μ-Meson Capture and Nuclear Shell Structure

The rate of absorption of negative $\mu$-mesons in lead and in calcium is calculated, using an interaction of the charge-exchange type described by Wheeler. A shell model of the nucleus is used in making explicit calculations of the various nuclear matrix elements. The ratio of the absorption rate in lead to that in calcium is found to be 6.1, in contrast to the value 14.3 predicted by the simpler Wheeler theory. Another result is that the absorption rate in mercury should be greater than that in lead by 20 percent in spite of its smaller atomic number. These results are in fair agreement with recent experimental measurements. The coupling constant for the charge-exchange reaction is $g\sim 3\times {10}^{-49\mathrm{}}$ erg ${\mathrm{cm}}^3$, approximately the same as the coupling constants for $\beta$-decay and $\mu$-decay.