Programs for computation of powder diffraction patterns

Programs for computation of
powder diffraction patterns
• W. Kraus, G. Nolze. Federal Institute for
Materials Research and Testing (BAM), Unter
den Eichen 87, D-12205 Berlin
• http://ccp14.minerals.csiro.au/ccp/webmirrors/powdcell/a_v/v_1/powder/e_cell.html
• http://ccp14.minerals.csiro.au/ccp/webmirrors/powdcell/a_v/v_1/powder/details/powcell
.htm
Structure Determination from
Powder Diffraction
• Course, experimental point of view
• http://www.cristal.org/course/index.html
• http://pcb4122.univlemans.fr/iniref/tutorial/indexa.html
• http://pcb4122.univ-lemans.fr/iniref.html
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Number of publications about
new structures determined
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•
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•
http://pcb4122.univ-lemans.fr/iniref.htm
1948-1987 28 structures
2000 520 structures
2006 1328 structures
Powcel
• unit cell parameters,
• atomic names, z, coordinates for the
asymmetric unit
• space group number
• information for each space group in file
pcwspgr.dat.
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BCC structured CsCl
CsCl input file for PowCell
cell 4.123 4.123 4.123 90 90 90
natom 2
Cs 55
0.0 0.0 0.0
Cl
17
0.5 0.5 0.5
rgnr 221
3
Powcell: diffraction pattern of Gold
space group 225
F4/m -3 2/m
a=b=c=4.078
Diamond
8 atoms in the unit cell: (0,0,0) and (1/4,1/4,1/4) + FCC translations
4
Diamond
Diamond
5
Diamond
Crystal structure determination
• Traditional approach or direct method:
using intensities and positions of
reflections to solve the structure
• Direct space methods: construction of
model for the structure (atomic positions)
and fitting the model intensity to the
diffraction pattern (whole pattern fitting)
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Structure determination
• Determination of lattice parameters
• Assignment of crystal symmetry and space
group
• Structure solution
– Initial model directly from experimental diffraction data
– Straightforward from single crystal data
– Solution from powder diffraction data under quite
active research
• Structure refinement
Crystal structure determination
• Problems
– Reflections overlap
– Background determination
– Preferred orientation
• Powerful numerical methods needed
• Prior information on the structure is useful
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Neutrons and x-rays
• X-rays: heavy elements
– Small sample needed when using synchrotron
radiation
• Neutrons: light elements
– Large samples
Pattern decomposition
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•
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•
•
Peak search
Fitting a model to diffraction lines
The diffraction pattern may be split in parts
Good profile shape function is important.
Taking into account K 1 and K 2 split
Needs determination of background
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Tasks: Indexing
• Example: Exhautive search of lattice
parameter space by varying lattice
parameters in discrete steps and isolating
volumes that contain possible solutions
(Dichotomy method).
• E.g. DICVOL91 (Boultif and Louër 1991)
• 84% succesful in triclinic structures
How many parameters?
• Lattice constants, max. 6
– Cubic: only a
– Triclinic: 3 lengths and 3 angles
• Experimental:
– angular offset
– shape of the reflections 3-4 parameters
• Number of parameters can be about 10
• 20-30 reflections are needed
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Equation in matrix form
• The equation Xhh h2 + Xkk k2 + Xll l2 + Xhk hk
+ Xhl hl + Xkl kl = 1/dhkl2
• Eq. in matrix form: HX = D
• H nxm matrix
• D column (nx1) containinf 1/d2 data
• X column (nx1)
SVD based algorithm
SVD singular value decomposition
• The equation Xhh h2 + Xkk k2 + Xll l2 + Xhk hk
+ Xhl hl + Xkl kl = 1/dhkl2 is solved iteratively
using SVD method (A. Coelho. J. Appl.
Cryst. 2003, 36, 86-95.
• Includes Monte Carlo search
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Singulaariarvohajotelma
• Singulaariarvohajotelma on
ominaisarvohajotelman yleistys, jolla on
käyttöä numeerisissakin tehtävissä.
• Hajotelman avulla voi tarkastella
esimerkiksi vektoreiden lineaarisen
riippuvuutta.
• Hajotelmaa voi käyttää myös ratkaistaessa
lineaarisia pienimmän neliösumman
tehtäviä.
Singular value decomposition
• In matrix formulation the singular value
decomposition is as follows: for each mxn matrix
A (even rectangular matrices, and especially for
column and row matrices) there exists unitary
mxm matrix U and nxn matrix V, and a mxn
matrix S so that A = U S VT.
