The intruder orbitals in superdeformed bands and alignment additivity of odd–odd nuclei in the A∼190 region

Nuclear Physics A 760 (2005) 263–273

The intruder orbitals in superdeformed bands and alignment additivity of odd–odd nuclei in the A ∼ 190 region X.T. He a,b,∗ , S.Y. Yu b,d , J.Y. Zeng a,c , E.G. Zhao a,b,d a Center of Theoretical Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China b Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China c School of Physics, Peking University, Beijing 100871, China d College of Science, Huzhou Normal University, Huzhou 313000, China

Received 15 February 2005; received in revised form 24 May 2005; accepted 14 June 2005 Available online 5 July 2005

Abstract The particle-number conserving (PNC) method for treating the cranked shell model with monopole and quadrupole pairing correlations is used to study the superdeformed (SD) bands observed in odd–odd nuclei in the A ∼ 190 mass region. Spins are assigned to the levels in these bands. The microscopic mechanism of the hω ¯ evolution of the dynamic moment of inertia J (2) for these SD bands are analyzed. In particular, the major roles of the j15/2 neutron and i13/2 proton orbitals played in the SD bands are investigated in detail by contributions to J (2) from each cranked orbital and interference terms between two cranked orbitals. Additivity in the A ∼ 190 mass region is investigated. The experimental evidence for additivity of alignments in 192 Tl can be reproduced by our calculations.  2005 Elsevier B.V. All rights reserved. PACS: 21.10.Re; 21.60.Cs; 27.80.+w Keywords: Dynamic moment of inertia; Angular momentum alignment; Cranked shell model; Pairing correlations

* Corresponding author.

E-mail address: [email protected] (X.T. He). 0375-9474/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2005.06.006

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1. Introduction More than 80 superdeformed (SD) rotational bands have been observed in the A ∼ 190 mass region (see Ref. [1]). For most SD bands in the A ∼ 190 mass region, the dynamic moment of inertia (MoI) exhibits a smooth rise with increasing hω ¯ [2], which is due to the gradual alignment of nucleons occupying high-N intruder orbitals (originating from the νj15/2 and the πi13/2 subshells) and the gradual disappearance of pairing with the collective rotation [3,4]. In this picture, Pauli blocking of high-N intruder orbitals is expected to induce a flat J (2) with hω. ¯ This effect is indeed observed in some nuclei, such as in 191 Hg 193 and Tl (see Fig. 1) where either the N = 7 neutron or N = 6 proton orbital is blocked. In some odd–odd nuclei, the J (2) even decreases with increasing h¯ ω (see Fig. 1) where both the νj15/2 and πi13/2 intruder orbitals are blocked. Also, the experimental data showed (see Fig. 1(b)) that at low frequency, the J (2) of 0-quasiparticle (qp) bands is lower than that of 1-qp bands with either neutron or proton orbital being blocked, and J (2) of 1-qp bands is lower than that of 2-qp bands with both the neutron and proton orbitals being blocked. The larger J (2) in odd–odd nuclei at low spin is due to the reduction of pairing correlations effected by blocked particles. The independent quasiparticle picture leads to additivity in some physics quantities, such as qp energy, MoI, angular momentum alignment, etc. In Ref. [5], it has been demonstrated that the picture of independent quasiparticle becomes a good approximation when pairing is sufficiently strong. Also the picture of independent particle (no pairing) implies additivity of alignments. Additivity will not hold in the transitional region between strong pairing and no pairing. Additivity of alignments often fails for normally deformed (ND) nuclei [6–8]. As for the SD nuclei in the A ∼ 150 region, the pairing is relatively weak due to the lower single particle level density, and additivity holds. In the A ∼ 190 region, the level density is such that pairing is in the transitional region between the strong and very weak. One might expect additivity should not hold. However, the experimental data

Fig. 1. Experimental J (1) and J (2) of the 0-qp band [192 Hg(1)], 1-qp bands [193 Tl(1), 191 Hg(1)] and 2-qp band [192 Tl(A)].

