Introduction

As a fundamental nuclear property, nuclear level density (NLD) plays a crucial role in many applications, such as the calculation of reaction cross sections with nucleosynthesis, the nuclear reaction calculation program TALYS [1], and the Hauser–Feshbach model for compound nucleus calculations [2, 3]. Owing to the complexity of nucleon interactions and the fact that the level density increases exponentially with an increase in the excitation energy, the accurate calculation of the NLD has long been a theoretical challenge.

Many methods for estimating NLD have been developed. The most common method is the Bethe formula based on the zero-order approximation of the partition function of the Fermi gas model [4, 5]. In attempting to reproduce the experimental data, various phenomenological modifications to Bethe’s original analytical formulation have been proposed—particularly to account for shell, pairing, and deformation effects—which led first to the constant-temperature formulation, then to the shifted Fermi gas model, and later to the popular back-shifted Fermi gas model [6, 2, 7].

There are many microscopic methods for calculating NLD, including the shell-model Monte Carlo method [8-10], the moments method derived from random matrix theory and statistical spectroscopy [11, 12], the stochastic estimation method [13], the Lanczos method using realistic nuclear Hamiltonians [14], the self-consistent mean-field approach based on the extended Thomas–Fermi approximation with Skyrme forces [15], and the exact pairing plus independent particle model at a finite temperature [16-19]. On the basis of the Hartree–Fock–Bogoliubov (HFB) model, S. Hilaire and S. Goriely developed a microscopic approach to describe NLD with great success [20-22]. Microscopic methods based on the self-consistent Hartree–Fock (HF) plus BCS model [23-25] have also been developed.

Recently, on the basis of the relativistic Hartree–Bogoliubov model [26-29], J. Zhao et al. developed a method for describing NLD [30]. In this model, the partition function is determined using the same two-body interaction employed in the HF plus BCS and HFB mean-field models [25], which includes shell, pairing, and deformation effects self-consistently. The total level densities are the convolution of the intrinsic level density and the collective level density. The intrinsic level density is obtained by an inverse Laplace transform of the partition function with the saddle-point approximation [31, 32]. Previously, the collective enhancement is considered via a phenomenological or semi-empirical multiplicative factor for rotational and vibrational degrees of freedom [24, 33-36] or more microscopically via a combinatorial method using single-particle level schemes obtained through HF plus BCS or HFB calculations [20, 22]. In Ref. [30], the collective enhancement is determined from the eigenstates of a corresponding collective Hamiltonian that considers quadrupole or octupole degrees of freedom. Both the intrinsic level density and the collective enhancement are determined by the same global energy density functional and pairing interaction.

The success of the microscopic description of NLD in even–even nuclei prompts us to extend it to odd-A nuclei. The core–quasiparticle coupling (CQC) model introduces a quasiparticle to the neighboring even–even core nuclei within the same covariant density functionals and achieved progress for describing the quantum phase transition in odd-A nuclei. It is based on the covariant density functional theory (CDFT), which has achieved considerable success in describing ground-state properties of both spherical and deformed nuclei all over the nuclear chart [26, 27, 37-40]. Its successful applications include superheavy nuclei [41-45], pseudospin symmetry [40, 46, 47], single-particle resonances [48-50], hypernuclei [51-56], thermal shape transition [57-59], and shell correction [60-64].

In this study, the CQC model is applied to the calculation of collective levels in even–odd uranium isotopes. For even–even isotopes, the collective levels are attained via the five-dimensional collective Hamiltonian (5DCH) model [65]. Similar to Ref. [30], the intrinsic level density is obtained using the finite-temperature CDFT [66, 59, 58]. Because the level density of the saddle point plays an important role in the compound nuclei reactions [67, 68], the level density of the saddle point ${\rho }_{\text{sd}}$ and the level density of the global minima ${\rho }_{\text{min}}$ are analyzed. The ratio of the level density at the saddle point to the level density at the global minimum is the quantity of interest.

The remainder of this paper is organized as follows. The theoretical framework is introduced in Sect. 2. The results for ${}^{234-240}$U are presented in Sect. 3. Sect. 4 presents a short summary.

