Low-lying level structure of $^{56}$Cu and its implications on the rp process

TL;DR

This study investigates the low-energy structure of $^{56}$Cu using advanced gamma-ray spectroscopy, providing new reaction rates that influence the understanding of the rp-process in x-ray bursts.

Contribution

It presents the first experimentally-constrained reaction rate for $^{55}$Ni(p,$$)$^{56}$Cu based on new level structure data, impacting rp-process modeling.

Findings

The reaction rate for $^{55}$Ni(p,$$)$^{56}$Cu has been experimentally constrained.

The rp-process can bypass the $^{56}$Ni waiting point with a branching of up to 40%.

Additional nuclear physics uncertainties remain to be addressed.

Abstract

The low-lying energy levels of proton-rich $^{56}$Cu have been extracted using in-beam $\gamma$-ray spectroscopy with the state-of-the-art $\gamma$-ray tracking array GRETINA in conjunction with the S800 spectrograph at the National Superconducting Cyclotron Laboratory at Michigan State University. Excited states in $^{56}$Cu serve as resonances in the $^{55}$Ni(p,$\gamma$)$^{56}$Cu reaction, which is a part of the rp-process in type I x-ray bursts. To resolve existing ambiguities in the reaction Q-value, a more localized IMME mass fit is used resulting in $Q=639\pm82$~keV. We derive the first experimentally-constrained thermonuclear reaction rate for $^{55}$Ni(p,$\gamma$)$^{56}$Cu. We find that, with this new rate, the rp-process may bypass the $^{56}$Ni waiting point via the $^{55}$Ni(p,$\gamma$) reaction for typical x-ray burst conditions with a branching of up to $\sim$40$\%$. We also identify additional nuclear physics uncertainties that need to be addressed before drawing final conclusions about the rp-process reaction flow in the $^{56}$Ni region.

Figures

Tables

Table 1. Table 1: Reconstructed level scheme of 56 Cu excitation levels with observed transition energies ( E γ subscript 𝐸 𝛾 E_{\gamma} ), relative intensities ( I γ subscript 𝐼 𝛾 I_{\gamma} ) normalized to the E γ subscript 𝐸 𝛾 E_{\gamma}  = 166 keV line, and tentative spin-parity assignments (see text for details).

$E_x$ (keV)	$E_{\gamma }$ (keV)	$I_{\gamma }$ ($\%$)	$J_i^{\pi }\to J_f^{\pi }$
166 (1)	166 (1)	100	$\left(3_1^{+}\right)\to \text{g.s.}$
572 (1)	572 (1)	122 (8)	$\left(5_1^{+}\right)\to \text{g.s.}$
826 (3)	660 (3)	28 (8)	$\left(4_2^{+}\right)\to \left(3_1^{+}\right)$
1037 (3)	871 (3)	50 (8)	$\left(2_1^{+}\right)\to \left(3_1^{+}\right)$
1224 (4)	1224 (4)	19 (10)	$(3_2^{+},5_2^{+})\to \text{g.s.}$

Table 2. Table 2: Summary of predictions for the Q-value of 55 Ni ( p , γ ) 𝑝 𝛾 (p,\gamma) .

Q-value (keV)	Method	Reference
560 $\pm$ 140	Mass extrapolation	AME2003 Wapstra et al. (2003)
190 $\pm$ 200	Mass extrapolation	AME2012 Audi et al. (2014)
600 $\pm$ 100	Coulomb Shift / Shell Model	Brown et al. Brown et al. (2002)
639 $\pm$ 82	IMME	This work

Table 3. Table 3: New measured and shell-model excitation energies for 56 Cu up to 3 MeV, resonance energies ( E r subscript 𝐸 𝑟 E_{r} ), and tentative spin-parity assignments. Spectroscopic factors C 2 ​ S superscript 𝐶 2 𝑆 C^{2}S used to calculate the partial proton and gamma widths ( Γ p subscript Γ 𝑝 \Gamma_{p} and Γ γ subscript Γ 𝛾 \Gamma_{\gamma} respectively) were calculated utilizing a shell model calculation with the GXPF1A interaction, using experimental energies when available.

