Delayed Muonium Formation

In insulators and semiconductors the radiolysis products left behind by a stopping µ+ include free electrons that can be attracted by the muon's positive charge and combine with it to form a muonium (Mu = µ+e-) atom. This process is known as delayed Mu formation.

Delayed Mu formation is readily distinguished on physical grounds from prompt Mu formation, in which the muon captures an electron while slowing down and is in the neutral atomic state as it thermalizes. Delayed Mu formation, by contrast, takes place after the muon has thermalized.

However, since the radiolysis electron often finds its way to the muon in a very short time, it is not always easy to distinguish between prompt and delayed Mu formation experimentally. Only when Mu formation is delayed by long enough for the initially diamagnetic µ+ spins to lag behind the fast-precessing prompt Mu spins is it possible to make a clear empirical distinction between the two from the µSR signals themselves. Such a "directly detectable" delay causes the Mu atoms to get out of phase, which produces both a reduction of the amplitude and a shift of the initial phase of the Mu precession signal.

On the other hand, it is often possible to distinguish prompt from delayed Mu formation by observing the effect of an applied electric field on the latter: let us assume provisionally that the muon continues in the "forward" direction after it produces the last radiolysis electron; then that electron is "behind" it and if we apply an electric field E in the "forward" direction (which we shall call the positive direction) then it will tend to pull apart the µ+ and the radiolysis electron. If E is stronger than the Coulomb field of the µ+ at the electron (which is always attractive), then the electron will be pulled away and muonium will not be formed. The geometry of the situation is illustrated in this Figure.

Naturally, not all radiolysis electrons are produced at the same distance from where the muon comes to rest. Those which are too close to be pulled away by E are captured by the muon, while those further away than some critical distance will escape. Since the electrons may also be "off to the side" relative to an axis through the muon parallel to E, we must consider (in 3 dimensions) the capture loci dividing the region from which the e- will be captured by the µ+ from the region where the e- will escape. The latter region is, of course, larger for a larger applied electric field.

Although we do not yet know the details of the spatial distribution of final radiolysis electrons about the stopped muon, the above considerations ensure that an applied E will influence the probability of delayed Mu formation. This is indeed the case. The first example was seen in solid nitrogen, where the electric field required to prevent Mu formation indicates that the mean distance between the muon and the last radiolysis electron is <R> ~ 50 nm, about 1000 times the radius of a hydrogen atom.

The result for s-N2 also shows a clear anisotropy between positive E (in the direction of the initial muon momentum) and negative E (in the opposite direction). This validates our provisional assumption that the muon stops (on average) downstream of its last radiolysis electron.

Jess H. Brewer
Last modified: Sat Nov 29 10:32:59 EST ±