Statistical spectroscopic calculation of expectation values and spin-cutoff factors

Recently we proposed a formalism, based in nuclear statistical spectroscopy, for efficient computation of nuclear level density, or densities of states, through a sum of partitioned binomial functions (SUPARB). In this Letter we extend the formalism to the calculation of locally averaged expectation values, with specific application to spin-cutoff factors and the angular momentum dependence of the nuclear density of states. Typeset using REVTEX 1 In a recent Letter [1] we proposed a computationally efficient method to compute nuclear level densities, based in the ideas of nuclear statistical spectroscopy [2–4]. While we refer interested readers to our original Letter for details, we briefly summarize here. Consider the density of states of a Hamiltonian Ĥ , ρ(E) = tr δ(E − Ĥ). (1) It is useful to partition the finite model space into subspaces, labeled by Greek letters α, β, etc., each with an associated projection operator Pα. Then one can define partial or configuration densities: ρα(E) = trPαδ(E − Ĥ). (2) The total density is the sum of the partial densities. (NB:We always include 2J + 1 degeneracies and so are formally considering state densities.) If the subspaces are single-particle configurations, e.g., (0d5/2) , (0d5/2) 2(1s1/2) , etc, then the partial, or configuration, moments up to fourth order for any system with Z protons and N neutrons in the valence space can be computed directly from the one+two-body matrix elements of Ĥ [5,6]. For any partition (configuration) α let dα = trPα be the dimension of the subspace α, and define the configuration average over the subspace to be 〈. . .〉α ≡ dα −1trPα(. . .). Then Ē(α) = 〈Ĥ〉α is the (configuration) centroid, γ(α) = 〈(Ĥ − Ēα)〉 α the configuration width, m3(α) = 〈(Ĥ − Ēα)〉α/γ(α) the scaled (dimensionless) third configuration moment, and m4(α) = 〈(Ĥ − Ēα)〉α/γ(α) the scaled fourth configuration moment. With these moments in hand, we model the partial densities as binomial distributions, following a recent suggestion of Zuker [7]. Starting with the binomial expansion of (1 +λ) and representing (