rietveld school 01 - Durham University...

Durham

ChemistryDepartment

DurhamUniversity

Welcome/IntroductionDr John S.O. Evans

Durham, January 2007

john.evans@durham.ac.uk www.durham.ac.uk/john.evans PCG workshop January 2007

Welcome

• Health and Safety/Fire procedures• Course will be lectures followed by

hands on practical• Emphasis on understanding not

learning a package• Please submit pre-course questionnaire

with name/pseudonym

Teachers/Tutors

• Jeremy Cockcroft UCL/Birkbeck• John Evans Durham Chemistry• Ivana Evans Durham Chemistry

• Will Bisson UCL• Sarah Lister Durham• Graham Stinton Durham• Lars Peters Durham

Thanks

• Tutors for time• Sponsors for cash• Alan Coelho for Topas software• Bob von Dreele/Alan Larson for gsas• Juan Rodriguez-Carvajal for fullprof

DurhamUniversity

Legalities

• By using ITS account you’re agreeing to University policy on use of the web

• You each have your own login id. Don’t change the password!!!

• Don’t try stealing the software! It won’t work anywhere else plus is time limited!

• All timetabled sessions are compulsory!

Tomorrow Morning

• First lecture will be in CG93 (Scarborough lecture theatre)

• Follow somebody!

• Who doesn’t know how to use solver in excel?

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Information in a Powder Pattern

2θ - degrees

1. Peak positions determined by size, shape, symmetry of unit cell – internal

structure

2 9 .52 95

2. Peak Intensities determined by

where atoms sit in unit cell – internal

structure

3. Peak widths influenced by size/strain of crystallites -

microstructure.

Timetable

Sunday Monday Tuesday Wednesday

09:003. Introduction to powder diffraction

and data collection.8.Structure factors, peak intensities

and Rietveld refinement13. Question and Answer session

09:454. jedit/topas academic/Rietveld

refinement9. Pawley and Rietveld refinement 14. Spot the errors

11:00 Coffee Coffee Coffee

11:305. Peak positions - indexing/cell

refinement exercises10. Intro to gsas/fullprof then

examples15. Free problems then wrap up

13:00 Lunch - Musgrave Room Lunch - Musgrave Room Departure

14:00 6. Peak shapes lecture 11. Restraints, Constraints, Rigid Bodies and Structure Solution

14:30 7. Peak shape tutorial 12. Structure solution and rigid body refinements

16:00 Registration in Trevelyan Tea Tea

16:30 7. Peak shape tutorial 12. Structure solution and rigid body refinements

18:00 Dinner - Trevelyan Close Close

18:30 Dinner - Trevelyan Dinner - Trevelyan

19:000.Welcome/Introduction

1.Symmetry lecture

20:00 2. Symmetry bar quiz Pub Quiz Free evening

Lecture slot Food/Drink Problems/Workshop

Durham

ChemistryDepartment

DurhamUniversity

Introduction to Software/ProblemsDr John S.O. Evans

Topas Academic

• Useful for teaching as can input equations, see everything in file very quickly

• Powder and single crystal• X-ray, neutron• Constant λ, time of flight, energy dispersive• Structural models, Pawley fitting, peak fitting• Restraints, constraints, penalty only fitting• Multi phase, multi histogram• Non crystallographic applications• Simulated Annealing for structure solution

Acknowledgement

• Alan Coelho (Topas/Topas-Academic author)• http://pws.prserv.net/Alan.Coelho/

Topas Graphics

Software Interface

• Command file driven for flexibility/speed• jedit – free customisable java editor• Interacts directly with software• Helps with formatting

http://www.dur.ac.uk/john.evans/topas_academic/topas_main.htmhttp://www.dur.ac.uk/john.evans/topas_workshop/pcg_workshop_menu.htm

jedit interface

Topas/jedit demo

Refining Parameters

str ‘cell params for an orthorhombic structure

a 7.31192

b 7.53699

c 7.69967

Refining Parameters – Using @

a @ 7.31192

b @ 7.53699

c @ 7.69967

Refining Parameters – Using names

a lpa 7.31192

b lpb 7.53699

c lpc 7.69967

Refining Parameters – Using Names

str ‘cell params for a cubic structure

a lpa 10.60992

b lpa 10.60992

c lpa 10.60992

Cubic(@ 10.60992)