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present additivity of alignments for SD bands in odd–odd nucleus 192 Tl and its neighboring odd-proton nucleus 191 Hg and odd-neutron nucleus 193 Tl [9]. In this paper, we use particle-number conserving (PNC) method [10] for treating the cranked shell model with monopole and quadrupole pairing correlations to study the SD bands in odd–odd nucleus 192 Tl, where both the intruder j15/2 neutron and i13/2 proton orbitals are blocked. This kind of configurations make the evolution of the kinematic and dynamic moment of inertia with frequency quite different from most SD bands in the A ∼ 190 region, which represents a challenge for theory to understand. The important roles of the j15/2 neutron and i13/2 proton orbitals played in the behavior of the SD bands observed in odd–odd nuclei are clearly exhibited by the contributions to MoI from each cranked orbital and the interference terms between two cranked orbitals. The bandhead spins are assigned to these bands in the paper since it have not yet been determined experimentally. Because the particle number is conserved and the Pauli blocking effect is treated consistently in the PNC method, it is suitable to use this method to test the additivity rule. The formalism of the approach will be briefly sketched (see Ref. [10] for details) in the next section. Calculated results and discussions are presented in Section 3. Conclusions and remarks are given in Section 4.

2. Formalism The CSM Hamiltonian with pairing reads: (1) HCSM = HSP − ωJx + HP = H0 + HP ,
where H0 = HSP − ωJx = i h0 (ω)i (i includes all the valence particles), h0 (ω) = hNilsson − ωjx , is the one-body part of HCSM , hNilsson the Nilsson Hamiltonian, −ωjx the Coriolis force. HP is the pairing including both monopole and quadrupole pairing correlations, HP = HP (0) + HP (2),  † †  † HP (0) = −G0 aξ aξ¯ aη¯ aη = −G0 (−)Ωξ −Ωη aξ† a−ξ a−η aη , (2) ξη

HP (2) = −G2



ξη

q2 (ξ )q2 (η)aξ† aξ†¯ aη¯ aη ,

(3)

ξη

√ with |ξ¯ (η) ¯ being the time-reversal state of |ξ(η) and q2 (ξ ) = 16π/5ξ |r 2 Y20 |ξ  the diagonal element of the stretched quadrupole operator. In our calculation, h0 (ω) is diagonalized to obtain the cranked Nilsson orbitals firstly. Then, HCSM is diagonalized in a sufficiently large cranked many-particle configuration (CMPC) space to obtain the yrast and low-lying eigenstates. The eigenstate of HCSM is expressed in terms of the CMPCs as  |ψ = Ci |i, (4) i

where |i denotes an occupation of particles in the cranked orbitals and Ci is the corresponding probability amplitude. The occupation probability of the cranked orbital µ (including both signature α = ±1/2) is

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nµ =



|Ci |2 Piµ ,

i

 Piµ =

1, 0,

|µ is occupied in CMPC |i, otherwise.

(5)

The angular momentum alignment of the state |ψ is given by   |Ci |2 i|Jx |i + 2 Ci∗ Cj i|Jx |j . ψ|Jx |ψ = i

(6)

i
Because Jx is an one-body operator, i|Jx |j  (i = j ) does not vanish only when |i and |j  differ by one particle occupation. After a certain permutation of creation operators, |i and |j  are expressed as |i = (−)Miµ |µ · · ·,

|j  = (−)Mj ν |ν · · ·,

(7)

where the ellipsis stands for the same particle occupation and = ±1 according to whether the permutation is even or odd. Then the dynamic moment of inertia is  dψ|Jx |ψ  (2) = j (µ) + j (2) (µν), J (2) = dω µ µ<ν (−)Miµ

dµ|jx |µ nµ , dω dµ|jx |ν  (−)Miµ +Mj ν Ci∗ Cj j (2) (µν) = 2 dω

= ±1, (−)Miν

j (2) (µ) =

(µ = ν).

