Intrinsic resolution of silicon (111) and (311) reflections.



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Intrinsic resolution of silicon (111) and (311) reflections.

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. gif 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. gif 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 gif. 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.



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Gwyndaf Evans
Fri Oct 7 15:42:16 MET 1994