Lorentz-Polarisation corrections.



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Lorentz-Polarisation corrections.

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. gif 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 gif.

  
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. gif 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. gif to gif we obtain

 

Similarly we can define as

 

which may be written as

 

Manipulation of Eq. gif 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. gif).



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