Microscopic Clustering in Light Nuclei

TL;DR

This paper reviews recent experimental and theoretical advances in understanding microscopic clustering phenomena in light nuclei, highlighting new findings and diverse modeling approaches.

Contribution

It provides a comprehensive overview of experimental results and theoretical models addressing nuclear clustering in light nuclei, including ab initio and advanced computational methods.

Findings

Experimental evidence of alpha-conjugate systems

Identification of molecular structures in neutron-rich nuclei

Constraints for ab initio nuclear theories

Abstract

We review recent experimental and theoretical progress in understanding the microscopic details of clustering in light nuclei. We discuss recent experimental results on $\alpha$-conjugate systems, molecular structures in neutron-rich nuclei, and constraints for ab initio theory. We then examine nuclear clustering in a wide range of theoretical methods, including the resonating group and generator coordinate methods, antisymmetrized molecular dynamics, Tohsaki-Horiuchi-Schuck-R\"opke wave function and container model, no-core shell model methods, continuum quantum Monte Carlo, and lattice effective field theory.

Tables

Table 1. Table 1: Comparison of the 0 + superscript 0 0^{+} energy spectra from experiments, 4 ​ α 4 𝛼 4\alpha OCM calculation  Funaki  et al. ( 2006b ) , and 4 ​ α 4 𝛼 4\alpha THSR calculation Funaki  et al. ( 2010 ) . Energies E 𝐸 E are measured in MeV from the 4 ​ α 4 𝛼 4\alpha threshold, and RMS radii R rms subscript 𝑅 rms R_{\rm rms} are in fm.

	4$\alpha$ THSR			4$\alpha$ OCM		Exp.
	$E$	$R_{\mathrm{rms}}$		$E$	$R_{\mathrm{rms}}$	$E$
$0_1^{+}$	$-$15.05	2.5	$0_1^{+}$	$-$14.37	2.7	$-$14.44
$0_2^{+}$	$-$4.7	3.1	$0_2^{+}$	$-$8.0	3.0	$-$8.39
			$0_3^{+}$	$-$4.41	3.1	$-$2.39
$0_3^{+}$	1.03	4.2	$0_4^{+}$	$-$1.81	4.0	$-$0.84
			$0_5^{+}$	$-$0.25	3.1	$-$0.43
$0_4^{+}$	3.04	6.1	$0_6^{+}$	2.08	5.6	0.66

Table 2. Table 2: Energies and rms radii of 0 2 + subscript superscript 0 2 0^{+}_{2} , 0 3 + subscript superscript 0 3 0^{+}_{3} , and 0 4 + subscript superscript 0 4 0^{+}_{4} states of 12 C and monopole transition values M ​ ( E ​ 0 ) 𝑀 𝐸 0 M(E0) between three 0 + superscript 0 0^{+} states calculated by extended THSR wave functions. Energies (MeV) are measured from the 3 ​ α 3 𝛼 3\alpha threshold, radii are in fm, and M ​ ( E ​ 0 ) 𝑀 𝐸 0 M(E0) values are in e 𝑒 e  fm 2 . Cal. A and Cal. B denote the results of Refs.  Funaki ( 2015 ) and Zhou  et al. ( 2016 ) , respectively.

	Cal. A	Cal. B
$E\left(0_2^{+}\right)$ ,   $R_{\mathrm{rms}}\left(0_2^{+}\right)$	0.23,   3.7	0,22,   3.9
$E\left(0_3^{+}\right)$ ,   $R_{\mathrm{rms}}\left(0_3^{+}\right)$	2.6,   4.7	1.7,   5.2
$E\left(0_4^{+}\right)$ ,   $R_{\mathrm{rms}}\left(0_4^{+}\right)$	3.9,   4.2	2.7,   4.0
$M\left(E0,0_2^{+}\to 0_1^{+}\right)$	$6.3-6.4$	6.2
$M\left(E0,0_2^{+}\to 0_3^{+}\right)$	$34-37$	47
$M\left(E0,0_2^{+}\to 0_4^{+}\right)$	$0.5-1.4$	7.7

Table 3. Table 3: NCSpM point rms matter radii and electric quadrupole moments for 12 C compared to experimental data. a Ref. Tanihata  et al. ( 1985 ) ; b Ref. Danilov  et al. ( 2009 ) ; c Ref. Ogloblin  et al. ( 2013 ) ; and d Ref. Ajzenberg-Selove ( 1990 ) . *Experimentally deduced, based on model-dependent analyses of diffraction scattering.

	matter radius (fm)		$Q$ ($e$fm2)
	Expt.	NCSpM	Expt.	NCSpM
$0_{gs}^{+}$	$2.43{\left(2\right)}^a$	$2.43\left(1\right)$
$0_2^{+}$ (Hoyle)	$2.89{\left(4\right)}^{b\ast }$	$2.93\left(5\right)$
$0_3^{+}$	N/A	$2.78\left(4\right)$
$2_1^{+}$	$2.36{\left(4\right)}^{b\ast }$	$2.42\left(1\right)$	$+6{\left(3\right)}^d$	$+5.9\left(1\right)$
$2^{+}$ above $0_2^{+}$	$3.07{\left(13\right)}^{c\ast }$	$2.93\left(5\right)$	N/A	$-21\left(1\right)$
$4_1^{+}$	N/A	$2.41\left(1\right)$	N/A	$+8.0\left(3\right)$
$4^{+}$ above $0_2^{+}$	N/A	$2.93\left(5\right)$	N/A	$-26\left(1\right)$

Table 4. Table 4: Ground state energies for α 𝛼 \alpha -cluster nuclei above 12 C. Shown are the results at LO and NNLO. NNLO* denotes the force supplied with a four-nucleon interaction. The experimental values are also given. Units are MeV.

	16O	20Ne	24Mg	28Si
LO	$-147.3\left(5\right)$	$-199.7\left(9\right)$	$-255\left(2\right)$	$-330\left(3\right)$
NNLO	$-138.8\left(5\right)$	$-184.3\left(9\right)$	$-232\left(2\right)$	$-308\left(3\right)$
NNLO*	$-131.3\left(5\right)$	$-165.9\left(9\right)$	$-198\left(2\right)$	$-233\left(3\right)$
Exp.	$-127.62$	$-160.64$	$-198.26$	$-236.54$

Table 5. Table 5: Ground state energies of various nuclei for interactions A and B. Shown are results for LO and LO + C(oulomb). All energies are in units of MeV.

Nucleus	A (LO)	B (LO)	A (LO + C)	B (LO + C)
3H	$-7.82\left(5\right)$	$-7.78\left(12\right)$	$-7.82\left(5\right)$	$-7.78\left(12\right)$
3He	$-7.82\left(5\right)$	$-7.78\left(12\right)$	$-7.08\left(5\right)$	$-7.09\left(12\right)$
4He	$-29.36\left(4\right)$	$-29.19\left(6\right)$	$-28.62\left(4\right)$	$-28.45\left(6\right)$
8Be	$-58.61\left(14\right)$	$-59.73\left(6\right)$	$-56.51\left(14\right)$	$-57.29\left(7\right)$
12C	$-88.2\left(3\right)$	$-95.0\left(5\right)$	$-84.0\left(3\right)$	$-89.9\left(5\right)$
16O	$-117.5\left(6\right)$	$-135.4\left(7\right)$	$-110.5\left(6\right)$	$-126.0\left(7\right)$
20Ne	$-148\left(1\right)$	$-178\left(1\right)$	$-137\left(1\right)$	$-164\left(1\right)$

Equations

$$E_{J,K}=\frac{{\hslash }^{2}J(J+1)}{2{\mathcal{I}}_{\mathrm{Be}}}-\frac{{\hslash }^2{K}^2}{4{\mathcal{I}}_{\mathrm{Be}}},$$

$$E_J={\hslash }^2\frac{J(J+1)}{4{\mathcal{I}}_{\mathrm{Be}}}.$$

$${\Psi }_{\text{RGM}}=\mathcal{A}\left\{\varphi (C_1)\varphi (C_2)\chi (\mathbf{\xi })\right\},$$

$${\Phi }_{\text{BB}}({\mathbf{S}}_1,\dots ,{\mathbf{S}}_k)=n_0\mathcal{A}\left\{\psi (C_1;{\mathbf{S}}_1)\cdots \psi (C_k;{\mathbf{S}}_k)\right\},$$

$${\Psi }_{\text{GCM}}$$

$${\Phi }_{\mathrm{AMD}}(\mathbf{Z})=\frac{1}{\sqrt{A!}}\mathcal{A}\{{\phi }_1,{\phi }_2,...,{\phi }_A\},$$

$${\phi }_i$$

$${\varphi }_{{\mathbf{X}}_i}(\mathbf{r})$$

$${\chi }_i$$

$$i\hslash \sum _{j\rho }{C}_{i\sigma ,j\rho }\frac{d{X}_{j\rho }}{dt}=\frac{\partial }{\partial X_{i\sigma }^{\ast }}\frac{⟨{\Phi }_{\mathrm{AMD}}(\mathbf{Z})|H|{\Phi }_{\mathrm{AMD}}(\mathbf{Z})⟩}{⟨{\Phi }_{\mathrm{AMD}}(\mathbf{Z})|{\Phi }_{\mathrm{AMD}}(\mathbf{Z})⟩}$$

$$C_{i\sigma ,j\rho }\equiv \frac{{\partial }^2}{\partial X_{i\sigma }^{\ast }\partial X_{j\rho }}\mathrm{In}⟨{\Phi }_{\mathrm{AMD}}(\mathbf{Z})|{\Phi }_{\mathrm{AMD}}(\mathbf{Z})⟩,$$

$$H=\sum _i{t}_i-T_{\text{c.m.}}+\sum _{i<j}{v}_{ij}^{\text{nuclear}}+\sum _{i<j}{v}_{ij}^{\text{Coulomb}},$$

$$v_{ij}^{\text{nuclear}}=v_{ij}^{\text{central}}+v_{ij}^{ls},$$

$$\Gamma =2P_{L=0}(a){\gamma }^2(a),$$

$$P_{L=0}(a)=ka/(F_{L=0}^2(ka)+G_{L=0}^2(ka))$$

$$\Phi (3\alpha )=\varphi ({\alpha }_1)\varphi ({\alpha }_2)\varphi ({\alpha }_3).$$

$${\Psi }_{3\alpha }^{\mathrm{THSR}}(B)$$

$$=\mathcal{A}\{\exp [-\frac{2}{B^2}({\vec{X}}_1^2+{\vec{X}}_2^2+{\vec{X}}_3^2)]\Phi (3\alpha )\}$$

$$=\exp (-\frac{6{\vec{\xi }}_3^2}{B^2})\mathcal{A}\{\exp (-\frac{4{\vec{\xi }}_1^2}{3B^2}-\frac{{\vec{\xi }}_2^2}{B^2})\Phi (3\alpha )\},$$

$${\vec{\xi }}_1={\vec{X}}_1-\frac{1}{2}({\vec{X}}_2+{\vec{X}}_3),$$

$${\vec{\xi }}_2={\vec{X}}_2-{\vec{X}}_3,$$

$${\vec{\xi }}_3=\frac{1}{3}({\vec{X}}_1+{\vec{X}}_2+{\vec{X}}_3).$$

$$\Phi (3\alpha \mathrm{THSR})=C{\Psi }_{3\alpha }^{\mathrm{THSR}}(B)/\exp (-\frac{6{\xi }_3^2}{B^2}),$$

$$|⟨\Phi (3\alpha \mathrm{THSR})|\Phi (3\alpha GCM/RCM)⟩{|}^2\approx 100\%.$$

$${\varphi }_{0s}(\vec{r})=(\pi b^2{)}^{-3/4}\exp (-\frac{{\vec{r}}^2}{2b^2}).$$

$$\exp (-\frac{2{\vec{X}}^2}{B^2})\varphi (\alpha )\propto \det |(0s{)}^4|.$$

$$\underset{B\to b}{\lim }{n}_{3\alpha }^{\mathrm{THSR}}(B)$$

$$=|(0s{)}^4(0p{)}^8;[444]L=0⟩,$$

$$\sum _B⟨{\Psi }_{3\alpha }^{\mathrm{THSR}}(B^{\prime })|(H-E_k)|{\Psi }_{3\alpha }^{\mathrm{THSR}}(B)⟩f_k(B)=0.$$

$$|⟨\Phi {(}^{16}\mathrm{O}+\alpha \mathrm{THSR})|\Phi {(}^{16}\mathrm{O}+\alpha GCM/RGM)⟩{|}^2\approx 100\%,$$

$${\Phi }_L{(}^{16}\mathrm{O}+\alpha \mathrm{THSR})$$

$$=\underset{|\vec{D}|\to 0}{\lim }{P}_L\mathcal{A}\{e^{-\frac{8(\vec{r}-\vec{D}{)}^2}{5B^2}}\varphi {(}^{16}\mathrm{O})\varphi (\alpha )\},$$

$$\vec{r}={\vec{X}}_{\mathrm{CM}}{(}^{16}\mathrm{O})-{\vec{X}}_{\mathrm{CM}}(\alpha ),$$

$$\mathcal{A}\{{\chi }^F(\vec{r})\varphi {(}^{16}\mathrm{O})\varphi (\alpha )\}=0,$$

$$\mathcal{A}\left\{\exp \left[-\sum _{k=1}^3[\frac{2(X_k^2+Y_k^2)}{b^2}+\frac{2Z_k^2}{B^2}]\right]\Phi (3\alpha )\right\},$$

Full text

1. I Introduction

2. II Recent experimental results

3. II.1 Experimental observables

4. II.2 Status of studies of light nuclei

5. II.2.1 Alpha-conjugate systems; $N$-alpha structures and chains

6. II.2.2 Molecular structures in neutron-rich nuclei

7. II.2.3 Key measurements that constrain ab initio theory

8. III Microscopic cluster models and antisymmetrized molecular dynamics

9. III.1 Overview of microscopic cluster models

10. III.2 Antisymmetrized molecular dynamics method

11. III.3 Time-dependent antisymmetrized molecular dynamics method

12. III.4 Effective nuclear interactions

13. III.5 Description of cluster and mean-field aspects in AMD models

14. III.5.1 Cluster breaking effects on $3\alpha$ structures in 12C

15. III.5.2 Cluster and mean-field modes in monopole excitations in 12C

16. IV Tohsaki-Horiuchi-Schuck-Röpke wave function and container model

17. IV.1 Introduction

18. IV.2 Alpha-condensate-like character of the Hoyle state

19. IV.2.1 $S$-wave dominance of $\alpha$-cluster motion in the Hoyle state

20. IV.2.2 Equivalence of the 3$\alpha$ RGM/GCM wave function to a single 3$\alpha$ THSR wave function

21. IV.3 Localized vs. nonlocalized clustering

22. IV.3.1 Shell-model limit of THSR wave function

23. IV.3.2 Inversion doublet bands of 20Ne and THSR wave function

24. IV.3.3 Localized clustering from inter-cluster Pauli repulsion

25. IV.3.4 Equivalence of prolate and oblate THSR wave functions after angular momentum projection

26. IV.4 Container picture of cluster dynamics

27. IV.5 Extended THSR wave function and examples of its application

28. IV.5.1 Breathing-like excitation of the Hoyle state

29. IV.5.2 Container evolution in 16O

30. IV.5.3 Neutron-rich Be isotopes

31. V No-core shell model

32. V.1 Symmetry-adapted no-core shell model approaches

33. V.2 Continuum no-core shell model approaches

34. VI Continuum quantum Monte Carlo

35. VI.1 Variational Monte Carlo

36. VI.2 Diffusion or Green’s function Monte Carlo

37. VI.3 Monte Carlo shell model

38. VII Nuclear lattice effective field theory

39. VII.1 Chiral effective field theory on a lattice

40. VII.2 Lattice formalism

41. VII.3 Adiabatic projection method

42. VII.4 Results

43. VII.4.1 Alpha-cluster nuclei

44. VII.4.2 Ab initio alpha-alpha scattering

45. VII.4.3 Nuclear binding near a quantum phase transition

46. VII.4.4 Clustering in neutron-rich nuclei

47. VIII Summary and outlook

I Introduction

Nuclear clustering describes the emergence of molecular-like structures in nuclear physics. In molecules there is a rich phenomenology of different chemical bonds, complex rotational and vibrational excitations, and intricate structural geometries. Could there be a similar level of complexity in nuclear systems? The possibilities are certainly there with strong binding among the four nucleons in an $\alpha$-particle and the consequences of a nearly bound di-neutron channel. However, the underlying physics is made more challenging by the democracy of particles involved in nuclear binding. Instead of heavy ions surrounded by light electrons, the protons and neutrons have nearly equal masses, and the clustering structures emerge from a delicate balance among repulsive short-range forces and Pauli blocking effects, attractive medium-range nuclear forces, and long-range Coulomb repulsion among protons.

The study of nuclear clustering really began with Rutherford’s discovery by alpha radiation Rutherford (1899) and the development of quantum mechanics. Gamow Gamow (1928) and, independently, Gurney and Condon Gurney and Condon (1928) described the $\alpha$-particle as undergoing quantum-mechanical tunneling from inside the decaying nucleus. About a decade later, Wheeler Wheeler (1937a) developed the resonating group method to describe $\alpha$-clusters and other cluster groupings within nuclei, while allowing protons and neutrons to maintain their fermionic quantum statistics. Afterwards came the work of Hafstad and Teller, which described even-even $N=Z$ nuclei in terms of an $\alpha$-particle model with bonds connecting clusters Hafstad and Teller (1938). Following along the same lines, Dennsion proposed a model of the low-lying states 16O in terms of four $\alpha$-clusters at the vertices of a regular tetrahedron Dennison (1940, 1954). At a more microscopic level, Margenau used a Slater determinant wave function for $\alpha$-clusters to compute an effective $\alpha$-$\alpha$ interaction Margenau (1941).

