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Variational calculation of the effect of isospin-mixing on superallowed Fermi β decay in the A≃70 mass region
Nuclear Physics A 747 (2005) 44–52
Variational calculation of the effect of isospin-mixing on superallowed Fermi β decay in the A 70 mass region A. Petrovici a,b , K.W. Schmid b,∗ , O. Radu a , 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 19 November 2003; received in revised form 24 September 2004; accepted 1 October 2004 Available online 19 October 2004
Abstract We study the influence of Coulomb-induced isospin-mixing on the superallowed Fermi β decay using the complex EXCITED VAMPIR variational approach for the description of the lowest excited 0+ states in the two isovector triplets of nuclei 70 Se, 70 Br, 70 Kr and 74 Kr, 74 Rb, 74 Sr, respectively. As expected, the effects are small but nevertheless non-negligible. The calculated strength for the 74 Rb → 74 Kr Fermi β decay to the first excited 0+ state turns out to be very weak, in agreement with the experimentally given upper limit. However, particular non-analog Fermi branches to excited 0+ states with considerable strength are predicted to coexist with the superallowed decay in some of the presently investigated nuclei. This may give some hope for possible experimental detection. 2004 Elsevier B.V. All rights reserved. PACS: 21.10.-k; 23.40.-s; 27.50.+e Keywords: Nuclear structure; Shape coexistence; Proton-rich nuclei; Superallowed β decay; Isospin mixing
* Corresponding author.
E-mail address: [email protected] (K.W. Schmid). 0375-9474/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.10.002
A. Petrovici et al. / Nuclear Physics A 747 (2005) 44–52
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1. Introduction The exotic nuclei near the N = Z line in the A 70 mass region display some interesting nuclear structure effects like shape-coexistence and -mixing as well as the competition between proton–neutron and like-nucleon pairing. Furthermore, these nuclei play an important role in nuclear astrophysics, since their weak decay determines details of the nucleosynthesis. And last but not least, it has been argued that the investigation of superallowed Fermi β decays between analog states, which provide a test of the validity of the conserved vector current (CVC) hypothesis and the unitarily of the Cabibbo–Kobayashi– Maskawa (CKM) matrix (see [1] and the references therein), would be of particular interest in nuclei with A 62, where the charge induced isospin-mixing is expected to be large, and this argument has indeed caused considerable theoretical [1–5] as well as experimental [6–11] effort within recent years. Obviously, microscopic nuclear structure calculations for such medium heavy nuclei are extremely involved. The adequate model spaces are far too large to allow for a complete diagonalisation of an appropriate effective many-body Hamiltonian and thus one has to rely on suitable approximate methods. Furthermore, the appropriate effective Hamiltonian itself is not known a priori and can only be determined by an iterative process of many time-consuming calculations. Both, the limitation of the particular approximate method used as well as the insufficient knowledge of the appropriate Hamiltonian, will leave some uncertainties in the quantitative results especially if small effects are to be investigated. However, within the last decade there have been many rather successful theoretical investigations about the structure of nuclei in the A 70 mass region, not only of those along the valley of β-stability, but also in some exotic nuclei close to the proton drip line [12–18]. For these calculations the so-called complex EXCITED VAMPIR approach (see [19] for a recent review and references therein) has been used. 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, total angular momentum) are restored by projection techniques and the resulting symmetryprojected configurations are then used as test wave functions in chains of successive variational calculations to determine the underlying HFB transformations as well as the configuration mixing. It can be shown [19] that HFB vacua of the above type account for arbitrary two-nucleon correlations and thus simultaneously describe, e.g., like-nucleon as well as isovector and isoscalar proton–neutron pairing. In contrast to the general EXCITED VAMPIR approach, in which no symmetries at all are imposed on the underlying HFB transformations, in the complex EXCITED VAMPIR approach, because of the assumption of time-reversal and axial symmetry some particular four- and more-nucleon correlations are missing. Inspite of this limitation, however, the complex EXCITED VAMPIR has been proven to be a rather useful tool for nuclear structure calculations in large model spaces. This holds especially for the A 70 mass region. Here the effective interaction was adjusted in many different calculations. Phenomena like shape-coexistence and -mixing as well as the competition between proton–neutron and like nucleon-pairing turned out to be very important and various experimental observations in many nuclei could be nicely
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reproduced. Thus it may be worthwhile to apply the same methods for the investigation of the influence of Coulomb-induced isospin-mixing on superallowed Fermi β decay in the A = 70 and A = 74 isovector triplets of nuclei, even more so, since some of these nuclei have already been studied in some detail recently [12,14,18]. We shall briefly describe the various calculations performed and their ingrediences in the next section. In Section 3 then the results on Coulomb-induced isospin-mixing effects on the superallowed Fermi β decays in the two isovector triplets of nuclei 70 Se, 70 Br, 70 Kr and 74 Kr, 74 Rb, 74 Sr, respectively, will be discussed. Finally we shall present some conclusions in Section 4.
