Collective 2+1 excitations in 206Po and 208,210Rn

In the present study, B(E2;21+→01+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(E2; 2^{+}_{1}\rightarrow 0^{+}_{1})$\end{document} values have been measured in the 208,210Rn and 206Po nuclei through Coulomb excitation of re-accelerated radioactive beams in inverse kinematics at CERN-ISOLDE. These nuclei have been proposed to lie in, or at the boundary of the region where the seniority scheme should persist. However, contributions from collective excitations are likely to be present when moving away from the N=126 closed shell. Such an effect is confirmed by the observed increased collectivity of the 21+→01+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2^{+}_{1}\rightarrow 0^{+}_{1}$\end{document} transitions. Experimental results have been interpreted with the aid of theoretical studies carried out within the BCS-based QRPA framework.


Introduction
One of the most fundamental concepts in nuclear structure are the magic numbers that are defined by the shell structure [1]. The tendency of like nucleons to pair to I π = 0 + drives nuclei with magic proton and neutron numbers to a more bound state than their immediate neighbours. The nuclei with magic proton and/or neutron number, such as the N = 126 isotones, can be reasonably well described with the nuclear shell model.
One of the successful models based on the excitations of unpaired nucleons to the shell-model orbitals is the seniority scheme [2]. If the valence nucleons reside at rela email: tuomas.grahn@jyu.fi b Present address: Argonne National Laboratory, USA atively high-j orbitals (j ≥ 7/2), the seniority ν, which is the number of unpaired nucleons, can be regarded as a good quantum number. In the even-mass N = 126 isotones with Z ≥ 82 the valence protons occupy the 1h 9/2 single-particle orbital. Indeed, the energies of low-lying levels up to I π = 8 + 1 can be well described as π1h 9/2 ν = 2 structures within the generalized seniority scheme [3]. The isomeric nature of the 8 + 1 states is characterized by the low energy of the 8 + 1 → 6 + 1 transitions. In the generalized seniority scheme these transitions, as well as all transitions in the ground-state band down to the 4 + 1 → 2 + ity changing since the 0 + ground states are ν = 0 states and, therefore, the B(E2; 2 + 1 → 0 + 1 ) values should follow a different trend compared to the seniority-conserving transitions. Hence, in the generalized seniority scheme the B(E2) values of the seniority-changing transitions are at their maximum at the mid-j shell.
However, when adding or removing protons/neutrons from a closed shell configuration, the particles/holes in the open shell start to interact via the quadrupole part of the residual interaction. Such a contribution is first observed for the low-spin states. This effect is clearly observed in the low-spin level-energy systematics as presented in Fig. 1 for the polonium isotopes. Immediately outside the N = 126 shell closure the energies of the 2 + 1 states decrease, followed by the 4 + 1 states when moving toward the lighter nuclei. The 2 + 1 level energies remain remarkably constant until the intruder states start to set in at around N = 114 when moving closer to the neutron mid shell at N = 104 [5,6].
In Ref. [7] the structure of neutron-deficient trans-Pb nuclei close to N = 126 has been discussed in terms of the seniority structures. However, as concluded in Ref. [8], the generalised seniority scheme with realistic interactions is inadequate to describe open-shell nuclei. In Ref. [9] simple shell-model calculations have been carried out for 208 Rn, in which two competing 4 + states were predicted, one originating from a proton ν = 2 multiplet and one from a neutron-hole configuration. In addition, the 2 + 1 states were assigned as neutron-hole excitations. In fact, the 4 + 2 states have been observed in Po (horizontal lines in Fig. 1) and Rn nuclei (see, e.g., Ref. [9] for 208 Rn) nuclei close to N = 126 as predicted by calculations [9].
In order to investigate the nature and collectivity of the 2 + 1 states near the N = 126 shell closure in 206 Po and 208,210 Rn, Coulomb-excitation measurements of radioactive ion beams in inverse kinematics were carried out at CERN-ISOLDE. In the present paper, the results and their impact on the understanding of the nuclear structure of the low-spin states near the N = 126 and Z = 82 closed shells are described.

