The phase ambiguity is first converted to a sign ambiguity, the latter is then resolved by a direct method. This was proposed in the 1960's [Fan Hai-fu, Acta Phys. Sin. (1965) 21, 1114-1118 (in Chinese); see also Fan, H.F., Chinese Physics (1965) 1429-1435 ]. Further development has been carried out from 1983 till now.

In SIR case, for a given reciprocal-lattice vector h, we have the structure-factor relation

    .     (1)

Here subscripts D, R and N denote respectively the derivative, the replacing-atom substructure and the native protein. From experiment we know the magnitude of F_D and F_N. Then we can locate the replacing atoms and calculate the vector F_R. Now, we have two different ways to draw the triangle of equation (1) leading to an enantiomorphous phase doublet for F_D and for F_N :

         (2)

and

    .    (3)

This is seen graphically in the following figure.

In SAD case on the other hand, we have the structure factor for a given reciprocal-lattice vector h as below.

    .    (4)

This breaks the Friedel law leading to

         (5)

and

    .    (6)

Hence

    .    (7)

where F" is the structure-factor contribution from the imaginary-part scattering of the heavy-atom substructure, i.e.

    .    (8)

The magnitude of F ⁺ and F ^-* can be obtained from experiment, while F" can be derived from the known anomalous-scattering substructure. Hence we have also two ways to draw the triangle of equation (7) leading to an enantiomorphous phase doublet for <F>, which is defined as

         (9)

and

    ,    (10)

while the phase doublet is expressed as

    .    (11)

Where φ" denotes the phase of F" (equation (8)). This is seen graphically below

In summary the phase ambiguity in both SIR and SAD cases can be expressed as

    .     (12)

For the SIR case

          (13)

and

          (14)

when we are deriving phases for the native protein, or

          (15)

when phases are derived for the derivative.

For the SAD case

          (16)

and

    ,    (17)

while the phases are derived for <F> (see equations (9) and (10)).

Now breaking the phase ambiguity is reduced to making choice of plus or minus Δφ in equation (12). A tangent formula with Δφ as argument and the probability formula for Δφ being positive have been worked out for this purpose [Fan & Gu, (1985). Acta Cryst. A41, 280-284 ].

    ,    (18)

              .    (19)

The definition of variables in formulae (18) and (19) can be found by clicking on the correspoding symbol listed below:

                        

In practice at the beginning, m_h and Δφ_h,best are calculated with P₊ set to 1/2. The results are substituted into (19) to obtain a new set of P₊.This is fedback to m_h and Δφ_h,best and then to P₊. Usually two cycles will be enough for breaking the sign ambiguity of Δφ_h. The program OASIS has been written in Fortran for the above calculation.