Theoretical framework

With the assumption of decoupling between intrinsic and collective degrees of freedom, the excitation energy of a nucleus can be written as $E^{*}=E_{\text{i}}+E_{\text{c}}$, where $E_{\text{c}}$ represents the collective excitation energy. The total level density is obtained as [36] ${\rho }_{\text{tot}}\left(E^{*}\right)={\displaystyle \int }{\rho }_{\text{i}}\left(E_{\text{i}}\right){\rho }_{\text{c}}\left(E^{*}-E_{\text{i}}\right)\text{d}{E}_{\text{i}}\mathrm{,}$ (1) with the collective level density given as ${\rho }_{\text{c}}(E)={\displaystyle \sum _{\text{c}}\delta }\left(E-E_{\text{c}}\right){\tau }_{\text{c}}\left(E_{\text{c}}\right)\mathrm{.}$ (2) For a collective state with the angular momentum $I_{\text{c}}$, the degeneracy is ${\tau }_{\text{c}}\left(E_{\text{c}}\right)=2I_{\text{c}}+1$.

The intrinsic level density ${\rho }_{\text{i}}$ can be obtained from the giant partition function of two types of particles via the inverse Laplace transform and saddle point approximation [32, 31]: ${\rho }_{\text{i}}=\frac{e^S}{{\left(2\pi \right)}^{3/2}{D}^{1/2}}\mathrm{,}$ (3) where $S$ represents the entropy, and $D$ is the determinant of a $3\times 3$ matrix that contains the second derivatives of the entropy with respect to the inverse temperature $\beta =1/\left(k_{B}T\right)$ and ${\mu }_{\tau }=\beta {\lambda }_{\tau }\left(\tau \equiv N\mathrm{,}Z\right)$ at the saddle point. The intrinsic excitation energy is calculated as $E_i(T)=E(T)-E(0)$, where $E(T)$ represents the binding energy of the nucleus at temperature $T$. The specific heat is defined by the relation $C_{\text{v}}=\partial E_i(T)/\partial T$.

According to the ideas presented in Ref. [32], the determinant $D$ can be simplified to the following form: $D=T^5{\frac{\partial S}{\partial T}|}_{NZ}{\frac{\partial N}{\partial {\lambda }_N}|}_{TZ}{\frac{\partial Z}{\partial {\lambda }_Z}|}_{T{\lambda }_N},$ (4) where $N$ and $Z$ represent the numbers of neutrons and protons respectively, and ${\lambda }_{\tau }\left(\tau \equiv N\mathrm{,}Z\right)$ denotes the neutron (proton) Fermi surface. For convenience, the temperature used is $k_{\text{B}}T$ (in units of MeV) and the entropy used is $S/k_{\text{B}}$ (dimensionless).

Entropy is extracted in the finite-temperature covariant density functional theory. In the covariant density functional theory, the Dirac equation for single nucleons is $\left[{\gamma }_{\mu }\left(i{\partial }^{\mu }-V^{\mu }\right)-\left(m+S\right)\right]{\psi }_k=\mathrm{0,}$ (5) where $m$ represents the nucleon mass, and ${\psi }_k(r)$ denotes the Dirac spinor field of a nucleon. The scalar $S(r)$ and vector potential $V^{\mu }(r)$ are $\begin{array}{c}S(r)={\alpha }_S{\rho }_S+{\beta }_S{\rho }_S^2+{\gamma }_S{\rho }_S^3+{\delta }_S\Delta {\rho }_S\\ V^{\mu }(r)={\alpha }_V{j}_V^{\mu }+{\gamma }_V{\left(j_V^{\mu }\right)}^3+{\delta }_V\Delta j_V^{\mu }\\ +{\tau }_3{\alpha }_{TV}{\overrightarrow{j}}_{TV}^{\mu }+{\tau }_3{\delta }_{TV}\Delta {\overrightarrow{j}}_{TV}^{\mu }+e{A}^{\mu }\end{array}$ (6)

