ORRUBA [7] is a high-solid-angle silicon detector array designed for the measurement of charged-particle reactions with radioactive beams. The position sensitivity of the array, which amounts to approximately 1 ° resolution in polar angle, was designed around the requirements of inverse-kinematics experiments with radioactive beams at Coulomb-barrier energies ( ∼ 5 MeV/u). The design was initially optimized for measuring (d,p) reactions on heavy fission fragment beams, which were a focus of the research program at the HRIBF at ORNL. The original array comprised two 12-fold rings of custom-designed resistive-strip X3 detectors from Micron Semiconductor, covering angles from ∼ 45 ° to 135 ° . In the downstream ring, detector telescopes were deployed (using 65- μ m-thick BB10 detectors) backed by 1000- μ m-thick X3 detectors) for particle identification. For the upstream barrel, which typically only detects particles lighter than the target (i.e., protons), a single layer of 1000- μ m-thick X3 detectors were deployed. Angles further upstream were subtended, when needed, by the SIDAR array of YY1 detectors, typically in a lampshade configuration.

In more recent years, the X3 detectors have been replaced with sX3 detectors (Figure 1), which include 4-fold non-resistive segmentation on the Ohmic contact, for improved energy resolution. Concurrently, the YY1 lampshade was replaced by an annular QQQ5 detector endcap to the sX3 barrel [8, 10], resulting in a more compact array, with near seamless polar angular coverage, enabling the array to be mounted inside major γ -ray detector arrays (see Section 3.2). The design of the QQQ5 detectors involves radial segmentation which is graded in pitch, placing increasingly finer segmentation away from the beam axis, to match the steepness of the kinematic shifts from either pickup or stripping reactions in inverse kinematics, in order to optimize resolution and channel count.

ORRUBA operates as a standalone detector using a fast ionization chamber as a recoil detector (Section 3.3), coupled to recoil separators such as the S800 at FRIB, and operates as the main particle detector for the JENSA gas-jet target [11, 12].

For many direct-reaction measurements, the detection of γ rays in coincidence with charged particles is either necessary (Coulomb-excitation measurements, for instance) or highly advantageous. For measurements such as particle transfer reactions, γ rays aid significantly in separating closely-spaced states populated in the reaction. In addition to the improved energy resolution of γ -ray detection, in many cases neighboring levels decay to different states, leading to better separation in γ -ray energy than the difference in excitation energy. Furthermore, γ rays carry additional information on the states populated (their decay paths, angular distributions, lifetimes, etc.) and can provide information on states not populated directly in the transfer reaction, but fed by decay.

Motivated by these advantages, there has been much investment across the globe in couplings of high-resolution and high-efficiency charged-particle and γ -ray detectors, with a focus on measuring direct reactions on radioactive beams in inverse kinematics. These include TIARA [13] coupled to EXOGAM (GANIL), the SHARC array [14] coupled to the TIGRESS (TRIUMF), TREX [15] and more recently HI-TREX [16] coupled to Miniball (ISOLDE), MUGAST coupled to AGATA [17] (GANIL), and the GODDESS coupling [18] of ORRUBA [7] to Gammasphere [19] and GRETINA [20–24].

GODDESS [18] (Pain et al., forthcoming) is a coupling of an upgraded version of ORRUBA to the large semiconductor γ -ray detector arrays in the US: Gammasphere and GRETINA (and, in the near future, GRETA [25]). GODDESS has been in routine operation with GRETINA since 2019 (see Figure 2), following its original deployment with Gammasphere in 2015. To date, GODDESS campaigns have been performed at ATLAS (2015, 2019, 2021, 2025), and FRIB (2024). In its default configuration, GODDESS provides near-seamless charged-particle coverage from ∼ 15 ° to ∼ 165 ° , with ∼ 1 ° of polar angular resolution and ∼ 80 % azimuthal coverage throughout this range, with particle identification in the forward hemisphere. GODDESS can be operated with a compact ionization chamber (Section 3.3) that mounts at zero degrees, or coupled to recoil separators such as the FMA and the S800.

In preparation for the deployment of GRETA at FRIB, at the time of writing GODDESS is being upgraded. A slightly smaller configuration, with new endcap detectors and a new vacuum chamber, will allow compatibility with the nearly full implementation of GRETA. This will provide a quasi-4 π particle- γ spectrometer with semiconductor resolution for FRIB.

For inverse-kinematics experiments with radioactive beams, detection and identification of the beam-like recoil is often desirable. Firstly, RIBs are often delivered with contaminants, so event-by-event identification of the recoil is needed to associate reaction ejectiles with the beam constituent of interest. Secondly, reactions are often performed on targets with undesired elements (such as the carbon component of polyethylene and deuterated polyethylene targets). Reactions on these nuclides, such as fusion-evaporation reactions, result in substantially different recoils (both nuclide and energy), which can be readily separated by measurement of the beam-like recoil downstream of the target.

