A primary goal of nuclear astrophysics is to understand the nucleosynthesis processes that created the elements and the environments in which these processes occur. For many environments, there are certain reaction rates that have a more significant impact than others on the resulting nucleosynthesis. To identify which nuclei are synthesized in a certain environment, it is then necessary to determine these significant reaction rates.
Many important nucleosynthesis processes occur in explosive, high-temperature environments such as novae, x-ray bursts, and supernovae. The higher the temperature, the higher the average interaction energy between nuclei. As the interaction energy increases, it becomes more likely that resonance reactions will occur, which proceed through excited states in the compound nucleus. For many nuclear reactions, it is not currently feasible to experimentally measure the rate directly. However, the strength of these resonances, and thus the overall nuclear reaction rate, will be determined by the structure of the compound nucleus. It is therefore possible to indirectly constrain the nuclear reaction rate by measuring structure properties of the nucleus of interest.
Different methods exist to indirectly constrain the reaction rate. The method best suited to constrain a particular reaction rate will partially be determined by the density of nuclear states within the region of excitation energy where resonances will most significantly contribute. If the level density is low (<10 levels/MeV) and the resonance width is sufficiently narrow, then the reaction rate can be experimentally constrained via the narrow resonance formalism. In this formalism it is assumed that the resonance is narrow enough to where the interaction energy will not vary over the width of the resonance. With this approximation, the reaction rate per particle pair (in cm 3 mole -1 s -1 ) takes the form [1]:
where the sum is over all narrow resonances, µ is reduced mass in amu, T 9 is temperature in GK, (ωγ) i and E i are strength and energy of resonance i in units of MeV, respectively. The resonance strength is defined as
where J r , J 1 , J 2 are the spins of the compound nucleus and reaction products. Γ r , Γ a , Γ b are the total width of the resonance, and the decay widths of entrance and exit channel, respectively.
In this formalism, the reaction rate is exponentially dependent on resonance energy, meaning it is exponentially dependent on the level energies of the compound nucleus. To effectively constrain the reaction rate, it is then crucial to precisely determine the level energies of the compound nucleus. Further structure information, such as the spin and decay widths of states in the compound nucleus are also important.
A specific example in which this type of constraint would be necessary is the 57 Ni(p,γ) 58 Cu reaction rate. This rate is significant because it has been shown to impact the production of 44 Ti, 57 Fe, and 59 Ni in core-collapse supernovae (CCSNe) [2][3][4].
The yield of 44 Ti in CCSNe can be inferred from the observation of gamma-rays in the CCSNe remnant, making it an important probe into the dynamics of these events. Magkotsios et. al. performed a nucleosynthesis sensitivity study using nuclear network calculations and demonstrated that increasing or decreasing the reaction rate of 57 Ni(p,γ) 58 Cu by a factor of 100 impacts the 44 Ti production by a factor of 10 or more at some point in the evolution of the network [2].
In a more recent sensitivity study, Hermansen et. al. also found that varying the rate of 57 Ni(p,γ) 58 Cu impacts the production of 44 Ti, but to a lesser extent than Magkotsios et. al. Varying the rate by a factor of 100 fell just under the threshold of what they considered significant, which was a change in the final abundance of 44 Ti by a factor of 1.1 or greater (essentially a 10% change) [3]. However, this study also noted that decreasing the 57 Ni(p,γ) 58 Cu reaction rate by a factor of 100 decreased the production of 59 Ni by a factor of 1.27. This is significant because determining the amount of 59 Ni produced in CCSNe is needed to test models of cosmic ray acceleration.
It should be noted that another recent sensitivity study by Subedi et. al. did not notice an impact on 44 Ti production when varying the 57 Ni(p,γ) 58 Cu reaction rate [5]. However, this is likely due to the fact that the authors assumed the reaction rate was well constrained by the theoretical calculations, and only varied the reaction rate by a factor of 10. As will be seen, this assumption is questionable for this scenario.
