Transcript Crystal Structure Determination and Refinement Using the Charles Campana

Crystal Structure Determination and
Refinement Using the
Bruker AXS SMART APEX II System
Charles Campana
Bruker AXS Inc.
Flowchart for Method
Select, mount, and opti call y ali gn a sui tabl e crystal
Eval uate crystal quali ty; obtain uni t cel l geometry
and prel iminary symmetry informati on
Measure intensity data
Data reducti on
Sol ve the structure
Adapted from William Clegg
“Crystal Structure Determination”
Oxford 1998.
Complete and refi ne the structure
Interpret the resul ts
Crystal Growing Techniques
 Slow
evaporation
 Slow
cooling
 Vapor
diffusion
 Solvent
diffusion
 Sublimation
http://laue.chem.ncsu.edu/web/GrowXtal.html
http://www.as.ysu.edu/~adhunter/YSUSC/Manual
/ChapterXIV.pdf
Examples of Crystals
Growing Crystals
Kirsten Böttcher and Thomas Pape
Select and Mount the Crystal

Use microscope

Size: ~0.4 (±0.2) mm

Transparent, faces, looks single

Epoxy, caulk, oil, grease to affix

Glass fiber, nylon loop, capillary
What are crystals ?
Crystallographic Unit Cell
Unit Cell Packing Diagram - YLID
7 Crystal Systems Metric Constraints

Triclinic – none

Monoclinic -  =  = 90,   90

Orthorhombic -  =  =  = 90

Tetragonal -  =  =  = 90, a = b

Cubic -  =  =  = 90, a = b = c

Trigonal -  =  = 90,  = 120, a = b (hexagonal setting)
 =  =  , a = b = c (rhombohedral setting)

Hexagonal -  =  = 90,  = 120, a = b
or
X-Ray Diffraction Pattern from
Single Crystal
Rotation Photograph
X-Ray Diffraction
X-ray beam
  1Å
(0.1 nm)
~ (0.2mm)3 crystal
~1013 unit cells, each ~
(100Å)3
Diffraction pattern on
CCD or image plate
Bragg’s law
n = 2d sin()


d
We can think of diffraction as reflection at sets of planes running through the
crystal. Only at certain angles 2 are the waves diffracted from different planes
a whole number of wavelengths apart, i.e. in phase. At other angles the waves
reflected from different planes are out of phase and cancel one another out.
Reflection Indices
z
These planes must intersect the cell
edges rationally, otherwise the
diffraction from the different unit
cells would interfere destructively.
y
x
We can index them by the number
of times h, k and l that they cut
each edge.
The same h, k and l values are used
to index the X-ray reflections from
the planes.
Planes 3 -1 2 (or -3 1 -2)
Diffraction Patterns
Two successive CCD detector images with a crystal rotation
of one degree per image
For each X-ray reflection (black dot) indices h,k,l can be assigned
and an intensity I = F 2 measured
Reciprocal space

The immediate result of the X-ray diffraction experiment is a list of X-ray
reflections hkl and their intensities I.

We can arrange the reflections on a 3D-grid based on their h, k and l
values. The smallest repeat unit of this reciprocal lattice is known as the
reciprocal unit cell; the lengths of the edges of this cell are inversely
related to the dimensions of the real-space unit cell.

This concept is known as reciprocal space; it emphasizes the inverse
relationship between the diffracted intensities and real space.
The structure factor F and
electron density 
Fhkl =
V xyz exp[+2i(hx+ky+lz)] dV
xyz = (1/V) hkl Fhkl exp[-2i(hx+ky+lz)]
F and  are inversely related by these Fourier transformations. Note that  is real
and positive but F is a complex number: in order to calculate the electron density
from the diffracted intensities I = F2 we need the PHASE ( ) of F. Unfortunately it is
almost impossible to measure  directly!
The Crystallographic Phase
Problem
The Crystallographic Phase Problem

In order to calculate an electron density map, we require both the intensities I
= F 2 and the phases  of the reflections hkl.

The information content of the phases is appreciably greater than that of the
intensities.

