 Solution

For $-z$, use a resonant half-turn pulse,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 givesThe 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.