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Madelung constant of NaCl

Okay, so the lattice of solid NaCl has the form of a cube, and thus when you start summing the interactions you get

-\frac{z_-z_+e^2}{4\pi\epsilon_0r_0}(6+...)

where r_0 is the distance Na-Cl from the central Na+ ion to the six closest Cl- ions.

The second term in the expansion is

-\frac{12}{\sqrt{2}}

third

\frac{8}{\sqrt{3}}

fourth

-\frac{6}{\sqrt{2}}

fifth

\frac{8\times3}{\sqrt{5}}

and I could calculate some more, but they get more and more geometrically tedious.

I know that it converges to 1.74... but want to firstly set up the general term for the series, and then prove that it converges (comparison test or similar depending on how the series looks). I've had a look at the wiki article but it didn't seem to help me much, or at least not in a way I understood.

edit: I've seen the series they give on wikipedia, (

\displaystyle\sum_{i,j,k=-\infty}^{\infty} \frac{(-1)^{i+j+k}}{\sqrt{i^2+j^2+k^2}}

) but don't directly see where this comes from. And why are they summing from -infty to +infty? That would mean we are looking at an infinitely large structure of ions? Also, wouldn't it be equally possible to just sum from 0 to infinity and multiply the result by 2, because cubes are symmetric...

Reply 1

Sinuhe

I'm afraid I don't know the answer to this, and at the moment I haven't got the time to look into it properly (finals are fast approaching!), but I've looked up some articles and a couple look promising - so if you let me know your email address I could send them to you and you can have a look at them yourself? (I don't want to post them here because they're copyrighted and also one of them was stamped with my IP address on downloading. Or are you on a university network and can download them yourself?)

But apparently the message to take home is that the 'normal' way that we use to 'show' what the series should converge to, the first few terms of which you've written above, doesn't actually converge! (I definitely remember writing 'it can be shown that this series converges' in an exam ... oh, the beauty of proof by omission!)
So this is definitely interesting, and I'll look into it more properly in the summer. Who knew that basically *all* the textbooks on this blatantly lie? :p:

Reply 2

nota bene

I'll PM you my email.

Basically, I was revising for something unrelated, but my book ("Why chemical reactions happen") mentioned this a bit, and I looked at it because I found it interesting.

edit: Thanks a lot for taking your time to look at this, I'm really grateful! And yes, proof by omission is a good one :wink: