Born Approximation Scattering Cross-Section

In the first Born approximation, the differential scattering cross-section ($\frac{d\sigma}{d\Omega}$) is proportional to the squared magnitude of the Fourier transform of the potential ($V_q$):

$ \frac{d\sigma}{d\Omega} \propto |V_q|^2 $

Potential Fourier Transform

For the given potential $V(r) = \frac{\beta}{r}e^{-\mu r}$ (a screened Coulomb potential), its Fourier transform is:

$ V_q = \frac{4\pi \beta}{q^2 + \mu^2} $

where $q$ is the magnitude of the momentum transfer.

Momentum Transfer Dependence

The momentum transfer magnitude $q$ relates to the scattering angle $\theta$ and incident momentum $p$ as:

$ q = 2p \sin\left(\frac{\theta}{2}\right) \implies q^2 = 4p^2 \sin^2\left(\frac{\theta}{2}\right) $

Cross-Section Calculation

Substitute $q^2$ into the expression for $V_q$ and calculate $|V_q|^2$:

$ |V_q|^2 = \left( \frac{4\pi \beta}{4p^2 \sin^2\left(\frac{\theta}{2}\right) + \mu^2} \right)^2 = \frac{(4\pi \beta)^2}{\left( 4p^2 \sin^2\left(\frac{\theta}{2}\right) + \mu^2 \right)^2} $

Thus, the dependence of the differential cross-section is:

$ \frac{d\sigma}{d\Omega} \propto \frac{1}{\left( 4p^2 \sin^2\left(\frac{\theta}{2}\right) + \mu^2 \right)^2} $

Rewrite the denominator term by factoring out $4p^2$:

$ \left( 4p^2 \sin^2\left(\frac{\theta}{2}\right) + \mu^2 \right)^2 = \left[ 4p^2 \left( \sin^2\left(\frac{\theta}{2}\right) + \frac{\mu^2}{4p^2} \right) \right]^2 $

$ = (4p^2)^2 \left( \sin^2\left(\frac{\theta}{2}\right) + \frac{\mu^2}{4p^2} \right)^2 $

Substituting this back yields:

$ \frac{d\sigma}{d\Omega} \propto \frac{1}{(4p^2)^2 \left( \sin^2\left(\frac{\theta}{2}\right) + \frac{\mu^2}{4p^2} \right)^2} $

Let $A^2 = \frac{\mu^2}{4p^2}$ (where $A$ is a constant since $\mu$ and $p$ are constants) and absorb the constant factor $\frac{1}{(4p^2)^2}$ into the definition of $A$ as per the options provided:

$ \frac{d\sigma}{d\Omega} \propto \left( A^2 + \sin^2\left(\frac{\theta}{2}\right) \right)^{-2} $

Final Dependence

The differential scattering cross-section depends on the scattering angle $\theta$ proportionally to $\left( A^2 + \sin^2\left(\frac{\theta}{2}\right) \right)^{-2}$.