Experiments
The radioactive 206 Po and 208,210 Rn nuclei were produced at CERN-ISOLDE [10] by bombarding an uranium carbide (UC X ) primary target with 1.4 GeV protons delivered by the PS Booster. The 206 Po beam was in fact extracted from the ISOLDE target after the proton irradiation stopped as the half-life of 206 Po is 8.8 days, which allowed sufficient yield for the present experiment from 206 Po activity accumulated during the previous irradiations of the primary target. Polonium atoms were ionized using the resonant-ionisation laser ion source (RILIS) [11] and mass selected by the ISOLDE High Resolution Separator (HRS). Radon atoms produced on-line at ISOLDE were ionized in the plasma ion source with the cooled Ta transfer line, and subsequently mass selected with the ISOLDE General Purpose Separator (GPS). After mass selection the 206 Po and 208,210 Rn nuclei were injected into the REX-ISOLDE post-accelerator complex [12] consisting of the REX-TRAP penning trap, the REX-EBIS charge breeder and the REX linear accelerator. REX-TRAP was used to cool, bunch, and purify the beams. The beam bunches were injected into the REX-EBIS charge breeder that matched the mass-to-charge (A/q) ratio of the ions to be suitable for post-acceleration. The REX linear postaccelerator delivered 2.85 MeV/u and 2.82 MeV/u 206 Po and 208,210 Rn beams, respectively, to the target position of the MINIBALL γ-ray spectrometer [13]. The beam energies were well below the limit to fulfill the criterion of "safe" Coulomb excitation [14]. The radioactive ion-beam yields at the MINIBALL target position were ∼ 5.6 · 10 5 pps ( 208 Rn 50+ ) and ∼ 2.1 · 10 5 pps ( 210 Rn 51+ ). The initial yield of 206 Po 49+ was ∼ 5.6 · 10 5 pps which decreased over the course of the measurement since 206 Po was extracted from the ion source without proton irradiation.
Coulomb excitation was performed using 2 mg/cm 2 thick 104 Pd and 114 Cd targets, respectively. The targets were chosen in a way that the excitation energy of the 2 + 1 state is not overlapping with the 2 + 1 state energies in the projectile nuclei, and is lower than the corresponding energies in the nuclei of interest in order to minimize the γ-ray background arising from the Compton scattering events. In addition, reaction kinematics allowed the separation of the target and projectile nuclei with the chosen targets. The MINIBALL γ-ray spectrometer consists of eight triple-cluster Ge detectors arranged in a close geometry around the target chamber. The present set up had a total photopeak efficiency of 7% for 1.3 MeV γ rays. MINIBALL was used to detect the γ rays de-exciting the states under investigation.
Both scattered projectiles and target recoils were detected using an annular double-sided silicon strip detector (CD) with 16 annular strips positioned downstream of the target. The present radioactive beams were found to be close to 100% pure by measuring the γ-ray spectra with the RILIS laser set on and off in the case of 206 Po. When the RILIS laser was set off, virtually no events in the particle-gated γ-ray spectra were observed. Since RILIS was not employed in the extraction of the 208,210 Rn beams, the same method could not be applied. Instead, the beam composition was measured using the ionization chamber downstream of the target and by using β-decay data from the beam-dump Ge detector. Events corresponding only to the Rn nuclei of interest were observed.
Identification of the beam and target nuclei detected in the CD was possible since both the scattering angle and the deposited energy of the recoiling particles were measured. A coincidence condition of exactly two particles in the CD (scattered beam and target recoil) and at least one γ-ray in MINIBALL was imposed. Figure 2 shows the spectrum of particle energy deposited in the CD as a function of scattering angle in the laboratory coordinates. In Fig. 4 γ-ray spectra respective to the three angular ranges (high, middle, low) are shown.
