XIX Conference on Applied Crystallography

Summer School on Polycrystalline Structure Determination

Kraków, September 2003

by

Wiesław Łasocha

Full Pattern Decomposition

Structure Solution from Powder Data. Where are we now ?- some numbers

• Inorganic Crystal Structure Data Base 2002 contains 62 382 entries, among which:

• in 11 316 entries powder data were used• in 11 150 cases the Rietveld method was applied• in 8646 structures neutron diffraction was used• in 519 cases synchrotron radiation was applied • in 186 entries electron powder diffraction was used• the biggest structure solved from the powder data

contains 112 atoms in a.u. [1]• most structures solved recently from powder data are

the structures of organic compounds[1] Wessels, T., Baerlocher, Ch., McCusker, L.B., Science, 284, 477

Number of crystal structures solved ‘ab initio’

0

100

200

300

400

500

600

700

800

years

1947-87198819891990199119921993199419951996199719981999200020012002

1987 1997 20021991

Structure determination

samplera

diat

ion neutron

laboratory

synchrotron

data collection

indexing

space group determinationin

tens

ity

extr

actio

n

Pawley

Le Bail

Treatment of overlap

chemical information

Rietveld refinement

structure completion

who

le p

atte

rn

Patterson & direct methods

FINAL STRUCTURE

new methods

chemical information

Mul

tiple

da

tase

t

equi

part

ition

Tri

p let

sFI

P S

Structure Determination from Powder Diffraction Data, ed. W.I.F.David, et all

Structure determination

Sample

Data collection

Indexing

Space group

Structure solution

Rietveld refinement

Pattern decomposition

Final Structure

Per aspera ad astra

Single crystal diffraction

2

Powder Diffraction Pattern - the basic source of information about the

investigated material

Powder diffraction pattern analysis without cell constraints

• Parish analysis - ‘peak hunting’ included in the APD software, NEWPAK program. characteristic -useful for indexing purposes -used in phase analysis -fast, no assumption about the cell parameters -rarely used for ab initio structure determination -broad peaks create problems, not suitable for overlapping reflections

Pattern Decomposition - general information

• Diffraction pattern can be described by the formula: Yi,c = M(i) = back(i) + {k}iAk qk (i) where: Ak = mk |Fk |2 mk - multiplicity factor, |Fk | - structure factor qk (i) = ck(i) Hk ck(i) - Lorentz-polarization & absorption terms Hk - normalized peak shape of kth reflection.

• Number of observed data in diffraction pattern Yi,o 10000 - 30000

• Number of parameters: cell parameters a,b,c, 6background b(i) 5 peak shape FWHM, Assym,

.... 10 number of intensities |Fk | to be found 1000 - ???

Pattern Decomposition - general information

• Aim: to find such a set of parameters for which

wiYi,o -Yi,c )2 = minimum {1} can be achieved by

minimisation of {1} using LS method or by other methods (genetic algorithm, simplex).

Source of trouble:

• number of points and parameters is large (computing problems)

• peaks overlap

The background

• The background intensity at the ith step: -an operator supplied file with the background intensities -linear interpolation between operator-selected points -a specified background function

• If background is to be refined -applied function can be phenomenological or based on physical reality, and include refineable model for amorphous component and thermal diffuse scattering. The function used most frequently: ybi=m=0,5Bm[(2i/BKPOS)-1]m

Peak shape

• Peak shape is a result of convolution of: -X-ray line spectrum, -all combined instrumental and geometric aberrations, -true diffraction effects of the specimen, that it is difficult to assign profile function which should be used in a particular case

• In practice (‘ab initio’ structure solution): -peak function which best fits to a selected fragment of the diffraction data is sought

• The most frequently used profile functions: Gaussian, Lorentzian, Pearson VII, Pseudo-Voight

• EXTRAC - ‘learned’ peak shape, selected peak is decomposed into series of base functions and stored in tabular form (for future use)

Profile functions

• Gaussian P(x)G =

• Lorentzian P(x)L =

• Voight P(x)V = L(x)G(x-u) du

• Pseudo-VoightP(x)p-V = L(x) + (1-)G(x), =f(2)

• Pearson VII P(x)PVII = a[1+(x/b)2]-m ,L{m=1},G{m} -where: Co =4ln2, C1 =4, C2h = (21/h -1)1/ , Hh = [w + vtg + utg21/,Assym. by adding, multiply,split

