Reorientation-effect measurement of the first 2$^+$ state in $^{12}$C: confirmation of oblate deformation

A Coulomb-excitation reorientation-effect measurement using the TIGRESS $\gamma-$ray spectrometer at the TRIUMF/ISAC II facility has permitted the first determination of the $\langle 2^+_1\mid\mid \hat{E2} \mid\mid 2^+_1\rangle$ diagonal matrix element in $^{12}$C from particle$-\gamma$ coincidence data. Required state-of-the-art no-core shell model calculations of the nuclear polarizability for the ground and first-excited (2$^+_1$) states in $^{12}$C using chiral NN N$^4$LO500 and NN+3NF350 interactions have been performed. Consistent predictions show a larger polarizability than previously anticipated. The polarizability of the 2$^+_1$ state is introduced into the current and previous Coulomb-excitation reorientation-effect analysis of $^{12}$C. Spectroscopic quadrupole moments of $Q_{_S}(2_1^+)= +0.053(44)$ eb and $Q_{_S}(2_1^+)= +0.08(3)$ eb are determined, respectively, yielding a weighted average of $Q_{_S}(2_1^+)= +0.071(25)$ eb, in agreement with recent ab initio calculations. The present measurement confirms that the 2$^+_1$ state of $^{12}$C is oblate and emphasizes the important role played by the nuclear polarizability in Coulomb-excitation studies of light nuclei.

Electric quadrupole matrix elements are key quantities in probing the collective structure of nuclei as they are a sensitive and direct measure of the quadrupole deformation. The precise determination of these matrix elements reveals the nuclear collectivity caused by the coherent motion of nucleons, and the associated nuclear wavefunctions. Modern nuclear theory is providing refined calculations of electric quadrupole matrix elements and related properties in light nuclei. Of particular interest is the testing-ground nucleus 12 C, as this is computationally accessible to most modern theoretical approaches, including ab initio [1][2][3][4][5][6][7][8][9] and cluster calculations [10][11][12][13][14][15]. Cluster calculations in 12 C suggest that the admixture of α-cluster wavefunctions may have a pronounced effect on the shape of mean-field states at lower energies. Considerable α-cluster triangle admixtures of 52% and 67% for the ground and 2 + 1 states, respectively, are predicted by fermionic molecular dynamics (FMD) calculations [12], whereas a meanfield contribution of 15% is predicted for the 0 + 2 Hoyle state [16]; the state crucial to fusion of three α particles in the core of massive stars. Moreover, cluster models predict a combination of triangular oblate shapes for the ground state and first 2 + 1 excitation in 12 C [10][11][12]. Mean-field calculations using a relativistic energydensity functional also show a cluster-like structure for the ground state of 20 Ne [17].
Experimentally, this strong mixing between the 0 + 2 Hoyle and 0 + 1 ground states is supported by the largest known electric monopole transition strength, 10 3 × ρ 2 (E0) = 500(81), determined from electron scattering measurements [18], which corresponds to about a 30% increase in the mean squared charge radius for the Hoyle state.
The reorientation effect [19,20,25] (RE) in Coulomb excitation measurements at energies well below the Coulomb barrier -so-called safe Coulomb excitation -provides a powerful spectroscopic probe for extracting 2 + 1 ||Ê2 || 2 + 1 diagonal matrix elements, which can be directly related to the Q S (2 + 1 ) value as Q S (2 + 1 ) = 0.75793 2 + 1 ||Ê2 || 2 + 1 [25]. The only RE measurement of the 2 + 1 state at 4.439 MeV in 12 C was performed at safe energies by Vermeer et al. [26] through a measurement of inelastically scattered 12 C ions by a 208 Pb target. The scattered 12 C ions were momentum analyzed using a magnetic spectrometer and detected at the focal plane using a position sensitive multi-wire proportional counter placed at a scattering angle in the laboratory frame of θ = 90 • . A value of Q S (2 + 1 ) = +0.06(3) eb was determined using the nominal nuclear polarizability parameter κ(g.s.) = 1 determined for the ground state of heavier nuclei [27]. This parameter represents the ratio of the observed isovector giant-dipole-resonance (GDR) effect to that predicted by the hydrodynamic model [27], and is a pivotal ingredient in the RE analysis of light nuclei, where κ > 1 values are generally observed [26][27][28][29][30][31][32][33][34].
