Discussion
In the Born approximation, scattering amplitude \(f(\theta)\) for Yukawa potential \(V(r) = \frac{\beta e^{-\mu r}}{r}\) is given by
(Given data : \(b = 2k\text{sin}\frac{\theta}{2}\), \(E = \frac{\hbar^{2}k^{2}}{2m}\))
Answer: A
The Born amplitude is \(-\frac{2m\beta}{\hbar^{2}(\mu^{2} + b^{2})}\).
For a central potential the first Born approximation is \(f(\theta) = -\frac{2m}{\hbar^{2}b}\int_{0}^{\infty} rV(r)\,\text{sin}(br)\,dr\), where b is the momentum transfer \(2k\,\text{sin}(\theta/2)\). With \(V = \beta e^{-\mu r}/r\):
\[ \begin{aligned} \int_{0}^{\infty} e^{-\mu r}\,\text{sin}(br)\,dr \\ &= \frac{b}{\mu^{2} + b^{2}} \\ f(\theta) &= -\frac{2m\beta}{\hbar^{2}(\mu^{2} + b^{2})} \end{aligned} \]
Letting \(\mu \to 0\) recovers the Rutherford amplitude, proportional to \(1/b^{2}\), a useful check. The cross-section \(|f|^{2}\) has the square of the bracket; the options with the bracket squared, cubed or square-rooted in the amplitude itself are wrong.
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