)XCexp(H

C 2ik0

k

0

12ik1

k

1 )XC(1πH

C

Lorentzian and Gaussian

FWHM

Pawley method - formulas

n1I h2

Ih

hk2i

n1ihk

h1

hkk

n1I

i}h{hhk2

Ik

2

n1I

22I

2

i}k{kk

)i(y)i(q1B

);i(q)i(q1H

B)H(A

0))i(qA)i(y)(i(q12A

))i(M)i(y(1

)i(qA)i(back)i(M

Programs applying this method: ALLHKL, SIMPRO, LSQPROF

Rietveld and Le Bail methods

))i(back)i(obs()obs(A)obs(A

))i(back)i(obs()i(q)calc(A)i(q)calc(A

)i(q)calc(A)obs(A

))i(back)i(obs()i(q)calc(A)i(q)calc(A

)i(q)calc(A)obs(A

i21

2211

22i2

2211

11i1

Rietveld method:

Le Bail method:

))i(back)i(obs()i(q)obs(A

)i(q)obs(A)obs(A i N1l l

nl

inm1n

m

ATRIB, EXTRA, EXTRAC, included in GSAS, FULLPROF

• Le Bail method

• Advantages: – fast, robust, easy to implementation in Rietveld programs -intensities always positive -prior knowledge easy to introduce (known fragment)

• Disadvantages: -e.s.ds of intensities not available

• Application: ‘ab initio’ structure determination

• Pawley method

• Advantages: –parameters are fitted by LS method -e.s.d’s of intensities are reported

• Disadvantages: -unstable calculations -negative intensities (removed by Wasser constraints) -complicated calculations (huge matrix to be inverted)

• Application: Lattice constants refinement, ab initio structure determination

Structure factors extraction in numbers

• Pawley method - 42• Le Bail method - 136• other methods - 34• pattern fitting without cell constraints - 14

• Programs most frequently used: FULLPROF - 46GSAS- 22 ARIT - 31 ALLHKL - 26

• Armel Le Bail http://www.cristal.org/iniref/progmeth.html

Diffraction pattern of propionic acid

small number of lines large number of lines

Lines’ positions depend on the lattice constants and the space group, peaks’ overlapping increase with 2angle

Peak Overlap in Powder Diffractometry

• Reflections overlap can be:

• exact (systematic) In tetragonal system, in s.g. P4; d(hkl)=d(khl), however intensity of I(hkl) & I(khl) are different d(120)=d(210). In cubic system d(340)=d(500); d(710)=d(550) but I(340) is not equal to I(500), and I(710) is different than I(550)

• accidental Some reflections (system orthorhombic-triclinic) have the same or nearly the same ds, but their Is are not related to each other.

2

222

2 alkh

d1 d= 222 lkh

a

Intensities of overlapping lines

• If two or more reflections are observed at 2 which differ by less than some critical value eps. these reflections belong to a group of overlapping (double) lines, the other reflections are called single lines.

• Critical eps. value is usually given as fraction of FWHM (full

width at half maximum): e.g.: eps. = 0.1-0.5FWHM

• With decrease of FWHM, number of single lines and possibility of structure solution increase. The lowest FWHMs are obtained using synchrotron radiation or focussing cameras, however, sometimes even such a good measurement does not lead to a successful structure solution.

Diffraction Patterns - powder diffractometer (red)

Guinier camera (green), synchrotron ESRF (blue)

Complex of DMAN with p-nitrosophenol: C14H19N2

+.C6H4(NO)O-.C6H4(NO)OH, measurement - ESRF, =0.65296A,SG:Pnma, a,b,c=12.2125, 10.7524, 18.6199(c/b=1.73)

Lasocha et al, Z.Krist. 216,117-121 (2001).

Overlapping reflections cont...

• Number of single reflections is 10-40% of the total number of the lines in a diffraction pattern.

• Due to peak overlapping in a diffraction pattern created by thousands of lines, few dozen of single lines are observed, so that by this method only very simple structures were determined (positions of heavy atoms)

• G. Sheldrick’s, rule ‘if less than 50% of theoretically observable reflections in the resolution range (d~1.2 – 1.0Ă) are observed (F>4F)), the structure is difficult to be solved by the conventional direct methods’.

G. Sheldrick’s, rule in practise

Single reflections Double reflections

Structure not solved Structure solved

•

Intensities of overlapping lines, basic approaches

• a) neglecting of overlapping lines• b) equipartition, intensity of a line cluster is divided

into n-components Ii = Itot/n• c) arbitrary intensity distribution

Itot = I1+I2 for two reflections 3 possibilities i) Itot = 2I1 = 2I2 ii) Itot = I1; I2 =0 iii) Itot = I2; I1=0 Methods very frequently used e.g. options of EXTRA program Altomare, Giacovazzo et al., J.Appl.Cryst. (1999) 32,339

Intensities of overlapping lines - DOREES method

• Reflections are divided into groups, in which there are single and overlapping lines. The groups of reflections could be triplets or quartets.– TRIPLETS: Three reflections create triplet H,K,H+K if:

H(h1,k1,l1), K(h2,k2,l2), H+K(h1+h2,k1+k2,l1+l2)– they represent three vectors forming triangle in reciprocal

space – examples of triplets: (004)(30-4)(300) ; (204)(10-4)(300) ;

one reflection e.g. (300) can be involved in many triplet relations.

