Rietveld

Report
Structure determination from X-ray
powder diffraction data
Alok Kumar Mukherjee
Department of Physics
Jadavpur University
Kolkata -700032
E-mail : [email protected]
Crystal structure determination using X-ray
powder diffraction is a far more difficult task
than several other applications of X-ray
powder diffraction, such as:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Phase identification
Quantitative phase analysis
Microstructure study
Texture analysis
Thin film study
Dynamic and nonambient diffraction
Single crystal versus powder crystal XRD
Although the single crystal and powder crystal XRD
patterns essentially contain the same information, but in
the former case the information is distributed in threedimensional space where as in the latter case the threedimensional data are “compressed” into one-dimension.
Single Crystal Diffraction
Powder Diffraction
As a consequence, there is generally considerable
overlap of peaks in the X-ray powder pattern
leading to ambiguities in extracting the peak
positions(2) and intensities, I(hkl), of individual
diffraction maxima.
The powder diffraction pattern
Why structure determination from powder XRD ?
Many crystalline solids including several
organic,
metal-organic,
pharmaceutical
compounds and nano-materials are available
only as polycrystalline powders. In such
cases, X-ray powder diffraction is the only
realistic option for structure elucidation.
Flow chart for structure determination from
X-ray powder diffraction data
High quality powder XRD
data collection
Measurement of step scanned
intensity data
Unit cell determination
Indexing the XRD pattern
Possible space group
assignment
Look for any obvious systematic
absences
Structure solution
Determination of approximate
atomic positions
Structure refinement
Refinement of atomic
coordinates, temperature
factors, occupancy factors
The whole procedure of structure determination
from X-ray powder data is not automatic. It
requires significant intelligent input from the
crystallographer at every step during the process.
For success in structure determination from
powder diffractometry (SDPD), the first step i.e.
data collection is crucial .
The quality of data sets necessary depends on
the type of analysis intended.
• Indexing requires accurate and precise peak
positions, specially at low 2θ values (large dspacings).
• Structure solution requires reliable intensities.
• Structure refinement requires good quality
high angle data (small d-spacings).
In addition, high-resolution data (narrow 2θ peak
width) is desirable.
High sample purity is required.
One potential adverse effect of using very high
resolution powder diffractometers is that any
impurity peaks from the sample are more easily
seen
→ problem in indexing !
Some useful data collection parameters
(i) Wave length of X-ray (usually CuK is used)
(ii) Monochromatization
(iii) Zero-shift alignment
(iv) Incident and diffracted beam apertures
Divergent slit ~ 0.1 mm – 0.6 mm
Receiving slit ~ 0.1 mm – 0.2 mm
(v) Data collection parameters (Step scan mode)
(a) Step size ~ 0.008º - 0.02º
(b) Counting time ~ 20 - 30 sec/step
Data acquisition variables can be adjusted to produce a
better quality data set.
Sample holder
X-ray tube
Detector
Slit boxes
The view of Bruker D8 Advance diffractometer with stationary X-ray
source and synchronized rotations of both the detector arm and
sample holder
Some criteria for good quality X-ray powder data
1. Peak to noise ratio
should be high
2. Full width half maxima
peak overlapping (~ 0.08º- 0.10º)
3. Number of peaks
4. Reproducibility of powder pattern
preferred orientation
Number of expected diffraction peaks
The number of reflections up to the diffraction angle θ in
a powder diffraction pattern is determined by the size
and symmetry of the unit cell and the wavelength of the
radiation used.
N=32πVSin2θ/3λ3Q ,
Where V is the unit cell volume, λ is the wavelength and
Q is the product of the average multiplicity of the
reflections and the number of lattice points per unit cell.
