- Email: [email protected]
Beyond mean field approach to the beta decay of medium mass nuclei relevant for nuclear astrophysics
Progress in Particle and Nuclear Physics 66 (2011) 287–292
Contents lists available at ScienceDirect
Progress in Particle and Nuclear Physics journal homepage: www.elsevier.com/locate/ppnp
Review
Beyond mean field approach to the beta decay of medium mass nuclei relevant for nuclear astrophysics A. Petrovici a,b,∗ , K.W. Schmid b , A. Faessler b a
National Institute for Physics and Nuclear Engineering, R-077125 Bucharest, Romania
b
Institut für Theoretische Physik, Universität Tübingen, D-72076 Tübingen, Germany
article
info
Keywords: Shape coexistence Gamow–Teller beta decay Proton-rich nuclei Neutron-rich nuclei
abstract The Gamow–Teller strength distributions for the β decay of the ground state as well as the lowest excited states of the rp-process waiting point nuclei 68 Se and 72 Kr are obtained within the complex Excited Vampir variational approach using realistic effective interactions and a rather large model space. The shape mixing is consistently described for both the states in the even–even parent and the states in the odd–odd daugther nucleus. The influence of the shape mixing accounted by the different effective interactions used and comparison with the available data are presented. The possible influence of the decay of the lowest excited states of the parent nuclei in the astrophysical environment of X-ray bursts is discussed. Gamow–Teller strength distributions, β -decay half-lives, and β -delayed neutron emission probabilities for neutron-rich Zr nuclei are investigated for the first time within the complex Excited Vampir approach using a large model space. Comparison with available data and predictions relevant for the astrophysical r process are presented. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Nuclear shape coexistence dominates the properties of proton-rich nuclei in the A = 60–90 mass region that are important for the rp-process and the understanding of the nucleosynthesis. Relevant for the Gamow–Teller (GT) beta decay of the waiting point nuclei could be the GT strength distributions for the low-lying excited 0+ states whose thermal population may result in a significant reduction of the effective lifetime at the high temperatures of X-ray bursts [1]. It is acknowledged that there are many nuclei and nuclear properties that are not within experimental reach. For an understanding of those systems, a robust nuclear theoretical capability is required. The description of the GT strength distributions for the beta decay of nuclei close to the proton-drip line in the A ∼ 70 region meets the difficulty of treating self-consistently the shape coexistence and mixing, which dominate the structure of both even–even parent and odd–odd daughter nucleus. Even more, drastic changes in structure are expected for small variations of the nucleon number. For the nuclei close to the N = Z line, the problem is complicated by the competition between the neutron–proton and like-nucleon pairing correlations expected to influence the behaviour of these nuclei significantly [2–4]. The r-process representing the main nucleosynthesis mechanism responsible for the production of heavy neutron-rich nuclei and for the existence of about half of the nuclei heavier than iron [5] is one of the most challenging questions of nuclear astrophysics. Relevant for the r-process are β -decay half-lives and the β -delayed neutron-emission probabilities. Reliable predictions for nuclear physics properties of extremly neutron-rich nuclei along the r-process path are required by the astrophysical scenarios.
∗
Corresponding author at: National Institute for Physics and Nuclear Engineering, R-077125 Bucharest, Romania. E-mail address: [email protected] (A. Petrovici).
0146-6410/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2011.01.022
288
A. Petrovici et al. / Progress in Particle and Nuclear Physics 66 (2011) 287–292
Table 1 The amount of mixing for the lowest 0+ states of 68 Se. I [h¯ ]
Bonn A
+
01 0+ 2 0+ 3
Bonn CD
o-mixing (%)
p-mixing (%)
o-mixing (%)
p-mixing (%)
58(2) 10(6) 16(7)(3)
22(10)(4) 73(5)(3) 38(20)(10)(2)
53(2) 5(5) 26
24(11)(4) 84(3) 32(16)(11)(10)(2)
We investigated the coexistence phenomena dominating the structure and dynamics of nuclei involved in the rp-process as well as r-process within the complex Excited Vampir approach. This approach uses Hartree–Fock-Bogoliubov (HFB) vacua as basic building blocks, which are only restricted by time-reversal and axial symmetry. The underlying HFB transformations are essentially complex and do mix proton- with neutron-states as well as states of different parity and angular momentum. The broken symmetries of these vacua (nucleon numbers, parity, and total angular momentum) are restored by projection techniques, and the resulting symmetry-projected configurations are used as test wave functions in chains of successive variational calculations to determine the underlying HFB transformations as well as the configuration mixing. The HFB vacua of the above type account for arbitrary two-nucleon correlations and thus simultaneously describe like-nucleon as well as isovector and isoscalar proton–neutron pairing. Furthermore, the complex Excited Vampir model (EXVAM) allows the use of rather large model spaces and realistic effective interactions. In the following sections, we shall present the complex Excited Vampir results on the structure and Gamow–Teller β decay of the rp-process waiting point nuclei 68 Se and 72 Kr and the β -decay properties of the neutron-rich 104 Zr and 106 Zr nuclei relevant for r-process. 2. rp-process waiting point nuclei 2.1. Structure and beta decay of
