# A fully microscopic model of total level density in spherical nuclei

N. Quang Hung Corresponding author<EMAIL_ADDRESS>N. Dinh
Dang<EMAIL_ADDRESS>L. Tan Phuc<EMAIL_ADDRESS>N. Ngoc Anh
<EMAIL_ADDRESS>T. Dong Xuan<EMAIL_ADDRESS>T. V. Nhan Hao
<EMAIL_ADDRESS>Institute of Fundamental and Applied Sciences, Duy Tan
University, Ho Chi Minh City 700000, Viet Nam  Faculty of Natural Sciences,
Duy Tan University, Da Nang City 550000, Viet Nam  Quantum Hadron Physics
Laboratory, RIKEN Nishina Center for Accelerator-Based Science, 2-1 Hirosawa,
Wako City, 351-0198 Saitama, Japan  Dalat Nuclear Research Institute, Vietnam
Atomic Energy Institute, 01 Nguyen Tu Luc, Dalat City 670000, Viet Nam
Faculty of Physics, University of Education, Hue University, 34 Le Loi Street,
Hue City 530000, Viet Nam

###### Abstract

A fully microscopic model for the description of nuclear level density (NLD)
in spherical nuclei is proposed. The model is derived by combining the
partition function of the exact pairing solution plus the independent-particle
model at finite temperature (EP+IPM) with that obtained by using the
collective vibrational states calculated from the self-consistent Hartree-Fock
mean field with MSk3 interaction plus the exact pairing and random-phases
approximation (SC-HFEPRPA). Two important factors are taken into account in a
fully microscopic way, namely the spin cut-off and vibrational enhancement
factors are, respectively, calculated using the statistical thermodynamics and
partition function of the SC-HFEPRPA without any fitting parameters. The
numerical test for two spherical 60Ni and 90Zr nuclei shows that the
collective vibrational enhancement is mostly dominated by the quadrupole and
octupole excitations. This is the first microscopic model confirming such an
effect, which was phenomenologically predicted long time ago and widely
employed in several NLD models. In addition, the influence of collective
vibrational enhancement on nuclear thermodynamic quantities such as excitation
energy, specific heat capacity and entropy is also studied by using the
proposed model.

###### keywords:

Nuclear level density, Statistical model calculation, Exact pairing solution,
Collective vibrational excitations

, , , , ,

The concept of NLD, defined as the number of levels per unit of excitation
energy, was first introduced a long time ago by Hans Bethe [1] by notifying
that the number of excited states in atomic nuclei increases rapidly with the
excitation energy. Consequently, it is impossible to individually treat those
states, even by using advanced experimental and theoretical techniques. Thus,
the NLD reflects the average properties of excited nuclei and has various
applications in the study of not only nuclear structure and reactions but also
nuclear engineering and astrophysics [2]. The NLD 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. [3]. Many
theoretical studies have been carried out during the last seven decades in
order to find a reliable and fully microscopic model of NLD.