respectively. The isoscalar density ${\rho }_S$, isoscalar current $j_V^{\mu }$, and isovector current ${\overrightarrow{j}}_{TV}^{\mu }$ have the following forms: $\begin{array}{l}{\rho }_S(r)={\displaystyle \sum _k{\overline{\psi }}_k}(r){\psi }_k(r)\left[v_k^2\left(1-2f_k\right)+f_k\right]\\ j_V^{\mu }(r)={\displaystyle \sum _k{\overline{\psi }}_k}(r){\gamma }^{\mu }{\psi }_k(r)\left[v_k^2\left(1-2f_k\right)+f_k\right]\\ {\overrightarrow{j}}_{TV}^{\mu }(r)={\displaystyle \sum _k{\overline{\psi }}_k}(r)\overrightarrow{\tau }{\gamma }^{\mu }{\psi }_k(r)\left[v_k^2\left(1-2f_k\right)+f_k\right],\end{array}$ (7) where $f_k$ represents the thermal occupation probability of quasiparticle states, having the form $f_k=1/\left(1+e^{E_k/k_{B}T}\right)$. $E_k$ represents the quasiparticle energy for the single-particle state $k$, and $E_k={\left[{\left({ϵ}_k-\lambda \right)}^2+{\Delta }_k^2\right]}^{\frac{1}{2}}$. The BCS occupation probabilities $v_k^2$ and the related $u_k^2=1-v_k^2$ are obtained as follows: $\begin{array}{l}{v}_k^2=\frac{1}{2}\left(1-\frac{{ϵ}_k-\lambda }{E_k}\right)\\ u_k^2=\frac{1}{2}\left(1+\frac{{ϵ}_k-\lambda }{E_k}\right),\end{array}$ (8) ${\Delta }_k$ is the pairing gap parameter, which satisfies the gap equation at a finite temperature: ${\Delta }_k=-\frac{1}{2}{\displaystyle \sum _{k^{\text{'}}>0}{V}_{k\overline{k}{k}^{\text{'}}{\overline{k}}^{\text{'}}}^{pp}}\frac{{\Delta }_{k^{\text{'}}}}{E_{k^{\text{'}}}}\left(1-2f_{k^{\text{'}}}\right)\mathrm{.}$ (9) The particle number $N(Z)$ is restricted by $N(Z)=2{\displaystyle \sum _{k>0}\left[v_k^2\left(1-2f_k\right)+f_k\right]}$. The entropy is computed using the relation $S=-k_{\text{B}}{\displaystyle \sum _k\left[f_k\ln f_k+\left(1-f_k\right)\ln \left(1-f_k\right)\right]}\mathrm{.}$ (10) For even–even nuclei, the collective levels are obtained via the five-dimensional collective Hamiltonian (5DCH) [65]. All the collective parameters, such as the inertia parameters and the collective potential, are extracted from the constrained CDFT+BCS in the triaxial deformation space.

For odd-$A$ nuclei, the collective levels are calculated using the CQC model [69], whose collective Hamiltonian is expressed as $\begin{array}{c}H=H_{\text{qp}}+H_{\text{c}}\\ =\left(\begin{array}{cc}\left({\epsilon }^{A-1}-\lambda \right)+{\Gamma }^{A-1}& {\Delta }^{A+1}\\ {\Delta }^{†A-1}& -\left({\epsilon }^{A+1}-\lambda \right)-{\Gamma }^{A+1}\end{array}\right)\\ +\left(\begin{array}{cc}{E}^{A-1}& 0\\ 0& E^{A+1}\end{array}\right),\end{array}$ (11) where $\lambda$ denotes the Fermi surface, and ${\epsilon }^{A\pm 1}$ and $E^{A\pm 1}$ represent the single-particle energy and the collective excitation energy for the even–even $A\pm 1$ core, respectively. $\Gamma$ and $\Delta$ denote the mean and pairing fields associated with long-range quadruple–quadruple particle–hole interactions and short-range monopole particle–particle interactions between the odd nucleon and core, respectively. The $\Gamma$ field can be expressed as $\begin{array}{c}{\Gamma }^{A\pm 1}=-\chi {\left(-1\right)}^{j+R+J}｛\begin{array}{ccc}j& 2& j^{\text{'}}\\ R^{\text{'}}& J& R\end{array}｝{\langle \mu j\Vert {\widehat{Q}}_2\Vert {\mu }^{\text{'}}{j}^{\text{'}}\rangle }^{A\pm 1}\\ \times {\langle \nu R\Vert {\widehat{Q}}_2\Vert {\nu }^{\text{'}}{R}^{\text{'}}\rangle }^{A\pm 1},\end{array}$ (12) where ${\langle \mu j\Vert {\widehat{Q}}_2\Vert {\mu }^{\text{'}}{j}^{\text{'}}\rangle }^{A\pm 1}$ and ${\langle \nu R\Vert {\widehat{Q}}_2\Vert {\nu }^{\text{'}}{R}^{\text{'}}\rangle }^{A\pm 1}$ are the reduced quadrupole matrix elements of the spherical hole (particle) and cores, respectively. The Fermi surface $\lambda$ and coupling strength of the quadrupole field $\chi$ are left as free parameters that are fit to data separately for positive- and negative-parity states.

