Reaction Rates of 64Ge(p,g)65As and 65As(p,g)66Se and the Extent of Nucleosynthesis in Type I X-Ray Bursts

arXiv:1505.02275v2 [astro-ph.SR] 29 Jan 2016 Summitted to ApJ 2015 May 15; accepted 2015 December 30 Preprint typeset using L A T EX style emulateapj v. 08/22/09 REACTION RATES OF 64GE(P,γ)65AS AND 65AS(P,γ)66SE 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 1Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China 2Departament de F´ısica i Enginyeria Nuclear, EUETIB, Universitat Polit`ecnica de Catalunya, Barcelona E-08036, Spain 3Institut d’Estudis Espacials de Catalunya, Barcelona E-08034, Spain 4Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824-1321, USA and 5China Institute of Atomic Energy, P. O. Box 275(10), Beijing 102413, China Summitted to ApJ 2015 May 15; accepted 2015 December 30 ABSTRACT 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 64Ge waiting point. Using the well known mass of 64Ge, the recently measured 65As mass, and largescale shell model calculations, we have determined new thermonuclear rates of the 64Ge(p,γ)65As and 65As(p,γ)66Se 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 64Ge waiting point in a number of representative onezone X-ray burst models. We find that in contrast to previous work, when all relevant uncertainties are considered, a strong 64Ge rp-process waiting point cannot be ruled out. The nuclear physics uncertainties strongly affect X-ray burst model predictions of the synthesis of 64Zn, the synthesis of nuclei beyond A= 64, energy generation, and burst light curve. We also identify key nuclear uncertainties that need to be addressed to determine the role of the 64Ge waiting point in X-ray bursts. These include the remaining uncertainty in the 65As mass, the uncertainty of the 66Se mass, and the remaining uncertainty in the 65As(p,γ)66Se reaction rate, which mainly originates from uncertain resonance energies. Subject headings: nuclear reactions, nucleosynthesis, abundances — stars: neutron — X-rays: bursts 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 Lpeak ≈104–105L⊙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 αp-process and the rp-process (Wallace & Woosley 1981;Schatz et al. 1998). These two processes involve α-particle-induced or proton-capture reactions on stable and radioactive nuclei, interrupted by occasional β-decays. When the rpprocess 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 β-decay at these “waiting points” (e.g., 60Zn, 64Ge, and 68Se) 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 A≈100 (Schatz et al. 2001;Elomaa et al. 2009). Model predictions depend, however, on astrophysical pa- Electronic address: [email protected]; [email protected]; sc[email protected] rameters 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 64Ge(p,γ)65As and 65As(p,γ)66Se 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 64Ge and 65As beams. Moreover, due to the unknown mass of 66Se and the lack of nuclear structure information for states within ≈1–2 MeV of the 64Ge+pand the (theoretical) 65As+p energy thresholds in 65As and 66Se, respectively, it is not possible to estimate rates for these reactions based on experimental nuclear structure data. As a result, XRB models use 64Ge(p,γ)65As and 65As(p,γ)66Se thermonuclear rates derived from theoretical calculations. Using such models it has been demonstrated that varying the 65As(p,γ)66Se rate by a factor of ten at the relevant temperatures affects the calculated abundances of nuclei between A≈65–100 by factors as large as about 5 (Parikh et al. 2008). For 64Ge(p,γ)65As, models have illustrated the importance of the Q-value (or proton separation energy Sp) adopted for this reaction, with variations by ±300 keV affecting final calculated abundances between A≈65–100 by factors as large as about 5 (Parikh et 2 Y.H. Lam et al. al. 2008,2009). In addition, the effective rp-process lifetime of the waiting-point nucleus 64Ge was investigated by Schatz (2006) based on the estimated proton separation energies of Sp(65As)=-0.36±0.15 MeV and Sp(66Se)=2.43±0.18 MeV, derived from Coulomb mass shift calculations (Brown et al. 2002). It was found that the effective lifetime of 64Ge for a given temperature and proton density is mainly determined by the Spvalues of 65As and 66Se and