Let's break down the problem **step by step**: ### **Given:** - Original lattice energy: \[ U(r) = -N \left[ \frac{Ae^2}{r} - \frac{B}{r^n} \right] \] - The repulsive term \(\frac{B}{r^n}\) is replaced by \(C \exp\left(-\frac{r}{\rho}\right)\). - **Find:** The equilibrium separation \(r_0\) at which the lattice energy is minimized (i.e., the two forms of repulsion give the same equilibrium). --- ### **Step 1: Write the Modified Potential** \[ U(r) = -N \left[ \frac{Ae^2}{r} - C \exp\left(-\frac{r}{\rho}\right) \right] \] --- ### **Step 2: Find the Equilibrium Separation \( r_0 \)** At equilibrium, the potential energy is minimized with respect to \(r\): \[ \left.\frac{dU}{dr}\right|_{r = r_0} = 0 \] Calculate the derivative: \[ U(r) = -N \left[ \frac{Ae^2}{r} - C \exp\left(-\frac{r}{\rho}\right) \right] \] \[ \frac{dU}{dr} = -N \left[ -\frac{Ae^2}{r^2} + C \cdot \left(\frac{1}{\rho}\right) \exp\left(-\frac{r}{\rho}\right) \right] \] Set to zero at \(r = r_0\): \[ -\frac{Ae^2}{r_0^2} + C \cdot \frac{1}{\rho} \exp\left(-\frac{r_0}{\rho}\right) = 0 \] \[ \frac{Ae^2}{r_0^2} = C \cdot \frac{1}{\rho} \exp\left(-\frac{r_0}{\rho}\right) \] --- ### **Step 3: Express \( r_0 \) in Terms of \( n \) and \( \rho \)** Recall, for the power-law repulsion, at equilibrium: \[ \frac{Ae^2}{r_0^2} = \frac{nB}{r_0^{n+1}} \] So, for the exponential repulsion, the equivalent equation is: \[ \frac{Ae^2}{r_0^2} = C \cdot \frac{1}{\rho} \exp\left(-\frac{r_0}{\rho}\right) \] Take natural logarithm on both sides: \[ \ln\left(\frac{Ae^2}{r_0^2} \cdot \frac{\rho}{C}\right) = -\frac{r_0}{\rho} \] \[ -\frac{r_0}{\rho} = \ln\left(\frac{Ae^2 \rho}{C r_0^2}\right) \] \[ r_0 = -\rho \ln\left(\frac{Ae^2 \rho}{C r_0^2}\right) \] This is a **transcendental equation** for \(r_0\), which generally requires numerical methods to solve. However, it expresses \(r_0\) in terms of \(\rho\) and the constants. --- ### **Step 4: Relate to Power-law Repulsion** For the **power-law repulsion**: \[ \frac{Ae^2}{r_0^2} = \frac{nB}{r_0^{n+1}} \] \[ r_0^{n-1} = \frac{nB}{Ae^2} \] \[ r_0 = \left( \frac{nB}{Ae^2} \right)^{\frac{1}{n-1}} \] --- ### **Step 5: Final Answer** **For the exponential repulsion:** \[ \boxed{ r_0 = -\rho \ln\left(\frac{Ae^2 \rho}{C r_0^2}\right) } \] **Or, equivalently:** \[ \boxed{ \frac{Ae^2}{r_0^2} = C \cdot \frac{1}{\rho} \exp\left(-\frac{r_0}{\rho}\right) } \] This is the equation for the equilibrium nearest neighbor separation \(r_0\) in terms of \(n\) and \(\rho\). --- **Summary Table:** | Repulsive Term | Equilibrium \( r_0 \) | |-------------------------|----------------------------------| | \( \frac{B}{r^n} \) | \( \left( \frac{nB}{Ae^2} \right)^{1/(n-1)} \) | | \( C e^{-r/\rho} \) | \( -\rho \ln\left(\frac{Ae^2 \rho}{C r_0^2}\right) \) or \( \frac{Ae^2}{r_0^2} = C \frac{1}{\rho} e^{-r_0/\rho} \) | --- If you need the value of \(r_0\) explicitly, you would need to solve the transcendental equation numerically for given values of the constants.