= Electrostatic model of ionic binding
{title2=$U_{ij}=z_i z_j e^2/(4\pi\epsilon_0r_{ij})$}
For two point <electric charges> in <vacuum>, <Coulomb's law> gives the interaction <potential energy>
$$
U_{ij}(r_{ij})=\frac{z_i z_j e^2}{4\pi\epsilon_0 r_{ij}},
$$
with zero at infinite separation. Here $r_{ij}$ is the distance between them and $\epsilon_0$ is the <vacuum permittivity>. Opposite signs give negative <potential energy> and attraction; equal signs give positive <potential energy> and repulsion. For a finite collection of distinct point <electric charges>, sum over $i<j$ to count each pair once. A bulk <ionic crystal> requires the extended lattice sum, including both signs, rather than multiplying one attractive pair by the <coordination number>.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[§8–1, equations (8.1)–(8.3); §8–3, many-neighbour lattice sum]{ref}
At short distances, overlapping <electron density> makes the simple point-charge picture inadequate. <Quantum mechanics>, including the <Pauli exclusion principle>, matters to the steep repulsive contribution. Pure attraction proportional to $-1/r$ would collapse as $r\to0$; it cannot set an equilibrium spacing on its own. A useful illustrative model is
$$
U(r)=-\frac{A}{r}+\frac{B}{r^n},\qquad A>0,\quad B>0,\quad n>1.
$$
Here $r$ scales an assumed fixed arrangement, $A$ packages its attractive contribution, and $B/r^n$ represents short-range repulsion. The constants have units <energy> times length and <energy> times length to the power $n$, respectively. This is a model, not a fitted description of a named <salt>.<The Feynman Lectures on Physics, Volume II by Richard Feynman and others>[§8–3, close-distance repulsion and limits of a purely electrostatic lattice calculation]{ref}
Differentiating gives $r_0=(nB/A)^{1/(n-1)}$ and $U''(r_0)=(n-1)A/r_0^3>0$, so the stationary spacing is a stable minimum. For the diagram's stipulated $n=9$, put $x=r/r_0$ and $E_0=A(1-1/n)/r_0>0$. Then
$$
\frac{U(r)}{E_0}=\frac{-9/x+1/x^9}{8},\qquad U(r_0)=-E_0.
$$
The <dimensionless> curves show how competing interactions select a spacing. Their <energy> scale and exponent are synthetic; neither is a measured <lattice energy> or a <melting> prediction.
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