[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 - 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.)
- The range of the inter-atomic potential in gaseous hydrogen is approximately $5\text{ \AA}$. In thermal equilibrium, the maximum temperature for which the atom-atom scattering is dominantly $s$-wave, is
