In this section we will look at the Patterson function, which was used
to solve many early crystal structures and is still important in the
location of heavy atoms.
Use to open a second copy of the applet in a new window.
The biggest problem in X-ray crystallography is our inability to measure phases directly. A single diffraction experiment only yields diffraction intensities, from which we can calculate the structure factor magnitudes, |F(h)|, but not their phases.
What happens if we ignore the phases and calculate a Fourier transform just using the information we have? For example, we could set all the phases to zero. It is convenient also to square the structure factor magnitudes.
The resulting map is called the Patterson function, P(x). It may be expressed by the Fourier transform as follows:
P(x) | = | 1/V Σ_{h}|F(h)|^{2}exp( -2πi h.x) |
= | 1/V Σ_{h}F(h)F(-h)exp( -2πi h.x) |
The convolution theorem states that the Fourier transform of a product is equal to the convolution of the Fourier transforms of the original functions. Since the Fourier transform of |F(h)| is ρ(x), and the Fourier transform of |F(-h)| is ρ(-x), P(x) must be the convolution of ρ(x) and ρ(-x):
P(x) | = | ∫_{y}ρ(y)ρ(x+y) dy |
For this exercise there are two copies of the Structure Factor applet - the second is in a new window or tab. This one shows the Patterson resulting from the two marked atoms, and the other shows the electron density itself.
The Patterson shows a big origin peak. This corresponds to the vector from every atom in the structure to itself (in this case two atoms). There are also peaks above and to the right, and below and to the left, representing the vectors from the first to the second atom and from the second to the first atom. Clearly the Patterson must always be centrosymmetric.
Check some of the structure factors to confirm that the Fourier coefficients of the Patterson are equal to the squares of the structure factor magnitudes, and that all the phases are zero.