XIX Conference on Applied Crystallography Summer School on Polycrystalline Structure Determination

1. XIX Conference on Applied CrystallographySummer School on Polycrystalline Structure Determination Full Pattern Decomposition Kraków, September 2003 by Wiesław Łasocha

2. 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

3. Number of crystal structures solved ‘ab initio’ 1987 1991 1997 2002

4. Structure determination chemical information chemical information Multiple dataset whole pattern Triplets FIPS Patterson & direct methods equipartition new methods Treatment of overlap structure completion FINAL STRUCTURE Le Bail intensity extraction Rietveld refinement Pawley data collection space group determination neutron indexing radiation synchrotron sample laboratory Structure Determination from Powder Diffraction Data, ed. W.I.F.David, et all

5. Structure determination Per aspera ad astra Final Structure Rietveld refinement Structure solution Pattern decomposition Space group Indexing Data collection Sample

6. Single crystal diffraction 2q

7. Powder Diffraction Pattern - the basic source of information about the investigated material

8. 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

9. Pattern Decomposition - general information • Diffraction pattern can be described by the formula: Yi,c = M(i) = back(i) + S{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,a,b,g 6 background b(i) 5 peak shape FWHM, Assym, h, .... 10 number of intensities |Fk | to be found 1000 - ???

10. Pattern Decomposition - general information • Aim: to find such a set of parameters for which Siwi(Yi,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

11. 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-appliedfunctioncan be phenomenological or based on physical reality, and include refineable model for amorphous component and thermal diffuse scattering. The function used most frequently:ybi=Sm=0,5Bm[(2qi/BKPOS)-1]m

12. 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)

13. Profile functions • Gaussian P(x)G = • Lorentzian P(x)L = • Voight P(x)V = L(x)G(x-u) du • Pseudo-Voight P(x)p-V = hL(x) + (1-h)G(x), h=f(2q) • Pearson VII P(x)PVII = a[1+(x/b)2]-m ,L{m=1},G{m} -where: Co =4ln2, C1 =4, C2h = (21/bh -1)1/2 , Hh = [w + vtgq + utg2q] 1/2,Assym. by adding, multiply,split

14. Lorentzian and Gaussian FWHM

15. Pawley method - formulas Programs applying this method: ALLHKL, SIMPRO, LSQPROF

16. Rietveld and Le Bail methods Rietveld method: Le Bail method: ATRIB, EXTRA, EXTRAC, included in GSAS, FULLPROF

17. 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

18. 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 - 46 GSAS - 22 ARIT - 31 ALLHKL - 26 • Armel Le Bail http://www.cristal.org/iniref/progmeth.html

19. Diffraction pattern of propionic acidsmall number of lines large number of lines Lines’ positions depend on the lattice constants and the space group, peaks’ overlapping increase with 2q angle

20. 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. d=

21. Intensities of overlapping lines • If two or more reflections are observed at 2q 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.

22. Diffraction Patterns - powder diffractometer (red) Guinier camera (green), synchrotron ESRF (blue)

23. Complex of DMAN with p-nitrosophenol: C14H19N2+.C6H4(NO)O-.C6H4(NO)OH, measurement - ESRF, l=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).

24. 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>4s(F)), the structure is difficult to be solved by the conventional direct methods’.

25. G. Sheldrick’s, rule in practise Structure not solved Structure solved Single reflections Double reflections

26. 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

27. 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/NTS K EKE-H-K. Jansen, Peschar, Schenk, J.Appl.Cryst., (1992)25,231

28. FIPS – Fast Iterative Patterson Squaring • Patterson function: P(u) = 1/V S h|Fh|2 exp(2pi(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(-2pi(hu)) du • the above procedure is repeated untill satisfatory results are obtainedEsterman,McCusker,Baerlocher, J.Appl.Cryst.(1992),25, 539

29. Experimental Methods • Method based on anisotropic thermal expansion • With temperature increase a,b,c,a,b,g 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 low This method was used in 1963 by Zachariasen to solved b-Pu structure. Zachariasen, Ellinger, Acta Cryst. (1963) 16, 369

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

31. A simplified texture-based method for intensity determination of overlapping reflections • Intensity affected by texture I0’ = I0f(G,a) • For a group of n overlapping reflections Ik’ = Si=1,nIi,0f(G,ai) • 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

32. 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 a angle between preferred orientation and the scattering vector are calculated • Reflections are divided into groups accordingly to the a angle • Assuming that I0’ = I0exp(Gcos2a) is the texture function, by weighted LS procedure from linear dependence of ln<E2> vs. < cos2a> , G parameter and its e.s.d, correlation coefficient were determined.

33. 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-ROMWessels, T., Baerlocher, Ch., McCusker, L.B., Science, 284, 477 Wessels, T., Ph.D. Thesis, ETH Zurich, Switzerland

34. 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,

35. 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

36. 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].

37. Successful structure solution Single reflections, known fragments, prior information, new experimental methods etc Double reflections