In this study, the 5DCH and CQC models are based on the CDFT calculation with the harmonic oscillator basis $N_f=16$.

Results and Discussion

The parameter sets of covariant density functional theory used in this study are PC-PK1 [70] and DD-LZ1 [71]. PC-PK1 is one of the most widely used point-coupling parameter sets, and DD-LZ1 is a density-dependent parameter set that aims to alleviate the spurious shell closure. The pairing effect is considered by the separable pairing force [72]. The nuclei considered are even–even ${}^{234-240}$U and even–odd ${}^{235-239}$U.

In the first step, large-scale finite-temperature CDFT calculations are performed for ${}^{234-240}$U in the temperature range of 0–2 MeV in the $({\beta }_2\mathrm{,}\gamma )$ plane. Fig. 1 shows the free energy surface evolution with respect to the temperature for ${}^{235}$U. The deformations of the global minimum and the saddle point change slightly with an increase in the temperature; i.e., with the increase in the temperature, the global minimum deformation ${\beta }_2$ decreases, while the saddle point deformation $\gamma$ slowly moves toward the prolate axis ($\gamma {=0}^{\circ }$) and ${\beta }_2$ remains nearly constant. The saddle point gradually becomes indistinct as the temperature approaches 1.6 MeV. The free energy surface evolution for other nuclei ${}^{\mathrm{234,236}-240}$U is similar to that for ${}^{235}$U. For the parameter set DD-LZ1, the energy surface shapes are similar, while the fission barrier heights are larger. Moreover, the free energy surfaces of ${}^{234-240}$U considering the axial octupole deformations are checked, and it is found that although the PESs of some nuclei are relatively soft in the octupole direction, there is no significant octupole deformation for the global minimum and first saddle point. Therefore, in the following calculation, the deformation space is limited to the (${\beta }_2$, $\gamma$) space.

Fig. 1Full-screen View

(Color online) Free energy surfaces in the (${\beta }_2$, $\gamma$) plane at temperatures of (a) 0, (b) 0.4, (c) 0.8, (d) 1.2, (e) 1.6, and (f) 2.0 MeV for ²³⁵U obtained via finite-temperature CDFT calculations using the parameter set PC-PK1. The global minimum and saddle points are represented by squares and stars, respectively. The energy separation between contour lines is 0.5 MeV.

It is shown that the intrinsic level density has an exponential relationship with the entropy in Eq. (3). To study the dependence of the entropy on the nuclear deformations, Fig. 2 shows the entropy surfaces at $T$=0.4 MeV and 0.8 MeV in the (${\beta }_2\mathrm{,}\gamma$) plane calculated using two parameter sets PC-PK1 and DD-LZ1. The global minima and the saddle points displayed in free energy potential surfaces Fig. 1 are also marked in Fig. 2. Comparing these two figures reveals that the entropy has a low value near the free energy global minimum, and it becomes high near the saddle point. There are numerous similarities between the entropy surface and the free energy surface. For the low temperature of $T$=0.4 MeV, there are sharp changes within a certain deformation range-particularly for DD-LZ1. For the high temperature of $T$=0.8 MeV, the entropy surfaces share substantial common features. Furthermore, several derivatives composing the $\sqrt{\left|D\right|}$ term, which appears as the denominator in Eq. (3), are extracted for all the deformation grid points, and the intrinsic level density ${\rho }_{\text{i}}$ is settled. The intrinsic level density surface on the logarithmic scale is analogous to the entropy surface here. It is omitted owing to space limitations.

