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 the β-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 micro-statistical 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 products ^133-137Xe and ^135-138Ba [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 parity-dependent 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 of ^94,96,98Mo and ^106,108Pd as well as the odd-A nuclei 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.

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 fields S and V derived 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: $\left[\alpha \cdot \text{p}+\beta \left(m+S\right)+V\right]{\psi }_i={\epsilon }_i{\psi }_i\mathrm{,}$ (1) where εi is the single-particle energy required to calculate the NLDs and S and V are the relativistic scalar field and time-like component of the vector field, respectively.

Upon obtaining the energy ε, spin projection m, and parity p of 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 density ρi; the generating function 𝒵 is expressed as: $\mathcal{Z}\left(x_1\mathrm{,}{x}_2\mathrm{,}{x}_3\mathrm{,}{x}_4\right)={\displaystyle \prod _{k=1}^4{\displaystyle \prod _{i=1}^{I_k}\left(1+x_k{p}_i^k{y}^{{\epsilon }_i^k}{t}^{m_i^k}\right)}}\mathrm{.}$ (2) The above generating function is a straightforward generalization of the function used previously [15] that accounts for not only the energy ${\epsilon }_i^k$ but also the spin projection $m_i^k$ and parity $p_i^k$. The variables xk (k = 1, ..., 4) enable us to count the number of particles and holes, y enables us monitor the excitation energies and t of the spin projections, and the indices I₁,...,I₄ denote the number of discrete states considered for each set of single-particle and single-hole states. In practice, I₂ = Z and I₄ = N for nuclei with Z protons and N neutrons. 𝒵 can be expanded into powers of xk as described below: $\mathcal{Z}\left(x_1\mathrm{,}{x}_2\mathrm{,}{x}_3\mathrm{,}{x}_4\right)={\displaystyle \sum _{\mathcal{N}}{\mathcal{F}}_{\mathcal{N}}}\left(y\mathrm{,}t\right){\displaystyle \prod _{k=1}^4{x}_k^{N_k}}\mathrm{.}$ (3) The symbol 𝒩 denotes any integer combination (N₁,N₂,N₃,N₄). The function $\mathcal{F}(y\mathrm{,}t)$ can be expanded into powers of y and t as follows: ${\mathcal{F}}_{\mathcal{N}}\left(y\mathrm{,}t\right)={\displaystyle \sum _U{\displaystyle \sum _M{\displaystyle \sum _{P=-\mathrm{1,}+1}{C}_{\mathcal{N}}}}}\left(U\mathrm{,}M\mathrm{,}P\right)y^U{t}^M\mathrm{,}$ (4) where U, M, and P denote the excitation energy, spin projection, and parity, respectively. The coefficients $C_{\mathcal{N}}\left(U\mathrm{,}M\mathrm{,}P\right)$ represent the number of solutions. Using the coefficients $C_{\mathcal{N}}\left(U\mathrm{,}M\mathrm{,}P\right)$, the simplest definition of the particle-hole state density ${\rho }_i$ is obtained as: ${\rho }_i\left(U\mathrm{,}M\mathrm{,}P\right)=\frac{1}{{\epsilon }_0}{C}_{\mathcal{N}}\left(U\mathrm{,}M\mathrm{,}P\right)\mathrm{.}$ (5) However, the state density ${\rho }_i$ is strongly dependent on the unit energy ε₀. Therefore, another method suggested by Williams [53] is employed in this study to limit the discretization effects, as explained below.

