Analysis methods to determine the bound-state beta-decay half-life of Thallium-205

. Bound-state β − -decay is an exotic decay mode that produces temperature-dependent stability in nuclei. A striking example is 205 Tl, in part because of its impact on the 205 Pb / 204 Pb cosmochronometer—a short-lived radionuclide clock that can provide unique constraints on s process material in the early solar system. The bound-state β − -decay of 205 Tl was measured at GSI, where fully stripped 205 Tl 81 + ions were produced and stored in the Experimental Storage Ring. Decay occurred during storage producing increased 205 Pb daughters with increased storage time. This contribution briefly outlines the experiment and describes analytical corrections required to extract the half-life.


Introduction
Bound-state β − -decay is a rare of example of how altering the electron density can change nuclear properties. Despite being characterised in 1947 [1], it was not measured experimentally until 1992 [2] with the development of the Experimental Storage Ring at GSI Helmholtzzentrum für Schwerionenforschung. The decay mode differs from normal β − -decay because the β-electron is created directly in a bound orbital of the daughter nucleus, thus increasing the Q-value of the reaction by the binding energy of the bound electronic orbital. Thallium-205 is particularly interesting because neutral 205 Tl is stable, with 205 Pb decaying via electron capture with a Q-value of 50.6 keV [3]. However, since the binding energy of a K-electron for Tl is 85.5 keV, the bound-state β − -decay of fully ionised or hydrogen-like 205 Tl can occur, primarily to the first excited state of 205 Pb as shown in Figure 1. This makes bare 205 Tl 81+ radioactive with a half-life first predicted to be 124 days [4].
The bound-state β − -decay of 205 Tl has a large impact on the termination of the s process. 205 Pb is one of 10 Short-Lived Radioactive (SLR) nuclei (short-lived on an astronomical timescale; 1-100 My) that decay into isotopic ratios that can preserve the abundance patterns of the Early Solar System (ESS). 205 Pb is particularly unique as it is the singular s-only SLR (only being produced by the s process) and also probes the final branching point of the s process at 204 Tl (half-life only 3.8 years). However, the first excited state of 205 Pb at just Figure 1. The low-lying nuclear structure for 205 Tl and 205 Pb shows the interplay between the neutral and fully stripped states. Note that because the bound-state β − -decay proceeds overwhelmingly to the first excited state of 205 Pb, it's half-life provides access to the nuclear matrix element of the two spin-1 2 states, which is a crucial astrophysical transition.
2.3 keV can be significantly thermally populated in stellar environments, drastically reducing the effective half-life due to the reduced spin difference of the EC decay. Takahashi and Yokoi [5,6] identified that the bound-state β − -decay of 205 Tl could provide a mechanism to counterbalance the 205 Pb thermal destruction. The predicted temperature at which boundstate β − -decay would become competitive was ∼ 180 MK (depending on electron density), which is reached both in the He-shell flashes of Thermally Pulsating AGB stars and the Cshell of massive stars. This idea was supported by the confirmation of live 205 Pb during the ESS, but inconsistent meteorite measurements have made it difficult to predict exactly what site (if any) could have provided this 205 Pb [7][8][9]. Thus an experimental value was required to provide clarity on the exact mechanism.
This contribution will briefly outline the experimental method used, but will focus on the analytic corrections required to extract the half-life. The intention is to outline the analysis methods used to derive the half-life, and thus will not comment on the result or its impact.