In this work, we perform a safe Coulomb-excitation RE study of the high-lying 2 + 1 state in 12 C using particle−γ coincidence measurements and state-of-theart NCSM calculations of the nuclear polarizabilities κ(g.s.) and κ(2 + 1 ). Although there seems to be a good agreement between previous theoretical and ex-perimental values of Q S (2 + 1 ), the present result emphasizes the crucial importance of determining κ in Coulomb-excitation studies of loosely-bound light nuclei. The main advantage of the particle-γ coincidence technique lies in the absence of target contaminants in the Doppler-corrected γ-ray spectrum.
For this measurement, a beam of 12 C 3+ ions, delivered to the TRIUMF/ISAC II facility [35] at 4.975 AMeV, has been used to populate the 2 + 1 state at 4.439 MeV in 12 C through Coulomb excitation. The beam energy was chosen in conformity with Spear's prescription of a minimum separation between nuclear surfaces of S(ϑ ) min 6.5 fm [36] to avoid Coulomb-nuclear interferences, where ϑ is the scattering angle in the centerof-mass frame. An average intensity of ≈ 5 × 10 8 particles/s was delivered to the TIGRESS array [37] over approximately three days, and impinged on a 3 mg/cm 2 thick 194 Pt target (96.45% enriched). The online data were collected in event-by-event mode using a highspeed digital data acquisition system with 100 MHz TIG-10 digital electronics modules.
The γ rays de-exciting the beam and target nuclei were detected using eight TIGRESS HPGe clover detectors positioned at 14.5 cm from the target, and covering around 15% of 4π. The scattered ions and recoiling particles were detected in a double-sided, CD-type silicon detector (S2 type from Micron Semiconductors [38]), which was mounted 19.4 mm downstream. The experimental set up is very similar to the one given in Ref. [29] apart from the use of an S2 detector, which is segmented into 48 rings and 16 azimuthal sectors on the ohmic side, and has the 12 outermost rings incomplete; hence, it does not present full azimuthal or φ symmetry. The scattered beam was fully stopped in the 500-µm thick S2 detector. Additional experimental details will be presented in a separate manuscript.
The energy calibration and relative photo-peak efficiency ε of the TIGRESS detectors were determined using standard radioactive 152 Eu and 56 Co sources. The calibration of all the silicon strips was done using a triple α source containing 239 Pu, 241 Am and 244 Cm, together with higher-energy calibration points provided by the elastically scattered beam particles simulated with GEANT4 [39], both including energy losses [40] in the 194 Pt target and the 0.58-mg/cm 2 thick aluminum coating on the ohmic side of the S2 detector facing the scattered 12 C beam. Typical particle-energy spectra for the innermost (a) and outermost (b) rings at average angles of θ = 31.7 • and 60.7 • are shown in Fig. 1.
Particle−γ coincidence events were selected by employing the condition that each hit in a TIGRESS detector has a hit in both a ring (θ ) and a sector (φ ) of the S2 detector within a coincidence time window of approximately 195 ns. The corresponding γ−ray spectra were further cleaned by subtracting random coincidence events outside the 195-ns time window. An additional energy-sharing condition of |E ring −E sector | ≤ 350 keV between each ring and sector yields a large background reduction in the particle spectra and enables a better selection of the inelastic peaks [29]. This energysharing condition was chosen to achieve the most optimum background reduction while conserving the area of the 4439-keV peak in the γ−ray spectrum. Figure 1 illustrates the effect with a large background reduction (black), as compared with no energy-sharing condition (brown), at low and intermediate energies. Finally, inelastic particle gates can be set on each ring particle spectrum to collect solely Coulomb-excitation events in coincidence with the γ ray of interest. Figure 2 shows the Doppler-corrected γ−ray energy spectrum obtained from the TIGRESS array after applying particle and time tagging conditions. The spectrum shows the 328-and 4439-keV γ−ray transitions depopulating the 2 + 1 level in 194 Pt and 12 C, respectively. Another second-order effect in Coulomb-excitation perturbation theory which may influence, particularly for light nuclei [26,28,29], the determination of both the sign and magnitude of 2 + 1 ||Ê2 || 2 + 1 is the E1 polarizability [41]. This involves virtual electric-dipole excitations via the GDR that polarize the shape of the 2 + 1 state through two-step processes of the type 0 + 1 → 1 − GDR → 2 + 1 . In particular, light nuclei present typical values of κ > 1 [28], which has a net effect of shift-  Figure 2: Doppler-corrected γ−ray energy spectrum generated by employing particle−γ, time, energy-sharing and inelastic-particle coincidence conditions. ing the measured Q S (2 + 1 ) values towards more prolate shapes.