the proton capture rate on 65As. Recently, precise mass measurements of nuclei along the rp-process path have become available. The mass of 64Ge 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 65As 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 65As of Sp=−90 ±85 keV (Tu et al. 2011), where the uncertainty is dominated by the uncertainty in the 65As mass. The mass of 66Se is not known experimentally. The extrapolated value predicted by AME2012 results in Sp(66Se) = 1720 ±310 keV. With the new mass of 65As, X-ray burst model calculations (Tu et al. 2011) suggested that 64Ge may not be a significant rp-process waiting point, contrary to previous expectations (Schatz et al. 1998;Woosley et al. 2004;Fisker et al. 2008;Parikh et al. 2009;Jos´e et al. 2010). We revisit this question here using our new nuclear reaction rates. Thermonuclear 64Ge(p,γ) and 65As(p,γ) reaction rates were first estimated by Van Wormer et al. (1994) based entirely on the properties of the mirror nuclei 65Ge and 66Ge, respectively. Spvalues of 65As and 66Se 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 65As and 66Se predicted by the finite-range droplet (FRDM) (M¨oller et al. 1995) and ETSFIQ (Pearson et al. 1996) mass models. Recently, the statistical model calculations have been updated using new predictions for the 65As and 66Se proton separation energies (see JINA REACLIB1(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 64Ge(p,γ), but also for 65As(p,γ). 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). 1http://groups.nscl.msu.edu/jina/reaclib/db Here we determine new thermonuclear 64Ge(p,γ)65As and 65As(p,γ)66Se reaction rates using the updated Sp values of 65As and 66Se 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 rp-process through 64Ge and reexamine the question of the 64Ge 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: NAhσvi=X i (NAhσvii r+NAhσvii DC)(2Ji+ 1)e−Ei/kT Pn(2Jn+ 1)e−En/kT (1) 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 i,NAhσvii r, can be calculated as a sum over all relevant compound nucleus states jabove the proton threshold (Rolfs & Rodney 1988;Iliadis 2007). It can be expressed by the following equation (Schatz et al. 2005): NAhσvii r= 1.54 ×1011(µT9)−3/2 ×X j ωγijexp −11.605Eij T9[cm3s−1mol−1], (2) where the resonance energy in the center-of-mass system, Eij =Ej−Sp−Ei, is calculated from the excitation energies of the initial Eiand compound nucleus Ejstate. For the ground-state capture, the resonance energy is represented by Ei r=Ej x−Sp.T9is the temperature in Giga Kelvin (GK) and µis the reduced mass of the entrance channel in atomic mass units (µ=AT/(1 + AT), with ATthe target mass number). In Eq. 2, the resonance energy and strength are in units of MeV. The resonance strength ωγ is defined by ωγij =2Jj+ 1 2(2Ji+ 1) Γij p×Γj γ Γj total .(3) where Jiis the target spin and Jj, Γij p, Γj γ, and Γj total are spin, proton decay width, γ-decay width, and total width of the compound nucleus state j, respectively. The total width is given by Γj total = Γj γ+PiΓij p, because other decay channels are closed (Audi et al. 2012) in the excitation energy range considered in this work. The proton width can be estimated by the following equation, Γp=X nlj C2S(nlj) Γsp(nlj),(4) Reaction rates of 64Ge(p,γ)65As(p,γ)66Se ... 3 TABLE 1 Properties of 65As for the ground-state capture utilized in the present 64Ge(p,γ)65As resonant rate calculation. The energy levels in the mirror nucleus 65Ge are listed in the 4th column for comparison. Jπ i[a] Ex[MeV] Er(MeV)[d] nlj C2SΓp[eV] Γγ[eV] ωγ [eV] Eexp x[a] Etheo x[b] Ex(65Ge)[c] 3/2− 10.000 0.000 0.000 0.090 2p3/20.196 1.19 ×10−34 0.00 0.00 5/2− 1[e] 0.187(3) 0.103 0.111 0.277 1f5/20.533 8.19 ×10−17 5.11×10−72.46×10−16 5/2− 20.501 0.605 0.591 1f5/20.010 3.76 ×10−10 4.73×10−51.13×10−9 5/2− 30.863 0.953 1f5/20.014 1.64 ×10−64.24×10−44.89×10−6 7/2− 10.947 0.890 1.037 1f7/20.013 1.28 ×10−52.43×10−44.88×10−5 7/2− 2[f] 1.070 1.155 1.160 1f7/20.002 8.50 ×10−62.11×10−43.27×10−5 [a] measured by Obertelli et al. (2011); [b] calculated by the present shell model; [c] compiled by Browne & Tuli (2010); [d] calculated by Er=Ex−Spwith Sp=−0.09 MeV (Tu et al. 2011); [e] calculated Erand Γpbased on the experimental value of Ex= 0.187 MeV for this