Unfortunately, it is almost impossible to measure the phases experimentally !
This is known as the crystallographic phase problem and
would appear to be insoluble
Real Space and Reciprocal Space
Real Space
 Unit Cell (a, b, c, , , )
 Electron Density, (x, y, z)
 Atomic Coordinates – x, y, z
 Thermal Parameters – Bij
 Bond Lengths (A)
 Bond Angles (º)
 Crystal Faces
Reciprocal Space
 Diffraction Pattern
 Reflections
 Integrated Intensities – (h,k,l)
 Structure Factors – F(h,k,l)
 Phase – (h,k,l)
Goniometer Head
3-Axis Rotation (SMART)
3-Axis Goniometer
SMART APEX II System
SMART APEX System
SMART APEX II System
APEX II detector
CCD Chip Sizes
X8 APEX, SMART APEX,
6000, 6500
4K CCD 62x62 mm
Kodak 1K CCD 25x25 mm
SMART 1000, 1500
& MSC Mercury
SITe 2K CCD 49x49 mm
SMART 2000
APEX II detector






transmission of fiber-optic taper
depends on 1/M2
APEX with direct 1:1 imaging
1:1 is 6x more efficient than 2.5:1
improved optical transmission by
almost an order of magnitude
allowing data on yet smaller microcrystals or very weak diffractors.
original SMART 17 e/Mo photon
APEX 170 e/Mo photon
project database
default settings
detector calibration
SMART
ASTRO
setup
sample screening
data collection strategy
data collection
SAINTPLUS
new project
change parameters
SAINT: integrate
SADABS: scale & empirical absorption correction
SHELXTL
new project
XPREP: space group determination
XS: structure solution
XL: least squares refinement
XCIF: tables, reports
George M. Sheldrick
Professor, Director of Institute and part-time programming technician
1960-1966: student at Jesus College and Cambridge University, PhD
(1966)
with Prof. E.A.V. Ebsworth entitled "NMR Studies of Inorganic
Hydrides"
1966-1978: University Demonstrator and then Lecturer at Cambridge
University; Fellow of Jesus College, Cambridge
Meldola Medal (1970), Corday-Morgan Medal (1978)
1978-now: Professor of Structural Chemistry at the University of
Goettingen
Royal Society of Chemistry Award for Structural Chemistry (1981)
Leibniz Prize of the Deutsche Forschungsgemeinschaft (1989)
Member of the Akademie der Wissenschaften zu Goettingen (1989)
Patterson Prize of the American Crystallographic Association (1993)
Author of more than 700 scientific papers and of a program called
SHELX
Interested in methods of solving and refining crystal structures (both
small
molecules and proteins) and in structural chemistry
email: [email protected]
fax: +49-551-392582
SHELXTL vs. SHELX*
http://shelx.uni-ac.gwdg.de/SHELX/index.html