The γ rays were recorded in coincidence with the twoparticle events (i.e. the scattered beam and the target recoil) observed in the CD. As the reaction kinematics can be reconstructed from the angular and energy information of the events recorded with the CD, and the γ-ray detection angle is known, event-by-event Doppler correction for the γ-ray energies can be applied, as described in detail in Ref. [13]. The background subtraction of the γ-ray spectra were carried out by subtracting the number of γ rays gated by the random events in the spectrum of time differences between the γ-ray events observed by MINIBALL and the particle events recorded by the CD detector. The subtracted background was normalized by the widths of the time windows. A spectrum of the time difference between the events observed in MINIBALL and the CD in the 208 Rn experiment is shown in Fig. 3. The procedure for the background subtraction was similar for the other two experiments. Sample γ-ray energy spectra, gated by the two-particle events observed in the CD with the three centre-of-mass scattering intervals as shown by Fig. 2c) is shown in Fig. 4.
The γ-ray energy spectra in coincidence with the twoparticle events in the CD were constructed similarly for 210 Rn and 206 Po. The sample spectra are shown in Figs. 5 and 6, respectively.
The transition probabilities were extracted from the measured γ-ray intensities according to the Coulomb-excitation theory. In order to extract matrix elements in the 206 Po and 208,210 Rn nuclei, the measured γ-ray intensities have to be converted to absolute Coulomb-excitation cross sections. The latter requires normalization to the excitation of the target nuclei with the known electromagnetic matrix elements. The Coulomb-excitation γ-ray intensities were extracted from the event-by-event Doppler-corrected γ-ray energy spectra as shown in Figs. 4, 5, and 6.
The data for each nuclei were subdivided into three independent groups, as illustrated by Fig. 2, each subdivision corresponding to low, middle and high scattering angles of the target nuclei observed with the CD. The angular ranges in the centre-of-mass coordinates are given in Table 1. The γ-ray intensity data, together with the known matrix elements of the target nuclei, were used as input for the Gosia2 Coulomb-excitation code [14]. The literature data used in the Coulomb-excitation analysis are shown in Table 2. Note that the 2 + 1 → 4 + 1 excitation is not observed in the data, but it is necessary to be included it in the Gosia2 analysis as a so-called buffer state. This does not have influence on the results as such but it is needed in the Gosia2 analysis as discussed in Ref. [15]. Some of the preliminary results of the present data were shown in Ref. [17].
In the Gosia2 analysis both the target and the projectile 2 + 1 excitations are treated simultaneously by minimizing the χ 2 function in parallel. In this way, the measured projectile γ-ray intensities, shown in Table 3, can be converted to absolute excitation cross sections using the measured target γ-ray intensities and known literature data, as described in detail in Ref. [15]. Two unknown matrix elements are needed to describe excitation of the projectile nucleus. Therefore, a two-dimensional χ 2 surface, in which the χ 2 value is plotted as a function of the transition (ME 02 ) and diagonal (ME 22 ) matrix elements was used in the analysis to find a global minimum corresponding to the solutions for both matrix elements. In Figs. 7, 8, and 9 such two-dimensional surfaces are shown for each nucleus of interest. The condition χ 2 < χ 2 min + 1 is applied to the graphs that represent the 1σ error bars of the resulting matrix-element values. The final results can therefore be extracted from the global minimum of χ 2 and are listed in Table 4. There is a strong correlation between the two matrix elements involved in the excitation process and therefore a subdivision of data and the present analysis technique are necessary to extract the B(E2) values [15]. The B(E2) values are given for the depopulating 2 + 1 → 0 + 1 transition, according to the relation B(E2; 2 + → 0 + ) = 1 5 × 0 + ||M (E2)||2 + 2 (see Eq. 3-31 in Ref. [16]).

