## The BSkG3 model

BSkG3 is a large-scale model of nuclear structure: the "large-scale" in this sentence refers to the number of nuclei (several thousands!) but also to our ambition to describe as much of nuclear structure as possible within a single framework. On this page, we provide some more explanation of the basic structure of this model and a link to a table containing a large amount of calculated ground-state properties for thousands of nuclei.

The model is based on the concept of a nuclear energy density functional (EDF), which starts from the total energy of a nucleus:

E_{tot } = E_{HFB}+ E_{corr} ,

which is calculated microscopically from a mean-field wavefunction of the Hartree-Fock-Bogoliubov (HFB) type. By minimizing the total energy, we find a HFB many-body wavefunction that represents the nuclear ground state and is used to calculate all kinds of properties. Our search for this minimal-enegy state is very general: in order to grasp as much correlations among nucleons as we can, we allow our HFB states to break several symmetries. In this way, we account consistently for (i) nuclear triaxiality, (ii) left-right reflection asymmetry and even (iii) time-reversal breaking in odd-mass and odd-odd systems due to the unpaired nucleons. In addition, we represent such nuclear configurations numerically on a rather fine three-dimensional coordinate grid, guaranteeing us a (very high) numerical accuracy of about 100 keV on the absolute values of the total energy.

One of the observables we calculate is the atomic mass M(N,Z) of a nucleus composed of N neutrons and Z protons as

M(N,Z) = E_{tot} + N M_{N} + Z(M_{P} + M_{e}) - B_{el}(Z) .

where B_{el}(Z) is a simple analytical estimate for the binding energy of Z electrons and M_{N/P/e} are the masses of a free neutron, proton and electron respectively.

Digging a little bit deeper, the two contributions to the total energy are

E_{HFB } = E_{kinetic} + E_{Coulomb} + E_{COM,1} + E_{Skyrme} + E_{pairing} ,

E_{corr }= E_{Wigner} + E_{vib}+ E_{rot} + E_{COM,2} .

The first of these equations is the mean-field energy, which is consistently optimized and contains the kinetic energy of the nucleons, the Coulomb energy, the one-body part of the centre-of-mass correction as well as the
Skyrme energy and the pairing energy. We refer to the original publication (see below) for more details but mention that the form of E_{Skyrme} is not standard: compared to the traditional Skyrme EDF,
BSkG3 employs extended density-dependencies to give us the freedom to combine an excellent description of the properties of finite nuclei and neutron stars. The second equation contains a set of perturbative corrections that aim to account for complicated beyond-mean-field effects that are only semivariationally optimized in our calculations:
the spurious energy due to rotational and vibrational motion, the two-body centre-of-mass correction as well as a Wigner energy.

Taken together, these terms depend on 29 free parameters: twenty are related to E_{HFB} and nine to E_{corr}. We optimized these parameters to an enormous set of nuclear data: the 2457 known masses of nuclei with Z$\geq$8 and 884 known charge radii, but also included explicit constraints on pairing properties, the 45 known fission barriers of
actinide nuclei as well as the models predictions for infinite nuclear matter. For the latter, we imposed the symmetry energy coefficient $J= 31$ MeV and restricted the incompressibility modulus $K_{\nu}$ and the isoscalar effective mass $M^*/M$
to [230,250] MeV and [0.8,0.86], respectively. For completenes' sake, we list here the complete set of model parameters and their values.

