β-decay half-lives and β-delayed neutron emission probabilities for several isotopes

Caballero-Folch, R.; Domingo-Pardo, C.; Agramunt, J.; Algora, A.; Ameil, F.; Ayyad, Y.; Benlliure, J.; Bowry, M.; Calviño, F.; Cano-Ott, D.; Cortès, G.; Davinson, T.; Dillmann, I.; Estrade, A.; Evdokimov, A.; Faestermann, T.; Farinon, F.; Galaviz, D.; García, A.R.; Geissel, H.; Gelletly, W.; Gernhäuser, R.; Gómez-Hornillos, M.B.; Guerrero, C.; Heil, M.; Hinke, C.; Knöbel, R.; Kojouharov, I.; Kurcewicz, J.; Kurz, N.; Litvinov, Yu.A.; Maier, L.; Marganiec, J.; Marta, M.; Martínez, T.; Montes, F.; Mukha, I.; Napoli, D.R.; Nociforo, C.; Paradela, C.; Pietri, S.; Podolyák, Zs.; Prochazka, A.; Rice, S.; Riego, A.; Rubio, B.; Schaffner, H.; Scheidenberger, Ch.; Smith, K.; Sokol, E.


I. INTRODUCTION
Very-neutron-rich nuclei may emit one or more neutrons when they disintegrate via β decay. This is the so-called βdelayed neutron (βn) emission process, which is energetically allowed when the Q β value of the decay exceeds the neutron separation energy S n of the daughter nucleus. The βn emission has been experimentally determined for about 230 neutron-rich nuclei, spanning from 8 He up to 150 La [1][2][3][4]. Most of these measurements took advantage of the large fission yields around the two fission peaks at A ∼ 95 and A ∼ 138 in thermal neutron-induced fission. However, β-delayed neutron emission has remained essentially inaccessible for nuclei heavier than A = 150, where only a minuscule value of 0.007% has been reported for the βn-emission probability of 210 Tl [5,6]. Because of the scarce or nonexistent βn data in the heavy-mass region, rapid neutron capture r-process [7] calculations have to rely entirely on theoretical models [8][9][10]. However, the performance of such models for reproducing the features of the β decay in r-process waiting-point nuclei has been tested with experimental data only for the two shell closures at N = 50 and N = 82 from measurements such as those reported in Refs. [11][12][13][14][15][16][17]. The comparison is much more limited in the N = 126 region, where only half-lives in the neighborhood of the doubly magic 208 Pb were available (see, e.g., Ref. [18]).
The neutron-rich nuclei "south" of 208 Pb are difficult to study experimentally because of the very small production cross sections and the large background induced by the heavy primary beam. In the present work it was possible to produce and identify reliably secondary neutron-rich nuclei in the region "south-east" of 208 Pb in the chart of nuclides thanks to the high-energy (1 GeV/u) 238 U beams available at the GSI facility. Extended motivation and results for this experiment were recently published in Ref. [19]. Here we present more details on the experimental apparatus in Sec. II, the analysis methodology and results, which are reported in Sec. III and, finally, Secs. IV and V summarize the main results and conclusions.