Fig. 2Full-screen View

(Color online) Entropy S for ${}^{235}$U at $T=$ 0.4 MeV (upper row) and 0.8 MeV (lower row) obtained via CDFT calculations using the parameter sets PC-PK1 (left) and DD-LZ1 (right) in the (${\beta }_2$, $\gamma$) plane.

In the nuclear reactions, the level density of the saddle point is critical. Fig. 3 shows several properties of the saddle points with respect to the temperature for ${}^{234-240}$U calculated with the parameter set PC-PK1, i.e., the excitation energies $E_i$, the entropy $S$, the specific heat $C_{\text{v}}$, the partial derivative of the entropy with respect to the temperature $\partial S/\partial T$, the pairing energy gap $\Delta$, and the intrinsic level density ${\rho }_{\text{i}}$. The temperature range ends at 1.4 MeV, which corresponds to the excitation energy of approximately 40 MeV. For higher temperatures up to 2 MeV, the trends do not change. In Fig. 3(e), pairing phase transitions occur at a temperature of approximately $0.6$ MeV for all nuclei. When $T\approx 0.4$ MeV, more curve details about $C_{\text{v}}$ and $\partial S/\partial T$ in Fig. 3(c) and (d) can be observed. For $C_{\text{v}}$, the curve reaches a local maximum and then decreases slightly, and the segments $T<0.4$ MeV and $T>0.6$ MeV are essentially two straight lines. Because the specific heat is the partial derivative of the excitation energy with respect to the temperature, it supports the fact that the excitation energy curves in Fig. 3(a) are actually two connecting quadratic parabolas. This is consistent with the fact that in the Fermi-gas model, the intrinsic excitation energy increases quadratically with respect to the temperature $E_i\propto T^2$ with slope changes around the pairing phase transition. For $\partial S/\partial T$, it reaches 60 MeV ${}^{-1}$, decreases to 40 MeV ${}^{-1}$, and then remains constant for $T>0.6$ MeV. Because of the direct relationship between $\partial S/\partial T$ and entropy $S$, the entropy should be quadratic for $T<0.4$ MeV and linear for $T>0.6$ MeV. This confirms the classic relation $S\propto T$ but only for high temperatures. The logarithmic intrinsic level density ${\rho }_{\text{i}}$ in Fig. 3(f) has analogous temperature dependence to the entropy $S$ in Fig. 3(b), representing $\ln ({\rho }_{\text{i}})\propto S$ in Eq. (3).

Fig. 3Full-screen View

(Color online) Excitation energy $E_i$ (a), entropy $S$ (b), specific heat $C_{\text{v}}$ (c), $\partial S/\partial T$ (d), pairing energy gap $\Delta$ (e), and intrinsic level density ${\rho }_{\text{i}}$ (f) with respect to the temperature for saddle points of ${}^{234-240}$U. The results were obtained via finite-temperature triaxial CDFT calculations with the parameter set PC-PK1.

For comparison, the corresponding properties of the global minimum of ${}^{234-240}$U are shown in Fig. 4. In Fig. 4(e), the pairing phase transition occurs at $T_{\text{c}}=0.6\sim 0.7$ MeV, while the proton gap is larger than the corresponding neutron gap at low temperatures. For the specific heat $C_{\text{v}}$ in Fig. 4 (c), the slopes before and after the phase transition are close, rendering the excitation energy curve in Fig. 4 (a) a smooth parabola. At high temperatures, the partial derivative $\partial S/\partial T$ in Fig. 4 (d) has nearly the same constant as that in Fig. 3(d). Because the intrinsic level density increases exponentially with respect to the entropy, this indicates that both the entropy $S$ in Fig. 3(b) and Fig. 4(b) and the logarithmic intrinsic level density in Fig. 3(f) and Fig. 4(f) have the same trends at a high excitation energy.