Experimental Method
The measurement of the bound-state β − -decay of 205 Tl was conducted at GSI using the full accelerator system [10] and the Experimental Storage Ring (ESR) [11]. Thallium cannot be used directly as a source as its vapours are highly toxic, so the primary beam was provided by a specifically-developed enriched-206 Pb source. The 206 Pb 67+ charge state was accelerated by the UNILAC and SIS-18 to 678 MeV/u with an intensity of 10 9 particles per second. This primary beam was impinged on a 9 Be target producing fully ionised 205 Tl 81+ at 592 MeV/u by projectile fragmentation, amongst a cocktail of contaminant species. 205 Tl 81+ was selected by the FRagment Separator (FRS) using Bρ-∆E-Bρ separation with an Al energy degrader and highly selective focal plane slits suppressing the amount of contaminant 205 Pb 81+ to < 0.2%. An ion optical simulation showing the separation of species along different trajectories through the FRS is shown in Figure 2. Each injection from the FRS produced ∼ 5 × 10 5 205 Tl 81+ ions at 400 MeV/u.
To achieve the required intensity of ∼ 10 6 205 Tl 81+ ions, up to 100 injections were stacked and cooled inside the ESR. The 205 Tl 81+ ions were then stored for periods of up to 10 hours to accumulate 205 Pb 81+ daughter ions. Because the mass difference of 205 Tl 81+ and 205 Pb 81+ ions is just 53 keV and bound-state β − -decay daughters have the same charge state as the parent ions, both species have the same mass-to-charge ratio and were stored on the same orbit in the ESR. To count the number of 205 Pb daughters, the bound β-electron was stripped away by interacting the stored ions with an Ar gas target for 10 minutes producing 205 Pb 82+ Figure 2. A MOCADI ion-optical simulation shows the trajectories of relevant projectile fragments through the FRS. Primary 206 Pb 81+ beam is well separated at the focal plane, whilst an energy degrader is required to separate the isobar 205 Pb 81+ from the desired 205 Tl 81+ . Image courtesy of Helmut Weick.
ions. Because the storage times were small compared to the half-life of the decay, daughter products grow linearly with storage time. Background contribution by 205 Pb 81+ contamination from the FRS and nuclear charge exchange reactions in the gas target was determined from the t s = 0 intercept, whilst the growth of 205 Pb 81+ intensity was directly attributable to bound-state β − -decay as all other parameters were maintained between storage runs.
Throughout the experiment, ions in the ring were monitored by a non-destructive 245 MHz resonant Schottky noise pickup [12] that measures the Schottky noise power from the revolution of ions. As the revolution frequency in the storage ring is determined by the massto-charge ratio, each ion in the ring can be identified and its intensity monitored. Ions that left the acceptance of the ring were monitored with particle detectors: the CsI-Silicon Particle detector for Heavy ions Orbiting in Storage rings (CsISiPHOS) [13] on the inside of the ring and a Multi-Wire Proportional Chamber (MWPC) [14,15] on the outside of the ring.
Each storage run can be characterised by 4 key stages: 1. the storage period after the accumulation of 205 Tl 81+ ions 2. the stripping stage where the gas target is active for 10 minutes 3. the electron cooling stage where ions were cooled into well-resolved peaks 4. the measurement window where the intensities of ions were taken Whilst the Pb and Tl Schottky intensities were measured during stage 4, the ratio that is required to determine the half-life is the value at the end of storage (stage 1), so a number of corrections need to be made before the ratios of different storage times can be compared.

Analysis Corrections
There are four corrections that need to be applied to the raw Schottky power intensities.

Resonance correction
The first is the Resonance Correction (RC), which accounts for the resonant response of the Schottky detector. Whilst the detector is sensitive between 241.6-248.4 MHz, a strong resonance response is centred at 243.6 MHz associated with the fundamental resonance frequency of the cavity. This amplifies the Schottky power of species closer to the resonance peak, and this amplification needs to be corrected for. Extensive details on this effect will be provided in an upcoming publication.

Schottky saturation correction
The second is the Saturation Correction (SC), which accounts for a saturation effect observed in the recorded Schottky data. An unexpected interaction between updated accelerator control systems and the recent New Time Capture (NTCAP) Schottky data acquisition [16] resulted in the adaptive amplifier on the NTCAP-DAQ being permanently activated. This caused a saturation effect above a threshold that meant noise power density was not proportional to ion number, causing Schottky lines to no longer decay exponentially. To identify the intensity threshold at which saturation set in, the decay constant of the beam measured by the MWPC was used in a fit to the Schottky noise power data such that the only free parameter was the final intensity at the end of stripping. This fit is shown in Figure 3, where the time coordinate is inverted to look at time from the end of stage 2. It is clear that the data deviates from exponential behaviour for noise power intensities over 600 × 10 3 (arb. units for the Tl peak), which was chosen to mark the saturated and non-saturated regions shown in Figure 3. This analysis revealed that for 12 shortest storage times, the Schottky noise power data was saturated for the 205 Tl 81+ peak at the measurement stage (note that the 205 Pb 82+ peak had a much smaller intensity, well below the saturation threshold). The Tl peak intensity during stage 2 (stripping) spans all values of the Tl peak at stage 4 (measurement). The first 10 hour storage (10hr-1) was best suited as a calibrant as it had the lowest final intensity and thus the largest portion of stage 2 in the non-saturated region. Two models were applied to the 10hr-1 data: where N exp (t) was the expected behaviour using the MWPC decay rate whilst N sat (t) was a phenomenological model that replicated the saturation effect. The ratio between these two models defined a Saturation Correction factor S C = N exp N sat > 1 for any given noise power density at stage 4. Statistical errors on this factor were determined by using a Monte Carlo calculation to propagate the fit errors.

Charge-changing cross-section ratio
The third correction factor has been termed the Cross-Section Ratio Correction (CSRC). This correction accounts for the fact that whilst 205 Pb 82+ ions stripped by the gas target are recorded, 205 Pb 80+ ions from electron recombination will leave the acceptance of the ring. Since the beam and target are identical for these two reaction channels, the correction factor is the ratio of the charge-changing reaction cross-sections: where N X is the number of ions counted in a detector X and R X is the rate in that detector. To measure this cross-section ratio, H-like primary beam 206 Pb 81+ was reacted with the gas target, and the CsISiPHOS and MWPC particle detectors captured the ionisation and recombination products respectively. Due to inexplicable effects in the CsISiPHOS data, the ratio was instead determined from the total ion loss rate, with ionisation to bare Pb being determined by the subtraction of the recombination rate from the total loss rate, as in the last equality of Equation 3. Because the Schottky data was saturated, a DC beam Current Transformer (DCCT) was used to measure the total ion current in the ring. The DCCT was recorded with a scaler counter, and the high event rate meant that the scalar counter didn't accrue many counts per event, so a stable derivative was very challenging. In particular, the derivative did not stabilise until a window of 50 seconds was used. The DCCT also has a zero-current offset to ensure non-zero scalar counting even if the ring is empty. This offset was measured in two ways, from 124 'stray' events when the ring was empty and from the offset of 14 exponential + constant fits from the 14 injections of primary beam during this measurement.
In the first, and most direct approach, the ions reacted (measured by the DCCT) over these 50 second windows were used to calculate 70 independent values for CSRC, each with uncertainty values from the underlying counting statistics from the scaler and MWPC events and the DCCT offset calibration. The correlation in the DCCT offset was handled by propagating errors using a Monte Carlo calculation, which confirmed the statistical errors were Gaussian. However, the resulting normalised residuals indicated the raw error bars underestimated the statistical spread around the mean, which was attributed to unquantified missing statistical uncertainty given that the DCCT was not intended for precision measurements. A Bayesian analysis was used to allow the data to constrain this missing uncertainty, which was assumed to be Gaussian from the Central Limit Theorem with a Jeffrey's prior. Integrating out the nuisance parameters lead to a Gaussian posterior distribution for the CRSC of R 1 = 1.433 (9).
In the second approach, the DCCT data was fit with an exponential decay function, circumventing the problem of an uncertain local derivative with a global fit. However, the decay rates determined from the DCCT were systematically lower than from the MWPC, despite measuring the same interaction with the gas target. A potential explanation was found in the Schottky spectrum where broad peaks next to the main beam were observed that grew over time, despite only having primary beam in the ring. Throughout this part of the experiment, the stochastic cooler [18] could not be turned off, so it was proposed that the stochastic cooler was providing a RF-kick to ions throughout the storage, building a side band over time which would reduce the measured decay constant of the DCCT rate explaining the behaviour. Whilst this theory wasn't able to be definitively confirmed, the decay rate from the MWPC was more reliable. Thus two values were calculated, R 2 = 1.425(5) from explicitly fitting a 'side band', and R 3 = 1.418(4) from simple exponential decay. This yielded a spread of analysis models producing a modelling systematic uncertainty of 0.005, taken from the variance of a uniform distribution. The side band model was chosen as the most accurate analysis.
Additionally, ∼ 5% of MWPC events demonstrated anomalous behaviour despite only primary beam being in the ring. A systematic error of 0.012 from the difference in including vs excluding these events was added. This resulted in a final value for the charge-changing cross-section ratio of R = 1.425(5) stat (13) syst .

