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
