Multiprobe study of excited states in ${}^{12}\mathrm{C}$: Disentangling the sources of monopole strength between the energy of the Hoyle state and $E_x=13$ MeV

Background: The Hoyle state is the archetypal $\alpha$-cluster state which mediates the $3\alpha$ reaction to produce ${}^{12}\mathrm{C}$ and is of great interest for both nuclear structure and astrophysics. Recent theoretical calculations predict a breathing-mode excitation of the Hoyle state at $E_x\approx 9$ MeV. Its observation is hindered by the presence of multiple broad states and potential interference effects. An analysis with Gaussian line shapes of measurements at the Research Center for Nuclear Physics (Osaka University) with the Grand Raiden spectrometer suggested that additional strength was needed at $E_x\approx 9$ MeV to reproduce the data; this analysis did not account for the well-known threshold effects observed in ${}^{12}\mathrm{C}$. Nevertheless, various theoretical studies have since concluded that this additional strength corresponds to the predicted breathing-mode excitation of the Hoyle state. To meaningfully identify a new source of monopole strength in this astrophysically significant region, a more appropriate phenomenological analysis which accounts for penetrability and interference effects must be used to determine whether the data can be explained with previously established states.

Purpose: We aim to investigate the monopole strength in the astrophysically important excitation-energy region of ${}^{12}\mathrm{C}$ between $E_x=7$ and 13 MeV to determine whether the previously established sources of monopole strength are able to reproduce the data.

Method: The ${}^{12}\mathrm{C}(\alpha ,{\alpha }^{\prime }){}^{12}\mathrm{C}$ and ${}^{14}\mathrm{C}(p,t){}^{12}\mathrm{C}$ reactions, which are expected to exhibit contrasting selectivity towards different monopole excitations, were employed at various detection angles and beam energies to populate states in ${}^{12}\mathrm{C}$. The inclusive excitation-energy spectra were simultaneously analyzed with multilevel, multichannel line shapes. Various scenarios with different sources of monopole strength and interference effects were considered to determine whether the ghost of the Hoyle state and the previously established broad $0_3^{+}$ state at $E_x\approx 10$ MeV are able to reproduce the observed monopole strength.

Results: Clear evidence was found for excess monopole strength at $E_x\approx 9$ MeV, particularly in the ${}^{12}\mathrm{C}(\alpha ,{\alpha }^{\prime }){}^{12}\mathrm{C}$ reaction at $0^{\circ }$. This additional strength cannot be reproduced by the previously established monopole states between $E_x=7$ and 13 MeV. Coincident charged-particle decay data suggest that the strength at $E_x\approx 9$ MeV is dominantly monopole, with no evidence of a $J>0$ contribution.

Conclusions: The data support a new source of monopole strength at $E_x\approx 9$ MeV, which cannot be described with a phenomenological parametrization of all previously established states. An additional $0^{+}$ state at $E_x\approx 9$ MeV yielded a significantly improved fit of the data and is a clear candidate for the predicted breathing-mode excitation of the Hoyle state. Alternatively, the results may suggest that a more sophisticated, physically motivated parametrization of the astrophysically important monopole strengths in ${}^{12}\mathrm{C}$ is required.

Figure 1

Schematic top-view diagram of the K600 spectrometer positioned at ${\theta }_{\text{lab}}=0^{\circ }$, with the focal-plane detectors located in the medium-dispersion focal plane as used for the ${}^{14}\mathrm{C}(p,t){}^{12}\mathrm{C}$ measurements of this work.

Figure 2

(a) The intrinsic line shapes for the $0_1^{+}$ and $2_1^{+}$ states of ${}^8\mathrm{Be}$, determined with an $\alpha –\alpha$ channel radius of $a_c=6$ fm. The (b) ${\alpha }_0$ and (c) ${\alpha }_1$ penetrabilities for ${}^{12}\mathrm{C}$, determined with a ${}^8\mathrm{Be}+{}^4\mathrm{He}$ channel radius of $a_c=6$ fm, and with Eq. (4) (colored dashed and dash-dotted curves) or Eq. (5) (colored solid curves). The red, vertical dashed lines indicate the resonance energies for the $0_1^{+}$ and $2_1^{+}$ states of ${}^8\mathrm{Be}$, which respectively correspond to the ${\alpha }_0$ and ${\alpha }_1$ thresholds for standard penetrabilities calculated with Eq. (4).

Figure 3

The feeding factors, $G_{ab}$, for (a) the ${}^{12}\mathrm{C}(\alpha ,{\alpha }^{\prime }){}^{12}\mathrm{C}$ measurement at ${\theta }_{\text{lab}}=0^{\circ }$ ($E_{\text{beam}}=160$ MeV) and (b) the ${}^{14}\mathrm{C}(p,t){}^{12}\mathrm{C}$ measurement at ${\theta }_{\text{lab}}=0^{\circ }$, arbitrarily normalized at $E_x=7.654$ MeV.

Figure 4

The coupling scheme for the ${}^{14}\mathrm{C}(p,t){}^{12}\mathrm{C}$ measurements analyzed in this work.

