65As(p, γ)66Se for type-I x-ray bursts

We present a new set of 64Ge(p, γ)65As and 65As(p, γ)66Se reaction rates based on recently evaluated proton separation energies S p(As) and S p(Se), and nuclear structure data from large-scale shell model calculations. Our new 64Ge(p,g) rate differs from those available in REACLIB by up to two orders of magnitude at temperatures encountered within type I X-ray bursts. We used one-zone post-processing type-I x-ray burst model to test our new rates, and present the astrophysical impact of these rates.


Introduction
Type I x-ray burst (XRB) is generated in the accreted envelope of a neutron star in a close binary star system during thermonuclear explosion.Presently, about 100 bursters have been discovered.Based on the characteristics of discovered bursters, their light curves have typical peak luminosities of roughly 10 4 − −10 5 L , timescales of 10-100 s, and recurrence time of one to several hours, see Ref. [1] and references therein.
Several models estimated that an accreted envelope rich in H/He develops to be intensely enriched in heavier mass nuclei through the α-particle-induced (αp) and rapid-proton-capture (rp) reactions on stable and radioactive nuclei during an XRB [2,3].In addition, β-decays may occasionally occur based on the properties of the created nuclear species during the abundance flow.When the rp-process advances to the proton dripline, capture of subsequent protons by nuclei is hindered by photodisintegration, (γ, p), which has an almost equal reaction rate as the respective rp-process.Hence, abundances accumulate at these particular proton-rich nuclei, namely "waiting points".In XRBs, beta decay of these waiting point nuclei may compete with the rate of proton capture.Such competitions play their role in extending nucleosynthesis to heavier nuclei up to mass A ≈ 100 during XRB.Some pivotal waiting points, e.g., 64 Ge, 68 Se, 72 Kr, were predicted by Schatz et al. [3] and the importance of long-lived 64 Ge was reassured by Woosley et al. [4] in a simulation of 15 sequential bursts.Various models based on multiple factors, like the accretion rate, the composition of the accreted material, the mass and equation of state assumed for the neutron star, and the nuclear masses and reaction rates, predict 65 Ar(p, γ) is one of the important reactions in XRB nucleosynthesis.These models varying the 65 As(p,γ) 66 Se rate by a factor of 10 at the respective temperature range cause the end-stage abundances of 65 A 100 to vary by factors as large as ∼5 [5].A recent review of Parikh et al. [6] also showed the importance of 64 Ge(p,γ) 65 As and 65 As(p,γ) 66 Se reactions with respect to uncertainty of rates and their impact on nucleosynthesis during XRBs.However, up to now, there is no direct measurement of these reactions at the respective energies in XRBs.Secondly, the mass of 66 Se is unmeasured.Thirdly, the energy levels of both 65 As and 66 Se nuclei up to 1 E x 2 MeV are also unmeasured.In such missing important nuclear physics input, most of the XRB simulations use 64 Ge(p,γ) 65 As and 65 As(p,γ) 66 Se reaction rates derived from statistical-model.Recently, the previously unknown proton separation energy of 65 As, S p ( 65 As) was measured to be −90 ± 85 keV [7] at the HIRFL-CSR (Cooler-Storage Ring at the Heavy Ion Research Facility in Lanzhou) [8] in an IMS (Isochronous Mass Spectrometry) mode.With the new S p ( 65 As) value, the X-ray burst model employed in Ref. [7] proposed that 64 Ge is not a significant rp-process waiting point.This suggestion is different from the previous expectations [3,4,[9][10][11].
This present work investigates the above controversy with a new set of thermonuclear 64 Ge(p,γ) 65 As and 65 As(p,γ) 66 Se reaction rates based on the updated S p ( 65 As), newly evaluated S p ( 66 Se), see Atomic Mass Evaluation (AME2012) [12], and the nuclear structure information from large-scale shell-model calculations.The astrophysical impact of our new rates and other available rates, e.g.rates from Van Wormer et al. [13], from statistical-model (Hauser-Feshbach formalism) NON-SMOKER code [14], and from JINA REACLIB [15] 1 , has been checked using the post-processing one-zone XRB model.

