Question 2 of 7: Ionic Cohesion — Repulsive/Coulomb Energy, Equilibrium Separation, and the 1-D Madelung Constant
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — Phys-A6: Solid State Physics — May 2015 (3 hours; closed book; useful equations and physical constants annexed to the paper). Any FIVE of the SEVEN questions constitute a complete exam paper (first five as they appear in the answer book are marked); all seven are solved below as a complete study resource.
Reference texts: C. Kittel, Introduction to Solid State Physics, 8th ed. (Ch. 1–3, 4, 6, 8, 14, 18); N. W. Ashcroft & N. D. Mermin, Solid State Physics, 1st ed. (Ch. 2, 4–7, 22, 28, 31–32).
Question 2: Ionic Cohesion — Repulsive/Coulomb Energy, Equilibrium Separation, and the 1-D Madelung Constant (20 marks)
Given. Figure P2a's own printed forms: repulsive energy $U_{\text{rep}}(R)=(2.5\times10^4)\,e^{-R/0.3}\ \text{eV}$, Coulomb energy $U_{\text{Coul}}(R)=-25/R\ \text{eV}$ (both with $R$ in \Å). Figure P2b: an infinite line of ions of alternating sign, equally spaced by $R$.
Fig. 2a — total energy $U(R)=U_{\text{rep}}+U_{\text{Coul}}$ (solid), matching Figure P2a's shape: a steep repulsive wall, a slowly-vanishing $-25/R$ attractive tail, and a minimum at $R_0$.
Fig. 2b — the 1-D alternating ionic chain of Figure P2b, spacing $R$, with the reference ion at the centre.
Approach. (a)–(b) recall the physical mechanism behind each term's sign and range; (c) set $dU/dR=0$ and solve numerically for the printed functional forms; (d) sum the alternating $\pm1/n$ Coulomb series along the chain.
Part (a) — origin of the repulsive term. As two ions approach, their filled electron orbitals begin to overlap. The Pauli exclusion principle forbids the overlapping electrons from occupying the same quantum states, forcing some of them into higher-energy states — this steeply rising energy cost is modelled by the printed $(2.5\times10^4)e^{-R/0.3}$ form. Its impact is to provide the short-range "hard wall" that stops the ions from collapsing into each other, setting a lower bound on how close they can approach.
Part (b) — origin of the Coulomb term. Ionic crystals are built from ions of opposite charge (e.g. Na$^+$ and Cl$^-$); their electrostatic attraction gives the long-range $-25/R$ term (negative = attractive, and it falls off slowly, as $1/R$, so many distant neighbours still contribute). Its impact is to provide essentially ALL of the crystal's cohesive (binding) energy — it is what holds the lattice together at all — while the short-range repulsion of part (a) only becomes significant at very small $R$.
Part (c) — equilibrium separation $R_0$. $R_0$ minimizes the total energy, $dU/dR=0$:
$$\frac{dU}{dR}=(2.5\times10^4)\left(-\frac1{0.3}\right)e^{-R/0.3}+\frac{25}{R^2}=0$$
This transcendental equation was solved numerically: the root is
$$\boxed{R_0\approx3.12\ \text{\Å}}$$
matching the dashed $R_0$ marker on Figure P2a, with $U(R_0)\approx-7.25\ \text{eV}$ — the depth of the well shown on the graph.
Part (d) — 1-D Madelung constant. Number the ions along the chain $n=1,2,3,\dots$ out from the reference ion at $R,2R,3R,\dots$, alternating sign starting with the opposite charge as nearest neighbour. The Coulomb energy of the reference ion (in the convention $U=-\alpha e^2/(4\pi\varepsilon_0 R)$, paper's eq. 12 style) sums both directions:
$$\alpha=2\left(\frac11-\frac12+\frac13-\frac14+\cdots\right)=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}$$
This is twice the standard alternating harmonic series, which converges to $\ln2$:
$$\boxed{\alpha=2\ln2=1.386}$$
the well-known 1-D Madelung constant (the factor of 2 counts the chain extending in both directions from the reference ion).