SHELXTL (Bruker Nonius)
 XPREP
 XS
 XM
 XE
 XL
 XPRO
 XWAT
 XP
 XSHELL
 XCIF











SHELX (Public Domain)*
None
SHELXS
SHELXD
SHELXE
SHELXL
SHELXPRO
SHELXWAT
None
None
CIFTAB
Chemistry 530
Structure and Spectroscopy
Crystallography Lectures
R. B. Wilson
What is crystallography?
Extraction of structural information from X-ray diffraction (XRD) data
Where does it fit in a course entitled ‘Structure and
Spectroscopy’?
Method for obtaining detailed and powerful structural information
about molecules of any complexity and about materials (bulk
samples)
Is diffraction a type of spectroscopy?
Spectroscopy and diffraction both involve the interactions of
light (electromagnetic radiation, emr) with matter.
Spectroscopy describes absorption or emission of
electromagnetic radiation (light)
Diffraction is a scattering phenomenon
Points of emphasis
•Model building exercise
•Statistical experiment
•Better crystals  better structures
Electrons diffract X-rays. Because electrons (associated with
atoms) are in specific locations in the crystal, analysis of the
diffraction pattern gives information about these locations
(atomic positions  structure).
Crystallography – 2
R. B. Wilson
Results from a “Small Molecule” Crystal Structure – Pictures
What does this picture portray?
Our final model, the one that gave best agreement
with the observed data
What structural information is conveyed
in this picture?
What does this picture portray?
What additional structural information is
conveyed in this picture?
Crystallography – 3
R. B. Wilson
Results – Distances and Angles
Table 1. Bond lengths [Å] and angles [°] for p17l_ra.
_________________________________________________________________________________________________
O(1)-C(13)
N(2)-C(1)
N(2)-C(3)
N(4)-C(5)
N(4)-C(3)
N(12)-C(13)
N(12)-C(5)
N(12)-H(12)
C(1)-C(10)
C(1)-C(11)
C(3)-C(8)
C(5)-C(6)
C(6)-C(7)
C(6)-H(6)
C(7)-C(8)
C(7)-H(7)
C(8)-C(9)
C(9)-C(10)
C(9)-C(15)
C(10)-H(10)
C(11)-H(11A)
C(11)-H(11B)
C(11)-H(11C)
C(13)-C(14)
C(14)-H(14A)
C(14)-H(14C)
C(14)-H(14B)
C(15)-H(15A)
C(15)-H(15B)
C(15)-H(15C)
1.2181(18)
1.3296(18)
1.3673(17)
1.3147(17)
1.3629(17)
1.3653(19)
1.4053(18)
0.891(18)
1.408(2)
1.496(2)
1.4156(19)
1.416(2)
1.363(2)
0.963(16)
1.414(2)
0.989(17)
1.419(2)
1.365(2)
1.509(2)
1.011(16)
0.96(2)
1.00(2)
0.95(2)
1.488(2)
0.93(2)
0.95(2)
0.86(2)
1.02(2)
0.95(2)
0.95(2)
C(1)-N(2)-C(3)
C(5)-N(4)-C(3)
C(13)-N(12)-C(5)
C(13)-N(12)-H(12)
C(5)-N(12)-H(12)
N(2)-C(1)-C(10)
N(2)-C(1)-C(11)
C(10)-C(1)-C(11)
N(4)-C(3)-N(2)
N(4)-C(3)-C(8)
N(2)-C(3)-C(8)
N(4)-C(5)-N(12)
N(4)-C(5)-C(6)
N(12)-C(5)-C(6)
C(7)-C(6)-C(5)
C(7)-C(6)-H(6)
C(5)-C(6)-H(6)
C(6)-C(7)-C(8)
C(6)-C(7)-H(7)
C(8)-C(7)-H(7)
C(7)-C(8)-C(3)
C(7)-C(8)-C(9)
C(3)-C(8)-C(9)
C(10)-C(9)-C(8)
C(10)-C(9)-C(15)
C(8)-C(9)-C(15)
C(9)-C(10)-C(1)
117.49(12)
117.86(12)
127.44(12)
118.7(11)
113.8(11)
122.58(13)
116.76(14)
120.62(14)
114.58(11)
122.70(12)
122.71(12)
113.66(12)
124.03(13)
122.29(13)
118.10(14)
122.9(10)
119.0(10)
120.24(14)
119.6(9)
120.2(9)
116.99(13)
124.38(13)
118.62(13)
117.11(13)
121.46(14)
121.42(14)
121.39(14)
C(9)-C(10)-H(10)
C(1)-C(10)-H(10)
C(1)-C(11)-H(11A)
C(1)-C(11)-H(11B)
H(11A)-C(11)-H(11B)
C(1)-C(11)-H(11C)
H(11A)-C(11)-H(11C)
H(11B)-C(11)-H(11C)
O(1)-C(13)-N(12)
O(1)-C(13)-C(14)
N(12)-C(13)-C(14)
C(13)-C(14)-H(14A)
C(13)-C(14)-H(14C)
H(14A)-C(14)-H(14C)
C(13)-C(14)-H(14B)
H(14A)-C(14)-H(14B)
H(14C)-C(14)-H(14B)
C(9)-C(15)-H(15A)
C(9)-C(15)-H(15B)
H(15A)-C(15)-H(15B)
C(9)-C(15)-H(15C)
H(15A)-C(15)-H(15C)
H(15B)-C(15)-H(15C)
120.2(9)
118.4(9)
111.9(11)
112.1(11)
101.6(16)
110.4(12)
112.1(17)
108.5(16)
123.77(15)
121.13(15)
115.09(14)
113.0(14)
109.9(13)
96.4(17)
117.0(15)
123(2)
91.3(19)
108.9(11)
109.6(12)
109.0(16)
110.4(12)
105.7(16)
112.9(17)
Crystallography – 4
R. B. Wilson
Results – Atomic positions
4
2
3
Table 2. Atomic coordinates ( x 10 ) and equivalent isotropic displacement parameters (Å x 10 ) for p17l_ra.
