The diffraction effect which occurs when X-rays are scattered from a crystalline material may be formulated
in a simple manner by considering the diffraction of a plane polarised X-ray from a small crystal placed at the
origin 
made up of 
unit cells defined by the three vectors 
, 
and 
.
The source is a large distance away form the crystal so that the incident
X-rays can be assumed to be plane waves with wavelength 
.
Similarly the point of observation 
is a large distance 
from .
Fig. 
shows the geometry of the scattering. The incident and scattered wave vectors are
labeled 
and 
respectively. The electric field vector of the incident
X-ray is assumed to be normal to the plane of the paper.
The 
th atom in the 
th unit cell has position vector 
and atomic scattering factor 
.
is given by
where 
is the atoms position vector with
respect to the origin of any unit cell and 
, 
and 
are fractional coordinated of the th
atom in the unit cell.
Figure: A plane wave represented by the beam vector is incident on a small crystal
containing N unit cells defined by the vectors , and . The th atom
in the th unit cell is represented by its position vector . The scattered radiation represented
by is observed at a point P, a large distance away,
such that 
. (taken from Warren [113])
The instantaneous electric field at is given by
since for 
, . The distances 
and 
are
equal to 
and 
respectively. Thus by
expanding with Eq. we may write 
as
To obtain the total electric field 
at we must sum over all atoms in the unit cell and over all
unit cells. Assuming the crystal to be a simple parallelopipedon with sides of length 
, 
and 
where 
, the total field is given by
The first summation term in Eq. is called the structure factor, 
, since it is dependent solely
on the positions of the constituent atoms in the unit cell.
Each of the last three summation terms in Eq. take the form of the geometric
progression 
with
and
Thus the sum in the 
-direction can then be written
with similar expressions for the 
and 
summations.
If we now multiply in Eq. by its complex conjugate 
then with some
manipulation of the expanded summation terms we obtain an expression for the intensity of the scattered radiation
for a polarised incident beam
For very large values of 
, 
and 
the intensity 
will be have very sharp maxima
when the equations
are satisfied.
These are the Laue equations stipulating the diffraction condition. The integers 
, 
and 
are the
corresponding Miller indices of the Bragg plane of reflection.
By considering a general reciprocal lattice
vector 
it can be shown that the Laue equations
are equivalent to the Bragg equation which in vector form may be written
where 
is the reciprocal lattice vector corresponding to the
Bragg plane with Miller indices 
. Given Eq. and our definition of 
we can now
rewrite the structure factor (Eq. ) in the more commonly seen form
In practice the incident X-ray beam is not parallel and the sample crystals are in general mosaic meaning that
they diffract as though they were made up of a very large number of smaller crystals packed together at slightly
differing orientations. The diffraction maxima are therefore much broader and lower than predicted by Eq. .
The measurable quantity of a diffraction experiments is the integrated intensity, i.e. the total intensity
resulting from integrating under a finite width diffraction profile.
The definition of the structure factor in Eq. is valid for a crystal in which all the atoms
are stationary and present in every single unit cell in the crystal. In reality the atoms are continually vibrating
about an average position due to the effects of temperature and particularly for the case of heavy atoms introduced
into protein crystals, the heavy atom may not be taken up by every protein molecule. Thermal vibration causes a
reduction in intensity of the Bragg peaks. If the th atoms thermal motion is described by an isotropic mean
squared displacement 
about its mean position then we can define a temperature factor 
for that atom such that
The second effect is accounted for by an atomic occupancy factor 
which is the fraction of unit cells in
the crystal which contain the atom in question.
The structure factor equation including the effects of isotropic thermal vibration and the atomic occupancy is
then written
