Microscopic description of the mirror nuclei 70Se and 70Kr

Nuclear Physics A 728 (2003) 396–414 www.elsevier.com/locate/npe

Microscopic description of the mirror nuclei 70Se and 70Kr A. Petrovici a,b , K.W. Schmid b,∗ , Amand Faessler b a National Institute for Physics and Nuclear Engineering, R-76900 Bucharest, Romania b Institut für Theoretische Physik, Universität Tübingen, D-72076 Tübingen, Germany

Received 24 June 2003; received in revised form 15 August 2003; accepted 9 September 2003

Abstract Results are presented on shape coexistence at low and high spins in the mirror nuclei 70 Se and 70 Kr obtained within the complex version of the Excited Vampir variational approach. Effects of oblate–prolate coexistence and mixing at low, intermediate and high spins are discussed. Strong M1, I = 0 transitions are predicted. A detailed comparison with the available experimental information is performed.  2003 Elsevier B.V. All rights reserved. PACS: 21.10.k; 21.60.n Keywords: Nuclear structure; Shape coexistence; Proton-rich mirror nuclei

1. Introduction The investigation of the structure of exotic nuclei around the N = Z line in the A  70 mass region is one of the most exciting challenges in low energy physics today. Apart from displaying some rather interesting nuclear structure effects these nuclei play an important role in nuclear astrophysics since the abundance flow of the rapid proton capture process (rp process) is determined mainly by the competition of proton capture reactions and β decays along the N = Z line. 70 The mirror nuclei 70 34 Se and 36 Kr play a particular role in this mass region. Recently 70 [1] the β-decay half-life of Kr, a bridge nuclide for the rp process beyond A = 70, was measured and turned out to be consistent with the assumption of a pure Fermi decay. The * Corresponding author.

E-mail address: [email protected] (K.W. Schmid). 0375-9474/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2003.09.004

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production of nuclei heavier than A = 68 in X-ray bursts can be strongly enhanced by the two-proton-capture reaction on 68 Se which represents another destruction channel in addition to β decay. In this reaction sequence proton scattering on 68 Se produces a small equilibrium abundance of proton-unbound 69 Br nuclei, which then capture another proton producing 70 Kr. Then the β decay out of the N = 34 isotone chain occurs at 70 Kr. The half-life and the structure of the lowest excitations in 70 Kr thus are very important for determining the role of the two-proton-capture on 68 Se. New experimental results on the high spin structure of the nucleus 70 Se have been recently reported [2]. By these experiments the known bands [3–5] could be extended considerably and some new bands could be identified. The charge independence of nuclear forces requires that the energy spectra in pairs of mirror nuclei are identical. Small differences between energy levels could then be interpreted as pure Coulomb effects. Introducing the Coulomb interaction for the valence protons we intend to make a comparison of the microscopic structure of the mirror nuclei 70 Se and 70 Kr. Models based on the variational approaches of the Vampir family have been successfully applied for the description of a variety of nuclear structure phenomena in the A  70 mass region, not only in nuclei along the valley of β-stability, but also in some exotic nuclei close to the proton drip line [6–13]. The complex Excited Vampir approach allows for a unified description of low and high spin states including in the projected mean fields neutron–proton correlations in both the T = 1 and T = 0 channels and general two-nucleon unnatural-parity correlations. The oblate–prolate coexistence and mixing, the variation of the deformation with mass number, increasing spin, as well as excitation energy have been compared with available experimental information. Since the Vampir approaches enable the use of rather large model spaces and of general two-body interactions, large-scale nuclear structure studies going far beyond the abilities of the conventional shell-model configuration-mixing approach are possible. Our previous investigations on microscopic aspects of shape coexistence in N  Z nuclei in the A  70 mass region indicated the presence of a strong competition between particular configurations based on large and small oblate and prolate quadrupole deformations. Furthermore, as expected, since in N  Z nuclei neutrons and protons fill the same single particle orbits, the neutron– proton pairing correlations were found to play an essential role [7]. In addition it was demonstrated, that the neutron and proton alignments with increasing angular momentum occur simultaneously in these nuclei [8–13]. On the other hand the theoretical results suggest that certain properties of these nuclei are extremely sensitive to small variations of particular parts of the effective Hamiltonian [8,9]. Thus, our results indicate that the oblate–prolate coexistence and mixing at low spins depend on the strengths of the neutron–proton T = 0 matrix elements involving nucleons occupying the 0f5/2 (0f7/2 ) and 0g9/2 single particle orbits. For example, the irregularities identified in the level scheme of the 72 Kr nucleus at low spins can be explained by a variable, in some cases very strong, oblate–prolate mixing. However, changes of only 15 keV in the strength of the mentioned matrix elements could modify the quantitative picture significantly [9]. Since our microscopic investigations on the coexistence phenomena in this mass region have already passed many experimental tests, we thought it worthwhile to apply the same

