Mass measurements show slowdown of rapid proton capture process at waiting-point nucleus 64Ge

X-ray bursts are among the brightest stellar objects frequently observed in the sky by space-based telescopes. A type-I X-ray burst is understood as a violent thermonuclear explosion on the surface of a neutron star, accreting matter from a companion star in a binary system. The bursts are powered by a nuclear reaction sequence known as the rapid proton capture process (rp process), which involves hundreds of exotic neutron-deficient nuclides. At so-called waiting-point nuclides, the process stalls until a slower β+ decay enables a bypass. One of the handful of rp process waiting-point nuclides is 64Ge, which plays a decisive role in matter flow and therefore the produced X-ray flux. Here we report precision measurements of the masses of 63Ge, 64,65As and 66,67Se—the relevant nuclear masses around the waiting-point 64Ge—and use them as inputs for X-ray burst model calculations. We obtain the X-ray burst light curve to constrain the neutron-star compactness, and suggest that the distance to the X-ray burster GS 1826–24 needs to be increased by about 6.5% to match astronomical observations. The nucleosynthesis results affect the thermal structure of accreting neutron stars, which will subsequently modify the calculations of associated observables. Rapid proton capture nucleosynthesis stalls at waiting-point nuclides, including 64Ge. Precision mass measurements in the vicinity of this nuclide influence state-of-the-art calculations of X-ray bursts from accreting neutron stars.

The present experiment was conducted at the Heavy Ion Research Facility in Lanzhou in operation at the Institute of Modern Physics of the Chinese Academy of Sciences 21,22 . Primary beams of 78 Kr were accelerated by a chain of machines (Fig. 2), including a sector-focusing cyclotron and a main cooler-storage ring (CSRm) operating as a heavy-ion synchrotron. Every 25 s, a new beam was extracted from the CSRm as a microsecond-long bunch of 78 Kr 19+ ions at a relativistic energy of 460 AMeV and focused onto a 15-mm-thick beryllium target in front of the fragment separator (RIBLL2).
Radionuclides were produced through projectile fragmentation nuclear reaction. In this reaction type, in principle, all the nuclei that can be obtained by removing nucleons from the projectile nucleus can be created, although the production probability rapidly decreases with the number of removed nucleons 23 . Owing to the high kinetic energies, the reaction fragments emerged from the target as fully stripped bare nuclei 24,25 . The fragments of interest were separated in flight from the remaining primary beam and intense contaminants using RIBLL2, and finally injected into the experimental cooler-storage ring (CSRe).
The RIBLL2-CSRe system was set to a fixed central magnetic rigidity of Bρ = mvγ/q = 5.528 Tm, where γ stands for the relativistic Lorentz factor; v and m/q are the velocity and mass-to-charge ratio of ions, respectively. In our experiment, q = Z. CSRe is characterized by a relatively large momentum acceptance of ±0.33%. However, due to the nuclear reaction process, the fragments have much broader momentum distributions. This setting was optimized for a mass-to-charge ratio of m/q ≈ 1.965. A cocktail beam of various produced species, including 63 Ge, 64,65 As and 66,67 Se, which match with RIBLL2-CSRe acceptance, was transmitted, injected and stored.