U(eq) is defined as one third of the trace of the orthogonalized Uij tensor.
________________________________________________________________________________
x
y
z
U(eq)
________________________________________________________________________________
O(1)
10905(2)
3807(2)
1239(1)
53(1)
N(2)
6742(2)
11218(2)
1570(1)
30(1)
N(4)
8092(2)
8453(2)
1692(1)
29(1)
N(12)
9325(2)
5696(2)
1913(1)
32(1)
C(1)
6167(2)
12572(2)
1167(1)
32(1)
C(3)
7501(2)
9777(2)
1248(1)
27(1)
C(5)
8770(2)
6989(2)
1415(1)
28(1)
C(6)
8896(2)
6692(2)
685(1)
34(1)
C(7)
8334(2)
8031(2)
241(1)
33(1)
C(8)
7632(2)
9654(2)
511(1)
28(1)
C(9)
7027(2)
11127(2)
92(1)
32(1)
C(10)
6327(2)
12571(2)
433(1)
35(1)
C(11)
5270(2)
14097(2)
1525(1)
40(1)
C(13)
10326(2)
4205(2)
1805(1)
34(1)
C(14)
10686(3)
3069(3)
2437(1)
47(1)
C(15)
7122(3)
11084(3)
-696(1)
41(1)
________________________________________________________________________________
Crystallography – 5
R. B. Wilson
Results – Displacement (thermal) Parameters
Table 3. Anisotropic displacement parameters (Å2x 103) for p17l_ra.
The anisotropic displacement factor exponent takes the form: -2p2[ h2 a*2U11 + ... + 2 h k a* b* U12 ]
______________________________________________________________________________
U11
U22
U33
U23
U13
U12
______________________________________________________________________________
O(1)
59(1)
70(1)
32(1)
-3(1)
8(1)
28(1)
N(2)
32(1)
33(1)
26(1)
-2(1)
1(1)
-2(1)
N(4)
31(1)
33(1)
22(1)
-1(1)
2(1)
0(1)
N(12)
37(1)
36(1)
22(1)
1(1)
6(1)
4(1)
C(1)
30(1)
33(1)
32(1)
0(1)
-3(1)
-3(1)
C(3)
25(1)
31(1)
24(1)
-1(1)
1(1)
-4(1)
C(5)
27(1)
33(1)
25(1)
0(1)
3(1)
-2(1)
C(6)
38(1)
38(1)
26(1)
-5(1)
5(1)
1(1)
C(7)
34(1)
43(1)
20(1)
-4(1)
3(1)
-3(1)
C(8)
27(1)
36(1)
22(1)
-1(1)
0(1)
-5(1)
C(9)
30(1)
41(1)
25(1)
3(1)
-3(1)
-9(1)
C(10)
34(1)
35(1)
34(1)
6(1)
-7(1)
-3(1)
C(11)
41(1)
38(1)
43(1)
-1(1)
-1(1)
4(1)
C(13)
33(1)
40(1)
29(1)
-2(1)
4(1)
3(1)
C(14)
52(1)
52(1)
37(1)
9(1)
10(1)
18(1)
C(15)
46(1)
51(1)
25(1)
6(1)
-2(1)
-6(1)
______________________________________________________________________________
Crystallography – 6
R. B. Wilson
Results – Hydrogen Bonding; Torsion Angles
Table 4. Hydrogen bonds for p17l_ra [Å and °].
______________________________________________________________
D-H...A
d(D-H)
d(H...A)
d(D...A)
______________________________________________________________
N(12)-H(12)...N(2)#1
0.891(18)
2.170(18)
3.0570(18)
______________________________________________________________
Symmetry transformations used to generate equivalent atoms:
#1 -x+3/2,y-1/2,-z+1/2
<(DHA)
173.9(16)
Table 5. Torsion angles [°] for p17l_ra.
__________________________________________________________________________________
C(3)-N(2)-C(1)-C(10)
-0.6(2)
N(4)-C(3)-C(8)-C(7)
3.2(2)
C(3)-N(2)-C(1)-C(11)
177.29(13)
N(2)-C(3)-C(8)-C(7)
-175.51(13)
C(5)-N(4)-C(3)-N(2)
177.11(12)
N(4)-C(3)-C(8)-C(9)
-178.05(12)
C(5)-N(4)-C(3)-C(8)
-1.72(19)
N(2)-C(3)-C(8)-C(9)
3.2(2)
C(1)-N(2)-C(3)-N(4)
178.76(12)
C(7)-C(8)-C(9)-C(10)
177.66(14)
C(1)-N(2)-C(3)-C(8)
-2.4(2)
C(3)-C(8)-C(9)-C(10)
-1.0(2)
C(3)-N(4)-C(5)-N(12)
-179.37(11)
C(7)-C(8)-C(9)-C(15)
-1.2(2)
C(3)-N(4)-C(5)-C(6)
-1.1(2)
C(3)-C(8)-C(9)-C(15)
-179.81(14)
C(13)-N(12)-C(5)-N(4)
-167.97(14)
C(8)-C(9)-C(10)-C(1)
-1.8(2)
C(13)-N(12)-C(5)-C(6)
13.8(2)
C(15)-C(9)-C(10)-C(1)
177.00(14)
N(4)-C(5)-C(6)-C(7)
2.3(2)
N(2)-C(1)-C(10)-C(9)
2.8(2)
N(12)-C(5)-C(6)-C(7)
-179.59(13)
C(11)-C(1)-C(10)-C(9)
-175.00(15)
C(5)-C(6)-C(7)-C(8)
-0.7(2)
C(5)-N(12)-C(13)-O(1)
1.0(3)
C(6)-C(7)-C(8)-C(3)
-1.9(2)
C(5)-N(12)-C(13)-C(14)
-179.30(15)
C(6)-C(7)-C(8)-C(9)
179.44(13)
Crystallography – 7
R. B. Wilson
Overview
Goal:
Picture of molecule
Implied:
3D coordinates, displacement (thermal) parameters
Result:
Molecular structure: distances, angles, planes, torsion or angles, H-bonding [intramolecular features]
Crystal structure: arrangement of molecules relative to each other [intermolecular features]
