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Once a muonic atom is created, various reactions can take place, which
include muon decay, elastic scattering, inelastic scattering, muon
transfer, muonic molecular formation followed by fusion or back decay, and
muon capture (after transfer to a heavy element). Some of the theoretical
input for density dependent reactions is given in terms of cross sections,
but it is convenient to convert them to rates. For the process *i*:

where *v* _{rel} is the relative velocity between the projectile and the
target,
and *n*_{i} is the number density of the relevant target species.
Inputs for other reactions are given by rates, some such as molecular
formation, dependent on the density and/or collision velocity, while
others such as muon decay independent of the above.
The simulation was done in an energy dependent manner, and for a given
state of the projectile with a particular energy, the probability for *i*th
reaction
,
out of *m* possible different reactions, is given
by:

where
is the reduced rate for each reaction,
given by Eq. 5.1 for cross section inputs, or by

for input by rates, where *C*_{i} is the relative
concentration of relevant species,
is the target number
density in the unit of liquid hydrogen atomic density
cm^{-3}, and
is the rate normalized to *n*_{0}.
The processes such as muon decay and nuclear muon capture are independent
of the target concentration *C*_{i}. For muon decay, for example,
is
,
where
is the free muon decay
rate and *Q*_{k} the Huff factor^{} in the *k*th element.

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