Kun-Peng Geng · Peng-Xiang Du · Jian Li · Dong-Liang Fang

Abstract In this study, a microscopic method for calculating the nuclear level density (NLD) based on the covariant density functional theory (CDFT) is developed.The particle-hole state density is calculated by a combinatorial method using single-particle level schemes obtained from the CDFT, and the level densities are then obtained by considering collective effects such as vibration and rotation.Our results are compared with those of other NLD models, including phenomenological, microstatistical and nonrelativistic Hartree–Fock–Bogoliubov combinatorial models.This comparison suggests that the general trends among these models are essentially the same, except for some deviations among the different NLD models.In addition, the NLDs obtained using the CDFT combinatorial method with normalization are compared with experimental data, including the observed cumulative number of levels at low excitation energies and the measured NLDs.The CDFT combinatorial method yields results that are in reasonable agreement with the existing experimental data.

Keywords Nuclear level density · Covariant density functional theory · Combinatorial method

1 Introduction

Nuclear level density (NLD) is the basic physical input for nuclear reactions.This is the key component for the calculation of reaction cross sections relevant to nucleosynthesis[1–6].The study of NLDs dates back to the 1930s, with Bethe’s pioneering work [7].Since then, many theoretical models such as the back-shifted Fermi gas model (BFM) [8],composite constant temperature model (CTM) [9] and generalized superfluid model (GSM) [10] have been adopted for NLD studies.These phenomenological models are widely used for nuclear reaction calculations.Phenomenological models rely on experimental data for adjusting the parameters; however, experimental data are limited, especially for nuclei far from theβ-stability line [11].To address these difficulties, many microscopic methods have been developed.

Over the past decades, various microscopic approaches for NLD have been proposed, including the equidistant spacing model [12–15], the shell-model Monte Carlo method[16–20], the spectral distribution calculation [21–23], the independent particle model at finite temperature [24–27],the microstatistical methods [28–31] and the random matrix method [32].Recently, a stochastic estimation method for the level density within the framework of the configuration-interaction shell model (CISM) was proposed [33]and applied in the calculation of NLDs of fission products133-137Xe and135-138Ba [34].Over the past two decades,microscopic methods based on the Hartree–Fock–Bogoliubov (HFB) combinatorial model [11] have been developed.The idea of using a combinatorial method to calculate the level densities was derived from the calculation of excitation-state densities [35].After successfully describing the excitation-state densities using combinations of nucleons occupying single-particle levels at the mean field, it was natural to further describe the level densities by considering the collective effects [36].This combinatorial approach is on par with statistical methods with respect to reproducing experimental data and can provide energy-, spin-, and paritydependent NLDs beyond the scope of statistical methods[37].The nonrelativistic HFB combinatorial methods based on Skyrme and Gogny effective interactions have successfully reproduced NLDs for various nuclei [37, 38] and have been applied to astrophysical reactions.The accuracy of NLD is related to the basic information about the nuclear structure, such as single-particle levels, deformation, and binding energy.In recent years, the covariant (relativistic)density functional theory has attracted considerable attention in the field of nuclear physics owing to its successful description of complex nuclear structures and reaction dynamics [39–45].For instance, it can satisfactorily reproduce the isotopic shifts in Pb isotopes [46] and can naturally explain the origin of the pseudospin and spin symmetries in the antinucleon spectrum [47], as well as provide a good description of the nuclear magnetic moments [48, 49].Recently, a microstatistical method based on the CDFT was developed to describe NLDs [50].The method was applied to the calculation of NLDs of94,96,98Mo and106,108Pd as well as the odd-Anuclei at saddle point [51], which were in good agreement with experimental data over the entire energy range of the measured values [50].While the microstatistical method can only calculate energy-dependent NLDs,the combinatorial method can calculate energy-, spin-, and parity-dependent NLDs.Therefore, it is meaningful to calculate the NLDs using the CDFT combinatorial method.

The theoretical framework and methods are introduced in Sect.2.The NLDs calculated using the CDFT combinatorial method are compared with other NLD predictions and experimental results in Sect.3.Finally, the conclusions and future prospects are presented in Sect.4.

2 Theoretical framework

Covariant density functional theory starts from a Lagrangian, and the corresponding Kohn-Sham equations have the form of a Dirac equation with effective fieldsSandVderived from this Lagrangian [39, 41, 49, 52].Specifically,the nucleons in the nucleus are described as Dirac particles moving in the average potential field provided by the meson and photon fields, interacting with each other through the exchange of mesons and photons.The Dirac equation is written as follows:

whereεiis the single-particle energy required to calculate the NLDs, andSandVare the relativistic scalar field and time-like component of the vector field, respectively.

Upon obtaining the energyε, spin projectionm, and paritypof the single-particle levels using CDFT, the level information is substituted into the generating function defined in the combinatorial method [11] to obtain the particle-hole state densityρi; the generating function Z is expressed as follows:

However, the state densityρiis strongly dependent on the unit energyε0.Therefore, another method suggested by Williams [53] is employed in this study to limit the discretization effects, as explained below.

Summing all theCNvalues up to a given excitation energyU, we first obtain the cumulative number of statesNN(U,M,P) , which represents the number of particle-hole states with excitation energyEsuch that 0 ≤E

The calculated particle-hole state densityρiis related only to the particle-hole excitation.Two special collective effects must be considered to obtain the level density, namely rotational and vibrational effects.If the nucleus under consideration displays spherical symmetry, the intrinsic and laboratory frames coincide, and the level density is trivially obtained using the following relation [7]:

The fraction 1/2 accounts for the fact that in mirror axially symmetric nuclei, the intrinsic states with spin projections+Kand -Kyield the same rotational levels.Moreover, in the second term of the summation, the symbolδ(x)(defined byδ(x)=1 ifxholds true and 0 otherwise) restricts the rotational bands built on intrinsic states with spin projectionK=0 and parityPto level sequences 0, 2, 4,...forP=+and 1, 3, 5,...forP=-.Finally, the rotational energy is obtained using the following expression [54]:

whereSdenotes entropy,Udenotes the excitation energy,andTdenotes the nuclear temperature.

Finally, a phenomenological damping function is introduced to avoid sharp transitions between the spherical and deformed level densities that affect the NLD predictions.The expression for the NLD after introduction of the damping function is [31]:

The damping functionfdamis expressed as follows [38]:

whereβis the quadrupole deformation parameter.Parameters 0.18 and 0.038 are adjusted according to the experimental data at the neutron separation energySn[56].The expression forfdamdeveloped previously [38] was simplified to depend only on the nuclear deformation, thereby reducing the number of phenomenological parameters.This damping function is used to suppress the discontinuities that occur between the spherical and deformed NLDs and ensure a smooth change of shape from deformed to spherical.

3 Results and discussion

In this section, we present our results for the NLDs obtained using the combinatorial method based on the CDFT and compare them with the results obtained using other NLD models [57] and experimental data [58, 59].The effective meson-exchange interaction parameter PK1 [60] is adopted throughout the CDFT calculations.For spherical CDFT calculations, we fixed the box size asRbox= 20 fm and the step size as Δr= 0.1 fm.In the present deformed CDFT calculations, the Dirac equation for nucleons and the Klein–Gordon equations for mesons are solved using an isotropic harmonic oscillator, and a basis for the 18 major oscillator shells is adopted.

Fig.1 (Color online) Comparison of NLDs calculated using different effective interactions under CDFT with experimental data [58, 59]

The combinatorial results rely on the properties of singleparticle levels.To understand the effect of using different effective interactions on the NLDs, the NLDs of112Cd and162Dy obtained with different CDFT effective interactions,namely PK1 [60], NL3 [61], DD-ME2 [62], and DD-PC1[63] are presented for comparison in Fig.1.For nuclei with small deformations (112Cd), the NLDs for the four effective interactions are close for excitation energies above 4 MeV.However, significant differences are observed for excitation energies below 4 MeV.This difference arises from the fact that the single-particle energies near the Fermi level are exceptionally sensitive to the choice of effective interaction,especially when the Fermi level is near the proton or neutron shells.Meanwhile, the CDFT calculations with the chosen interactions do not adequately reproduce the single-particle levels of the magic nuclei132Sn [64], which explains the deviations of our results from the measurements presented in Fig.1.For the well-deformed162Dy , the NLDs calculated with the four effective interactions deviate from each other in the entire region of the excitation energy, although the overall trend is consistent.This deviation at low excitation energies is mainly caused by differences in the singleparticle energies around the Fermi level.Meanwhile, the entropySobtained from the four effective interactions is significantly different for nuclei with large deformations, which ultimately leads to deviation of the NLDs at high excitation energies.Compared with the experimental data of112Cd [58]and162Dy [59] extracted by the Oslo group based on the analysis of particle-γcoincidence in the (3He ,αγ) and (3He ,3He ”γ) reactions, the deviation between the NLDs obtained by using PK1 effective interaction and the results obtained by using other effective interactions is within a reasonable range.

The nonrelativistic HFB combinatorial methods, including Skyrme and Gogny interactions, have been widely used in NLD predictions [55, 58]; the NLDs of spherical nuclei (55Co ,132Sn , and208Pb ) and deformed nuclei (94Mo ,95Mo ,and166Er ) calculated using the CDFT combinatorial method are compared with the corresponding experimental data, as illustrated in Fig.2.

For55Co , the CDFT combinatorial method yields a lower NLD than the experimental value, whereas the Skyrme combinatorial method yields a higher NLD.Although the results of the Gogny combinatorial method are closer to the experimental data to a certain extent, a strong oscillation exists.For132Sn and208Pb , the results of the CDFT combinatorial method are similar to those of the Skyrme combinatorial method; however, the results of the Gogny combinatorial method are significantly lower than those of the two methods.Compared to the experimental data, the CDFT combinatorial method can better describe the NLD of208Pb.The NLDs of55Co ,132Sn , and208Pb obtained using the three combinatorial methods are oscillatory to a certain extent because the three nuclei are spherical with highly degenerate single-particle levels with a strong shell effect.

Fig.2 (Color online) Comparison of NLDs calculated based on the CDFT combinatorial method with results from the HFB combinatorial methods and microstatistical method (HFB+Sta) [31, 37, 38].The experimental data are taken from Refs.[65–68]

For94Mo , the results of both the CDFT and Skyrme combinatorial methods are closer to the experimental data than those of the Gogny combinatorial method.For95Mo and166Er , the NLDs obtained using the three combinatorial methods are similar and slightly higher than the experimental data.It should be emphasized that the deformed nuclei94Mo ,95Mo , and166Er break the single-particle level degeneracy, yielding smoother NLDs, as shown in Fig.2.It is noteworthy that the CDFT combinatorial method yields higher NLDs for94Mo and95Mo.One possible reason for this is that the CDFT combinatorial method uses a rigidbody value when considering the rotational effect, which is inappropriate for soft nuclei [69].

In addition, the NLDs of55Co and94Mo can be adequately described by the microstatistical model [31] relative to the CDFT combinatorial method, while both models provide similar descriptions for the NLDs of95Mo and166Er.In comparison, the microstatistical model yields relatively smooth results for spherical and deformed nuclei, which do not satisfactorily reflect the shell effect.Therefore, it can be concluded that the CDFT combinatorial method is as capable of describing NLDs as the other two combinatorial methods and the microstatistical model.