Reaction rates 2.1 The formalism
The total reaction rate includes the resonant and direct-capture rates of proton capture on ground state and all thermally excited states in the target nucleus weighted with their individual population factors [16,17].We calculated the direct-capture rates for the 64 Ge(p,γ) 65 As and 65 As(p,γ) 66 Se reactions by using the Woods-Saxon nuclear potential and a Coulomb potential of uniform-charge distribution embedded in RADCAP code [18].We found that the ratio of direct-capture contributions to resonant contributions are less than about 0.01 for T ≤ 0.05 GK, and for T ≥ 0.05 GK, the total rates are dictated by resonant contributions for both reactions.Therefore, we only assume the resonant contributions to be the total reaction rates for both rp processes.
We employed the narrow resonance formalism [17,19], to obtain the resonant rate.The reduced mass μ is defined as A T /(1+A T ), with A T the target mass, and resonant strength ωγ and the resonant energy E r are in units of MeV.The resonant strength ωγ is defined by of which the J T and J are the spins of the target and resonant state, respectively.Γ p is the proton width for the entrance channel, and Γ γ is the gamma (γ) width for the exit channel, and other decay channels are closed [20] thus the total width Γ tot ≈ Γ p + Γ γ .The proton width is defined as of which θ 2 (nl j) is proton-transfer spectroscopic factors, and Γ sp is single-proton widths [21].The Γ sp are obtained from proton scattering cross sections calculated with Woods-Saxon potential well [22].
The other alternative to compute the proton partial widths is, of which R = r 0 × (1 + A T ) 1/3 fm (with r 0 = 1.25 fm) is the nuclear channel radius, see Refs [13,23] for definition of notations.Both methods (i.e., by Eq. 2 and Eq. 3) obtained the proton widths in a maximum difference of about 35%.
The main constituents needed to obtain the resonant 64 Ge(p,γ) and 65 As(p,γ) rates are energy levels of 65 As and 66 Se, proton transfer spectroscopic factors, and proton and γ-ray partial widths.However, only has a single level been observed at E x = 187(3) keV [24] for 65 As; whereas one level has been confirmed at E x = 929 keV, and two other levels were tentatively assigned at 2064 keV (4 + ) and 3520 keV (6 + ) for 66 Se [24,25].We obtained other levels up to Gamow energy window, spectroscopic factors and γ widths with the large-scale shell model using NuShellX@MSU [26].The nuclear wave functions have been computed from numerical diagonalization of isospin-conserving Hamiltonian of p f -shell nuclei, namely GXPF1a [27,28], without truncation.The total γ width [17,23] for every considered level consists of B(E2) and B(M1) matrix elements.Both matrix elements of every electromagnetic transition have been obtained based on the empirical effective charges and empirical quenching factors defined in Ref. [27].For the 65 As(p,γ) 66 Se reaction rates, we have also considered proton captures on the first few excited states of 65 As.The first set of the properties of every resonance of both reactions were shown in Tables 1 and 2 of Ref. [29].
Recent hydrodynamic XRB models have approached maximum temperatures in the range of 1.5 T [GK] 2 [4,11].Resonant rates of the 64 Ge(p,γ) and 65 As(p,γ) reactions at this temperature range tend to be dominated by levels below ≈ 2.5 MeV of the respective proton thresholds.There are some high-spin levels below this thresholds referring to the mirror nuclei of 64 Ge and 65 As, i.e. 65 Ge and 66 Ge, respectively.However, the total contribution from these high-spin states to the total resonant rates is negligible because of the small proton partial widths.

2.2
The Present 64 Ge(p, γ) 65 As and 65 As(p, γ) 66 Se reaction rates Both thermonuclear 64 Ge(p,γ) and 65 As(p,γ) reaction rates were first computed by Van Wormer et al. [13] by referring to the properties of the mirror nuclei 65 Ge and 66 Ge, including the S p values.Then, both rates have been revisited by Rauscher and Thielemann [30] with a statistical-model (Hauser-Feshbach formalism) [14] using the masses of 65 As and 66 Se estimated by the finite-range droplet macroscopic model (FRDM) [31] and ETSFIQ mass model [32].The other set of theoretical rates are recently compiled and available online, i.e.JINA REACLIB [15].However, the estimated rates above differ from one another by up to several orders of magnitude over the respective XRB temperature range mainly due to the selected proton separation energies.Also, the level densities of excited states in both mirrors 65 Ge and 66 Ge are not high, and thus such low-density properties are also expected in 65 As and 66 Se near their proton thresholds.Hence, the reliability of the statistical-model calculations for both reactions may be questionable.The uncertainty in the Present 64 Ge(p,γ) rate arises from the uncertainty in S p ( 65 As) along with an uncertainty of +/-100 keV for every 65 As energy level calculated with the shell model.The uncertainty in the Present 65 As(p,γ) rate arises simply from the large uncertainty of +/-310 keV in S p ( 66 Se).