Though recoil separators provide numerous benefits for recoil detection, they are not always available, or necessary. Furthermore, their use is complicated in many cases by the energy, angle and charge-state distributions of beam-like recoils after the reaction target. Alternatively, for beam intensities below ∼ 1 0 6 ions/second, a zero-degree detector that sees the entire beam flux can often be used. Ionization chambers can be very effective as such detectors. They can be segmented along the stopping axis, and easily tuned in pressure to match the required stopping power to the energy and charge of the incoming ion, to optimize particle identification via Δ E-E. They are radiation hard, and can deliver good (typically a few percent) resolution at reasonable count rates ( < 1 0 4 − 6 , depending on design).

Conventional transverse-field gridded ionization chambers have been used as zero-degree detectors (e.g., [26]), but they are rate limited to ∼ 1 0 4 ions/second, due to the long drift times. More recently, a number of axial-field ionization chambers have been developed with increased count-rate capability. The TEGIC detector [27], was designed for high-energy ions ( ∼ 100 MeV/u), using a series of aluminized Mylar foils, stacked along the beam axis, to form parallel transmission electrodes, alternately biased to form a series of anodes and cathodes. In this manner, the electron and ion drift distance can be substantially reduced (typically to 1–2 cm) compared to a more conventional transverse-field gridded ionization chamber (where drifts are typically 5–10 cm). The TEGIC electrodes are tilted from the beam axis at 30 ° , to help reduce recombination. The reduced drift distance results in substantially increased counting rate capacity, enabling this detector to operate at up to 1 0 6 ions/second.

Because the foils provide too much dead material for this design to be used for low-energy ions (such as in the 5–10 MeV/u direct-reaction experiments discussed herein), an axial-field ionization chamber was built in support of the ORRUBA program. This detector was based upon the concept of the TEGIC detector, but replaced the foils with a series of high-transmission wire grids (using ∼ 18 μ m diameter wires, spaced by 2 mm, giving > 99 % transmission per grid) [9], so that the ions do not traverse dead material between electrodes (instead, a few percent of the ions stop prematurely, by hitting a wire as they traverse the detector). In this detector, groups of anodes were electrically connected together and read out via charge-sensitive preamplifiers to provide Δ E and residual energy signals, while the cathodes were grounded. This provided particle identification at rates up to 1 0 6 ions/second [9].

Subsequently, a number of other axial-field ionization chamber detectors have been built upon the wire-grid design, incorporating various improvements. A more compact tilted-grid ionization chamber was built, to operate in the much more confined space of GODDESS [18]. An ionization chamber for ANASEN [28] simplified the design by removing the tilt from the grids, and along with it broadening due to tilted windows and asymmetric dead gas lengths, with minimal impact on resolution or count rate capacity [29]. This larger detector also introduced individual wire readout on the entrance anodes, with the rotation of the grids oriented for XY measurement of position of the ion as it enters the detector, with 3 mm resolution. The TRIFIC detector [30] was developed at TRIUMF, using the tilted-grid approach, but biasing the anodes and cathodes symmetrically (rather than grounding the cathodes) for reduced fringe-field effects and enabling operation at higher electric fields.

The most recent detector in this series, MAGIC (Multi Axial-field Gridded Ionization Chamber), is purposefully built for GODDESS (Pain et al., forthcoming). In order to operate in the small space available, while maintaining maximum acceptance and easy reconfiguration, the perpendicular grids are self-supporting and stacked using electric headers (see Figure 3), which provide both mechanical support, and electrical connections from each of the grids to the back flange, where signals are brought out of vacuum. This design makes the detector easily adjustable and serviceable. In this detector, the front two anodes use individual wire readout, for XY position measurement, with 2 mm resolution. The remaining anode signals are brought out of vacuum individually, and can easily be recombined (via a custom preamplifier motherboard) to optimize the anode groupings for particle identification. Furthermore, this is the first detector that provides readout of the cathode signal in addition to the anodes, which facilitates gain matching and improved sensitivity (Pain et al., forthcoming).

To constrain the reaction rate from a single isolated low-lying resonance, three things are needed: the resonance energy E r , the J π of the resonance, and its resonance strength ω γ . Determination of the energy of the resonance is most critical; for a given resonance strength, it impacts the reaction rate exponentially (Equation 1). It also impacts the resonance strength itself (along with the J π of the resonance, which constrains the orbital angular momentum of the captured particle) by impacting barrier penetrabilities; at low energies, the barrier penetrabilities, and hence resonance strength, also exhibit an exponential dependence on resonance energy.

The combination of high resolution charged-particle and γ -ray detection can enable the use of transfer reactions as a mechanism to populate states of astrophysical interest, using high-resolution γ -ray spectroscopy to obtain precise energies of the states populated. In such measurements, the detection, identification and energy measurement of the outgoing ejectile can give unambiguous determination of the nucleus populated, and the approximate formation energy. By detecting angle of the outgoing proton not only helps with the kinematic reconstruction of the two-body reaction, but also enables determination of the momentum vector of the recoiling nucleus, which can be used for a more-precise Doppler correction to the γ rays, and can provide angular distributions that can be used for J π assignments.