The spread of revolution times, ΔT/T, of the stored ions with a mass-to-charge ratio difference, Δ(m/q)/(m/q), is given as 26 ΔT T = α p Δ(m/q) (m/q) + ( γ 2 where Δv/v is the velocity spread. The momentum compaction factor α p = 1/γ 2 t , where γ t is the transition energy, is a machine parameter that depends on the ring lattice and ion optical tuning 27 . According to equation (1), the ion's revolution time is directly proportional to its mass-to-charge ratio if the second term containing the velocity spread is made negligibly small. This is the basic equation of storage-ring mass spectrometry 26,28 .
One of the ways to compensate for the inevitable velocity spread of secondary ions is to employ a special isochronous ion optical mode of the ring. In this mode, a faster particle with a given mass-to-charge ratio, m/q, revolves on a longer orbit in the ring, whereas a slower particle with the same m/q moves on a correspondingly shorter orbit, such that their revolution times are the same. The isochronous condition is fulfilled for particles that have energies corresponding to γ = γ t . In this experiment, γ t = 1.352. In the first order, this condition makes the second term on the right-hand side of equation (1) become zero. This is the basis for conventional isochronous mass spectrometry (IMS) 29,30 .
However, the condition γ = γ t is fulfilled only for a small range of mass-to-charge ratios, namely, the isochronicity window, which features the maximum resolving power 31 . To attain high mass-resolving power in the entire range of nuclear species, we succeeded in measuring the velocity of every stored ion in addition to its revolution time 32 . This enabled us to correct for 'non-isochronicity' effects and led to dramatically boosted sensitivity, efficiency and precision. We named this technique Bρ-defined IMS or Bρ-IMS (refs. 33,34), which is applied here to measure previously unknown masses (Methods).
The revolution times were measured with a couple of dedicated time-of-flight (ToF) detectors 35 (Methods). The measured revolution a sequence of proton captures and β decays along the proton drip line (Fig. 1). This line defines the limit beyond which the nuclei are unbound with respect to proton emission 4 . Thus, to unveil the mechanisms of the rp process, the properties of very exotic neutron-deficient nuclei need to be known. However, the relevant nuclei are typically characterized by short half-lives and tiny production yields in nuclear reactions utilized in terrestrial laboratories, which makes their experimental investigation extremely challenging 5 .
Waiting-point (WP) nuclei on the rp process play a particularly important role [6][7][8][9][10][11][12][13][14] . These are the nuclei where the fast proton capture reactions cannot energetically proceed further and the process stalls until a much slower β + decay enables a bypass. A sequential capture of two protons can, in some cases, allow the process to bridge the WP.
The key quantities defining the matter flow are one-proton (S p ) and two-proton (S 2p ) separation energies, which are derived from nuclear masses of the relevant nuclei (Methods).
Three long-lived WP nuclei beyond 56 Ni (proton number Z = 28), namely, 64 Ge (Z = 32), 68 Se (Z = 34) and 72 Kr (Z = 36), play a crucial role. Their long effective stellar lifetimes give rise to the observed tails in the burst light curve. Comparing the modelled light curve profiles with observations provides quantitative information about the corresponding stellar environments [15][16][17][18][19] . The rp process first encounters the exotic 64 Ge nucleus, which has a β + half-life of 63.7(25) s (ref. 20) comparable with the overall X-ray burst duration. It is, therefore, decisive to constrain the matter flow through 64 Ge.
In this work, we present the high-precision measurements of masses of 63 Ge, 64,65 As and 66,67 Se. With our new results, all the relevant separation energies around WP 64 Ge (Fig. 1) are precisely determined.