2. The calculations We calculated the lowest 0+ states in 70 Se, 70 Br, 70 Kr, 74 Kr, 74 Rb, and 74 Sr. For this purpose the complex version of the EXCITED VAMPIR approach (see Ref. [19]) was used. This method uses time-reversal invariant and axially symmetric Hartree–Fock–Bogoliubov (HFB) vacua constructed from essentially complex quasi-particle-transformations as basic building blocks and determines the underlying HFB-transformations by successive variational calculations after the restoration of angular momentum, nucleon-numbers and parity via projection methods. Finally, the residual interaction between the created configurations is diagonalized. For each nucleus of the A = 70 triplet the energetically lowest 13, for the A = 74 triplet the lowest 18 I π = 0+ configurations have been taken into account. It should be stressed here, that isospin is not considered as a good quantum number in this approach. Thus, even if a charge symmetric interaction is used, good total isospin can only be expected, if the configurations form a complete set under isospin-rotations. This is obviously not the case, if only the 13 or 18 lowest states are considered. Furthermore, as has been discussed in detail in [19], symmetry-projected configurations created by the complex EXCITED VAMPIR approach account for arbitrary two-nucleon correlations, however, because of the assumption of time-reversal and axial symmetry some four- and more-nucleon correlations are missing. Thus even if the number of configurations would be drastically increased, there is still some “spurious isospin impurity” to be expected (see also [20,21]). We shall come back to this problem later. As in our earlier calculations for nuclei in the A 70 mass region [18] we used a 40 Ca core and included 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. As corresponding single particle energies (in units of the oscillator energy h¯ ω = 41.2A−1/3) −0.015, −0.301, 0.215, −0.625, 0.118 and −0.007 for both protons and neutrons were taken. These are the averages of the slightly differing proton- and neutron-energies used in the earlier calculations. 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 a short-range (0.707 fm) Gaussian with strength −35 MeV in the T = 1 (proton–proton, neutron–neutron and proton–neutron) channel and a Gaussian of the same range with strength −180 MeV in the T = 0 channel in order to increase the pairingcorrelations in the corresponding channels. In addition the isoscalar interaction was modified by monopole shifts of −315 keV for all T = 0 matrix elements of the form 0g9/2 0f ;
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ˆ 9/20f ; I T = 0 with 0f denoting either the 0f5/2 or the 0f7/2 orbit and of I T = 0|G|0g ˆ −500 keV for those of the form 1p1d5/2 ; I T = 0|G|1p1d 5/2 ; I T = 0, where 1p denotes either the 1p1/2 or the 1p3/2 orbit. These shifts had been introduced in the earlier calculations in order to influence the deformation. We shall refer to this Hamiltonian, which has been applied rather successfully to many nuclei in the A = 70 mass region, as H0 in the following. Obviously, the above defined model space and effective Hamiltonian are charge symmetric. Thus (if the configuration space would be complete with respect to isospinrotations) one expects degenerate isovector excitation spectra for any triplet of nuclei with proton and neutron numbers (Z − 1, Z + 1), (Z, Z) and (Z + 1, Z − 1), respectively. Here Z is an odd number. Furthermore, one expects for the superallowed Fermi-transitions from the ground state of the (Z + 1, Z − 1)- to the ground state of the (Z, Z)-system and from the ground state of the (Z, Z)- to the ground state of the (Z − 1, Z + 1)-system both a total strength of two and (because of orthogonality) vanishing strengths for all the transitions from the ground state of the (Z + 1, Z − 1)- to all excited states of the (Z, Z)-system as well as from the ground state of the (Z, Z)- to all the excited states of the (Z − 1, Z + 1)nucleus. Deviations from these values as well as from the degeneracy of the spectra can then be attributed to isospin-mixing effects. In order to study those, the Coulomb-interaction has been included. For this purpose we need not only the two-body matrix elements in the model space but also the Coulombcontributions to the single particle energies resulting from the 40 Ca core. In order to determine the latter, we performed spherically symmetric Hartree–Fock calculations (with partial occupations of the unfilled j -shells) for the two doubly-even nuclei (Z − 1, Z + 1) and (Z + 1, Z − 1) of each triplet using the Gogny-interaction D1S and a 21 major-shell model space as described, e.g., in Ref. [19]. The contribution of the core to the proton single particle energies of the above valence orbits was calculated. The averages of the results for the (Z − 1, Z + 1) and (Z + 1, Z − 1) systems (which are almost identical) were then used for the full triplet of nuclei. We add the Coulomb-matrix elements and -single particle contributions to the charge-symmetric effective Hamiltonian defined above. The resulting Hamiltonian will be denoted by H1 in the following. Furthermore, the radial overlaps of the proton (21-major shell) Hartree–Fock wavefunctions of the (Z − 1, Z + 1)- and the neutron wave functions for the (Z + 1, Z − 1)-system and vice versa have been computed for the above defined valence orbits. The deviations of these overlaps from unity were in all cases less than half a percent. Obviously, inspite of the large basis, the tails of the wavefunctions may not be correctly described by the expansion in harmonic oscillators. This problem needs further investigation. In the present work these corrections have been neglected. Two different approaches have been considered: first, the Hamiltonian H1 was diagonalized for each considered nucleus within the complex EXCITED VAMPIR solutions obtained with the charge symmetric Hamiltonian H0 (“perturbative” approach). Second, the complex EXCITED VAMPIR calculations are repeated with H1 right from the beginning (“variational” approach). We would like to stress, that (because of the charge symmetry of H0 ) the isospin-mixing effects discussed in the next chapter are entirely due to the Coulomb-interaction. Possible effects due to charge symmetry breaking in the strong interaction (as included, e.g., in
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the Bonn CD-potential) are expected to be much smaller than those due to the Coulombinteraction. Thus no attempt has been made in the present investigation to include these additional effects.
