Single-particle shell strengths near the doubly magic nucleus $^{56}$Ni and the $^{56}$Ni(p,$\gamma$)$^{57}$Cu reaction rate in explosive astrophysical burning

Angle-integrated cross-section measurements of the $^{56}$Ni(d,n) and (d,p) stripping reactions have been performed to determine the single-particle strengths of low-lying excited states in the mirror nuclei pair $^{57}$Cu-$^{57}$Ni situated adjacent to the doubly magic nucleus $^{56}$Ni. The reactions were studied in inverse kinematics utilizing a beam of radioactive $^{56}$Ni ions in conjunction with the GRETINA $\gamma$-array. Spectroscopic factors are compared with new shell-model calculations using a full $pf$ model space with the GPFX1A Hamiltonian for the isospin-conserving strong interaction plus Coulomb and charge-dependent Hamiltonians. These results were used to set new constraints on the $^{56}$Ni(p,$\gamma$)$^{57}$Cu reaction rate for explosive burning conditions in x-ray bursts, where $^{56}$Ni represents a key waiting point in the astrophysical rp-process.

Doubly magic nuclei represent special cornerstones in our understanding and exploration of nuclear structure (see, e.g., [1,2]). These nuclei, and the nuclei in their immediate vicinity, should be well described by very pure shell-model configurations. In the case of the doubly magic, self-conjugate nucleus 56 Ni, the major shell closure at N = Z = 28 arises from the spin-orbit splitting of the 0 f 5/2 and 0 f 7/2 orbitals. There is evidence that 56 Ni has a relatively soft core with significant configuration mixing in the shell structure [3,4,5]. It exists on the cusp between stability, having a (terrestrial) half-life of 6.08 d, and particle instability; the neighboring nucleus, 57 Cu, is only proton bound in its ground state. This special location means 56 Ni can form a key waiting point in explosive astrophysical burning scenarios such as x-ray bursters, impeding the rate of flow of material further along the proton drip-line via the 56 Ni(p,γ) 57 Cu reaction in the rp (rapid proton capture) process [6,7]. 56 Ni is the longest-lived waiting point nucleus and historically was originally thought to represent the termination point of the rp-process [6]. A direct measurement of the 56 Ni(p,γ) breakout reaction is not currently feasible with existing radioactive beam intensities. Therefore, to constrain the 56 Ni(p,γ) 57 Cu reaction rate used in astrophysical models, an indirect approach is mandated. This requires a knowledge of the shell structure and properties of states in 57 Cu involved in the explosive astrophysical temperature burning range from T ∼ 0.5-2.0 GK.
The mirror nucleus of 57 Cu, 57 Ni, has a ground-state spinparity of 3/2 − , corresponding to a 1p 3/2 neutron shell-model configuration, and 5/2 − and 1/2 − excited states corresponding to the occupation 0 f 5/2 and 1p 1/2 neutron shells, respectively [8]. A pioneering study of the 56 Ni(d,p) 57 Ni transfer reaction, measuring the differential cross section in inverse kinematics, found these three lowest-lying states were described with relatively pure single-neutron configurations with spectroscopic factors C 2 S ∼ 0.9, with an approximate factor of two uncertainties [9]. This information was then used to estimate the resonance contributions of the two low-lying analog excited states in 57 Cu [9,10] for the 56 Ni(p,γ) 57 Cu reaction rate, assuming isospin symmetry. Subsequently, an attempt was made to measure the single-particle strengths of these states directly in 57 Cu using the 56 Ni( 3 He,d) 57 Cu proton transfer reaction; however, due to the limited resolution of ∼ 700 keV in excitation energy and low statistics, definite conclusions could not be drawn [11]. Two higher-lying 5/2 − and 7/2 − states in 57 Cu are expected to dominate the 56 Ni(p,γ) 57 Cu reaction rate for burning temperatures T > 1 GK (see, e.g., ref. [12]) but only limits were set for the experimental C 2 S values in 57 Ni [9]. Calculations predict the combined resonant capture reaction rate on the four lowestlying excited states in 57 Cu is expected to dominate (by 3 orders of magnitude) over the direct-capture contribution in explosive astrophysical burning conditions [12,7].
