One-dimensionality in atomic nuclei: A candidate for linear-chain $\alpha$ clustering in ${}^{14}\mathrm{C}$

The clustering of $\alpha$ particles in atomic nuclei results in the self-organization of various geometrical arrangements at the femtometer scale. The one-dimensional alignment of multiple $\alpha$ particles is known as linear-chain structure, evidence of which has been highly elusive. We show via resonant elastic and inelastic $\alpha$ scattering of a radioactive ${}^{10}\mathrm{Be}$ beam that excited states in the neutron-rich nucleus ${}^{14}\mathrm{C}$ agree with recent predictions of linear-chain structure based on an antisymmetrized molecular dynamics model.

Figure 1

(a) Trajectories of a ${}^{10}\mathrm{Be}+\alpha$ scattering event. (b) ${\theta }_{\mathrm{lab}}^{{}^{10}\mathrm{Be}}$ vs ${\theta }_{\mathrm{lab}}^{\alpha }$ plot with the kinematical curves for elastic and inelastic scattering to the ${}^{10}\mathrm{Be}{2}_1^{+}$ state. Data shown have been selected via energy-loss gating for ${}^{10}\mathrm{Be}$ beam particles.

Figure 2

Differential cross sections of ${}^{10}\mathrm{Be}$ scattering off $\alpha$ particles: (a) elastic scattering and (c) inelastic scattering to the $2_1^{+}$ state of ${}^{10}\mathrm{Be}$. The color scale is given in mb/sr. Excitation functions for (b) elastic and (d) inelastic scattering also are shown. The shaded spectrum is gated by ${\theta }_{\mathrm{c}.\mathrm{m}.}={70}^{\circ }\text{–}{90}^{\circ }\left({45}^{\circ }\text{–}{55}^{\circ }\right)$, and the blank spectrum by ${90}^{\circ }\text{–}{110}^{\circ }\left({70}^{\circ }\text{–}{80}^{\circ }\right)$ for elastic (inelastic) scattering. The lines indicate identified resonances.

Figure 3

Differential elastic cross sections for resonances identified in $\alpha +{}^{10}\mathrm{Be}$ scattering (Fig. 2). The data (crosses) are compared to $P_L^2\left({\theta }_{\mathrm{c}.\mathrm{m}.}\right)$ functions (curves) and $R$-matrix calculations (histograms).

Figure 4

Comparison with theoretical predictions: (a) Energy levels of positive-parity states in ${}^{14}\mathrm{C}$ relative to $S_{\alpha }$; shown are (I) present results, (II) a previously suggested positive-parity band [21], and (III) triaxially deformed and (IV) linear-chain levels predicted by the $\beta \text{−}\gamma$ constraint AMD method [8]. Predicted distributions of proton density ${\rho }_p$, neutron density ${\rho }_n$, and their differential ${\rho }_p-{\rho }_n$ in $10\times 10 {\mathrm{fm}}^2$ boxes also are shown. (b) $P_{\mathrm{B}.\mathrm{B}.}$ vs ${\theta }_{\mathrm{B}.\mathrm{B}.}$ plot of the Brink–Bloch wave function for the ${}^{14}\mathrm{C}{4}^{+}$ state. The results for ${}^{10}\mathrm{Be}\left(0^{+}\right)$ with $L=4$ (solid line), ${}^{10}\mathrm{Be}\left(2^{+}\right)$ with $L=2$ (dotted line), ${}^{10}\mathrm{Be}\left(2^{+}\right)$ with $L=4$ (dashed line), and ${}^{10}\mathrm{Be}\left(2^{+}\right)$ with $L=6$ (dash-dotted line) are displayed. The geometrical arrangements of ${}^{10}\mathrm{Be}+\alpha$ clusters for an arbitrary ${\theta }_{\mathrm{B}.\mathrm{B}.}$ are schematically shown above the $L$ labels. (c) Inelastic cross section gated by $E_{\mathrm{c}.\mathrm{m}.}=7.0\text{–}7.5$ MeV. The data are compared to $R$-matrix calculations assuming only $L=2$ (dotted line) and mixing $L=2,4$ (dashed line).