Two corrections are applied to the profile fitted intensities. The desired intensity measurement
is related to the experimentally observed intensity 
by the equation
where 
is the Lorentz kinematical factor for the particular diffraction geometry. The Lorentz
factor accounts for the differing times spent in the diffraction condition by different
reflections. This can be equivalently thought of as correcting for the differing times taken by the
reciprocal lattice points to traverse the Ewald sphere. Reciprocal lattice points which cross the
Ewald sphere with trajectories normal to the sphere surface and/or are far from the rotation axis
giving them greater velocities will pass quicker through the sphere compared with points which have
trajectories with small grazing angles relative to the sphere surface and/or smaller velocities
because they are closer to the rotation axis. For rotation geometry with the rotation axis
perpendicular to the beam direction the Lorentz correction takes the form
where 
is the Bragg angle of the reflection and 
is the axial coordinate of the
corresponding reciprocal lattice point, defined in cylindrical polar coordinates about the rotation
axis, as it intersects the Ewald sphere.
The polarisation correction 
is dependent on the polarisation content of the X-ray beam and the
path followed by the X-ray up to its point of detection. Expressions for the polarisation
correction for an X-ray beam diffracted from a single reflection crystal monochromator have been
derived by Azaroff [4] and later Kahn et al [63]. An equivalent expression
for the case of a double reflection monochromator is now derived. Fig. 
shows
the path of the X-ray beam and the three reflections which it undergoes before it is incident on the
detector. The situation shown is that for the X31 beam-line which is described in more detail in
Chapter .
Figure: Schematic diagram showing the path of the X-ray beam after reflection from a double face
monochromator and also from the planes of a crystalline sample. The polarisation state of the X-ray
beam is represented by the relative magnitudes of the perpendicular components, 
and
, of the electric field vector.
We can consider the X-ray beam travelling from the left to be made up of two components which are
represented by two mutually perpendicular electric field vectors and
. is defined in the vertical i.e. normal to the plane of the synchrotron
orbit. The total intensity of the incident X-ray beam is given by the sum of intensities
arising from two individual components. i.e.
or in terms of the electric field vectors
The X-ray beam is first diffracted twice in the vertical plane from the monochromator set at an
angle 
to the incident beam direction. The 
and 
components of the X-ray
beam are modified by the first two reflections to produce a beam described by 
and 
. The beam is then diffracted by the sample crystal and the polarisation
content of each diffracted beam takes a form dependent on the scattering angle 
and the
orientation of the scattered ray w.r.t. to the components of the incident X-rays electric field
vector components and described by the angle 
.
Let us first consider the component of the incident X-ray beam. Using Eq. and
defining 
we can write down the electric field vector after two
reflections from the monochromator as
After diffraction from the sample crystal,
the component of the X-ray beam may be split further into two components
and 
where the subscripts 
and 
indicate components normal to and parallel to the sample scattering plane. Defining 
, the two components may be written as
and
We note that the component of the incident X-ray beam is perpendicular to the plane of
scattering of the monochromator. The analogous expressions for 
and
are therefore independent of 
and are found to be
and
The resultant X-ray intensity after diffraction from the sample crystal is given by
We now follow the derivation of Kahn et al. and define 
to be
Combining Eqs. to we obtain
Similarly we can define 
as
which may be written as
Manipulation of Eq. using the identities
and
now yields
which is analogous to Eq.12 of Kahn et al.
The reflection dependent term is enclosed in the second pair of curly brackets. We note that it is composed
of two parts. A term which is exactly equal to the polarisation factor for an unpolarised
beam, 
and a modifying
term 
which includes the polarisation state of the
X-ray beam incident on the crystal sample. The overall polarisation factor is therefore
The variation of the polarisation factor with the angle is most significant when 
(100%
horizontally polarised radiation incident on the sample crystal) which is almost the case for synchrotron
radiation (see Sec. ).
