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.