First measurement of $^{30}$S+$\alpha$ resonant elastic scattering for the $^{30}$S($\alpha$,p) reaction rate

TL;DR

This study provides the first experimental measurement of $^{30}$S+$\alpha$ resonant elastic scattering, identifying new resonant states in $^{34}$Ar that significantly influence the $^{30}$S($\alpha$,p) reaction rate critical for modeling type I x-ray bursts.

Contribution

First measurement of $^{30}$S+$\alpha$ resonant elastic scattering, discovering new resonant states and refining the $^{30}$S($\alpha$,p) reaction rate for astrophysical models.

Findings

Identified three new resonant states in $^{34}$Ar between 11.1 and 12.1 MeV.

Calculated an upper limit for the $^{30}$S($\alpha$,p) reaction rate that exceeds statistical model estimates.

Showed that the new reaction rate significantly impacts x-ray burst energy predictions.

Abstract

Background: Type I x-ray bursts are the most frequent thermonuclear explosions in the galaxy, resulting from thermonuclear runaway on the surface of an accreting neutron star. The $^{30}$S($\alpha$,p) reaction plays a critical role in burst models, yet insufficient experimental information is available to calculate a reliable, precise rate for this reaction. Purpose: Our measurement was conducted to search for states in $^{34}$Ar and determine their quantum properties. In particular, natural-parity states with large $\alpha$-decay partial widths should dominate the stellar reaction rate. Method: We performed the first measurement of $^{30}$S+$\alpha$ resonant elastic scattering up to a center-of-mass energy of 5.5 MeV using a radioactive ion beam. The experiment utilized a thick gaseous active target system and silicon detector array in inverse kinematics. Results: We obtained an excitation function for $^{30}$S($\alpha$,$\alpha$) near $150^{\circ}$ in the center-of-mass frame. The experimental data were analyzed with an $R$-Matrix calculation, and we observed three new resonant patterns between 11.1 and 12.1 MeV, extracting their properties of resonance energy, widths, spin, and parity. Conclusions: We calculated the resonant thermonuclear reaction rate of $^{30}$S($\alpha$,p) based on all available experimental data of $^{34}$Ar and found an upper limit about one order of magnitude larger than a rate determined using a statistical model. The astrophysical impact of these two rates has been investigated through one-zone postprocessing type I x-ray burst calculations. We find that our new upper limit for the $^{30}$S($\alpha$,p)$^{33}$Cl rate significantly affects the predicted nuclear energy generation rate during the burst.

Tables

Table 1. Table 1: Coupling schemes for states in 34 Ar for J π ≤ 4 + superscript 𝐽 𝜋 superscript 4 J^{\pi}\leq 4^{+} for the 33 Cl+p channel. The lowest ℓ p subscript ℓ p \ell_{\rm p} is assumed, and not all possible linear combinations are denoted. See the text.

$J^{\pi }$	${\ell }_p$	$s$	$s_1\oplus s_2$	$s\oplus \ell$
$0^{+}$	2	2	$\uparrow \uparrow$	$\uparrow \downarrow$
$1^{-}$	1	2	$\uparrow \uparrow$	$\uparrow \downarrow$
$2^{+}$	0	2	$\uparrow \uparrow$	—
$3^{-}$	1	2	$\uparrow \uparrow$	$\uparrow \uparrow$
$4^{+}$	2	2	$\uparrow \uparrow$	$\uparrow \uparrow$

Table 2. Table 2: Best fit level parameters of 34 Ar determined by the present work. All levels are newly proposed. The table is arranged such that the corresponding physical property of each state in 34 Ar precedes the corresponding R 𝑅 R -Matrix fit parameter. As we could not uniquely constrain the spin-parity of the 11.09-MeV level, two possible assignments are given, as well as the corresponding widths. The 12.08-MeV level is shown in italic letters as there is a large systematic uncertainty associated with it. See the text.

$E_{\mathrm{ex}}$ (MeV)	$E_r$ (MeV)	$J^{\pi }$	${\ell }_{\alpha }$	${\Gamma }_{\alpha }$ (keV)	${\theta }_{\alpha }^2$ (%)	${\Gamma }_p$ (keV)	$\xi$ (%)
$11.092\left(85\right)$	$4.353\left(85\right)$	$(2^{+},4^{+})$	2, 4	${20}_{-17}^{+80}$, ${0.5}_{-0.4}^{+1.4}$	${40}_{-33}^{+180}$, $8_{-6}^{+10}$	25${}_{}{}^{+500}{}_{-20}{}^{}$, ${0.3}_{-0.3}^{+3.5}$	1, 0.1
$11.518\left(89\right)$	$4.779\left(89\right)$	$2^{+}$	2	${100}_{-60}^{+120}$	${90}_{-55}^{+110}$	210${}_{}{}^{+600}{}_{-170}{}^{}$	4
12.079(95)	5.340(95)	(2+)	2	260${}_{}{}^{+400}{}_{-120}{}^{}$	100${}_{}{}^{+150}{}_{-45}{}^{}$	340${}_{}{}^{+550}{}_{-200}{}^{}$	6

Table 3. Table 3: Resonance parameters of 34 Ar adopted in the calculation of the 30 S( α 𝛼 \alpha , p) stellar reaction rate calculation. Resonances with E r < 0.7 subscript 𝐸 r 0.7 E_{\rm r}<0.7  MeV are not tabulated as they fall below the x-ray burst Gamow window and the Wigner limit for the α 𝛼 \alpha channel rapidly vanishes. Parameters shown in boldface are based on experimental data. Level energies below E ex < 11 subscript 𝐸 ex 11 E_{\rm ex}<11  MeV are taken from Ref.  O’Brien  et al. ( 2009 ) , where we assumed J π = 0 + superscript 𝐽 𝜋 superscript 0 J^{\pi}=0^{+} and Γ α = 1 2 ​ W Γ ​ α = γ subscript Γ 𝛼 1 2 subscript 𝑊 Γ 𝛼 𝛾 \Gamma_{\alpha}=\frac{1}{2}W_{\Gamma{\alpha}}=\gamma ; the relative dependence of J 𝐽 J and γ 𝛾 \gamma for small J 𝐽 J in our framework is exemplified in Fig.  10 . The higher energy resonances are from the present work and are separated from the others by a horizontal rule. We stress that the tabulated properties which are not taken from experimental data may not be correct for individual resonances, but rather that the sum of these contributions to the stellar reaction rate can be considered an upper limit under an extreme assumption, which is interesting to investigate. See the text.

$E_{\mathrm{ex}}$ (MeV)	$E_r$ (MeV)	$J^{\pi }$	$\omega$	${\Gamma }_{\alpha }$ (keV)	$W_{{\Gamma }_{\alpha }}$ (keV)	$W_{{\Gamma }_p}$ (MeV)	$\omega \gamma$ (keV)
7.47	0.73	($0^{+}$)	1	$2\times {10}^{-15}$	$4\times {10}^{-15}$	$2\times {10}^{-1}$	$2\times {10}^{-15}$
7.88	1.14	($0^{+}$)	1	$2\times {10}^{-9}$	$3\times {10}^{-9}$	$4\times {10}^{-1}$	$2\times {10}^{-9}$
7.96	1.22	($0^{+}$)	1	$1\times {10}^{-8}$	$2\times {10}^{-8}$	$4\times {10}^{-1}$	$1\times {10}^{-8}$
8.15	1.41	($0^{+}$)	1	$4\times {10}^{-7}$	$8\times {10}^{-7}$	$6\times {10}^{-1}$	$4\times {10}^{-7}$
8.30	1.56	($0^{+}$)	1	$4\times {10}^{-6}$	$8\times {10}^{-6}$	$6\times {10}^{-1}$	$4\times {10}^{-6}$
8.55	1.81	($0^{+}$)	1	$1\times {10}^{-4}$	$2\times {10}^{-4}$	$8\times {10}^{-1}$	$1\times {10}^{-4}$
8.74	2.0	($0^{+}$)	1	$7\times {10}^{-4}$	$1\times {10}^{-3}$	$9\times {10}^{-1}$	$7\times {10}^{-4}$
8.89	2.15	($0^{+}$)	1	$3\times {10}^{-3}$	$5\times {10}^{-3}$	1	$3\times {10}^{-3}$
8.99	2.25	($0^{+}$)	1	$7\times {10}^{-3}$	0.01	1	$7\times {10}^{-3}$
9.42	2.68	($0^{+}$)	1	0.1	2	1	0.1
9.75	3.01	($0^{+}$)	1	0.7	1	2	0.7
10.32	3.58	($0^{+}$)	1	7	10	2	7
10.81	4.07	($0^{+}$)	1	30	50	3	30
11.09	4.35	2+	5	20	50	5	100
11.52	4.78	2+	5	100	110	5	500
12.08	5.34	2+	5	260	260	6	1300

Equations

$$\begin{array}{r}3\alpha \to {}^{12}\mathrm{C}(\mathrm{p},\gamma {)}^{13}\mathrm{N}(\mathrm{p},\gamma {)}^{14}\mathrm{O}(\alpha ,\mathrm{p}{)}^{17}\mathrm{F}(\mathrm{p},\gamma )\\ {}^{18}\mathrm{Ne}(\alpha ,\mathrm{p}{)}^{21}\mathrm{Na}(\mathrm{p},\gamma {)}^{22}\mathrm{Mg}(\alpha ,\mathrm{p}{)}^{25}\mathrm{Al},\end{array}$$

$$\begin{array}{r}{}^{25}\mathrm{Al}(\mathrm{p},\gamma {)}^{26}\mathrm{Si}(\alpha ,\mathrm{p}{)}^{29}\mathrm{P}(\mathrm{p},\gamma ){}^{30}\mathrm{S}(\mathbf{\alpha },\mathrm{p}{)}^{33}\mathrm{Cl}(\mathrm{p},\gamma )\\ {}^{34}\mathrm{Ar}(\alpha ,\mathrm{p}{)}^{37}\mathrm{K}(\mathrm{p},\gamma {)}^{38}\mathrm{Ca}(\alpha ,\mathrm{p}{)}^{41}\mathrm{Sc}.\end{array}$$

$$E_{c.m.}=\frac{M+m}{4M{\cos }^2{\vartheta }_{\mathrm{lab}}}{E}_{\alpha },$$

$$\frac{d\sigma }{d\Omega }=\frac{Y_{\alpha }S(E_{\mathrm{b}})}{I_{\mathrm{b}}n\Delta E\Delta {\Omega }_{c.m.}}\frac{m}{M+m},$$

$$\frac{\Delta E_{\rm c.m.}}{E_{\rm c.m.}}=\sqrt{\Bigg{(}\frac{\Delta E_{\alpha}}{E_{\alpha}}\Bigg{)}^{2}+4\Bigg{(}\frac{\cos(\vartheta_{\rm lab})-\cos(\vartheta_{\rm lab}^{\prime})}{\cos(\vartheta_{\rm lab})}\Bigg{)}^{2}},$$

$$W_{{\Gamma }_i}=\frac{2{\hslash }^2}{{\mu }_i{R}_i^2}{P}_{{\ell }_i},$$

$$\omega \equiv \frac{2J_{\mathrm{r}}+1}{(2J_{\mathrm{A}}+1)(2J_{\mathrm{a}}+1)},$$

$$\gamma \equiv \frac{{\Gamma }_{\mathrm{a}}{\Gamma }_{\mathrm{b}}}{{\Gamma }_{\mathrm{tot}}},$$

Full text

Type I x-ray bursts are the most frequently observed thermonuclear explosions in the galaxy, resulting from thermonuclear runaway on the surface of an accreting neutron star. The 30S($\alpha$, p) reaction plays a critical role in burst models, yet insufficient experimental information is available to calculate a reliable, precise rate for this reaction.

Purpose

Our measurement was conducted to search for states in 34Ar and determine their quantum properties. In particular, natural-parity states with large $\alpha$-decay partial widths should dominate the stellar reaction rate.

Method

We performed the first measurement of 30S+$\alpha$ resonant elastic scattering up to a center-of-mass energy of 5.5 MeV using a radioactive ion beam. The experiment utilized a thick gaseous active target system and silicon detector array in inverse kinematics.

Results

We obtained an excitation function for 30S($\alpha$, $\alpha$) near $150^{\circ}$ in the center-of-mass frame. The experimental data were analyzed with $R$-Matrix calculations, and we observed three new resonant patterns between 11.1 and 12.1 MeV, extracting their properties of resonance energy, widths, spin, and parity.

Conclusions

We calculated the resonant thermonuclear reaction rate of 30S($\alpha$, p) based on all available experimental data of 34Ar and found an upper limit about one order of magnitude larger than a rate determined using a statistical model. The astrophysical impact of these two rates has been investigated through one-zone postprocessing type I x-ray burst calculations. We find that our new upper limit for the 30S($\alpha$, p)33Cl rate significantly affects the predicted nuclear energy generation rate during the burst.

I Introduction

Type I x-ray bursters (XRBs) are a class of astronomical objects observed to increase in luminosity by factors of typically tens to several hundreds Galloway et al. (2008) for a short period of time (tens of seconds) with the photon flux peaking in the x ray and a total energy output of about $10^{39}$–$10^{40}$ ergs Lewin (1977); Ayasli and Joss (1982). The sources of such emissions repeat these outbursts typically on time scales of hours to days, allowing for the extensive study of the burst morphology of individual XRBs. In our galaxy, over ninety such sources are presently known since their initial discovery some forty years ago. XRBs are modelled very successfully as a neutron star accreting material rich in hydrogen and/or helium from a low-mass companion. The accretion mechanism causes the formation of an electron-degenerate envelope around the neutron star, where the thin-shell instability triggers a runaway thermonuclear explosion at peak temperatures of $1.3-2.0$ GK Schatz et al. (2001); Woosley et al. (2004); Schatz and Rehm (2006); Fisker et al. (2008); Parikh et al. (2013), which we observe as an x-ray burst.

The sharp rise of the x-ray fluence is understood to be powered by explosive helium burning on the neutron-deficient side of the Segrè chart Champagne and Wiescher (1992); Lewin et al. (1993); Wiescher et al. (1999); Schatz and Rehm (2006); Parikh et al. (2013). In a mixed hydrogen and helium shell, the explosive nucleosynthesis initially manifests as a series of ($\alpha$, p)(p, $\gamma$) reactions on oxygen seed nuclei near the proton drip line (14,15O), called the $\alpha$p-process Wallace and Woosley (1981). One such sequence in this burning pathway is

[TABLE]

which continues as

[TABLE]

In this sequence, the ($\alpha$, p) reactions proceed through $T_{z}=\frac{N-Z}{2}=-1$ compound nuclei. The $\alpha$p-process gives way to the rapid proton-capture process (rp process) near the $Z\approx 20$ region owing to the ever increasing Coulomb barrier and decreasing ($\alpha$, p) $Q$ values. Aside from the two protons consumed in the nuclear trajectory from 12C to 14O, the $\alpha$p process is schematically pure helium burning (since the abundance of hydrogen is constant), and it does not include any $\beta^{+}$ decays which tend to hamper the energy generation rate in explosive nucleosynthesis.

While a plethora of nuclear processes tend to take place in a given regime of stellar nucleosynthesis, typically the precise rates of only a handful of these processes influence the predicted nature and magnitude of actual astrophysical observables. It is these specific nuclear quantities which should be well constrained by laboratory experimentation. This general picture is confirmed in XRBs, where the nuclear reaction network includes hundreds of species and thousands of nuclear transmutations. Studies have shown that it is only a small subset of these nuclear transmutations which need to be known precisely, as they make a predominant contribution to the nuclear trajectory to higher mass and energy generation Parikh et al. (2008), at least for the examined models.

The 30S($\alpha$, p) reaction is identified as one such important reaction, contributing more than 5% to the total energy generation Parikh et al. (2008), influencing the elemental abundances in the burst ashes Parikh et al. (2008) relevant to compositional inertia (see, e.g., Taam (1980) for a description of this phenomenon), moving material away from the 30S waiting point Iliadis et al. (1999), and possibly accounting for double peaked XRBs Fisker et al. (2004). A recent study found the 30S($\alpha$, p) reaction sensitivity in XRBs among the top four in a single zone model Cyburt et al. (2016), as well as having a prominent (but unquantified) impact on the burst light curve in a multizone model.

A firmer understanding of the input nuclear physics for XRB models will allow for more reliable comparison with observations to constrain neutron star binary system properties, such as accretion rate and metallicity, as well as the neutron star radius itself Heger et al. (2007); Galloway et al. (2004); Özel (2006); Özel et al. (2012); Güver et al. (2010); Zamfir et al. (2012).

The ($\alpha$, p) reactions occurring on lower mass nuclei such as 14O and 18Ne have been measured directly Notani et al. (2004); Fu et al. (2007); Kim et al. (2015); Bradfield-Smith et al. (1999); Groombridge et al. (2002), and the properties of resonances in the compound nuclei 18Ne and 22Mg have been the subject of a number of indirect studies (see, e.g., Refs. Hu et al. (2014); Zhang et al. (2014) and references therein). In spite of these extensive works, those cross sections still remain quite uncertain. Unfortunately, the situation is much more dire in the case of the ($\alpha$, p) reactions induced on higher mass targets such as 30S. The only experimental information on the structure of 34Ar above the $\alpha$ threshold and the 30S($\alpha$, p) stellar reaction rate is limited to a preliminary report on a transfer reaction study of the compound nucleus 34Ar at high excitation energy O’Brien et al. (2009) and a time-reversal study Deibel et al. (2011). The present work is the first experimental investigation using the entrance channel 30S+$\alpha$.