Some years later, Morinaga suggested that non-spherical and even linear chains of $\alpha$-clusters could describe some states of $\alpha$-like nuclei Morinaga (1956). One of the candidates for such a description was the second $0^{+}$ state of 12C postulated by Hoyle Hoyle (1954) as responsible for enhancing the triple-$\alpha$ reaction in stars and experimentally observed soon after Cook et al. (1957). Concurrent with these theoretical developments, new experiments provided high-quality data on elastic $\alpha$-$\alpha$ scattering Heydenburg and Temmer (1956); Nilson et al. (1958); Afzal et al. (1969). This is in turn led to the development of an effective $\alpha$-$\alpha$ interaction Ali and Bodmer (1966).

At around the same time, Brink used Margneau’s Slater determinant wave function for the $\alpha$-cluster and the generator coordinate method to simplify calculations that were difficult in the more general formalism of the resonating group method Brink (1966a). The equivalence of the generator coordinate method and resonating group method was later clarified by Horiuchi Horiuchi (1970). On the topic of $\alpha$-decays, Clark and Wang computed the probability of $\alpha$-clusters to form near the surface of heavy nuclei Clark and Wang (1966). Meanwhile Ikeda, Takigawa, and Horiuchi noticed that $\alpha$-clustering appeared close to $\alpha$-decay thresholds, and these were denoted schematically with the so-called Ikeda diagrams Ikeda et al. (1968). Following these same concepts, the study of clustering has been extended to proton-rich and neutron-rich systems with nearby open thresholds. The corresponding states are weakly-bound systems of clusters and excess neutrons or protons.

There have been a number of reviews on clustering in nuclei Akaishi et al. (1986); von Oertzen et al. (2006); Freer (2007); Beck (2010, 2012); Horiuchi et al. (2012); Beck (2014); Funaki et al. (2015). The purpose of this review is to give a broad overview of the exciting developments in the past few years. Due to space limitations, it is not possible to cover all areas of research in depth. Nevertheless, we try to give a balanced view of the field as seen by a team of practitioners covering a range of methods and expertise. In the review of theoretical methods, we focus on microscopic clustering where clusters emerge from nucleonic degrees of freedom. As the field is dynamic and evolving, several key issues are not resolved at present, and there are disagreements among different methods. Furthermore, some of the most interesting results will likely come in the near future. This is to be expected in a growing field with important open questions and active research being pursued by many.

It is useful to briefly summarize the strengths and challenges of the various theoretical approaches. Most of the methods we discuss are variational calculations using some prescribed ansatz for the nuclear wave function. These include antisymmetrized molecular dynamics, fermionic molecular dynamics, the Tohsaki-Horiuchi-Schuck-Röpke wave function and container model, and microscopic cluster models using the resonating group or generator coordinate methods. These variational approaches often yield good agreement with experimental data as well as an intuitive picture of the underlying nuclear wave functions. The main challenges are to incorporate first principles nuclear forces and remove systematic errors associated with the choice of variational basis states.

Some variational methods have also been combined with Monte Carlo techniques. Variational Monte Carlo uses stochastic sampling to compute overlap integrals. It is also often used as a starting point for diffusion or Green’s function Monte Carlo simulations. These calculations have used first principles nuclear forces, and the systematic errors can be estimated by allowing unrestricted evolution of the quantum wave function. The major challenge for these calculations is that the computational effort increases exponentially with the number of particles. Another method called Monte Carlo shell model uses auxiliary-field Monte Carlo to select optimized variational basis states. As with other variational methods, the challenges are systematic errors due to the choice of basis states.

No-core shell model with continuum calculations start from first principles nuclear forces described by chiral effective field theory and have shown impressive agreement for the continuum properties of light nuclei. Similar to Green’s function Monte Carlo, the challenge for this method is the exponential scaling of effort when treating larger systems. The symmetry-adapted no-core shell model provides some very promising ideas for mobilizing computational resources in an efficient manner based on symmetries. Nevertheless difficulties remain in reaching larger systems accurately with first principles nuclear forces.

Nuclear lattice effective field theory uses chiral effective field theory and lattice Monte Carlo techniques to determining nuclear structure, scattering, and reactions. It has the advantage of relatively mild scaling with system size and a common platform in which to treat few-body and many-body systems at zero and nonzero temperature. However there is the added difficulty of working on a lattice with broken rotational symmetry, and the lattice spacing must be decreased to reduce systematic errors.

We also mention several other recent studies. In one recent work the states of 12C are considered in a Skyrme model Lau and Manton (2014). While the calculations produce good agreement with the measured experimental spectrum, the detailed connection to the underlying nuclear forces is not yet fully realized. While the inadequacies of the shell model in describing cluster structures have been known since the early years, the explanation of nuclear clustering as an emergent collective phenomenon near open thresholds is provided in Ref. Okolowicz et al. (2013) by treating the nucleus as an open quantum system coupling through nearby continuum states.

The review begins with an account of recent experimental results and future directions. We then discuss several theoretical approaches, including the resonating group and generator coordinate methods, antisymmetrized molecular dynamics, Tohsaki-Horiuchi-Schuck-Röpke wave function and container model, no-core shell model methods, continuum quantum Monte Carlo, and lattice effective field theory. We then conclude with a summary and outlook for the future.

II Recent experimental results

II.1 Experimental observables

The experimental study of the role of clustering in nuclei dates back to the earliest observations of $\alpha$-decay of heavy nuclei. In the early models of nuclei, it was assumed by many that the $\alpha$-particle may play an important role, e.g. the paper by Hafstad and Teller in 1938 Hafstad and Teller (1938) nicely describes the possible structures of nuclei such as 8Be, 12C and 16O as constructed from $\alpha$-particles. This early work also speculated on the existence of molecular structures in light nuclei, where neutrons, or even neutron holes, might be exchanged among $\alpha$-particle cores. These basic ideas remain the drivers for much of the present experimental program. The “modern” era of nuclear clustering was catalyzed by the ideas of Morinaga in 1956, who had suggested that the 7.65 MeV Hoyle state in 12C, which had recently been experimentally measured, might be a linear arrangement of 3$\alpha$-particles Morinaga (1956). The concept that linear chain structures might exist in nuclei has stuck with the subject until the present and remains to be resolved. Experiment has been substantially motivated by the desire to provide evidence for the types of structures envisaged by Morinaga and those calculated by Brink using the Bloch-Brink Alpha Cluster Model Brink and Boeker (1967); Brink (2008). For example, in the case of 12C, the $\alpha$-cluster model finds two structures. The first is an equilateral triangular arrangement which historically has been associated with the ground-state, and the second is a linear arrangement (or chain).

The ability of experiments to elucidate the cluster structures of light and heavy nuclei is determined by the range of experimental observables that may be extracted. From a simplistic starting point, the moment of inertia of a rotating nucleus gives an insight into the deformation which can be at least shown to be consistent with a cluster structure, even if not direct evidence. If 8Be is used as an example, then the ground-state rotational band has 0+, 2+ and 4+ states at 0, 3.06 and 11.35 MeV. The ratio of the 4+ to 2+ energy is 3.7, very close to that one would expect for a rotational nucleus, 3.33. The moment of inertia that one extracts from $E_{\rm rot}=J(J+1)\hbar^{2}/2\mathcal{I}$ is commensurate with that found in ab initio Green’s function Monte Carlo (GFMC) calculations, which strongly reveal the cluster structure Wiringa et al. (2000). We discuss calculations using Green’s function Monte Carlo in subsection VI.2. As a simple guide, the value of $\hbar^{2}/2\mathcal{I}$ associated with the 2+ state is 0.51 MeV, which even in a simple calculation yields a separation of two $\alpha$-particles by twice the $\alpha$-particle radius. The observation of a series of states which lie on a rotational sequence is not watertight evidence of either clustering or deformation. Here measurements of electromagnetic transition strengths provide tests of the overlaps of initial and final-state structures and the degree of collectivity. For the case of 8Be, a measurement of the $B(E2)$ transition strength from the 4+ to the 2+ state provides a consistent description with both the rotational picture and the GFMC calculations Datar et al. (2013).

However, and as noted above, this simplistic interpretation needs to be treated with care. Firstly, all of the states in 8Be are unbound and hence are embedded in the continuum and hence will have continuum contributions. Second, the widths of the states are significant (see section II.2.1), and correspondingly the lifetimes short, and thus an understanding of what collectivity means on such short timescales is unclear. Finally, many calculations use bound-state approximations and hence cannot be completely accurate. There is an interesting discussion of the meaning of rotational bands where the resonances are embedded in the continuum, with a focus on 8Be by Garrido et al. Garrido et al. (2013). The conclusion is that rotational bands embedded in the continuum may still be a meaningful concept, but that the continuum affects properties such as transition probabilities and hence here the continuum needs to be treated carefully. This is particularly important for the comparison with ab initio methods.

The width of a state reveals a significant amount of detail regarding the structure and the decay. The greater the overlap of initial structure with the decay partition then the shorter the lifetime and the greater the width. In the case of the 2+ excitation of 8Be, the width is tabulated as 1.5 MeV. The decay width is also affected by the barrier through which the decay must proceed, but if the Coulomb and centrifugal barriers are removed, then the reduced width may be compared with the Wigner limit. This is the value the reduced width should adopt if the $\alpha$-particles are fully preformed. For this particular state, it is found that the experimental width is very close to the Wigner limit, again indicating the existence of the cluster structure Overway et al. (1981); Cerny (1974). A further signature, not available to the decay of the example states in 8Be, is the measurement of the dominant decay channel. States with strong cluster-like properties should preferentially decay by cluster emission as opposed to proton or neutron decay, for example. In reactions, this structural similarity would be described in terms of a spectroscopic factor or an asymptotic normalization coefficient (ANC).

In the following sections we explore many of the recent developments in the experimental study of nuclear clustering. In many cases the recent work builds on significant historical work. There are many review articles which describe the development of the subject and we refer the reader to the following references: Freer and Merchant (1997); von Oertzen et al. (2006); Freer (2007); Beck (2010, 2012, 2014); Freer and Fynbo (2014).

II.2 Status of studies of light nuclei

II.2.1 Alpha-conjugate systems; $N$-alpha structures and chains

By far the most experimental attention has been devoted to the study of the cluster structure of $\alpha$-conjugate nuclei. Here the challenges have been to first provide a deeper insight into the nature of the cluster structures and ultimately to determine if the chain-states really exist in light nuclei or not. The eventual aim is to determine experimental characteristics such that they may be tested against ab initio or other microscopic calculations.

8*Be

As already described, one of the best examples of the comparison between ab initio theory and experiment, is the measurement of the gamma decay of the 4+ state in 8Be to the 2+ state Datar et al. (2013). This was a tour-de-force where a gamma decay branch of $\sim 10^{-7}$ was observed. The experiment involved the use of a helium gas-jet target, and the 4+ state was resonantly populated with a 4He beam. The emitted gamma-ray and the subsequent emission of the two $\alpha$-particles from the decay of the 2+ state were detected in a triple coincidence. A cross section of 165(54) nb was observed which translated to a $B(E2)$ of $25\pm 8$ $e^{2}$fm4. This is remarkably close to the value most recently calculated in the GFMC approach of $26.0\pm 0.6$ $e^{2}$fm4 Datar et al. (2013) (see reference [16] in this paper and Wiringa et al. (2000)). These latter calculations had famously found the ground state of 8Be to be highly clustered and predicted with significant precision the excitation energy spectrum Wiringa et al. (2000). Given that the $B(E2)$ is sensitive to both the overlap of the charge distribution and the collective behavior, such a result could be taken as evidence of both the cluster and collective behaviors. However, that being the case, this raises a rather interesting conundrum.111W. Nazarewicz, private communication at the 2015 Gordon Research Conference, New Hampshire, USA The widths of both the 2+ and 4+ states are large (1.5 and 3.5 MeV, respectively). From the uncertainty principle, these would correspond to lifetimes of the order of 10-22* seconds. This is the transit time of a nucleon with the Fermi energy to cross the nucleus. How is it possible for collective processes to develop and for rotational behavior to occur given the apparent mismatch in timescales, and what do rotations mean in such systems Fossez et al. (2016)? It is therefore possible that what is observed experimentally are simply patterns more generally linked to the underlying symmetry of a dumbbell-like structure. When it comes to precisely describing the properties of such states embedded in the continuum, the influence of the continuum on transition properties need to be fully accounted for Garrido et al. (2013), and it is vital that ab initio methods be developed for such unbound systems.

*12**C ground-state and rotational band

Similar questions are pertinent for the next $\alpha$-conjugate system, 12C. The effect of the continuum on the rotational bands in 12C is discussed in Ref. Garrido et al. (2016). Here the transitions between states are found to be consistent with the rotational picture. For 8Be all the states lie above the $\alpha$-decay threshold and hence, by the definition for the emergence of clustering developed by Ikeda, have the ingredients for the formation of clusters Ikeda et al. (1968). However, the ground state of 12C lies $\sim 7.3$ MeV below the decay threshold, and hence the cluster structure would be suppressed. However, as shown in Fig. 1, antisymmetrized molecular dynamics (AMD) calculations indicate that states above the decay threshold (Hoyle-band) clearly have a cluster structure, but even within the ground state this component may not be insignificant Kanada-En’yo (2007). This is supported by recent calculations using nuclear lattice simulations Epelbaum et al. (2012).

We discuss AMD methods in some detail in section III and lattice methods in section VII. The experimental $B(E2)$ for the transition from the first 2${}^{+}_{1}$ state at 4.4 MeV to the ground state has been determined to be $7.6\pm 0.4$ $e^{2}$fm4, which compares favourably with that calculated within the AMD framework of 8.5 $e^{2}$fm4 Kanada-En’yo (2007). The calculated value for the transition from the 4${}^{+}_{1}$ state is similar with a value of 16 $e^{2}$fm4. To date, there is no experimental measurement, but this would in principle confirm if these states are rotationally linked.

This raises the question whether it might be experimentally possible to observe the intrinsic cluster structure shown in the AMD calculations for the 12C ground state. One possibility might be via ultra-relativistic 12C+208Pb collisions where differences between the $\alpha$-clustered and uniform 12C nucleus may be visible in quantities such as the triangular flow, event-by-event fluctuations, or the correlations of the elliptic and triangular flows Broniowski and Ruiz Arriola (2014). A similar approach, e.g. examination of the properties of the fragmentation of 12C at high energy have been explored in Artemenkov et al. (2017). Another possibility is via $\alpha$-particle knockout from the ground state. The measurement of the 12C($p$,$p\alpha$) reaction using polarized beams found analyzing powers which were strongly indicative of $\alpha$-particles being preformed in the ground state Mabiala et al. (2009). This provides no information on any geometric arrangement or otherwise. Alternatively, it may be possible to exploit the dynamical symmetries associated with the triangular arrangement of the three $\alpha$-particles. The early work of Hafstad and Teller Hafstad and Teller (1938) paved the way for the more recent work of Bijker and Iachello Bijker and Iachello (2014). The dynamical symmetries of the 3$\alpha$-system correspond to a spinning top with a triangular point symmetry ($D_{3h}$). The rotational properties of these states are given by

[TABLE]

where $\mathcal{I}_{\rm Be}$ is the moment of inertia corresponding to two touching $\alpha$-particles, which can be determined from the 8Be ground-state rotational band Hafstad and Teller (1938). $K$ is the projection of the angular momentum onto the symmetry axis of the 3$\alpha$ system. One would expect that there should be a number of rotational bands with different values of $K$. For $K^{\pi}=0^{+}$, the rotations will be around an axis which lies in the plane of the three $\alpha$-particles, generating a series of states 0+, 2+, 4${}^{+},\ldots\,$. These correspond to the rotation of a 8Be nucleus, the rotation axis passing through the center of the third $\alpha$-particle. The next set of rotations corresponds to the rotation around an axis perpendicular to the plane of the triangle, with each $\alpha$-particle having one unit on angular momentum, thereby giving $L=3\times 1\hbar$; $K^{\pi}=3^{-}$. Rotations around this axis and those parallel to the plane combine to give a series of states 3-, 4-, 5${}^{-},\ldots$ .

The ground-state band described above, with 2+ and 4+ states at 4.4 and 14.1 MeV, would correspond to the $K^{\pi}=0^{+}$ rotational band. A candidate for the $K^{\pi}=3^{-}$ band head is the 9.6 MeV 3- state. The 13.3 MeV state presently tentatively labelled with $J^{\pi}=2^{-}$ in tabulations has recently been shown to almost certainly have $J^{\pi}=4^{-}$ Freer et al. (2007); Kirsebom et al. (2010). Moreover, the 3- state has been shown to have a reduced $\alpha$-width which indicates a cluster structure Kokalova et al. (2013). The observation of a candidate for a 5- state at 22.5 MeV Marin-Lambarri et al. (2014), would appear to complete the systematics and are also consistent with the AMD calculations Kanada-En’yo (2007). Widths of the negative-parity states have not been calculated with AMD. However, Uegaki’s $3\alpha$GCM calculations describe well the widths of the $3^{-}$ at 9.64 MeV and $4^{-}$ at 13.35 MeV Uegaki et al. (1979). The former and the latter are dominated by the ${}^{8}{\rm Be}(0^{+})+\alpha$ and ${}^{8}{\rm Be}(2^{+})+\alpha$ partial decay widths. The width of the $5^{-}$ may be dominated by the ${}^{8}{\rm Be}(2^{+})+\alpha$ partial decay width, but as yet there are no calculations to confirm this.

As with 8Be, the widths of the unbound states in 12C influence the possible collective interpretation. The 14.1 MeV, 4+, state has a width of 270 keV and the 9.6 MeV $3^{-}$ state has a width of 46 keV, both of which may not affect the collective timescale. However, the states associated with the Hoyle state (see below) have large widths of the order of MeV or greater and a simple rotational picture may be an over simplification.

The Hoyle state and collective excitations

The Hoyle state in 12C is one of the best known states in nuclei given its rather crucial role in the synthesis of carbon through the triple-$\alpha$ process. The recent review of this state Freer and Fynbo (2014) provides a comprehensive description of its role in synthesis and its experimental properties. Suffice to say, from an experimental perspective those properties have been well characterized. On the other hand, its structure is less well understood.

