Analytic proof of partial conservation of seniority in $j=9/2$ shells

TL;DR

This paper provides a complete analytic proof for the partial conservation of seniority in j=9/2 shells, revealing a unique symmetry related to specific quantum states and matrix element properties.

Contribution

It extends previous numerical findings by deriving an explicit analytic proof of seniority conservation in j=9/2 shells, highlighting the role of fractional parentage coefficients.

Findings

All relevant matrix elements are proportional to one-particle cfp's.

The partial conservation is due to the unique properties of v=3 and 5 states in j=9/2 shells.

The proof reveals a partial dynamic symmetry specific to this shell.

Abstract

A partial conservation of the seniority quantum number in $j=9/2$ shells has been found recently in a numerical application. In this paper a complete analytic proof for this problem is derived as an extension of the work by Zamick and P. Van Isacker [Phys. Rev. C 78 (2008) 044327]. We analyze the properties of the non-diagonal matrix elements with the help of the one-particle and two-particle coefficients of fractional parentage (cfp's). It is found that all non-diagonal (and the relevant diagonal) matrix elements can be re-expressed in simple ways and are proportional to certain one-particle cfp's. This remarkable occurrence of partial dynamic symmetry is the consequence of the peculiar property of the $j=9/2$ shell, where all $v=3$ and 5 states are uniquely defined.