Furthermore, it is necessary to compare the NLDs obtained in the study with those obtained using phenomenological models (BFM, CTM, and GSM) [57].Phenomenological models can adequately describe the experimental data at neutron separation by fitting the experimental data.The level densities at the neutron separation energy are the most commonly used experimental data and are obtained using the mean distanceD0of thes-wave resonance.The NLDs of the even-even nuclei (112Cd and162Dy ), odd-Anuclei (51V ,97Mo , and119Sn ), and odd-odd nucleus (60Co )calculated using the CDFT combinatorial method are compared with those obtained using phenomenological models and the available experimental data, as demonstrated in Fig.3.As shown in Fig.3, the CDFT combinatorial method is as capable of reproducing the experimental data on the neutron separation energy as the phenomenological models; in particular, the CDFT combinatorial method can adequately describe the NLD of51V.In addition, the overall description of the experimental data below the neutron separation energy provided by the CDFT combinatorial method is similar to that of the phenomenological methods.In conclusion, although there are few differences in the results for each NLD model, the overall ability to describe the experimental data is similar.

Fig.3 (Color online) Comparison of NLDs obtained from the CDFT combinatorial method with other NLD predictions [57] and experimental data [58, 58, 59, 59, 66, 70].The full asterisk corresponds to the experimental data at the neutron separation energy Sn [56]

The most extensive and reliable sources of experimental information on NLD are the s-wave neutron resonance spacingD0[56] and observed low-energy excited levels[56].To measure the dispersion between the theoretical and experimentalD0, thefrmsfactor is defined as follows:

whereDth(Dexp.) is the theoretical (experimental) resonance spacing, andNeis the number of nuclei in the compilation.The results for the ratio ofDthtoDexp.are presented in Fig.4.Upon calculation, we find thatfrms=3.62 , with a total of 66 nuclei.Under the same calculation conditions,frmsof the Gogny (D1S) interaction based on the HFB combinatorial method is measured as 7.25 [11].At present, the result of the CDFT combinatorial method is larger than thefrms=1.8 deviation of the phenomenological BFM [10]and thefrms=2.1 value obtained using the microstatistical method [31].However, the HFB combinatorial method based on the Skyrme effective interaction improves the pairing correlation [55] and collective effects, yielding the valuefrms=2.3 [37].Future studies can extend the same approach to improve the CDFT combinatorial method.

Fig.4 Ratio of the s-wave neutron resonance spacing for the CDFT combinatorial method ( Dth ) and the experimental data ( Dexp. ), as compiled in Ref.[56]

When phenomenological NLDs are used in nuclear physics applications, such as nuclear data evaluation or accurate and reliable estimation of reaction cross-sections,a few parameters are used considering the dependence of the phenomenological expressions [71].The results of the combinatorial method can also normalize both the experimental-level scheme at a low excitation energy and the neutron resonance spacing atU=Snin a manner similar to what is typically used for analytical formulas.More precisely, the normalized level density can be obtained as follows: [37]

where the energy shiftδis extracted from the analysis of the cumulative number of levels andαcan be obtained using the following expression:

With this normalization, the experimental low-lying states andD0values can be reproduced reasonably well.

Fig.6 Normalized parameters α (left) and δ (right) are plotted as functions of the atomic mass number

Fig.5 (Color online) NLDs obtained using the CDFT combinatorial method with (red) and without (black) normalization.The experimental data are taken from Refs.[56, 66, 70]

Finally, the NLDs calculated using the CDFT combinatorial method are compared with the observed low-energy excited levels, which constitute the most extensive and reliable source of experimental information on NLDs [56].The cumulative number of nuclear levelsN(U) indicates the sum of the number of levels below the excitation energyU(includingU)

Fig.7 (Color online) Comparison of the results obtained from the CDFT combinatorial method (red lines) with the cumulative number of observed levels [56] (black lines) as a function of the excitation energy

Fig.8 (Color online) Comparison between the calculated NLDs based on the CDFT combinatorial method (red lines) and experimental data [58,59, 66–68, 70, 72–75] (black dot).The full asterisk corresponds to the experimental data at the neutron separation energy Sn [56]

Here,N(U,M,P) is the cumulative number of levels under spinMand parityPat the excitation energyE.The predicted cumulative number of levelsN(U) is compared with the experimental value [56], as shown in Fig.7, and the results include both light and heavy nuclei, as well as spherical and deformed nuclei.Overall, the results of the CDFT combinatorial method are in reasonable agreement with the experimental data at lower excitation energies, especially for light nuclei.At high excitation energies, the theoretically calculated cumulative number of nuclear levels is higher than the experimental value.The NLDs increase rapidly with increasing excitation energy, andN(U) is expected to gradually increase at high excitation energies.However, the experimental levels of the cumulative number gradually stabilize.This may be attributed to the limitations imposed by the experimental conditions owing to which the number of levels cannot be counted completely.

In Fig.8, the CDFT combinatorial method predictions after normalization are compared with the experimental data extracted using the Oslo method [58, 59, 66–68, 70, 72, 73]and the particle evaporation spectrum [74, 75].The Oslo method is model-dependent.To extract the absolute value of the total level density from the measured data, the experimental NLDs must be normalized by the total level density at the neutron binding energy, which in turn is derived from the neutron resonance spacing.For a meaningful comparison between the CDFT combinatorial predictions and data obtained using the Oslo method, it is important to normalize the NLDs obtained using the CDFT combinatorial method to the level density value atU=Snconsidered by the Oslo group.As shown in Fig.8, the results of the CDFT combinatorial method after normalization agree well with the experimental data belowSn, except for the small NLDs of111Cd and161Dy at low excitation energies.This low result is due to the larger energy spacing of the theoretically calculated single-particle levels near the Fermi level.Overall,the results obtained using the CDFT combinatorial method are reliable.

4 Summary and prospects

A combinatorial method was adopted to describe the nuclear level densities for nuclear reaction calculations.The particle-hole state density was obtained using a combinatorial method with the single-particle level provided by the PK1 effective interaction based on CDFT.Energy-,spin-, and parity-dependent NLDs were obtained after accounting for collective effects, including vibration and rotation.Our results were compared with those obtained using other NLD models, including phenomenological,microstatistical, and nonrelativistic HFB combinatorial models.The comparison suggests that bar some small deviations among the different NLD models, the general trends among these models are essentially the same.In conclusion, the CDFT combination method can reproduce experimental data at or below the neutron separation energy.This implies that the CDFT combinatorial method is as reliable as the other models at describing NLDs.Finally, the comparison of the NLDs of the CDFT combinatorial method with normalization with the experimental data exhibited excellent agreement between the observed cumulative number of levels at low excitation energies and the measured NLDs below the neutron separation energy.Our results demonstrate the predictive power of the CDFT combinatorial method.However, in our approach,pairing correlations are not considered and the collective effects are empirical.In our future work, we aim to improve upon the CDFT combinatorial method by considering the inclusion of energy-dependent pairing correlations and adopting a partition function approach to the treatment of vibrational enhancement.These results can potentially help our study of important neutron capture processes, such as the r-process.

Author ContributionsAll authors contributed to the study conception and design.Material preparation, data collection and analysis were performed by Kun-Peng Geng, Peng-Xiang Du, Jian Li and Dong-Liang Fang.The first draft of the manuscript was written by Kun-Peng Geng and Peng-Xiang Du, and all authors commented on previous versions of the manuscript.All authors read and approved the final manuscript.Data availabilityThe data that support the findings of this study are openly available in Science Data Bank at https:// www.doi.org/ 10.57760/ scien cedb.j00186.00211 and https:// cstr.cn/ 10.57760/ scien cedb.j00186.00211.

Declarations

Conflict of interestThe authors declare that they have no conflict of interest.