Figure 5

Excited states from the ${}^{14}\mathrm{C}(p,t){}^{12}\mathrm{C}$ reaction at ${\theta }_{\text{lab}}=0^{\circ }$ observed on opposite sides of the focal-plane detector. In panel (a) the effect of kinematic broadening as well as small contributions of higher-order aberrations can be seen, which are corrected in panel (b). The influence on the resulting line shapes are shown in panel (c).

Figure 6

The excitation-energy spectra from the ${}^{12}\mathrm{C}(\alpha ,{\alpha }^{\prime }){}^{12}\mathrm{C}$ measurement at ${\theta }_{\text{lab}}=0^{\circ }$ ($E_{\text{beam}}=160$ MeV), focused on modeling the VDC response and target-related attenuation for (a) the $J^{\pi }=2_1^{+}$ and (b) $J^{\pi }=0_2^{+}$ states of ${}^{12}\mathrm{C}$. The spectrum presented in panel (a) corresponds to a subset of the data around the $2_1^{+}$ state. The small contaminant peak below the Hoyle state corresponds to the $E_x=7.547\left(3\right)$ MeV state of ${}^{13}\mathrm{C}$ from the ${}^{13}\mathrm{C}(\alpha ,{\alpha }^{\prime })$ reaction. The experimentally observed line shapes (${\sigma }_{\text{obs}}$) are produced through circular convolution (denoted $⊛$) of the intrinsic line shape with a combination of Gaussian ($G$), Landau ($\ell$), and truncated Landau ($L^{\prime }$) distributions.

Figure 7

Full-range excitation-energy spectra of all analyzed inclusive measurements. The instrumental background spectra and the associated scaled contributions to the spectra of interest are displayed in orange and green, respectively. The excitation energies, spins, and parities for well-resolved ${}^{12}\mathrm{C}$ states are indicated. Contaminant peaks are labeled according to Table 3. The red histogram (i) corresponds to contaminant events from the ${}^1\mathrm{H}(\alpha ,\alpha )$ reaction. The pair of blue, vertical dashed lines on each spectrum indicates the excitation-energy range of full acceptance by the spectrometer, as summarized in Table 1. For details on the violet line shapes denoted by ($\varepsilon$) and (h), see Sec. 3c.

Figure 8

The matrices of particle energy (measured with the CAKE) versus the excitation energy of ${}^{12}\mathrm{C}$ for the measurements of (a) ${}^{12}\mathrm{C}(\alpha ,{\alpha }^{\prime }){}^{12}\mathrm{C}$ at ${\theta }_{\text{lab}}=0^{\circ }$ ($E_{\alpha }=200$ MeV) and (b) ${}^{14}\mathrm{C}(p,t){}^{12}\mathrm{C}$ at ${\theta }_{\text{lab}}=0^{\circ }$. The matrices in panels (a) and (b) are gated on a subset of the CAKE detector channels, corresponding to decay-angle ranges of ${140}^{\circ }<{\theta }_{\text{lab}}<{161}^{\circ }$ and ${153}^{\circ }<{\theta }_{\text{lab}}<{165}^{\circ }$, respectively. The red lines indicate the observed ${\alpha }_0$ and proton decay modes of ${}^{12}\mathrm{C}$. The orange-highlighted regions indicate $\alpha$ particles corresponding to either ${\alpha }_1$ decay or the ${}^8\mathrm{Be}\to 2\alpha$ breakup channel. The violet, horizontal lines in panels (a) and (b) indicate the approximate electronic thresholds for the silicon detectors of 600 and 750 keV, respectively. For panel (a), loci corresponding to ${\alpha }_0$ decay from contaminant states of ${}^{16}\mathrm{O}$ are labeled and a display in color threshold of $>2$ events is imposed.

Figure 9

Angular-correlation functions of ${\alpha }_0$ decay across the ring detector channels of the CAKE, relative to the beam axis, for the excitation-energy regions of (a) $E_x=10.0$ to 10.3 MeV and (b) $E_x=10.6$ to 11.1 MeV from the measurement of ${}^{12}\mathrm{C}(\alpha ,{\alpha }^{\prime }){}^{12}\mathrm{C}$ at ${\theta }_{\text{lab}}=0^{\circ }$ with $E_{\alpha }=200$ MeV. The sums of angular correlations with different $\ell$ values are incoherent, with the exception being the combination of $\ell =0$, 1, and 2 in panel (b).

Figure 10

The angular correlations of ${\alpha }_0$ decay across the ring detector channels of the CAKE, relative to the beam axis, for the excitation-energy regions of (a) $E_x=8.5$ to 9.0 MeV and (b) $E_x=9.8$ to 10.0 MeV as well as (c) the narrow $2^{+}$ state at $E_x=16.106$ MeV from the measurement of ${}^{14}\mathrm{C}(p,t){}^{12}\mathrm{C}$ at ${\theta }_{\text{lab}}=0^{\circ }$ with $E_{\text{beam}}=100$ MeV. The sums of angular correlations with different $\ell$ values are incoherent, with the exception being the combination of $\ell =0$, 1, and 2 in panel (d).