68
Se
We present the first attempt at a completely self-consistent calculation of the Gamow–Teller beta decay of the ground state and the lowest two excited 0+ states in 68 Se to 68 As using the complex Excited Vampir approach. For nuclei in the A ≃ 70 mass region, we use a 40 Ca core and include the 1p1/2 , 1p3/2 , 0f5/2 , 0f7/2 , 1d5/2 and 0g9/2 oscillator orbits for both protons and neutrons in the valence space. The effective two-body interaction is constructed from a nuclear matter G-matrix based on the Bonn one-boson-exchange potential (Bonn A/Bonn CD) [6]. We calculated the lowest 0+ states of the even–even parent 68 Se and the 1+ states in the beta window in the odd–odd daughter 68 As. For the description of the lowest three 0+ states of 68 Se, we included in the Excited Vampir many-nucleon basis up to eighteen 0+ configurations. The final solutions have been obtained diagonalizing the residual interaction between the considered Excited Vampir configurations. The results concerning the oblate-prolate mixing in the wave functions of the lowest three 0+ -states of 68 Se are presented in Table 1. In this table, we indicate the amount of mixing for the Excited Vampir configurations contributing more than 2% of the total amplitude. It turns out that 62% of the ground state wave function results from oblate-deformed configurations, while the remaining 38% are made up by various prolatedeformed configurations using Bonn A. Using Bonn CD, the oblate components represent only 55%. The main components of the wave function are characterized in the intrinsic system by β2 = −0.29 and β2 = 0.31 for the oblate and prolate configurations, respectively. The first excited 0+ state is dominated by an almost spherical configuration with β2 = 0.04 that represent 73% of the total amplitude for Bonn A and 84% using Bonn CD. The excitation energy of this 0+ state is 1.117 MeV and 0.843 MeV using Bonn A and Bonn CD, respectively. For the second excited 0+ state, 72% of the total amplitude is represented by prolate configurations and 28% by oblate ones using Bonn A, while using Bonn CD a different contribution of the similar projected configurations is adding to an almost identical oblate and prolate content of the final wave function. The almost spherical prolate components of the ground state wave function represent 4% while for the second excited 0+ state 30% and 22% of the total amplitude for Bonn A and CD, respectively. The second excited 0+ state is obtained at an excitation energy of 2.155 (2.004) MeV for Bonn A (Bonn CD), while experimentally no excited 0+ state was identified up to now. In the 68 As nucleus, we calculated up to 139 1+ Excited Vampir configurations in order to describe the states within the QEC window. As in 68 Se in 68 As, strong mixing of differently deformed oblate and prolate configurations in the final solutions was obtained. In Fig. 1, we present the accumulated GT strength for the ground state and the lowest two excited 0+ of 68 Se using Bonn A and Bonn CD potential. The calculated GT decay half-life is obtained by summing up the contributions up to 1+ states in 68 As with excitation energies lying within the experimental QEC : 1 T1/2
=
gA2 − D
+ + 2 f (Z , Ei )|⟨1+ i ‖β ‖0 ⟩| .
(1)
i
The Fermi integrals f (Z , Ei ) are taken from Ref. [7]. We use D = 6146 s and gA = 1.26. The experimental value of the half-life for the decay of the ground state of 68 Se is 35.5(7) s [8], while the Excited Vampir result is 48.8 (33.5) s using Bonn A (Bonn CD) [6]. The half-life of the first excited 0+ state calculated using the theoretical excitation energy is 60.7 s using Bonn A and 62.8 s using Bonn CD. In the astrophysical environment, the decay rate will be an average weighted
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289
Fig. 1. The Gamow–Teller accumulated strength for the decay of the ground state and the lowest two excited 0+ -states of 68 Se obtained within the complex Excited Vampir model using Bonn A (left figure) and Bonn CD potential (right figure).
Fig. 2. The Gamow–Teller accumulated strength for the decay of the ground state of 72 Kr obtained within the complex Excited Vampir model using Bonn A (Bonn CD) potential and different model spaces is compared to data [10,11].
