Question

In a very simple model of a crystal, point-like atomic ions are regularly spaced along an infinite one-dimensional row with spacing \(R\). Alternate ions carry equal and opposite charges \(\pm e .\) The potential energy of the ith ion in the electric field due to the \(j\) th ion is $$ \frac{q_{i} q_{j}}{4 \pi \epsilon_{0} r_{i j}} $$ where \(q_{k}\) is the charge on the \(k\) th ion and \(r_{i j}\) is the distance between the \(i\) th and \(j\) th ions. Write down a series giving the total contribution \(V_{i}\) of the ith ion to the overall potential energy. Show that the series converges, and, if \(V_{i}\) is written as $$ V_{i}=\frac{\alpha e^{2}}{4 \pi \epsilon_{0} R} $$ find a closed-form expression for \(\alpha\), the Madelung constant for this (unrealistic) lattice.

Short answer

The Madelung constant, \( \alpha \), for this lattice is approximately -2.

Step-by-step solution

- Understanding the problem and setup

The potential energy of the ith ion in the electric field due to the jth ion is given by \( \frac{q_{i} q_{j}}{4\pi\epsilon_{0} r_{i j}} \) where the charges alternate as \( +e \) and \( -e \). The distance between ions is given by \( r_{i j} = |i-j|R \).

- Write down the series for the potential energy

We write the total potential energy contribution of the ith ion, \( V_i \), as a series where it interacts with all other ions: \[ V_i = \sum_{j eq i} \frac{q_i q_j}{4\pi\epsilon_{0} r_{ij}} = \frac{e}{4\pi\epsilon_{0}} \sum_{j eq i} \frac{(-1)^{i+j} e}{|i-j|R} \] where the term \( (-1)^{i+j} \) captures the alternating charges.

- Simplifying the series

By pulling out constants and focusing on the summation part, we get: \[ V_i = \frac{e^2}{4\pi\epsilon_{0}R} \sum_{j eq i} \frac{(-1)^{i+j}}{|i-j|} \]

- Establishing convergence of the series

To show convergence, note that \( \sum_{j eq i} \frac{(-1)^{i+j}}{|i-j|} \) alternates in sign and grows more slowly than the harmonic series, which is conditionally convergent. Specifically, each term decreases in magnitude as \( |i-j| \) increases, ensuring the series converges.

- Finding the Madelung constant (\( \alpha \))

Given \( V_i \) in the form \( \frac{\alpha e^2}{4\pi\epsilon_{0}R} \), set this equation equal to our simplified series: \[ \frac{\alpha e^2}{4\pi\epsilon_{0}R} = \frac{e^2}{4\pi\epsilon_{0}R} \sum_{j eq i} \frac{(-1)^{i+j}}{|i-j|} \] The Madelung constant is thus \( \alpha = \sum_{j eq i} \frac{(-1)^{i+j}}{|i-j|} \).

- Evaluating the series for \( \alpha \)

The series \( \sum_{j eq i} \frac{(-1)^{i+j}}{|i-j|} \) is known to approximate to approximately -2. Thus, \( \alpha \approx -2 \).

Key concepts

Potential Energy

Potential energy is the energy that is stored in an object due to its position relative to other objects. In the context of ionic crystals, potential energy arises from the electrostatic interactions between ions. For two ions with charges \(q_i\) and \(q_j\) separated by a distance \(r_{ij}\), the potential energy is given by: $$ \frac{q_i q_j}{4\pi\epsilon_{0} r_{ij}}. $$ Here, \(\epsilon_{0}\) is the permittivity of free space and \(r_{ij}\) is the distance between the ions. In a crystal lattice, we sum up these interactions to find the total potential energy contributed by the ith ion. This involves considering the influence of all other ions in the lattice.

Convergence of Series

The convergence of a series is crucial in determining whether the sum of an infinite sequence of terms results in a finite value. For the potential energy series of an ion in an ionic lattice, we have: $$ V_i = \frac{e^2}{4\pi\epsilon_0 R} \sum_{j eq i} \frac{(-1)^{i+j}}{|i-j|}. $$ The series alternates in sign thanks to the factor \((-1)^{i+j}\), which reflects the alternating charges in the lattice. The series converges because each term's magnitude decreases as \(|i - j|\) increases, similar to the harmonic series, making it conditionally convergent. This property ensures that despite the infinite number of terms, the sum approaches a finite limit.

Lattice Structure

A lattice structure refers to a regular, repeating arrangement of atoms or ions in a crystal. In this problem, we have a one-dimensional lattice where ions are spaced a distance \(R\) apart, with alternating positive and negative charges \(\pm e\). The regularity in spacing and the pattern of alternating charges lead to specific interactions between ions, which are essential for calculating the total potential energy. Knowing the lattice structure allows us to systematically compute the interactions and understand the macroscopic properties of the crystal.

Electric Field

The electric field is a vector field that represents the force that a charge would experience per unit charge at any point in space. In an ionic lattice, the electric field experienced by an ion is due to the superposition of fields from all other ions. Since the charges alternate, the field contributions also alternate in sign. The potential energy of an ion in an electric field is derived from the work needed to move the charge within the field. This relationship helps in setting up the series for the ion's potential energy as a function of its interactions with the surrounding ions.

Alternating Series

An alternating series is a series where the signs of the terms alternate, either positive to negative or vice versa. In our problem, the series for the potential energy includes a term \((-1)^{i+j}\) which causes the terms to alternate in sign: $$ \sum_{j eq i} \frac{(-1)^{i+j}}{|i-j|}. $$ Alternating series can converge even when the individual terms do not approach zero rapidly. The key here is that the magnitude of the terms decreases, ensuring that the overall sum converges to a finite value. This property is crucial for calculating the Madelung constant, which encapsulates the total electrostatic interaction within the lattice.