How do we get this information?
X-ray Diffraction Experiment
Structure Determination and Analysis
What is needed for this experiment?
•Crystal (~75% probability)
•Means to collect or obtain single-crystal data
•PC and software to deconvolute data (solve structure) (~90-95% probability)
Crystallography – 8
R. B. Wilson
Overview
A
Calculate Reflections for Model of a Crystal
Measure Reflections from a Crystal
1. Attach crystal (< 0.4 mm) to cryo-loop using oil
2. X-rays + Crystal = “Reflections” (really diffraction)
3. 3-D repeat pattern defines the Unit Cell and
metric symmetry of cell gives Bravais Lattice
Equation A: electron density in cell (atoms) =
Σ (± phase) (F measured) (cos …)
Equation B: F calculated (with ± phase) =
Σ (atoms in model) (cos …)
(sum over reflections)
(sum over atoms)
1. Start model – Find the first few atom(s) in the unit cell using
“Direct”, “Patterson”, “Dual-Space” or “Monte-Carlo” methods to
assign some Phases (± signs)
h
3
3
3
3
k
4
4
5
5
l F measured Phase assigned
7
139
+
8
21
-2
67
-1
221
+
Equation A
2. a. Refine atoms positions (least squares)
b. Calculate F calculated using Model and Equation B
c. Improve electron map using phases of F calculated and Equation A
4. Measure Intensities
of reflections
(1000’s of reflections)
h
3
3
3
3
k
4
4
5
5
5. Confirm Lattice using symmetry
of Intensities then look for
systematic absences
to give probable Space Group
P21/n
(for this reciprocal plot)
l
7
8
-2
-1
F measured = ± (Intensity)½
± 139
± 21
± 67
± 221
h
3
3
3
3
k
4
4
5
5
l F measured F calculated
7
139
+133
8
21
+17
-2
67
-75
-1
221
+209
Equation A
3. Repeat 2a–2b–2c until all atoms are located and parameters converged
4. Agreement factor “R” shows progress: R = Σ |F calc – F meas| / Σ F meas
5. Calculate bond lengths and angles, plot picture, evaluate model
Crystallography – 9
R. B. Wilson
The X-ray Diffraction Experiment – Measuring Data
1.
Start with a crystal, max dimension < 0.4 mm, (less than diameter of incident X-ray beam)
min dim, depends of crystal volume, beam intensity, atomic contents
X-rays, high energy, short  (~1x10-10 m, 1 Å, 0.1 nm, 100 pm)
2.
X-ray + Crystal = ‘reflections’ (diffracted intensities)
Why X-rays? “Any radiation can be diffracted by appropriate scatterers, but as a rule, the most important information
arises when the wavelength of the radiation is similar to, or smaller than, the size of the spacings between the objects
being studied.” www.matter.org.uk/diffraction/
What size are molecular dimensions we want to observe? 0.5-100 Å, 10–11 – 10–8 m, X-rays have  appropriate to
interact with objects of molecular dimensions
Why crystal? Need regular, repeating pattern of diffracting objects to get well-defined diffraction pattern
(like evenly spaced lines on diffraction grating)
Diffraction – one interaction of light with matter; others are absorption, emission, scattering, etc
What do X-rays actually interact with?
Electrons in molecules (or atoms) in crystral; more electrons, greater diffracted intensity
Bright areas called spots, reflections, diffracted intensities, each has position and intensity
Diffraction is a kind of scattering, coherent, diffracted wavelength = incident wavelength
Light is redirected (deflected), bright spots due to constructive interference of diffracted waves
dark areas due to destructive interference of diffracted waves
Crystallography – 10
R. B. Wilson
Measuring Diffraction Data
3.
Does diffraction pattern look like a molecule – NO
Since diffraction (orderly pattern of spots) due to periodic (repeating) pattern in crystal, no surprise that the
pattern reveals details of repeating structure in crystal, spacing of spots gives unit cell dimensions, size and
shape
Unit Cell – postulated by Bergman and Hauy, characterized by Frankenheim and Bravais (pre X-rays)
