Reaction rates of {}^{64}Ge(p,\gamma){}^{65}As and {}^{65}As(p,\gamma){}^{66}Se and the extent of nucleosynthesis in type I X-ray bursts

Reaction rates of Ge(,)As and As(,)Se and the extent of nucleosynthesis in type I X-ray bursts

Y.H. Lam1, J.J. He1, A. Parikh2,3, H. Schatz4, B.A. Brown4
M. Wang1, B. Guo5, Y.H. Zhang1, X.H. Zhou1, H.S. Xu1;;

The extent of nucleosynthesis in models of type I X-ray bursts and the associated impact on the energy released in these explosive events are sensitive to nuclear masses and reaction rates around the Ge waiting point. Using the well known mass of Ge, the recently measured As mass, and large-scale shell model calculations, we have determined new thermonuclear rates of the Ge(,)As and As(,)Se reactions with reliable uncertainties. The new reaction rates differ significantly from previously published rates. Using the new data we analyze the impact of the new rates and the remaining nuclear physics uncertainties on the Ge waiting point in a number of representative one-zone X-ray burst models. We find that in contrast to previous work, when all relevant uncertainties are considered, a strong Ge -process waiting point cannot be ruled out. The nuclear physics uncertainties strongly affect X-ray burst model predictions of the synthesis of Zn, the synthesis of nuclei beyond , energy generation, and burst light curve. We also identify key nuclear uncertainties that need to be addressed to determine the role of the Ge waiting point in X-ray bursts. These include the remaining uncertainty in the As mass, the uncertainty of the Se mass, and the remaining uncertainty in the As(,)Se reaction rate, which mainly originates from uncertain resonance energies.

nuclear reactions, nucleosynthesis, abundances — stars: neutron — X-rays: bursts

1Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China \move@AU\move@AF\@affiliation  2Departament de Física i Enginyeria Nuclear, EUETIB, Universitat Politècnica de Catalunya, Barcelona E-08036, Spain \move@AU\move@AF\@affiliation  3Institut d’Estudis Espacials de Catalunya, Barcelona E-08034, Spain \move@AU\move@AF\@affiliation  4Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824-1321, USA \move@AU\move@AF\@affiliation  5China Institute of Atomic Energy, P. O. Box 275(10), Beijing 102413, China

1 Introduction

A type I X-ray burst (XRB) arises from a thermonuclear runaway in the accreted envelope of a neutron star in a close binary star system (for reviews, see, e.g., Lewin et al. (1993); Schatz et al. (1998); Strohmayer & Bildsten (2006); Parikh et al. (2013)). Roughly 100 bursting systems have been discovered to date, with light curves exhibiting peak luminosities of  10–10 and timescales of 10–100 s. During an XRB, models predict that a H/He-rich accreted envelope may become strongly enriched in heavier nuclei though the -process and the -process (Wallace & Woosley 1981; Schatz et al. 1998). These two processes involve -particle-induced or proton-capture reactions on stable and radioactive nuclei, interrupted by occasional -decays. When the -process approaches the proton dripline, successive capture of protons by nuclei is inhibited by a strong reverse photodisintegration reaction rate. The competition between the rate of proton capture and the rate of -decay at these “waiting points” (e.g., Zn, Ge, and Se) determines the extent of the synthesis of heavier mass nuclei during the burst (Schatz et al. 1998). Peak temperatures during the thermonuclear runaway may approach or exceed 1 GK, resulting in the synthesis of nuclei up to mass  (Schatz et al. 2001; Elomaa et al. 2009). Model predictions depend, however, on astrophysical parameters such as accretion rate, the composition of the accreted material, and the neutron star surface gravity, as well as on nuclear physics quantities such as nuclear masses and reaction rates.

The Ge(,)As and As(,)Se reactions have been demonstrated to have a significant impact on nucleosynthesis during XRBs. (See Parikh et al. (2014) for a recent review of the impact of nuclear physics uncertainties on predicted yields and light curves from XRB models.) Direct measurements of these reactions at the relevant energies in XRBs are not yet possible due to the lack of sufficiently intense radioactive Ge and As beams. Moreover, due to the unknown mass of Se and the lack of nuclear structure information for states within  1–2 MeV of the Ge+ and the (theoretical) As+ energy thresholds in As and Se, respectively, it is not possible to estimate rates for these reactions based on experimental nuclear structure data. As a result, XRB models use Ge(,)As and As(,)Se thermonuclear rates derived from theoretical calculations. Using such models it has been demonstrated that varying the As(,)Se rate by a factor of ten at the relevant temperatures affects the calculated abundances of nuclei between  65–100 by factors as large as about 5 (Parikh et al. 2008). For Ge(,)As, models have illustrated the importance of the -value (or proton separation energy ) adopted for this reaction, with variations by 300 keV affecting final calculated abundances between  65–100 by factors as large as about 5 (Parikh et al. 2008, 2009). In addition, the effective -process lifetime of the waiting-point nucleus Ge was investigated by Schatz (2006) based on the estimated proton separation energies of (As)=-0.360.15 MeV and (Se)=2.430.18 MeV, derived from Coulomb mass shift calculations (Brown et al. 2002). It was found that the effective lifetime of Ge for a given temperature and proton density is mainly determined by the values of As and Se and the proton capture rate on As.

Recently, precise mass measurements of nuclei along the -process path have become available. The mass of Ge has been measured at the Canadian Penning Trap at Argonne National Laboratory (Clark et al. 2007) and the LEBIT Penning Trap facility at Michigan State University (Schury et al. 2007). More recently, the mass of As has been measured at the HIRFL-CSR (Cooler-Storage Ring at the Heavy Ion Research Facility in Lanzhou) (Xia et al. 2002) using IMS (Isochronous Mass Spectrometry). The measurements can be combined to obtain an experimental proton separation energy for As of  keV (Tu et al. 2011), where the uncertainty is dominated by the uncertainty in the As mass. The mass of Se is not known experimentally. The extrapolated value predicted by AME2012 results in Se keV. With the new mass of As, X-ray burst model calculations (Tu et al. 2011) suggested that Ge may not be a significant -process waiting point, contrary to previous expectations (Schatz et al. 1998; Woosley et al. 2004; Fisker et al. 2008; Parikh et al. 2009; José et al. 2010). We revisit this question here using our new nuclear reaction rates.

Thermonuclear Ge(,) and As(,) reaction rates were first estimated by Van Wormer et al. (1994) based entirely on the properties of the mirror nuclei Ge and Ge, respectively. values of As and Se were estimated to be 0.169 MeV and 1.909 MeV, respectively. Later on, both rates have been calculated (Rauscher & Thielemann 2000) with the statistical Hauser-Feshbach formalism (NON-SMOKER (Rauscher & Thielemann 1998)) using the masses of As and Se predicted by the finite-range droplet (FRDM) (Möller et al. 1995) and ETSFIQ (Pearson et al. 1996) mass models. Recently, the statistical model calculations have been updated using new predictions for the As and Se proton separation energies (see JINA REACLIB111 (Cyburt et al. 2010)). The predicted rates differ from one another by up to several orders of magnitude over typical XRB temperatures. Moreover, the reliability of statistical model calculations for these rates is questionable due to the low compound nucleus level densities, especially for Ge(,), but also for As(,).

In this work we refer to previously available rates using the nomenclature adopted in the JINA REACLIB database. The laur rate refers to the rate estimated by Van Wormer et al. (1994); the rath rate was calculated by Rauscher & Thielemann (2000). The rath, thra, rpsm rates are the statistical-model calculations with FRDM, ETSFIQ, as well as Audi & Wapstra (1995) estimated masses, respectively. The recent ths8 rate is from Cyburt et al. (2010).

Here we determine new thermonuclear Ge(,)As and As(,)Se reaction rates using the updated values of As and Se together with new nuclear structure information from large-scale shell-model calculations. Using the new data we fully characterize the nuclear physics uncertainties that affect the -process through Ge and reexamine the question of the Ge waiting point.

2 Reaction rate calculations

The total thermonuclear proton capture reaction rate consists of the sum of resonant- and direct-capture (DC) on ground state and thermally excited states in the target nucleus, weighted with their individual population factors (Fowler et al. 1964; Rolfs & Rodney 1988). It can be calculated by the following equation:

with the parameters defined by Schatz et al. (2005).