The polarization potential V pol generated by the E1 polarizability is incorporated into Coulomb-excitation analyses by a reduction in the quadrupole interaction, V 0 (t), which results in an effective potential, V e f f (t) [30], where a is the half-distance of closest approach in a head-on collision and r(t) the magnitude of the projectile-target position vector. For the case of projectile excitation, z is given by [25], with R = 1.2A 1/3 fm being the nuclear radius, T p the kinetic energy (in MeV) in the laboratory frame, α = hc 2π 2 σ −2 the nuclear polarizability, where α = 2P 0 as defined by Alder and Winther [25], and κ the polarizability parameter. The (−2) moment of the total photoabsorption cross section, σ −2 , and κ are related by [28], The value of κ can accordingly be modified in modern Coulomb-excitation codes such as GOSIA [42]. For light nuclei, values of κ > 1 have been determined by Coulomb-excitation measurements for a few favorable cases where Q S (J) = 0 [30,32,34], i.e., for J = 1/2 excited states, and shell-model calculations [22,43].
For the case of arbitrary spins, Häusser and collaborators developed an expression for κ in terms of E1 and E2 matrix elements [30], κ = X X 0 , where X 0 = 0.0004 A Z eMeV −1 and X is given by, where the sum extends over all intermediate states |n connecting the initial |i and final | f states with E1 transitions and W (11J i J f , 2J n ) is the Racah Wcoefficient [44] with J i = 0, J f = 2 and J n = 1 for the case at hand. It is important to note here that the product of twoÊ1 operators yields anÊ2 operator; hence, some of the isoscalar giant quadrupole resonance strength may appear in the sum given in Eq. 4. In the present work, NCSM calculations have been performed to estimate κ for the ground and 2 + 1 states in 12 C. Previous SM calculations of κ(2 + 1 ) = 0.77 presumed that all the E1 strength from the ground state was concentrated at the GDR energy [22]. Our NCSM calculations used the chiral NN + 3NF350 interaction [45][46][47], including the N 3 LO NN interaction [45] and the local N 2 LO 3N interaction [46] with the cutoff of 350 MeV [47], and considered model spaces with basis sizes of N max = 4 for the natural and N max = 5 for the unnatural parity states. From Eq. 4, which included the E1 matrix elements from all the transitions connecting 28 1 − states up to 30 MeV, values of κ(g.s.) = 1.6(2) and κ(2 + 1 ) = 2.2(2) are predicted. As shown in Fig. 3, the E1 strength is concentrated at an energy of about 24 MeV -the centroid energy of the GDR [48]. The lowest calculated 1 − state energy was set to the lowest found 1 − 1 state at 10.84 MeV. In order to study convergence and determine uncertainties, predictions with the NN + 3NF350 interaction have been validated by additional NCSM calculations using the NN N 4 LO500 interaction [49,50] SRG evolved [51] to 2.4 fm −1 at the same N max = 4/5 space and at a smaller N max = 2/3 space at varied harmonic-oscillator frequencies, as well as at a larger N max = 6 space for natural parity and N max = 7 for unnatural parity states, which included 22 1 − states up to 30 MeV. The latest calculations yield similar results of κ(g.s.) = 1.5(2) and κ(2 + 1 ) = 2.1 (2). In general, to improve on the present NCSM description, one should include RGM-like cluster states with explicit α particles, e.g., 8 Be+α and couple them with the currently used NCSM basis. Such approach called NCSM with continuum is now under development. However, the good stability of all the 1 − states with N max demonstrates that our expansion is adequate.
