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Particle-number conserving analysis of the moments of inertia of high-K multiquasiparticle bands in 179Ta
Nuclear Physics A 816 (2009) 19–32 www.elsevier.com/locate/nuclphysa
Particle-number conserving analysis of the moments of inertia of high-K multiquasiparticle bands in 179Ta Z.H. Zhang, X. Wu, Y.A. Lei ∗ , J.Y. Zeng State Key Lab of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China Received 15 July 2008; received in revised form 20 October 2008; accepted 21 October 2008 Available online 29 October 2008
Abstract The experimental one-, three-, and five-quasiparticle bands in 179 Ta are analyzed by the particle-number conserving (PNC) method for the cranked shell model with pairing interaction, in which the particle-number is conserved and the blocking effects are considered exactly. Once the quasiparticle configurations, particularly those of the high-K three- and five-quasiparticle bands, are determined, the experimental moments of inertia and their variations with rotational frequencies can be reproduced very well by PNC calculations without any free parameter. © 2008 Elsevier B.V. All rights reserved. PACS: 21.60.-n; 21.60.Cs; 23.20.Lv; 27.70.+q Keywords: Particle-number conserving method; Moment of inertia; Pairing correlation; High-K multiquasiparticle band; Blocking effect
1. Introduction A striking feature of axially symmetric deformed nuclei around the mass A ∼ 180 (Z ∼ 71–75, N ∼ 98–108) is the observation of a number of low-lying high-K, high-seniority rotational bands [1,2]. These high-K bands arise from the coupling of a few orbitals near both proton and neutron Fermi surfaces, with large angular momentum projections Ωi on the nuclear symmetry axes (K = i Ωi ); e.g. the proton orbitals 7/2+ [404](g7/2 ), 9/2− [514](h11/2 ), 5/2+ [402](d5/2 ), and the neutron orbitals ν7/2+ [633](i13/2 ), 7/2− [514](h9/2 ), ν9/2+ [624] (i13/2 ), etc. (the bold face letter indicates the high-j intruder orbitals). These high-K and * Corresponding author. Tel.: +86 10 62755208; fax: +86 10 62751615.
E-mail addresses: [email protected] (Y.A. Lei), [email protected] (J.Y. Zeng). 0375-9474/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2008.10.008
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high-seniority bands are particularly favorable in studying the blocking effects of the pairing correlation of the unpaired nucleons at low rotational frequencies. It is well known that pairing correlation is extremely important in the low angular momentum region, where they are manifested by reducing the nuclear moment of inertia (MOI) of the rigid-body estimation [3]. Usually nuclear pairing correlation is treated by Bardeen–Cooper–Schrieffer (BCS) or Hartree–Fock–Bogolyubov (HFB) quasiparticle (qp) formalism. There is no doubt that the BCS quasiparticle formalism achieved great successes in the superconductivity theory of metals. However, when it is applied to nuclear pairing, along with all its benefits it suffers from some serious problems, the nonconservation of the number of particles and the related spurious states being one of them. Since the number of nucleons (∼ 102 ), particularly the number of valence nucleons (∼ 10) which dominate the feature of nuclear low-lying excited states, is very limited, n/n is non-negligible [4]. The remedy in terms of the particle number projection techniques considerably complicates the algorithm, yet not helping in improving the description of the higher-excited part of the spectrum of the pairing Hamiltonian [5]. The most serious problem of BCS formalism in treating nuclear pairing is that it is unable to treat the blocking effects properly [6]. As Rowe had emphasized [6], while the blocking effects are straightforward, it is very difficult to treat them in BCS formalism because they introduce different quasiparticle bases for different blocked levels. As a typical example, the observed low-lying 1-qp, 3-qp, and 5-qp bands in 179 Ta [7,8] are analyzed by the particle-number conserving (PNC) method [9,10], in which the particle number is conserved from beginning to end and the Pauli blocking effects are taken into account exactly. The key point of the PNC method is that, instead of the usual single-particle levels (SPL) truncation, the many-particle configuration (MPC) truncation is adopted, which makes the PNC calculations numerically workable and sufficiently accurate in describing nuclear low-lying excited states. The advantages of the MPC truncation over the SPL one had been discussed in [11]. The stability of the calculation with MPC basis (Fock space) cutoff was illustrated in detail in [5,12]. A brief description of the PNC treatment for the cranked shell model (CSM) with pairing is presented in Section 2. The calculated results for 179 Ta and discussions are given in Section 3. The experimental MOIs of the 1-, 3-, and 5-qp bands can be satisfactorily reproduced by the PNC calculation in which no free parameter is involved. A brief summary is given in Section 4. 2. A brief introduction to PNC method for the CSM The CSM Hamiltonian of an axially symmetric nucleus in the rotating frame is [9,10] HCSM = HNil − ωJx + HP = H0 + Hp , H0 = HNil − ωJx ,
(1)
where HNil is the Nilsson Hamiltonian, −ωJx is the Coriolis interaction with cranking frequency ω about the x axis, perpendicular to the nuclear symmetrical z axis, H0 = HNil − ωJx is the one-body part of HCSM , and HP is the pairing interaction HP = −G aξ+ aξ+ (2) ¯ aη¯ aη ξη
where ξ¯ (η) ¯ is the time-reversal state of the Nilsson state ξ (η), and G is the effective strength of nuclear pairing interaction. For the details of the solution to the eigenvalue problem of HCSM , please see [9,10].
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Assuming an eigenstate of HCSM is Ci |i (Ci real) |ψ =
21
(3)
i
where |i is an eigenstate of H0 , i.e. a cranked many-particle configuration (CMPC), then the angular momentum alignment of |ψ is Ci2 i|Jx |i + 2 Ci Cj i|Jx |j (4) ψ|Jx |ψ = i
i
and the kinematic moment of inertia of state |ψ is 1 ψ|Jx |ψ. (5) ω Because Jx is a one-body operator, i|Jx |j (i = j ) may not vanish when two CMPCs |i and |j differ by only one particle occupation. After a certain permutation of creation operators, |i and |j can be recast into J (1) =
|i = (−1)Miμ |μ · · ·,
|j = (−1)Mj ν |ν · · ·
(6)
where the ellipsis · · · stands for the same particle occupation, and (−1)Miμ = ±1, (−1)Mj ν = ±1 according to whether the permutation is even or odd. Therefore, the kinematic moment of inertia of |ψ is (1) jμ(1) + jμν (7) J (1) = μ
μ<ν
where jμ(1) = μ|jx |μnμ /ω, 2 (1) jμν = μ|jx |ν (−1)Miμ +Mj ν Ci∗ Cj ω
(μ = ν)
i
and nμ =
|Ci |2 Piμ
(8)
i
is the occupation probability of the cranked orbital |μ, Piμ = 1 if |μ is occupied in |i, and Piμ = 0 otherwise. In the following PNC calculations, the CSM Hamiltonian (1) is diagonalized in a sufficiently large CMPC space (i.e. the Fock space spanned by the eigenstates of H0 , the one-body part of HCSM ) to obtain the low-lying excited eigenstates |ψ’s. The dimension of the CMPC space is about 700 for proton and 800 for neutron. As we are only interested in the yrast and low-lying excited states, the numbers of the important CMPCs involved (weight > 1%) is not very large (usually < 20) and almost all the CMPCs with weight > 0.1% are counted in, so the solutions to the low-lying excited states are quite accurate. The effective pairing interaction strength is determined by the experimental odd–even differences in nuclear binding energies, and the corresponding values are Gp = 0.28 MeV (proton), Gn = 0.26 MeV (neutron), so there is no free parameter in the following PNC calculation. The experimental kinematic MOI for each band is extracted by
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J (1) (I ) h¯ 2
=
2I + 1 . Eγ (I + 1 → I − 1)
(9)
