An Introduction to the Rietveld Method › workshops › 08 › Introduction to Rietveld 1.pdf ·...

transcript

An Introduction to the Rietveld Method

James A. KadukSenior Research Associate

Analytical Sciences Research ServicesINEOS Technologies

James.Kaduk@ineos.com

Use all the crystallography and diffraction physics we know to model the powder pattern

( )2hkl o c

hklw F k F−∑ ( )2

i oi cii

w y y−∑

Quantity Minimized in the Least Squares

Single Crystal vs. Rietveld

What we’ll try to do

Introduction to practiceUnderstanding the instrumentCommon global parametersQuality measuresMinimize the fear

What we’ll not try to cover

Basic crystallographyLeast-squares theoryDiffraction theoryProgram tutorial (GSAS, Fullprof, Topas,

Riquas, HighScore Plus…)

The Rietveld Method is a refinement technique, not a

structure solution method. A good starting model is required!

How good is good?

• All atoms within 0.5-0.6 Å of their true positions, but…

• x% of the total scattering power

How do you get the model?

• Powder Diffraction File (PDF/LPF)• Inorganic Crystal Structure Database (ICSD)• Metals Data File (CRYSTMET)• Cambridge Structural Database (CSD)• (Protein Data Bank) (PDB)• Crystal Data Identification File (CDIF)• Primary literature• ab initio structure determination

What determines the intensities?(1) Structure Factors

Atomic positionsOccupanciesAtomic scattering factorsDisplacement coefficientsLattice parametersSymmetry

What determines the intensities?(2) Global Parameters

ConcentrationIncident intensityBackgroundDiffuse scatteringExtinctionAbsorption

Preferred orientationMultiplicityLp factorProfile functionDiffractometer parameters…

To get accurate results, we must model all these quantities correctly!

The advantage of the Rietveld method is that it uses all the

information in the powder pattern, and yields the most information.

We need to remember that we are fitting a model to data, and that

our answers will only be as good as the model is appropriate.

Two StepsPreparation and Refinement

In some programs (like GSAS) these two steps are separate,and in others, they are combined into a single operation.

The Sample

• Need a powder, but…• Random is best, but…• Resolution – more is better, but can generate

size/strain by trying to get powder• Phase purity• An advantage of the Rietveld method is that

ideal samples are rare, and the method provides a way of dealing with real samples

The Instrument

• Alignment/systematic errors • (zero, shift, trns)• Wavelength(s)• Profile function

Data Collection• Compromises!• Fixed step sizes (but “new” GSAS format)• Wide 2θ range• ≥ 5 steps across FWHM of sharpest peaks• Constant or variable counting time• Step or continuous scan• ~10,000 counts for strongest peaks• Programs assume fixed slits!

Background

• Crucial to get right – affects integrated intensities (and thus the structure) – especially the displacement coefficients

• Interacts with the profile function• Peak tails• Use a few parameters as possible• Crystalline sample – slowly varying• Background parameters are highly-correlated

Background Functions

Shifted Chebyshev Polynomials of the First Kind

b j jj

I B T x−=

=∑T0(x) = 1, T1(x) = x, T2(x) = 2x2-1, Tn+1(x) = 2xTn(x) - Tn-1(x)

max min

2(2 2 ) 12 2

x θ θθ θ

−= −

Cosine Fourier Series

cos[ ( 1)]N

I B B x j=

= + −∑

Polynomial

I BBKPOS

⎛ ⎞= −⎜ ⎟⎝ ⎠

Diffuse ScatteringThe Debye Equation

4 sinsin

( ) ( ) 2 ( ) ( ) 4 sin

n i jijn i j

I f f f r

π θλ

θ θ θ θ π θλ

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥= +⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∑∑

P. Debye, Annalen der Physik, 351, 809 (1915)

Diffuse Scattering in GSAS (#1)

