Some Considerations Concerning the Nuclear Matrix ElementΣ|Bij|2in Beta Decay

Abstract

Evidence that the nuclear matrix element $\ensuremath{\Sigma}{|{B}_{\mathrm{ij}}|}^{2}$ contributes to first-forbidden transitions may be deduced from the electron-capture-positron-emission ratios observed in these transitions and from an analysis of the energy and $Z$ dependence of the coefficients of all the nuclear matrix elements involved. The analysis of these coefficients for the capture and for the emission processes shows that it is reasonable to expect that, except for $\ensuremath{\Sigma}{|{B}_{\mathrm{ij}}|}^{2}$, these transitions would be characterized by essentially the same capture-positron ratios as those characteristic of allowed transitions. Consequently the deviations of the observed ratios from the allowed values would be due to contributions from $\ensuremath{\Sigma}{|{B}_{\mathrm{ij}}|}^{2}$; and the magnitude of the deviation, in a particular case, together with lifetime information may be used to compute values for this matrix element and for the sum of the others. Values are thus computed from available experimental data for three 2- \ensuremath{\rightarrow} 2+ transitions, and these are compared with values derived directly from theory of Rose and Osborne. In each case the value of $\ensuremath{\Sigma}{|{B}_{\mathrm{ij}}|}^{2}$ computed from experiment is nearly as large as the theoretical $\ensuremath{\Sigma}{|{B}_{\mathrm{ij}}|}^{2}$ value calculated for nucleon spin change two and is considerably larger than those calculated for spin change zero or one; furthermore, the "observed" value for the sum of the other matrix elements is considerably smaller than the largest single matrix element calculated for spin change zero or one. These results are consistent with the conclusion that the nuclear states involved are of such a nature that the transitions proceed unhindered only via $\ensuremath{\Sigma}{|{B}_{\mathrm{ij}}|}^{2}\ensuremath{\Delta}j=2$. One arrives at the same conclusion on the basis of the shell model description of these nuclear states. For 2- \ensuremath{\rightarrow} 0+ transitions, which can be effected only by $\ensuremath{\Sigma}{|{B}_{\mathrm{ij}}|}^{2}$, the "observed" and theoretical values of the matrix element are in good agreement; and it is of interest to note that both the theoretical and "observed" values of $\ensuremath{\Sigma}{|{B}_{\mathrm{ij}}|}^{2}$ in each of these 2- \ensuremath{\rightarrow} 0+ transitions are about the same or smaller than the values for the 2- \ensuremath{\rightarrow} 2+ transition in the same nucleus.