Competition between Allowed and First-Forbidden β Decay: The Case of 208 Hg → 208 Tl

The β decay of 208 Hg into the one-proton hole, one neutron-particle 20881 Tl 127 nucleus was investigated at CERN-ISOLDE. Shell-model calculations describe well the level scheme deduced, validating the proton-neutron interactions used, with implications for the whole of the N > 126 , Z < 82 quadrant of neutron-rich nuclei. While both negative and positive parity states with spin 0 and 1 are expected within the Q β window, only three negative parity states are populated directly in the β decay. The data provide a unique test of the competition between allowed Gamow-Teller and Fermi, and first-forbidden β decays, essential for the understanding of the nucleosynthesis of heavy nuclei in the rapid neutron capture process. Furthermore, the observation of the parity changing 0 þ → 0 − β decay where the daughter state is core excited is unique, and can provide information on mesonic corrections of effective operators.

the abundances of nuclei with magic neutron numbers are enhanced. Thus the A ∼ 195 r-process abundance peak is the consequence of the N ¼ 126 neutron shell closure. β-decay half-lives are basic nuclear physics input in r-process calculations. While half-lives of a large number of fission products were recently measured [4,5], the N ¼ 126 r-process path nuclei are experimentally unreachable [6,7], and we have to rely on theoretical calculations. In this context the nuclei in the N ¼ 126 region are of particular interest [8] because first-forbidden (FF) β decays successfully compete [9][10][11][12][13][14][15] with allowed Gamow-Teller (GT) and Fermi decays, and this impacts on the calculations of r-process nucleosynthesis abundances [16]. However, the calculation of FF β decay is notoriously difficult and subject to debate.
An ideal nucleus to study the competition between FF and allowed β decay should have a small number of both negative and positive parity levels below the Q β value, with simple and well-understood wave functions. The β decay of 208 Hg into the one-proton-hole, one-neutron-particle 208 Tl nucleus with a Q β ¼ 3. 48 (3) MeV [17] provides this ideal testing ground. Q β is low due to the vicinity of the stability line, and the wave functions are simple due to the small number of valence nucleons outside the doubly magic 208 Pb core. Furthermore, states with spin I ¼ 0 and I ¼ 1 of both positive and negative parities are available by combining a neutron above the N ¼ 126 core with a proton hole below Z ¼ 82. In addition, 208 Tl, with one proton hole and one neutron outside 208 Pb, provides directly the neutron-proton two-body matrix elements for the shell-model calculations [18]. Therefore the understanding of excited states in 208 Tl is essential for the successful prediction of properties of nuclei in the little studied N>126, Z<82 region [19].In this region excited states were observed in only a handful of nuclei: 208 Tl [20] and 209 Tl [21][22][23], and more recently, with the advent of radioactive-beam facilities, γ-ray spectroscopy following internal decays provided information on the yrast structures of 208 Hg [24], 209 Tl [24], and 210 Hg [25]. Single-neutron states in 207 Hg [26] and γ-ray transitions in 211;213 Tl [27] were also identified. In this Letter we present results from the β decay of 208 Hg into 208 Tl, providing information on the competition between allowed and first-forbidden β decay and validating the proton-neutron interaction "south-east" of 208 Pb.
Experimental details.-Experiments to measure the β decay of 208 Hg to 208 Tl were performed at the ISOLDE decay station (IDS) at CERN. 208 Hg nuclei were produced by impinging 1.4 GeV protons on a molten lead target. These were extracted using a FEBIAD VADIS ion source [28], accelerated to 30-50 keV, mass selected with a dipole magnet, and finally implanted in a tape at the IDS. Two experiments were performed [29] in 2014 and 2016. In both cases the IDS consisted of plastic scintillation detectors for β-particle detection surrounding the implantation point, and five composite Ge detectors for γ-ray measurements. In 2014 four Clover Ge and one Miniball Cluster detector were used, while in 2016 five clover Ge detectors were employed. The γ-ray detection efficiency at 1 MeV was 8% and 4% in 2014 and 2016. The β-particle detection efficiencies were ∼30% in 2014 and ∼85% (with a ∼4π detector placed in the vacuum chamber) in 2016. Data were time stamped to a precision of 10 ns and recorded using a triggerless data acquisition system. Correlations between detectors were made in software using the GRAIN software package [30].