II. EXPERIMENTAL SETUP AND ION IDENTIFICATION
The present measurements were carried out at the GSI Helmholtz Center for Heavy Ion Research. A 238 U beam was accelerated to an energy of 1 GeV/u by the UNILAC linear accelerator coupled to the SIS 18 synchrotron. The average beam intensity was 2 × 10 9 ions/spill, with a pulsed beam structure of 1 s spill duration (SIS extraction) and a repetition cycle of 4 s. The beam impinged on a 9 Be target with a thickness of 1.6 g/cm 2 at the entrance of the fragment separator (FRS) [20]. The selection of the ions of interest, from this point to the detection system, was done with the Bρ-E-Bρ method by using the FRS as an achromatic spectrometer. Neutron-rich nuclei "south-east" of 208 Pb in the chart of nuclides were produced by using two FRS settings centered on 211 Hg and 215 Tl. To minimize the number of ionic charge states of the secondary beam a niobium layer with a thickness of 223 mg/cm 2 was placed behind the Be target. In addition, a homogeneous Al degrader with a thickness of 2.5 g/cm 2 was placed at the first FRS focal plane (S1) in combination with thick Cu slits, which served to reduce the contribution of fission fragments and primary beam charge states with magnetic rigidity Bρ initially similar to that of the setting isotope. A wedge-shaped Al degrader with a thickness of 3874 mg/cm 2 was employed as achromatic degrader at the intermediate focal plane S2 (see Fig. 1).
Ion species were identified by means of standard FRS tracking detectors. Two plastic scintillators located at S2 and at the final focal plane S4 were used to measure the time of flight (t TOF ) of the ions. Two systems of time projection chambers (TPCs) [21] placed at S4 and S2 allowed us to determine accurate Bρ values for each ion by measuring their trajectory with respect to the central fragment. The measured t TOF in combination with the Bρ provided the necessary information to calculate the mass-to-charge ratio (A/q) on an event-by-event basis. The resolution thus obtained in A/q was 2.5‰ full width at half maximum (FWHM).
To determine the atomic number Z, two fast multiple sampling ionization chambers (MUSICs) [22] were placed in the S4 experimental area. Although the detected nuclei were mainly bare, H-and He-like charge states were also detected in the MUSICs. These events were corrected by combining the information of the two MUSICs and calculating the energy loss in the S2 degrader following the method applied in previous studies in this mass region [23][24][25][26]. In addition, it was needed to treat the gain fluctuations in the MUSIC detectors caused by the variations of the temperature in the experimental hall, and consequent changes in gas pressure, which were also corrected numerically [27]. The final resolution obtained in Z for the Pb-Bi region was of 6‰ FWHM. Finally, the Z versus A/q particle identification diagram (PID) was experimentally validated with a dedicated run. 205 Bi ions were implanted into a passive plastic stopper and the decays of well-known isomeric transitions [28] were measured with HPGe detectors.
The PID obtained, including nuclei produced in both FRS settings with all the statistics accumulated during the  experiment, is shown in Fig. 2. All identified nuclei were already reported in previous experiments [29,30].

A. Implantation and decay-detection system
The secondary beam of neutron-rich isotopes was focused at the final focal plane S4 of the FRS. A third Al degrader with variable thickness was adjusted in order to slow down the ions of interest and to implant them into the central region of an active stopper named "silicon implantation beta absorber" (SIMBA) [31,32]. SIMBA enabled position and energy measurement of heavy charged ions as well as charged particles from α and β decays. The main difference between the SIMBA system employed here and previous versions was the smaller number of Si detectors required to stop and implant the ions in the present experiment, which had a higher atomic number (Z ∼ 82) than those measured in the past (Z ∼ 50, [31]). In addition, the geometry and overall size of SIMBA were also modified in order to optimize neutron detection (see FIG. 3. Picture of SIMBA without its cylindrical coverage of 11.5 cm diameter. below). The present SIMBA version consisted of a stack of nine highly segmented Si detectors (see Figs. 3 and 4).
The first two layers, called XY -tracking detectors, consisted of single-sided Si-strip detectors (SSSDs) with their strips in orthogonal orientation with respect to each other. They were used for determining accurately the ion position in the transversal plane of the beam. A center-of-gravity method applied to the charge shared over all the strips allowed us to determine the ion position with an accuracy of ±1 mm 2 [27,34], corresponding to one pixel in a silicon layer of SIMBA. The implant and decay-sensitive region consisted of two SSSD layers (front absorbers), three double-sided silicon stripped detectors (DSSSDs) designated as implantation layers A, B, and C, and two SSSD layers (rear absorbers). The energy deposited by the ions along these seven Si detectors was used to detect whether the ion was implanted or if it punched through, as well as to determine the corresponding implant layer or depth. The segmentation of the DSSSDs was 60-fold in the X and 40-fold in the Y direction, with a strip width of 1 mm. Figure 5 shows the total number of implanted ions for each isotope in the DSSSDs of SIMBA.
The energy deposited by β decays in each Si-electrode showed, as expected, a continuum spectrum which extended up to ∼2-3 MeV. The energy deposited in the DSSSDs was determined from the response of the Y strips, which were readout via logarithmic preamplifiers. The latter allowed clear separation of the ion implants and decays in the spectrum [see Table I and Figs. 6(a) and 6(b)]. A pulse generator was used to perform the gain-matching of the Y strips. An accurate energy calibration of layers A and B was accomplished using well-known α decays from several At, Bi, Rn, and Po isotopes produced in the decay of implanted Tl, Pb, and Bi nuclei [see Figs. 6(a) and 6(b)]. No α emitter was implanted in layer C and, therefore, a coarse energy calibration of the latter was made by using only the broad β spectrum [ Fig. 6(d)]. Nevertheless, for implant-β time correlations only an energy window covering the broad β spectrum is required and, therefore, an accurate energy calibration is of secondary relevance for this data analysis.
SIMBA was placed inside the cylindrical hole (23 cm diameter) of the beta delayed neutron (BELEN) detector [46][47][48][49]. BELEN consisted of an array of 30 3  2.54 cm diameter, embedded in a high-density polyethylene (PE) matrix (Fig. 7). The 3 He tubes were distributed in two rings: an inner one with a radius of 14.5 cm and 10 tubes of 10 atm, and the outer ring with a radius of 18.5 cm and 20 tubes of 20 atm. This configuration was designed by means of GEANT4 [50,51] and MCNPX [52][53][54] Monte Carlo (MC) simulations in order to achieve a high and flat detection efficiency (see Fig. 8). At a neutron energy up to 1 MeV the detection efficiency was 40(2)%, and it decreased to 25% at 5 MeV. The MC codes were experimentally validated at E n = 2.3 MeV with a dedicated measurement of a wellcalibrated 252 Cf source.
The Q βn window of the exotic nuclei involved in the present measurement spans neutron energies up to 2.5 MeV [55,56]. In this energy range, the average neutron detection efficiency  Fig. 9.
The energy window considered in the data analysis as neutron events comprises the range from a low threshold of the 191 keV peak up to the end of the main peak of the reaction [see Eq. (1)] at 765 keV. The latter peak corresponds to the kinetic energy of the two reaction products, a triton and a proton: To reduce the neutron background in BELEN two additional elements were implemented in the setup. On one hand, a PE wall with a thickness of 30 cm was installed upstream from BELEN to shield the detector from neutron background induced by the secondary beam (see Fig. 7). This wall had a central hole of 180 mm in the X direction and 70 mm in the Y direction to let the beam into SIMBA. In addition, a layer of borated rubber was attached to the rear of the PE wall in order to absorb thermalized neutrons that could eventually reach the rear side of the wall. On the other hand, the BELEN detector was surrounded by 20 cm of PE shielding (see Fig. 7) in order to moderate and absorb scattered neutrons from the surroundings. The GSI multibranch system (MBS) [57] was used to acquire data from the FRS tracking detectors and SIMBA. This data-acquisition system (DACQ) was triggered by a scintillator at S4 with an efficiency of ∼100% for heavy ions. The MBS was also triggered by high-energy implant and low-energy β-decay events in SIMBA. On the other hand, the neutron data from BELEN were acquired by using a digital self-triggered system [58] based on SIS3302 VME digitizers from Struck Innovative Systems [59]. Each module had 8 input channels running at 100 MHz sampling rate with an ADC resolution of 16 bit. A common clock was used for time stamping the events acquired in the BELEN and MBS DACQ systems with 20 ns resolution. In this way, ion-implant versus β-decay time correlations and ion-β-neutron time correlations could be built over an arbitrarily long time window and in both forward (increasing) and backward (decreasing) time directions. The latter aspect was a key feature in the analysis stage to determine reliably the background level (see Sec. III).