For example:
λ= 1.5418Å, V= 1500 Å3 , 2θ= 25°
N
=32πVSin2θ/3λ3Q
= 637/Q
For orthorhombic system :
Q
and
N
= average multiplicity X no. of lattice points
per unit cell
=5X1
=5
= 637/5
≈ 127
Effect of counting time on statistical error during
a single measurement
Counts Counting Number (N) of
Spread (√N) Error () at 90% Error () at 99%
per sec time (s) registered counts
confidence(%)
confidence(%)
100
1
100
25
100
2500
10
16.4
25.9
50
3.3
5.2
=[Q/(N)½] x 100%, Q=1.64 for 90% confidence level
Q=2.59 for 99% confidence level
Good quality XRD data is the primary requirement
Particular problem with organic and pharmaceutical
compounds ~ diffraction data often contain very few
peaks
Non-reproducible data due to preferred orientation
Effect of the divergence slit width
Slit with 0.05mm
Slit with 0.6mm
Effect of the divergence slit width in peak asymmetry
Slit with 0.05mm
Slit with 0.6mm
Powder pattern indexing
The basic equation used for indexing a powder diffraction
pattern is
Qhkl = 1/dhkl2 = h2a*2+k2b*2+l2c*2+2klb*c*cos *+
2hlc*a*cos*+2hka*b*cos*
where, dhkl is the interplanar spacing corresponding to
the (hkl) plane and a*, b*, c*, *, *, * are the reciprocal
lattice cell parameters.
The 2θ positions corresponding to a reasonable number of
diffraction maxima (say 20-30) are extracted from the
observed pattern by fitting individual peaks using
appropriate profile function.
Different Indexing programs:
TREOR → A semi-exhaustive trial-and-error indexing
program, which is based on the permutation of Miller
indices in a selected basis set of lowest Bragg angle peaks.
DICVOL → An exhaustive trial-and-error indexing program
based on the variation of the lengths of cell edges and
inter-axial angles over finite ranges, followed by a
progressive reduction of these intervals by means of a
dichotomy procedure.
ITO → A zone search indexing method with the
provision for the reduction of the most probable
unit cell. It is based on specific relations among Qhkl
values in reciprocal space.
MCMAILLE → Based on the Monte Carlo and grid
search method.
X-cell→ An indexing algorithm which uses an
extinction-specific dichotomy procedure to perform
an exhaustive search of parameter space to
establish a complete list of all possible unit cell
solutions.
Commonly used figures of merit
To assess the reliability of indexing, two figures of merit are
commonly used.
The de Wolff ’s figure of merit (M20)
M20= Q20 / 2N20 x (ΔQ)av
where,
Q20 is the Q value (1/d2) for the 20th observed line, N20 is
the number of different diffraction lines possible up to the
20th observed line, (ΔQ)av is the average absolute
discrepancy between the observed and calculated Q values
for the 20 observed lines.
The Smith and Snyder figure of merit (FN)
FN= N / Nposs x (Δ2)av
Where Nposs is the number of calculated diffraction
lines up to the Nth observed line and (Δ2)av is the
average absolute discrepancy between the observed
and calculated 2 values.
Higher the accuracy of the data collected and more
complete the pattern, the larger will be M20 and FN
values.
Comments on the value of M20
• A ‘solution’ with FOM value less than 5.0 is worthless.
• Any ‘solution’ that leaves more than two very weak lines
unexplained is not useful. However, if the FOM is greater
than 10.0 it might be worthwhile to examine the input
data and the ‘solution’ more closely.
• Even ‘solutions’ that index all lines with a FOM > 10.0
should not be accepted uncritically without further
investigation.
Experience shows that M20 values greater than 15 are likely to
correspond to correct solutions for laboratory X-ray powder data.
Pitfalls in indexing:
• Attempting to index a poor data is a
non-starter
→ trying to solve a jigsaw puzzle with half
the pieces missing !!
• Pseudo-symmetry: when certain lattice parameters have
values that result in the symmetry of the lattice appearing
to be higher than reality e.g. ,
→
Monoclinic angle β very close to 90°, the metric
tensor for the lattice will be similar to that of an
orthorhombic lattice.
→
unit cell has a monoclinic symmetry, but a≈c and γ
close to 120°; the symmetry of the lattice is then
pseudo-hexagonal.
• Instrumental errors:
Satisfactory solution may not be obtained if 2θ zero
error greater than 0.08° .
• Dominant Zones
Indexing a powder pattern with dominant zones
(one cell axis is much shorter than the other two) is
often a problem.
If most of the first few lines are of
h0l type → the d-spacings intrinsically lack
3-dimensional unit cell information.