Experiment			Shell Model			$C^{2}S$
$E_x$ (keV)	$E_r$ (keV)		$E_x$ (keV)	$E_r$ (keV)	$J^{\pi }$	$l=1$	$l=3$	${\Gamma }_{\gamma }$ (eV)	${\Gamma }_p$ (eV)
166(1)			146		$\left(3_1^{+}\right)$	$0.84$	$9.1\times {10}^{-3}$	$8.4\times {10}^{-5}$
572(1)			483		$\left(5_1^{+}\right)$	0.70	0.16	$1.1\times {10}^{-3}$
826(3)	187(82)		1066	427	$\left(4_2^{+}\right)$	0.12	0.69	$4.5\times {10}^{-4}$	$1.2\times {10}^{-16}$
1037(3)	398(82)		1023	384	$\left(2_1^{+}\right)$	0.64	0.16	$1.2\times {10}^{-2}$	$1.7\times {10}^{-7}$
1224(4)	585(82)	$\{$	1146 1474	507 835	$\left(5_2^{+}\right)$ $\left(3_2^{+}\right)$	0.15 0.10	0.71 0.68	$2.0\times {10}^{-3}$ $1.9\times {10}^{-3}$	$4.3\times {10}^{-6}$ $3.9\times {10}^{-5}$
			1582	943	$0_1^{+}$		$3.8\times {10}^{-2}$	$1.6\times {10}^{-6}$	$1.5\times {10}^{-4}$
			1913	1274	$2_2^{+}$	$0.15$	0.57	$1.4\times {10}^{-2}$	$3.7$
			2036	1397	$1_1^{+}$		$1.3\times {10}^{-2}$	$4.5\times {10}^{-4}$	$8.1\times {10}^{-3}$
			2066	1427	$3_3^{+}$	0.59	$9.1\times {10}^{-2}$	$2.2\times {10}^{-2}$	48
			2226	1587	$2_3^{+}$	$1.7\times {10}^{-3}$	$3.9\times {10}^{-2}$	$3.8\times {10}^{-3}$	0.53
			2272	1633	$4_3^{+}$	0.63	0.13	$5.3\times {10}^{-2}$	210
			2350	1711	$7_1^{+}$		$9.9\times {10}^{-3}$	$1.2\times {10}^{-4}$	$5.8\times {10}^{-2}$
			2393	1754	$6_1^{+}$		0.72	$3.1\times {10}^{-2}$	5.5
			2419	1780	$1_2^{+}$		$1.0\times {10}^{-2}$	$9.8\times {10}^{-3}$	$8.8\times {10}^{-2}$
			2483	1844	$3_4^{+}$	$5.5\times {10}^{-2}$	$1.3\times {10}^{-2}$	$8.9\times {10}^{-3}$	59
			2505	1866	$1_3^{+}$		$7.3\times {10}^{-3}$	$2.0\times {10}^{-2}$	0.11
			2543	1904	$2_4^{+}$	$1.6\times {10}^{-2}$	$7.5\times {10}^{-3}$	$9.2\times {10}^{-3}$	23
			2630	1991	$3_5^{+}$	$8.8\times {10}^{-3}$	$3.6\times {10}^{-3}$	$9.6\times {10}^{-3}$	19
			2723	2084	$4_4^{+}$	$2.3\times {10}^{-2}$	$2.0\times {10}^{-2}$	$9.9\times {10}^{-3}$	75
			2762	2123	$6_2^{+}$		$5.0\times {10}^{-2}$	$1.5\times {10}^{-2}$	2.6
			2914	2275	$5_3^{+}$	$1.1\times {10}^{-2}$	$1.0\times {10}^{-2}$	$1.7\times {10}^{-2}$	77

Table 4. Table 4: The recommended reaction rate N A ​ ⟨ σ ​ υ ⟩ subscript 𝑁 𝐴 delimited-⟨⟩ 𝜎 𝜐 N_{A}\langle\sigma\upsilon\rangle as a function of temperature T 𝑇 T (GK) from this work, together with 1- σ 𝜎 \sigma uncertainties (higher and lower).

