Basics of Muon Physics
Introduction to µSR --> here
Basic Muon Properties
The Muon
Follow the above link for a very brief summary of the muon's properties such as charge, spin, mass etc. The figure also shows abbreviated cartoons of the reactions in which muons are created and destroyed, described in more detail below. If you want some really detailed and authoritative information about the properties of the muon, go visit the Lepton page of the Particle Data Group, where you can find essentially everything that is known about elementary particles!
Pion Decay
Muons are produced in various high energy processes and elementary particle decays such as kaon decay, but µSR requires low energy muons in order to stop the beam in samples of convenient thickness, and these are available in the required intensities only from the ordinary two-body decay of charged pions, 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. The most remarkable feature of positive pion decay is that it maximally violates Parity symmetry, causing the µ+ to be emitted with perfect spin polarization. This is the greatest advantage of µSR as a magnetic resonance technique: whereas NMR and ESR rely upon a thermal equilibrium spin polarization, usually achieved at low temperatures in strong magnetic fields, µSR begins with a perfectly polarized probe, regardless of conditions in the medium to be studied. It also implies that muon spin degrees of freedom usually start their evolution as far from thermal equilibrium as conceivable.
Most µ+ beams today are literally emitted from positive 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.
Muon Decay
The propensity of the muon decay positron to be emitted along the spin of the µ+ is another consequence of P-violation in the weak interaction which allows us to read out the information encoded in the evolution of an initially polarized muon spin ensemble. The information is delivered to the experimenter in the form of rather high energy (up to 52 MeV) positrons which readily penetrate sample holders, cryostats or ovens and the detectors used to establish the time and direction of the muon decay.
The Asymmetry
The decay probability of the muon depends as shown here upon the fraction x of the maximum possible total relativistic e+ energy of 52.83 MeV and the angle between the muon spin direction and the direction of e+ emission.
The asymmetry factor a increases monotonically with the e+ energy and is 100% for the maximum energy. Note that a changes sign at low energy; however, very few positrons are emitted with such low energies (see above) and those which are will usually not be detected.
In any real experiment, some of the lower energy positrons do not penetrate intervening material to trigger the detectors, or are "curled up" by applied magnetic fields, so that the efficiency f(x) for detecting positrons is energy dependent, forcing an integration over E(x) a(x) f(x) dx to obtain the average asymmetry. This, combined with the finite solid angle of any real detectors, renders the experimental maximum asymmetry Ao an empirical parameter to be determined by measurement on a sample known not to produce any muon depolarization but otherwise identical in every respect to that under investigation. Obviously, this calibration can never be perfect; in general one should not believe any absolute calibration of Ao to better than about 5% of itself.
It is perhaps unfortunate that a is traditionally known as the "asymmetry" (rather than perhaps the "anisotropy") since this term does not connote polarization (of a spin ensemble) to most people in the magnetic resonance community. However, at this point we are stuck with the term, much as we are stuck with the technical misnomer "muonium."