3. Results and discussion 3.1. The A = 70 nuclei As already mentioned, for each of the nuclei in the A = 70 triplet 70 Se, 70 Br and the lowest 13 0+ configurations have been successively constructed by the complex EXCITED VAMPIR approach and finally the residual interaction between these configurations has been diagonalized. The first three columns of Fig. 1 present the resulting spectra, if the charge-symmetric Hamiltonian H0 is used. As can be seen the spectra of the two doubly-even nuclei are perfectly degenerate. The spectrum of the doubly-odd nucleus, however, displays slight deviations of the order of about 200 keV. These differences of the energies in the doubly-odd system with respect to the neighbouring doubly-even nuclei indicate (as discussed above) that the configuration space is not complete with respect to isospin rotations partly because of truncation, partly because of “missing” four- and morenucleon couplings in the EXCITED VAMPIR solutions. As expected, this deficiency is reflected in the strengths of the Fermi-transitions, too. For the total strength (ST ) of the transitions from the Kr ground state to all the calculated 13 0+ states of the Br nucleus we obtain 1.975 with 1.967 for the ground to ground transition (Sg−g ), while the total strength for the transitions from the Br ground state to the 13 lowest 0+ states in Se is 1.977 with again 1.967 for the ground to ground transition. Thus in between 1.3 ((2 − ST )/2) and 1.7 ((2 − Sg−g )/2) percent of the expected strength is not accounted for due to truncation and/or inherent isospin impurity effects. Now H1 is diagonalized within these solutions. The Coulomb interaction removes the degeneracy and nicely reproduces the correct energy differences between the ground states of the three nuclei. The corresponding relative excitation energies are displayed in the 3 middle columns of Fig. 1. As can be seen the deviations from degeneracy are here considerably larger than in the charge symmetric case. For the total Fermi-transition strength for the Kr ground state decays to the Br states one obtains 1.970 and 1.979 for the Br ground state decays to the Se states. Thus the total strengths are almost identical to those obtained without the Coulomb-interaction. For the ground to ground transitions one obtains now 1.935 and 1.967, respectively. Thus the effect of the perturbative treatment of the Coulomb-interaction (α = (Sg−g (H0 ) − Sg−g (H1 ))/2) is negligible for the Br to Se ground to ground transition while for the Kr to Br ground to ground transition a depletion of α = 1.6 percent is obtained. About one third of this depleted strength is concentrated in a single excited state. Finally, the results of using H1 already in the EXCITED VAMPIR variational calculations are presented in the 3 rightmost columns of Fig. 1. As can be seen the main effect of the non-perturbative (“variational”) treatment of the Coulomb-interaction is that it lowers some of the excited states (e.g., the second excited ones) by a considerable amount. In the variational treatment one obtains now only 1.946 and 1.959 for the total Kr to Br
70 Kr
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Fig. 1. The theoretical spectra of the lowest 10 0+ states in 70 Se, 70 Br and 70 Kr obtained within the complex EXCITED VAMPIR approach using the charge symmetric Hamiltonian H0 (“no Coulomb”) are compared to those resulting from perturbative (“perturbative”) and non-perturbative (“variational”) treatment of the Coulomb-interaction via the Hamiltonian H1 . Table 1 The total (ST ) and analog (Sg−g ) Fermi β decay strengths of selected A = 70 nuclei for the “no-Coulomb” (H0 ) and “Coulomb” (H1 ) effective Hamiltonian. The “perturbative” (p) and the “variational” (v) approaches are described in the text Parent nucleus 70 Kr 70 Br
p
H0
H1v
H1
ST
Sg−g
ST
Sg−g
ST
Sg−g
1.975 1.977
1.967 1.967
1.970 1.979
1.935 1.967
1.946 1.959
1.917 1.951
and Br to Se transition strength, respectively, while for the corresponding ground to ground transitions 1.917 and 1.951 are computed. A summary of the results is presented in Table 1. Estimating the error by summing the missing strengths between all and the analog transitions for the charge symmetric case (1 = (ST (H0 ) − Sg−g (H0 ))/2) and the missing total strengths between the variational calculations performed with H0 and H1 (2 = (ST (H0 ) − ST (H1 ))/2) one may conclude that the upper limit for the isospin mixing effect on the Br to Se ground to ground transition is about 0.8 percent with this strength distributed over many excited states, while for the Kr to Br ground to ground transition a depletion of at least 0.7 and at most 2.5 percent is obtained. In this latter case a nonanalog branch feeding the fourth excited 0+ state in Br with an upper limit of 0.7 percent is obtained.
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Fig. 2. Same as in Fig. 1, but for the A = 74 isovector triplet of nuclei.