T9	$N_A⟨\sigma \upsilon ⟩$ (cm3/s/mole)
	Recommended	Lower	Upper
0.1	1.497e-19	8.583e-20	2.422e-19
0.2	8.661e-12	8.121e-12	1.082e-11
0.3	1.666e-08	1.183e-08	2.084e-08
0.4	1.244e-06	7.796e-07	1.746e-06
0.5	1.859e-05	1.237e-05	3.606e-05
0.6	1.156e-04	8.254e-05	2.911e-04
0.7	4.677e-04	3.197e-04	1.357e-03
0.8	1.529e-03	8.877e-04	4.423e-03
0.9	3.880e-03	2.009e-03	1.149e-02
1.0	8.951e-03	4.287e-03	2.551e-02
1.5	1.583e-01	8.221e-02	3.294e-01
2.0	9.620e-01	6.177e-01	1.655e+00

Table 5. Table 5: REACLIB fit coefficients for our recommended 55 Ni(p, γ 𝛾 \gamma ) reaction rate.

$E_x$	$a_0$	$a_1$	$a_2$	$a_3$	$a_4$	$a_5$	$a_6$
1224	1.052854	-6.805068	7.127737E-01	-1.049583	5.849955E-02	-3.234916E-03	-9.787774E-01
Other	-5.223069E+01	-9.902812	1.336866E+02	-7.623392E+01	-8.335959E-01	2.019964E-01	6.914259E+01
1038	-5.177171	-4.627019	-7.755680E-02	8.817104E-02	-4.086783E-03	1.981643E-04	-1.549327
826	-2.601956E+01	-2.170262	4.521332E-04	-6.347735E-04	3.674535E-05	-2.248394E-06	-1.499681

Equations

$$N_A⟨\sigma \upsilon ⟩\propto \sum _i(\omega \gamma {)}_i\exp (-E_i/kT)$$

$$\omega \gamma =\frac{2J+1}{(2J_p+1)(2J_{{}^{55}\mathrm{Ni}}+1)}\frac{{\Gamma }_p{\Gamma }_{\gamma }}{\Gamma }.$$

$$\Delta M=a+b{T}_z+c{T}_z^2.$$

$$b$$

$$c$$

$$\begin{array}{rl}{N}_A<\sigma \upsilon >& =\sum _i\exp (a_{0i}+a_{1i}{T}_9^{-1}+a_{2i}{T}_9^{-1/3}+a_{3i}{T}_9^{1/3}\\ & +a_{4i}{T}_9+a_{5i}{T}_9^{5/3}+a_{6i}\ln T_9)\end{array}$$

Full text

††preprint: APS/123

I Introduction

Accreting neutron stars in binary systems undergo episodes of explosive hydrogen and helium burning, observed as Type-I x-ray bursts. The main observable of these events, the x-ray burst light-curve, is shaped by the nuclear energy generation during the rapid proton-capture process (rp-process) Wallace and Woosley (1981); Schatz et al. (1998). This process involves a series of proton captures and $\beta$-decays that proceed near the proton-drip line.