• If A is a square matrix, S is diagonal. The
diagonal elements of S are called singular
values.
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Space group determination
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Forbidden reflections
Problem: overlapping reflections
Trial and error
Structure: comparing model and
experimental intensity
• Iterative improvement
Structure analysis: R-factor
• A measure of agreement of the structural
model with the experiment
• R = ( ||Fhkl|obs - |Fhkl|calc| )/ |Fhkl|obs
• For some purposes also
• R = whkl (|Fhkl|obs - |Fhkl|calc )2 where whkl
are weighting factors.
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Patterson function from
intensities of the reflections hkl
Generally
I(q) = P(r) exp(i q·r) d3r
P(r) ~ I(q) exp(-i q·r) d3q
If the electron density is periodic, P is called
the Patterson function. It is then
P(r) ~1/V hkl | F(hkl) |2 exp(-iqhkl·r)
~ 1/V hkl | F(hkl) |2 cos (hx+ky+lz)
Phase problem: special cases
• The structure contains heavy atoms,
whose positions are known. The phase of
these heavy atoms determine phases of
other structure factors.
• General constraints as >0.
• Known sructures
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Minimization methods
• Plenty of fitting parameters
• Powerful fitting method needed
– Monte Carlo
– Simulated annealing
– Genetic algorithms
• Reliable intensity information essential
Crystal structure solution
• Patterson method: only structure factor
amplitudes are used.
• P(r) = 1/V |F(h)|2 exp(-2 h·r).
– Each peak in P corresponds to an interatomic
distance within the unit cell.
– Heights of the peak are proportional to the
scattering power of the atoms
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Crystal structure solution
• Direct methods. Attempts to solve phases
directly from the diffraction data.
Constraint: electron density is positive in
the unit cell.
• Example. P-methoxy benzoic acid.
Monoclinic structure from 169 nonoverapping and 203 overlapping
reflections.
Rietvelt refinement programs
• FullProf: Rodríguez-Carvajal. Laboratoire
Léon Brillouin (CEA-CNRS) CEA/Saclay
http://www.ccp14.ac.uk/
http://www.ccp14.ac.uk/ccp/ccp14/ftpmirror/fullprof/pub/divers/fullprof.2k/Windo
ws/
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Rietvelt refinement programs
• GSAS: Larson and von Dreele. Los Alamos
National Laboratory
• Scheme in GSAS Indexation
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–
–
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Structure factor amplitude extraction
Method for structure factors selection
Conventional Patterson or direct methods
Model building, molecule location, unconventional
methods...
– Completion of the structure
– Final Rietveld refinement
http://www.ccp14.ac.uk/about.htm
Example. Kaolinite Al2Si2O5(OH)4
b
c
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Kaolinite (triclinic C1)
• Platelike orientation (110)
8215
-1-11
kaolinite1993 (pref.Or)
25
30
003
-201
-1-31
-131 200
022
0-22
-112
1-11
-1-12
002
021
0-21
20
-111
020
15
110
001
0
10
111
1-10
4107
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Example. Kaolinite
• Neutron powder diffraction 1.5 K
• Wavelength 1.9102 Å
• Most thermal contraction occur in 001
direction due to decrease in interlayer
distance.
• H atom positions?
•DL Bish. Clays and Clay Minerals. 41, 6, 738-744, 1993
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Refinement
• Missing reflections h+k = odd
• Young and Hewat kaolinite structure
model 1988
• Program GSAS
• Anisotropic displacement parameters for H
• H1 random positional disorder
• OH bond lengths between 0.976 and
0.982 Å
•DL Bish. Clays and Clay Minerals. 41, 6, 738-744, 1993
Example. New study on kaolinite
• Single crystal refinement
• Synchrotron radiation, microfocus beam line at
ESRF
• Wavelength 0.6883 Å
• Room temperature
• Space group C1
• Positions of intralayer H atoms could not be
refined.
• The diffraction patterns showed diffuse
scattering in streaks parallel to [001] direction
caused by stacking faults.
Neder et al. Clays and clay minerals 1999, 47, 4, 487-494.
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Example. K3Ba3C60
• The crystal
structure of bodycentered cubic
K3Ba3C60. The
Ba2+ and K+ ions
are shifted from
the tetrahedral
sites, (0, 1/2, 1/4).
• Fullerenes in the
corners and in the
middle of the cell.
Margadonna et al. Chem. Mater., 12 (9), 2736 -2740, 2000.