II Experiment

We performed the first measurement of $\alpha$ resonant elastic scattering on a 30S radioactive isotope beam (RIB) using a thick target in inverse kinematics Artemov et al. (1990). The experiment was carried out at the CNS Radioactive Ion Beam separator (CRIB) Kubono et al. (2002); Yanagisawa et al. (2005), owned and operated by the Center for Nuclear Study (CNS), the University of Tokyo, and located in the RIKEN Nishina Center. The CRIB facility has been a workhorse for measurements of elastic scattering of primarily astrophysical interest Teranishi et al. (2003, 2007); Yamaguchi et al. (2009); He et al. (2007, 2008, 2009); Kim et al. (2010); Yamaguchi et al. (2011); Jung et al. (2012); Chen et al. (2012); Yamaguchi et al. (2013); Jin et al. (2013); He et al. (2013); Chen et al. (2012); Jung et al. (2012, 2014), schematically using similar techniques to the one adopted in the present study.

The 30S RIB was produced inflight using the 3He(28Si, 30S)n transfer reaction. A 28Si9+ primary beam was extracted from an electron cycltron resonance ion source and accelerated to 7.3 MeV/u by the RIKEN AVF cyclotron ($K\approx 70$) with a typical intensity of 80 pnA. We impinged the 28Si beam on the production target located at the entrance focal plane to CRIB, comprised of a windowed, cryogenic gas cell Yamaguchi et al. (2008). 3He gas at 400 Torr was cooled to an effective temperature of 90 K with LN2; the gas was confined by 2.5-$\mu$m Havar windows in a cylindrical chamber with a length of 80 mm and a diameter of 20 mm, yielding a 3He target thickness of approximately 1.7 mg cm*-2*. As the fully stripped species 30S16+ is the easiest to separate and distinguish from the primary beam, we used Be (2.5 $\mu$m) and C (300 $\mu$g cm*-2*) stripper foils immediately after the production target; when the Be (C) stripper foil was new, the 30S16+ purity was 88% (67%), but decreased dramatically within hours as the beam degraded the foils. Studies are ongoing to investigate the effects of stripper foil degradation on beam purity and intensity. The resulting cocktail beam was separated by a double achromatic system (set to $\frac{\Delta p}{p}=1.875\%$ with slits at the dispersive focal plane) and further purified with a Wien (velocity) filter. The 30S RI beam arrived on target with typical purity of 28% and an intensity of $8\times 10^{3}$ pps, successfully injecting $1.6\times 10^{9}$ 30S ions during the main measurement over two days.

The setup at the experimental focal plane, shown in Figs. 1 & 2, consisted of two beamline monitors, an active target system (see below), and an array of silicon strip detectors (SSDs). The beamline monitors were parallel plate avalanche counters (PPACs, enumerated ‘a’ and ‘b’, respectively) Kumagai et al. (2001), which served to track the beam ions event by event and to produce the trigger signal for the data acquisition system (DAQ). The DAQ triggered when both PPACs fired in coincidence with an SSD to obtain the physics of interest; the DAQ also triggered in a downscaled mode for $\frac{1}{n}$ PPAC coincidences ($n=2.2\times 10^{4}$) regardless of the SSD signals. This trigger setup is standard in CRIB experiments, and it allows for event-by-event analysis of scattering events as well as simultaneous diagnosis of the RIB for systematic behavior while keeping a total trigger rate $<1$ kHz. Due to a DAQ error, we could only fully analyze the data from the forward right-side SSD.

During the RIB production, the efficiency of each PPAC was determined to be quite high ($>\!\!99$%) considering the number of events recorded by each detector; however, at the start of the scattering measurement PPACb was damaged by several discharges, and its efficiency became somewhat lower ($\sim 90$%) for most of the experiment. Each SSD was 0.5 mm thick and had an active area of $91\times 91$ mm2, eight strips on one side, and a single pad on the reverse. A scattering chamber filled with about $\frac{1}{4}$ atm 90% natHe + 10% CO2 gas mixture housed both the active target system and the SSDs.111Gas mixture percentages are quoted by volume. The He + CO2 gas pressure was monitored continuously throughout the scattering measurement and managed by a dedicated system; we set the gas flow controller to circulate fresh gas into the chamber at 20 standard liters per minute with the evacuation rate regulated to keep a constant gas pressure of $194.2\pm 0.5$ Torr during the entire measurement. The gas-filled chamber was sealed off from the beamline vacuum with a 7.4-$\mu$m Kapton foil; the entrance window was 40 mm in diameter.

An active target is a device where a material serves simultaneously as a target and part of a detector, in principle allowing one to perform direct measurements at a beam interaction position. The readout section of our active target is an etched copper plate placed under the field cage, opposite to the cathode top plate, so that electrons created in the electric field of the cage by ionizing radiation drift towards it. The readout pads are separated into four sections: one for detecting the beam or heavy recoils and three for detecting outgoing light ions. Forty-eight pads comprise the beam readout section, while the regions for detecting light ions are comprised of eight rectangular pads each. The pads are 3.5 mm in depth, surrounded by 0.25 mm of insulation on all sides (making 0.5 mm of insulation between each pad). Each pad is also bisected diagonally into two congruent right triangles, so that the collected charge can be read out from two opposing sides (backgammon pads). The section for detecting the beam ions is the largest and located at the center, slightly shifted towards the beam upstream direction after installation in the scattering chamber. The regions for detecting light ions surround the beam section on the left, right, and downstream sides. Gas electron multiplier (GEM) foils were used to set different effective gains over the beam and light-ion regions. Over the center of the downstream high-gain GEMs was a bridge to prevent the unscattered beam ions from saturating the light ion signals.

We quantified the measurement capabilities of the active target using both online and offline measurements. For the low-gain region, we compared the position of 30S ions derived from the active target to those determined by extrapolation of the PPAC data. For the high-gain region, we analyzed the aggregate track width of radiation emitted from a standard $\alpha$ source in a fixed position as measured by the active target; the tracks were software gated to be in coincidence with a geometrically-central SSD strip. By assuming a standard CRIB PPAC resolution of $\sim\!\!1~{}{\rm mm}$ based on our experience and the known finite strip size of the SSD, we varied the active target resolution in a Monte Carlo simulation until the calculations agreed with the data. The performance of the active target depended on the type of measurement, quoted below at $1\sigma$. The $Y$ position, determined by the electron drift time, was the most precise being $\leq 0.5$ mm; the high precision of the drift-time measurement enabled us to confirm the PPAC resolution, which was found to be 0.9 mm. The $X$-position resolution, determined by charge division in the backgammon pads, was 3 mm.

In the present work, the typical 30S scattering laboratory angle and change in energy loss was difficult to reliably distinguish from the unscattered beam given the above resolution for the low-gain GEM in $X$.

Considering the close spacing of the high-gain GEM data and their relatively large distance from typical scattering locations, extrapolating the vector of an $\alpha$ particle’s track results in a large uncertainty. Instead, we found that averaging the pad $X$ and $Y$ data over the center (in $Z$) of the high-gain GEM reduced the uncertainty and was sufficient for our purposes.

The 30S energy on target was measured to be $48.4\pm 2.0$ MeV. The stopping power for 30S in the He+CO2 gas mixture was determined by both a direct measurement of the beam energy at five target pressures and by a comparison of the shape of the Bragg curve and stopping position of the unscattered ions in the active target as shown in Fig. 3. Excellent agreement was found between the measurements and the prediction using Ziegler’s method; the maximum difference between the measured and calculated 30S residual energy was 700 keV or less than 4%. The energy loss and the Bragg curve of the contaminant 29P were also reproduced using an identical approach, giving us confidence in our treatment of the energy loss in the PPACs, entrance window, and He+CO2 gas mixture. The event-by-event particle identification of the cocktail beam is shown in Fig. 4.

We confirmed the energy loss of $\alpha$ particles using a standard triple $\alpha$ source and an $\alpha$ beam created by CRIB, checking that both their Bragg curves and residual energies agreed with the calculations. A calibrated $\Delta E$–$E$ spectrum from the 30S+$\alpha$ scattering measurement is shown in Fig. 5; the figure shows clearly that the measured locus is consistent with the theoretical trend for $\alpha$-particles. The dynamic range of the high-gain GEMs was optimized to be 10–100 keV corresponding to the energy deposit of $\alpha$ particles, which would always be stopped in the first SSD layer unlike high energy protons. As protons with enough energy to reach an SSD deposit $<5$ keV per pad, they could not be detected by the active target system.