Summing all the $C_{\mathcal{N}}$ values up to a given excitation energy U, we first obtain the cumulative number of states $N_{\mathcal{N}}\left(U\mathrm{,}M\mathrm{,}P\right)$, which represents the number of particle-hole states with excitation energy E such that 0≤E<U. The particle-hole state density is defined as: ${\rho }_i=\frac{d{N}_{\mathcal{N}}\left(U\mathrm{,}M\mathrm{,}P\right)}{dU}\mathrm{.}$ (6) The calculated particle-hole state density ρi is 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]: ${\rho }_{\text{sph}}\left(U\mathrm{,}M\mathrm{,}P\right)={\rho }_i\left(U\mathrm{,}M=J\mathrm{,}P\right)-{\rho }_i\left(U\mathrm{,}M=J+\mathrm{1,}P\right)\mathrm{.}$ (7) For deformed nuclei, within the hypothesis of axial symmetry, the NLD, after accounting for the rotational effect, is expressed as: $\begin{array}{c}{\rho }_{\text{def}}\left(U\mathrm{,}M\mathrm{,}P\right)=\frac{1}{2}\left[{\displaystyle \sum _{K=-J\mathrm{,}K\ne 0}^J{\rho }_i\left(U-E_{\text{rot}}^{J\mathrm{,}K}\mathrm{,}K\mathrm{,}P\right)}\right]\\ +({\delta }_{\left(J\text{even}\right)}{\delta }_{\left(p=+\right)}{\rho }_i\left(U-E_{\text{rot}}^{J\mathrm{,}K}\mathrm{,0,}P\right)\\ +{\delta }_{\left(J\text{odd}\right)}{\delta }_{(p=-)}{\rho }_i\left(U-E_{\text{rot}}^{J\mathrm{,}K}\mathrm{,0,}P\right)).\end{array}$ (8) The fraction 1/2 accounts for the fact that in mirror axially symmetric nuclei, the intrinsic states with spin projections +K and -K yield the same rotational levels. Moreover, in the second term of the summation, the symbol δ_(x) (defined by δ_(x)=1 if x holds true and 0 otherwise) restricts the rotational bands built on intrinsic states with spin projection K=0 and parity P to level sequences 0, 2, 4,... for P=+ and 1, 3, 5,... for P=-. Finally, the rotational energy is obtained using the following expression [54]: $E_{\text{rot}}^{J\mathrm{,}K}=\frac{J\left(J+1\right)-K^2}{2{\mathcal{J}}_{\perp }}\mathrm{,}$ (9) where ${\mathcal{J}}_{\perp }$ denotes the moment of inertia of a nucleus rotating around the axis perpendicular to the symmetry axis. In this study, ${\mathcal{J}}_{\perp }$ is approximated by the rigid-body value ${\mathcal{J}}_{\perp }^{\text{rigid}}$, which is expressed as: ${\mathcal{J}}_{\perp }^{\text{rigid}}=\frac{2}{5}m{R}^2\left(1+\sqrt{\frac{5}{16\pi }}{\beta }^2\right)$ (10) for an ellipsoidal shape with the quadrupole deformation parameter β. The vibration enhancement is approximated as [55]: $K_{\text{vib}}=\text{exp}\left[\delta S-\left(\delta U/T\right)\right]\mathrm{,}$ (11) where S denotes entropy, U denotes the excitation energy, and T denotes 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]: $\begin{array}{c}\rho \left(U\mathrm{,}M\mathrm{,}P\right)=\left[1-f_{\text{dam}}\left(U\right)\right]K_{\text{vib}}{\rho }_{\text{sph}}\left(U\mathrm{,}M\mathrm{,}P\right)\\ +f_{\text{dam}}\left(U\right)K_{\text{vib}}{\rho }_{\text{def}}\left(U\mathrm{,}M\mathrm{,}P\right).\end{array}$ (12) The damping function f_dam is expressed as follows [38]: $f_{\text{dam}}=1-\frac{1}{1+\exp \left(\beta -0.18\right)/0.038}\mathrm{,}$ (13) where β is the quadrupole deformation parameter. Parameters 0.18 and 0.038 are adjusted according to the experimental data at the neutron separation energy Sn [56]. The expression for f_dam developed 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.

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 as R_box = 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.

The combinatorial results rely on the properties of single-particle levels. To understand the effect of using different effective interactions on the NLDs, the NLDs of ¹¹²Cd and ¹⁶²Dy 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 nuclei ¹³²Sn [64], which explains the deviations of our results from the measurements presented in Fig.1. For the well-deformed ¹⁶²Dy, 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 single-particle energies around the Fermi level. Meanwhile, the entropy S obtained 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 of ¹¹²Cd [58] and ¹⁶²Dy [59] extracted by the Oslo group based on the analysis of particle-γ coincidence 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.

Fig. 1Full-screen View

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

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, and ²⁰⁸Pb) and deformed nuclei (⁹⁴Mo, ⁹⁵Mo, and ¹⁶⁶Er) calculated using the CDFT combinatorial method are compared with the corresponding experimental data, as illustrated in Fig. 2.

Fig. 2Full-screen View

(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].

For ⁵⁵Co, 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. For ¹³²Sn and ²⁰⁸Pb, 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 of ²⁰⁸Pb. The NLDs of ⁵⁵Co, ¹³²Sn, and ²⁰⁸Pb 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.