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methods to study the structure of the lowest few even-spin positive-parity states in the mirror nuclei 70 Se and 70 Kr. We shall briefly describe the complex Excited Vampir variational procedure and define the effective Hamiltonian in the next section. In Section 3 we shall discuss the results on shape coexistence in 70 Se and 70 Kr at low and high spins. Finally we shall present some conclusions in Section 4. 2. Theoretical framework We calculated the lowest few positive-parity states up to spin 20+ in 70 Se and 18+ in First the Vampir solutions, representing the optimal mean-field description of the yrast states by single symmetry-projected HFB determinants, were obtained. Then the Excited Vampir approach was used to construct additional excited states by independent variational calculations. Finally, for each considered spin the residual interaction between the various orthogonal configurations was diagonalized. We define the model space and the effective Hamiltonian as in our earlier calculations for nuclei in the A  70 mass region [8]: a 40 Ca core is used and the valence space consists out of the 1p1/2 , 1p3/2 , 0f5/2 , 0f7/2 , 1d5/2 and 0g9/2 oscillator orbits for both protons and neutrons. For the corresponding single-particle energies we take (in units of the oscillator energy h¯ ω = 41.2 A−1/3) 0.040, −0.270, 0.300, −0.560, 0.157 and 0.029 for the proton, and −0.070, −0.332, 0.130, −0.690, 0.079 and −0.043 for the neutron levels, respectively. The effective two-body interaction is a renormalized nuclear matter G-matrix based on the Bonn one-boson-exchange potential (Bonn A). The Coulomb matrix elements are added for the valence protons. It is worthwhile to mention a few particular aspects of the renormalization procedure which are relevant to the present investigations. The G-matrix is modified by three short range (0.707 fm) Gaussians for the isospin T = 1 proton–proton, neutron–neutron and neutron–proton matrix elements with strengths of −35 MeV. The isoscalar spin 0 and 1 particle–particle matrix elements are enhanced by an additional Gaussian with the same range and the strength of −180 MeV. In addition, the interaction contains monopole shifts of −315 keV for all the diagonal isospin T = 0 matrix elements
9/20f ; I T = 0 with 0f denoting either the 0f5/2 or of the form 0g9/2 0f ; I T = 0|G|0g the 0f7/2 orbit. These shifts have been introduced in our earlier calculations in order to influence the onset of deformation. For the present report we used the same G-matrix (A) as in all our previous investigations in the A = 70 mass region starting from the renormalization that enabled us to get a rather good description of the low and high spin states in the even mass nuclei. More data on the electromagnetic properties of the nuclei in this mass region could justify a new endeavor on the renormalization procedure starting from the CD Bonn which includes charge symmetry breaking and charge independence breaking. 70 Kr.

3. Results and discussion In a first step we calculated the lowest even-spin positive-parity states up to spin 8+ varying the strength of the above mentioned monopole shifts for the neutron–proton

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interaction in between −275 and −350 keV in order to understand the evolution of the oblate–prolate mixing as a function of these shifts. For this purpose we used the (onedeterminant) Vampir approximation to construct the yrast states for each spin starting from intrinsically oblate and prolate deformed trial configurations. The final choice for the monopole shifts inside the range specific for the A  70 mass region was then determined by adjusting it to the relative position of the lowest two bands experimentally observed in the nucleus 70 Se. For 70 Kr unfortunately no experimental data on excited states do exist. Therefore the same monopole shifts of −315 keV (as adjusted to the Se nucleus) have been used in the Kr-nucleus, too. In the following we shall report our results for the structure of always the lowest few states up to spin 20+ in 70 Se and 18+ in 70 Kr. 3.1. The nucleus 70 Se Since the density of the orthogonal projected configurations obtained using the Excited Vampir procedure is rather different for low and high spins, different total numbers of configurations were taken into account for the various spin-values: in the nucleus 70 Se for the spins 2+ , 4+ and 6+ a 10-dimensional basis was used. For the other spins in the final diagonalization 15 up to 19 projected configurations were taken into account. In 70 Se the first minimum is oblate deformed (β2 = −0.33) in the intrinsic system for the spins 0+ and 2+ . Starting from spin 4+ the first minimum prefers prolate deformation (β2 = 0.36) and the oblate deformed configuration becomes the second minimum up to spin 8+ . For higher spins then more and more prolate deformed configurations become energetically preferred leaving for the oblate minimum only the third place for 10+ , the fifth one for 12+ , the eleventh one for 14+ and the highest one in the 17-dimensional many-nucleon basis used in the present calculations for 16+ . After the diagonalization of the residual interaction the oblate and prolate configurations become strongly mixed for the lowest three states up to spin 4+ as can be seen from Table 1. The o–p mixing then is decreasing already for spin 6+ and becomes negligible for the higher spin-values except for angular momentum 12+ . For the higher angular momenta the lowest states are entirely dominated by the mixing of various prolate deformed configurations. For medium and high angular momenta a very high density of configurations was obtained. The lowest 16 orthogonal minima for each spin from 12+ up to 20+ are bunched in 3.5 MeV up to 3.8 MeV. This high density of projected configurations explains the strong mixing of the states and also their decay by many significant E2 branches (see also Fig. 1). In Table 1 only the amplitudes of configurations contributing more than 3% to the wave function of a given state are presented. As can be seen the main projected determinant contributes less than 50% for many of the investigated states and for the states with I π > 10+ up to 9 configurations become important. We organized the states in bands according to the B(E2) values connecting them. In Fig. 1 we present the lowest bands calculated in 70 Se and compare them with the available experimental data for the positive-parity states. Due to the mixed character of the states the B(E2) strengths for some medium and high spin states are strongly fragmented as it is indicated in Table 2 and Fig. 1. The lowest 5 bands are labeled according to the nature of the underlying configurations for the states building the corresponding band. For example, the states of the yrast band (o–p(p)1) are characterized by the oblate–