Separately for each signature sequence within a rotational band (I = α mod 2). The relation between the rotational frequency ω and nuclear angular momentum I is Eγ (I + 1 → I − 1) , (10) Ix (I + 1) − Ix (I − 1) where Ix (I ) = (I + 1/2)2 − K 2 , K is the projection of nuclear total angular momentum along the symmetry z axis of an axially symmetric nuclei. It is noted that because Rx (π) = e−iπJx , [Jx , Jz ] = 0, the signature scheme breaks the K quantum number. However, it has been pointed out in [9] and [13] that, though [Jx , Jz ] = 0, we have [Jx , Jz2 ] = 0. Thus we can construct the simultaneous eigenstates of (Rx (π), Jz2 ). Each CMPC |i in (3) is chosen as a simultaneous eigenstate of (H0 , Jz2 ), for the details please see [9]. For the ground state band of an even– even nucleus (described by the qp band |0 in the BCS formalism), K π = 0+ , I = α mod 2 = 0, 2, 4, . . . (α = 0). For the 1-qp band (denoted by α1+ |0 in the BCS formalism), K = Ω1 (> 0), which has two sequences, α = ±1/2. For the 2-qp band (denoted by α1+ α2+ |0 in the BCS independent qp formalism), for treating nuclear pairing, there are four sequences, i.e. K = |Ω1 + Ω2 |, α = α1 + α2 = 0, 1, and K = |Ω1 − Ω2 |, α = α1 + α2 = 0, 1, or their four linear combinations. In the PNC treatment for the CSM with pairing, the PNC many-particle wave function is chosen as the simultaneous eigenstates of (H0 , Jz2 ). Corresponding to the 2-qp band α1+ α2+ |0 in the BCS formalism, we have four sequences in the PNC formalism with K = |Ω1 + Ω2 |, α = 0, 1, and K = |Ω1 − Ω2 |, α = 0, 1, and the former is the high-K one. Similarly, for the 3-qp band (denoted by α1+ α2+ α3+ |0 in the BCS formalism), we have eight sequences h¯ ω(I ) =
K = |Ω1 + Ω2 + Ω3 |,
α = ±1/2,
K = |Ω1 + Ω2 − Ω3 |,
α = ±1/2,
K = |Ω1 − Ω2 + Ω3 |,
α = ±1/2,
K = |−Ω1 + Ω2 + Ω3 |,
α = ±1/2
and the first one is the high-K band (α = ±1/2). The situation is similar for the 5-qp band. The 3-qp and 5-qp bands addressed in Section 3 are all high-K bands (K = |Ω1 + Ω2 + Ω3 |). However, it should be noted that, though the projection K of nuclear total angular momentum of a deformed spheroidal nucleus is a constant of motion, because of the Coriolis anti-pairing interaction in the CSM Hamiltonian (1) and [Jx , Jz ] = 0, K cannot keep constant with increasing ω. The K structure of the wave function in a single-j cranked shell model was analyzed in detail in [13]. P.M. Walker et al. [1,14] pointed out that some forms of K-mixing must exist to enable K-forbidden transition observed in a lot of low-lying rotational bands of axially symmetric nuclei (e.g. Hf isotope chain). In fact, to some extent collective rotation itself destroyes the axial symmetry even for the ground state (K = 0) band of an even–even nuclei, i.e. with increasing ω, K = 0 components gradually enter into the ground state band. The situation is similar for the other multiquasiparticle bands. Thus, the meaning of the quantum number K usually used to label a rotational band is vague. However, by convention, K is still used as a convenient quantum number to describe rotational bands of deformed spheroidal nuclei.
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Fig. 1. Cranked Nilsson levels near the Fermi surface of 179 Ta (Lund systematics). (a) Proton, (b) neutron. The Nilsson parameters κ, μ are taken from Ref. [15] and the deformation parameters from Ref. [16], ε2 = 0.242, ε4 = 0.052.
3. Calculation of the multiquasiparticle bands of 179 Ta 3.1. Cranked Nilsson levels The cranked Nilsson levels near the Fermi surface of 179 Ta are given in Fig. 1(a) (proton) and (b) (neutron). The Nilsson parameters κ and μ, and the deformation parameters ε2 = 0.242, ε4 = 0.052 are taken from the Lund systematics [15,16]; i.e. an average of the neighboring even– even Hf and W isotopes (also see Table 5 of Ref. [7]). However, this level scheme is unable to correctly reproduce the experimental bandhead energies of the low-lying excited 1-qp bands of 179 Ta, particularly the ground state band (gsb) configuration, π7/2+ [404]. As for the discussion about the deviation of the observed bandhead energies of the 1-qp bands from the Nilsson single particle level scheme (Lund systematics), please see Ref. [7], particularly Fig. 12 of [7]. Thus, a more proper single-particle level scheme near the Fermi surface is needed for reliable calculations of the bandhead energies and MOIs of each bands, particularly for the 3-qp and 5-qp bands. A slightly modified Nilsson level scheme is given in Fig. 2. The change in μ will cause the change of relative position of Nilsson levels in the same N shell, and the change in κ will lead to the change of relative position of neighboring N shell. The details are given in the caption of Fig. 2. According to this scheme, the calculated bandhead energies of 1-qp bands (pairing interaction not considered) is shown in Fig. 3(a). It is noted that for the gsb, the odd proton occupies π9/2− [514] according to Fig. 1(a), but π7/2+ [404] according to the modified scheme Fig. 2(a), which is in agreement with the observed data (Fig. 3(c)). In Fig. 3(b) we show the bandhead energies of the 1-qp bands calculated by PNC method with pairing, in which the effective proton pairing interaction strength Gp = 0.28 MeV is determined by the experimental odd–even difference in binding energies, and no free parameter is involved. It is seen that the experimental bandhead energies of the 1-qp bands can be well reproduced by PNC calculations. In the follow-
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Fig. 2. A slightly modified Nilsson level scheme to account for the experimental bandhead energies of 1-qp bands (see Fig. 3). The deformation parameters ε2 , ε4 , remain unchanged [16]. (a) For proton, κ4 = 0.060 (N = 4 major shell), κ5 = 0.0606 (N = 5 major shell), μ4 = 0.540, μ5 = 0.659. In addition, the Nilsson level 1/2+ [411] is further shifted upward by 0.085h¯ ω0 , 5/2+ [402] is shifted upward by 0.032h¯ ω0 . (b) For neutron, κ5 = 0.0521, κ6 = 0.0650, μ5 = 0.34, μ6 = 0.40, and 1/2− [521] is shifted upward by 0.060h¯ ω0 .
ing, we will adopted the modified Nilsson level scheme to calculate the MOIs of the observed 1-qp and high-K 3-qp and 5-qp bands. 3.2. MOIs of the 1-qp bands The comparison between the experimental and the calculated MOIs by PNC method for the 1-qp bands are presented in Figs. 4(a)–8(a), and the corresponding occupation probability nμ for each cranked Nilsson level near the Fermi surface is given in Figs. 4(b)–8(b). The Nilsson levels far above Fermi surface (nμ ∼ 0) and levels far below (nμ ∼ 2) are not shown. It is encouraging that all the experimental MOIs and their variation with rotational frequency ω are reproduced quite well by PNC method with no free parameter involved, because the effective pairing interaction strength, Gp = 0.28 MeV, Gn = 0.26 MeV, are determined by the experimental odd–even mass differences. It is expected that for a more realistic single-particle level scheme, the calculated J (1) (ω) could be improved further. The PNC calculations for MOIs also confirm the configuration assignments for the 1-qp bands in 179 Ta in [7]. The MOIs of these bands and their variations with ω can be understood from the occupation probability nμ of each cranked Nilsson level near Fermi surface. In the PNC calculations, while the total number of particles ( μ nμ ) is strictly conserved, each nμ may vary with ω. Here are some observations: (a) Both the experimental and calculated MOIs of the 1-qp bands, the gsb K π = 7/2+ (Fig. 4(a)), the K π = 5/2+ band at 238 keV (Fig. 6(a)), and K π = 1/2+ band at 520 keV (Fig. 7(a)), are approximately “identical” [17] to that of the reference K π = 0+ gsb of 178 Hf
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Fig. 3. The bandhead energies of 1-qp bands in 179 Ta. (a) The calculated bandhead energies according to the Nilsson level scheme shown in Fig. 2(a). (b) The same as (a) but the pairing interaction is taken into account (Gp = 0.28 MeV). (c) The experimental results.
