Microscopic clustering in light nuclei

This review examines the tendency of light nuclei to exhibit clustering, where correlations between nucleons result in the formation of precipitates, typically $\alpha$ particles. The observation of clustering dates to the earliest days of the subject, where $\alpha$ particles were the building blocks of some nuclear models. The description of a nucleus in terms of clusters was attractive in terms of simplifying the computationally challenging problem through the reduction of the degrees of freedom. However, more recently it has been possible to develop ab initio methods which seek to build nuclei not from the clusters, but from the individual nucleons with a full account of the Pauli exclusion principle. This review links the development of the subject from the assumption of preformed $\alpha$ particles, through to the development of models which demonstrate the appearance of clustering from the $A$-nucleon wave function with realistic but effective interactions, to finally first principle approaches using interactions based on chiral effective field theory and the symmetries of quantum chromodynamics. This places the understanding of clustering as a cornerstone of the development of nuclear theory as it attempts to develop a complete understanding of light nuclei from the fundamental strong force.

Figure 1

(Upper) Experimental energy-spin systematics of states in ${}^{12}\mathrm{C}$. Filled symbols are strong assignments, and open symbols are tentative assignments which are yet to be confirmed. The squares correspond to the ground-state rotational band $0^{+}$, $2^{+}$, and $4^{+}$. Triangles are the $3^{-}$, $4^{-}$, and $5^{-}$ states. Circles are states associated with the Hoyle band ($0^{+}$, $2^{+}$, and $4^{+}$), and diamonds are the $1^{-}$ and $2^{-}$ states. The various lines correspond to best fits to the rotational systematics. (Lower) Energy levels of the $0^{+}$, $2^{+}$, and $4^{+}$ states in ${}^{12}\mathrm{C}$ and matter density distributions obtained by antisymmetrized molecular dynamics (AMD) with variation after projection using the modified Volkov No. 1 (MV1) force ([191]) (asterisk symbols) are compared with the experimental energy spectra from [3], [122], [124], and [178]. The intrinsic density distributions are shown together with percentages of the dominant component in the final wave functions. The $4_2^{+}$ state has dominantly the same intrinsic component as that of the $2_2^{+}$ state. The states with strong $E2$ transitions are connected by solid lines. Dashed lines correspond to the tentative assignments of experimental levels in the upper panel.

Figure 2

The calculated inelastic form factor for electron inelastic scattering from the $0_1^{+}$ ground state to the $0_2^{+}$ excited state ([137]) for the Bose-Einstein condensate approach (red), compared with the experimental data from [318], [314], [162], and [257].

Figure 3

(a) The measured $E1$ and $E2$ cross sections of the ${}^{12}\mathrm{C}(\gamma ,{\alpha }_0){}^8\mathrm{Be}$ reaction. (b) The measured $E1\text{−}E2$ relative phase angle (${\varphi }_{12}$) together with the phase angle calculated from a two-resonance model. A reanalysis of the data by Fynbo et al. indicates for a channel radius $R_0=1.6\mathrm{f}\mathrm{m}$ the resonance energy is $E_R=10.025\pm 0.050$, $\mathrm{\Gamma }=1.6\pm 0.13\mathrm{M}\mathrm{e}\mathrm{V}$, and ${\mathrm{\Gamma }}_{\gamma }=220_{-36}^{+26}\mathrm{m}\mathrm{e}\mathrm{V}$. From [368].

Figure 4

Theoretical and experimental energy levels $E0$ and $E2$ transitions, radii of ${}^{12}\mathrm{C}$. The energy levels of $0^{+}$ and $2^{+}$ states are shown by solid and dashed lines, respectively. $B(E2)$ values ($e^2{\mathrm{fm}}^4$) are shown by the corresponding arrows (cyan text and arrows). $M(E0)$ values for $0_2^{+}\to 0_1^{+}$ are shown by the arrows from $0_2^{+}$ to $0_1^{+}$ (purple text and arrows). Root-mean-square radii of point-proton distribution in $0^{+}$ states are shown at the right side of the energy levels (black text). Experimental data are from [3], [56], [178], [11], and [368]. Theoretical results of the microscopic $3\alpha$ models, $3\alpha$ RGM ([188]), the extended THSR ([132]; [134]), $3\alpha$ GCM(D) ([76]), $3\alpha$ GCM(U) ([334]), and $3\alpha$ GCM(S) ([323]) are shown in the upper panel, and those of the $3\alpha$ $\mathrm{GCM}+p_{3/2}$ ([323]), the $\mathrm{FMD}+3\alpha$ ([56]), AMD ([190], [191]), NCSM ([260]), NCSpM ([83]), and NLEFT ([104]) calculations are shown in the lower panel. For comparison, the $3\alpha$ $\mathrm{GCM}+p_{3/2}$ results are also shown in the upper panel. The NLEFT energies have about 2 MeV errors and can be improved in future work using new methods as described by [220].