3.2. The A = 74 nuclei For each of the nuclei in the A = 74 triplet 74 Kr, 74 Rb and 74 Sr the lowest 18 0+ configurations have been successively constructed by the complex EXCITED VAMPIR approach and finally the residual interaction between these configurations has been diagonalized. The first three columns of Fig. 2 present the resulting spectra, if the charge-symmetric Hamiltonian H0 is used. Again, as in the A = 70 triplet, the spectra of the two doubly-even nuclei are perfectly degenerate. The deviations in the spectrum of the doubly-odd nucleus are again of the order of about 200 keV. For the total strength of the transitions from the Sr ground state to all the calculated 18 0+ states in Rb we obtain here 1.954 with 1.947 for the ground to ground transition, while the total strength for the transitions from the Rb ground state to the 18 lowest 0+ states in Kr is 1.957 with 1.948 for the ground to ground transition. Thus here in between 2.3 and 2.7 percent of the strength is not accounted for due to truncation and/or inherent isospin impurity effects. Again the diagonalization of H1 removes the degeneracy and nicely reproduces the correct energy differences between the ground states of the three A = 74 nuclei, too. The corresponding relative excitation energies are presented in the 3 middle columns of Fig. 2. As in the A = 70 triplet also here the deviations from degeneracy due to the Coulombinteraction are considerably larger than the “spurious” ones in the charge symmetric case. For the total Fermi-transition strength for the Sr decays to the Rb states one obtains 1.940 and 1.948 for the Rb decays to the Kr states. Within half a percent thus the total strengths
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Table 2 The total (ST ) and analog (Sg−g ) Fermi β decay strengths of selected A = 74 nuclei for the “no-Coulomb” (H0 ) and “Coulomb” (H1 ) effective Hamiltonian. The “perturbative” (p) and the “variational” (v) approaches are described in the text Parent nucleus 74 Sr 74 Rb
p
H0
H1v
H1
ST
Sg−g
ST
Sg−g
ST
Sg−g
1.954 1.957
1.947 1.948
1.940 1.948
1.918 1.929
1.932 1.946
1.893 1.924
are again the same as obtained without the Coulomb-interaction. For the ground to ground transitions one obtains now 1.918 and 1.929, respectively. Finally, the results of using H1 already in the EXCITED VAMPIR variational calculations are presented in the 3 rightmost columns of Fig. 2. Again the effect of the non-perturbative (“variational”) treatment with respect to the “perturbative” treatment of the Coulomb-interaction on the spectra is of similar size that in the A = 70 triplet discussed above. In the variational treatment one obtains here 1.932 and 1.946 for the total Sr to Rb and Rb to Kr transition strength, respectively, while for the corresponding ground to ground transitions 1.893 and 1.924 do result. The discussed results are presented in Table 2. Estimating the errors in the same way as for the A = 70 triplet, one obtains a depletion of the ground to ground decay from Sr to Rb in between 1.3 and 2.7 percent of the sum rule strength and a non-analog branch with an upper limit of about 1 percent from the ground to the second excited state. For the Rb to Kr ground to ground decay the depletion is in between 0.2 and 1.2 percent. Here an upper limit of only 0.3 percent is obtained for the branch feeding the second excited state in Kr. Experimentally, the non-analog Fermi-decay to the first exited 0+ state in Kr has been investigated [8,10,11] and has turned out to be very weak in agreement with the result of our calculations.
4. Conclusions In the present paper we discussed results on the effect of isospin-mixing on superallowed Fermi β decay for the A = 70 and A = 74 isovector triplets with angular momentum and parity 0+ in the frame of the complex EXCITED VAMPIR approach. The size of the effects is consistent with those obtained with the shell-model reported in Refs. [1,4]. Only Coulomb effects have been considered and a possible charge symmetry breaking of the strong interaction has been neglected. Special effort has been made to study the isospin impurity induced by truncation of the configuration space and/or “missing” correlations in the configurations themselves. It was shown that these effects are definitely smaller than the isospin-mixing effects induced by the Coulomb interaction. However, even the effects of the latter on the superallowed Fermi transitions are rather small and thus difficult to describe by any microscopic many-body theory quantitatively. Small changes in the effective interaction and/or the size of the model space could yield considerable changes in the quantitative results. However, we think that (with a conservative estimation of the errors) we have obtained at least the rough magnitude of the effects to be expected. Furthermore, since it turned out that a large fraction of the depleted strength of the ground to ground
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transition can be attributed to particular non-analog decay branches at least in some cases (74 Sr → 74 Rb and 70 Kr → 70 Br), there is some hope for experimental detection. Whether microscopic calculations in medium heavy nuclei, however, can really help to check the conserved vector current hypothesis and to pin down the mixing between up and down quarks [1] is in our opinion rather doubtful. First of all, at least in model spaces being adequate for medium heavy nuclei, only approximate treatments of the nuclear many body problem (like, e.g., the variational methods applied here) are numerically feasible and hence the truncation problem will always remain even though the problem of “missing” couplings can be overcome by using the general instead of the complex EXCITED VAMPIR approach [19]. Second, there is no “perfect” effective interaction appropriate for the mass region under consideration. Thus there will always remain some quantitative uncertainty, which, unfortunately, may be larger than the effect under consideration. References [1] [2] [3] [4] [5] [6]
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Variational calculation of the effect of isospin-mixing on superallowed Fermi β decay in the A 70 mass region A. Petrovici a,b , K.W. Schmid b,∗ , O. Radu a , 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 19 November 2003; received in revised form 24 September 2004; accepted 1 October 2004 Available online 19 October 2004
Abstract We study the influence of Coulomb-induced isospin-mixing on the superallowed Fermi β decay using the complex EXCITED VAMPIR variational approach for the description of the lowest excited 0+ states in the two isovector triplets of nuclei 70 Se, 70 Br, 70 Kr and 74 Kr, 74 Rb, 74 Sr, respectively. As expected, the effects are small but nevertheless non-negligible. The calculated strength for the 74 Rb → 74 Kr Fermi β decay to the first excited 0+ state turns out to be very weak, in agreement with the experimentally given upper limit. However, particular non-analog Fermi branches to excited 0+ states with considerable strength are predicted to coexist with the superallowed decay in some of the presently investigated nuclei. This may give some hope for possible experimental detection. 2004 Elsevier B.V. All rights reserved. PACS: 21.10.-k; 23.40.-s; 27.50.+e Keywords: Nuclear structure; Shape coexistence; Proton-rich nuclei; Superallowed β decay; Isospin mixing
* Corresponding author.