Experimental procedure
The nuclei of interest are extremely neutron deficient and have very short half-lives, spreading from 54.0(4.0) ms for 66 Se to 153.6(1.1) ms for 63 Ge (ref. 20) (Table 1). Such short-lived nuclides have to be produced at a specialized radioactive ion beam facility and promptly studied, which   The nuclides are organized according to neutron (horizontally) and proton (vertically) numbers. Nuclides whose masses were taken from the latest AME'20 database 36 , whose masses were experimentally determined or whose mass uncertainties were improved in this work are indicated in black, red and blue colours, respectively. The one-proton (S p ) and two-proton (S 2p ) separation energies (values expressed in keV) follow the same colour code. The pathway of the rp process nucleosynthesis is shown with the black arrows. The legend provides more details.
Article https://doi.org/10.1038/s41567-023-02034-2 time spectrum is illustrated in Fig. 3a. It contains all the valid particles measured within the about two-week-long experiment. Nuclides of interest with unknown masses as well as those with well-known masses are indicated in different colours. Well-known mass values from the literature with uncertainties of σ < 5 keV have been used as calibrants ( Fig. 3; blue peaks) to precisely determine the masses of 63 Ge, 64,65 As and 66,67 Se ( Fig. 3; red peaks).

Masses of 63 Ge, 64,65 As and 66,67 Se
The mass excess (in keV) is defined as ME = m − A, where m is the atomic mass (in atomic units with u = 931,494.10242 (28) keV c −2 (ref. 36)) and A = Z + N is the mass number of the nucleus. The ME values determined in this work are given in Table 1, where we employ natural units with c = ℏ = 1. The numbers of measured ions (Events) are also listed. The literature half-lives (T 1/2 ) are from the 2020 NUBASE evaluation (ref. 20) and mass values (ME AME'20 ) are from the 2020 Atomic Mass Evaluation (AME'20) (ref. 36). In addition to a compilation of all the experimentally known masses, AME'20 provides extrapolated values for yet-unknown masses. Such extrapolations are indicated with the symbol # in the table.
It is striking that only six ions of the short-lived 64 As were detected, which corresponds to a rate of about three ions per week. Still, its mass could be determined with a relative mass uncertainty of δm/m ≈ 1.8 × 10 −6 .
The mass-to-charge ratios of 66 Se 34+ and 33 Cl 17+ ions are nearly the same. These ions cannot be resolved by their revolution times and are seen as a common revolution time peak at about 635.5 ns (Fig. 3a).  (21) The following data are obtained from this work: the number of measured ions (Events), mass excess values (ME IMS ) and proton separation energies (S p ). The literature half-lives (T 1/2 ) are from the NUBASE' 20 (ref. 20). ΔME (ME IMS -ME AME '20 ) values are the differences from literature mass values (ME AME'20 ) taken from AME20 36 ). The symbol # indicates the extrapolated values in AME '20 (refs. 20,36).

SSC RIBLL2
From ion source

Fig. 2 | Schematic of the Heavy Ion Research Facility in Lanzhou.
In this experiment, the primary beam of 78 Kr 19+ was successively accelerated by the sector-focusing cyclotron (SFC) and heavy-ion synchrotron (CSRm) and then extracted to bombard the production target to produce exotic nuclei. Beams of various nuclear species including 63 Ge, 64,65 As and 66,67 Se were collected and separated with RIBLL2 and injected into CSRe, where the masses of the nuclei of interest were determined with high precision through the measurement of their revolution times and velocities applying Bρ-IMS spectrometry 33,34 . The separated-sector cyclotron (SSC) was not used in this experiment. The inset shows a schematic of the ToF detector. The secondary electrons released from the carbon foil are guided by the crossed electrostatic and magnetic fields to the MCP. The electrostatic field E is produced by three potential plates, namely, carbon foil, MCP and equalizing ring, and the magnetic field B is produced by an external dipole magnet.
The uncertainties of the ME values of 63 Ge, 65 As and 67 Se were improved by more than a factor of two. The new ME value of 65 As is in agreement, within 1.4σ uncertainty, with our previous measurement 37,38 . The previously known S p ( 65 As) = −90 (85) keV is now replaced by −221(42) keV, which firmly establishes this nucleus as proton unbound.
To check the reliability of the measured masses, we employed a standard procedure described in detail elsewhere [39][40][41][42][43][44] . Each of the reference masses (i) was supposed, one by one, to be unknown, and its mass was redetermined by using the remaining reference nuclides as calibrants. The redetermined mass excesses (ME exp,i ) and the corresponding statistical uncertainties (σ exp,i ) were utilized to calculate the normalized χ n defined as where N c is the number of calibrants and ME AME20,i and σ AME20,i denote the mass excesses and 1σ uncertainties from AME'20, respectively 20,36 . The calculated χ n = 1.066 is within the expected range of χ n = 1 ± 1/√2N c = 1 ± 0.139 at the 1σ confidence level, indicating that no additional systematic errors need to be considered.

