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 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
- 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.)
