Elucidating the nature of the proton radioactivity and branching ratio on the first proton emitter discovered ^53mCo

Abstract

The observation of a weak proton-emission branch in the decay of the 3174-keV ^53mCo isomeric state marked the discovery of proton radioactivity in atomic nuclei in 1970. Here we show, based on the partial half-lives and the decay energies of the possible proton-emission branches, that the exceptionally high angular momentum barriers, ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$ and ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7$, play a key role in hindering the proton radioactivity from ^53mCo, making them very challenging to observe and calculate. Indeed, experiments had to wait decades for significant advances in accelerator facilities and multi-faceted state-of-the-art decay stations to gain full access to all observables. Combining data taken with the TASISpec decay station at the Accelerator Laboratory of the University of Jyväskylä, Finland, and the ACTAR TPC device on LISE3 at GANIL, France, we measured their branching ratios as b_p1 = 1.3(1)% and b_p2 = 0.025(4)%. These results were compared to cutting-edge shell-model and barrier penetration calculations. This description reproduces the order of magnitude of the branching ratios and partial half-lives, despite their very small spectroscopic factors.

Introduction

The observation of proton emission in the decay of the 3174-keV isomeric state in ⁵³Co marked the discovery of proton radioactivity in 1970^1,2. This new form of radioactive decay had already been predicted based on the estimated decay energy, the Q_p value, and the calculated probability for a proton to tunnel through the Coulomb and centrifugal barriers^3,4 but it took years before it was experimentally verified. Since then, over 60 proton emitters have been discovered^5,6, and the region near doubly-magic N = Z = 28, ⁵⁶Ni, has continued to exhibit discovery potential for exotic decay modes. For example, discrete-energy proton branches competing with γ-ray emission have been found stemming from the 10⁺ isomer in ⁵⁴Ni^7,8 and from a rotational state at about 10 MeV excitation energy in ⁵⁶Ni^9,10.

Until today, only one weak proton decay branch in ^53mCo was known, estimated to have a branching ratio b_p ≈ 1.5%¹¹, connecting the isomeric (19/2⁻) state with the 0⁺ ground state of ⁵²Fe. A second branch was predicted to occur with a much weaker relative intensity of 1/250 into the first excited 2⁺ state in ⁵²Fe¹¹. A direct experimental measurement of either of the proton-emission branching ratios has not been available for ^53mCo prior to this work. In addition, this experimental input is required to allow theory to elucidate the nature of these rare decay branches having exceptionally high angular momenta, ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$ and ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7$. Theoretically, an explanation of proton-emitting states requires a description of the wave functions of the initial and final states as well as a static or advanced time-dependent approach to the quantum tunnelling process. Model calculations typically infer values for the decay energies to derive (partial) half-lives or spectroscopic factors to be compared with experiments. The determination of the proton decay width offers a powerful means to characterize the isomeric state because of its sensitivity to the fine details of the wave function, of particular interest in ^53mCo due to its peculiar structure (full alignment in the angular momentum) and its proximity to doubly-magic ⁵⁶Ni.

Apart from the branching ratios, all the required information to determine the partial half-life for the proton radioactivity of ^53mCo has been already measured. At the time of its discovery, the reported decay energies for proton emission from this state were Q_p = 1560(40) keV² and 1590(30) keV¹¹. Since then, several measurements have improved the precision and values of 1558(8) keV¹² and 1559(7) keV¹³ were determined. The most precise value thus far has been obtained in an experiment at the Accelerator Laboratory of the University of Jyväskylä, using a double Penning trap, Q_p = 1558.8(17) keV¹⁴, yielding an overall weighted average of 1558.9(16) keV.