E-mail address: [email protected] (K.W. Schmid). 0375-9474/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.10.002
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1. Introduction The exotic nuclei near the N = Z line in the A 70 mass region display some interesting nuclear structure effects like shape-coexistence and -mixing as well as the competition between proton–neutron and like-nucleon pairing. Furthermore, these nuclei play an important role in nuclear astrophysics, since their weak decay determines details of the nucleosynthesis. And last but not least, it has been argued that the investigation of superallowed Fermi β decays between analog states, which provide a test of the validity of the conserved vector current (CVC) hypothesis and the unitarily of the Cabibbo–Kobayashi– Maskawa (CKM) matrix (see [1] and the references therein), would be of particular interest in nuclei with A 62, where the charge induced isospin-mixing is expected to be large, and this argument has indeed caused considerable theoretical [1–5] as well as experimental [6–11] effort within recent years. Obviously, microscopic nuclear structure calculations for such medium heavy nuclei are extremely involved. The adequate model spaces are far too large to allow for a complete diagonalisation of an appropriate effective many-body Hamiltonian and thus one has to rely on suitable approximate methods. Furthermore, the appropriate effective Hamiltonian itself is not known a priori and can only be determined by an iterative process of many time-consuming calculations. Both, the limitation of the particular approximate method used as well as the insufficient knowledge of the appropriate Hamiltonian, will leave some uncertainties in the quantitative results especially if small effects are to be investigated. However, within the last decade there have been many rather successful theoretical investigations about the structure of nuclei in the A 70 mass region, not only of those along the valley of β-stability, but also in some exotic nuclei close to the proton drip line [12–18]. For these calculations the so-called complex EXCITED VAMPIR approach (see [19] for a recent review and references therein) has been used. 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, total angular momentum) are restored by projection techniques and the resulting symmetryprojected configurations are then used as test wave functions in chains of successive variational calculations to determine the underlying HFB transformations as well as the configuration mixing. It can be shown [19] that HFB vacua of the above type account for arbitrary two-nucleon correlations and thus simultaneously describe, e.g., like-nucleon as well as isovector and isoscalar proton–neutron pairing. In contrast to the general EXCITED VAMPIR approach, in which no symmetries at all are imposed on the underlying HFB transformations, in the complex EXCITED VAMPIR approach, because of the assumption of time-reversal and axial symmetry some particular four- and more-nucleon correlations are missing. Inspite of this limitation, however, the complex EXCITED VAMPIR has been proven to be a rather useful tool for nuclear structure calculations in large model spaces. This holds especially for the A 70 mass region. Here the effective interaction was adjusted in many different calculations. Phenomena like shape-coexistence and -mixing as well as the competition between proton–neutron and like nucleon-pairing turned out to be very important and various experimental observations in many nuclei could be nicely
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reproduced. Thus it may be worthwhile to apply the same methods for the investigation of the influence of Coulomb-induced isospin-mixing on superallowed Fermi β decay in the A = 70 and A = 74 isovector triplets of nuclei, even more so, since some of these nuclei have already been studied in some detail recently [12,14,18]. We shall briefly describe the various calculations performed and their ingrediences in the next section. In Section 3 then the results on Coulomb-induced isospin-mixing effects on the superallowed Fermi β decays in the two isovector triplets of nuclei 70 Se, 70 Br, 70 Kr and 74 Kr, 74 Rb, 74 Sr, respectively, will be discussed. Finally we shall present some conclusions in Section 4.
