Question

Match the LIST-I with LIST-II 

LIST-I (Parameters)	LIST-II (Expressions)
A. Potential energy of a system of two atoms	I. \( \alpha = \sum_{j} \pm \frac{1}{p_{ij}} \)
B. Madelung constant	II. \( U_{ij} = \pm \frac{q^2}{r} \)
C. Coulomb electrostatic energy	III. \( U = -\frac{A}{r^2} + \frac{B}{r^{10}} \)
D. Cohesive energy	IV. \( U_{ij} = \lambda e^{-r_{ij}/\rho} \pm \frac{q^2}{r_{ij}} \)

Choose the correct answer from the options given below:

• A. A - II, B - I, C - III, D - IV
• B. A - III, B - II, C - I, D - IV
• C. A - I, B - II, C - IV, D - III
• D. A - III, B - I, C - II, D - IV

Hint:

Start with the most fundamental and easily recognizable pairs. The Coulomb energy (C-II) and the general form of interatomic potential (A-III) are often the easiest to identify, which can help eliminate incorrect options quickly.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question requires matching fundamental concepts in solid-state physics with their corresponding mathematical expressions.
Step 2: Detailed Explanation:
A. Potential energy of a system of two atoms: The potential energy \(U\) between two atoms is the sum of an attractive potential (negative term) and a repulsive potential (positive term). Expression III, \( U = -\frac{A}{r^2} + \frac{B}{r^{10}} \), represents such a potential (a form of the Mie or Lennard-Jones potential). Thus, A matches III.
B. Madelung constant: The Madelung constant (\(\alpha\)) is a geometric factor that accounts for the entire electrostatic interaction of an ion with all other ions in a crystal lattice. It is defined as a sum over the lattice points. Expression I, \( \alpha = \sum_{j} \pm \frac{1}{p_{ij}} \), where \(p_{ij}\) is the normalized distance to other ions, is the definition of the Madelung constant. Thus, B matches I.
C. Coulomb electrostatic energy: The fundamental expression for the electrostatic potential energy between two point charges is given by Coulomb's law. Expression II, \( U_{ij} = \pm \frac{q^2}{r} \), represents this energy between two ions of charge \(q\) separated by distance \(r\). Thus, C matches II.
D. Cohesive energy: The cohesive energy of an ionic crystal is related to the total potential energy per ion pair at equilibrium. This total energy includes the attractive Coulomb interaction and a short-range repulsive interaction (often modeled by an exponential term, as in the Born-Mayer model). Expression IV, \( U_{ij} = \lambda e^{-r_{ij}/\rho} \pm \frac{q^2}{r_{ij}} \), represents this total potential energy, which is directly used to find the cohesive energy. Thus, D matches IV.
Step 3: Final Answer:
The correct matching is A-III, B-I, C-II, D-IV.