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

(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

(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. *A**41**,
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.

Equation (18) is an enantiomorph-sensitive
tangent formula, it may be useful for refining the sign as well as the
magnitude of *Δφ*_{h}.
Practical applications of equation (18) is still to be explored.