= Solution
For $-z$, use a resonant half-turn pulse,
$$
\boxed{\tau_\pi=\pi/\Omega.}
$$
The up amplitude vanishes because $\cos(\Omega\tau_\pi/2)=0$, while the down amplitude has unit magnitude. Its <phase> is a common <phase> of the final state, so any RF <phase> gives
$$
\boxed{|\psi(\tau_\pi)\rangle\sim|\downarrow\rangle,
\qquad \langle s_z\rangle=-\hbar/2.}
$$
The final population inversion is insensitive to a free-precession <phase>, unlike the orientation of a transverse <superposition>. This is a <Rabi pi pulse> with the drive-amplitude convention specified above.
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