Here, we present a first study of the 56 Ni(d,n) 57 Cu proton transfer reaction aimed at the first direct experimental determinations of the proton single-particle strengths of the key lowlying excited states in 57 Cu by angle-integrated cross-section measurements. The method exploits the high resolution and efficiency of the GRETINA (Gamma-Ray Energy Tracking Inbeam Nuclear Array) device [13] and an intense 56 Ni radioactive beam produced in flight by the National Superconducting Cyclotron Laboratory (NSCL). This new approach has been shown to work successfully in determining spectroscopic factors, including key astrophysical resonances [14,15], and is described for the present application in detail below. In addition, we have performed a measurement of the 56 Ni(d,p) 57 Ni transfer reaction to study analog states in the mirror nucleus 57 Ni, and explore evidence for isospin symmetry breaking effects. Furthermore, the 56 Ni(d,p) data were also used to measure for the first time the single-particle strength of a high-lying excited state in 57 Ni whose analog likely determines the explosive astrophysical burning rate above T ∼ 1 GK in the 56 Ni(p,γ) 57 Cu reaction.
The 56 Ni(d,n) 57 Cu and 56 Ni(d,p) 57 Ni reaction studies were performed in inverse kinematics. A 33.6 MeV/u 56 Ni 28+ beam was produced by in-flight fragmentation of a 28-pnA 160-MeV/u primary beam of 58 Ni 27+ ions which impinged upon a 1316mg/cm 2 thick 9 Be production target. The A1900 fragment separator [16] selected the ions of interest based on their magnetic rigidity and used a 150-mg/cm 2 thick Al achromatic wedge to provide isotopic separation at its focal plane. The resulting 56 Ni 28+ beam had a purity of 47% (contaminated mainly by 55 Co with traces of 52 Mn and 51 Cr ions) and an average intensity of 3 × 10 5 56 Ni particles per second. The 56 Ni beam impinged on a 10.7(8)-mg/cm 2 thick deuterated polyethylene target, (CD 2 ) n , which was surrounded by the GRETINA detectors [13] positioned in two rings at laboratory angles of 58 • and 90 • with respect to the beam direction. Beam-like residues were collected and analyzed with the S800 spectrograph [17] positioned at 0 • scattering angle. The analysis line to the S800 was operated in achromatic mode to obtain a total acceptance of nearly 100% for 57 Cu 29+ and 50(10)% for 57 Ni 28+ run settings (the lower value for 57 Ni reflects the need to block out scattered 56 Ni 28+ ions from part of the focal plane). Using the S800, we measured the intensity of 56 Ni 28+,27+,26+ species after the CD 2 target relative to the number of incident 56 Ni 28+ beam ions, where we observed that 80% of the incident ions emerged in the fully-stripped charge state. Given the similar properties of energy, mass, and nuclear charge of the 56 Ni beam and heavy residues 57 Cu and 57 Ni, we estimated an 80(5)% and 40(11)% collection and detection efficiency for 57 Cu 29+ and 57 Ni 28+ ions, respectively. To account for reactions on carbon producing the residues of interest, measurements were also performed for approximately half the duration of the CD 2 runs with an 8.8(15)mg/cm 2 thick (CH 2 ) n target. The data acquisition system was triggered either by a residueγ coincidence or downscaled (by a factor of 5) residue singles events. The GRETINA absolute singles efficiency was cali- brated with 56 Co and 152 Eu sources as described in ref. [18]. An efficiency of 5.5% for E γ = 1332 keV was achieved for the nine-module setup employed here. The in-beam GRETINA efficiency was boosted by a factor 1.06(5) compared to a stationary source. Doppler-adjusted γ-ray energy spectra gated on 57 Cu and 57 Ni ions are shown in Figs. 1a) and b), respectively. Events induced by reactions on carbon atoms have already been subtracted, following the procedure in refs. [14,15]. In the 57 Cu spectrum, two strong peaks are observed at energies of 1028(1) and 1109(2) keV, which are assigned to the γ decays to the ground state of the first and second excited states at 1028(4) and 1106(4) keV observed by Zhou et al. in a study of the 1 H( 58 Ni, 57 Cu)2n reaction [10]. There is no evidence for the γ decay at 2398(10) keV reported in the same study, or for previously unobserved γ decays from the excited state at 2525 (17) keV [19,20]. In Fig. 1b, three peaks are observed at energies of 768(1), 1122(5), and 2579(4) keV. The first two are assigned to the known decays of the 5/2 − and 1/2 − levels at 769 and 1113 keV in 57 Ni [8,9], respectively. The third peak is assigned to the known decay of the 7/2 − state at 2578 keV [8]. Angle-integrated cross sections for the four lowest excited states in 57 Cu and 57 Ni are shown in Table 1 assuming 100% γ branches to the ground states (consistent with nuclear compilation data for 57 Ni [8]), and insignificant γ-decay feeding from higher-lying excited states (there are no other γ-ray lines observed). The overall errors for the cross sections in