Basic building block of crystal, smallest volume that can be used to generate complete crystal using only
translation
4.
Measure diffracted intensities (there are thousands of these)
Need a way to ‘name’ and catalog them, use a set of 3 integers, called indices, (h, k, l)
Indices based on diffraction geometry relative to the unit cell; related to location
Record indices (h, k, l) and intensity (I), Imeasured or Iobserved or Iobs
Since electrons diffract X-rays, expect Iobs to contain information about electron density pattern that generated
the diffraction pattern
Electron density and (Iobs )1/2 are closely related; (Iobs )1/2 = Fobs ;
F = structure factor, structure amplitude, form factor
5.
Symmetry of diffracted intensities and systematic absences (+ unit cell info from 3)  ‘space’ group, tells 3D
packing pattern (Z from 5 helps)
Crystallography – 11
R. B. Wilson
Determining Molecular Structures – Building the Model
Now, we have collected data (location and intensity of ‘reflections’), what do we do with it?
Know unit cell dimensions, space group, how do we deconvolute the measured intensities into atom positions?
II. Build a Structural Model
Compare electron density from measured intensities, (Iobs )1/2 = Fobs , to electron density from model, Fcalc
That is, compare Fobs to Fcalc, using Equations A and B
Eq. A: measured electron density in unit cell = Σ (±phase)(Fcalc)(cos……..)
Sum over all reflections; cos term involves coordinates of point in unit cell where you are calculating electron density
Eq. B: Fcalc( with ±phase) = Σ (atoms in model) (cos….)
Sum over all atoms; cos term involves coordinates of atoms
F is fourier transform of unit cell contents (atoms, electron density) sampled at lattice points h,k,l
1.
Start the model – ‘find’ the first atom or atoms in the unit cell
Think of diffracted intensities (Iobs) as wave with amplitude and phase
When we calculate Fobs from Iobs [(Iobs )1/2 = Fobs ], phase info lost in sq. rt.
We would like to calculate an electron density map (Eq. A), but, we need phases to go with Fobs
This is commonly called ‘The Phase Problem’.
•Use ‘direct methods’ to guess a trial set of phases; (program XS)
This is a statistical (brute force) process enabled by the power of computers
•Use Eq. A to plot approximate electron density in the unit cell
[Really an E map, normalized structure factors adjusted for decrease in scattering due to sin θ]
Look for molecular pattern or fragment
OR, if there is one or more ‘heavy’ atoms present, use Patterson method (program XS)
•Calculate Patterson (vector) map, relate highest peaks to heavy atom position (guess heavy-atom
position)
•Use (x,y,z) of heavy atom in Eq. B to get a set of phases
OR, if the molecule can be accurately modeled, use ‘Dual Space’ methods (program XM)
•Model molecule, rotate and translate model inside the unit cell until F calc and Fobs match
(also enabled by power of computers)
Crystallography – 12
R. B. Wilson
Complete the Model
2.
Complete the model
Calculate Fcalc from Eq. B
Compare to Fobs
Least Squares: refine atom positions (x, y, z) to improve agreement between F obs and Fcalc
Difference Fourier: calculate new electron density map using phases from Eq. A, using phases from F calc above
Find remaining atoms
3.
Repeat as often as needed to complete structure
Least-squares, difference Fourier cycling
4.
Compare Fobs to Fcalc using “R”
There are different formulations of R, fundamentally all compare model (calc) to observations (measured data)
5.
Use good atomic positions to calculate structural parameters: distances, angles, planes, torsion angles
Draw a picture, evaluate the model
Conclusions
1.
Build a model and compare it with the measured data
2.
Space group is part of the model
3.
All atoms contribute to F(hkl); Each intensity contains information about all atom positions
Therefore, all atoms must be included in the model