The fact that no-core shell model calculations fail to reproduce the energy of the Hoyle state Navrátil et al. (2000b, 2007), without resorting to a significantly expanded harmonic oscillator basis, indicates already that the structure lies beyond that described readily by the shell model. The first ab initio calculation of the Hoyle state was performed only a few years ago in Ref. Epelbaum et al. (2011). These latter calculations were able to explicitly capture the $\alpha$-clusterization that appears in this state. The AMD calculations, Fig. 1, indicate that the Hoyle state is an extended three $\alpha$-system and that the associated 2+ and 4+ excited states are not rigid, rotational, excitations and that a loose assembly of $\alpha$-particles, an $\alpha$-gas, may be a better description. A similar conclusion was reached in the fermionic molecular dynamics (FMD) calculations for the same states Neff and Feldmeier (2014). Here it was suggested that the 2+ and 4+ resonances might be considered as members of a rotational band built on the 8Be ground state with the third $\alpha$-particle orbiting around the 8Be nucleus with relative orbital angular momentum 2 or 4, respectively. The origin of nuclear clustering with relevance to the formation of the Hoyle state is also discussed by Okolowicz et al. Okolowicz et al. (2013).

It was observed by Barker and Treacy Barker and Treacy (1962) that in order to reproduce the width of the Hoyle state, one has to use an unusually large radius: with a radius of $1.6~{}{\rm fm}\,A^{1/3}$, a width of 9.3 eV corresponds to a dimensionless reduced width, $\theta^{2}={\gamma_{\lambda}}^{2}M_{red}R^{2}/\hbar^{2}$, as large as 1.5. Hence, the width of the Hoyle state is very large; this can only be understood if there is a large degree of $\alpha$-clustering. The presence of this cluster structure enhances the $\alpha$-capture cross section. But its existence within the Gamow window results in the overall capture cross section being boosted by a factor 108. Without the precise location of this state the abundance of carbon-12 would be greatly reduced, and thus it is intimately related to the existence of organic life. The rather deep question is if this is a happy accident, or if there is some reason why states with strongly-developed cluster structure should exist close to the corresponding decay thresholds Okolowicz et al. (2013); Freer and Fynbo (2014); Epelbaum et al. (2013a, b).

Beyond the fact that the Hoyle state has a 3$\alpha$-cluster structure, the nature of that structure remains to be resolved. The AMD calculations in Fig. 1 indicate a dominance of ${}^{8}{\rm Be}+\alpha$ configurations in a loose assembly such that the 2+ and 4+ excitations do not possess a clear rotational behavior. The fermionic molecular dynamics (FMD) calculations of the Hoyle state yield similar conclusions Chernykh et al. (2007). An extension of these ideas is that the state may be described by a gas/condensate of $\alpha$-particles Funaki et al. (2009). In principle, it may be possible to gain an insight into the structure through the decay properties of the state. In this instance there are two decay modes open; sequential and direct. In the latter the system does not decay through the 8Be ground-state. An upper limit for non-sequential $\alpha$-decay of 4% was first determined in 1994 Freer et al. (1994). Subsequently, a measurement of the ${}^{40}{\rm Ca}+{{}^{12}{\rm C}}$ reaction at 25 MeV/nucleon suggested that the branching ratio was in fact higher at $7.5\pm 4\%$. This was challenged by further measurements where upper limits as low as $5\times 10^{-3}$ (95% C.L.) Kirsebom et al. (2012); Manfredi et al. (2012) and 9(2)$\times$10*-3* has been put forward Rana et al. (2013). This was improved to be 0.2$\%$ Itoh et al. (2014). These measurements have now reached a sensitivity at which the phase space effects cease to be the dominant factor and it may be possible to probe the structure with limits of 0.047$\%$ Smith et al. (2017) and 0.043$\%$ Dell’Aquila et al. (2017), compared with the predicted phase space limit of 0.06$\%$ Smith et al. (2017).

A second approach is to probe the charge distribution through electron inelastic scattering Sick and Mccarthy (1970); Horikawa et al. (1971); Nakada et al. (1971); Strehl and Schucan (1968). In such measurements the transition form factor is determined, which probes the overlap of the ground state with the Hoyle state. To interpret such measurements a model is required which can describe both the ground and excited states. Both the condensate Funaki et al. (2006a) and FMD descriptions Chernykh et al. (2007) indicate that the Hoyle state is associated with a radius larger than that of the ground state by a factor of 1.35 to 1.60 (depending on the model used to analyze the data), which would correspond to an increase in volume by a factor of 2.5 to 4. Fig. 2 shows the calculated electron inelastic scattering distribution for the condensate model Funaki et al. (2006a).

A third approach to deduce the structure of the Hoyle state is to search for collective excitations, in particular the 2+ excitation. Inelastic scattering measurements Freer et al. (2009); Zimmerman et al. (2011); Itoh et al. (2011) were the first to provide evidence for such an excitation. A common analysis of the evidence for a 2+ resonance from the proton- and $\alpha$-particle scattering data is given in Ref. Freer et al. (2012a), and a discussion of the impact of these measurements is given in Ref. Fynbo and Freer (2011). The 2+ lineshape, which is found in the inelastic scattering measurements 12C($\alpha,\alpha^{\prime}$) and 12C($p,p^{\prime}$) Freer et al. (2012a), determined the properties to be $E_{x}$ = 9.75(0.15) MeV with a width of 750(150) keV. The existence of the 2+ resonance was confirmed by a measurement of the 12C($\gamma,3\alpha$) reaction at the HI$\gamma$S facility Zimmerman et al. (2013). The excitation function for these measurements are shown in Fig. 3 and gives resonant parameters of $E_{x}$ = 10.13(6) MeV and $\Gamma$ = 2.1(3) MeV Zimmerman (2013).

These measurements have now been extended to higher energies and continue the expected trend for the 2+ excitation.222M. Gai, private communication

If the state has a rotational behavior then there should also be a 4+ state close to 14 MeV. There exists tentative evidence for such a state at 13.3 MeV, with a width of 1.7 MeV  Freer et al. (2011); Jyväskylä (2013); Ogloblin et al. (2014). The existence of this latter state has yet to be definitively confirmed. It appears to decay strongly to the 8Be ground state as opposed to the 2+ excited state, which might provide an insight into the way the angular momentum is constructed, i.e., through the orbiting of the $\alpha$-particle around a 8Be($0^{+}$) core. Although much progress has been made in terms of understanding the structure of 12C, the measurements are typically challenging and often far from unambiguous. As such, the need for detailed spectroscopy continues. Here the approach of the Aarhus group Kirsebom et al. (2014) in measuring electromagnetic properties points the way for those future studies. The $p+{}^{11}$B capture reaction is used to resonantly populate states in 12C, and their decay after emitting an unobserved gamma decay is recorded through the subsequent charged particle channel.

The Hoyle state, though extended, is not consistent with a linear chain structure arrangement that would require the 2+ state to lie $\sim 1$ MeV lower than observed experimentally. Guidance from theory Kanada-En’yo (2007) suggests that the 10.3 MeV, 0${}^{+}_{3}$ state is the best possibility. This state has a width of 3 MeV and a 2+ state corresponding to a linear chain structure would be expected close to 11.5 MeV and would have a very large width. As yet, such a state remains to be observed. Recently, the possibility of two $0^{+}$ states around 10 MeV was experimentally reported by Itoh et al. Itoh et al. (2011) and supported by the extended Tohsaki-Horiuchi-Schuck-Röpke (THSR) calculation Funaki (2015); Funaki et al. (2015).

Figure 4 shows the compilation of theoretical spectra and transitions for $0^{+}$ and $2^{+}$ states compared with the experimental data. Although there are many non- and semi-microscopic $3\alpha$ calculations, we only show microscopic calculations with fully antisymmetrized wave functions and nucleon-nucleon interactions. It is difficult to directly compare the reproduction quality of microscopic calculations with non-microscopic calculations where interactions (or the Hamiltonian) are usually phenomenologically adjusted to fit the energy spectra of 12C. It should be also noted that we should not discuss ab initio calculations obtained from the realistic nuclear forces on the same footing with the calculations using phenomenological effective nuclear interactions. Details of the theoretical frameworks and interactions are explained in later sections. In the $3\alpha$RGM Kamimura (1981), extended THSR Funaki (2015); Funaki et al. (2015), $3\alpha$GCM Descouvemont and Baye (1987); Uegaki et al. (1979); Suhara and Kanada-En’yo (2015), and $3\alpha$+$p_{3/2}$ Suhara and Kanada-En’yo (2015) calculations, phenomenological effective nuclear interactions of the Volkov forces Volkov (1965) are used. The interaction parameters of the Volkov forces are tuned to reproduce $\alpha$-$\alpha$ scattering, though there are minor differences in the parameters among these calculations. The AMD results Kanada-En’yo (1998a, 2007) are obtained by using the MV1 forceAndo et al. (1980), which is a phenomenological effective nuclear interactions modified from the Volkov force to describe the saturation properties, whereas the FMD+$3\alpha$ results Chernykh et al. (2007) are obtained based on the realistic Argonne V18 potential with phenomenological tuning. For the NCSM Navrátil et al. (2007) and nuclear lattice effective field theory (NLEFT) Epelbaum et al. (2012) calculations, the results obtained with the realistic $NN$ and $NNN$ forces derived from the chiral effective theory are shown. In the no-core symplectic model (NCSpM) calculation Dreyfuss et al. (2013), a simplified effective Hamiltonian is used.

In general, the $3\alpha$ calculations describe well the energy spectra of cluster states above the $3\alpha$ threshold and the electron scattering form factors for the $0^{+}_{1}$ state and $0^{+}_{2}\to 0^{+}_{1}$ transition, but they are not sufficient in describing some properties of low-lying states such as the $0^{+}_{1}$-$2^{+}_{1}$ level spacing, $E2$ transition strength for $2^{+}_{1}\to 0^{+}_{1}$ and $0^{+}_{2}\to 2^{+}_{1}$. Hybrid calculations of the $3\alpha$+$p_{3/2}$ and FMD+$3\alpha$ models as well as the AMD can reasonably describe the ground band properties and excited spectra for cluster states. The NCSM calculation fails to describe the excited cluster states above threshold since those states are beyond the model space, whereas the NCSpM, which contains higher shell configurations for cluster excitations, and the NLEFT calculations describe cluster structures in excited states above the threshold. The ab initio calculations (NCSpM and NLEFT) tend to much underestimate the size of the ground state, and also give small values of the size and $E0$ matrix element for the Hoyle state. The $\alpha$-decay widths are calculated in the $3\alpha$RGMKamimura (1981) and $3\alpha$GCM(D) in Ref. Descouvemont and Baye (1987) by solving ${}^{8}{\rm Be}+\alpha$ scattering, and evaluated in the extended THSR Funaki (2015); Funaki et al. (2015), $3\alpha$GCM(U) in Ref. Uegaki et al. (1979), and AMD Kanada-En’yo (2007) within bound state approximations using reduced width amplitudes. The available data for the $\alpha$-decay widths are reproduced quantitatively or qualitatively by theoretical calculations.

Though much progress has been made in understanding the structure of 12C, it is apparent there is both need and scope for measurements to more precisely constrain the properties of the states presented in Fig. 1. In particular, this requires the measurement of electromagnetic transition rates where possible.

*States in 16O, dynamical symmetries and chains

One way of further testing our understanding of cluster correlations and the structure of 12C is through the extension of that understanding to 16O, which now is within the reach of ab initio approaches, see e.g., Ref. Epelbaum et al. (2014). Though much work has been done in both experiment and theory for this nucleus, here we provide some historical perspective first and then reflect on the most recent developments.

The work by Hafstad and Teller Hafstad and Teller (1938) indicates the collective properties of the 4$\alpha$ system should be described by the tetrahedral symmetry group, $T_{d}$. Here the characteristics are those of a spherical top, with equal moments of inertia and independent of rotation axis. If one assumes the separation between the $\alpha$-particles is that which is associated with the 8Be ground state, $\mathcal{I}_{\rm Be}$, then the rotational energies are given by

[TABLE]

The rotation of the tetrahedral structure corresponds to the equivalent rotation of two 8Be nuclei around their symmetry axis, and hence the ${4\mathcal{I}_{\rm Be}}$ in the denominator. The symmetry then dictates that all values of $J$ are permitted except $J=1$, 2 and 5; states with $J=0$, 4 and 8 have even parity and $J=3$, 7 and 11 have negative parity. A key feature of this structure would be degenerate 6+ and 6- states. A similar conclusion can be found in the recent work of Bijker and Iachello Bijker and Iachello (2014) as shown in Fig. 5, which was triggered by related work performed using lattice simulations in Ref. Epelbaum et al. (2014). The algebraic cluster model of Bijker and Iachello Bijker and Iachello (2014) generates energy eigenvalues and eigenvectors obtained by diagonalizing a finite-dimensional matrix, rather than by solving a set of coupled differential equations in coordinate space Bijker (2015). The model then describes the relative motion of the clusters.

The potential similarity between the structural properties of 12C and 16O and the underlying dynamical symmetries is compelling. However there are other models, e.g., the $\alpha$-cluster model Bauhoff et al. (1984) that provide a good description of the energy spectrum of 16O states. Hence, it is important to examine the decay properties. The experimentally observed states at 6.130 MeV, 3-; 10.356 MeV, 4+ ($\Gamma$=26 keV); and 21.052 MeV, 6+ ($\Gamma$=205 keV) have been linked in the work of Ref. Bijker and Iachello (2014) to the collective excitations of the tetrahedral structure. These same calculations predicted states at 6.132, 10.220 and 21.462 MeV and electromagnetic transition strengths $B(E3)$ and $B(E4)$ of 181 and 338 $e^{2}$fm2L compared with experimental values of 205(10) and 378(133) $e^{2}$fm2L. The comparison between experiment and theory is clearly compelling. The widths of the unbound 4+ and 6+ states are similar to those for the ground-state band in 12C. Nevertheless, caution is required when interpreting transition rates for such states.

The alternative theoretical approach provided by the $\alpha$-cluster model (ACM) calculations of Bauhoff, Shultheis and Shultheis Bauhoff et al. (1984) offer a different perspective. These calculations identify a number of cluster structures, including a tetrahedral arrangement of the four $\alpha$-particles in the ground-state. In addition, a planar arrangement of $\alpha$-particles is found for the first excited 0+ state, which can be associated with a ${}^{12}{\rm C}+\alpha$ structure. We note the similarity to the lattice results from Ref. Epelbaum et al. (2014). The main difference between the ACM and algebraic cluster model (ACM’) of Ref. Bijker and Iachello (2014) is evident in the assignment of the 10.356 MeV 4+ state to rotational bands. The ACM assigns it to the planar rotational structure, whereas the ACM’ links it to the tetrahedral ground-state.

The algebraic cluster model reproduces the $B(E4)$ for the 10.356 MeV to ground state transition, while the $\alpha$-cluster model would place this state in a different band. This is clearly contradictory. What is clear from measurements of the $\alpha$-decay branching ratios for decay to the 12C ground state and first excited states is that the states in the ACM planar band, above the $\alpha$-decay threshold, all have very similar decay properties. They predominantly decay to the ground state Tilley et al. (1993); Wheldon et al. (2011). Moreover there is also a negative parity band built on the 7.12 MeV, 1- state with very similar decay properties. This similar structure of this group of states conflicts with the tetrahedral interpretation and indicates a collective excitation built around a ${}^{12}{\rm C}+\alpha$ cluster structure where the total angular momentum of the state is generated by the orbital motion of the $\alpha$-particle around the 12C core. These two different perspectives on the nature of the low-lying states in 16O need to be resolved. Precision measurements of the complete electromagnetic decay patterns are likely to be the way forward. Measurements of the ANCs of the states close to the decay threshold in 16O have recently been reported Avila et al. (2015). ANCs provide a model-independent assessment of the cluster structure and as such are also a key ingredient in refining the understanding of 16O as a test case for nuclear theory.

One of the longest-standing questions related to 16O is the existence of the 4$\alpha$ chain-state. The 4$\alpha$ decay threshold is at 14.4 MeV and thus a chain state would exist close to or above this energy. The 15.1 MeV 0+ state is a potential candidate, though the state has been identified as the analogue of the Hoyle state Funaki et al. (2009). The proximity of this state to the decay threshold means that the 15.1 MeV state cannot decay strongly to the 4$\alpha$ final state. There are, however, a number of resonances that decay to the ${}^{8}{\rm Be}+{}^{8}{\rm Be}$ or ${}^{12}{\rm C(Hoyle)}+\alpha$ final states. The pioneering measurements of Chevallier et al. Chevallier et al. (1967) revealed both the excitation energy and dominant angular momenta of the 16O resonances that decayed to ${}^{8}{\rm Be}+{}^{8}{\rm Be}$ as populated in the 12C(4He,8Be)8Be reaction. Remarkably, the energy-spin systematics of selected narrow resonances fell onto a $J(J+1)$ trajectory with moment of inertia commensurate with a structure where the $\alpha$-particles are arranged in a linear fashion; an $\alpha$-particle chain. This work was published in 1967 and until the present has been held up as an example of extreme $\alpha$-clustering.

Confirmation of such an exotic structure is clearly vital. There are a number of possible approaches. One is to confirm the details of the excitation function, and the second is to search for higher spin members of the 4 $\alpha$-particle chain band. The band was only observed up to spin 6. Subsequent measurements by Brochard et al. Brochard et al. (1976) found no evidence for the 8+ member. The measurements of Chevalier et al. have been revisited Curtis et al. (2013), as displayed in Fig. 6. The highly detailed excitation functions for the 12C($\alpha$,8Be)8Be and 12C($\alpha$,12C[7.65 MeV])$\alpha$ reactions presented in Ref. Curtis et al. (2013) show that the original structure that was interpreted as resonances in the earlier work Chevallier et al. (1967) was more complex and that no evidence for an 8+ state could be identified. This most recent study contained over 400 measurements at different energies, with significant coverage of the angular distributions which should permit the components from resonances and transfer-like processes to be disentangled. A measurement of 13C($\alpha$,8Be+8Be)$n$ has recently been published which provides some insight as to what are resonant features in the ${}^{12}{\rm C}+\alpha$ excitation function Curtis et al. (2016). With the existence of excitation functions for ${}^{12}{\rm C}+\alpha$ leading to 4$\alpha$ unbound final-states as well as bound states Ames (1982), there is in principle sufficient data to perform a complete R-matrix analysis of the data to constrain states with enhanced 4$\alpha$ reduced decay widths. This is likely to be a key component in constraining the structure of 16O above the 4$\alpha$-decay threshold. Recent measurements of $\alpha$ inelastic scattering populating 0+ states in this region also indicates the spectrum of states may be more complicated than has been previously been concluded Li K.C.W. (2017).

