Low collectivity of the first 2+ states of 212,210Po

The lifetimes of the first 2+ excited states of 212,210Po were measured in two transfer reactions 208Pb(12C,8Be)212Po and 208Pb(12C,10Be)210Po by the Recoil Distance Doppler Shift (RDDS) method and by the Doppler Shift Attenuation method (DSAM), respectively. The derived absolute B(E2) values of 2.6(3) W.u. for 212Po and 1.83(28) W.u. for 210Po indicate low collectivity. It is shown that the properties of the yrast 21+, 41+, 61+ and 81+ states in both nuclei cannot be described consistently in the framework of nuclear shell models. It is also demonstrated in the case of 210Po that Quasi-particle Phonon Model (QPM) calculations cannot overcome this problem thus indicating the existence of a peculiarity which is neglected in both theoretical approaches.


Introduction
Understanding nuclear structure in terms of single-particle motion and collective behaviour is one of the challenges of nuclear physics. The nuclear shell model provides the basic framework for understanding of the single-particle motion in nuclei [1].The collective behaviour is understood as a result of coherent movement of valence nucleons caused by the residual interaction [2]. In this regard, properties of open-shell nuclei in the immediate vicinity of double-magic cores are of particular importance. Such nuclei can often be understood well within the framework of the shell models and at the same time their valence particles can induce the onset of collective behaviour. With two and four valence nucleons in respect to the double-magic core 208 Pb the nuclei 210 Po and 212 Po provide a good testing ground for studying the beginning of the evolution from single-particle to collective motion in the mass A ≈ 208 region.
In a recent study, it has been shown that a phenomenological single-j shell model accounts very well for the energies of the low-lying states of 212 Po (cf. Fig. 3 in Ref. [3]). In addition, Auerbach and Talmi [4] have suggested that the yrast sequence 2 + 1 − 4 + 1 − 6 + 1 − 8 + 1 of 212 Po follows a seniority-like energy pattern resulting in an isomeric 8 + 1 state. All of this raises the question whether shell models can describe consistently the electromagnetic properties of the 2 + 1 -4 + 1 -6 + 1 -8 + 1 sequence as well. To address these questions experimental information on the absolute B(E2) strengths for the 4 + 1 → 2 + 1 and 2 + 1 → 0 + 1 transitions is needed. In the case of 210 Po, data on its absolute strengths for E2 transitions between its yrast states are available [5,6]. Large-scale shell-model studies of 210 Po using realistic interactions [7,8] well reproduce the energies of the yrast 2 + , 4 + , 6 + , and 8 + levels. The E2 strengths for the transitions between the 4 + , 6 + and 8 + states are also almost perfectly reproduced (cf. Table  VII in Ref. [7] and Table III in Ref. [8]). However, in both studies [7,8] the B(E2; 2 + 1 → 0 + 1 ) value is overestimated by a factor of six. Such a significant discrepancy between the shell-model calculations and the data is an indication for either an inaccurate experimental value [7,8] or for deficiencies in the model, as suggested in Ref. [8].
All of this has motivated us to perform two experiments especially designed to measure the lifetimes of the 2 + 1 states of 212,210 Po. Both experiments were performed at the FN Tandem facility at the University of Cologne, Germany.
2. Lifetime measurement of the first 2 + state of 212 Po 2.1. Experimental set-up The lifetime of the 2 + 1 state of 212 Po was measured by utilizing the RDDS method [9,10]. The excited states of 212 Po were populated using the α-transfer reaction 208 Pb( 12 C, 8 Be) 212 Po. The target consisted of a 0.6 mg/cm 2 thin layer of Pb evaporated on a 2 mg/cm 2 thick Au backing foil and was placed with the Au facing the beam. The beam energy of 64 MeV was chosen in such a way that the energy at which the reaction takes place after the Au backing to be about ∼ 62 MeV. The reaction was induced in the reaction chamber of the Cologne coincidence plunger device [11]. The stopper was a self-supporting 2 mg/cm 2 thick Au foil. Data were taken at six plunger distances: 25µm, 35µm, 43µm, 55µm, 70µm and 100µm.