Reaction rates connected to the so-called waiting point nuclei van Wormer et al. (1994), where the reaction flow slows down significantly, have the most significant impact on the observed light curve. The doubly-magic nucleus 56Ni has been identified as one of a few major waiting points in the rp-process. This is due to the combination of its long stellar electron capture half-life of $\sim$3 hrs Fuller et al. (1982) (which for the fully ionized ion differs from its terrestrial half-life and depends on the stellar electron density), and its low proton-capture Q-value (690 keV) Audi et al. (2014). The effective lifetime of 56Ni under typical x-ray burst conditions, which depends steeply on temperature, has been constrained by experimental data related to the 56Ni(p,$\gamma$) Rehm et al. (1998); Forstner et al. (2001) and 57Cu(p,$\gamma$) Langer et al. (2014) reaction rates. However, large uncertainties exist in the nuclear physics of more neutron-deficient nuclei in the 56Ni region. In particular, a sequence of proton-capture reactions in the 55Ni, 56Cu, 57Zn isotonic chain may be strong enough for the rp-process to bypass 56Ni (Fig. 1). In this case, 56Ni would not be an rp-process waiting point, reducing the sensitivity of burst models to the 56Ni(p,$\gamma$) rate. The 57Cu(p,$\gamma$) reaction rate remains important because the bypass exits the N=27 isotonic chain through $\beta$-decay of 57Zn to 57Cu. Consequently, the reaction flow would proceed more rapidly into the Ge-Se-Kr mass region and a lower amount of A = 56 material would be produced in the ashes.

The 55Ni(p,$\gamma$) reaction determines the branching at 55Ni into the 56Ni bypass reaction sequence. Here, we address uncertainties in this reaction rate experimentally, and reanalyze theoretical predictions of the reaction Q-value. We then use the new data to determine, in the context of the remaining nuclear physics uncertainties, the conditions under which the rp-process bypasses 56Ni.

The 55Ni(p,$\gamma$)56Cu reaction proceeds through a few isolated narrow resonances, and the astrophysical rate can be approximated by

[TABLE]

where $E_{i}=E_{i}^{x}-Q$ is the resonance energy with reaction Q-value $Q$ and 56Cu exitation energy $E_{i}^{x}$. The resonance strength is given by

[TABLE]

Here, $J$ is the resonance spin, $J_{p}$ the proton spin, $J_{{}^{55}\mathrm{Ni}}$ is the ground-sate spin of 55Ni, $\Gamma_{p}$ the proton partial width, $\Gamma_{\gamma}$ the $\gamma$ partial width and $\Gamma=\Gamma_{p}+\Gamma_{\gamma}$.

Only scarce experimental data for the odd-odd 56Cu nucleus exist in the literature. 56Cu, as well as its well-understood mirror nucleus 56Co, are part of the $A=56$, $T=1$ isospin triplet. Based on this, the ground-state of 56Cu is assumed to be $J^{\pi}=4^{+}$ with a measured terrestrial $\beta$-decay half-life of 93(3) ms Junde et al. (2011). To date, no low-lying excited states have been observed experimentally. In a recent $\beta$-delayed proton decay study of 56Zn, several higher-lying 56Cu resonances above 1391 keV excitation energy were observed Orrigo et al. (2014). Under astrophysical conditions, however, these resonances are too high in energy to be of relevance. In the absence of knowledge of spectroscopic information, shell-model calculations using the KB3 interaction in the $pf$-shell performed with the code ANTOINE have been used in the past Fisker et al. (2001). However, uncertainties in shell-model predictions of excitation energies can amount up to 200 keV, leading to orders of magnitude uncertainty in the resonant-capture rate. Here we experimentally determine, for the first time, the excitation energies of low-lying states in 56Cu that serve as resonances in the 55Ni(p,$\gamma$)56Cu reaction

For a precise determination of the 55Ni(p,$\gamma$)56Cu rate, both the low-lying level scheme of 56Cu and the reaction Q-value need to be well-known since the resonance energies enter the rate exponentially. While the mass of 55Ni is experimentally well-known with an error of 0.75 keV Kankainen et al. (2010), the mass of 56Cu is not experimentally known. Conflicting predictions for the 56Cu mass exist in the literature. The extrapolated 56Cu mass in the AME2003 compilation Wapstra et al. (2003) results in a 55Ni proton-capture Q-value of 560(140) keV. A similar result of 600(100) keV is obtained with Coulomb shift calculations Brown et al. (2002). Using the 56Cu mass in the most recent AME2012 compilation, however, results in a Q-value of 190(200) keV Audi et al. (2014). We obtain a new prediction for the reaction Q-value by using the isobaric multiplet mass equation (IMME).