In general, theoretical approaches to NLD are classified into the
phenomenological and microscopic models. Phenomenological NLD models such as
the back-shifted Fermi gas (BSFG) and constant temperature [4, 5] were derived
based on simple analytical formulas containing some phenomenological
parameters such as the level density parameter, shell correction, pairing
energy, temperature, energy shift, spin cut-off factor, etc. The values of
these parameters are obtained from the local and/or global fittings to a
limited number of experimental data such as the experimental cumulative number
of discrete levels at low-excitation energy $E^{*}$ and the neutron resonance
data at $E^{*}=B_{n}$ with $B_{n}$ being the neutron binding energy [6]. For
nuclei, whose experimental NLD data are completely unknown, the prediction of
the above models becomes questionable (see e.g., Ref. [7]). In this case, the
development of microscopic methods should be more favorable. Several
microscopic NLD models have been developed such as the Hartree-Fock BCS
(HFBCS) [8], static path approximation (SPA) [9], SPA plus random-phase
approximation (SPA+RPA) [10, 11], finite-temperature shell model Monte Carlo
(SMMC) [12, 13], and Hartree-Fock-Bogolyubov plus combinatorial method (HFBC)
[14, 15]. They were often derived based on a Hamiltonian, which consists of a
realistic or phenomenological single-particle/mean-field potential of Woods-
Saxon (SPA+RPA or SMMC) or Skyrme Hartree-Fock (HFBCS) or Hartree-Fock-
Bogolyubov (HFBC) combined with the residual interactions (pairing correlation
and collective excitations) beyond the single-particle mean field. However,
some ambiguities within the above microscopic models still remain. Firstly,
pairing is approximately treated within the HFBCS and HFBC, which both violate
the particle-number conservation. Consequently, the NLD obtained within the
HFBCS and HFBC has to be normalized by using two additional parameters, whose
values are extracted by fitting to the experimental cumulative number of
discrete levels at low $E^{*}$ and neutron resonance data at $E^{*}=B_{n}$
(see e.g., Eq. (9) of Ref. [15]), losing certainly their microscopic nature.
Secondly, the spin cut-off factor $\sigma$, which is important for determining
the spin distribution of the NLD, is empirically determined from the rigid-
body limit within the HFBCS, HFBC, SPA, and SPA+RPA, whereas it is
microscopically calculated based on the exact ratio between the total
$\rho(E^{*})$ and spin projected $\rho_{J}(E^{*})$ NLDs within the SMMC.
Thirdly, the residual correlations are either not taken into account (within
the HFBCS and HFBC) or taken into account in a simplified way (within the
SPA+RPA). For instance, the HFBC considers only excitations built on all
uncorrelated particle-hole configurations, whereas the SPA+RPA employs very
simple two-body interactions of the quadrupole-quadrupole and/or higher
excitations, (see e.g., Eq. (1) of Ref. [11]). Lastly, the numerical
calculations within the SPA+RPA and SMMC are time consuming, in particular for
heavy nuclei.

Recently, we have proposed a microscopic NLD method based on the exact pairing
plus independent-particle model at finite temperature (EP+IPM) [16]. This
model, which contains no fitting parameters to the experimental NLD data and
has very short computing time, has provided a good description for the NLDs
and thermodynamic properties of not only hot 170-172Yb [16] and 60-62Ni [17]
nuclei but also hot rotating 96Tc [18], 184Re, 200Tl, 211Po, and 212At [19]
isotopes. However, it still contains two shortcomings. The first shortcoming
is the spin cut-off factor taken from the empirical formula in the rigid-body
limit for axially deformed nuclei, namely $\sigma_{\perp}\approx
0.015A^{5/3}T$ and
$\sigma_{\parallel}=\sigma_{\perp}\sqrt{(3-2\beta_{2})/(3+\beta_{2})}$, where
$T,\beta_{2}$, $\sigma_{\perp}$, and $\sigma_{\parallel}$ are nuclear
temperature, quadrupole deformation parameter, perpendicular and parallel spin
cut-off factors, respectively [16]. The second shortcoming is the collective
enhancement factor $k_{\rm coll}$ consisting the vibrational $k_{\rm vib}$ and
rotational $k_{\rm rot}$ excitations, which are also empirically described as

$\displaystyle k_{\rm vib}=\exp[0.055A^{2/3}T^{4/3}]~{},$ (1) $\displaystyle
k_{\rm rot}=(\sigma_{\perp}^{2}-1)/[1+e^{(E^{*}-U_{C})/D_{C}}]+1~{},$ (2)

where $D_{C}=1400\beta_{2}^{2}A^{-2/3}$ and $U_{C}=120\beta_{2}^{2}A^{1/3}$.
The second shortcoming is the most difficult problem of the present NLD
models. Regarding the spin cut-off factor, it can be microscopically
calculated by using the statistical thermodynamic method e.g., in Refs. [20,
21], namely,

$\sigma^{2}=\frac{1}{2}\sum_{k}m_{k}^{2}{\rm sech}^{2}\frac{1}{2}\beta
E_{k}~{},$ (3)

where $m_{k}$ is the single-particle spin projection (written in the deformed
basis), $\beta=1/T$ is the inverse of temperature $T$, and the quasiparticle
energy $E_{k}$ is calculated from
$E_{k}=\sqrt{(\epsilon_{k}-\lambda)^{2}+\Delta^{2}}$ if pairing is included or
$E_{k}=\epsilon_{k}-\lambda$ if no pairing is considered ($\Delta$,
$\epsilon_{k}$, and $\lambda$ are pairing gap, single-particle energies and
chemical potential, respectively). Within the EP, the pairing gap $\Delta$ and
quasiparticle energy $E_{k}$ are exactly calculated by using e.g., Eqs. (11)
and (12) of Ref. [22]. As for the vibrational enhancement, there exists
another approximate formula given in Refs. [14, 23, 24]

