1 Introduction

Type I X-ray bursts (XRBs) are triggered by thermonuclear runaways in the accreted envelopes of neutron stars in close binary systems [1, 2]. During the thermonuclear runaway, the accreted envelope enriched in H and He can be transformed to matter strongly enriched in heavier species (up to A≃100 [3, 4]) via the rapid proton capture process (rp-process) [5-7]. The physical theory of the XRBs has been reviewed in Refs. [8-10].

The rp-process is mostly characterized by localized (p,γ)–(γ,p) equilibrium within particular isotonic chains near the proton drip-line. The abundance distribution within an isotonic chain exponentially depends on the difference in nuclear masses because the abundance ratio between two neighboring isotones is proportional to exp[S_p/kT], where S_p is the proton separation energy. In particular, isotonic chains with sufficiently small S_p values (relative to XRB temperatures at 1 GK and kT≈100 keV) need to be be known with a precision of at least ∼50–100 keV [6, 11]. To compare model predictions with observations of the light curves [12], reliable nuclear physical inputs, such as the precise S_p values and thermonuclear reaction rates, are needed for those nuclei along the rp-process path.

⁵⁶Cu is a very important nucleus, because it is close to the waiting-point nucleus ⁵⁶Ni and lies on the rp-process path [6, 12]. In this work, a new thermonuclear reaction rate is derived for ⁵⁵Ni(p,γ)⁵⁶Cu in the temperature region relevant to XRBs based on the recent experimental results of energy levels and proton separation energy of S_p = 579.8(7.1) keV in ⁵⁶Cu, together with the shell-model calculation and mirror nuclear structure in ⁵⁶Co.

2 Proton separation energy of S_p(⁵⁶Cu)

Previously, four theoretical proton separation energy, S_p, values have been predicted for ⁵⁶Cu, as S_p(⁵⁶Cu) = 573±54 keV based on the charge-symmetry formula of Kelson–Garvey [13], 560±140 keV from Coulomb displacement energy (CDE) mass relation [14], 526±1 keV using the improved Kelson–Garvey (ImKG) mass relation, [15], and 647±88 keV based on the mirror symmetry and known data from beta-delayed proton spectroscopy [16]. Using the smoothness trends of the mass surface, analysis by atomic mass evaluations (AME) provided S_p(⁵⁶Cu) values as 459±200 keV in AME85 [17], 560±140 keV in both AME95 [18] and AME03 [19], and 190±200 keV in AME12 [20]. The large uncertainty in the results is due to the theoretically estimated mass of ⁶⁵Cu.

The first measurement of the nuclear mass of ⁶⁵Cu was first performed in 2016, with a relative precision of (1–4)×10^-7 using isochronous mass spectrometry (IMS) at the Cooler Storage Ring (CSRe), at the Institute of Modern Physics IMP in Lanzhou, China. Together with the unpublished mass excess value of ME(⁶⁵Cu)=-38643(15) keV, the value of S_p(⁵⁶Cu) = 596±15 keV was compiled into AME16 [21] (referenced as "private communication"). This experimental value was formally published in 2018 [22]; however, during the review process of this paper, a more accurate value of ME(⁶⁵Cu)=-38626.7(7.1) keV was measured [23] using the Low-Energy Beam and Ion Trap (LEBIT) 9.4-T penning trap mass spectrometer in the National Superconducting Cyclotron Laboratory (NSCL) at the Michigan State University (MSU). Then, an accurate value of S_p = 579.8(7.1) keV was deduced. Based on the weighted average S_p values obtained from the MSU and IMP measurements, the value of S_p = 582.8(6.4) keV was adopted in their thermonuclear ⁵⁵Ni(p,γ)⁵⁶Cu rate calculations [23]. This shows that two experimental ME values agree well within the uncertainties, and the S_p values also agree well with the three theoretical predictions mentioned above, except for the highly accurate ImKG prediction with an uncertainty of ±1 keV [15].

In this work, in the reaction rate calculations we use the value of S_p = 579.8(7.1) keV derived at MSU, because the MSU mass value of ⁶⁵Cu has been the most accurate value until now.