X-ray burst simulations
The impact of the new masses has been investigated through state-of-the-art multizone X-ray burst simulations. The accreting neutron-star system properties were selected to best reproduce the 2007 bursting epoch of the well-studied 'clock burster' GS 1826- 24 (refs. 45-47). Methods provides the details. For the calculations, we replaced the corresponding nuclear masses in the AME'20 database 36 (indicated as AME'20 hereafter) with our new results obtained in this work (indicated as 'Updated').
The simulated X-ray burst light curves are shown in Fig. 4. The new nuclear masses, particularly the less bound 65 As and more bound 66 Se, result in a stalled rp process at the WP 64 Ge (Fig. 1). Single-zone calculations in another work 48 indicate that a slower flow reduces energy generation but preserves hydrogen to be burnt later in the burst, thus leading to longer bursts. We emphasize that the somewhat increased peak X-ray flux (Fig. 4) is not seen in single-zone calculations 48 , and therefore, we speculate that this is related to the convective mixing of different burning layers.
The new distributions of the elements produced through the rp nucleosynthesis ('ashes') are also modified (Fig. 4, inset). The ash abundances of Urca nuclides are particularly notable 49,50 , which may impact the interpretation of model-observation comparisons for X-ray superbursts and neutron-star crust cooling 51,52 , as well as of important electron-capture heating sources 53 . Of these, we note that our new masses result in a 17% increase in the A = 64 ash mass fraction and 14% decrease in the A = 65 ash mass fraction, which is consistent with the rp process stalling near 64 Ge. The former will result in increased electron-capture heating, whereas the latter will result in reduced Urca cooling, overall implying a somewhat warmer accreted neutron-star crust.
The X-ray burst luminosity measured by a telescope is directly proportional to the emitted light flux and inversely proportional to the square of the distance (d 2 ) and the surface gravitational redshift (1 + z) 2 of the burster, corrected by the electromagnetic wave transport efficiency [54][55][56] . Our new light curve enables us to set new constraints on the optimal d and (1 + z) parameters (Methods), which fit the observational data. The results are shown in Fig. 5a. The increased peak luminosity m/q (u e -1 ) T (ns) Counts (per 20 ps) 1  which reflects the mean number of secondary electrons released from the foil surface (equation (8)), for 37 Cl 17+ and 66 Se 34+ ions, see the peak at about 635.5 ns in a. The events belonging to each of the ion species can be clearly separated. c, Histogram of calculation with equation (7) involving the mass-to-charge ratios (m/q) of the peaks resolved in b, namely, 37 Cl 17+ and 66 Se 34+ ions.   Table 1.
The constraints on (1 + z) can be further converted into limits on neutron-star compactness (M NS /R NS ) following the approach introduced elsewhere 57 . The general relativistic neutron-star mass M GR and radius R GR are determined by (1 + z) = 1/√1 − 2GM GR /(R GR c 2 ), where c is the speed of light and G is the gravitational constant. For the choice that the Newtonian mass is equal to the general relativistic mass, M NS = M GR , then R GR = √(1 + z)R NS . The compactness constraints corresponding to the 95% confidence intervals (Fig. 5a) are shown in Fig. 5b, along with mass and radius constraints from other observational probes and theoretical limits. Our results suggest a less-compact neutron star in GS 1826-24, which will require an adjustment of the equation of state for dense matter 58 . It should be noted that several model uncertainties contribute to this compactness constraint, for example, possible alterations of the light curve due to flame spreading. As such, it is most interesting to compare the relative change in compactness bands due to a change in nuclear physics, as opposed to focusing on the absolute constraints themselves.

Outlook
Previously, compatibility with zero proton separation energy in 65 As used in a single-zone model showed that 64 Ge is not a significant WP nucleus in the rp process 37 . Within the framework of the employed multizone X-ray burst model, we arrive here at a different result. In this context, we note that although the uncertainties due to nuclear masses are largely reduced through this work, the ones related to astrophysical simulations remain. This requires further developments, for example, by accounting for three-dimensional convection 59 .
Systematic studies of the sensitivity of single-zone X-ray burst simulations to mass uncertainties 48 concluded that masses of only a few nuclei have a major effect, with 65 As and 66 Se causing the largest impact. The masses for both nuclei were measured in this work. A consequence is an increased importance to constrain nuclear physics around the following 68 Se and 72 Kr WPs, where the mass measurements of the relevant nuclides are not yet accomplished.

Online content
Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/s41567-023-02034-2. The limits for neutron-star compactness determined using other astrophysical observables are also shown, that is, from individual merger events 62,63 , quiescent low-mass X-ray binaries 64 and pulsar radio timing combined with X-ray pulse-profile modelling 65 , along with the unphysical region excluded by causality 66 (shown in black).
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Nucleon separation energies
Energy required to remove one or several nucleons from a nucleus is termed the separation energy 4 . It is given by the difference in the mass of the parent atom and the sum of the masses of the removed nucleons and daughter atom. It is noted that the atomic electrons need to be taken into account. For instance, the proton separation energy, S p (Z, N), of a nucleus with Z protons and N neutrons is defined as where m(Z, N) and m(Z − 1, N) are the masses of the parent and daughter atoms, respectively, and m H is the mass of the hydrogen atom. Correspondingly, the two-proton separation energy, S 2p , is given by Other separation energies can be analogously constructed.