2. The calculations We calculated the lowest 0+ states in 70 Se, 70 Br, 70 Kr, 74 Kr, 74 Rb, and 74 Sr. For this purpose the complex version of the EXCITED VAMPIR approach (see Ref. [19]) was used. This method uses time-reversal invariant and axially symmetric Hartree–Fock–Bogoliubov (HFB) vacua constructed from essentially complex quasi-particle-transformations as basic building blocks and determines the underlying HFB-transformations by successive variational calculations after the restoration of angular momentum, nucleon-numbers and parity via projection methods. Finally, the residual interaction between the created configurations is diagonalized. For each nucleus of the A = 70 triplet the energetically lowest 13, for the A = 74 triplet the lowest 18 I π = 0+ configurations have been taken into account. It should be stressed here, that isospin is not considered as a good quantum number in this approach. Thus, even if a charge symmetric interaction is used, good total isospin can only be expected, if the configurations form a complete set under isospin-rotations. This is obviously not the case, if only the 13 or 18 lowest states are considered. Furthermore, as has been discussed in detail in [19], symmetry-projected configurations created by the complex EXCITED VAMPIR approach account for arbitrary two-nucleon correlations, however, because of the assumption of time-reversal and axial symmetry some four- and more-nucleon correlations are missing. Thus even if the number of configurations would be drastically increased, there is still some “spurious isospin impurity” to be expected (see also [20,21]). We shall come back to this problem later. As in our earlier calculations for nuclei in the A 70 mass region [18] we used a 40 Ca core and included 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. As corresponding single particle energies (in units of the oscillator energy h¯ ω = 41.2A−1/3) −0.015, −0.301, 0.215, −0.625, 0.118 and −0.007 for both protons and neutrons were taken. These are the averages of the slightly differing proton- and neutron-energies used in the earlier calculations. 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 a short-range (0.707 fm) Gaussian with strength −35 MeV in the T = 1 (proton–proton, neutron–neutron and proton–neutron) channel and a Gaussian of the same range with strength −180 MeV in the T = 0 channel in order to increase the pairingcorrelations in the corresponding channels. In addition the isoscalar interaction was modified by monopole shifts of −315 keV for all T = 0 matrix elements of the form 0g9/2 0f ;
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ˆ 9/20f ; I T = 0 with 0f denoting either the 0f5/2 or the 0f7/2 orbit and of I T = 0|G|0g ˆ −500 keV for those of the form 1p1d5/2 ; I T = 0|G|1p1d 5/2 ; I T = 0, where 1p denotes either the 1p1/2 or the 1p3/2 orbit. These shifts had been introduced in the earlier calculations in order to influence the deformation. We shall refer to this Hamiltonian, which has been applied rather successfully to many nuclei in the A = 70 mass region, as H0 in the following. Obviously, the above defined model space and effective Hamiltonian are charge symmetric. Thus (if the configuration space would be complete with respect to isospinrotations) one expects degenerate isovector excitation spectra for any triplet of nuclei with proton and neutron numbers (Z − 1, Z + 1), (Z, Z) and (Z + 1, Z − 1), respectively. Here Z is an odd number. Furthermore, one expects for the superallowed Fermi-transitions from the ground state of the (Z + 1, Z − 1)- to the ground state of the (Z, Z)-system and from the ground state of the (Z, Z)- to the ground state of the (Z − 1, Z + 1)-system both a total strength of two and (because of orthogonality) vanishing strengths for all the transitions from the ground state of the (Z + 1, Z − 1)- to all excited states of the (Z, Z)-system as well as from the ground state of the (Z, Z)- to all the excited states of the (Z − 1, Z + 1)nucleus. Deviations from these values as well as from the degeneracy of the spectra can then be attributed to isospin-mixing effects. In order to study those, the Coulomb-interaction has been included. For this purpose we need not only the two-body matrix elements in the model space but also the Coulombcontributions to the single particle energies resulting from the 40 Ca core. In order to determine the latter, we performed spherically symmetric Hartree–Fock calculations (with partial occupations of the unfilled j -shells) for the two doubly-even nuclei (Z − 1, Z + 1) and (Z + 1, Z − 1) of each triplet using the Gogny-interaction D1S and a 21 major-shell model space as described, e.g., in Ref. [19]. The contribution of the core to the proton single particle energies of the above valence orbits was calculated. The averages of the results for the (Z − 1, Z + 1) and (Z + 1, Z − 1) systems (which are almost identical) were then used for the full triplet of nuclei. We add the Coulomb-matrix elements and -single particle contributions to the charge-symmetric effective Hamiltonian defined above. The resulting Hamiltonian will be denoted by H1 in the following. Furthermore, the radial overlaps of the proton (21-major shell) Hartree–Fock wavefunctions of the (Z − 1, Z + 1)- and the neutron wave functions for the (Z + 1, Z − 1)-system and vice versa have been computed for the above defined valence orbits. The deviations of these overlaps from unity were in all cases less than half a percent. Obviously, inspite of the large basis, the tails of the wavefunctions may not be correctly described by the expansion in harmonic oscillators. This problem needs further investigation. In the present work these corrections have been neglected. Two different approaches have been considered: first, the Hamiltonian H1 was diagonalized for each considered nucleus within the complex EXCITED VAMPIR solutions obtained with the charge symmetric Hamiltonian H0 (“perturbative” approach). Second, the complex EXCITED VAMPIR calculations are repeated with H1 right from the beginning (“variational” approach). We would like to stress, that (because of the charge symmetry of H0 ) the isospin-mixing effects discussed in the next chapter are entirely due to the Coulomb-interaction. Possible effects due to charge symmetry breaking in the strong interaction (as included, e.g., in
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the Bonn CD-potential) are expected to be much smaller than those due to the Coulombinteraction. Thus no attempt has been made in the present investigation to include these additional effects.