III Analysis

III.1 Determination of cross section

We measured (1) the residual energy of $\alpha$ particles using an SSD, (2) the beam trajectory using the two PPACs, and (3) the recoiling $\alpha$ particle position using the high-gain portion of the active target. These three pieces of information are sufficient to determine the center-of-mass energy $E_{\rm c.m.}$ for elastic scattering, defined as

[TABLE]

where $M$ and $m$ are the masses of 30S and 4He, respectively, $\vartheta_{\rm lab}$ is the laboratory scattering angle, and $E_{\alpha}$ is the laboratory energy of the scattered $\alpha$ particle. Using the experimentally verified stopping power of the He+CO2 gas for 30S ions and $\alpha$ particles, we numerically solved the above kinematic equation event-by-event. We selected test points along the extrapolated 30S ion trajectory in 1 mm steps, calculating $E_{\rm c.m.}$ according to the beam’s energy loss up to each point. Each tested scattering depth fixes the value of $\vartheta_{\rm lab}$ by the geometric measurements of the PPACs and high-gain GEM. Finally, the initial $\alpha$ energy was determined with Eq. (3) and its residual energy was calculated considering its energy loss along said path, which we then compared with the energy recorded by the SSD. The algorithm continued until the 30S ion came to rest with no solution being found or when the calculated and measured residual $\alpha$ particle energy disagreed by less than 100 keV, which we define as a true scattering event at that $E_{\rm c.m.}$. In the case that more than one test point satisfied these conditions, we select the $E_{\rm c.m.}$ with the smallest disagreement between the measured and calculated residual $\alpha$ energy. The resulting $\alpha$ spectrum is shown in Fig. 6.

The differential cross section was then calculated using

[TABLE]

where $Y_{\alpha}$ is the yield of $\alpha$ particles at each energy bin, $S(E_{\rm b})$ is the stopping power of 30S in He+CO2, $I_{\rm b}$ is the number of 30S beam ions injected into the target, $n$ is the number density of 4He atoms, $\Delta E$ is the energy bin size (100 keV), and $\Delta\Omega_{\rm c.m.}$ is the center-of-mass solid angle at each energy bin. The number of beam ions injected into the target $I_{\rm b}$ was defined as the coincidence between the two PPACs, recorded as a scaler during the run, multiplied by the average 30S purity, which includes a cut for the active target entrance window for successful injection into the target. The lower efficiency of PPACb cancels out in the deduction of the cross section because we demanded beam particle detection with the PPACs for counting both the number of scattering events and injected beam ions. Since the scattering could take place over a range of target depths, we calculated the solid angle $\Omega_{\rm c.m.}$ from the vantage point of each actual scattering event and fit the trend with an empirical function. The yield of $\alpha$ particles $Y_{\alpha}$ was scaled universally by a factor of $2.0\pm 0.1$ to match the calculated magnitude of Rutherford scattering at lower energies; a similar deficiency was observed in the number of $\alpha$ particles (produced in the cocktail beam by CRIB) detected by the high-gain GEM compared to the SSD in a low-statistics test run giving a scaling factor 1.8$\pm$0.3 (see Secs. II B and III D below and Ref. Kahl (2015) for further details). A series of measurements are already planned to further investigate and constrain this scaling factor. The resulting excitation function is shown in Fig. 9(a).

III.2 Sources of background

Detected $\alpha$ particles might originate from a source other than elastic scattering of 30S with the helium nuclei in the target gas. We applied software gates to the PPAC data event-by-event to ensure the incident beam ions were consistent with the properties of 30S, which removed contributions to the $\alpha$ spectrum induced by other heavy ion species within the cocktail beam.

One might imagine various reactions with the PPACs, Kapton window (stoichiometry C22H10N2O5), or the CO2 used as a quenching gas in the active target. The standard PPACs used at CRIB are each filled with 9 Torr C3F8 over a length of $\approx 35$ mm ($\approx 0.3~{}{\rm mg}~{}{\rm cm}^{-2}$) confined with 2 $\mu$m aluminized Mylar windows (H8C10O4) and interspaced with a further three 1.5-$\mu$m similar foils (8.5 $\mu$m in total).

The 30S beam profile on PPACa does not have a line of sight to the high-gain GEM owing to the active target entrance window combined with the bridge over the downstream high-gain GEM, although the edge of the 30S profile on PPACb does have such a line of sight. Thus, we can geometrically rule out PPACa (but not PPACb) as a source of background $\alpha$ particles.

Although the CNO-group elements require some consideration, we can immediately rule out hydrogen as a background source of $\alpha$ particles, because the 30S(p, $\alpha$) reaction $Q$ value is $-8.47$ MeV, and the 30S+p system $E_{\rm c.m.}<4$ MeV anywhere after the dispersive focal plane.

As for the entrance window and quenching gas, the Coulomb barriers for 30S+12C, 30S+14N, and 30S+16O are 24.4, 28.0, and 31.3 MeV, respectively. The 30S beam energy impinging on the Kapton window is about 2.34 MeV/u, yielding $E_{\rm c.m.}=20.0,22.3,24.3$ MeV for nuclear interactions with 12C, 14N, and 16O, respectively. As for the incident 30S beam energy impinging on the He+CO2 gas, it is about 1.62 MeV/u, which yields $E_{\rm c.m.}=13.9,16.9$ MeV for nuclear interactions with 12C and 16O, respectively. Considering that the center-of-mass energies are always below the respective Coulomb barriers for the entrance window and quenching gas, this implies that the heavy-ion fusion cross sections should be many orders of magnitude lower than that of $\alpha$ elastic scattering.

Although we are not aware of any experimental data concerning 30S-induced heavy-ion reactions, the fusion study with 12C and 16O on the stable isotopes 28,29,30Si by Jordan et al. Jordan et al. (1979) is analogous if we accept isospin symmetry. Their center-of-mass energies broadly overlap with ours sufficiently to make a germane comparison. In that work, the authors see smooth behavior of the excitation functions except in the case of 12C+28Si, where they report oscillatory behavior between $21\sim 26$ MeV in 12C bombarding energies, which would correspond to maximum $E_{\alpha}$ in the range of $5.5\sim 7$ MeV from the 28Si(12C, 4He) reaction. $E_{\alpha}$ corresponding to the lowest region of interest in our study of 30S+$\alpha$ is around 15 MeV ($E_{\rm c.m.}\approx 4$ MeV). Thus, the only structure seen by Jordan et al. is quite far away from the structure we report here.

As relevant to the present study, Jordan et al. importantly find smooth cross sections for 12C and 16O with 30Si, the mirror nucleus of 30S. Such behavior implies that a background source of $\alpha$ particles in the present work induced by CNO-group elements should have a relatively flat energy distribution. Figure 9(a) shows that our observed resonant structure is manifested as destructive interference with pure Rutherford scattering. It means that any unaccounted for background of $\alpha$ particles arising from the 30S beam interacting with CNO-group elements would tend to decrease our observed resonance dips and thus our deduced partial widths $\Gamma_{i}$ could be modestly smaller than the true values. If we consider the relative differences in the maximum ($\approx$ 60 mb/sr) and minimum ($\approx$ 5 mb/sr) differential cross sections around 5 MeV center-of-mass energy, then a smooth background cannot comprise more than 8% of the measurement in the region of interest. This uncertainty turns out to be smaller than the statistical error and as such can be reasonably neglected.

The main sources of energy-dependent background could be $\alpha$ particles originating from the RIB production target satisfying the $B\rho$ selection as well as contributions from inelastic scattering. The bumps seen in the excitation function around 3.5 MeV in Fig. 9(a) correspond to the region where $\alpha$ particles magnetically selected by CRIB are expected to appear. These background ions are observed clearly in the spectrum of the SSD residual energy against the system time of flight (ToF) in Fig. 7. The ToF is the time between PPACa and the SSD, following Ref. Hayakawa et al. (2016). The figure shows all SSD events gated on incoming 30S ions, about $\approx 80\%$ of which are discarded by the requirements of the kinematic solution. The true elastic scattering events fall within a small locus on the histogram with a specific ToF, depicted by a narrow gate. Conversely, the beamlike $\alpha$ particles span the entire ToF range with temporal spacing exactly corresponding to the inverse of the cyclotron radiofrequency signal, because these ions do not deposit enough energy to trigger the PPAC and merely arrive at the SSD in chance coincidence with a 30S ion at the PPAC. Ordinarily, the relation of the PPAC signal and cyclotron radiofrequency corresponds to a relative flight time of a beam ion within the cocktail beam (see Fig. 4), but this relation does not hold for such a chance coincidence. Although most of such events are easily discarded, there is a small region of overlap where the beamlike $\alpha$ particles may contribute as much as 20% of the data in this region.

