Investigation of 22 Mg levels via resonant scattering of 18 Ne + α

The α resonant scattering on 18 Ne was measured in inverse kinematics to understand the α -clustering of proton-rich 22 Mg nucleus, performed at the CNS Radio-Isotope Beam Separator (CRIB) of Center for Nuclear Study, University of Tokyo, located at the RIBF of RIKEN Nishina Center. The excitation function of 22 Mg was obtained for the excitation energies of 10 – 16 MeV by adopting the thick-target method. Several resonances were evident in the present work, which implies the existence of energy levels with large α widths. Since energy levels were not clearly observed at the astrophysically important energy range, upper limits on the 18 Ne ( α , α ) 18 Ne cross section were set. The astrophysical impact was also investigated by estimating the 18 Ne ( α , p ) 21 Na cross section.


Introduction
The α-cluster structure in atomic nuclei has been one of the most interesting topics in nuclear physics.The α-clusterization of self-conjugate (N = Z) and A = 4n (n = 2, 3, 4, /) nuclei including 8 Be, 12 C, 16 O, and 20 Ne has been studied extensively for decades [1][2][3].The strong evidence for the α-cluster structure was found through the studies.Observations of a series of levels with large α reduced widths that form a rotational band can provide a convincing probe for α-cluster configuration in the nuclei.The development of theoretical models and rare isotope beams has provided significant opportunities to extend our knowledge of the α-cluster structure in exotic nuclei.Experimental investigations on the α-clusterization of neutron-rich nuclei, including 10 Be, 12 Be, and 14 C, have been performed [4][5][6].These studies have substantially improved our understanding of the αcluster structure in neutron-rich nuclei; however, α-cluster studies of proton-rich (neutron-deficient) nuclei have not been sufficiently established.Considering the isobaric invariance of the nuclear force, the characteristics of the α-cluster structure for a proton-rich nucleus are similar to those for its mirror nucleus.For example, experimental results indicate that some of the observed resonances in 11 C may originate from the negative-parity cluster band, which is analogous to its mirror nucleus 11 B [7,8].
The α-cluster structure of neutron-rich 22 Ne has been extensively investigated both experimentally and theoretically [9][10][11][12][13].Rogachev et al. observed a splitting of 1 − , 3 − , 7 − , and 9 − α-cluster states into doublets [10], which could be theoretically explained by the extended two cluster model (ETCM) calculation assuming the α + 18 O two-cluster configuration [11].Recently, Kimura suggested the presence of two kinds of α-cluster structures in 22 Ne using the hybrid-generator coordinate method (GCM) calculation.The first is the molecular orbital bands with the α + 16 O core and two valence neutrons, which correspond to the observed α-cluster states below the α + 18 O threshold energy reported in [14,15].The other is the α + 18 O molecular bands, which correspond to the observed states above the threshold energy reported in Ref. [10].
Studies on the α-cluster structure of proton-rich 22 Mg are still very rare.The GCM calculation predicted the existence of the 1 − and 3 − doublet states located at excitation energies of 12-13 MeV, assuming the α + 18 Ne two-cluster model [11].Considering the lower energy level density of proton-rich nuclei than that of neutron-rich nuclei, observing doublets should be easier in the case of the 22 Mg nucleus.However, the experimental data obtained by Goldberg et al. [12] show no clear evidence of the doublets.Although the excitation function of mirror nucleus 22 Ne in [12] show the existence of the strong 1 − and 3 − doublets originating from the α-clustering, the excitation function of 22 Mg is rather featureless.The authors could not specify the reason for the absence of the doublets since the quality of the 22 Mg data was not sufficient for independent analysis.
A powerful approach for investigating the α-cluster structure in the 22 Mg nucleus is to populate α-cluster states by resonant elastic scattering of 18 Ne and α.Therefore, we measured 18 Ne + α resonant scattering using 18 Ne rare isotope beam to identify predicted 1 − and 3 − doublets.The excited states of 22 Mg have been extensively investigated by various nuclear reactions, including 12   -24].The α partial widths are, however, not known for most levels, which cannot provide a clear evidence of the α-clusterization in the 22 Mg nucleus.In the present study, the excitation function of 22 Mg was obtained for excitation energies of E x ~10-16 MeV, which can provide important information about the energy level properties including α partial widths.
Investigating the spectroscopic information of 22 Mg also plays a crucial role in understanding the astrophysically important 18 Ne (α, p) 21 Na reaction because the reaction occurs through the resonances in the compound nucleus.The 18 Ne (α, p) 21 Na reaction is known as one of the possible breakout routes from the hot-CNO cycles, which leads the rapid proton capture (rp) process [25][26][27].In a recent sensitivity study by Cyburt et al., the 18 Ne (α, p) 21 Na reaction was identified as one of the most important reactions which impact on the X-ray burst light curve and the composition of burst ashes [28].Considering the typical X-ray burst temperature of T ~2 GK, the Gamow window corresponds to the excitation energy E x 9.56-10.96MeV.Therefore, it is important to study the energy level properties of 22 Mg in this energy region.The 18 Ne (α, p) 21 Na reaction has been studied by direct measurements [19-21] and timereversal reaction measurement [29].The experimental results, however, show a large discrepancy.In the present work, the 18 Ne (α, p) 21 Na reaction cross section was estimated based on the experimental level structure of the 22 Mg nucleus.

