The structure factor equation (Eq. 
) may be written so as to separate out the X-ray
energy independent and dependent terms as follows (Karle [64])
Assuming that all the anomalous scattering heavy atoms are of the same type this can be written as
where 
is defined as that part of the structure factor which is X-ray energy independent.
We define the Thompson scattering component of the heavy atom scattering factor as
By equating the exponent terms in Eqs. and we obtain
The square of the amplitude of 
is found by multiplying Eq. by its complex
conjugate to give
which reduces to
where the 
sign indicates the Bijvoet pairs and the constants 
, 
and 
are
defined as
and
The set of equations arising from measurements of Bijvoet pairs at two or more X-ray energies may
be solved for 
, 
and 
. The phase 
of the heavy atom structure
factor may be found by Patterson synthesis or direct methods approaches using either or the Bijvoet
differences 
as coefficients. Given , the phase 
of the
normal scattering part of the structure factor may be calculated and subsequently a Fourier synthesis with
coefficients 
may be calculated to produce an electron density map.
