Transcript XIX Conference on Applied Crystallography Summer School on

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
• 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’
1947-87
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
800
700
600
500
400
300
200
100
0
years
1987
1991
1997
2002
chemical information
equipartition
chemical information
Patterson & direct methods
new methods
Treatment of overlap
Le Bail
Pawley
intensity
extraction
whole pattern
Structure determination
FINAL
STRUCTURE
structure
completion
Rietveld refinement
data collection
space group determination
neutron
indexing
synchrotron
sample
laboratory
Structure Determination from Powder Diffraction Data, ed. W.I.F.David, et all
Structure
determination
Final Structure
Rietveld refinement
Structure solution
Pattern decomposition
Space group
Indexing
Data collection
Sample
Single crystal diffraction
2q
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) + 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
background
b(i)
peak shape FWHM, Assym, h, ....
number of intensities |Fk | to be found
6
5
10
1000 - ???
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
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=Sm=0,5Bm[(2qi/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
Profile functions
• Gaussian
P(x)G =
• Lorentzian
P(x)L =
C0
 Hk
C1
πH
exp(  C 0 X ik )
2
(1  C 1 X ik )
2
1
k
• 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
Lorentzian
and
FWHM
Gaussian
Pawley method - formulas
M ( i )  back ( i )   A k q k ( i )
{ k }i
   nI  1
2

1
I
2
( y ( i )  M ( i ))
2
A k
1
  2  nI  1

2
I
2
q k ( i )( y ( i )   A h q h ( i ) )  0
{ h }i
1
A k  ( H hk ) B h
1
H hk   ni  1
Bh  
n
I 1

1

2
I
2
i
q k ( i ) q h ( i );
q h (i )y (i )
Programs applying this method: ALLHKL, SIMPRO, LSQPROF
Rietveld and Le Bail methods
Rietveld method:
A 1 ( obs )   i
A 2 ( obs )   i
A 1 ( calc ) q 1 ( i )
A 1 ( calc ) q 1 ( i )  A 2 ( calc ) q 2 ( i )
A 2 ( calc ) q 2 ( i )
A 1 ( calc ) q 1 ( i )  A 2 ( calc ) q 2 ( i )
( obs ( i )  back ( i ))
( obs ( i )  back ( i ))
A 1 ( obs )  A 2 ( obs )   i ( obs ( i )  back ( i ))
Le Bail method:
A
n 1
m
n
m
A ( obs ) q i ( i )
( obs )   i N
( obs ( i )  back ( i ))
n
 l  1 A l ( obs ) q l ( i )
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
Le Bail method
other methods
pattern fitting without cell constraints
• Programs most frequently used:
FULLPROF
GSAS
ARIT
ALLHKL
- 46
- 22
- 31
- 26
• Armel Le Bail http://www.cristal.org/iniref/progmeth.html
- 42
- 136
- 34
- 14
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 2q angle
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)
1
d
2
h k l
2

2
a
2
2
d=
a
h k l
2
2
2
• accidental
Some reflections (system orthorhombic-triclinic) have the same or nearly the
same ds, but their Is are not related to each other.
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.
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,
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).
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>4(F)), the structure is difficult to
be solved by the conventional direct methods’.
G. Sheldrick’s, rule in practise
•
Structure not solved
Single reflections
Structure solved
Double reflections
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/NTS K EKE-H-K.
Jansen, Peschar, Schenk, J.Appl.Cryst., (1992)25,231
FIPS – Fast Iterative Patterson Squaring
– Patterson function: P(u) = 1/V S h |Fh|2 exp(2i(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(-2i(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,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
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,a)
• For a group of n overlapping reflections
Ik’ = Si=1,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 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 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.
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
•
Double reflections
Single reflections,
known fragments,
prior information,
new experimental
methods etc