[You may use $\int x^n e^{ax} dx = \frac{d^n}{da^n} \int e^{ax} dx$]
Born Approximation Calculation
The differential scattering cross-section in the first Born approximation is given by:
$ \frac{d\sigma}{d\Omega} = \left(\frac{2m}{\hbar^2}\right)^2 |\tilde{V}(q)|^2 $where $m$ is the particle mass, $\hbar$ is the reduced Planck constant, $q$ is the magnitude of the momentum transfer, and $\tilde{V}(q)$ is the Fourier transform of the potential $V(r)$. For a central potential, the Fourier transform is:
$ \tilde{V}(q) = \int V(\vec{r}) e^{-i\vec{q}\cdot\vec{r}} d^3r = \frac{4\pi}{q} \int_0^\infty V(r) r \sin(qr) dr $Evaluating the Integral
Substitute the given potential $V(r) = V_0 r e^{-\mu r}$:
$ \tilde{V}(q) = \frac{4\pi V_0}{q} \int_0^\infty r^2 e^{-\mu r} \sin(qr) dr $The integral can be evaluated using standard methods (e.g., differentiation under the integral sign or complex analysis). The result is:
$ \int_0^\infty r^2 e^{-\mu r} \sin(qr) dr = \frac{2q(q^2 - 3\mu^2)}{(\mu^2 + q^2)^3} $Substituting this back into the expression for $\tilde{V}(q)$:
$ \tilde{V}(q) = \frac{4\pi V_0}{q} \left[ \frac{2q(q^2 - 3\mu^2)}{(\mu^2 + q^2)^3} \right] = \frac{8\pi V_0 (q^2 - 3\mu^2)}{(\mu^2 + q^2)^3} $Asymptotic Behavior for Large q
We need the behavior as $q \to \infty$, given $q \gg \mu$. In this limit, terms involving $\mu$ are negligible compared to terms involving $q$.
Approximating the expression for $\tilde{V}(q)$:
$ \tilde{V}(q) \approx \frac{8\pi V_0 q^2}{(q^2)^3} = \frac{8\pi V_0 q^2}{q^6} = 8\pi V_0 q^{-4} $Therefore, the Fourier transform $\tilde{V}(q)$ behaves as $q^{-4}$ for large $q$.
Cross-Section Dependence
The scattering cross-section depends on the square of the Fourier transform:
$ \frac{d\sigma}{d\Omega} \propto |\tilde{V}(q)|^2 $Substituting the asymptotic behavior:
$ \frac{d\sigma}{d\Omega} \propto (q^{-4})^2 = q^{-8} $Thus, the scattering cross-section depends on $q$ as $q^{-8}$ as $q \to \infty$. This corresponds to Option A.
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
- 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
- The differential scattering cross-section $\frac{d\sigma}{d\Omega}$ for the central potential $V(r) = \frac{\beta}{r}e^{-\mu r}$, where $\beta$ and $\mu$ are positive constants, is calculated in the first Born approximation. Its dependence on the scattering angle $\theta$ is proportional to ($A$ is a constant below.)
