Microscopic model for the collective enhancement of nuclear level densities
aa r X i v : . [ nu c l - t h ] N ov Microscopic model for the collective enhancement of nuclear level densities
Jie Zhao ( 赵 杰 ), Tamara Nikˇsi´c, and Dario Vretenar
2, 3 Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518055, China Physics Department, Faculty of Science, University of Zagreb, Bijeniˇcka Cesta 32, Zagreb 10000, Croatia State Key Laboratory of Nuclear Physics and Technology,School of Physics, Peking University, Beijing 100871, China (Dated: November 4, 2020)A microscopic method for calculating nuclear level density (NLD) is developed, based on theframework of energy density functionals. Intrinsic level densities are computed from single-quasiparticle spectra obtained in a finite-temperature self-consistent mean-field (SCMF) calculationthat takes into account nuclear deformation, and is specified by the choice of the energy densityfunctional (EDF) and pairing interaction. The total level density is calculated by convoluting theintrinsic density with the corresponding collective level density, determined by the eigenstates ofa five-dimensional quadrupole or quadrupole plus octupole collective Hamiltonian. The parame-ters of the Hamiltonian (inertia parameters, collective potential) are consistently determined bydeformation-constrained SCMF calculations using the same EDF and pairing interaction. Themodel is applied in the calculation of NLD of , , Mo, , Pd, , Cd, , , Dy,
Er,and , Yb, in comparison with available data. It is shown that the collective enhancement of theintrinsic level density, consistently computed from the eigenstates of the corresponding collectiveHamiltonian, leads to total NLDs that are in very good agreement with data over the entire energyrange of measured values.
I. INTRODUCTION
Level density is a basic nuclear property that alsoplays a crucial role in many applications, from calcula-tion of reaction cross sections relevant for nucleosynthe-sis to energy production. An accurate computation ofnuclear level density (NLD) is a challenging theoreticaltask because of the complexity of the inter-nucleon inter-action and the fact that the number of levels increasesexponentially with excitation energy. The most widelyused methods for calculating NLDs have been based onthe Bethe formula, formulated with the partition func-tion in the zeroth-order approximation of the Fermi-gasmodel [1]. In realistic applications to finite nuclei variousphenomenological modifications to the original analyticalformula have been suggested [2, 3]. The extensions of theBethe formula and its limitations are discussed in Ref. [4].A number of microscopic approaches to modellingNLD have been reported, such as the Shell-ModelMonte Carlo method [5–7], the moments method de-rived from random matrix theory and statistic spec-troscopy [8, 9], the stochastic estimation method [10],the Lanczos method using realistic nuclear Hamiltoni-ans [11], the self-consistent mean-field approach based onthe extended Thomas-Fermi approximation with Skyrmeforces [12], and the exact pairing plus independent par-ticle model at finite temperature [13–16]. Microscopicmethods based on the self-consistent Hartree-Fock (HF)plus BCS model [17–19] and Hartree-Fock-Bogoliubov(HFB) model [20–22] have also been developed to de-scribe NLD. In this framework the partition function isdetermined using the same two-body interaction as in theHF plus BCS or HFB mean-field models [18] and, there-fore, shell, pairing, and deformation effects are includedself-consistently. The intrinsic level density is obtained by an inverse Laplace transform of the partition func-tion with the saddle-point approximation [23]. A collec-tive enhancement of the NLD can be taken into accountby a phenomenological or semi-empirical multiplicativefactor for rotational and vibrational degrees of freedom[19, 24–27], or more microscopically by a combinatorialmethod using single-particle level schemes obtained inHF plus BCS or HFB calculations [21, 22]. The successof the microscopic self-consistent HFB-based approach toNLDs has also motivated calculations of fission cross sec-tions [28, 29], and studies of nuclear shape evolution inthe fission process [30].In a recent calculation of level densities in Dy and Moisotopes [25], with single-particle spectra obtained froma Woods-Saxon potential, the collective enhancement ofthe level densities has been determined using availableexperimental levels at low excitation energies, and alsocompared with a phenomenological macroscopic model.However, in many cases, and especially in nuclei far fromstability, the phenomenological and semi-empirical ap-proaches cannot be applied on a quantitative level. Inthis work we develop a microscopic method for calculat-ing NLDs, in which the single-quasiparticle spectrum isobtained using a finite-temperature self-consistent mean-field (SCMF) method, while the collective enhancementis determined from the eigenstates of a corresponding col-lective Hamiltonian that takes into account quadrupoleand octupole degrees of freedom. Both the intrinsic leveldensity and the collective enhancement are determinedby the same global energy density functional and pairinginteraction.For the finite-temperature (FT) and deformation-constrained SCMF calculations we employ the relativisticHartree-Bogoliubov (RHB) model [31–34]. This modelhas been applied to structure studies over the wholemass table, and its beyond-mean-field extension, espe-cially the collective Hamiltonian approach [35], used in anumber of calculations of low-energy excitation spectra.Nuclear thermodynamics [36–38] and induced fission dy-namics [39, 40] have also been explored with the FT-RHBmodel.The theoretical framework and methods are introducedin Sec. II. The details of the calculation and the resultsfor , , Mo, , Pd, , Cd, , , Dy,
Er,and , Yb are discussed in Sec. III. Sec. IV containsa short summary of the principal results.
II. THEORETICAL FRAMEWORK
Assuming that a nucleus is in a state of thermal equilib-rium at temperature T , it can be described by the finitetemperature (FT) Hartree-Fock-Bogoliubov (HFB) the-ory [41, 42]. In the grand-canonical ensemble, the expec-tation value of any operator ˆ O is given by the ensembleaverage h ˆ O i = Tr [ ˆ D ˆ O ] , (1)where ˆ D is the density operator:ˆ D = 1 Z e − β ( ˆ H − λ ˆ N ) . (2) Z is the partition function, the inverse temperature β =1 /k B T with the Boltzmann constant k B , ˆ H is the Hamil-tonian of the system, λ denotes the chemical potential,and ˆ N is the particle number operator. The entropy ofthe compound nuclear system is S = − k B h ˆ D ln ˆ D i . Inthis work we employ the multidimensionally-constrained(MDC) RHB model [33, 43–45] at finite temperatureto calculate the single-nucleon quasiparticle states. Theminimization of the grand-canonical potential Ω = h ˆ H i + T S − µ h ˆ N i , where µ = βλ , yields the FT-RHB equation Z d r ′ (cid:18) h − λ ∆ − ∆ ∗ − h + λ (cid:19) (cid:18) U k V k (cid:19) = E k (cid:18) U k V k (cid:19) . (3) E k is the quasiparticle energy and ˆ h denotes the single-particle Hamiltonianˆ h = α · p + β [ M + S ( r )] + V ( r ) + Σ R ( r ) , (4)where, for the relativistic energy-density functional DD-PC1 [46], the scalar potential, vector potential, and re-arrangement terms read S = α S ( ρ ) ρ S + δ S △ ρ S ,V = α V ( ρ ) ρ V + α T V ( ρ ) ~ρ T V · ~τ + e − τ A , Σ R = 12 ∂α S ∂ρ ρ S + 12 ∂α V ∂ρ ρ V + 12 ∂α T V ∂ρ ρ T V , (5)respectively. M is the nucleon mass, α S ( ρ ), α V ( ρ ),and α T V ( ρ ) are density-dependent couplings for differ-ent space-isospace channels, δ S is the coupling constant of the derivative term, and e is the electric charge. Thesingle-nucleon densities ρ S (scalar-isoscalar density), ρ V (time-like component of the isoscalar current), and ρ T V (time-like component of the isovector current), are de-fined by the following relations: ρ S ( r ) = X k> V † k ( r ) γ (1 − f k ) V k ( r ) + U Tk ( r ) γ f k U ∗ k ( r ) ,ρ T V ( r ) = X k> V † k ( r ) τ (1 − f k ) V k ( r ) + U Tk ( r ) τ f k U ∗ k ( r ) ,ρ V ( r ) = X k> V † k ( r )(1 − f k ) V k ( r ) + U Tk ( r ) f k U ∗ k ( r ) , (6)where f k is the thermal occupation probability of thequasiparticle state k f k = 11 + e βE k , (7)The pairing potential reads∆( r σ , r σ ) = Z d r ′ d r ′ X σ ′ σ ′ V pp ( r σ , r σ , r ′ σ ′ , r ′ σ ′ ) × κ ( r ′ σ ′ , r ′ σ ′ ) , (8)where V pp is the effective pairing interaction and κ is thepairing tensor, κ = X k> V ∗ k (1 − f k ) U k + U k f k V † k . (9)In the particle-particle channel we use a separable pairingforce of finite range [47]: V ( r , r , r ′ , r ′ ) = G δ ( R − R ′ ) P ( r ) P ( r ′ ) 12 (1 − P σ ) , (10)where R = ( r + r ) / r = r − r denote the center-of-mass and the relative coordinates, respectively. P ( r )reads P ( r ) = 1(4 πa ) / e − r / a . (11)The two parameters of the interaction were originally ad-justed to reproduce the density dependence of the pair-ing gap in nuclear matter at the Fermi surface calculatedwith the D1S parameterization of the Gogny force [48].The entropy is computed using the relation: S = − k B X k [ f k ln f k + (1 − f k ) ln(1 − f k )] . (12)Employing the saddle point approximation [23], one ob-tains the following expression for the intrinsic level den-sity ρ i ρ i = e S (2 π ) / D / , (13)where D is the determinant of a 3 × β and µ τ = βλ τ ( τ ≡ p, n ) at the saddlepoint. The intrinsic excitation energy is calculated as U i ( T ) = E ( T ) − E (0), with E ( T ) the binding energy ofthe nucleus at temperature T .With the assumption of a decoupling between intrinsicand collective degrees of freedom, the excitation energyof a nucleus can be written as U = U i + U c , where U c is the collective excitation energy [25]. The total leveldensity is obtained as ρ tot ( U ) = Z ρ i ( U i ) ρ c ( U − U i ) dU i , (14)with the collective level density ρ c ( U ) = X c δ ( U − U c ) τ c ( U c ) . (15)For a collective state with the angular momentum I c , thedegeneracy is τ c ( U c ) = 2 I c + 1.In the microscopic model used in the present workthe collective levels are eigenstates either of the five-dimensional quadrupole [49] or the axial quadrupole-octupole Hamiltonians [50]. In the former case in whichone considers only quadrupole degrees of freedom, thecollective Hamiltonian readsˆ H coll ( β, γ ) = − ~ √ wr (cid:26) β (cid:20) ∂∂β r rw β B γγ ∂∂β − ∂∂β r rw β B βγ ∂∂γ (cid:21) + 1 β sin 