For ⁹⁴Mo, the results of both the CDFT and Skyrme combinatorial methods are closer to the experimental data than those of the Gogny combinatorial method. For ⁹⁵Mo and ¹⁶⁶Er, the NLDs obtained using the three combinatorial methods are similar and slightly higher than the experimental data. It should be emphasized that the deformed nuclei ⁹⁴Mo, ⁹⁵Mo, and ¹⁶⁶Er 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 for ⁹⁴Mo and ⁹⁵Mo. One possible reason for this is that the CDFT combinatorial method uses a rigid-body value when considering the rotational effect, which is inappropriate for soft nuclei [69].

In addition, the NLDs of ⁵⁵Co and ⁹⁴Mo can be adequately described by the microstatistical model [31] relative to the CDFT combinatorial method, while both models provide similar descriptions for the NLDs of ⁹⁵Mo and ¹⁶⁶Er. 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 distance D₀ of the s-wave resonance. The NLDs of the even-even nuclei (¹¹²Cd and ¹⁶²Dy), odd-A nuclei (⁵¹V, ⁹⁷Mo, and ¹¹⁹Sn), 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 of ⁵¹V. 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. 3Full-screen View

(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 spacing D₀ [56] and observed low-energy excited levels [56]. To measure the dispersion between the theoretical and experimental D₀, the f_rms factor is defined as: $f_{\text{rms}}=\text{exp}{\left[\frac{1}{N_e}{\displaystyle \sum _{i=1}^{N_e}{\text{ln}}^2}\frac{D_{\text{th}}^{\text{i}}}{D_{\text{exp}\text{.}}^{\text{i}}}\right]}^{1/2}\mathrm{,}$ (14) where D_th(D_exp.) is the theoretical (experimental) resonance spacing and Ne is the number of nuclei in the compilation. The results for the ratio of D_th to D_exp. are presented in Fig. 4. Upon calculation, we find that f_rms=3.62, with a total of 66 nuclei. Under the same calculation conditions, f_rms of 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 the f_rms=1.8 deviation of the phenomenological BFM [10]and the f_rms=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 value f_rms=2.3 [37]. Future studies can extend the same approach to improve the CDFT combinatorial method.

Fig. 4Full-screen View

Ratio of the s-wave neutron resonance spacing for the CDFT combinatorial method (D_th) and the experimental data (D_exp.), 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 at U=Sn in a manner similar to what is typically used for analytical formulas. More precisely, the normalized level density can be obtained as: [37] $\rho \left(U\mathrm{,}J\mathrm{,}P\right)=e^{\alpha \sqrt{U-\delta }}\times \rho \left(U-\delta \mathrm{,}J\mathrm{,}P\right)\mathrm{,}$ (15) where the energy shift δ is extracted from the analysis of the cumulative number of levels and α can be obtained using the following expression: ${\rho }_{\text{th}}\left(S_n\right)\times e^{\alpha \sqrt{S_n}}={\rho }_{\text{exp}\text{.}}\left(S_n\right)\mathrm{.}$ (16) With this normalization, the experimental low-lying states and D₀ values can be reproduced reasonably well.

As an illustration, the variation in the NLDs obtained using the CDFT combinatorial method before and after normalization is shown in Fig. 5. When normalization is applied, the NLDs pass through the experimental data at the neutron separation energy Sn. As shown in Fig. 5(a), if the theoretically calculated value is close to the experimental value at the neutron separation energy Sn, the results before and after normalization do not change significantly. As shown in Fig. 5(b), the results after normalization are in better agreement with the experimental data [70], particularly above 2 MeV. This is because the coefficient $e^{\alpha \sqrt{U-\delta }}$ is related to the excitation energy U, and when U is small, normalization has a negligible effect. α and δ values of some nuclei are presented in Fig. 6.

Fig. 5Full-screen View

(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].

Fig. 6Full-screen View

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

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 levels N(U) indicates the sum of the number of levels below the excitation energy U (including U) $N(U)={\displaystyle \sum _M{\displaystyle \sum _{P}N}}\left(U\mathrm{,}M\mathrm{,}P\right)\mathrm{.}$ (17) Here, N(U,M,P) is the cumulative number of levels under spin M and parity P at the excitation energy E. The predicted cumulative number of levels N(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, and N(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.

Fig. 7Full-screen View

(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.

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 at U=Sn considered by the Oslo group. As shown in Fig. 8, the results of the CDFT combinatorial method after normalization agree well with the experimental data below Sn, except for the small NLDs of ¹¹¹Cd and ¹⁶¹Dy 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.

Fig. 8Full-screen View

(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].

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 non-relativistic 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.