The dominant decay mode of ^53mCo is β⁺ decay to its isobaric analogue state in ⁵³Fe. This decay supports the I^π = (19/2⁻) assignment for the ^53mCo state implying a full alignment of the angular momenta of one proton hole and two neutron holes in the f_7/2 shell with respect to doubly-magic ⁵⁶Ni. The branching ratio b_p ≈ 1.5% for proton radioactivity was estimated based on comparisons with model-dependent cross-sections for various products of the reaction p + ⁵⁴Fe. Measuring such a low proton branch is challenging because most of the decays proceed via β⁺ decay, and in addition, the half-lives of the ground (240(9) ms) and the isomeric (245(10) ms) state of ⁵³Co are nearly identical. The situation is illustrated in Fig. 1. The dominant β⁺ branches have practically the same energy distribution, because both ⁵³Co and ^53mCo decay primarily into their respective isobaric analogue states in the daughter nucleus ⁵³Fe. Secondly, any nuclear reaction aimed at producing ⁵³Co in the high-spin isomeric state leads to a population of both ⁵³Co and ^53mCo with an a priori unknown production ratio.

Fig. 1: Decay scheme of ^53mCo.

The scheme is based on previous^13,47,48 and present results. Level energies are given in keV and are relative to the ground state of ⁵³Co. The half-life of the isomer is based on the ACTAR-TPC measurement and literature⁴⁷. The absolute proton branching ratio, b_p1, was measured with TASISpec. The relative branching ratio, b_p2/b_p1, was determined with ACTAR TPC.

In this work, we studied the proton radioactivity from ^53mCo to the ground state of ⁵²Fe and determined its absolute branching ratio, b_p1 (see Fig. 1) using a pure beam of ^53mCo delivered by the JYFLTRAP Penning trap¹⁵ to the TASISpec decay station¹⁶. All other ions, including ⁵³Co in its ground state, were removed in the trapping and purification process. The relative branching ratio, b_p2/b_p1, of proton radioactivity from ^53mCo into the first excited state and the ground state in ⁵²Fe was measured using the ACtive TARget and Time Projection Chamber, ACTAR TPC device^17,18 at the LISE3 separator¹⁹ of GANIL. In this work, we have combined these state-of-the-art methodologies of ion manipulation as well as sensitive decay detection using complementary methods at world-leading facilities with cutting-edge detector setups to provide the complete description of proton radioactivity out of ^53mCo, 50 years after its discovery.

Results

One of the two experiments included in this study was carried out at the Ion-Guide Isotope Separator On-Line (IGISOL) facility²⁰ in the Accelerator Laboratory of the University of Jyväskylä in Finland. The isomeric-state ions, ^53mCo⁺, were produced in the fusion-evaporation reaction of protons on a ⁵⁴Fe target. A quantum-state selection for the decay-spectroscopy measurements was implemented using the JYFLTRAP double Penning trap. Purified ions were implanted into the TASISpec high-resolution charged particle-γ coincidence set-up. Altogether 42 h of data were collected for ^53mCo, resulting in around 150000 implanted ^53mCo⁺ ions.

As seen in Fig. 2, a proton branching ratio of b_p1 = 1.3(1)% was found to best describe the experimental energy spectrum. The best match was determined for E_p1,LAB = 1537(1) keV, which agrees well with the experimental value derived from the ^53mCo and ⁵²Fe mass differences from the Penning-trap mass spectrometry once the dead layer of the implantation detector is accounted for using a self-consistent method between the experiment and Geant4 simulations²¹.

Fig. 2: Proton branching ratio, b_p1, determination.

a Experimental and simulated energy spectra recorded at the TASISpec implantation detector for ${\beta^+}$ particles and protons, the error bars represent the standard error in the number of counts. The simulated spectrum was normalized to the experimental one in the energy range 500 keV–1000 keV. The experimental data are best described when an absolute proton branching ratio of b_p1 = 1.3(1)% is used. The difference between the experimental and simulated spectra can be seen at the bottom. b Result of the Figure-of-Merit (FoM) from a minimization algorithm between experimental and simulated results.

According to Geant4 simulations, the TASISpec experiment has a sensitivity of b_p2 ≥ 0.5% for the 709-keV p2 branch. The β⁺ background is too high for the identification of protons below 900 keV (see Fig. 2a), and it could not be significantly improved by requiring a coincidence with the 849-keV γ-ray connecting the 2⁺ state and the 0⁺ ground state in ⁵²Fe (see Fig. 1). This is consistent with an upper limit of b_p2/b_p1 = 1/250 estimated in the early experiment¹¹.