Fixing Parameters – Using !

a !lpa 10.60992

b !lpa 10.60992

c !lpa 10.60992

• N.B. jedit column editting – hold down ctrl key and type all !’s at once

Refining Parameters - Macros

Zero_Error( , 0) ‘fixed zero point

Zero_Error( @ , 0.013) ‘refine zero point

Zero_Error( !zero, 0) ‘fixed zero point

Zero_Error( zero, 0.013) ‘refined zero point

User Defined Equations

‘topas zero point correction in input file

prm zero 0.01

th2_offset = zero;

‘topas zero point correction in input file

prm zero 0.01

prm corr 0.003

prm corr2 0.001

th2_offset = corr2*X^2 + thcorr*X + zero;

‘silly(?) example of topas equations

prm zero 0.01

prm var1 0.003

prm !date 21 ‘st of April

th2_offset = zero + var1 * date;

• Flexible system for defining your own equations• Fully “programmable program”• e.g. gsas vs topas:

• shft = 3600*height/π*radius • 2θobs = 2θcalc + zero – 2*shft*Cos(θ)

‘topas height/zero point correction in input file

prm zero 0.01

prm height 0.15

th2_offset = zero - 2*height*Cos(Th)/radius;

zero 0.01 y shft -10.00 y

Simulated Annealing

• Flexible Simulated Annealing approach for complex structures:

• Flexible definition of restraints, constraints, rigid bodies, etc

‘topas annealing expression in input fileval_on_continue = Val + Val * Rand(-0.1,0.1);val_on_continue = ideal_coord + Rand(-0.1,0.1);

Running Tutorial Problems – Session 4

• You should all be able to log in to computer• Work on the j: drive where you have full read/write

privileges• Use folder j:\school_work• Run jedit from icon on desktop• Launch topas-academic from inside jedit• Access tutorials

Initial Software Setup

• First time you log in you’ll need to double click on:

• I:\licence\rietveld\rietveld_setup.bat• Wait a few minutes for software to be

copied• Use desktop icons created

• [For fullprof will need to double click on rietveld_setup.bat each time you log in]

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Session 5 – Peak Positions

2θ - degrees

structure

2 9 .52 95

structure

microstructure.

Session 5 – Peak Positions

Bragg’s Law: nλ=2dsinθ

d-spacing formulae

• Triclinic

• Cubic

( ) ( ) ( )[ ]βγααγβγβαγβα coscoscos2coscoscos2coscoscos2sinsinsin11 22222222222222222 −+−+−+++= chlabbcklahkabcbalcakcbh

Bragg’s Law: nλ=2dsinθ

Indexing Tutorials – Session 5

• Folders should contain print outs of powder patterns for tutorials 4 and 5

• Try to index powder pattern for tutorial 4 by hand/with excel

• Use excel to refine cell parameter of diffraction pattern for tutorial 5

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2-Theta11 20 30 40 50 60 70

2-Theta

11 20 30 40 50 60 70

Durham

ChemistryDepartment

DurhamUniversity

Session 6 – Peak ShapesDr John S.O. Evans

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Session 6 – Peak Shapes

2θ - degrees

structure

2 9 .52 95

structure

microstructure.

