Exact expressions for the number of levels in single-j orbits for three, four, and five fermions
Abstract
We propose closed-form expressions of the distributions of magnetic quantum number and total angular momentum for three and four fermions in single- orbits. The latter formulas consist of polynomials with coefficients satisfying congruence properties. Such results, derived using doubly recursive relations over and the number of fermions, enable us to deduce explicit expressions for the total number of levels in the case of three-, four-, and five-fermion systems. We present applications of these formulas, such as sum rules for six- and nine- symbols, obtained from the connection with fractional-parentage coefficients, an alternative proof of the Ginocchio-Haxton relation, or cancellation properties of the number of levels with a given angular momentum.