Results.-The rate of 208 Hg delivery to the implantation position was ∼5 and 25 Hz in 2014 and 2016, respectively. Better statistics on the 208 Hg → 208 Tl decay were obtained in 2016, although the data were dominated by the β decays of 208 At to 208 Po [31]. The 2014 data were cleaner, so the spectra presented here are from the 2014 measurement, while the intensities and lifetimes are from 2016.γ-ray spectra, with and without β coincidence requirements, are shown on Fig. 1. By selecting the well-established 453 keV PHYSICAL REVIEW LETTERS 125, 192501 (2020) γ ray in 208 Tl [20], the coincidence spectrum presented in Fig. 1(c) is clean, showing the 72 and 82 keV K α and K β thallium x rays and a small number of γ rays. Based on γ − γ coincidences the level scheme of 208 Tl was obtained and it is given in Fig. 2. The properties of the γ-ray transitions assigned to 208 Tl are listed in Table I.
The 208 Tl level scheme deduced is in agreement with the one obtained following the α decay of 212 Bi [20]. The spin parities of the 5 þ ground state, the 4 þ 40, the (4 þ ) 473, and the ð3Þ þ 493 keV excited states are adopted from these works. The rest of the level scheme is new. We do not confirm the level scheme presented following a previous β-decay study, where 208 Hg was produced in multinucleon reactions followed by chemical separation [33,34].
Almost all of the γ rays assigned to 208 Tl are in prompt coincidence with β particles. The exceptions are the 221 and 1314 keV transitions originating from the excited state at 1807 keV. By examining the β-γ time spectrum, shown in Fig. 3, a lifetime of T 1=2 ¼ 1.3ð1Þ μs was obtained for the 1807 keV level.
By comparing the intensity of the 453 keV M1 transition with that of the 157 keV in the 936 keV gated spectrum, the electron conversion coefficient of the 157 keV line was obtained as α ¼ 2.1ð3Þ. This proves its M1 character [α theor ðM1Þ¼2.38 [35] ]. Consequently, the excited states at 1429 and 1586 keV have the same parity. In addition, the sum (γ plus conversion electron) intensity of the 221, 1314, 374, and 533 keV transitions cannot be larger than the total intensity reaching the ground state. This requires that the electron conversion coefficients of all the above listed γ rays are small. This rules out M1 character for the 221 keV transition [α theor ðM1Þ¼0.92 [35] ]. E2 is favored by the isomerism as E1 yields the extreme hindrance of 10 −8 W:u:.
Two rather different values have been published for the lifetime of the 208 Hg ground state. T 1=2 ¼ 41 þ5 −4 min was reported from the earlier mentioned β-decay measurements [36,37], but recently a much shorter value of 132(50) s was published following a projectile fragmentation experiment [7,38]. We measured the half-life by implanting 208 Hg nuclei for a fixed time, than observing their decay. After removing the activity with the tape system, we repeated this sequence several times. In our first experiment in 2014 we aimed to be sensitive to 30 min lifetime. The nonobservation of the 208 Hg decay indicated that its lifetime is much shorter than 30 min. In 2016, we measured the half-life 157.0(4) 48 (11) 1), respectively. This restricts the spin of these states to 0 or 1, based on the log ft systematics of Ref. [39]. Lower-lying levels need to have increasing spins as they decay towards the 5 þ ground state. Ultimately the spin parities of the three states fed directly in β decay are assigned by comparison with shell-model calculations.
Discussion.-In order to understand the structure of 208 Tl, shell model calculations have been performed, using the OXBASH [40] code. Level energies and transition rates were calculated in the πð0g 7=2 ; 1d; 2s 1=2 ; 0h 11=2 Þ νð0i 11=2 ; 1g; 2d; 3s 1=2 ; 0j 15=2 Þ model space using the Kuo-Herling interaction [41] for ππ and νν and H7B [42] for πν. For 208 Tl this reduces to H7B only. The calculations were done in particle-particle mode relative to a hypothetical 132 Sn core. In an extended model space neutron particlehole (ph) excitations from the 2p 1=2 and 2p 3=2 orbits across the N ¼ 126 shell closure were considered to account for the 2p2h content in the ð0-1Þ − states and γ-ray transitions between them. The other transitions were calculated in the valence space only. The inclusion of the core breaking excitation is needed in order to account for the 1=2 − state with νp −1 1=2 νg 2 9=2 configuration at the relatively low energy of 2149 keV in 209 Pb [23]. γ-decay transition rates were calculated using effective operators e π ¼ 1.5e, e ν ¼ 0.85e for E2 transitions and g s ¼ 0.7g free s for M1 transitions. No E1 transitions are allowed in this model space.
