Pairing-excitation versus intruder states in Ni and Zr
A discussion on the nature of the states in Ni (, ) is presented and a comparison is made with its valence counterpart Zr (, ). Evidence is given for a proton intruder state at only MeV excitation energy in Ni, while the analogous neutron intruder states in Zr reside at keV and keV. The application of a shell-model description of intruder states reveals that many pair-scattered neutrons across have to be involved to explain the low excitation energy of the proton-intruder configuration in Ni.
pacs:21.10.-k, 21.60.Cs, 27.50.+e, 27.60.+j
The nucleus Ni was initially considered as a semi-magic nucleus arising from a major proton-shell closure and a neutron subshell closure. This interpretation was inferred from the high energy of the first-excited state ( keV Broda et al. (1995)) in contrast with the low energy of the first-excited state ( keV Bernas et al. (1982)). Conflicting observations arose, however, as mass measurements do not reveal a clear neutron shell gap at Rahaman et al. (2007); Guénaut et al. (2007) and the mean value of W.u. Sorlin et al. (2002); Bree et al. (2008) is too large for a pronounced subshell gap Langanke et al. (2003).
Currently, it is qualitatively understood that the apparent semi-magic properties of Ni are not caused by a strong subshell closure and a corresponding large energy gap, but rather follow from the parity change between the shell and the orbital across , prohibiting quadrupole excitations Grawe et al. (2001). The value is explained by strong pair scattering across Sorlin et al. (2002), which indicates that the stabilizing effect is subtle.
Despite these qualitative insights, the structure of Ni and the region around is not yet fully understood. While the focus was, so far, mainly on neutron excitations across , little is known about proton excitations across . Although separation energies give evidence for a major shell closure at , a proton two-particle-two-hole (2p-2h) state could appear nonetheless at lower excitation energies due to pairing correlations and proton-neutron - residual interactions Heyde et al. (1983, 1987); Wood et al. (1992). Its excitation energy will depend critically, however, on the stabilizing properties of the gap as the quadrupole part of the - interaction depends on the number of valence neutron particles or holes.
Since the valence counterpart of Ni, Zr (, ), is a stable isotope, it has been investigated in numerous transfer reactions and thus its structure is better known than the one of Ni. In the present paper, the low-energy structures of both nuclei, and the states in particular, are compared based on experimental information available in the literature (see Fig. 1). While most properties are similar in Ni and Zr, possible (2p-2h) excitations in Ni will behave different from the (2p-2h) excitations in Zr. In the following, a candidate for a (2p-2h) is discussed on the basis of a shell-model approach of intruder states Heyde et al. (1987), after which implications for the stabilizing properties of the gaps are discussed.
Ii Low-energy structure of Ni and Zr
ii.1 The active valence nucleons
At low excitation energies, the Ni and Zr valence nucleons (neutrons and protons, respectively) are expected to be predominantly active in the and space, and to a smaller degree in the and space. The energy difference between the and orbitals constitutes the energy gap in Ni and Zr, respectively.
In a simplified picture, the ground state of Ni and Zr is expected to exhibit a character, while excited states could be created by promoting a nucleon pair from the or to the orbital. It has been observed that the state in Ni and Zr feature remarkable similarities. Their respective excitation energies of and keV are almost identical, as well as their respective monopole (E) transition strengths of and Kibédi and R. H. Spear (2005).
Using spectroscopic factors from transfer reactions Cates et al. (1969), and mean-square radii and determined from the Zr(,)Y reaction Warwick et al. (1979), the measured (E) transition strength in Zr can be reproduced with a simple two-component model allowing for strong () and () configuration mixing Wood et al. (1999). This gives substantial evidence for a strongly mixed ground state and excited state. Recent shell-model calculations confirm these observations Sieja et al. (2009a, b). Although the similar (E) transition strength in Ni is not understood, the similar excitation energy suggests comparable and configurations, involving now the neutrons.
The state arising from has not been identified in Ni nor in Zr. The state, which has been observed in Ni at keV Mueller et al. (2000), might be a possible candidate, although such a state is not observed in Zr in spite of the more extensive spectroscopic information.
