Slowing-down times and stopping powers for∼2−MeVμ+in low-pressure gases

Abstract

The times taken by positive muons $({\ensuremath{\mu}}^{+})$ to slow down from initial energies in the range $\ensuremath{\sim}3$ to $1\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$, to the energy of the last muonium formation, $\ensuremath{\approx}10\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, at the end of cyclic charge exchange, have been measured in the pure gases ${\mathrm{H}}_{2}$, ${\mathrm{N}}_{2}$, $\mathrm{Ar}$, and in the gas mixtures $\mathrm{Ar}\text{\ensuremath{-}}\mathrm{He}$, $\mathrm{Ar}\text{\ensuremath{-}}\mathrm{Ne}$, $\mathrm{Ar}\text{\ensuremath{-}}{\mathrm{CF}}_{4}$, ${\mathrm{H}}_{2}\text{\ensuremath{-}}\mathrm{He}$, and ${\mathrm{H}}_{2}\text{\ensuremath{-}}{\mathrm{SF}}_{6}$, by the muon spin rotation $(\ensuremath{\mu}\mathrm{SR})$ technique. At $1\phantom{\rule{0.3em}{0ex}}\mathrm{atm}$ pressure, these slowing-down times, ${\ensuremath{\tau}}_{\mathrm{SD}}$, in $\mathrm{Ar}$ and ${\mathrm{N}}_{2}$, vary from $\ensuremath{\sim}14\phantom{\rule{0.3em}{0ex}}\mathrm{ns}$ at the highest initial energies of $2.8\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$ to $6.5\phantom{\rule{0.3em}{0ex}}\mathrm{ns}$ at $1.6\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$, with much longer times, $\ensuremath{\sim}34\phantom{\rule{0.3em}{0ex}}\mathrm{ns}$, seen at this energy in ${\mathrm{H}}_{2}$. Similar variations are seen in the gas mixtures, depending also on the total charge and nature of the mixture and consistent with well-established (Bragg) additivity rules. The times ${\ensuremath{\tau}}_{\mathrm{SD}}$ could also be used to determine the stopping powers, $dE∕dx$, of the positive muon in ${\mathrm{N}}_{2}$, $\mathrm{Ar}$, and ${\mathrm{H}}_{2}$, at kinetic energies near $2\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}$. The results demonstrate that the ${\ensuremath{\mu}}^{+}$ and proton have the same stopping power at the same projectile velocity, as expected from the historic Bethe-Bloch formula, but not previously shown experimentally to our knowledge for the muon in gases at these energies. The energy of the first neutralization collision forming muonium (hydrogen), which initiates a series of charge-exchanging collisions, is also calculated for $\mathrm{He}$, $\mathrm{Ne}$, and $\mathrm{Ar}$. The formalism necessary to describe the stopping power and moderation times, for either muon or proton, in three energy regimes---the Bethe-Bloch, cyclic charge exchange, and thermalization regimes---is developed and discussed in comparison with the experimental measurements reported here, and elsewhere. The slowing-down times through the first two regimes are controlled by the relevant ionization and charge-exchange cross sections, whereas the final thermalization regime is most sensitive to the forwardness of the elastic scattering cross sections. In this regime the slowing-down times (to $kT$) at nominal pressures are expected to be $\ensuremath{\lesssim}100\phantom{\rule{0.3em}{0ex}}\mathrm{ns}$.