plot absolute intensities; plot at isoDo not plot absolute intensities; plot at iso--statistical errors. Small peaks are statistical errors. Small peaks are important like big peaks.important like big peaks.Use all the indices and check parameter errors.Use all the indices and check parameter errors.First get a good experiment/spectrumFirst get a good experiment/spectrum

Grain size effectGrain size effect

Variations in observed intensities Variations in observed intensities (bad statistics)(bad statistics)

Figure: Effect of specimen rotation and Figure: Effect of specimen rotation and particle size on particle size on SiSi powder intensity powder intensity using conventional using conventional diffractometerdiffractometer and and CuKCuKαα radiation.radiation.

International Tables forInternational Tables forCrystallography, Vol. C,Crystallography, Vol. C,ed. A.J.C. Wilson,ed. A.J.C. Wilson,KluwerKluwer Academic Publishers, 1992. Academic Publishers, 1992.

In the Rietveld refinement In the Rietveld refinement don’tdon’t

refine parameters which are fixed by the structure refine parameters which are fixed by the structure relations (fractional corelations (fractional co--ordinates, lattice parameters)ordinates, lattice parameters)refine all the parameters describing the line broadening refine all the parameters describing the line broadening simultaneouslysimultaneouslyrefine the anisotropic temperature factors from Xrefine the anisotropic temperature factors from X--ray ray powder diffraction data, or when highpowder diffraction data, or when high--θθ range has not range has not been measuredbeen measureduse diffraction patterns measured in a narrow rangeuse diffraction patterns measured in a narrow rangeforget that the number of structure parameters being forget that the number of structure parameters being refined cannot be larger than the number of linesrefined cannot be larger than the number of lines

Corundum, neutron dataCorundum, neutron data

20°-140° 20°-120° 20°-100° 20°-80° Reference

a[Å] 4.7584(2) 4.7585(2) 4.7587(2) 4.7591(3) 4.7586(1)

c[Å] 12.9895(3) 12.9899(4) 12.9905(4) 12.9914(5) 12.9897(1)

z(Al) 0.3521(1) 0.3521(1) 0.3520(1) 0.3521(1) 0.35216

B(Al) [Å2] 0.41(1) 0.38(2) 0.51(3) 0.59(4)

x(O) 0.3060(2) 0.3062(2) 0.3061(3) 0.3063(3) 0.30624

B(O) [Å2] 0.43(3) 0.34(3) 0.36(4) 0.33(5)

2θ−zeroshift

0.055(9) 0.075(9) 0.090(12) 0.117(25)

z-Displacement -0.091(8) -0.110(8) -0.124(11) -0.150(23)

R(Bragg) 3.8% 3.6% 3.2% 2.7% 0.6%

Indexing of the diffraction patternIndexing of the diffraction patternin unknown phasesin unknown phases

The most critical parameters for the convergence of the The most critical parameters for the convergence of the RietveldRietveldrefinement refinement -- lattice parameterslattice parameters

If you know the indexing (crystal system)If you know the indexing (crystal system)

If not: DICVOL (dichotomy), TREOR (trial and error)If not: DICVOL (dichotomy), TREOR (trial and error)

AutoxAutox, Ito, , Ito, TaupTaup/Powder, /Powder, LzonLzon, , LoshLosh, Kohl, , Kohl, ScanixScanix, , XrayscanXrayscan, , EFLECH/Index, EFLECH/Index, SupercellSupercell, , CrysfireCrysfire suite, suite, CheckcellCheckcell ……

Quality of the experimentQuality of the experiment

A good refinement, a successful analysis depends A good refinement, a successful analysis depends strongly on the quality of the experiment:strongly on the quality of the experiment:

•• Instrument:Instrument:instrument characteristics and assessmentinstrument characteristics and assessmentchoice of instrument optionschoice of instrument options

•• Collection strategiesCollection strategiesrangerangestep sizestep sizecollection timecollection timeetc.etc.