The , , , and levels in Ni at respective excitation energies of , , , and keV are good candidates for the seniority levels. In Zr, a similar structure is observed with the respective seniority levels at excitation energies of , , , and keV.
ii.2 Intruders across the or gap
The excitation energy of 2p-2h intruder states in nuclei at a major closed shell (or ) can be estimated from summing the ()(2p-1h) and ()(1p-2h) intruder excitation energies in the () and () nuclei Van Duppen et al. (1984) (see Ref. Heyde et al. (1988) for details). Using this prescription, the excitation energies of, e.g., (2p-2h) states in lead and tin nuclei are generally reproduced within keV.
In Zr, it is shown by the Zr(p,t) reaction Ball (1972) that the (1p-2h) configuration is mainly distributed over two states at excitation energies of and keV NND (). The Sr(,n) Glenn et al. (1971), (,), and Zr(,) reactions Blok et al. (1976) show that the major fraction of the (2p-1h) configuration in Zr resides in the -keV state NND (). By using the above mentioned prescription and averaging the excitation energies of the two (1p-2h) Zr levels, an expected excitation energy of keV for the (2p-2h) state in Zr can be deduced. The situation is depicted by the dashed lines in Fig. 1.
It has been shown by a Zr(,) reaction Ball et al. (1971) that the (2p-2h) configuration is mainly concentrated in the states at and keV excitation energy. The average of both excitation energies is keV, which differs only by keV from the estimate.
The same reasoning can be applied to Ni. The (2p-1h) character of the -keV level in Cu is suggested by a large spectroscopic factor in the Zn(,He) reaction Zeidman and J.A. Nolen, Jr (1978) and a small B(E2) transition strength to the Cu ground state observed in Coulomb excitation Stefanescu et al. (2008). From a recent Fe -decay study Pauwels et al. (2008), the (1p-2h) state in Co was identified at keV, giving rise to an estimated excitation energy of the (2p-2h) state in Ni at only keV.
A good candidate for a (2p-2h) configuration would be the state in Ni, which is also a possible candidate for a state. From the presently available experimental data, however, it is not possible to differentiate between the two possible configurations. Although extremely challenging, future transfer and multi-Coulomb-excitation experiments can deliver crucial information to investigate this state and other low-energy levels in Ni.
In spite of their very similar excitation spectrum, there is thus a large difference in excitation energy of the 2p-2h intruder states across or in respectively Ni and Zr. The possible reasons for this difference will now be investigated.
Iii Shell-model description for 2p-2h states in Ni and Zr
The excitation energies in Ni and Zr suggest nearly identic structures of their and states. On the other hand, the summed excitation energy of the (1p-2h) and (2p-1h) levels in Co and Cu, respectively, is very different from the (2p-2h) excitation energy in Zr. The shell-model approach of Ref. Heyde et al. (1987) provides a quantitative description of (2p-2h) and (2p-2h) states, which can explain this apparent paradox in Ni and Zr.
iii.1 Framework and results
Intruder states result from particle-hole excitations across major closed shells. Nevertheless, they appear at low excitation energy because of both strong pairing and - correlations. For the 2p-2h intruder states, this is expressed Heyde et al. (1987) as
where is the excitation energy of the intruder state, the single-particle shell-gap energy with the respective subscripts and denoting particles and holes, the nucleon pairing energy, and the - residual-interaction energy.
The shell-gap and pairing energies for Ni and Zr are deduced from measured one- and two-nucleon separation energies Audi et al. (2003); Rahaman et al. (2007) (see Ref. Heyde et al. (1987) for details). Starting from the experimental 2p-2h excitation energies, the respective - residual energies can be extracted, using equation 1. These values are listed in Table 1. It is important to note that the values in the table are subject to mixing, and no transfer data are known for Ni. The excitation energy of the Zr (2p-2h) configuration, e.g., is taken as the average of the - and -keV levels, which are strongly populated in the Zr(,) reaction Ball et al. (1971).
|Ni||2202111Estimate from summing (2p-1h) and (1p-2h) excitation energies.||5270(320)||4500(700)||-3838(1000)|
The extracted - residual-interaction energy mainly results from quadrupole correlations. Fig. 2 shows a schematic representation of the quadrupole - energy Heyde et al. (1987) as a function of neutron (proton) number between the closed shells at and assuming two extreme cases: is a closed (dashed lines) and open (full line) shell configuration. In the latter case, the contribution of quadrupole correlations is strongest around , and intruder states are expected lowest in excitation energy. On the other hand, if represents a shell closure, the contribution of quadrupole correlations becomes negligible around , and pairing-excitation states at high excitation energy might be observed.