state; [f] negligible contribution to the rate for temperatures up to 2 GK. where C2S(nlj) denotes a proton-transfer spectroscopic factor, while Γsp is a single-proton width for capture of a proton on an (nlj) quantum orbital. The Γsp 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), Γp=3~2 µR2Pℓ(E)C2S. (5) Here, R=r0×(1 + AT)1/3fm (with r0= 1.25 fm) is the nuclear channel radius. The Coulomb penetration factor Pℓis given by Pℓ(E) = kR F2 ℓ(E) + G2 ℓ(E),(6) where k=√2µE/~is the wave number with energy Ein the center-of-mass (c.m.) system; Fℓand Gℓare 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 64Ge(p,γ) and 65As(p,γ) rates are energy levels in 65As and 66Se, proton transfer spectroscopic factors, and proton and gamma-ray partial widths. For 65As, only a single level has been observed at Ex= 187(3) keV (Obertelli et al. 2011). For 66Se, one level has been confirmed at Ex= 929(7) 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 γwidths, Γγ, have been calculated from the electromagnetic reduced transition probabilities B(Ji→ Jf;L), 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 M1 and E2 types. The relations are (Herndl et al. 1995): ΓE2[eV] = 8.13 ×10−7E5 γ[MeV]B(E2)[e2fm4],(7) and ΓM1[eV] = 1.16 ×10−2E3 γ[MeV]B(M1)[µ2 N].(8) The B(E2) values have been obtained from empirical effective charges, ep= 1.5e,en= 0.5e, whereas the B(M1) values have been obtained with a four-parameter set of empirical g-factors, i.e. gs p= 5.586, gs n=−3.826 and gl p= 1, gl n= 0 (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 64Ge is quite high (Ex= 902 keV), thermal excitation can be neglected for typical X-ray burst temperatures. For the 65As(p,γ)66Se rate, we included proton capture on the first four thermally excited states of 65As (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 65As and 66Se for the ground-state captures are summarized in Table 1and Table 2, respectively. In addition, the properties of 66Se for the first-excited-state capture (the major thermally excited-state contribution) are summarized in Table 3. Peak temperatures in recent hydrodynamic XRB models have approached 1.5–2 GK (Woosley et al. 2004;Jos´e et al. 2010). At such temperatures, resonant rates for the 64Ge(p,γ) and 65As(p,γ) reactions are expected to be dominated by levels with Er≤2.5 MeV (i.e., Gamow energy (Rolfs & Rodney 1988)). This means that excitation energy regions of up to Ex≤2.5 MeV for 65As, 4 Y.H. Lam et al. TABLE 2 Properties of 66Se for the ground-state capture utilized in the present 65As(p,γ)66Se resonant rate calculation. Jπ i[a] Ex[MeV] Er[MeV][c] C2S7/2C2S3/2C2S5/2C2S1/2Γγ[eV] Γp[eV] ωγ [eV] Eexp x[b] Etheo x[a] (l= 3) (l= 1) (l= 3) (l= 1) 0+ 10.000 0.000 — 0.746 — — — 2+ 10.929(7) 0.982 — 0.037 0.069 0.119 0.045 2.765 ×10−4— — 0+ 21.130 — 0.013 2.020 ×10−8— — 2+ 21.552 — 0.002 0.011 0.071 0.051 1.402 ×10−4— — 0+ 31.575 — 0.002 2.056 ×10−6— — 2+ 31.952 0.232 0.001 0.001 0.180 0.000 4.040 ×10−43.079 ×10−20 1.924 ×10−20 0+ 41.989 0.269 0.002 1.441 ×10−84.797 ×10−18 5.996 ×10−19 2+ 42.053 0.333[e] 0.002 0.005 0.027 0.025 2.218 ×10−43.517 ×10−14 2.198 ×10−14 4+ 12.064(3)[d] 2.110 0.344 0.008 0.004 8.382 ×10−42.301 ×10−16 2.589 ×10−16 3+ 12.102 0.382 0.001 0.000 0.017 3.835 ×10−51.057 ×10−14 9.249 ×10−15 2+ 52.277 0.557[e] 0.000 0.004 0.015 0.001 1.751 ×10−42.205 ×10−91.378 ×10−9 3+ 22.419 0.699 0.000 0.004 0.063 2.361 ×10−41.781 ×10−71.557 ×10−7 1+ 12.474 0.754[e] 0.016 0.000 0.052 3.579 ×10−31.152 ×10−54.306 ×10−6 4+ 22.498 0.778 0.001 0.115 2.653 ×10−41.888 ×10−72.122 ×10−7 2+ 62.556 0.836[e] 0.001 0.021 0.033 0.032 9.915 ×10−46.723 ×10−53.935 ×10−5 3+ 32.668 0.948 0.000 0.065 0.145 1.550 ×10−47.969 ×10−41.136 ×10−4 4+ 32.740 1.020 0.002 0.040 8.404 ×10−48.287 ×10−69.232 ×10−6 1+ 22.781 1.061[e] 0.060 0.351 0.002 5.338 ×10−34.483 ×10−39.137 ×10−4 2+ 72.865 1.145 0.000 0.000 0.000 0.028 1.175 ×10−35.315 ×10−36.014 ×10−4 1+ 32.867 1.147 0.065 0.001 0.222 2.170 ×10−35.887 ×10−27.847 ×10−4 3+ 42.882 1.162 0.002 0.003 0.002 1.209 ×10−39.439 ×10−44.639 ×10−4 0+ 52.888 1.168 0.629 1.716 ×10−41.985 ×10−12.143 ×10−5 4+ 42.907 1.187 0.000 0.005 3.139 ×10−41.001 ×10−51.091 ×10−5 2+ 82.949 1.229 0.000 0.003 0.005 0.016 9.006 ×10−41.085 ×10−25.197 ×10−4 3+ 52.969 1.249 0.000 0.006 0.000 7.804 ×10−44.817 ×10−35.877 ×10−4 4+ 52.998 