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Fig. 1. The theoretical spectrum of 70 Se for even-spin positive-parity states calculated within the complex Excited Vampir approximation is compared to the experimental results [2]. The labels o and p are for states based on intrinsically oblate and prolate deformed configurations, respectively. The M1, I = 0 transitions are indicated by dashed lines.

prolate mixing, dominated by the oblate components at low spin, and built by prolate deformed configurations only at the highest spins. The lowest two states of spin 2+ and 4+ are connected by strong E2 transitions: 1469 e2 fm4 and 1482 e2 fm4 , respectively. The corresponding B(E2) strength is much smaller for the 6+ states: 290 e2 fm4 . These strong B(E2) values support the similar collective structure of the corresponding states. For the 2+ and 6+ states these transitions are also found experimentally. The oblate–prolate mixing, the nature and the deformation of the states are also reflected by the spectroscopic quadrupole moments presented in Table 3. The complex feeding pattern of the lowest few states of a given spin includes strong M1 transitions connecting states of the same spin. In Table 4 we present the strongest M1 transitions and the decomposition of the total strengths in proton-orbital (π -o), protonspin (π -s) and neutron-spin (ν-s) components. The major contribution to the strongest transitions is brought by particles occupying the spherical 0g9/2 orbital for both protons and neutrons. Other important contributions come from rearrangements of particles occupying the 0f5/2 and 1p3/2 orbitals that increase (for the large B(M1)) or cancel (for the small B(M1)) the 0g9/2 contribution. Since the strong M1 transitions represent in some cases the fastest decay path, it would be useful to identify them experimentally in order to construct the correct E2 links of states in the collective bands.

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Table 1 The amount of mixing for the states in 70 Se presented in Fig. 1 I [h] ¯ 0+ 1 0+ 2 0+ 3 2+ 1 2+ 2 2+ 3

4+ 1 4+ 2 4+ 3 6+ 1 6+ 2 6+ 3

o-mixing

35

64

39

40(16)

43

54

16+ 1

56

44

26

47(19)(3)

57

36(3)

40

58

46(3)

21(21)(3)

63

33

(87)6

18+ 1

93

5

13

48(32)(3)

93(3) 59(22)(7)(7) 50(20)(18)(5)(5) 55(12)(9)(8)(4)(4)(3)(3) 32(24)(7)(7)(7)(6)(5)(4)(3) 98 85(4)(3) 81(5)(4)(3)(3) 71(12)(4)(3) 33(26)(15)(11)(9) 51(16)(9)(4)(4)(3)(3)(3) 38(17)(9)(8)(7)(6)(4)(4)(3) 40(19)(10)(8)(8)(8)(4) 33(21)(10)(9)(4)(4)(3)(3)(3)(3) 99

96

8+ 4

85(8)

10+ 1

96

12+ 1 12+ 2 12+ 3 12+ 4 12+ 5 12+ 6

p-mixing (%)

87(4)(3)

16+ 3

96

92

o-mixing (%)

16+ 2 16+ 4 16+ 5 16+ 6 16+ 7 16+ 8 16+ 17

5

8+ 3

10+ 2 10+ 3 10+ 4 10+ 5 10+ 6

I [h] ¯ 14+ 1 14+ 2 14+ 3 14+ 4 14+ 5 14+ 6 14+ 11

8+ 1

8+ 2

p-mixing (%)

89(3) 94 83(8) 3

49(31)(6)(5)

18+ 2 18+ 3 18+ 4 18+ 5 18+ 6 18+ 7 18+ 8

66(15)(12)(3) 49(21)(13)(10)(3) 64(16)(9)(4) 74(17)(3) 44(21)(15)(6)(5)(3) 44(15)(13)(13)(6)(3) 22(15)(15)(13)(9)(8)(7)(3)(3) 41(29)(6)(6)(4)(4)(3)

