Short Rietveld Course, Atlanta May 2001 The Rietveld Methodw3.esfm.ipn.mx/~fcruz/ADR/Cristalografia/Roisnel-Carvajal/Rietveld... · May 2001 Short Rietveld Course, Atlanta Least squares:

May 2001 Short Rietveld Course, Atlanta

Short Rietveld Course, Atlanta May 2001

The Rietveld Method

What is Rietveld refinement?Structural and profile parameters,standard deviations and R-factors


Juan Rodríguez-CarvajalLaboratoire Léon BrillouinCEA/Saclay FRANCE


Juan Rodríguez-CarvajalService de Physique Statistique, Magnétisme et SupraconductivitéDRFMC-CEA/Grenoble, France


A powder diffraction pattern can be recorded in numerical form for a discrete set of scattering angles, times of flight orenergies. We will refer to this scattering variable as : T. The experimental powder diffraction pattern is usually given as two arrays :

The profile can be modelled using the calculated counts: yciat the ith step by summing the contribution from neighbouring Bragg reflections plus the background.

1,...,,i i i n

T y=

A powder diffraction pattern from the data treatment point of view


yi

Position “i”: Ti

Bragg position Th

yi-ycizero

Powder diffraction profile:scattering variable T: 2θ, TOF, Energy


profileThe model to calculate a powder diffraction pattern is:

( )h hh

ci i iy I T T b= Ω − +∑( ) 1x dx

+∞

−∞Ω =∫

Profile function characterized by its full width at half maximum (FWHM=H)and shape parameters (η, m, ...)

( ) ( ) ( )x g x f x instrumental intrinsic profileΩ = ⊗ = ⊗

The profile of powder diffraction patterns


The profile of powder diffraction patterns

( )h hh

c i iiy I T T b= Ω − +∑Contains structural information: atom positions, magnetic moments, etc( )h h II I= β

( , )h PixΩ = Ω β Contains micro-structural information: instr. resolution, defects, crystallite size, ..

( )Bi ib b= β Background: noise, diffuse scattering, ...


, ,h

( )h hci i iy s I T T bφ φ φφ

= Ω − +∑ ∑

Several phases (φ = 1,nφ) contributing to the diffraction pattern

, ,h

( )h h

p p p p pci i iy s I T T b

φφ φφ

= Ω − +∑ ∑

Several phases (φ = 1,nφ) contributing to several (p=1,np) diffraction patterns


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

22

1

( )n

i i cii

w y yχ β=

= −∑

21i

iwσ

=2iσ : is the variance of the "observation" yi


Least squares: Gauss-Newton (1)Minimum necessary condition:

A Taylor expansion of around allows the application of an iterative process. The shifts to be applied to the parameters at each cycle for improving χ2 are obtained by solving a linear system of equations (normal equations)

2

0∂=

∂χβ

( )icy β 0β

0

0 0

0

( ) ( )

( )( )

A b

ic ickl i

i k l

ick i i ic

i k

y yA w

yb w y y

β β

β

=

∂ ∂=

∂ ∂∂

= −∂

∑

∑

βδ

β β

β


Least squares: Gauss-Newton (2)

The new parameters are considered as the starting ones in the next cycle and the process is repeated until a convergence criterion is satisfied. The variance of the adjusted parameters are calculated by the expression:

The shifts of the parameters obtained by solving the normal equations are added to the starting parameters giving rise to a new set

01 0= + ββ β δ

1( ) ( )Ak kk

N - P+C

2 2ν

22ν

σ β χ

χχ

−=

=


Least squares: a local optimisation method

• The least squares procedure provides (when it converges) the value of the parameters constituting the local minimum closest to the starting point

• A set of good starting values for all parameters is needed

• If the initial model is bad for some reasons the LSQ procedure will not converge, it may diverge.


The structural information contained in the integrated

intensities


2h h

I jL pO ACF=Integrated intensities are proportional to the square of the structure factor F. The factors are: Multiplicity (j), Lorentz-polarization (Lp), preferred orientation (O), absorption (A), other “corrections” (C) ...