Theoretical investigations
A theory designed to describe collective excitations in spherical open-shell even-even nuclei is the quasiparticle random-phase approximation (QRPA) [21,22]. The QRPA approach is based on the Bardeen-Cooper-Schrieffer (BCS) quasiparticles that are obtained by solving the BCS equations of motion within a chosen single-particle model space [22]. In the present calculations the single-particle space consisted of 12 proton states (0g 9/2 , 0g 7/2 , 1d 5/2 , 0h 11/2 , 1d 3/2 , 2s 1/2 , 0h 9/2 , 1f 7/2 , 0i 13/2 , 2p 3/2 , 1f 5/2 , 2p 1/2 , in ascending order of energy) and 13 neutron states (0h 9/2 ,1f 7/2 , 0i 13/2 , 2p 3/2 ,1f 5/2 , 2p 1/2 ,1g 9/2 , 0i 11/2 , 0j 15/2 , 2d 5/2 , 3s 1/2 , 1g 7/2 , 2d 3/2 , in ascending order of energy). These singleparticle spaces were chosen such that the respective proton and neutron Fermi energies were well contained inside the model space. The single-particle energies were obtained from a Coulomb-corrected Woods-Saxon potential with the fitted parametrization taken from Ref. [16]. The adopted two-nucleon interaction was the Bonn-A oneboson exchange potential transferred to nuclear matter by 1 Table 3. Efficiency corrected γ-ray intensities measured in coincidence with two events in the CD for the projectile and target nuclei. Low, middle and high refer to the angular ranges given in Table 1 the G-matrix techniques [23]. The two-nucleon potential was adapted to finite nuclei by a simple parametrization [24,25] where the pairing monopole matrix elements were scaled by one parameter for the protons and another one for the neutrons. These parameters were fixed by adjusting the lowest quasiparticle energies to the empirical pairing gaps obtained from the tabulated separation energies for protons and neutrons [26]. After solving the BCS equations the two-quasiparticle combinations were formed. These served as the building blocks for the QRPA matrices that were diagonalized in a standard way [22]. The particle-hole part of the twonucleon interaction was adjusted to reproduce the experimental energy of the 2 + 1 state in the discussed 208,210 Rn and 206 Po nuclei in the way described in Refs. [24,25]. This adjustment guarantees the optimum collectivity of the 2 + 1 state from the theory point of view. After the adjustment the wave function of the 2 + 1 state is a coherent combination of two-quasiparticle pairs, a characteristic feature of a collective wave function [22]. The wave function obtained in this way can be used to produce a theoretical estimate of the reduced transition probability B(E2) of an electric quadrupole transition from the 2 + 1 state to the ground state in a way as described in Ref. [22]. The B(E2; 2 + 1 → 0 + 1 ) value can be expressed in Weisskopf units (W.u.) [22] and it depends on the adopted effective charges for protons and neutrons. In the present calculations the bare charges, i.e. 1e for protons and 0e for neutrons were chosen. The corresponding computed results (B(E2; 2 + 1 → 0 + 1 ) values and diagonal matrix elements) are listed in the two last columns of Table 4.  Fig. 3. Time difference histogram of γ rays observed in MINI-BALL (tγ) and particles in the CD (tp) in the 208 Rn experiment. The solid red arrow marks the prompt coincidence time window and the dashed arrows show the random coincidence windows that are used for the background subtraction of the γ-ray energy spectra.

Discussion
Earlier studies of the transition probabilities in the N = 122 isotones have suggested that the seniority structures might persist in these isotopes even beyond the closed shells [27]. This was based on the hindered transition probability of the 8 + 1 → 6 + 1 transition. In Fig. 10 the partial level-energy systematics are plotted for the N = 122 isotones. Indeed, the low excitation-energy difference of the 8 + 1 and 6 + 1 states suggests a proton multiplet-type structure. However, the evolution of the 2 + 1 state energy suggests the opposite as it goes down in energy as a function of A. This is regarded as a sign of increasing collectivity.
In Fig. 11 the experimental B(E2) values for the 2 + 1 → 0 + 1 and 8 + 1 → 6 + 1 transitions in the N = 122 isotopes have been plotted. The argument for the seniority structure arises from the hindered 8 + 1 + → 6 + 1 transition probabilities at mid-j = 9/2 proton sub shell. Such an evolution is indeed clearly visible and may indicate seniority structure at high spin. However, at low spin and in particular for the 2 + 1 states the structure is different. The measured B(E2) values indicate moderate collectivity that sets in immediately when moving away from the closed proton shell.
