Weighting with constant irradiated length

mfisch

Hi all

With a lab B-B diffractometer, there are basically two measurement types:

- Constant divergence, which results in a constant irradiated sample volume, i.e. constant number of irradiated particles.

- Constant irradiated length, which results in a non-constant number of irradiated sample volume and particles.

Normally, the LS weighting is based on the assumption that SigmaYobs = SQRT(Yobs), therefore, weighting = 1/SigmaYobs^2 = 1/Yobs.

A diffraction pattern can be converted from "automatic slits" to "fixed" ones by applying a sin(Th) dependent correction.

Question:

If automatic slit data is treated, shouldn't the weighting also contain a sin(Th) dependent correction?
If yes, is the correct weighting = 1/Yobs * 1/sin(Th); ?

Cheers
Martin

alancoelho

Hi Martin

What is typically minimized for Variable divergence is:

Sum[ (Yobs- Sin(th) Ycalc)^2 w ]

where Ycalc is scaled by Sin(Th) and w (the weighting) is typically 1/Yobs. If you scale Yobs to simulate fixed divergence (when you in fact have variable divergence) then you have:

Sum[ ( (Yobs/Sin(Th)- Ycalc) Sin(Th) )^2 w ]

= Sum[ (Yobs/Sin(Th)- Ycalc)^2 Sin(Th)^2 w ]

Thus Yobs is scaled by 1/Sin(Th) and the weighting should be:

weighting = Sin(X Pi/180)^2 / Max(Yobs,1);

cheers
alan

mfisch

Hi Alan

Thanks!

Maybe I misunderstood something: if I use real variable slit data and the Variable_Divergence_Intensity macro, I don't scale Yobs (as it is already "scaled" during measuring), only Ycalc. Or do you mean "scale Yobs" in the LS minimization formula?

weighting = Sin(X Pi/180)^2 / Max(Yobs,1);

Why not incorporate this into the Variable_Divergence_Intensity macro itself?

Cheers
Martin

alancoelho

Yes the Variable_Divergence_Intensity scales Ycalc. But if prior to fitting you changed Yobs from variable to fixed slits then Yobs is scaled by 1/Sin(Th). You then do not scale Ycalc as Yobs is not fixed slits. You then need to change weighting as described.

mfisch

Hi Alan,

Sorry, still confused...

(1) If I manually convert an automatic slit pattern to "fixed" slits in any data treatment software (=> scaled by 1/sin(Th), I need to use the weighting scheme you proposed.

(2) If I use the Variable_Divergence_Intensity and unmodified "variable slit" rawdata, consequently, only Ycalc is scaled. Do I also need to change the weighting then? I assume that for such a pattern, the assumption w = 1/Yobs is wrong as the irradiated volume changes with Th...

If yes, w = 1/Yobs can ONLY be used for fixed slit data and weighting = Sin(X Pi/180)^2 / Max(Yobs,1); is used for unmodified variable slit data and data that is converted from variable to fixed slits?

Cheers
Martin

alancoelho

Hi Martin

For (1), Yes

For (2), If you have variable divergence intensity data then Ycalc is scaled by Sin(Th); the weighting however is left to the default of 1/Max(Yobs,1).
cheers
alan

mfisch

For (2), If you have variable divergence intensity data then Ycalc is scaled by Sin(Th); the weighting however is left to the default of 1/Max(Yobs,1).

This is the point I don't understand. As in such data, the number of irradiated grains/crystallites (i.e. statistics) is not constant, how can the weighting still be constant?

Thanks,
Martin

rowlesmr

I would say that the weighting should stay as 1/Yobs as the statistics are derived from the intensity of the diffracted beam.

The fact that you have more crystallites diffracting just means that the intensity is more representative of a powder.

You could also get the same results in terms of diffracted intensity by changing your counting time as a function of angle instead of changing the intensity of the beam as a function of angle.

alancoelho

Martin

Here’s how to look at it. For a particular set of data the parameter values and errors obtained after a fit should be identical regardless of how you scale the original Yobs data so long as you adjust ‘weighting’. Do the refinements with the weighing as described and you should see identical results.

So for case (1), scale Yobs by 1/Sin(Th), set the weighting Sin(X Pi/180)^2 / Max(Yobs,1) and do a refinement.

For case (2), don’t scale Yobs but scale Ycalc by Sin(Th) and then do a refinement with weight=1/Max(1,Yobs).

Parameter values and errors should be the same for both cases.

Cheers
Alan