Question

For a covalently bonded solid consisting of ions of mass \(m\), the binding potential can be assumed to be given by \[ U(r) = -\varepsilon \left( \frac{r}{r_0} \right) e^{-\frac{r}{r_0}}, \] where \(\varepsilon\) and \(r_0\) are positive constants. What is the Einstein frequency of the solid in Hz?

• A. \(\frac{1}{2\pi} \sqrt{\frac{\varepsilon \varepsilon}{m r_0^2}}\)
• B. \(\frac{1}{2\pi} \sqrt{\frac{\varepsilon}{m e r_0^2}}\)
• C. \(\frac{1}{2\pi} \sqrt{\frac{\varepsilon}{m r_0^2}}\)
• D. \(\frac{1}{2\pi} \sqrt{\frac{\varepsilon \varepsilon}{2 m r_0^2}}\)

Hint:

The Einstein frequency is related to the parameters of the potential energy function. This formula can be derived by considering the harmonic approximation for small oscillations around the equilibrium position.

Answer 1

The Correct Option is B

- The Einstein frequency of a solid can be derived from the potential energy function. Using the standard procedure for harmonic oscillators and the form of the given potential energy, the Einstein frequency is found to be: \[ f_E = \frac{1}{2\pi} \sqrt{\frac{\varepsilon}{m r_0^2}} \] - By analyzing the problem in terms of known constants and simplifying, the correct Einstein frequency formula is obtained as shown in option (B).