To overcome this problem XRD pattern should be
collected of an unground powder sample.
• Impurities and other phases:
Which peak(s) due to impurity ?
• Samples that change phase during data
acquisition
Indexing of the XRD data collected with step size
0.02º and counting time 20s per step results a
triclinic cell with low FOM.
Crystal system : Triclinic
a=4.734(4) Å,
α=92.1(1)º
b=8.248(6) Å
β=85.8(1)º
c=10.811(7) Å
γ=98.0(1)º
Vol=416.7 Å3
F20= 23, M20= 15
No corresponding
calculated position
WPPD plot
Indexing of the same XRD data collected with step
size 0.008º and counting time 30s per step results a
orthorhombic cell with high FOM.
Crystal system : Orthorhombic
a=16.666(2) Å
b=10.548(2) Å
c=9.476(2) Å
Vol=1665.8 Å3
F20= 88, M20= 52
WPPD plot
Severely overlapping peaks may results no indexing at all
Space group assignment- most tricky part!
The choice of space group is possibly the difficult
part to automate.
The presence or absence of screw axes or glide
planes from indexing is not always obvious since
the average powder pattern is likely to contain
only a few reflections of the type h00 (or 0k0, 00l)
and hk0 (or h0l , 0kl etc.). The paucity of data
results in real ambiguities in the choice of space
groups.
If there is a choice between a commonly
observed space group and a rare one based on
occurrence in the Cambridge Crystallographic
Database (e.g P2/c vs P21/c), then try to solve
the structure using the most common space
group first.
Example:
A molecular structure indexed
orthorhombic system showed Z=8.
in
the
The reflection conditions are ambiguous and
indicate either Pbca or Pnma (this situation can
easily arise due to overlap of key reflections in
the powder pattern). Both these space groups are
found to occur frequently for molecular
compounds.
For Pnma, however, the presence of mirror plane
would suggest either two molecules related by
mirror symmetry or two molecules per
asymmetric unit with the mirror plane passing
through both molecules
→ unlikely situation from packing consideration.
So, in this case the first choice should be Pbca
A better way to overcome this problem is to employ
Whole pattern fitting approach in which the diffraction
profile is fitted in the absence of a structural model, but
in the presence of the reflection conditions of the
various space groups under consideration.
In EXPO-2004, for each crystal system, the probabilities
of different extinction symbols are calculated. A list of
possible space groups compatible with each extinction
symbol is presented.
Successful structure solution will only ascertain the
correctness of the assigned space group!
Structure was solved
in P21/a - which had a
lower FOM
Structure solution
Traditional methods (Patterson, Direct methods)
Using the extracted intensities, the structure solution
procedures are similar to those in the single crystal case.
Direct space approach (Monte Carlo, Simulated annealing,
Grid search etc.)
All direct space approaches avoid the problematic step of
Ihkl extraction from the experimental powder pattern
Trial structural models generated in direct space. Suitability
of a model assessed by comparing the calculated powder
pattern based on the model with the observed powder
pattern.
Monte Carlo approach
In the Monte Carlo approach a sequence of structures
is generated as potential structure solution. The first
structure (x1) is generally chosen as a random position
of the structural fragment in the unit cell. Starting
from the structure xi, the structural fragment is
subjected to a random displacement to generate a
trial structure (xtrial).
The trial structure is then accepted or rejected by
considering the difference z between the Rwp values
corresponding to structures xtrial and xi.
z = Rwp (xtrial) – Rwp (xi)
if z ≤ 0, the trial structure is automatically accepted,
whereas if z > 0, the trial structure is accepted with
probability exp(-z/s) and rejected with probability [1exp(-z/s)], where s is a scaling factor.
After a sufficiently extensive range of structural
space has been explored, the structure
corresponding to lowest Rwp is considered as the
starting model for refinement
The main factor limiting the efficiency of Monte
Carlo calculation is the number of structural
degrees of freedom varied during calculation.
Thus in the Monte Carlo approach, the number of
degrees of freedom in the structural fragment is a
more important consideration than the number of
atoms in the asymmetric unit.
