Strong enhancement of level densities in the crossover from spherical to deformed neodymium isotopes

Understanding the evolution of level densities in the crossover from spherical to well-deformed nuclei has been a long-standing problem in nuclear physics. We measure nuclear level densities for a chain of neodymium isotopes $^{142,144-151}$Nd which exhibit such a crossover. These results represent to date the most complete data set of nuclear level densities for an isotopic chain between neutron shell-closure and towards mid-shell. We observe a strong increase of the level densities along the chain with an overall increase by a factor of $\approx 170$ at an excitation energy of 7.5 MeV and saturation around mass 150. Level densities calculated by the shell model Monte Carlo (SMMC) are in excellent agreement with these experimental results. Based on our experimental and theoretical findings, we offer an explanation of the observed mass dependence of the level densities in terms of the intrinsic single-particle level density and the collective enhancement.


Introduction
Compound-nucleus reaction cross sections are indispensable in a variety of applications such as understanding stellar nucleosynthesis [1], designing next-generation nuclear reactors [2], and optimizing transmutation of nuclear waste [3]. Such reactions are well understood by Hauser-Feshbach theory [4], but this theory requires as input statistical nuclear properties, such as the nuclear level density (NLD). However, the microscopic calculation of NLDs in the presence of correlations is a challenging many-body problem. Furthermore, experimental data are usually limited to low excitation energies [5] and neutron resonance measurements at the neutron separation energy [6], making it difficult to benchmark theoretical models.
Understanding the effects of deformation on the NLD is a long-standing open problem in nuclear physics; see, e.g., Ref. [7] and references therein. Compared to a spherical nucleus of similar mass, the NLD of a deformed nucleus is determined by two competing effects: (i) in mean-field theory, the onset of deformation breaks the magnetic degeneracy of the spherical single-particle levels, leading to an effective decrease of the average single-particle level density at the Fermi energy [8], and thus lowers the NLD of intrinsic states; and (ii) rotational bands built on top of each of these intrinsic configurations lead to enhancement of the NLD [9,10].
Bjørnholm et al. [9] predicted a collective enhancement factor ≈ 10 for vibrations and ≈ 100 for rotations, altogether a factor of ≈ 1000. Various phenomenological NLD models [11][12][13][14][15][16] were modi-fied to take into account these enhancement factors. Yet modern combinatorial and mean-field methods must still have to be augmented by phenomenological collective enhancement factors [17][18][19]. Shell effects and pairing correlations were included empirically in an energy-dependent level density parameter [20]. Recent shell-model Monte Carlo (SMMC) calculations [21,22] indicated a significantly smaller collective enhancement than suggested by Bjørnholm et al. [9]. However, as stated by Junghans et al. [23], experimental information on the NLD is mandatory for quantifying this effect. A reliable theoretical model should reproduce the NLD well not only at the low-lying discrete levels and at the neutron resonance energy, but also for a broad excitationenergy region. The Oslo method used in this work provides the functional form of the NLD in the energy range between the low-lying discrete levels and the neutron separation energy, where often there is no other experimental data available.
We present a systematic study of the NLD for a chain of neodymium isotopes, starting from 142 Nd at the N = 82 shell closure and up to the welldeformed 151 Nd, probing the effect of collectivity on the NLD in an unprecedented way. We have also performed microscopic SMMC calculations for 142 Nd up to 152 Nd and find them to be overall in excellent agreement with experiment. We show that the combined effect of a decrease in the singleparticle level density with mass number and a collective enhancement results in an increase of the NLD with deformation that saturates around mass 150.

Experiment
The light-ion reactions were performed at the Oslo Cyclotron Laboratory. The targets were metallic foils of 142,144,146,148,150 Nd with thicknesses of ≈ 2 mg/cm 2 and enrichments of ≈ 97%. The targets were bombarded with proton and deuteron beams of energies 16.0 MeV and 13.5 MeV, respectively. The SiRi particle-telescope system [24] was applied to determine the outgoing particle type and energy. The 64 particle telescopes were located ≈ 5 cm from the target in eight angles between 126 • and 140 • with respect to the beam direction. The particle energy resolution was ≈150 keV (FWHM). The γ rays following the reactions were measured with the NaI(Tl) scintillator array CACTUS [25] and the LaBr 3 (Ce) scintillator array OSCAR. Additional details are provided in the Supplemental Material [26] .

The Oslo method
We extract the NLD for 142,144−151 Nd applying the Oslo method for a set of particle-γ ray coincidences. From the measured ejectile, we obtain information on the initial excitation energy E of the residual nucleus. The γ rays detected in coincidence with the ejectile reveal the decay properties from this specific excitation energy. Figure 1a shows how the data are sorted into a matrix of initial excitation energies E versus the γ-ray energy E γ . This raw matrix is unfolded for each excitation energy bin ( Fig. 1b) using the known detector response functions [27,29]. Finally, the first generation (primary) γ-ray matrix P (E γ , E) is obtained, see Fig. 1c. The first-generation procedure is based on an iterative subtraction technique [28] which separates the distribution of the first emitted γ rays from all available γ cascades.
The next step in the Oslo method is to factorize the primary γ-ray matrix by Here, we have applied the Brink hypothesis [30]: the γ-ray transmission coefficient T is approximately independent of excitation energy and spin/parity. The factorization is justified by Fermi's golden rule [31,32], which states that the decay rate is proportional to the NLD at the final excitation energy after emitting the primary γ ray. The fitting procedure performed using Eq. (1) enables the simultaneous extraction the NLD and the γ-ray transmission coefficient. However, it has been shown [33] that any transformation of the form (2) gives the same fit to P (E γ , E). To determine the parameters A and α in (2), we use other experimental data. At low excitations, we normalize the NLD to known discrete levels [34]. At high excitations, we used the measured average neutron s-wave resonance spacing D 0 [35] at the neutron separation energy S n .
To convert the measured D 0 to total level density, we use the spin cutoff model [36,37]. The values σ(S n ) of the spin cutoff parameter at the 0.65(5) 9.828 6.6(7) 19 (4) (9) 7.278 6.3(6) a Interpolated between even-A neighbors. b Scaled from systematics [26]. c Adjusted to reproduce ρ(S n ). neutron separation energy S n are estimated based on a rigid-body moment of inertia (RMI) [26] and are tabulated in Table 1. Table 1 also includes the quadrupole deformation β 2 from Ref. [38] and the temperature T CT extracted by fitting the constanttemperature formula to the high-energy data points, where E 0 is a shift parameter to match ρ(S n ) [26].

SMMC calculations
The SMMC method [39,40] enables the exact calculation (up to statistical errors) of NLDs in the framework of the configuration-interaction (CI) shell model. This method allows us to use manyparticle model spaces that are many orders of magnitude larger than those that can be treated by conventional shell model methods [41]. In contrast to combinatorial and mean-field approaches, the SMMC approach does not require any empirical enhancement factors, and is therefore a suitable approach for studying the deformation dependence of the NLD.
We carried out SMMC calculations in the protonneutron formalism [42] for the chain of neodymium isotopes 142−152 Nd. The CI shell model space includes the complete 50 − 82 shell plus the 1f 7/2 orbital for protons, and the complete 82 − 126 shell plus the 0h 11/2 and 1g 9/2 orbitals for neutrons. The effective interaction parameters are given in Ref. [21]. For the odd-mass isotopes, there is a sign problem associated with the projection on an odd number of neutrons at low temperatures and the ground-state energies were taken from Ref. [43]. The latter estimated ground-state energies for all the odd neodymium isotopes in the chain with the exception of 151 Nd. In contrast to state densities that count the 2J +1 degeneracy of each level with spin J, the measured level densities count each such level only once. In SMMC, the level densities are obtained by projection on M = 0 (M = 1/2) for even-mass (oddmass) nuclei [44,45]. SMMC state densities for the neodymium isotopes were presented in Ref. [46]. We provide more details for the SMMC calculations in the Supplemental Material [26].

