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Although muons are produced in a variety of high-energy processes and elementary particle decays such as K+ --> µ+ vµ decay [Yamazaki, 1984], µSR requires low energy muons in order to stop the beam in samples of convenient thickness (< 1 cm) and these are available in the required intensities only from ordinary two-body pion decay, from which the muon emerges (in the rest frame of the pion) with a momentum of 29.79 MeV/c and a kinetic energy of 4.119 MeV. The lifetime of a free charged pion is 26.03 ns.
Most µ+ beams today are literally emitted from pion decay at rest in the surface layer of the primary target where the pions themselves are produced by collisions of high energy protons with target nuclei - hence the common mnemonic name, surface muons [Bowen, 1985]. Unfortunately, this mode is not available for negative muons because a negative pion stopping in the production target almost always undergoes nuclear capture from low-lying orbitals of pionic atoms before it has a chance to decay. This problem was actually solved long before the surface muon beam was invented, when many fundamental physics experiments with muons were primarily concerned with the ``heavy electron'' behaviour of the µ-: in so-called ``conventional'' muon channels, pions are allowed to decay in flight down a relatively long straight section where the decay muons are collected by axial or alternating-gradient magnetic fields; the muons emitted ``backward'' in the pion rest frame have quite different momenta from those emitted forward (or from the pions themselves), and can thus be selectively extracted by a bending magnet. The disadvantages of such backward muon beams are their relatively higher momentum (usually ~ 50-100 MeV/c), their larger momentum spread (and therefore lower stopping density) and their much larger phase space (and therefore lower luminosity). Today they are rarely used for µ+SR, but for µ-SR there is no alternative.
Numerous attempts have been made to produce ultra-slow muon beams, using a number of ingenious methods. That technology is beyond the scope of the present treatment (meaning I don't have time to put it in right now); but to date no one has produced large numbers of polarized muons of very low momentum, and without this the µSR applications will remain limited. But stay tuned! This will be the next qualitative beakthrough in the field!