III. DETERMINATION OF β-DECAY HALF-LIVES AND P 1n VALUES
The methodology followed here for the analysis of the β-decay half-lives is similar to the approach successfully applied in previous experiments at isotope separator on-line (ISOL) and fragmentation facilities, such as described in Refs. [16,31]. There are two fundamental aspects to this data analysis; namely, a reliable background characterization of β and neutron events and the spatial-and temporal-correlation approach. We describe first the analysis methodology from a rather general perspective and afterwards we show in detail its application to one of the largest implant statistics case, which is 213 Tl. The latter is also used to illustrate the background treatment in the analysis. Let us consider one specific isotope i. The Bateman analytical expression [60] describes the time evolution of its abundance N i (t). In this analysis we assumed that only the parent (N 1 ) and daughter (N 2 ) decays are contributing to the decay curve. This assumption is justified because all granddaughters of the analyzed nuclei are either stable nuclei or have a half-life much longer than that of the parent and daughter nuclei. The resulting expression based on this assumption is given by where λ 1 = ln(2)/T 1/2 is the decay constant for the implanted nucleus (i = 1), with unknown half-life T 1/2 , and λ 2 is the decay constant of the daughter nucleus (i = 2). As shown below, in many cases even the contribution of the daughter nucleus was very small. Regarding the spatial correlation, we considered an implant and a decay event to be spatially associated when the β-decay position measured in layers A, B, or C of SIMBA was within a correlation region of 3 mm 2 around the implant position measured in the same layer. Extending this condition to neighboring layers did not improve noticeably the statistics and was therefore disregarded. In general, smaller and larger correlation areas were not found to provide a better result in terms of statistics and signal-to-background ratio in the time-correlation diagrams [27].
Regarding the time correlation, the methodology followed here consisted of building a correlation histogram for every implant of a certain species i containing its time difference with respect to all, forward and backward, β events within a broad time window t T 1/2 . In this analysis we used ten times the expected half-life, t 10T 1/2 . As demonstrated below, the uncorrelated background rate is a constant value, which can be referred to as b. In this case, the probability density function describing the time dependence of the correlation distribution is given by [61] where ε β is the β-detection efficiency in SIMBA. Thus, for a certain number N 1 (0) of implanted events, the total number N β of β particles detected at a time t with respect to the implantation time (t = 0) is given by where N Allβ β (t) is the total number of detected decays at a time t, b is the β background normalized and corrected by ε β , and t corresponds to the bin time width used in the implant-β time-correlation histogram. The time evolution of the parent abundance is described by N 1 (t) = N 1 (0)e −λ 1 t , whereas the contribution of the daughter is given by analysis, the common factor N 1 (0)ε β in Eq. (4) is derived from the first bin in the correlation diagram [27]. In this way the quantity of interest (T 1/2 ) can be reliably determined without knowing explicitly the β-detection efficiency.

A. Reference analysis of 213 Tl half-life
The thallium isotope 213 Tl was implanted with large statistics (1015 implants), so we used this case to establish the analysis methodology on a reliable statistical basis. The β background showed a dependency with the time structure of the pulsed primary beam. Indeed, during beam extraction from SIS (1 s) the overall background level of β-like events in SIMBA was ∼40% higher than during the time-interval between spills [27]. This feature led to a better signal-tobackground ratio in the implant-β time-correlation histograms when only β-events outside of the spill time intervals were considered in the correlations, when compared with the same diagram including β-events inside and outside the spill. With this restriction in mind, the background level evaluation was based on time backward (t < 0) implant-β correlations, i.e., the time difference between each implant and all the β events occurring before it, within a broad time window ( t 10T 1/2 ) and in the same correlation area used in the forward analysis of 3 × 3 mm 2 . The background level thus determined allows one to adjust the parameter b in Eq. (3). This approach is illustrated in Fig. 10 for the case of 213 Tl, which shows backward (negative) and forward (positive) implant-β correlations. The contribution to the measured β activity from decays of other nuclei can be assumed to be negligible due to the very low average implantation rate of 2 × 10 −5 ions s −1 pixel −1 . By using Eq. (4), a binned maximum likelihood (ML) [62] analysis of the time correlation histogram was carried out, which yielded a half-life for 213

B. Neutron branching ratio for 213 Tl
The implant-β neutron correlations were analyzed by selecting a correlation window of n t = 400 μs forward and backward in time, following each β detection. This time interval was determined according to the expected neutron moderation time in polyethylene. Considering the β efficiency ε β as a constant value over the range of energies of interest, the P 1n value can be directly obtained from the subtraction of time-forward and time-backward β-neutron correlated events, where ε n is the BELEN neutron efficiency, N fwd βn is the number of forward correlated implant-β-neutron events, and N β is the number of parent β decays. N bkd βn designates the backward β-neutron correlations, which were used to define the uncorrelated neutron background level. The efficiency can be considered flat over the energy range of interest according to the Q βn values of the implanted isotopes (100 keV to 2.5 MeV; [55,56]), with a constant value of 38% and a relative uncertainty of 5% (see Fig. 8). This overall uncertainty includes the contributions of statistical errors for N fwd βn , N bkd βn , and N β together with the uncertainty for the BELEN detector efficiency from the fluctuations over the energy range. For 213 Tl we measured five forward and no backward correlated neutrons (Fig. 11), which yielded a P 1n value of 7.6% ± 3.4%.