The calculations indicate that the only possible isomer at ≈2 MeV excitation energy is the 0 − state with νp −1 1=2 g 2 9=2 πs −1 1=2 character. Accordingly we associate this with the 1807 keV level. The other member of the multiplet is predicted to be the lowest lying 1 − state, therefore we assign this to the 1960 keV state. These assignments are supported by a comparison to the 206 Tl ground state and 305 keVexcited state with hh configuration νp There is good agreement between the experimental and theoretical level schemes (see Fig. 4). The small discrepancy for negative parity states is due to omission of the octupole phonon coupled to the low-lying positive-parity states, which by mixing would lower the energies of the yrast states. The highly retarded transition strengths of the γ rays depopulating the 0 − isomer are also well reproduced. The experimental transition strengths BðE2Þ¼6.8ð12Þ × 10 −3 and BðE3Þ¼ 11.5ð20Þ × 10 −3 W:u: for the 221 and 1314 keV transitions, respectively, compare well with the theoretical values of BðE2Þ¼1.8 × 10 −3 and BðE3Þ¼7.9 × 10 −3 W:u: The experimental branching ratios are in good agreement with the theoretical values, as shown in Table I While we assign negative parity to all three states directly populated by β decay, we examined other scenarios. In particular we looked into the possibility that the directly populated states are of positive parity 0 þ with νs 1=2 πs −1 1=2 configuration or 1 þ with νs 1=2 πs −1 1=2 , νs 1=2 πd −1 3=2 , and νd 5=2 πd −1 3=2 (see Fig. 4). None of the scenarios, with the exception of that presented in Fig. 2 is compatible with the shell model calculations.
We now examine the β decay of 208 Hg. First-forbidden β decays populate negative parity states in 208 Tl. These correspond to νg 9=2 → πh 11=2 and νi 11=2 → πh 11=2 decays. The measured log ft values in the range of 5.2-6.0 are in line with those observed for first-forbidden decays in this mass region [20,46]. We refrain from a shell model calculation of rank L ¼ 0,1FF 0 þ → ð0; 1Þ − transitions as a consistent treatment requires full inclusion of Δl ¼ 1 πν orbitals in 1p1h core excitations, i.e., 2p2h states in 208 Tl [47,48]. We note that the shell-model calculations available for the β decay of N ¼ 126 nuclei [9,10] do not consider neutrons above the 126 shell closure, therefore cannot provide reliable estimates for the decay of 208 Hg.
Positive parity states could be populated by the allowed β decay. For neutrons above N ¼ 126 there are neither allowed Gamow-Teller (GT) nor Fermi transitions to protons below Z ¼ 82. This is because the number of nodes between the corresponding orbitals change (Δn ¼ 1), and hence the orthogonality of the radial wave functions zeroes the GT matrix element as its operator contains no radial dependence [49,50]. Besides, the Fermi operator does not act on the radial part of the wave functions. For N>126 the only allowed transition is the GT νi 11=2 → πi 13=2 , which requires at least 1p1h proton core excitations. For N<126 the allowed GT/Fermi transitions are νh 9=2 → πðh 11=2 ;h 9=2 Þ, νðf 5=2 ;f 7=2 Þ → πðf 5=2 ;f 7=2 Þ, νi 13=2 → πi 13=2 , and νðp 3=2 ;p 1=2 Þ → πðp 3=2 ;p 1=2 Þ. The νh 9=2 → πh 9=2 GT decay would populate 1p1h neutron core excitations at E x ∼ 8 MeV, higher than the aforementioned νi 11=2 → πi 13=2 . All others need 2p2h core excitations, consequently lying even higher in energy, well outside the β-decay window and beyond the neutron separation energy, therefore they cannot be populated in β decay. The several 1 þ and 0 þ states expected in 208 Tl to lie below the Q β value (see Fig. 4) could be mixed with the lowest same spin GT resonances. Specifically, π −2 πi 13=2 νi 11=2 states, if mixed with the lower lying 1 þ and 0 þ configurations, could act as a "doorway" to populate them in (weak) GT, while their γ decay will proceed via their main configurations. The experimental log ft values would correspond to an estimated 1% core-excited admixture in the wave functions. The inclusion of only oneparticle one-hole proton or neutron core excitation into the shell model calculations does not provide the required mixing for these decays. However, calculations including one-particle one-hole core excitations for both protons and neutrons are not feasible for 208 Tl.
In summary, the β decay of 208 Hg was studied. The level scheme of the single proton-hole single neutron-particle 208 Tl nucleus was established, providing the first direct test of the proton-neutron residual interaction in the N>126, Z<82 quadrant. 208 Hg provides a unique testing ground of the competition between allowed and first-forbidden β decay. However, with a half-life of 135(10) s, it populates directly only negative parity states via first-forbidden decays. The strongest branch establishes a 0 þ → 0 − decay to a core excited daughter state. This is the first such β decay observed and provides information on meson corrections of effective operators. The present data provide important constraints on theoretical models addressing the competition between allowed and first-forbidden β decays, important for the detailed understanding of the nucleosynthesis of heavy r-process elements.