A full description of the fundamentals of the direct methods approach may be found in [39], from which some of the derivations given here are taken.
The theory of direct methods approaches to the solution of crystallographic structure stems from two properties of the electron density distribution within the crystal structure. Firstly, the electron density is positive, and secondly it is composed of discrete atoms whose positions may be considered to be random in the unit cell.
The aim of direct methods is to obtain phase information given only the measured
amplitudes 
of structure factors with indices 
. The method
relies on being able to identify linear combinations of the phases of specific groups of
reflections which are invariant or semi invariant with respect to the choice of origin in the
crystal lattice. Structure semi-invariants are invariant only with respect to
a number of permissible origins which display the same point symmetry whereas structure invariants
are completely independent of the choice of origin.
Each reflection in a data set may contribute to many structure invariants. It is therefore possible to build up a network of these invariants which link the phases of all reflections. Thus by selecting one or more semi-invariant phases and fixing their values, one defines the origin to be at a point in the lattice which possesses some point symmetry. One is then able to use the network of structure invariants and semi-invariants to calculate the phases of other reflections in the data set and so obtain enough information to enable the calculation of an electron density map.
Sayre [99] in 1952 derived an expression for one such structure invariant. Using the properties of
positivity and atomicity of the electron density along with the assumption that the structure consists of equal
atoms he noticed that the electron density function 
and its square 
have
similar forms.
We can now define corresponding atomic scattering factors (
and 
) and structure
factors (
and 
) for normal and squared
electron density functions of the 
equal atoms at positions 
in the unit cell such that
and
The convolution theorem states that the Fourier transform of the two functions multiplied together in real space
is equivalent to convoluting the Fourier transforms of the two individual functions in reciprocal space. We can
therefore write in terms of the convolution integral,
The structure factor is however a discrete function implying that we can sum over 
instead
of integrating w.r.t. , giving
If we now divide Eq. 
by Eq. we obtain in terms
of . Substituting for in Eq. gives the basic form
of Sayre's equation,
Multiplying this equation through by 
gives finally
We observe that is real and positive implying that the resultant of the addition
of all vectors on the right hand side is also real and positive. The usefulness of this equation now relies
on the probabilistic argument that for large values of , 
and 
,
i.e. the largest terms in the summation which contribute to the resultant `real' vector (which has zero phase) are
also likely to
have a phase close to zero. Eq. defines what is known as a triplet invariant.
An improvement in the atomicity condition is generally achieved by using so-called normalised
structure factors which correspond to the scattering from a structure consisting of point atoms at rest,
i.e. atoms with atomic scattering factors which are constant as a function of scattering angle.
The normalised or sharpened structure factors 
are given by
where 
is a factor accounting for the effects of symmetry elements on specific
groups of reflections. The 
's are normalised such that 
.
The use of 's also simplifies the form of the probability
distribution 
for a triplet invariant which was derived by
Cochran [24]. For a structure containing N non-equal atoms the probability
that a phase 
is correctly indicated is given by
where 
is a normalisation term, the magnitude 
is
and 
.
The distribution has its maximum at 
and
becomes narrower as
increases. i.e. for larger values of , 
and 
In general phase will be indicated by more than one triplet phase
relationship. Thus the overall probability distribution is the product of the individual
distributions [39]. For 
triplet phase relationships giving an indication of
the phase by the expressions
each having a probability distribution 
, the resultant probability distribution
will be
where 
. 
will have its
maximum when the exponent is at its maximum. By setting
and
Eq. may be rewritten as
It is immediately clear from Eq. that has its maximum
when 
. Larger values of 
tend to sharpen the
distribution, i.e lower the variance. Eq. is the Tangent formula first developed by
Karle and Hauptman [65] and gives the most probable value, 
, of the
phase as indicated by triplet phase invariants.
