máster en cristalografía y cristalización sevilla, 12 de...

Difraction powder pattern indexing...

Máster en Cristalografía y cristalización

Diffraction powder pattern indexing &

Intensity extraction

Máster en Cristalografía y cristalizaciónSevilla, 12 de Diciembre de 2012

Vicente Esteve CanoDpto. de Química Inorgánica y OrgánicaUniversitat Jaume I de Castellón

Máster en Cristalografía y cristalización Sevilla, 2012


1.- Introduction. What is indexing? Basic equations

2.- Figures of Merit. Indexation methods. Accuracy o f data

3.- Consideration about space group determination

4.- Checks on the validity of indexation

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powder pattern sample

angular position of diffraction peaks

crystal lattice

intensity of the peaks

unit cell content phase content

peak shape microstructure

Information contained in a powder pattern

λλλλ = 2 dhkl sin θθθθ

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What is indexing?

Unit cell parameters

( a ,b, c, αααα, ββββ, γγγγ )

Positions 2 θθθθ of diffraction peaks

lattice symmetry

systematic absences

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Basic equation

Qhkl =[d*] 2 = h2 A + k2 B + l 2 C + hk D + hl E + kl F(triclinic general case)

A = a*2 D = 2 a* b* cos αααα*

B = b*2 E = 2 a* c* cos ββββ*

C = c*2 F = 2 b* c* cos γγγγ*

- One equation per diffraction line

- h k l are integers (Miller indices)

Optimal number of equations ?

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Qhkl for special lattices

cubic

tetragonal

hexagonal

rhombohedral/triclinic

orthorhombic

monoclinic

Qhkl = (h2+k2+l2)A

Qhkl = (h2+k2)A + l 2C

Qhkl = (h2+k2+hk)A + l 2C

(with -h+k+l=3n+1 absent)

Qhkl = h2A + k 2B + l 2C

Qhkl = h2A + k 2B + l 2C + hlE

Number of observations N: 15 + number of parameters to determine

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Cubic system

Qhkl = (h2+k2+l2) A

sin 2θhkl = (λ2/4a2) ( h2 + k2 + l2 )

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The observational data

λλλλ = 2 dhkl sin θθθθhkl

Qhkl = 4 sin 2θθθθhkl / λλλλ2

Qhkl = 1 / dhkl2(a,b,c, αααα,ββββ,γγγγ)

Accuracy?

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Complexity = f (number of reflections N)

N ≈≈≈≈ (4ππππ/3) V m-1 dN-3

V is the volume of the cell

m is the multiplicity factor of Bragg reflections: 2 for triclinic; 4 for monoclinic, 8 for orthorhombic; 16 for tetragonal; 24 for hexagonal; 36 for trigonal; 48 for cubic (de Wolff, 1961)

Actually the number of observed reflections is smaller than N. We lost weak (or not observed) reflections and systematic extinctions

Large N means severe peak overlap

Compare triclinic and cubic situations!

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How to estimate the unit cell volume of a triclinic unit cell ?

V = 13.39 d320 (Smith, 1977)

N ≈≈≈≈ (4ππππ/3) V m-1 dN-3

because of the overlap N ≈≈≈≈1.4·Nobs

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Some historical milestones- Runge (1917) finds Q relations useful for constructing the reciprocal space

- Ito (1949) and de Wolff (1957) report a more general method for the geometrical reconstruction from Q relations

- Visser (1969) Computer program using this algebraic procedure ITO

- Werner (1964) Method based on the permutation of Miller indices TREOR

- Louër & Louër (1972) Method based in successive dichotomies DICVOL

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Present trends

Increase of the robustness against:

- the presence of small impurities

- zero-point displacement errors

Active inclusion of systematic absences due to space group symmetry

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The figures of merit

Q20 = Q value of the 20th line. The larger unit cell, the smaller Q 20. The higher symmetry, the larger Q 20

Ncalc = number of calculated reflections with Q < Q 20(systematic absences not taken into account). It is good signal if N calc is only slightly larger than 20

<∆∆∆∆Q>= <|Qobs -Qcalc |> The discrepancy should be as small as possible. Q calc refers to the nearest line

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M20 (de Wolff, 1968)

M20 = Q20 / (2 <∆∆∆∆Q> Ncalc )

For M20 > 10 the indexing is substantially correct

With right measurements in good diffractometersM20 >20 in routine work

Principal problem: Presence of a dominant zone

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FN (Smith & Snyder, 1979)

FN = (N/Nposs ) / <|2θθθθobs -2θθθθcalc |>

If <∆∆∆∆2θθθθ> and N/ Nposs are identical for a cubic and a triclinic indexing, the cubic indexing is obviously to be preferred (Werner).