Example. K3Ba3C60
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•
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High-resolution synchrotron X-ray
diffraction measurements were
performed on the K3Ba3C60
sample sealed in a 0.5 mm
diameter glass capillary.
Data were collected in continuous
scanning mode using nine
Ge(111) analyzer crystals on the
BM16 beamline at ESRF at 10
and 295 K (wavelength 0.83502
Å).
Analysis of the diffraction data at
both temperatures was performed
with the GSAS suite of Rietveld
analysis programs.
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•
•
Temperature-dependent
synchrotron X-ray diffraction
measurements were also
performed on the SwissNorwegian beamline (BM1A) at
the ESRF with a 300 mm diameter
Mar Research circular image
plate.
The sample was cooled from 320
to 105 K at a rate of 25 K/h.
Diffraction patterns were
measured every 5 min with a
sample to detector distance of 200
mm and with an exposure time of
40 s.
Data analysis was performed with
the Fullprof suite of Rietveld
analysis programs.
Margadonna et al. Chem. Mater., 12 (9), 2736 -2740, 2000.
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Example. K3Ba3C60
• The neutron diffraction experiment was undertaken with
the high-resolution diffractometer D2b (1.5944 Å) at the
Institute Laue Langevin, Grenoble, France.
• The sample (0.58 g) was loaded in a cylindrical
vanadium can (diameter = 5 mm) sealed with indium
wire and then placed in a standard ILL liquid helium
cryostat.
• The data were collected in the scattering angle range 0164.5 in steps of 0.05 deg.
• A full diffraction profile was measured with counting time
of 10 h at 10 K.
• The data analysis was performed with the GSAS
software.
Margadonna et al. Chem. Mater., 12 (9), 2736 -2740, 2000.
Example. Structures of the Polymer
Electrolyte Complexes
• Polymer electrolytes consist of
salts, e.g., LiCF3SO3,
dissolved in solid highmolecular-weight polymers,
such as PEO (CH2CH2O)n.
• Neutron diffraction experiment
• Refinement was carried out
using the GSAS program
package.
Rietveld refinement
• The refinements involved
1734 data points, 50 atoms in
the asymmetric unit, 162
variables, and 135 soft
constraints.
• For both PEO6:LiPF6 and
PEO6:LiSbF6 the powder
data could be indexed on
monoclinic cells and with
systematic absences that
unambiguously determined
the space group as P21/a.
Gadjourova et al. Structures of the Polymer Electrolyte Complexes PEO6:LiXF6 (X
= P, Sb), Determined from Neutron Powder Diffraction Data. Chem. Mater., 13 (4),
1282 -1285, 2001.
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Rietvelt refinement programs
• A. A. Coelho Whole-profile structure
solution from powder diffraction data using
simulated annealing. J. Appl. Cryst. (2000). 33,
899-908
• Starting point: space group and lattice
parameters
• Solution of structures by simulated annealing
• The simulated annealing control parameters
have been systematically investigated.
– Most significant: electrostatic-potential penalty
functions.
Rietvelt refinement programs
• EXPO (Altomare et al. J. Appl. Cryst.
1999, 32, 338-340, and J. Appl. Cryst. 35,
182-184.
• Electron density modification procedure
• Diagonal least squares refinement
• Ratio of the number of observations to the
number of parameters should be greater
than 3.
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References
• JI Langford and D Louëb. Powder diffraction. Rep.Prog.
Phys. 59, 1996, 131-234
• K Harris and M Tremayne. Crystal structure
determination from powder diffraction data. Chem.
Mater. 1996, 8, 2554-2570.
• BM Kariuki et al. The application of genetic algorithm for
solving crystal structures from powder diffraction data.
Chem. Phys. Lett. 280, 1997, 189-195.
• H Putz, JC Schön, M Jansen. Combined method for ab
initio structure solution from powder diffraction data. J.
Appl. Cryst. 32, 1999, 864-870
– cost function E + R, simulated annealing
– E: Lennard-Jones –type potential
Model independent solution
• MEM
• A model-independent maximum-entropy method is
presented which will produce a structural model from
small-angle X-ray diffraction data of disordered systems
using no other prior information. In this respect, it differs
from conventional maximum-entropy methods which
assume the form of scattering entities a priori. The
method is demonstrated using a number of different
simulated diffraction patterns, and applied to real data
obtained from perfluorinated ionomer membranes, in
particular Nafion(TM), and a liquid crystalline copolymer
of 1,4-oxybenzoate and 2,6-oxynaphthoate (B-N).
• Elliot and Hanna. J. Appl. Cryst. 1999, 32(6), 1069-1083.
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