II.2.2 Molecular structures in neutron-rich nuclei

The idea that light nuclei might have a molecular structure where typically the valence neutron is exchanged between $\alpha$-particle cores has been explored extensively Seya et al. (1981b); von Oertzen (1996a, 1997); Oertzen (1997); Itagaki and Okabe (2000). In essence, it is possible to form linear combinations of the neutron wave function around the $\alpha$-particle cores and obtain, for example, two-centered molecules with delocalized neutrons in $\pi$ and $\sigma$-orbitals von Oertzen et al. (2006); Freer (2007). Here the single-center orbitals both have $p$-type character. It is also possible to build more complex molecular structures with non-identical cores, for example, in nuclei such as 21,22Ne von Oertzen et al. (2006). This is illustrated in Fig. 7 Kimura and Horiuchi (2004).

The simplest example of this molecular behavior is found in the rotational bands of 9Be. The ground state band ($K^{\pi}=3/2^{-}$) is well-understood in terms of its $\pi$-type characteristics. The 1/2+ excited state at 1.68 MeV has a sequence of positive parity states ($3/2^{+}$, $5/2^{+}$, $7/2^{+}$…) which may be connected to $\sigma$-type molecular structures, as shown in Fig. 8.

These two bands have spin and parity values consistent with molecular structures. Furthermore, as indicated in the figure, the moments of inertia extracted from the gradients of the bands are similar to the moment of inertia found for 8Be, i.e., indicating the $\alpha$-$\alpha$ core structure is largely preserved. There are few ways to observe directly the ground state structure of the nucleus 9Be, however measurements of the decay correlations of 7Be and 6Li nuclei following the interaction with a 9Be target showed strong and unexpected alignment. This was concluded to be evidence for the $\pi$-type molecular structure of the 9Be ground-state Charity et al. (2015). The right hand panel of Fig. 8 shows the systematics of a negative parity band in 10Be. Here this would correspond to a mixed $\pi$-$\sigma$ configuration for the valence neutrons. Again the deformation is found to be consistent with the molecular picture, though it is apparent the moment of inertia has increased. Part of the origin of this effect is the proximity of this band to the $\alpha$-decay threshold, such that, as in the case of 12C, the cluster structure is enhanced.

The most pronounced example of molecular behavior studied to date is that associated with a series of states close to the $\alpha$- and neutron-decay thresholds in 10Be. The 6.179 MeV, 0${}^{+}_{2}$ state has a suppressed gamma decay, with a lifetime of the order of 1 ps. This isomeric behavior does not arise due to the lack of possible decay paths, but may be understood in terms of the small overlap of its structure and that of the more compact lower energy, 3.36 MeV, 2${}^{+}_{1}$ state. This already signals an unusual structure, in analogy to the Hoyle state in 12C. The excited state at 7.542 MeV, 2${}^{+}_{2}$, is believed to be a collective excitation of this state. This state lies above, but very close to, the $\alpha$-decay threshold (7.409 MeV), and thus its decay to this channel is strongly suppressed by the Coulomb and centrifugal barriers. Nevertheless, the $\alpha$-decay has been found to correspond to a very large reduced width Liendo et al. (2002), representative of the large degree of clusterization, although there is disagreement in the absolute value Milin et al. (2005). The 4+ member of the band has been identified to lie at 10.15 MeV Milin et al. (2005). An unambiguous measurement of the spin and parity of the state was found in the resonant scattering of ${}^{6}{\rm He}+{}^{4}{\rm He}$ Freer et al. (2006). This result has also been confirmed through a second resonant scattering measurement performed at Notre Dame Suzuki et al. (2013). If a collective model is applied, the moment of inertia associated with the rotational band would indicate that the state has a rather extreme deformation associated with the two valence neutrons occupying $\sigma$-like orbitals, with a density maximum between the two $\alpha$-particles which, via the Pauli exclusion principle, forces an increased separation of the two $\alpha$-particles. There are also indications from as yet unpublished measurements of a possible 6+ state at higher energy.333G. Rogachev, private communication Such a structure would be the analogue of the 3$\alpha$-chain state, but with two proton holes.

Valence neutrons in $\pi$- and $\sigma$-orbitals play an important role also in structure change of the ground states along Be isotopes. Because of the lowering mechanism of the $\sigma_{1/2}$-orbital in a well-clusterized $2\alpha$ system, the $N=8$ shell gap vanishes in neutron-rich Be. As a result, the ground states of 11Be and 12Be have $\sigma$-type molecular structures characterized by intruder configurations with large deformation (enhanced clustering), as supported by experimental observations such as Gamov-Teller and $E2$ transitions as well as the low-lying energy spectra Suzuki and Otsuka (1997); Iwasaki et al. (2000b, a); Navin et al. (2000); Shimoura et al. (2003); Pain et al. (2006); Imai et al. (2009); Meharchand et al. (2012). Contrary to the enhanced clustering in 11Be and 12Be, the ground state of 10Be has weak clustering because of the attractive role of the $\pi$-orbital neutrons. The systematic change of cluster structures along the Be isotope chain is reflected in the $N$ dependence of charge radii, which have been precisely determined by isotope shift measurements Nörtershäuser et al. (2009); Krieger et al. (2012). The charge radius is smallest at $N=6$ for 10Be indicating a possible new magic number at $N=6$ instead of $N=8$. This trend is described well by the weakening and enhancement of the cluster structures in AMD and FMD calculations Krieger et al. (2012); Kanada-En’yo (2015).

The experimental efforts to extend the systematics from dimers to trimers has seen a focus on trying to understand the systematics of three-centered molecules. Milin and von Oertzen had performed some pioneering work which established a set of candidate bands in 13C Milin and von Oertzen (2002) and 14C von Oertzen et al. (2004). In the case of 13C the experimental situation remains unclear as the rotational systematics proposed in Ref. Milin and von Oertzen (2002) are inconsistent with measurements of ${}^{9}{\rm Be}+\alpha$ resonant scattering Freer et al. (2012b).444Also measurements from the Naples group, yet unpublished. There are other studies of the 13C system, e.g. Soic et al. (2003); Rodrigues et al. (2010), but these are inconclusive in terms of the molecular structure of this nucleus. There have been a number of studies of ${}^{10}{\rm Be}+\alpha$ resonant scattering which populate resonances above the $\alpha$-decay threshold (12 MeV) Freer et al. (2014); Fritsch et al. (2016); Yamaguchi et al. (2017). This is higher energy than the 0+ band head identified by von Oertzen and co-workers von Oertzen et al. (2004), 9.75 MeV, and as such resonances may be associated with higher nodal cluster structures. Nevertheless, these latest measurements provide some tentative evidence for linear chain structures in 14C, as the level spacing and relative energies to the ${}^{10}{\rm Be}+\alpha$ threshold of the observed states agree well with the AMD prediction Suhara and Kanada-En’yo (2010a). However, it is clear that a definitive conclusion has yet to be reached here.

II.2.3 Key measurements that constrain ab initio theory

Clustering reveals much about the nature of the force through which the constituent components of the nucleus interact and the symmetries that result. This provides a crucial connection with ab initio theory. The nuclear strong interaction is clearly complex and this is revealed in the details of the unbound and bound light nuclei. The $\alpha$-particle is one of the most highly bound light nuclei with a very high-lying, $\sim 20$ MeV, first excited state. And here the array of correlations include not only $n-n$ and $p-p$ but also $n-p$ to maximize the binding energy. The tendency of other nuclei to optimize their own binding by generating spatial and momentum correlations induces the formation of clusters. This is responsible for clustering in $\alpha$-conjugate nuclei, Borromean and molecular systems, alike.

Nuclei that display extreme or exotic behavior where the effects of the correlations are maximal are an excellent test of theory. Good examples of this are the ground and excited states of 8Be and the Hoyle state in 12C, which have both been described above. To be useful in constraining theory and providing discrimination between approaches, high precision measurements are often required. One of the best examples of this is the study of the $T=1$ analogue states in 10Be, 10C and 10B∗.

Precision measurements of the lifetime of first 2+ state in 10C using the Doppler Shift Attenuation Method deduced a lifetime of $\tau$ = 219 $\pm$(7)stat $\pm$(10)sys fs, corresponding to a $B(E2)$ of 8.8(3) $e^{2}$fm4 McCutchan et al. (2012a). Similar measurements of the same transition in 10Be found a $B(E2)$ of 9.2(3) $e^{2}$fm4 McCutchan et al. (2009). The ground and 2+ states of these nuclei are believed to possess a molecular structure where two valence particles (2 neutrons or 2 protons) orbit the 2$\alpha$-particle cores. These measurements were compared with both the Green’s function Monte Carlo and no-core shell model (NCSM) calculations. The reproduction of the experimental results, especially the GFMC calculations, was not satisfactory and showed significant sensitivity to the details of the 2– and 3–body forces employed. A subsequent measurement of the $B(E2)$ for the transition from the $J=2,$ $T=1$ state at 5.164 MeV to the $J=0,$ $T=1$ state at 1.740 MeV in 10B found a value of 6.1(22) e2fm4  McCutchan et al. (2012b). This is much lower than the simple average of the 10Be and 10C measurement, which may not simply be understood and stands as an important test of ab initio theory.

This set of measurements is a fine example of the need for precision experimental data to properly understand the nature of the strong interactions in light nuclei and the ability of first principles approaches to reproduce experimental properties. This must be a significant area of effort for experiment and theory over the next decade.

III Microscopic cluster models and antisymmetrized molecular dynamics

Various cluster phenomena in stable nuclei have been theoretically investigated using microscopic cluster models such as resonating group methods (RGM) and generator coordinate methods (GCM). In the progress of understanding the physics of unstable nuclei, cluster models have been extended to deal with cluster structures with valence neutrons. Moreover, more flexible methods such as antisymmetrized molecular dynamics (AMD) Ono et al. (1992a, b); Ono and Horiuchi (2004); Kanada-Enyo et al. (1995); Kanada-En’yo and Horiuchi (2001); Kanada-En’yo et al. (2012) and fermionic molecular dynamces (FMD) Feldmeier (1990); Feldmeier et al. (1995); Feldmeier and Schnack (2000); Neff and Feldmeier (2003, 2004) have been developed which do not rely on the a priori assumption of existence of clusters. The AMD and FMD wave functions are based not on cluster degrees of freedom but on nucleon degrees of freedom. In this sense, these models are not cluster models. Nevertheless, since they can express various kinds of cluster structures as well as shell-model features, they are powerful approaches for the study of cluster features in general nuclei.

In this section we give a brief overview of cluster models, the AMD method, and their extensions. Then we discuss some topics focusing on how these models describe the coexistence of cluster and mean-field aspects. It should be commented that the AMD and FMD methods were originally developed in the time-dependent form for nuclear reaction studies. However, we here describe the models for structure studies. For details of the AMD method for nuclear structure and reaction studies, see Refs. Kanada-En’yo and Horiuchi (2001); Kanada-En’yo et al. (2012); Kimura et al. (2016) and references therein.

III.1 Overview of microscopic cluster models

Since the 1960’s, microscopic cluster models have been applied to investigate cluster phenomena such as nuclear scattering and cluster structures. In the early history, scattering between light nuclei such as $\alpha+\alpha$ scattering has been intensively studied with the RGM Wheeler (1937b, a); Wildermuth and Kanellopoulos (1958); Tang et al. (1978).

Despite of the success of the RGM in microscopic description of relative motion between composite particles, practical application of the RGM is limited to the light mass region because of the computational costs of antisymmetrization in the treatment of the norm and Hamiltonian kernels. We discuss some recent developments in subsection V.2 in ab initio no-core shell model calculations.

Since 1970’s, owing to application of the GCM Hill and Wheeler (1953); Griffin and Wheeler (1957) using the Bloch-Brink cluster wave function Brink (1966b), further progress of microscopic studies of cluster phenomena has been made for heavy mass and many-cluster systems as well as unstable nuclei Fujiwara et al. (1980). The RGM and GCM are microscopic cluster models, in which antisymmetrization of all nucleons composing clusters are fully taken into account, and the Hamiltonian is composed of nucleon kinetic energies and nucleon-nucleon interactions based on nucleon degrees of freedom. Clusters are usually written in terms of simple shell-model configurations with/without excitation, and the inter-cluster motion is solved within the model wave functions.

The model wave function of the RGM for a single-channel case of two clusters $C_{1}$ and $C_{2}$ is given as

[TABLE]

where ${\cal A}$ is the nucleon antisymmetrizer, $\phi(C_{i})$ is the internal wave function of the $C_{i}$-cluster, and $\bm{\xi}$ is the relative coordinate between the centers of mass of the clusters. The inter-cluster wave function $\chi(\bm{\xi})$ is determined by solving the RGM equation derived from the projection of the Schrödinger equation onto the RGM model space. Distortion of clusters and multi-channel systems can be taken into account in the RGM by extending the single-channel to coupled-channel problems.

To describe the inter-cluster motion with the GCM approach, Brink adopted the following multi-center cluster wave function (called the Bloch-Brink cluster wave function) as a basis wave function Brink (1966b),

[TABLE]

where the $i$th cluster ($C_{i}$) is localized around $\bm{S}_{i}$, and $n_{0}$ is a normalization constant. The wave function $\psi(C_{i};\bm{S}_{i})$ for the $i$th cluster is written in terms of the harmonic oscillator shell-model wave function located at $\bm{S}_{i}$. When the clusters are far from each other and feel weak antisymmetrization effects between clusters, the parameter $\bm{S}_{i}$ indicates the mean center position of the cluster, and hence, the spatial configuration of the parameters $\{\bm{S}_{1},\ldots,\bm{S}_{k}\}$ specifies the geometry of cluster structures. It means that the single Bloch-Brink cluster wave function expresses a cluster wave function, in which centers of clusters are localized around certain positions. In the small distance ($|\bm{S}_{i}|$) case that clusters largely overlap with each other, the Bloch-Brink cluster wave function becomes a specific shell-model wave function of the SU(3) shell model because of antisymmetrization of nucleons among clusters.

For the detailed description of inter-cluster motion, the superposition of the Bloch-Brink wave functions is considered by adopting the cluster center parameters $\{\bm{S}_{1},\ldots,\bm{S}_{k}\}$ as generator coordinates in the GCM approach,

[TABLE]

where $P^{J\pi}_{MK}$ is the total-angular momentum and parity projection operator, and coefficients $f(\bm{S}_{1},\ldots,\bm{S}_{k})$ are determined by solving the Hill-Wheeler equation Hill and Wheeler (1953). In principle, the GCM with full model space of the basis Bloch-Brink wave functions is equivalent to the RGM Horiuchi (1970). With the GCM approach it became possible to practically calculate heavy mass systems and also many-cluster systems microscopically.

For scattering problems, the RGM can be applied rather straightforwardly because the inter-cluster wave function is explicitly treated. On the other hand, in the application of the GCM to scattering problems, it is necessary to connect the basis wave functions in the internal region with continuum states in the asymptotic region at a chosen channel radius Kamimura (1977); Descouvemont and Baye (2010).

Based on the GCM, Bay and Descouvemont studied various low-energy reactions of astrophysical interest Descouvemont and Baye (2010). Following the progress in the physics of unstable nuclei, the microscopic cluster approaches have been extended and applied to study cluster structures of unstable nuclei. One of the main interests in the study of unstable nuclei are properties of valence neutrons surrounding one core or two clusters in neutron-rich nuclei. Microscopic three-body calculations for two valence neutrons around a core nucleus have been achieved by many groups to investigate the neutron halo and two-neutron correlation in drip-line nuclei such as 6He, 11Li, and 14Be Varga et al. (1994); Arai et al. (1999); Descouvemont (1995, 1997). Baye and Descouvemont have studied cluster features of the Be isotopes using a GCM approach with the Bloch-Brink wave functions of two $\alpha$-clusters and valence neutrons Descouvemont (2002). They have also applied the coupled-channel GCM of ${}^{6}{\rm He}+{}^{6}{\rm He}$ and ${}^{8}{\rm He}+\alpha$ channels to study cluster structures of 12Be Descouvemont and Baye (2001); Dufour et al. (2010).

Ito et al. have applied a more generalized approach of the coupled-channel GCM to 10Be and 12Be Ito et al. (2004, 2008); Ito and Ikeda (2014). The method is successful in describing gradual changes of valence neutron configurations from strong-coupling clustering with a molecular orbital structure to weak-coupling clustering in the asymptotic region with the increase of the $\alpha$-$\alpha$ distance. Varga and his collaborators have performed accurate calculations for many cluster systems in unstable $p$-shell nuclei with the stochastic variational method (SVM) Varga et al. (1994, 1995); Varga and Suzuki (1995); Arai et al. (2001). The SVM is a microscopic cluster model with the RGM-type cluster wave function written as a linear combination of stochastically chosen basis wave functions. Because of the stochastic procedure in choosing the basis wave functions, it is a powerful approach to treat many cluster systems. For instance, it has been applied to accurately solve four-cluster problems in the study of unstable $p$-shell nuclei such as the $2\alpha+2n$ system of 10Be Varga et al. (1995); Ogawa et al. (2000); Arai et al. (2001); Arai (2004).

