XIX Conference on Applied Crystallography Summer School on Polycrystalline Structure Determination

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XIX Conference on Applied CrystallographySummer
School on Polycrystalline Structure Determination
Full Pattern Decomposition

• Kraków, September 2003
• by
• Wieslaw Lasocha

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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
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Number of crystal structures solved ab initio
1987
1991
1997
2002
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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
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Structure determination
Per aspera ad astra
Final Structure
Rietveld refinement
Structure solution

Pattern decomposition
Space group
Indexing
Data collection
Sample
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Single crystal diffraction

2q
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Powder Diffraction Pattern - the basic source of
information about the investigated material
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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

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Pattern Decomposition - general information

• Diffraction pattern can be described by the
  formula Yi,c M(i) back(i) SkiAk 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 - ???

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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

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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 ybiSm0,5Bm(2qi/BKPOS)-1m

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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)

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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),
  hf(2q)
• Pearson VII P(x)PVII a1(x/b)2-m
  ,Lm1,Gm -where Co
  4ln2, C1 4, C2h (21/bh -1)1/2 , Hh w
  vtgq utg2q 1/2,Assym. by adding, multiply,split

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Lorentzian and Gaussian
FWHM
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Pawley method - formulas
Programs applying this method ALLHKL, SIMPRO,
LSQPROF
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Rietveld and Le Bail methods
Rietveld method
Le Bail method
ATRIB, EXTRA, EXTRAC, included in GSAS, FULLPROF
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• 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.ds 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

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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/progme
  th.html

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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
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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
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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.

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Diffraction Patterns - powder diffractometer
(red) Guinier camera (green), synchrotron ESRF
(blue)
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Complex of DMAN with p-nitrosophenol
C14H19N2.C6H4(NO)O-.C6H4(NO)OH, measurement -
ESRF, l0.65296A,SGPnma, a,b,c12.2125, 10.7524,
18.6199(c/b1.73)
Lasocha et al, Z.Krist. 216,117-121 (2001).
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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. Sheldricks, rule if less than 50 of
  theoretically observable reflections in the
  resolution range (d1.2 1.0A) are observed
  (Fgt4s(F)), the structure is difficult to be
  solved by the conventional direct methods.

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G. Sheldricks, rule in practise

Structure not solved
Structure solved
Single reflections
Double reflections
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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 I1I2 for two reflections 3
  possibilities i) Itot 2I1
  2I2 ii) Itot I1 I2 0 iii) Itot I2
  I10 Methods very frequently used e.g.
  options of EXTRA program Altomare,
  Giacovazzo et al., J.Appl.Cryst. (1999) 32,339

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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,HK if H(h1,k1,l1), K(h2,k2,l2),
  HK(h1h2,k1k2,l1l2)
• 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 EH1/NTS
  K EKE-H-K. Jansen, Peschar, Schenk,
  J.Appl.Cryst., (1992)25,231

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FIPS Fast Iterative Patterson Squaring

• Patterson function P(u) 1/V S h Fh2
  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) Fh2 ?VP(u) exp(-2pi(hu)) du
• the above procedure is repeated untill
  satisfatory results are obtained
  Esterman,McCusker,Baerlocher, J.Appl.Cryst.(1992),
  25, 539

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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
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Different preferred orientation (flat sample
holder (red), sample in capillary (green)
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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
  Si1,n Ii,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 mgtn measurements set of n linear equations
  are created and solved

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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 lnltE2gt vs. lt cos2agt , G parameter
  and its e.s.d, correlation coefficient were
  determined.

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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

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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,

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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/progme
  th.html

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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.

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Successful structure solution

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