Formula (1) implies that the structure factors which we have successfully calculated in the pervious lesson, can be used as Fourier coefficients in an Fourier Summation (or synthesis) to generate the electron density. To be complete, the summation would go from - infinite to + infinite for all indices h,k,l. In reality we have limitations due to the extent to which the diffraction pattern is observed, and the synthesis will be approximate only and may show some truncation effects.

(1)

We will caclulate the electron density for our previously calculated structure You need to have the structure saved with a unique file name or the default will be used. In our 1-dimensional case (1) reduces to

which is a real function caclulated from structure factor amplitudes and the corresponding phases. We expect this electron density to show peaks at the atoms positions.

**Calculation of Patterson maps**

We also introduce another very important function, the Patterson function, which is essentially an autoconvolution function of the structure factor. The Patterson function does not need phases and uses only the easily accessible intensities (square of the structure amplitude) as Fourier coefficients :

The question arises, what is the physical meaning of a Patterson map? Look
at the calculated Patterson function and you will see that the peaks refer to the
interatomic distances. Such a map has N*N - N peaks, and will be very crowded. So why
bother for large molecules? Well, apparently the square weighting of F (and thus of *z*,
number of electrons) makes distances betwen heavy atoms very prominent (see exercises).
Such maps are used to locate the heavy atoms in a protein structure in isomorphous
replacement phasing. They are also used to loacte anomalous scatterers, with anomalous
differences as map coefficients. Patterson maps can be solved by hand, direct methods,
Patterson vector superpostion methods or correlation searches. They are also very useful
for locating non-crystallographic symmetry
and in molecular replacement techniques.

All you need for the Fourier calculation is a file containing the structure factor amplitudes (or their square, the intensities) and their phases (the Patterson does not need the phases). Pick a grid spacing and a maximum resolution (which of course can never be larger than the resolution of the data) and the file name you stored the list of reflections under. Execute the default case and familiarize youself with the ouput.

- Run the default values on the previous structure factor calculation. How does the magnitude of the Fourier peaks relate to the size of the atom? Are they where you expect them?
- Look at the patterson map. Disregard the prominent self-peak at 0 and 1.0. What do the two peaks refer to? Why are they of the same size even if the 2 atoms are very different?

- Now reduce the resolution (say, 4 A) and look what happens to the peaks. Can you see some artefacts around 0.4 ? Reduce the resolution to 8 A. Do you notice that even the peak positions shift?
- What happened to the 2 Patterson peaks at low resolution?

- Now run following structure factor calculation : defaults, resolution 0.8 (why not 0.7?), C in 1, Fe in position 2. Go to Fourier calculationa and increase the grid to 80 points. set the resolution to 2.5. Run it. Scroll down to the iron atom and note the ripples (resulting from the summation truncation) around the peak. Increase the resolution to 0.8. Rerun. What happened to the ripples?

Once you have understood the concept of fourier transforms, structure factors, and how they relate, you will be better equipped to understand the methods to derive experimental phases and to solve the phase problem.

Back to new structure factor calculation

**Back to Introduction**

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Bernhard Rupp**

Last revised
Dezember 27, 2009 01:40