To understand cluster structures of low-lying states of neutron-rich Be isotopes, molecular orbitals (MO) for surrounding neutrons around the $2\alpha$ core were proposed von Oertzen et al. (2006); Seya et al. (1981a); von Oertzen (1996b); Itagaki and Okabe (2000); Ito and Ikeda (2014). Microscopic MO models Okabe et al. (1977, 1978) have been developed and applied to 10Be by Itagaki et al. Itagaki and Okabe (2000). The model is based on the GCM for $2\alpha+2n$ using a truncated model space. Neutron configurations are restricted to the MOs, which are covalent bond orbitals written as linear combinations of $p$-orbits around each $\alpha$-cluster, whereas the $\alpha$-$\alpha$ distance is treated as the generator coordinate. The molecular orbital models have been also applied to neutron-rich C isotopes with the $3\alpha$ core and valence neutrons Itagaki et al. (2001).

The relation of the cluster wave functions with shell model ones was described based on the harmonic oscillator basis expansion and discussed from the SU(3) group symmetry Wildermuth and Kanellopoulos (1958); Bayman and Bohr (1958); Elliott (1958a, b). The concept has been followed by symmetry-adapted models such as symplectic (no-core) shell models Rowe and Rosensteel (1980); Rowe and Wood (2010); Draayer and Rosensteel (1983); Dytrych et al. (2007) and algebraic cluster models Bijker and Iachello (2002); Cseh (1992, 2014).

To describe competition between the cluster and $jj$-coupling shell model states, Itagaki et al. extended the Bloch-Brink Alpha cluster model wave function by adding spin-dependent imaginary parts to the cluster center parameters. This is essential for spin-orbit interactions in the $jj$-coupling shell-model Itagaki et al. (2005); Suhara et al. (2013). The model is called antisymmetrized quasi-cluster model (AQCM) and can efficiently describe the smooth transition from the $\alpha$-cluster wave function to the $jj$-coupling shell model wave function in 12C with the cluster breaking parameter $\Lambda$ from $\Lambda=0$ to $\Lambda=1$ as shown in Fig. 10.

III.2 Antisymmetrized molecular dynamics method

The AMD method is an approach which treats nucleon degrees of freedom independently without assuming any clusters. Nevertheless, the AMD can describe various cluster structures because the Bloch-Brink cluster wave functions for any cluster channels are contained in the AMD model space.

An AMD wave function is given by a Slater determinant of single-nucleon Gaussian wave functions,

[TABLE]

where the $i$th single-particle wave function $\varphi_{i}$ is written by a product of spatial, spin, and isospin wave functions as

[TABLE]

$\phi_{{\bm{X}}_{i}}$ and $\chi_{i}$ are the spatial and spin functions, respectively, and $\tau_{i}$ is the isospin function fixed to be up (proton) or down (neutron). The width parameter $\nu$ is fixed to be an optimized value for each nucleus.

In the AMD wave function, the $i$th single-particle wave function is expressed by the Gaussian wave packet localized around the position $\bm{X}_{i}$. The Gaussian center positions $\bm{X}_{i}$ and the intrinsic-spin orientations $\xi_{i}$ for all nucleons are treated independently as variational parameters which are determined by energy variation.

It should be noted that the AMD wave function can be interpreted as an extended version of the Bloch-Brink wave function in a sense that all clusters are resolved completely to single nucleons. The AMD model covers the Bloch-Brink cluster model space as well as the AQCM. Indeed, by choosing a specific configuration of the Gaussian center positions $\{\bm{X}_{i}\}$, the AMD wave function can express the Bloch-Brink and AQCM wave functions. If a system favors a specific cluster structure, that structure is automatically obtained in the AMD model space after energy variation. The AMD wave function can also describe shell-model configurations because of the antisymmetrization between nucleons. It means that formation and dissolution of clusters are taken into account owing to the flexibility of the model wave function. This is a great advantage superior to cluster models in description of both cluster and mean-field features in the ground and excited states of exotic nuclei. As an extension of the AMD wave function, triaxially-deformed Gaussian wave packets were proposed by Kimura et al. instead of the spherical Gaussian wave packets Kimura (2004). The deformed basis AMD is efficient to describe the coexistence of deformed mean-field states and cluster states in $sd$- and $pf$-shell nuclei. See Refs. Kanada-En’yo et al. (2012); Kimura et al. (2016) and references therein. It should be commented that deformed Gaussian wave packets have been proposed in Ref. Bauhoff et al. (1985) for the time-dependent cluster model (TDCM) Drozdz et al. (1982); Caurier et al. (1982).

For structure studies in the AMD framework, the energy variation is performed after parity projection. For the angular-momentum projection, the variation before the projection (VBP) is performed in the simple AMD Kanada-Enyo et al. (1995), whereas the variation is performed after the projection (VAP) in the AMD+VAP method Kanada-En’yo (1998a).

For the description of excited states, the AMD wave functions obtained by the energy variation are superposed. For instance, mixing of different basis AMD wave functions (multiconfiguration mixing) is usually done in the AMD+VAP method. In the AMD+GCM method, many AMD wave functions are superposed by means of the GCM with constraint parameters as generator coordinates. In the $\beta$- and $\beta\gamma$-constraint AMD Kimura (2004); Suhara and Kanada-En’yo (2010b), the energy variation is done under the constraints on the deformation parameters $\beta$ and $(\beta,\gamma)$, respectively. In the $d$-constraint AMD Taniguchi et al. (2004), the constraint for the distance between two (or three) centers of subgroups is adopted. After the energy variation with the constraints, the obtained AMD wave functions are superposed with the GCM treatment. Namely, coefficients of wave functions are determined by solving the Hill-Wheeler equation, i.e., by diagonalizing the norm and Hamiltonian matrices with respect to the adopted basis AMD wave functions. In the AMD+GCM, large amplitude dynamics along the generator coordinates are microscopically taken into account.

Although the AMD+GCM is useful for large amplitude collective motion, it is not efficient to describe single-particle excitations on a mean-field state because the lowest state is chosen in the energy variation procedure. To overcome this problem, the shifted basis AMD (sAMD) Kanada-En’yo (2016b, a); Chiba et al. (2016) has been constructed to describe small amplitude modes on top of the ground state. A small shift of the Gaussian center position of each single-particle wave function is prepared on the ground state AMD wave function and all the shifted bases wave functions are superposed to describe linear combinations of one-particle and one-hole (1p-1h) excitations. The method combined with the cluster GCM was applied to monopole and dipole excitations in light nuclei and described coexistence of low-energy cluster modes and high-energy giant resonances.

Since the basis AMD wave function is written as a Slater determinant of single-particle wave functions, the simplest case of the single AMD wave function without projections can be regarded as a Hartree-Fock approach simplified in the restricted model space. However, because of the linear superpositions as well as the parity and angular momentum projections of AMD wave functions, higher correlations beyond mean-field approaches are taken into account even in the ground state in the AMD framework. As mentioned previously, the AMD model contains mean-field states as well as various cluster states in its model space and therefore it is able to describe the coexistence of mean-field and cluster aspects in the ground and excited states of nuclear systems.

III.3 Time-dependent antisymmetrized molecular dynamics method

The AMD wave function was originally used for nuclear reaction studies in a time-dependent framework Ono et al. (1992a, b); Ono and Horiuchi (2004). In the time-dependent AMD, the spin functions $\chi_{i}$ are usually fixed to be $\chi_{\uparrow}$ or $\chi_{\downarrow}$, and time evolution of a system is described by the time-dependent Gaussian center positions $\bm{X}_{i}$ determined by the time-dependent variational principle as

[TABLE]

where $\sigma,\rho=x,y,z$. The time-dependent AMD can be regarded as an extended version of the TDCM Drozdz et al. (1982); Caurier et al. (1982); Bauhoff et al. (1985) in the sense that all clusters are resolved completely into single nucleons.

In applications of the AMD and extended versions to heavy-ion collisions, the stochastic two-nucleon collision term is added to the equation of motion. The model successfully described multifragmentations at intermediate energy. Feldmeier has proposed a wave function quite similar to the AMD wave function for nuclear reactions and structure studies Feldmeier (1990); Feldmeier et al. (1995); Feldmeier and Schnack (2000) and named it fermionic molecular dynamics (FMD). The model wave function of the FMD is also given by a Slater determinant of single-nucleon Gaussian wave packets. The major difference in the wave function between the FMD and AMD is that the width parameter $\nu$ can be independently chosen for each nucleon as $\nu_{i}$ and treated as variational parameters in the FMD, whereas it is common for all nucleons in the AMD. Instead, the diffusion and the deformation of wave packets are stochastically incorporated in an extended version (AMD+Vlasov) for reaction studies Ono and Horiuchi (1996).

In structure studies the flexible treatment of the width parameters in the FMD is efficient, for example, for the neutron-halo structure of neutron-rich nuclei. The variation of width parameters in the time-dependent FMD is also effective in description of the giant monopole resonance Furuta et al. (2010), whereas the giant resonances are described with superposition of shifted single-particle Gaussian wave packets in the sAMD framework.

III.4 Effective nuclear interactions

In the cluster model and the AMD calculations, phenomenological effective nuclear interactions composed of the two-body central and spin-orbit ($ls$) forces are usually used. The Hamiltonian consists of the kinetic energies, the effective nuclear interactions, and Coulomb interaction as

[TABLE]

where the center-of-mass kinetic energy $T_{\textrm{c.m.}}$ is subtracted. Note that the center-of-mass motion can be easily separated from the wave functions in the cluster and AMD models when a common width parameter is used. For the central forces of the effective nuclear interactions, Gaussian finite-range interactions with and without the zero-range density-dependent term (or the zero-range three-body term) are adopted in most cases. The central forces are supplemented with the finite-range or zero-range $ls$ forces.

For light mass nuclei, density-independent interactions such as the Minnesota Thompson et al. (1977) and Volkov Volkov (1965) interactions are often used. The Minnesota force is originally adjusted to fit the $S$-wave nucleon-nucleon scattering as well as scattering between light nuclei. For the Volkov force, the standard parameter set reproduces $\alpha$-$\alpha$ scattering. The Volkov force can be adjusted to fit the $S$-wave nucleon-nucleon scattering lengths by tuning parameters for the Bartlett and Heisenberg terms. In general, these density-independent effective interactions cannot describe the saturation properties of nuclear matter and have an overbinding problem in heavy mass nuclei. Therefore, interaction parameters are sometimes readjusted to reproduce energies for the mass number region of interest, though the original parameter sets reproduce the properties of nucleon-nucleon and $\alpha$-$\alpha$ scattering.

To overcome the overbinding problem, the central forces with the zero-range density dependent or zero-range three-body term are used for heavy mass nuclei. Examples are the Gogny forces Berger et al. (1991) and the Modified Volkov (MV) forces Ando et al. (1980). The central force of the Gogny forces consists of finite-range two-body terms and a zero-range density dependent term, whereas that of the MV forces contains a zero-range three-body term instead of a zero-range density-dependent term. These interactions systematically reproduce the binding energies over a wide mass number region. However, they cannot quantitatively reproduce the scattering and structure properties of very light systems such as the nucleon-nucleon and $\alpha$-$\alpha$ scattering as well as the size of the $\alpha$-particle.

There are many cluster model calculations for light nuclei using the Minnesota and Volkov forces. In the AMD calculations for $p$-shell, $sd$-shell, and $pf$-shell nuclei, the Volkov, Modified Volkov No.1 (MV1), and Gogny forces are used. As already mentioned, these interactions used in the cluster model and AMD calculations are effective nuclear interactions that are phenomenologically adjusted to properties of nuclear structures and/or scattering.

In the FMD calculations, effective nuclear interactions derived from the realistic nuclear interactions are usually used with the unitary correlation operator method (UCOM), in which the short-range and tensor correlations are taken into account in the interaction operator of the Hamiltonian Feldmeier et al. (1998); Neff and Feldmeier (2003); Roth et al. (2010). Therefore the FMD+UCOM calculation is a first principles method starting from realistic nuclear interactions.

III.5 Description of cluster and mean-field aspects in AMD models

III.5.1 Cluster breaking effects on $3\alpha$ structures in 12C

Despite the success of $3\alpha$-cluster models for many excited states 12C, microscopic $3\alpha$-cluster models are not sufficient to describe the large level spacing between the $0^{+}_{1}$ and $2^{+}_{1}$ states because $\alpha$-cluster breaking is not taken into account in the models. Moreover, it is difficult to confirm the $3\alpha$ cluster formation in the 12-nucleon dynamics because clusters are a priori assumed in the models. These problems have been overcome by the AMD and FMD models. In the AMD and FMD calculations for 12C Kanada-En’yo (1998a, 2007); Neff and Feldmeier (2004); Chernykh et al. (2007), $3\alpha$-cluster structures are formed in the calculated results without assuming the existence of $\alpha$-clusters.

As mentioned previously, the model spaces of the AMD and FMD contain the Bloch-Brink cluster wave functions and also cluster breaking configurations. In the 12C($0^{+}_{1}$), the cluster breaking component, i.e., the $p_{3/2}$ closed-shell component is significantly mixed in the dominant $3\alpha$ cluster structure as seen in the compact intrinsic density distribution in Fig. 1. Due to the mixing of the cluster breaking component, the band-head 12C($0^{+}_{1}$) gains extra energy of the spin-orbit attraction resulting in stretching of the $0^{+}$-$2^{+}$ level spacing consistently with the experimental energy spectra. It is also the case in the FMD calculation (see Fig. 4). The significant mixing of the cluster breaking component in 12C($0^{+}_{1}$) is clearly indicated in the AQCM calculation in Fig. 10 by the finite value of the cluster breaking parameter $\Lambda$ at the energy minimum.

The cluster breaking component does not give drastic effects to excited $3\alpha$ cluster states. However, excited $0^{+}$ structures are more or less affected by the cluster breaking component mixed in the 12C($0^{+}_{1}$) through the orthogonality, and therefore, quantitative differences can be seen between model calculations with and without the cluster breaking. For example, in the calculated energy spectra shown in Fig. 4, the AMD and FMD calculations show a trend of the larger $0^{+}_{2}$-$2^{+}_{2}$ level spacing than that obtained by $3\alpha$ calculations without the cluster breaking, because the cluster breaking in the 12C($0^{+}_{1}$) induces the global energy gain of excited $0^{+}$ states. Moreover, the AMD calculation shows the larger $E2$ strength for $2^{+}_{2}\to 0^{+}_{3}$ than that for $2^{+}_{2}\to 0^{+}_{2}$, differently from the $3\alpha$ calculations that give dominant $E2$ strength for $2^{+}_{2}\to 0^{+}_{2}$. Suhara and Kanada-En’yo investigated the cluster breaking effects on $3\alpha$ cluster structures in 12C by explicitly adding the $p_{3/2}$ closed-shell configuration (cluster breaking component) into the $3\alpha$ model space Suhara and Kanada-En’yo (2015). Comparison of the results with and without the $p_{3/2}$ configuration is shown in the right two columns in the upper panel of Fig. 4. The figure shows increasing of the $2^{+}_{1}$ and $2^{+}_{2}$ energies relative to the $0^{+}_{1}$, $0^{+}_{2}$, and $0^{+}_{3}$ ones, and also the inversion of the dominant $E2$ strengths between $2^{+}_{2}\to 0^{+}_{2}$ and $2^{+}_{2}\to 0^{+}_{3}$. This may indicate that cluster breaking should not be ignored for detailed discussions of band assignments in model calculations.

III.5.2 Cluster and mean-field modes in monopole excitations in 12C

In experimental and theoretical studies of nuclear clustering, isoscalar monopole (ISM) and dipole (ISD) transitions are good probes to pin down cluster states Kawabata et al. (2007); Funaki et al. (2006a); Yamada et al. (2012); Kanada-En’yo (2016b); Chiba and Kimura (2015). Yamada et al. pointed out that two different modes of ISM excitations coexist in 16OYamada et al. (2012): one is the isoscalar giant monopole resonance (ISGMR) known to be the collective breathing mode, and the other is the low-energy ISM strengths for cluster states. The low-energy ISM strengths were experimentally observed also for 12C Youngblood et al. (1998); John et al. (2003). A hybrid model of the shifted basis AMD (sAMD) and $3\alpha$-GCM was applied to the ISM excitations in 12C and described the low-energy ISM strengths for cluster modes separating from high-energy ISGMR strengths Kanada-En’yo (2016b). The separation of the low-energy and high-energy parts of the ISM strengths qualitatively agrees with the experimental data (see Figs. 11 (a) and (b)). As explained in the previous section, the sAMD bases describe coherent 1p-1h excitations for the GMR, whereas the $3\alpha$-GCM bases are essential for the large amplitude cluster modes which contribute to the low-energy strengths. Figure 11 (e) shows the ISM strengths obtained only by the sAMD bases without $3\alpha$ configurations, and Figures 11 (c) and (d) show the ISM strengths calculated using specific $3\alpha$ configurations in addition to the sAMD bases. As clearly seen, the sAMD describes only the high-energy ISM strengths for the ISGMR but fails to describe significant low-energy ISM strengths. As $3\alpha$ configurations are added to the sAMD bases, a peak grows up and comes down to the low-energy region (see Fig. 11 (c)). Then, the low-energy peak finally splits into the $0^{+}_{2}$ and $0^{+}_{3}$ in the full sAMD+$3\alpha$GCM calculation because of the coupling of the radial motion with the rotational motion of clusters. Namely, the large amplitude cluster motion is essential for the low-energy ISM strengths and the fragmentation of the ISM strengths occurs by the coupling of the radial and rotational motions in the $3\alpha$ dynamics.

