II-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 54 (2021)

RIKEN Accel. Prog. Rep. 54 (2021)

A fully microscopic model of total level density in spherical nuclei†
N. Quang Hung,∗1,∗2 N. Dinh Dang,∗3 L. Tan Phuc,∗1,∗2 N. Ngoc Anh,∗4 T. Dong Xuan,∗1,∗2 and T. V. Nhan Hao5

†
∗1
∗2
∗3
∗4
∗510 6108 Exp. data
106k Vib =110 410(a) 60Ni

21000

Exp. data

Ref. 4)
Ref. 5)
Ref. 6)

λ = 0,+ 1λ = 0,+ 1,- 2+λ = 0,+ 1,- 2,+ 3+ + -ρ (MeV -1)

ρ (MeV -1)

The nuclear level density (NLD) is one of important inputs for the calculations of low-energy nuclear
reactions and astrophysics. It also contains various information on the internal structure of atomic nuclei
such as single-particle levels, pairing correlations, spin
distributions, collective (vibrational and/or rotational)
excitations, nuclear thermodynamics, etc. Althoughmany theoretical studies have been carried out during
the last seven decades to ﬁnd a reliable and fully microscopic model of NLD, there has still been lacking a fully
microscopic NLD model. Recently, we have proposeda microscopic NLD model based on the exact pairing
plus independent-particle model at ﬁnite temperature(EP+IPM).1) The latter, which has a very short computing time and contains no ﬁtting parameters to theexperimental NLD data, can describe quite well theNLDs and thermodynamic properties of several hotnuclei. Nevertheless, the EP+IPM still contains twoshortcomings, which limit its fully microscopic foundation. These shortcomings come from the use of empirical formulas for the spin cut-oﬀ σ and collective
enhancement factors. The latter is a multiplication ofthe vibrational kvib and rotational krot enhancement
factors.1)
In this paper, we develop a fully microscopic version of the EP+IPM, limited to spherical nuclei (rotational enhancement factor krot = 1), in which threesigniﬁcant improvements are included. First, thesingle-particle spectra are taken from the HartreeFock mean ﬁeld plus exact pairing (HF+EP) with aneﬀective MSk3 interaction,2) instead of the WoodsSaxon potential in the original version of EP+IPM.
Second, the spin cut-oﬀ parameter is directly calculated by∑using the statistically() thermodynamic method
σ 2 = 12 k m2k sech2 12 Ek /T , where mk is the singleparticle spin projection (in the deformed basis) andEk is the quasiparticle energy. Last, the vibrational enhancement of NLD is directly calculated
based on the vibrational partition function Zvib (T ) =∑−Eiλ /T, where Eiλ are all the eigenvalues
λ,i (2λ + 1)eobtained by solving the self-consistent HF+EP+RPA(SC-HFEPRPA) equation3) for the corresponding multipolarity λ, which runs from 0 to 5. The total partition function of the system is then calculated by combining this vibrational partition function with that

λ = 0, 1, 2, 3, 4+
λ = 0,+ 1,- 2,+ 3,- 4,+ 5-5

10 15 20 25
E* (MeV)

104Ref. 7)
Ref. 8)10 2(b) 90Zr

100

λ = 0,+ 1λ = 0,+ 1,- 2+
λ = 0,+ 1,- 2,+ 3- +λ = 0,+ 1,- 2,+ 3,- 4
λ = 0,+ 1,- 2,+ 3,- 4,+ 5-0k Vib =148

E* (MeV)

12Fig. 1. Total NLDs obtained within the fully microscopic
EP+IPM by gradually adding higher collective vibrational modes λ to the vibrational partition function versus the experimental data for 60 Ni (a) and 90 Zr (b).
Experimental data are taken from Refs. 4–8).

of the EP+IPM (without vibrational enhancement),′′′= ln ZEP+IPM+ ln Zvib, where Z ′ (T )namely ln Ztotal
is the excitation partition function.1)
The numerical tests for two spherical 60 Ni and 90 Zrnuclei (see Fig. 1) show that the monopole and dipole
excitations (λ = 0+ and 1− ) have little enhancement tothe total NLD. By adding the quadrupole λ = 2+ and
octupole 3− excitations, the EP+IPM NLDs are in excellent with the experimental data. However, adding
higher hexadecapole λ = 4+ and quintupole λ = 5−
excitations enhances the NLD further but the results
obtained slightly overestimate the experimental data
because these states are always located at a very highexcitation energy, which goes beyond the vibrational
excitation region. The results shown in Fig. 1 are ofparticular valuable as they are the ﬁrst microscopic
calculation, which conﬁrms the important role of thequadrupole and octupole excitations in the descriptionof NLD. We also found that the vibrational enhancement factor obtained within our fully microscopic approach is smaller than that calculated using the empirical and phenomenological formulas. In addition,
the spin cut-oﬀ factor obtained within our microscopic
approach are larger than that obtained by using theempirical formula.
References

1) N. Q. Hung, N. D. Dang, L. T. Q. Huong, Phys. Rev.
Lett. 118, 022502 (2017).
2) L. T. Phuc, N. Q. Hung, N. D. Dang, Phys. Rev. C 97,024331 (2018).
3) L. T. Phuc, N. Q. Hung, N. D. Dang, Phys. Rev. C 99,064322 (2019).
4) A. V. Voinov et al., Phys. Rev. C 76, 044602 (2007).
5) J. R. Huizenga et al., Phys. Rev. 182, 1149 (1969).
6) A. V. Voinov et al., EPJ Web Conf. 21, 05001 (2012).
7) https://www-nds.iaea.org/RIPL-3/ .
8) Y. Byun et al., Phys. Rev. C 90, 044303 (2014).

Condensed from the article in Phys. Lett. B 811, 135858(2020)
Institute of Fundamental and Applied Sciences, Duy TanUniversity
Faculty of Natural Sciences, Duy Tan University
RIKEN Nishina Center
Dalat Nuclear Research Institute, Vietnam Atomic EnergyInstitute
Faculty of Physics, Hue University of Education- 46 -1