Figure 5

The calculated spectrum of ${}^{16}\mathrm{O}$ states assuming a $T_d$ dynamical symmetry, obtained using the algebraic cluster model. From [37].

Figure 6

Measurements of the ${}^{12}\mathrm{C}({}^4\mathrm{He},{}^8\mathrm{Be}){}^8\mathrm{Be}$ reaction for ${\theta }_{\mathrm{c}.\mathrm{m}.}=90°$ ([65]) data points. The dashed and dotted lines correspond to the measurements of [58] and [46]. See [65] for more details.

Figure 7

The formation of molecular orbitals for neon isotopes from the valence orbitals of neutrons around the cores of ${}^{16}\mathrm{O}$ and ${}^4\mathrm{He}$. From [201].

Figure 8

Rotational bands of ${}^8\mathrm{Be}$, ${}^9\mathrm{Be}$, (left panel) and ${}^{10}\mathrm{Be}$ (right panel). The excitation energies are plotted as a function of angular momentum $J(J+1)$. The Coriolis decoupling parameter $a$ for the $K=1/2$ band is indicated. From [128].

Figure 9

Comparison between experimental $B(E2)$ values and GFMC and NCSM calculations using the AV18 potential with the IL7 three-nucleon interactions. The dashed lines show the corresponding isospin-symmetric results; see [246] for the full details. The black square at ${}^{10}\mathrm{B}$ illustrates the experimental value of $6.1e^2{\mathrm{fm}}^4$. From [247].

Figure 10

$0^{+}$-projected energy surface on the $\mathrm{\Lambda }\text{−}D$ plane for ${}^{12}\mathrm{C}$ calculated by the AQCM. The interaction and width parameters are the same as those in [319].

Figure 11

The energy weighted ISM strength distributions obtained by the $\mathrm{sAMD}+3\alpha$ GCM and those measured by $(\alpha ,{\alpha }^{\prime })$ scattering ([183]). The experimental $E0$ strength for the $0_2^{+}$ measured by electron scattering ([57]) is also shown in (b). (c), (d) Those calculated with truncated model space of $3\alpha$ configurations: (c) calculation using $3\alpha$ configurations at $\theta =\pi /2$ and sAMD bases. (d) Same as (c) but only compact $3\alpha$ configurations with $|{\mathit{S}}_i|<2\mathrm{fm}$. (e) Strengths obtained by the sAMD bases without the $3\alpha$ configurations. (a), (b), and (e) From [194].

Figure 12

(a), (b) Monopole excitations in ${}^{12}\mathrm{C}$ calculated with the time-dependent FMD. From [140]. Oscillation frequencies of the root-mean-square radius are plotted against the Fourier component in the small amplitude limit in (a), and those are plotted as a function of the excitation energy given by the initial amplitude in (b). (c) Sketches for width oscillation and intercluster modes in the time-dependent FMD. (d) $0^{+}$ energy surface of the $3\alpha$ THSR wave function on the $B\text{−}b$ plane ($B$ is denoted by $R_0$). (e) Sketches for $b$ and $B$ modes in the THSR. (a), (b) From [194]. (d) From [332].

Figure 13

Nucleon density distribution of the ${}^{16}\mathrm{O}+\alpha$ intrinsic THSR wave function of Eq. (5.16) for $D=0.6\mathrm{fm}$. A large distance of about 3.6 fm between ${}^{16}\mathrm{O}$ and $\alpha$ clusters is seen.

Figure 14

Nucleon density distribution for a strongly prolate THSR wave function of $3\alpha$s.

Figure 15

Contour map of the squared overlap between a $0^{+}$ wave function with ${\beta }_x={\beta }_y=0.9\mathrm{fm}$, ${\beta }_z=2.5\mathrm{fm}$ and $0^{+}$ wave functions with various deformations ${\beta }_x={\beta }_y$ and ${\beta }_z$. Numbers attached to the contour curves are squared overlap values. From [362].