C. Thallium isotopes: 211-216 Tl
Five more thallium isotopes were measured and their β-decay half-lives and neutron-branching ratios were analyzed Tl. An almost negligible contribution from the much slower decay of the daughter nuclei ( 211,212,214 Pb) can be observed in these three cases. The most exotic thallium species implanted, 215,216 Tl, were measured with rather limited statistics of only 281 and 99 implants, respectively. In this case, we noticed a slight improvement in the signal-to-background ratio in the correlation histograms when the correlation area was enlarged from 3 × 3 mm 2 to 5 × 5 mm 2 . The ML analysis also shown in diagrams of Fig. 12 yielded a half-lives T 1/2 = 9.7 ± 3.8 s for 215 Tl and T 1/2 = 5.9 ± 3.3 s for 216 Tl.
The neutron-data analysis showed the presence of only one correlated implant-β-neutron event for each of 211,212,215 Tl, which yields neutron-branching ratios of 2.2(2.2)%, 1.8(1.8)% and 4.6(4.6)%, respectively. For these three cases, given the low number of events compatible with the physical boundary, we have alternatively calculated a conservative upper limit based on the Bayesian approach [63], which yields upper limits of 10%, 8%, and 20% at a confidence level (CL) of 95%. On the other hand, with BELEN we were able to observe a rather large number (10) of implantβ-neutron correlated events for 214 Tl (histogram also shown in Fig. 12), resulting in a P 1n value of 34.3% ± 12.2%. No single correlated or uncorrelated event was detected for 216 Tl which, according to the implantation statistics obtained, led to an upper limit of 11.5% and a Bayesian upper limit of P 1n < 52% at the 95% CL. Concerning the other implanted species, 209,210 Tl, the implantation statistics was not enough to determine either their half-lives or the neutron-branching ratios. can be observed on its diagram, contributions from 215 Pb and its daughter 215 Bi are present in the correlation histogram. For the case of the measurement of 216 Pb, it allowed us to carry out a cross-check of the aforementioned analysis methodology, because its half-life can be determined by means of two different methods: implant-β and implant-α correlations. The former is also illustrated in Fig. 13, which shows the implant-β correlation and the result of the ML analysis. On the other hand, the peak of α particles at 6778.3 keV, clearly identified in the energy spectrum of SIMBA [see Fig. 6

E. Mercury isotopes: 208-211 Hg
From the eight mercury isotopes identified, 206-213 Hg, implant statistics were high enough to analyze four of them reliably, 208-211 Hg. The decay curve is strongly determined by the number of ion implants, as well as by the value of the half-life. Thus, for 208 Hg, with low implantation statistics (220 events) and a relatively long half-life obtained from the ML analysis, T 1/2 = 132.2 ± 50 s, the statistical uncertainty was 38%. On the other hand, 209 Hg has a factor of about two more implants (583 events) and a shorter half-life (T 1/2 = 6.3 ± 1.1 s), which leads to an uncertainty much lower, 17%. An intermediate situation is found for the remaining two mercury isotopes, 210,211 Hg, for which the half-life analysis yielded T 1/2 = 63.7 ± 11.6 s and T 1/2 = 26.4 ± 8.1 s, respectively. In the latter case, the value obtained in the present work for the half-life of 211 Tl was employed in the analysis. All of the implant-β correlation diagrams for these cases are shown in diagrams of Fig. 15. Regarding the neutron emission branching ratio, for 210 Hg and 211 Hg one implant-β-neutron event has been detected in the forward (moderation) time window for each nucleus. This measurement yields neutron-branching ratios of 2.2(2.2)% and 6.3(6.3)%, respectively. For these cases the calculated conservative upper limit based on the Bayesian approach [63] yields upper constraints of 10%, and 28% at a CL of 95%.

