Question

Neutrons are spin \(\frac{1}{2}\) fermions. An unpolarized beam of neutrons has an equal number of spins in the \(+1 / 2\) and \(-1 / 2\) states. When a beam of unpolarized neutrons is passed through unpolarized \({ }^{3}\) He, the neutrons can be absorbed by the \({ }^{3}\) He to create \({ }^{4} \mathrm{He}\). If the ${ }^{3} \mathrm{He}$ is then polarized, so that the spins of the neutrons in the nucleus of the \({ }^{3} \mathrm{He}\) are all aligned, will the same number of neutrons in the unpolarized neutron beam be absorbed by the polarized ${ }^{3}$ He? How well is each of the two spin states of the unpolarized neutron beam absorbed by the \({ }^{3} \mathrm{He}\) ?

Short answer

Answer: No, the same number of neutrons will not be absorbed from both spin states. The absorption of neutrons in each of the two spin states will be different since the interaction between the neutron and the \({ }^{3} \mathrm{He}\) is spin-dependent. Neutrons with the spin state aligned with the \({ }^{3} \mathrm{He}\) spins will be absorbed more effectively compared to those in the opposite spin state.

Step-by-step solution

Unpolarized \({ }^{3} \mathrm{He}\)

When the neutrons in the beam are unpolarized, they have an equal number of spins in the +1/2 and -1/2 states. In the given context, an unpolarized \({ }^{3} \mathrm{He}\) would absorb neutrons equally from both spin states (since it's unpolarized).

Polarized \({ }^{3} \mathrm{He}\)

When \({ }^{3} \mathrm{He}\) is polarized, the spins of the neutrons in the nucleus of the \({ }^{3} \mathrm{He}\) are all aligned in one direction. This means that the effective potential energy due to the interaction between the two particles (neutron and \({ }^{3} \mathrm{He}\)) will depend on the relative spin orientations of the neutron and \({ }^{3} \mathrm{He}\).

Neutron absorption

If the \({ }^{3} \mathrm{He}\) is polarized, the absorption of neutrons will be different based on their spins since we have a spin-dependent interaction. One of the spin states may have higher absorption, meaning it is more favorable energetically. This can be explained using the Pauli Exclusion Principle, which states that two identical fermions (in our case, neutrons) cannot occupy the same quantum state simultaneously.

Two spin states absorption

When the polarized \({ }^{3} \mathrm{He}\) interacts with the unpolarized neutron beam, neutrons in the spin state that is aligned with the \({ }^{3} \mathrm{He}\) spins are more likely to be absorbed as they have lower potential energy compared to those in the opposite spin state. In this case, the absorption of neutrons in the aligned spin state (with \({ }^{3} \mathrm{He}\) spins) will be higher than those in the opposite spin state. In conclusion, if the \({ }^{3} \mathrm{He}\) is polarized, the same number of neutrons in the unpolarized neutron beam will not be absorbed from both spin states. The absorption of neutrons in each of the two spin states of the unpolarized neutron beam will be different since the interaction between the neutron and the \({ }^{3} \mathrm{He}\) is spin-dependent. The neutrons with the spin state aligned with the \({ }^{3} \mathrm{He}\) spins will be absorbed more effectively compared to those in the opposite spin state.