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Rietveld Analysis of X-ray and neutron diffraction patterns

Analysis of the whole diffraction pattern Profile fitting is included Not only the integrated intensities

Refinement of the structure parameters from diffraction data Quantitative phase analysis Lattice parameters Atomic positions and occupancies Temperature vibrations Grain size and micro-strain (in the recent versions)

Not intended for the structure solution The structure model must be known before starting the Rietveld

refinement

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Non-refinable parameters in the Rietveld method

Space group Chemical composition Analytical function describing the shape of the

diffraction profiles Wavelength of the radiation (can be refined in Fullprof

or in LHRL; suitable for the synchrotron data) Intensity ratio in K1, K2 doublet Origin of the polynomial function describing the

background

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Rietveld analysis

History

H.M. Rietveld - neutron data, fixed wavelength

D.E. Cox - X-ray data R.B. Von Dreele - neutron data, TOF D.B. Wiles & R.A. Young - X-ray data,

2 wavelengths, more phases Helsinki group - spherical functions for

preferred orientation but a single wavelength

Fullprof, LHRL - surface absorption BGMN - automatic calculation,

crystallite size and microstrain in form of ellipsoids

P. Scardi et at - size, strain

Computer programs

H.M. Rietveld DBW2.9, DBW3.2 (Wiles &

Young) University of Helsinki Fullprof (J. Rodriguez-Carvajal) BGMN (R. Bergmann) LHRL (C.J. Howard & B.A. Hunter) P. Scardi et al.

Bärlocher GSAS

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Integral intensity

Calculated intensity:

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

The intensity of the Bragg reflections

p k

kpikibic IGyy

kkkkkkk EAPFLSmI2

5

Scattering by one elementary cell

Structure factor

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

khhkkh

zkyhxifNF

ifNF

n

jjjjjjk

n

jkj

tkj

tkjjk

2313122

332

222

11

1

1

2

222exp

2exp

22exp

hBhrh

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Temperature vibrations

Atomic displacement (in Cartesian co-ordinates)

c

bb

aaa

uuuuu

uuuuu

uuuuu

t

tjjj

00

cossin10

coscot1

;2

1 **

***

2

233231

322221

312121

FβFFB

uuB

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Crystal symmetry restrictions

Six anisotropic temperature factors per atom in a general case (symmetrical matrix)

For an atom in a site of special symmetry the B-matrix must be invariant to the symmetry operations (in the Cartesian axis system)

An example - rotation axes parallel with z

BBPP t

100

0cossin

0sincos

P

8

Temperature vibrations - special cases

Isotropic atomic vibrations

Overall temperature factor

jj

n

jjjjjjjk

uB

BzkyhxifNF

22

12

2

8

sinexp2exp

n

jjjjjjk zkyhxifNuF

12

222 2exp

sin8exp

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Scattering by one atom

Atomic scattering factor

a, b, c are from the “International Tables for Crystallography” f’, f” must be checked and changed for synchrotron radiation

Another possibility Include our set of the atomic scattering factors

ffcbafi

ii

4

12

2sinexp

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Preferred orientation of grains (texture)

Gauss-like distribution

March-Dollase correction

Spherical functions

kk

kk

kk

GGGP

GGGP

GGGP

3122

2122

2122

sinexp1

sinexp1

exp1

2

3

2

1

221 sin

1cos

kkk G

GP

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Absorption correction

For flat samples - micro-absorption and surface absorption (Hermann & Ermrich)

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

1sin

1sin

11

)(1

00

0

P

PPA sk

0.0 0.1 0.2 0.3-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

# 1

# 2

ln (

Inte

ns

ity

ra

tio

)

(sin/)2

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Absorption correction

For thin samples (powder on glass) in symmetrical arrangement

thick sample, high absorption

thin sample, low absorption

t A : ( )1 2

t A t 0: sin0.00 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

experimental data

absorption factor

apparent temperature

log

(In

ten

sit

y r

ati

o)