Figure 16

Rotation average of a prolate THSR wave function around an axis ($x$ axis) perpendicular to the symmetry axis ($z$ axis) of the prolate deformation.

Figure 17

Size parameters $B_1$ and $B_2$ for $2\alpha$ and $3\alpha$ containers, respectively, where $B_1^2=b^2+{\beta }_1^2$ and $B_2^2=(3/4)b^2+{\beta }_2^2$.

Figure 18

Energy spectrum obtained by extended $4\alpha$ THSR of [133] which is compared with those by $4\alpha$ OCM and experiment.

Figure 19

Results for the ${}^6\mathrm{He}$ wave function using a no-core shell model with continuum. The horizontal axis is the separation between the two halo neutrons $r_{\mathrm{nn}}$, and the vertical axis is the separation between the $\alpha$-particle core and the center of mass of the two halo neutrons $r_{\alpha ,\mathrm{nn}}$. Adapted from [299].

Figure 20

The density distributions $r^2\rho (r)$ of the ground state ($0_1^{+}$) and the Hoyle state ($0_2^{+}$) of ${}^{12}\mathrm{C}$ ([49]). The variational Monte Carlo results are indicated by ${\mathrm{\Psi }}_V$ while Green’s function Monte Carlo results are labeled as GFMC. Adapted from [49].

Figure 21

The ${}^8\mathrm{Be}$ proton densities for $|\mathrm{\Phi }⟩$ and $|{\mathrm{\Phi }}^{\mathrm{intr}}⟩$ ([310]; [359]). Results are shown for $N_b={10}^0$, ${10}^1$, and ${10}^2$ basis states. Each density distribution shows the $y\text{−}z$ plane for intercepts $x=0$ and 1 fm. Adapted from [310].

Figure 22

Possible configurations of four nucleons on the lattice (shown here in a two-dimensional sketch).

Figure 23

Lattice results for the ${}^{12}\mathrm{C}$ spectrum at leading order (LO). Panel I shows the results using initial states $A$, $B$, and $\mathrm{\Delta }$, each of which approaches the ground-state energy. Panel II shows the results using initial states $C$, $D$, and $\mathrm{\Lambda }$. These trace out an intermediate plateau at an energy $\simeq 7\mathrm{MeV}$ above the ground state. Here $A-D$ are configurations that start with delocalized nucleons, where as $\mathrm{\Delta }$ refers to a compact triangle, while $\mathrm{\Lambda }$ denotes an obtuse triangular configuration. For details, see [104].

Figure 24

An initial state formed of two clusters $|\overrightarrow{R}⟩$. The two clusters are separated by the displacement vector $\overrightarrow{R}$.

Figure 25

A sketch of the lattices for the cluster-cluster calculations in the overlapping and the noninteracting regions. $R_{\mathrm{in}}$ is the largest radial distance that is free of systematic errors due to the periodic boundary in the cubic box with volume $L^{\prime 3}$, where $L^{\prime 3}\ll L^3$. $R_{\mathrm{w}}$ indicates the radius of the spherical wall discussed in Sec. 8b.

Figure 26

Schematic illustration of the $\alpha$-cluster initial states in ${}^{16}\mathrm{O}$ with tetrahedral and planar configurations.

Figure 27

Top panel: $S$-wave phase shifts for $\alpha \text{−}\alpha$ scattering at LO, NLO, and NNLO. Inset: The calculation based on an EFT with pointlike $\alpha$ particles is shown. From [159]. Bottom panel: $D$-wave phase shifts at LO, NLO, and NNLO. From [158], [269], and [2]).

Figure 28

Zero-temperature phase diagram as a function of the parameter $\lambda$ in the strong interaction $V_{\lambda }=(1-\lambda )V_{\mathrm{A}}+\lambda V_{\mathrm{B}}$. A first-order quantum phase transition from a Bose gas to a nuclear liquid at the point appears where the scattering length $a_{\alpha \alpha }$ crosses zero. This is very close to the value $\lambda =0$. Also shown are the $\alpha$-like nuclear ground-state energies $E_A$ for $A$ nucleons up to $A=20$ relative to the corresponding multi-alpha threshold $E_{\alpha }A/4$. The last $\alpha$-like nucleus to be bound is ${}^8\mathrm{Be}$ at the unitarity point where $|a_{\alpha \alpha }|=\infty$. This unitarity point is very close to the value $\lambda =1$.