3 Previous results for reaction rates

The reaction rate of ⁵⁵Ni(p,γ)⁵⁶Cu was estimated for the first time by van Wormer et al. [24], based on the properties of 10 resonances in the mirror nucleus ⁵⁶Co, at S_p = 459 keV determined by AME85. This rate is referred to as laur in the JINA Reaclib Database [25]. Later, Fisker et al. [26] performed a shell-model calculation at the value of S_p = 563 keV estimated by AME95. The following comments need to be added about the study by Fisker et al. [26]: (1) the direct-capture (DC) rate, expressed by Eq. (15) in the paper, need to be multiplied by a factor of two; (2) considering the ⁵⁵Ni(p,γ)⁵⁶Cu reaction, certain resonance strength values listed in Table I in the paper (p. 270) cannot be reproduced by the relationship of ωγ=ωΓ_pΓ_γ/(Γ_p+Γ_γ), based on their Γ_p and Γ_γ values; and (3) the resonant rate cannot be reproduced by the analytical formula in Eq. (7) in the paper using the listed strengths. Unfortunately, we cannot find the exact source of these errors. In addition, shell-model levels predicted by Fisker et al. for ⁵⁶Cu (with excitation energy up to E_x = 3561 keV) were only valid for a temperature region up to ∼3.5 GK by considering the Gamow window. Therefore, the rates listed in Table II in Ref. [26] are typically underestimated above ∼3.5 GK. In the JINA Reaclib Database, the rate calculated by Fisker et al. [26] was revised at the value of S_p = 648 keV, and denoted as nfis.

In addition, certain theoretical rates based on the statistical model are available in the JINA Reaclib Database [25]. For instance, the rath, thra rates are obtained from statistical model calculations [27] using the finite-range droplet macroscopic (FRDM) [28](S_p = -448 keV) and the ETSFIQ mass models [29] (S_p = 1919 keV), respectively. The ths8 rate is the theoretical value by Rauscher [25] (S_p = 648 keV). These indicated that the rates from statistical models differ from one another by up to several orders of magnitude in the typical XRB temperature range. Therefore, experimental data for ⁵⁶Cu, such as the S_p value and level structure information, are strongly required.

In 2017, the low-lying energy levels of ⁵⁶Cu (E_x = 166, 572, 826, 1037, and 1224 keV) were obtained by in-beam γ-ray spectroscopy using the state-of-art Gamma-Ray Energy Tracking In-beam Nuclear Array (GRETINA) detector together with the S800 spectrograph at MSU [30]. With the newly obtained value of S_p = 639±82 keV using a more localized isobaric multiplet mass equation (IMME) fit, the ⁵⁵Ni(p,γ)⁵⁶Cu rate was calculated together with the shell-model results. In this study, this rate is denoted as ong.

As mentioned in the previous section, the high-precision mass measurement of ⁵⁶Cu was performed at MSU in 2018 [23]: Valverde et al. obtained a weighted average value of S_p = 582.8(6.4) keV and applied it in the reaction rate calculations. In that study, the energy levels observed in Ref. [30] were utilized, and the resonance energies and corresponding proton width, Γ_p, were scaled to the revised S_p value. Although the reaction rates and the associated uncertainties were calculated and presented in a figure, no numerical rates were provided for the readers. In this study, this rate is denoted as valverde.

4 Energy levels in ⁵⁶Cu

As mentioned in the previous section, Ong et al. [30] observed five low-lying energy levels, E_x = 166, 572, 826, 1037, and 1224 keV, in ⁵⁶Cu, using in-beam γ-ray spectroscopy at MSU [30]. Orrigo et al. [31] performed a β-delayed-proton decay experiment of ⁵⁶Zn at the Grand Accélérateur National d’Ions Lourds (GANIL), and observed six proton-decay branches. Based on these events, the level scheme of ⁵⁶Cu was constructed in the range of E_x = 1391–3508 keV. The associated large uncertainty of 140 keV was due to the uncertainty of S_p=560±140 keV estimated by AME03 at that time. In this work, the excitation energies obtained by Orrigo et al. are shifted by 19.8 keV (i.e. to 579.8–560.0 keV) upward, to E_x = 1411, 1711, 2557, 2680, 3443, and 3528 keV, according to the new MSU ground-state mass of ⁵⁶Cu applied here. The associated uncertainty in the level energies was reduced to 12.3 keV (in this study 13 keV was adopted), which could be attributed to that adopted at 10 keV proton energy, E_p, and 7.1 keV S_p value. The energy levels in ⁵⁶Cu obtained experimentally and calculated by the shell-model are shown in Fig. 1. For comparison, the experimental data of the mirror nucleus ⁵⁶Co are also shown, and the mirror assignments are suggested accordingly.