Fig. 4. (a) The MOI of the ground state band, K π = 7/2+ . The experimental results are denoted by 1 (α = −1/2) and 2 (α = 1/2). No significant signature splitting is observed. The calculated results by PNC method are shown as a solid line (α = 1/2) and a dotted line (α = −1/2), and both nearly coincide with each other. The experimental MOIs of the reference band (the gsb K π = 0+ of 178 Hf, the qp vacuum band) are denoted by solid circles ". (b) The occupation probability nμ of each cranked Nilsson proton level near Fermi surface of 179 Ta. Due to the pairing interaction the Nilsson levels above Fermi surface (5/2+ [402], 1/2− [541], etc.) are partially occupied, and the levels below (9/2− [514], 1/2+ [411], etc.) are partially vacuous.
(qp-vacuum band). This is because for these 1-qp bands the unpaired proton blocks the normal Nilsson orbitals, 7/2+ [404](g7/2 ), 5/2+ [402](d5/2 ), and 1/2+ [411](d3/2 ), respectively, which have a negligibly small Coriolis response [9,18]. (b) On the contrary, the experimental and calculated J (1) for the K π = 1/2− , π1/2− [541](h9/2 ) band is much larger than that of the reference gsb band of 178 Hf (see Fig. 8(a)), because π1/2− [541](h9/2 ) is a high-j intruder proton orbital (see Fig. 2(a)) with a large Coriolis response. However, with increasing ω the difference between J (1) (π1/2− [541]) and J (1) (gsb
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Fig. 5. The same as Fig. 4, but for the 1-qp band at 31 keV, K π = 9/2− (9/2− [514]). The experimental and calculated MOI of this high j (but high Ω) 9/2− [514](h11/2 ) band are slightly larger than the experimental MOI of the reference gsb of 178 Hf (qp vacuum band).
Fig. 6. The same as Fig. 4, but for the 1-qp band at 238 keV, π 5/2+ [402]. The MOI of this K π = 5/2+ band is nearly identical to that of the reference K π = 0+ gsb of 178 Hf.
of 178 Hf) becomes smaller and smaller as the Coriolis anti-pairing interaction increases. The K π = 9/2− band at 31 keV (Fig. 5(a)) is the intermediate case, because the π9/2− [514](h11/2 ) is a high-j , but high-Ω (deformation aligned) orbital with only a small Coriolis response. (c) It should be noted that the concept of “blocked level” only has an vague meaning even at the bandhead. For example, for the gsb K π = 7/2+ and the K π = 9/2− band at 31 keV, the orbitals above the Fermi surface (π5/2+ [402], π1/2− [541], etc.) are also partially occupied, whereas the orbitals below the Fermi surface (π9/2− [514], π1/2+ [411], etc.) are partially vacuous (see Figs. 4(b) and 5(b)). In fact, the occupation probability distributions are similar for both the gsb K π = 7/2+ band and the 31 keV K π = 9/2− band, except that the nμ ’s for the Nilsson levels π7/2+ [404] and (π9/2− [514] are interchanged (compare Figs. 4(b) and 5(b)).
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Fig. 7. The same as Fig. 4, but for 1-qp band at 520 keV, K π = 1/2+ (α = −1/2), π 1/2+ [411](d3/2 ).
Fig. 8. The same as Fig. 4, but for the high j and low Ω band K π = 1/2− (α = 1/2), π 1/2− [541](h9/2 ).
3.3. MOIs of the 3-qp bands 3.3.1. The 3-qp band at 1252 keV, K π = 21/2− . Four possible qp configurations are suggested in [7]: π 3 9/2− [514]7/2+ [404]5/2+ [402], π9/2− [514] ⊗ ν 2 5/2− [512]7/2− [514], π7/2+ [404] ⊗ ν 2 5/2− [512]7/2+ [633], π7/2+ [404] ⊗ ν 2 9/2+ [624]7/2− [514]. From the gK -factor analysis in [7], the last two configurations are not appropriate, but the mixing of the first two is possible. The calculated J (1) by PNC method for the first configuration is given in Fig. 9(a), and the second in Fig. 9(c). Comparing the calculated and experimental J (1) , the first configuration is more favorable. Fig. 9(b) shows that for the K π = 21/2− band
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Fig. 9. The 3-qp band K π = 21/2− at 1252 keV. (a) The calculated J (1) for the 3-qp configuration π 3 9/2− [514] 7/2+ [404]5/2+ [402] is denoted by a solid line (α = 1/2) and a dotted line (α = −1/2) (α = ±1/2, no obvious signature splitting is found), which agrees well with the experimental J (1) (1, α = −1/2; 2, α = 1/2). The J (1) of the reference band (gsb K π = 0+ of 178 Hf) are also presented by ". (b) For this 3-qp configuration the three proton orbitals near Fermi surface (see Fig. 2(a)) are completely blocked. (c) The calculated J (1) for the 3-qp configuration π 9/2− [514] ⊗ ν 2 5/2− [512]7/2− [514] disagrees with the experiments for ω > 0.20 MeV/h¯ . (d) The neutron occupation probability nμ for this 3-qp configuration.
at 1252 keV, the three proton orbitals in the vicinity of Fermi surface, 9/2− [514], 7/2+ [404] and 5/2+ [402] (see Fig. 2(a)) are almost completely blocked by unpaired protons, thus there appears a large gap near the proton Fermi surface. In this particular case, nμ ≈ 0 for all the Nilsson orbitals above the Fermi surface and nμ ≈ 2 for those below, and the proton gap parameter p ∼ 0. 3.3.2. The 3-qp K π = 25/2+ band at 1318 keV, and K π = 23/2− band at 1327 keV. Based on the gK -factor analysis for the 3-qp band at 1318 keV, the configuration is assigned to be π9/2− [514] ⊗ ν 2 9/2+ [624]7/2− [514] (see Table 4 of Ref. [7]). The calculated J (1) for this configuration is shown in Fig. 10(a), which agrees well with the experimental results, thus confirms this qp configuration. Because the high j intruder neutron orbital 9/2+ [624](i13/2 ) and proton orbital π9/2− [514](h11/2 ) are blocked, the MOI of this band is much larger than that of the qp-vacuum band (K π = 0+ ) of 178 Hf. However, with increasing ω, the difference between the MOIs of both bands gradually decreases due to the strong Coriolis anti-pairing effect.
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Fig. 10. The 3-qp band, K π = 25/2+ at 1318 keV, the configuration is π 9/2− [514] ⊗ ν 2 9/2+ [624]7/2− [514]. No significant signature splitting is observed. (a) The calculated MOI (solid line, α = 1/2; dotted line α = −1/2) agrees very well with the experimental results (1, α = −1/2; 2, α = 1/2). The experimental J (1) for the reference K π = 0+ band of 178 Hf is denoted by ". (b) The calculated neutron occupation probability nμ for this 3-qp band.
