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### alberto.martinelli

Mar 31, 2021, 3:21:05 AMMar 31
to diffpy-users

Dear all, I've got a question that is puzzling me about the radial distribution function.

Let's consider for example a metal oxide MO; we can have a peak in the G(r) function purely originated by the nearest M-O pair. If we calculate the corresponding radial distribution function, by integrating this peak we can obtain the coordination number. But the problem is: the calculated coordination number refers to oxygen or to the metal?

### Simon Billinge

Mar 31, 2021, 7:29:09 AMMar 31
Thanks Alberto, this is a great question.  You caught us out being lazy!!!  If you look carefully in all our papers we tend to define the rho(r), g(r), G(r) and R(r) functions properly, but then when we integrate R(r) to get N(r) we miraculously say "and if you have a simple elemental material this integration gives the coordination number" and move on.  The reason is that actual answer is quite complicated and we usually want to just give a really simple intuitive idea of the power of that R(r) function in some simple case and not make it complicated off the bat.

If I find time in the next few days I will write something that takes one through it in an understandable way, but in a kind of "Fermatian marginal note" way  the answer is that, on the one hand that it gives the "average" coordination, but on the other hand, if you decompose the sum into a sum over distinct sites (i.e., partials) and you treat each partial separately, it yields the correct coordination with respect to each partial, summed up.

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Simon Billinge
Professor, Columbia University
Physicist, Brookhaven National Laboratory

### alberto.martinelli

Mar 31, 2021, 9:10:39 AMMar 31
to diffpy-users
Alberto

### Simon Billinge

Apr 3, 2021, 1:48:29 PMApr 3
OK, I finally got to this.  I just jotted down the logic with pen and paper.  I hope you can follow.  please see here:

I did it for atoms of the same type but on sites with different coordination number, and I kept it to the 1D case to avoid pesky 4 pi r^2 and things.  The extension to atoms of different type and 3D should be straightforward once you have the logic down.

It's also a bit confusing because I used N_1 as the "number of atoms on site-type 1" and also N_1(r) for the function that is the integral of R(r) over a range r.  Not great choice of notation....sorry!  Hopefully it is not too confusing.  Maybe the full definition could go into the next "Underneath" book.....

### alberto.m...@spin.cnr.it

Apr 7, 2021, 4:27:30 AMApr 7
Dear Simon,

many thanks!

Alberto
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