F. Gold isotopes: 204-206 Au
The 203-209 Au isotopes were identified in this measurement, but the implantation statistics was only high enough to analyze three of them, [204][205][206] Au. In the case of 204 Au (see Fig. 16) we also found a slight improvement of the decay curve when the correlation area was enlarged from 1 pixel (9 mm 2 ) to 2 pixels (25 mm 2 ) around the implant location and included in the correlation those implant-β events detected during the spill time. These provide a better sensitivity for the analysis, which yields a half-life of T 1/2 = 33.7 ± 14.9 s. For the cases of 205 Au and 206 Au, both with ∼100 implants, the ML analysis of the implant-β correlation diagrams yielded half-lives of T 1/2 = 35.4 ± 16.9 s and T 1/2 = 55.7 ± 16.7 s, respectively. Figure 17 shows the analysis for these two gold isotopes.

G. Bismuth isotopes: 218-220 Bi
Bismuth was the heaviest element implanted in SIMBA and we were able to determine the half-lives of three isotopes, 218-220 Bi. The ML analysis of 218 Bi yields a half-life of T 1/2 = 38.5 ± 21.6 s and, as can be seen in the diagram in Fig. 18, the β contribution comes only from its own decay, as its daughter ( 218 Pb) is an α emitter. The half-life analysis for 219 Bi yields T 1/2 = 8.7 ± 2.9 s (see the diagram also on Fig. 18) and it includes the recent published half-life for its daughter, 219 Po, T 1/2 = 620 ± 59 s [66].
For the β-decay analysis of 220 Bi one has to take into account that the half-life of its daughter nucleus 220 Po is still unknown. Thus, our analysis provides a range of possible half-life values for 220 Bi, which spans between 4 and 15 s. The bold marker in Fig. 19 represents the 220 Bi half-life using the theoretical prediction calculated by FRDM+QRPA model [67] for the half-life of 220 Po, T 1/2 = 138.47 s.