For detecting the light reaction fragments six solar cells (10 mm × 10 mm) were used. The array of solar cells was mounted in the plunger chamber at backward angles with respect to the beam axis, covering an angular range between 116.8 • and 167.2 • . The solar cells were placed at a distance of about 15 mm between their centres and the target. The γ rays from the decay of the excited states of 212 Po were registered by 11 HPGe detectors mounted outside the plunger chamber in two rings at distance of, on average, 12 cm from the target. Five detectors were positioned at backward angles (142 • with respect to the beam axis) and the other six detectors were placed at forward angles (45 • with respect to the beam axis). Data were taken in coincidence mode of at least one solar cell and one HPGe detector (particle-γ) or when at least two HPGe detectors (γ-γ) were in coincidence.

Data analysis and results
The particle-γ coincidence data were sorted in twelve matrices depending on the positions of the HPGe detectors and the plunger distances. A particle projection of the particle-γ matrix is shown in Fig. 1(a) as an example. The γ rays in coincidence with the group of particles indicated as " 212 Po & 200 Tl" in Fig. 1(a) are shown in Fig. 1(b). This spectrum is dominated by transitions from excited states of 200 Tl which is produced by the 197 Au( 12 C,2αn) transfer reaction in the backing or in the stopper. However, the 727-, 405-, and 223-keV lines which are the γ-ray transitions depopulating the first three yrast states of 212 Po [12,13,3] are also clearly The RDDS data for this transition was analysed by utilizing the Differential Decay Curve Method (DDCM) [15,16]. The standard application of DDCM requires the I sh γ and I un γ components (for each distance) to be measured from spectra in coincidence with Doppler-shifted components of transitions that feed directly the excited state of interest. Then the lifetime τ i of the level of interest for the i-th target-to-stopper distance depends on I sh and I un in the simple way [15,16]: as here the derivative of the Doppler shifted intensities as a function of the target-to-stopper distance, d dx I sh , is determined by a piecewise polynomial fit to the measured intensities I sh . The presented particle-gated γ-ray spectra in Fig. 2(a) are, in fact, γ-ray singles spectra which, in principal, only contain information for the effective lifetime of the 2 + 1 state of 212 Po. Therefore, the intensities of the I sh and I un derived from these spectra cannot be used directly for extracting the lifetime of the 2 + 1 state by means of the DDCM. Special care has to be taken into account for analyzing the impact of the transitions directly feeding the 2 + 1 state on the I sh and I un components of the 727-keV transition (see the inset in Fig. 1(b)).
The lifetimes of the 2 + 2,3 states of 212 Po were measured in our previous study [3] to be below 1 ps which means that they contribute only to the fast feeding of the 2 + 1 state. The lifetime of the 3 − state at 1537 keV is not known and in order to simplify the discussion at this moment we assume that its lifetime is sufficiently short so that it decays only in flight. Under this assumption the only essential feeder to the 2 + 1 state remains the 405-keV transition which depopulates the 4 + 1 state of 212 Po. It is expected that the 4 + 1 state has a long lifetime of about 140 ps, or longer [3]. Indeed, as can be seen from the insets in Fig. 2 Fig. 2(a)). Under the considerations above, it is obvious that the intensities of the shifted components of the 727-keV transition being directly determined from the particle-gated spectra are also related only to the lifetime of the 2 + 1 state of 212 Po. In this respect, both I un γ and I sh γ can be considered as effectively derived from γ-ray spectra in coincidence with the shifted components of all transitions directly feeding the state of interest. Therefore, they can be used directly in Eq. (1).