3. Results and discussion 3.1. The A = 70 nuclei As already mentioned, for each of the nuclei in the A = 70 triplet 70 Se, 70 Br and the lowest 13 0+ configurations have been successively constructed by the complex EXCITED VAMPIR approach and finally the residual interaction between these configurations has been diagonalized. The first three columns of Fig. 1 present the resulting spectra, if the charge-symmetric Hamiltonian H0 is used. As can be seen the spectra of the two doubly-even nuclei are perfectly degenerate. The spectrum of the doubly-odd nucleus, however, displays slight deviations of the order of about 200 keV. These differences of the energies in the doubly-odd system with respect to the neighbouring doubly-even nuclei indicate (as discussed above) that the configuration space is not complete with respect to isospin rotations partly because of truncation, partly because of “missing” four- and morenucleon couplings in the EXCITED VAMPIR solutions. As expected, this deficiency is reflected in the strengths of the Fermi-transitions, too. For the total strength (ST ) of the transitions from the Kr ground state to all the calculated 13 0+ states of the Br nucleus we obtain 1.975 with 1.967 for the ground to ground transition (Sg−g ), while the total strength for the transitions from the Br ground state to the 13 lowest 0+ states in Se is 1.977 with again 1.967 for the ground to ground transition. Thus in between 1.3 ((2 − ST )/2) and 1.7 ((2 − Sg−g )/2) percent of the expected strength is not accounted for due to truncation and/or inherent isospin impurity effects. Now H1 is diagonalized within these solutions. The Coulomb interaction removes the degeneracy and nicely reproduces the correct energy differences between the ground states of the three nuclei. The corresponding relative excitation energies are displayed in the 3 middle columns of Fig. 1. As can be seen the deviations from degeneracy are here considerably larger than in the charge symmetric case. For the total Fermi-transition strength for the Kr ground state decays to the Br states one obtains 1.970 and 1.979 for the Br ground state decays to the Se states. Thus the total strengths are almost identical to those obtained without the Coulomb-interaction. For the ground to ground transitions one obtains now 1.935 and 1.967, respectively. Thus the effect of the perturbative treatment of the Coulomb-interaction (α = (Sg−g (H0 ) − Sg−g (H1 ))/2) is negligible for the Br to Se ground to ground transition while for the Kr to Br ground to ground transition a depletion of α = 1.6 percent is obtained. About one third of this depleted strength is concentrated in a single excited state. Finally, the results of using H1 already in the EXCITED VAMPIR variational calculations are presented in the 3 rightmost columns of Fig. 1. As can be seen the main effect of the non-perturbative (“variational”) treatment of the Coulomb-interaction is that it lowers some of the excited states (e.g., the second excited ones) by a considerable amount. In the variational treatment one obtains now only 1.946 and 1.959 for the total Kr to Br
70 Kr
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Fig. 1. The theoretical spectra of the lowest 10 0+ states in 70 Se, 70 Br and 70 Kr obtained within the complex EXCITED VAMPIR approach using the charge symmetric Hamiltonian H0 (“no Coulomb”) are compared to those resulting from perturbative (“perturbative”) and non-perturbative (“variational”) treatment of the Coulomb-interaction via the Hamiltonian H1 . Table 1 The total (ST ) and analog (Sg−g ) Fermi β decay strengths of selected A = 70 nuclei for the “no-Coulomb” (H0 ) and “Coulomb” (H1 ) effective Hamiltonian. The “perturbative” (p) and the “variational” (v) approaches are described in the text Parent nucleus 70 Kr 70 Br
p
H0
H1v
H1
ST
Sg−g
ST
Sg−g
ST
Sg−g
1.975 1.977
1.967 1.967
1.970 1.979
1.935 1.967
1.946 1.959
1.917 1.951
and Br to Se transition strength, respectively, while for the corresponding ground to ground transitions 1.917 and 1.951 are computed. A summary of the results is presented in Table 1. Estimating the error by summing the missing strengths between all and the analog transitions for the charge symmetric case (1 = (ST (H0 ) − Sg−g (H0 ))/2) and the missing total strengths between the variational calculations performed with H0 and H1 (2 = (ST (H0 ) − ST (H1 ))/2) one may conclude that the upper limit for the isospin mixing effect on the Br to Se ground to ground transition is about 0.8 percent with this strength distributed over many excited states, while for the Kr to Br ground to ground transition a depletion of at least 0.7 and at most 2.5 percent is obtained. In this latter case a nonanalog branch feeding the fourth excited 0+ state in Br with an upper limit of 0.7 percent is obtained.
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Fig. 2. Same as in Fig. 1, but for the A = 74 isovector triplet of nuclei.