Peak Shapes – View 1

• Peak shapes are a nuisance. For Rietveld refinement we only fit peaks to get an good agreement between yobs and ycalc to give us an accurate structural model

• We’re therefore not interested in the mathematical details of the peak shape model

• Failure to fit the finest details of the peak shape (e.g. tails of the peaks) aren’t very important

Peak Shapes – View 2

• Peak shapes in a powder diffraction pattern result from a combination of instrumental effects (the optics you use) and sample effects (size/strain)

• There is a wealth of fascinating information contained in experimental peak shapes

Peak Shapes – 3 Approaches

• Empirical Peak Shapes– Used by most Rietveld packages– Whatever function fits the data is good

• Fundamental Parameters– Instrumental contribution to peak shape– Sample contribution to peak shape– Excellent fits with very few parameters

• Semi Empirical– Define instrument with empirical function– Convolute with sample contribution

Gaussian/Lorentzian Functions

x 2θ-2θhkl – where 2θhkl is position of reflectionfwhm full width at half maximumη mixing parameter for composite function

Gaussian vs Lorentzian

www.phys.unsw.edu.au/~mgb/pics/gausscauchy.gif

• Lorentz sharp near maximum but has long tails away from peak

• Gauss smaller tails but rounded maximum

• PV mixes two functions

• Mixing η 0 to 1

Gaussian, Lorentzian, Pseudo Voigt

2Th Degrees58.458.458.3558.358.2558.258.1558.158.055857.9557.957.8557.857.7557.757.6557.657.5557.557.4557.457.3557.357.2557.257.1557.157.055756.9556.956.85

7,0006,5006,0005,5005,0004,5004,000

3,5003,0002,5002,0001,5001,000

-1,000-1,500

2Th Degrees58.458.458.3558.358.2558.258.1558.158.055857.9557.957.8557.857.7557.757.6557.657.5557.557.4557.457.3557.357.2557.257.1557.157.055756.9556.956.85

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2Th Degrees58.458.458.3558.358.2558.258.1558.158.055857.9557.957.8557.857.7557.757.6557.657.5557.557.4557.457.3557.357.2557.257.1557.157.055756.9556.956.85

-1,000

Gauss wRp=30.5%

Lorentz wRp=19.9%

η=0.68 wRp=17.6%Individual peak fitting on simulated Y2O3 data 3/2/07

fwhm vs 2θ

Gauss: ( ) 212 tantan WVUfwhm ++= θθ

Lorentz: θθ

tancos

YXfwhm +=

0 20 40 60 80 100 120 140 160

2-theta

U, V, W, X, YRefineableParameters

GSAS TCHZ type function

• Modified Thompson-Cox-Hastings pseudo-Voigt

GSAS/Topas

‘topas TCHZ peak shape function

TCHZ_Peak_Type( pku,-0.15636`,pkv, 0.24248`, pkw, -.16266`, !pkx, 0.00000, pky, 0.03861`, !pkz, 0.0000)

Look in topas.log to see what’s happening

TCHZ_Peak_Type(pku,-0.02510`,pkv, 0.03611`,pkw, -0.01238`,pkx, 0.00000`,pky, 0.16545`,pkz, 0.00010`)

prm pku -0.02510` min = Max(-1, Val-.1); max = Min(2, Val+.1); del 1.0e-4 prm pkv 0.03611` min = Max(-1, Val-.1); max = Min(2, Val+.1); del 1.0e-4 prm pkw -0.01238` min = Max(-1, Val-.1); max = Min(2, Val+.1); del 1.0e-4 prm pkx 0.0000 min = Max(-1, Val-.1); max = Min(2, Val+.1); del 1.0e-4 prm pky 0.16545` min = Max(0.0001, Val-.1); max = Min(2, Val+.1); del 1.0e-4 prm pkz 0.0000 min = Max(0.0001, Val-.1); max = Min(2, Val+.1); del 1.0e-4 peak_type pv pv_lor = 1.36603 (((pky) Tan(Th) + (pkz) /Cos(Th)) /(

