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 $\left( A^2 + \sin^2\left(\frac{\theta}{2}\right) \right)^{-2}$.
Important Questions from Scattering Theory
- A phase shift of $30^\circ$ is observed when a beam of particles of energy $0.1$ MeV is scattered by a target. When the beam energy is changed, the observed phase shift is $60^\circ$. Assuming that only s-wave scattering is relevant and that the cross-section does not change with energy, the beam energy is
- Consider the potential
$$V(\vec{r}) = \sum_i V_0 a^3 \delta^{(3)}(\vec{r} - \vec{r}_i)$$
where $\vec{r}_i$ are the position vectors of the vertices of a cube of length $a$ centered at the origin and $V_0$ is a constant. If $V_0 a^2 \ll \frac{\hbar^2}{m}$, the total scattering cross-section, in the low-energy limit, is - A sphere G of radius $b$ is fixed mid-air and several spheres identical to the first one are shot at it with their velocities parallel to each other. If the shot spheres fall within an imaginary cylinder of radius $a \ (b \ll a)$ then the fraction of spheres that will hit G is
- A particle is scattered by a central potential $V(r) = V_0 r e^{-\mu r}$, where $V_0$ and $\mu$ are positive constants. If the momentum transfer $\vec{q}$ is such that $q = |\vec{q}| \gg \mu$, the scattering cross-section in the Born approximation, as $q \to \infty$, depends on $q$ as
[You may use $\int x^n e^{ax} dx = \frac{d^n}{da^n} \int e^{ax} dx$] - When a neutron of $1 \text{ keV}$ kinetic energy impinges on a $^{12}\text{C}$ target, the total scattering cross section is $1000 \text{ barns}$. The approximate value of the phase shift $\delta_0$ is