Figure 11

The optimized global fit for model $M_{0_2^{+}}$: it is assumed that the broad monopole strength above the primary peak of the $0_2^{+}$ Hoyle state and below $E_x\approx 13$ MeV corresponds only to the ghost of the Hoyle state. See text for details.

Figure 12

A comparison of the fits which only include the previously established sources of monopole strength. The $5\sigma$ band for the $0_2^{+}$ Hoyle state corresponds to the range between the fits where the total width of the Hoyle state was set at$\pm 5\sigma$ corresponding to $\mathrm{\Gamma }=9.3\left(9\right)$ eV. See text for details.

Figure 13

The optimized global fit for submodel , which accounts for all of the previously established sources of monopole strength between $E_x=7$ and 13 MeV: the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ resonance at $E_x\approx 10$ MeV. This particular fit is used in the comparison in Fig. 12 and the results are summarized in Table 4. See text for details.

Figure 14

The optimized global fit for submodel , which accounts for all of the previously established sources of monopole strength between $E_x=7$ and 13 MeV: the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ resonance at $E_x\approx 10$ MeV. In this particular case, the Wigner limit is not applied to any broad states between $E_x=7$ to 13 MeV. See text for details.

Figure 15

The optimized global fit for model , which accounts for the previously established monopole strengths between $E_x=7$ and 13 MeV (the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ at $E_x\approx 10$ MeV) and introduces an additional source of monopole strength at $E_x\approx 9$ MeV, denoted $0_{\mathrm{\Delta }}^{+}$.

Figure 16

The optimized global fit for model , which accounts for the previously established monopole strengths between $E_x=7$ and 13 MeV (the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ at $E_x\approx 10$ MeV) and introduces an additional source of monopole strength at $E_x\approx 9$ MeV, denoted $0_{\mathrm{\Delta }}^{+}$. The contribution of the $0_{\mathrm{\Delta }}^{+}$ response is superimposed on the combined monopole strength from the previously established $0_2^{+}$ and $0_3^{+}$ states.

Figure 17

The average global fit for model , which accounts for the previously established monopole strengths between $E_x=7$ and 13 MeV (the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ at $E_x\approx 10$ MeV) and introduces an additional source of monopole strength at $E_x\approx 9$ MeV, denoted $0_{\mathrm{\Delta }}^{+}$.

Figure 18

The average global fit for model , which accounts for the previously established monopole strengths between $E_x=7$ and 13 MeV (the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ at $E_x\approx 10$ MeV) and introduces an additional source of monopole strength at $E_x\approx 9$ MeV, denoted $0_{\mathrm{\Delta }}^{+}$.

Figure 19

The optimized global fit for model $M_{0_2^{+}}$ [performed with the penetrability prescription of Eq. (6)]: it is assumed that the broad monopole strength above the primary peak of the $0_2^{+}$ Hoyle state and below $E_x\approx 13$ MeV corresponds only to the ghost of the Hoyle state. See the Appendix for details.

Figure 20

The optimized global fit for submodel [performed with the penetrability prescription of Eq. (6)], which accounts for all of the previously established sources of monopole strength between $E_x=7$ and 13 MeV: the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ resonance at $E_x\approx 10$ MeV. The results of this fit are summarized in Table 7. See the Appendix for details.

Figure 21

The optimized global fit for model [performed with the penetrability prescription of Eq. (6)], which accounts for the previously established monopole strengths between $E_x=7$ and 13 MeV (the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ at $E_x\approx 10$ MeV) and introduces an additional source of monopole strength at $E_x\approx 9$ MeV, denoted $0_{\mathrm{\Delta }}^{+}$.

Figure 22

The optimized global fit for model [performed with the penetrability prescription of Eq. (6)], which accounts for the previously established monopole strengths between $E_x=7$ and 13 MeV (the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ at $E_x\approx 10$ MeV) and introduces an additional source of monopole strength at $E_x\approx 9$ MeV, denoted $0_{\mathrm{\Delta }}^{+}$. The contribution of the $0_{\mathrm{\Delta }}^{+}$ response is superimposed on the combined monopole strength from the previously established $0_2^{+}$ and $0_3^{+}$ states.

Figure 23

The averaged global fit for model [performed with the penetrability prescription of Eq. (6)], which accounts for the previously established monopole strengths between $E_x=7$ and 13 MeV (the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ at $E_x\approx 10$ MeV) and introduces an additional source of monopole strength at $E_x\approx 9$ MeV, denoted $0_{\mathrm{\Delta }}^{+}$.

Figure 24

The averaged global fit for model [performed with the penetrability prescription of Eq. (6)], which accounts for the previously established monopole strengths between $E_x=7$ and 13 MeV (the $0_2^{+}$ Hoyle state and a broad $0_3^{+}$ at $E_x\approx 10$ MeV) and introduces an additional source of monopole strength at $E_x\approx 9$ MeV, denoted $0_{\mathrm{\Delta }}^{+}$.