(Sqrt( Sqrt( ((pku) Tan(Th)^2 + (pkv) Tan(Th) + (pkw) + (pkx) /Cos(Th)^2)^2 ) )^5 +2.69269 Sqrt( Sqrt( ((pku) Tan(Th)^2 + (pkv) Tan(Th) + (pkw) + (pkx) /Cos(Th)^2)^2 ) )^4 ((pky) Tan(Th) + (pkz) /Cos(Th)) +2.42843 Sqrt( Sqrt( ((pku) Tan(Th)^2 + (pkv) Tan(Th) + (pkw) + (pkx) /Cos(Th)^2)^2 ) )^3 ((pky) Tan(Th) + (pkz) /Cos(Th))^2 +4.47163 Sqrt( Sqrt( ((pku) Tan(Th)^2 + (pkv) Tan(Th) + (pkw) + (pkx) /Cos(Th)^2)^2 ) )^2 ((pky) Tan(Th) + (pkz) /Cos(Th))^3 +0.07842 Sqrt( Sqrt( ((pku) Tan(Th)^2 + (pkv) Tan(Th) + (pkw) + (pkx) /Cos(Th)^2)^2 ) ) ((pky) Tan(Th) + (pkz) /Cos(Th))^4 +((pky) Tan(Th) + (pkz) /Cos(Th))^5

)^0.2)) - 0.47719 (((pky) Tan(Th) + (pkz) /Cos(Th)) /(

)^0.2))^2 + 0.1116 (((pky) Tan(Th) + (pkz) /Cos(Th)) /(

)^0.2))^3; pv_fwhm = (

)^0.2);

Fundamental Parameters Approach

• Peak Widths depend on– X-ray source– Instrument– Sample

• Convolution or folding “blends” one function with another

( ) ( ) ( ) ( ) ( ) ( ) ττττττ dtfgdtgftgtf −=−=⊗ ∫∫∞

∞−

( ) ( ) SampleInstrumentSourceY ⊗⊗=θ2(or empirical instrumental function)

Convolution Approach

( ) ( )θ2YSampleInstrumentSource =⊗⊗

Contributions to peak shape

Sample Contributions - Size

• Scherrer formula• β full width half maximum; k a constant• LVol is volume weighted mean column height; only for

cubic crystals and h00 reflections is it equal to L0 in figures below

Sample Contributions - Strain

• d-spacings of d+∆d and d-∆d• e0=∆d/d• Assume Braggs law gives different 2θ for different

d-spacings

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2-theta10 20 30 40 50 60 70 80 90 100 110 120 130 140

X-ray Particle Size from peak shape

Size: 324(110) nmStrain: 0.1637(40)

Size: 20.4(8) nmStrain: 0.138(7)

0.00.51.01.52.02.53.03.54.04.5

0 20 40 60 80 100 120 140 160 180

2-theta

1/cos(th)tan(th)

sizestrain

• Size broadening – “broad over whole 2-theta range”• Strain broadening – “narrow at low 2-theta”

Sample Contributions - Caveat

• Be careful!• See links in tutorial

• http://www.dur.ac.uk/john.evans/topas_workshop/size_strain.htm

Source Profile

N.B. source profile only much sharper then observed peak

Source ⊗ Equatorial

N.B. peak now considerably broadened by instrument

Source ⊗ Equatorial ⊗ Axial

N.B. peak now has asymmetry

Source ⊗ Equatorial ⊗ Axial ⊗ Sample

N.B. peak maximum not at calculate 2θ

Topas Language

Radius(217) ‘diffractometer radius

Divergence(1) ‘divergence slit deg

axial_conv

filament_length 12

sample_length 15

receiving_slit_length 12

primary_soller_angle 5.1

secondary_soller_angle 5

Slit_Width(0.1) ‘receiving slit

CS_L(size_lor, 329.49186) ‘size term nm

Strain_G(strain_g, 0.04939) ‘strain term

The tutorial (hard!)

• Take an experimental data set• Determine fwhm for all peaks and plot in excel• Use excel to fit fwhm=f(2θ) functions

– fwhm = U tan2θ + V tanθ + W– fwhm = U 2θ2 + V 2θ + W

• Try these same functions in a Rietveld refinement in topas• Try a fundamental parameters approach in topas and get

a better fit with fewer parameters.