– If two planes forming triplet are strong, it is possible that the third line from triplet is also strong. If more than one such triplets are found, this relation seems to be more probable EH=1/NTK EKE-H-K. Jansen, Peschar, Schenk, J.Appl.Cryst., (1992)25,231

FIPS – Fast Iterative Patterson Squaring

– Patterson function: P(u) = 1/V h|Fh|2 exp(2i(hu)) {1}is obtained from available data (equipartitioned

dataset)

– a non-linear modification is applied to Patterson function (e.g. squaring)

– intensities for the reflections of interest (overlapping) are obtained by back-transformation of the modified map (single lines remain unchanged): |F’h|2 = VP’(u) exp(-2i(hu)) du

– the above procedure is repeated untill satisfatory results are obtained Esterman,McCusker,Baerlocher, J.Appl.Cryst.(1992),25, 539

Experimental Methods

• Method based on anisotropic thermal expansion

• With temperature increase a,b,c,are changed,

The lines which overlap at temp. T1 can be separated at temp. T2. It should be no phase transitions between T1 & T2, and symmetry ought to be sufficiently lowThis method was used in 1963 by Zachariasen to solved -Pu structure.

Zachariasen, Ellinger, Acta Cryst. (1963) 16, 369

Different preferred orientation (flat sample holder (red), sample in capillary (green)

A simplified texture-based method for intensity determination of overlapping reflections

• Intensity affected by texture I0’ = I0f(G,)

• For a group of n overlapping reflections Ik’ =

i=1,nIi,0f(G,i)• The basic idea is to find a set of the most appropriate

intensities (including overlapping) which corresponds to all patterns with different texture

• Assumptions:• intensity of a cluster of n reflections is accurately

measured • preferred orientation function and its coefficients are

determined• for m>n measurements set of n linear equations are

created and solved

A simplified texture-based method for intensity...

• The measured patterns are decomposed into intensities, single intensities (within 0.5FWHM limit) are normalised.

• Few of the most probable texture directions are selected, and for each direction the angle between preferred orientation and the scattering vector are calculated

• Reflections are divided into groups accordingly to the angle

• Assuming that I0’ = I0exp(Gcos2) is the texture

function, by weighted LS procedure from linear dependence of ln<E2> vs. < cos2> , G parameter and its e.s.d, correlation coefficient were determined.

A simplified texture-based method for intensity...

• the difference in the texture should be sufficient for different measurements

• n overlapping reflections are resolved in orientation space

• To conclude: Texture which is obstacle to structure solution may be helpful in the intensity determination of overlapping lines Lasocha, Schenk (1997). J. Appl. Cryst. 30, 561 Cerny R. Adv. X-ray Anal. 40. CD-ROM Wessels, T., Baerlocher, Ch., McCusker, L.B., Science, 284, 477 Wessels, T., Ph.D. Thesis, ETH Zurich, Switzerland

State of art and new perspectives for ab initio structure solution from powder data

• New procedures for decomposition of powder pattern -positivity constraints( positivity of electron density and Patterson map, Bayesian approach to impose Is positivity) -prior knowledge (known fragment, pseudo-transitional symmetry, texture)-already options in EXPO program

• Combination of simulated annealing with direct methods• Real space techniques for phase extension and refinement

• C.Giacovazzo, Plenary lectures, ECM-21, Durban,• C.Giacovazzo, XIX Conference on Applied Crystallography, Kraków• W.David, Plenary lectures, ECM-21, Durban,

Methods used for estimation of intensities of overlapping reflections in numbers

• Full data, equipartitioning - 141• partial data set, overlapping lines excluded - 80• DOREES - 6• FIPS and other new methods - a few successful applications

• positivity constraints,Bayesian approach David & Sivia)- 2

• known fragment, positivity constraints (Giacovazzo et al.,) - great number of results recently published In some, new, very promising methods, full pattern decomposition is not required.

• Armel Le Bail http://www.cristal.org/iniref/progmeth.html

Conclussions• treatment of overlapping reflections - potential of

experimental methods, possibilities of anisotropic broadening, or different peak shape in the same pattern

• design of experiment accordingly to the problem to be solved

• new theoretical achievements - new perspectives for the ‘ab initio’ structure solution ‘powder diffraction methods work perfectly with good data, with bad ones do not work at all...’ ‘The rules are simple to write, but often difficult in practise’ [Gilmore 1992].

Successful structure solution

Single reflections,known fragments,prior information, new experimental methods etc

Double reflections

•