2sin( ) 1exp2DS

RQI A UQRQ

⎛ ⎞= −⎜ ⎟⎝ ⎠

Q = 2π/d

GSAS Diffuse Scattering Function #1 Terms

2θ, deg

0 20 40 60 80 100 120 140 160

R = 1.6, U = 0.05R = 2.4, U = 0.05R = 1.6, U = 0.2

Differences in Diffuse Scattering Terms

2θ, deg

0 20 40 60 80 100 120 140 160

1.6/0.05-2.4/0.051.6/0.2-2.4/0.2

χ2 = 1.112

Profile Coefficients

• Crucial to getting the right answers• Structure/intensities/overlap/tails• Valuable information in profile coefficients

X-ray Profile Functions

• Gaussian• Lorentzian (Cauchy)• Modified Lorentzians• Split Pearson VII• (pseudo)Voigt• Empirical “learned”• Stacking fault model – DIFFax, …• Fundamental parameters

Instrument Profile Function

• Some programs (GSAS) require one• Helps interpret refined values• Use a sample free of size and strain broadening

– SRM 660a/b (LaB6)– SRM1976/a (corundum plate)– SRM 640c/d (Si)

Peak Position Errors Bruker D8 AdvanceSRM 660a LaB6

New Tube 13 August 2006

X Data

0 20 40 60 80 100 120 140 160

KADU1044Δ2θ = 0.009(2) + 0.012(2)cosθ + 0.016(2)sin2θ

Bruker D8 Advance Resolution FunctionSRM 660a LaB6

New Tube 13 August 2006

X Data

0 20 40 60 80 100 120 140

KADU1044FWHM = 0.0006(295)tan2

θ + 0.021(4)tanθ + 0.0441(295) + 0.005(29)/cos2θ

Bruker D8 Advance Peak ShapesPearson VII Exponent

SRM 660a LaB6 New Tube 13 August 2006

2θ, deg

0 20 40 60 80 100 120 140 160

123456789012345678901234567890123456789012345678901234567890 INS BANK 1 INS HTYPE PXCR INS 1 IRAD 3 INS 1 ICONS 1.540629 1.544451 -0.990 0 0.5 0 0.5 INS 1I HEAD NIST SRM 660a LaB6 VANTEC-1 0.3 mm div slit 29 Apr 2004 INS 1I ITYP 0 5.0000 150.0000 1 INS 1PRCF1 2 18 0.01 INS 1PRCF11 0.287900E+00 0.000000E+00 1.124000E+00 2.477000E+00 INS 1PRCF12 2.103000E+00 0.442000E+00 2.052000E+00 -4.818000E+00 INS 1PRCF13 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF14 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF15 0.000000E+00 0.000000E+00INS 1PRCF2 3 13 0.01 INS 1PRCF21 0.336500E+00 0.000000E+00 1.032000E+00 0.000000E+00 INS 1PRCF22 2.526000E+00 2.051000E+00 0.269500E-01 0.005000E-01 INS 1PRCF23 0.444100E+00 -5.024000E+00 0.000000E+00 0.000000E+00INS 1PRCF24 0.000000E+00 INS 1PRCF3 4 12 0.01 INS 1PRCF31 2.000000E+00 -2.000000E+00 5.000000E+00 0.000000E+00 INS 1PRCF32 0.100000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF33 0.000000E+00 0.150000E-01 0.150000E-01 0.750000E+00