0 20 40 60 80 100 120 140 160

2-theta

Semi Empirical Size/Strain

• See:– tutorial 9 - Size/Strain Analysis: Shows how

size/strain can be determined in topas using the CeO2 round robin data and an empirical instrumental function

– tutorial 10 – determining the size of nanoparticles

Durham

ChemistryDepartment

DurhamUniversity

Session 9 – Pawley/Rietveld ProblemsDr John S.O. Evans

Tomorrow’s Programme

• Wednesday morning open questions session– Either submit questions on paper before hand

or ask on the morning– Will limit to 30-45 minutes

• Spot the errors tutorials– Deliberate errors in data for trouble shooting

• Free problems/play with your data• Wrap up session (if required)

Today’s Programme

• Next – workshop tutorials• After coffee

– Gsas demonstration– Workshop tutorials

• Split into “fullprof-bias” room (on the right)?• After lunch

– intro to problems lecture– Restraints/rigid bodies– Structure solution problems (simple)

Rietveld/Pawley Problems

• Tutorial 1 – TiO2 Rietveld• Tutorial 2 – TiO2 Pawley• Tutorial 3 – ZrW2O8 X-ray/neutron/neutron time of

flight• Tutorial 10.5 – Jeremy’s PbSO4 data in .xye

format for you to try in topas• Gsas 2 (later) – Jeremy’s PbSO4 data• Tutorial 11 – Combined X-ray and neutron

refinement (also gsas3/gsas4)• Tutorial 12 – Multiphase Rietveld refinement

Durham

ChemistryDepartment

DurhamUniversity

Session 10 – Other SoftwareDr John S.O. Evans

• General Structural Analysis System• Bob Von Dreele/Alan Larson• Very widely used• Magnetic refinements• Run via expedt/expgui• Actually has a .exp file (like .inp file) in

background

Fullprof Suite

• Juan Rodriguez-Carvajal• Has a .pcr file (like .inp file) in background• Very widely used• Magnetic refinements• Use winplotr interface

Mistakes

• Hardest thing in practice is spotting mistakes• You don’t know what the “right” answer is• Are there errors in model or data?

• mistake_02 to mistake_08 contain data with “errors” in them – solve the errors and win a prize!

• Stick to isotropic temperature factors• Stick to a TCHz analytical peak shape function• Fix pkx and pkz at 0

Durham

ChemistryDepartment

DurhamUniversity

Session 11 – Constraints, Rigid Bodies and Structure Solution

Dr John S.O. Evans

Data Compression

2-theta

(a) (b)

(d)(c)

Single crystal

Four differently oriented single crystals

Polycrystalline material

3D Information compressed onto 1D

‘…data compressed into one dimension…’

Insufficient information

• Powder pattern 3D diffraction data compressed onto 1D

• Loss of “information” relative to a single crystal experiment

• Single crystal people like 10 observations (10 hkl reflections) per refined parameter

• How many “observations” do we have in a powder pattern?

alvo4_tch.inp

• 5-80 0.02 step 3750 observations• 530 hkl reflections in 2θ range

2Th Degrees8075706560555045403530252015105

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AlVO4 100.00 %

STR(P_-1, AlVO4)a @ 6.54138`b @ 7.75971`c @ 9.13591`al @ 96.18561`be @ 107.23757`ga @ 101.40086`

How many peaks?

2Th Degrees1918171615

3,5003,0002,5002,0001,5001,000

AlVO4 100.00 %

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AlVO4 100.00 %

How many peaks?

2Th Degrees2524.824.624.424.22423.823.623.423.2

AlVO4 100.00 %

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How many peaks?

2Th Degrees62.56261.56160.560

AlVO4 100.00 %

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Information in a powder pattern

• How much information is there in a powder pattern?