with the thermal population decreasing exponentially with the excitation energy. Even at the highest temperature of the X-ray bursts (2 GK), since the first two excited 0+ states are situated relatively high in energy and the corresponding halflife is longer than the ground-state half-life the effective half-life will be determined by the decay of the ground state of 68 Se. 2.2. Structure and beta decay of
72
Kr
In this subsection, we present the results of the first attempt at a completely self-consistent calculation of the Gamow–Teller β decay of the lowest two 0+ states and the yrast 2+ state in 72 Kr to 72 Br using the complex Excited Vampir approach. In the experiments at CERN/ISOLDE, the GT strength distribution for the 72 Kr beta decay could only be established up to 2 MeV excitation energy in 72 Br while the whole beta window amounts to QEC = 5.040(375) MeV [9]. The lowest 0+ and 2+ states of 72 Kr and the 1+ , 2+ , and 3+ states in 72 Br have been calculated using the same effective interaction and model space as for the investigation of the 68 Se β decay. The results are compared with the ones obtained using the Bonn A potential and a larger model space containing in addition 2s1/2 , 1d3/2 and 0g7/2 orbitals for both protons and neutrons. The structure of the wave functions for parent and daughter states indicate variable, in some cases strong oblate-prolate shape mixing. In Fig. 2, we present the accumulated Gamow–Teller strength for the decay of the ground state of 72 Kr and compare it to the available data [9]. The experimental value of the half-life for the decay of the ground state of 72 Kr is 17.1(2) s [9], while the Excited Vampir result is 20.8 (18.9) s using Bonn A (Bonn CD). For the decay of the first excited 0+ , the calculated half-life is 17.3 (12.9) s using Bonn A (Bonn CD) and the experimental excitation energy for the 0+ state. Using Bonn A (Bonn CD), the partial half-life of the yrast 2+ of 72 Kr is 18.7 (21.6) s for the decay to the 1+ states and 19.8 s for the decay to the 3+ states in 72 Br. Even at the highest temperature of the X-ray bursts (2 GK) since the half-life obtained for the decay of the first excited 0+ state is close to the half-life of the ground state, the effect is within the uncertainty of the ground-state half-life and astrophysically will make no difference. Furthermore, the first excited 0+ state is an isomeric state which could be much longer lived at high temperatures as the 72 Kr will be completely ionized and the decay via conversion electrons will not work. It probably has to transition into the 2+ state via gamma absorption or to capture a proton which would then
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Fig. 3. The Gamow–Teller strength distribution for the decay of the ground state of 104 Zr (left figure) and 106 Zr (right figure) obtained within the complex Excited Vampir model using Bonn A potential.
get re-emitted by the unbound 69 Br and could end up in the 72 Kr ground state. All these options could be faster than the beta decay and in this case an equilibrium population of the excited states would be achieved from which the beta decay would occur. The excitation energy of the 2+ state of 72 Kr is even higher and the E2 decay is a fast collective one. Of course, the theoretical results on the Gamow–Teller strength distributions could be changed, at least for the high-excitation energy region, by the higher-lying configurations not included in the complex Excited Vampir many-nucleon basis and changes in the renormalization of the effective interaction. 3. β-decay half-lives and β-delayed neutron emission probabilities of Zr nuclei relevant for r-process The isotopic chain of neutron-rich zirconium nuclei offers an example of rapid transition from spherical to deformed shape with a possible identification of the sudden onset of quadrupole deformation between N = 58 and 60. For nuclei in the A ≃ 100 mass region, we use a large model space above a 40 Ca core built out of 1p1/2 , 1p3/2 , 0f5/2 , 0f7/2 , 2s1/2 , 1d3/2 , 1d5/2 , 0g7/2 , 0g9/2 , and 0h11/2 oscillator orbits for both protons and neutrons in the valence space. The effective two-body interaction is constructed from a nuclear matter G-matrix based on the Bonn one-boson-exchange potential (Bonn A). This G-matrix was modified by adding short-range (0.707 fm) Gaussians in the T = 1 and T = 0 channels in order to enhance the pairing correlations. In addition, the isoscalar interaction was modified by monopole shifts for all ˆ |0g9/2 0f ; IT = 0⟩ involving the 0f5/2 and 0f7/2 orbitals, ⟨0g7/2 0g9/2 ; T = 0 matrix elements of the form ⟨0g9/2 0f ; IT = 0|G