The ratio of the proton-emission branches of ^53mCo to the first excited state and to the ground state of ⁵²Fe was measured using the ACTAR TPC detection system at the LISE3 beam line of GANIL in another experiment. Secondary beams of ⁵³Co ions were produced by projectile fragmentation of a ⁵⁸Ni primary beam at 75 MeV/nucleon. Approximately 8% of the ⁵³Co ions were produced in the 19/2⁻ isomeric state. During 19 h of data taking, 3.6 × 10^{6 53}Co ions were implanted, close to 12,000 decay protons were identified, and 2167 could be further analysed. Protons leaving the active volume or directed to the cathode were disregarded in the analysis. The proton-energy spectra for two gas pressure settings of ACTAR TPC, extracted from the proton ranges in the active volume of the detector, are presented in Fig. 3. This spectrum is conditioned by a 4 s coincidence window after a ⁵³Co implantation and a 300-ms anti-coincidence window after ⁵⁶Cu and ⁵⁷Zn implantation to avoid β-delayed protons from these nuclei in the data set. Both decays to the 0⁺ and to the 2⁺ states of ⁵²Fe are visible in the spectra. For the high-pressure setting, the number of analysed p1 and p2 protons is 1563 and 45 respectively, whereas the respective numbers are 532 and 27 for the low-pressure setting. The half-life of the state was measured for the two transitions by correlating each decay event with ⁵³Co implantations within a ± 4-s time window. The resulting half-life spectra are presented in Fig. 3 and yield an average value of T_1/2 = 239(21) ms. This number agrees with the literature values, and we obtain a total average of T_1/2 = 245(10) ms.

Fig. 3: Experimental proton energy and half-life measurement using ACTAR TPC.

Left: Proton energy spectra associated with the decay of ^53mCo for the two different pressure settings (380 and 292 mbar). Right: Half-life spectra obtained by correlating the proton emission, identified by their characteristic energy-loss profile, with ^53mCo implantation for both pressure regimes. The line represents a least-squares fit of an exponential decay to the data.

The proton-detection efficiency, determined using a dedicated Monte-Carlo simulation, was 18.6(10)% for p1 and 25.7(3)% for p2 for the high-pressure setting and 7.7(23)% and 25.3(6)% for the low-pressure one, respectively. The resulting relative branching ratios for the ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7$ and the ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$ decays are b_p2/b_p1 = 2.04(34)% and 1.52(56)% for the different pressure settings, respectively. This yields a final ratio of b_p2/b_p1 = 1.90(29)%. From the above numbers, we calculate an absolute branching ratio of b_p2 = 0.025(4)% and ${{{{{{\rm{b}}}}}}}_{{{{{{{\rm{\beta }}}}}}}^{+}/{{{{{\rm{EC}}}}}}}=98.67(10)\%$. The partial half-life for the proton decay of the isomer is determined to be 18.5(16) s, which yields partial half-lives of 18.8(16) s and 980(162) s for proton radioactivity to the ground and first excited state in ⁵²Fe, respectively.

Discussion

Experimentally measured partial half-lives can be used to probe the wave function of the isomeric state via comparison to theoretical calculations. The partial half-lives are calculated from the inverse of the decay widths multiplied by ℏ. The 19/2⁻→ 0⁺ decay (p1 branch) involves ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$ proton emission, and the 19/2⁻→ 2⁺ decay (p2 branch) involves ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7$ proton emission (see Fig. 1). The decay rates are products of large barrier-penetration factors and tiny spectroscopic factors; in the original paper of Cerny et al.², a single-particle half-life for 19/2⁻→ 0⁺, ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$, of 60 ns was obtained from a standard barrier penetration calculation and a spectroscopic factor of 1 × 10⁻⁶ was estimated by Peker et al.²² in a simple model for the wave functions and residual interaction; none of them provided a satisfactory description. In this work we factorized the decay width into two components: a many-body nuclear structure part that gives the spectroscopic factors, S_p, and a potential barrier penetration part that gives the single-particle decay widths, Γ_sp, Γ = S_p · Γ_sp.