• Rarely enough

• See literature by e.g. Giacovazzo/David/Di Sivia

Restraints

• Bring in chemical information• Dealt with in least squares in the same way

as data• “Soft Restraints”

restdatatot KK 22

2 χχχ +=

Restraints

• e.g. you might know that Zr1-O2 distance should be ~2.075 Å

• Apply a “penalty” if it’s not that value• Penalty = (value-2.075)2

Distance_Restrain(Zr1 O1, #ideal_dist, #actual dist, #tolerance, #weight)

Distance_Restrain(Zr1 O1, 2.075, 2.037, 0.01, 1)

Angle_Restrain(O1 Zr2 02, 90, 91, 1, 1)

Penalty = weight * (2.075-2.037)^2;

Topas bond lengths/angles

append_bond_lengths

Zr1:0 O1:3 0 1 0 2.12510 O1:10 0 0 0 2.12510 90.053 O1:7 -1 0 1 2.12510 90.053 90.053 O2:7 -1 1 1 2.12575 179.898 89.897 90.036 O2:3 0 1 -1 2.12575 90.015 89.897 90.036 179.898

O2:10 -1 0 0 2.12575 90.015 90.015 90.036 179.898 89.897

O1:3 0 1 0

Restraints

• See tutorials

Constraints

• Force sub-sections of structure to be rigid• Dealt with in least squares in the same way

as symmetry• “Hard Constraints”

Constraints/Rigid Bodies

• e.g. force the Zr1-O2 bond to be exactly 2.075 Å• e.g. rigid body• How many parameters?

macro Octahedra(Zr1, o1, o2, o3, o4, o5, o6, 2.075)

Point_for_site(s0, 0, 0, 0)

Point_for_site(s1, r, 0, 0)

Point_for_site(s2, -r, 0, 0)

Point_for_site(s3, 0, r, 0)

Point_for_site(s4, 0, -r, 0)

Point_for_site(s5, 0, 0, r)

Point_for_site(s6, 0, 0, -r)

TLS Matrices

• Atoms on both sides of a rigid group ought to vibrate in related ways

• Atoms/adps independently – how many parameters?

• Rigid body/TLS e.g. 8 parameters

Rigid Bodies

• P2O7 groups are well understood – 2 parameters per P2O7?

Typically1.5 Å

1.55 1.65

P-O(-P) Distance

Structure Solution

• Simulated annealing type approach• Refine, randomise, refine

‘topas annealing expression in input fileval_on_continue = Val + Val * Rand(-0.1,0.1);val_on_continue = ideal_coord + Rand(-0.1,0.1);

Tutorials – Restraints/Constraints

• Tutorial 3 – ZrW2O8

• GSAS 4 – restraints/combined refinement oxide • GSAS 5 – Rigid bodies Sc2(WO4)3 using TLS matrices• GSAS 7 – organic using restraints• GSAS 9 – Ni coordination polymer using restraints• Tutorial 14 – refining an organic using restraints then as a

rigid group

Tutorials – Structure Solution

• Tutorial 13 – Structure solution of an inorganic oxide

• Tutorial 15 – Organic structures and structure solution

• Tutorial 3 – data suitable for structure solution

Guided Tours?

• Bragg-Brentano/Transmission instruments• Incident beam mono/no mono• d8’s/d5000’s• High temperature furnace/cryostat• Vantec/Lynx-Eye psd’s• Sol-X energy dispersive

• Single crystal instruments (Judith’s)

Durham

ChemistryDepartment

DurhamUniversity

Session 14 – MistakesDr John S.O. Evans

Group Refinement

• Group discussion of order of refining parameters• e.g. if you refine just scale parameter when peak

positions are wrong the peaks will “disappear”• Look at obs/calc patterns before thinking what to

refine• Don’t be afraid to e.g. fix scale parameter to a

sensible value at the start• Never refine a parameter if you don’t understand

its meaning!

Mistakes

• Hardest thing in practice is spotting mistakes• You don’t know what the “right” answer is• Are there errors in model or data?• Ivana’s “Rietveld Crimes”

• mistake_02 to mistake_08 contain data with “errors” in them – solve the errors and win a prize!

• Stick to isotropic temperature factors• Stick to a TCHz analytical peak shape function• For this instrument you can fix pkx and pkz at 0

rietveld school 01 - Durham University...

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