Integrated intensities


The Structure Factor contains the structural parameters

( ) ( ) 1

2h h h t rn

j j j jsj s

F O f T exp i Sπ=

= ∑ ∑

( , , ) ( 1, 2, ... )r j j j jx y z j n= =

( ) ( ) ( ), , , , 1, 2, ...h Ts s Gs

h k l h k l S s N= = =

sinexp( )j jT B2

2

θλ

= −


Structural Parameters(simplest case)

( , , )r j j j jx y z= Atom positions (up to 3nparameters)

jO Occupation factors (up to n-1 parameters)

jB Isotropic displacement (temperature) factors (up to n parameters)


Structural Parameters(complex cases)

As in the simplest case plus additional (or alternative) parameters:

• Anisotropic temperature (displacement) factors

• Anharmonic temperature factors

• Special form-factors (Symmetry adapted spherical harmonics ), TLS for rigid molecules, etc.

• Magnetic moments, coefficients of Fourier components of magnetic moments , basis functions, etc.


The peak shape function of powder diffraction patterns contains the Profile Parameters

(constant wavelength case)

( , ) ( , )h P h Pi ix T Tβ βΩ = Ω −

( ) 1x dx+∞

−∞Ω =∫

The cell parameters are included, through Th, within the profile function. They determine the peak positions in the whole diffraction pattern.


2( ) exp( )G GG x a b x= −

2

2 ln2 4ln2G Ga b

H Hπ= =

Gaussian function

Integral breadth:1

2 ln 2GG

Ha

πβ = =


2( )1

L

L

aL xb x

=+

2

2 4L La b

H Hπ= =

Lorentzian function

Integral breadth:1

2LL

Ha

πβ = =


Fwhm

BG

x0

x

I

Comparison of Gaussian and Lorentzian peak shapes of the same peak height “I” and same width “Fwhm”


Convolution properties of Gaussianand Lorentzian functions

1 2 1 2

2 21 2 1 2

( , ) ( , ) ( , )

( , ) ( , ) ( , )

L x H L x H L x H H

G x H G x H G x H H

⊗ = +

⊗ = +

( , ) ( , ) ( , , )L G L GL x H G x H V x H H⊗ =


( ) ( ) (1 ) ( )pV x L x G xη η′ ′= + −The pseudo-Voigt function

The Voigt function

( ) ( ) ( ) ( ) ( )V x L x G x L x u G u du+∞

−∞= ⊗ = −∫( ) ( , , ) ( , , )L G L GV x V x H H V x β β= =

( ) ( , , )pV x pV x Hη=


Mapping: Pseudo-Voigt ⇔ VoigtThomson-Cox-Hasting formulation

( , ) ( , )G LH F H Hη =

1( , ) ( , )G LH H F H η−=

5 4 3 2 2 3 4 5( 2.69269 2.42843 4.47163 0.07842 )G G L G L G L G L LH H H H H H H H H H H= + + + + +2 3

1.36603 0.47719 0.11116L L LH H HH H H

η = − +

2 30.72928 0.19289 0.07783LHH

η η η= + +

2 3 1/ 2(1 0.74417 0.24781 0.00810 )GHH

η η η= − − −


2 22tan tan

cosG

GIH U V Wθ θθ

= + + +

tancosL

YH X θθ

= +

Profile Parameters(simple cases)

Parameters controlling the Full-Width at half maximumU, V, W, IG, X, Y


2 2 2 22( (1 ) ) tan tan

cosG

G STIH U D V Wξ θ θθ

= + − + + +

[ ( )]( ) tancos

ZL ST

Y F SH X Dξ θθ

+= + +

Modeling the Gaussian and Lorentziancomponents of the profile function in terms of microstructural parameters


Profile R-factors used in Rietveld Refinements


R-factors and Rietveld Refinements (1)