$k_{\rm vib}=\exp[\delta S-\delta U/T)~{},$ (4)

where $\delta S=\sum_{i}(2\lambda_{i}+1)[(1+n_{i}){\rm ln}(1+n_{i})-n_{i}{\rm
ln}n_{i}]$ and $\delta U=\sum_{i}(2\lambda_{i}+1)\omega_{i}n_{i}$ are the
changes in the entropy and excitation energy, respectively, due to vibrational
modes with $\omega_{i}$, $\lambda_{i}$, and $n_{i}$ being the energies,
multipolarities, and temperature-dependent occupation numbers, respectively.
The occupation numbers $n_{i}$ are defined as
$n_{i}=\exp(-\gamma_{i}/2\omega_{i})/[\exp(\omega_{i}/T)-1]$ with
$\gamma_{i}=0.0075A^{1/3}(\omega_{i}^{2}+4\pi^{2}T^{2})$ being the spreading
widths of the vibrational excitations. As for the phonon energies
$\omega_{i}$, the modified equations including shell correction $E_{\rm
shell}$ are considered, namely $\omega_{2}=65A^{-5/6}/(1+0.05E_{\rm shell})$
for the quadrupole and $\omega_{3}=100A^{-5/6}/(1+0.05E_{\rm shell})$ for the
octupole [14]. It is clear to see that Eq. (4) adopts only the lowest energies
$\omega_{2,3}$ of the quadrupole ($\lambda=2$) and octupole ($\lambda=3$)
vibrations and neglects higher vibrational energies and other multipolarities
such as monopole ($\lambda=0$), dipole ($\lambda=1$), hexadecupole
($\lambda=4$), etc. Indeed, such vibrational energies and multipolarities can
be microscopically calculated within the random-phase approximation (RPA), one
of the most extensive approximations for nuclear collective vibrational
excitations. In this case, one should construct a vibrational partition
function of the following form

$Z_{\rm
vib}(T)=\sum_{\lambda}(2\lambda+1){\sum_{i}}e^{-E_{i}^{\lambda}/T}~{},$ (5)

in the canonical ensemble, where $E_{i}^{\lambda}$ are all the eigenvalues
(energies) obtained by solving the RPA equation for the corresponding
multipolarity $\lambda$, which runs from 0 to 4 or 5. Similarly, one can
construct a partition function for rotational excitation $Z_{\rm rot}(T)$
similar to Eq. (5) by replacing the energies $E_{i}^{\lambda}$ with rotational
states obtained from different rotational bands.

In the present Letter, we develop a fully microscopic method for the
description of NLD, limited to spherical nuclei (no rotational enhancement or
$k_{\rm rot}=1$), although the proposed idea is applicable also for deformed
systems. The model is derived based on the EP+IPM in Ref. [16], however, three
significant improvements have been proposed. First, the single-particle
spectra are taken from the Hartree-Fock mean field plus exact pairing (HF+EP)
with an effective Skyrme interaction (MSk3) as developed in Ref. [25], instead
of the phenomenological Woods-Saxon potential. This HF+EP with MSk3 force has
provided a very good description of not only binding and two nucleon-
separation energies but also nucleon densities and single-particle occupation
numbers of light and spherical 22O and 34Si nuclei at zero and finite
temperatures. Second, the spin cut-off parameter is calculated using Eq. (3),
instead of empirical formula. Third, the vibrational enhancement is
microscopically treated by combining the EP+IPM partition function with that
given in Eq. (5), in which the self-consistent Hartree-Fock+EP+RPA (SC-
HFEPRPA) with the same MSk3 force [26] is used instead of the conventional
HF+RPA, namely

$\ln Z^{\prime}_{\rm total}(T)=\ln Z^{\prime}_{\rm EP+IPM}(T)+\ln
Z^{\prime}_{\rm SC-HFEPRPA}(T)~{},$ (6)

where $Z^{\prime}(T)$ denotes the excitation partition function [27]. Knowing
the total partition function (6), one can easily calculate the excitation
energy $E(T)$, entropy $S(T)$, and heat capacity $C(T)$, which are later used
together with the spin cut-off factor $\sigma$ to calculate the total NLD
$\rho_{\rm tot}(E^{*})$ as has been done in Ref. [16]. Thus, the collective
vibrational enhancement is directly included in the total partition function
and the vibrational enhancement factor $k_{\rm vib}$ can be calculated via
$k_{\rm vib}(E^{*})=\rho_{\rm tot}(E^{*})/\rho_{\rm int}(E^{*})$, where
$\rho_{\rm int}(E^{*})$ is the intrinsic NLD, that is, the NLD obtained by
using $Z_{\rm EP+IPM}$ only. The model in this case is fully microscopic as it
does not contain any empirical expressions and fitting parameters to the NLD
data.