63(8)(7)(7)(6)(3) 20+ 1

95 87(3)(3) 73(9)(8) 46(16)(16)(11)(5) 66

20(6)

27(6)

43(9)(5)(3)(3)

20+ 2 20+ 3 20+ 4 20+ 5 20+ 6 20+ 7 20+ 8 20+ 9

68(19)(8) 63(24)(4)(4) 33(21)(16)(13)(10) 68(9)(7)(4)(3)(3) 69(7)(5)(4)(4)(3) 25(24)(23)(12)(8) 51(14)(13)(5)(4)(3) 27(27)(26)(7)(5) 39(21)(15)(11)(5)

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Table 2 B(E2; I → I − 2) values (in e2 fm4 ) for some states of the nucleus 70 Se presented in Fig. 1. The strengths for the secondary branches are given in parentheses and the labels indicate the end point of the transitions. In Fig. 1 the transitions corresponding to the strengths given in brackets are not shown. As effective charges ep = 1.5 and en = 0.5 have been used I [h] ¯

o–p(p)1

p–o(o)2

p–o(p)3

2+

1124 1600

1186 1640

1037 1508

(90)p–o(o)2

(47)o–p(p)1

1497 (480)p–o(o)2

1371 (388)o–p(p)1

1572

1992 (60)p–o(o)2

1771 (48)o–p(p)1

941 [264]

4+ 6+ 8+ 10+ 12+

p4

p5

2020

1706

1425

1543

2007

(80)p–o(p)3 1136

(94)p–o(o)2 1498

[86] 1332

1246

[149]p5

(132)p4

(380) 14+

1880

[75]p5 1068

1386

1304

[125]p–o(o)2 1331

16+

1343

(464) 1311

855

602

[86]p–o(o)2 655

(246)o–p(p)1

(214)p–o(p)3

(433)p4

1092

[276][153] 1114

[297][84] 905

(173)o–p(p)1

[65]p5

[370]

954 (43)o–p(p)1

1034 (56)p–o(p)3

1040 (146)p–o(p)3

(102)p–o(p)3 18+ 20+

1493 1370

(77)p4 [246]p5

(167)

Table 3 sp Spectroscopic quadrupole moments Q2 (in e fm2 ) of selected states in 70 Se. As effective charges ep = 1.5 and en = 0.5 have been used I [h] ¯ 2+ 4+ 6+ 8+ 10+ 12+ 14+ 16+ 18+ 20+

o–p(p)1 17.31 −12.76 −90.87 −104.44 −107.25 −107.01 −103.95 −97.90 −93.74 −90.96

p–o(o)2 −21.76 5.20 80.57 90.19 87.00 29.77 85.66 80.95

p–o(p)3 −14.40 −40.70 −65.02 −85.55 −89.70 −93.57 −95.04 −95.89 −96.88 −91.28

p4

−101.02 −100.67 −99.03 −97.11 −97.56 −97.35 −97.68

p5

−90.41 −93.38 −98.31 −93.53 −89.03 −89.11

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Table 4 B(M1; I = 0) values (in µ2N ) in 70 Se for the transitions indicated by dashed lines in Fig. 1. The proton-orbital (π -o), proton-spin (π -s) and neutron-spin (ν-s) contributions are obtained using as efective g-factors (in µN ) gπl = 1.0, gπs = 5.5857 and gνs = −3.8263 Iinit → Ifin [h] ¯ 8+ 3 8+ 4 10+ 2 10+ 4 10+ 6 10+ 4 10+ 5 10+ 6 12+ 2 12+ 5 12+ 6 12+ 4 12+ 6 14+ 2 14+ 4 14+ 5 14+ 6 16+ 2 16+ 3 16+ 8 16+ 7

→ 8+ 1 → 8+ 1 → 10+ 1 → 10+ 1 → 10+ 1 → 10+ 2 → 10+ 2 → 10+ 2 → 12+ 1 → 12+ 1 → 12+ 1 → 12+ 2 → 12+ 2 → 14+ 1 → 14+ 2 → 14+ 2 → 14+ 2 → 16+ 1 → 16+ 1 → 16+ 1 → 16+ 2

π -o (%)

π -s (%)

ν-s (%)

B(M1)