Peak Position Errors, PANalytical X'Pert Pro MPDSRM 660a, LaB6, 25 January 2008

1/2 deg divergence, 0.02 rad Sollers

2θ, deg

0 20 40 60 80 100 120 140 160

LAB6_05x column 2 vs y column 2

PANalytical X'Pert Pro MPD Resolution FunctionSRm 660a LaB6 25 January 2008

2θ, deg

0 20 40 60 80 100 120 140 160

LAB6_05FWHM = -0.016(57) + 0.0537(32)/cosθ, r2 = 0.93

PANalytical X'Pert Pro MPD Peak ShapesSRM 660a LaB6 25 January 2008

2θ, deg

0 20 40 60 80 100 120 140 160

123456789012345678901234567890123456789012345678901234567890 INS BANK 1 INS HTYPE PXCR INS 1 IRAD 3 INS 1 ICONS 1.540629 1.544451 -0.990 0 0.5 0 0.5 INS 1I HEAD NIST SRM 660a LaB6 PIXCEL 1/2 deg div 0.02 Soller 25 Jan 2008 INS 1I ITYP 0 5.0000 150.0000 1 INS 1PRCF1 2 18 0.01 INS 1PRCF11 0.756500E+00 0.000000E+00 3.646000E+00 2.428000E+00 INS 1PRCF12 1.906000E+00 1.308000E+00 1.063000E+00 0.000000E+00 INS 1PRCF13 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF14 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF15 0.000000E+00 0.000000E+00INS 1PRCF2 3 13 0.01 INS 1PRCF21 1.153000E+00 -0.928000E+00 4.161000E+00 0.000000E+00 INS 1PRCF22 2.472000E+00 1.814000E+00 0.157700E-01 0.005000E-01 INS 1PRCF23 1.232000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF24 0.000000E+00 INS 1PRCF3 4 12 0.01 INS 1PRCF31 2.000000E+00 -2.000000E+00 5.000000E+00 0.000000E+00 INS 1PRCF32 0.100000E+00 0.000000E+00 0.000000E+00 0.000000E+00INS 1PRCF33 0.000000E+00 0.150000E-01 0.150000E-01 0.750000E+00

GSAS Profile Function #2 (3-5)pseudo-Voigt

2 22tan tan

cosPU V Wσ θ θθ

= + + +

( )cos cos tancos

X ptec Y stecϕγ ϕ θθ

+= + +

2 cos sin 2tan 2

if asymzero shift trnsθ θ θθ

⎛ ⎞Δ = + + +⎜ ⎟⎝ ⎠

Size Broadening

18000( )iso

18000( )inst

KLX ptec X

=+ −

18000( )inst

λπ⊥ = −

Strain Broadening - Isotropic

( )100%18000 instS Y Yπ

Strain Broadening - Anisotropic

( )100%18000 instS Y stec Yπ

= + −

( )100%18000 instS Y Yπ

⊥ = −

( ) 100%18000

H K LS

dS hkl h k lπ= ∑

Constant Microstrain Surface

How do you know when you’re finished?

“A Rietveld refinement is never finished, only abandoned”

P. W. Stephens

(or when you’ve answered the question you set out to answer)

J. A. Kaduk

There is no one measure of the quality of a Rietveld refinement!

Statistical Measures

( )22 2/wp i oi ci i oii i

R w y y w y= −∑ ∑

/p oi ci oii i

R y y y= −∑ ∑

2exp 2

N PRw y−

( )1( )

y yN P y

χσ−

=− ∑

χ⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

At the end of refinement

• χ2 < 1: the σ(yoi) are wrong• χ2 = 1: the errors are dominated by statistics• χ2 > 1:

– The model is reasonable, but the σs are underestimated

– The model contains systematic errors– The model is wrong

Graphical Measures

The Rietveld Plot

The Weighted Plot

The Error Plot

The Normal Probability Plot

Low Angle Plot

High Angle Plot

Background Plot

Cumulative χ2 Plot

A Case Study

Effluorescenceat the base of a door

601 Door Effluorescence

00-041-1476> Sylvite - KCl01-070-2509> Halite - NaCl

01-078-1064> Trona - Na3H(CO3)2(H2O)2

10 20 30 40 50 60 70 80 90Two-Theta (deg)

[kadu1220.raw] 601 door effluorescence (40,40,0.3) JAK

Use 17-100°. Refine 3 phase fractions and 3 background terms.

Get in the right neighborhood, then deal with the largest errors.

χ2 = 71.61

Add shift and 3 cells.

χ2 = 29.75

Add profile X and Yfor all three phases.

X < 0 for phases 1 and 2.Phase 3 Y = 7.1(23).

Fix all three at instrumental values.

χ2 = 10.58

Add a common Uiso for KCl.

χ2 = 8.619Uiso = 0.075!

Add five C-C diffuse scattering terms.

χ2 = 5.94

Has the feel of granularity, so micronize the sample.