Fig. 11. The same as Fig. 10, but for the 3-qp band at 1327 keV, K π = 23/2− , π 7/2+ [404] ⊗ ν 2 9/2+ [624]7/2− [514], which is similar to the 1318 keV K π = 25/2+ band, except the blocked proton intruder orbital π 9/2− [514] is replaced by the proton normal orbital π 7/2+ [404]. As a consequence the J (1) (K π = 23/2− ) is a little smaller than J (1) (K π = 25/2+ ). (b) The calculated neutron occupation probability nμ for this 3-qp band.
Based on the gK -factor analysis (see Table 4 of Ref. [7]), the 3-qp configuration π7/2+ [404]⊗ 2 ν 9/2+ [624]7/2− [514] is assigned for the K π = 23/2− band at 1327 keV. The calculated J (1) for this configuration is shown in Fig. 11(a), which also agrees well with the experiment. Comparing Figs. 10(a) and 11(a), it is seen J (1) (K π = 23/2− band) is a little smaller than J (1) (K π = 25/2+ band), which is understandable because the intruder proton orbital π9/2− [514] for the K π = 25/2+ band is replaced by the normal proton orbital π7/2+ [404] (deformation aligned) for the K π = 23/2− band.
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Fig. 12. The observed 3-qp band at 1628 keV, K π = 19/2+ or K π = 21/2− (see §3.5.2 of Ref. [7]). Two possible configurations are K π = 19/2+ , π 9/2− [514]ν 2 9/2+ [624]1/2− [521], and K π = 21/2− , π 7/2+ [404] ⊗ ν 2 9/2+ [624]7/2− [514]. The calculated J (1) for the first configuration K π = 19/2+ is given in Fig. 12(a) (no obvious signature splitting is found), and the occupation probability nμ for each neutron orbitals is given in Fig. 12(b). The calculated J (1) for the second configuration K π = 21/2− is given in Fig. 12(c) and the occupation probability nμ for each neutron orbitals is given in Fig. 12(d).
3.3.3. The 3-qp band at 1628 keV, K π = 19/2+ or K π = 21/2− Two possible 3-qp configurations are suggested in [7] for the band at 1628 keV: K π = 19/2+ ,
π9/2− [514] ⊗ ν 2 9/2+ [624]1/2− [521],
K π = 21/2− ,
π7/2+ [404] ⊗ ν 2 9/2+ [624]7/2− [514].
Based on the gK -factor analysis (see Table 4 of Ref. [7]), both configurations seem possible. The calculated J (1) by PNC method for the two configurations are given in Figs. 12(a) and 12(c), respectively. The calculated J (1) for the first configuration agrees with the experiment. Thus, the first configuration is more reasonable. 3.4. MOIs of the 5-qp bands 3.4.1. The five-quasiparticle band at 2640 keV, K π = 37/2+ Based on the gK -factor analysis in [7], the 5-qp configuration π 3 9/2− [514]7/2+ [404] 5/2+ [402] ⊗ ν 2 9/2+ [624]7/2− [514] is suggested. The calculated J (1) by PNC method for this
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Fig. 13. The 2640 keV 5-qp band, K π = 37/2+ , π 3 9/2− [514]7/2+ [404]5/2+ [402] ⊗ ν 2 9/2+ [624]7/2− [514]. (a) Comparison between the calculated (solid line, α = 1/2, dotted line α = −1/2, both are almost coincide with each other.) and experimented MOIs (1, α = −1/2; 2, α = 1/2). (b) The three proton Nilsson orbitals near Fermi surface (see Fig. 2(a)) are completely blocked. (c) The two neutron Nilsson orbitals near Fermi surface (see Fig. 2(b)) are almost completely blocked.
configuration is given in Fig. 13(a) (no signature splitting is found), which is in agreement with the experimental results, thus confirms this 5-qp configuration assignment. It is seen that the three proton orbitals near Fermi surface, 9/2− [514], 7/2+ [404] and 5/2+ [402] are completely blocked (Fig. 13(b)) and the two neutron orbitals in the vicinity of Fermi surface are also almost completely blocked (Fig. 13(c)). It is noted that at lower frequency, J (1) (K π = 37/2+ band) is much larger than J (1) (K π = 0+ band of 178 Hf), which is understandable because both the intruder neutron orbital 9/2+ [624](i13/2 ) and proton orbital π9/2− [514](h11/2 ) are completely blocked. However, with increasing ω, the difference in J (1) between both bands gradually decreases due to the strong Coriolis anti-pairing interaction. 3.4.2. The five-quasiparticle band at 2791 keV, K π = 33/2− Two possible 5-qp configurations are suggested in [7] for the K π = 33/2− band at 2791 keV: π 3 9/2− [514]7/2+ [404]1/2− [541] ⊗ ν 2 7/2− [514]9/2+ [624], π 3 9/2− [514]7/2+ [404]5/2+ [402] ⊗ ν 2 7/2− [514]5/2− [512]. From the gK -factor analysis, the first configuration seems to be more reasonable. The calculated J (1) for the two configurations are given in Figs. 14(a) and 14(d), respectively. Thus, we favor the first configuration and the corresponding occupation probability nμ is shown in Figs. 14(b) (proton) and 14(c) (neutron). 4. Summary The experimental 1-qp and high-K 3- and 5-qp bands in 179 Ta are analyzed by PNC method for the CSM with pairing, in which Pauli blocking effects are taken into account exactly. A slightly modified Nilsson level scheme is adopted to reproduce the experimental bandhead energies of the 1-qp bands. The effective pairing interaction strength is determined by the experimental odd–even differences in binding energies. The experimental MOIs of all these bands are satisfactorily reproduced by PNC calculations with no free parameter, once the quasiparticle configurations bands are reasonably assigned. From the occupation probability nμ of each cranked Nilsson orbitals, the large differences in MOIs and their variations with rotational frequency between different quasiparticle bands can be understood.
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Fig. 14. The 2791 keV K π = 33/2− 5-qp band. Two possible configurations are suggested in Ref. [7]; i.e. π 3 9/2− [514] 7/2+ [404]1/2− [541] ⊗ ν 2 7/2− [514]9/2+ [624] and π 3 9/2− [514]7/2+ [404]5/2+ [402] ⊗ ν 2 7/2− [514]5/2− [512], no signature splitting is found. The calculated J (1) for the two configurations are given in Fig. 14(a) and (d) respectively (solid line, α = 1/2; dotted line, α = −1/2), and the corresponding proton and neutron occupation probabilities for each orbitals are given in Fig. 14(b), (c) and (e), (f), respectively.
Acknowledgement This work is supported by the Natural Science Foundation of China under the Nos. 10675007, 10778613, 10675006, and 10435010; 973 program 2008CB717803. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
P. Walker, G. Draculis, Nature 399 (1999) 35. S. Singh, S.S. Malik, A.K. Jain, B. Singh, At. Data Nucl. Data Tables 92 (2006) 1. A. Bohr, B.R. Mottelson, Nuclear Structure, vol. 2, Benjamin, MA, 1975. J.Y. Zeng, T.S. Cheng, Nucl. Phys. A 405 (1983) 1. H. Moligue, J. Dudek, Phys. Rev. C 56 (1997) 1795. D.J. Rowe, Nuclear Collective Motion, Methuen, London, 1974, p. 194. F.G. Kondev, G.D. Dracoulis, A.P. Byrne, T. Kibédi, S. Bayer, Nucl. Phys. A 617 (1997) 91. F.G. Kondev, et al., Eur. Phys. J. A 22 (2004) 23. J.Y. Zeng, T.H. Jin, Z.J. Zhao, Phys. Rev. C 50 (1994) 1388. X.B. Xin, S.X. Liu, Y.A. Lei, J.Y. Zeng, Phys. Rev. C 62 (2000) 067303. C.S. Wu, J.Y. Zeng, Phys. Rev. C 39 (1989) 666. S.X. Liu, J.Y. Zeng, E.G. Zhao, Phys. Rev. C 66 (2002) 024320. C.S. Wu, J.Y. Zeng, Phys. Rev. C 41 (1990) 1822. P.M. Walker, et al., Phys. Rev. Lett. 65 (1990) 416. S.G. Nilsson, et al., Nucl. Phys. A 131 (1969) 1. R. Bengtsson, S. Fraundorf, F.R. May, At. Data Nucl. Data Tables 35 (1986) 15. C. Baktash, J.D. Garrett, D.F. Winchell, A. Smith, Phys. Rev. Lett. 69 (1992) 1500. J.Y. Zeng, Y.A. Lei, T.H. Jin, Z.J. Zhao, Phys. Rev. C 50 (1994) 746.
