State densities of heavy nuclei in the static-path plus random-phase approximation

P. Fanto and Y. Alhassid

Phys. Rev. C 103, 064310 – Published 10 June 2021

Abstract

Nuclear state densities are important inputs to statistical models of compound-nucleus reactions. State densities are often calculated with self-consistent mean-field approximations that do not include important correlations and must be augmented with empirical collective enhancement factors. Here, we benchmark the static-path plus random-phase approximation (SPA + RPA) to the state density in a chain of samarium isotopes ${}^{148 – 155}{Sm}$ against exact results (up to statistical errors) obtained with the shell-model Monte Carlo (SMMC) method. The SPA + RPA method incorporates all static fluctuations beyond the mean field together with small-amplitude quantal fluctuations around each static fluctuation. Using a pairing plus quadrupole interaction, we show that the SPA + RPA state densities agree well with the exact SMMC densities for both the even- and odd-mass isotopes. For the even-mass isotopes, we also compare our results with mean-field state densities calculated with the finite-temperature Hartree-Fock-Bogoliubov (HFB) approximation. We find that the SPA + RPA repairs the deficiencies of the mean-field approximation associated with broken rotational symmetry in deformed nuclei and with the violation of particle-number conservation in the pairing condensate. In particular, in deformed nuclei the SPA + RPA reproduces the rotational enhancement of the state density relative to the mean-field state density.

Received 20 September 2020
Revised 28 January 2021
Accepted 30 April 2021

DOI:https://doi.org/10.1103/PhysRevC.103.064310

Physics Subject Headings (PhySH)

Energy levels Nuclear many-body theory

150 ≤ A ≤ 189

Many-body techniques Shell model Thermal & statistical models

Nuclear Physics

Authors & Affiliations

P. Fanto^* and Y. Alhassid ^†

Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06511, USA

^*paul.fanto@yale.edu
^†yoram.alhassid@yale.edu

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Vol. 103, Iss. 6 — June 2021

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Images

Figure 1
The canonical entropy as a function of inverse temperature $β$ for the even-mass samarium isotopes ${}^{148, 150, 152, 154}{Sm}$ . For each isotope, the SPA + RPA entropy (orange circles) is compared with the SMMC entropy (blue squares) and the HFB entropy (green dashed-dotted lines). The error bars on the SMMC and SPA + RPA results are statistical errors due to the Monte Carlo sampling. The insets show an expanded scale at large values of $β$ .
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Figure 2
The state density $ρ$ as a function of excitation energy $E_{x}$ for the even-mass samarium isotopes. The SPA + RPA state density (orange circles) is compared with the SMMC density (blue squares) and the HFB density (green dashed-dotted lines) for each isotope. The gray dashed lines show the composite formula (45) for ${}^{148, 150}{Sm}$ and the BBF (46) for ${}^{152, 154}{Sm}$ , with the parameters from Table 2 (see text). Experimental neutron resonance data (red triangles) and low-energy level counting data (black histograms) are also shown. The error bars show statistical errors in the SPA + RPA and SMMC results.
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Figure 3
The energy $E$ (top panel) and excitation energy $E_{x}$ (bottom panel) as functions of inverse temperature $β$ for ${}^{152}{Sm}$ . The SPA + RPA energies (solid orange circles) are compared with the SMMC energies (solid blue squares) and HFB energies (green dashed-dotted lines). The gray dashed-dotted line in the bottom panel indicates the neutron separation energy $S_{n}$ .
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Figure 4
The enhancement factor $K = ρ / ρ_{HFB}$ for the SPA + RPA (orange circles), the SMMC (blue squares), and the neutron resonance data (red triangles) for the even-mass samarium isotopes. The error bars indicate statistical errors from the Monte Carlo sampling.
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Figure 5
The canonical entropy as a function of inverse temperature $β$ for the odd-mass samarium isotopes ${}^{149, 151, 153, 155}{Sm}$ . The SPA + RPA entropy (orange circles) is compared with the SMMC entropy (blue squares) for each isotope. The error bars show statistical errors arising from the Monte Carlo sampling. The insets show an expanded scale at large values of $β$ .
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Figure 6
The state density $ρ$ as a function of excitation energy $E_{x}$ calculated with the SPA + RPA (orange circles) and SMMC (blue squares) for the odd-mass samarium isotopes ${}^{149, 151, 153, 155}{Sm}$ . The BBF calculated with parameters obtained from fitting the SPA + RPA ground-state energies (gray dashed lines) and fitting the SMMC ground-state energies (black dotted lines) are also shown. Experimental neutron resonance data (red triangles) and low-energy level counting data (black histograms) are shown as well.
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Physical Review C

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