ˆ |0g7/2 0g9/2 ; IT = 0⟩ and ⟨1d5/2 0h11/2 ; IT = 0|Gˆ |1d5/2 0h11/2 ; IT = 0⟩. IT = 0|G In agreement with the experimental data, our results indicate shape transition from moderate deformation in 98 Zr to large deformation in 104,106,110 Zr, shape coexistence and variable shape mixing at low as well as intermediate and high spins specific for each investigated Zr isotope [12]. The present study is the first attempt at a completely self-consistent calculation of the Gamow–Teller β -decay properties of 104 Zr and 106 Zr nuclei. We calculated the lowest 10 0+ states in the even–even 104,106 Zr nuclei and the lowest 50 1+ states in the odd–odd 104,106 Nb nuclei. Our investigations indicated that the ground states of 104 Zr and 106 Zr are dominated (99%) by a strongly deformed configuration [12]. In 104 Zr beginning with spin 6+ the yrare states manifest significant mixing of two or three prolate deformed projected configurations, whereas for spins 10+ , 12+ , and 14+ even for the lowest band the states manifest mixing. In 106 Zr, significant mixing of different prolate deformed configurations was found starting with spin 10+ . The calculated 1+ states are all situated within 6.2 MeV excitation energy in 104 Nb and 6.0 MeV in 106 Nb. In both nuclei, the states manifest strong mixing of differently deformed configurations in the intrinsec system. The Excited Vampir configurations are strongly or moderately deformed and more than 75% manifest prolate deformation. Some states are dominated by strong prolate-oblate mixing. The theoretical results for the GT strength distribution for the decay of the ground state of 104 Zr and 106 Zr are presented in Fig. 3. The strong GT β -decay branches for both 104 Zr and 106 Zr nuclei indicate essential contribution from the g9π/2 g7ν/2 , dπ5/2 dν3/2 , and dπ5/2 dν5/2 matrix elements. In Fig. 4, we present the accumulated Gamow–Teller strength for the decay of the ground state of 104 Zr and 106 Zr. The involved β window (Qβ ) and the one-neutron separation energy in the daughter nucleus (Sn ) are the measured ones in 104 Nb, but the predicted ones by mass models in 106 Nb [13,14]. The lowest 1+ state was considered the ground state according to the experimental assignment [13,14]. The experimental value of the half-life for the decay of the ground state of 104 Zr of 1200(300) ms [15] was changed recently to 870(50)(30) ms [14]. The Excited Vampir result obtained calculating the lowest 50 1+ states is 2040 ms. Recent experimental data [14] for the half-life of 106 Zr indicate 260(20)(30) ms, while the Excited Vampir result is 230 ms involving the lowest 50 1+ states. Of course, the theoretical results on the Gamow–Teller strength distributions could be changed, more for the high-excitation energy region, by the higher-lying configurations not included in the complex Excited Vampir many-nucleon basis. The probability of β -delayed neutron emission (Pn ) was calculated taking
A. Petrovici et al. / Progress in Particle and Nuclear Physics 66 (2011) 287–292
291
Fig. 4. The Gamow–Teller accumulated strength for the decay of the ground state of 104 Zr (left figure) and 106 Zr (right figure) obtained within the complex Excited Vampir model using Bonn A potential.
into account the GT contributions from the 1+ states with excitation energy larger than Sn : Qβ ∑
Pn =
f (Z , Qβ − Eex )B(GT , Eex )
Sn Qβ ∑
.
(2)
f (Z , Qβ − Eex )B(GT , Eex )
0
The experimental results indicate for Pn an upper limit of 1% for the decay of 104 Zr and an upper limit of 7% for the decay of 106 Zr [14]. The Excited Vampir Pn value based on the lowest 50 1+ states is smaller than 1% for 104 Zr decay and amounts to 2% for 106 Zr. Of course, these results are strongly dependent on the 1+ states higher in energy than Sn and require a much larger EXVAM basis than the presently involved one. 4. Conclusion In the present paper, we presented the results obtained within the complex Excited Vampir variational approach using realistic effective interactions and large model spaces on the β decay of the rp-process waiting point nuclei 68 Se and 72 Kr as well as the β -decay properties of the neutron-rich 104 Zr and 106 Zr relevant for the r-process. This approach going beyond mean field approximation allows us to describe self-consistently the shape coexistence and mixing dominating the structure of the involved states in both parent and daughter nucleus. Our results indicate that in the astrophysical environment of the X-ray bursts the decay of the low-lying states of 68 Se and 72 Kr will not influence the effective half-life that will be determined by the decay of the ground state of these nuclei. The complex Excited Vampir model reveals that the excited 1+ states in the odd–odd daugther nuclei 104 Nb and 106 Nb contributing significantly to the accumulated GT strength manifest strong mixing of oblate and prolate differently deformed configurations, while the ground state of the even–even parent nuclei 104 Zr and 106 Zr, respectively, is dominated by one prolate strongly deformed configuration. The theoretical results on the β -decay properties are in agreement with the available experimental information for both the proton-rich and the neutron-rich investigated nuclei. Of course, the theoretical results could be changed, at least for the high-excitation energy region, by even higher-lying configurations not included in the complex Excited Vampir many-nucleon basis. Also changes in the renormalization of the effective interaction could influence the shape mixing in the structure of the involved wave functions. Acknowledgement This work was partly supported by the ANCS (Romania) under the PNCDI2 program and the DFG (Germany) under contract 436RUM 113/20/0–3. References [1] H. Schatz, A. Aprahamian, J. Görres, M. Wiescher, T. Rauscher, J.F. Rembges, F.-K. Thielemann, B. Pfeiffer, P. Möller, K.-L. Kratz, Phys. Rep. 294 (1998) 167. [2] A. Petrovici, K.W. Schmid, A. Faessler, Nuclear Phys. A 647 (1999) 197. [3] A. Petrovici, K.W. Schmid, A. Faessler, Nuclear Phys. A 665 (2000) 333. [4] A. Petrovici, K.W. Schmid, A. Faessler, Nuclear Phys. A 710 (2002) 246. [5] J.J. Cowan, F.-K. Thielemann, J.W. Truran, Phys. Rep. 208 (1991) 267.
292 [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
A. Petrovici et al. / Progress in Particle and Nuclear Physics 66 (2011) 287–292 A. Petrovici, K.W. Schmid, O. Andrei, A. Faessler, Phys. Rev. C 80 (2009) 044319. N.B. Gove, M.J. Martin, Nucl. Data Tables 10 (1971) 205. P. Baumann, et al., Phys. Rev. C 50 (1994) 1180. I. Piqueras, et al., Eur. Phys. J. A 16 (2003) 313. A. Petrovici, K.W. Schmid, O. Radu, A. Faessler, Phys. Rev. C 78 (2008) 044315. A. Petrovici, J. Phys. G: Nucl. Part. Phys. 37 (2010) 064036. A. Petrovici, International Nuclear Physics Conference 2010, Vancouver, Canada, 2010. S. Rinta-Antila, et al., Eur. Phys. J. A 31 (2007) 1. J. Pereira, et al., Phys. Rev. C 79 (2009) 035806. A. Schmitt, N. Kaffrell, N. Trautmann, Jahresbericht 1979 (1980) 34.
Contents lists available at ScienceDirect
Progress in Particle and Nuclear Physics journal homepage: www.elsevier.com/locate/ppnp
Review
Beyond mean field approach to the beta decay of medium mass nuclei relevant for nuclear astrophysics A. Petrovici a,b,∗ , K.W. Schmid b , A. Faessler b a
National Institute for Physics and Nuclear Engineering, R-077125 Bucharest, Romania
b
Institut für Theoretische Physik, Universität Tübingen, D-72076 Tübingen, Germany
article
info
Keywords: Shape coexistence Gamow–Teller beta decay Proton-rich nuclei Neutron-rich nuclei
abstract The Gamow–Teller strength distributions for the β decay of the ground state as well as the lowest excited states of the rp-process waiting point nuclei 68 Se and 72 Kr are obtained within the complex Excited Vampir variational approach using realistic effective interactions and a rather large model space. The shape mixing is consistently described for both the states in the even–even parent and the states in the odd–odd daugther nucleus. The influence of the shape mixing accounted by the different effective interactions used and comparison with the available data are presented. The possible influence of the decay of the lowest excited states of the parent nuclei in the astrophysical environment of X-ray bursts is discussed. Gamow–Teller strength distributions, β -decay half-lives, and β -delayed neutron emission probabilities for neutron-rich Zr nuclei are investigated for the first time within the complex Excited Vampir approach using a large model space. Comparison with available data and predictions relevant for the astrophysical r process are presented. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Nuclear shape coexistence dominates the properties of proton-rich nuclei in the A = 60–90 mass region that are important for the rp-process and the understanding of the nucleosynthesis. Relevant for the Gamow–Teller (GT) beta decay of the waiting point nuclei could be the GT strength distributions for the low-lying excited 0+ states whose thermal population may result in a significant reduction of the effective lifetime at the high temperatures of X-ray bursts [1]. It is acknowledged that there are many nuclei and nuclear properties that are not within experimental reach. For an understanding of those systems, a robust nuclear theoretical capability is required. The description of the GT strength distributions for the beta decay of nuclei close to the proton-drip line in the A ∼ 70 region meets the difficulty of treating self-consistently the shape coexistence and mixing, which dominate the structure of both even–even parent and odd–odd daughter nucleus. Even more, drastic changes in structure are expected for small variations of the nucleon number. For the nuclei close to the N = Z line, the problem is complicated by the competition between the neutron–proton and like-nucleon pairing correlations expected to influence the behaviour of these nuclei significantly [2–4]. The r-process representing the main nucleosynthesis mechanism responsible for the production of heavy neutron-rich nuclei and for the existence of about half of the nuclei heavier than iron [5] is one of the most challenging questions of nuclear astrophysics. Relevant for the r-process are β -decay half-lives and the β -delayed neutron-emission probabilities. Reliable predictions for nuclear physics properties of extremly neutron-rich nuclei along the r-process path are required by the astrophysical scenarios.
∗
Corresponding author at: National Institute for Physics and Nuclear Engineering, R-077125 Bucharest, Romania. E-mail address: [email protected] (A. Petrovici).
0146-6410/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ppnp.2011.01.022
288
A. Petrovici et al. / Progress in Particle and Nuclear Physics 66 (2011) 287–292
Table 1 The amount of mixing for the lowest 0+ states of 68 Se. I [h¯ ]
Bonn A
+
01 0+ 2 0+ 3
Bonn CD
o-mixing (%)
p-mixing (%)
o-mixing (%)
p-mixing (%)
58(2) 10(6) 16(7)(3)
22(10)(4) 73(5)(3) 38(20)(10)(2)
53(2) 5(5) 26
24(11)(4) 84(3) 32(16)(11)(10)(2)
We investigated the coexistence phenomena dominating the structure and dynamics of nuclei involved in the rp-process as well as r-process within the complex Excited Vampir approach. This approach uses Hartree–Fock-Bogoliubov (HFB) vacua as basic building blocks, which are only restricted by time-reversal and axial symmetry. The underlying HFB transformations are essentially complex and do mix proton- with neutron-states as well as states of different parity and angular momentum. The broken symmetries of these vacua (nucleon numbers, parity, and total angular momentum) are restored by projection techniques, and the resulting symmetry-projected configurations are used as test wave functions in chains of successive variational calculations to determine the underlying HFB transformations as well as the configuration mixing. The HFB vacua of the above type account for arbitrary two-nucleon correlations and thus simultaneously describe like-nucleon as well as isovector and isoscalar proton–neutron pairing. Furthermore, the complex Excited Vampir model (EXVAM) allows the use of rather large model spaces and realistic effective interactions. In the following sections, we shall present the complex Excited Vampir results on the structure and Gamow–Teller β decay of the rp-process waiting point nuclei 68 Se and 72 Kr and the β -decay properties of the neutron-rich 104 Zr and 106 Zr nuclei relevant for r-process. 2. rp-process waiting point nuclei 2.1. Structure and beta decay of
68
Se
We present the first attempt at a completely self-consistent calculation of the Gamow–Teller beta decay of the ground state and the lowest two excited 0+ states in 68 Se to 68 As using the complex Excited Vampir approach. For nuclei in the A ≃ 70 mass region, we use a 40 Ca core and include the 1p1/2 , 1p3/2 , 0f5/2 , 0f7/2 , 1d5/2 and 0g9/2 oscillator orbits for both protons and neutrons in the valence space. The effective two-body interaction is constructed from a nuclear matter G-matrix based on the Bonn one-boson-exchange potential (Bonn A/Bonn CD) [6]. We calculated the lowest 0+ states of the even–even parent 68 Se and the 1+ states in the beta window in the odd–odd daughter 68 As. For the description of the lowest three 0+ states of 68 Se, we included in the Excited Vampir many-nucleon basis up to eighteen 0+ configurations. The final solutions have been obtained diagonalizing the residual interaction between the considered Excited Vampir configurations. The results concerning the oblate-prolate mixing in the wave functions of the lowest three 0+ -states of 68 Se are presented in Table 1. In this table, we indicate the amount of mixing for the Excited Vampir configurations contributing more than 2% of the total amplitude. It turns out that 62% of the ground state wave function results from oblate-deformed configurations, while the remaining 38% are made up by various prolatedeformed configurations using Bonn A. Using Bonn CD, the oblate components represent only 55%. The main components of the wave function are characterized in the intrinsic system by β2 = −0.29 and β2 = 0.31 for the oblate and prolate configurations, respectively. The first excited 0+ state is dominated by an almost spherical configuration with β2 = 0.04 that represent 73% of the total amplitude for Bonn A and 84% using Bonn CD. The excitation energy of this 0+ state is 1.117 MeV and 0.843 MeV using Bonn A and Bonn CD, respectively. For the second excited 0+ state, 72% of the total amplitude is represented by prolate configurations and 28% by oblate ones using Bonn A, while using Bonn CD a different contribution of the similar projected configurations is adding to an almost identical oblate and prolate content of the final wave function. The almost spherical prolate components of the ground state wave function represent 4% while for the second excited 0+ state 30% and 22% of the total amplitude for Bonn A and CD, respectively. The second excited 0+ state is obtained at an excitation energy of 2.155 (2.004) MeV for Bonn A (Bonn CD), while experimentally no excited 0+ state was identified up to now. In the 68 As nucleus, we calculated up to 139 1+ Excited Vampir configurations in order to describe the states within the QEC window. As in 68 Se in 68 As, strong mixing of differently deformed oblate and prolate configurations in the final solutions was obtained. In Fig. 1, we present the accumulated GT strength for the ground state and the lowest two excited 0+ of 68 Se using Bonn A and Bonn CD potential. The calculated GT decay half-life is obtained by summing up the contributions up to 1+ states in 68 As with excitation energies lying within the experimental QEC : 1 T1/2
=
gA2 − D
+ + 2 f (Z , Ei )|⟨1+ i ‖β ‖0 ⟩| .
(1)
i
The Fermi integrals f (Z , Ei ) are taken from Ref. [7]. We use D = 6146 s and gA = 1.26. The experimental value of the half-life for the decay of the ground state of 68 Se is 35.5(7) s [8], while the Excited Vampir result is 48.8 (33.5) s using Bonn A (Bonn CD) [6]. The half-life of the first excited 0+ state calculated using the theoretical excitation energy is 60.7 s using Bonn A and 62.8 s using Bonn CD. In the astrophysical environment, the decay rate will be an average weighted
A. Petrovici et al. / Progress in Particle and Nuclear Physics 66 (2011) 287–292
289
Fig. 1. The Gamow–Teller accumulated strength for the decay of the ground state and the lowest two excited 0+ -states of 68 Se obtained within the complex Excited Vampir model using Bonn A (left figure) and Bonn CD potential (right figure).
Fig. 2. The Gamow–Teller accumulated strength for the decay of the ground state of 72 Kr obtained within the complex Excited Vampir model using Bonn A (Bonn CD) potential and different model spaces is compared to data [10,11].
with the thermal population decreasing exponentially with the excitation energy. Even at the highest temperature of the X-ray bursts (2 GK), since the first two excited 0+ states are situated relatively high in energy and the corresponding halflife is longer than the ground-state half-life the effective half-life will be determined by the decay of the ground state of 68 Se. 2.2. Structure and beta decay of
72
Kr
In this subsection, we present the results of the first attempt at a completely self-consistent calculation of the Gamow–Teller β decay of the lowest two 0+ states and the yrast 2+ state in 72 Kr to 72 Br using the complex Excited Vampir approach. In the experiments at CERN/ISOLDE, the GT strength distribution for the 72 Kr beta decay could only be established up to 2 MeV excitation energy in 72 Br while the whole beta window amounts to QEC = 5.040(375) MeV [9]. The lowest 0+ and 2+ states of 72 Kr and the 1+ , 2+ , and 3+ states in 72 Br have been calculated using the same effective interaction and model space as for the investigation of the 68 Se β decay. The results are compared with the ones obtained using the Bonn A potential and a larger model space containing in addition 2s1/2 , 1d3/2 and 0g7/2 orbitals for both protons and neutrons. The structure of the wave functions for parent and daughter states indicate variable, in some cases strong oblate-prolate shape mixing. In Fig. 2, we present the accumulated Gamow–Teller strength for the decay of the ground state of 72 Kr and compare it to the available data [9]. The experimental value of the half-life for the decay of the ground state of 72 Kr is 17.1(2) s [9], while the Excited Vampir result is 20.8 (18.9) s using Bonn A (Bonn CD). For the decay of the first excited 0+ , the calculated half-life is 17.3 (12.9) s using Bonn A (Bonn CD) and the experimental excitation energy for the 0+ state. Using Bonn A (Bonn CD), the partial half-life of the yrast 2+ of 72 Kr is 18.7 (21.6) s for the decay to the 1+ states and 19.8 s for the decay to the 3+ states in 72 Br. Even at the highest temperature of the X-ray bursts (2 GK) since the half-life obtained for the decay of the first excited 0+ state is close to the half-life of the ground state, the effect is within the uncertainty of the ground-state half-life and astrophysically will make no difference. Furthermore, the first excited 0+ state is an isomeric state which could be much longer lived at high temperatures as the 72 Kr will be completely ionized and the decay via conversion electrons will not work. It probably has to transition into the 2+ state via gamma absorption or to capture a proton which would then
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Fig. 3. The Gamow–Teller strength distribution for the decay of the ground state of 104 Zr (left figure) and 106 Zr (right figure) obtained within the complex Excited Vampir model using Bonn A potential.
get re-emitted by the unbound 69 Br and could end up in the 72 Kr ground state. All these options could be faster than the beta decay and in this case an equilibrium population of the excited states would be achieved from which the beta decay would occur. The excitation energy of the 2+ state of 72 Kr is even higher and the E2 decay is a fast collective one. Of course, the theoretical results on the Gamow–Teller strength distributions could be changed, at least for the high-excitation energy region, by the higher-lying configurations not included in the complex Excited Vampir many-nucleon basis and changes in the renormalization of the effective interaction. 3. β-decay half-lives and β-delayed neutron emission probabilities of Zr nuclei relevant for r-process The isotopic chain of neutron-rich zirconium nuclei offers an example of rapid transition from spherical to deformed shape with a possible identification of the sudden onset of quadrupole deformation between N = 58 and 60. For nuclei in the A ≃ 100 mass region, we use a large model space above a 40 Ca core built out of 1p1/2 , 1p3/2 , 0f5/2 , 0f7/2 , 2s1/2 , 1d3/2 , 1d5/2 , 0g7/2 , 0g9/2 , and 0h11/2 oscillator orbits for both protons and neutrons in the valence space. The effective two-body interaction is constructed from a nuclear matter G-matrix based on the Bonn one-boson-exchange potential (Bonn A). This G-matrix was modified by adding short-range (0.707 fm) Gaussians in the T = 1 and T = 0 channels in order to enhance the pairing correlations. In addition, the isoscalar interaction was modified by monopole shifts for all ˆ |0g9/2 0f ; IT = 0⟩ involving the 0f5/2 and 0f7/2 orbitals, ⟨0g7/2 0g9/2 ; T = 0 matrix elements of the form ⟨0g9/2 0f ; IT = 0|G