1.278 0.000 0.079 1.392 ×10−34.711 ×10−43.960 ×10−4 4+ 63.114 1.394 0.000 0.157 1.374 ×10−33.262 ×10−31.088 ×10−3 2+ 93.217 1.497 0.000 0.003 0.021 0.036 1.534 ×10−33.221 ×10−19.539 ×10−4 0+ 63.231 1.511 0.002 2.561 ×10−32.244 ×10−22.873 ×10−4 3+ 63.237 1.517 0.000 0.000 0.006 1.586 ×10−35.258 ×10−31.066 ×10−3 1+ 43.274 1.554 0.005 0.072 0.010 3.113 ×10−32.111 ×10−11.151 ×10−3 4+ 73.299 1.579 0.000 0.008 9.407 ×10−41.075 ×10−35.644 ×10−4 3+ 73.328 1.608 0.001 0.004 0.022 1.139 ×10−38.762 ×10−29.842 ×10−4 2+ 10 3.330 1.610 0.008 0.083 0.001 0.012 2.089 ×10−32.222 ×10+0 1.304 ×10−3 2+ 11 3.357 1.637 0.001 0.000 0.027 0.114 1.348 ×10−32.761 ×10+0 8.423 ×10−4 5+ 13.394 1.674 0.000 1.355 ×10−31.460 ×10−41.812 ×10−4 4+ 83.424 1.704 0.000 0.018 1.417 ×10−35.639 ×10−31.274 ×10−3 3+ 83.487 1.767 0.000 0.000 0.056 4.881 ×10−32.792 ×10−23.635 ×10−3 2+ 12 3.499 1.779 0.006 0.060 0.001 0.041 4.335 ×10−47.440 ×10+0 2.709 ×10−4 [a]calculated by the present shell model; [b]measured by Obertelli et al. (2011) and Ruotsalainen et al. (2013); [c]calculated by Er=Ex−Spwith Sp= 1.720 MeV (AME2012); [d]calculated Erand Γpbased on the experimental value of Ex= 2.064 MeV for this state; [e]resonances dominantly contributing to the rate within temperature region of 0.2–2 GK. and up to Ex≤4.2 MeV for 66Se should be considered in the resonant rate calculations for 64Ge(p,γ) and 65As(p,γ), respectively. In the present shell-model calculations, the maximum excitation energies considered for 65As and 66Se are 1.07 MeV and 3.50 MeV, respectively (see Tables 1and 2). For 64Ge(p,γ) only resonances up to Er=1.035 MeV contribute significantly to the reaction rate up to 2 GK; for 65As(p,γ) only five resonances (at Er=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 Er=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 1and 2should 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), NAhσvii DC = 7.83 ×109ZT µT 2 91/3 Si DC(E0) ×exp "−4.249 Z2 Tµ T91/3#[cm3s−1mol−1], (9) with ZTbeing the atomic number of either 64Ge or 65As. The effective astrophysical S-factor at the Gamow energy E0, i.e., Si DC(E0), can be expressed by (Fowler et al. Reaction rates of 64Ge(p,γ)65As(p,γ)66Se ... 5 TABLE 3 Properties of 66Se for the first-excited-state capture utilized in the present 65As5/2− 1(p,γ)66Se resonant rate. Jπ i[a] Ex[MeV] Er[MeV][c] C2S7/2C2S3/2C2S5/2C2S1/2Γγ[eV] Γp[eV] ωγ [eV] Eexp x[b] Etheo x[a] (l= 3) (l= 1) (l= 3) (l= 1) 2+ 31.952 0.045 0.002 0.035 0.000 0.013 4.040 ×10−44.401 ×10−56 1.834 ×10−56 0+ 41.989 0.082 0.244 1.441 ×10−82.398 ×10−40 1.998 ×10−41 2+ 42.053 0.146 0.000 0.019 0.088 0.098 2.218 ×10−44.499 ×10−26 1.875 ×10−26 4+ 12.064(3)[d] 2.110 0.157 0.001 0.016 0.002 8.382 ×10−41.261 ×10−25 9.458 ×10−26 3+ 12.102 0.195 0.001 0.011 0.000 0.001 3.835 ×10−54.039 ×10−22 2.356 ×10−22 2+ 52.277 0.370 0.001 0.038 0.020 0.184 1.751 ×10−44.549 ×10−12 1.895 ×10−12 3+ 22.419[e] 0.512 0.001 0.083 0.004 0.226 2.361 ×10−41.862 ×10−81.086 ×10−8 1+ 12.474 0.567 0.000 0.000 0.002 3.579 ×10−35.149 ×10−11 1.287 ×10−11 4+ 22.498 0.591 0.000 0.043 0.208 2.653 ×10−46.133 ×10−84.599 ×10−8 2+ 62.556 0.649 0.001 0.026 0.191 0.008 9.915 ×10−43.292 ×10−71.371 ×10−7 3+ 32.668[e] 0.761 0.004 0.107 0.000 0.002 1.550 ×10−42.983 ×10−51.459 ×10−5 4+ 32.740 0.833 0.002 0.011 0.001 8.404 ×10−41.353 ×10−59.987 ×10−6 1+ 22.781 0.874 0.001 0.032 0.002 5.338 ×10−39.809 ×10−52.408 ×10−5 2+ 72.865[e] 0.958 0.000 0.002 0.324 0.054 1.175 ×10−36.788 ×10−41.793 ×10−4 1+ 32.867 0.960 0.000 0.000 0.004 2.170 ×10−32.502 ×10−76.254 ×10−8 3+ 42.882[e] 0.975 0.001 0.008 0.001 0.035 1.209 ×10−37.087 ×10−42.607 ×10−4 0+ 52.888 0.981 0.089 1.716 ×10−48.636 ×10−66.852 ×10−7 4+ 42.907 1.000 0.000 0.000 0.518 3.139 ×10−46.881 ×10−54.233 ×10−5 2+ 82.949 1.042 0.000 0.006 0.012 0.007 9.006 ×10−46.740 ×10−41.606 ×10−4 3+ 52.969[e] 1.062 0.002 0.003 0.001 0.025 7.804 ×10−41.754 ×10−33.151 ×10−4 4+ 52.998[e] 1.091 0.003 0.025 0.000 1.392 ×10−32.795 ×10−36.970 ×10−4 4+ 63.114[e] 1.207 0.001 0.061 0.103 1.374 ×10−33.138 ×10−29.872 ×10−4 2+ 93.217 1.310 0.000 0.019 0.016 0.001 1.534 ×10−33.314 ×10−26.107 ×10−4 0+ 63.231 1.324 0.025 2.561 ×10−32.545 ×10−41.929 ×10−5 3+ 63.237 1.330 0.001 0.007 0.000 0.053 1.586 ×10−31.026 ×10−19.112 ×10−4 1+ 43.274 1.367 0.002 0.002 0.002 3.113 ×10−34.840 ×10−34.737 ×10−4 