(8)

i
Calculations show that the interference term, j (2) (µν) which has no counterpart in the mean-field (BCS) treatment, plays an important role in the odd–even difference [11,12]. The expression for the kinematic moment of inertia J (1) = ψ|Jx |ψ/ω is similar.

3. Calculated results and discussions 3.1. Parameters In our calculation, the Nilsson parameters (κ, µ) are taken from the Lund systematics [13] (for neutron N = 6 shell, their values are shifted slightly), and the deformation parameters are taken as ε2 = 0.48 and ε4 = 0.048. For ND bands in the rare-earth nuclei, pairing correlation strengths are determined by the experimental odd–even differences in binding energies and bandhead MoI. For SD bands in the A ∼ 190 region, no experimental binding energies are available. The effective pairing strengths are determined by fitting the values of J (1) and J (2) for 192 Tl(A) over the frequency range h¯ ω ≈ 0.10–0.35 MeV. The effective pairing strengths also depend on the dimension of the truncated CMPC space. In the following calculations, the dimensions of CMPC space are about 700 for proton (the truncated CMPCs energy Ec is about 0.72h¯ ω0 , hω ¯ 0 = 41A−1/3 MeV), and 1000 for neutron (Ec is about 0.48 h¯ ω0 ). In such a CMPC space, the effective pairing correlation

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strengths (G0 for monopole and G2 for quadrupole pairing correlations) in units of MeV are G0p = 0.3,

G0n = 0.2,

G2p = 0.01,

G2n = 0.011.

3.2. Moment of inertia The configurations of the five SD bands in 191 Hg, 192 Tl and 193 Tl are given in Table 1. The calculations confirm the configurations assigned to these bands in experiments [9,14,15]. The comparisons of J (1) and J (2) moments of inertia between the experimental and calculated SD bands of 192 Tl(A) and 192 Tl(B) are presented in Fig. 2. It shows that (i) J (2) presents flat (band B) or U-shaped curve (band A) with increasing frequency hω ¯ while J (2) exhibits a smooth rise with increasing h¯ ω for most SD bands in this region, and (ii) J (2) is smaller than J (1) over part of frequency range and crosses with J (1) at a certain h¯ ω. The calculated results reproduce experimental data very well. The properties of bands A and B are very similar to each other. Thus, we will take band A for example to discuss. The proton and neutron occupation probabilities of 192 Tl(A) are shown in Fig. 3(a) and (b), respectively. The blocking of individual high-j proton orbital [642]5/2(α = −1/2) is obvious in the frequency range 0 < h¯ ω < 0.35 MeV. While at hω ¯ > 0.35 MeV there is an exchange of the occupation probabilities between [642]5/2(α = −1/2) and [651]1/2(α = Table 1 The configurations of the SD bands in 192,193 Tl and 191 Hg SD band

Configuration

192 Tl(A)

(π [642]5/2, α = −1/2) ⊗ (ν[761]3/2, α = −1/2) (π [642]5/2, α = +1/2) ⊗ (ν[761]3/2, α = −1/2) (π [642]5/2, α = −1/2) (π [642]5/2, α = +1/2) (π [761]3/2, α = −1/2)

192 Tl(B) 193 Tl(1) 193 Tl(2) 191 Hg(1)

Fig. 2. Experimental and theoretical J (1) and J (2) of the SD bands A and B in 192 Tl.

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Fig. 3. Occupation probabilities nµ of each proton and neutron cranked orbital µ near the Fermi surface. The blocked orbitals are denoted by dot-dashed line.