•• samplesamplesample sizesample sizesample preparationsample preparationsample conditionsample condition

InstrumentInstrument

Rietveld refinements does not require at all the most powerful Rietveld refinements does not require at all the most powerful instrument but the one suitable for the analysis:instrument but the one suitable for the analysis:

•• quantitative analyses of samples with big grain sizes (metal?, hquantitative analyses of samples with big grain sizes (metal?, high crystal igh crystal symmetries) require a diffracting volume of statistical significsymmetries) require a diffracting volume of statistical significance => ance => large sampling volume, large beam, with not too low divergence =large sampling volume, large beam, with not too low divergence => a > a medium to low resolution diffractometermedium to low resolution diffractometer

•• structural refinements of low symmetries compounds (monoclinic, structural refinements of low symmetries compounds (monoclinic, triclinic) require often a high resolution diffractometertriclinic) require often a high resolution diffractometer

Low and linear background is always preferredLow and linear background is always preferredNo additional lines (beta lines) is also in general preferredNo additional lines (beta lines) is also in general preferredLarge collectable ranges are importantLarge collectable ranges are importantHigher diffraction intensities are also always goodHigher diffraction intensities are also always goodSmaller broadening helps the analysisSmaller broadening helps the analysisSimple geometries are betterSimple geometries are betterThere is no perfect instrument to get everythingThere is no perfect instrument to get everything

High resolution instrumentsHigh resolution instrumentsThese instruments put the emphasis on the smaller line These instruments put the emphasis on the smaller line width obtainable: width obtainable:

•• Pro: Pro: less overlapped peaks (more details for structural refinements)less overlapped peaks (more details for structural refinements)higher accuracy for microstructural analyseshigher accuracy for microstructural analysesbetter separation for multiple phases or pseudobetter separation for multiple phases or pseudo--symmetriessymmetriessmaller sampling volumessmaller sampling volumeshigher cell determination accuracyhigher cell determination accuracy

•• Cons: Cons: smaller sampling volumessmaller sampling volumeslow divergence (less grain statistic) => less accuracy in intenslow divergence (less grain statistic) => less accuracy in intensityitysmaller intensities => higher collection timessmaller intensities => higher collection timesmore difficult to fit, more difficult to fit, more sensitive to modelsmore sensitive to models

Good for structural refinements when high precision is Good for structural refinements when high precision is requestedrequested

Low resolution instrumentsLow resolution instruments

Pro:Pro:•• higher intensitieshigher intensities•• better statistics (higher sampling volumes, more grains diffractbetter statistics (higher sampling volumes, more grains diffracting)ing)•• faster collection timesfaster collection times•• easier to fiteasier to fit

Cons:Cons:•• less details for complicated structures or samplesless details for complicated structures or samples•• less precision (not always less accuracy)less precision (not always less accuracy)•• not suitable for low symmetry compounds or determination of sizenot suitable for low symmetry compounds or determination of size--strain strain

for highly crystallized samplesfor highly crystallized samplesThese instruments are good for normal quantitative and qualitatiThese instruments are good for normal quantitative and qualitative ve analyses or when good statistics of grains is required (texture analyses or when good statistics of grains is required (texture etc.).etc.).

A good overall instrumentA good overall instrument

For quantitative phase analysis:For quantitative phase analysis:•• medium resolutionmedium resolution•• monochromator on the diffracted beammonochromator on the diffracted beam•• Adequate radiation (fluorescence)Adequate radiation (fluorescence)

Structural refinements or structure determinationStructural refinements or structure determination•• high resolutionhigh resolution•• monochromatormonochromator•• no no ΚαΚα

22preferred (structure determination)preferred (structure determination)

Microstructural analysesMicrostructural analyses•• high resolutionhigh resolution

Texture analysesTexture analyses•• medium to low resolution, except for medium to low resolution, except for polyphasedpolyphased samplessamples•• fast collection timefast collection time•• good grain statisticgood grain statistic

Instrument assessmentInstrument assessment

In most cases the instrument alignment and setting is more imporIn most cases the instrument alignment and setting is more important than tant than the instrument itselfthe instrument itself

Be paranoid on alignment, the beam should pass through the rotatBe paranoid on alignment, the beam should pass through the rotation center ion center and hit the detector at zero 2and hit the detector at zero 2θθ

The background should vary regularly, no strange bumps, no additThe background should vary regularly, no strange bumps, no additional linesional lines

Check the omega zeroCheck the omega zero

Collect some times a standard for line positions and check if thCollect some times a standard for line positions and check if the positions e positions are good both at low and high diffraction angleare good both at low and high diffraction angle

Data collectionData collection

The range should always be the widest as possible, compatible wiThe range should always be the widest as possible, compatible with the th the instrument and collection time (no need to waste time if no reliinstrument and collection time (no need to waste time if no reliable able information is coming out from a certain range)information is coming out from a certain range)