Table 1 were obtained from a combination of statistical uncertainties for individual transitions, and an uncertainty of 20% (28%) for 57 Cu ( 57 Ni) estimated by combining contributions to the uncertainty from target thicknesses, Doppler-corrected γ-ray detection efficiency, residual charge-state distributions, and the momentum acceptance, in quadrature. The theoretical angle-integrated single-particle (C 2 S = 1) cross sections for the 56 Ni(d,n) 57 Cu and 56 Ni(d,p) 57 Cu reactions shown in Table 1 were calculated using the finite-range adiabatic approximation [21], which incorporates deuteron breakup. A laboratory beam energy of 32 MeV/u was used corresponding to the approximate center-of-target energy. Nucleon-target interactions used the CH89 [22] optical potential and the nucleonnucleon interaction from ref. [23]. The n(p) + 56 Ni final states were described by a real Woods-Saxon potential with central and spin-orbit terms. The radius and diffuseness of these potentials were set to 1.23 fm and 0.67 fm, respectively. The spin-orbit term was given the standard depth of V so = 6 MeV. For the (d,p) calculations, the central potential depth was adjusted to reproduce the final bound-state binding energies. For the (d,n) calculations, the final states are resonances. Here, we applied a bound-state approximation in which we adjusted the depth of the central potential to produce a final state bound by just E = 0.001 MeV, as was done in [14]. For low-lying resonances, this approximation introduces less than 1% error, but for the two higher-lying resonances, this approximation can introduce an error of ∼ 6%. Repeating the calculation with the Becchetti-Greenlees potential [24] changed the total cross section by about 15%, dominating the error introduced by the bound-state approximation. Based on this and other studies [25], we estimate an error of up to 30% in the total cross section calculations. The effective adiabatic potentials for (d,p) and (d,n) were computed with twofnr [26] and the transfer calculations were performed with the reaction code, fresco [27]. Shell-model wavefunctions for 56,57 Ni and 57 Cu were obtained in the full p f model space with the GPFX1A Hamiltonian [28] for the isospin-conserving strong interaction plus the Coulomb and charge-dependent Hamiltonians from ref. [29]. The spectroscopic factors (shown in Table 1) were derived from the overlap of these wavefunctions.
Experimental spectroscopic factors for the two lowest-lying excited states in 57 Cu are reported here for the first time. The values agree very well with shell-model calculations and are consistent with strong single particle states. A level hierarchy with the 5/2 − level below the 1/2 − is confirmed by the relative difference in cross sections for these states, which are consistent with our reaction theory calculations. Previously this hierarchy had been assumed largely on the basis of mirror energy level shift arguments [10,12]. This relative difference in cross section is also reflected for the analog states in 57 Ni. The spectroscopic factors for these states in 57 Ni are found to be very similar to the analog states in 57 Cu, and show no evidence of significant isospin symmetry breaking. The 57 Ni C 2 S values are lower, and more precise, than those first reported by Rehm et al. [9], but agree well within errors. A clear difference between the spectra in Fig. 1 is the strong presence of the decay of the 7/2 − state in 57 Ni which is not present for 57 Cu. From this we can infer that proton decay represents the dominant branch of the 7/2 − state in 57 Cu (we therefore do not present a C 2 S value for this state in 57 Cu in Table 1). The C 2 S (ℓ = 3) value for the 7/2 − state in 57 Ni represents a first measurement (rather than an upper limit) and is an interesting test of the shell-model calculations. It is found to be much weaker in single particle strength than the lower-lying states and is broadly consistent with the shell-model prediction. The shell-model calculations suggest the neighboring 5/2 − state has a smaller single-particle strength consistent with its non-observation here in both the 57 Ni and 57 Cu data. Interestingly, the γ decay of this state was clearly observed in the 1 H( 58 Ni, 57 Cu)2n reaction [10], which suggests the γ branch for this state in 57 Cu is at least comparable to the proton branch, otherwise it would not have been seen in that study. Table 2 shows resonance parameters for the four lowestlying excited states in 57 Cu used to calculate the 56 Ni(p,γ) 57 Cu astrophysical reaction rate. Shell-model calculations of the γdecay widths used the M1 and E2 effective operators from ref. [30]. The uncertainty in the γ-decay widths is about a factor of two [30]. Proton-decay widths were calculated from Γ = C 2 S · P · 2W, where P are the penetration factors obtained from the Coulomb wavefunction, and W is the Wigner single-particle width. The