# Total number of $J$ levels for identical particles in a single-$j$ shell using coefficients of fractional parentage

## TL;DR

This paper derives analytical formulas for counting total $J$ levels of three and four identical fermions in a nuclear shell, using sum rules for fractional parentage and special identities without involving $6j$ or $9j$ symbols.

## Contribution

It introduces new analytical expressions for total $J$ levels in nuclear shells, utilizing sum rules and identities that bypass complex angular momentum coupling symbols.

## Key findings

- Derived formulas for three and four fermions in a shell
- Used sum rules and identities to simplify calculations
- Provides a new approach avoiding $6j$ and $9j$ symbols

## Abstract

Analytical expressions of the total number of $J$ levels for three and four fermions in a nuclear $j^n$ shell are provided. The formulas were derived using a combination of sum rules for coefficients of fractional parentage, and "unusual" identities, i.e. which do not contain the weighting factor $(2J + 1)$ involving $6j$ and $9j$ symbols.