3 γ (cid:20) − ∂∂γ r rw sin 3 γB βγ ∂∂β + 1 β ∂∂γ r rw sin 3 γB ββ ∂∂γ (cid:21)(cid:27) + 12 X k =1 ˆ J k I k + V ( β, γ ) , (16)where B ββ , B βγ , B γγ are the mass parameters, I k is themoment of inertia, V ( β, γ ) denotes the collective poten-tial, w and r are functions of the mass parameters andmoments of inertia.In the case that includes octupole correlations, thecurrent implementation of the collective Hamiltonian isrestricted to axial symmetry, that is, only the axialquadrupole and octupole deformations are considered ascollective coordinates. This approximation is justifiedin heavy, axially deformed nuclei that will be examinedin this work. The axial quadrupole-octupole collective Hamiltonian takes the formˆ H coll ( β , β ) = − ~ √ w I " ∂∂β r I w B ∂∂β − ∂∂β r I w B ∂∂β − ∂∂β r I w B ∂∂β + ∂∂β r I w B ∂∂β + ˆ J I + V ( β , β ) . (17)The mass parameters B , B , B , and the moment ofinertia I are functions of the quadrupole β and octupole β deformations. w = B B − B .The mass parameters, moments of inertia, and collec-tive potentials as functions of the collective coordinates q ≡ ( β, γ ) or ( β , β ), are completely determined by thedefomation-constrained self-consistent RHB calculationsat zero temperature for a specific choice of the nuclearenergy density functional and pairing interaction. In thepersent version of the model, the mass parameters de-fined as the inverse of the mass tensor B ij ( q ) = M − ij ( q ),are calculated in the perturbative cranking approxima-tion [51] M C p = ~ M − M (3) M − , (18)where (cid:2) M ( k ) (cid:3) ij = X µν D (cid:12)(cid:12)(cid:12) ˆ Q i (cid:12)(cid:12)(cid:12) µν E D µν (cid:12)(cid:12)(cid:12) ˆ Q j (cid:12)(cid:12)(cid:12) E ( E µ + E ν ) k . (19) | µν i are two-quasiparticle wave functions, and E µ and E ν the corresponding quasiparticle energies. ˆ Q i denotesthe multipole operators that correspond to the collectivedegrees of freedom. The collective potential V is obtainedby subtracting the vibrational zero-point energy (ZPE)from the total RHB deformation energy. Following theprescription of Refs. [52–55], the ZPE is computed usingthe Gaussian overlap approximation, E ZPE = 14 Tr h M − M (1) i . (20)The microscopic self-consistent solutions of the con-strained RHB equations, that is, the single-quasiparticleenergies and wave functions on the entire energy surfaceas functions of the deformations, provide the microscopicinput for the calculation of both the collective inertia andzero-point energy. The Inglis-Belyaev formula is used forthe rotational moment of inertia. From the diagonaliza-tion of the collective Hamiltonian one obtains the collec-tive energy spectrum.The deformation-dependent energy landscape ismapped in a self-consistent RHB calculation with con-straints on the mass multipole moments Q λµ = r λ Y λµ .The nuclear shape is parameterized by the deformationparameters β λµ = 4 π AR λ h Q λµ i . (21) (a) Mo Mo Mo Pd Pd Cd Cd E ne r g y ( M e V ) -10-50 (b) P a i r i ng ( M e V ) (c) E n t r op y Temperature (MeV) 0.2 0.4 0.6 0.8 1.0 (d) r i ( M e V - ) Temperature (MeV)
FIG. 1. (Color online) The energy of the equilibrium (global)minimum (a), the pairing energy (b), entropy (c), and in-trinsic level density ρ i (d), as functions of temperature for , , Mo, , Pd, and , Cd. The results are obtainedin finite-temperature triaxial RHB calculations with the DD-PC1 energy density functional and finite-range pairing inter-action, as described in the previous section.