The (2p-2h) and (2p-2h) states in Ni reside at respective excitation energies of keV and MeV, which are rather similar, even though the shell gap is about MeV larger than the subshell gap. For both gaps, a large gain in pairing energy ( and keV, respectively) exists. For , it fully explains the low (2p-2h) excitation energy. The low excitation energy of the (2p-2h) state, on the other hand, requires a strong gain in binding energy from the - residual interactions ( MeV). This means that many valence neutrons must be available, i.e., tends to behave rather as an open shell configuration, as given by the full line in Fig. 2.
As noticed already, the (2p-2h) state in Zr appears at a remarkably similar excitation energy to the (2p-2h) state in Ni. Table 1 reveals, however, that the larger shell-gap energies at compared to is mainly compensated by a stronger gain in pairing energy. So, although both excitation energies are almost identical, the situations at and are different. Like the (2p-2h) state in Zr, the low excitation energy of the (2p-2h) state in Ni is explained by the gain in pairing energy, which is consistent with a good shell closure.
The (2p-2h) configuration in Zr is centered at a significantly higher excitation energy ( keV) than the (2p-2h) state in Ni ( MeV), despite a -MeV smaller shell gap and similar pairing energy. This implies a much weaker - residual interactions in the (2p-2h) states of Zr: the average excitation energy of keV is consistent with essentially no - residual interaction. In contrast to in the nickel isotopes, behaves as a closed shell configuration, as depicted in Fig. 2 by the dashed lines.
It can be seen from Table 1 that the open and closed character as observed in the properties of the and subshell is caused by a stronger pair scattering of neutrons across than protons across : at , the pairing energy is about MeV larger than the shell gap, while at , this amounts only to about MeV. Moreover, the difference in pairing energies compensates the difference in unperturbed shell-gap energies giving rise to almost identical excitation energies of the states in Ni and Zr.
In Cu, the (2p-1h) levels are identified at and keV, respectively, based on the particle-core model Oros-Peusquens and Mantica (2000) and the small B(E2) transition strength Stefanescu et al. (2008). This is keV lower in excitation energy with respect to the (2p-1h) state in Cu. Extrapolating this trend to the nickel and cobalt isotopes, means that the intruder configuration might reside at even lower excitation energies in Ni and become even the ground state in Co.
The Ni and Zr low-energy structures have been compared in the framework of 2p-2h configurations across the , and , (sub)shell gaps. The discussion was triggered by recent experimental data obtained in Cu Stefanescu et al. (2008) and Co Pauwels et al. (2008). Strong similarities are observed between the two valence counterparts, but also important differences. The states in the respective nuclei feature almost identical excitation energies and monopole (E) transition strengths. Based on the summing prescription of the (1p-2h) and (2p-1h) levels in Co and Cu, respectively, the (2p-2h) state in Ni is estimated at only -MeV excitation energy, while the (2p-2h) state in Zr is centered around keV.
In an attempt to understand the origin of this difference in Ni and Zr, the shell-model description of intruder states Heyde et al. (1987) has been applied. It shows that the excitation energies of the states in Ni and Zr are similar in spite of the difference in the unperturbed single-particle shell-gap energies, as it is compensated to a large extent by the difference in pairing energies. Moreover, it shows that stronger neutron pair scattering in Ni gives rise to more active valence neutrons, which strongly interact with proton excitations across . As a result, the (2p-2h) state in Ni is strongly pushed down by - residual interactions by as much as MeV, while the (2p-2h) state in Zr is hardly affected by - residual interactions. These findings highlight the fact that neutron pair scattering across around Ni is far more important than proton pair scattering across around Zr.
Acknowledgements.This work was supported by FWO-Vlaanderen (Belgium), GOA/2004/03 (BOF-K.U.Leuven), the Interuniversity Attraction Poles Programme – Belgian State – Belgian Science Policy (BriX network P6/23), the European Commission within the Sixth Framework Programme through I3-EURONS (contract no. RII3-CT-2004- 506065).
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