00-041-1476> Sylvite - KCl01-070-2509> Halite - NaCl

01-078-1064> Trona - Na3H(CO3)2(H2O)2

10 20 30 40 50 60 70 80 90Two-Theta (deg)

[kadu1220.raw] 601 door effluorescence (40,40,0.3) JAK[kadu1221.raw] 601 door effluorescence, micronized (40,40,0.3) JAK

Copy KADU1220.EXP to KADU1221.EXP, change the data file, and add profile Y for phase 3.

χ2 = 3.789Uiso = 0.0290(4) Å2

Extra phase(s)?

01-077-0114> Aluminum - Al(OH)3

00-001-0909> Nahcolite - NaHCO3

00-046-1045> Quartz - SiO2

10 20 30 40 50 60Two-Theta (deg)

[kadu1221.raw] 601 door effluorescence, micronized (40,40,0.3) JAK

Add trona 2nd-order preferred orientation.

χ2 = 3.040

Corundum?

Quantitative Phase Analysis

43.0(1)31.5(1)25.4(1)wt%

TronaHaliteSylvitePhase

Profile Y before and after micronizing

0.21(1)0.62(1)0.26(1)Strain, %

14.0(10)37.5(2)17.1(1)Micronized

2.10326.1(3)13.8(2)Hand ground

TronaHaliteSylvitePhase

A Case Study

IUCr CPD QPA RRSample 1H

Sample 1H

29.4(2)34.5(1)36.1(1)INEOS

30.0334.2635.35XRF

30.1934.6935.12Weight, %

ZinciteZnO

FluoriteCaF2

Corundumα-Al2O3

QPA by the Rietveld Method

S M ZXSMZ

α α αα =

“Star” Quality Structures from the PDF-4+ 2007

04-008-819704-005-4266 (I)04-004-2852PDF Entry

⅓,⅔,0.38190.006

¼,¼,¼0.004

0.30624,0,¼0.003

Atom 2, xyzUiso

⅓,⅔,00.007

0,0,00.005

0,0,0.352160.003

Atom 1, xyzUiso

3.2505.207

5.4634.760

12.993Cell a

P63mcFm mR cSpace Group

ZinciteFluoriteCorundumPhase

Use 20-148°. Three phase fractions and 3 background terms.

χ2 = 19.08

Change to 25-148°. Add 3 cells and shift.

χ2 = 17.54

Add profile X and Yfor each phase.

X = 2.428, Y = 1.906

χ2 = 2.835

Change peak cutoffs from 1% to 0.2%.

Add the structural parameters.

28.43(7)33.67(8)37.90(16)Observed

30.1934.6935.12Expected, wt%

ZinciteFluoriteCorundumPhase

AKLD = 0.055

Absolute Concentration ErrorsIUCr CPD QPARR Samples 1n

2008 Results, Triplicate Analyses

Expected Concentration, wt%0 20 40 60 80 100

Al2O3CaF2ZnO

Another Case Study: Alka-Seltzer

02-061-2110> C6H8O7 - Citric acid

02-060-0435> C9H8O4 - 2-(Acetyloxy)-benzoic acid

00-015-0700> Nahcolite - NaHCO3

10 20 30 40 50 60 70Two-Theta (deg)

[alka-seltzerm.raw] Alka-Seltzer, micronized (40,40,0.3) JAK

The package says Alka-Seltzereach tablet contains:

30.861000C6H8O7

Citric acid

10.03325C9H8O4

Acetylsalicylic acid

59.121916NaHCO3

Sodium bicarbonate

Concentration, wt%Amount, mgCompound

Sum = 3241 mg. The tablet used weighed 3223.3 mg (99.45%).

Refine 3 background terms and 3 phase fractions

χ2 = 139.9

To get peaks in the right places, refine cells and a common

specimen displacement term

χ2 = 96.48

Add profile Y (strain) for each phase

χ2 = 22.44

Change to anisotropic strain broadening (Profile #4)

for NaHCO3 and citric acid

χ2 = 16.10

Add Uiso for NaHCO3(guessed initially)

Add diffuse scattering terms for Kapton background

χ2 = 9.147

Get the aspirin CIF from CCDC to use the experimental Uiso, and refine

(grouped) the Uiso for citric acid(used default values initially)

Some citric acid Uiso go < 0. Reset to “reasonable” values.

