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 of energy $E$ scatters off a repulsive spherical potential
$V(r) = \begin{cases} V_0 & \text{for } r < a \\ 0 & \text{for } r \ge a \end{cases}$
where $V_0$ and $a$ are positive constants. In the low energy limit, the total scattering cross-section is $\sigma = 4\pi a^2 \left(\frac{1}{ka}\tanh ka - 1\right)^2$, where $k^2 = \frac{2m}{\hbar^2}(V_0 - E) > 0$. In the limit $V_0 \to \infty$ the ratio of $\sigma$ to the classical scattering cross-section off a sphere of radius $a$ is - In the partial wave expansion, the differential scattering cross-section is given by
$$ \frac{d\sigma}{d(\cos\theta)} = \left| \sum_l (2l+1)e^{i\delta_l} \sin\delta_l P_l(\cos\theta) \right|^2 $$
where $\theta$ is the scattering angle. For a certain neutron-nucleus scattering, it is found that the two lowest phase shifts $\delta_0$ and $\delta_1$ corresponding to $s$-wave and $p$-wave, respectively, satisfy $\delta_1 \approx \delta_0/2$. Assuming that the other phase shifts are negligibly small, the differential cross-section reaches its minimum for $\cos\theta$ equal to - 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
A particle of mass $m$ and energy $E > 0$, in one dimension is scattered by the potential shown below.
If the particle was moving from $x = -\infty$ to $x = \infty$, which of the following graphs gives the best qualitative representation of the wavefunction of this particle?