Parameters of the Skyrme energy | |||
---|---|---|---|

t_{0} [MeV fm^{3} ] | -2325.35 | x_{0} | 0.558834 |

t_{1} [MeV fm^{5} ] | 749.82 | x_{1} | 2.940880 |

t_{2} [MeV fm^{5} ] | 0.01 | t_{2}x_{2} [MeV fm^{5}] | -432.256954 |

t_{3} [MeV fm^{3+3$\alpha$}] | 14083.45 | x_{3} | 0.628699 |

t_{4} [MeV fm^{5+3$\beta$}] | -498.01 | x_{4} | 5.657990 |

t_{5} [MeV fm^{5+3$\gamma$}] | 266.52 | x_{5} | 0.396933 |

W_{0} [MeV fm^{5}] | 119.735 | W$'$_{0} [MeV fm^{5}] | 78.988 |

$\alpha$ | 0.2 | $\beta$ | 0.8333333 |

$\gamma$ | 0.25 | ||

Parameters of the pairing energy | |||

$\kappa$_{n} | 123.2 | $\kappa$_{p} | 129.07 |

E_{cut} | 7.961 | ||

Parameters of the collective correction | |||

b | 0.810 | c | 7.756 |

d | 0.289 | $\ell$ | 5.499 |

$\beta$_{vib} | 0.827 | V_{W} [MeV] | -1.716 |

$\lambda$ | 437.20 | V^{'}_{W} [MeV] | 0.502 |

A_{0} | 37.801 |

The resulting parameter set, i.e. the BSkG3 model, achieves very good accuracy on the known nuclear masses with a root-mean-square (rms) of $\sigma(M)$ = 0.627 MeV. The rms for mass differences such as the neutron separation energy ($\sigma(S_N)$ = 0.442 MeV) and beta-decay energies ($\sigma(Q_{\beta})$ = 0.534 MeV) are even lower, while the global reproduction of nuclear charge radii is also excellent ($\sigma(R_c)$ = 0.0237 fm). The model also offers an unprecedented accuracy for fission properties: it reproduces both the inner and outer barries of 45 actinide nuclei (including odd-mass and odd-odd ones!) with an accuracy of $\sigma(E_I)$= 0.33 MeV and $\sigma(E_{II})$ = 0.51 MeV, respectively. This performance for finite nuclei is combined with excellent properties for neutron star studies: BSkG3 predicts an equation of state that is entirely compatible with the most recent theoretical and observational constraints on neutron stars, such as the existence of pulsars with masses above twice the solar mass.

We provide a table with the calculated ground state (g.s.) properties of 8486 nuclei, i.e. all systems within the predicted driplines up to Z=110. The formatting of this table should allow for easy data manipulation with a range of programming languages, in particular FORTRAN and Python. Its columns contain the following data:

- proton number Z
- mass number A
- $\beta_2$, the *total* quadrupole deformation.
- $\beta_4$, the *total* hexadecapole deformation.
- $R_c$, the root-mean-square charge radius in fm.
- $\gamma$, the triaxiality angle.
- $S_n$, the neutron separation energy in MeV.
- $S_p$, the proton separation energy in MeV.
- $Q_{\beta}$, the $\beta^-$ decay energy in MeV.
- $M$
_{cal}, the calculated atomic mass in MeV. - $M$
_{cal}- $M$_{cal}, the difference between experimental and calculated atomic mass in MeV. - $J$
_{exp}, the experimental spin of the g.s in units of $\hbar$. - $J$
_{th}, the calculated g.s. spin in units of $\hbar$. - $P$
_{exp}, the experimental parity of the g.s. - $P$
_{th}, the calculated parity of the g.s. - $\beta_{20}$, first component of the nuclear quadrupole deformation
- $\beta_{22}$, second component of the nuclear quadrupole deformation
- $\beta_{30}$, first component of the nuclear octupole deformation
- $\beta_{32}$, second component of the nuclear octupole deformation
- $S$
_{2n}, the two-neutron separation energy in MeV. - $S$
_{2p}, the two-proton separation energy in MeV. - $\Delta
^{(3)}_{n}, the three-point neutron gap in MeV. - $\Delta
^{(3)}_{p}, the three-point proton gap in MeV. - $\Delta
^{(5)}_{n}, the five-point neutron gap in MeV. - $\Delta
^{(5)}_{p}, the five-point proton gap in MeV. - $\langle r_c^4 \rangle^{1/4}$, the fourth moment of the nuclear charge radius in units of fm.
- $\mathcal{I}_1$, the smallest of three Belyaev moments of inertia in units of $\hbar^2$ MeV
^{-1}. - $\mathcal{I}_2$, the middle of three Belyaev moments of inertia in units of $\hbar^2$ MeV
^{-1}. - $\mathcal{I}_3$, the largest of three Belyaev moments of inertia in units of $\hbar^2$ MeV
^{-1}.