This approach is quite popular with organic or
pharmaceutical compounds where the molecular
compositions are known a priori.
Rietveld method
In the Rietveld method, the integrated intensities of the
reflections (Ihkl) are calculated from the atomic parameters
of the model. The equations for y(2i)cal now becomes
y(2i)cal = smhkl (Lp)hkl |Fhkl|2 Phkl hkl + b(2i)
Where, s is the scale factor, mhkl is the reflection
multiplicity, (Lp)hkl is the Lorentz-polarization factor, P is the
preferred orientation function,  is the reflection profile
function, Fhkl is the structure factor for the reflection, b(2i) is
the background intensity at the ith step.
Criteria of fit
The agreement between the observed and calculated
powder patterns is judged by several indicators. The
profile R-factors are,
R- pattern (profile)
RP =  |yi (obs) - yi (calc)| /  |yi (obs)|
R weighted pattern (profile)
RwP = [  wi {yi (obs) - yi (calc)}2 / wi { yi (obs)}2]1/2
R-Structure factor
RF =  |(IK(obs))1/2 - (IK(calc))1/2| / (IK(obs))1/2
R-Bragg factor
RB =  |IK(obs) - IK(calc)| /  |IK(obs)|
R expected
RE = [(N- P)/  wi { yi (obs)}2]1/2
Where, N and P are the number of profile points and
refined parameters, respectively
Goodness-of-fit
2 = (Rwp / RE)2
Steps in structure determination from powder data
Examples from our recent work
Example-1:
5-5′- disubstituted hydantoins
Imidazolidine-2,4-dione, or hydantoin,
is a five-membered heterocyclic ring
containing a reactive cyclic urea
nucleus.
Hydantoins can serve as useful
intermediates in the synthesis of
amino acids.
Dipropyl glycine hydantoin
The indexing with TREOR showed an
orthorhombic unit cell with
a= 7.167(1), b= 13.964(1), c =10.957(3)Å
[M(20)= 49, F(20)= 84(0.003343, 72)]
Statistical analysis of the using the
FINDSPACE module of EXPO2004
indicated Pnma as the most probable
space group (Z’= 0.5)
ORTEP view
Hydantoin moeity
lying on the mirror
plane and side chains
are above and below
the plane
Final Rietveld plot
Formation of C11(4)C11(4)[R22(8)] network
Crystal data
Chemical Formula
C9 H16 N2 O2
Mr
184.24
Temperature (K)
293(2)
system,
Space group, Z
Orthorhombic,
Pnma, 4
a, b, c(Å)
7.1577(5), 13.9721(9), 11.0211(10)
Volume (Å3)
1102.2(2)
D (Mg/m3)
1.110
Wavelength (Å)
1.54056
Diffractometer
Bruker D8 Advance
Rp
0.0477
Rwp
0.0687
χ2
1.671
Cyclohexanespiro-5′-hydantoin
Synthesized as
microcrystalline powder of
cyclohexanespiro-5′ –
hydantoin monohydrate(I)
The compound (I) was slowly heated to
115°C and kept at the elevated
temperature for 1 hr. The sample was
cooled to room temperature (25°C) to
obtain
Anhydrous
cyclohexanespiro-5′–
hydantoin (II)
Comparison of observed powder profiles of
anhydrous and hydrated phases
Normalized Intensity
Anhydrous Phase-II
Hydrated Phase-I
2θ(º)
Intensity(counts)
Final Rietveld Plot of hydrated form-I
2θ(º)
Intensity(counts)
Final Rietveld Plot of anhydrous form-II
2θ(º)