II Experimental Determination of the 56Cu level scheme

Excited states of 56Cu were populated in inverse kinematics in an experiment performed at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University Langer et al. (2014). A stable 160 MeV/$u$ 58Ni primary beam impinged on a 752 mg/cm2 9Be target placed at the entrance of the A1900 fragment separator Morrissey et al. (2003). After purification by the A1900 using the $B\rho-\Delta E-B\rho$ method, the produced 56Ni secondary beam had a rate of $\sim 10^{5}$ pps with a beam purity of $\sim 75\%$. The 56Ni beam (E = $\sim 75$ MeV/u) was then incident upon a 225 mg/cm2 CD2 target, producing 56Cu through various reaction channels. The CD2 target was located in the center of the $\gamma$-ray energy tracking array GRETINA Paschalis et al. (2013), which was used to measure energies of the prompt $\gamma$-rays emitted from the de-excitation of the excited states in 56Cu. GRETINA consists of 28 coaxial HPGe detector crystals, which are closely-packed to cover roughly 1$\pi$ in solid angle. Kinematical reconstruction of the momentum, angle, and position of each 56Cu recoil at the target based on observables at the S800 focal plane, combined with the high position resolution for $\gamma$-ray detection in GRETINA allow for accurate Doppler-shift corrections for $\gamma$-rays emitted in-flight. The recoil velocity $\beta=\upsilon/c$ used for the Doppler-shift correction was extracted using momentum information, and was determined for each individual event to correct for energy loss in the target. The 56Cu recoils, after leaving the target, were identified using detectors situated in the focal plane of the S800 spectrograph Bazin et al. (2003) located downstream from GRETINA. The S800 focal plane contained a set of two cathode readout drift counters that were used to determine the particle trajectory, a gas-filled ionization chamber that measured energy loss $\Delta E$, and a plastic scintillator that, along with the thin timing scintillator at the A1900 focal plane and the scintillator at the S800 object position, were used for time-of-flight (TOF) analysis. The measured time-of-flight between the A1900 focal plane and S800 object scintillators was used to uniquely identify the 56Cu recoil by $\Delta E$-TOF (Fig. 2).

The low-lying level scheme of 56Cu was constructed using observed $\gamma$-ray transitions, $\gamma-\gamma$ coincidences and guidance from the experimentally based level scheme of the mirror nucleus 56Co. The Doppler-corrected spectrum of the $\gamma$-rays detected by GRETINA, in coincidence with the 56Cu recoils in the S800, shows five $\gamma$-ray transitions (Fig. 3). An additional line at $E_{\gamma}$ = 1027 keV stems from contamination from a well-known $\gamma$-ray transition in 57Cu which is located next to 56Cu in the particle identification spectrum (Fig. 2). We confirmed that this $\gamma$-ray line disappears from the $\gamma$ spectrum when the particle identification gate in Fig. 2 is tightened to only include the most centrally located events in the 56Cu recoil region.

The left half of figure 5 shows the reconstructed 56Cu level scheme. The strongest observed line at $E_{\gamma}$ = 166(1) keV is close in energy to the first excited state at 158 keV ($J^{\pi}=3^{+}$) in the mirror nucleus 56Co. Based on experimental information from the 56Co mirror nucleus, we expect the first excited state to be the most intense transition as it is fed from several higher-lying states. This line is observed to be in coincidence with two other $\gamma$-transitions, supporting its assignment as direct decay from the the first excited $3^{+}$ state (Fig. 4).