3.2. The A = 74 nuclei For each of the nuclei in the A = 74 triplet 74 Kr, 74 Rb and 74 Sr the lowest 18 0+ configurations have been successively constructed by the complex EXCITED VAMPIR approach and finally the residual interaction between these configurations has been diagonalized. The first three columns of Fig. 2 present the resulting spectra, if the charge-symmetric Hamiltonian H0 is used. Again, as in the A = 70 triplet, the spectra of the two doubly-even nuclei are perfectly degenerate. The deviations in the spectrum of the doubly-odd nucleus are again of the order of about 200 keV. For the total strength of the transitions from the Sr ground state to all the calculated 18 0+ states in Rb we obtain here 1.954 with 1.947 for the ground to ground transition, while the total strength for the transitions from the Rb ground state to the 18 lowest 0+ states in Kr is 1.957 with 1.948 for the ground to ground transition. Thus here in between 2.3 and 2.7 percent of the strength is not accounted for due to truncation and/or inherent isospin impurity effects. Again the diagonalization of H1 removes the degeneracy and nicely reproduces the correct energy differences between the ground states of the three A = 74 nuclei, too. The corresponding relative excitation energies are presented in the 3 middle columns of Fig. 2. As in the A = 70 triplet also here the deviations from degeneracy due to the Coulombinteraction are considerably larger than the “spurious” ones in the charge symmetric case. For the total Fermi-transition strength for the Sr decays to the Rb states one obtains 1.940 and 1.948 for the Rb decays to the Kr states. Within half a percent thus the total strengths
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Table 2 The total (ST ) and analog (Sg−g ) Fermi β decay strengths of selected A = 74 nuclei for the “no-Coulomb” (H0 ) and “Coulomb” (H1 ) effective Hamiltonian. The “perturbative” (p) and the “variational” (v) approaches are described in the text Parent nucleus 74 Sr 74 Rb
p
H0
H1v
H1
ST
Sg−g
ST
Sg−g
ST
Sg−g
1.954 1.957
1.947 1.948
1.940 1.948
1.918 1.929
1.932 1.946
1.893 1.924
are again the same as obtained without the Coulomb-interaction. For the ground to ground transitions one obtains now 1.918 and 1.929, respectively. Finally, the results of using H1 already in the EXCITED VAMPIR variational calculations are presented in the 3 rightmost columns of Fig. 2. Again the effect of the non-perturbative (“variational”) treatment with respect to the “perturbative” treatment of the Coulomb-interaction on the spectra is of similar size that in the A = 70 triplet discussed above. In the variational treatment one obtains here 1.932 and 1.946 for the total Sr to Rb and Rb to Kr transition strength, respectively, while for the corresponding ground to ground transitions 1.893 and 1.924 do result. The discussed results are presented in Table 2. Estimating the errors in the same way as for the A = 70 triplet, one obtains a depletion of the ground to ground decay from Sr to Rb in between 1.3 and 2.7 percent of the sum rule strength and a non-analog branch with an upper limit of about 1 percent from the ground to the second excited state. For the Rb to Kr ground to ground decay the depletion is in between 0.2 and 1.2 percent. Here an upper limit of only 0.3 percent is obtained for the branch feeding the second excited state in Kr. Experimentally, the non-analog Fermi-decay to the first exited 0+ state in Kr has been investigated [8,10,11] and has turned out to be very weak in agreement with the result of our calculations.
4. Conclusions In the present paper we discussed results on the effect of isospin-mixing on superallowed Fermi β decay for the A = 70 and A = 74 isovector triplets with angular momentum and parity 0+ in the frame of the complex EXCITED VAMPIR approach. The size of the effects is consistent with those obtained with the shell-model reported in Refs. [1,4]. Only Coulomb effects have been considered and a possible charge symmetry breaking of the strong interaction has been neglected. Special effort has been made to study the isospin impurity induced by truncation of the configuration space and/or “missing” correlations in the configurations themselves. It was shown that these effects are definitely smaller than the isospin-mixing effects induced by the Coulomb interaction. However, even the effects of the latter on the superallowed Fermi transitions are rather small and thus difficult to describe by any microscopic many-body theory quantitatively. Small changes in the effective interaction and/or the size of the model space could yield considerable changes in the quantitative results. However, we think that (with a conservative estimation of the errors) we have obtained at least the rough magnitude of the effects to be expected. Furthermore, since it turned out that a large fraction of the depleted strength of the ground to ground
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transition can be attributed to particular non-analog decay branches at least in some cases (74 Sr → 74 Rb and 70 Kr → 70 Br), there is some hope for experimental detection. Whether microscopic calculations in medium heavy nuclei, however, can really help to check the conserved vector current hypothesis and to pin down the mixing between up and down quarks [1] is in our opinion rather doubtful. First of all, at least in model spaces being adequate for medium heavy nuclei, only approximate treatments of the nuclear many body problem (like, e.g., the variational methods applied here) are numerically feasible and hence the truncation problem will always remain even though the problem of “missing” couplings can be overcome by using the general instead of the complex EXCITED VAMPIR approach [19]. Second, there is no “perfect” effective interaction appropriate for the mass region under consideration. Thus there will always remain some quantitative uncertainty, which, unfortunately, may be larger than the effect under consideration. References [1] [2] [3] [4] [5] [6]
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