The spectroscopic factor²³ is given by the reduced matrix element

$${{{{{\rm{S}}}}}}_{{{{{\rm{p}}}}}}=\frac{{|\langle \Psi {({\,}^{52}{{\mbox{Fe}}})}_{f}{J}_{f}|{\tilde{a}}_{n,\,\ell,\,j}|\Psi {({\,}^{53}{{\mbox{Co}}})}_{i}\,{J}_{i}\rangle|}^{2}}{(2{J}_{i}+1)}$$

(1)

where ${\tilde{a}}_{n,\,\ell,\,j}$ is a single-proton destruction operator, J_i = 19/2 and J_f = 0 or J_f = 2. The spectroscopic factors summed over all final states (f, J_f) gives the orbital occupation number for the orbital $({n,\,\ell,\,j})$ in the initial state (i, J_i). For the single-particle decay width we use a Woods-Saxon potential²⁴. For a given separation energy, the high-${{{{{\mathcal{l}}}}}}$ single-particle wave functions are constrained to have the correct asymptotic behaviour of the decay by increasing the Woods-Saxon well depth. This calculation is combined with a Coulomb plus angular momentum barrier-penetration model with a radius parameter in agreement with results from proton scattering from a Woods-Saxon potential.

The Coulomb potential used with the Woods-Saxon calculation was obtained from a uniform charge density distribution with radius r_c·A^1/3. For mass number A = 52, the parameter r_c = 1.22 fm was chosen to reproduce the experimental Coulomb displacement energy between ⁵³Fe and ⁵³Co of 9.07 MeV. With a diffuseness parameter of 0.67 fm, the potential radius r₀ = 1.26 fm was chosen to reproduce the experimental root-mean-square charge radius of ⁵²Fe of 3.73 fm²⁵. The magnitude of the 0f_7/2 proton single-particle energy of 7.15 MeV is close to the experimental proton separation energy of ⁵²Fe, 7.38 MeV²⁶.

This potential was then used to calculate proton scattering from ⁵²Fe with the code WSPOT²⁷. The potential depth was adjusted to give fixed resonance energies for ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7$ and ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$. The widths for these scattering states could numerically be obtained down to about 5 × 10⁻¹¹ MeV. For example, for Q_p = 2.0 MeV, ${\Gamma }_{{{{{{\rm{sp}}}}}}}({{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7)=4.9\times {10}^{-9}$ ${{{{{\rm{MeV}}}}}}$, and for Q_p = 4.0 MeV, ${\Gamma }_{{{{{{\rm{sp}}}}}}}({{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9)=8.6\times {10}^{-11}$ ${{{{{\rm{MeV}}}}}}$.

The single-particle proton widths were also calculated from²⁸

$${\varGamma }_{sp}=2{\gamma }^{2}P(\ell,\,{R}_{c},\,{Q}_{p})$$

(2)

with ${{\gamma }}^{2}=\frac{{{{\hslash }}}^{2}{{{{{{\rm{c}}}}}}}^{2}}{2{{{{{\rm{\mu }}}}}}{{{{{{\rm{R}}}}}}}_{{{{{{\rm{c}}}}}}}^{2}}$ and we obtain the Coulomb penetration, P, from Barker²⁹. The channel radius, R_c, was chosen to match the decay widths obtained from the Woods-Saxon potential. A value of R_c = 5.46 fm works for both ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7$ and ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$. With this, the barrier penetration model gives the same result as the Woods-Saxon scattering calculation over the Γ_sp range from 5 × 10⁻¹¹ MeV to 1 × 10⁻⁸ MeV to within about one percent. The barrier-penetration model can be extrapolated down to the Q values needed for proton emission of ^53mCo that have single-particle decay widths of the order of 10⁻¹⁵ MeV. The results for ${{{{{{\rm{T}}}}}}}_{1/2{{{{{\rm{;sp}}}}}}}={{\hslash }}\cdot {{{{{\rm{ln}}}}}}(2)/{\Gamma }_{{{{{{\rm{sp}}}}}}}$ are given in Table 1. The uncertainties come from the uncertainty in the experimental Q_p values. Relative to results for proton emission with an angular momentum of ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=1$, the ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$ decay is hindered by a factor of about 10¹², and the ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7$ decay is hindered by a factor of about 10⁹.