Fig. 4Full-screen View

(Color online) Same as Fig. 3 but for the global minimum.

In addition, the result for the parameter set DD-LZ1 is obtained. The specific heat $C_{\text{v}}$ varies more gently near the pairing phase transition, while other properties are roughly the same as those for PC-PK1.

In the second step, the collective level densities of odd-$A$ nuclei are calculated via the CQC model, while those of even–even nuclei are obtained via the 5DCH model. The collective Hamiltonian of the odd-$A$ nuclei is obtained by coupling the core parts of the two adjacent even–even nuclei and one particle or hole in the spherical case. The two free parameters of the model, i.e., the Fermi surface $\lambda$ and coupling strength $\chi$, are adjusted according to the experimental values of the low excitation spectrum. Details can be found in Ref. [69]. The CQC parameters for the odd-$A$ ${}^{235-239}$U nuclei corresponding to two density functional parameter sets PC-PK1 and DD-LZ1 are presented in Table 1.

Take ²³⁵U as an example. Its low excitation spectra obtained using PC-PK1 and DD-LZ1 together with the experimental data from the NNDC [73] are shown in Fig. 5. The coupling strength and Fermi surface are finely tuned to reflect the collective enhancement induced by negative parity according to the corresponding experimental data of the low excitation spectrum. The calculated levels exhibit good qualitative agreement with the experimental results.

Fig. 5Full-screen View

Calculated low-energy positive-parity bands (panels a and b), of ²³⁵U based on triaxial CDFT calculations with the parameter sets PC-PK1 (red) and DD-LZ1 (blue) together with the available experimental data (black) [73].

In the final step, when performing the convolution using the Eq. (1) to obtain the total level density, it is found that the total level density ${\rho }_{\text{tot}}$ depends on the number of collective levels considered. This implies that a specific collective truncation parameter should be introduced and adjusted manually. To alleviate this problem, inspired by Ref.[74, 36], a factor $\exp \left(-E_c/T\right)$ is inserted into Eq. (1), and the total level density is rewritten as ${\rho }_{\text{tot}}\left(E^{*}\right)={\displaystyle \int }\exp \left(-\frac{E^{*}-E_i}{T}\right){\rho }_{\text{i}}\left(E_{\text{i}}\right){\rho }_{\text{c}}\left(E^{*}-E_i\right)\text{d}{E}_{\text{i}}\mathrm{.}$ (13) This factor simulates the coupling between the collective levels and intrinsic levels: higher collective levels correspond to a weaker contribution to the convolution; thus, a fixed truncation parameter is not needed. According to the highest excitation energy considered, tests reveal that an angular momentum up to 18 $\hslash$ with the first 20 collective levels for each angular momentum is sufficient.

In Fig. 6, the total level density and the intrinsic level density at the global minima and saddle points are compared with available experimental data below 5 MeV [75], available calculations in the TALYS-1.95 package [1] directory density/ground/hilaire or density/fission/hilaire/Max1 and directory density/ground/goriely or density/fission/goriely/inner, or the Reference Input Parameter Library (RIPL) [20, 22]. Clearly, the intrinsic level densities deviate from the experimental data, indicating the necessity of collective degrees of freedom. For excitation energies up to 10 MeV, the total level density calculated via the two parameter sets PC-PK1 and DD-LZ1 agrees well with the two sets of theoretical calculations performed by S. Goriely and S. Hilaire in TALYS. However, for energies above 10 MeV, the total level density obtained via the relativistic density functional agrees well with Hilaire’s calculation, whereas Goriely’s calculation exhibits a sharp increase. The ratio of the total level density to Hilaire’s calculation result ${\rho }_{\text{tot}}/{\rho }_{\text{Hilaire}}$ is well within an order of magnitude.