A few caveats are in order to properly use these data.

- All experimental values were taken from W. J. Huang et al., Chin. Phys. C 45, 03002 (2020).
- When experimental values are not available, the tables still lists a number to ensure easy (computer-)reading. The listed numbers are easily recognisable: 99.99 for energy quantities and 99.9 for the experimental spins and parities.
- Our symmetry-broken mean-field approach does not allow for a straightforward extraction of angular momentum quantum numbers. The calculated g.s. quantum numbers have thus been extracted in an ad-hoc way by rounding the angular momentum expectation values of any unpaired nucleons to the nearest half-integer value.
- For left-right asymmetric systems, the remark above also extends to the g.s. parity. For such cases, we simply guess positive parity.

For a precise definition of all these quantities and more details on the BSkG series of models, please see:

- G. Grams, W. Ryssens, G. Scamps, S. Goriely and N. Chamel (2023), arXiv:2307.14276,

- W. Ryssens, G. Scamps, S. Goriely and M. Bender (2023), Eur. Phys. J. A 59, 96,

- W. Ryssens, G. Scamps, S. Goriely and M. Bender (2022), Eur. Phys. J. A 58, 246,.

- G. Scamps, S. Goriely, E. Olsen, M. Bender and W. Ryssens (2021), Eur. Phys. J. A 57, 333.

The BSkG models share many properties with the preceding BSk-series, detailed in these publications:

- S. Goriely, N. Chamel and J. M. Pearson (2016) Phys. Rev. C93, 034337,

- S. Goriely (2015) Nucl. Phys. A933, 68,

- S. Goriely, N. Chamel, and J.M. Pearson (2013) Phys. Rev. C88, 061302,

- S. Goriely, N. Chamel, and J.M. Pearson (2013) Phys. Rev. C88, 024308,

- S. Goriely, N. Chamel, and J.M. Pearson (2010) Phys. Rev. C82, 035804,

- S. Goriely, N. Chamel, and J.M. Pearson (2009) Eur. J. Phys. A42, 547,

- N. Chamel, S. Goriely, and J.M. Pearson (2009) Phys. Rev. C80, 065804,

- S. Goriely, N. Chamel, and J.M. Pearson (2009) Phys. Rev. Let. 102, 152503,

- N. Chamel, S. Goriely, and J.M. Pearson (2008) Nucl. Phys. A812, 72,

- S. Goriely and J.M. Pearson (2008) Phys. Rev. C77, 031301,

- S. Goriely, M. Samyn, and J.M. Pearson (2007) Phys. Rev. C75, 064312,

- S. Goriely, M. Samyn, and J.M. Pearson (2006) Nucl. Phys. A773, 279,

- S. Goriely, M. Samyn, J.M. Pearson, and M. Onsi (2005) Nucl. Phys. A750, 425,

- M. Samyn, S. Goriely, M. Bender, and J.M. Pearson (2004) Phys. Rev. C70, 044309,

- S. Goriely, M. Samyn, M. Bender, and J.M. Pearson (2003) Phys. Rev. C68, 054325,

- M. Samyn, S. Goriely, and J.M. Pearson (2003) Nucl. Phys. A725, 69,

- S. Goriely, M. Samyn, P.-H. Heenen, J.M. Pearson, and F. Tondeur (2002) Phys. Rev. C66, 024326,

- M. Samyn, S. Goriely, P.-H. Heenen, J.M. Pearson, and F. Tondeur (2002) Nucl. Phys. A700, 142.

**Last update in October 2023.**