Exact expressions for the number of levels in single-$j$ orbits for three, four, and five fermions

We propose closed-form expressions of the distributions of magnetic quantum number $M$ and total angular momentum $J$ for three and four fermions in single-$j$ orbits. The latter formulas consist of polynomials with coefficients satisfying congruence properties. Such results, derived using doubly recursive relations over $j$ 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-$j$ and nine-$j$ 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.

Figure 1

Magnetic quantum number distribution $P(M;j,4)$ for a 4-fermion system with spin $j=7/2$. The red (resp. blue) curve is the approximation (3.52) with $\pi \left(n\right)=0$ (resp. 1). The black circles are the exact values.

Figure 2

Magnetic quantum number distribution $P(M;j,4)$ for a four-fermion system with spin $j=15/2$. See Fig. 1 for details.

Figure 3

Angular momentum distribution $Q(J;j,4)$ for a four-fermion system with spin $j=7/2$. The red (resp. blue) curve is the approximation (3.60) with $\pi \left(n\right)=0$ (resp. 1). The black circles are the exact values.

Figure 4

Angular momentum distribution $Q(J;j,4)$ for a four-fermion system with spin $j=15/2$. See Fig. 3 for details.