The comparison of rp-reaction rates
In the following discussion, we use the nomenclature mentioned in JINA REACLIB database: the laur rate -the rate estimated by Van Wormer [13]; the rath rate -the rate calculated by Raucher and Thielemann [30]; the rath, thra, rpsm rates -the statistical-model calculations with FRDM, ETSFIQ, and estimated masses from Audi and Wapstra [33], respectively; and the ths8 rate -the recent theoretical rate from Rauscher [15].

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The comparison of the Present 64 Ge(p,γ) 65 As rate with others compiled in JINA REACLIBrpsm, rath, thra, laur, and ths8, is shown in Fig. 1.Significantly, the Present rate deviates from others in the temperature range of interest in XRBs.Although the S p value adopted in rpsm is almost within 1σ error of the value that we used for the Present rate based on the recent measurement [7], the rpsm is not close to our Present rate.This deviation manifests the weakness of statistical-model in describing reaction that has low-density of excited states, particularly 65 As.
A similar comparison of the Present 65 As(p,γ) 66 Se rate with other rates from JINA REACLIB is presented in Fig. 2. The S p values adopted in rath, thra, and ths8 are beyond 1σ error of the data compiled in AME2012, except those adopted in laur and rpsm.In the temperature regime beyond 1 GK, the Present rate deviates significantly from others, whereas laur is the lowest rate because only were three excited states taken into account [13].

Astrophysical implication
We used the one-zone K04 model [5,10] to check the astrophysical impact of our Present 64 Ge(p,γ) and 65 As(p,γ) rates, especially the final abundances (as mass fractions X) and the nuclear energy generation rate, E gen , during an XRB.In this proceeding, we compare the final abundances and E gen produced from Present rates to results obtained from rates available in JINA REACLIB: laur, rath, rpsm, thra, ths8, c.f. Figs.3(c) and 4(c) of Ref. [29], respectively.
Fluxes of reactions are rather similar before the 64 Ge waiting point, but tremendous increments happen on net fluxes toward higher masses for rates of JINA REACLIB.The model using both Present 64 Ge(p,γ) and 65 As(p,γ) rates predicts a very remarkable lowest final abundances at the highest masses, c.f. Fig. 3(c) of Ref. [29].Comparing to predictions using rates from JINA REACLIB, the differences are as large as a factor of ≈ 7 at individual values of mass A. Consequently, the estimated E gen using both Present rates is among the lowest at late times, c.f. Fig. 4(c) of Ref. [29].
Furthermore, the model predicted a large depletions of A = 64 using other rates, except using the Present or the ths8 rates.The underlying reason is because using the Present rates predicts A = 64 to be the largest mass fraction of all XRB nucleosynthesis products.José et al. also predicted an almost similar outcome using 1-D hydrodynamic XRB models, and the largest predicted mass fractions are A = 60 and 64 [11].

Summary and perspectives
We have obtained new thermonuclear rates for the 64 Ge(p,γ) 65 As and 65 As(p,γ) 66 Se reactions based on large-scale shell model calculations and proton separation energies, S p ( 65 As) and S p ( 66 Se) derived from a recent mass measurement of the 65 As [7] and AME2012 [12].The 64 Ge(p,γ) 65 As and 65 As(p,γ) 66 Se rates are lower than other rates up to a factor of ≈ 6 and differs by up to a factor of ≈ 3 from other rates presented in the literature, respectively.
Secondly, we have used an one-zone type-I X-ray burst model to check the impact of the Present rates and to compare the end-stage abundances and E gen with results from different available rates.Our new rates strongly quench the production of nuclide toward A ≈ 100, which reverses the recent claim that 64 Ge is not a significant rp-process waiting point in Ref. [7], but agrees with the previous predictions [3,4,[9][10][11].
In the following improvement of the new rates, we will fold the uncertainty of S p values with the root-mean-square deviation value of the employed shell-model Hamiltonian and will also check with other possibly dominant uncertainties which may alter the upper and lower limits of the Present rates.Moreover, we will also examine the new rates with more type-I X-ray burst models.

DOI: 10
.1051/ C Owned by the authors, published by EDP Sciences, 201

Figure 1 .
Figure 1. 64Ge(p,γ) 65 As reaction rates.The Present rate is shown as a green band with the upper and lower limits.See details in the text.