, ,

,

100obs i calc i

ip

obs ii

y yR

y

−=

∑∑

1/ 22, ,

2,

100i obs i calc i

iwp

i obs ii

w y yR

w y

−

=

∑

∑

R-pattern

R-weighted pattern

1/ 2

2,

( )100expi obs i

i

N P CRw y

− + =

∑

Expected R-weighted pattern



Reduced Chi-square

Goodness of Fit indicator

2

2 wp

exp

RRνχ

=

wp

exp

RS

R=



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


Crystallographic R-factors used in Rietveld Refinements

, ,

,

' '100

' '

obs k calc kk

Bobs k

k

I IR

I

−=

∑∑

Bragg R-factor

, ,

,

' '100

' '

obs k calc kk

Fobs k

k

F FR

F

−=

∑∑

Crystallographic RF-factor.


Crystallographic R-factors used in Rietveld Refinements

Provides ‘observed’ integrates intensities for calculating Bragg R-factor

In some programs the crystallographic RF-factor is calculated using just the square root of ‘Iobs,k’

,, ,

,

( )( )' '

( )i k obs i i

obs k calc ki calc i i

T T y BI I

y B Ω − − = −

∑

,,

' '' ' obs k

obs k

IF

jLp=


Simulation of systematic errors in Rietveld refinements

•What is the effect of the resolution and the systematic errors in the profile peak shape function in the quality of the refined structural parameters?

•Are the estimated standard deviations a measure of the accuracy of the refined parameters?

•Are the R-factors good indicators of the quality of a structural model?

Important questions:


Complexity of a structural problem:effective number of reflections and solvability

indexIf one is interested in “structural parameters” the number of independent observations is not the number of points in the pattern N.What is the number of “independent” observations? (No rigorous answer …)Points to be considered:

• Signal-to-noise ratio, statistics.• Number of independent Bragg reflections: NB• Number of structural free parameters: NI=Nf• Degree of reflection overlap: resolution versus separation

between consecutive reflections.

• Effective number of observations (resolution weighted): Neff• “Solvability” index: ratio between the effective number of

observations and the number of structural parameters: r = Neff/NI


Effective number of reflections Neff

Two reflections separated by ∆(Q) can be discriminated properly if the following relation holds:

∆(Q) = 2π2j/(Q2Vo) ≥ p DQ

DQ is the FWHM in Q-space, p is of the order of the unity A single reflection at Qo contributes to Neff as 1/(1+Nn), where Nn is the number of reflections in the neighbourhood of Qo , verifying:

Qo - p DQ ≤ Qn ≤ Qo + p DQ

1,

11

B

effi N i

NN=

=+∑The formula for calculating Neff

is:


Plot of ∆Q and DQ versus Q


Simulation of systematic errors in Rietveld refinements (1)

(1) For a hypothetical (or real) compound characterised by a given crystal structure, scale factor (related to a “ counting time ”), resolution function, peak shape, angular range and background level, calculate a theoretical (deterministic) diffraction pattern: yi(theo)

(2) Nr versions of the “ observed ” diffraction pattern are generated by corrupting the deterministic pattern by a noise calculated with a generator of pseudo-random numbers according to the Poisson distribution : yi(obs) = Pyi(theo).

(3) The process is repeated for Nt different values of the “ counting time ” (scale factor), thus producing Nr x Ntobserved diffraction patterns for the same compound.

Method:


Simulation of systematic errors in Rietveld refinements (2)

(4) Each pattern is refined by the RM by using either the “ true ” model or a biased model (e.g. wrong peak shape).

(5) The values of the refined parameters are then compared to the true valued (bias and dispersion) and the computed standard deviations are compared to the empirical ones obtained from the Nr versions of the same diffraction pattern, as a function of the counting time.

(6) Finally we study the behaviour of the R-factors versus counting time for the different conditions of refinement.

(7) Refine the diffraction patterns without structural model to obtain the “ expected ” R-factors for the best structural model by profile-matching iterating the Rietveld formula for the Bragg R-factor (Lebail fit)


Results of simulations: Refinements of simulated

powder diffraction patterns using correct and biased models


0

2000

4000

6000

8000

10000

0.0 25.0 50.0 75.0 100.0 125.0 150.0

Refinement of simulated patternwith biased peak shape and correct

structural model.NIntdp=39, Nprof=5, Nref=1507

Neff=(70, 137, 248)/Solv=(1.8,3.5,6.4)

Inte

nsity

(a.u

.)