Comparison of the packing of the hydrated and
the anhydrous phase
R22(8) ring
formed by
N-H…O
hydrogen
bonds
Hydrated Phase
Anhydrous Phase
Crystal data and Rietveld refinement parameters for
hydrated and anhydrous phases
Formula weight
Temperature
Crystal system
Space group
Unit cell
Dimensions
(I)
186.21
293(2)K
Orthorhombic
Pna21
a=16.887(2) Å
b=9.245(2) Å
c=6.267(1) Å
(II)
168.20
293(2) K
Monoclinic
P21/c
a=12.7468(4) Å
b=7.0777(2) Å
c=10.3348(3) Å
β=110.891(2) °
Volume
Z
Density (calculated)
Rp
Rwp
RF2
χ2
978.3(4)Å3
4
1.264g/cm3
0.0478
0.0778
0.1029
4.786
871.09(4) Å3
4
1.283 g/cm3
0.0399
0.0557
0.0815
0.9655
Example-2:
Two nimesulide derivatives
1
2
Nimesulide, N-(4-nitro-2phenoxyphenyl)
methanesulfonamide, is an
effective non-steroidal antiinflammatory drug (NSAID),
which can inhibit
cyclooxygenase-2 (COX-2)
enzyme selectively
To the best of our knowledge, this is the first example of
structure determination of nimesulide analogues from Xray powder diffraction data
Rietveld plot and Ortep view of 1
Rietveld plot and Ortep view of 2
Example-3:
Three o-Hydroxyacetophenone derivatives with
varied degrees of flexibility
Indexing and Space group determination
1
2
3
Final Rietveld plot and ORTEP diagram
Final Rietveld plot and ORTEP diagram
Final Rietveld plot and ORTEP diagram
Crystal Data
(1)
(2)
(3)
Chemical Formula
C10H12O3
C17H18O
C14H18O5
Mr
180.20
270.33
266.29
Temperature (K)
295(2)
295(2)
295(2)
system,
Space group, Z
Monoclinic,
P21/c, 4
Monoclinic,
P21/c, 4
Monoclinic,
P21/c, 4
a, b, c(Å)
9.8702 (15), 13.7735 (19), 12.207 (6), 16.487 (8),
8.1460 (12)
7.495 (4)
8.0139 (5), 7.2616 (3),
23.9540 (18)
 ()
123.368 (5)
102.952 (5)
95.738 (5)
Volume (Å3)
924.9 (3)
1470.0 (13)
1386.99 (16)
D (Mg/m3)
1.294
1.221
1.275
Wavelength (Å)
1.5406
1.5406
1.5406
Data/ restraints/
parameters/
Rp
4750 / 78/ 123
4790/ 122/ 173
4800/ 126 / 152
0.0465
0.0579
0.0496
Rwp
0.0659
0.0855
0.0751
R(F2)
0.0978
0.1208
0.1208
χ2
9.68
6.66
3.35
Example-4:
3-phenylpropionic acid derivatives
Final Rietveld plot and ORTEP diagram
Compound 2
Final Rietveld plot and ORTEP diagram
Compound 3
Final Rietveld plot and ORTEP diagram
Compound 4
Crystal Data
Molecular Weight
Temperature (K)
Crystal System
C10H12O3 (2)
180.2
293
Monoclinic
C10H12O3 (3)
180.2
293
Monoclinic
C10H12O3 (4)
180.2
293
Monoclinic
Space Group
a(Å)
b(Å)
c(Å)
β(º)
P21/n
22.8913(14)
4.8938(27)
8.2326(6)
99.227(3)
P21/c
7.8243(4)
19.6301(7)
6.7344(4)
114.385(4)
P21/n
12.2392(5)
7.6727(2)
11.1229(2)
114.168(3)
Volume (Å3)
Z
910.33(14)
4
942.08(6)
4
952.97(3)
4
Dcalc (gcm-3)
Wavelength (Å)
2θ interval (º)
No. of parameters
No. of data points
No. of restraints
1.315
1.5418
5-100
64
4624
51
1.27
1.5418
5-100
55
4780
47
1.256
1.5418
5-100
58
4780
46
Rp
0.0527
0.0541
0.0476
Rwp
0.0748
0.078
0.0662
R(F2)
χ2
0.1425
5.514
0.1009
3.204
0.0669
1.416
Example-5:
3-phenylpropionic acid and its derivatives with
Z´= 2 .