Fig. 6Full-screen View

(Color online) Total level densities (solid) and intrinsic level density (dash-dotted) at the global minima ($\text{a}$-$\text{g}$) and saddle points (a'-g') of the ^234-240U with respect to the excitation energy $E^{*}$. The pink and blue lines correspond to calculations with the parameter sets PC-PK1 and DD-LZ1, respectively. The experimental data (black squares) from Ref. [75] and calculations from Refs. [20, 22] are compared.

Furthermore, we compare the total level densities at the saddle points ${\rho }_{\text{sd}}$ and those at the global minima ${\rho }_{\text{min}}$ and plot their ratio in Fig. 7. For low excitation energies up to 5 MeV, the majority of ${\rho }_{\text{sd}}$ values are smaller than ${\rho }_{\text{min}}$, which may be related to pairing association [76]. The pronounced peaks at 0–5 MeV for ${}^{237-239}$U are actually caused by a small drop in the intrinsic level density at the global minimum (Fig. 6). For intermediate energies in the range of 5 MeV $<E^{*}<$ 30 MeV, ${\rho }_{\text{sd}}$ increases more quickly than ${\rho }_{\text{min}}$. In particular, for ${}^{235}$U, this ratio increases linearly with respect to the excitation energy. When the atomic number increases from ${}^{234}$U to ${}^{240}$U, the curve of this ratio tends to shift downward.

Fig. 7Full-screen View

(Color online) Ratio of the total level density at the saddle point to that at the global minima ${\rho }_{\text{sd}}/{\rho }_{\text{min}}$ in the logarithmic scale with respect to the excitation energy for ${}^{234-240}$U based on triaxial CDFT calculations with the parameter set PC-PK1. The embedded subfigure is for ²³⁵U in the linear scale.

Finally, it is convenient to study the ratio of the total level density to the intrinsic level density $K_{\text{coll}}(E^{*})={\rho }_{\text{tot}}(E^{*})/{\rho }_{\text{i}}(E^{*})$ [30] as a collective enhancement factor and attribute it to the inclusion effect of collective levels. Fig. 8 shows this collective enhancement factor $K_{\text{coll}}$ with respect to the excitation energy $E^{*}$ for ^234-240U. Clearly, the curves can be divided into two groups: one group consists of low-lying concentrated curves for even–even nuclei ^234-240U, and the other group consists of scattered curves for three odd-$A$ nuclei ${}^{235-239}$U. This implies different collective spectrum patterns for even–even nuclei and odd-$A$ nuclei. For even–odd nuclei, the factor $K_{\text{coll}}$ obtained via the parameter set DD-LZ1 exceeds that for PC-PK1. The magnitude of $K_{\text{coll}}$ is similar to that of the nuclei ${}^{\mathrm{94,96,98}}$Mo, ${}^{\mathrm{106,108}}$Pd, and ${}^{\mathrm{106,112}}$Cd discussed in Fig. 5(a) of Ref. [30] at the low excitation energy. The collective enhancement factor shown in this figure undoubtedly indicates the fact that the collective levels of odd-$A$ nuclei are lower and denser than those of even–even nuclei. For level densities at saddle points, this factor $K_{\text{coll}}$ is almost indistinguishable from those of the global minimum.

Fig. 8Full-screen View

(Color online) Collective enhancement factor $K_{\text{coll}}$ with respect to the excitation energy for global minima of ^234-240U based on triaxial CDFT calculations with the parameter sets PC-PK1 and DD-LZ1.

Summary

In this study, according to the finite-temperature covariant density functional theory, the total level densities were obtained by convolving the intrinsic level densities and the collective levels achieved using the parameter sets PC-PK1 and DD-LZ1. For saddle points on the free energy surface in the $({\beta }_2\mathrm{,}\gamma )$ plane, the entropy and several derivatives composing the $\sqrt{\left|D\right|}$ term were extracted, and the intrinsic level density, which has an exponential relationship with the entropy, was determined. The collective levels of even–even nuclei were calculated using the five-dimensional collective Hamiltonian model, and those of odd-$A$ nuclei were calculated using the CQC model. The total level densities of ${}^{234-240}$U agreed well with the available experimental data and Hilaire’s result. The behavior of even–even nuclei and odd-$A$ nuclei can be easily distinguished from the collective enhancement factor $K_{\text{coll}}$.