Decay Rates of Bound Negative Muons

Abstract

The decay rate of negative muons bound to nuclei of atomic number $Z$, ${\ensuremath{\Lambda}}_{d}(Z)$, has been investigated experimentally by two independent methods: (a) the "sandwich" method, and (b) the "calibrated efficiency" method. Both methods are based on the fact that the negatron yield per muon, ${y}^{\ensuremath{-}}(Z)$, is proportional to $\frac{{\ensuremath{\Lambda}}_{d}(Z)}{{\ensuremath{\Lambda}}_{t}(Z)}$, where ${\ensuremath{\Lambda}}_{t}(Z)$ is the total disappearance rate of negative muons for element $Z$, and are designed to avoid absolute measurements of ${y}^{\ensuremath{-}}(Z)$. In method (a), ${\ensuremath{\mu}}^{\ensuremath{-}}$ are stopped in a multilayer "sandwich" target made by alternately stacking sheets of two elements $Z$, ${Z}^{\ensuremath{'}}$, and the resultant ${e}^{\ensuremath{-}}$ time distribution is decomposed into components due to $Z$ and ${Z}^{\ensuremath{'}}$. The ratio of muon stops in $Z$ and ${Z}^{\ensuremath{'}}$ is established empirically; knowing ${\ensuremath{\Lambda}}_{d}({Z}^{\ensuremath{'}})$, ${\ensuremath{\Lambda}}_{d}(Z)$ can be computed. This method was applied to Al, Fe, Zn, Cd, Mo, W, and Pb. In method (b) ${\ensuremath{\mu}}^{\ensuremath{-}}$ and ${\ensuremath{\mu}}^{+}$ of identical range distributions are stopped in a given target, and the ${e}^{+}$ yield, ${y}^{+}$, is used as a calibration of the ${e}^{\ensuremath{-}}$ counting efficiency. This method has been applied to C, Ca, Ti, V, Mn, Fe, Co, Ni, Zn, I, and Pb. The sources of error of either method are discussed in detail. The results indicate:(1) In the range $20<Z<30$, ${\ensuremath{\Lambda}}_{d}(Z)>{\ensuremath{\Lambda}}_{d}(0)$, i.e., the bound decay rate exceeds the vacuum (i.e., ${\ensuremath{\mu}}^{+}$) decay rate; ${\ensuremath{\Lambda}}_{d}(Z)$ presents a sharp peak near $Z=26$.(2) For $Z>30$, one finds ${\ensuremath{\Lambda}}_{d}(Z)<{\ensuremath{\Lambda}}_{d}(0)$, i.e., the decay is inhibited by binding. The effect is very marked for the heaviest elements, e.g., $\frac{{\ensuremath{\Lambda}}_{d}(82)}{{\ensuremath{\Lambda}}_{d}(0)}=0.34\ifmmode\pm\else\textpm\fi{}0.04$.These results are compared with the predictions of simplified theoretical models. The peak near $Z=26$ is tentatively attributed to the Coulomb enhancement of the outgoing electron wave function at the point of decay.