Finally, in a sensitivity study on the impact of nuclear reaction rates on νp-process nucleosynthesis, Nishimura et. al. found that the 57 Ni(p,γ) 58 Cu reaction significantly impacts 57 Fe production. While the astrophysical site of the νp-process is still unknown, candidate cites include CCSNe. This study varied reaction rates for 23 parameterized thermodynamic trajectories that covered a wide range of possible νp-process conditions. In 13 of these 23 trajectories, the 57 Ni(p,γ) 58 Cu reaction rate was shown to impact 57 Fe production [4]. By number of trajectories impacted, this tied for the third most significant reaction identified in the study.
Understanding how the isotopes 44 Ti, 57 Fe, and 59 Ni are produced is important in determining the nucleosynthesis processes that occur in CCSNe. However, despite this importance, no experimental rates exist for the 57 Ni(p,γ) 58 Cu reaction. Current theoretical rates for the reaction are based off of Hauser-Feshbach statistical model calculations, but the validity of these calculations is questionable. Low spin states in the compound nucleus, 58 Cu, between 3 and 6 MeV are expected to contribute to resonances in the reaction. Table 1 lists the current known levels in 58 Cu between 3 and 6 MeV [6]. As can be seen, the level density of low spin states in this region falls below 10 levels/MeV, which is generally taken to be the threshold of Hauser-Feshbach model validity. Additional study into the structure of 58 Cu in the 3 to 6 MeV excitation range is thus necessary to determine the validity of the theoretical calculations.
To provide the capabilities to measure structure properties of 58 Cu, and other nuclei significant to nuclear astrophysics, an Enge split-pole spectrograph was installed at the University of Notre Dame's Nuclear Science Laboratory [7]. The Enge split-pole spectrograph is an ideal tool to perform this study, as it will provide precise measurements of the 58 Cu level energies to be used in constraining the rate via the narrow resonance formalism if the level density does fall below 10 levels/MeV.
The Enge split-pole spectrograph that was installed at the Nuclear Science Laboratory is based on the design of H. A. Enge [8]. The spectrograph is a Scanditronix model ESP 90. Charged particles that are ejected from a nuclear reaction and enter the spectrograph will be separated and focused onto the focal plane based on their magnetic rigidity. The split-pole design creates a magnetic field that provides second-order transverse focusing and vertical focusing. In practice, this ensures that particles entering the spectrograph with the same magnetic rigidity but diverging angles will still be focused at the same point on the focal plane. This increases the angular acceptance of the device (8-12 msr) while still providing good resolving power (p/∆p > 5000) [7,8]. A diagram of the device is shown in Figure 1.
By measuring the position of particles on the focal plane, the energy of the ejectile can be determined. From the ejectile energy and reaction kinematics, the level energies of the residual nucleus can be measured to high precision. While the resolution (and thus the precision) of the excitation spectrum will depend on the ejectile energy and the reaction in question, measurements of level energies to a precision of ∼ 2-5 keV are expected.
In addition to measurements of level energies, the spectrograph is also capable of constraining other structure properties. The spectrograph can be rotated between 0 • and 60 • to measure angular distributions, which can be used to constrain the spin of states in the residual nucleus. Detectors can also be placed in the target chamber to detect charged particle decays of the residual nuclei in coincidence with the ejectiles. This allows for the measurement of particle branching ratios and the determination of particle widths of states in the residual nuclei. All of these capabilities make the Enge split-pole spectrograph an ideal tool for probing the structure of nuclei in order to constrain reaction rates relevant to nuclear astrophysics.
To effectively utilize the resolving power of the Enge splitpole spectrograph, a detector is required at the focal plane that is capable of a position resolution on the order of mm. To this end, a position sensitive ionization chamber was designed and constructed. This focal plane detector (FPD) was based on the design of the detector that was used with the Enge split-pole spectrograph originally located at Yale University's Wright Nuclear Structure Laboratory, though it is now at Florida State University's John D. Fox Accelerator Laboratory [9,10]. While the detector was designed to measure light particles (such as H and He), in principle it could also be used to measure heavier particles.
The FPD consists of an entrance window, exit window, cathode, and two position-sensitive anode sections (one at the entrance and one at the exit). A diagram of the detector and its major components is shown in Figure 2. The position-sensitive sections each consist of three anode wires and a printed circuit board (PCB) assembly. The assembly consists of a PCB board that runs parallel to the plane created by the anode wires. On this PCB are a row of 220 conducting pads, each with a width of 2.286 mm and separated by 0.254 mm.