Particle-number conserving analysis of the moments of inertia of high-K multiquasiparticle bands in 179Ta Z.H. Zhang, X. Wu, Y.A. Lei ∗ , J.Y. Zeng State Key Lab of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China Received 15 July 2008; received in revised form 20 October 2008; accepted 21 October 2008 Available online 29 October 2008
Abstract The experimental one-, three-, and five-quasiparticle bands in 179 Ta are analyzed by the particle-number conserving (PNC) method for the cranked shell model with pairing interaction, in which the particle-number is conserved and the blocking effects are considered exactly. Once the quasiparticle configurations, particularly those of the high-K three- and five-quasiparticle bands, are determined, the experimental moments of inertia and their variations with rotational frequencies can be reproduced very well by PNC calculations without any free parameter. © 2008 Elsevier B.V. All rights reserved. PACS: 21.60.-n; 21.60.Cs; 23.20.Lv; 27.70.+q Keywords: Particle-number conserving method; Moment of inertia; Pairing correlation; High-K multiquasiparticle band; Blocking effect
1. Introduction A striking feature of axially symmetric deformed nuclei around the mass A ∼ 180 (Z ∼ 71–75, N ∼ 98–108) is the observation of a number of low-lying high-K, high-seniority rotational bands [1,2]. These high-K bands arise from the coupling of a few orbitals near both proton and neutron Fermi surfaces, with large angular momentum projections Ωi on the nuclear symmetry axes (K = i Ωi ); e.g. the proton orbitals 7/2+ [404](g7/2 ), 9/2− [514](h11/2 ), 5/2+ [402](d5/2 ), and the neutron orbitals ν7/2+ [633](i13/2 ), 7/2− [514](h9/2 ), ν9/2+ [624] (i13/2 ), etc. (the bold face letter indicates the high-j intruder orbitals). These high-K and * Corresponding author. Tel.: +86 10 62755208; fax: +86 10 62751615.
E-mail addresses: [email protected] (Y.A. Lei), [email protected] (J.Y. Zeng). 0375-9474/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2008.10.008
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high-seniority bands are particularly favorable in studying the blocking effects of the pairing correlation of the unpaired nucleons at low rotational frequencies. It is well known that pairing correlation is extremely important in the low angular momentum region, where they are manifested by reducing the nuclear moment of inertia (MOI) of the rigid-body estimation [3]. Usually nuclear pairing correlation is treated by Bardeen–Cooper–Schrieffer (BCS) or Hartree–Fock–Bogolyubov (HFB) quasiparticle (qp) formalism. There is no doubt that the BCS quasiparticle formalism achieved great successes in the superconductivity theory of metals. However, when it is applied to nuclear pairing, along with all its benefits it suffers from some serious problems, the nonconservation of the number of particles and the related spurious states being one of them. Since the number of nucleons (∼ 102 ), particularly the number of valence nucleons (∼ 10) which dominate the feature of nuclear low-lying excited states, is very limited, n/n is non-negligible [4]. The remedy in terms of the particle number projection techniques considerably complicates the algorithm, yet not helping in improving the description of the higher-excited part of the spectrum of the pairing Hamiltonian [5]. The most serious problem of BCS formalism in treating nuclear pairing is that it is unable to treat the blocking effects properly [6]. As Rowe had emphasized [6], while the blocking effects are straightforward, it is very difficult to treat them in BCS formalism because they introduce different quasiparticle bases for different blocked levels. As a typical example, the observed low-lying 1-qp, 3-qp, and 5-qp bands in 179 Ta [7,8] are analyzed by the particle-number conserving (PNC) method [9,10], in which the particle number is conserved from beginning to end and the Pauli blocking effects are taken into account exactly. The key point of the PNC method is that, instead of the usual single-particle levels (SPL) truncation, the many-particle configuration (MPC) truncation is adopted, which makes the PNC calculations numerically workable and sufficiently accurate in describing nuclear low-lying excited states. The advantages of the MPC truncation over the SPL one had been discussed in [11]. The stability of the calculation with MPC basis (Fock space) cutoff was illustrated in detail in [5,12]. A brief description of the PNC treatment for the cranked shell model (CSM) with pairing is presented in Section 2. The calculated results for 179 Ta and discussions are given in Section 3. The experimental MOIs of the 1-, 3-, and 5-qp bands can be satisfactorily reproduced by the PNC calculation in which no free parameter is involved. A brief summary is given in Section 4. 2. A brief introduction to PNC method for the CSM The CSM Hamiltonian of an axially symmetric nucleus in the rotating frame is [9,10] HCSM = HNil − ωJx + HP = H0 + Hp , H0 = HNil − ωJx ,
(1)
where HNil is the Nilsson Hamiltonian, −ωJx is the Coriolis interaction with cranking frequency ω about the x axis, perpendicular to the nuclear symmetrical z axis, H0 = HNil − ωJx is the one-body part of HCSM , and HP is the pairing interaction HP = −G aξ+ aξ+ (2) ¯ aη¯ aη ξη
where ξ¯ (η) ¯ is the time-reversal state of the Nilsson state ξ (η), and G is the effective strength of nuclear pairing interaction. For the details of the solution to the eigenvalue problem of HCSM , please see [9,10].