The self-consistent RHB equations are solved by expand-ing the single-nucleon spinors in a harmonic oscillator(HO) basis. The present calculations have been per-formed in a HO basis truncated to N f = 20 oscillatorshells for the axially symmetric case (heavy Dy, Er, andYb nuclei), while N f = 16 has been used for the triax-ial case (medium-heavy Mo, Pd, and Cd isotopes). Fordetails of the MDC-RHB model we refer the reader toRefs. [44, 56]. III. ILLUSTRATIVE CALCULATIONS IN THEMASS A=100 AND A=160 REGIONS
The microscopic approach and the particular modeldeveloped in this work will be illustrated with calcula-tions of the total level densities for , , Mo, , Pd, , Cd, and , , Dy,
Er, , Yb. The rela-tivistic energy density functional DD-PC1 [46] is used inthe particle-hole channel, while particle-particle correla-tions are described by the separable pairing force (10) inthe Bogoliubov approximation.In the first step, for each nucleus a FT-RHB calcula-tion is performed at the equilibrium (global) minimumto determine the intrinsic level density. In Fig. 1 we dis-play the calculated energies of the equilibrium minima,the pairing energies, entropies, and intrinsic level densi-ties as functions of temperature for , , Mo, , Pd, , Cd. Most of these nuclei exhibit deformation en-ergy surfaces that are soft in γ deformation (cf. Fig. 2),while the octupole deformation β does not play a sig-nificant role at low energies. The RHB calculation has,therefore, been restricted to triaxial quadrupole deforma-tions. As shown in Fig. 1 (b), pairing correlations decreaserapidly as temperature increases and the pairing energyvanishes at the critical temperature T c = 0 . ∼ . U i = aT ,where a is the level density parameter. From Fig. 1 (a)one notices that the energy indeed increases quadrati-cally with temperature, and a change of slope can beassociated with the pairing phase transition at the criti-cal temperature. Fig. 1 (c) shows that below the criticaltemperature T c the entropy also increases quadraticallywith temperature. After the pairing phase transition theentropy increases linearly with T , in agreement with theBethe formula S = 2 aT . The intrinsic level density in-creases exponentially with the entropy (cf. Eq. (13)),and a change of slope, or even a discontinuity, is foundaround T c , as shown Fig. 1 (d).In the second step a large scale zero-temperatureMDC-RHB calculation is performed to generate thecollective potential energy surface (PES), single-quasiparticle energies and wave functions in the ( β, γ )plane. Fig. 2 displays the resulting deformation en-ergy surfaces of , , Mo, , Pd, , Cd. Atzero temperature the ground state shape for Mo isalmost spherical and the PES is soft in both β and γ directions. The equilibrium deformation of Mo is at( β, γ ) ∼ (0 . , ◦ ), and ( β, γ ) ∼ (0 . , ◦ ) for Pd. Theisotopes Mo,
Pd, , Cd exhibit β -deformed min-ima at β = 0 . ∼ .
25. As noted above, the PESs forall these nuclei are rather soft in the γ direction. Withthe single-quasiparticle energies and wave functions de-termined in self-consistent RHB calculations, the corre-sponding mass parameters, moments of inertia, and ZPEover the entire PES can be computed. These quantitiesspecify the collective Hamiltonian Eq. (16).To illustrate the level of agreement with low-energyexperimental levels, in Fig. 3 we compare the calculatedlow-spin collective levels of Mo with the available datafrom Ref. [57]. The experimental levels are shown in theupper panel, and the eigenstates of the quadrupole collec-tive Hamiltonian in the lower panel. The calculated levelsare in good qualitative agreement with experiment, ex-cept for the fact that the calculated excitation spectrumis somewhat stretched out compared to data. In particu-lar, the moment of inertia of the theoretical yrast band issmaller than the empirical one. This is because the collec-tive inertia is calculated from the Inglis-Belyaev formulawhich does not include Thouless-Valatin rearrangementcontributions and, therefore, predicts effective momentsof inertia that are smaller than empirical values. Thepredicted energies of 2 +1 and 0 +2 are 0.92 MeV and 1.94MeV, respectively, are compared to the experimental val-ues: 0.87 MeV and 1.74 MeV. The predicted energy of4 +1 is 2.07 MeV, which is again considerably above theexperimental value of 1.57 MeV. Here we note that, whilethe theoretical states are purely collective, there are indi-cations of non-collective components in the 4 +1 state [58].For some levels at higher energies, for instance, the ex- b g (deg) Mo b g (deg) Mo b g (deg) Mo b g (deg) Pd b g (deg) Pd b Cd g (deg) b g (deg) Cd FIG. 2. (Color online) Self-consistent