4+ 73.299 1.392 0.000 0.003 0.026 9.407 ×10−41.039 ×10−26.470 ×10−4 3+ 73.328 1.421 0.001 0.073 0.003 0.019 1.139 ×10−34.284 ×10−16.629 ×10−4 2+ 10 3.330 1.423 0.000 0.015 0.036 0.001 2.089 ×10−37.650 ×10−28.473 ×10−4 2+ 11 3.357 1.450 0.000 0.020 0.001 0.006 1.348 ×10−31.496 ×10−15.568 ×10−4 5+ 13.394 1.487 0.002 0.000 1.355 ×10−33.592 ×10+3 1.242 ×10−3 4+ 83.424 1.517 0.000 0.013 0.145 1.417 ×10−31.591 ×10−11.054 ×10−3 3+ 83.487 1.580 0.000 0.051 0.000 0.001 4.881 ×10−39.800 ×10−12.833 ×10−3 2+ 12 3.499 1.592 0.001 0.006 0.041 0.008 4.335 ×10−42.659 ×10−11.803 ×10−4 [a]calculated by the present shell model; [b]measured by Obertelli et al. (2011) and Ruotsalainen et al. (2013); [c]calculated by Er=Ex−Sp−0.187 in units of MeV, with Sp= 1.720 MeV (AME2012); [d]calculated Erand Γpbased on the experimental value of Ex= 2.064 MeV for this state; [e]resonances dominantly contributing to the rate within temperature region of 0.2–2 GK. 1964;Rolfs & Rodney 1988), Si DC(E0) = Si(0) 1 + 5 12τ,(10) where Si(0) is the S-factor at zero energy, and the dimensionless parameter τis given numerically by τ= 4.2487(Z2 Tµ/T9)−1/3for the proton capture. In this work, we have calculated the DC S-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 V0was determined by matching the bound-state energies. The spectroscopic factors were taken from the shell model calculation and are listed in Table 1and Table 2. The optical-potential parameters (Huang et al. 2010) are R0=Rs.o.=RC= 1.25 ×(1 + AT)1/3fm, a0=as.o.= 0.65 fm, with a spin-orbit potential depth of Vs.o.=−10 MeV. Here, R0,Rs.o., and RCare the radii of central potential, the spin-orbit potential and the Coulomb potential, respectively; a0and as.o.are the corresponding diffuseness parameters in the central and spin-orbit potentials, respectively. For the 65As(p,γ)66Se reaction, S(0) values for DC captures into the ground state and the first-excited state (Ex=929 keV) in 66Se are calculated to be 8.3 and 3.5 MeV·b, respectively. The total DC rate for this reaction is only about 0.1% that of the resonant one at 0.05 GK. For the 64Ge(p,γ)65As reaction, we find a DC S(0) value for this reaction of about 35 MeV·b. 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 64Ge(p,γ)65As and 65As(p,γ)66Se reactions, respectively. The probabilities of populating the first-excited states in 64Ge (Ex=902 keV) and 65As 6 Y.H. Lam et al. (Ex=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 64Ge(p,γ)65As and 65As(p,γ)66Se rates are listed in Table 4as functions of temperature. The present (Present, hereafter) rates can be parameterized by the standard format of Rauscher & Thielemann (2000). For 64Ge(p,γ)65As, we find: NAhσvi= exp(−78.204 −13.819 T9 +12.211 T1/3 9 + 81.566T1/3 9−13.138T9+ 1.717T5/3 9−16.149 lnT9) + exp(−93.260 −10.059 T9 +15.189 T1/3 9 + 61.887T1/3 9+ 21.717T9−8.625T5/3 9−22.943 lnT9) + exp(−75.104 −3.788 T9 +19.347 T1/3 9 + 52.700T1/3 9−32.227T9+ 14.766T5/3 9+ 1.270 lnT9), with a fitting error of less than 0.3% at 0.1–2 GK; for 65As(p,γ)66Se, we find: NAhσvi= exp(−111.177 −2.639 T9 +50.997 T1/3 9 + 106.669T1/3 9−64.623T9+ 13.521T5/3 9+ 31.256 lnT9) + exp(−124.702 −12.436 T9 +52.765 T1/3 9 + 89.593T1/3 9−12.219T9+ 0.456T5/3 9−2.886 ln T9) + exp(−116.814 −5.202 T9 +63.424 T1/3 9 + 48.281T1/3 9+ 83.320T9−188.849T5/3 9+ 21.362 lnT9), 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)). Figure 1shows the comparison of the Present 64Ge(p,γ)65As rate with others compiled in JINA REACLIB: rpsm,rath,thra,laur, and ths8. Note that only the rpsm rate uses an Sp(65As) value that is within 1σof the recently determined experimental value. The Present rate differs significantly from others in the temperature region of interest in XRBs. The disagreement, in particular with the rpsm rate, demonstrates that the statisticalmodel is not applicable for this reaction owing to the low density of excited states in 65As. Similarly, the comparison of the Present 65As(p,γ)66Se rate with other rates available in the JINA REACLIB: rpsm,rath,thra,laur, and ths8, is presented in Fig. 2. Only the laur and rpsm rates use Sp(66Se) values that are within 1σof the currently accepted value. Although the Present rate differs significantly from the others, especially at lower and higher temperature regions, it is consistent with all others within the remaining large uncertainties. At T >1 GK, the laur rate is the lowest rate simply because only three excited states were considered by Van Wormer et al. (1994). It should be noted that Fig. 1.