−1/2) orbitals, which is due to the intruding of another high-j orbital [651]1/2 at high h¯ ω [16]. The high-j neutron orbital [761]3/2 is almost fully blocked at low h¯ ω, and another high-j intruder orbital [752]5/2 above the Fermi surface is partially occupied at high hω ¯ due to pairing correlations. More detailed information about contribution to J (2) from each high-j intruder cranked orbital µ, j (2) (µ), and the interference term j (2) (µν) between orbitals µ and ν [see Eq. (8)] is presented in Fig. 4. In Fig. 4(a), it is shown that the intruder of orbital [651]1/2 changes the contributions coming from orbital [642]5/2. The contributions from orbital [642]5/2 decrease over the frequency range 0.1 < h¯ ω < 0.25 MeV and increase rapidly when frequency hω ¯ > 0.25 MeV. The increase of J (2) for band A at frequency h¯ ω > 0.3 MeV is mainly from the contributions of [642]5/2. Orbital [642]5/2 also contributes to the decrease of J (2) over the range in frequency 0.1 < hω ¯ < 0.2 MeV. In Fig. 4(b), we find that the decrease of J (2) in the frequency range 0.1 < hω ¯ < 0.3 MeV mainly comes from the contributions of the interference term j (2) ([752]5/2[761]3/2). It is the blocking of [761]3/2 that causes another high-j orbital [752]5/2 partially occupied (see Fig. 3), and causes a decrease of contributions of the interference term [752]5/2[761]3/2 to band A. The total contributions to J (2) of interference terms [J (2) (µν)] and direct terms [J (2) (µ)] from neutron j15/2 and proton i13/2 orbitals at fixed frequencies are displayed in Table 2. We can see that for direct terms, while the contributions decrease with increasing frequency for neutron, it decreases and then increases (U-shaped curve) for proton with the minimum around h¯ ω ≈ 0.25 MeV. For interference terms, the contributions present U-shaped curve for both neutron and proton, except with minimum at hω ¯ ≈ 0.15 MeV and hω ¯ ≈ 0.35 MeV, respectively. Here we can see that not only the contributions of direct term, but also the contributions of interference terms from high-j intruder orbitals are important to the behavior of J (2) of 192 Tl(A).

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Fig. 4. The contributions to J (2) from each high-j intruder orbital near the Fermi surface. j (2) (µ) is the direct contributions from orbital µ and j (2) (µν) the interference contributions between orbitals µ and ν. The blocked orbitals are denoted by dot-dashed line.

Table 2 The total contributions to J (2) of interference terms [J (2) (µν)] and direct terms [J (2) (µ)] (in units of h¯ 2 MeV−1 ) from neutron j15/2 and proton i13/2 orbitals h¯ ω (MeV)

Proton

0.05 0.15 0.25 0.35 0.45

Neutron

J (2) (µν)

J (2) (µ)

J (2) (µν)

J (2) (µ)

−0.8011 −2.1212 −8.9743 −13.7241 5.9781

27.6846 23.8781 19.8923 28.1634 57.9889

−8.6515 −10.2508 −7.24 2.2419 6.8575

35.3734 31.6448 25.9053 21.4047 20.7008

3.3. Spin assignments The bandhead spin assignments are shown in Table 3, where I01 denotes the values taken exp from Ref. [17], I02 from Ref. [18], I0 from our calculations and I0 from experimental data [9,14,15]. All the bandhead spins of the five SD bands in our calculations agree with the experimental proposition. We extract the kinematic and dynamic moments of inertia by using the experimental intraband E2 transition energies as follows: J (1) (I − 1) h2 ¯

J (2) (I ) h2 ¯

=

=

2I − 1 , Eγ (I → I − 2)

4 . Eγ (I + 2 → I ) − Eγ (I → I − 2)

(9) (10)

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Table 3 The bandhead spins of the SD bands in 192,193 Tl and 191 Hg exp

SD band

I01 (h¯ )

I02 (h¯ )

I0 (h¯ )

I0 (h¯ )

192 Tl(A)

17 20 9.5 8.5 17.5

13 16 9.5 8.5 13.5

15 18 9.5 8.5 15.5

15 18 9.5 8.5 15.5

192 Tl(B) 193 Tl(1) 193 Tl(2) 191 Hg(1)

Fig. 5. Comparison between J (1) ’s determined by different spin assignments. The symbols denote the experimental data and the lines denote the theoretical results.