Revolution time measurement
The Each detector was composed of a carbon foil, electrostatic potential electrodes and a set of microchannel plates (MCPs), directly inserted into the ultrahigh-vacuum environment of the storage ring, as well as Helmholtz coils placed outside the vacuum chamber 35,71,72 . The self-supporting foil had a thickness of 20 μg cm -2 , corresponding to about 90 nm, and a diameter of 40 mm. A sketch of the detector is given in Fig. 2 (inset).
Each ion circulating in the ring successively passed through the foils of both TOF detectors at each revolution. Secondary electrons released from the surfaces of the foils were isochronously guided by the crossed electrostatic and weak magnetic fields to the MCP. The anode signal from the MCP was continuously digitized with 50 gigasamples per second. The measurement was triggered by the injection of fresh ions into the CSRe. The measurement time was 400 μs, corresponding to about 600 revolutions of ions in the ring.
On average, 36 ions were stored in the ring in each injection. The quantities of various ion species depended only on their production yields and transmission efficiencies. The digitized data were processed to determine all the passage times. To extract timing information independent from the signal amplitudes, a constant fractional discrimination algorithm 38 has been employed. All the timing signals were sorted and unambiguously assigned to individual ions. As the final result, for every stored ion, we obtained two trains of timing signals from two detectors. By fitting these time sequences, revolution time and velocity of every ion at every revolution was determined using the procedure described in another work 32 . Depending on the number of signals, the revolution times of individual ions are typically determined with sub-picosecond precision. All the individual revolution times were put into a histogram forming an integrated time spectrum (Fig. 3a). The measured spectrum was compared with the one simulated according to equation (1) (ref. 73). Since the pattern of mass-to-charge ratios is unique, unambiguous particle identification was achieved.

Bρ-IMS technique
The modification of the present technique 33,34 , compared with conventional IMS 29,30 , is the enabled velocity measurement for each stored ion. The momentum of a particle, p, with mass m and velocity v is defined as p = mvγ, where γ = 1/√1 − v 2 /c 2 is the relativistic Lorentz factor and c is the speed of light in a vacuum. Numerous fragments are produced in the nuclear reaction and their relative momentum distributions are a few per cent broad 5 . Therefore, many different combinations of mass-to-charge ratios and velocities match the magnetic rigidity of Bρ = mvγ/q = 5.528 Tm of the RIBLL2-CSRe system. However, the isochronous condition is satisfied only for particles with γ = γ t = 1.352 (equation (1)), defining the centre of the isochronous window. The attainable mass-resolving power rapidly deteriorates outside of the isochronous window 26,27,43,74,75 . By obtaining the information on the velocity of every stored ion at each revolution, we are now able to correct for 'non-isochronicity' effects. As a result, we succeeded in improving the mass-resolving power in the isochronous window by about a factor of 2. Furthermore, this high mass-resolving power is maintained for the entire set of measured nuclei. The standard deviation of the peaks in the spectrum shown in Fig. 3a is about 1 ps, which can be translated to m/q spreads of about 5/q keV. For our mass measurements in the A ≈ 65 and Z ≈ 33 region, this corresponds to the mass-resolving power of about 160,000 (full-width at half-maximum) (refs. 33,34).
The achieved improvements boost the sensitivity, efficiency and precision of the IMS and are decisive to enable mass measurements on the very rare species reported here.
By knowing the revolution time T and velocity v of every ion, magnetic rigidity Bρ and orbit length C can be straightforwardly determined via Bρ = m q γv (5) and C = Tv. (6) Since the particles with the same Bρ must have the same mean C, a universal dependence, that is, Bρ(C), can be constructed, which is valid for all the ions. By using nuclei with well-known masses (σ < 5 keV), the correspondence of Bρ to C has been determined. We note that the reference masses typically belong to not very exotic nuclear species and are thus abundant in the spectrum (Fig. 3a). Therefore, the calibration Bρ(C) function is statistically solid and robust. Every unknown mass-to-charge ratio (m/q) i can then be derived via where index i runs over all the nuclear species of interest. Equation (7) is the basic formula of Bρ-IMS (refs. 33,34).

Mass of 66 Se
The mass-to-charge ratios of 66 Se 34+ and 33 Cl 17+ are nearly identical. Their revolution time distributions completely overlap. To obtain the m/q value of 66 Se, we utilized the dependence of timing-signal amplitudes on the ion charge 41 . The ions with a larger charge cause, on average, more electrons to be emitted from the foil surface, and thus, the amplitudes of the individual signals increase with the ion charge. We introduced a charge-dependent parameter U defined as where ϵ is the detection efficiency for a specific ion, A is the average amplitude of the timing signals, N is the number of accomplished revolutions and A i is the amplitude of each timing signal. Index i runs over all the timing signals for this specific ion. A scatter plot of U versus m/q is shown in Fig. 3c. The group of counts corresponding to 66 Se 34+ can be clearly separated from the ones for 37 Cl 17+ (Fig. 3b), and hence, the mass-to-charge ratio can be precisely determined.