IV. SUMMARY AND DISCUSSION
This section provides a summary of the main results obtained in this work, as well as a comparison with all previous experimental results and the theoretical predictions of FRDM + QRPA [67] and DF3 + cQRPA [68] models. A detailed comparison with the more recent calculations of Refs. [69,70] can be found in Ref. [19]. In summary, half-life values for 20 neutron-rich isotopes of Au, Hg, Tl, Pb, and Bi have been determined experimentally, as well as neutron-branching ratios (or upper limits) for eight of them. The results are displayed in Figs. 20 and 21, and listed in Tables II and III. As some of the analyzed isotopes are not expected to be neutron emitters according to their Q βn values (see the last column of Table II), due to the low statistics available, the neutron-emission analysis has been focused on those with large-enough Q βn . For comparison purposes, previously published theoretical and experimental half-life values have also been included in Fig. 20.
Apart from the discrepancies found for the 208,209 Hg isotopes, a rather good agreement is found between the present results and previously published experimental data. In the case of gold isotopes, all half-lives are fully compatible with those reported in previous measurements [18,[73][74][75][76], which are in reasonable agreement with DF3 + cQRPA for N 126. For 206 Au (N = 126) the half-life obtained confirms the recent value published in Ref. [76] and both differ from the DF3 + cQRPA value and the trends predicted by the available theoretical models. Whether this "gold anomaly" is related to effects changing the occupation of the νi 11/2 orbital, or due to a weakening of the spin-orbit field caused by the tensor force [80], or might be due to the recently proposed three-body force mechanism [81,82] remains an open question   which calls for more detailed theoretical studies and further specific experiments allowing for reconstruction of the decay scheme.
Concerning Hg isotopes, as discussed in Ref. [19], recent measurements at the CERN On-Line Isotope Mass Separator (ISOLDE) facility [83] indicate that the half-lives of the 208,209 Hg nuclei are much shorter than the values reported in Ref. [77] and, therefore, this discrepancy will not be discussed further. Theoretical half-life predictions by FRDM + QRPA show a fairly good agreement with the measured values, with the only exception being 209 Hg. Interestingly, the opposite behavior is found regarding DF3 + cQRPA predictions. This feature might reflect that first-forbidden transitions still play a dominant role in the β decay of 209 Hg. Further high-resolution spectroscopy measurements are needed in order to shed light on this aspect.
Regarding the thallium isotopes, previous half-lives were obtained in another experiment by using a similar experimental setup, but with a completely different analysis approach [78,84]. In summary, a good agreement is found for 211 Tl, whereas the half-lives of 212 Tl and 213 Tl differ by factors of two to three. It is worth emphasizing the overall good agreement for the thallium chain between the present results and FRDM + QRPA predictions, including the case of the most exotic nuclei reported here for the first time, [214][215][216] Tl. This result seems to indicate the rather low relevance of first forbidden (FF) transitions in the N 126 mass region, as discussed in Ref. [19].
The analysis of the lead isotopes includes three new halflives, 216-218 Pb. The half-life obtained for 215 Pb is in reasonable agreement with the two previous measurements [65,79]. Along the lead isotopic chain, the FRDM + QRPA model overestimates the experimental values by factors between three to nine. The values predicted by DF3 + cQRPA are, on average, slightly closer to the measured half-lives, but the mass (neutron number) dependency of the half-life is not satisfactorily reproduced by either of these two models.
For the bismuth chain the FRDM + QRPA predictions agree reasonably well with the heaviest measured nuclei 219-220 Bi, whereas almost one order-of-magnitude difference 064322-13   Experimental neutron-branching ratios and upper limits obtained in this work (red dots) and conservative upper-limit based on the Bayesian approach at a CL of 95% (red dashed lines). The blue triangles correspond to the theoretical predictions of the FRDM + QRPA [67] and DF3 + cQRPA [68] models. Blue squares and stars show KTUY [69] and RHB + RQRPA [70], respectively, and blue circles show recent new calculations, QRPA-HF, by Ref. [72]. See text and Table II for details. is found for 218 Bi. The latter half-life is, however, rather well established experimentally [78]. The result quoted in this work for the half-life of 220 Bi, 4-15 s (see Fig. 19), can be redetermined more accurately once the half-life of 220 Po is measured.
The neutron-branching ratios determined in this work represent the first set of experimental data available in this mass region. Therefore, the values reported here can only be compared with theoretical predictions. Both theoretical and experimental P 1n values are listed in Table II and displayed in Fig. 21.
In summary, the agreement between theory and experiment is rather good for the Hg and Tl isotopes with masses between 208 and 214. However, for the two heaviest thallium nuclei 215,216 Tl, KTUY [69] and RHB + RQRPA [70] models are in agreement but the other theoretical predictions of the neutron emission overestimate substantially the experimental results obtained.

V. CONCLUSIONS
Since both the half-life and the neutron emission probability are integral quantities of the β decay, it is difficult to explain why the FRDM + QRPA reproduces so well the average decay strength over the full Q β window (see Fig. 20) along the chain of measured Tl isotopes, whereas it seems to fail dramatically in the upper energy range of 215,216 Tl, beyond the neutron separation energy of the daughter nuclei, as deduced from their neutron-branching ratios. At first sight, one is tempted to attribute such a discrepancy to the possible contribution of high-energy first forbidden (FF) transitions populating low-lying levels in the daughter nuclei and thus hindering the emission of neutrons. Nevertheless, this interpretation is at  variance with the overall systematics found in the N 126 region [18,19,65,76,78], where Gamow-Teller (GT) transitions seem to play a dominant role in general. At present, the only plausible explanation for such a feature could be that the overall strength of the β decay is overestimated in FRDM + QRPA, both in the full Q β and in the upper energy window Q βn . As reported in Ref. [19], more advanced microscopic models such as RHB + RQRPA [70] and KTUY [69] show an inverted behavior, yielding good predictions for the P 1n values, but discrepant values for the half-lives, thus not improving the situation. Recent calculations based on an improved QRPA and HF theory [72], included in Fig. 21, also show large discrepancies. Clearly, more β-decay measurements and theoretical efforts are needed in this mass region to gain a better understanding of the underlying nuclear structure effects, as well as to guide global theoretical models far-off stability.