— Reaction rates of the 64Ge(p,γ)65As reaction (in units of cm3mol−1s−1). The Present rate (red line) together with the upper and lower limits deduced from uncertainties are shown by a (green) colored band. Other available rates from JINA REACLIB (Cyburt et al. 2010) are shown for comparison. See details in the text and Table 4. (A color version of this figure is available in the online journal.) Reaction rates of 64Ge(p,γ)65As(p,γ)66Se ... 7 TABLE 4 Thermonuclear 64Ge(p,γ)65As and 65As(p,γ)66Se rates in units of cm3s−1mol−1. 64Ge(p,γ)65 As 65As(p,γ)66Se T9NAhσvilower upper NAhσvilower upper 0.2 4.79 ×10−17 3.99 ×10−20 1.23 ×10−15 1.86 ×10−16 9.97 ×10−18 1.45 ×10−15 0.3 1.33 ×10−13 8.44 ×10−15 7.66 ×10−13 2.21 ×10−12 1.79 ×10−12 1.92 ×10−11 0.4 3.08 ×10−11 9.92 ×10−13 6.56 ×10−11 1.96 ×10−98.15 ×10−10 8.26 ×10−9 0.5 1.89 ×10−91.98 ×10−10 4.30 ×10−91.51 ×10−73.03 ×10−84.65 ×10−7 0.6 5.47 ×10−81.42 ×10−88.93 ×10−83.08 ×10−63.31 ×10−77.55 ×10−6 0.7 6.86 ×10−72.61 ×10−79.38 ×10−72.93 ×10−51.73 ×10−65.91 ×10−5 0.8 4.62 ×10−62.15 ×10−66.52 ×10−61.69 ×10−46.47 ×10−62.94 ×10−4 0.9 2.02 ×10−58.05 ×10−63.52 ×10−56.86 ×10−42.18 ×10−51.08 ×10−3 1.0 6.53 ×10−52.17 ×10−51.39 ×10−42.15 ×10−37.04 ×10−53.19 ×10−3 1.1 1.69 ×10−44.69 ×10−54.30 ×10−45.55 ×10−32.12 ×10−47.96 ×10−3 1.2 3.69 ×10−48.83 ×10−51.09 ×10−31.23 ×10−24.67 ×10−41.74 ×10−2 1.3 7.10 ×10−41.50 ×10−42.38 ×10−32.42 ×10−29.89 ×10−43.41 ×10−2 1.4 1.24 ×10−32.34 ×10−44.62 ×10−34.33 ×10−21.90 ×10−36.11 ×10−2 1.5 1.98 ×10−33.42 ×10−48.16 ×10−37.18 ×10−23.33 ×10−31.02 ×10−1 1.6 2.98 ×10−34.73 ×10−41.33 ×10−21.12 ×10−15.69 ×10−31.59 ×10−1 1.7 4.25 ×10−36.28 ×10−42.04 ×10−21.65 ×10−19.38 ×10−32.34 ×10−1 1.8 5.78 ×10−38.02 ×10−42.97 ×10−22.34 ×10−11.49 ×10−23.31 ×10−1 1.9 7.58 ×10−39.93 ×10−44.13 ×10−23.18 ×10−12.22 ×10−24.50 ×10−1 2.0 9.63 ×10−31.20 ×10−35.52 ×10−24.19 ×10−13.21 ×10−25.93 ×10−1 the shell model calculation provides the first reliable estimate of the uncertainty of the 65As(p,γ) reaction rate, especially as the Hauser-Feshbach rates may suffer from unknown systematic errors due to the limited applicability of the statistical model near the proton drip line. Uncertainties for the Present 64Ge(p,γ)65As and 65As(p,γ)66Se rates were estimated by considering the uncertainties in the Spvalues (±85 keV for 65As and ±310 keV for 66Se) and estimated uncertainties in the calculated level energies (±168 keV for both 65As and Fig. 2.— As Fig. 1, but for the 65As(p,γ)66Se reaction. Note the rpsm rate is quite close to the rath,ths8 and thra rates at different temperature regions (Cyburt et al. 2010), and hence the corresponding line is not clearly visible. See details in the text and Table 4. (A color version of this figure is available in the online journal.) 66Se (Honma et al. 2002)2). These were added in quadrature to give uncertainties of ±188 keV and ±353 keV for the resonance energies Erof 64Ge(p,γ)65As and 65As(p,γ)66Se, respectively. Note that for the two known levels, i.e., Ex=187 keV in 65As and Ex=2064 keV in 66Se, an experimental excitation energy uncertainty of ±3 keV is used instead. For 64Ge(p,γ)65As, all resonance strengths ωγ are proportional to Γpsince Γp≪Γγ (see Table 1and Eq. 3). The uncertainties in Γp(or ωγ) owing to the uncertainties in Erare calculated based on the energy dependence expressed in Eq. 5. In the case of 65As(p,γ)66Se, only five resonances (at Er=0.333, 0.557, 0.754, 0.836 and 1.061 MeV dominate the resonant rate within the temperature region of 0.2–2 GK. Here, the resonant strengths ωγ are proportional to Γpfor the first four resonances, while the strength of the last resonance at 1.061 MeV (dominating over ∼0.9–2 GK) depends on both, Γpand Γγ. Uncertainties in Γγcan be neglected because of the much larger rate uncertainties caused by Erand Γp. The uncertainties for all resonances listed in Table 2are considered in the present calculations. 