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Crystallographic Questions
Andrea Thorn
Crystallographic MasterclassCrystallographic Masterclass
• Please fill in your name and email address in the list.
• There will be 8 lectures, 3 on the following Thursdays, and then on Tuesday, same room, same time.
• Most lectures will be mirrored on Diamond.
• Next lecture: Experimental Phasing Basics Thursday 15th of September 16:00
What is crystallography?
Crystallography is the science that examines the arrangement of atoms in solids.
How does crystallography work?What can I do with it?
We take a crystal. We shoot it with X‐rays.The electrons in the crystal diffract the X‐rays, and the diffracted X‐rays are photographed.From the photos we can determine the positions of the atoms in the crystal.
Why can atomic structures be measured with X‐rays?
X‐rays
On X‐rays:
xkcd.com
How to make X‐rays?
How to make X‐rays. The boring way: X‐ray tube
Deutsches Museum
X‐rays
How to make X‐rays. The exciting way: Synchrotron
Wikimedia Commons – SOLEIL Synchrotron, France
What is a crystal?
A crystal is a solid material whose constituents, such asatoms, molecules or ions, are arranged in a highlyordered microscopic structure, forming a crystal latticethat extends in all directions.
Crystals are made up ofidentical ‘bricks’ (unit‐cells) that constitute athree‐dimensionaltranslation lattice. Thecell is defined by the 3vectors a, b and c; the 3angles between them are, and as shown in thediagram.
a b
c
Image courtesy of George M. Sheldrick
For good a good tutorial on point groups and space groups, look at: http://symmetry.otterbein.edu/
A diffraction experimentA diffraction experiment
laser
slide with 2D pattern
screen
The sides of the unit‐cell on the slide areabout 105 times longer than those in thecrystal; the wavelength of the laser isabout 104 times larger than that of the X‐ray beam, so the diffraction angles in thisdemonstration are about 10 timessmaller.
Image courtesy of George M. Sheldrick
What does the diffraction look like?
Sodium chlorideStructure discovered byWilliam Lawrence Braggwhen he was 21 yearsold!One of the first fourcrystal structureselucidated.It was not believed atfirst that each ion wascoordinated by sixcounter ions, and hence,the crystal representedone molecule, was notbelieved at first. Wikimedia Commons
This X‐ray photographwas taken by RosalindFranklin in 1951.Watson and Cricktried to understandthe structure of DNAfor some time.When Maurice Wilkins showed this photo to them, they understood that DNA must be a double helix.
Wikimedia Commons
How are the spots related to the structure?
Wikimedia Commons
Wikimedia Commons
This is the Fourier duck:
Kevin Cowtan: http://www.ysbl.york.ac.uk/~cowtan/fourier/
This is the Fourier transform of the Fourier duck:
Kevin Cowtan: http://www.ysbl.york.ac.uk/~cowtan/fourier/
What is the ‘phase problem’?
Fourier duck’s amplitudes and the Fourier cat’s phases:
Kevin Cowtan: http://www.ysbl.york.ac.uk/~cowtan/fourier/
What happens after the reflections were recorded?
Workflow
Data collection
Integration: Where are the reflections, and how black are they?
Phasing: Getting an initial model
Structural Refinement: Adapting the model to the data
How do I get a structural model?
Let’s find a black board…and talk about data/parameter ratio and least squares.
Measured data points with uncertainties:
Model : m ∙ x + bm, b: parameters
Model : m ∙ x + bm, b: parameters
Model : m ∙ x + bm, b: parameters
Model : m ∙ x + bm, b: parametersParameter refinement by least squares:
Model : m ∙ x + bm, b: parametersParameter refinement by least squares:
Least Squares
The best solution has the smallest squared sum (yellow).
+ + + =
Model : m ∙ x + bm, b: parametersParameter refinement by least squares:
Model : m ∙ x + bm, b: parametersParameter refinement by least squares:
Data : parameter ratioThe more data there is, the better can a given number of parameters be determined.
The fewer parameters there are, the better they can be determined by a given number of data.
More data!Less parameters!
(More Parameters than measurements: No unique solution. )
Least squares solution for 4 data points and 2 parameters:
... more data ...
...better least squares fit!
...better least squares fit!
To make the data/parameter ratio bigger, constraints and restraints are applied.
Restraints are dependencies which have not be exactly fulfilled. They are treated like data and have a target value and a standard uncertainty
› › › More data!
Constraints are values or dependencies which have to be fulfilled exactly
› › › Less independent parameters!
Restraints (blue) are used for the refinement:
Restraints (blue) are used for the refinement:
Restraints (blue) are used for the refinement:
A possible constraint would be:b = zero (b has not to be refined anymore)
A possible constraint would be:b = zero(b has ot to be refined anymore)
What is the resolution?What is R‐VALUE?
This is the Fourier duck from low resolution data:
Kevin Cowtan: http://www.ysbl.york.ac.uk/~cowtan/fourier/
R‐value:
Xkcd.com
Image plate X‐ray detectors usually stand perpendicular to the primary beam that passes through the middle of the circular (340 mm diameter) active area of the detector. Calculate the maximum resolution that can be reached with a distance of 45 mm between crystal and detector (a) with CuK radiation, = 1.5418 Å and (b) with MoK, = 0.7107 Å.
Use Bragg's law:
nλ = 2d∙sinθ(n is usually 1)