2Θ (°)

RM (systematic errors): Biased peak shape+ correct structural model NI=39 + bad resolution


0

4000

8000

12000

0.0 25.0 50.0 75.0 100.0 125.0 150.0

Refinement of simulated patternwith biased peak shape and correct

structural model.NIntdp=39, Nprof=5, Nref=1507

Neff=(185, 352, 611)/Solv=(4.8,9.0,15.7)

Inte

nsity

(a.u

.)

2Θ (°)

RM (systematic errors): Biased peak shape+ correct structural model NI=39 + better resolution


Behaviour of Rietveld R-factors, and other indicators,

versus counting statistics for perfect and biased models


0

5

10

15

20

25

30

35

40

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

R-factors (RP, R

WP, R

Bragg and R

F)

cRp(B)cRwp(B)RbRF

R(%

)

Log(Counting Time)

NIntdp=16, Nprof=8, Nref=393Neff=(173, 244, 301)

Correct model

RM (systematic errors): Behaviour of R-factors versus counting time (correct model)


RM (systematic errors): Behaviour of RWP factors versus counting time (biased peak shape)

0

10

20

30

40

50

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

R-factors (RWP

) RwpcRwpRwp(B)cRwp(B)R

WP(%

)

Log(Counting Time)

NIntdp=16, Nprof=8, Nref=393Neff=(173, 244, 301)Biased peak shape


0

2

4

6

8

10

12

14

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

R-factors (RBragg

and RF)

RbRF

R(%

)

Log(Counting Time)


RM (systematic errors): Behaviour of RBragg and RFfactors versus counting time (biased peak shape)


0,50

1,00

1,50

2,00

2,50

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

Durbin-Watson DW DWDW(S)

Dur

bin-

Wat

son

stat

istic

s

Log(Counting Time)


DW(S): Biased peak shape

RM (systematic errors): Behaviour of DW indicator versus counting time (biased/unbiased peak shape)


RM (systematic errors): Behaviour of reduced Chi-square versus counting time (correct model)

0,600

0,800

1,000

1,200

1,400

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

R-factors(χ 2)

Chi2Chi2(B)

Red

uced

χ2

Log(Counting Time)


Correct model


0

10

20

30

40

50

60

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

R-factors(χ2)

Chi2Chi2(B)

Red

uced

χ2

Log(Counting Time)


RM (systematic errors): Behaviour of reduced Chi-square versus counting time (biased peak shape)


Behaviour of positional parameters (atom co-ordinates), versus counting

statistics, resolution, ... for perfect and biased models


RM (systematic errors): Behaviour of positional parameters versus counting time (biased peak shape)

0,1380

0,1400

0,1420

0,1440

0,1460

0,1480

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

Coordinate "x" of the first atom x1

Red

uced

coo

rdin

ate

Log(Counting Time)



Atom co-ordinates versus counting statistics


-0,0300

-0,0200

-0,0100

0,0000

0,0100

0,0200

0,0300

0 2 4 6 8 10 12 14

NIntdp=16, Nprof=8, Nref=393Neff=(173, 244, 301)Biased peak shapeLow counting time

Del

ta (Å

)

Positional Parameter

RM (systematic errors): Positional parameters NI=16 Low counting time (biased peak shape)


RM (systematic errors): Positional parameters, NI=16 Intermediate counting time (biased peak shape)

-0,0300

-0,0200

-0,0100

0,0000

0,0100

0,0200

0,0300

0 2 4 6 8 10 12 14


Intermediate counting time

Del

ta (Å

)



RM (systematic errors): Positional parameters, NI=16 High counting time (biased peak shape)

-0,0300

-0,0200

-0,0100

0,0000

0,0100

0,0200

0,0300

0 2 4 6 8 10 12 14


High counting time

Del

ta (Å

)



