The angular range over which total reflection occurs in a perfect crystal [113] [6] when absorption effects are neglected is given by
where 
is the volume of the unit cell of the crystal. (For
silicon 
Å
.) By differentiating the Bragg equation, dividing the
resulting expression through by the original and rearranging we obtain the useful result
into which we may substitute 
from Eq. 
for 
to give
The magnitude of the structure factor 
was calculated from the structure factor
equation for a zinc blend Fcc type structure using values for the atomic scattering
factor of silicon taken from [60]. For the Si(311) reflection the energy
resolution calculated in this way using Eq. was
.
The value of 
for the Si(111) reflection was 
[6].
Figure: Diagram showing the X-ray source, first slits, and monochromator
crystal from right to left.
In (a) the observed energy resolution is dominated by the vertical beam divergence. If the slits
are closed down the range of angles incident on the monochromator crystal is
reduced resulting in an improved energy resolution. However as the situation shown in (b) is
reached there is still a small residual range of angles hitting the monochromator
due to the finite vertical source size. The lower limit for the instruments energy resolution
is set by the vertical source size and position and by the intrinsic resolution of the
monochromator crystal and reflection in use.
The presence of a single slit before the monochromator crystal means that the measured energy
resolution of X31 may be controlled to a limited extent by varying the range of divergence
of X-rays travelling from the source which are incident on the crystal. The numerical upper and
lower limits for the measured resolution of the instrument are set by the vertical beam divergence
and the vertical source size respectively. This can be understood with the aid of
figure . In the limiting case where the vertical slits are nearly closed
X-rays covering a range of angles defined by the vertical source size are incident on the
monochromator implying that the intrinsic resolution for the particular reflection
can never be reached. Opening the slits from this position increases the range of angles as
contributions from the natural vertical beam divergence are allowed to strike the crystal. The
maximum value of the instrument resolution is reached when the slits are wide enough to accept
the whole range of divergent X-rays.