In this approach, it is not important whether the reaction proceeds via the component of the wavefunction important for the capture reaction; that is, the resonance strength is not constrained from the cross sections, only by the energy and J π assignments derived from the analysis. This can often reduce uncertainties on resonance strength from experiments with relatively simple analyses, without the concerns pertaining to efficiencies, acceptances, deadtime and normalization that must be addressed in order to extract reaction cross sections for spectroscopic factors (as discussed in Section 4.2). To this end, a series of (³He,t) experiments have been performed with GODDESS; examples of such experiments are given in Sections 6.1 and 6.6.

For low-lying resonances, Γ p is a particularly important parameter for determining ω γ . Indeed, in very low-lying resonances (below ∼ 500 keV), Γ p is generally much smaller than Γ γ . In this case, the resonance strength can be approximated as: ω γ = 2 J f + 1 2 J t + 1 2 J p + 1 Γ p , where Γ p depends critically on E r and J π of the resonance.

In the absence of further information, the maximum strength of a pure single-particle resonance at E r and with J π can be calculated. At such low energies, the value of this width, Γ s p , is dominated by the penetrability of the Coulomb and angular-momentum barriers, so is strongly dependent on E r and J π (or, more strictly, the orbital angular momentum of the proton, ℓ p ). The single-proton width can be determined by calculating single-particle wavefunctions corresponding to the elastic scattering of a proton by a realistic diffuse potential, such as a Woods-Saxon well [31], at the resonance energy of interest [32, 33]. Tuned by adjusting the potential depth, the width of the pure single-proton resonance is determined by the energy dependence of the phase shift δ , and can be calculated by codes such as DWUCK and WSPOT [34, 35].

The proton width of a given resonance is further dependent on the overlap between the many-body nuclear wavefunction of the resonance and the pure single-proton wavefunction - i.e., the proton spectroscopic factor, C 2 S : Γ p = C 2 S Γ s p . (3)

The many-body wavefunction is, a priori, unknown for a given resonance. However, it can be constrained by a nuclear structure model, such as shell-model calculations, or ideally by experimental data, such as from a transfer reaction [3].

The proportionality between cross sections (i.e., spectroscopic factors) from proton-transfer reactions and radiative proton direct-capture [36] and resonant capture reactions [33, 37, 38] is documented. It is important to note that in the extraction of resonance strengths from transfer reactions, the same potential should be used for the calculation of the transfer-reaction cross sections as for the calculation of the single-particle proton widths, as was suggested by the late John Schiffer [32]. Particularly, a strong dependency between the geometry of the single-particle binding potential and reaction cross sections is well known; providing a consistent potential is employed between the two reactions, much of the uncertainty associated with this potential choice cancels [32, 39–41].

The use of transfer reactions to obtain resonance strengths has a number of advantages. Firstly, it can be used to study multiple resonances in a single measurement. Secondly, because the transfer reactions are measured at energies above the Coulomb barrier (typically, several MeV/u upward), the cross sections are not hindered by barrier penetrability. This allows transfer reactions to be used to study very low-lying resonances that are out of reach for direct measurements in the foreseeable future.

Though proton-transfer reactions, such as (³He,d) and (d,n), are the reactions of choice for extracting proton spectroscopic factors, the application of these reactions to experiments in inverse kinematics with radioactive beams remains a challenge. Both (³He,d) and (d,n) reactions are experimentally complicated, by target requirements and the complexities of spectroscopic neutron detection, respectively. Recently, the technique of measuring angle-integrated cross sections by γ-ray tagging the final state, such as a number of recent measurements using GRETINA and the S800 [42, 43], has been employed. Though this approach can be effective, it relies on knowledge of proton- γ branching ratios, corrections for feeding, and on the spins of the final states being known in order to infer ℓ p . Furthermore, for transfer onto states with non-zero spins, even knowledge of J π assignments of final states is insufficient for a conclusive interpretation, as the total angle-integrated cross section for a given final J π can be composed of a sum of multiple proton orbital angular momentum couplings. Without a method to deconvolve these components, it is impossible to assign more than limits on spectroscopic factors for the individual orbital angular components, which inherently have vastly different contributions to the astrophysical reaction rate, due to barrier penetrabilities. Combining these experiments with neutron detection helps address this issue, at the cost of efficiency and the complexity of neutron detection.

However, the isospin independence of the nuclear force can be exploited to constrain proton spectroscopic factors from their neutron counterparts, by using the mirrored reaction (for example, the (d,p) reaction) to extract neutron spectroscopic factors for the equivalent state in the mirror system. There are several experimental advantages to using this technique of measuring (d,p) on proton-rich nuclei, including simple targets, high particle-detection efficiency that is well understood, a compact setup that can be fielded with large germanium detector arrays, and positive Q values which reduce kinematic compression in inverse-kinematic stripping reactions. This approach has been benchmarked for a number of astrophysically-interesting cases in the s d shell, finding general systematic agreement between proton and neutron spectroscopic factors (extracted from (³He,d) and (d,p) reactions, respectively), and associated direct measurements of proton resonance strengths, to within about 30 % [37]. The (d,p) reaction has recently exploited using isobaric analog states in neighboring isobars, at the NSCL using the ²⁶Si(d,p γ )²⁷Si to constrain the ^26mAl(p, γ )²⁷Si reaction [44], and at TRIUMF using the ²³Ne(d,p γ )²⁴Ne reaction to constrain the ²³Al(p, γ )²⁴Ne reaction [45].