Fig. 1.Full-screen View

(Color online) Energy levels in ⁵⁶Cu obtained experimentally and calculated by the shell-model. The experimental data are obtained from Ref. [30], as well as from Ref. [31] with corrections (shifted upward by 19.8 keV) discussed in the text; the exact results of the shell-model calculations with a GXPF1A interaction are obtained from Ref. [30]. The experimental data of the mirror nucleus ⁵⁶Co (obtained from NNDC website [32], see Ref. [33]) are shown for comparison, and the mirror assignment are suggested accordingly (the provisional values are indicated by dotted lines).

Figure 1 shows an exceptional mirror symmetry of the excited states between ⁵⁶Cu and ⁵⁶Co. Therefore, the energies of the missing high-lying levels in ⁵⁶Cu can be assumed identical to the corresponding energies in the mirror ⁵⁶Co. Thus, for the missing levels in ⁵⁶Cu, we calculated their resonant rates using the level energies from ⁵⁶Co. To estimate the uncertainty in the level energies (E_x) of ⁵⁶Cu calculated by this approach, the neighboring mirror pairs in the pf-shell region [32] were analyzed. Figure 2 shows the level energy difference between the mirror pairs of Tz = 1. The horizontal axis E_mean indicates the average level energies of the mirror pairs, and the vertical axis represents the level energy difference. The root mean square (rms) value was found to be ∼84 keV. Thus, in this study, we assumed an uncertainty of ±84 keV in the energies estimated for the missing states in ⁵⁶Cu, for which a provisional correspondence was demonstrated between the mirror ⁵⁶Co and shell-model calculated levels, shown in Fig. 1 by the dotted blue lines. In addition, we also adopted these shell-model level energies for six states of E_x gt;2470 keV (see Table 1) in the reaction rate calculations, and an rms uncertainty of 184 keV was estimated for these calculated levels, based on the energy difference between the experimental and shell-model levels in ⁶⁵Cu (the correspondence is shown in Table 1). However, their contribution to the total rate is negligible in the temperature region relevant to XRB, which is discussed in the next section.

Fig. 2.Full-screen View

(Color online) Level energy difference between mirror pairs of Tz = 1 for the pf-shell nuclei. The experimental level energies are obtained from the NNDC website [32].

5 Revised reaction rate

For a typical (p,γ) reaction, the total rate consists of the resonant and DC rates of proton capture on the ground state, and all thermally excited states in the target nucleus are weighted with their individual population factors [35, 40]. Here, the excitation energies in ⁵⁵Ni are rather high (E_x = 2089 keV for the first excited state), and therefore the contributions from the thermally populated excited states can be entirely neglected in the temperature region relevant to XRB. The DC rate is also negligible compared to the resonant rate as demonstrated in previous works [24, 26]. Thus, only the resonant rate (equivalent to the total rate) was calculated in the temperature region relevant to XRB. In this work, we calculated the ⁵⁵Ni(p,γ)⁵⁶Cu resonant rate using the most accurate experimental value of S_p(⁵⁶Cu) = 579.8±7.1 keV [23], the recent experimental level energies in ⁵⁶Cu, the mirror information for the missing high-lying states in ⁵⁶Cu, and the previous shell-model results for six levels above E_x=2470 keV (see Table 1). A similar approach was used in Refs. [24, 34, 35].

The resonant ⁵⁵Ni(p,γ)⁵⁶Cu rate was calculated using Eq. (7) in Ref. [26], which is the well-known narrow resonance formalism given as [24, 34, 35]

where the unit of the resonant energy E_r and the strength ωγ is in MeV. The E_r value can be calculated by the relation of E_r = E_x-S_p, where E_x denotes the level energies in ⁵⁶Cu (see Sect. 4), and the MSU value is S_p(⁵⁶Cu) = 579.8±7.1 keV. For the proton capture reaction, the reduced mass μ is defined by A_T/(1+A_T) (here, the target mass is A_T = 55 for ⁵⁵Ni). The resonant strength ωγ is defined as (Eq. (8) in Ref. [26])

where J_T and J are the spins of the target (J_T = 7/2 for the ground state of ⁵⁵Ni) and resonant state, respectively. The Γ_p and Γ_γ parameters are the partial widths for the entrance and exit channels, respectively, and Γ_tot is the total decay width of a resonance (Γ_tot Γ_p + Γ_γ).

