Emergent geometry and duality in the carbon nucleus

Abstract

The carbon atom provides the backbone for the complex organic chemistry composing the building blocks of life. The physics of the carbon nucleus in its predominant isotope, 12 C, is similarly full of multifaceted complexity. Some nuclear states of 12 C can be preferentially treated as a collection of independent particles held by the mean field of the nucleus, while other states behave more as a collection of three alpha-particle clusters. But these two pictures are not mutually exclusive, and some states can be described in either fashion. 1, 2 In this work, we provide the first model-independent tomographic scan of the three-dimensional geometry of the nuclear states of 12 C using the ab initio framework of nuclear lattice effective field theory. We find that the well-known but enigmatic Hoyle state is composed of a "bent-arm" or obtuse triangular arrangement of alpha clusters. We identify all of the low-lying nuclear states of 12 C as having an intrinsic shape composed of three alpha clusters forming either an equilateral triangle or an obtuse triangle. From these basic structural formations, the various nuclear states correspond to different rotational and vibrational excitations as well as either distortions or large-amplitude displacements of the alpha clusters. The states with the equilateral triangle formation also have a dual description in terms of particle-hole excitations in the mean-field picture. We compare our theoretical calculations with experimental data for binding energies, quadrupole moments, electromagnetic transitions, charge densities, and form factors. The overall agreement is good, and further studies using higher-fidelity interactions are planned.

Cluster excitation and single-nucleon excitation in 12C

“…The four parameters in the interaction are determined by fitting to the ground-state energy of 4 He and 12 C, the ground-state charge radius of 12 C, and to several electromagnetic transition rates [17].…”

“…For the choices of initial trial states see the discussion in Refs. [15,17]. One can calculate the Euclidean projection amplitudes at time step 𝑁 𝑡 as…”

“…𝐴, for all the nucleons, we can study the density distribution, transition form factors, and geometric structure as illustrated in Ref. [17].…”

“…Thus, in nature we see the 8 Be nucleus as a prominent example of clustering with two weakly interacting 𝛼 clusters. Similarly, the famous "Hoyle state" of 12 C, which is crucial for the synthesis of carbon and heavier elements in stars, is a three 𝛼-cluster state and its geometric shape has been debated for a long time [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].…”

“…Either the cluster picture or the single-particle picture would likely be too simple for realistic nuclear systems, therefore the duality and interplay of these is an interesting topic [20]. In a recent work using nuclear lattice effective field theory (NLEFT) [17], the geometric shape has been studied for low-lying states of 12 C, and some interesting features of cluster excitation and single-nucleon excitation can be seen from the 3D density distribution. In this contribution, we will show a more quantitative description of the excitation modes by looking at the distribution of distances between two 𝛼 clusters and of two nucleons inside a cluster.…”