To test the proposed model, we select the spherical 60Ni nucleus, which is the
only nucleus whose experimental NLD data are presently available from
$E^{*}\sim 0$ up to $\sim 23$ MeV, well above $E^{*}=B_{n}=11.358$ MeV [7, 28,
29]. The high-energy part of the NLD is very important to test the validity of
the microscopic calculations. The average binding energy BE/A and energy
$E_{2_{1}^{+}}$ of the first $2^{+}$ state obtained within the SC-HFEPRPA
calculation for 60Ni are -8.736 and 1.336 MeV, respectively, which are in
excellent agreement with the experimental data (BE/A = -8.780 MeV and
$E_{2_{1}^{+}}$ = 1.333 MeV). Moreover, the energy weighted sum rules for the
isoscalar (IS) and isovector (IV) excitations are perfectly conserved for all
the multipolarities $\lambda^{\pi}=0^{+},1^{-},2^{+},3^{-},4^{+}$, and $5^{-}$
with natural parities. The results obtained are shown in Figs. 1 and 3. A
similar test for another spherical 90Zr nucleus is also performed and the
results are illustrated in Fig. 2. The pairing interaction $G$ is selected, as
usual, so that the neutron and proton pairing gaps at $T=0$ agree with those
obtained from the experimental odd-even mass differences for 60Ni and 90Zr,
respectively [16].

Figure 1: (Color online) (a) The total NLD obtained within the fully
microscopic EP+IPM by gradually adding higher collective vibrational modes
$\lambda$ to the vibrational partition function (5) versus the experimental
data for 60Ni. (b) The best total NLD obtained from (a) with
$\lambda=0^{+},1^{-},2^{+}$, and $3^{-}$ (thick solid line) in comparison with
those obtained within the HFBCs (for positive $+\pi$ and negative $-\pi$
parities and HFBCS as well as those obtained within the EP+IPM with
phenomenological ($k_{\rm vib}^{\rm Phen.}$), empirical ($k_{\rm vib}^{\rm
Emp.}$), and without ($k_{\rm vib}=1$) vibrational enhancements. (c) The
vibrational enhancement factor ($k_{\rm vib}$) obtained by using empirical
(1), phenomenological (4), and microscopic formulas. (d) The spin cut-off
parameter $\sigma^{2}$ obtained by using empirical (rigid-body limit) and
microscopic (3) formulas.  Figure 2: (Color online) Same as Fig. 1 but for
90Zr.

Fig. 1(a) depicts the total NLDs $\rho(E^{*})$ obtained within our fully
microscopic EP+IPM by gradually adding the higher multipolarities $\lambda$ to
the vibrational partition function (5). It is clear to see in this figure that
the monopole and dipole excitations ($\lambda=0^{+}$ and $1^{-}$) (dotted
line) have very small enhancement to $\rho_{\rm int}$ (thin solid line).
Adding the quadrupole $\lambda=2^{+}$ vibration significantly increases the
NLD and the result obtained (dash line) agrees well with not only the low-
energy ($E^{*}<B_{n}$) but also the high-energy ($E^{*}\geq B_{n}$) NLD data.
Adding the octupole $\lambda=3^{-}$ excitation (thick solid line) does not
enhance much the NLD. However, adding higher hexadecapole $\lambda=4^{+}$
(dash dotted line) and tetrahedral $\lambda=5^{-}$ (dash-dot dotted line)
excitations enhances the NLD further but the results obtained slightly
overestimates the experimental data, in particular the data at $E^{*}\geq
B_{n}$. This is because their excitations always take place at a very high
excitation energy, which goes beyond the vibrational excitation region and
thus, should not be included in the vibrational partition function. The
analysis in Fig. 1(a) strongly indicates that the vibrational enhancement of
NLD is mainly due to the quadrupole and octupole excitations. Indeed, this
suggestion was initiated long time ago in Ref. [23] but without any
microscopic justification. Since then, it was widely used in various NLD
models, e.g., Refs. [14, 24]. Therefore, the results shown in Fig. 1(a) are of
particular valuable as they are the first microscopic calculation, which
confirms the important role of the quadrupole and octupole excitations in the
NLD.