The lowering mechanism of the ISM strengths by the large amplitude cluster motion was also demonstrated in the time-dependent FMD calculation by Furuta et al. Furuta et al. (2010). In applications of time-dependent approaches to nuclear excitations, the response functions are calculated by the Fourier transform (frequency) of the time evolution of the system. For the ISM excitations, the initial state is prepared by imposing an external field (operator) $\sum_{i}\bm{r}^{2}_{i}$ to the ground state, and starting from the initial state the time evolution of the system is solved with time-dependent FMD. In the FMD framework, the single-particle excitations are expressed by the time-dependent width parameters of single-nucleon Gaussian wave packets, whereas the radial cluster motion is described by the time-dependent Gaussian center positions. The Fourier transform of the root-mean-square radius shows two modes with different frequencies corresponding to the width oscillation mode and the radial cluster (inter-cluster) mode (see Fig. 12). They analyzed the dependence of frequencies of two modes on the oscillation amplitude and found that the higher frequency for the width mode, corresponding to the breathing mode, does not depend on the amplitude. Rather, the lower peak frequency for the cluster mode moves down significantly to lower energy as the amplitude becomes larger. This result is consistent with the sAMD+$3\alpha$GCM result discussed previously, although the GMR mode is expressed by linear combinations of shifted Gaussian wave packets with a fixed width in the sAMD instead of the variational width in the FMD wave function. Note that the quantization of excitation modes and spin-parity projections are performed in the sAMD+$3\alpha$GCM but they are not done in the TD-FMD. It should be remarked that monopole vibrations in 8Be have been investigated using the TDCM and shows similar features for the width oscillation and radial cluster modes Drozdz et al. (1982).

It is also valuable to consider a link to the THSR wave function for two kinds of monopole modes. In the THSR model, the width oscillation mode is expressed by the parameter $b$ for the $\alpha$-cluster size, and the radial cluster mode is described by the parameter $B$ for the $\alpha$ distribution size as shown in Fig. 12(e). The $0^{+}$ energy surface on the $B$-$b$ plane in Fig. 12(d) shows the coexistence of two modes. The energy surface is very soft along the $B$ mode and it is steep along the $b$ mode. It indicates that the origin of the low-energy ISM strengths is the large amplitude cluster motion decoupled from the width (coherent single-particle excitation) mode for the ISGMR.

IV Tohsaki-Horiuchi-Schuck-Röpke wave function and container

model

IV.1 Introduction

Cluster model studies in the 1970’s showed that the Hoyle state of 12C has a gas-like structure of three $\alpha$ clusters which are weakly bound with predominantly $S$-wave correlations among the $\alpha$-particles  Horiuchi (1974); Uegaki et al. (1977); Kamimura (1981). The gas-like structure of the Hoyle state was reconsidered in a new light in Ref. Tohsaki et al. (2001). In this paper it was proposed that the Hoyle state has a 3$\alpha$-condensate-like structure, and the Tohsaki-Horiuchi-Schuck-Röpke (THSR) wave function was presented for the sake of expressing the $\alpha$-condensate-like structure. It was soon discovered Funaki et al. (2003) that the 3$\alpha$ THSR wave function was nearly identical to the 3$\alpha$ cluster-model wave functions obtained in 1970’s, namely the 3$\alpha$ Brink-GCM (generator coordinate method) wave function of Ref. Uegaki et al. (1977) and the 3$\alpha$ RGM (resonating group method) wave function of Ref. Kamimura (1981).

About 10 years later it was found that the THSR wave function for ${}^{16}{\rm O}+\alpha$ clustering in 20Ne was nearly identical to the Brink-GCM wave function for ${}^{16}{\rm O}+\alpha$ clustering  Zhou et al. (2013). This finding was striking since the ${}^{16}{\rm O}+\alpha$ Brink-GCM wave functions with spatially-localized ${}^{16}{\rm O}+\alpha$ structures describe accurately the states of the so-called inversion-doublet bands of 20Ne where the even parity and odd parity levels are split into two separate bands. The THSR wave function was found to describe well the spatially-localized cluster structures even though it was originally designed to describe gas-like delocalized cluster wave functions. The fact that the THSR wave function can describe both localized and delocalized clustering led to the introduction of the container model of cluster dynamics Zhou et al. (2014a).

Here we discuss the THSR wave function and its history, starting from its initial introduction to the container model of cluster dynamics Funaki et al. (2015); Schuck et al. (2016); Tohsaki et al. (2017). We explain some characteristics of the THSR wave function that might appear contradictory, such as the nucleon-density distribution showing localized clustering despite the nonlocalized character of the THSR wave function and the equivalence of prolate and oblate THSR wave functions after angular momentum projection. The container model is deeply connected to the evolution of cluster structure, and we demonstrate this in 12C and 16O.

IV.2 Alpha-condensate-like character of the Hoyle state

IV.2.1 $S$-wave dominance of $\alpha$-cluster motion in the Hoyle state

The Hoyle state is located slightly above the 3$\alpha$ and 8Be$(0^{+}_{1})$ + $\alpha$ thresholds. The small excitation energy of this state, 7.66 MeV, is very difficult to explain by the shell model. The decay width $\Gamma$ of the Hoyle state is very small (8.7 eV) because the energy is well below the Coulomb barrier. Assuming for the moment two subsystems in a relative $S$ wave, the R-matrix calculation of the width yields

[TABLE]

where $\gamma^{2}(a)$ is the reduced width,

[TABLE]

is the Coulomb barrier penetrability, $a$ is the channel radius, $F_{L}$ and $G_{L}$ are the regular and irregular Coulomb functions, respectively, and $k$ is the wave number. See Ref. Descouvemont and Baye (2010) for a review. The observed value $\gamma^{2}_{obs}(a)$ for the Hoyle state is very large. It is comparable to or larger than the Wigner-limit value $\gamma^{2}_{W}(a)$ = $3\hbar^{2}/(2\mu a^{2})$ that corresponds to an $\alpha$ cluster with uniform density at radial distances less than $a$. The very large value of $\gamma^{2}_{obs}(a)$ suggests that the structure of the Hoyle state is composed of an 8Be$(0^{+}_{1})$ core and loosely attached $\alpha$ cluster in an $S$ wave. This conclusion does not support the idea of the 3$\alpha$ linear-chain structure proposed by Morinaga Morinaga (1966, 1956), since the 3$\alpha$ linear-chain structure would produce a reduced width $\gamma^{2}(a)$ that is significantly smaller than $\gamma^{2}_{W}(a)$ Suzuki et al. (1972).

The $S$-wave dominance of the 8Be$(0^{+}_{1})$-$\alpha$ relative wave function indicated by the observed $\alpha$-width was confirmed theoretically by solving the 3$\alpha$ problem by the use of 3$\alpha$ OCM (orthogonality condition model) Horiuchi (1974). Since 8Be$(0^{+}_{1})$ consists of two $\alpha$ clusters weakly coupled in a relative $S$ wave, the Hoyle state was concluded to have a weakly-coupled 3$\alpha$ structure in relative $S$ waves with large spatial extent. It was therefore described as a gas-like state of $\alpha$ clusters. A few years later, the results of the 3$\alpha$ OCM study were confirmed by fully microscopic 3$\alpha$ calculations by two groups, namely the 3$\alpha$ GCM calculation of Ref. Uegaki et al. (1977), and 3$\alpha$ RGM calculation of Ref. Kamimura (1981). These calculations nicely reproduced not only the excitation energy of the Hoyle state but also other experimental properties including the $\alpha$-decay width, the inelastic electron-scattering charge form factor, and $E0$ and $E2$ transition properties.

IV.2.2 Equivalence of the 3$\alpha$ RGM/GCM wave function to a single 3$\alpha$ THSR wave function

More than 20 years after the 3$\alpha$ OCM, GCM, and RGM studies mentioned above, the Hoyle state was reconsidered in a new light in Ref. Tohsaki et al. (2001). The authors of this paper proposed, for the description of the Hoyle state, the following new model wave function $\Psi^{\rm THSR}_{3\alpha}$ called the THSR wave function. Let $\Phi(3\alpha)$ be a simple product of three $\alpha$-cluster wave functions,

[TABLE]

The THSR wave function has the form

[TABLE]

where $\vec{X}_{i}$ is the center of mass of cluster $i$ and $\vec{\xi}_{k}$ are Jacobi coordinates defined as

[TABLE]

With the center-of-mass dependence removed, the wave function has the form

[TABLE]

where $C$ is a normalization constant. As shown in Eq. (4.16), the THSR wave function can be regarded as expressing the ${{}^{8}{\rm Be}(0^{+}_{1})}+\alpha$ cluster structure, where a ${{}^{8}{\rm Be}(0_{1}^{+})}$-like cluster ${\cal A}\{\exp(-\vec{\xi}_{2}^{2}/B^{2})\phi(\alpha_{2})\phi(\alpha_{3})\}$ and the $\alpha_{1}$ cluster couple via S-wave with inter-cluster wave function $\exp[-4\vec{\xi}_{1}^{2}/(3B^{2})]$. On the other hand, Eq. (4.15) shows that the THSR wave function represents the state where three $\alpha$-clusters occupy the same single 0S-orbit $\exp(-2\vec{X}^{2}/B^{2})$, namely a 3$\alpha$ condensate state which is a finite-size counterpart of the macroscopic $\alpha$-particle condensation in infinite nuclear matter at low density Röpke et al. (1998). What the authors of Ref. Tohsaki et al. (2001) proposed was that the ${{}^{8}{\rm Be}(0_{1}^{+})}+\alpha$ structure of the Hoyle state can be regarded as being a 3$\alpha$ condensate-like state. Furthermore one can in general expect the existence of an $n\alpha$ condensate-like state in the vicinity of the $n\alpha$ threshold in $\alpha$-conjugate nuclei.

An important and striking fact is that both the $3\alpha$ GCM wave function of Ref. Uegaki et al. (1977) and the $3\alpha$ RGM wave function of Ref. Kamimura (1981) are each nearly equivalent to a single $3\alpha$ THSR wave function Funaki et al. (2003):

[TABLE]

Hence the 3$\alpha$ THSR wave functions reproduce the same Hoyle state experimental data well described by the 3$\alpha$ RGM/GCM wave functions. We refer the reader to the recent review in Ref. Tohsaki et al. (2017) for applications of the THSR wave function to the Hoyle state and discussions of electric transitions, $\alpha$-condensation probabilities, and comparisons with quantum Monte Carlo calculations.

IV.3 Localized vs. nonlocalized clustering

IV.3.1 Shell-model limit of THSR wave function

Let $b$ be the single-nucleon oscillator size parameter for the $0s$ H.O. (harmonic oscillator) orbit,

[TABLE]

When $B=b$, we have

[TABLE]

Therefore $\Psi^{\rm THSR}_{3\alpha}(B=b)$ = 0 by the Pauli exclusion principle. By expressing the normalization constant of $\Psi^{\rm THSR}_{3\alpha}(B)$ with $B>b$ as $n^{\rm THSR}_{3\alpha}(B)$, one can prove the relation

[TABLE]

where [444] refers to the spatial-symmetry Young diagram Hutzelmeyer and Hackenbroich (1970). This relation means that $\Psi^{\rm THSR}_{3\alpha}(B)$ for $B$ close to $b$ is close to the shell-model wave function of the 12C ground state. So while the THSR wave function for large $B$ is a gas-like state of clusters, the THSR wave function with small $B$ is a shell-model-like state.

In Ref. Tohsaki et al. (2001), the ground and Hoyle states of 12C were obtained as the lowest and second lowest energy states of the GCM equation with the basis function $\Psi^{\rm THSR}_{3\alpha}(B)$,

[TABLE]

The GCM wave functions of the ground and Hoyle states were found to have about 93% and 98% squared overlaps with single THSR wave functions.

IV.3.2 Inversion doublet bands of 20Ne and THSR wave function

In 20Ne the even-parity $K^{\pi}=0^{+}$ rotational band upon the ground state and odd-parity $K^{\pi}=0^{-}$ rotational band upon the $J^{\pi}=1^{-}$ state at 5.80 MeV constitute inversion-doublet bands having the same intrinsic ${}^{16}{\rm O}+\alpha$ cluster structure Horiuchi and Ikeda (1968). The splitting between the even-parity and odd-parity bands can be understood as arising from tunneling of the $\alpha$ through the ${}^{16}{\rm O}$ core to form the corresponding mirror configuration. The empirical success of this description constitutes evidence of spatial localization of the clusters. Much later it was discovered that the GCM/RGM wave functions describing the inversion-doublet bands were found to be almost equivalent to a single ${}^{16}{\rm O}+\alpha$ THSR wave functions Zhou et al. (2012, 2013),

[TABLE]

where the 16O + $\alpha$ THSR wave function has the form

[TABLE]

where

[TABLE]

and $P_{L}$ is the projection operator for angular momentum $L$.

The relation in Eq. (4.27) casts doubt on the use of the energy curve for the Brink wave function in determining the spatial localization of clusters. The 16O + $\alpha$ Brink wave function ${\cal A}\{\exp[-8(\vec{r}-\vec{D})^{2}/(5b^{2})]\phi(^{16}{\rm O})\phi(\alpha)\}$ has the inter-cluster separation parameter $\vec{D}$. The optimum value of $D=|\vec{D}|$ is obtained from the minimum energy point of the energy expectation value as a function of $D$. It was shown in Ref. Zhou et al. (2013) that this way of determining the inter-cluster distance is misleading because if one uses as the size parameter of the inter-cluster motion not $8/(5b^{2})$ given by the Brink wave function, but the much smaller value $8/(5B^{2})$, the energy minimum point resides at $D$ = 0.

IV.3.3 Localized clustering from inter-cluster Pauli repulsion

The limit $D\to 0$ appearing in Eq. (4.27) would seem to suggest that the distributions of the 16O and $\alpha$ clusters are overlapping and therefore not localized. The resolution of this apparent contradiction becomes clear after calculating the nucleon density distribution of the 16O + $\alpha$ THSR wave function, as was performed in Ref. Zhou et al. (2014a). In Fig. 13 we show the nucleon density distribution of the 16O + $\alpha$ intrinsic THSR wave function of Eq. (4.27) for the small value $D=0.6\,{\rm fm}$. We do not directly set $D=0$ as this would force the intrinsic THSR wave function to be symmetric under parity. We see in Fig. 13 that there exists a large separation of about 3.6  fm between the 16O and $\alpha$ clusters. This spatial localization of clusters is due to the nucleon antisymmetrizer ${\cal A}$. The antisymmetrizer ${\cal A}$ generates Pauli-forbidden states $\chi^{F}(\vec{r})$ for the ${}^{16}{\rm O}-\alpha$ relative motion which have the property

[TABLE]

where $\chi^{F}(\vec{r})$ are H.O. eigenstates with Gaussian factor $\exp[-8\vec{r}^{2}/(5b^{2})]$ and oscillator quanta $2n+L$ smaller than 8. Thus the probability for two clusters 16O and $\alpha$ to be very close together is small. This is nothing more than Pauli repulsion.

We can say that the dynamics favors non-localized clustering but the constraints of antisymmetrization make the system exhibit localized clustering in the intrinsic frame. While this conclusion holds for two-cluster systems in general, the pairwise Pauli repulsion between clusters generally does not produce static localization in the intrinsic frame for more than two clusters. This is why a nonlocalized gas-like structure of three $\alpha$ clusters arises in the Hoyle state even though a localized dumbbell-like structure appears in the $2\alpha$ system.

If there are additional constraints, however, there can be localized clustering even in systems with three or more clusters. An excited state can be viewed as the minimum energy state under the requirement of orthogonality to all energy eigenstates at lower energies. This requirement of orthogonality can constrain the possible deformations of the excited state. Consider, for example, an excited state of the $3\alpha$ system that is orthogonal to the ground state, the Hoyle state, and also the next excited state above the Hoyle state. Suppose furthermore that these constraints energetically favor a strongly prolate THSR wave function of the form,

[TABLE]

where with $B\gg b$. The nucleon density distribution for this THSR wave function is shown in Fig. 14 Zhou et al. (2014a). We see three localized $\alpha$ clusters forming a linear-chain structure.

IV.3.4 Equivalence of prolate and oblate THSR wave functions after angular momentum projection

One unusual feature of THSR wave functions is that prolate and oblate wave functions can become equivalent after angular momentum projection. Fig. 15 shows a contour map of the squared overlaps between a prolate 0+ THSR wave function with 0+ THSR wave functions with various deformations in 20Ne Zhou et al. (2014a). The deformed THSR wave function $\Phi_{\rm Ne}$ of 20Ne has the form ${\cal A}[\chi(\vec{r})\phi(\alpha)\phi(^{16}{\rm O})]$ where $\chi(\vec{r})$ is $\exp[-\sum_{k=x,y,z}(8/5B_{k}^{2})r_{k}^{2}]$ and $B_{k}^{2}=b^{2}+2\beta_{k}^{2}$. We see in this figure that the prolate THSR wave function with $\beta_{x}$ = $\beta_{y}$ = 0.9 fm, $\beta_{z}$ = 2.5 fm is almost 100% equivalent to oblate THSR wave functions with $\beta_{x}$ = $\beta_{y}$ $\approx$ 2.1 fm and $\beta_{z}$ between 0 and 1.2 fm after angular momentum projection onto $0^{+}$. The equivalence of prolate and oblate THSR wave functions after angular momentum projection is true for all the spin-parity states of 20Ne.

Despite this equivalence of prolate and oblate wave functions after angular momentum projection, we can say that the 20Ne states expressed by the THSR wave functions all have prolate deformation as the actual deformation. This conclusion is obtained from the fact that the expectation values of the quadrupole moments of all the 20Ne states expressed by THSR wave functions have negative sign. From the well-known formula

[TABLE]

we know that when the expectation value $Q(J)$ of the quadrupole moment of the wave function with good spin $J$ is negative, the quadrupole moment of the intrinsic state, $Q({\rm intrinsic}),$ is positive and therefore prolate. The THSR wave function after angular momentum projection has the form $\Phi^{J}_{\rm Ne}$ = ${\cal A}[\chi_{J}(\vec{r})\phi(\alpha)\phi(^{16}{\rm O})]$ and we can prove that this type of wave function $\Phi^{J}_{\rm Ne}$ gives us the following formula for $Q(J)$ Zhou et al. (2014a):

[TABLE]

where

[TABLE]

and

[TABLE]

This shows that $Q(J)$ has negative value and explains why the calculated values of $Q(J)$ by THSR wave functions have all negative sign. Of course the negative sign of $Q(J)$ by THSR wave functions is in accordance with the prolate distribution of nucleon density shown in Fig. 13.

