Divergent issue of Madelung's constant

Question

This is a question triggered by this post

Madelung's constant is defined to the coefficient of electrostatic potential energy in a ionic crystal. In the example of $NaCl$, \begin{equation} M = \sum_{ijk}{}^{'}\frac{(-1)^{i+j+k}}{\sqrt{i^2+j^2+k^2}} \end{equation} is conditionally convergent.

Since this sum is conditionally convergent it is not suitable as definition of Madelung's constant unless the order of summation is also specified. There are two "obvious" methods of summing this series, by expanding cubes or expanding spheres. The latter, though devoid of a meaningful physical interpretation (there are no spherical crystals) is rather popular because of its simplicity. Thus, the following expansion is often found in the literature:[2] $$ M = -6 +12/ \sqrt{2} -8/ \sqrt{3} +6/2 - 24/ \sqrt{5} + \dotsb = -1.74756\dots. $$ However, this is wrong as this series diverges as was shown by Emersleben in 1951.[3][4] The summation over expanding cubes converges to the correct value. An unambiguous mathematical definition is given by Borwein, Borwein and Taylor by means of analytic continuation of an absolutely convergent series.

I have the following questions.

1) Expanding sphere leads to a divergent series. OK, what if I make a perfect spherical sample of $NaCl$, will the experimentally measured Madelung's constant to be infinity?

(EDIT: I maybe have some misunderstanding of divergence: it could be the case that the series is bounded but don't have a definite limit. So is the series for expanding sphere bounded? and if there is no definite limit, what's the experimentally measured value for a perfect spherical crystal?)

2) What physical principle dictates the order of summation? or why does finite number obtained by analytic continuation should be consistent with the observed value?

3) What is the role of charge neutrality here? I ever programed to compute Madelung's constant using the fractional charge idea((assign $\frac{1}{8}$ charge to the corner, $\frac{1}{4}$ to the edge, $\frac{1}{2}$ to the face ), which makes the expanding cube charge neutral. Does that mean charge neutrality is one of conditions that must be enforced? or it is just for the sake of computational efficiency?

Answer 1

This is not a rigorous answer to OP's questions, as we have not done any calculations or estimates.

The Madelung constant for the $NaCl$ crystal (of infinite size) is formally the sum

$$\tag{1} \sum_{\vec{r}\in \mathbb{Z}^3\backslash\{\vec{0}\}}\frac{(-1)^{x+y+z}}{||\vec{r}||_2} .$$

Here the $p$-norm is defined as

$$\tag{2} || \vec{r} ||_p ~:=~ \sqrt[p]{|x|^p+|y|^p+|z|^p}, \qquad || \vec{r} ||_{\infty}~:=~\max(|x|,|y|,|z|). $$

The procedure of arbitrarily truncating the summation (1) after completing a cube or ball of a certain size $a$ (possibly with an extra condition of only allowing electrically neutral sizes $a$), and then letting the size $a\to \infty$, is from a physical perspective rather crude.

Such truncation method seems ripe for systematic errors. If such truncation method happens to produce the correct answer, it seems more like an accident than a reliable scientific method.

Intuitively, it seems better to introduce a sufficiently smooth (or at least continuous) regulator function $\eta:[0,\infty[\to [0,\infty[$ with

$$\tag{3} \eta(0)~=~1\qquad \text{and}\qquad \lim_{r\to\infty}\eta(r)~=~0,$$

next consider the sum

$$\tag{4} \sum_{\vec{r}\in \mathbb{Z}^3\backslash\{\vec{0}\}}\frac{(-1)^{x+y+z}}{||\vec{r}||_2}\eta( ||\vec{r}||_p ~\varepsilon), $$

and finally removing the regulator $\varepsilon\searrow 0^{+}$.

In the spirit of Terence Tao blog entry, one would expect that the regularized sum (1) would not depend on the regulator function $\eta$ (or the $p\geq 1$ in the $p$-norm) once the regulator function $\eta$ belongs to a sufficiently nice class, and that such definition would agree with the (somewhat un-intuitive) definition by Borwein, Borwein and Taylor via analytic continuation.