In the programs it is given in the format:

FN (<∆∆∆∆2θθθθ>, Nposs )

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ITO-method (Visser, 1969)

• A triclinic unit cell is assumed

• The smallest Q is taken as A: A= Q1 (one checks for the presence of h2A= 4A, 9A…

• Next small Q: B= Q2 (check for harmonics)

• Search for hk0 reflections: One looks for that D value explaining the maximum number of Qh±k0= h

2A+k2B±hkD

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• Next unexplained Q: C= Q3

• Repeat the process carried out for hk0, to reflexions h0l (E) and to 0kl (F)

• Once A, B, C, D, E, F are known, the remaining Qhkl should be explained in terms of them

• If not, start again the procedure, assuming that Q1=4A (the first order could be absent)

• Useful for symmetries down to monoclinic and unit cells less than 1000A3

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Permutation method (TREOR)

a) Set-up of a system of linear equations

Qi = h i2 A + k i

2 B + l i2 C

for the base reflections (with smallest 2 θθθθ values). Different least square trials are performed with permutations of h k l from 0 to 4.

b) Only the best solutions are tested with the Q values of the rest of N lines (used passively).

Werner (1964) Z.Kristallogr. 120, 275

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TREOR (4)- 84 10 02

s40-va2 7.995000 12.232000 15.670000 15.997000 17.542000 18.492001 20.915001 21.722000 24.254999 24.582001 25.202999 25.565001 26.427999 27.124001 27.267000 27.834999 28.863001 29.186001

FIGURE OF MERIT REQUIRED= 10 MAX NUMBER OF UNINDEXED LINES= 1THE 7 FIRST LINES ADJUSTED BY THEIR HIGHER ORDERSCUBIC,TETRAGONAL,HEXAGONAL AND ORTHORHOMBIC SYMMETR YMAX CELL EDGE= 60.0 MAX CELL VOLUME= 2500.0D1= 0.000200 SSQTL= 0.050000 D2= 0.000400 WAVE= 1.540598NUMBER OF TEST LINES= 18 IQ REQUIRED= 15

Input data (TREOR90):

1) Peak positions 2 θθθθ (or d or Q)

2) Maximum cell volume

3) Maximum cell edge

4) Indicate monoclinic (maximum ββββangle) and triclinic tests

5) Possible errors on observed data

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Successive dichotomy method (DICVOL)

Louër & Louër (1972) J.Appl.Cryst. 5, 271

Exhaustive search of solutions in nD parameter space, working from the cubic side of the symmetry sequence: n = 1 (cubic) to 6 (triclinic), according to a dichotomy algorithm

Computer program: DICVOL06 ( J. Appl. Cryst. 37, 724, 2004; Z. Krist. 26, 191, 2007)

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# Input file for WDICVOL06 (created by WinPLOTR)! DATA FILE: BARBITAL.DAT! INSTRM: 0! WAVE: 1.54060 1.54439 0.5

DICVOL input parameters:20 2 1 1 1 1 0 0 ! N,ITYPE,JC,JT,JH,JO,JM,JTR 25.0 25.0 25.0 0.0 2500.0 90.0 125.0 ! AMAX,BMAX,CMAX,VOLMIN,VOLMAX,BEMIN,BEMAX1.54060 0.000 0.000 0.000 ! WAVE,POIMOL,DENS,DELDEN

0.030 10.000 0 0 1 0 2 ! EPS, FOM, N_IMP, ZERO_search, ZERO_refinem ent, DV06_option, ISUP10.39893 14.61714 14.82710 15.12112 16.53855 16.80712 19.71762 20.86397 21.09645 21.20449 23.49739 24.74766 25.70824 25.97648 26.60000 26.76568 27.53413 28.53317 29.44824 29.80722

...

...

...

Input data (DICVOL06):

1) Peak positions 2 θθθθ (or d or Q)

2) Symmetry system to check

3) Maximum cell edges

4) Maximum ββββ angle if monoclinic test

5) Possible error on observed data

6) Minimum FOM (M 20)

7) Maximum number of impurity lines

8) Zero error and possibility zero error refinement

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How to use DICVOL?

1. Run DICVOL with the first 20 lines. These lines should be not too weak to avoid the introduction of one peak belonging to an impurity. Select the solution with the best figure of merit M 20

2. Perform DICVOL runs by adding sequentially the weak lines. If the weak lines are indexed, this represents a further confirmation that the solution is correct.

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Hull method (hexagonal)

sin 2θ = A (h2+k2+hk)+ C l 2

log sin 2θhkl = log A + log [h 2+k2+hk + C/A l 2 ]

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What makes indexing difficult?Large asymmetrical unit = severe peak overlap

Shifted zero point

Lines measured with poor accuracy

Presence of impurities

Pseudosymmetry

Unequal lattice parameters:

1) but equal within experimental error;

2) one much longer than the other two;

3) one much shorter than the other two (dominant zo ne)

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Accuracy of data

The absolute error value for every diffraction line should be <0.03º2 θθθθ (CuK αααα radiation) in order to achieve meaningful FoM’s values.