All F(hkl) contriubute to electron density(xyz)
Therefore, must include all measured intensities in electron density calculation
Crystallography
R. B. Wilson
•Glusker JP, Lewis M, Rossi M. Crystal Structure Analysis for Chemists and Biologists. VCH
Publishers. NY:1994, ISBN 0471185434.
•Rhodes G. Crystallography Made Crystal Clear. Academic Press. CA: 2000, ISBN
0125870728.
Retrieved from "http://en.wikipedia.org/wiki/X-ray_crystallography"
Chemistry 530
Structure and Spectroscopy
Crystallography – 13
R. B. Wilson
What is diffraction?
Basic principles, sample/pattern relationships, implications for your structure
Diffraction is the bending and spreading of waves when they meet an obstruction.
In the case of atoms in a crystal, core electrons associated with each atom are the
‘obstruction’. Electrons diffract X-rays.
In a crystal, atoms are arranged in a very regular pattern or repeat. It is this
regular repeat that enables the diffraction pattern. A random arrangement of
atoms would scatter waves that would randomly add but also randomly cancel.
There would be no directional reinforcement and no distinct diffraction pattern.
The direction of the diffracted beams is described by Bragg's law:
n λ = 2dsin θ
where λ is the wavelength, d is the distance between scattering centers, θ is
the angle of diffraction and n is an integer known as the order of the
diffracted beam.
The locations (θ values) of the diffracted intensities produce information about the
size and shape of the unit cell (and thus crystal lattice, all the unit cells in
translational combination).
Diffraction data
Diffracted intensities, reflections, spots
Intensity (Iobs, Fobs) directly related to brightness
From photon counting statistics,
s.u.(Iobs) is related to (Iobs)1/2
As collected on a modern diffractometer equipped with a CCD
area detector (Bruker Apex)
Each “frame” (plane) is a slice of data space (reciprocal space).
Crystallography – 14
R. B. Wilson
Diffraction Patterns
On the following pages, we will consider some 2-D arrangements of “atoms” and “molecules” and the diffraction
patterns that these arrangements produce. The left set of blocks represent atomic arrangements (A), the right
blocks the resulting diffraction patterns (D). A numbering scheme is shown below.
A
D
A
1
2
3
1
2
3
4
5
6
4
5
6
7
8
9
7
8
9
10
11
12
10
11
12
Crystallography – 15
R. B. Wilson
Diffraction Patterns
Consider A1-A3 and D1-D3. What is the relationship between distances between atoms in A and distances between intensities in D?
Crystallography – 16
R. B. Wilson
Diffraction Patterns
•What does the progression in A7-9 represent? What effect does this progression have on diffraction patterns, D7-9?
•Compare A3 and A9. What structural feature is the same, and what is different? On diffraction patterns D3 and D9, what is the same
and what is different? [There is a similar relationship between A11&12 and D11 and 12.
Crystallography – 17
R. B. Wilson
Diffraction Patterns
•Compare A10-12. What structural feature is the same, and what is different? How do these similarities and difference translate
into diffraction patterns D10-12?
Crystallography – 18
Diffraction Patterns
R. B. Wilson
Crystallography – 19
R. B. Wilson
Diffraction Patterns
The following questions refer to the patterns on page 18.
•Compare A1-3. What feature is the same, and what is different? How do these similarities and differences translate into diffraction patterns D1-3?
•Compare A4-6. What feature is the same, and what is different? How do these similarities and differences translate into diffraction patterns D4-6?
•Compare A7-9. What feature is the same, and what is different? How do these similarities and difference translate into diffraction patterns D7-9?
•Compare A9, 10 and 12.. What feature is the same, and what is different? How do these similarities and difference translate into diffraction patterns
