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What is x-ray crystallography?

X-ray crystallography allows us to be able to determine the precise atomic positions and therefore the bond lengths and angles of molecules within a single crystal. X-ray crystallography is a technique in which the pattern produced by the diffraction of x-rays through the closely spaced lattice of atoms in a crystal is recorded and then analysed to reveal the nature of that lattice. Like other techniques, crystallography has its limitations as it only produces an average picture of a structure. However it is extremely powerful in that the results are very accurate, giving bond lengths to a few tens of a pico-metre; (1 Å = 100 pm).

UNIT CELL: Simplest repeating unit - array of lattice points joined together, is fundamental to a particular structure.

CRYSTAL: Stack of unit cells.

X-RAY DIFFRACTION

• Diffraction depends on spacings between scattering bodies and wavelengths if incident radiation.
• Diffraction patterns depend on orientation of the crystal; the intensities of spots yield information on atomic positions
• A crystal acts as a 3D diffraction grating for x-rays as the interatomic spacings in a crystal are the same size as the wavelengths of x-rays.
• Diffraction from crystals - oscillating electrons emit x-rays in all directions. X-ray diffraction locates electron density.

Here are some of the main issues arising from x-ray crystallography:

• Powder diffraction
• Thermal motion
• Bragg's Law
• Fourier Synthesis and the Phase Problem

CRYSTAL STRUCTURE DETERMINATION

2 stages:

1. Solution of the phase problem yields partial structure and approximate phases.
2. Completion of structure by Fourier synthesis and refinement of structural parameters to optimise fit between observed and calculated intensities in diffraction pattern.

R-factors

R=0% - Perfect agreement between observed and calculated intensities (but this is never achieved due to random errors in experimental measurements).

A good structure has R<5%

[Refining vibrational parameters for each atom in a structure is important - reduces R-factor greatly].

Anisotropic Refinement

An atom does not vibrate equally in all directions therefore they vibrate anisotropically.

R(anisotropic) = 5%

Low temperatures

1. Collection at low T decreases thermal motion (thermal motion attenuates high resolution data - hence more precise structure at low T).
2. Reactive/unstable compounds have a longer lifetime at lower temperatures.

Standard deviations

Example: 2.62(1) Å bond length indicates 2.62 Å with a standard deviation of 0.01. [Where (1) refers to the last figure of decimals quoted].

REFINEMENT:

• Once all the atoms in a structure have been located, the final part of the process is to refine it, (ie:get the fit between the observed and calculated structure as accurate as possible). The positions of atoms are refined using the least-squares method, and corrections to the data are made.
• An R-factor is then computed for the molecule under investigation which gives an overall measure of how correct the structure is.
• For powders - several reflections lie under a single profile therefore the structure is refined against the whole pattern, (Rietveld Refinement).

Parameters that need to be considered for refinement:

Positions,

Thermal motion,

Partial occupancy of atomic sites,

Background parameters, (powder only),

Preferred orientation. (powder only).

Click here for more information regarding x-ray crystallographic techniques

Quiz yourself on x-ray crystallography now:

1. What is a unit cell?
2. Discuss what is meant by the phase problem.
3. Discuss what is meant by refinement.
4. What is the standard deviation of a bond length of 2.620(1) Å?
5. Describe the advantages and disadvantages of powder diffraction.
6. What are the advantages of performing a crystal structure determination at low T.
7. What does the R-factor represent?
8. What are anisotropic displacement parameters?
9. Why are hydrogen atoms difficult to locate using diffraction data?
10. Sodium tungsten bronze, Na_0.8WO₃ is cubic. Its powder diffraction pattern (λ = 1.54 Å) is as follows:

θ /°: 11.60, 16.52, 20.38, 23.71, 26.71, 29.50, 34.64, 37.09, 39.49

Assign lattice type P, I or F, index the pattern and find the unit cell dimension.

ANSWERS TO QUIZ

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Andrew Gray, Emma Downie,
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Thanks to; Prof Tasker, Prof Yellowlees, Dr. Paul Murray and all those members of staff that gave up time to assist us.
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