Experiment
The α resonant elastic scattering of 18 Ne was measured in inverse kinematics at the CNS Radio-Isotope Beam Separator (CRIB) [30,31] of the Center for Nuclear Study, University of Tokyo, located at the RIBF of RIKEN Nishina Center.A schematic of the experimental setup is shown in Figure 1.The 18 Ne rare isotope beam was produced by using the in-flight (IF) method.A primary 16 O beam with an Schematic view of the CRIB separator (top) and the experimental setup for the 18 Ne + α resonant scattering measurement at the F3 focal plane (bottom).The 18 Ne beam particles were identified and monitored by two F3 PPACs. 4 He gas at 470 Torr was filled to the chamber, which was sealed with a 23-μm-thick Mylar foil as a beam entrance window.The energy and position of the recoiling α particle were measured by ΔE-E silicon detector telescopes (Tel 1 and Tel 2).energy of 8.026 MeV/u from the AVF cyclotron [32] was delivered to the F0 focal plane of CRIB and bombarded a 3 He gas target.Then, a secondary 18 Ne beam was produced by the 3 He ( 16 O, 18 Ne)n reaction.The 3 He gas was contained in the cell at a pressure of 360 Torr.The 3 He gas atoms were isolated from the beam line kept at a high-level vacuum by using 2.5-μm-thick Havar foils as the entrance and exit windows.To increase the density of the 3 He gas target and the intensity of the secondary 18 Ne beam, a cryogenic system using liquid nitrogen was used [33].The areal thickness of the 3 He gas target was achieved to be 1.54 mg/cm 2 by keeping the temperature at T ~90 K.
18 Ne 10+ ions were selected by a double achromatic system with a proper magnetic rigidity (Bρ) value of 0.5920 Tm, which is optimized to obtain the maximum 18 Ne beam production rate.A slit of ±15 mm was installed at the momentum dispersive focal plane (F1) to remove the beam contaminations produced due to various nuclear reactions, yielding the momentum dispersion Δp/p ~1%.The secondary beam particles were further purified using the Wien Filter (WF) system by applying a high voltage of ±59.5 kV.Two delay-line-type PPACs [34] were installed downstream of the WF to measure the time and the two-dimensional position information for each secondary beam particle.By using PPACs, the secondary beam identification after the WF was performed.The 18 Ne beam intensity and purity were ~2.6 × 10 5 pps and ~65%, respectively.The impurities were 17 F 9+ (~28%) and 16 O 8+ (~4%), respectively.A small amount of other beam species including 15 O 8+ , 13 N 7+ , and 4 He 2+ was also observed.These contaminants were clearly excluded by the time-of-flight information in the final analysis.
The F3 target chamber was filled with 4 He gas for α scattering measurement.The 4 He gas was at a pressure of 470 Torr and was sealed with a 23-μm-thick aluminized Mylar foil as a beam entrance window.The 4 He target pressure was selected to stop the 18 Ne beam before it reaches the detector.The 18 Ne beam energy after passing through the target entrance window was measured as 45.2 ± 1.1 MeV, which is consistent with an energy loss calculation result considering the Bρ value of 0.5920 Tm and the effective thickness of ~45 μm Mylar foil for the two PPACs and the beam entrance window.By adopting the thick-target method in inverse kinematics [35], a wide range of excitation energies of 22 Mg was investigated with a single 18 Ne beam energy.
The energy and position of the recoiling α particles were obtained using two sets of ΔE-E silicon detector telescopes.The central telescope (Tel 1) was installed at 430 mm downstream from the entrance window of the target chamber along the beam axis.The other telescope (Tel 2) was located at 10.15 °off from the beam axis, as viewed from the center of the entrance window.Tel 1 (Tel 2) consisted of 20-μm-, 496-μm-, and 485-μm-thick (20-μm-and 1500-μm-thick) silicon detectors.The most energetic α particles from the 18 Ne (α, α) 18 Ne reaction could be entirely stopped under these conditions.Each detector had 16 strips and an active area of 50 × 50 mm 2 .Energy calibration of each strip was carried out by using an α-emitting source composed of 148 Gd (3.148 MeV), 241 Am (5.462 MeV), and 244 Cm (5.771 MeV).Since the energy range of recoiling α particles is much wider (0-27 MeV) than that of the α particles from the source, additional energy calibration was required for the high-energy region.α beams at various energies (13, 15, 20, and 25 MeV) were used for this purpose.
Even after the beam purification using the CRIB spectrometer, beam-like α particles were transported to the F3 reaction target chamber.These beam-like α particles were produced at the upstream of the beam line and selected by the Bρ value which was set for the 18 Ne beam particles of interest.The number of the beam-like α particles was much less than the number of 18 Ne beam particles (≪ 0.01%); however, an amount comparable to that of the reaction products reached the central telescope.Therefore, the argon target was used for the background measurement.The argon gas pressure of about 87 Torr was selected so that the incident particles exhibit energy losses similar to those of 4 He gas.By comparing the two α spectra obtained with and without the 4 He gas target, the contribution from beam-like α particles was identified.
3 Data analysis

