Gross Theory of Nuclear β-Decay

TLDR: In this paper, a theory for the gross properties of the nuclear a-decay is developed, where summations over final states are replaced by integrations, and the average of the squared absolute value of a nuclear matrix element times the final level density is investigated instead of individual matrix elements.

Abstract: A theory for the gross properties of the nuclear a-decay is developed. In order to treat the gross features, summations over final states are replaced by integrations, and the average of the squared absolute value of the nuclear matrix element times the final level density Cthis quantity is denoted by IMaCE) \2 where -E is the decay energy) is investigated instead of individual matrix elements. First, the general slowness of the allowed a-decay is qualitatively demonstrated by the use of sum rules. Next, a model is set up in order to make quantitative calculations. In this model, an existence of "single-nucleon energies E" is assumed, and each nucleon is assumed to make a "transition" with probability DQ CE, E) as a result of the opera­ tion of the single-particle a-decay operator. IMaCE) 12 is given as an integral with respect to E, whose integrand is the product of DaCE, E) and the distribution function of nucleons over E. Some interference effects are neglected, and the exclusion principle is introduced not in the integrand but in the lower limit of the integration domain. The half-lives of allowed a-decays are calculated with this model. At first, the Fermi gas model is used to evaluate the ene;gy distribution of the single-nucleons. With some trial forms of Da CE, E), a reasonable agreement with experiment is obtained for odd-mass nuclei, especially for nuclei with high Q-values. Secondly, the even-odd mass difference is taken into account in a simplified way to refine the treatment of even-mass nuclei. The results show that the majority of allowed a-decays can be explained to a considerable degree by the gross theory which is utterly dif­ ferent from current theories of a-decay.