There is another particle group to the left of the $\alpha$ background in Fig. 7 which is quite unusual, as its residual energy is positively correlated with its system flight time. The nature of this locus is not certain but is most likely associated with light ejecta resulting from heavy ions striking the Wien filter electrodes in chance coincidence with a 30S beam ion. Considering the number of events in this locus per ns, we estimate it may contribute 5% of the data in Fig. 9(a) near 3.3 MeV.

As described below in Sec. III.4, introducing individual resonances in this region with widths equal to the theoretical limit made no discernible change to the calculated excitation function given our energy resolution. Changing the fill-gas to perform background measurements with an active target necessitates adjusting the GEM high voltage settings (which would still not guarantee identical operating conditions) as well as replication of all tuning and calibration measurements, which requires a significantly larger investment of time compared to changing the target gas in an ordinary target. Although we prepared to make such a measurement with an Ar+CO2 gas mixture (as in our previous measurements, e.g., Yamaguchi et al. (2013)), unfortunately we did not have enough time. Using the Wien filter to steer the beam, we determined that a vast majority of the beamlike $\alpha$ particles are confined to a narrow energy region.

Further analysis of Fig. 7 also sheds some light on the origin of the scaling factor of 2 applied to obtain the correct magnitude of the absolute value of the elastic scattering differential cross section. As stated above, about 20% of the events fall within a narrow locus and are consistent with bona fide elastic scattering events according to the kinematic solution algorithm. More than three times this many events ($\sim 60\%$ of the entire spectrum) fall within the depicted gate, but over 70% of the events within this gate have no corresponding high-gain GEM data ($>40\%$ of the entire spectrum), thus $\vartheta_{\rm lab}$ is unknown and Eq. LABEL:(eq:kinematics) cannot be solved. The solid angle of the right side of the downstream high-gain GEM is about 50% that of the SSD, and the Rutherford cross section is known to go as $(\sin^{4}\frac{\theta}{2})^{-1}$, indicating it does not change appreciably over a small change ($\approx 10^{\circ}$) in $\vartheta_{\rm lab}$. This implies that about $\frac{1}{2}$ of the events in the ToF-E locus of Fig. 7 should be in coincidence with the high-gain GEM, but only about $\frac{1}{4}$ actually have high-gain GEM data. These results are quite consistent with our scaling factor of 2.

As for possible contributions from inelastic scattering, the first excited state of 30S is relatively high at $E_{1{\rm x}}=2.21$ MeV and with a spin-parity of $2^{+}$. The increased scattering threshold as well as the requirement for $\ell\geq 2$ from the angular momentum selection rules indicates that the widths, which decrease with increasing $\ell$, suggest a significantly lower cross section than elastic scattering. For example, in other studies of $\alpha$ elastic scattering, this contribution was found to be less than 10% Yamaguchi et al. (2011, 2013), where the first excited states are much lower in energy. Moreover, as the resonances we analyzed were in the region of $4.0\leq E_{\rm c.m.}\leq 5.6$ MeV, contributions from inelastic scattering would show up near $1.8\leq E_{\rm c.m.}\leq 3.4$ MeV in the elastic spectrum, where resonances were neither resolved nor analyzed in our data. Therefore, it is reasonable to neglect any possible contribution from inelastic scattering in the present analysis; thus, we assume the total width can be expressed as $\Gamma=\Gamma_{\alpha_{0}}+\Gamma_{\rm p}$.

III.3 Experimental error

A number of different factors can influence the determination of the center-of-mass energy $E_{\rm c.m.}$ for a given event: the spread in the beam energy from the momentum selection as well as straggling, the SSD resolution for measurement of the $\alpha$ particle residual energy, the straggling of the $\alpha$ particle, and the position determinations of both the recoiling $\alpha$ and beam ion. However, since we use the geometric measurements to determine $\vartheta_{\rm lab}$ and the residual energy of the outgoing $\alpha$ particle to deduce $E_{\alpha}$, these have the most profound effect on the determination of $E_{\rm c.m.}$. Based on Eq. (3), the uncertainty in the center-of-mass energy $\Delta E_{\rm c.m.}$ can be expressed as

[TABLE]

where $\Delta E_{\alpha}$ is the uncertainty in the measured $\alpha$-particle energy, $\vartheta_{\rm lab}$ is the average measured angle, and the uncertainty in the measured angle is $\Delta\vartheta_{\rm lab}=|\vartheta_{\rm lab}-\vartheta_{\rm lab}^{\prime}|$. In the following illustrative calculations, $E_{\rm c.m.}$ was varied in 1 MeV increments over the range of 2–6 MeV.

Under the experimental conditions, the energy resolution of the SSD for 4.78-, 5.48-, and 5.795-MeV $\alpha$ particles from a standard source was 103, 98, and 87 keV, respectively. For higher energy $\alpha$ particles, we assumed the resolution of 1.5% as measured for the 5.795-MeV $\alpha$ particles, which should be an overestimate. In an offline test, the SSD resolution for the 5.48-MeV line was as good as 29 keV under vacuum which broadened to 70 keV when the chamber was filled with He+CO2 gas; by folding an assumed 64 keV of broadening from energy straggling with the intrinsic SSD resolution, we were able to reproduce the measured width. Considering the position of the $\alpha$ source was nearly 40 cm from the SSD in offline tests and $\alpha$ particles scattered at an initial laboratory energy of 5.5 MeV would be nearly twice as close to the SSD, 64 keV can be considered the maximum uncertainty for straggling, with higher energy $\alpha$ particles straggling much less as well as originating much closer to the detectors. We finally adopted values for $\Delta E_{\alpha}$ by adding the above SSD resolution and the assumed straggling in quadrature, except for the highest energy $\alpha$ particles where we simply adopted an uncertainty of 1.5% since summing the overestimated uncertainties from both resolution and straggling effects is unreasonable.

In order to estimate the uncertainty in $\vartheta_{\rm lab}$ arising from the experimental determination of the scattering position, we need to first estimate the average $\vartheta_{\rm lab}$ as a function of $E_{\rm c.m.}$. We plotted both the laboratory scattering angle $\vartheta_{\rm lab}$ and the center-of-mass angle $\vartheta_{\rm c.m.}$ event by event in order to determine their average values as functions of the center-of-mass energy; the average $\vartheta_{\rm c.m.}$ is shown in Fig. 8. While the precision of each PPAC to determine a beam particle’s position is 0.9 mm in both $X$ and $Y$, the position resolution becomes 4 mm in both dimensions when extrapolated to a typical scattering depth. The resolution achieved for the $\alpha$ particle’s position with the backgammon pads was 3 mm in $X$ and 0.5 mm in $Y$. All these uncertainties were added together in quadrature to yield a final uncertainty of 6.4 mm in the determination of $\vartheta_{\rm lab}$. A new angle $\vartheta_{\rm lab}^{\prime}$ was calculated by shifting the position of the $\alpha$ particle by the above 6.4 mm, assuming a standard scattering depth $Z$ representative of each of the five center-of-mass energies. The resulting range of $\Delta\vartheta_{\rm lab}$ was found to be 1.3–$2.1^{\circ}$, increasing with decreasing $E_{\rm c.m.}$.

Finally, we obtained an estimate for the uncertainty of the center-of-mass energy of about 60–100 keV as shown in Fig. 8; the intrinsic resolution of the SSD had the predominant effect, which was more pronounced at the higher energies. Thus, it can be seen that the energy binning choice of 100 keV is consistent with our achieved resolution.

We confirmed with a simple calculation that the above geometric uncertainties dominate over the uncertainty in the beam energy. Suppose we have two identical measurements, but we know that the incident energy differs between the two beam ions. The result of the kinematic solution is that the optimized scattering depth will be larger for the higher energy beam ion and vice versa for the low energy beam ion, because it is the scattering depth combined with the incident beam energy together that finally determines $E_{\rm c.m.}$. Assuming a nominal scattering energy of $E_{\rm c.m.}=4.0$ MeV, changing the transverse scattering position by the 6.4 mm uncertainty mentioned above is equivalent to $\Delta\vartheta_{\rm lab}=1.7^{\circ}$, changing the scattering depth $\Delta Z$ by 27 mm, or changing the beam energy by 5.6 MeV. Thus, the uncertainties of these measurements dominate over the intrinsic spread in the beam energy of 2.0 MeV.