Kinematics reconstruction
The particle identification was performed by the standard energy loss techniques.A typical particle identification plot obtained at the central telescope is shown in Figure 2. The total energy deposition of the particles is plotted as a function of energy deposition in the ΔE detector.As shown in the figure, α particles were clearly separated without significant contamination from other charged particle groups.The α particles with E tot ~13 MeV were observed in the background run with argon gas, as shown in Figure 2B, indicating that those α particles were contaminants in the secondary beams.The beam-like α particles were clearly distinguished by using the time-of-flight information between the PPAC and the second layer of the telescope, as shown in Figure 3.
The α particles in coincidence with the 18 Ne beams incident in the target chamber were selected for further analysis.The measured energy of the α particle (E α ) was converted to the center-of-mass energy of 18 Ne + α system (E c.m. ) by assuming the elastic scattering kinematics using where M Ne and M α are the nuclear masses of the 18 Ne and α particle, respectively, and θ lab is the scattering angle in the laboratory frame.The value of θ lab was determined using the trajectories of the recoiling α particle and corresponding 18 Ne beam particle at the reaction vertex.The reaction vertex in the extended gas target was reconstructed by considering the energy losses of the 18 Ne beam and recoiling α particle in 4 He gas.The energy loss functions were obtained using the SRIM code [36].Direct measurement of the energy loss of the 18 Ne beam at six different target pressures in the present study was in good agreement with the SRIM calculation result.