The reason why prolate and oblate THSR wave functions are almost equivalent after angular momentum projection is explained by the fact that the rotation-average of a prolate THSR wave function is almost equivalent to an oblate THSR wave function. The rotation-averaged wave function $\Phi^{\rm ave}(\beta_{x}=\beta_{y},\beta_{z})$ generated from a prolate THSR wave function $\Phi^{\rm prolate}(\beta_{x}=\beta_{y},\beta_{z})$ is defined as

[TABLE]

If we rotate a prolate THSR wave function around an axis ($x$ axis) perpendicular to the symmetry axis of the prolate deformation ($z$ axis) and construct a wave function by taking an average over this rotation, the density distribution of the rotation-average wave function will be oblate (see Fig. 16). In the case of the $0^{+}$ state, when we construct the rotation-average wave function from the prolate THSR wave function with $(\beta_{x},\beta_{y},\beta_{z})$ = (0.9, 0.9, 2.5 fm) which gives the minimum energy for $0^{+}$, it is almost 100% equivalent to the oblate THSR wave function with $(\beta_{x},\beta_{y},\beta_{z})$ = (0.9, 2.1, 2.1 fm).

[TABLE]

IV.4 Container picture of cluster dynamics

The ground and Hoyle states in 12C are obtained as the eigenstates with lowest and second lowest energies of Eq. (4.25). This equation is the GCM equation with respect to the size parameter $B$ of the THSR wave function. The excitation of the system is described by the dynamics of the system size. This description is very different from the traditional description of the system excitation by RGM/GCM equation, which treats the dynamics of inter-cluster motion. In Ref. Zhou et al. (2014a), this new description of the cluster dynamics is called the container model of cluster dynamics. The container refers to the self-consistent field with size $B$ in which clusters are accomodated and make nonlocalized motion.

The GCM equation with respect to the container size parameter $B$ was also solved in the $4\alpha$ system Funaki et al. (2010). Table 1 shows the energy spectra of $0^{+}$ states obtained by $4\alpha$ THSR-GCM. In this table we show the energy spectra of $0^{+}$ states obtained by $4\alpha$ OCM Funaki et al. (2006b) together with experimental spectra. This $4\alpha$ OCM study confirmed the assignments by former 12C + $\alpha$ cluster model studies that the observed $0_{2}^{+}$ and $0_{3}^{+}$ states are dominantly 12C$(0_{1}^{+})$ + $\alpha$ ($S$ wave) and 12C$(2_{1}^{+})$ + $\alpha$ ($D$ wave) configurations respectively. On the other hand, the $4\alpha$ OCM study newly assigned the dominant configurations for the $0_{4}^{+}$ and $0_{5}^{+}$ states to be 12C$(0_{1}^{+})$ + $\alpha$ (higher-nodal $S$ wave) and 12C$(1_{1}^{-})$ + $\alpha$ ($P$ wave) respectively. What is interesting about this $4\alpha$ OCM study is the assignment of the $4\alpha$ condensate-like structure to the observed $0_{6}^{+}$ state at 15.1 MeV excitation. The reason for this assignment is that the reduced-width amplitude of the calculated $0_{6}^{+}$ state is large only in the 12C(Hoyle) + $\alpha$ ($S$ wave) channel Funaki et al. (2006b). The good accordance of the calculated decay width $\Gamma_{\rm cal}$ = 136 keV with the observed width $\Gamma_{\rm exp}$ = 166 keV for this $0_{6}^{+}$ state gives high reliability to this OCM assignment. In Ref. Yamada et al. (2012), it is reported that the fine structures of the observed isoscalar monopole strength function up to about 16 MeV in 16O are well reproduced by this $4\alpha$ OCM. Compared to the OCM calculation, the THSR-GCM calculation gives us only four $0^{+}$ states in the excitation-energy region of observed six $0^{+}$ states. However the fourth $0^{+}$ state of the THSR-GCM calculation can be considered to correspond to the $0_{6}^{+}$ state of the $4\alpha$ OCM and hence to the observed $0_{6}^{+}$ state at 15.1 MeV excitation. This is because the reduced-width amplitude of the $0_{4}^{+}$ state of the THSR-GCM calculation is markedly large only in the channel 12C(Hoyle) + $\alpha$ ($S$ wave) Funaki et al. (2010). Thus also in the case of the $4\alpha$ system, the THSR-GCM calculation describes the excitation of the system from the ground state to the $4\alpha$ gas-like excited state, although the description of the excitation to other states is incomplete. In order to remedy the incompleteness of the description of Ref. Funaki et al. (2010) we have to extend the $4\alpha$ THSR wave function so that the THSR wave function includes not the single size parameter $B$ but two or more $B$ parameters. In the case of two $B$ parameters, one $B$ is for the container containing three $\alpha$ clusters and the other $B$ is for the container for the relative motion between $3\alpha$ system and fourth $\alpha$ cluster. In the next subsection we discuss the extension of the THSR wave function so that it includes two or more $B$ parameters.

The GCM wave functions $\Phi_{\lambda}^{\rm THSRGCM}$ of the obtained four $0^{+}$ states with $\lambda=1,2,3,4$ are found to have almost 100% squared overlaps with single orthogonalized THSR wave functions $\widehat{\Phi}_{\lambda}(\beta_{0})$ for a certain value of $\beta_{0}$ satisfying $B^{2}=b^{2}+2\beta_{0}^{2}$. $\widehat{\Phi}_{\lambda}(\beta_{0})$ is defined by

[TABLE]

where

[TABLE]

for $\lambda=2,3,4$. Here, $N_{\lambda}$ is a normalization constant and $P_{0}=1$. Since the orthogonalization operator $P_{\lambda-1}$ expresses the necessary property which any excited state should satisfy, the essential character of $\Phi_{\lambda}^{\rm THSRGCM}$ is expressed by $\Phi_{4\alpha}^{\rm THSR}(\beta_{0})$. Thus, although $\Phi_{\lambda}^{\rm THSRGCM}$ is constructed by a linear combination of many $4\alpha$ THSR wave functions, its essential character is described by only a single $4\alpha$ THSR wave function. The optimum values of $\beta_{0}$ for four $0_{\lambda}^{+}$ states are 1.2 fm, 2.5 fm, 4.0 fm, 6.5 fm, for $\lambda=1,2,3,4,$ respectively, which means that the system size becomes larger with increasing $\lambda$.

IV.5 Extended THSR wave function and examples of its application

IV.5.1 Breathing-like excitation of the Hoyle state

The container model of cluster dynamics uses the system size parameter as the generator coordinate for clustering motion. In the case of $3\alpha$ system, we can introduce size parameters $B_{1}$ and $B_{2}$ for $2\alpha$ and $3\alpha$ containers as shown in Fig.17. The extended THSR wave function for this double container system is given as Zhou et al. (2014b)

[TABLE]

When $B_{2}^{2}=(3/4)B_{1}^{2}$, $\Phi^{\rm exTHSR}_{3\alpha}(B_{1},B_{2})$ = $\Phi^{\rm THSR}_{3\alpha}(B_{1}).$ As has been done in the traditional THSR wave function, we can use deformed containers. In this case, we replace $\vec{\xi}_{1}^{2}/B_{2}^{2}$ by $\sum_{i=x}^{z}\xi_{1i}^{2}/B_{2i}^{2}$ and also $\vec{\xi}_{2}^{2}/B_{1}^{2}$ by $\sum_{i=x}^{z}\xi_{2i}^{2}/B_{1i}^{2}$.

The extended THSR wave function for $3\alpha$ system has been applied to the studies of the ground state Zhou et al. (2014b) and the positive-parity excited states in 12C Funaki (2015); Zhou et al. (2016). Refs. Funaki (2015) and  Zhou et al. (2016) supported the existence of two $0^{+}$ states ($0_{3}^{+}$ and $0_{4}^{+}$) around 10 MeV excitation energy above the Hoyle state ($0_{2}^{+}$) that was proposed by the $3\alpha$ OCM studies combined with the CSM (complex scaling method) in Refs. Kurokawa and Katō (2005); Ohtsubo et al. (2013). The $0_{4}^{+}$ state is the state whose existence had been long known since 1970’s, and this state is considered to have a bent-chain structure of $3\alpha$ with a large component of 8Be($2_{1}^{+}$) + $\alpha$($D$ wave) Uegaki et al. (1977); Kanada-En’yo (1998b); Neff and Feldmeier (2004). On the other hand the $0_{3}^{+}$ state is the state whose existence was newly proposed and which was suggested to be a breathing excitation of the Hoyle state in Ref. Kurokawa and Katō (2005); Kurokawa and Kato (2007). The theoretical proposal of the existence of two $0^{+}$ states ($0_{3}^{+}$ and $0_{4}^{+}$) around 10 MeV excitation energy was soon supported experimentally by Itoh et al. Itoh et al. (2013). In Ref. Itoh et al. (2013), it is reported that the observed broad $0^{+}$ state at 10 MeV consists of two components. The lower $0^{+}$ state do $\alpha$-decay to the ground state of 8Be only while the higher $0^{+}$ state has a distinct peak at 10.8 MeV with a width of 0.4 MeV in the coincidence spectrum for the first excited state of 8Be channel. These two $0^{+}$ states were considered to have consistent properties predicted for a higher nodal state of the Hoyle state and a linear-like 3$\alpha$ state, respectively.

Table 2 shows energies and rms radii of $0^{+}_{2}$, $0^{+}_{3}$, and $0^{+}_{4}$ states of 12C and monopole transition values $M(E0)$ between three $0^{+}$ states which are calculated by extended $3\alpha$ THSR wave functions Funaki (2015); Zhou et al. (2016). We see that the Hoyle state ($0^{+}_{2}$) and other two $0^{+}$ states have very large rms radii. The very large value of the calculated $E0$ strength $M(E0;0_{3}^{+}\to 0_{2}^{+})$ = 35 $e$ fm2 or 47 $e$ fm2 supports the idea to regard the $0^{+}_{3}$ state as a breathing-like excited state of the Hoyle state $0^{+}_{2}$. In Ref. Funaki (2015), the total width of the $0^{+}_{4}$ state is calculated to be 0.7 MeV which is to be compared with the observed width 1.42 MeV. As for the $0^{+}_{3}$ state the calculated $\alpha$ width is 1.1 MeV which is rather close to the observed width 1.45 MeV Itoh et al. (2011).

In Ref. Zhou et al. (2016), two kinds of inter-cluster relative wave functions were analysed which are contained in the four $0^{+}$ states ($0^{+}_{1}\sim 0^{+}_{4}$) obtained by the extended $3\alpha$ container model. The first kind is an $S$-wave relative wave function between 8Be($0^{+}_{1}$) and the remaining $\alpha$ cluster, and the second kind is an $S$-wave relative wave function between two $\alpha$ clusters after the integration over the $\vec{\xi}_{1}$ Jacobi coordinate by using a single Gaussian weight. It was found that both kinds of relative wave functions have one more node in the $0^{+}_{3}$ state than in the Hoyle state. This result implies that $0^{+}_{3}$ state is the breathing-like excited state of the Hoyle state. It is because the generating operator for the breathing excitation

[TABLE]

can be rewritten as

[TABLE]

IV.5.2 Container evolution in 16O

The extended THSR wave function was applied recently to study the evolution of cluster structure in 16O Funaki (2018). As in the $3\alpha$ system in subsection IV.5.1, two deformed containers are adopted where the first container is for the $3\alpha$ subsystem and the second container is for the relative motion between the $3\alpha$ subsystem and the fourth $\alpha$ cluster. Fig. 18 shows the energy spectrum obtained by the extended $4\alpha$ THSR (denoted as eTHSR) compared with the $4\alpha$ OCM and experiment Funaki (2018). The fifth $0^{+}$ state ($0^{+}_{5}$) is just above the $4\alpha$ threshold and the two size parameters of its eTHSR wave function are nearly the same. This means that $4\alpha$ clusters in this state are accommodated approximately in a single container. Since the size parameters of this state are large, $\beta_{x}=\beta_{y}\approx 5.6\,{\rm fm}$, $\beta_{z}\approx 2.0\,{\rm fm}$, the $0^{+}_{5}$ state represents a Hoyle-analogue state in 16O.

The ground state ($0^{+}_{1}$) has the smallest radius and its intrinsic shape is reported to be terahedral. It is based on the calculated result that while the $3\alpha$ sub-container has oblate shape the container describing the relative motion between the $3\alpha$ subsystem and the 4th $\alpha$ cluster is of prolate shape. The calculated second $0^{+}$ state ($0^{+}_{2}$) is reported to have 12C$(0^{+}_{1})$ + $\alpha$ ($S$ wave) structure. The reason for this identification is that the size parameters of the $3\alpha$ sub-container are close to those of the $3\alpha$ container of the 12C ground state and are nearly spherical. Also, the container describing the relative motion between the $3\alpha$ subsystem and the 4th $\alpha$ cluster is also nearly spherical but with a much larger radius. This structure of the second $0^{+}$ state is in good accordance with previous cluster model studies Suzuki et al. (1972); Funaki et al. (2006b). Similarly, the structure of the calculated third $0^{+}$ state ($0^{+}_{3}$) is in good accordance with previous cluster model studies Suzuki et al. (1972); Funaki et al. (2006b); namely the $0^{+}_{3}$ state is reported to have 12C$(2^{+}_{1})$ + $\alpha$($D$ wave) structure. The structure of the calculated fourth $0^{+}$ state ($0^{+}_{IV}$) is reported to have 12C$(0^{+}_{1})$ + $\alpha$($S$ wave), where the container describing the relative motion between the $3\alpha$ subsystem and the 4th $\alpha$ cluster is much larger than that of the second $0^{+}$ state.

The container model and the extended THSR wave function describes the evolution of the cluster structure from the ground state up to the $4\alpha$ condensate-like state (Hoyle-analogue state) through the various 12C + $\alpha$ structures and is therefore well-suited for studying the evolution of cluster structure. In describing this evolution, we go from a single container to several containers, a process called container evolution Funaki (2018).

IV.5.3 Neutron-rich Be isotopes

Extended THSR wave functions have also been applied to neutron-rich nuclei. Since the container describes valence neutrons with size parameters different from the container for the core part of the system, the use of the extended THSR wave function for neutron-rich nuclei is quite natural. Here we report the works of Refs. Lyu et al. (2015) and Lyu et al. (2016) which treat 9Be and 10Be, respectively.

In the case of 9Be, the valence-neutron wave function $F(\vec{r})$ in the extended THSR wave function should have negative parity, and it is given by

[TABLE]

The phase factor $\exp(i\phi_{R})$ makes the parity of $F_{n}(\vec{r})$ negative. In Ref. Lyu et al. (2015), the ground rotational-band levels, $3/2^{-}$, $5/2^{-}$, $7/2^{-}$, were treated, and it was found that the extended THSR wave functions of these levels have about 95% squared overlaps with the wave functions obtained by GCM calculation by using $2\alpha+n$ three-body Brink wave functions.

In the case of 10Be, the energy spectra of two rotational bands upon the ground state and the $0_{2}^{+}$ state were calculated using single extended THSR wave functions and were compared with those obtained by AMD calculations Kobayashi and Kanada-Enyo (2012); Suhara and Kanada-Enyo (2010). For the ground band, the extended THSR wave functions where two valence neutrons occupy the orbit $F_{n}(\vec{r})$ were used. The modification of these extended THSR wave functions were also made by introducing the distance parameter $\vec{R}_{\rm pair}$ between the center of mass of the $2\alpha$ system and center of mass of the $2n$ system. It was reported that both kinds of extended THSR wave functions give very similar energy spectra to that of the AMD calculations Kobayashi and Kanada-Enyo (2012). For the excited band, the extended THSR wave functions were constructed by accommodating two valence neutrons into the $\sigma$-type single-neutron orbit. The obtained energy spectrum is very similar to but a little higher than the AMD energy spectra in Refs. Kobayashi and Kanada-Enyo (2012); Suhara and Kanada-Enyo (2010). The extended THSR wave function of the $0_{2}^{+}$ state is not orthogonalized to that of the ground state, but the squared overlap between them is as small as 1.4%. We see thus that the wave functions as simple as the single extended THSR wave functions give good results that are quite similar to AMD calculations.

V No-core shell model

In contrast with the traditional shell model approach which starts with an inert core of nucleons filling a closed shell, the no-core shell model treats all nucleons as active. The many-body basis states are the energy eigenstates of the spherical harmonic oscillator,

[TABLE]

with some finite truncation imposed in the total oscillator excitation energy Navrátil et al. (2000b, a). Here $m$ is the nucleon mass and $\Omega$ is the oscillator frequency. The truncation of the basis in terms of the total sum of oscillator excitation energies allows for an exact factorization of the wave function into separated center-of-mass and relative-coordinate degrees of freedom.

In these no-core shell model calculations the interactions among nucleons include a nucleon-nucleon potential fitted to experimental nucleon-nucleon scattering data as well as higher-nucleon interactions fitted to few-nucleon observables. Some take the approach of using a high-quality phenomenological potential Wiringa et al. (1995), while others apply the organizational principles of chiral effective field theory to produce effective chiral interactions for nucleons Epelbaum et al. (2009); Machleidt and Entem (2011).

The method has had many remarkable successes in recent years in describing nuclear structure from first principles, e.g., Navrátil et al. (2007); Maris et al. (2009); Roth et al. (2011); Barrett et al. (2013). For the study of nuclear clustering, however, the no-core shell model in its basic form is typically not efficient in describing spatial correlations among nucleons forming localized clusters.

V.1 Symmetry-adapted no-core shell model approaches

The symmetry-adapted no-core shell model overcomes the problem of efficiently describing clustering by making use of exact and dynamical symmetries of the spherical harmonic oscillator Hamiltonian associated with collective mode excitations Dytrych et al. (2007); Draayer et al. (2011); Dreyfuss et al. (2013); Dytrych et al. (2013); Dreyfuss et al. (2016). We can rewrite the single-particle spherical harmonic oscillator Hamiltonian in terms of the usual ladder operators,

[TABLE]

We see there is a U(3) symmetry group associated with unitary $3\times 3$ rotations of the $x,y,z$ quanta, and the component continuously connected to the identity forms an SU(3) symmetry group Elliott (1958a, b). But the symmetry group can be expanded further by also allowing SU(1,1) transformations of the form

[TABLE]

The transformation can also be applied to $c_{y,i}$, $c_{z,i}$, and any set of real orthogonal linear combinations of $c_{x,i}$, $c_{y,i}$, and $c_{z,i}$, and thus we also have an ${\rm SU}(1,1)\otimes{\rm O}(3)$ symmetry. It can be shown that the full dynamical group of the spherical harmonic oscillator is the real symplectic group ${\rm Sp}(6,{\bf R})$ for $6\times 6$ matrices Rowe and Wood (2010).