In Fig. 1(b), the best NLD obtained within the EP+IPM with
$\lambda=0^{+},1^{-},2^{+}$, and $3^{-}$ is compared with those obtained
within other microscopic HFBCS and HFBC (for positive $+\pi$ and negative
-$\pi$ parities) approaches taken, respectively, from RIPL-2 [30] and RIPL-3
[31] nuclear database as well as those calculated within the EP+IPM without
vibrational enhancement ($k_{\rm vib}=1$) and with empirical (1) and
phenomenological (4) formulas for $k_{\rm vib}$. It is seen that the HFBC over
estimates the data, whereas the HFBCS agrees only with the data in Ref. [7]
(open squares) at $E^{*}>$ $\sim$4 MeV because this method was normalized to
fit to these data. Once the data are updated and extended to higher $E^{*}$ as
in Ref. [29], the HFBCS NLD deviates from the new data. This result clearly
shows the main drawback of the HFBC and HFBCS models. The EP+IPM NLDs, which
employ the empirical $k_{\rm vib}^{\rm Emp.}$ and phenomenological $k_{\rm
vib}^{\rm Phen.}$, agree with the data at 5 MeV $<E^{*}<B_{n}$ only due to the
fact that the values of $k_{\rm vib}^{\rm Emp.}$ and $k_{\rm vib}^{\rm Phen.}$
are larger than the corresponding microscopic calculation $k_{\rm vib}^{\rm
Micr.}$ [Fig. 1(c)]. The later is calculated by taking the ratio between the
EP+IPM NLD with $\lambda=0^{+},1^{-},2^{+}$, and $3^{-}$ and its state density
($k_{\rm vib}=1$).111It is seen in Fig. 1(c) that the value of $k_{\rm
vib}^{\rm Micr.}$ does not start from 1 at $E^{*}\rightarrow 0$ as those of
$k_{\rm vib}^{\rm Emp.}$ and $k_{\rm vib}^{\rm Phen.}$. This is due to the
well-known unphysical divergence of the saddle-point approximation at very low
$T$ or $E^{*}$, which has been widely employed in most microscopic NLD models
[34]. Moreover, the microscopic spin cut-off factor calculated within the
EP+IPM using Eq. (3) is found to be larger than that calculated from the
empirical formula of rigid body [Fig. 1(d)]. Similar results can be seen in
Fig. 2 but for the 90Zr nucleus, whose NLD data are available below
$E^{*}<B_{n}$ [32]. The EP+IPM NLD with the quadrupole and octupole
excitations is found in the best agreement with the experimental data [Fig.
2(a)], whereas the HFBCS (HFBC) overestimates (underestimates) the data [Fig.
2(b)]. The values of $k_{\rm vib}^{\rm Emp.}$ and $k_{\rm vib}^{\rm Phen.}$
are higher than that of $k_{\rm vib}^{\rm Micro.}$ [Fig. 2(c)], while the
microscopic spin cut-off factor is higher than the empirical one [Fig. 2(d)],
similar to those obtained for 60Ni in Fig. 1. All the results shown in Figs. 1
and 2 are particularly important because it ensures the validity of our fully
microscopic NLD model, in which pairing is exactly treated and the spin cut-
off and collective vibrational enhancement factors are microscopically
calculated.

Figure 3: (Color online) The neutron pairing gap $\Delta_{N}$ (a), excitation
energy $E^{*}$ (b), heat capacity $C$ (c), and entropy $S$ (d) as functions of
$T$ obtained within the EP+IPM for 60Ni with (solid line) and without (dotted
line) collective vibrational enhancement. In (b), the excitation energy
$E^{*}$ obtained within the HFBC and HFBCS are also plotted, for comparison
with the EP+IPM calculations (with and without collective enhancement).