(sin/)2

sin

2exp1

2

10

tII

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Extinction correction(for large crystallites)

Extinction for the Bragg case (= 90)

Extinction for the Laue case (= 0)

kLkBk EEE 22 sincos

xEB

1

1

1for1024

15

128

3

8

11

2

1for48

5

421

32

32

xxxxx

xxxx

EL

2

e

k

V

FDx

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Profile functions

Gauss

Lorentz (Cauchy)

Pearson VII

Pseudo-Voigt

2ln4;22exp 02

200

C

CCG ki

kk

4;

221

120

2

20

0

CC

CL

kik

k

5.0

122;22

1241

2

1

02

20

m

mC

CP

mm

kik

m

kVII

GLpV 1

WVU kkk tantan22

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Background

Subtraction of the background intensities

Interpolation of the background intensities

Polynomial function (six refinable parameters) Origin of the background - improves the pivoting of the

normal matrix

A special function for amorphous components

n

m m

mmib QB

QBBQBBy

1 12

12210

sin

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Minimisation routine

Uses the Newton-Raphson algorithm to minimise the quantity

Normal matrix

iii

icioi ywyywR 1;2

iicio

m

icim

i n

ic

m

icimn

yyx

ywy

x

y

x

ywM

0

yxM

PN

yywM i

icioi

mmm

2

1

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Reliability factors

The profile R-factor ………

The weighted Rp ………………………………………

The Bragg R-factor ………

The expected Rf ………………………………………

The goodness of fit

iio

iicio

p y

yyR

2

1

2

2

iioi

iicioi

wp yw

yywR

iko

ikcko

B I

IIR

2

1

2exp

i

ioi yw

PNR

2

exp

2

R

R

PN

yywGOF wpi

icioi

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Connecting parameters, constrains

Young - parameter coupling Coding of variables: number of the parameter in the normal

matrix + weight for the calculated increment Lattice parameters in the cubic system: 41.00 41.00 41.00 Fractional co-ordinates at 12k in P63/mmc, (x 2x z): 20.50 21.00

31.00

Fullprof - constrains Inter-atomic distances may be constrained

BGMN - working with molecules Definition of the molecule (in Cartesian co-ordinates) Translation and rotation of the whole molecule

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Structure of the input file(Fullprof for anglesite)

COMM PbSO4 D1A(ILL),Rietveld Round Robin, R.J. Hill,JApC 25,589(1992) !Job Npr Nph Nba Nex Nsc Nor Dum Iwg Ilo Ias Res Ste Nre Cry Uni Cor 1 7 1 0 2 0 0 0 0 0 0 0 0 0 0 0 0!!Ipr Ppl Ioc Mat Pcr Ls1 Ls2 Ls3 Syo Prf Ins Rpa Sym Hkl Fou Sho Ana 0 0 1 0 1 0 0 0 0 1 6 1 1 0 0 1 1!! lambda1 Lambda2 Ratio Bkpos Wdt Cthm muR AsyLim Rpolarz 1.54056 1.54430 0.5000 70.0000 6.0000 1.0000 0.0000 160.00 0.0000!NCY Eps R_at R_an R_pr R_gl Thmin Step Thmax PSD Sent0 5 0.10 1.00 1.00 1.00 1.00 10.0000 0.0500 155.4500 0.000 0.000!! Excluded regions (LowT HighT) 0.00 10.00 154.00 180.00! 34 !Number of refined parameters!! Zero Code Sycos Code Sysin Code Lambda Code MORE -0.0805 81.00 0.0000 0.00 0.0000 0.00 0.000000 0.00 0! Background coefficients/codes 207.37 39.798 65.624 -31.638 -90.077 47.978 21.000 31.000 41.000 51.000 61.000 71.000