III.4 $R$-Matrix analysis

To extract the resonance parameters of interest, we performed a multilevel, multichannel $R$-Matrix calculation Lane and Thomas (1958) with the sammy8 code Larson (2000). Succinctly, the $R$-Matrix method calculates the interference between the regular and irregular Coulomb functions with physical resonances. The resonances are parameterized by their energy $E_{\rm r}$ (the same as $E_{\rm c.m.}$ from elastic scattering as $Q=0$), channel $i$ partial widths $\Gamma_{i}$, and the angular momenta transfer $\ell_{i}$. The resonance shape is determined by the entrance channel $\ell_{\alpha}$, the resonance height from the entrance channel $\Gamma_{\alpha}$, and the resonance width depends on total width $\Gamma$. The total width is a sum of the proton and $\alpha$ partial widths, as both channels are open; the gamma partial widths $\Gamma_{\gamma}$ are negligibly small for these highly excited, particle-unbound states. For the case of 30S+$\alpha$ elastic scattering, the situation is simplified for the entrance channel, as both the nuclei have a ground-state spin-parity $J^{\pi}=0^{+}$, and so the quantum selection rules dictate a unique resonance $J^{\pi}$ for each $\ell_{\alpha}$ value—namely that $J=\ell_{\alpha}$ and the parity is always natural for populated states in 34Ar.

The calculated excitation function was broadened based on the experimental energy resolution and performed at an average angle of $\vartheta_{\rm c.m.}=150^{\circ}$ as evaluated above in Sec. III.3. We quantified the quality of a fit by the reduced chi-square $\chi^{2}_{\nu}$, which is the chi-square $\chi^{2}$ divided by the number of degrees of freedom $\nu$. Fitting the data with pure Coulomb scattering resulted in $\chi^{2}_{\nu}=4.17$ with 35 degrees of freedom, indicating the possibility for significant improvement could be expected by including the interference effect of resonances in an $R$-Matrix fit. As there are no known levels in 34Ar with $E_{\rm ex}>11$ MeV, we had to carefully introduce new resonances until the experimental data were reasonably reproduced. The maximum width of a resonance can be estimated with the Wigner limit Rolfs and Rodney (1988) as

[TABLE]

where $\mu$ is the channel reduced mass, $R$ is the channel radius, and $P_{\ell}$ is the channel penetrability, respectively, for channel $i$. We calculate the penetrability as $P_{\ell}=\frac{\rho}{F_{\ell}^{2}+G_{\ell}^{2}}$, where $\rho=\frac{kR}{\hbar}$ includes the phase space factor $k$, and $F_{\ell}$ and $G_{\ell}$ are the regular and irregular Coulomb functions, respectively. Such a physical constraint is particularly relevant when introducing new resonances to help limit the parameter space. We adopted the channel radius given by $R_{i}=1.45(A^{1/3}_{1}+A^{1/3}_{2})$ fm, where $A_{1}$ and $A_{2}$ are the mass numbers of the two species in channel $i$; an identical parametrization was used in the studies of 21Na+$\alpha$ Binh (2010) and 26Si+p Jung et al. (2012), which are two of the most similar experiments to the present work. For consistency, the same $\alpha$-channel radius was also used in the $R$-Matrix calculation.

At the outset, we began with a single channel ($\Gamma=\Gamma_{\alpha}$), single level manual analysis starting with the lowest-energy features and slowly moving to higher excitation energies in discrete steps of 100 keV. The width was set to the Wigner limit ($\Gamma_{\alpha}=W_{\Gamma_{\alpha}}$) to determine which features could be resolved by assuming the existence of a physical resonance. At this time we also checked possible values of the angular momentum transfer $\ell_{\alpha}$; although the experimental setup allowed for values of up to $\ell_{\alpha}=6$, $\ell_{\alpha}\geq 5$ never gave good fits, since resonances with these higher transfers are essentially not visible within the present resolution. Only under this condition where $\Gamma_{\rm p}=0$ and $\Gamma_{\alpha}$ was at the Wigner limit was it possible to observe a change of any kind in the calculation for $E_{\rm c.m.}\leq 3.8$ MeV, and even so the calculated deviation from pure Coulomb scattering was of a smaller magnitude than the experimental uncertainty, particularly near 3.5 MeV. The calculations were consistent with our interpretation that the fluctuations below 3.8 MeV are statistical or background induced. The subsequent multilevel, multichannel analysis thus focuses on the region of 3.9–5.6 MeV and assumes $\ell_{\alpha}\leq 4$; three resonancelike structures could be resolved near $E_{\rm c.m.}\approx 4.35$, 4.78, and 5.34 MeV. Although resonances observed by transfer reactions always appear as peaks in the differential cross section, in the case of elastic scattering the interference pattern caused by a resonance can be observed as a diplike structure rather than as a peak, particularly below the Coulomb barrier.

For the proton channel, we assumed the lowest $\ell_{\rm p}$ allowed would have the predominant contribution. The spin of the proton $s_{1}=\frac{1}{2}$ and the spin of the 33Cl ground state $s_{2}=\frac{3}{2}$, which can align ($\uparrow\uparrow$) or anti-align ($\uparrow\downarrow$) to give the total spin $s=s_{1}\oplus s_{2}$, and the same is true for the resulting spin $s$ coupling with $\ell_{\rm p}$ to sum $J=\ell_{\rm p}\oplus s$. An example of the lowest-$\ell_{\rm p}$ coupling schemes are shown in Table 1 for up to $4^{+}$ natural-parity states in 34Ar.

For convenience, we introduced the dimensionless reduced partial width $\theta_{i}^{2}=\Gamma_{i}/W_{\Gamma_{i}}$, in order to easily ensure that, regardless of $\ell$, $\theta_{i}^{2}\leq 1$. Resonant elastic scattering is often analyzed by a single-channel formalism because the resonance shape and height are not affected by the other channels; thus at the outset we simplified our model by controlling the proton width via a universal ratio of the dimensionless reduced partial widths $\xi\equiv\theta_{\rm p}^{2}/\theta_{\alpha}^{2}$, which was found to be 3% in a similar work Binh (2010). Although the value of $\Gamma_{\rm p}$ derived this way may have a large uncertainty as well as model dependence, it is physically unrealistic to perform a single-channel analysis so far above the proton threshold.

Starting with the first resonance near 4.35 MeV and truncating the excitation function towards higher energies, a computer code optimized $E_{\rm r}$, $\ell$, $\theta_{\alpha}^{2}$, and $\xi$, until all three resonances were introduced and the fit took into account the entire energy range of the experimental excitation function. Once we had such a reasonable fit ($\xi\approx 4$%), we then allowed $\theta_{\rm p}^{2}$ to vary individually for each resonance and again covaried sets of resonance parameters to search for the best fit for the entire spectrum. In summary, $E_{\rm r}$ was covaried over 200 keV in 1 keV steps, $\ell$ was covaried for values over the range of 0–4, $\theta_{\alpha}^{2}$ was covaried in 1% steps up to $>99$%, and $\theta_{\rm p}^{2}$ was varied in small increments up to 10% (past where $\xi$ showed poor behavior) in our search for the best fit, shown in Fig. 9(a), where the horizontal errors are from the 100-keV binning and the vertical errors are purely statistical; the absolute magnitude of the statistical error was scaled by the same factor of 2.0 as the data. All possible $\ell$ values for the 4.78-MeV state are shown in Fig. 9(b) for illustrative purposes.

A number of systematic uncertainties that might affect the differential cross section were carefully considered. These include contributions to each of the parameters in Eq. LABEL:(eq:dsigma-domega), namely the number density of helium atoms $n$, the stopping power of the beam $S(E_{\rm b})$, the number of injected beam ions $I_{\rm b}$, and the changing solid angle $\Delta\Omega$. The number density of 4He atoms in the target employed cannot be changed by any physical argument, since the pressure gauge in the gas flow controller described in Sec. II was consistent with two other pressure gauges. The gas density calculated from the nominal laboratory conditions with the ideal gas law is consistent with the density utilized in all energy loss calculations. Any error in the density of helium would apply equally to CO2 which induced significant energy loss for the heavy ions. Such energy loss calculations are generally known to be accurate to the order of 10% or better, and moreover we experimentally verified the stopping position and Bragg curve of 30S in the target gas. Although we initially assumed the energy distribution of the 30S ions should be gaussian, we found the centroid was skewed to the low energy side by 1%; calculating the 30S ion incident energy event-by-event based on the $rf$ data rather than assuming a gaussian distribution did not result in any noticeable difference in the elastic scattering excitation function. The number of incident 30S ions is determined in part by the PPAC scalers, the magnitude of which can be confirmed as we recorded a downscaled spectrum of the cocktail beam; although the latter is less accurate, the two methods agreed within 6%. We checked the method of calculating the solid angle, as well as the absolute efficiency of the silicon detectors, considering the known intensity of the standard $\alpha$ source used in off-line calibration runs; within the errors of these calculations, we found the efficiency $\eta>99\%$ for the relevant silicon detectors before and after the experimental run, indicating the detectors were not damaged during the experiment. As the scaling factor of $2.0\pm 0.1$ is the main source of systematic error, we take its associated uncertainty of 5% as the systematic uncertainty in the present work. Because the statistical error is $\geq 25$% over the resonant-dominated region, the systematic error makes a negligible contribution to the final error evaluation.