Furthermore, a number of astrophysically-interesting nuclides for proton capture lie on or close to the N = Z line (See Section 6). The technique described above is further simplified in the case of the N = Z nuclides, as the ‘target’ nucleus is self-conjugate, and hence identical for the two mirror systems. ²⁶Al represents an important testing ground for this approach, as it is a radioactive N = Z nuclide with strong astrophysical interest [46], yet the ground state (^26gAl) has a long enough lifetime that it can be fabricated into target. Consequently, numerous resonances in the ^26gAl(p, γ ) reaction have been well studied in both normal [47] and inverse [48] kinematics, as well as with indirect techniques such as ^26gAl (³He,d) [47, 49] using an ^26gAl target, and inverse kinematics (d,p) measurements using beams of ²⁶Al [50–52]. The lowest resonance for which there have been direct measurements of the resonance strength via ^26gAl(p, γ ) is the 189-keV resonance. This has been measured [47] using an ^26gAl target to be 0.055 (9) meV [47] and, via inverse kinematics using a 1 0 9 ions/s radioactive beam at TRIUMF using the DRAGON recoil separator to be 0.035 (7) meV [48]. An indirect measurement of the ^26gAl(³He,d)²⁷Si reaction using an ^26gAl target and a Q3D spectrograph yielded a resonance strength of 0.064 for an ℓ p = 1 transfer, with unreported uncertainty (statistical errors on the differential cross section amount to ∼ 20 % ) ¹ . This experiment was unable to probe lower-energy resonances due to the strong backgrounds from the ²⁷Al component of the target.

More recently (d,p) experiments using radioactive ²⁶Al beams have been used to determine the strengths of resonances out of current reach of direct (p, γ ) measurements [50, 51], which are unconstrained by target impurities that limited the ^26gAl(³He,d)²⁷Si experiment. These experiments are described in Section 6.3. Concurrently, as a benchmarking of the (d,p) technique, spectroscopic factors for the mirrors to high-lying resonances were extracted, and resonance strengths determined and compared to direct measurements using (p, γ ) reactions [50].

When constraining resonance strengths via mirror symmetry, it is important to note that the mirror states in the two systems lie at different energies with respect to the separation energy. For example, the low-lying resonances in the ²⁶Al + p system lie hundreds of keV above S p in ²⁷Si, whereas the mirror states in the ²⁶Al + n system lie 2–3 MeV below S n in ²⁷Al. Though the wavefunctions of the mirror states are of a highly similar structure, there are differences due to the effects of different couplings to the continuum. In some studies (e.g., [51]), the spectroscopic factors are assumed to be equal in the two systems to about 30 % , which is often a reasonable approximation, compared to other uncertainties in the experiments, and certainly to the potential orders of magnitude uncertainty in resonance strengths prior to experimental constraint. However, a rigorous approach is to explicitly account for the systematic difference in spectroscopic factors in the two systems due to coupling to the continuum. In [50], spectroscopic factors were calculated for bound ²⁷Al states and unbound ²⁷Si in the Shell Model Embedded in the Continuum (SMEC) formalism [53, 54], using the USDb interaction [55]. The ratio of spectroscopic factors between the two mirror systems was then used to scale the experimentally-determined spectroscopic factors between ²⁷Al and ²⁷Si. This scaling, which is dependent on the energies of the states with respect to their separation energies, and the orbital angular momentum, is typically of the order 10–40 % .

Recent developments at the ReA facility at FRIB are enabling the delivery of beams containing isomeric states in which the GS:IS composition can be controlled without impact on the other properties of the reaccelerated beam [65, 66]. This is achieved by completely stopping the fragmentation beam, and reaccelerating it to energies appropriate for either direct measurements at astrophysical energies, or to Coulomb-barrier energies that are appropriate for indirect techniques for constraining astrophysical reaction rates, such as direct reactions. The GS:IS ratio of the reaccelerated beam can be controlled by two mechanisms. Firstly, the tuning of the fragment separator can be employed to change the GS:IS content of the fragmentation beam before stopping and reacceleration. Secondly, if the lifetimes of the two states are conducive, the adjustable hold-up times in the reacceleration system can be used to further modify the GS:IS composition of the reaccelerated beam. Crucially, because of the stopping and reacceleration, the final beam properties (energy, emittance, etc.) are largely isolated from these adjustments to the GS:IS ratio. These two mechanisms are discussed in Section 5.1.1 and Section 5.1.2.

