Nilsson Model Calculations of Mu Capture Rates in2s−1dNuclei

Theoretical calculations of mu capture rates in $2s-1d$ nuclei are compared with experiment in the hope of elucidating the coupling constants of the interaction. Working from Primakoff's closure-approximation expression for the total average capture rate, the nuclear matrix element is treated in the context of the Nilsson unified model. A Hill-Wheeler integration must be performed to avoid extraneous coordinates in the $A$-particle wave function. The one- and two-particle parts of the matrix element are broken up into the various shell contributions, since all of the angular momentum properties reside in the shell wave function for the nucleons outside the ${\mathrm{O}}^{16}$ core. The closed-shell matrix elements are easily treated with standard angular-momentum techniques. The method for reducing the outer-shell matrix elements to a form amenable to evaluation by a computer is given in an appendix. Radial integrals are obtained from the Ford-Wills muon wave functions. The average neutrino momentum $\overline{\nu }$ is chosen on the basis of Kaplan's Fermi-gas model for the capture process and the subsequent comparison with neutron evaporation rates. The choice of nuclear parameters for ${\mathrm{F}}^{19}$, ${\mathrm{Ne}}^{20}$, ${\mathrm{Si}}^{28}$, ${\mathrm{Cl}}^{35}$, and ${\mathrm{Cl}}^{37}$ is discussed and numerical results are given. Comparing with experimental rates, one cannot exclude the possibility that the Fermi part of the interaction is absent. If a $V-A$ theory is assumed, however, we conclude the induced pseudoscalar coupling is probably present. The induced pseudoscalar with the "wrong" sign, $g_P=-8\mathrm{}{g}_A$, is definitely excluded, and the "large" pseudoscalar, $g_P=16\mathrm{}{g}_A$, seems to fit the data better at $\overline{\nu }=0.75\mathrm{}$.