Z.H. Zhang et al. / Nuclear Physics A 816 (2009) 19–32
Assuming an eigenstate of HCSM is Ci |i (Ci real) |ψ =
21
(3)
i
where |i is an eigenstate of H0 , i.e. a cranked many-particle configuration (CMPC), then the angular momentum alignment of |ψ is Ci2 i|Jx |i + 2 Ci Cj i|Jx |j (4) ψ|Jx |ψ = i
i
and the kinematic moment of inertia of state |ψ is 1 ψ|Jx |ψ. (5) ω Because Jx is a one-body operator, i|Jx |j (i = j ) may not vanish when two CMPCs |i and |j differ by only one particle occupation. After a certain permutation of creation operators, |i and |j can be recast into J (1) =
|i = (−1)Miμ |μ · · ·,
|j = (−1)Mj ν |ν · · ·
(6)
where the ellipsis · · · stands for the same particle occupation, and (−1)Miμ = ±1, (−1)Mj ν = ±1 according to whether the permutation is even or odd. Therefore, the kinematic moment of inertia of |ψ is (1) jμ(1) + jμν (7) J (1) = μ
μ<ν
where jμ(1) = μ|jx |μnμ /ω, 2 (1) jμν = μ|jx |ν (−1)Miμ +Mj ν Ci∗ Cj ω
(μ = ν)
i
and nμ =
|Ci |2 Piμ
(8)
i
is the occupation probability of the cranked orbital |μ, Piμ = 1 if |μ is occupied in |i, and Piμ = 0 otherwise. In the following PNC calculations, the CSM Hamiltonian (1) is diagonalized in a sufficiently large CMPC space (i.e. the Fock space spanned by the eigenstates of H0 , the one-body part of HCSM ) to obtain the low-lying excited eigenstates |ψ’s. The dimension of the CMPC space is about 700 for proton and 800 for neutron. As we are only interested in the yrast and low-lying excited states, the numbers of the important CMPCs involved (weight > 1%) is not very large (usually < 20) and almost all the CMPCs with weight > 0.1% are counted in, so the solutions to the low-lying excited states are quite accurate. The effective pairing interaction strength is determined by the experimental odd–even differences in nuclear binding energies, and the corresponding values are Gp = 0.28 MeV (proton), Gn = 0.26 MeV (neutron), so there is no free parameter in the following PNC calculation. The experimental kinematic MOI for each band is extracted by
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J (1) (I ) h¯ 2
=
2I + 1 . Eγ (I + 1 → I − 1)
(9)
Separately for each signature sequence within a rotational band (I = α mod 2). The relation between the rotational frequency ω and nuclear angular momentum I is Eγ (I + 1 → I − 1) , (10) Ix (I + 1) − Ix (I − 1) where Ix (I ) = (I + 1/2)2 − K 2 , K is the projection of nuclear total angular momentum along the symmetry z axis of an axially symmetric nuclei. It is noted that because Rx (π) = e−iπJx , [Jx , Jz ] = 0, the signature scheme breaks the K quantum number. However, it has been pointed out in [9] and [13] that, though [Jx , Jz ] = 0, we have [Jx , Jz2 ] = 0. Thus we can construct the simultaneous eigenstates of (Rx (π), Jz2 ). Each CMPC |i in (3) is chosen as a simultaneous eigenstate of (H0 , Jz2 ), for the details please see [9]. For the ground state band of an even– even nucleus (described by the qp band |0 in the BCS formalism), K π = 0+ , I = α mod 2 = 0, 2, 4, . . . (α = 0). For the 1-qp band (denoted by α1+ |0 in the BCS formalism), K = Ω1 (> 0), which has two sequences, α = ±1/2. For the 2-qp band (denoted by α1+ α2+ |0 in the BCS independent qp formalism), for treating nuclear pairing, there are four sequences, i.e. K = |Ω1 + Ω2 |, α = α1 + α2 = 0, 1, and K = |Ω1 − Ω2 |, α = α1 + α2 = 0, 1, or their four linear combinations. In the PNC treatment for the CSM with pairing, the PNC many-particle wave function is chosen as the simultaneous eigenstates of (H0 , Jz2 ). Corresponding to the 2-qp band α1+ α2+ |0 in the BCS formalism, we have four sequences in the PNC formalism with K = |Ω1 + Ω2 |, α = 0, 1, and K = |Ω1 − Ω2 |, α = 0, 1, and the former is the high-K one. Similarly, for the 3-qp band (denoted by α1+ α2+ α3+ |0 in the BCS formalism), we have eight sequences h¯ ω(I ) =
K = |Ω1 + Ω2 + Ω3 |,
α = ±1/2,
K = |Ω1 + Ω2 − Ω3 |,
α = ±1/2,
K = |Ω1 − Ω2 + Ω3 |,
α = ±1/2,
K = |−Ω1 + Ω2 + Ω3 |,
α = ±1/2
and the first one is the high-K band (α = ±1/2). The situation is similar for the 5-qp band. The 3-qp and 5-qp bands addressed in Section 3 are all high-K bands (K = |Ω1 + Ω2 + Ω3 |). However, it should be noted that, though the projection K of nuclear total angular momentum of a deformed spheroidal nucleus is a constant of motion, because of the Coriolis anti-pairing interaction in the CSM Hamiltonian (1) and [Jx , Jz ] = 0, K cannot keep constant with increasing ω. The K structure of the wave function in a single-j cranked shell model was analyzed in detail in [13]. P.M. Walker et al. [1,14] pointed out that some forms of K-mixing must exist to enable K-forbidden transition observed in a lot of low-lying rotational bands of axially symmetric nuclei (e.g. Hf isotope chain). In fact, to some extent collective rotation itself destroyes the axial symmetry even for the ground state (K = 0) band of an even–even nuclei, i.e. with increasing ω, K = 0 components gradually enter into the ground state band. The situation is similar for the other multiquasiparticle bands. Thus, the meaning of the quantum number K usually used to label a rotational band is vague. However, by convention, K is still used as a convenient quantum number to describe rotational bands of deformed spheroidal nuclei.
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Fig. 1. Cranked Nilsson levels near the Fermi surface of 179 Ta (Lund systematics). (a) Proton, (b) neutron. The Nilsson parameters κ, μ are taken from Ref. [15] and the deformation parameters from Ref. [16], ε2 = 0.242, ε4 = 0.052.
3. Calculation of the multiquasiparticle bands of 179 Ta 3.1. Cranked Nilsson levels The cranked Nilsson levels near the Fermi surface of 179 Ta are given in Fig. 1(a) (proton) and (b) (neutron). The Nilsson parameters κ and μ, and the deformation parameters ε2 = 0.242, ε4 = 0.052 are taken from the Lund systematics [15,16]; i.e. an average of the neighboring even– even Hf and W isotopes (also see Table 5 of Ref. [7]). However, this level scheme is unable to correctly reproduce the experimental bandhead energies of the low-lying excited 1-qp bands of 179 Ta, particularly the ground state band (gsb) configuration, π7/2+ [404]. As for the discussion about the deviation of the observed bandhead energies of the 1-qp bands from the Nilsson single particle level scheme (Lund systematics), please see Ref. [7], particularly Fig. 12 of [7]. Thus, a more proper single-particle level scheme near the Fermi surface is needed for reliable calculations of the bandhead energies and MOIs of each bands, particularly for the 3-qp and 5-qp bands. A slightly modified Nilsson level scheme is given in Fig. 2. The change in μ will cause the change of relative position of Nilsson levels in the same N shell, and the change in κ will lead to the change of relative position of neighboring N shell. The details are given in the caption of Fig. 2. According to this scheme, the calculated bandhead energies of 1-qp bands (pairing interaction not considered) is shown in Fig. 3(a). It is noted that for the gsb, the odd proton occupies π9/2− [514] according to Fig. 1(a), but π7/2+ [404] according to the modified scheme Fig. 2(a), which is in agreement with the observed data (Fig. 3(c)). In Fig. 3(b) we show the bandhead energies of the 1-qp bands calculated by PNC method with pairing, in which the effective proton pairing interaction strength Gp = 0.28 MeV is determined by the experimental odd–even difference in binding energies, and no free parameter is involved. It is seen that the experimental bandhead energies of the 1-qp bands can be well reproduced by PNC calculations. In the follow-
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Fig. 2. A slightly modified Nilsson level scheme to account for the experimental bandhead energies of 1-qp bands (see Fig. 3). The deformation parameters ε2 , ε4 , remain unchanged [16]. (a) For proton, κ4 = 0.060 (N = 4 major shell), κ5 = 0.0606 (N = 5 major shell), μ4 = 0.540, μ5 = 0.659. In addition, the Nilsson level 1/2+ [411] is further shifted upward by 0.085h¯ ω0 , 5/2+ [402] is shifted upward by 0.032h¯ ω0 . (b) For neutron, κ5 = 0.0521, κ6 = 0.0650, μ5 = 0.34, μ6 = 0.40, and 1/2− [521] is shifted upward by 0.060h¯ ω0 .