The experimental and theoretical B(E2; 2 + 1 → 0 + 1 ) values extracted in the present study for the N = 122 isotones under investigation have been plotted in Fig. 12 as a function of the proton number. The values follow a typical pattern when filling a sub shell. The B(E2) value is at the minimum at the closed proton shell nucleus 204 Pb and increases towards the middle of the π0h 9/2 sub shell. In these isotones the neutrons are occupying mainly the 1f 5/2 sub shell and protons mainly the 0h 9/2 sub shell immediately after 204 Pb. These sub-shell orbitals have ∆n = 1, ∆ = 2 and ∆j = 2 being quadrupole partners and therefore proton-neutron interaction is moderately strong (see, e.g., [29]). . Gamma-ray energy spectrum following Coulomb excitation of the 208 Rn beam impinging on the 114 Cd target, in coincidence with the two-particle events in the CD such that the target recoil is detected in the (a) high, (b) middle and (c) low angular range (cf. Fig. 2a), b) and c)). The event-by-event Doppler correction is performed for the target recoils (red) and for the scattered beam (blue). The plotted surface is the region where χ 2 < χ 2 min + 1 corresponding to the 1σ error bars [15]. The χ 2 scale is given on right.
In Fig. 13 the present experimental and theoretical B(E2; 2 + 1 → 0 + 1 ) values are plotted for the radon nuclei as a function of the mass number A. The theoretical B(E2) value for the closed neutron-shell nucleus 212 Rn is very similar to that for the closed proton-shell nucleus 206 Pb. Immediately, when removing neutrons from the closed N = 126 2p 1/2 sub shell and entering into the 1f 5/2 sub shell, the proton-neutron interaction starts to generate collectivity and thus increase the B(E2) values. Evidence for this behaviour are from the high B(E2) values, both experimental and theoretical, of the lighter Rn isotopes. 1 208 Rn and 210 Ra extracted in the present work apart from the experimental value for 204 Pb (Z = 82), which has been taken from Ref. [28]. Some of the values are offset of their actual Z location for the clarity of the presentation.

Diagonal matrix elements
In the present work, only single-step excitations to the 2 + 1 states were observed both for the projectile and the target nuclei. The analysis method applied, described in detail in Ref. [15], in principle allows one to extract also the diagonal matrix elements of the 2 + 1 state. Since no additional data such as mean lifetime values exist for the nuclei under investigation, the present experiments are not very sensitive to the diagonal matrix elements, which results in large error bars as shown in Figs. 7, 8, and 9. On the basis of the vicinity of the closed neutron and protons shells at N = 126 and Z = 82 one could argue that the quadrupole moments of the 2 + 1 states would be close to zero. The present theoretical calculations suggest considerable collectivity of the 2 + 1 → 0 + 1 transitions, which indeed is observed, and, therefore, the zero quadrupole moments of the 2 + 1 states seem unjustified. The present Coulomb-excitation measurements, to a certain extent, suggest that the nuclei under investigation have non-vanishing quadrupole moments of the 2 + 1 state (see Table 4). In case of 208 Rn the quadrupole moment can be fixed to a positive value within 1σ confidence. However, in order to confirm this observation, complementary data are needed.

Conclusions
In the present work the Coulomb-excitation measurements of the radioactive 206 Po, 208,210 Rn beams in inverse kinematics have been carried out at CERN-ISOLDE. In addition, theoretical studies within the BCS-based QRPA calculations have been performed in this region of the nuclear chart. The data reveal the increased collectivity of the 2 + 1 → 0 + 1 transitions in these nuclei. Such an observation is also reproduced by the theory. While the higher-spin level patterns, namely the 8 + 1 state, suggest the presence of the seniority structures, the 2 + 1 state has a dominantly collective character which is likely to originate from an increase in proton-neutron interaction as the spatial overlap of the wave function gets stronger when entering the 1f 5/2 and 0h 9/2 sub shells for the neutrons and protons, respectively.