The step sizeThe step size

The step size should be compatible with the line broadening charThe step size should be compatible with the line broadening characteristics acteristics and type of analysisand type of analysisIn general 5In general 5--7 points in the half upper part of a peak are sufficient to defi7 points in the half upper part of a peak are sufficient to define ne its shape.its shape.Slightly more points are preferred in case of overlapping.Slightly more points are preferred in case of overlapping.More for sizeMore for size--strain analysis.strain analysis.Too much points (too small step size) do not increase the resoluToo much points (too small step size) do not increase the resolution, tion, accuracy or precision, but just increases the noise at equal totaccuracy or precision, but just increases the noise at equal total collection al collection timetimeThe best solution is to use the higher step size possible that dThe best solution is to use the higher step size possible that do not o not compromise the information we need.compromise the information we need.Normally highly broadened peaks => big step size => less noise aNormally highly broadened peaks => big step size => less noise as we can s we can increase the collection time per step (> 0.05° in 2increase the collection time per step (> 0.05° in 2θθ))very sharp peaks => small step size (from 0.02 to 0.05 for Braggvery sharp peaks => small step size (from 0.02 to 0.05 for Bragg--Brentano)Brentano)

Total collection timeTotal collection time

Ensure the noise is lower than the intensity of small peaksEnsure the noise is lower than the intensity of small peaksIf the total collection time is limited, better a lower noise thIf the total collection time is limited, better a lower noise than a an a smaller step size.smaller step size.Better to collect a little bit more than to have to repeat an exBetter to collect a little bit more than to have to repeat an experiment.periment.If collection time is a problem go for 1D or 2D detectors:If collection time is a problem go for 1D or 2D detectors:

•• CPS 120: 2 to 5 minutes for a good spectrum of 120 degrees (goodCPS 120: 2 to 5 minutes for a good spectrum of 120 degrees (good for for quantitative phase analyses or follow reactions, transformationsquantitative phase analyses or follow reactions, transformations, , analyses in temperature)analyses in temperature)

•• Image plates or CCDs: very fast collection times when texture isImage plates or CCDs: very fast collection times when texture isneeded or is a problemneeded or is a problem

Data quality (not related to intensity) of these detectors is a Data quality (not related to intensity) of these detectors is a little bit little bit lower than the one from good point detectors. But sometimes lower than the one from good point detectors. But sometimes intensity rules!intensity rules!

Respecting statisticsRespecting statistics

In principle the measurement should be done at isoIn principle the measurement should be done at iso--statistical values:statistical values:

For practical reasons this is not always possible.For practical reasons this is not always possible.Scattering factors and LScattering factors and L--P effects decrease the intensity at high angle.P effects decrease the intensity at high angle.In many cases, peaks at low angles are more sensible to heavy atIn many cases, peaks at low angles are more sensible to heavy atoms and oms and peaks at high angles to light atoms.peaks at high angles to light atoms.A good strategy is to divide the range in different part and useA good strategy is to divide the range in different part and use a different a different collection time reducing the noise for the high angles part.

1Ii

collection time reducing the noise for the high angles part.

Sample characteristicsSample characteristics

The sample should be sufficiently large The sample should be sufficiently large iniini order that the beam is always order that the beam is always entirely inside its volume/surface.entirely inside its volume/surface.

Sample position is critical for good cell parameters (along withSample position is critical for good cell parameters (along with perfect perfect alignment of the instrument) determination.alignment of the instrument) determination.

The number of diffracting grains at each position should be signThe number of diffracting grains at each position should be significant (> ificant (> 10000 grains). Remember that only a fraction is in Bragg conditi10000 grains). Remember that only a fraction is in Bragg conditions and ons and diffract. Higher beam divergence or size increases this number. diffract. Higher beam divergence or size increases this number. So the So the sample should have millions grains in the diffracting volume.sample should have millions grains in the diffracting volume.