The well-known total photo-absorption cross section measured for the ground state of 12 C can be used to benchmark our NCSM calculations. A value of σ −2 = 244 µb/MeV in the 1985 evaluation by Fuller [52], yields κ(g.s) = 1.6 using Eq. 3, in excellent agreement with our NCSM polarizability calculations for the ground state. The consistency of our calculations further supports the value of κ(2 + 1 ) = 2.2(2) implemented in our GOSIA analysis [42] throughout this work. The integrated γ−ray yields for the 2 + 1 → 0 + 1 transitions in 12 C and 194 Pt have been calculated using the semi-classical coupled-channel Coulomb-excitation least-squares code GOSIA [42]. The semi-classical approximation is confirmed from Rutherford scattering cross sections and the calculated Sommerfeld parameter, η = a Ż = 31 ≫ 1, where Ż is the de Broglie wavelength. Calculations consider the known spectroscopic information such as level lifetimes, branching ratios 4 and matrix elements, kinematics, detector geometry and beam energy losses. The effect of higher-lying states in 12 C has been estimated using GOSIA and considered negligible (< 0.1%). Figure 4 shows the experimental and theoretical heavy-ion angular distributions from the eight clover yields for the 2 + 1 → 0 + 1 transitions in 194 Pt (a) and 12 C (b). Predictions of the cross sections for populating states in 12 C were calculated at fixed values of 2 + 1 ||Ê2 || 0 + 1 = 0.0630 eb [53], 2 + 1 ||Ê2 || 2 + 1 = +0.070 eb and κ = 2.2, the intersection point of the centroid of the two bands in Fig. 5, and normalized to the experimental yields with a common normalization factor. The shape of the angular distributions predicted by GOSIA for both 194 Pt and 12 C are in good agreement with experiment.
The highprecision NCSM calculation using the CDB2k NN potential is given in Fig. 5 by the square data point [6], Q S (2 + 1 ) = +0.060(4) eb and B(E2; 2 + 1 → 0 + 1 ) = 8.8(7) e 2 fm 4 . Similar values of Q S (2 + 1 ) = +0.0591(25) eb and Q S (2 + 1 ) = +0.059(1) eb are calculated, respectively, using chiral NN+3N interactions [7] and the no-core symplectic model [3]. Calculations are in agreement with the weighted average presented in this work. Unfortunately, the model space that can currently be reached with the NCSM does not allow a calculation of the nuclear polarizability for the 2 + 2 state built on the Hoyle state. One could, however, speculate that if κ further increases with excitation energy, as the nucleus becomes more loosely bound, a more pronounced prolate shape might be expected for the shape of the Hoyle state rotational band. This is in concordance with the prolate bent-arm configuration -with Q S (2 + 2 ) = −0.07(2) eb -predicted by ab initio calculations using chiral perturbation theory on a lattice [2], and, although with an extremely large prolate deformation, the Q S (2 + 2 ) = −0.21(1) eb value predicted by the no-core symplectic model [3]. Such an enhanced polarizability might explain the sudden change in the shape of the Hoyle state, which seems to be in disagreement with early models of cluster formation such as that of Morinaga, where αcluster structures gradually emerge with increasing excitation energy and are fully realized at the α threshold [60,61].
In conclusion, the Coulomb-excitation analysis performed in this work using the TIGRESS array and the new value of κ(2 + 1 ) calculated with the NCSM have permitted the determination of the 2 + 1 ||Ê2 || 2 + 1 diagonal matrix element in 12 C from particle−γ coincidence data. The present work confirms an oblate deformation for the 2 + 1 state in 12 C, in agreement with recent ab initio and cluster-model calculations.
Finally, it is important to emphasize that NCSM calculations show that the polarizability parameter for excited states can be very different from the ground state value determined from total photo-absorption cross-section data. This unanticipated change of the nuclear polarizability from the ground state to the first excitation in 12 C may not only affect Coulomb-excitation analyses of light nuclei, but in general, as nuclei become less bound, Coulomb-excitation studies of states at high excitation energies (e.g., superdeformed bands found in nuclei with spherical ground states). This possibility clearly needs further investigations.