Results
In Fig. 2 we compare the experimentally extracted NLDs of 142,144−151 Nd with the SMMC results. Above an excitation energy of ∼ 2 − 3 MeV, the experimental NLDs are almost linear in a logarithmic scale and are well-described by the constant-temperature formula (3). It was conjectured that this behavior emerges once the first pair of nucleons is broken [47][48][49], i.e., for an excitation energy E > 2∆, where ∆ is the pairing gap. In contrast to recent findings in 167,168,169 Tm [50], we do not observe any experimental or theoretical signatures of irregular bumps in the NLD curves. Figure 3 shows the experimental and SMMC NLDs from Fig. 2 at three excitation energies of 2.5, 5 and 7.5 MeV as a function of deformation β 2 . The deformation of the even-mass isotopes is determined from the compilation of Pritychenko et al. [38], using the measured B(E2) values between the ground state and the first excited 2 + state. For the odd-mass isotopes, we assume a deformation that is the average of their even-mass neighbors. These values of β 2 are listed in Table 1; see also the Supplemental Material [26].
At excitation energies of 2.5 MeV and 7.5 MeV, the NLD is determined, respectively, by a complete set of known low-lying discrete levels and by the average neutron resonance spacing D 0 , while at the intermediate excitation energy of 5 MeV, the NLD is determined by the Oslo method. We find that the deformation dependence of the experimental NLDs at these three excitation energies follow closely the empirical form where C and η are fit parameters and β max 2 = 0.25. The resulting fits of Eq. (4) to the experimental data are shown by the curves in Fig. 3. We obtain similar values of the parameter η for the evenand odd-mass isotopes with η = 118, 136 and 166 at E = 2.5, 5.0 and 7.5 MeV, respectively. There   Table 1. The curves are calculated from Eq. (4); see text.
is a strong odd-even effect where the NLD of an odd-mass nucleus is higher than the NLDs of its even-mass neighbors, which can be attributed to the blocking effect of the odd neutron [51].
We observe overall excellent agreement between the experimental and SMMC NLDs. In particular, the trend with mass number is well reproduced for both the even-and odd-mass isotopes. For 152 Nd there are no experimental results to compare with, but the SMMC results deviate from the curves fitted to the experimental systematics. In general, we find larger deviations for the odd-mass isotopes, in particular at the highest energy of E = 7.5 MeV. We note, however, that the experimental results at this energy are estimated by extrapolating the Oslo data beyond the neutron separation energy using the constant-temperature formula (3), and thus the comparison is not as conclusive as for the even isotopes. Figure 4 shows the total enhancement of the NLD at E = 7.5 MeV of the even-mass neodymium isotopes relative to the NLD of 144 Nd, the lightest isotope in the chain for which an experimental value of D 0 exists. Overall, we observe excellent agreement between experiment and the SMMC results. We find a large enhancement in the experimental NLD by a factor of ≈ 170 for 150 Nd (relative to 142 Nd).

Explanation of the mass dependence
The intrinsic mean-field state density ρ mf , which is calculated using the same effective interaction as in SMMC, is affected mostly by the average single-particle level density g n ( F ) of the neutrons at the Fermi energy F . In deformed nuclei, rotational bands built on top of intrinsic band heads lead to an enhancement of the total state density, described by a collective enhancement factor K = ρ state /ρ mf [21]. This collective enhancement of ρ state is also reflected in the total level density, since these two densities are related by a spin cutoff parameter that is only weakly dependent on mass. Thus, the mass dependence of the NLD is determined by two factors: ρ mf and K. With 82 neutrons, 142 Nd is semi-magic and its neutron Fermi energy is in the middle of the shell gap and is thus characterized by a relatively low g n ( F ). As we start filling the 82 − 126 major shell in 144 Nd, the neutron Fermi energy rises, so that it is close to the 2f 7/2 orbital, leading to a sharp increase in g n ( F ) and thus in ρ mf . The g n ( F ) of the spherical meanfield solution continues to increase in 146 Nd, though at a more moderate rate. The increase in ρ mf is shown by the red circles in Fig. 4. The mean-field solution for our shell model interaction becomes deformed starting in 148 Nd. Deformation lifts the spherical degeneracy of the single-particle levels and decreases g n ( F ) in the deformed isotopes, leading to a decrease in ρ mf compared to 146 Nd. This decrease is compensated by the rise of K with deformation. As a result, the total NLD enhancement increases with mass but saturates around 150 Nd, for which the SMMC NLD is very similar to that in 152 Nd. This is shown by the black circles in Fig. 4, describing the product of K and the increase of ρ mf (relative to 144 Nd), and follow closely the observed total enhancement of the NLD with mass number A.

Conclusions
We extracted experimental NLDs for a long chain of neodymium isotopes using the Oslo method. We observed a large total enhancement of the NLD in the crossover from spherical to deformed isotopes, which saturates around A = 150. The availability of experimental data for such a long isotopic chain makes these isotopes an excellent benchmark for testing the quality of current and future NLD models. We calculated SMMC NLDs of these isotopes in the framework of the CI shell model and found them to be overall in excellent agreement with the experimental NLDs. We explained the mass dependence of the NLDs by the combined effects of the intrinsic single-particle level density and of the collective enhancement.