The resonance parameters deduced from the $R$-Matrix analysis are shown in Table 2. The uncertainties in the adopted level parameters were calculated in the following ways. For the excitation energy $E_{\rm ex}$, we used the experimental energy resolution as discussed in Sec. III.3 and shown in Fig. 8. The error of the remaining level parameters was evaluated considering the range where an individual parameter is allowed to vary within one standard deviation of the best fit $\chi^{2}_{\nu}$; the same method was used over the range of $E_{\rm c.m.}$ 1.9–4.1 MeV with pure Rutherford scattering to determine the scaling factor and its error of $2.0\pm 0.1$. The recommended spin parity $J^{\pi}$ is given, and any other spin-parity which is possible is listed, as are the associated widths in their respective columns separated by commas. The error in $\Gamma_{\rm p}$ is seen to be generally larger than in $\Gamma_{\alpha}$, because the $\alpha$ elastic scattering resonant structure is much less sensitive to the proton channel compared to the $\alpha$ channel.

The resonance parameters obtained in the present study appear to be reasonable except for the widths for the 12.08-MeV state. In particular, the 12.08-MeV state’s structure cannot be a pure $\alpha$ cluster which also has a non-negligible proton decay branch. Our favored interpretation is that there are one or more unresolved resonances with substantial $\alpha$-cluster configuration in this region. Moreover, the behavior of the resonance tail is unconstrained by the data, and any interference effects from unknown physical resonances outside the energy range cannot be accounted for. Thus, there are large systematic uncertainties associated with the resonance parameters extracted from an $R$-Matrix fit for states near the boundary of the experimental energy range. However, it cannot be doubted that the data indicate one or more very strong $\alpha$-cluster resonance(s) in this region of excitation energy, which is a point we emphasize in our discussion of these results below.

IV Discussion

We observed the signature interference patterns of several resonances in 34Ar with large $\alpha$ partial widths $\Gamma_{\alpha}$ via $\alpha$ elastic scattering on 30S. The cluster threshold rule predicts the existence of these states, which have a large overlap of the cluster configuration to the nuclear wavefunction nearby the respective cluster’s separation energy Kubono (1994); Kubono et al. (2010). Such $\alpha$-cluster resonances have typically dominated the stellar rate of exothermic ($\alpha$, n) and ($\alpha$, p) reactions on $T_{z}=\pm 1$ nuclei, respectively, when they fall within the astrophysical Gamow burning window Aprahamian et al. (2005). $\alpha$ resonant elastic scattering has long been known as a powerful tool to selectively observe states with large $\Gamma_{\alpha}$. The effect is especially pronounced in inverse kinematics, where measurements at large backward angles are possible and the nonresonant cross sections are minimized; under these conditions, one expects to observe states with $\Gamma_{\alpha}$ comparable to the experimental energy resolution Artemov et al. (1990). According to calculations of the Wigner limit [see Eq. (6)], the maximum theoretical width shrinks rapidly as the energy is reduced towards the threshold.

Our observation that all three resonances are consistent with a $2^{+}$ assignment may make one wonder if there is a reason for such behavior. These may be regarded as a triplet if the tentative assignments of $2^{+}$ are correct for both the 11.09- and 12.08-MeV states. Alternatively, our 11.51-MeV state could be regarded as a $2^{+}$ doublet paired with either these other two states. We note that all three are observed as diplike structures, so it may not be surprising that features in the differential cross section with similar interference patterns can result from physical resonances with the same $J^{\pi}$. A system of $\alpha$-cluster doublets was observed in the $T_{z}=+1$ nucleus 22Ne Rogachev et al. (2001); Dufour and Descouvemont (2004) with $J^{\pi}$ correlated with increasing energy, albeit for states of negative rather than positive parity. Unfortunately, comparison with model predictions is still a challenge for the 30S mass region.

IV.1 Reaction rate

The peak temperature of x-ray bursts is expected to be in the range of 1.3–2 GK corresponding to the Gamow burning windows of $1.7\lesssim E_{\rm c.m.}\lesssim 3.8$ MeV. To make a meaningful evaluation of the stellar reaction rate in XRBs, we therefore need to consider not only the resonances discovered in the present work, but also 34Ar states at lower $E_{\rm ex}$. In fact, before the present work there has never been an evaluation of the 30S($\alpha$, p) cross section based on experimental level structure of 34Ar owing to the paucity of such data and the experimental challenges of studies in this region of the periodic table.

The 36Ar(p, t)34Ar measurement performed at the Research Center for Nuclear Physics (RCNP), Osaka University observed resonances above the $\alpha$ threshold at a relatively smooth interval—about four resonances per MeV—over four MeV in excitation energy O’Brien et al. (2009). Considering the significant difference in energy resolution, our observation of three (or four, depending on the interpretation of our 12.08 MeV state) resonances per MeV over the range of our experimental energy and as the resolution allows, there is a basic consistency between the two studies, which both only populate natural-parity states.222While a single step (p, t) transfer reaction can only populate natural-parity states, a multistep process allows for the population of unnatural-parity states (see, e.g., Ref. Setoodehnia et al. (2013)). However, the cross section to populate unnatural-parity states by a multistep process is significantly smaller than the cross section for a single step process as a general rule.

However, the results from the RCNP spectroscopic study only provides us with preliminary resonance energies, and some assumptions are required before we may apply them. Firstly, we naïvely assumed that each state has $J^{\pi}=0^{+}$. As for the partial widths, based on the present results and the similar level density between the two studies, we set $\Gamma_{\alpha}=\frac{1}{2}W_{\Gamma_{\alpha}}$. Although our $\theta_{\alpha}^{2}$ are generally larger than 0.5 according to Table 2, setting $\theta_{\alpha}^{2}\approx 1$ for such a large series of resonances would be unusual considering $\Gamma_{\rm p}\neq 0$ and thus $\Gamma>\Gamma_{\alpha}$ and hence $\theta_{\alpha}^{2}<1$. We believe a factor of 0.5 is still rather extreme but more reasonable.

It should be noted a thorough analysis of the RCNP experimental data would only improve the situation with regard to the precision of the excitation energies (or the removal of any states which are background induced) and not the spin-parities nor the partial widths. The limited angular distribution available from their spectrometer does not cover a full phase for reliable comparison with a DWBA calculation Matic et al. (2009, 2011).

We calculated the resonant reaction rate per particle pair $\langle\sigma v\rangle$ using the standard formulation Rolfs and Rodney (1988) which depends only on the resonance energies $E_{\rm r}$, spins $J$, and the channel partial widths $\Gamma_{i}$. The spin comes into play in the spin statistical factor $\omega$ as:

[TABLE]

where $J_{\rm A,a}$, the spins of the two nuclei in the entrance channel, are both zero in the case of 30S+$\alpha$. The reduced width $\gamma$ is defined as:

[TABLE]

for the entrance and exit channel partial widths $\Gamma_{\rm a,b}$, respectively. Their product $\omega\gamma$ is called the resonance strength as it is proportional to the integral of the resonance cross section. We also use the standard simplification that $\gamma=(\Gamma_{\alpha}\Gamma_{p})/(\Gamma_{\alpha}+\Gamma_{p})\approx\Gamma_{\alpha}$ when $\Gamma_{\alpha}\ll\Gamma_{p}$, which is a realistic assumption considering the vastly different Wigner limits of the two channels.333According to our data from Table 2, the best fit $\Gamma_{\rm p}$ and $\Gamma_{\alpha}$ are of comparable size. However, we note that the upper-limit errors for $\Gamma_{\rm p}$ are extremely large, which is not inconsistent with the assumption that $\gamma\approx\Gamma_{\alpha}$. Even in the case that $\Gamma_{\alpha}=\Gamma_{\rm p}$, $\gamma=\frac{1}{2}\Gamma_{\alpha}$, giving a similar factor of 2 error to $\omega\gamma$ as the assumption of $J=0$ as shown presently. The resulting $\omega\gamma$ for each resonance in the 36Ar(p, t) study based on our assumptions that $J=0$ and $\gamma=\Gamma_{\alpha}=\frac{1}{2}W_{\Gamma_{\alpha}}$ varies by only a factor of around two for $J\leq 3$ (although it quickly drops off for $J\geq 5$) as shown in Fig 10, vindicating our arbitrary treatment of the spin in our framework; $\omega\gamma$ is independent of low $J$ to first order because we parametrize the width based on the Wigner limit, which decreases with increasing $J$, whereas for the 30S+$\alpha$ entrance channel $\omega=2J_{\rm r}+1$. While the properties of an individual resonance calculated in this manner will be unreliable, the sum of these contributions can be considered an upper limit under extreme assumptions on the nuclear structure of 34Ar.