When a particle passes under the anode wires, it liberates electrons that are collected at the anode wires, and a charge is induced on the pick-up pads. Each one of these pick-pads is attached to a delay tap that delays the signal by 5 ns. Each delay tap is connected in series to form a delay line. The induced charge on the pick-up pads splits and travels to both ends of the delay line. By taking the timing difference between the signals on either end of the delay line, the position of the particle can be determined. A partial diagram of the anode wires and pick-up pads is shown in Figure 3. Additionally, a plastic scintillator is placed behind the FPD to measure the residual energy of ejectiles after passing through the FPD. This residual energy measurement aids in particle identification, which is necessary to select the reaction channel of interest.
An array of silicon diode detectors has been designed and constructed to be mounted on the target chamber flanges. Each flange-mounted array consists of 2 rows of 3 individual diode detectors with an angular spacing of 10 • . The flanges are positioned on the beam-right side of the target chamber in 30 • increments, allowing for the diode detectors to cover angles from 20 • to 160 • in 10 • increments. These diode arrays will detect charged particle decays from the residual nucleus in coincidence with the ejectiles detected in the FPD. This will allow for charged particle branching ratios to be measured and charged particle widths of the resonance to be determined. A picture of a single diode array is shown in Figure 4.
As of early 2024, the Enge Split-Pole Spectrograph at the University of Notre Dame is fully operational and the first scientific measurement has been performed. In order to probe the structure of 58 Cu in the 3-6 MeV excitation region, the 58 Ni( 3 He,t) 58 Cu reaction was measured.
A 3 He beam of 21 MeV was impinged on a (∼200 µg/cm 2 58 Ni target with a thin 12 C backing, and the charge exchange reaction was used to populate states in 58 Cu. The tritons ejected in the reaction were separated using the spectrograph. From the position of the tritons on the focal plane, their energies could be determined and the level energies of states in 58 Cu could be measured. Data were collected at 6 different entrance angles of the spectrograph ranging from 14 • to 35 • in the laboratory frame. The analysis procedure and preliminary results are described in the following section.
To determine characteristics of both the spectrograph and the FPD, the 3 He elastic scattering peak was measured at an angle of ∼14.5 • and four different magnetic field settings. Because the bending radius of a particle is proportional to the magnetic field strength, these measurements were able to provide a calibration between the position on the focal plane and particle bending radius. The bending radius of the particle can then ultimately be used to determine its energy. A plot of the position of the peaks at the four different magnetic field strengths is shown in Figure 5. The peaks have been normalized to a height of 1 for plotting purposes. At each magnetic field setting, a doublet was observed. This corresponded to 3 He scattering on 58 Ni (large peak) and 12 C (small peak). The width of the peak was consistent across the position range of the detector, which is evidence that our detector was parallel to the focal plane. In our calibration, we find a slight quadratic shape between the position and bending radius. This is consistent with the fact that the focal plane is slightly curved. Furthermore, fitting these peaks with a Gaussian suggests a FWHM of 2.1 mm, which we take to be the position resolution of our detector. As previously stated, the energy resolution that corresponds to this position resolution will depend on the reaction, but is expected to be ∼20-50 keV for most reactions of interest.
To select ejectiles from the reaction channel of interest, particle identification (PID) is necessary. A plot of the energy deposited in the cathode vs. the focal plane position was found to be a particularly useful PID. A typical PID plot is shown in Figure 6a. The cathode energy has an associated non-linearity across the detector, which is thought to be caused by incomplete charge collection at certain points of the detector. The non-linearity of the alpha group is more pronounced, and this is likely due to the alpha particles liberating more electrons as they pass through the detector, making their cathode energy measurement more sensitive to the incomplete charge collection. However, this non-linear shape is consistent throughout all runs, and we can therefore correct the cathode energy to create a more linear PID plot. An example of the PID plot with the cathode correction is shown in Figure 6b.
