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In gases, Mu is always formed in a collision between the µ+ and an atom or molecule from which it “steals” an electron. This may take place “promptly” at high energy (~keV) or “delayed” (thermally, with impurity species). Ionization products from the muon's track do not play a role.
In metals, the µ+ is screened by conduction electrons; there is no Mu in the usual sense of a freely evolving 2-spin system. Ionization etc. “heals” before it can affect the µSR signal.
In semiconductors & insulators (including most liquids) Mu is most likely to be formed by capture of radiolysis electrons by the µ+ that created them in its ionization track (DMF). Some “prompt” Mu formation may also occur.
The electronic structure of Mu in matter is quite varied. In gases, liquids and insulators it resembles the free Mu atom in vacuum, but in semiconductors there are Mu states ranging from isotropic, deeply bound Mu through intermediate bond-centred Mu* to true “shallow” states in which the electron is entirely “out in the lattice”.
Mu- may form in many oxides like calcite; this would explain the absence of Mu.
So far (to my knowledge) no Mu signals have been detected in the complicated oxide insulators where µSR is frequently applied to studies of exotic magnetism. This is peculiar, in view of the clear evidence for DMF in almost all simple oxide insulators like quartz and sapphire, and in the ionic alkali halides.