Understanding Neutron Scattering and Phase Shift
This question involves calculating the approximate s-wave phase shift ($\delta_0$) for a neutron scattering off a Carbon target. We are given the neutron's kinetic energy ($E$) and the total scattering cross section ($\sigma_{tot}$). For low-energy neutrons, where the de Broglie wavelength is much larger than the nuclear size, the scattering is predominantly s-wave (l=0). The total scattering cross section can be related to the s-wave phase shift using the formula:
$ \sigma_{tot} = \frac{4\pi}{k^2} \sin^2(\delta_0) $
where '$k$' is the wave number of the neutron and '$\delta_0$' is the s-wave phase shift.
Calculating the Wave Number ($k$)
First, we need to calculate the wave number '$k$' using the neutron's kinetic energy '$E$'. The formula for '$k$' is:
$ k = \sqrt{\frac{2mE}{\hbar^2}} $
Let's define the constants and convert the given values to SI units:
- Kinetic energy, $E = 1 \text{ keV} = 1 \times 10^3 \text{ eV} = 1 \times 10^3 \times (1.602 \times 10^{-19} \text{ J}) = 1.602 \times 10^{-16} \text{ J}$.
- Mass of neutron, $m \approx 1.675 \times 10^{-27} \text{ kg}$.
- Reduced Planck constant, $\hbar \approx 1.054 \times 10^{-34} \text{ J s}$.
Now, substitute these values into the formula for $k^2$:
$ k^2 = \frac{2 \times (1.675 \times 10^{-27} \text{ kg}) \times (1.602 \times 10^{-16} \text{ J})}{(1.054 \times 10^{-34} \text{ J s})^2} $
$ k^2 = \frac{5.363 \times 10^{-43} \text{ kg J}}{1.111 \times 10^{-68} \text{ J}^2 \text{ s}^2} \approx 4.827 \times 10^{25} \text{ m}^{-2} $
Calculating $\sin^2(\delta_0)$
Next, we use the given total scattering cross section ($\sigma_{tot}$) and the calculated $k^2$ to find $\sin^2(\delta_0)$.
- Total scattering cross section, $\sigma_{tot} = 1000 \text{ barns}$.
- Convert barns to square meters: $1 \text{ barn} = 10^{-28} \text{ m}^2$.
- So, $\sigma_{tot} = 1000 \times 10^{-28} \text{ m}^2 = 10^{-25} \text{ m}^2$.
Rearranging the cross section formula to solve for $\sin^2(\delta_0)$:
$ \sin^2(\delta_0) = \frac{\sigma_{tot} k^2}{4\pi} $
Substitute the values:
$ \sin^2(\delta_0) = \frac{(10^{-25} \text{ m}^2) \times (4.827 \times 10^{25} \text{ m}^{-2})}{4\pi} $
$ \sin^2(\delta_0) = \frac{4.827}{4\pi} \approx \frac{4.827}{12.566} \approx 0.3841 $
Determining the Phase Shift ($\delta_0$)
Now, we find the value of $\sin(\delta_0)$ and then $\delta_0$ itself.
$ \sin(\delta_0) = \sqrt{0.3841} \approx 0.6197 $
To find $\delta_0$, we take the arcsine:
$ \delta_0 = \arcsin(0.6197) $
Calculating the angle in radians:
$ \delta_0 \approx 0.671 \text{ radians} $
Convert the phase shift to degrees:
$ \delta_0 (\text{degrees}) = 0.671 \text{ rad} \times \frac{180^\circ}{\pi} \approx 38.5^\circ $
Conclusion
The calculated approximate value for the phase shift $\delta_0$ is $38.5^\circ$. Comparing this result to the given options:
- $18^\circ$
- $108^\circ$
- $90^\circ$
- $36^\circ$
The value $38.5^\circ$ is closest to $36^\circ$. Therefore, the approximate value of the phase shift $\delta_0$ is $36^\circ$. Small discrepancies can arise due to the approximations used for constants and the simplified scattering formula.
Important Questions from Scattering Theory
- 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 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?
- 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 - 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