The transitions at $E_{\gamma}$ = 660(3) keV and $E_{\gamma}$ = 871(3) keV are observed to be in coincidence with the $E_{\gamma}$ = 166(1) keV transition as shown in Fig. 4, but not with each other. Based on the prior assignment of the 166 keV first excited state, two states are placed at $E_{x}=826(3)$ keV and $E_{x}=1037(3)$ keV, respectively. No ground state decays are observed for either of these states. There are three known states in the 56Co mirror at similar energies of $E_{x}=830,970$ and $1009$ keV. Of those, the 1009 keV state decays predominantly to the ground state. Both the $J^{\pi}=4^{+}$ 830 keV and $J^{\pi}=2^{+}$ 970 keV states decay primarily to the first excited state at 158 keV with only a 34$\%$ and 0.3$\%$ direct transition to the ground state, respectively. Based on the decay modes and similarities in energies, the two observed states at $E_{x}=826(3)$ keV and $E_{x}=1037(3)$ keV are tentatively assigned as $J^{\pi}=4^{+}$ and $J^{\pi}=2^{+}$, respectively.

The observed line at $E_{\gamma}$ = 572(1) keV is not seen in coincidence with the 166 keV line. The mirror 56Co has a $J^{\pi}~{}=~{}5^{+}$ state at $E_{x}~{}=~{}577$ keV that decays only to the ground state. Based on the similar energies and similar decay modes, we tentatively assign the 572 keV transition to be the second excited $J^{\pi}~{}=~{}5^{+}$ state.

The $E_{\gamma}=1224(4)$ keV line is not observed to be in coincidence with any other $\gamma$-ray transition, and it is therefore assigned to a level at that energy. The analog states in the mirror with the closest energies are 1009 keV ($J^{\pi}=5_{2}^{+}$) and 1115 keV ($J^{\pi}=3_{2}^{+}$) which both decay largely to the ground state. Other higher lying states in 56Co (the next one is at 1450 keV) decay predominantly through cascades, which is not supported by our measurement. We tentatively assign $E_{x}~{}=~{}1224(4)$ keV as either the $J^{\pi}$ = 3${}_{2}^{+}$ or the $J^{\pi}=5_{2}^{+}$ state. The observed transitions, intensities and assignments are tabulated in Table 1. A comparison to the mirror nucleus is shown in Fig. 5.

III Mass Estimate of 56Cu using the Isobaric Multiplet Mass Equation

We use the isobaric mass multiplet equation (IMME) to predict a new 56Cu mass, which is needed to derive the reaction Q-value and the resonance energies. The 56Cu ground state (J${}^{\pi}=4^{+}$) is part of the $A=56$, $T=1$ triplet, and its mass excess can be calculated using

[TABLE]

The $a$ coefficient for integer triplets is the mass excess of the isobaric analogue state (IAS) of the $T_{z}$ = 0 member of the triplet, in this case the $J^{\pi}=4^{+}$ state in 56Ni, and can be calculated from the reported IAS excitation energy of 6432 keV Borcea et al. (2001). The IMME $b$ and $c$ coefficients for the $A=56$ triplet have not been published, but can be estimated using fits to coefficients of triplets in the vicinity of $A=56$. Global fit functions of IMME parameters have been discussed in MacCormick and Audi (2014), where the authors treat the nucleus as a homogeneous charged sphere, and coefficients $a$, $b$ and $c$ are reported for the $A=4n$ subgroups. Here, we fit only to coefficients for a local region with A$=$32, 36, 40 and 48. As per the homogeneous charged sphere approximation of Jänecke (1966), the $b$ and $c$ coefficients can be parametrized in the following manner:

[TABLE]

where $C^{1}_{b},C^{2}_{b},C^{1}_{c},C^{2}_{c}$ are fit parameters. The fits obtained for $b$ and $c$ in the local vicinity are then used for the $A=56$, $T=1$ subgroup. The resulting fit extrapolated to $A=56$ results in $c$ = 110(95) keV and $b$ = -8680(109) keV. Along with the result for the $a$ coefficient from Borcea et al. (2001) of 6431.9 (7)   keV, this provides a mass excess prediction for 56Cu of -38685(82) keV and, thus, a Q-value of 639 $\pm$ 82 keV. The error is taken from the largest deviation between a measured mass and the predicted value from the fit function in the local region of interest.