4. ASTROPHYSICAL IMPLICATION We examine the impact of our new 64Ge(p,γ) and 65As(p,γ) rates and their uncertainties on the rp-process using one-zone XRB models. Post processing calculations using temperature and density trajectories from the literature enable a quick assessment of the impact of nuclear physics changes on the strength of the 64Ge waiting point using the A= 64 abundance, and on the burst energy generation rate. We use the post-processing approach for the K04 X-ray burst model (Parikh et al. 2The 66Zn case is studied with the present model space and interaction, and an rms deviation between the experimental and calculated level energies is found to be about 140 keV (Lam et al. 2016). 8 Y.H. Lam et al. 0.1 1 10 D (Present) laur rath rpsm thra ths8 0.1 1 10 100 9080706050 Mass Number A 0.1 1 10 X / X Present (b) (c) (a) Fig. 3.— Final abundances, as mass fractions X, following one-zone XRB calculations using the K04 thermodynamic history (Parikh et al. 2008,2009). Results using rates determined in the Present work and in JINA REACLIB: rpsm,rath,thra,laur, and ths8 are indicated. Panels (a) and (b) show the effects of using either different 64Ge(p,γ) or different 65As(p,γ) rates, respectively, while panel (c) shows the effect of using different 64Ge(p,γ) and 65As(p,γ) rates together. The impact of the uncertainties in the Present 64Ge(p,γ) and 65As(p,γ) rates (see Figs. 1and 2) is indicated as ∆(Present) in panels (a) and (b). (A color version of this figure is available in the online journal.) 2008,2009). However, postprocessing calculations do not take into account the changes in temperature and density that result from the energy generation changes. They can therefore not predict reliably the quantitative impact on produced abundances and light curves. To account for this effect, we also use the full one-zone X- ray burst model (Schatz et al. 2001), which represents a more extreme burst with very hydrogen rich ignition. 4.1. Post processing results for K04 With the representative K04 thermodynamic history (Parikh et al. 2008,2009), final abundances (as mass fractions X) and the nuclear energy generation rate Egen during a burst have been studied by performing separate XRB model calculations with different rates. In the K04 model, the peak temperature Tpeak = 1.4 GK is similar to those reached at the base of the envelope in comparable hydrodynamic XRB models (e.g., 1.3 GK in Jos´e et al. (2010)). Figs. 3and 4compare results for Xand Egen using rates from the present work to results using rates available in JINA REACLIB: laur, rath, rpsm, thra, ths8. The impact of (a) using different 64Ge(p,γ) rates (with the 65As(p,γ) rate held constant at the Present value), (b) using different 65As(p,γ) rates (with the 64Ge(p,γ) rate held constant at the Present value) and (c) using different 64Ge(p,γ) and 65As(p,γ) rates together, is indicated in each of the two figures. For each change in reaction rate, the corresponding inverse reaction rate is also changed to maintain detailed balance. This inverse rate strongly depends on the adopted reaction Sp-value for the respective forward rate. As we compare the impact of different rates that have been determined using very different Spvalues (see Sect. 2 and Figs. 1 and 2), the results illustrate not only the influence of the rate calculation, but also the influence of different forward to reverse rate ratios due to different Sp-values. The results of Figs. 3and 4are interesting, but not entirely unexpected. For the case of the 64Ge(p,γ) reaction, an equilibrium between the rates of the forward (p,γ) and reverse (γ,p) processes is quickly established due to the small (p,γ)Q-value relative to kT at XRB temperatures: at 1 GK, kT ≈100 keV. As a result, it is the (p,γ)Q-value, rather than the actual 64Ge(p,γ) rate, that is the most important nuclear physics quantity needed to characterize the equilibrium abundance of 65As (and the subsequent flow of material to heavier nuclei through the 65As(p,γ) reaction), c.f. Schatz et al. (1998); Iliadis (2007); Parikh et al. (2009). This is nicely illustrated in Fig. 3(a): the 64Ge(p,γ) rates adopting positive (p,γ)Q-values (thra, laur, rath) give relatively lower final abundances around A= 64 and larger abundances at higher masses precisely because of the larger equilibrium abundances of 65As during the burst, allowing for increased flows of abundances to higher masses via the 65As(p,γ) reaction. On the other hand, the