Understanding the reaction flow breaking out of the hot CNO cycle, and the abundance of ¹⁸F produced in novae (a major source of 511-keV radiation, and hence a potential prompt γ -ray observable), is influenced by the strengths of resonances in ¹⁹Ne, which impact the ¹⁸F(p, α )¹⁵O and ¹⁸F(p, γ )¹⁹Ne reactions. An experiment was performed to measure the ¹⁸F(d,p)¹⁹F reaction to determine spectroscopic factors in the mirror system, to constrain the proton widths in ¹⁹Ne [76]. An isotopically-pure ¹⁸F beam was from the HRIBF was impinged at 6 MeV/u upon a 160 μ g/cm² deuterated polyethylene target. In this pioneering experiment using a (d,p) reaction in inverse-kinematics with a radioactive beam, and an application of the mirror-symmetry approach, proton ejectiles were detected at backward angles in the SIDAR array. In coincidence, the very forward-focused ¹⁹Ne recoils were detected in coincidence at the focal plane of the Daresbury Recoil Separator, while the ¹⁵N recoils from the α emission channel were detected in an annular silicon detector at forward angles. Angular distributions for the protons were measured, and spectroscopic factors extracted from a DWBA analysis, from which proton widths were determined using mirror symmetry. These were used to calculate the astrophysical rate of the ¹⁸F(p, α )¹⁵O reaction, which was found to be reduced by approaching an order of magnitude over the temperature range of classical novae [76, 77]. However, systematic uncertainties remained pertaining to the precise mirror assignments between ¹⁹Ne and ¹⁹F.

More recently, uncertainties in this reaction rate stemming from uncertainties in the energies of low-lying resonances ¹⁹Ne have been addressed, using the ¹⁹F(³He, t γ )¹⁹Ne reaction, measured using GODDESS at ATLAS [10, 78]. The experiment used a 30-MeV ³He beam incident on a ∼ 1 mg/cm² CaF 2 target, and outgoing tritons were detected and identified in the ORRUBA silicon detectors. De-excitation γ rays were measured in coincidence in the Gammasphere array of Compton-suppressed HPGe detectors. Over forty decays from 21 energy levels were identified. In particular, the positions of two 3/2⁺ states near the proton threshold were determined, and the location of an 11/2⁺ state, previously thought to be unbound, was found to be sub-threshold. This reduced the upper limit to the ¹⁸F(p, α )¹⁵O reaction rate by a factor of 1.5–17 across the nova temperature range, reducing the nova detection probability uncertainty by a factor of two [10, 78].

The radioisotope ²²Na is one of the most promising targets of discrete γ -ray astronomy, producing a 1.275 MeV γ -ray line with a lifetime ( t 1 / 2 = 2.6 years) that is sufficiently short to maintain spatial correlation with localized sources, yet long enough to last beyond the opaque conditions at the peak of a nova explosion. ²²Na is predicted to be produced in sufficient quantity in novae to be detectable, and therefore ²²Na is a leading candidate for a prompt γ -ray signature for nova nucleosynthesis in our galaxy. Current and previous instruments may be close in sensitivity to detecting the ²²Na line [79–82], and planning has begun for future missions with greater sensitivity [83]. Understanding the quantity of ²²Na produced in a typical nova explosion is crucial to informing such missions, by predicting the number of novae that are within detectable distance, and enabling a quantitative interpretation of this potential signature, should it be detected. This requires reliable models of nova nucleosynthesis, including reaction rates which affect the ²²Na abundance ejected.

The ²²Na(p, γ )²³Mg reaction is one of the main destruction mechanisms affecting ²²Na yields in nova models. Despite decades of direct and indirect study of the proton resonances important for ²²Na(p, γ ), there is no consensus on the resonance strengths for this important reaction. The resonance believed to be most important at nova temperatures is the 205 keV (7/2)⁺ resonance. There have been two absolute determinations of the strength of this resonance through direct measurements of the (p, γ ) reaction [84–86], performed in 1990 and 2010. ² These experiments differ in resonance strength by about a factor of about 2.5, resulting in a discrepancy in ²²Na yields from novae, and hence the possibility of prompt γ detection, of a factor of 1.4–2.0. It should be noted that all of these challenging direct (p, γ ) measurements utilized ²²Na-implanted targets. Such measurements are subject to possible systematic uncertainties associated with beam/target overlap due to non-uniformities of implanted ²²Na ions (in both depth and transverse profiles) and the beam profile, and uncertainties in the stopping power of the target. This is further complicated by target degradation from irradiation with intense proton beams (typically tens of μ A), and backgrounds from the mCi-activity of the targets.