As seen in Table 2, the more precise estimate from this work agrees within errors with the Coulomb-shift calculation from Brown et al. (2002), favoring a higher Q-value compared to the lower extrapolated value reported in the AME2012 compilation. A recent IMME-based estimate using the T=2 quintet Tu et al. (2016) reported a Q-value of 651(88) keV. Moreover, requiring reasonable Coulomb shifts for higher-lying mirror states, as extracted experimentally in Orrigo et al. (2014) between 56Cu and 56Co, also favors a higher Q-value.

IV Thermonuclear reaction rate

With our measurement and our predicted 56Cu mass, we have determined the resonance energies of the 55Ni(p,$\gamma$)56Cu reaction. In order to determine the astrophysical reaction rate, proton- and $\gamma$-widths ($\Gamma_{p}$ and $\Gamma_{\gamma}$ respectively) were calculated for each state using a shell-model with the GXPF1A interaction Honma et al. (2005) (Table LABEL:table1). These calculations allowed up to 3-particle 3-hole excitations in the $pf$-shell.

Reaction-rate uncertainties were calculated with a Monte-Carlo approach, similar to that of Iliadis et al. (2015), to properly account for the uncertainties in the excitation energies. Resonance energies and the reaction Q-value were allowed to vary assuming a Gaussian distribution within the uncertainties given in Table LABEL:table1. The uncertainty in the spin assignment for the 1224 keV state was also taken into account, but this represented only a small percentage of the uncertainty. The sampled resonance energy and corresponding rescaled proton-widths are used as input to Eq. 1, producing a sample of rates. At a given temperature, the 50th, 16th and 84th percentiles of the distribution of rate values provides the median, and 1-$\sigma$ uncertainty, respectively. The results are shown in Fig. 6. To assess the reaction rate uncertainty prior to our measurement, we used the shell-model calculation and assumed a 200 keV uncertainty for the resonance energies. The resulting rate uncertainty (the light blue band in Fig. 6) ranges from 4 orders of magnitude at 0.1 GK to about an order of magnitude at 2.0 GK. This is reduced at low temperatures to less than two orders of magnitude by our measurement (the gray band in Fig. 6). The additional uncertainty from the calculated proton and $\gamma$ partial widths is estimated to be significantly smaller, about of a factor of 2 Langer et al. (2014). Thus, the dominant remaining source of uncertainty is the $\sim$80 keV error in the 56Cu mass, with smaller contributions from the uncertainties of the experimentally-unmeasured proton and $\gamma$ partial widths.

Table LABEL:reaclibtab gives the corresponding REACLIB rate fit coefficients, using the parametrization given in Eqn. 6, for our updated 55Ni(p,$\gamma$) reaction rate.

[TABLE]

V Consequences on the rp-process flow around 56Ni

The astrophysical conditions that would lead the rp-process flow to bypass the 56Ni waiting point were investigated using a limited reaction network that includes the nuclides in Fig. 1. The network was seeded with 55Ni, where the rp-process enters the A = 56 region. 56Ni was treated as a sink in the network calculation, with only flow into this nuclide being allowed. In this case, the ratio of the abundance of all other nuclei (57Ni, 57,58Cu, and 58Zn) to the total abundance in the $N~{}=~{}28$ and $N~{}=~{}29$ chains is a measure of the fraction of the rp-process reaction flow that bypasses 56Ni, as it measures the amount of material trapped in neither 55Co nor 56Ni. The reaction network was run at constant temperature and proton density for 1 s, approximately 5 half-lives of 55Ni. A constant proton density was ensured by keeping the mass density constant, and by using a large proton-to-seed ratio of $\sim$400 such that the change in the proton abundance due to the comsumption of protons is negligible.

