Our aim

Single Crystal Diffraction

Xray diffraction

Cheiron School, Spring8, 2012Claire Wilson• To explore getting from a crystal
to a structural model

Single Crystals
Material
i l iin solid
lid state withi h 3‐ddtranslational symmetryyy• Crystals may be:– elements (e.g.
(e g diamond),diamond) minerals,minerals inorganic salts
salts,
molecular materials (organic, metal complexes),
extended lattice structures (MOFs),
macromolecules (proteins, DNA, viruses)ca

For synchrotron single crystal diffraction
experiments crystals are typically
10 – 100 microns in size (1μm= 1x10‐6m=10,000Å)m=10 000Å)

Microscope analogy• With visible light look at objects similar size
to wavelength (400‐700nm)• For atoms and molecules radiation withwavelength of similar size is xrays• Reform image mathematically and not inreal timeWhy do we need a crystal?
• Identical repeat unit repeated infinitely in 3
dimensions leads to diffraction• Measure position and intensity of diffracted
beams and work back to the structure

Lattice• Array of points equivalent by translation – lattice• Unit cell describes geometry of lattice

Crystal Systems

Symmetry

Miller indices hkl

Definition– A crystallographic
yg p symmetry
yy operation
pis asymmetry operation, which mapsa (p(periodic)) crystal
ystructure onto itself.

• Description oforientation of planesslicing through crystalcb
http://www.york.ac.uk/depts/maths/histstat/symmetry/welcome.htm

Reciprocal lattice• Valuable construct indiffraction• Reciprocal geometric
relationshipl i hibetween crystal
l tti andlatticed itits
diffraction pattern
http://escher.epfl.ch/rlattice/aReciprocal lattice relationships• a*.b* b = a*.c* = b*.ab* = b*.cb* = c*.a* = c*.b*b=0• a*.a = b*.b = c*.c = 1• a* = (b × c) / V(4)• b* = (c × a) / V(5)• c* = (a × b) / V(6)• V* = a.b × c
Reciprocal lattice vectorh = ha* + kb* + lc*
a x b =|a||b|sinγ a.b =|a||b|cosγ

Diffraction• Radiation scattered in all directions• only measurable intensity where constructive
interference – scattered waves are in phase• xrays scatteredd byb different
diffpoints
i must hhave
path length differences equal to nλ• Laue Conditions – 3 conditions in 3D must be
simultaneously satisfied to have diffraction

Scattering vector• In plane of and bisecting the incident andg
1/d or d*diffracted beams with magnitude• Rearrange Bragg equationsini θ = (λ/2) ((1/d)/d)
scatteringg angle
g related to 1/d/The Bragg Equation• Consider
C id diffraction
diff ti as if
reflection from sets of planes
passing through lattice points• Path length difference of thescattered rays from adjacentplanes is 2Δnλ=2d
λ 2dhklsinθ
i θ

Usuallyy ignore
gn and consider
n=2 as scattering from planes
2h,2k,2l

Ewald sphere• For monochromatic beam ‐spherepradius 1/λ/ centered on
crystal (C)• Incident beam enters at Q
• Origin of reciprocal space at O
• Bragg equation is satisfied whennode P on surface of sphere thesinθ =OP/QO = d*/(2/λ) = λ¥2dEwald sphere• Crystal rotated to bring reciprocal lattice
points into diffractingg ppposition• Multiple diffraction• Lorentz
LFFactor• Blind regions
gLimiting Sphere• Boundary between observableand unobservable reflections fora given wavelength,wavelength λ• Long wavelengths – small limiting
sphereh– lower
lresolution
l i d
data
obtainable

Intensity of Diffracted beams

L is the limiting sphereradius 2/λ ,R the reflecting sphereradius 1/λ• Lattices
L iandd unit
i cells
ll ddescribe
ib geometry off diffdiffractioni• Diffracted intensities differ for different reflections –ddependd on orientationi t ti off planelini crystalt l• Weak and strong intensities – both contain information• Weakerk at highh h scattering anglesl – atomic scatteringfactor and temperature factor• DependDd on unit
it cellll contentst t – atomt
ttypes anddpositions• I(h) = k |F(h)|2Structure factor F(h)Structure factor representation• Structure factor for a given plane or reflection (h)• SummedSd over allll j atoms ini uniti cell,ll withihpositions xj and that have scattering power fj –atomict i scattering
tt i ffactort• xj = xja + yjb + zjc position of atom j• h = ha* + kb* + lc*• hxj = hxj + kyj + lzj( ) = A(h) + iB(h) =F(h) = |F(h)| eiφ(h)φ(h)+sin φφ(h))|F(h)|(cos φ• Each reflection (diffracted wave) has amplitude |F|and phase φ

Xray atomic scattering factors

Neutron scattering
141000• Not point scatterer –electron cloud• 2θ
2θ=0°0 electrons scatterin phase = atomicnumber