Table 1 Results for the proton-decay calculations compared to experiment

For the calculation of the spectroscopic factors, we use the ($0{{{{{{\rm{f}}}}}}}_{7/2}$, $0{{{{{{\rm{f}}}}}}}_{5/2}$, $1{{{{{{\rm{p}}}}}}}_{3/2}$, $1{{{{{{\rm{p}}}}}}}_{1/2}$) (or in shorthand notation, fp) model space with the GPFX1A Hamiltonian³⁰. To this, we add the configurations where one proton is moved into the high-ℓ orbitals that are involved in the decay. For the two-body matrix elements that connect the fp and high-${{{{{\mathcal{l}}}}}}$ orbitals, we used the M3Y interaction³¹, and harmonic-oscillator radial wave functions with ℏω = 10 MeV. To keep the basis dimensions tractable, the fp part of the wave function was truncated to allow for only up to one proton or one neutron to be excited from the $0{{{{{{\rm{f}}}}}}}_{7/2}$ shell to one of the ($0{{{{{{\rm{f}}}}}}}_{5/2}$, $1{{{{{{\rm{p}}}}}}}_{3/2}$, $1{{{{{{\rm{p}}}}}}}_{1/2}$) orbitals. The single-particle energies for the high-${{{{{\mathcal{l}}}}}}$ orbitals were placed $({{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}-3){{\hslash }}{{{{{\rm{\omega }}}}}}$ above the $0{{{{{{\rm{f}}}}}}}_{7/2}$ orbital. The configuration-mixing calculations were carried out with the Oxbash code³². The spectroscopic factors, S_p, obtained from these calculations and the results for the partial half-lives are given in Table 1.

The spectroscopic factors are sensitive to the model-space truncation. If we use the minimal basis of just the $0{{{{{{\rm{f}}}}}}}_{7/2}$ orbital, the spectroscopic factor for ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7$ is increased by about a factor of 10 and the spectroscopic factor for ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$ is increased by about a factor of 1.5. We have also calculated the M3Y two-body matrix elements using the Woods-Saxon radial wave functions for the artificially bound high-${{{{{\mathcal{l}}}}}}$ states. With this change the spectroscopic factor for ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=7$ is increased by about a factor of two, and the spectroscopic factor for ${{{{{{\mathcal{l}}}}}}}_{{{{{{\rm{p}}}}}}}=9$ is decreased by about a factor of two. The calculated results are of the same order of magnitude as the experimental values; a maximum discrepancy of a factor of two to four was found.

The decay of ^53mCo, the first proton emitter ever observed, was studied in two experiments, the first at the IGISOL facility of the Accelerator Laboratory of the University of Jyväskylä and the second at the LISE3 separator of GANIL. The IGISOL data was used to determine the absolute branching ratio for proton emission of this isomeric state to the ground state of ⁵²Fe to be $1.3(1)\%$. The ratio between proton emission to the first excited state relative to the ground state of ⁵²Fe was determined with the ACTAR TPC device at GANIL yielding an absolute branching ratio to the excited state of 0.025(4)%. All decay branches of the isomer have been experimentally measured and a theoretical description has been proposed that reproduces the order of magnitude of the branching ratios and partial half-lives, despite their extremely small spectroscopic factors.

Methods

The ions of interest for the TASISpec setup were produced using a 10μA, 40 MeV proton beam impinging into an enriched 1.8 mg/cm^{2 54}Fe target at the IGISOL facility²⁰ in the JYFL Accelerator Laboratory. The reaction products were stopped in helium gas (P = 200 mbar) in the IGISOL light ion guide³³ and extracted using a sextupole ion guide³⁴ before acceleration through a 30 kV potential. Most of the ions end up as singly charged in the helium gas. A 55° dipole magnet was used to select ions with a mass-to-charge ratio of A/q = 53. The separated ion beam was cooled and bunched in a gas-filled radio-frequency quadrupole cooler and buncher (RFQ)³⁵. The ion bunches were then injected into the JYFLTRAP double Penning trap¹⁵.

