Systematic investigation of the high-K isomers and the high-spin rotational bands in the neutron rich Nd and Sm isotopes by a particle-number conserving method

# Systematic investigation of the high-K isomers and the high-spin rotational bands in the neutron rich Nd and Sm isotopes by a particle-number conserving method

Zhen-Hua Zhang Mathematics and Physics Department, North China Electric Power University, Beijing 102206, China
July 19, 2019
###### Abstract

The rotational properties of the neutron rich Nd and Sm isotopes with mass number are systematically investigated using the cranked shell model with pairing correlations treated by a particle-number conserving method, in which the Pauli blocking effects are taken into account exactly. The 2-quasiparticle states in even-even Nd and Sm isotopes with excitation energies lower than 2.5 MeV are systematically calculated. The available data can be well reproduced and some possible 2 and 4-quasiparticle isomers are also suggested for future experiments. The experimentally observed rotational frequency variations of moments of inertia for the even-even and odd- nuclei are reproduced very well by the calculations. The effects of high-order deformation on the 2-quasiparticle excitation energies and moments of inertia of the ground state bands in even-even Nd and Sm isotopes are analyzed in detail. By analyzing the occupation probability of each cranked Nilsson orbitals near the Fermi surface and the contribution of each major shell to the angular momentum alignments, the alignment mechanism in these nuclei is understood clearly.

## I Introduction

For the neutron rich rare-earth nuclei with mass number , especially Nd () and Sm () isotopes, there are many novel phenomena, e.g., nuclear quantum phase transition from spherical to deformed shape Iachello and Zamfir (2004); Nikšić et al. (2007), octupole vibration Phillips et al. (1986, 1988); Ibbotson et al. (1993), isomers Walker and Dracoulis (1999), etc. From neutron number , the nuclei are well-deformed and possess prolate ground state rotational bands. In this mass region, there are several high- orbitals around the proton and neutron Fermi surface, e.g., , , , , and . Therefore, this may give rise to the formation of various high- multi-quasiparticle (qp) isomers, which are particularly favorable for studying the blocking effects of the pairing correlations.

Due to the high statistics, the spontaneous fission of the actinide nuclei has been used to populate the isomeric and high-spin states of neutron rich nuclei in mass region Greenwood et al. (1987); Hamilton et al. (1995, 1997). Up to now, using the spontaneous fission of Cf Zhu et al. (1995); Babu et al. (1996); Zhang et al. (1998); Gautherin et al. (1998); Hwang et al. (2008a, b); Simpson et al. (2009); Urban et al. (2009); Hwang et al. (2010); Wang et al. (2014) and in-flight fission of a U beam on a Be target Patel et al. (2014, 2016); Ideguchi et al. (2016), various high- isomers and high-spin rotational bands for the neutron rich Nd and Sm isotopes, including both the even-even and the odd- nuclei, have been established. Most recently, the lightest 4-qp high- isomer in this mass region has been observed in Sm Patel et al. (2016). These data can reveal detailed information on the single-particle structure, shell structure, the high- isomerism, etc., thus providing a benchmark for various nuclear models.

Several nuclear models have been used to investigate the properties of these neutron rich nuclei, including quasiparticle rotor model Simpson et al. (2009); Urban et al. (2009), a mean-field type Hartree-Fock-Bogoliubov theory with Gogny force D1S Gautherin et al. (1998), projected shell model Yang et al. (2010); Yang and Sun (2011) and potential energy surface calculations Patel et al. (2014, 2016). However, most of these models focus on the even-even nuclei. Only the projected shell model and the quasiparticle rotor model were used to investigate the odd- nuclei Sm Urban et al. (2009); Yang et al. (2010). Therefore, it is necessary to perform a systematic investigation including both the even-even and the odd- Nd and Sm isotopes, which can improve our understanding for these neutron rich nuclei.

In the present work, the cranked shell model (CSM) with pairing correlations treated by a particle-number conserving (PNC) method Zeng and Cheng (1983); Zeng et al. (1994a) is used to investigate systematically the neutron rich Nd and Sm isotopes with mass number , including both even-even and odd- nuclei. In contrary to the conventional Bardeen-Cooper-Schrieffer or Hartree-Fock-Bogoliubov approaches, the many-body Hamiltonian is solved directly in a sufficiently large truncated Fock-space in the PNC method Wu and Zeng (1989). Therefore, the particle-number is conserved and the Pauli blocking effects are treated exactly. The PNC-CSM has been employed successfully for describing various phenomena, e.g., the odd-even differences in moments of inertia (MOIs) Zeng et al. (1994b), identical bands Liu et al. (2002); He et al. (2005), nuclear pairing phase transition Wu et al. (2011), antimagnetic rotation Zhang et al. (2013a); Zhang (2016a), rotational bands and high- isomers in the rare-earth Liu et al. (2004); Zhang et al. (2009a, b); Zhang (2016b, c) and actinide nuclei He et al. (2009); Zhang et al. (2011, 2012, 2013b), etc. The PNC scheme has also been used both in relativistic and non-relativistic mean field models Meng et al. (2006); Pillet et al. (2002); Liang et al. (2015) and the total-Routhian-surface method with the Woods-Saxon potential Fu et al. (2013a, b). Recently, the PNC method based on the cranking covariant density functional theory has been developed Shi et al. (2018). Similar approaches to treat pairing correlations with exactly conserved particle number can be found in Refs. Richardson and Sherman (1964); Pan et al. (1998); Volya et al. (2001); Jia (2013a, b); Chen et al. (2014).

This paper is organized as follows. A brief introduction to the PNC treatment of pairing correlations within the CSM is presented in Sec. II. The numerical details used in PNC calculations are given in Sec. III. In Sec. IV, the 2-qp energies and MOIs are calculated and compared with the data. The 2-qp states in even-even Nd and Sm isotopes with excitation energies lower than 2.5 MeV are systematically investigated. The effects of high-order deformation and alignment mechanism in these nuclei are discussed in detail. A brief summary is given in Sec. V.

## Ii PNC-CSM formalism

The cranked shell model Hamiltonian of an axially symmetric nucleus in the rotating frame can be written as

 HCSM = H0+HP=HNil−ωJx+HP , (1)

where is the Nilsson Hamiltonian Nilsson et al. (1969), is the Coriolis interaction with the cranking frequency about the axis (perpendicular to the nuclear symmetry axis). is the pairing interaction,

 HP(0) = −G0∑ξηa†ξa†¯ξa¯ηaη , (2) HP(2) = −G2∑ξηq2(ξ)q2(η)a†ξa†¯ξa¯ηaη , (3)

where () labels the time-reversed state of a Nilsson state (), is the diagonal element of the stretched quadrupole operator, and and are the effective strengths of monopole and quadrupole pairing interaction, respectively.

Instead of the usual single-particle level truncation in conventional shell-model calculations, a cranked many-particle configuration (CMPC) truncation (Fock space truncation) is adopted, which is crucial to make the PNC calculations for low-lying excited states both workable and sufficiently accurate Molique and Dudek (1997); Wu and Zeng (1989). Usually a CMPC space with the dimension of 1000 should be enough for the calculations of rare-earth nuclei. By diagonalizing the in a sufficiently large CMPC space, sufficiently accurate solutions for low-lying excited eigenstates of can be obtained, which can be written as

 |Ψ⟩=∑iCi|i⟩(Cireal) , (4)

where is a CMPC (an eigenstate of the one-body operator ).

The angular momentum alignment for the state is

 ⟨Ψ|Jx|Ψ⟩=∑iC2i⟨i|Jx|i⟩+2∑i

and the kinematic MOI of state is

 J(1)=1ω⟨Ψ|Jx|Ψ⟩ . (6)

Because is a one-body operator, the matrix element () may not vanish only when and differ by one particle occupation Zeng et al. (1994a). After a certain permutation of creation operators, and can be recast into

 |i⟩=(−1)Miμ|μ⋯⟩ ,|j⟩=(−1)Mjν|ν⋯⟩ , (7)

where and denotes two different single-particle states, and , according to whether the permutation is even or odd. Therefore, the angular momentum alignment of can be expressed as

 ⟨Ψ|Jx|Ψ⟩=∑μjx(μ)+∑μ<νjx(μν) . (8)

where the diagonal contribution and the off-diagonal (interference) contribution can be written as

 jx(μ) = ⟨μ|jx|μ⟩nμ , (9) jx(μν) = 2⟨μ|jx|ν⟩∑i

and

 nμ=∑i|Ci|2Piμ , (11)

is the occupation probability of the cranked orbital , if is occupied in , and otherwise.

## Iii Numerical details

In this work, the deformation parameters (, and ) of Nd and Sm isotopes used in PNC-CSM calculations are taken from Ref. Möller and Nix (1995), which are shown at Table 1. The Nilsson parameters ( and ) are taken as the traditional values Nilsson et al. (1969). The experimental data show that the ground state of isotones (e.g., Nd and Sm) is  Hwang et al. (1997); Reich (2005). However, the calculated ground state using the traditional Nilsson parameters is  Nilsson et al. (1969). Therefore, to reproduce the experimental level sequence, the neutron orbital is shifted upwards slightly by for all these nuclei.

The effective pairing strengths for each nuclei, in principle, can be determined by the experimental odd-even differences in nuclear binding energies Wang et al. (2012),

 Pp = 12[B(Z+1,N)+B(Z−1,N)]−B(Z,N) = Eg(Z,N)−12[Eg(Z+1,N)+Eg(Z−1,N)] , Pn = 12[B(Z,N+1)+B(Z,N−1)]−B(Z,N) (12) = Eg(Z,N)−12[Eg(Z,N+1)+Eg(Z,N−1)] ,

where is the ground state energy of the nucleus, and are connected with the dimension of the truncated CMPC space. In this work, the CMPC space is constructed in the proton major shells and the neutron major shells, respectively. The CMPC truncation energies are about 0.85 for protons and 0.80 for neutrons, respectively. The dimensions of the CMPC space are about 1000 for both protons and neutrons in the present calculation. For all Nd and Sm isotopes, the corresponding effective monopole and quadrupole pairing strengths are chosen as  MeV and  MeVfm for protons,  MeV and  MeVfm for neutrons. Figure 1 shows the comparison between experimental (black solid circles) and calculated (red open circles) neutron odd-even difference for Nd (upper panel) and Sm (lower panel) isotopes. It can be seen that the data can be well reproduced. In principal, the pairing strengths should be different for each nucleus. Note that for some neutron rich Nd and Sm nuclei, the experimental binding energy is not accurate Wang et al. (2012). Therefore, in the present work the pairing strengths for all nuclei are chosen as the same value to get a global fit. Previous investigations have shown that after the quadrupole pairing being included, the description of experimental band-head energies and the level crossing frequencies can be improved Diebel (1984). As for the quadrupole pairing, the strength is determined by the bandhead MOIs in the present work. The quarople pairing is also included in the projected shell model when investigating the Nd and Sm isotopes, in which BCS method is used to treat the pairing correlations Yang et al. (2010). In the projected shell model, the quadrupole pairing strength is chosen to be proportional to the monopole pairing strength with proportionality constant 0.18 Yang et al. (2010). As for this point, this proportionality is much smaller in the present work (less than 0.1). However, the PNC method is different from the BCS method. Therefore, the effective pairing strength should be different.

The stability of the calculations against the change of the dimension of the CMPC space has been investigated in Refs. Molique and Dudek (1997); Zeng et al. (1994a); Zhang et al. (2012). In present calculations, almost all the CMPCs with weight in the many-body wave functions are taken into account, so the solutions to the low-lying excited states are accurate enough. A larger CMPC space with renormalized pairing strengths gives essentially the same results.

## Iv Results and discussion

### iv.1 Cranked Nilsson levels

As an example of Nd and Sm isotopes around mass region, the calculated cranked Nilsson levels near the Fermi surface of Sm are shown in Fig. 2. The positive (negative) parity levels are denoted by blue (red) lines. The signature () levels are denoted by solid (dotted) lines. It can be seen from Fig. 2 that, there are several high- orbitals around the proton and neutron Fermi surface, e.g., , , , , and . Therefore, this may lead to the formation of various high- multi-qp isomers in Nd and Sm isotopes around mass region. It also can be seen that there are two sub-shells at proton number and neutron number , respectively. So for Nd () isotopes, the excitation energies of the proton 2-qp states should be a little higher. The experimentally favored 2-qp states should base on neutron configurations. Indeed, no proton 2-qp state in neutron rich Nd isotopes has been observed experimentally up to now. The energy gap at is much smaller than that of . So for Sm () isotopes, the proton 2-qp states should exist.

### iv.2 2-qp excitation energies in even-even Nd and Sm isotopes

A series of 2-qp isomers have been observed experimentally in even-even Nd and Sm isotopes at mass region Simpson et al. (2009); Patel et al. (2014); Ideguchi et al. (2016); Wang et al. (2014); Patel et al. (2016), which provide detailed information for these neutron rich nuclei. It should be noted that in this mass region, the high-order deformation is remarkable Möller and Nix (1995), and have a measurable effect on the structure of these nuclei, e.g., the inclusion of will alter the the 2-qp excitation energy about 250 keV Patel et al. (2014). Systematically calculated 2-qp states in even-even Nd isotopes with excitation energies lower than 2.5 MeV are shown in Table 2. and denote the calculated results with and without deformation, respectively. In addition, the energy differences are also shown in the last column. It can be seen that, the data are reproduced quite well by the PNC-CSM calculations no matter whether the deformation is considered or not. This indicates that the adopted single-particle level scheme is suitable for the PNC-CSM calculations. The energy differences for these four observed 2-qp states are all less than 100 keV. It seems that the deformation has small effects on the excitation energies of these 2-qp states. However, if one see through Table 2, the effects of deformation are prominent in some 2-qp states. For example, the excitation energy of state with configuration in Nd is lowered by 439 keV after the deformation being neglected. This is because after the deformation is switched off, the sequence of the single-particle levels is changed. The root-mean-square deviation between and is about 130 keV. Therefore, the deformation still has remarkable effects on the excitation energies of the 2-qp states. Due to the large shell gap at , the energies of the proton 2-qp states in Nd isotopes are all quite high. However, with increasing neutron number, the energy of the lowest proton 2-qp state with in each Nd isotopes decreases from more than 2.0 MeV to about 1.6 MeV, which may be observed in future experiments. The lowering of the excitation energy of state is caused by the decreasing of the shell gap with increasing neutron number. Since the proton-neutron residual interaction is neglected in the PNC-CSM calculations, the excitation energies of the 4-qp states with two quasi-protons and two quasi-neutrons can be simply obtained by summing the energies of the corresponding 2-qp states. Especially for Nd, the excitation energy of the 4-qp state () is only about 2899 keV from PNC-CSM calculation. Therefore, I hope this state can be observed by future experiments.

In Table 3, similar results are shown for Sm isotopes. Different from Nd isotopes, the shell gap at is much smaller, so the proton 2-qp states should exist. Indeed, 2-qp states with based on proton configuration have been observed in Sm Wang et al. (2014) and Sm Patel et al. (2016). The available data are also reproduced quite well by the PNC-CSM calculations, especially after the deformation being considered. From Table 3 it can be seen that, the effects of deformation on Sm isotopes are more prominent than Nd isotopes. The root-mean-square deviation between and is about 260 keV, which is consistent with the potential energy surface calculations in Ref. Patel et al. (2014). In Ref. Simpson et al. (2009), the calculated lowest 2-qp state in Sm using the quasiparticle rotor model is . However, in their calculation, the state is yrast with increasing spin. Therefore, they assigned the 1398keV state in Sm as . In the PNC-CSM calculations, the excitation energy of is much higher than 1398 keV, whereas the calculated is very close to the data. In addition, the excitation energies of the two states in Sm are quite close to each other, so their configurations need further investigation. These will be discussed later. Recently, one 4-qp isomer with excitation energy 2757 keV has been observed in Sm, which is assigned as (). It can be seen that the proton 2-qp state and neutron 2-qp state are all the lowest-lying 2-qp excitations in Sm. The calculated excitation energy for this 4-qp state is 2918 keV, which is very close to the data. Due to the low excitation energy of the proton 2-qp states in Sm isotopes, possible 4-qp states with the lowest 2-quasi-proton and 2-quasi-neutron configurations may exist.

### iv.3 Rotational properties in Nd and Sm isotopes

Furthermore, the rotational bands observed in Nd and Sm isotopes are analyzed. Figure 3 shows the experimental (solid circles) and calculated (solid black lines) kinematic MOIs for the ground state bands (GSBs) in even-even Nd (upper panel) and Sm (lower panel) isotopes from to . The calculated results without high-order deformation are also shown as red lines. The experimental kinematic MOIs for each band are extracted by

 J(1)(I)ℏ2=2I+1Eγ(I+1→I−1) (13)

separately for each signature sequence within a rotational band ( mod 2). The relation between the rotational frequency and nuclear angular momentum is

 ℏω(I)=Eγ(I+1→I−1)Ix(I+1)−Ix(I−1) , (14)

where , is the projection of nuclear total angular momentum along the symmetry axis of an axially symmetric nuclei. It can be seen that the MOIs and their variations with the rotational frequency are well reproduced by the PNC-CSM calculations. The data show that there is no sharp upbending in all Nd and Sm isotopes, which is consistent with the PNC-CSM calculations except for Nd and Sm. It can be seen that obvious upbendings exist in the calculated MOIs of Nd and Sm when deformation is considered. After the deformation being switched off, the upbendings become less prominent and the results are more consistent with the data. This indicate that with the rotational frequency increasing, the deformation may become smaller. It also can be seen that at the low rotational frequency region, deformation has little effect on the MOIs, while with increasing rotational frequency, it will change the behavior of MOIs. This is because the deformation will change the position of the high- orbitals, and then influence the alignment process of these orbitals in high-spin region Zhang et al. (2013b); Liu et al. (2012). Note that the deformation parameter is fixed in the present PNC-CSM calculation, while it may change with the rotational frequency. I expect that after considering this effect, the results can be improved further.