To proceed with the DDCM analysis the mean velocity of the recoiling nuclei v has to be known. This value was experimentally determined from the centroids of the shifted and the unshifted components of 727-keV transition to be v =0.72(5)%c. The DDCM fits using v =0.72(5)%c and intensities (I un γ and I sh γ ) extracted with the procedure described above are shown in Fig. 2(b). These fits result in a weighted mean value for the lifetime of the 2 + 1 state of 21.8 (19) ps.
It has to be noted that the only assumption in the derivation of the above result which is not directly supported by experimental observations, is that the feeding from the 3 − 1 state is fast. To investigate the influence of this feeding to the lifetime of the 2 + 1 state further, we have also considered the alternative limit, i.e. we have assumed that the feeding from the 3 − 1 state is very slow and contributes only to the unshifted component of the 727-keV transition. In this case, besides the intensity of the 405-keV transition, the intensity of the unshifted component of the 727-keV transition has to be reduced by additional 10% which accounts for the intensity of the 810-keV transition (3 − 1 → 2 + 1 , cf. the inset of Fig. 1(b)). This alternative approach reduces the deduced lifetime of the 2 + 1 to 19.2 (18) ps. For the final value for the lifetime of the 2 + 1 state we conservatively adopt the average value between the two limits which is: τ (2 + 1 , E x = 727 keV) = 20.5(26) ps. Taking into account the known electron conversion coefficient for the 2 + 1 → 0 + 1 transition of 212 Po [12] and α-branching ratio [13], the newly derived lifetime of the 2 + 1 state translates to absolute transition strength B(E2; 2 + 1 → 0 + 1 ) = 193(24) e 2 fm 4 = 2.6(3) Wu.
3. Lifetime measurement of the first 2 + state of 210 Po 3.1. Experimental set-up For measuring the lifetime of the first 2 + state of 210 Po by means of DSAM (cf. Ref. [9] and references therein) the 208 Pb( 12 C, 10 Be) 210 Po transfer reaction was used. The target was a selfsupporting 10 mg/cm 2 thick Pb foil enriched to 99.14 % with the isotope 208 Pb. Besides the placement of the ring at forward angles at 35 • , the experiment was performed with the same detectors' set-up as the previous one.

Data analysis and results
The particle-γ coincidence data were sorted in two matrices depending on the position of the HPGe detectors. A projection of the particle-γ matrix obtained with γ-ray detection at 142.3 • is shown in Fig. 3(a). The γ rays in coincidence with the group of particles indicated as " 210 Po" in Fig. 3(a) are shown in Fig. 3(b). This spectrum is dominated by the 1181-keV and the 245-keV lines which are the γ-ray transitions depopulating the first two yrast states of 210 Po [17]. Besides some contaminants from 211 Po (which are shown in purple), all other γ rays originate from the decay of excited states of 210 Po. The 1181-keV γ-ray line shows Doppler shape which allow us to extract the lifetime of the 2 + 1 state of 210 Po (cf. Fig 4).  line-shape fits of the 1181-keV (2 + 1 → 0 + 1 ) transition observed at forward (a) and at backward (b) angles. The solid (red) line represents the total fit. The 1181-keV line is fitted simultaneously with the 245-keV (4 + 1 → 2 + 1 ) line which is always emitted from a stopped nucleus (the insets). The dotted and dashed lines represent the individual contributions of 1181-keV (red) and 245-keV (purple) lines, respectively, to the total fit. An unidentified stopped contaminant with E γ = 1174 keV is taken into account (brown). The line-shape analysis was performed with the integrated software package APCAD (Analysis Program for Continuous Angle DSAM) [19]. Details about the used approach can be found in Ref. [3] where the software was verified.
