S-Wave Scattering: Relating Phase Shift and Energy
The problem involves s-wave scattering, where the phase shift ($\delta_0$) is the primary parameter. We are given two scenarios with different energies and corresponding phase shifts. A key condition is that the scattering cross-section ($\sigma$) remains constant with energy.
Physics Principles
- For s-wave scattering ($l=0$), the total cross-section is proportional to $\frac{\sin^2(\delta_0)}{k^2}$, where $k$ is the wave number.
- The wave number $k$ is related to the particle's kinetic energy $E$ by $E = \frac{\hbar^2 k^2}{2m}$, which implies $k^2 \propto E$.
- Therefore, the s-wave cross-section is $\sigma \propto \frac{\sin^2(\delta_0)}{E}$.
Applying the Constant Cross-Section Condition
Since the cross-section ($\sigma$) is constant, we can write the relationship between the two scenarios (1 and 2) as:
$ \frac{\sin^2(\delta_1)}{E_1} = \frac{\sin^2(\delta_2)}{E_2} $Here:
- $E_1 = 0.1$ MeV
- $\delta_1 = 30^\circ$
- $E_2$ is the unknown energy
- $\delta_2 = 60^\circ$
Calculation
- Calculate the sine squared values for the phase shifts:
- $ \sin^2(\delta_1) = \sin^2(30^\circ) = (0.5)^2 = 0.25 $
- $ \sin^2(\delta_2) = \sin^2(60^\circ) = (\frac{\sqrt{3}}{2})^2 = \frac{3}{4} = 0.75 $
- Substitute the values into the constant cross-section equation: $ \frac{0.25}{0.1 \text{ MeV}} = \frac{0.75}{E_2} $
- Solve for $E_2$: $ E_2 = \frac{0.75 \times 0.1 \text{ MeV}}{0.25} $ $ E_2 = \frac{0.075}{0.25} \text{ MeV} $ $ E_2 = 0.3 \text{ MeV} $
Conclusion
The beam energy corresponding to the $60^\circ$ phase shift, under the condition of constant cross-section and only s-wave scattering, is $0.3$ MeV.
Important Questions from Scattering Theory
- 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 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
- 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