42

36

22

0.10

55

6

39

0.17

40

40

20

0.31

60

–4

44

0.18

48

20

32

0.61

39

38

23

0.16

31

50

19

0.17

46

30

24

0.92

39

42

19

0.52

49

15

36

0.19

48

17

35

0.41

23

58

19

0.18

44

32

24

0.40

32

54

14

0.53

3

88

9

0.08

49

12

39

0.08

41

52

7

0.18

8

98

–6

0.11

35

40

25

0.12

42

28

30

0.32

46

22

32

0.32

The alignment plot representing the angular momentum contribution of the nucleons occupying the 0g9/2 spherical orbital in the direction of the total angular momentum presented in Fig. 2 shows similar contributions from protons and neutrons for the yrast line, while for the third band which is built on the second minimum starting from spin 10+ up to 18+ a faster proton alignment is observed. Faster proton alignment above spin 10+ up to 18+ is obtained for the fourth band, too. The smallest alignment is found in the second band dominated by the oblate configurations. The particle occupation of the 0g9/2 orbital presented in Fig. 3 corroborates the trend manifested in the alignment plot. Comparing the theoretical results with the available experimental information one can present a microscopic interpretation of the particular behavior identified in 70 Se. Based on the irregularities observed in the yrast band at low spin it is considered that this band undergoes a band crossing in the spin region I = 4 → 8 [2]. The Excited Vampir calculations indicate that for the spins 0+ and 2+ the oblate minimum is more bound than

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Fig. 2. The alignment plot for some bands calculated in 70 Se.

the prolate one by 258 and 170 keV, respectively. At spin 4+ the prolate minimum is 17 keV below the oblate one and already at spin 6+ this distance increases to 350 keV. The relative position of these two minima is responsible for the o–p mixing illustrated in Table 1, the sign and the absolute values of the spectroscopic quadrupole moments of the lowest two 2+ , 4+ and 6+ states presented in Table 3 and the E2 transitions connecting them. The theoretical picture for the decay of the lowest 2+ , 4+ and 6+ states is in agreement with the experimental E2 decay branches connecting the states of the lowest two positive-parity bands. At higher spins the structure of the states building the p–o(o)2 band is dominated by the oblate components going higher in energy with respect to the prolate minima. This behavior is not found in the experimental spectrum for the band built on the second 2+ in Ref. [2]. The third theoretical band p–o(p)3 is a candidate for the experimental band built on the highest identified 8+ state. The mixing of the high spin states belonging to this band illustrated in Table 1 supports the strong fragmentation of the total B(E2) strength for the decay of a given state presented in Table 2. This theoretical result is in agreement

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Fig. 3. The occupation of the 0g9/2 spherical orbital for some bands in 70 Se.

with the experimental observation that at medium spins 40% of the intensity of this band populates states of the ground-state band. However, one should keep in mind the strong M1 transitions connecting the medium spin states which offer an alternative way to feed the yrast states from higher bands like p4 or p5 or even from higher but very close lying states as the 16+ states indicated in Fig. 1. Furthermore, one should also mention the high density of 20+ states which are decaying by many E2 branches feeding the lowest few 18+ states. We presented many more side bands than the presently identified ones in order to reveal different possible ways to feed the yrast line (including M1, I = 0 transitions) and to guide the next experimental investigations. At high spins the theoretical spectrum is compressed as compared with the experiment. Of course a larger model basis would be more adequate for the description of the nuclei in the A = 70 mass region, but this would increase too much the computing time for the moment. Furthermore, the corresponding effort is not justified by the available data on electromagnetic properties that could guide the renormalization of the two-body interaction. Definitely, the variable mixing of the different projected configurations creates irregularities in the theoretical bands. However, a direct connection with the experimental bands

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is not possible for the moment. Data on electromagnetic properties would here simplify the comparison between experiment and theoretical predictions considerably. 3.2. The nucleus 70 Kr The results obtained for the 70 Kr nucleus display many similarities with those obtained for the 70 Se nucleus, but there are also significant differences. The lowest minimum for the spin 0+ was found to be prolate deformed in the intrinsic system and remains the dominant configuration of the ground state even after the diagonalization of the residual interaction as can be seen from the oblate–prolate mixing illustrated in Table 5. The energy separation between the lowest prolate and the first excited oblate configuration at spin 0+ , 2+ , 4+ and 6+ amounts to 335, 116, 121 and 343 keV, respectively. Strong mixing of the oblate and prolate deformed configurations to the final wave functions was found for the lowest two 2+ and 4+ states. At spin 6+ , however, the oblate mixing into the lowest state decreases and becomes negligible already at spin 8+ . The second state from the spin 0+ up to 8+ is dominated by the oblate deformed configurations. As for 70 Se starting with spin 10+ the oblate minimum is moving up in energy and due to the high density of orthogonal configurations the total oblate amplitude is distributed over several close lying states. Accounting for the density of the orthogonal projected configurations the dimension of the many-nucleon basis was increased from 10 (for the spins 2+ , 4+ , 6+ ) up to 16 for the 0+ and the I π > 12+ states. In Table 5 we illustrate the structure of the wave functions in terms of amplitudes for the projected configurations representing more than 3% in the final wave functions. The states have been organized in bands according to the B(E2) values connecting them. The lowest few positive-parity bands, labeled according to the structure of the main underlying configurations are presented in Fig. 4. The yrast band (p–o(p)1) is dominated by oblate–prolate mixing at low spins, but different than the situation found in 70 Se the maximum contribution is brought by the prolate deformed configurations. The band built on the second 0+ state (o–p(p)4) is dominated by the oblate deformed configurations up to spin 10+ , but the structure is changed to a mixing of prolate deformed configurations at high spins, in contrast to the behavior of the band built on the second 0+ in 70 Se. The band dominated by the oblate components at high spins (p–o(o)6) is the highest one presented in Fig. 4. The oblate–prolate mixing is clearly illustrated by the spectroscopic quadrupole moments given in Table 7 for the states building the bands presented in Fig. 4. The effect of the mixing of different configurations in the structure of the discussed states is reflected by the fragmentation of the B(E2) strengths as can be seen from Table 6. Significant B(E2) strengths characterize the transitions connecting the lowest two states up to spin 8+ . They amount to 1602, 1464, 694 and 205 e2 fm4 for the 2+ , 4+ , 6+ and 8+ states, respectively. As it was found in 70 Se at intermediate spins the decay pattern includes M1, I = 0 transitions, even stronger in some cases in 70 Kr. The B(M1) strengths and their decomposition in proton-orbital, proton-spin and neutron-spin contributions are presented in Table 8 and indicated by dashed lines in Fig. 4. Much faster than the E2 decay path for some particular states these M1 transitions should be considered in the experimental investigations.