Because of this, and its astrophysical importance, the 205-keV (7/2)⁺ resonance has also been the subject of a number of indirect studies, based on measurements of branching ratios of this state, fed by β decay. However, rather than resolving the situation, these indirect studies yield resonance strengths with even greater disagreement, spanning over an order of magnitude, and typically with lower strengths than the direct measurements. A recent result yields the lowest number yet [88]. This stems from an experiment using the new gas-filled detector GADGET [89], which is designed to substantially suppress β backgrounds compared to silicon-based setups, to aid in measuring low-energy β -delayed protons. This experiment yielded a substantially smaller proton branching ratio for the 205-keV resonance, despite being in agreement with previous results for the 275-keV and 583-keV resonances [90]. Nominally, the result from the GADGET measurement suggests a resonance strength that is a factor of 7 and 22 below the two direct measurements.

However, in all the β -delayed proton experiments, the branching ratios obtained must be combined with the absolute lifetime of the state in order to determine a resonance strength, and it has been suggested that the lifetime of the state be revisited [91]. The universally-adopted value for the lifetime of the 205-keV (7/2)⁺ resonance is 10 (3) fs, stemming from a fusion-evaporation measurement using Gammasphere [92, 93], in which the lifetime was determined from the fractional Doppler-shift technique. A more recent measurement at TRIUMF has placed a 3 σ upper limit on the lifetime of 12 fs [94] using the Doppler-shift Attenuation Method. However, shell-model calculations suggest a shorter lifetime (0.6–1.7 fs) [91, 95], which would systematically raise all the resonance strengths from indirect measurements by about an order of magnitude.

A detailed systematic study of the spectroscopic strengths of single-proton states in ²³Mg would considerably enlighten the situation, as the large variations in resonance strengths correspond to equally large variations in proton spectroscopic factors for these states, as in Equation 3. Though some proton spectroscopic factors have been determined using the ²²Na(³He,d)²³Mg reaction [96] in an experiment using an implanted ²²Na target and a Q3D spectrometer, only upper limits were obtained for most of the resonances in the astrophysically interesting region. Despite the excellent resolution afforded by the spectrometer, the experiment was hampered by strong background lines and less distinct angular distribution shapes from (³He,d), and the need for substantial shielding to cope with the activity of the ²²Na target.

To address this, the GODDESS collaboration undertook a measurement of the ²²Na(d,p)²³Na reaction in inverse kinematics, to determine single-particle spectroscopic factors of the neutron mirror states, and thereby inform the resonance strengths in ²³Mg independently of the systematics of the previous measurements. Spectroscopic strengths in N = Z mirror systems are typically preserved to ∼ 20%–30% level, a considerably smaller uncertainty than the discrepancies in the resonance strengths, and the differences in mirror systems can further be addressed by SMEC calculations [50] (see Section 4.3). The measurement utilized a 10-MeV/A²²Na beam produced by the ²¹Ne(d,p)²²Na reaction at 10 MeV/A using the RAISOR facility at ATLAS at Argonne National Laboratory. The beamline was tuned for ²²Na( q = 11 + ), providing suppression of Ne (fully stripped Ne ions have q = 10 + ), with further suppression of scattered beam by the RF sweeper. This energy is well-suited to the population of the astrophysically-important low-angular-momentum states, and produces distinctive angular distributions. Such measurements in inverse kinematics involve substantially smaller quantities of ²²Na, thereby avoiding the radiological complications of experiments with ²²Na targets. As with a many s d -shell N = Z nuclei, ²²Na has a 1⁺ isomer a few hundred keV above the ground state. Though the ²¹Ne(d,n)²²Na reaction populates both the ground state and isomer, the ²²Na isomer γ decays with a 243 ns half life to the ²²Na ground state. The flight path between the production target and the experimental target is ∼ 150 feet - which is equivalent to ∼ 5 half lives. Therefore, 97% of the isomeric component of the beam decayed to the ground state by the time the ions reached the experimental target.

The GODDESS position-sensitive fast ionization chamber provided real-time beam diagnostics, including beam composition, rates by particle type, and mm-precision spatial feedback, to aid in tuning of the beamline and optimization of the RF sweeper phase. Protons emitted in the ²²Na(d,p γ )²³Na reaction were detected in the ORRUBA silicon detectors, and de-excitation γ rays in GRETINA. The data from this experiment are currently under analysis.

The 1.8 MeV γ -ray line from the decay of ²⁶Al is a major target of γ -ray astronomy. Its distribution in galactic coordinates has been extensively mapped, and its Doppler shift studied, indicating that it is co-rotating with, and hence pervasive across, the galaxy. Though it is likely that multiple sites contribute to the 1.8-MeV signature, clues to possible sources can be garnered from its correlation with other signatures that have also been directionally-mapped [97]. Notably, the 1.8-MeV γ ray shows strong correlation with 53 GHz free-free microwave emission, which is an indicator of ionized dust clouds, and hence regions of massive star formation [98]. It is likely that massive stars contribute substantially to the ²⁶Al signature, and the rate ^26gAl(p, γ )²⁷Si reaction at stellar temperatures impacts the net production of ²⁶Al. Although many direct measurements constraining ^26gAl (p, γ ) resonance strengths have been performed, at such low temperatures ( < 0.1 GK) the main resonances within the Gamow window for massive stars are out of reach. As discussed in Section 4.3, the lowest energy for which a direct (p, γ ) measurement has been possible is the 189-keV resonance, studied in both normal [47] and inverse kinematics [48].