D9, 10 and 12?
The following questions refer to the patterns on page 20.
•What is the general feature common to A4-12 that does not appear in A1? Which of these produces the “worst” diffraction pattern?
•The general term that describes this phenomenon is mosaicity or mosaic spread. Is it better to have a small or a large mosaic spread in your
crystal?
The following questions refer to the patterns on page 21.
•Which of these patterns most closely represents a crystalline solid? An amorphous solid? A “polycrystalline powder?
•In general, what do patterns D7, 8, 10 and 11 indicate about their corresponding arrangements, A7, 8, 10 and 11.?
Crystallography – 20
Diffraction Patterns
R. B. Wilson
Crystallography – 21
Diffraction Patterns
R. B. Wilson
Crystallography – 22
R. B. Wilson
Diffraction Patterns
General crystallography references:
•Glusker JP, Lewis M, Rossi M. Crystal Structure Analysis for Chemists and
Biologists. VCH Publishers. NY:1994.
•Rhodes G. Crystallography Made Crystal Clear. Academic Press. CA: 2000.
Diffraction patterns taken from:
•G. Harburn, CA Taylor, Welberry TR, Atlas of Optical Transforms. Cornell University
Press. NY:1975
X-ray Diffraction Essentials (rate limiting)
•
Chemist isolates condensed matter, ideally a well-ordered single crystal
•
Generate X-ray beam (currently constant energy Mo 0.71 Å or Cu 1.54 Å)
•
Set appropriate sample conditions (temperature, pressure, etc.)
•
Expose bulk sample for diffraction experiment (transmission or reflection)
•
Sample’s periodic electron density generates diffraction pattern (lattice)
•
Pattern resolution 0.4 to 35 Å (wide angle) and 30 to 500 Å (small angle)
•
Digital area detector (CCD or multi-wire) counts scattered photons
•
Experiment takes minutes to hours (it took days to weeks in 1994)
•
Observations improve with sample ordering (in 1, 2, or 3-dimensions)
•
Observed intensities transform to electron density (after phasing)
•
Chemist optimizes (refines) structural model against observed data
The Ideal Sample
The ideal sample, like this 40 x 70 x 360 μm crystal mounted on a 300 μm
cryo-loop, is attached using an inert oil, frozen in place using a cold gas
stream, and recovered after the diffraction experiment is finished.
Like many spectroscopic techniques, the overall amplitude of the diffraction
pattern is proportional to sample volume. There are no limits on crystal
morphology, but the practical range of linear dimensions is 5 to 500 μm.
Air, temperature, and light sensitive samples are routinely used.
The Results: Crystal and Molecular Structure
The Lexicon is in Play
Today X-ray crystallography is used extensively to determine the
structures of molecules in crystals. The instruments used to
measure X-ray diffraction, known as X-ray diffractometers, are
now computer-controlled, making the collection of diffraction data
highly automated. The diffraction pattern of a crystal can be
determined very accurately and quickly (sometimes in a matter of
hours) even though thousands of diffraction points are measured.
Computer programs are then used to analyze the diffraction data
and determine the arrangement and structure of the molecules in
the crystal. (Chemistry: The Central Science, Brown, LeMay,
Bursten, Pearson Education, Inc., 2006)
Courseware: Chem 483
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Introduce state-of-the-art instrument
Align single crystal and determine unit cell
Initiate single crystal data collection
Initiate wide angle (powder) data collection
Initiate small angle data collection
History of crystallography (from ‘48)
History of Illinois X-ray adventures (from ’27)
Process single crystal data
Refine cell parameters from powder data
Discuss small angle scattering curve
Solve and refine crystal structure
small angle
single crystal
wide angle
Theory
Practice
Confirmation