Because of the collective excitations, not only the NLD but also the
thermodynamic quantities of excited nuclei are enhanced. We show in Fig. 3 the
thermodynamic quantities such as neutron pairing gap $\Delta_{N}$ (a),
excitation energy $E^{*}$ (b), heat capacity $C$ (c), and entropy $S$ (d) as
functions of $T$ obtained within the EP+IPM with and without the contribution
of collective vibrational enhancement. For $E^{*}(T)$, the predictions of the
HFBC and HFBCS taken from Refs. [30, 31] are also plotted in Fig. 3(b).
Obviously, the neutron pairing gap $\Delta_{N}$ (proton pairing gap is zero
because 60Ni has a proton magic number) calculated within the exact pairing
decreases with increasing $T$ and remains finite even at $T=3$ MeV, in
agreement with many microscopic calculations (see e.g., Ref. [33]).
Consequently, one can see an $S$-shaped heat capacity, which indicates the
signature of superfluid-normal phase transition in finite nuclear systems. The
collective vibrational enhancement is seen to enhance all the thermodynamic
quantities, except the exact pairing gap, which is the concept of the mean
field, whereas the collective enhancement goes beyond the mean-field concept.
It is also seen that the HFBC predicts a rather small excitation energy in
comparison with the HFBCS and our EP+IPM [Fig. 3(b)]. This result of HFBC can
be easily understood because this method is constructed based only on all the
combinations of uncorrelated particle-hole excitations, meaning that some
correlated excitations are not taken into account [14, 15]. The HFBCS
excitation energy is very close to that of the EP+IPM calculated without
collective enhancement. This result is reasonable because the HFBCS does not
explicitly treat the collective enhancement [8]. The results shown in Fig. 3
are very interesting because they are obtained from the first microscopic
model exploring the effect of collective enhancement on nuclear thermodynamic
quantities.

The present Letter proposes a fully microscopic model for the description of
total NLD. The model is proposed by combining the thermodynamic partition
function of the exact pairing solution with that obtained within the finite-
temperature independent-particle model (EP+IPM) and collective vibrational
excitation modes. The later are calculated from the Hartree-Fock mean field
with MSk3 interaction and self-consistently combined with the exact pairing
solution and random-phase approximation (SC-HFEPRPA). In addition, the spin
cut-off parameter is also microscopically calculated within the EP+IPM using
the statistical thermodynamics. The numerical test has been carried out for
the spherical 60Ni nucleus, the only nucleus whose NLD data are available from
the excitation energy of 0 to about 23 MeV. A similar test has also been
performed for the spherical 90Zr nucleus, whose NLD data are provided below
the neutron binding energy. The results obtained show that by combining the
EP+IPM partition function with that obtained using the collective vibrational
states taken from the SC-HFEPRPA calculation, which excellently reproduces the
experimental binding energy and energy of the first $2^{+}$ state, we are able
to study the contributions of different vibrational modes from the monopole
($0^{+}$) to tetrahedral ($5^{-}$) states, from which the quadrupole and
octupole excitations are found to be the most importance. This is, indeed, the
first microscopic model capable to confirm the important role of the
vibrational quadrupole and octupole excitations in the NLD. The NLD obtained
within this model is found in a much better agreement with the experimental
data than those calculated within other approaches such as HFBCS and HFBC as
well as those obtained by using the empirically/phenomenologically vibrational
enhancement and spin cut-off factors. It has also been found that the
vibrational enhancement factor obtained within our fully microscopic approach
is lower than that calculated using the empirical and phenomenological
formulas. Regarding the spin cut-off factor, its value obtained within our
microscopic model is larger than that obtained by using the empirical formula.
The effect of vibrational enhancement on nuclear thermodynamic quantities such
as excitation energy, entropy, and heat capacity is also studied, for the
first time, within our fully microscopic model. Finally, the present model
does not consume much computing time, namely the EP+IPM calculation takes less
than 5 min [16], whereas the SC-HFEPRPA takes about 1 hour for the calculation
of each multipolarity. Therefore, one calculation for one nucleus, even a
heavy one, takes less than 6 hours and thus can be performed on a PC. Although
the present Letter is restricted to the description of NLD in spherical
nuclei, the proposed method can also be applied to deformed isotopes, that is,
the rotational enhancement factor can also be microscopically calculated from
the partition function of all the excited states coming from the rotational
bands. The result obtained for deformed nuclei will be reported in the
forthcoming papers.

## Acknowledgement

This work is funded by the National Foundation for Science and Technology
Development (NAFOSTED) of Vietnam through Grant No. 103.04-2019.371.

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