Even with the constraint on the 55Ni(p,$\gamma$) rate from this work, there remain additional uncertainties that affect the rp-process flow. The proton-capture rate on 56Cu determines the branching at 56Cu, where $\beta$ decay leads back to 56Ni, and also determines the total proton-capture flow at 55Ni in the case of $(p,\gamma)-(\gamma,p)$ equilibrium between 55Ni and 56Cu. In addition, the mass of 57Zn has not been measured and its uncertainty affects the 57Zn($\gamma$,p) rate, which hampers the flow bypassing 56Ni at high temperatures. Finally, the uncertain 78 $\pm$ 17 $\%$ $\beta$-delayed proton branch of 57Zn Blank et al. (2007) directs the reaction flow back to 56Ni and needs to be better constrained. To explore the effect of these uncertainties, we considered two scenarios of maximal and minimal favorability for the bypass. In the case of the maximal (minimal) favorability: (1) the 56Cu(p,$\gamma$) rate was increased (decreased) by a factor of 100, the expected uncertainty of a shell-model rate; (2) the 55Ni(p,$\gamma$) rate was increased (decreased) by the uncertainty reported in this work; (3) the $\beta$-delayed proton-emission rate of 57Zn was decreased (increased) by the uncertainty reported by Blank et al. (2007).

Fig. 7 shows the resulting fraction of the reaction flow that bypasses 56Ni as a function of temperature and proton density for the two scenarios. In the scenario with the most favorable nuclear physics assumptions, 56Ni is significantly bypassed for temperatures in the range of about 0.4 - 1.2 GK and proton densities above 104 g/cm3. These are within the range of typical X-ray burst conditions, with peak temperatures of 1-2 GK and proton densities up to 106 g/cm3. On the other hand, for the most unfavorable scenario proton densities in excess of 106 g/cm3 are required for the reaction flow to bypass 56Ni. Therefore, in the favorable scenario, 56Ni would be partially bypassed by the rp-process in all X-ray bursts, while in the unfavorable scenario the full rp-process would always pass through 56Ni.

VI Conclusion

This work presents the first experimentally-constrained 55Ni(p,$\gamma$)56Cu thermonuclear reaction rate, utilizing 5 newly identified excited states in 56Cu, a new theoretically-constrained reaction Q-value, and a new shell-model calculation of $\gamma$- and proton-widths. Below a temperature of 0.5 GK, the experimental data reduce the rate uncertainty from a factor of 105 to 102 at 0.1 GK and by almost an order of magnitude at 0.5 GK . The dominant remaining uncertainty is the reaction Q-value due to the unknown mass of 56Cu. For temperatures above 0.5 GK, the reaction rate is dominated by higher-lying resonances that have not been determined experimentally. With the new data, and using a detailed network analysis, we find that within remaining uncertainties the rp-process can bypass the 56Ni waiting point for typical x-ray burst conditions with a bypass branch as high as $\sim$40$\%$. We also identify additional nuclear physics uncertainties in the 56Cu(p,$\gamma$) reaction rate, the 57Zn mass, and the 57Zn $\beta$-delayed proton emission branch that need to be addressed.

The authors want to thank the staff and the beam operators at the NSCL for their effort during the experiment. This work is supported by NSF Grants No. PHY11-02511, No. PHY10-68217, No. PHY14-04442, No. PHY08-22648 (Joint Institute for Nuclear Astrophysics), and No. PHY14-30152 (JINA Center for the Evolution of the Elements). GRETINA was funded by the U.S. DOE Office of Science. Operation of the array at NSCL is supported by NSF under Cooperative Agreement PHY11-02511 (NSCL) and DOE under Grant No. DE-AC02-05CH11231 (LBNL).