triaxial quadrupoledeformation-constrained energy surfaces of , , Mo, , Pd, and , Cd in the β - γ plane (0 ≤ γ ≤ ◦ ). Foreach nucleus the energies are normalized with respect to thebinding energy of the global minimum. The contours joinpoints on the surface with the same energy, and the spacingbetween neighbouring contours is 0.5 MeV. perimental values for 0 +3 and 6 +1 are 2.78 MeV and 2.87MeV, respectively, while the calculation gives the valuesof 3.78 MeV and 3.54 MeV. In addition to the perturba-tive cranking approximation used to calculate the massparameters, we also note that, in particular for the ex- Mo (a)Exp. (b)Cal. + + + + + E ne r g y ( M e V ) + FIG. 3. (Color online) The calculated positive-parity low-spinstates of Mo (b) and their possible experimental counter-parts (a). Not all known levels of these spins are shown inpanel (a). For a detailed discussion see Ref. [57]. cited 0 + states, another effect that is not included in themodel is the coupling of nuclear shape oscillations withpairing vibrations, that is, vibration of the pairing den-sity. However, the aim of the present study is not a de-tailed reproduction of the low-energy spectra and, there-fore, the qualitative level of agreement between modelcalculations and experiment, illustrated in Fig. 3, shouldbe sufficient for an estimate of the collective enhancementof the level density.Employing the collective levels obtained by diagonal-ization of the quadrupole Hamiltonian Eq. (16), the to-tal level densities can now be computed from Eqs. (14)and (15). In Fig. 4 we compare, for , , Mo, , Pd,and , Cd, the intrinsic level densities calculated withthe FT-RHB model Eq. (13) (dash-dotted blue) and thecorresponding total level densities (solid red), with theavailable data below . ρ tot ( U ) = K coll ( U ) ρ i ( U ) , (22)as functions of excitation energy for , , Mo, , Pd,and , Cd. In general, K coll ( U ) exhibits an increasewith energy in the interval below . ≈ Mo, and at ≈ -1 (a) Mo (b) Mo Exp. Intrinsic Total -1 (c) Mo (d) Pd -1 (e) Pd Cd Energy (MeV) Le v e l D en s i t y ( M e V - ) -1 (g) Cd FIG. 4. (Color online) The calculated intrinsic level densi-ties (dash-dotted blue) and total level densities (solid red), asfunctions of excitation energy for , , Mo, , Pd, and , Cd. The data (black squares) are from Refs. [59–61]. Mo are actually caused by the dips of the intrinsic leveldensities (cf. Fig. 4), and can be related to a collapse ofpairing correlations at these energies. This is an artefactof the SCMF calculation that does include projection ongood particle number and is, therefore, unphysical. Inthe lower panel we plot the ratios ρ tot /ρ exp . Except forthe oscillations at very low energies below 1 MeV wherethere are only a few levels, for most of these nuclei theratio is actually close to 1 over the entire low-energy in-terval.Several studies based on the non-relativistic GognyHFB and relativistic RHB models have shown that heav-ier nuclei in the mass A ≈ −
170 region, such asDy, Er and Yb isotopes, exhibit axially symmetric equi-librium shapes, but their potential energy surfaces arerather soft in the octupole β direction. This is illus-trated in Fig. 6, where we display the two-dimensionalRHB deformation energy surfaces of , , Dy,
Er,and , Yb in the ( β , β ) plane calculated at zero tem-perature. One notices that, although the global minimaare located at β = 0 . ∼ . β = 0, the minima areextended in the direction of axial octupole deformation β . For this reason we expect a significant contributionof octupole vibrations to the low-energy collective states. K c o ll Mo Mo Mo Pd Pd Cd Cd Mo Pd Mo Cd Mo Cd Pd (b) r t o t / r e x p Energy (MeV)
FIG. 5. (Color online) The collective enhancement factors K coll (a), and ratios ρ tot /ρ exp of calculated and experimen-tal level densities (b), as functions of excitation energy for , , Mo, , Pd, and , Cd. Dy (a) (b) Dy Dy (d) Er -0.5 0.0 0.50.00.10.20.30.4 (f)(e) Yb b b -0.5 0.0 0.5 Yb FIG. 6. (Color online) Self-consistent RHB axially symmet-ric deformation energy surfaces of , , Dy,
Er, and , Yb in the ( β , β ) plane. For each nucleus the ener-gies are normalized with respect to the binding energy of theglobal minimum. The contours join points on the surfacewith the same energy, and the spacing between neighbouringcontours is 1.0 MeV. (a) Dy Dy Dy Er Yb Yb E ne r g y ( M e V ) -10-50 (b) P a i r i ng ( M e V ) (c) E n t r op y Temperature (MeV) 0.2 0.4 0.6 0.8 1.0 (d) r i ( M e V - ) Temperature (MeV)
FIG. 7. (Color online) Same as in the caption to Fig. 1 but forthe axially-symmetric and reflection-asymmetric RHB calcu-lations of , , Dy,
Er, and , Yb.
As the current implementation of our collective Hamil-tonian does not allow the simultaneous breaking of axialand reflection symmetries, in this case we will employ theaxially symmetric and reflection asymmetric quadrupole-octupole Hamiltonian of Eq. (17) to calculate the collec-tive enhancement of the RHB intrinsic level densities.Axially symmetric and reflection asymmetric FT-RHBcalculations are performed for the equilibrium minimato compute the intrinsic level densities. The bindingenergy, pairing energy, entropy, and intrinsic level den-sity as functions of nuclear temperature are displayed inFig. 7. Just as in the case of the mass A ≈
100 region,the binding energies increase quadratically with temper-ature, while the entropy first increase quadratically with T below the critical temperature of pairing phase tran-sition, and linearly for higher tempertaures. Fig. 7 (b)shows that the pairing collapse occurs at the critical tem-perature T c = 0 . ∼ . T is characterized by a discontinuity at T c .The self-consistent RHB energy surfaces and the corre-sponding ZPEs, the mass parameters, and the momentsof inertial in the ( β , β ) plane at zero temperature, de-termine the axial quadrupole-octupole collective Hamil-tonian Eq. (17). The eigenstates of this Hamiltonian areused to compute the total level densities (Eqs. (14) and(15)). The calculated intrinsic level densities and the to-tal level densities of , , Dy,
Er, and , Yb,as functions of the excitation energy, are compared inFig. 8 with the experimental values from Refs. [62–66].Similar to the result obtained in the mass A ≈
100 re-gion, the intrinsic level densities are systematically belowthe experimental values for all isotopes and all excita-tion energies. In the energy interval of measured values,the consistent microscopic calculation of the collectiveenhancement, using the axial quadrupole and octupolecollective degrees of freedom relevant for this mass re-gion, produces total level densities that are in very good -1 Dy (a) Exp. Intrinsic Total (b) Dy -1 (c) Dy (d) Er -1 (f)(e) Yb Le v e l D en s i t y ( M e V - ) Energy (MeV) Yb FIG. 8. (Color online) The calculated intrinsic level densi-ties (dash-dotted blue) and total level densities (solid red),as functions of excitation energy for , , Dy,
Er, and , Yb. The data (black squares) are from Refs. [62–66]. K c o ll Dy Dy Dy Er Yb Yb Dy Er Dy Yb Dy Yb r t o t / r e x p Energy (MeV)
FIG. 9. (Color online) Same as in the caption to Fig. 5 but forthe axially-symmetric and reflection-asymmetric calculationsof , , Dy,
Er, and , Yb. agreement with available data.Figure 9 diplays the collective enhancement factors K coll (a), and ratios ρ tot /ρ exp of calculated and exper-imental level densities (b), as functions of excitation en-ergy for , , Dy,
Er, and , Yb. Comparedto the case of A ≈
100 nuclei, K coll exhibits a more pro-nounced increase with energy. For Dy and
Yb onenotices two strong peaks at ≈ ρ tot /ρ exp in panel (b). For , Dy we cancompare the values of K coll at neutron separation energywith the recent prediction of the rotational enhancementfactor R averaged over angular momentum of Ref. [27].The predicted values of R : 45.3 at S n = 8 .
20 Mev for
Dy, and 46.1 at S n = 7 .
