2.1.2 Production of Spin Polarized Muons and $\mu {\cal SR}$

In conventional magnetic resonance experiment spin polarization is achieved by a combination of high field and low temperature, kT can be made on the order of or less than the relevant magnetic level splitting, and thermal equilibrium will ensure some spin-polarization. Other methods for polarizing spins include nuclear reactions, tilted foil methods, and optical excitation. In $\mu {\cal SR}$, the parity violating decay of the pion is the process by which the muon spin is polarized. The two features of $\pi^+$ decay that yield the spin-polarization of the product $\mu^+$ are i) it is 2 body final state ii) it is a weak decay.

The pion is a boson, so its initial angular momentum is zero; Hence the total final angular momentum of the products must also be zero. Because the neutrino is always produced in a helicity (spin$\cdot$momentum) state of -1, and, in the rest frame of the pion, the muon and neutrino are emitted ``back-to-back'', the muon must also be in a -1 helicity eigenstate, i.e. it is spin-polarized.

This scheme is used in practice in the following way:

• A beam of intermediate energy ($\sim$500 MeV) protons is trained on a production target of some light nucleus material such as carbon or beryllium.
• Nuclear reactions occur in the production target which produce positive pions. Some pions remain in the target, and some are emitted with net kinetic energy.
• The pions quickly decay, emitting muons via 2.3 The muons have a distribution of momenta some with high momentum from high energy pions decaying in flight, and some with low momentum muons that are produced by pions deep within the production target.
• Muons emitted in a particular direction are guided down a beamline to a Wein velocity filter where crossed electric and magnetic fields select a particular momentum. The momentum used corresponds to muons from pions which decay at rest on the surface of the production target. Such muons have a well-defined kinetic energy which is quite small (4.1 MeV) and are called ``surface muons''. Other muons (and positrons from muons that have already decayed) with different momenta are thus bent out of the beam direction, and play no role in the experiment.
• At low fields, the Wein filter (or ``separator'') can be used simply as a momentum selector, but at high fields, it also rotates the muon's spin. Typically two settings of the separator are used, one that rotates the spin very little, and one that rotates it by 90$^\circ$.
• After filtering the beam is focused electromagnetically on the target material of interest.

Once a beam of spin-polarized muons is produced in the above manner, it can be used in a $\mu {\cal SR}$ experiment. Such experiments are described in the following section.