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Table 5 The amount of mixing for the states in 70 Kr presented in Fig. 4 I [h] ¯ o-mixing (%) 0+ 1 0+ 2 0+ 3 2+ 1 2+ 2

37

59

56

36(3)(3) 91

40

58

58

39

2+ 4 4+ 1 4+ 2 4+ 3 6+ 1 6+ 2 6+ 3 6+ 4 6+ 5 6+ 6 8+ 1 8+ 2

8+ 3 8+ 4 8+ 5 8+ 6 8+ 7 10+ 1 10+ 2 10+ 3 10+ 4 10+ 5 10+ 6 12+ 1 12+ 2 12+ 3 12+ 4 12+ 5 12+ 6

p-mixing (%)

I [h] ¯ 14+ 1 14+ 2 14+ 3 14+ 4 14+ 5 14+ 7

o-mixing (%)

p-mixing (%) 90 47(27)(13)(6)(4)

7

26(25)(18)(8)(6)(3)

5

29(18)(17)(13)(9)(3)(3)

28

28(11)(8)(6)(6)(4)(4)(3)

43

21(10)(7)(4)(4)(3)(3)

94(3) 16+ 1 37

60

61

37 82(10)(5)

13

82

85

12

64

24(7)

9

76(8)(4)

20

66(4)(4)(3)

6

84(5)(4)

16+ 2 16+ 3 16+ 4 16+ 5 16+ 6 16+ 8

82(7)(3) 48(24)(13)(6) 33(24)(16)(9)(5)(3) 3

26(24)(10)(10)(9)(6)(4)(4)(3) 8

47(15)(13)(7)(4)(4)

62

10(6)(3)(3)(3)(3)

18+ 1

89(3)(3)

18+ 2

18+ 3

40(13)(13)(9)(9)(8) 3

18+ 4

4

91

18+ 5

80(9)(5) 5

53(33)(3)(3)

58

22(10)(3)

33(5)

23(18)(15) 61(12)(10)(7)(4)(4) 95 78(16)

8

63(15)(4)(4)(3)

74

8(6)(4)(3)

6

52(14)(9)(7)(5)

7

62(9)(8)(5)(4) 93 60(29)(4) 46(24)(14)(6)(3)

39

28(13)(8)(6)(3)

27

24(22)(10)(10)(3)

24

31(13)(9)(5)(5)(5)(4)

18+ 7 18+ 11 18+ 12

36(35)(16)(5)(3) 55(19)(6)(5)(4)(3)(3)(3) 52(14)(11)(5)(4)(3)(3)(3)

18+ 6

84(4)

18(17)(13)(8)(7)(6)(6)(4)(4)(3)

45(19)(12)(9)(5) 44(23)(12)(9)(6)(3) 34

36(11)(10)(5)

44

15(12)(11)(4)(3)(3)(3)

408

A. Petrovici et al. / Nuclear Physics A 728 (2003) 396–414

Fig. 4. The theoretical spectrum of 70 Kr for even spin positive parity states calculated within the complex Excited Vampir approximation. The labels o and p are for states based on intrinsically oblate and prolate deformed configurations, respectively. The M1, I = 0 transitions are indicated by dashed lines.