ℓ-dependent radius for the evaluation of P was chosen to reproduce the single-particle width obtained for proton scattering from a Woods-Saxon potential. The Woods-Saxon parameters were consistent with the geometry used in the cross section calculations and were chosen to reproduce the proton separation energy of 56 Ni to 55 Co and the rms charge radius of 58 Ni. The potential depth for the scattered proton was adjusted to give a resonance Q-value of 1 MeV. The uncertainty in these calculations of the single-particle proton-decay widths is ∼ 20%. A proton separation energy of 690.3(4) keV [31] was used for 57 Cu to calculate the resonance energies, which incorporate the new more precise excitation energy values reported here for the two lowest-lying excited states. The proton widths  Table 2 (shown in orange). The total rates from refs. [7,10,9] are shown for comparison; additionally, as the uncertainty bounds were presented in ref. [10], these are also depicted. See text for details. for these states were calculated for the first time using the experimentally constrained C 2 S values obtained for 57 Cu in the present work. For the 7/2 − state we derive the proton width value using the C 2 S (ℓ = 3) value reported here for the first time for the analog state in 57 Ni. For the proton width of the 2398-keV 5/2 − level, we take the C 2 S value from the shell-model calculation which is compatible with the experimental observational limit in the present 57 Ni data (see Table 1).
The present 56 Ni(p,γ) 57 Cu reaction rate and uncertainty bounds incorporating our new experimental results and shell-model calculations is shown in Fig. 2. Our rate for T < 1 GK is dominated by the ℓ = 1 capture resonance at 417.7(1.8) keV for which we derive a resonance strength, ωγ = 1.9(8) × 10 −7 eV. For the region with T > 1 GK, both the ℓ = 3 captures on the higher-lying 5/2 − and 7/2 − states are found to contribute to, and collectively dominate, the reaction rate. The 5/2 − state is the only lower-lying resonance for which we have only an experimental upper limit on the spectroscopic factor. If this value is significantly lower than the shell-model calculation this would reduce its strength/significance relative to the 7/2 − state. A much higher value for C 2 S , and therefore the proton partial width, is deemed unlikely as γ decay has been observed from this 5/2 − state in 57 Cu [10]. Figure 2 also shows some previous reaction rate calculations for comparison [7,10,9]. At lower temperatures, the van Wormer theoretical calculation is orders of magnitude lower and reflects the then unknown resonance energies in 57 Cu. The later rate of Zhou et al. used known experimental energies but the proton and γ widths were entirely estimated from theory; the large reaction rate difference, particularly for T < 1 GK, is caused by the relatively low proton partial width derived for the 1/2 − resonance in that estimate [10]. The rate of Rehm et al. [9] incorporates the results for the spectroscopic factors for the two lowest-lying excited states in 57 Ni. This central rate agrees well at low temperatures, as the same proton width and resonance strength values are obtained for the 1/2 − state. One should note that the spectroscopic factor used for the calculation in ref. [9] was based on the analog state in 57 Ni, and had a higher C 2 S value (and uncertainty) than that measured directly in the present study on 57 Cu. For the two higher-lying resonance contributions refs. [9] and [10] adopted the same partial width values so their total rates for T > 1 GK are nearly the same. Our result is significantly lower in this region, reflecting the lower partial widths obtained (particularly for the 5/2 − level) in the present shell-model calculations.
In summary, we report angle-integrated cross-section measurements for the 56 Ni(d,n) 57 Cu and 56 Ni(d,p) 57 Ni transfer reactions. Comparisons with reaction theory calculations allow definitive assignments of the first two excited states in 57 Cu to their analogs in the mirror partner, 57 Ni. First measurements of the spectroscopic factors for these two states in 57 Cu show they have a strong single-particle character with values agreeing well with the new shell-model calculations obtained using a full p f model space with the GPFX1A interaction plus Coulomb and charge-dependent Hamiltonians. From a comparison with their analog states in 57 Ni we find no evidence for significant isospin symmetry breaking effects. The spectroscopic factor of a highlying 7/2 − state is determined for the first time in 57 Ni and is found to have a much weaker single particle character in reasonable agreement with the new shell-model calculations. We use these new results to re-evaluate and significantly constrain the 56 Ni(p,γ) 57 Cu astrophysical reaction rate required for modeling of explosive burning in x-ray bursts where the astrophysical rp-process is thought to occur.