4.1 Models of Muonium Formation

Muonium is a light hydrogen-like neutral atom composed of an electron bound to a positive muon (Mu = e$^- + \mu^+$), commonly produced when muons are implanted in non-metals including gases, liquids and solids ranging from semiconductors to fullerenes. One of the oldest unanswered questions in the field of $\mu{\cal SR}$ is ``How is muonium formed when a muon stops in one of these materials?'' The literature on this subject records a number of experiments devoted to answering this and the related question, ``What properties of the sample influence muonium formation?'' Until recently only reasoned guesses based on indirect evidence have been possible.

Following the first observation of muonium in highly purified water by $\mu{\cal SR}$ [7] the diamagnetic fraction (free $\mu^{+}$ and $\mu^{+}$-substituted molecules) was measured in the presence of electron scavengers [8]. The increase in diamagnetic fraction with the concentration of NO₃^- ions was attributed to an increased probability of thermal, unsolvated muons becoming hydrated and subsequently forming MuOH by fast proton transfer, instead of capturing an electron to form muonium. One model proposed to explained this behavior pictured the muon losing kinetic energy near the end of the track by the creation of free electrons, ions and radicals in a radiation spur [8,9]. The thermal muon in the vicinity of the terminal spur could then form muonium by simply capturing a free electron from among the spur products. This model was borrowed from the analogous theory of positronium formation when positrons are injected into condensed media [10].

Another model of muonium formation proposed that the eventual distribution of muons among various states is determined by processes that occur while the $\mu^{+}$is still losing its initial kinetic energy [18,11,12,13]. At high velocity (as from a surface muon beam) the muon should behave like any fast charged particle and undergo energy loss by Bethe-Bloch ionization of the medium; no significant amount of muonium should form until the kinetic energy has dropped to several tens of keV, where the muon velocity becomes comparable to the orbital velocity of electrons of the medium. Then charge exchange collisions become important as the muon undergoes a rapid series of several hundred electron pickup and stripping cycles, shedding energy each time atoms of the medium or the muonium is ionized.

At an energy of order 100 eV charge exchange is no longer dominant and the fraction of muonium at these energies is expected to be influenced by the relative electron affinities of the muon and atoms of the sample. In materials with ionization potentials smaller than that of muonium (13.5 eV) most muons are expected to emerge from this stage as hot muonium atoms. Further thermalization of both muons and muonium atoms will continue by elastic and inelastic collisions with neutral atoms and hot atom reactions which may, depending on the stopping medium, produce $\mu^{+}$-substituted molecules and/or molecular ions. The final distribution of muon charge states will be determined by the reactions that $\mu^{+}$ and Mu undergo in the last few steps. In low pressure rare gases there is a strong correlation between muonium formation and ionization potential of the gas, which seems to support this model.

The principal distinction between these two models lies chiefly in when and at what energy muonium is formed. ``Hot'' muonium formation is a ``prompt'' process, occurring during a time when the muon is rapidly losing energy, within a few tens of ps after entering a condensed sample. Muonium formation after the muon has come to thermal equilibrium with its surroundings requires time for electrons to diffuse to the muon, and is therefore termed ``delayed'' muonium formation.