ˆ |0g7/2 0g9/2 ; IT = 0⟩ and ⟨1d5/2 0h11/2 ; IT = 0|Gˆ |1d5/2 0h11/2 ; IT = 0⟩. IT = 0|G In agreement with the experimental data, our results indicate shape transition from moderate deformation in 98 Zr to large deformation in 104,106,110 Zr, shape coexistence and variable shape mixing at low as well as intermediate and high spins specific for each investigated Zr isotope [12]. The present study is the first attempt at a completely self-consistent calculation of the Gamow–Teller β -decay properties of 104 Zr and 106 Zr nuclei. We calculated the lowest 10 0+ states in the even–even 104,106 Zr nuclei and the lowest 50 1+ states in the odd–odd 104,106 Nb nuclei. Our investigations indicated that the ground states of 104 Zr and 106 Zr are dominated (99%) by a strongly deformed configuration [12]. In 104 Zr beginning with spin 6+ the yrare states manifest significant mixing of two or three prolate deformed projected configurations, whereas for spins 10+ , 12+ , and 14+ even for the lowest band the states manifest mixing. In 106 Zr, significant mixing of different prolate deformed configurations was found starting with spin 10+ . The calculated 1+ states are all situated within 6.2 MeV excitation energy in 104 Nb and 6.0 MeV in 106 Nb. In both nuclei, the states manifest strong mixing of differently deformed configurations in the intrinsec system. The Excited Vampir configurations are strongly or moderately deformed and more than 75% manifest prolate deformation. Some states are dominated by strong prolate-oblate mixing. The theoretical results for the GT strength distribution for the decay of the ground state of 104 Zr and 106 Zr are presented in Fig. 3. The strong GT β -decay branches for both 104 Zr and 106 Zr nuclei indicate essential contribution from the g9π/2 g7ν/2 , dπ5/2 dν3/2 , and dπ5/2 dν5/2 matrix elements. In Fig. 4, we present the accumulated Gamow–Teller strength for the decay of the ground state of 104 Zr and 106 Zr. The involved β window (Qβ ) and the one-neutron separation energy in the daughter nucleus (Sn ) are the measured ones in 104 Nb, but the predicted ones by mass models in 106 Nb [13,14]. The lowest 1+ state was considered the ground state according to the experimental assignment [13,14]. The experimental value of the half-life for the decay of the ground state of 104 Zr of 1200(300) ms [15] was changed recently to 870(50)(30) ms [14]. The Excited Vampir result obtained calculating the lowest 50 1+ states is 2040 ms. Recent experimental data [14] for the half-life of 106 Zr indicate 260(20)(30) ms, while the Excited Vampir result is 230 ms involving the lowest 50 1+ states. Of course, the theoretical results on the Gamow–Teller strength distributions could be changed, more for the high-excitation energy region, by the higher-lying configurations not included in the complex Excited Vampir many-nucleon basis. The probability of β -delayed neutron emission (Pn ) was calculated taking
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291
Fig. 4. The Gamow–Teller accumulated strength for the decay of the ground state of 104 Zr (left figure) and 106 Zr (right figure) obtained within the complex Excited Vampir model using Bonn A potential.
into account the GT contributions from the 1+ states with excitation energy larger than Sn : Qβ ∑
Pn =
f (Z , Qβ − Eex )B(GT , Eex )
Sn Qβ ∑
.
(2)
f (Z , Qβ − Eex )B(GT , Eex )
0
The experimental results indicate for Pn an upper limit of 1% for the decay of 104 Zr and an upper limit of 7% for the decay of 106 Zr [14]. The Excited Vampir Pn value based on the lowest 50 1+ states is smaller than 1% for 104 Zr decay and amounts to 2% for 106 Zr. Of course, these results are strongly dependent on the 1+ states higher in energy than Sn and require a much larger EXVAM basis than the presently involved one. 4. Conclusion In the present paper, we presented the results obtained within the complex Excited Vampir variational approach using realistic effective interactions and large model spaces on the β decay of the rp-process waiting point nuclei 68 Se and 72 Kr as well as the β -decay properties of the neutron-rich 104 Zr and 106 Zr relevant for the r-process. This approach going beyond mean field approximation allows us to describe self-consistently the shape coexistence and mixing dominating the structure of the involved states in both parent and daughter nucleus. Our results indicate that in the astrophysical environment of the X-ray bursts the decay of the low-lying states of 68 Se and 72 Kr will not influence the effective half-life that will be determined by the decay of the ground state of these nuclei. The complex Excited Vampir model reveals that the excited 1+ states in the odd–odd daugther nuclei 104 Nb and 106 Nb contributing significantly to the accumulated GT strength manifest strong mixing of oblate and prolate differently deformed configurations, while the ground state of the even–even parent nuclei 104 Zr and 106 Zr, respectively, is dominated by one prolate strongly deformed configuration. The theoretical results on the β -decay properties are in agreement with the available experimental information for both the proton-rich and the neutron-rich investigated nuclei. Of course, the theoretical results could be changed, at least for the high-excitation energy region, by even higher-lying configurations not included in the complex Excited Vampir many-nucleon basis. Also changes in the renormalization of the effective interaction could influence the shape mixing in the structure of the involved wave functions. Acknowledgement This work was partly supported by the ANCS (Romania) under the PNCDI2 program and the DFG (Germany) under contract 436RUM 113/20/0–3. References [1] H. Schatz, A. Aprahamian, J. Görres, M. Wiescher, T. Rauscher, J.F. Rembges, F.-K. Thielemann, B. Pfeiffer, P. Möller, K.-L. Kratz, Phys. Rep. 294 (1998) 167. [2] A. Petrovici, K.W. Schmid, A. Faessler, Nuclear Phys. A 647 (1999) 197. [3] A. Petrovici, K.W. Schmid, A. Faessler, Nuclear Phys. A 665 (2000) 333. [4] A. Petrovici, K.W. Schmid, A. Faessler, Nuclear Phys. A 710 (2002) 246. [5] J.J. Cowan, F.-K. Thielemann, J.W. Truran, Phys. Rep. 208 (1991) 267.
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