The alpha particles originating from the ( 3 He,α) reaction on both 12 C and 58 Ni populate well known states in 11 C and 57 Ni, respectively. Peaks from these states provide points that can be used to simultaneously determine the magnetic field and entrance angle of the spectrograph. This is done in the following manner.
First, a grid of reasonable magnetic field strengths and opening angles are selected. From the calibration provided by the elastic scattering data, the positions of the particles are converted to bending radii. Then, for each magnetic field and angle pair, the bending radii of these states are converted to excitation energies. The squares of the residuals between the calculated level energies and known level energies are then determined. The magnetic field and angle pair that is chosen is the one with the minimum sum of these squares.
Once the magnetic field and entrance angle are determined for a given set of runs, a particle group can be selected by gating on cathode energy, and the ejectile energy can be determined. From the ejectile energy and reaction kinematics, the excitation energy of the compound nucleus can be determined. Preliminary results from the reaction of interest, 58 Ni( 3 He,t) 58 Cu, are shown in the following section.
By gating on tritons in the cathode vs. position plot, the tritons from the 58 Ni( 3 He,t) 58 Cu reaction can be selected. From the calibration method described above the triton energies, and therefore the excitation energies of 58 Cu, can be determined. An example spectrum of 58 Cu excitation energy taken at a laboratory entrance angle of 24.7 • is shown in Figure 7. The black dotted lines represent states in 58 Cu that have been previously measured with low uncertainty (<10 keV), while the red dotted lines represent states in 58 Cu that were previously measured but with high uncertainty (>10 keV). There is good agreement between the experimental data and states with low uncertainty. Furthermore, the FWHM of the triton peaks is on the order of 30 keV, which will likely allow us to reduce the uncertainty in states with high uncertainty to ∼ 3 keV.
Furthermore, peaks can be seen that do not correspond to known levels, most notably above 4 MeV. While the analysis is still ongoing, this could suggest additional levels in 58 Cu that may lead to significant resonances in the 57 Ni(p,γ) 58 Cu reaction.
It should also be noted that the diode detectors were used in the present experiment. It is still too early in the analysis to determine whether they were able to detect a significant number of proton decays in 58 Cu, but, if they were, this could provide proton branching ratios for higher lying states. Triton spectra were also taken at multiple angles, and work is underway to extract angular distributions that could help constrain the spins of the states. The additional constraints on structure properties such as level energies, spins, and proton branching ratios should ultimately impact the experimental constraint of the rate.
To effectively constrain certain reaction rates that significantly impact nucleosynthesis in explosive astrophysical environments, it is necessary to measure the level energies of states in the compound nucleus that may lead to resonances. Further constraints can be placed on the reaction rate by determining the spins of these states, as well as their particle widths. The Enge split-pole spectrograph is an ideal tool for this task, and one has been installed in the University of Notre Dame's Nuclear Science Laboratory. To measure the position of particles on the focal plane, an ionization chamber has been constructed. This ionization chamber is capable of measuring the position of ejectiles with a resolution of ∼2.1 mm and provides particle identification that can distinguish between isotopes. This position resolution was determined by measuring elastic scattering of 3 He at 21 MeV, but should not be strongly dependent on particle energy. It may, however, depend on the particle species, as particles with a higher Z will liberate more electrons in the detector and produce stronger electric signals.
The first scientific measurement of the Enge split-pole spectrograph has taken place to probe the structure of 58 Cu using the 58 Ni( 3 He,t) 58 Cu reaction. Tritons ejected in the reaction were separated in the Enge split-pole spectrograph based on their magnetic rigidity. The energy of the tritons was determined by their position on the focal plane. From the energy of the tritons and reaction kinematics, the level energies of states in 58 Cu could be determined. Preliminary analysis is ongoing, but the measurement is expected to reduce the level energy uncertainty in states that will likely contribute to significant resonances, thereby constraining the astrophysical rate of 57 Ni(p,γ) 58 Cu.
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (https://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 304, 02002 (2024) https://doi.org/10.1051/epjconf/202430402002
EPJ Web of Conferences 304, 02002 (2024) https://doi.org/10.1051/epjconf/202430402002
EPJ Web of Conferences 304, 02002 (2024) https://doi.org/10.1051/epjconf/202430402002 HINPw7