J (1) depends on the spin values while J (2) does not. For 191 Hg(1), the spins are taken from the determination of the most recent experiment (Ref. [14]). Fig. 5(a) shows the behavior of J (1) in 191 Hg(1). It can be seen that (i) J (1) presents a U-shaped curve with increasing frequency hω, ¯ and (ii) J (1) is larger than J (2) during part of the frequency range, (2) then crosses with J at a certain h¯ ω. Our calculations reproduce all these properties very well [see Fig. 5(a)]. The J (1) ’s of 192 Tl(A) and 192 Tl(B) determined by different spin assignments are displayed in Fig. 5(b) and (c), respectively. It is presented that our spin assignments make the h¯ ω evolution of the J (1) shows the similar properties as that of 191 Hg(1). Considering the same neutron configurations which response to the behavior of J (1) mentioned above, we adopt the bandhead spin values as 15h¯ and 18h¯ for 192 Tl(A) and 192 Tl(B) respectively, which confirm the spins assigned to the bands in Ref. [9]. 3.4. Alignments and additivity In Ref. [9], the experimental data showed that with respect to yrast SD band 192 Hg(1) reference, the alignments for band A and B in 192 Tl agree well with the sums of alignments for νj15/2 (α = −1/2) band in 191 Hg and two signature πi13/2 (α = ∓1/2) bands in 193 Tl,

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Fig. 6. (a) The experimental alignments for νj15/2 (α = −1/2) band 191 Hg(1) (solid line), π i13/2 (α = ±1/2) bands 193 Tl(1, 2) (solid line) and the sums of these alignments (dotted and dot-dashed lines), and the corresponding alignments for 192 Tl(A, B) (solid and open circles). (b) As for (a), except for theoretical results. (c) The experimental alignments for band 193 Hg ([512]5/2) (solid line), 193 Hg ([642]9/2) (solid line), and the sums of these alignments (dotted line), and the alignment for band 194 Hg ([512]5/2[642]9/2) (solid circle).

respectively. This is shown in Fig. 6(a). The theoretical alignment i is given by i(ω) = Jx (SD band) − Jx (reference band).

(11) 192 Tl

Additivity of alignments concerning SD bands in odd–odd nucleus and their neighboring odd-neutron nucleus 191 Hg and odd-proton nucleus 193 Tl (or 191 Tl) has been studied in Refs. [17,19]. Some different conclusions were drawn. While the cranked Hartree– Fock–Bogoliubov (HFB) with the Lipkin–Nogami method leads to additivity in Ref. [17], additivity is not hold when studied by the projected shell model [19]. In present work, alignments for the same bands as that shown in Fig. 6(a) are studied. In Fig. 6(b), it can be see that although the additivity rule in our results is not applied so well as that in experimental data, it is still obeyed to a great extent. In the A ∼ 190 region, the observed frequency range of the SD bands is lower than that in the A ∼ 150 region, and the single particle level density is higher. Both these differences make pairing more important in the A ∼ 190 region than that in the A ∼ 150 region. In the A ∼ 150 region, pairing is very weak, and additivity holds well. By contrast, pairing in the A ∼ 190 region is in the transitional region between strong and very weak. Thus, additivity was not expected to hold. In Ref. [8], the nonadditivity of alignments in the A ∼ 190 region was investigated. It is claimed that the additivity of MoI and alignment in SD nuclei, 194 Hg, is lost mainly because of the destructive interference between blocking effects. However, not only the experimental data but also our calculation results show the additivity of alignments in 192 Tl(A, B). In the PNC calculations, we only considered the neutron–neutron or proton–proton interaction, but did not consider the neutron–proton interaction. The fact that the experimental evidence for additivity of alignments in 192 Tl is approximately reproduced by our calculation may be understood as following: because the

