Distribution of nuclei in equilibrium stellar matter from the free-energy density in a Wigner-Seitz cell

G. Grams, S. Giraud, A. F. Fantina, and F. Gulminelli

Phys. Rev. C 97, 035807 – Published 29 March 2018

Abstract

The aim of the present study is to calculate the nuclear distribution associated at finite temperature to any given equation of state of stellar matter based on the Wigner-Seitz approximation, for direct applications in core-collapse simulations. The Gibbs free energy of the different configurations is explicitly calculated, with special care devoted to the calculation of rearrangement terms, ensuring thermodynamic consistency. The formalism is illustrated with two different applications. First, we work out the nuclear statistical equilibrium cluster distribution for the Lattimer and Swesty equation of state, widely employed in supernova simulations. Secondly, we explore the effect of including shell structure, and consider realistic nuclear mass tables from the Brussels-Montreal Hartree-Fock-Bogoliubov model (specifically, HFB-24). We show that the whole collapse trajectory is dominated by magic nuclei, with extremely spread and even bimodal distributions of the cluster probability around magic numbers, demonstrating the importance of cluster distributions with realistic mass models in core-collapse simulations. Simple analytical expressions are given, allowing further applications of the method to any relativistic or nonrelativistic subsaturation equation of state.

Received 27 September 2017
Revised 22 December 2017

DOI:https://doi.org/10.1103/PhysRevC.97.035807

Physics Subject Headings (PhySH)

Equations of state Equations of state of nuclear matter Novae & supernovae Nuclear astrophysics Nuclear density functional theory

Nuclear Physics

Authors & Affiliations

G. Grams¹, S. Giraud², A. F. Fantina², and F. Gulminelli³

¹Universidade Federal de Santa Catarina, Brazil
²Grand Accélérateur National d'Ions Lourds (GANIL), CEA/DRF - CNRS/IN2P3, Bvd Henri Becquerel, 14076 Caen, France
³LPC (CNRS/ENSICAEN/Université de Caen Normandie), UMR6534, 14050 Caen Cédex, France

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 97, Iss. 3 — March 2018

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Nuclear contribution to the Gibbs free energy, with and without the rearrangement term, for two different thermodynamic conditions, as a function of the number of nucleons in the dense phase $A_{r}$ : $n_{B} = 3 \times 10^{- 6} {fm}^{- 3}, T = 0.74$ MeV, $Y_{e} = Y_{p} = 0.43$ (left panel) and $n_{B} = 1.22 \times 10^{- 2} {fm}^{- 3}, T = 4.13$ MeV, $Y_{e} = Y_{p} = 0.31$ (right panel). The fraction $x_{i} = Z_{r} / A_{r}$ is fixed to the one of the most probable nucleus. The dots (squares) correspond to the most probable nucleus obtained without (with) the rearrangement term; the crosses indicate the $A_{r}$ of the average nucleus obtained with the LS EoS. See text for details.
Reuse & Permissions
Figure 2
Ratio $x_{i} = Z / A$ corresponding to the minimum of ${\tilde{G}}_{nuc}$ at a fixed $A$ , as a function of $A$ , for two thermodynamic conditions: $n_{B} = 3 \times 10^{- 6} {fm}^{- 3}, T = 0.74$ MeV, $Y_{e} = 0.43$ (solid line) and $n_{B} = 1.22 \times 10^{- 2} {fm}^{- 3}, T = 4.13 MeV, Y_{e} = 0.31$ (dashed line). Triangles indicate the values obtained with the LS EoS.
Reuse & Permissions
Figure 3
Baryon number density inside the nucleus, $n_{i}$ , versus $A$ for three thermodynamic conditions: (1) $n_{B} = 3 \times 10^{- 6} {fm}^{- 3}, T = 0.74$ MeV, $Y_{e} = 0.43$ , (2) $n_{B} = 2 \times 10^{- 3} {fm}^{- 3}, T = 2.3$ MeV, $Y_{e} = 0.34$ , and (3) $n_{B} = 1.22 \times 10^{- 2} {fm}^{- 3}, T = 4.13$ MeV, $Y_{e} = 0.31$ . Solid lines correspond to the solution of Eq. (43), dots correspond to $n_{i}$ for the average nucleus predicted by the LS EoS, and dashed lines represent the cluster distribution (in arbitrary units).
Reuse & Permissions
Figure 4
Distribution of nuclei predicted by the NSE model for the thermodynamic condition $n_{B} = 8.6 \times 10^{- 4} {fm}^{- 3}, T = 1.83$ MeV, $Y_{e} = 0.36$ . (b) shows the contour plot of the cluster probabilities (red to blue, from more to less abundant), while the black dot corresponds to the average nucleus ( $Z_{SNA} = 38, N_{SNA} = 62$ ) obtained with the LS EoS. (a) displays the probability distributions for $Z = 32, 38$ , and 41 (from left to right curve), while (c) displays the probability distribution for $N = 40, 62$ , and 80 (from bottom to top curve).
Reuse & Permissions
Figure 5
$Z$ (left panel) and $N$ (right panel) along the collapse trajectory (time is given in milliseconds), at the center of the star. Dashed lines correspond to the most probable $Z$ (left panel) and $N$ (right panel) obtained with the LS mass functional, while solid lines correspond to the variance. Red dots represent the most probable $Z$ (left panel) and $N$ (right panel) obtained implementing the HFB-24 mass model. Insets show a zoom of the final part of the plot. Arrows indicate four chosen points along the trajectory before the core bounce. See text for details.
Reuse & Permissions
Figure 6
Distribution of nuclei $(N, Z)$ for four chosen thermodynamic conditions along a collapse trajectory at the center of the star: (a) $n_{B} = 3.86 \times 10^{- 6} {fm}^{- 3}, T = 0.79$ MeV, $Y_{e} = 0.43$ , (b) $n_{B} = 3.35 \times 10^{- 4} {fm}^{- 3}, T = 1.51$ MeV, $Y_{e} = 0.38$ , (c) $n_{B} = 3.01 \times 10^{- 3} {fm}^{- 3}, T = 2.68$ MeV, $Y_{e} = 0.33$ , and (d) $n_{B} = 5.85 \times 10^{- 3} {fm}^{- 3}, T = 3.33$ MeV, $Y_{e} = 0.31$ . Contour lines correspond to the cluster normalized probabilities (red to blue, more to less probable) for the original LS model and for the HFB-24 nuclear mass model. Dots correspond to the average single nucleus predicted by the LS EoS. See text for details.
Reuse & Permissions

Physical Review C

covering nuclear physics