The alignment plot is presented in Fig. 5 and the occupations for the 0g9/2 spherical orbital are shown in Fig. 6. In order to illustrate the effect of the Coulomb interaction we present some results concerning the oblate–prolate mixing and the corresponding influence on the electromagnetic properties of the 0+ , 2+ , 4+ , 6+ and 8+ states in the two mirror nuclei 70 Se and 70 Kr. These calculations have been performed using for the above discussed monopole shifts the value −345 keV for both cases: with and without Coulomb interaction. The Excited Vampir basis contains 8 orthogonal projected configurations for each spin. In Fig. 7 are compared the corresponding ground-state bands up to spin 8+ in both nuclei 70 Se and 70 Kr. The two spectra in each nucleus are very much similar, but the structure of the wave functions in terms of the oblate–prolate (o–p) mixing is significantly different for the spins manifesting strong o–p mixing. For the states 0+ , 2+ , 4+ , 6+ , 8+ in 70 Se nucleus presented in Fig. 7 the oblate mixing amounts to 60%, 64%, 45%, 6% and < 1%, respectively in the case of no Coulomb and

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409

Table 6 B(E2; I → I − 2) values (in e2 fm4 ) for some states of the nucleus 70 Kr presented in Fig. 4. The strengths for the secondary branches are given in parentheses and the labels indicate the end point of the transitions. In Fig. 4 the transitions corresponding to the strengths given in brackets are not shown. As effective charges ep = 1.5 and en = 0.5 have been used I [h] ¯

p–o(p)1

2+

1264 1764 1766 (174)o–p(p)4 1720 (58)o–p(p)4

4+ 6+ 8+

10+

1848 [107]p2

p2

[1025] (184)p–o(p)1 [498][260] 1857 (131)p–o(p)p1 [90]o-p(p)4 1369 (630)p3 1492 (249)p3

p3

o–p(p)4 1170 1681 1674 (159)p–o(p)1 1710 [147]

1328 (211)o(p)1 (510)p(o)2 1936 (88)o(p)1 1848

12+

1840

14+

1746

16+

1473

1673

1848

18+

1448

1338 [175]

1848

1107 (352)p3 (286) 1267 (225)p–o(p)5 1174 (292)p3 (108)p–o(p)5 (95)p–o(o)6 573 (177)p–o(p)1 (256)p3 (122)p–o(p)5 1244 (318)p3

p–o(p)5

p-o(o)6

[1970]

792 (648) [249][222] 1219 (619)

1848

1459 (202)o–p(p)4 1117 (204)o–p(p)4 [106]

1459

737 (245)o–p(p)4 [309][273]

944 (276)p–o(p)5 [325]

1238

649 [144]

Table 7 sp Spectroscopic quadrupole moments Q2 (in e fm2 ) of selected states in 70 Kr. As effective charges ep = 1.5 and en = 0.5 have been used I [h] ¯ 2+ 4+ 6+ 8+ 10+ 12+ 14+ 16+ 18+

p–o(p)1 −10.67 −22.19 −72.08 −94.65 −98.44 −100.51 −98.00 −91.10 −89.59

p2

−48.80 −107.75 −102.13 −104.65 −103.27 −99.10

p3

−88.48 −86.39 −100.27 −72.12 −95.59 −98.01

o–p(p)4 7.89 14.87 63.47 71.64 45.63 −26.54 −91.25 −98.15 −90.17

p–o(p)5

−103.93 −88.75 −48.71 −46.68 −84.67

p–o(o)6

−87.81 −25.92 −93.88 −54.23 −17.12 18.29 −15.46

39%, 30%, 9%, 1%, <1% with Coulomb interaction. The corresponding mixing from 0+ to 8+ in 70 Kr is 44%, 52%, 46%, 17%, 4%, respectively for no Coulomb calculations and 33%, 28%, 20%, 10%, 4%, respectively with Coulomb interaction.

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A. Petrovici et al. / Nuclear Physics A 728 (2003) 396–414

Table 8 B(M1; I = 0) values (in µ2N ) in 70 Kr for states indicated in Fig. 4. The proton-orbital (π -o), proton-spin (π -s) and neutron-spin (ν-s) contributions are obtained using as effective g-factors (in µN ) gπl = 1.0, gπs = 5.5857 and gνs = −3.8263 Iinit → Ifin [h] ¯

π -o (%)

π -s (%)

ν-s (%)

B(M1)

+ 6+ 5 → 61

41

25

34

0.12

42

47

11

0.19

49

18

32

0.52

46

15

39

0.89

40

50

10

0.52

45

15

40

0.70

43

42

15

0.96

48

37

15

0.30

42

31

27

1.54

42

59

–1

0.20

49

37

14

0.18

46

21

33

0.77

53

23

24

0.30

53

39

8

0.31

40

28

32

0.73

44

36

20

0.59

49

45

6

0.14

43

47

10

0.40

43

45

12

0.28

54

18

28

0.44

39

28

33

0.27

41

44

15

0.35

36

26

38

0.42

27

45

28

0.13

51

8

41

0.17

6+ 6 6+ 5 8+ 3 8+ 4 10+ 2 10+ 3 10+ 6 12+ 2 12+ 3 12+ 5 12+ 5 12+ 6 12+ 6 14+ 2 14+ 3 14+ 5 14+ 3 14+ 7 14+ 7 16+ 2 16+ 4 16+ 3 16+ 6 16+ 8