Gas Stripping Efficiency
The fourth correction factor accounts for the interaction with the gas target. In particular, 205 Tl 81+ ions are lost to electron recombination whilst 205 Pb 82+ daughters are uncovered by stripping (recombination is already accounted for by the CSRC), so the intensity of the two species throughout gas stripping is given by: where N X,2 is the ion number at the start of stage 2 and λ str,X is the stripping decay constant, which is specific to individual species. An MWPC was inserted during most storage times, so individual values for λ str,Tl were determined for 11 of the 18 storage times by fitting an exponential to the event rate measured in the detector. For 205 Tl 81+ ions at 400 MeV/u into a gas target of ∼ 3 × 10 12 atoms/cm −2 , the average decay constant was λ str,Tl = 0.00312(9) s -1 . The value of λ str,Pb was determined from the MWPC rate during the 206 Pb 81+ primary beam measurements, ensuring the same systematics. For 206 Pb 81+ ions at 400 MeV/u into a gas target of ∼ 3 × 10 12 atoms/cm −2 , the average decay constant was λ str,Pb = 0.00757(2) s -1 . This means that for a 10 minute stripping duration, the stripping efficiency for revealing 205 Pb 82+ ions was ∼ 99%.
These four correction factors can be applied to the measured raw Schottky intensities at stage 4 to provide a ratio at the end of storage: .

Half-life Fit
With the ratios at the end of storage determined, the increase in 205 Pb 82+ daughters can be fit using linear regression. Because the storage losses due to electron recombination in the electron cooler and collisions with the rest gas in the ring determine the ion intensity throughout storage, they need to be incorporated when solving the differential growth equation. The full solution to this differential equation is given by: where R(t s ) = N Pb (t s )/N T l (t s ) is the ratio, R 0 the background contribution, λ β b the bound-state β − -decay decay rate, γ the Lorentz factor for conversion into the laboratory frame, and λ cc the respective loss rates during storage. Using a Taylor series, the exponential can be expanded, and noting that λ β b ≪ λ cc , this expression can be simplified to: R(t s ) = λ β b γ t s 1 + 1 2 (λ cc T l − λ cc Pb )t s + O(t 2 s ) + R 0 exp (λ cc T l − λ cc Pb )t s .
The approximation in Equation 8 differs from the full result in Equation 7 by < 0.15%. The four corrections described in Section 3 consist of a mix of individual and average values depending on measurement constraints and the plausibility of varying experimental conditions. The uncertainties on the average values used will thus be correlated across different storage times, making a simple χ 2 -minimisation very challenging to correctly handle the correlations. To solve this a Monte Carlo calculation was done to propagate the correlated errors consistently, where 10 5 linear regressions were applied to resampled data producing a distribution in the determined β − b -decay rate λ β b . At present, the collaboration is still finalising how to handle the background contamination consistently, so the final half-life result is not ready at this stage. However, the analysis is almost finalised and important experimental clarification on the role of 205 Pb produced during the s process will be published soon.