Discussion
For a covalently bonded solid consisting of ions of mass \(m\), the binding potential can be assumed to be given by
\[ U(r) = -\epsilon\left(\tfrac{r}{r_{0}}\right)e^{-r/r_{0}} , \]
where \(\epsilon\) and \(r_{0}\) are positive constants. What is the Einstein frequency of the solid in Hz?
Answer: B
The Einstein frequency is \(\frac{1}{2\pi}\sqrt{\frac{\epsilon}{m e r_{0}^{2}}}\).
Each ion vibrates about the minimum of \(U(r)\) with spring constant \(U''\) there. With \(x = r/r_{0}\), \(U = -\epsilon x e^{-x}\):
\[ \begin{aligned} \frac{dU}{dx} &= -\epsilon(1 - x)e^{-x} = 0 \Rightarrow x = 1 \\ \frac{d^{2}U}{dx^{2}} &= \epsilon(2 - x)e^{-x} = \frac{\epsilon}{e} \text{ at } x = 1 \end{aligned} \]
So \(U''(r_{0}) = \epsilon/(e r_{0}^{2})\), and
\[ \begin{aligned} \nu_{E} &= \frac{1}{2\pi}\sqrt{\frac{U''}{m}} \\ &= \frac{1}{2\pi}\sqrt{\frac{\epsilon}{m e r_{0}^{2}}} \end{aligned} \]
The options with \(e\) in the numerator have the exponential factor inverted, and the factor 2 in the other options comes from evaluating \(2 - x\) at \(x = 0\) instead of at the minimum.
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