Excitation function of 18 Ne + α elastic scattering
The differential cross section of 18 Ne + α resonant elastic scattering in the center-of-mass frame was calculated by where Y is the yield of recoiling α particles, θ lab is the scattering angle in the laboratory frame, I is the number of 18 Ne beam particles incident on the target, N is the number of 4 He target atoms, and ΔΩ lab is the solid angle covered by the detector.The number of incident 18 Ne beam particles was counted using two F3 PPACs.To obtain the precise number of beam particles entering into the target chamber through the entrance window, an additional cut was applied to the 18 Ne beam events.The positions of the beam particles at the target entrance were reconstructed eventby-event by extrapolating the beam trajectory obtained at two F3 PPACs, and then 18 Ne beam events falling in the diameter of the entrance of the target chamber were selected.A total of ~1.33 × 10 1018 Ne beam ions impinged on the target during the runs.
The excitation function of 18 Ne + α resonant elastic scattering was extracted by selecting α events with an angular range of 0 °≤ θ lab ≤ 7 °(166 °≤ θ c.m. ≤ 180 °).The solid angle was calculated using the known detector geometry and reaction vertex of each event as a function of E c. m .Due to the finite angular range of θ lab , an average value of solid angle at the reaction vertex (or E c. m. ) was used.The areal number density of 4 He atoms was obtained by considering the effective target thickness as a function of E c. m. .The uncertainty in E c. m. was measured to be approximately 50-100 keV, depending on the energy.The uncertainty originates from the energy resolution of a silicon detector (30-90 keV) and energy straggling of the 18 Ne beam and α particles in the gas (20-60 keV).

Upper limits on the cross section
The 18 Ne (α, α) 18 Ne cross section extracted in the present work is rather smooth in the energy region below E c. m. < 3 MeV.In this region, however, two small bumps were observed at E c. m. ~2.6 MeV and 2.7 MeV in the cross section spectrum.Several resonances in 22 Mg have been identified in the energy range through the previous 18 Ne (α, p) 21 Na reaction study as reported in [20].For instance, two energy levels located at the resonance energies of E r = 2.52 ± 0.14 MeV and E r = 2.72 ± 0.14 MeV have been reported in the previous direct measurement by Groombridge et al. [20].However, the existence of these resonances is not obvious in our data, possibly due to the insufficient statistics.Therefore, the upper limits on the cross section were set to indicate the possible maximum resonant cross section that is consistent with our experimental spectrum.Figure 4 shows the obtained upper limits by assuming hypothetical levels located at E c. m. = 2.63 MeV (top) and E c. m. = 2.75 MeV (middle).The upper limit with both hypothetical levels is also plotted in the figure (bottom).The black circles represent the empirical cross sections obtained at 0 °≤ θ lab ≤ 7 °.The blue solid lines represent the best fit curves for the observed bumps.The A typical particle identification plot is shown.The total energy of a recoiling particle is plotted as a function of the energy deposition in the first layer of the telescope.Events in the region with thick solid lines are identified as α particles from (A) 4 He and (B) argon gas run.

FIGURE 3
The ToF between the F3 PPAC and the second layer of the Tel 1 is plotted as a function of the energy of α particles.The beam-like α particles from the upstream were clearly separated from the recoiling α particles.The slope-like dependence observed at high energies is due to the slewing effect (which is not corrected in this plot).reduced χ 2 value of the best fit curve is 0.893.The shape of the resonance was assumed to be Gaussian, as plotted as a red dotted line in the figure, because the broadening by the experimental resolution is expected to dominate the width.The width of the Gaussian was assumed to be 50 keV to fit the observed bumps.The normalization factor of the distribution was then increased until the χ 2 value decreased to a prescribed amount, resulting in a one-sigma confidence level.The red dashed lines represent the upper limits of the 18 Ne (α, α) 18 Ne cross section.A fluctuation of experimental data points was also observed at E c. m. ~2.9 MeV.This bump could be a corresponding resonance at E r = 2.87 ± 0.14 MeV reported in a previous work [20]; however, it does not fall into the Gamow window at T ~2 GK.Therefore, the upper limit was not evaluated for this resonance.