! Data for PHASE number: 1 ==> Current R_Bragg: 4.16 PbSO4 !Nat Dis Mom Pr1 Pr2 Pr3 Jbt Irf Isy Str Furth ATZ Nvk Npr More 5 0 0 0.0 0.0 0.0 0 0 0 0 0 0.00 0 7 0P 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 0 11.00000 0.00 0.00 0.00 0.00 0.00! U V W X Y GauSiz LorSiz Size-Model 0.15485 -0.46285 0.42391 0.00000 0.08979 0.00000 0.00000 0 121.00 131.00 141.00 0.00 151.00 0.00 0.00! a b c alpha beta gamma 8.480125 5.397597 6.959482 90.000000 90.000000 90.000000 91.00000 101.00000 111.00000 0.00000 0.00000 0.00000! Pref1 Pref2 Asy1 Asy2 Asy3 Asy4 0.00000 0.00000 0.28133 0.03679-0.09981 0.00000 0.00 0.00 161.00 331.00 341.00 0.00

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Quantitative phase analysis

Volume fraction

Weight fraction

Use the correct occupanciesUse the correct occupancies : N = occupancy / max # of Wyckoff positions

ppe

e

SV

SVV

2

2

ppe

e

SZMV

SZMVm

2

2

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Tips and tricks (on the course of the refinement)

Instrumental parameters

Scale factor (always) Background (1) Line broadening and shape

(3) Zero shift (4) Sample displacement or

transparency (5) Preferred orientation (7) Surface absorption (7) Extinction (7)

Structure parameters

Scale factor (always) Lattice parameter (2) Atomic co-ordinates (6) Temperature factors (8) Occupancies (8), N =

occ/max(N) important for quantitative phase analysis

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Tips and tricks (how to obtain reliable data)

Use only good adjusted diffractometer Bad adjustment causes the line shift and broadening; the

latter cannot be corrected in the Rietveld programs Use only fine powders

Coarse powder “randomises” the integral intensities Coarse powder causes problems with rough surface

Use sufficient counting time The error in intensity is proportional to sqrt(N) as for the

Poisson distribution Apply dead-time correction

For strong diffraction lines, the use of the dead-time correction is strongly recommended

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Effect of the grain size

Variations in observed intensities (bad statistics)

Figure: Effect of specimen rotation and particle size on Si powder intensity using conventional diffractometer and CuK radiation.

International Tables for Crystallography, Vol. C, ed. A.J.C. Wilson, Kluwer Academic Publishers,

1992.

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In the Rietveld refinement don’t

refine parameters which are fixed by the structure relations (fractional co-ordinates, lattice parameters)

refine all three parameters describing the line broadening concurrently

refine the anisotropic temperature factors from X-ray powder diffraction data

use diffraction patterns measured in a narrow range forget that the number of structure parameters being

refined cannot be larger than the number of lines

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Corundum

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Auxiliary methodsand computer programs

The most critical parameters for the convergence of the Rietveld refinement - lattice parameters

FIRESTAR

Only the crystal system must be known (not the space group) The diffraction pattern must be indexed

*cos**2*cos**2

*cos**2***1 2222222

ahccbk

bhkacbkahdhkl

2

2

sin21

hkld

min1

2sin

2

2

i hkli d

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Problems with positions of diffraction lines

Residual stresses in bulk materials Anisotropic deformation of crystallites (anisotropy of mechanical

properties) Presence of errors in the structure (stacking faults, …)

Use of the programs working with net integral intensities (POWOW, POWLS) is recommended

How to get the net intensities? Numerical integration (not for the overlapped lines) Profile fitting using analytical functions (for overlapped lines) -

DIFPATAN

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Indexing of the diffraction patternin unknown phases

Computer program TREOR (Trials and Errors)

Requirements A single phase in the specimen High-quality data (particularly, the error in the positions of

diffraction lines must not exceed 0.02° in 2) Very good alignment of the diffractometer or the use of an

internal standard (mixed to the specimen)