Final Rietveld plot and ORTEP diagram
Final Rietveld plot and ORTEP diagram
Final Rietveld plot and ORTEP diagram
Crystal Data
C9H10O2 (1)
C10H12O2 (5)
C10H12O3 (6)
Molecular weight
Temperature
Wavelength(Å)
Crystal system
Space group
a (Å)
150.18
293
1.5418
Monoclinic
P21/a
31.6676(22)
164.20
293
1.5418
Monoclinic
P21/n
17.3804 (12)
180.20
293
1.5418
Monoclinic
P21/a
13.8863(14)
b (Å)
9.8510(6)
6.1389 (5)
10.6470(13)
c (Å)
5.4829(4)
16.9998 (12)
13.9018(17)
β (°)
98.776(5)
91.025 (4)
114.787(3)
Volume(Å3)
Z / Z'
Dcalc
2 interval (˚)
Stepsize,counting time
1690.4(3)
8, 2
1.180
5-65
0.02, 10
1813.5 (4)
8, 2
1.203
5-100
0.02, 10
1866.0(6)
8, 2
1.282
5-100
0.02, 10
No. of data points
No. of parameters
No. of background points
3014
116
20
4774
121
10
4774
126
10
Rp
Rwp
R(F2)
χ2
0.0346
0.0489
0.0788
4.285
0.0501
0.0715
0.0729
2.220
0.0609
0.0853
0.1435
1.742
Example-6:
Ramipril -Tris(hydroxymethyl)amino methane
Ramipril, an
angiotensin-converting
enzyme (ACE) inhibitor,
belongs to a family of
drugs used in
cardiovascular
problems
In the absence of suitable single-crystals of
ramipril-tris co-crystal , structure solution was
achieved from laboratory X-ray powder diffraction
data
Indexing of powder pattern showed a monoclinic system,
a=24.334 Å, b=6.460 Å, c=9.532 Å, β=96.95˚
V=1474.5 Å3
The space group indicated was P 21
After several unsuccessful attempts crystal structure of
ramipril-tris was finally solved in direct-space using the
program FOX
View of the asymmetric unit of ramipril-tris co-crystal
Final Rietveld plot of ramipril-tris
Crystal data and Rietveld refinement parameters of
Ramipril-tris
Empirical formula
Formula weight
Temperature
Wavelength
Crystal system
Space group
Unit cell dimensions
Volume
Z
Density (calculated)
2 range for data collection
Step size
Counting time per step
No. of profile data steps
No. of variable parameters
No. of background points
Rp
Rwp
RF2
2
C23H32N2O5. C4H11NO3
537.65
293(2) K
1.5418 Å
Monoclinic
P21
a= 24.382(2)Å, b= 6.477 (1) Å,
c= 9.554(1) Å; = 96.92 (1) °
1497.7 (4) Å3
2
1.214 g/cm3
6 to 60°
0.01°
10 s
5726
218
10
0.0520
0.0680
0.1735
24.66
References:
1. A. Bhattacharya, S. Ghosh, K. Kankanala, V. R. Reddy, K. Mukkanti, S. Pal & A. K.
Mukherjee. Chem. Phys. Lett. (2010), 493, 151-157.
2. A. Bhattacharya, K. Kankanala, S. Pal & A. K. Mukherjee. J. Mol. Struc. (2010), 975, 40-46.
3. B. Chattopadhyay, A. K. Mukherjee , N. Narendra, H. P. Hemantha, V.V. Sureshbabu, M.
Helliwell, & M.Mukherjee. Cryst. Growth & Des. (2010) 10, 4476- 4484
4. U. Das, B. Chattopadhyay, M. Mukherjee & A. K. Mukherjee. Chem. Phys. Lett. (2011) 501,
351-357.
5. B. Chattopadhyay, M.Mukherjee, Kantharaju, V.V. Sureshbabu & A.K.Mukherjee. Z. Krist
(2008), 223, 591-597.
6. B. Chattopadhyay, S. Ghosh, S. Mondal, M.Mukherjee & A.K.Mukherjee. CrystEngComm,
(2011) , DOI:10.1039/C1CE05920C
7. U. Das, B. Chattopadhyay, M. Mukherjee & A. K. Mukherjee. Cryst. Growth & Des. (2011) ,
DOI: 10.1021/cg201290g.
Acknowledgements
Prof . Monika Mukherjee
Dr. Soumen Ghosh
Dr. Basab Chattopadhyay
Dipak K. Hazra
Abir Bhattacharya
Uday das

similar documents