how to interpret the scale factor for a multi-phase refinement?

[Yiyi Yao]

Hello,

I am trying to do a multiphase refinement on nanostructured samples of MoS2. The 2 phases/CIF files I'm using are the standard 2H and 3R polymorph, which are quite similar except they differ in the layer stacking pattern. 

I'm wondering how to interpret the numbers that come out of refinement of the "scale factor" of each phase? For example, if scale factor for phase 1 = 0.65, and scale phase for phase 2 = 0.30, can I say that 65% of the material looks like phase 1 and 30% looks like phase 2, while 5% is not well matching either phase? The numbers do not always add up to 1.0. 

The default scale factor is 1.0, but sometimes after refinement, the value can exceed 1.0, like phase 1 = 1.05 and phase 2 = 0.37? Do I simply take the sum of the numbers obtained and calculate a percentage of that total sum?

Thanks very much for any help!

[Yiyi Yao]

I have attached the 2 CIF files and an experimental sample G(r) if that helps

[Simon Billinge]

Hi Yiyi,

Thanks for the question.  Broadly speaking the (mathematical) model has two tasks regarding the scale factor.  The first is to find a global scale factor that makes all the curves bigger or smaller.  If we did absolute normalization of data without errors, this scale factor would always be 1, but it rarely happens so we let the model vary that.  If you have a two-phase structural model the mathematical model's other job is to find a relative scale factor for the two phases.  We sometimes call this a mixing fraction.   If our two components are A and B we might write something like G_tot = f*G_A + (1-f)*G_B.

Now, there is more than one way to set things up in PDFgui. They are mathematically equivalent so it doesn't matter how you do it.  However, if you want to follow the logic above, the way that will give the most intuitive results would be to set a global scale factor (which is an attribute of the data, so select the dataset and make a variable @27 or whatever in the scale-factor field there) and then for the scale factors in each of the structural phases give one phase, say A, its own variable (say @50) and the other phase, B, give it (1-@50) in the scale factor position.  Your @50 variable will then give you the "proportion" of thing A directly.

Be careful with "proportion", it is strictly the proportion of the signal, which is some kind of weird scattering power weighted thing.  If you look in the results section of PDFgui, I think it actually gives the atomic and mass and per unit cell percents of each phase that it extracts from this number. 

S

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Simon Billinge

Professor, Columbia University

Physicist, Brookhaven National Laboratory

[Yiyi Yao]

Hi Prof Billinge,

Thank you for the response! If I set up the scaling as you suggest, then the scale factors of each phase should sum to 1.0, but then what does the scale factor of the data set itself mean (variable @27 in your example), esp if it does not equal 1.0?

[Simon Billinge]

Hi Yiyi,

If you did a perfect data reduction, with a perfect normalization, the scale will be 1.  There are so many unknown factors when you do the data reduction that the chances of that happening are quite small.  This scale factor corrects the model for these effects.   If you used PDFgetX3 to do the data reduction, it uses an arbitrary scale factor and all bets are off, basically (though it often gets a normalization close to 1).

Having a scale factor that is not 1 doesn't affect the quality of a structural refinement (the structural parameters are not affected).  This is all discussed here:

 P. F. Peterson, E. S. Bozin, Th. Proffen, and Billinge, S. J. L. “Improved measures of quality for
atomic pair distribution functions”. In: J. Appl. Crystallogr. 36 (2003), p. 53. doi:
10.1107/S0021889802018708. url: http://dx.doi.org/10.1107/S0021889802018708

S

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