The mass-selective buffer gas-cooling technique³⁶ was employed to select either the isomeric-state ions (^53mCo⁺) or the ground-state ions (⁵³Co⁺) for the TASISpec post-trap decay spectroscopy setup, see Fig. 4. The ions were first held in the trap for 30 ms to cool their axial and cyclotron motions. After this initial cooling, all ions were removed from the centre of the trap by exciting them to a larger radius using a dipolar magnetron excitation for 10 ms. The mass-selective quadrupolar excitation was then applied for 80 ms at the ion’s cyclotron resonance frequency. The quadrupolar excitation converts the slow magnetron motion of the ions into a much faster cyclotron motion. As a result, the ions with a cyclotron resonance frequency matching with the applied quadrupolar excitation frequency are cooled and centred in the trap due to collisions with the helium buffer gas.

Fig. 4: Number of ions per second detected at the microchannel-plate detector behind JYFLTRAP.

The number is displayed as a function of the quadrupolar excitation frequency in the purification trap, the error bars represent the standard error in the number of counts. The relevant frequency region covering the ground and isomeric states of ⁵³Co and ⁵³Fe is shown. The red dotted line indicates one of the frequencies used for selecting the isomer ^53mCo for the TASISpec measurements, the dashed blue line gives the frequency for the ground state of ⁵³Co.

To select the isomeric state ^53mCo, the quadrupolar excitation frequency in the first trap was set at values of 2 029 210 Hz (first part of the data taking) and 2 029 217 Hz (second part, red dotted line in Fig. 4). For the ground state of ⁵³Co, a frequency of 2 029 348 Hz was applied (dashed blue line in Fig. 4). The mass-selected ion bunches were extracted from the trap, accelerated to 30 keV and sent to the TASISpec setup every 141 ms.

TASISpec is composed of two subsystems, an inner silicon cube with about 80% efficiency for charged-particle detection and a surrounding array of high-purity germanium detectors for γ-ray detection. It has been extensively modelled using Geant4³⁷ and has been proven to allow for self-consistent cross checks of decay schemes derived from experimental spectra^38,39.

The inner subsystem covers five sides of a (~6 cm)³ cube with pixelated double-sided silicon strip detectors: four ‘box detectors’ (16 × 16 strips, 0.97 mm thick) and one ‘implantation detector’ in the direction of the beam (32 × 32 strips, 0.52 mm thick). The surrounding germanium array consisted of two EUROGAM II four-fold Clover detectors⁴⁰ and one EUROBALL seven-fold Cluster detector⁴¹. This lowered the nominal ∼40% detection efficiency at 150 keV photon energy¹⁶ to about ∼30%, which was verified using standard calibration sources of ¹³³Ba, ¹⁵²Eu, and ²⁰⁷Bi.

For this experiment, we performed Geant4 parameter scans of (i) the proton-emission branching ratio, ${{{{{{\rm{b}}}}}}}_{{{{{{\rm{p}}}}}}1}$, and (ii) the ${{{{{{\rm{E}}}}}}}_{{{{{{\rm{p}}}}}}1,{{{{{\rm{LAB}}}}}}}$ value of the decay. The aim is to find the combination of parameters where experiment and simulation match best. Although the Q value for the decay was determined very precisely¹⁴, the inclusion of the proton decay energy as a free parameter avoids any bias of the results due to any imperfection of the energy calibration of the silicon detectors. First, all simulated spectra were normalized to the experimental ^53mCo spectrum in the energy range 500 keV–1000 keV (see Fig. 2a). Second, a comparison algorithm employing the Anderson Darling test tool from ROOT⁴² was used as our Figure-of-Merit to probe the match between experimental and simulated spectra for the energy range from 1000 keV to 1500 keV, which includes the proton peak p1. The result of the comparison is shown in Fig. 2b.

The second experiment was performed at the LISE3 beam line of GANIL. The ⁵³Co ions were produced by the fragmentation of a ⁵⁸Ni beam at 75 MeV/nucleon on a 660 μm thick beryllium target. The fragments were selected by the LISE3 spectrometer and identified using the energy loss versus time-of-flight method on an event-by-event basis. The time-of-flight was generated with the cyclotron radiofrequency and the timing signal of a fast gas detector located just before the entrance window of the ACTAR TPC indicating an ion entering the chamber.