Atom co-ordinates versus resolution (“solvability index”)

r=Neff / NI


RM (systematic errors): Positional parameters NI=39, Counting Time: 12500 (correct model, r=3.5)

-0,0400

-0,0200

0,0000

0,0200

0,0400

0 5 10 15 20 25 30 35 40

NIntdp=39, Nprof=5, Nref=1507Neff=(70, 137, 248)/Solv=(1.8,3.5,6.4)

Correct model-(Counting time:12500)

Del

ta (Å

)



-0,0400

-0,0200

0,0000

0,0200

0,0400

0 5 10 15 20 25 30 35 40

NIntdp=39, Nprof=5, Nref=1507Neff=(102, 192, 362)/Solv=(2.6,4.9,9.3)Correct model-(Counting time:12500)

Del

ta (Å

)




RM (systematic errors): Positional parameters NI=39, Counting Time: 12500 (correct model, r=9)

-0.0400

-0.0200

0.0000

0.0200

0.0400

0 5 10 15 20 25 30 35 40


Del

ta (Å

)



Atom co-ordinates versus statistics (correct model, NI=72)



-0,3000

-0,2000

-0,1000

0,0000

0,1000

0,2000

0,3000

0 10 20 30 40 50 60 70 80


Correct model-(Counting time:125)

Del

ta (Å

)




-0,1000

-0,0500

0,0000

0,0500

0,1000

0 10 20 30 40 50 60 70 80


Del

ta (Å

)




-0,1000

-0,0500

0,0000

0,0500

0,1000

0 10 20 30 40 50 60 70 80


Del

ta (Å

)




-0,1000

-0,0500

0,0000

0,0500

0,1000

0 10 20 30 40 50 60 70 80


Del

ta (Å

)



Atom co-ordinates versus statistics (biased peak shape, NI=39)


-0,1000

-0,0500

0,0000

0,0500

0,1000

0 5 10 15 20 25 30 35 40


Biased peak shape-(Counting time:12500)

Del

ta (Å

)


RM (systematic errors): Positional parameters NI=39 Counting Time: 12500 (biased peak shape, r=9)


RM (systematic errors): Positional parameters NI=39 Counting Time: 12500 (biased peak shape, r=4.9)

-0,1000

-0,0500

0,0000

0,0500

0,1000

0 5 10 15 20 25 30 35 40



Del

ta (Å

)



-0,1000

-0,0500

0,0000

0,0500

0,1000

0 5 10 15 20 25 30 35 40



Del

ta (Å

)


RM (systematic errors): Positional parameters NI=39 Counting Time: 12500 (biased peak shape, r=3.5)


Conclusions (1)

(1) When the refined model is unbiased (no systematic errors), least squares estimates of the standard deviations represent not only the precision but also the accuracy. For biased models (usual practical situation) the standard deviations do not represent accuracy.

(2) For biased models (a known systematic error in profile is introduced) in simple structures some fitted parameters (profile and thermal parameters) show a bias while others are barely affected(position parameters).

(3) Concerning accuracy, nothing is gained repeating the measurement several times if the systematic error is expected to be the same. The standard deviation obtained from different measurement is not, in such a case, a measure of the accuracy.


Conclusions (2)

(4) For complex structures high statistical accuracy and high resolution is required for getting the true parameter values even is the refined model is unbiased. Suggestion: the solvability index (r = Neff/Nf for p=1/2) should be largely greater than 4-5 to be sure that the structural parameters are accurate enough. More experience is needed to establish precise rules.

(5) 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 the structural model. The R-factors obtained by a refinement of the whole pattern without structural model provide the “expected” values for the best structural model.

(6) Well behaved peak shape could be more important than resolution in some cases.

Documents

Short Rietveld Course, Atlanta May 2001 The Rietveld Methodw3.esfm.ipn.mx/~fcruz/ADR/Cristalografia/Roisnel-Carvajal/Rietveld... · May 2001 Short Rietveld Course, Atlanta Least squares: