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In recent years, due to the extensive application of radioactive detecting devices in nuclear experiments, it has become possible to study the structure of neutron-rich nuclei far from the
β -stability line. The in-beam spectroscopy of neutron-rich nuclei in the rare earth region has been successfully investigated [1-10]. These studies revealed detailed information of the nuclear structure in exotic nuclei. With the increase of the neutron number, changes in the shell structure are expected. Recently, for example, Hartley et al. presented evidence of the existence of a deformed sub-shell gap atN=98 [11], contrary to the previous understanding that such a gap occurred atN=100 . Odd-mass nuclei can provide additional information for the Nilsson configuration assignment of the rotational band. However, most of the existing studies concentrate on the properties of even-even nuclei. Experimental data for odd-mass neutron-rich rare-earth nuclei are rare due to the experimental difficulties.A level scheme based on isomer depopulation in odd-Z neutron-rich nucleus 163Eu was presented recently by Yokoyama et al. [1] , and independently by Patel et al. [2]. The newly observed
γ− ray spectrum was assigned to the ground-state band based on the protonπ52+ [413] state. The 964(1) keV isomer was interpreted as a three-particle state, while its configuration is still an open issue. So far, there have been no detailed theoretical calculations of these experimental investigations.In the present work, the newly observed isomer and rotational band built on the ground-state of odd-Z neutron rich nucleus 163Eu are investigated by the cranked shell model, with pairing treated by the particle-number conserving method. This is the first time that a spectroscopical study including both the isomer and ground-state band in 163Eu is performed theoretically. To discuss the effect of the unpaired odd nucleon on the rotational properties of 163Eu, calculations of the neighboring even-even nuclei 162Sm and 164Gd are carried out as well.
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The cranked shell model Hamiltonian with pairing correlation is
HCSM=HSP−ωJx+HP .HSP=∑ξ(hNil)ξ is the single-particle Hamiltonian, wherehNil is the Nilsson Hamiltonian andξ (η ) the eigenstate of the single-particle Hamiltonianhξ(η) , andˉξ (ˉη ) its time-reversed state.−ωJx is the Coriolis interaction with the cranking frequencyω about the x axis (perpendicular to the nuclear symmetry z axis). The pairingHP=HP(0)+HP(2) includes monopoleHP(0)=−G0∑ξηa†ξa†¯ξa¯ηaη and quadrupoleHP(2)=−G2 ∑ξηq2(ξ)q2(η)a†ξa†¯ξa¯ηaη pairing correlations, whereq2(ξ)= √16π/5⟨ξ|r2Y20|ξ⟩ is the diagonal element of the stretched quadrupole operator.In the PNC method, the cranked shell model Hamiltonian
HCSM is diagonalized in the cranked many-particle configuration (CMPC) space without particle quasi-particle transformation. Thus the particle-number is conserved and the Pauli blocking effect is taken into account exactly. The CMPC space is a Fock space. Therefore, the HamiltonianHCSM can be diagonalized in a comparatively small space to obtain sufficiently accurate solutions of the ground and low-lying states.The eigenstate of
HCSM is|ψ⟩=∑iCi|i⟩ , with the configuration|i⟩ defined by the occupation of the cranked single-particle orbitals by real particles. The solution|ψ⟩ can always be obtained even for a pair-broken state, and provides a reliable way of assigning the configuration of a multi-particle state.nμ=∑i|Ci|2Piμ is the occupation probability of the cranked orbital|μ⟩ , wherePiμ=1 if|μ⟩ is occupied in|i⟩ , andPiμ=0 otherwise. The particle number isN=∑μnμ . The kinematic moment of inertia in the state|ψ⟩ is given byJ(1)=1ω⟨ψ|Jx|ψ⟩ , where the angular momentum alignment⟨ψ|Jx|ψ⟩=∑i|Ci|2⟨i|Jx|i⟩+ 2∑i<jC∗iCj⟨i|Jx|j⟩ . Details of the PNC-CSM method can be found in Refs. [12-14]. -
In the present work, the Nilsson states are calculated for the valence single-particle states for proton
N=0−5 and neutronN=0−6 major shells. The Nilsson parameters(κ,μ) are taken from the Lund systematics [15]. The deformation parametersε2 ,ε4 andε6 are taken from the Möller table [16] , except forε4 in 163Eu. Instead ofε4=0 ,ε4=−0.02 is taken to reproduce better the experimental excitation energy and moment of inertia of 163Eu.The effective pairing strengths
G0 andG2 can be determined, in principle, by the odd-even differences of the nuclear binding energies. For the neutron-rich rare-earth nuclei, their values are determined by the odd-even differences of moments of inertia. The effective pairing strengths are connected with the dimension of the truncated CMPC space. In the present calculations, the CMPC spaces are constructed for the protonN=3,4,5 and neutronN=4,5,6 shells. The dimension of the CMPC space is about 700, and the corresponding effective pairing strengths areG0=0.20 MeV andG2=0.02 MeV for both neutrons and protons. Effective monopole and quadrupole pairing forces with similar strengths are used in the Projected Shell Model [17]. The stability of the PNC-CSM calculations with the variation of the dimension of the CMPC space was investigated in Refs. [18-20].The cranked proton single particle level structure near the Fermi surface of 163Eu is very similar to the neighboring even-even nuclei [21], and is presented in Fig. 1. The signature
α=+1/2(−1/2) levels are denoted by solid (dashed) lines. Results with and without high-order deformationε6 are compared at rotational frequencyℏω=0 . A deformed energy gap atZ=62 arises in calculations with non-zeroε6 , which leads to a significant effect on the energy and configuration assignment of the multi-particle states, especially for the newly discovered 964(1) keV isomer. This will be discussed in detail later. In addition, compared to the results with zeroε6 , the deformed energy gap atZ=60 is reduced and the one atZ=68 enlarged. Our results in Fig. 1 show that the inclusion of theε6 deformation can change the order of some single particle levels, resulting in appearance of new sub-shell gaps. To reflect the shell changes in the exotic mass region, an early attempt was made by empirically adjusting the Nilsson parameters from their standard values, see for example Refs. [20, 22, 23].Figure 1. Cranked proton Nilsson levels near the Fermi surface of 163Eu with signature
α=+1/2 (solid) andα=−1/2 (dash).ε2=0.275 ,ε4=−0.02 andε6 = 0.042.The configuration of each multi-particle state is explicitly assigned through the occupation probability
nμ of each cranked Nilsson orbitalμ . The proton orbital occupation probabilitiesnμ versus rotational frequencyℏω of the ground-state band in 162Sm, 163Eu and 164Gd are shown in Fig. 2. As shown, the proton orbitalπ52+[413] is blocked(nμ≈1) in 163Eu while it is either almost fully occupied(nμ≈2) or empty(nμ≈0) in even-even nuclei 162Sm and 164Gd. Therefore, the configuration of the ground-state band in 163Eu is assigned asπ52+[413] . Configurations of the other multi-particle states, listed in Table 1, are assigned similarly.Kπ configuration Ex /keV(ε6≠0 )Ex /keV(ε6=0 )Eexpx /keV52+ π52+[413] 0 0 0 32+ π32+[411] 312.1 273.7 52− π52−[532] 778.9 400.9 72− π72−[523] 851.0 1050.2 12+ π12+[411] 1617.9 1630.5 72+ π72+[404] 1738.0 12+ π12+[420] 1858.5 32− π32−[541] 2016.5 1842.2 12− π12−[541] 2523.6 132− ν72+[633]⊗ν12−[521]⊗π52+[413] 1001.3 1057.0 964(1) 112− ν72+[633]⊗ν12−[521]⊗π32+[411] 1313.4 1330.7 132− π52+[413]⊗π52−[532]⊗π32+[411] 1509.9 1134.9 132+ ν72+[633]⊗ν12−[521]⊗π52−[532] 1780.2 1457.9 172− ν72+[633]⊗ν52−[512]⊗π52+[413] 1839.2 1499.3 172+ π52+[413]⊗π52−[532]⊗π72−[523] 2024.5 1872.5 172+ ν72+[633]⊗ν52−[512]⊗π52−[532] 2618.1 1900.2 Table 1. Low-lying multi-particle states in 163Eu predicted by the PNC-CSM calculations.
Figure 2. (color online) Occupation probabilities
nμ of each cranked proton orbitalμ (including bothα=±1/2 ) near the Fermi surface of 162Sm, 163Eu and 164Gd for the ground-state bands. The solid blue (short dash red) line denote positive (negative) parity orbital. Fully occupiednμ≈2 and emptynμ≈0 orbitals are not labelled.The 964(1) keV isomer in 163Eu was observed recently by Yokoyama et al. [1] and independently by Patel et al. [2] . The spin and parity of this isomer is in both works given as
132− . However, its configuration is disputed. It was interpreted as the coupling of theKπ=4−(ν212−72+) neutron excitation and theπ52+ [413] odd proton by the deformed Hartree-Fock model with angular momentum projection [1] , while it was referred to as the three-proton excitation state with a configurationπ52+ [413]⊗π52− [532]⊗π32+ [411] in the Nilsson-BCS calculations [2].The low-lying multi-particle states of 163Eu predicted by the PNC-CSM calculations are listed in Table 1. A significant influence of the
ε6 deformation is demonstrated by the energy and configuration assignments of the multi-particle states. As shown in Table 1, the lowest three-particle excitation state of 163Eu is theν72+[633]⊗ ν12−[521]⊗π52+[413] configuration state. Its energy is 1001.3 keV and 1057.0 keV as given by calculations with non-zero and zeroε6 , respectively. Both values reproduce well the experimental measurement. As for the second three-particle excitation state, if we do not consider theε6 effect, it is aπ52+[413]⊗π52−[532] ⊗π32+[411] configuration state, which is predicted as the 964(1) keV isomer by the Nilsson-BCS calculations. Its energy is 1134.9 keV in PNC-CSM calculations, which is very close to the experimental data as well. If we take into account the effect ofε6 deformation, the second three-particle state is aν72+[633]⊗ν12−[521]⊗π32+[411] configuration state. The three-proton state with a configurationπ52+[413]⊗π52−[532] ⊗π32+[411] becomes the third three-particle state with a much higher energy of 1509.9 keV, which can not reproduce well the experimental data. Thus the 964(1) keV132− isomer can be assigned aν72+[633]⊗ ν12−[521]⊗π52+[413] configuration with high confidence. The three-proton state in non-zeroε6 calculations results from the enlarged energy gap atZ=62 (see Fig. 1).In general, compared to the calculations with zero
ε6 , the state energies are higher in the non-zeroε6 calculations for most of the three-particle states (see Table 1). Normally, the calculations with more degrees of freedom give lower energy. In the present calculations with non-zeroε6 , the ground-state is lower in energy by about 5 MeV for protons and 15 MeV for neutrons. The eigen-energies of the excited states are lower as well. The higher energies of the multi-particle states in non-zeroε6 calculations are mainly caused by the reduced protonZ=62 and enlarged neutronN=102 deformed energy gaps when theε6 deformation is included. Several investigations have already given evidence that the high-orderε6 deformation is important for the structure of the neutron-rich rare-earth nuclei [5, 8, 21].Figure 3 shows the experimental and calculated kinematic moment of inertia
J(1) for the ground-state bands in 162Sm, 163Eu and 164Gd. No significant signature of splitting is found in these bands, and only the calculated moments of inertia for favored signature bands are shown. It is seen that the experimental data are reproduced quite well by the PNC calculations. Compared to the neighboring even-even nuclei 162Sm and 164Gd, a 10%~15% increase ofJ(1) can be seen for the one-particle ground-state band in 163Eu. This can be explained by the pairing reduction due to blocking. Calculations without pairing, which are not shown here, result in moments of inertiaJ(1) that are almost the same for these three bands.J(1) of the ground-state bands in 162Sm and 164Gd are very similar, although there are two more protons in 164Gd than in 162Sm. Additional information is given by the contributions of protonsJ(1)p (dashed dot lines) and neutronsJ(1)n (dashed lines).J(1)n for all three bands is almost the same, whileJ(1)p for the one-particle ground-state band in 163Eu is larger than in 162Sm and 164Gd by ~18% in the low frequency range. The 10%~15% increase ofJ(1) in 163Eu comes from the contribution of protons, namely, from the Pauli blocking effect of the protonπ52+[413] orbital.Figure 3. (color online) Comparison of the theoretical kinematic moment of inertia
J(1) for the ground-state bands in 162Sm (navy), 163Eu (olive) and 164Gd (red) with the experimental data. Theoretical total moments of inertiaJ(1) are denoted by the solid lines, the contribution from protons (neutrons)J(1)p (J(1)n ) are denoted by dashed dot (dashed) lines and the experimental data are denoted by symbols. -
The recently observed isomer and ground-state band in odd-Z neutron-rich rare-earth nucleus 163Eu are investigated by using the cranked shell model, with pairing treated by a particle-number conserving method. The experimental energy of the isomer and the kinematic moment of inertia
J(1) of the ground-state band are reproduced very well by the theoretical calculations. This is the first time a detailed spectroscopical investigation of the observed 964(1) keV isomer and ground-state of 163Eu is performed theoretically. To investigate the pairing and Pauli blocking effects, the rotational bands in the neighboring even-even nuclei 162Sm and 164Gd were calculated as well.The configuration of the isomer is assigned as the two-neutron excitation of
Kπ=4−(ν212−72+) coupled with the odd-protonπ52+ [413] state, i.e.,132−(ν72+[633]⊗ν12− [521]⊗π52+[413]) configuration state, which is consistent with the deformed Hartree-Fock model with angular momentum projection. The high-orderε6 deformation plays an important role in the configuration assignment due to its effect on the nuclear mean field. More low-lying multi-particle states in 163Eu are predicted.Compared to the neighboring even-even nuclei 162Sm and 164Gd, a 10%~15% increase of
J(1) is found in the ground-state band of 163Eu. Detailed theoretical investigations show that the increase ofJ(1) comes from the contribution of protons. It can be explained by the pairing reduction due to the blocking of the odd-proton on theπ52+ [413] orbital in 163Eu.
High-K isomer and the rotational properties in the odd-Z neutron-rich nucleus 163Eu
- Received Date: 2019-03-06
- Available Online: 2019-06-01
Abstract: The newly observed isomer and ground-state band in the odd-Z neutron-rich rare-earth nucleus 163Eu are investigated by using the cranked shell model (CSM), with pairing treated by the particle-number conserving (PNC) method. This is the first time detailed theoretical investigations are performed of the observed 964(1) keV isomer and ground-state rotational band in 163Eu. The experimental data are reproduced very well by the theoretical results. The configuration of the 964(1) keV isomer is assigned as the three-particle state