Astrophysical implication
Two small bumps observed in the present work fall within the Gamow window of the astrophysically important 18 Ne (α, p) 21 Na reaction relevant to a temperature of T ~2 GK.The contribution of the observed bumps to the 18 Ne (α, p) 21 Na reaction was then investigated.The cross section of the 18 Ne (α, p) 21 Na reaction was calculated using the Breit-Wigner formula [37].
where λ is the de Broglie wavelength; E r is the resonance energy; J, J Ne , and J α are the spins of resonance, 18 Ne, and 4 He, respectively.Γ α , Γ p , and Γ tot are the α partial width, proton partial width, and total width, respectively.Γ tot = Γ α + Γ p was assumed in the calculations.Two hypothetical levels used in the upper limit calculation were considered to be the resonances for the 18 Ne (α, p) 21 Na reaction, where the resonance energies were adopted as E r = 2.63 and 2.75 MeV.Γ α values of both resonances were determined by further analyzing with the R-matrix code SAMMY8 [38,39], where the experimental energy increase of ~50 keV was assumed.The best fit yielded values of Γ α = 15 keV for both resonances.The spin and parity of both resonances was adopted as J π = 0 + while calculating the cross section.Several J π values for the resonances located in the astrophysically important energy region have been suggested in the previous direct measurement by Groombridge et al. [20], including J π = 0 + .However, the choice of other J π values such as J π = 1 − or 2 + for observed bumps would imply our Γ α which exceed the Wigner limit.Table 1 summarizes the resonance parameters used in the 18 Ne (α, p) 21 Na cross section calculations.
Although proton resonant scattering on the 21 Na nucleus has been measured in literatures [22][23][24], Γ p in the corresponding energy TABLE 1 The resonance parameters used in the 18 Ne (α, p) 21 Na cross section calculation are summarized.Two sets of parameters ("COM1" and "COM2") are used in the calculation to illustrate the sensitivity of the Γ p for the reaction cross section.All energies are expressed in MeV.region has not been reported so far.To approximate Γ p values for the resonances, the Wigner limit for the proton was calculated by Γ w = 2Z 2 /μR 2 P l , where μ is the reduced mass, R is the interaction radius, and P l is the penetrability of a given orbital angular momentum l.An interaction radius of R = 1.35 (1 + 21 1/3 ) fm [24] was adopted in the calculation.By considering the global mean reduced proton width θ 2 p = 0.0045, suggested in [40], Γ p was estimated to be ~10 keV for the resonances at E r = 2.63 and 2.75 MeV, which is summarized as "COM1" in Table 1.To illustrate the sensitivity of the proton widths for the 18 Ne (α, p) 21 Na reaction cross section, another set of Γ p values for the resonances was approximated by assuming Γ p = Γ tot − Γ α ("COM2" in Table 1).We adopted Γ tot = 0.1 MeV (Γ tot = 0.21 MeV) for the resonance at E r = 2.63 MeV (E r = 2.75 MeV), as reported in the previous direct measurement by Groombridge et al. [20].
The calculation results for the 18 Ne (α, p) 21 Na reaction cross section are shown in Figure 5, in comparison with the previous experimental results.The cross sections deduced by the Breit-Wigner formula using two sets of Γ p values are shown as the green dashed and black solid lines, respectively.The blue triangles represent the experimental data obtained by Salter et al. [29], which were determined from the time-reversal reaction measurement.The 18 Ne (α, p 0 ) 21 Na reaction cross section was inferred by using the principle of the detailed balance theorem [41]; therefore, their data can provide a lower limit for the cross section.The red squares represent the 18 Ne (α, p) 21 Na cross section derived from the resonance parameters reported by Groombridge et al. [20].The black squares represent the recent direct measurement results by Anastasiou et al. [21], which shows a lower cross section by almost an order of magnitude compared with that of Groombridge et al.As shown in the figure, our calculation results indicate that the 18 Ne (α, p) 21 Na reaction cross section depends critically on the proton widths of the resonances in the astrophysically important energy region.Thus, experimental studies of the Γ p are highly required for a conclusive understanding of the 18 Ne (α, p) 21 Na reaction cross section.