χ2 = 8.741

Add 2nd-order spherical harmonic preferred orientation coefficients

for each phase

χ2 = 5.355

Alka-Seltzer Analysis

28.3(1)30.86Citric Acid

8.6(1)10.03Acetylsalicylic

63.1(1)59.12Sodium

Bicarbonate

RefinedExpectedwt%

AKLD = 0.080

Take to extreme

XPS anode deposit on rag

A Challenging Case StudyMullite

Kyanite Mining CompanySupplied by Dilip Jang

Data collection by Fangling Needham, ICDD

01-070-3755> Quartz - SiO2

01-074-4143> Mullite - Al4.44Si1.56O9.78

01-079-1456> Mullite - Al4.54Si1.46O9.73

04-008-9528> Al2.13Si0.87O4.93 - Aluminum Silicon Oxide

00-015-0776> Mullite - Al6Si2O13

10 20 30 40 50 60 70 80 90 100 110 120 130Two-Theta (deg)

[Fangling_1_mullite_vslit_9hrs_Mul_A.RAW] Mullite

Mullite Crystal Structure

Mullite and quartz phase fractions, and 3-term shifted

Chebyshev background

Add lattice parameters for both phases, shift, and profile X and Y for each phase.

Quartz profile X refines to 1.71(32) < 2.477; fix.

χ2 = 239.1

There are additional peaks (phases)

2021628810033I

1.7651.9633.1893.2494.0486.652d

51.7646.2127.9627.4321.9413.332θ

The strongest two peaks correspond to the strongest peaks of cristobalite and rutile. Add as phases 3 and 4 (phase fractions).

Change excluded region to 0-13°, and add 3 more background terms.

χ2 = 106.6

Some amorphous material is present

POWPLOT/R

Add 3 Type 1 diffuse scattering terms – amplitudes only

χ2 = 90.37

Refine the mullite structure with constraints and restraints

Al1-O5 = Al1-O6 = 1.91(2) ÅAl2/Si3-O5, O6, O7 = 1.67(2) Å

O-O = 2.73 (3) ÅAl4-O = 1.80(2) Å

χ2 = 73.37

Composition of Al2/Si3

M-O = 1.693. Interpolate between 1.61 and 1.74 to get Al/Si = 64/36.

Set fracs to 0.66/0.33 and constraint 2/1. Add Al4, O7, O8 fracs, as well as M and O Uiso.

wRp = 0.0733, χ2 = 63.33

Check the chemistrya = 7.5460(2), b = 7.7026(2), c = 2.88511(5) Å

ICSD 2008/1 has 24 mullites ±0.03 Å

Al5.14Si1.20O10.050.186(3)0.300(1)0.599(2)Current

Al4.59(6)Si1.41(6)O9.71(3)0.15(2)0.35(2)0.50(1)Average

FormulaAl4 fracSi3 fracAl2 frac

+20.22/-20.10, unrestrained

Look at the difference plot.Mineral-related strong

and/or 13.33° among long.Kyanite (!), Al2Si2O5

01-071-6298, ICSD 77538.

Add kyanite as phase 5.

Change mullite profile to #4, anisotropic strain broadening

Add quartz structure with restraints, and 2nd-order spherical harmonic preferred

orientation for mulllite

Add two more distances to the diffuse scattering function

Add profile U for mullite and quartz

Quartz not significant, so fix.

Add cells and profile Yfor phases 3, 4, and 5.