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neutron and proton Fermi surfaces lie in different major shells, the addition of a valence neutron does not change the proton pair field significantly, and vice versa. In case of a even– even nucleus, addition of the second neutron changes the neutron pair field dramatically, and additivity lost. For example, as show in Fig. 6(c), the sum of the experimental alignments for 1-quasineutron band 193 Hg ([512]5/2) and 193 Hg ([624]9/2) is much larger than the alignment of 2-quasineutron band 194 Hg ([512]5/2[624]9/2). Additivity is lost here due to the important contributions from the interference between the last two neutrons [8].

4. Summary In summary, the PNC method for treating the cranked shell model with monopole and quadrupole pairing correlations has been used to investigate the microscopic mechanism of the hω ¯ variation of the SD bands in odd–odd nucleus 192 Tl. Both the blocked proton orbital [642]5/2 and neutron orbital [761]3/2 play a very important role in the evolution of the dynamic moment of inertia J (2) as a function of the rotational frequency h¯ ω. The contributions from the interference terms [j (2) (µν)], which have no counterpart in the mean-field (BCS) treatment, are important and cannot be neglected. The additivity of alignments is normally thought not applicable in the A ∼ 190 mass region due to the destructive interference between blocking effects. However, it holds in the case of the SD bands in 192 Tl(A, B). This is because the interaction between the last neutron and proton is not so important as that of proton–proton and neutron–neutron, and is negligible.

Acknowledgements The authors would like to express their gratitude to Dr. S.X. Liu in Peking university for valuable discussions. This work was supported by National Natural Science Foundation of China under Grant No. 10375001, the Major State Basic Research Development Program under Grant No. G2000-0774-07 and the Knowledge Innovation Project of the Chinese Academy of Sciences under Grant No. KJCX2-SW-N02.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

B. Singh, R. Zywina, R.B. Firestone, Nucl. Data Sheets 97 (2002) 241. R.V.F. Janssens, T.L. Khoo, Annu. Rev. Nucl. Part. Sci. 41 (1991) 321, and references therein. D. Ye, et al., Phys. Rev. C 41 (1990) R13. M. Rilry, et al., Nucl. Phys. A 512 (1990) 178. R. Bengtsson, S. Frauendorf, Nucl. Phys. A 314 (1979) 27. S. Frauendorf, et al., Nucl. Phys. A 431 (1984) 511, and references therein. S.M. Mullins, et al., Phys. Rev. C 61 (2000) 044315. S.X. Liu, J.Y. Zeng, Phys. Rev. C 66 (2002) 067301. S.M. Fischer, et al., Phys. Rev. C 53 (1996) 2126. J.Y. Zeng, et al., Phys. Rev. C 63 (2001) 024305. S.X. Liu, J.Y. Zeng, L. Yu, Nucl. Phys. A 735 (2004) 77. S.X. Liu, J.Y. Zeng, Nucl. Phys. A 736 (2004) 269.

X.T. He et al. / Nuclear Physics A 760 (2005) 263–273

[13] [14] [15] [16] [17] [18] [19]

T. Bengtsson, I. Ragnarsson, Nucl. Phys. A 436 (1985) 14. S. Siem, et al., Phys. Rev. C 70 (2004) 014303. S. Bouneau, et al., Phys. Rev. C 53 (1996) R9. X.T. He, et al., Eur. Phys. J. A 23 (2005) 217. P.-H. Heenen, R.V.F. Janssens, Phys. Rev. C 57 (1998) 159. S.X. Liu, J.Y. Zeng, Phys. Rev. C 58 (1998) 3266. J. Zhang, et al., Phys. Rev. C 58 (1998) 868.

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