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Ge Monochromator Emission Profile

[From the Rietveld mailing list 18/6/2012]

A question to the specialists: In case of an asymmetric monochromator
like typical Ge, can we expect an homogeneous distribution of intensity
within the beam bunch?

I agree with your assessment; I have also sometimes observed alpha 1 and alpha 2 separation that is different to what is expected when a pre-monochromator is used. Like you suggest it is due to an inhomogeneous wavelength spread across the beam in the equatorial plane. For a flat pre-monochromator then it would be expected as physically the alpha 1 and alpha 2 would be spatially separated in the equatorial plane. A bent crystal attempts to fix this but of course misalignment could change matters.

Thus there are two problems when using a Ge pre-monochromator:

1) The emission profile changes due to the filter of the crystal; this can be modelled by fitting enough Voigts to fit the emission profile shape; the Tan(Th) broadening dependence of the emission profile allows for such refinement.

2) The non-standard change in alpha 1 and alpha 2 separation as a function of 2Th can be modelled as follows:

The alpha 1 and alpha 2 components of the primary beam hits the sample off axis. For a off axis ray the change in 2Th measured as:

Delta_2Th =  (1/2) divergence^2 / Tan(Th)

where divergence is the angle primary ray makes with the axis in the axial plane (small angle approximations used). Alpha 1 and alpha 2 would both have different Delta_2Th's but it's the difference we are interested in. We can change the wavelength of alpha 2 to reflect this change and the correction becomes:

Aplha_2_wavelength_new = Aplha_2_wavelength (1 - (Pi/360)^2 divergence^2 / Tan(Th)^2)

Implementing this into an emission profile, using TOPAS for example, is as follows:

lam
ymin_on_ymax  0.0001
la  0.0159 lo  1.534753 lh  3.6854
la  0.5791 lo  1.540596 lh  0.437
la  0.0762 lo  1.541058 lh  0.6
prm al_in_degrees 0 min 0 max = 2 Val + .1;
prm alpha2_intensity 1 min 1e-6 max 2
la = alpha2_intensity 0.32417;
lo  = 1.5444493 (1 - (al_in_degrees  (Pi / 360)  / Tan(Th))^2 );
lh  0.52
la  = alpha2_intensity 0.0871;
lo  = 1.544721 (1 - (al_in_degrees  (Pi / 360)  / Tan(Th))^2 );
lh  0.62

The two parameters of al_in_degrees and alpha2_intensity are refined to change the emission profile.

Alpha 1 will also be shifted but that is taken up by lattice parameters, zero error, specimen displacement etc… and would be difficult to refine in the presence of the others.

The above seems to work but there may well be other affects not taken into consideration and IMO it would be difficult to discern other effects.

Cheers Alan

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