A 9/2⁺ resonance at 127 keV is likely to dominate in massive stars, which can be populated via ℓ p = 0 proton capture on the 5^{+ 26g}Al. However, due to the lower energy, this resonance is several orders of magnitude weaker than the 189-keV resonance, and hence out of reach for direct measurements. An indirect measurement of the ^26gAl(³He,d)²⁷Si reaction [49] using an ^26gAl target and a Q3D spectrograph was unable to provide more than an upper limit on this resonance, due to the strong backgrounds from reactions on the ²⁷Al component of the target. More recently, (d,p) experiments using radioactive ²⁶Al beams have been used to determine the strengths of resonances out of current reach of direct (p, γ ) measurements [50, 51], which are unconstrained by target impurities that limited the ^26gAl(³He,d)²⁷Si experiment. These experiments, performed at the HRIBF with ORRUBA [50] and at TRIUMF [51], are in remarkable agreement, constraining the strength of the 127-keV resonance ( ω γ = 2.6 − 0.9 + 0.7 × 1 0 − 5 meV [50] and 2.5 (5) × 1 0 − 5 meV [51]). The resonance strength was found to be 4 times higher than the previously adopted upper limit, and to dominate the reaction rate at temperatures between 0.04 GK and 0.1 GK [50]. The experiments also placed upper limits on the strength of the even lower-lying 68 keV resonance ( ω γ ≤ 3.0 × 1 0 − 12 meV [50] and 8 × 1 0 − 13 meV [51]).

³⁰P is of particular interest for understanding classical nova nucleosynthesis on ONe white dwarfs [99], due in part to the long lifetime of ³⁰P ( ∼ 2.5 min) with respect to the timescale of a nova outburst. The ³⁰P(p, γ )³¹S reaction is a potential bottleneck, affecting the reaction flow into the A = 30–40 mass range during the nova [100]. As a consequence, the rate affects the abundances of isotopes of phosphorus, sulphur and silicon - critical elements for observational constraints on novae. The ³⁰P(p, γ )³¹S reaction rate directly affects the isotopic ratio of ³⁰Si/²⁸Si, which is an important nova identifier in the analysis of pre-solar grains [101]. Furthermore, the O/S, S/Al, O/P and P/Al elemental ratios have recently been shown to be particularly sensitive probes of nova peak temperatures, with final abundance ratios varying by 2–3 orders of magnitude due to peak temperature changes between ∼ 230 and ∼ 310 MK. In this detailed study, in which these ratios were found to be one-to-two orders of magnitude more sensitive than ratios solely of elements lighter than phosphorus [102], the impact of various reaction rates on the ratios was examined. The uncertainty in the ³⁰P(p, γ )³¹S rate was highlighted as the major nuclear physics uncertainty in interpreting these ratios, currently hampering the use of these ratios to constrain the energetics of novae.

The rate of this reaction depends critically on the spectroscopic strengths of levels between 6 and 7 MeV excitation in ³¹S. The ³⁰P(d,p γ )³¹P reaction was measured with GODDESS in inverse kinematics, using mirror symmetry to inform state in ³¹S. An 8 MeV/u beam of ³⁰P ( ∼ 80 % pure) was produced via the ²⁹Si(d,p)³⁰P reaction, using the RAISOR facility at ATLAS, and delivered to the GODDESS particle- γ spectrometer.

The protons emitted from the ³⁰P (d,p)³¹S reaction on a ∼ 600 μ g/cm² C 2 D 4 target were measured in the ORRUBA detectors. The GODDESS position-sensitive ionization chamber was used to identify P recoils, and aid in the kinematic reconstruction using the recoil position to help account for the large in-flight beam spot. Using coincident γ rays to aid in resolution, angular distributions were measured and spectroscopic factors determined. As in the case of ²⁶Al, states of a particular J π can be populated via multiple ℓ n transfers, due to the 1⁺ ground-state spin. A manuscript on these results is currently in preparation (Ghimire et al., forthcoming).

The elemental and isotopic composition of dust grains formed during the cooling of nova outflows can provide a signature of the nova origin of these grains, and furthermore provide metrics against which nova models can be tested. Such pre-solar grains can be found in primitive meteorites within the solar system [103]. However, the majority of grains originate from supernovae and massive stars and, although a number of isotopic ratios (including C, N and Si isotopes) are indicators of nova origins, none provide an unambiguous nova signature. A promising candidate for pre-solar grain classification is the ³⁴S/³²S ratio, which recent studies have suggested is constrained to a narrow range in nova grains [101], limit its usefulness. The ³⁴Cl(p, γ ) reaction impacts the nucleosynthetic flow in the sulfur region; if the rate proceeds fast enough with respect to ³⁴Cl β decay, the reaction flow bypasses ³⁴S. However, the ³⁴Cl(p, γ ) reaction rate is subject to substantial uncertainties. In nova sensitivity studies [100], a statistical Hauser-Feshbach calculation is adopted for the ³⁴Cl(p, γ ) reaction rate, and assigned factor of 100 uncertainty due to the lack of experimental constraint, resulting in × 5 variations of final ³⁴S abundance. However, in addition to the 0^{+ 34}Cl ground state ( t 1 / 2 = 1.53 s), the uncertainties are compounded by a low-lying long-lived 3 + isomer at 146 keV ( t 1 / 2 = 32 min), which can both be produced directly via the nucleosynthesis network, and by thermal population at nova temperatures [57, 75]. Notably, the reaction network in [100] did not treat the isomer explicitly. It is necessary to assess the reaction rate on both ^34gCl and ^34mCl, and include both explicitly in network calculations.