ACTAR TPC is a gas detector that was filled for the present experiment with an Ar (95%) + CF₄ (5%) mixture, with 16384 read-out pads and an active volume of 25 cm × 25 cm × 20 cm working as a time-projection chamber. Each pad of the detection plane is connected to the GET electronics⁴³, allowing the reconstruction of the 3-dimensional tracks of the implanted ions or the emitted protons. Due to difficulties in the production process of the pad plane, some pads were grounded, creating blind zones on the detection plane. To minimize the effect of these zones on the measurement of the ratio of the proton decay branches, the measurement was performed with two pressure settings of ACTAR TPC: a high pressure (P = 380 mbar) and a low pressure (P = 292 mbar) setting.

Data from the LISE3 beam line and ACTAR TPC were correlated with a common dead time as well as an event-number counter and a common time stamp. The data acquisition was triggered by a signal from the fast gas detector for ion implantation, and by the pad plane multiplicity signal (with a threshold of 11 pads) for the decay events. The implantation and decay were registered as separate events by the data acquisition. When a data acquisition trigger is issued, the ACTAR TPC pad plane collected charges for 10 μs. The signal for each pad is sampled at 25 MHz after being passed through a preamplifier and a shaper. The processing of the GET electronics is already described in the literature^44,45, and more details for the present experiment can be found elsewhere^8,46.

Decay events are distinguished from implantation events by the absence of a signal in the fast gas detector and in the silicon detector upstream ACTAR TPC. Protons from the decay of the ^53mCo isomer are identified by their characteristic energy-loss profile, namely the Bragg peak, and are thus distinguished from α particles from the natural radioactivity of the ACTAR TPC material and from cosmic rays⁴⁵.

A non-negligible but undetermined fraction of the implanted ⁵³Co nuclei were not neutralised in the gas of ACTAR TPC and drifted towards the cathode of the detector with a drift time of the order of 3 ms, much shorter than the half-life of ^53mCo. While drifting, a proportion of the ions may neutralize and will stop drifting. To obtain the correct proton-track length used to determine the proton energy in all cases, only protons directed towards the pad plane (i.e., opposite to their drift direction) with angles >20° with respect to the cathode plane were considered in the analysis to avoid “grazing” angles.

High-energy protons have a significant probability to leave the active volume of ACTAR TPC. To be considered for further analysis, the protons must be stopped within the active volume of the detector. This is ensured for the lateral sides of the detector by the pad pattern (no signal on the external pads). To remove protons hitting the pad plane, we imposed angular cuts. Simulations allowed us to determine that p1 protons with angles <50° (high-pressure setting) or 40° (low-pressure setting) will stop in the active volume, before reaching the pad plane, independent of their starting position, i.e., the implantation height if there is no drift or higher above the pad plane for drifting ions. For p2 protons, only angles <70° are considered. This is a consequence of the pad multiplicity threshold that is not reached when the track is (close to) perpendicular to the pad surface.

To determine the impact of these restrictions and to obtain the branching ratio, b_p, of each proton line, simulations were performed with the Geant4 tool kit²¹. The simulated data were subjected to the above-mentioned experimental limitations to determine the proton-detection efficiencies under the same conditions. The simulations have the following ingredients: (1) an event generator for experimentally determined proton emission points from all ⁵³Co events and random proton-emission directions with chosen energy, (2) the simulation of the proton energy loss along its trajectory in ACTAR TPC using Geant4 with the gas pressure adjusted to reproduce the measured track length, (3) the drift of the ionization signal, with dispersion and amplification on the collection plane, and (4) the processing of the signal collected on the pads. In addition, to qualitatively determine the angular cuts necessary to ensure a proton detection efficiency independent of the height of the emission point, a fraction of the decay events was generated from the cathode plane.

Applying the same selection criteria as for the experimental data allows the determination, for each proton energy, of the global detection efficiency that combines the selection of the events as a function of the observed proton signal and the escape probability from the detection volume. As many events can be generated, the efficiency uncertainties are dominated by systematic effects due to uncertainties of the simulation parameters, rather than statistics⁴⁴.