ing, we will adopted the modified Nilsson level scheme to calculate the MOIs of the observed 1-qp and high-K 3-qp and 5-qp bands. 3.2. MOIs of the 1-qp bands The comparison between the experimental and the calculated MOIs by PNC method for the 1-qp bands are presented in Figs. 4(a)–8(a), and the corresponding occupation probability nμ for each cranked Nilsson level near the Fermi surface is given in Figs. 4(b)–8(b). The Nilsson levels far above Fermi surface (nμ ∼ 0) and levels far below (nμ ∼ 2) are not shown. It is encouraging that all the experimental MOIs and their variation with rotational frequency ω are reproduced quite well by PNC method with no free parameter involved, because the effective pairing interaction strength, Gp = 0.28 MeV, Gn = 0.26 MeV, are determined by the experimental odd–even mass differences. It is expected that for a more realistic single-particle level scheme, the calculated J (1) (ω) could be improved further. The PNC calculations for MOIs also confirm the configuration assignments for the 1-qp bands in 179 Ta in [7]. The MOIs of these bands and their variations with ω can be understood from the occupation probability nμ of each cranked Nilsson level near Fermi surface. In the PNC calculations, while the total number of particles ( μ nμ ) is strictly conserved, each nμ may vary with ω. Here are some observations: (a) Both the experimental and calculated MOIs of the 1-qp bands, the gsb K π = 7/2+ (Fig. 4(a)), the K π = 5/2+ band at 238 keV (Fig. 6(a)), and K π = 1/2+ band at 520 keV (Fig. 7(a)), are approximately “identical” [17] to that of the reference K π = 0+ gsb of 178 Hf
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Fig. 3. The bandhead energies of 1-qp bands in 179 Ta. (a) The calculated bandhead energies according to the Nilsson level scheme shown in Fig. 2(a). (b) The same as (a) but the pairing interaction is taken into account (Gp = 0.28 MeV). (c) The experimental results.
Fig. 4. (a) The MOI of the ground state band, K π = 7/2+ . The experimental results are denoted by 1 (α = −1/2) and 2 (α = 1/2). No significant signature splitting is observed. The calculated results by PNC method are shown as a solid line (α = 1/2) and a dotted line (α = −1/2), and both nearly coincide with each other. The experimental MOIs of the reference band (the gsb K π = 0+ of 178 Hf, the qp vacuum band) are denoted by solid circles ". (b) The occupation probability nμ of each cranked Nilsson proton level near Fermi surface of 179 Ta. Due to the pairing interaction the Nilsson levels above Fermi surface (5/2+ [402], 1/2− [541], etc.) are partially occupied, and the levels below (9/2− [514], 1/2+ [411], etc.) are partially vacuous.
(qp-vacuum band). This is because for these 1-qp bands the unpaired proton blocks the normal Nilsson orbitals, 7/2+ [404](g7/2 ), 5/2+ [402](d5/2 ), and 1/2+ [411](d3/2 ), respectively, which have a negligibly small Coriolis response [9,18]. (b) On the contrary, the experimental and calculated J (1) for the K π = 1/2− , π1/2− [541](h9/2 ) band is much larger than that of the reference gsb band of 178 Hf (see Fig. 8(a)), because π1/2− [541](h9/2 ) is a high-j intruder proton orbital (see Fig. 2(a)) with a large Coriolis response. However, with increasing ω the difference between J (1) (π1/2− [541]) and J (1) (gsb
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Fig. 5. The same as Fig. 4, but for the 1-qp band at 31 keV, K π = 9/2− (9/2− [514]). The experimental and calculated MOI of this high j (but high Ω) 9/2− [514](h11/2 ) band are slightly larger than the experimental MOI of the reference gsb of 178 Hf (qp vacuum band).
Fig. 6. The same as Fig. 4, but for the 1-qp band at 238 keV, π 5/2+ [402]. The MOI of this K π = 5/2+ band is nearly identical to that of the reference K π = 0+ gsb of 178 Hf.
of 178 Hf) becomes smaller and smaller as the Coriolis anti-pairing interaction increases. The K π = 9/2− band at 31 keV (Fig. 5(a)) is the intermediate case, because the π9/2− [514](h11/2 ) is a high-j , but high-Ω (deformation aligned) orbital with only a small Coriolis response. (c) It should be noted that the concept of “blocked level” only has an vague meaning even at the bandhead. For example, for the gsb K π = 7/2+ and the K π = 9/2− band at 31 keV, the orbitals above the Fermi surface (π5/2+ [402], π1/2− [541], etc.) are also partially occupied, whereas the orbitals below the Fermi surface (π9/2− [514], π1/2+ [411], etc.) are partially vacuous (see Figs. 4(b) and 5(b)). In fact, the occupation probability distributions are similar for both the gsb K π = 7/2+ band and the 31 keV K π = 9/2− band, except that the nμ ’s for the Nilsson levels π7/2+ [404] and (π9/2− [514] are interchanged (compare Figs. 4(b) and 5(b)).
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Fig. 7. The same as Fig. 4, but for 1-qp band at 520 keV, K π = 1/2+ (α = −1/2), π 1/2+ [411](d3/2 ).
Fig. 8. The same as Fig. 4, but for the high j and low Ω band K π = 1/2− (α = 1/2), π 1/2− [541](h9/2 ).
3.3. MOIs of the 3-qp bands 3.3.1. The 3-qp band at 1252 keV, K π = 21/2− . Four possible qp configurations are suggested in [7]: π 3 9/2− [514]7/2+ [404]5/2+ [402], π9/2− [514] ⊗ ν 2 5/2− [512]7/2− [514], π7/2+ [404] ⊗ ν 2 5/2− [512]7/2+ [633], π7/2+ [404] ⊗ ν 2 9/2+ [624]7/2− [514]. From the gK -factor analysis in [7], the last two configurations are not appropriate, but the mixing of the first two is possible. The calculated J (1) by PNC method for the first configuration is given in Fig. 9(a), and the second in Fig. 9(c). Comparing the calculated and experimental J (1) , the first configuration is more favorable. Fig. 9(b) shows that for the K π = 21/2− band
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Fig. 9. The 3-qp band K π = 21/2− at 1252 keV. (a) The calculated J (1) for the 3-qp configuration π 3 9/2− [514] 7/2+ [404]5/2+ [402] is denoted by a solid line (α = 1/2) and a dotted line (α = −1/2) (α = ±1/2, no obvious signature splitting is found), which agrees well with the experimental J (1) (1, α = −1/2; 2, α = 1/2). The J (1) of the reference band (gsb K π = 0+ of 178 Hf) are also presented by ". (b) For this 3-qp configuration the three proton orbitals near Fermi surface (see Fig. 2(a)) are completely blocked. (c) The calculated J (1) for the 3-qp configuration π 9/2− [514] ⊗ ν 2 5/2− [512]7/2− [514] disagrees with the experiments for ω > 0.20 MeV/h¯ . (d) The neutron occupation probability nμ for this 3-qp configuration.
at 1252 keV, the three proton orbitals in the vicinity of Fermi surface, 9/2− [514], 7/2+ [404] and 5/2+ [402] (see Fig. 2(a)) are almost completely blocked by unpaired protons, thus there appears a large gap near the proton Fermi surface. In this particular case, nμ ≈ 0 for all the Nilsson orbitals above the Fermi surface and nμ ≈ 2 for those below, and the proton gap parameter p ∼ 0. 3.3.2. The 3-qp K π = 25/2+ band at 1318 keV, and K π = 23/2− band at 1327 keV. Based on the gK -factor analysis for the 3-qp band at 1318 keV, the configuration is assigned to be π9/2− [514] ⊗ ν 2 9/2+ [624]7/2− [514] (see Table 4 of Ref. [7]). The calculated J (1) for this configuration is shown in Fig. 10(a), which agrees well with the experimental results, thus confirms this qp configuration. Because the high j intruder neutron orbital 9/2+ [624](i13/2 ) and proton orbital π9/2− [514](h11/2 ) are blocked, the MOI of this band is much larger than that of the qp-vacuum band (K π = 0+ ) of 178 Hf. However, with increasing ω, the difference between the MOIs of both bands gradually decreases due to the strong Coriolis anti-pairing effect.