A recent spectrograph measurement [104] located levels in ³⁵Ar but was unable to constrain the J π or widths relevant to the ³⁴Cl(p, γ )³⁵Ar reaction rate. A theoretical study of rp-process nuclei with low-lying isomers [105] used shell-model calculations of spectroscopic factors and γ widths to estimate stellar enhancement factors for radiative capture rates on the isomeric states due to thermal population. For ³⁴Cl(p, γ ), an enhancement factor of 1 0 3 was found, peaked at 0.2 GK. As calculations were performed using the USD interaction [106], only positive-parity states were included. However, in other mid s d -shell nuclei, substantial spectroscopic strength is expected for ℓ p = 1 resonances, as the lowest states from the f p -shell are typically located close to the proton separation energy in these nuclei [50, 52, 107–109]. A more recent (2020) study [107] in ( 0 + 1 ℏ ω ) space using the s d p f - m u interaction [110] predicted negative parity states in the astrophysically interesting energy range. However, with no experimental constraint on these levels, 200-keV uncertainties were assumed on excitation energies, and factor 2 uncertainties on spectroscopic factors (partial widths), leading to large uncertainties in the reaction rate. As the authors note: ‘In a study by Fry et al., 17 ³⁵Ar levels have been detected in the energy region Ex = 5.9–6.7 MeV and their excitation energies have been determined, but not spins, parities, widths, or branching ratios. Because of the paucity of such information, it is not yet possible to derive meaningful experimental ^34g,mCl(p, γ )³⁵Ar reaction rates’ [107].

A systematic experimental determination of the distribution of single-proton spectroscopic strengths as a function of excitation energy in ³⁵Ar, for both ^34gCl and ^34mCl, would considerably enlighten the situation. A^34g,mCl(d,p)³⁵Cl experiment has been approved by the FRIB PAC [111], using the techniques outlined in Sections 5 and 6.6, and is awaiting scheduling at the time of writing.

The ³⁸K(p, γ )³⁹Ca reaction is an important bottleneck to the end-point of the rp-process chain. The reaction rate has been estimated to be uncertain by a factor of 10 4 [100], leading to large uncertainties in abundances from novae [100, 112]. Further, in Type I x-ray bursts on neutron stars, the rp process branches and proceeds either via ³⁶K ( β + , t 1 / 2 = 0.342 s)³⁶Ar(p, γ )³⁷K(p, γ )³⁸Ca or ³⁶K(p, γ )³⁷Ca( β + , t 1 / 2 = 0.175 s)³⁷K(p, γ )³⁸Ca. Since ³⁹Sc is almost proton unbound, the rp flow must wait for ³⁸Ca( β + , t 1 / 2 = 0.440 s)³⁸K(p, γ )³⁹Ca [113, 114]. The ³⁸K(p, γ )³⁹Ca reaction is thus an important path to the formation of heavier elements. However, there is limited experimental constraint; the current rate widely used for nucleosynthesis calculations (JINA REACLIB v2.0) is a theoretical rate based on a Hauser-Feshbach statistical model calculation [115]. Furthermore, the relatively short-lived ( t 1 / 2 = 924.3 ms) ³⁸K isomer is important, as it is the endpoint of 76.5 % of the ³⁸Ca decays [116] and the rise times of x-ray bursts typically fall in the range of 1–10 s. As such, capture on both ground ( ^38g K, J π = 3 + ) and isomeric ( ^38m K, J π = 0 + ) states plays an important role in the astrophysical network, and needs experimental constraint.

Though a direct measurement of the ^38gK(p, γ )³⁹Ca reaction has been reported in recent papers [112, 117], many significant questions remain open. This experiment targeted three known 5 / 2 + states (at 386 ± 10 keV, 515 ± 10 keV, and 689 ± 10 keV) in ³⁹Ca, assumed to be populated by ℓ = 0 protons coupled to the 3^{+ 38}K ground state. Only upper limits were set for the lower two resonances. A strength for the 689-keV resonance was extracted, but its energy was found to lie 10 keV lower than the adopted energy, at 679 keV. This reaction rate has been addressed by two recent ORRUBA/GODDESS experiments, as detailed in Sections 6.6.1 and 6.6.2.