2.1 Resonant rates

For isolated narrow resonances, the resonant reaction rate for capture on a nucleus in an initial state , , can be calculated as a sum over all relevant compound nucleus states above the proton threshold (Rolfs & Rodney 1988; Iliadis 2007). It can be expressed by the following equation (Schatz et al. 2005):

where the resonance energy in the center-of-mass system, , is calculated from the excitation energies of the initial and compound nucleus state. For the ground-state capture, the resonance energy is represented by . is the temperature in Giga Kelvin (GK) and is the reduced mass of the entrance channel in atomic mass units (, with the target mass number). In Eq. LABEL:eq2, the resonance energy and strength are in units of MeV. The resonance strength is defined by


where is the target spin and , , , and are spin, proton decay width, -decay width, and total width of the compound nucleus state , respectively. The total width is given by , because other decay channels are closed (Audi et al. 2012) in the excitation energy range considered in this work.

  [a] [MeV] (MeV)[d] [eV] [eV] [eV]
[a] [b] (Ge)[c]
  3/2 0.000 0.000 0.000 0.090 2 0.196 0.00 0.00
  5/2[e] 0.187(3) 0.103 0.111 0.277 1 0.533 5.1110 2.4610
  5/2 0.501 0.605 0.591 1 0.010 4.7310 1.1310
  5/2 0.863 0.953 1 0.014 4.2410 4.8910
  7/2 0.947 0.890 1.037 1 0.013 2.4310 4.8810
  7/2[f] 1.070 1.155 1.160 1 0.002 2.1110 3.2710
measured by Obertelli et al. (2011); calculated by the present shell model; compiled by Browne & Tuli (2010); calculated by with MeV (Tu et al. 2011); calculated and based on the experimental value of  MeV for this state; negligible contribution to the rate for temperatures up to 2 GK.
TABLE 1: Properties of As for the ground-state capture utilized in the present Ge(,)As resonant rate calculation. The energy levels in the mirror nucleus Ge are listed in the 4 column for comparison.

The proton width can be estimated by the following equation,


where denotes a proton-transfer spectroscopic factor, while is a single-proton width for capture of a proton on an quantum orbital. The are obtained from proton scattering cross sections calculated with a Woods-Saxon potential (Richter et al. 2011; Brown 2014). Alternatively, the proton partial widths may also be calculated by the following expression (Van Wormer et al. 1994; Herndl et al. 1995),


Here, fm (with  fm) is the nuclear channel radius. The Coulomb penetration factor is given by


where is the wave number with energy in the center-of-mass (c.m.) system; and are the regular and irregular Coulomb functions, respectively. The proton widths given by these two methods (i.e., by Eqs. 4 and 5) agree well with each other, with a maximum difference of about 35%.

The key ingredients necessary to estimate the resonant Ge(,) and As(,) rates are energy levels in As and Se, proton transfer spectroscopic factors, and proton and gamma-ray partial widths. For As, only a single level has been observed at  keV (Obertelli et al. 2011). For Se, one level has been confirmed at  keV, and indications for two other levels at 2064(3) keV and 3520(4) keV have been reported, with tentative assignments of (4) and (6), respectively (Obertelli et al. 2011; Ruotsalainen et al. 2013). There are no more experimental data available for these two nuclei. In this work, we have calculated the energy levels, spectroscopic factors and gamma widths within the framework of a large-scale shell model, without truncation, using the shell-model code NuShellX@MSU (Brown & Rae 2014). The effective interaction GXPF1a (Honma et al. 2004, 2005) has been utilized for these two pf-shell nuclei.

The widths, , have been calculated from the electromagnetic reduced transition probabilities (), which carry the nuclear structure information of the resonant states and the final bound states (Brussaard & Glaudemans 1977). The reduced transition rates are computed within the shell model. Most of the transitions in this work are of and types. The relations are (Herndl et al. 1995):




The values have been obtained from empirical effective charges, , , whereas the values have been obtained with a four-parameter set of empirical -factors, i.e. , and ,  (Honma et al. 2004).