Our goal here is to provide an evaluation of the 30S($\alpha$, p) reaction rate in x-ray bursts by assuming broad widths for all the known states in 34Ar near the astrophysically interesting region. According to Fig. 10, the choice of $J^{\pi}=0^{+}$ yields a median value of the resonance strength for low $J$ in our model and thus represents the physically realistic case where there are a multitude of different spins among the levels in 34Ar over the alpha threshold. It can be seen in the case of $J=3$ or higher that the reaction rate could be half or less than our suggested upper limit. To demonstrate, we considered the case when $\gamma=\Gamma_{\alpha}=2W_{\Gamma_{\alpha}}$ and $J^{\pi}=4^{+}$; between $T=1and3$ GK the reaction rate is a factor of 2 lower than our evaluation in spite of the fact that the reduced width was twice rather than half the Wigner limit. This feature of the Wigner limit is satisfyingly consistent with the fact that resonant thermonuclear reaction rates tend to be dominated by lower partial wave contributions because the angular momentum barrier is smaller. Importantly, the rate only exceeds our evaluation by 0.2% at $T=1.5$ GK if we allow the maximum $\Gamma_{\alpha}$ from our experimental uncertainties in Table II, which we consider to be a trivial difference.

The 34Ar resonance parameters adopted for our 30S($\alpha$, p) stellar reaction rate calculation are listed in Table 3 along with calculations of the proton and $\alpha$ Wigner limits. We used the graph digitizing system gsys from the Hokkaido University Nuclear Reaction Data Centre, which was developed specifically for extracting numerical nuclear data from published spectra, to obtain the resonance energies from the 36Ar(p, t) study O’Brien et al. (2009). For the known resonances, we found deviations from the compiled level scheme Endt (1990) of at most around 30 keV, which is the same as the experimental error quoted by O’Brien et al. The resulting stellar reaction rate is shown in Fig. 11 in comparison to a statistical model (SM) rate Rauscher (2008). As this is the first 30S($\alpha$, p) reaction rate based on the experimental level structure of 34Ar, there are not many other studies to compare ours against. The two most relevant studies are the 33Cl(p, $\alpha$) measurement Deibel et al. (2011) and a recent survey of $\alpha$-induced cross sections for masses $A\approx 20$–50 Mohr (2015).

In order to compare the present results with the time-reversal study by Deibel et al. Deibel et al. (2011), one should keep in mind that the 30S($\alpha$, p) $Q$ value is 2.080 MeV, and thus their energy range in 30S+$\alpha$ is $4.09\leq E_{\rm c.m.}\leq 5.35$ MeV (as shown in their Table I), quite similar to the range of resonances observed in the present work. The previous study includes only the ($\alpha$, p0) ground-state component, whereas the present $\Gamma_{\rm p}$ does not exclude the summed contribution from all states where the assumption $\Gamma_{\alpha}\ll\Gamma_{\rm p_{i}}$ remains valid for at least one pi; the SM rate implicitly includes transitions to allowed states. The present work shows an enhancement of around a factor of 5 over the SM rate as an upper limit, and the work of Deibel et al. shows a 30S($\alpha_{0}$, p0) cross section which is more comparable to the SM rate and is considered as a lower limit to the total ($\alpha$, p) cross section. Even if one only includes the resonances we observed at CRIB, it can be seen that near 3 GK our three resonances alone are quite similar to the SM rate.

The reduction scheme presented by Mohr generally shows a global behavior of the cross sections for ($\alpha$, p) and ($\alpha$, n) reactions over a large energy range for medium mass nuclei Mohr (2015). Specifically, most of the experimental data can be reproduced by a SM calculation. However, deviations higher than the expected cross sections were found in some of the measurements with 23Na and 33S, whereas the species 36Ar and 40Ar were seen to be much lower (at least for the available data). For the case of 23Na, the work of Mohr motivated the community to re-investigate the 23Na($\alpha$, p) cross-section, which was finally found to be consistent with the SM calculation within the experimental uncertainties Almaraz-Calderon et al. (2014, 2015); Tomlinson et al. (2015); Howard et al. (2015). In the outstanding cases, these discrepancies certainly warrant further investigation to determine if they are real or artificial (see also the discussion in the recent work by Anderson et al. Anderson et al. (2017)). If the effects are real, 30S is seen to fall within the mass range where there is a cross section enhancement over the SM, which supports the findings of the present study.

IV.2 Astrophysical implications

The impact of our new upper limit for the 30S($\alpha$, p)33Cl rate was examined within the framework of one-zone XRB postprocessing calculations using the K04 ($T_{\rm peak}=1.4$ GK) model Parikh et al. (2008). As shown in Fig. 12, striking differences in the profiles of nuclear energy generation rates between approximately 1 and 10 s are seen when comparing XRB calculations using the present upper limit and a statistical model rate calculation Rauscher (2008) (see Fig. 11). Indeed, $E_{\rm gen}$ differs by as much as 25% between calculations using these two rates. Nucleosynthesis predictions are also affected by the particular 30S($\alpha$, p) rate adopted in the calculations. Comparing results using the present upper limit and the statistical model rate, abundance differences of up to 30% are observed for species with mass fractions $>10^{-6}$ (summed over mass number), for $A$ over the rather broad range of approximately 20–$80$. Further tests using full hydrodynamic XRB models are needed to explore in detail the possible dramatic impact of the 30S($\alpha$, p)33Cl rate on predictions of XRB observables.

V Summary

We observed several resonances with large $\alpha$ widths in the energy range $E_{\rm ex}=11.1$–12.1 MeV for the first time in 34Ar via the $\alpha$ resonant elastic scattering of 30S and determined their properties of spin, parity and widths. Using our new data, we were able to make the first-ever calculation of the astrophysical 30S($\alpha$, p) cross section based on the experimental level structure of 34Ar. Although these resonances do not seem to have a large effect for the astrophysically interesting energies important for XRBs, we could set a reasonable upper limit on the stellar reaction rate of about one order of magnitude greater than the Hauser-Feshbach statistical model. The resonances we observed correspond very well to the energy range covered in the time-reversal study. These two studies complement each other nicely, as our work provides an upper limit to the cross section, which we determined to be somewhat above the existing 33Cl(p0, $\alpha_{0}$) lower limit to the total 30S($\alpha$, p) cross section. Although the present knowledge of the level structure of 34Ar, as well as the 30S($\alpha$, p) cross section, at the most astrophysically interesting temperatures remains elusive, our new upper limit can, for the first time, conclusively rule out the artificial cross section enhancement of a factor of a hundred over the SM used in one XRB model Fisker et al. (2004). This can, in turn, rule out the influence of the 30S($\alpha$, p) reaction in explaining such double-peaked burst morphology, consistent with the theoretical findings of a recent study Cyburt et al. (2016).

From a technical perspective, we developed the highest quality 30S radioactive ion beam for astrophysical studies yet in the world. Our analysis also showed that active target systems must be designed with extremely high precision capabilities and that reports of $\alpha$ scattering with such systems must be viewed with scrutiny. However, the active target system enabled us to understand the energy loss properties of the beam very clearly, which is often a challenge for experiments performed using a thick target in inverse kinematics.

Further work is required to elucidate the behavior of the 30S($\alpha$, p) stellar reaction rate over the energy ranges applicable to XRBs so that its predicted influence on the energy generation, compositional inertia, and burst light curve can be reliably extracted from theoretical models. Of course, a direct measurement of the 30S($\alpha$, p) reaction at the relevant energies would be the ideal approach, but it is unclear when a sufficiently intense, low energy 30S RIB will become available. An intensity like $10^{4}$ pps is insufficient, and it took us four years to develop such an RIB for the present study. Instead, the community should continue to exploit indirect methods as in the present study in the near future. In particular, it is critical to obtain more experimental knowledge of the quantum properties, particularly $\Gamma_{\alpha}$, of states in 34Ar over $E_{\rm ex}$ from 8.0 to 11.5 MeV.

Acknowledgements

This experiment was performed at RI Beam Factory operated by RIKEN Nishina Center and CNS, University of Tokyo. We appreciate the professional operation of the AVF cyclotron and the ion source by the RIKEN and CNS staff which made this work possible. This work was partly supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), National Natural Science Foundation of China (Grants No. 11135005 and No. 11021504), the Major State Basic Research Development Program of China (2013CB834406), JSPS KAKENHI (No. 21340053 and No. 16K05369) and the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, and the UK Science and Technology Facilities Council (STFC). A.A.C. was supported in part by an Ontario Premier’s Research Excellence Award (PREA) and by the DFG cluster of excellence “Origin and Structure of the Universe” (www.universe-cluster.de).