opposite is true for those rates adopting negative (p,γ)Q-values (ths8, Present, rpsm) because of the larger equilibrium abundances of 64Ge and lower relative abundances of 65As during the burst. Indeed, the summed mass fractions of species with A > 70, X70, vary considerably for different choices of Q-values. For example, when the thra, ths8 or Present rates are adopted, X70 = 0.58, 0.21, or 0.33, respectively. A consequence of the increased flow of abundances to heavier nuclei is seen in Fig. 4(a), where the models adopting the thra, laur, and rath rates give the largest Egen at late times due to energy released from the decay of the larger amounts of heavy nuclei produced during the burst. As expected, the opposite is true for Egen in the models using the ths8, Present and rpsm rates. We note that the predictions for Egen at late times vary rather significantly between the models using the different rates, with differences as large as a factor of ≈2. For the case of the 65As(p,γ) reaction, where a large positive (p,γ)Q-value is adopted in all rate estimates, the importance of the actual rate is illustrated in Fig. 3(b). The model adopting the largest rate at the most relevant temperatures (T > 0.9 GK, c.f. Jos´e et al. (2010)), Reaction rates of 64Ge(p,γ)65As(p,γ)66Se ... 9 10 16 10 17 10 18 1 10 10 16 10 17 10 18 10 16 10 17 10 18 Present laur rath rpsm thra ths8 10 16 10 17 10 10 16 10 17 10 16 10 17 (a) (b) (c) Time (s) E gen (erg g -1 s -1 ) Fig. 4.— As Fig. 3, but for nuclear energy generation rates Egen during the burst in the K04 XRB model. Panels (a) and (b) show the effects of using either different 64Ge(p,γ) or different 65As(p,γ) rates, respectively, while panel (c) shows the effect of using different 64Ge(p,γ) and 65As(p,γ) rates together. The impact of the uncertainties in the Present 64Ge(p,γ) and 65As(p,γ) rates (see Figs. 1and 2) is indicated in panels (a) and (b). Panels to the right show expanded views of the panels to the left. (A color version of this figure is available in the online journal.) rath, gives the lowest abundances around A= 64 and the largest abundances at higher masses. Again, as expected, the opposite is true for the model using the lowest 65As(p,γ) rate, laur. The variation in X70 for different choices of the 65As(p,γ) rate is significant, with X70 = 0.49 with the rath rate, and 0.29 with the laur rate. The behavior of Egen for these models is again in accord with the distributions of the final abundances, with the largest Egen at late times arising from the model using the rath rate, and the lowest Egen arising from the model using the laur rate. Finally, the effects of using different 64Ge(p,γ) and 65As(p,γ) rates from the same theoretical model calculation are shown in Figs. 3(c) and 4(c). This reveals the impact of competing influences from these two rates. For example, the model using the ths8 64Ge(p,γ) rate gave the lowest relative abundances at higher masses (see Fig. 3(a)), while the model using the ths8 65As(p,γ) rate gave among the highest relative abundances at higher masses (see Fig. 3(b)). When these two rates are used together, the combined impact on the final abundances (and Egen) is, not surprisingly, moderated. Fig. 3(c) also shows that using the Present 64Ge(p,γ) and 65As(p,γ) rates results in the strongest 64Ge waiting point, and the lowest final abundances at the highest masses. Differences with respect to predictions using rates from JINA REACLIB are as large as a factor of ≈7 at individual values of A.X70 calculated with the two Present rates differs by as much as factor of 1.8 from models using the other rates. We have also examined the impact on XRB model predictions of our uncertainties in the Present 64Ge(p,γ) and 65As(p,γ) rates, as shown in Figs. 1and 2. Reverse rates for the lower and upper forward rates were determined using exactly the Q-values adopted for the corresponding forward rate calculations. Panels (a) and (b) of Figs. 3 and 4show how these rate uncertainties affect final abundances and Egen in the K04 model, with mass fractions above A= 64 varying by up to a factor of 3 due to the individual uncertainties in the rates, X70 varying by up to a factor of 2, and Egen varying by up to 35% at late times. The impact on Xand Egen of the uncertainties