In addition, we have also included proton resonant captures on the thermally excited target states. Since the first-excited state in Ge is quite high ( keV), thermal excitation can be neglected for typical X-ray burst temperatures. For the As(,)Se rate, we included proton capture on the first four thermally excited states of As (i.e., on the 0.187, 0.501, 0.863 and 0.947 MeV states listed in Table 1). Capture on thermally excited states contributes at most 38% to the total capture rate at 2 GK (Lam et al. 2016). The properties of As and Se for the ground-state captures are summarized in Table 1 and Table 2, respectively. In addition, the properties of Se for the first-excited-state capture (the major thermally excited-state contribution) are summarized in Table 3.

  [a] [MeV] [MeV][c] [eV] [eV] [eV]
[b] [a]
   0.000 0.000 0.746
   0.929(7) 0.982
   1.130 0.013
   1.952 0.232
   1.989 0.269
   2.053 0.333[e]
   2.064(3)[d] 2.110 0.344
   2.102 0.382
   2.277 0.557[e]
   2.419 0.699
   2.474 0.754[e]
   2.498 0.778
   2.556 0.836[e]
   2.668 0.948
   2.740 1.020
   2.781 1.061[e]
   2.865 1.145
   2.867 1.147
   2.882 1.162
   2.888 1.168
   2.907 1.187
   2.949 1.229
   2.969 1.249
   2.998 1.278
   3.114 1.394
   3.217 1.497
   3.231 1.511
   3.237 1.517
   3.274 1.554
   3.299 1.579
   3.328 1.608
   3.330 1.610
   3.357 1.637
   3.394 1.674
   3.424 1.704
   3.487 1.767
   3.499 1.779
calculated by the present shell model; measured by Obertelli et al. (2011) and Ruotsalainen et al. (2013); calculated by with MeV (AME2012); calculated and based on the experimental value of  MeV for this state; resonances dominantly contributing to the rate within temperature region of 0.2–2 GK.
TABLE 2: Properties of Se for the ground-state capture utilized in the present As(,)Se resonant rate calculation.
  [a] [MeV] [MeV][c] [eV] [eV] [eV]
[b] [a]
   1.952 0.045 0.002 0.035 0.000 0.013
   1.989 0.082 0.244
   2.053 0.146 0.000 0.019 0.088 0.098
   2.064(3)[d] 2.110 0.157 0.001 0.016 0.002
   2.102 0.195 0.001 0.011 0.000 0.001
   2.277 0.370 0.001 0.038 0.020 0.184
   2.419[e] 0.512 0.001 0.083 0.004 0.226
   2.474 0.567 0.000 0.000 0.002
   2.498 0.591 0.000 0.043 0.208
   2.556 0.649 0.001 0.026 0.191 0.008
   2.668[e] 0.761 0.004 0.107 0.000 0.002
   2.740 0.833 0.002 0.011 0.001
   2.781 0.874 0.001 0.032 0.002
   2.865[e] 0.958 0.000 0.002 0.324 0.054
   2.867 0.960 0.000 0.000 0.004
   2.882[e] 0.975 0.001 0.008 0.001 0.035
   2.888 0.981 0.089
   2.907 1.000 0.000 0.000 0.518
   2.949 1.042 0.000 0.006 0.012 0.007
   2.969[e] 1.062 0.002 0.003 0.001 0.025
   2.998[e] 1.091 0.003 0.025 0.000
   3.114[e] 1.207 0.001 0.061 0.103
   3.217 1.310 0.000 0.019 0.016 0.001
   3.231 1.324 0.025
   3.237 1.330 0.001 0.007 0.000 0.053
   3.274 1.367 0.002 0.002 0.002
   3.299 1.392 0.000 0.003 0.026
   3.328 1.421 0.001 0.073 0.003 0.019
   3.330 1.423 0.000 0.015 0.036 0.001
   3.357 1.450 0.000 0.020 0.001 0.006
   3.394 1.487 0.002 0.000
   3.424 1.517 0.000 0.013 0.145
   3.487 1.580 0.000 0.051 0.000 0.001
   3.499 1.592 0.001 0.006 0.041 0.008
calculated by the present shell model; measured by Obertelli et al. (2011) and Ruotsalainen et al. (2013); calculated by in units of MeV, with MeV (AME2012); calculated and based on the experimental value of  MeV for this state; resonances dominantly contributing to the rate within temperature region of 0.2–2 GK.
TABLE 3: Properties of Se for the first-excited-state capture utilized in the present As(,)Se resonant rate.