The lifetime of the first excited 2 + state of 210 Po was obtained from the line shape of the 1181-keV (2 + 1 → 0 + 1 ) transition. The feeding history of the state (see the inset in Fig. 3(b)) was accounted. A special care was taken to account for the impact of the 245-keV (4 + 1 → 2 + 1 ) transition. The 4 + 1 state of 210 Po is a long-lived state with lifetime τ = 2.21(10) ns [5]. Consequently, it always decays at rest in the present experiment. Indeed, as can be seen from the insets in Fig. 4 the 245-keV γ-ray line show no indication of Doppler-shifted components in its shape. Hence, when the 2 + 1 state is fed from the 4 + 1 → 2 + 1 transition it also always decays at rest which gives extra counts into the fully stopped component of the 1181-keV transition. In order to extract correctly the lifetime of the first 2 + state of 210 Po by means of the Dopplershift attenuation method, the contribution of the γ rays coming from the 245-keV transition to the fully stopped component of the 1181-keV transition has to be eliminated. That procedure could be automatically carried out with APCAD by simultaneously fitting the 1181-and the 245-keV lines. Under this assumption, the final value of the lifetime of the 2 + 1 is extracted to be 2.6(4) ps. Taking into account the known electron conversion coefficient for the 2 + 1 → 0 + 1 transition of 210 Po [20], the revised lifetime of the 2 + 1 state translates to absolute transition strength B(E2; 2 + 1 → 0 + 1 ) = 136(21) e 2 fm 4 = 1.83(28) Wu.

Discussion
The derived absolute B(E2) values between the first 2 + and ground states in both ( 212,210 Po) nuclei indicate very low collectivity. In fact, the revised value for 210 Po is a factor of three times larger than the adopted one but still two times smaller than the calculated one in the framework of the single-j shell model [14]. In the case of 212 Po, the obtained value is more than a factor of two times smaller than the calculated one in the framework of the single-j shell model [3]. The results from the single-j shell-model calculations for 210,212 Po and 210 Pb (for completeness) are presented in Table 1. The labelling of the columns reflects the approach in choosing the effective charges. In the case of SM1-gh, the effective proton and neutron charges in the E2 transition operator were determined from the measured B(E2; 8 + 1 → 6 + 1 ) values for 210 Pb and 210 Po [20] This approach yields effective charges of e ν = 1.04e and e π = 1.52e. Another approach is to determine the effective charges from the measured B(E2; 2 + 1 → 0 + 1 ) values for 210 Pb and 210 Po which leads to effective charges of e ν =0.83e and e π =1.09e. The results from these calculations are presented in the column labelled as SM2-     Table 2 and the details about these calculations are presented in [14]. The realistic shell model (SM) reproduces almost perfectly the energies of the yrast states in 210 Pb and 210 Po. However, in both cases the description of the B(E2) values is only marginally improved with respect to the ones obtained in the single-j shell-model calculation (SM2-gh).
It is also interesting to check whether the problem is specific for shell models only. For this purpose we have performed Quasi-particle Phonon Model (QPM) calculations [21] for 210 Po. The results are presented in Table 2 and the details about these calculations can be found in [18]. The energies of the states of interest are reasonably well reproduced. It has to be noted that in the chosen approach to fix the strength parameters to the electric strengths of the 2 + 1 state, the result for the energies of the states should be considered as a prediction of the model. The major discrepancy between the QPM calculations and the experimental data appears in the E2 transition strengths for the cascade 8 + 1 → 6 + 1 → 4 + 1 . Overall, the model underestimates these values by a factor of 8. The problem existing in the shell model description also appears in a different form in the present QPM calculations.

Summary
In the present study we have measured the lifetimes of the 2 + 1 states of 212,210 Po. The derived absolute B(E2) values indicate low collectivity in the structure of these states. No consistent description of the properties of the yrast 2 + 1 , 4 + 1 , 6 + 1 and 8 + 1 states in both nuclei is observed in the framework of nuclear shell models. The additional QPM calculations which have been done for the nucleus 210 Po show that the problem existing in the shell model description also appears in a different form in the QPM calculations. More thorough theoretical investigations of this problem are needed.