xix conference on applied crystallography summer school on polycrystalline structure determination
DESCRIPTION
XIX Conference on Applied Crystallography Summer School on Polycrystalline Structure Determination. Full Pattern Decomposition. Kraków, September 2003 by Wiesław Łasocha. Structure Solution from Powder Data. Where are we now ?- some numbers. - PowerPoint PPT PresentationTRANSCRIPT
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XIX Conference on Applied Crystallography
Summer School on Polycrystalline Structure Determination
Kraków, September 2003
by
Wiesław Łasocha
Full Pattern Decomposition
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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’
0
100
200
300
400
500
600
700
800
years
1947-87198819891990199119921993199419951996199719981999200020012002
1987 1997 20021991
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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
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Structure determination
Sample
Data collection
Indexing
Space group
Structure solution
Rietveld refinement
Pattern decomposition
Final Structure
Per aspera ad astra
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Single crystal diffraction
2
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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) + {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 - ???
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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
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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: ybi=m=0,5Bm[(2i/BKPOS)-1]m
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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-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
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Lorentzian and Gaussian
FWHM
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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
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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
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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.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
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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 - 46GSAS- 22 ARIT - 31 ALLHKL - 26
• Armel Le Bail http://www.cristal.org/iniref/progmeth.html
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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
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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.
2
222
2 alkh
d1 d= 222 lkh
a
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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.
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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, =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).
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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. 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’.
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G. Sheldrick’s, rule in practise
Single reflections Double reflections
Structure not solved Structure solved
•
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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 = 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
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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,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
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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
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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
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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,)
• 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
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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 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.
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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/progmeth.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
•