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Fig. 10. The 3-qp band, K π = 25/2+ at 1318 keV, the configuration is π 9/2− [514] ⊗ ν 2 9/2+ [624]7/2− [514]. No significant signature splitting is observed. (a) The calculated MOI (solid line, α = 1/2; dotted line α = −1/2) agrees very well with the experimental results (1, α = −1/2; 2, α = 1/2). The experimental J (1) for the reference K π = 0+ band of 178 Hf is denoted by ". (b) The calculated neutron occupation probability nμ for this 3-qp band.
Fig. 11. The same as Fig. 10, but for the 3-qp band at 1327 keV, K π = 23/2− , π 7/2+ [404] ⊗ ν 2 9/2+ [624]7/2− [514], which is similar to the 1318 keV K π = 25/2+ band, except the blocked proton intruder orbital π 9/2− [514] is replaced by the proton normal orbital π 7/2+ [404]. As a consequence the J (1) (K π = 23/2− ) is a little smaller than J (1) (K π = 25/2+ ). (b) The calculated neutron occupation probability nμ for this 3-qp band.
Based on the gK -factor analysis (see Table 4 of Ref. [7]), the 3-qp configuration π7/2+ [404]⊗ 2 ν 9/2+ [624]7/2− [514] is assigned for the K π = 23/2− band at 1327 keV. The calculated J (1) for this configuration is shown in Fig. 11(a), which also agrees well with the experiment. Comparing Figs. 10(a) and 11(a), it is seen J (1) (K π = 23/2− band) is a little smaller than J (1) (K π = 25/2+ band), which is understandable because the intruder proton orbital π9/2− [514] for the K π = 25/2+ band is replaced by the normal proton orbital π7/2+ [404] (deformation aligned) for the K π = 23/2− band.
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Fig. 12. The observed 3-qp band at 1628 keV, K π = 19/2+ or K π = 21/2− (see §3.5.2 of Ref. [7]). Two possible configurations are K π = 19/2+ , π 9/2− [514]ν 2 9/2+ [624]1/2− [521], and K π = 21/2− , π 7/2+ [404] ⊗ ν 2 9/2+ [624]7/2− [514]. The calculated J (1) for the first configuration K π = 19/2+ is given in Fig. 12(a) (no obvious signature splitting is found), and the occupation probability nμ for each neutron orbitals is given in Fig. 12(b). The calculated J (1) for the second configuration K π = 21/2− is given in Fig. 12(c) and the occupation probability nμ for each neutron orbitals is given in Fig. 12(d).
3.3.3. The 3-qp band at 1628 keV, K π = 19/2+ or K π = 21/2− Two possible 3-qp configurations are suggested in [7] for the band at 1628 keV: K π = 19/2+ ,
π9/2− [514] ⊗ ν 2 9/2+ [624]1/2− [521],
K π = 21/2− ,
π7/2+ [404] ⊗ ν 2 9/2+ [624]7/2− [514].
Based on the gK -factor analysis (see Table 4 of Ref. [7]), both configurations seem possible. The calculated J (1) by PNC method for the two configurations are given in Figs. 12(a) and 12(c), respectively. The calculated J (1) for the first configuration agrees with the experiment. Thus, the first configuration is more reasonable. 3.4. MOIs of the 5-qp bands 3.4.1. The five-quasiparticle band at 2640 keV, K π = 37/2+ Based on the gK -factor analysis in [7], the 5-qp configuration π 3 9/2− [514]7/2+ [404] 5/2+ [402] ⊗ ν 2 9/2+ [624]7/2− [514] is suggested. The calculated J (1) by PNC method for this
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Fig. 13. The 2640 keV 5-qp band, K π = 37/2+ , π 3 9/2− [514]7/2+ [404]5/2+ [402] ⊗ ν 2 9/2+ [624]7/2− [514]. (a) Comparison between the calculated (solid line, α = 1/2, dotted line α = −1/2, both are almost coincide with each other.) and experimented MOIs (1, α = −1/2; 2, α = 1/2). (b) The three proton Nilsson orbitals near Fermi surface (see Fig. 2(a)) are completely blocked. (c) The two neutron Nilsson orbitals near Fermi surface (see Fig. 2(b)) are almost completely blocked.
configuration is given in Fig. 13(a) (no signature splitting is found), which is in agreement with the experimental results, thus confirms this 5-qp configuration assignment. It is seen that the three proton orbitals near Fermi surface, 9/2− [514], 7/2+ [404] and 5/2+ [402] are completely blocked (Fig. 13(b)) and the two neutron orbitals in the vicinity of Fermi surface are also almost completely blocked (Fig. 13(c)). It is noted that at lower frequency, J (1) (K π = 37/2+ band) is much larger than J (1) (K π = 0+ band of 178 Hf), which is understandable because both the intruder neutron orbital 9/2+ [624](i13/2 ) and proton orbital π9/2− [514](h11/2 ) are completely blocked. However, with increasing ω, the difference in J (1) between both bands gradually decreases due to the strong Coriolis anti-pairing interaction. 3.4.2. The five-quasiparticle band at 2791 keV, K π = 33/2− Two possible 5-qp configurations are suggested in [7] for the K π = 33/2− band at 2791 keV: π 3 9/2− [514]7/2+ [404]1/2− [541] ⊗ ν 2 7/2− [514]9/2+ [624], π 3 9/2− [514]7/2+ [404]5/2+ [402] ⊗ ν 2 7/2− [514]5/2− [512]. From the gK -factor analysis, the first configuration seems to be more reasonable. The calculated J (1) for the two configurations are given in Figs. 14(a) and 14(d), respectively. Thus, we favor the first configuration and the corresponding occupation probability nμ is shown in Figs. 14(b) (proton) and 14(c) (neutron). 4. Summary The experimental 1-qp and high-K 3- and 5-qp bands in 179 Ta are analyzed by PNC method for the CSM with pairing, in which Pauli blocking effects are taken into account exactly. A slightly modified Nilsson level scheme is adopted to reproduce the experimental bandhead energies of the 1-qp bands. The effective pairing interaction strength is determined by the experimental odd–even differences in binding energies. The experimental MOIs of all these bands are satisfactorily reproduced by PNC calculations with no free parameter, once the quasiparticle configurations bands are reasonably assigned. From the occupation probability nμ of each cranked Nilsson orbitals, the large differences in MOIs and their variations with rotational frequency between different quasiparticle bands can be understood.
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Fig. 14. The 2791 keV K π = 33/2− 5-qp band. Two possible configurations are suggested in Ref. [7]; i.e. π 3 9/2− [514] 7/2+ [404]1/2− [541] ⊗ ν 2 7/2− [514]9/2+ [624] and π 3 9/2− [514]7/2+ [404]5/2+ [402] ⊗ ν 2 7/2− [514]5/2− [512], no signature splitting is found. The calculated J (1) for the two configurations are given in Fig. 14(a) and (d) respectively (solid line, α = 1/2; dotted line, α = −1/2), and the corresponding proton and neutron occupation probabilities for each orbitals are given in Fig. 14(b), (c) and (e), (f), respectively.
Acknowledgement This work is supported by the Natural Science Foundation of China under the Nos. 10675007, 10778613, 10675006, and 10435010; 973 program 2008CB717803. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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