Total and Parity-Projected Level Densities of Iron-Region Nuclei in the Auxiliary Fields Monte Carlo Shell Model

We use the auxiliary-fields Monte Carlo method for the shell model in the complete (pf + 0g9/2)-shell to calculate level densities. We introduce parity projection techniques which enable us to calculate the parity dependence of the level density. Results are presented for 56Fe, where the calculated total level density is found to be in remarkable agreement with the experimental level density. The parity-projected densities are well described by a backshifted Bethe formula, but with significant dependence of the single-particle level-density and backshift parameters on parity. We compare our exact results with those of the thermal Hartree-Fock approximation. PACS numbers: 21.10.Ma, 21.60.Cs, 21.60.Ka, 27.40.+z Typeset using REVTEX 1 Nuclear level densities are important for theoretical estimates of nuclear reaction rates in nucleosynthesis. The sand r-processes that involve medium-mass and heavier nuclei are determined by the competition between neutron-capture and β-decay, and the neutroncapture cross-sections are strongly affected by the level density around the neutron resonance region. Reliable estimates of nuclear abundances often require accurate level densities. For example, the abundance of s-process nuclei with non-magic neutron number is (in the local approximation) inversely proportional to the neutron-capture cross-section [1] which in turn is proportional to the level density. Most conventional calculations of the nuclear level density are based on the Fermi gas model within the grand-canonical ensemble [2]. For a gas of free nucleons one obtains the well-known Bethe formula. A simple but useful phenomenological modification is often adopted, in which the excitation energy Ex is backshifted [3], giving a total nuclear level density of ρ(Ex) = g √ π 24 a 1 4 (Ex −∆) 5 4 e √ a(Ex−∆) (1) with g = 2. The backshift ∆ originates in pairing correlations and shell effects, while the parameter a is determined by the single-particle level-density at the Fermi energy. By adjusting the value of a for each nucleus, the backshifted Bethe formula (BBF) (1) fits well a large volume of experimental data. The value of the parameter, however, is not well understood; the Fermi-gas model grossly underestimates the value of a, and cannot account for its exact mass and nucleus dependence. Consequently, it is difficult to predict the level density to an accuracy better than an order of magnitude. Much less is known about the parity-dependence of the level density. The finite-temperature mean-field approximation [4] offers an improvement over the Fermi gas model but still ignores important two-body correlations, especially at low temperatures. In this paper we study the nuclear level density in the framework of the interacting shell model, in which the two-body correlations are fully taken into account within the model space. It should be noted, however, that the finite size of the model space limits the validity of such calculations to below a certain excitation energy. The size of the valence shells 2 required to describe the neutron-resonance region for medium-mass and heavier nuclei is too large for conventional diagonalization techniques to be practical. However, the recently proposed shell model Monte Carlo (SMMC) method [5] makes it possible to calculate thermal averages in much larger model spaces by using fluctuating auxiliary-fields. As shown below, these methods are particularly suitable for calculations of level densities. Nuclei in the iron region play a special role in nucleosynthesis. They are the heaviest that can be produced inside normal massive stars, and the starting point of the synthesis of heavier nuclei. These nuclei are in the middle of the pf -shell, and are just beyond the range of nuclei where conventional shell model techniques can be applied in a complete pf -shell model space [6,7]. Truncated shell calculations [8] were successfully used to describe the low-lying states in these nuclei. However, their neutron separation energy, typically Ex ∼ 5– 15 MeV, is too high to justify such truncation. The SMMC method was used to calculate thermal properties of Fe in a full pf -shell [9] with the Brown-Richter Hamiltonian. The Monte Carlo sign problem of this realistic interaction is overcome through the techniques of Ref. [10]. However, the statistical errors were too large to obtain accurate level densities. Furthermore, the energy range of interest in the iron region (Ex ∼ 5–15 MeV) contains negative-parity states which are not included in the pf -shell model space. In this letter we introduce parity-projection methods for the auxiliary fields, and use the SMMC within the full pf and 0g9/2-shell to calculate total and parity-projected level densities in the iron region. This model space is sufficient to describe both positiveand negative-parity states for excitation energies up to 20 MeV. To keep the statistical errors small, we construct an interaction which is free from the Monte Carlo sign problem, yet realistic enough to describe collective features that affect the level density. In particular we present results for Fe, for which experimental data are available. We adopt an isoscalar Hamiltonian of the form [11]