R-matrix analysis and α-cluster structure
The excitation function of 18 Ne + α elastic scattering obtained at the higher excitation energy region E x 11-16 MeV is shown in Figure 6.The black circles represent the differential cross sections of the 18 Ne (α, α) 18 Ne obtained at 0 °≤ θ lab ≤ 7 °.Several peaks were evident in the spectrum, which implies the existence of resonances with large Γ α .Since the strength of the α-clustering feature of a resonance state is reflected by its α width, observed peaks in the present work are possible candidates of α-cluster states.To constrain the energy level properties of 22 Mg including Γ α , an analysis using the R-matrix calculation code SAMMY8 [38,39] has been in progress.A channel radius of R c = 5.0 fm was adopted in the calculation, which is the same value used in the GCM calculation [11].We calculated the excitation function at an average angle of θ c.m. = 173 °, and the result was then broadened considering the experimental energy resolution.
By introducing three resonances in the R-matrix calculation, the fitting curve was obtained at E x ≤ 12.3 MeV, which is plotted as red solid line in Figure 6.The resonance parameters are summarized in Table 2. Since the spectroscopic information of 22 Mg nucleus in this energy region is very limited, an intensive R-matrix analysis with all possible spin and natural parity combinations for observed resonances should be carefully performed until the experimental excitation function is wellreproduced.The resonance parameters with best fitting result will be provided in the future.The χ 2 analysis will be performed to deduce possible parameters for each peak.The dimensionless partial width θ 2 a for each level will be calculated by θ 2 a = Γ α /Γ W , where Γ W is the Wigner limit of Γ α , which can provide a direct comparison with theoretical predictions in [11].More detailed calculations would be necessary to reveal the nature of those levels, if they are shell-model-like or cluster-like states.We measured α resonant elastic scattering on 18 Ne using the thick-target method in inverse kinematics technique to improve our knowledge of the α-cluster structure of proton-rich 22 Mg nucleus.The excitation function for 18 Ne (α, α) 18 Ne in the energy range of E x ~10-16 MeV was obtained at 0 °≤ θ lab ≤ 7 °.Several levels with large α widths were evident in our result, which can be candidates for the αcluster states.To clarify the energy level properties of 22 Mg and to investigate α-clustering features, the R-matrix analysis is in progress.The resonance parameters will be extracted considering all possible combinations of spin and parity for observed peaks.The first experimental constraints on spectroscopic information of 22 Mg above E x ~13 MeV will be provided.To better understand the experimental results, complete theoretical descriptions are required in the future.
No levels were evident at E x < 11 MeV in the present work, even though two small bumps were observed in the cross section spectrum.Therefore, we set upper limits on the 18 Ne (α, α) 18 Ne cross section, which indicate the possible maximum resonant cross section assuming the hypothetical levels at E x = 10.772 and 10.892 MeV (E c.m. = 2.63 and 2.75 MeV).We also estimated the astrophysically important 18 Ne (α, p) 21 Na reaction cross section based on our experimental data.The calculation indicates that the 18 Ne (α, p) 21 Na cross section depends on the proton widths of the resonances as well.Experimental studies on Γ p are necessary to evaluate the 18 Ne (α, p) 21 Na reaction cross section conclusively.

FIGURE 1
FIGURE 1Schematic view of the CRIB separator (top) and the experimental setup for the 18 Ne + α resonant scattering measurement at the F3 focal plane (bottom).The 18 Ne beam particles were identified and monitored by two F3 PPACs.4He gas at 470 Torr was filled to the chamber, which was sealed with a 23-μm-thick Mylar foil as a beam entrance window.The energy and position of the recoiling α particle were measured by ΔE-E silicon detector telescopes (Tel 1 and Tel 2).

FIGURE 2
FIGURE 2A typical particle identification plot is shown.The total energy of a recoiling particle is plotted as a function of the energy deposition in the first layer of the telescope.Events in the region with thick solid lines are identified as α particles from (A)4 He and (B) argon gas run.

FIGURE 5
FIGURE 5 Comparison of the 18 Ne (α, p) 21 Na reaction cross section calculations in the present work with the previous measurements by Salter et al. [29], Groombridge et al. [20], and Anastasiou et al. [21] is shown.

FIGURE 4
FIGURE 4The upper limits of the 18 Ne (α, α)18 Ne cross section by assuming hypothetical levels located at E c. m. = 2.63 MeV (top) and E c. m. = 2.75 MeV (middle) are plotted as red dashed lines.The upper limit with both resonances is also plotted (bottom).The one-sigma confidence level was considered to obtain the upper limit.The black circles represent the experimental excitation function obtained at 0 °≤ θ lab ≤ 7 °, the blue solid line represents the best fitting result of it, and the red dotted line represents each hypothetical level.

FIGURE 6
FIGURE 6The excitation function of the elastic scattering of α particles on18 Ne obtained at 0 °≤ θ lab ≤7 °(black circles) with the R-matrix fit with three resonances (red solid line) is shown.

TABLE 2
Resonance properties of 22 Mg extracted from the present work are summarized.Results from the previous works are listed for comparison.