In practice, <| ∆∆∆∆2θθθθ |> use to be less than 0.01º2 θθθθfor well crystallised materials

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D. LouërProceedings of the Int. Conference “Accuracy in powder Diffraction II” held in Gaithersburg (1992)

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How to improve the accuracy?- adding an internal standard for position

- adjusting the zero-2 θθθθ position (<0.005º2 θθθθ) or using parallel beam geometry

-adjusting the θθθθ/2θθθθ angular range using highly oriented crystallites of a layer-type material, deposited in a thin layer on a silicon-plate to reduce transparency eff ects (e.g. fluorophlogopite mica, 10 harmonics)

- with a correct adjustment no internal standard is required (very smooth front surface of the sample a nd thickness control for materials with only light ato ms; no problem if transmission, flat sample or capillar y geometries)

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PseudosymmetryIt happens when the apparent lattice symmetry is

higher than the true one

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Crystal data: a=6.8576(3) b=6.8831(5) c=19.988(2) Å

Confirmation when space group was found,

P212121 with Z = 4

Tetragonal as lattice pseudosymmetry

True symmetry is orthorhombic

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Considerations about space group determination

Lattice symmetry and 14 Bravais lattices

Triclinic

Monoclinic

Orthorhombic

Trigonal

Hexagonal

Tetragonal

Cubic

P

P, C, ( I )

P, C, (A, B), I, F

R

P

P, I

P, F, I

These absences extend uniformly in the reciprocal space

-1

2/m

mmm

-3m

6/mmm

4/mmm

m-3m

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Reflection conditions for lattice centerings

P: no systematic absences

C: only with h+k = 2n

B: only with h+l = 2n

A: only with k+l = 2n

I: only with h+k+l = 2n

F: only if conditions for A, B and C are simultaneously fulfilled.

R: only with -h+k+l =3n (hexagonal axes and obversedsetting)


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Present in all the crystalline systems but triclinic.

In triclinic crystal possible space groups symmetry, P1 and P-1, belong to the same diffraction class P1


Reflection conditions due to the presence of translational symmetry: Bravais centered lattices; glide planes and helicoidal axes

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Reflection conditions. Monoclinic system

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Reflection conditions. Orthorhombic system

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Reflection conditions. Trigonal system

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Reflection conditions. Hexagonal system

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Reflection conditions. Tetragonal system

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Reflection conditions. Cubic system

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Checks on the validity of the indexation1) Reviewing all the discrete data

DICVOL run

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Checks on the validity of the indexation1) Reviewing all the discrete data

SEARCH OF ORTHORHOMBIC SOLUTION(S)**********************************

VOLUME DOMAIN BEING SCANNED :===========================

LOWER BOUND = 0.00 A**3 HIGHER BOUND = 400.00 A**3

O R T H O R H O M B I C S Y S T E M DIRECT PARAMETERS : A= 7.97263 B= 5.74605 C= 4.96433 VOLUME=

227.42STANDARD DEVIATIONS : .00039 .00038 .00037REFINED ZERO-POINT SHIFT : 0.0163 deg. 2-theta

.

.

.

* NUMBER OF LINES.- LINES INPUT = 20.- LINES INDEXED = 20.- LINES CALCULATED = 31

* MEAN ABSOLUTE DISCREPANCIES<Q> =0.2501E-04

<DELTA(2-THETA)> =0.2990E-02MAX. ERROR ACCEPTED (DEG. 2-THETA) =0.4500E-01

* FIGURES OF MERIT1.- M( 20) = 181.82.- F( 20) = 215.8(0.0030, 31)

THE SOLUTION IS NOW USED TO TRY TO INDEX ALL INPUT 111 LINES :==============================================================DIRECT PARAMETERS : A= 7.97179 B= 5.74459 C= 4.96251 VOLUME=

227.26STANDARD DEVIATIONS : .00023 .00012 .00016REFINED ZERO-POINT SHIFT : 0.0283 deg. 2-theta

.

.

.

* NUMBER OF LINES.- LINES INPUT = 111.- LINES INDEXED = 111.- LINES CALCULATED = 229

* AVERAGE 2-Theta DIFFERENCE FOR INDEXED LINES =-0.0001* MEAN ABSOLUTE DISCREPANCIES

<Q> =0.1005E-03<DELTA(2-THETA)> =0.7640E-02

MAX. ERROR ACCEPTED (DEG. 2-THETA) =0.4500E-01* FIGURES OF MERIT

1.- M( 20) = 105.22.- F( 20) = 122.8(0.0053, 31)3.- F(111) = 63.4(0.0076, 229)

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Checks on the validity of the indexation2) Use of observed (or known) density to check

whether a solution corresponds to an integer number of chemical formula in the cell (Z)

cAVN

MZ=ρ

3) Diffraction symbol (space group) determination

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Checks on the validity of the indexation4) Pattern matching (e.g. Le Bail method). Visual

inspection of the quality of the fit.

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Checks on the validity of the indexation5) Structure determination

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This presentation is entirely based on the presentation prepared

by

Aknowlegments and References

Jordi Rius PalleiroInstitut de Ciència de Materials de Barcelona (CSIC)&Xavier AlcobéCentres Cientítics i Tecnològics, Universitat de Barc elona

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And the paper entitled “Automatic Indexing: Procedures and Applications”, by Daniel Louër, 1992.





Thanks so much for yourattention!

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