Grain statisticsGrain statistics

sufficient poor

Ambient conditionsAmbient conditions

In some cases constant ambient conditions are important:In some cases constant ambient conditions are important:•• temperature for cell parameter determination or phase transitiontemperature for cell parameter determination or phase transitionss•• humidity for some organic compounds or pharmaceuticalshumidity for some organic compounds or pharmaceuticals•• can your sample be damaged or modify by irradiation (normally Cocan your sample be damaged or modify by irradiation (normally Copper pper

or not too highly energetic radiations are not influencing much)or not too highly energetic radiations are not influencing much)

Special attachmentsSpecial attachments

Non classical Rietveld applicationsNon classical Rietveld applications

Quantitative analysis of crystalline/nonQuantitative analysis of crystalline/non--crystalline mixtures (crystalline mixtures (LutterottiLutterotti et al, 1997)et al, 1997)Using Le Bail model for amorphous (needs a pseudo crystal structUsing Le Bail model for amorphous (needs a pseudo crystal structure)ure)

Anisotropic Microstructure:Anisotropic Microstructure:

-- Le Bail, 1985. Profile shape parameters computed from the crystLe Bail, 1985. Profile shape parameters computed from the crystallite allite size and size and microstrainmicrostrain values (<M> and <values (<M> and <εε22>1/2)>1/2)

More stable than More stable than CagliotiCaglioti formulaformulaInstrumental function neededInstrumental function needed

-- PopaPopa, 1998 (J. , 1998 (J. ApplAppl. . CrystCryst. 31, 176). General treatment for anisotropic . 31, 176). General treatment for anisotropic crystallite and crystallite and microstrainmicrostrain broadening using harmonic expansion.broadening using harmonic expansion.-- LutterottiLutterotti & & GialanellaGialanella, 1998 (, 1998 (ActaActa Mater. 46(1), 101). Stacking, Mater. 46(1), 101). Stacking, deformation and twin faults (Warren model) introduced.deformation and twin faults (Warren model) introduced.

Rietveld Stress and Texture Analysis (RiTA)Rietveld Stress and Texture Analysis (RiTA)

•• Characteristics of Texture Analysis:Characteristics of Texture Analysis:•• Powder DiffractionPowder Diffraction•• Quantitative Texture Analysis needs single peaks for pole figureQuantitative Texture Analysis needs single peaks for pole figure meas.meas.•• Low symmetries Low symmetries --> too much > too much overlapedoverlaped peakspeaks•• Solutions: Groups of peaks (WIMV, done), peak separation (done)Solutions: Groups of peaks (WIMV, done), peak separation (done)

•• What else we can do? What else we can do? --> Rietveld like analysis?> Rietveld like analysis?•• 1992. Popa 1992. Popa --> harmonic method to correct preferred orientation in one > harmonic method to correct preferred orientation in one

spectrum.spectrum.•• 1994. Ferrari & Lutterotti 1994. Ferrari & Lutterotti --> harmonic method to analyze texture and > harmonic method to analyze texture and

residual stresses. Multispectra measurement and refinement.residual stresses. Multispectra measurement and refinement.•• 1994. Wenk, Matthies & Lutterotti 1994. Wenk, Matthies & Lutterotti --> Rietveld+WIMV for Rietveld Texture > Rietveld+WIMV for Rietveld Texture

analysis.analysis.•• 1997. GSAS got the harmonic method (wide acceptance?).1997. GSAS got the harmonic method (wide acceptance?).•• 2001. 2001. ChateignerChateigner & & LutterottiLutterotti: “Combined Analysis”, also incorporating : “Combined Analysis”, also incorporating

separate independent measurements, like reflectivityseparate independent measurements, like reflectivity

TextureTexture

From pole figures

Orientation Distribution Function (ODF)

From spectra

strains & stressesstrains & stresses

Fe Cu

• Macro elastic strain tensor (I kind)• Crystal anisotropic strains (II kind)

C

Macro and micro stresses

Applied macro stresses

Why the Rietveld refinement is widely used?Why the Rietveld refinement is widely used?

•• No single crystal availableNo single crystal available•• ProPro

•• It uses directly the measured intensities pointsIt uses directly the measured intensities points•• It uses the entire spectrum (as wide as possible)It uses the entire spectrum (as wide as possible)•• Less sensible to model errorsLess sensible to model errors•• Less sensible to experimental errorsLess sensible to experimental errors

•• ConsCons•• It requires a modelIt requires a model•• It needs a wide spectrumIt needs a wide spectrum•• Rietveld programs are not easy to useRietveld programs are not easy to use•• Rietveld refinements require some experience (1Rietveld refinements require some experience (1--2 years?)2 years?)

•• Can be enhanced by:Can be enhanced by:•• More automatic/expert mode of operation More automatic/expert mode of operation •• UserUser--friendly friendly softwaressoftwares (dangerous)(dangerous)