The structure factor equation (Eq. ) may be written so as to separate out the X-ray energy independent and dependent terms as follows (Karle )
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
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.