Cl‐

12900

10Cl80087006600

4500

K+400

2O300

005101520253035200-2100-401-623456789 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
ZNeutron scattering length (fm)– random variation with Z

scattered intensity for Xrays – proportional to Z2

Atomic scattering factors and atomic
displacements• Atoms in crystals notstationary but vibrating• Intensity reduced at
higher T and higher angles
by temperature or Debye‐
Waller factor, B• U, mean displacementd loffatomStructure factor• F(h)( ) integratedgelectron densityy ρ at everyy pointpx
in the unit cell
• Or summation atomic scattering factors fj (all jatoms in unit cell)• Structure factor (phase and amplitude) ‐Fourier transform of the electron density

B 8π2<u>2
B=8πElectron density

The ‘Phase
Phase problem
problem’V = unit
nit cell volumeol me• ρ(x) inverse Fourier transform of F(h)• With all F(h) (phase and amplitude) ‐completel t description
di ti off unit
it cellll contentst t

But do not have all information to reconstruct ρ(x)9 V ‐ unit cell volume
9 |F(h)| ‐ structure factor amplitude (proportionalto measured intensities)
X φ(h) – phase information – dondon’tt measureIs the phase important?
A duckd k andd iits FFourieritransform
and a cat and its Fouriertransform

From Kevin Cowtan
Cowtan’ss Book of Fourier
http://www.ysbl.york.ac.uk/~cowtan/fourier/magic.html

Is the phase important?
• The phase contains much more information
about the atomic positions than theamplitudes• Need some way to obtain estimated values forthe phase and to ‘solve’ the structure

Is the phase important?
• Take amplitudes (magnitudes)from duck transform with thephases from the cat transformand when you reconstruct animage it is recognisable as acat
• The image that contributed tothe magnitudes has gone

From Kevin Cowtan
Cowtan’ss Book of Fourier
http://www.ysbl.york.ac.uk/~cowtan/fourier/magic.html

Data collection• Xray source• Diffractometer – circles to move crystal• Detectors ‐ CCD, image plate, pixel detectors
to measure position andd intensity off ddiffractedffdintensities

DiffractometerExperimental considerations• Some examples of experimental choices:– choice of crystal
y– wavelength to use– sample environment ee.g.
g temperature– exposure time
– resolution of data

• Depending on material and nature of study

http://phillips‐lab.biochem.wisc.edu/xrayviewuse.html

Data reduction• Convert
Ct measured
d intensities
i t iti tto |Fobs| ‐ ‘observed‘ bd
structure factor amplitudes’ with associatedstandard uncertainty
uncertainty, σ(Fobs)• Corrections applied e.g.
– Lorenz‐polarisation effects– Absorption correction – absorption by crystal depends
on ppath length
g throughg crystal
yfor a given
greflection,
unit cell contents, wavelength• Unit cell parameters• Space group determination• Result – list of reflections as h, k, l, |Fobs|, σ(Fobs)

Structure Solution methods• Direct methods• Heavy‐atomy((Patterson)) methods• Charge flipping– Fragment searches– Inference– isostructural relationships

Result may be one heavy atom site orcomplete nonnon‐hydrogenhydrogen structure
depending on method and success level

Model development• Calculate amplitudes |Fcalc(h)| and phases as now‘know’know some atom positions xj from trial
structure• Fourier transform |Fobsb (h)| with calculated
phases – new information ‐ model density plus
additional atoms not included in model• Difference Fourier maps using |Fobs(h)| ‐|Fcalc(h)|missing atoms/misassigned atoms clearer

Structure model
Typically consists of• Atom type e.g. C, N, Fe (which atomic form
factor to use)
• AtomApositionsii
ini fractional
fil coordinates
dix,y,zp y• Site occupancy• Atomic displacement parameters – usuallyi t i Uiso
isotropic
Ui or anisotropici t i Uij
• Unit cell and space group

Model development and structurerefinement• Include ‘new’ atoms in model and repeat until
all atoms located• Model improvements – displacement
parameter modeling – isotropic‐anisotropic,
y g atoms
disorder,, hydrogen• Least squares refinement ‐ minimise |Fobs(h)|‐|F|Fcalc(h)| to givegi e best fit of model

Results and interpretation• Result ‐ electron density map• Refined model– coordinates of atoms, x,y,z with estimatedperror, at centre of electron densityy peaks– Atomic displacement parameters– Unit cell pparameters and spacepgroupg p• Interpret ‐ bond lengths and angles for ‘bonded’
atoms torsion angles
atoms,• Intermolecular distances e.g. H‐bonding.

Web resources• Kevini Cowtan
C
http://www.ysbl.york.ac.uk/~cowtan/• Gervais Chapuis
http://escher.epfl.ch/eCrystallography/pp
yg p y• Xrayview http://phillips‐
lab.biochem.wisc.edu/xrayviewuse.html• Joe Reibenspies
http://xray.tamu.edu/courses.php