Peak temperatures in recent hydrodynamic XRB models have approached 1.5–2 GK (Woosley et al. 2004; José et al. 2010). At such temperatures, resonant rates for the Ge(,) and As(,) reactions are expected to be dominated by levels with  MeV (i.e., Gamow energy (Rolfs & Rodney 1988)). This means that excitation energy regions of up to  2.5 MeV for As, and up to  4.2 MeV for Se should be considered in the resonant rate calculations for Ge(,) and As(,), respectively. In the present shell-model calculations, the maximum excitation energies considered for As and Se are 1.07 MeV and 3.50 MeV, respectively (see Tables 1 and 2). For Ge(,) only resonances up to =1.035 MeV contribute significantly to the reaction rate up to 2 GK; for As(,) only five resonances (at =0.333, 0.557, 0.754, 0.836 and 1.061 MeV) dominate the total resonant rate within the temperature region of 0.2–2 GK. Those 21 resonances above =1.061 MeV make only negligible contributions to the total reaction rate up to 2 GK. Therefore, the contributions from the levels presented in Tables 1 and 2 should be adequate to account for these two resonant rates at XRB temperatures.

2.2 Direct-capture rates

The nonresonant direct-capture (DC) rate for proton capture can be estimated by the following expression (Angulo et al. 1999; Schatz et al. 2005),

with being the atomic number of either Ge or As. The effective astrophysical -factor at the Gamow energy , i.e., , can be expressed by (Fowler et al. 1964; Rolfs & Rodney 1988),


where (0) is the -factor at zero energy, and the dimensionless parameter is given numerically by for the proton capture.

In this work, we have calculated the DC -factors with the RADCAP code (Bertulani et al. 2003). The Woods-Saxon nuclear potential (central + spin orbit) and a Coulomb potential of uniform-charge distribution were utilized in the calculation. The nuclear central potential was determined by matching the bound-state energies. The spectroscopic factors were taken from the shell model calculation and are listed in Table 1 and Table 2. The optical-potential parameters (Huang et al. 2010) are  fm,  fm, with a spin-orbit potential depth of MeV. Here, , , and are the radii of central potential, the spin-orbit potential and the Coulomb potential, respectively; and are the corresponding diffuseness parameters in the central and spin-orbit potentials, respectively.

For the As(,)Se reaction, (0) values for DC captures into the ground state and the first-excited state (=929 keV) in Se are calculated to be 8.3 and 3.5 MeVb, respectively. The total DC rate for this reaction is only about 0.1% that of the resonant one at 0.05 GK. For the Ge(,)As reaction, we find a DC (0) value for this reaction of about 35 MeVb. The DC contribution is only about 0.3% even at the lowest temperature of 0.06 GK. Even when considering estimated upper limits to the DC contribution (He et al. 2014), the resonant contributions still dominate the total rates above 0.06 GK and 0.05 GK for Ge(,)As and As(,)Se reactions, respectively. The probabilities of populating the first-excited states in Ge (=902 keV) and As (=187 keV) relative to the ground states at temperatures below 0.1 GK are extremely small, and hence contributions of the direct-capture from these excited states can be neglected.

3 Results

The resulting total thermonuclear Ge(,)As and As(,)Se rates are listed in Table 4 as functions of temperature. The present (Present, hereafter) rates can be parameterized by the standard format of Rauscher & Thielemann (2000). For Ge(,)As, we find:

with a fitting error of less than 0.3% at 0.1–2 GK; for As(,)Se, we find:

with a fitting error of less than 0.4% at 0.1–2 GK. We emphasize that the above fits are only valid with the stated error over the temperature range of 0.1–2 GK. Above 2 GK, one may, for example, match our rates to statistical model calculations (see e.g., NACRE by Angulo et al. (1999)).

Ge(,)As As(,)Se
   lower upper lower upper