Question

Two identical atoms of mass \( m \) are bound to each other by the Lennard-Jones potential,
\[ V = \varepsilon \left[ \left( \frac{r_0}{r} \right)^{12} - 2 \left( \frac{r_0}{r} \right)^6 \right] \] The frequency of small oscillations about the equilibrium is:

• A. \( \sqrt{\frac{\varepsilon}{m r_0^2}} \)
• B. \( \sqrt{\frac{2\varepsilon}{m r_0^2}} \)
• C. \( \sqrt{\frac{12\varepsilon}{m r_0^2}} \)
• D. \( \frac{\pi}{2} \sqrt{\frac{\varepsilon}{m r_0^2}} \)

Hint:

The Lennard-Jones potential describes interatomic interactions, including bonding forces.

Answer 1

The Correct Option is C

Expanding \( V(r) \) about equilibrium \( r = r_0 \) and using harmonic approximation:
\[ \omega = \sqrt{\frac{V''(r_0)}{m}} \] After differentiating and solving:
\[ \omega = \sqrt{\frac{12\varepsilon}{m r_0^2}} \]