In this work, the gamma width Γ_γ for the part of the unbound state in ⁵⁶Cu was calculated by the available half-lives T_1/2 of the corresponding bound state in the mirror ⁵⁶Co using Γ_γ = ln2× /T_1/2, as shown in Table 1, where the previous shell-model gamma widths (Γ_γ) [30] are indicated in the parentheses of the corresponding Γ_γ column. It can be seen that the gamma widths calculated by the shell-model deviate from the mirror values by a factor of no more than ∼3, except for a factor of 5.5 for the 1.411 MeV (0⁺) state; however, this is not a noticeable parameter in the rate calculation, as discussed below. For the other remaining states listed in Table 1, the previous shell-model gamma widths (Γ_γ) [30] are adopted. In Ref. [30], the uncertainty for the calculated γ widths were estimated to be approximately a factor of two [37]. We conservatively estimated the uncertainty in the gamma widths adopted here as a factor of three; however, the uncertainties in the gamma widths obtained from the mirror half-lives T_1/2 are less than a factor of two.

The proton widths were calculated by the appropriate scaling to those in Ref. [30] enabled by the energy difference of E_r in the calculation of the penetrability factor P≤ll. The leading uncertainties in Γ_p are due to those in E_r from the P≤ll factors, and they are significantly larger than those resulting from the uncertainty due to the choice of channel radius r₀ and diffuseness a (∼20% [36]). In addition, the uncertainties in the C²S factors calculated by the shell-model are estimated to be ∼20% [37], which can be neglected in the present calculations. In Table 1, the corresponding uncertainties are indicated in the parentheses of the Γ_γ column. For example, the number 2.3 indicates an uncertainty factor of 2.3.

The resonant parameters for calculating the ⁵⁵Ni(p,γ)⁵⁶Cu reaction rate are listed in Table 1. The peak temperature in the recent hydrodynamic XRB models can be as high as ∼1.4 GK [11, 38, 39]. In this study, we consider the reaction rate that is valid for a temperature region up to 2 GK, corresponding to a Gamow peak of E_r≈1.77 MeV with a width of Δ≈1.28 MeV [40]. Therefore, we believe that the resonances listed in Table 1 are sufficient for the description of the reaction rates of the relevant XRB region.

The percentage contributions of each resonance to the total resonant rate were calculated. Only five resonances contribute significantly to the total rate in the temperature region up to ∼2 GK: at E_x = 826, 1037, 1224, 1930, and 2060 keV. The contributions of these five special resonances to the total rate are shown in Fig. 3. Ong et al. provisionally assigned Ex = 1224 keV to either the Jπ = 3⁺ or the 5⁺ state. The cases of 3⁺ and 5⁺ are shown in Figs 3(a) and 3(b), respectively. Only three resonances, at E_x = 1037(2⁺), 1224(3⁺ or 5⁺), and 1930(3⁺) keV dominate most of the temperature region relevant to XRB (∼0.4–1.4 GK). If the E_x = 1224 keV state is of 3⁺, it nearly dominates the entire temperature region relevant to XRB. If the E_x = 1224 keV state is of 5⁺, the total rate can be reduced by a factor of ∼3 compared with the 3⁺ case. Thus, the experimental determination of the Jπ value is strongly required for the E_x = 1224 keV state. It should be noted, that other resonances can also have noticeable contribution, which are not shown in Fig. 3, by considering the uncertainties in their E_r and ωγ values. A realistic reaction rate and the associated uncertainties were calculated and discussed in the following paragraph. It should be noted, that the rate by Ong et al. assumed the E_x = 1224 keV state as a doublet (with 3⁺ and 5⁺), which might not be correct, according to the mirror information and shell-model calculations. This assumption probably results in an overestimation of the rate by different degrees depending on the assignment of this state to 3⁺ or 5⁺. Here, for comparison we also adopt the same assumption.

