A8 - neutron polarization

each neutron has a spin, $s = 1 / 2$ , and an angular momentum, $\pm ℏ / 2$
for a bean pf neutrons, each having a spin vector, ${\vec{s}}_{n}$ , the polarization is the ensemble average over all the neutron spin vectors, normalized to their modulus:

\vec{P} = \frac{⟨ \vec{s_{n}} ⟩}{1 / 2} = 2 ⟨ {\vec{s}}_{n} ⟩

if an external field is applied, there can only be two possible orientations, spin up and down, with $N_{+}$ and $N_{-}$ in each state

P = \frac{N_{+} - N_{-}}{N_{+} + N_{-}} = \frac{F - 1}{F + 1}

where, $F = N_{+} / N_{-}$ is called the flipping ratio