Add 4th-order spherical harmonics for mullite and quartz preferred orientation

wRp = 0.0340, χ2 = 14.27

Largest errors at non-mullite peaksΔF = 0.41/-0.30 eÅ-3

Lattice constants are a = 7.54693(10) b = 7.70404(9) c = 2.885182(23) Alpha = 90 Beta = 90 Gamma = 90 Cell volume = 167.7497(33)

Name X Y Z Ui/Ue*100 Site sym Mult Type Seq Fractn Al1 0.000000 0.000000 0.000000 1.187(17) 2/M(001) 2 AL 1 1.0000 Al2 0.14783(12) 0.33997(11) 0.500000 1.187(17) M(001) 4 AL 2 0.599(1) Si3 0.14783(12) 0.33997(11) 0.500000 1.187(17) M(001) 4 SI 3 0.300(1) Al4 0.26732(80) 0.20424(64) 0.500000 1.187(17) M(001) 4 AL 4 0.142(1) O5 0.35892(23) 0.42105(16) 0.500000 1.020(32) M(001) 4 O 5 1.0000 O6 0.12749(23) 0.22258(20) 0.000000 1.020(32) M(001) 4 O 6 1.0000 O7 0.500000 0.000000 0.500000 1.020(32) 2/M(001) 2 O 7 0.60(1) O8 0.4426(22) 0.0493(21) 0.500000 1.020(32) M(001) 4 O 8 0.126(7)

Al1-O5 1.893 ×4Al1-O6 1.966(2) ×2

Al4-O5 1.808(5)Al4-O6 1.793(4) ×2Al4-O8 1.782(19)Al4-O5 2.381(5)Al4-O7 2.358(5)

Al2/Si3-O5 1.711(2)Al2/Si3-O6 1.709(1) ×2

Al2-Si3-O7 1.663(1)average = 1.697

Al2-Si3-O8 1.751(14), 1.768(13)

Chemical Reasonableness

Al2/Si3-O and fracs: 2/3 AlAl4-O: Al

Al4.96Si1.20O9.70+19.68/-19.40

77.81 wt % Al2O3, 22.19 wt% SiO2a: 75 wt% Al2O3, V: 25% SiO2

2.15(6)Kyanite0.60(2)Rutile0.61(3)Cristobalite7.15(6)Quartz

89.38(3)MulliteConcentration, wt%Phase

Typically 10-12% glassQuartz granularity?Exceptionally high-quality data

Chemical Reasonableness

Chemical Reasonableness“Chemical reasonableness in Rietveld analysis; organics”,

J. A. Kaduk, Powder Diffraction, 22(1), 1-9 (2007).

“Chemical reasonableness in Rietveld analysis; inorganics”, J. A. Kaduk, Powder Diffraction, 22(3), 268-278 (2007).

“Structure Refinement”, J. A. Kaduk, Chapter 8 in Practice and Applications of Powder Diffraction, A. Clearfield, J. H.

Reibenspies, and N. Bhuvanesh, editors. Blackwell (2008).

Chemical Reasonableness• Convergence (symmetry)• Molecular Geometry

– Bonded and Non-Bonded Distances– Angles– Torsions– Planarity– Hydrogen Bonds– Displacement Coefficients– Atomic Valences

• Magnitude of σs• Difference Fourier• Bulk and Individual Phase Compositions

Organic Structures

• Cambridge Structural Database• ConQuest/Vista• Mogul (intramolecular)• Gold (intermolecular)

Inorganic Structures

• PDF-4+• ICSD• CRYSTMET• Pearson’s Crystal Data• Bond Valence• Ionic Radii (R. D. Shannon and C. T. Prewitt),

“Effective ionic radii in oxides and fluorides”, Acta Cryst., B25(5), 925-946 (1969).

Bond ValenceThe Chemical Bond in Inorganic Chemistry, I. David Brown.

IUCr Monographs on Crystallography 12. Oxford University Press (2002).

“Bond valence parameters for solids”, N. E. Brese and M. O’Keefe, Acta Cryst., B47, 192-197 (1991).

http://www.ccp14.ac.uk/ccp/web-mirrors/i_d_brown/bond_valence_parm

0exp ijij

B−⎛ ⎞

= ⎜ ⎟⎝ ⎠ i ij

jV S=∑

Implement chemistry knowledge inRestraints (soft constraints)

distances Dangles Atorsions Tplanar groups Pchemical composition Cchiral volume Kϕ/ψ Rmagnetic moments M

Rigid Bodies

An Introduction to the Rietveld Method › workshops › 08 › Introduction to Rietveld 1.pdf ·...

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