How to Derive the Born-Mayer Equation?

Question

Born-Mayer Equation: $E_{P,min}=-A\frac{N_A|z_1z_2|e^2}{4\pi\varepsilon_0d}(1-\frac{d^*}{d})$

Here is where I have gotten:

$$ E_P=-A\frac{N_A|z_1z_2|e^2}{4\pi\varepsilon_0d}, E_P^*=N_AC'e^{-d/d^*}\\ $$ Energy minimized when $\frac{d(E_P+E_P^*)}{dd}=0$

$$A\frac{N_A|z_1z_2|e^2}{4\pi\varepsilon_0d^2}-\frac{C'N_A}{d^*}e^{-d/d^*}=0\\ A\frac{N_A|z_1z_2|e^2}{4\pi\varepsilon_0d^2}=\frac{C'N_A}{d^*}e^{-d/d^*}$$

I tried taylor expanding $\frac{C'N_A}{d^*}e^{-d/d^*}\approx \frac{C'N_A}{d^*}(1-\frac{d}{d^*})$, but I don't think that this approximation is appropriate since d/d* is not very small, and it didn't take me anywhere.

Answer 1

There's no need to Taylor expand; the exponential cancels exactly.

Starting with the following (lattice-constant dependent) expression for the total ionic energy:

\begin{equation} E_P(r) = -\frac{AN_Ae^2|z_1z_2|}{4\pi{}\epsilon_0r} + CN_A{}e^{-\left(\frac{r}{d^{*}}\right)} \end{equation}

where $N_A$ is the Avogadro constant, $e$ is the electron charge, $\epsilon_0$ is the permittivity of free space, $z_1$ and $z_2$ are the charges of the interacting ions, $A$ is the (geometric) Madelung constant, and $r$ is the lattice parameter, we aim to compute the relative factor $C$ between the attractive term and the repulsive term by imposing the condition that the derivative of the total interaction energy with respect to coordinate (lattice constant) vanishes at equilibrium ($r = d$).

\begin{equation} \frac{\mathrm{d}E_P}{\mathrm{d}r}\bigg|_{r=d} = \frac{AN_Ae^2|z_1z_2|}{4\pi{}\epsilon_0{}d^2} - \frac{1}{d^{*}}CN_A{}e^{-\left(\frac{r}{d^{*}}\right)} = 0 \end{equation}

This results in the following expression for $C$:

\begin{equation} C = \frac{Ae^2|z_1z_2|}{4\pi{}\epsilon_0{}d}\left(\frac{d^{*}}{d}\right) e^{\left(\frac{r}{d^{*}}\right)} \end{equation}

and substituting into the original expression for $E_P$ evaluated at equilibrium, we arrive at

\begin{equation} E_P(d) = -\frac{AN_Ae^2|z_1z_2|}{4\pi{}\epsilon_0d} \left( 1 - \left( \frac{d}{d^{*}} \right) e^{-\left(\frac{d}{d^{*}}\right)} e^{\left(\frac{d}{d^{*}}\right)} \right) \end{equation}

\begin{equation} E_P(d) = -\frac{AN_Ae^2|z_1z_2|}{4\pi{}\epsilon_0d} \left( 1 - \left( \frac{d}{d^{*}} \right) e^{\left(-\frac{d}{d^{*}} + \frac{d}{d^{*}} \right)} \right) \end{equation}

which simplifies to the following:

\begin{equation} E_P(d) = -\frac{AN_Ae^2|z_1z_2|}{4\pi{}\epsilon_0d} \left( 1 - \frac{d}{d^{*}} \right) \end{equation}