Number of states with given spin I and isospin T for n fermions in a j orbit*

Abstract

In this paper we study formulas of the number of states with given total spin I and isospin T, for n nucleons in a single-j shell, denoted by DIT (j, n). The Talmi's recursion formulas for the number of states with given z-axis projection of total spin are generalized by further considering the isospin couplings, and are applied to derive explicit formulas of DIT (j, n).

Exact expressions of the distributions of total magnetic quantum number and angular momentum in single- j orbits: A general technique for any number of fermions

“…It must be noticed that this formula do not apply for negative M, i.e., if p > 2 j. From the relation (27), setting p = 2l, one gets the total number of levels for four fermions with integer j,…”

“…The sum (29) involves in its generic term the four-fermion distribution P( j − p; j, 4) with j = j − s/2, p = p − 4s + 1. We may use the expression (27). If p 0, then one has also p 0 so that the relation ( 27) applies, ignoring the part factored by H (p).…”

“…the difference P 5,2 (p − 2 j) − P 5,2 (p − 2 j − 3) depends only on four-fermion functions P 4,0 , P 4,1 , P(2 j − p + 2; j − 1/2, 4). When evaluating this last element, one must take care that 2 j − p + 2 is negative and that the expression (27) does not hold for negative M. We use the M → −M symmetry and the known value for the four-fermion distribution, P(2 j − p + 2; j − 1/2, 4) = P(p − 2 j − 2; j − 1/2, 4) = P 4,0 (6 j − p) − P 4,1 (4 j + 1 − p).…”

“…In a general way, the relations established for the piecewise polynomials can be implemented in an algorithm that would enable one to determine the distribution of angular momentum M whatever the number of fermions. The techniques presented here can be generalized to the case of two angular momenta, for instance to determine the number of states with a given total spin I and isospin T for N nucleons in a singlej shell [27].…”

“…In the N = 3 case, an alternative derivation much simpler than the one previously published and relying on the use of fractional parentage coefficients [26] was obtained. In a similar approach, Yin and Zhao generalized Talmi's recursion formulas by further considering the isospin couplings and derived explicit formulas of the number of states with a given total spin I and isospin T for N nucleons in a singlej shell [27].…”