→ 6+ 1 → 6+ 3 → 8+ 1 → 8+ 1 → 10+ 1 → 10+ 1 → 10+ 3 → 12+ 1 → 12+ 1 → 12+ 3 → 12+ 4 → 12+ 4 → 12+ 5 → 14+ 1 → 14+ 1 → 14+ 1 → 14+ 2 → 14+ 4 → 14+ 5 → 16+ 1 → 16+ 1 → 16+ 2 → 16+ 2 → 16+ 4

The total B(E2) strength for the decay of a given state is almost not changed, but the strength of the secondary branch is significantly modified for the states manifesting strong o–p mixing. In 70 Se, for example, the ratio B(E2; 61 → 42 )/B(E2; 61 → 41 ) is 0.30 for no Coulomb, but 0.04 with Coulomb; B(E2; 41 → 22 )/B(E2; 41 → 21 ) is 0.05 without Coulomb, but 0.10 with Coulomb; for the 2+ and 8+ states the fragmentation of the B(E2) is not significant in both cases. In 70 Kr the only significant change is manifested for the B(E2; 61 → 42 )/B(E2; 61 → 41 ) from 0.12 (no Coulomb) to 0.03 (with Coulomb). Since the o–p mixing is smaller if the Coulomb interaction is introduced the B(E2, I = 0) transition strengths are stronger for no Coulomb case.

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Fig. 5. The alignment plot for some of the theoretical bands in 70 Kr.

In 70 Kr the B(E2, I =0) (Coulomb)/B(E2, I = 0) (no Coulomb) from the first excited to the yrast state of a given spin amounts to 0.86, 0.66, 0.60, and 0.58 for the spins 2+ , 4+ , 6+ , 8+ , respectively. In 70 Se the corresponding B(E2, I = 0) ratio amounts to 0.92, 0.34, 0.24 and 0.40 for the spin 2+ , 4+ , 6+ and 8+ , respectively. As it is expected, the maximum effect of the changes in the o–p mixing is obtained for the spectroscopic quadrupole moments. In 70 Se significant changes are obtained sp for the lowest two states of spin 2+ and 4+ . With Coulomb interaction Q2 (2+ 1) = sp sp + + 2 2 −29.09 e fm , Q2 (22 ) = +23.30 e fm , but without Coulomb Q2 (21 ) = +15.80 e fm2 , sp 2 + + + 70 Q2 (2+ 2 ) = −19.90 e fm . Small changes are obtained for the spins 4 , 6 , 8 . In Kr sp + sp + 2 2 without Coulomb (with Coulomb) Q2 (21 ) = −2.20 e fm (−28.17 e fm ), Q2 (22 ) = sp sp + 2 2 +0.50 e fm2 (+23.58 e fm2 ), Q2 (4+ 1 ) = −11.10 e fm (−53.35 e fm ), Q2 (42 ) = sp sp + 2 2 +4.30 e fm2 (+45.53 e fm2 ), Q2 (6+ 1 ) = −67.10 e fm (−80.50 e fm ), Q2 (62 ) = +59.10 e fm2 (+68.51 e fm2 ) and almost no change is observed for the 8+ states.

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Fig. 6. The occupation of the 0g9/2 spherical orbital for some bands in 70 Kr.

The B(M1, 82 → 81 ) transition strengths are almost two times stronger with Coulomb interaction included in both nuclei.

4. Conclusions In the present paper we did present new microscopic results on the shape coexistence at low and high spins in the mirror nuclei 70 Se and 70 Kr. The complex Excited Vampir results indicate a strong influence of particular T = 0 matrix elements of the effective Hamiltonian involving neutrons and protons occupying the 0f5/2 (0f7/2 ) and 0g9/2 spherical orbitals on the structure of the yrast and first excited band at low and intermediate spins. The oblate–prolate coexistence and mixing are very sensitive to small changes of the strengths of these matrix elements around the value used in this mass region. The coexistence phenomena are dominating the structure of the investigated mirror nuclei 70 Se and 70 Kr. Strong similarities, but also significant differences characterize the behavior of the two mirror nuclei. The results obtained for the nucleus 70 Se have been compared with recent experimental data. Unfortunately the majority of spin-parity assignments are only tentative. Nevertheless

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413

Fig. 7. The theoretical spectra for the ground state bands in 70 Se and 70 Kr calculated with and without Coulomb interaction.

we tried to compare the complex decay pattern identified experimentally for the positiveparity states with the Excited Vampir results. Most of the experimental characteristics are reproduced at least qualitatively. Strong M1 I = 0 transitions connecting intermediate spin states are predicted that would influence the complex decay pattern significantly. Data on the electromagnetic properties would be useful in order to fix the matrix elements that influence the oblate– prolate mixing and by that the similarity in the behavior of the two mirror nuclei at low spins.

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