The majority of anomalous scattering in protein crystallography concerns structures containing a very large number of light atoms for which the anomalous scattering contribution is neglected and only a few heavy atoms of the same type where the anomalous scattering corrections are significant. For brevity, we shall from now on until the end of this chapter neglect thermal effects and the occupancy factor in our discussion of the structure factor and continue to use the definition given in Eq. .

If we consider a structure containing light atoms or normal scatterers and heavy anomalous scatterers of the same type then using our definition of the atomic scattering factor including anomalous effects (Eq. ) we can write down the structure factor equation for a reflection from a plane with Miller indices as

where the subscripts and indicate contributions from the protein atoms and the heavy atoms respectively.

Eq. maybe be conveniently expressed in graphical form as an Argand diagram. Fig. shows the structure factors for an arbitrary reflection and its so called Friedel mate (with Miller indices ) for a structure containing anomalous scatterers. For the case where there are no significant anomalous scatterers present in a structure the magnitudes of any Friedel pairs of reflections and are equal and their respective phases are and as can be seen from Eq. ; this is known as Friedel's law. However when anomalous scatterers are present in a structure the effect of the imaginary part of their atomic scattering factors is to introduce asymmetry into Friedel related reflections thus making their magnitudes in general different (see figure caption).

**Figure:** Argand diagram representation of the structure factor equation for a protein containing
anomalously scattering heavy atoms. (a) shows the structure factors of two reflections and .
The individual contributions to the structure factors from the protein atoms, , and the heavy
atoms, are shown.
The imaginary part of the heavy atom structure factor always leads the real part by in phase. This has
the effect of making the two resultant structure factors and different
in magnitude. (b) shows the same situation after reflection of through the real axis. This
convention will be used in future for convenience.

We can use Fig. to define a few important quantities which are useful to the analysis of macromolecular crystal structure. The vector is defined by

and is significant since it defines a structure dependent quantity which is dependent only on the contributions from the real parts of the atomic scattering factors of all atoms. This is important as it facilitates separate refinement of the heavy atom structure factors and . In general for the case of protein structures containing only a few heavy atoms and is usually taken to be the average of the magnitudes of the Friedel mates, i.e.

Of importance in the determination of the positions of heavy atoms in a structure are the anomalous or Bijvoet differences [10] . The cosine rule for Fig. (b) gives

and

Subtracting Eq. from Eq. gives the result

Using the identity

and Eq. we can rearrange Eq. to obtain

Fri Oct 7 15:42:16 MET 1994