[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

From the Rietveld mailing list 5/10/2012:

Yaroslav Thank you for the MYTHEN data.

And thank you Lubo for also sending the data and for pointing out that the splitting increases at high angles and hence the opposite effect to a capillary.

I took the liberty of trying to fit to the data in a purely empirical manner; it's a little naive of course as many have no doubt spent a lot of time looking at the MYTHEN detector in detail. I myself would favour alignment such that splitting does not occur as Francois explained.

FWIW however and when desperate a 'perfect' empirical fit is possible with four Gaussians for an emission profile with an Rwp of 3.52% for a structural fit and 3.80% for a Pawley fit. It also seems that a Gaussian convolution that is constant with 2Th is also necessary. The main components of the peak shape are:

macro LL { min 1e-5 max 1 val_on_continue = Rand(.001, .1); } prm xx  0.58476` min .3 max 1.5 macro Fn(x) { / Tan(x)^xx }
prm w1  0.00017`	min -.01 max .01	val_on_continue = Val + Rand(-1, 1) 0.0001;	
prm w2 -0.00047`	val_on_continue = Val + Rand(-1, 1) 0.0001;	
 
prm w3 -0.00038`	min -.01 max .01	val_on_continue = Val + Rand(-1, 1) 0.0001;	
prm w4  0.00040`	val_on_continue = Val + Rand(-1, 1) 0.0001;	
 
prm w5  0.00000`	min -.01 max .01	val_on_continue = Val + Rand(-1, 1) 0.0001;	
prm w6 -0.00022`	val_on_continue = Val + Rand(-1, 1) 0.0001;	
lam
	ymin_on_ymax 0.001 
	la 1 
		lo  0.82257
		lg @  0.21693` LL
		lo_ref
 
	la  @  0.17840` min .1 max 10
		lo =  0.82257 + w1 + w2 Fn(Th); 
		lg @  0.19588` LL
	la  @  0.28193` min .1 max 10
		lo =  0.82257 + w3 + w4 Fn(Th); 
		lg @  0.07650` LL
 
	la  @  1.24781` min .1 max 10
		lo =  0.82257 + w5 + w6 Fn(Th); 
		lg @  0.17791` LL
 
gauss_fwhm @  0.0150418688` min 1e-5

I apologize if the TOPAS script is not understandable to some. The w2, w4 and w6 parameters offsets emission profile lines as a function of 1/Tan(Th) which then offsets the emission profile lines in 2Th space proportional to 2Th. The xx parameter if set to 1 increases Rwp by around 1%.

In any case if desperate then an empirical fit is possible using an emission profile comprising 4 Gaussians.