63 Mev for
Dy (Table I of ofRef. [27]), are very close to the corresponding collectiveenhancement factors obtained in the present microscopiccalculation: K coll = 39 . Dy, and K coll = 42 . Dy.
IV. SUMMARY
A fully self-consistent microscopic approach for calcu-lating nuclear level densities has been developed, basedon global nuclear energy density functionals. The in-trinsic level densities are computed in the thermody-namical approach using the saddle point approximation,with single-quasiparticle spectra obtained in a finite-temperature self-consistent mean-field (SCMF) calcula-tion. In the present work we have used the finite-temperature relativistic Hartree-Bogoliubov (FT-RHB)model based on the DD-PC1 energy density functionaland a finite-range pairing interaction. The total leveldensities are obtained by convoluting the intrinsic densi-ties with the corresponding collective level densities. Thecollective levels are calculated as eigenstates of a five-dimensional quadrupole or quadrupole-octupole Hamil-tonian, with parameters (mass parameters, momentsof inertia, collective potential) fully determined by theSCMF calculation of the deformation energy surfaces andthe corresponding single-quasiparticle levels as functionsof the collective coordinates (shape variables). There-fore, in this approach both the intrinsic and collectivelevel densities are completely determined by the choiceof a global energy density functional and pairing interac-tion. One has to choose, however, the coordinates of thecollective Hamiltonian depending on the specific nucleusunder consideration. This is done for practical reasons,as the collective Hamiltonian can only take into account asmall number of most relevant coordinates. For instance,quadrupole or quadrupole plus octupole shape variableswill typically be used as collective coordinates. The model has been tested in several illustrative cal-culations in the A ≈
100 and A ≈ −
170 mass re-gions, where accurate experimental level densities areavailable in the energy interval below the neutron separa-tion energy. In the former region we have computed thelevel densities of , , Mo, , Pd, and , Cd. Ingeneral these nuclei exhibit equilibrium minima at mod-erate quadrupole deformation, and the deformation en-ergy surfaces are rather soft in the γ degree of freedom.Thus we have used the five-dimensional Hamiltonian inthe quadrupole variables β and γ to calculate the lev-els that determine the collective enhancement of the in-trinsic level densities. In the mass region of heavier nu-clei level densities have been calculated for , , Dy,
Er, and , Yb. To a good approximation the equi-librium minima of these nuclei are axially quadrupoledeformed, but also extended (soft) in the octupole defor-mation. In this case we have used an axially symmetricquadrupole-octupole Hamiltonian to calculate the collec-tive level densities.In both mass regions it has been shown that, while thecalculated intrinsic level densities reproduce the energydependence of the data, their values are systematicallytoo small and, therefore, additional degrees of freedomrelated to the shape of a nucleus have to be taken intoaccount. The collective enhancement computed usingthe eigenstates of the five-dimensional quadrupole (mass A ≈ A ≈ − ACKNOWLEDGMENTS
This work has been supported by the Inter-Governmental S&T Cooperation Project between Chinaand Croatia. It has also been supported in part bythe QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and EuropeanUnion through the European Regional DevelopmentFund - the Competitiveness and Cohesion OperationalProgramme (KK.01.1.1.01) and by the Croatian ScienceFoundation under the project Uncertainty quantificationwithin the nuclear energy density framework (IP-2018-01-5987). J.Z. acknowledges support by the NationalNatural Science Foundation of China under Grant No.12005107 and No. 11790325. Calculations have been per- formed in part at the HPC Cluster of KLTP/ITP-CASand the Supercomputing Center, Computer Network In-formation Center of CAS. [1] H. A. Bethe, Rev. Mod. Phys. , 69 (1937).[2] A. Koning, S. Hilaire, and S. Goriely,Nucl. Phys. A , 13 (2008).[3] S. Goriely, Nucl. Phys. A , 28 (1996).[4] D. Gross and R. Heck, Phys. Lett. B , 405 (1993).[5] Y. 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