Fig. 3.Full-screen View

(Color online) Percentage contribution of resonances to the total resonant rate. The five resonances (listed in Table 1) with significant contribution (gt; 5%) are shown: (a) 3⁺ for E_x = 1224 keV and (b) 5⁺ for E_x = 1224 keV.

Table 2 summarizes the thermonuclear ⁵⁵Ni(p,γ)⁵⁶Cu rates calculated analytically using Eq. 1 with the resonant parameters listed in Table 1. The listed 1σ uncertainties (lower and upper limits) were calculated by a Monte-Carlo method, which simultaneously sampled the uncertainties in E_r, Γ_p, and Γ_γ. Our (mean) rate can be parameterized very well by the standard format of [27]

with a fitting error less than 3.2% in the range of 0.1–2.0 GK.

Figure 4 shows a comparison of the rate in this study with two recently obtained results given in Refs. [23, 30]. It can be seen that our revised rate and associated uncertainties are rather different from those in the previous studies. The present uncertainties are significantly smaller than those by Ong et al. [30] mainly because we used a considerably more accurate experimental S_p value (and hence E_r) in the reaction rate calculations. Here, two remarks need to be made: 1) The recommended rate and uncertainties listed in Table IV in the paper by Ong et al. are not in exact agreement with those shown in Fig. 6 in the same paper, and the exact reason for this is not known and 2) according to the method of rate calculation described by Valverde et al. [23], we cannot reproduce the rate shown in Fig. 4 in their paper in any form. They used S_p = 582.8(6.4) keV, which is only ∼3 keV higher than our value of S_p = 579.8(7.1) keV. Our rate is expected not to be very different from that by Valverde et al. In the work by Valverde et al., if it is assumed that they used the exactly same Γ_p and Γ_γ parameters (instead of the scaled ones), we need to be able to approximately reproduce their rate within one or two factor. This suggests that Valverde et al. presented an incorrect rate and associated uncertainties in the figure. As there are no numerical rates listed in their tables, the exact reason for this is not known. Our rates a re listed explicitly in Table 2.

Fig. 4.Full-screen View

(Color online) Comparison of the rate in this study and two recent results, Ong et al. [30] and Valverde et al. [23, 30]. The associated uncertainties are indicated by the colored bands. The rate in our study and the rate by Ong et al. (shown in Fig. 6 in their paper) are overlaid with that in the Fig. 4 in the paper by Valverde et al. since the latter did not provide numerical rates.

Figure 5 shows a comparison of the rate in this study with five different rates available in the JINA Reaclib Database: rath, thra, ths8, nifs, laur, ths8 rates. The uncertainties of the present rate are indicated by error bands in the figure. All rates for the ⁵⁵Ni(p,γ)⁵⁶Cu reaction are summarized in Table 2. It can be seen that neither the rates from the statistical model (rath, thra, ths8), nor that from the shell-model nifs, or the simple laur rate (based on the mirror information), agree with the rate in this study within the well-constrained uncertainties over the entire temperature region relevant to XRB. This implies that the experimental information of Sp, level structures, and ⁵⁶Cu properties are very important to determine the ⁵⁵Ni(p,γ)⁵⁶Cu rate more accurately.

Fig. 5.Full-screen View

(Color online) Ratios between the rate in this study and those compiled in the JINA Reaclib Database. The reference of unity is indicated by a solid black line. See text for details.

6 Summary

The thermonuclear ⁵⁵Ni(p,γ)⁵⁶Cu rate was recalculated using the most accurate proton separation energy of S_p(⁵⁶Cu) = 579.8±7.1 keV, the recent experimental levels in ⁵⁶Cu, together with the shell-model calculation and the mirror nuclear structure in ⁵⁶Co. Our revised rate deviates significantly from those available in the literature. To further reduce the uncertainties of the rate, the experimental determination of the Jπ value (3⁺ or 5⁺) is strongly required for the crucial E_x = 1224 keV state in ⁵⁶Cu. In addition, we found that the previous reaction rate curve (Fig. 4 in Ref. [23]) is incorrect. Therefore, we recommend our revised rate to be incorporated in the future astrophysical network calculations, as it is based on a more fundamental experimental background. The astrophysical application of our revised rates in Type I XRB calculations is now under progress, which is beyond the scope of this work.