The experimental MOIs of even-even Nd and Sm isotopes show that the upbending is weak and becomes less and less obvious with increasing neutron number. To understand this, the occupation probability of each orbital (including both ) near the Fermi surface for the GSBs in Nd isotopes is shown in Fig. 4. The top and bottom rows are for protons and neutrons, respectively. The positive (negative) parity levels are denoted by blue solid (red dotted) lines. The Nilsson levels far above the Fermi surface () and far below () are not shown. In the PNC-CSM, the total particle number is exactly conserved, whereas the occupation probability for each orbital varies with rotational frequency. By examining the -dependence of the orbitals near the Fermi surface, one can get some insights on the band crossing. It can be seen from the upper panel of Fig. 4 that for the proton of Nd, the orbitals above the Fermi surface are nearly all empty, and the orbitals below the Fermi surface are nearly all occupied, and they are nearly unchanged with increasing rotational frequency. This is due to the large shell gap at , which makes the proton pairing correlations very weak. While with neutron number increasing, especially for Nd, and become partly occupied and partly empty, respectively. This is caused by the decreasing of the shell gap with increasing neutron number. With rotational frequency increasing, the occupation of and becomes nearly occupied and nearly empty at  MeV, respectively. Therefore, these two proton high- orbitals may contribute to the upbending. It can be seen from the lower panel of Fig. 4 that for neutrons, only the occupation probabilities of in Nd changes drastically around the upbending frequency. So this neutron orbital may contribute to the upbending in Nd. While in other Nd isotopes, the occupation probabilities of all the orbitals either keep unchange or change gradually with increasing rotational frequency. Therefore, the contribution to the upbending from neutron in these nuclei may be little. The present calculations show that the proton shell gap is smaller than the shell gap. Therefore, the proton occupation probabilities of Sm isotopes must be a little different from those in Nd isotopes. While for neutrons, the occupation probabilities are very close to each other when the nuclei with the same neutron number being considered. Moreover, it can be seen from Fig. 3 that the behavior of the MOIs for Nd and Sm are quite similar, so only the occupation probabilities for Nd isotopes are given to illustrate the alignment process.

It is well known that the upbending is caused by the alignment of the high- intruder orbitals Stephens and Simon (1972), which corresponds to the neutron and proton orbitals in rare-earth nuclei. In order to have a more clear understanding of the alignment mechanism in these neutron rich nuclei, the contribution of each proton and neutron major shell to the total angular momentum alignment for the GSBs in Nd isotopes are shown in Fig. 5. It can be seen that for proton, the main contribution to the angular momentum alignment comes from the major shell ( orbitals). Moreover, the contribution gradually increases with increasing neutron number. While for neutron, the contribution from major shell ( orbitals) is prominent only in Nd and Nd. In Nd, the contribution from major shell gets as smaller as major shell. This is due to the fact that with neutron number increasing, the high- but high- orbital gets close to the Fermi surface, which contributes not very much to the alignment. Therefore, one can get that different from a typical nucleus, in which the upbending is caused by whether the neutron or the proton alignment, both neutron and proton alignments contribute to the upbending in these neutron rich Nd and Sm isotopes. In the lighter Nd and Sm isotopes, the alignment is due to both neutron and proton orbitals. Meanwhile, the proton orbitals play a more and more important role in the alignment process with neutron number increasing. The competition between the alignment of proton and neutron high- orbitals makes the upbending in these Nd and Sm isotopes very weak and less obvious with increasing neutron number.

Figure 6 shows the experimental and calculated MOIs of 2-qp bands in Nd and Sm isotopes. The experimental MOIs are denoted by full black cicles (signature ) and open circles (signature ), respectively. The calculated MOIs by the PNC-CSM are denoted by black solid lines (signature ) and red dotted lines (signature ), respectively. The data are taken from Refs. Simpson et al. (2009); Wang et al. (2014); Patel et al. (2016). It can be seen that the data can be reproduced very well by the PNC-CSM calculations, except two bands in Sm. The agreement between the calculation and the data also supports the configuration assignments for these 2-qp states. Note that in Refs. Simpson et al. (2009); Wang et al. (2014), the 1279 keV level is assigned as and the 1322 keV level is assigned as , respectively. However, the present PNC-CSM calculations show that, if the configuration assignments for these two bands are changed, the MOIs can be reproduced quite well. In addition, as I mentioned before, the 1398 keV state in Sm was previously assigned as in Ref. Simpson et al. (2009). It can be seen that the calculated MOIs for this band with and configurations are similar. Therefore, it is difficult to distinguish these two configuration assignments by their MOIs. Due to the fact that the calculated excitation energy of is very close to the data, this state is attentively assigned as . More detailed experimental information is needed to give a solid configuration assignment for this state.