The symmetry-adapted no-core shell model uses the real symplectic group ${\rm Sp}(6,{\bf R})$ and its subgroup ${\rm SU}(3)$ to generate linear combinations of spherical harmonic basis states which form complete representations of the $\rm{SU}(3)$ subgroup for some selected quantum numbers $(\lambda,\mu)$ of the Cartan subalgebra of $\rm{SU}(3)$. As the quantum numbers $(\lambda,\mu)$ correspond to different deformation geometries, the problem of capturing the collective behavior induced by clustering can be considerably more efficient in the symplectic basis. One of the future challenges for the symmetry-adapted no-core shell model approach is to handle realistic nuclear forces with significant terms breaking symplectic or SU(3) symmetry.

A specific version of the symmetry-adapated no-core shell model called the no-core symplectic model (NCSpM) was used to compute the low-lying even parity states of 12C Dreyfuss et al. (2013). The results for the rms matter radii and electric quadrupole moments are shown in Table 3 Dreyfuss et al. (2013). The NCSpM calculation gives a point matter rms radius for the ground state in agreement with experiment. The calculation yields a point matter radius of $r_{\rm rms}=2.93$ fm for the Hoyle state, which is slightly larger than that of the ground state. While this result is smaller than the results typically obtained in cluster model calculations, it is close to a recent value deduced from experiment, $2.89(4)$ fm Danilov et al. (2009), and is similar to ab initio lattice EFT results at leading order, 2.4(2) fm Epelbaum et al. (2011).

The NCSpM calculations yield an electric quadrupole moment for the $2_{1}^{+}$ state in agreement with the experimental value. Similarly, a positive quadrupole moment is found for the $4_{1}^{+}$ state, and the $0_{1}^{+}$, $2_{1}^{+}$, and $4_{1}$ are consistent with a rotational band with an oblate structure. On the other hand, a large negative result is found for the $2^{+}_{2}$ state above the Hoyle state and the same for the $4^{+}$ state above the Hoyle state. The results are consistent with a rotational band associated with the Hoyle state with a substantial prolate deformation. Such a prolate deformation has also been found in ab initio lattice EFT results Epelbaum et al. (2011, 2012).

We note that SU(3)-symmetry has been also used to study clustering in shell model calculations with a core. The cluster-nucleon configuration interaction model is one such approach Volya and Tchuvil’sky (2015). This method has recently been used to probe the cluster structure of 20Ne resonances in elastic ${}^{16}\rm{O}+\alpha$ scattering Nauruzbayev et al. (2017).

V.2 Continuum no-core shell model approaches

Another way to incorporate clustering in the no-core shell model is to consider spherical harmonic oscillator states corresponding to more than one center. This is done by combining the no-core shell model formalism with the resonating group method (RGM). A review article summarizing recent developments can be found in Ref. Navrátil et al. (2016). In the following we discuss the case with two clusters.

Let the binary-cluster state of interest have total angular momentum $J$, parity $\pi$, and isospin $T$. We start with binary-channel basis states of the form Quaglioni et al. (2010)

[TABLE]

Here $\left|A{-}a\,\alpha_{1}I_{1}^{\,\pi_{1}}T_{1}\right\rangle$ and $\left|a\,\alpha_{2}I_{2}^{\,\pi_{2}}T_{2}\right\rangle$ are the internal wave functions of the first and second clusters, containing $A{-}a$ and $a$ nucleons respectively. They carry angular momentum quantum numbers $I_{1}$ and $I_{2}$ which are coupled together to form spin $s$, and the clusters have orbital angular momentum $\ell$. Their parity, isospin and additional quantum numbers are written as $\pi_{i},T_{i}$, and $\alpha_{i}$, respectively, with $i=1,2$. The separation vector between the cluster centers is

[TABLE]

where $\vec{r}_{i}$ are the single-particle coordinates for $i=1,\cdots A$. It is convenient to group all relevant quantum numbers into a collective index $\nu=\{A{-}a\,\alpha_{1}I_{1}^{\,\pi_{1}}T_{1};\,a\,\alpha_{2}I_{2}^{\,\pi_{2}}T_{2};$ $\,s\ell\}$. In order to enforce the correct fermionic statistics, one uses the inter-cluster antisymmetrizer,

[TABLE]

where the sum runs over all possible permutations $P$ that can be carried out among nucleons, and ${\rm sgn}(P)$ is the sign of the permutation.

The antisymmetrized basis states can be used to expand the many-body wave function as

[TABLE]

The coefficient functions $g^{J^{\pi}T}_{\nu}(r)$ correspond to the relative-motion radial wave functions between the clusters. These unknown coefficient functions are solved by the non-local integral-differential coupled-channel equations

[TABLE]

where $E$ is the total energy in the center-of-mass frame, and the two integration kernels are the Hamiltonian kernel,

[TABLE]

and the norm kernel,

[TABLE]

The nontrivial norm kernel is the result of the non-orthogonality of the basis states (5.49). Furthermore, the exchange terms in the antisymmetrizer give rise to non-local terms in the two kernels.

This no-core shell model with resonating group formalism has been used very successfully to calculate many elastic scattering processes and inelastic reactions involving light nuclei Quaglioni and Navrátil (2008); Navrátil et al. (2010); Navrátil and Quaglioni (2012). The method has recently been improved further by also including basis states corresponding to the regular no-core shell basis with the full $A$-body space in one cluster. This has the advantage of encoding the short-range interactions between clusters more efficiently than the resonating group method would otherwise. This approach, known as the no-core shell model with continuum approach, has been used to describe two-body reactions Dohet-Eraly et al. (2016); Raimondi et al. (2016), unbound states Baroni et al. (2013), and even three-body reactions Quaglioni et al. (2013). Quite recently there have also been no-core shell model with continuum studies of the cluster structure of 6Li Hupin et al. (2015) as an $\alpha$-cluster and deuteron and also of 6He Romero-Redondo et al. (2014, 2016) in terms of an $\alpha$-cluster and two neutrons.

In Fig. 19 we show results for the 6He wave function using no-core shell model with continuum Romero-Redondo et al. (2016). The horizontal axis is the separation between the two halo neutrons, $r_{\rm nn}$, and the vertical axis is the separation between the alpha-particle core and the center of mass of the two halo neutrons, $r_{\alpha,{\rm nn}}$. The plots shows the dominance of a di-neutron configuration where the two neutrons are about 2 fm apart and the $\alpha$-particle about 3 fm away. There is also a smaller contribution from a much smaller contribution from a split configuration where the two neutrons are far from each other with the $\alpha$-particle situated in between.

The no-core shell model with continuum can be viewed as one of several continuum shell model methods with a long history Mahaux and Weidenmüller (1969). Some other recent developments are the shell model embedded in the continuum Okolowicz et al. (2003), continuum shell model Volya and Zelevinsky (2005), and no-core Gamow shell model Papadimitriou et al. (2013).

One recent work with particular relevance for nuclear clustering is Ref. Kravvaris and Volya (2017), which uses the no-core shell model and resonating group method for clusters, but also applies the harmonic oscillator expansion for the relative separation between clusters. In this work they compute spectroscopic amplitudes for the low-lying even parity states of 8Be, 10Be, 12C into open $\alpha$-separation thresholds.

VI Continuum quantum Monte Carlo

A recent review on continuum Quantum Monte Carlo methods in nuclear physics has been recently been published Carlson et al. (2015). Here we give an overview of the methods and studies which have been used to investigate clustering in nuclei.

VI.1 Variational Monte Carlo

Variational Monte Carlo (VMC) relies on the variational principle that the energy of any trial wave function will be greater than or equal to the ground state energy. We are of course assuming only physical states antisymmetrized with respect to the exchange of all identical fermions. The strategy is to start with some general functional form for the trial wave function $\Psi_{T}^{\{\alpha_{i}\}}$ which depends on some set of unknown parameters $\{\alpha_{i}\}$. One then computes the energy expectation $E_{T}^{\{\alpha_{i}\}}$ for the trial state

[TABLE]

and minimizes with respect to $\{\alpha_{i}\}$. Instead of minimizing the energy, one can also minimize the expectation value of the variance operator $(H-\lambda I)^{2}$, which vanishes only when $\lambda$ is an exact energy eigenvalue.

Since the trial wave function is typically a function with many degrees of freedom, the inner products in Eq. (6.56) are computed using Monte Carlo integration. If the interactions in $H$ have a local structure in position space, then the required integration can be performed quite simply by selecting points in the space of the particle coordinates, ${\bf r}_{1},{\bf r}_{2},\cdots$, chosen according to the squared absolute value of the trial wave function, $|\Psi_{T}^{\{\alpha_{i}\}}({\bf r}_{1},{\bf r}_{2},\cdots)|^{2}$ McMillan (1965); Ceperley et al. (1977). For each set of points, the expectation of $H$ correspond to the value of the function $\Psi_{T}^{\{\alpha_{i}\}*}H\Psi_{T}^{\{\alpha_{i}\}}({\bf r}_{1},{\bf r}_{2},\cdots)$. If one divides by the relative probability of selecting the points ${\bf r}_{1},{\bf r}_{2},\cdots$, then the value one records in the Monte Carlo integration of this observable is $H\Psi_{T}^{\{\alpha_{i}\}}({\bf r}_{1},{\bf r}_{2},\cdots)/\Psi_{T}^{\{\alpha_{i}\}}({\bf r}_{1},{\bf r}_{2},\cdots).$

The quality of the variational Monte Carlo result depends entirely on the functional form used for the trial wave function. Therefore it is important to incorporate particle correlations into $\Psi_{T}^{\{\alpha_{i}\}}$. In variational Monte Carlo calculations of the structure of 16O Pieper et al. (1992), the trial wave function included non-central two-body and three-body correlations acting on Slater determinants of $S$-wave and $P$-wave one-body wave functions. The expectation values of operators were calculated using a cluster expansion for the spin- and isospin-dependent terms up to four-body order. In many cases variational Monte Carlo is also used to optimize the trial wave function serving as a starting point for other Monte Carlo calculations such as diffusion or Green’s function Monte Carlo.

VI.2 Diffusion or Green’s function Monte Carlo

Diffusion or Green’s function Monte Carlo (GFMC) starts with a trial wave function $|\Psi_{T}\rangle$ and uses Euclidean time evolution to extract the ground state wave function Kalos (1962). Originally diffusion Monte Carlo and Green’s function Monte Carlo referred to slightly different algorithms. However in today’s usage, they refer to the same method. The ground state wave function is obtained in the large time limit as

[TABLE]

The parameter $\lambda$ is used to stabilize the normalization of the wave function and gives an estimate of the ground state energy $E_{0}$. A more direct calculation of the ground state energy is given by the ratio

[TABLE]

When exponentiated over a short time step $\Delta\tau$, the kinetic energy term in $H$ gives rise to a diffusion process which is modeled as a random walk in the space of all possible particle coordinates. Meanwhile, the particle interactions result in an exponential growth or decay for each possible spin and isospin channel.

One of the main computational challenges in GFMC is the sign oscillation problem associated with the exchange of identical fermions. These sign oscillations will render the numerator and denominator to be vanishingly small in the limit of large time $\tau$. For real-valued wave functions the fixed-node approximation gives a remedy for this problem by restricting the random walk in the space of particle coordinates to a region where the trial wave function remains positive. For complex-valued wave functions as one finds in nuclear physics, a generalization of the approach called the constrained path approximation is used Wiringa et al. (2000). In the constrained path approximation one restricts the random walk to a region where the overlap of the propagated state with the trial wave function is positive Carlson et al. (2015).

GFMC has been used to compute the spectra of many light nuclei Wiringa et al. (2000); Pieper and Wiringa (2001); Pieper et al. (2002). This includes a well-known study of the $\alpha$-cluster structure of the 8Be ground state Wiringa et al. (2000). There have also been also recent studies of the Hoyle state of 12C Carlson et al. (2015) and its transitions to the ground state. These calculations find a radius for the Hoyle state of more than 3.1 fm, which is much larger than the ground state radius 2.43 fm. Fig. 20 shows the density distributions $r^{2}\rho(r)$ of the ground state ($0^{+}_{1}$) and the Hoyle state ($0^{+}_{2}$) of 12C Carlson et al. (2015). Similar results have been obtained using the THSR wave function Tohsaki et al. (2017).

In order to the improve the computational scaling of the diffusion Monte Carlo simulations with the number of particles, one approach being pursued is introducing an auxiliary field to rewrite the spin-dependent interactions in terms of one-body spin operators. This method is called auxiliary-field diffusion Monte Carlo Gandolfi et al. (2007); Gezerlis et al. (2013).

VI.3 Monte Carlo shell model

The Monte Carlo Shell Model (MCSM) approach is a variational method which uses auxiliary-field Monte Carlo simulations to determine a set of low-energy basis states $|\phi_{n}\rangle$ Abe et al. (2012). In this discussion we focus on the no-core version of MCSM where all nucleons are active. Each $|\phi_{n}\rangle$ is a Slater determinant of deformed single-particle shell model states. The resulting states are given good angular momentum and parity quantum quantum numbers by explicit projection. In order to remove residual errors due to the basis truncation, extrapolations are performed as a function of the energy variance Shimizu et al. (2010, 2012b). This method has been used to study the alpha-two-neutron cluster structure of 6He, two-alpha structure of 8Be, and two-alpha-two-neutron structure of 10Be Shimizu et al. (2012a); Yoshida et al. (2013).

For the case with total angular momentum $J=0$, the projected wave function is

[TABLE]

The linear combination of the unprojected basis states, $|\Phi\rangle$, cannot be considered as an intrinsic state since the principal axis of each basis state, $|\phi_{n}\rangle$, are not all aligned in the same direction. This is fixed by performing a rotation $R(\Omega_{n})$ so that the quadrupole moment is diagonalized, and $Q_{zz}\geq Q_{yy}\geq Q_{xx}$ so that the principal axis is aligned with the $z$-axis. The intrinsic wave function $|\Phi^{{\rm intr}}\rangle$ is then defined as

[TABLE]

In Fig. 21 we show the 8Be proton densities for $|\Phi\rangle$ and $|\Phi^{{\rm intr}}\rangle$ Shimizu et al. (2012a); Yoshida et al. (2013). We show results for $N_{b}=10^{0},10^{1},10^{2}$ basis states. Each density distribution shows the the $yz$ plane for intercepts $x=0$ fm and $x=1$ fm.

VII Nuclear lattice effective field theory

VII.1 Chiral effective field theory on a lattice

The basic idea of Nuclear Lattice Effective Field Theory (NLEFT) is to merge the successful chiral EFT for nuclear forces pioneered by Weinberg Weinberg (1990, 1991) with lattice Monte Carlo methods, that allow for numerically exact solutions of the nuclear $A$-body problem. First, the ingredients to construct the chiral nuclear EFT are briefly discussed. The EFT is formulated in terms of the asymptotically observed states, the nucleons and the pions, the latter being the Goldstone bosons of the spontaneously broken chiral symmetry of QCD. The basic idea of the Weinberg approach is to use chiral perturbation theory to construct the potential between two, three and four nucleons. The various contributions are organized according to the power counting based on the small parameter $Q$, with $Q\in\{p/\Lambda,M_{\pi}/\Lambda\}$. Here, $p$ denotes some soft external momentum, $M_{\pi}$ the pion mass and $\Lambda$ the hard scale that accounts for all physics integrated out. Usually, this scale is set by the appearance of the first resonance, like the $f_{0}(500)$ in pion-pion scattering or the $\Delta(1232)$ in pion-nucleon scattering. For the nuclear force problem, the leading order (LO) contributions are of order ${\mathcal{O}}(Q^{0})$, comprising the leading one-pion exchange (OPE) and two local four-nucleon contact interactions without derivatives. At next-to-leading order (NLO), ${\mathcal{O}}(Q^{2})$, one has the leading two-pion exchange (TPE) interactions and seven further four-nucleon terms with two derivatives (for on-shell scattering) as well as two isospin symmetry-breaking terms that account for the dominant strong interaction difference between the proton-proton, proton-neutron and neutron-neutron systems. Finally, at next-to-next-to-leading order (N2LO), ${\mathcal{O}}(Q^{3})$, that is the accuracy to which most NLEFT calculations have been carried out so far, one has further TPE corrections proportional to the dimension-two low-energy constants (LECs) $c_{i}$ of the effective pion-nucleon Lagrangian that can be precisely determined from the dispersive Roy-Steiner equation analysis of pion-nucleon scattering Hoferichter et al. (2015). At this order, three-nucleon forces start to contribute. These fall in three topologies. The two-pion exchange diagram is entirely given in terms of the LECs $c_{1,2,4}$. The one-pion exchange coupling to a four-nucleon term and the local six nucleon contact term are parametrized by the LECs $D$ and $E$, respectively. These are commonly determined from the triton binding energy and the axial-vector current contribution to triton decay Gazit et al. (2009). For further details, we refer the reader to the reviews Epelbaum et al. (2009); Machleidt and Entem (2011).

In the lattice formulation, Euclidean space-time is given by a finite hypercubic volume, with $L$ the length in any of the spatial directions and $L_{t}$ the extension in the temporal direction. Further, the lattice is defined by a minimal spatial distance $a$, the lattice spacing, and similarly by $a_{t}$ in the temporal direction. In most calculations discussed in what follows, a coarse spatial lattice with $a=1/(100~{}{\rm MeV})=1.97$ fm was used, while $a_{t}$ is chosen to be $a_{t}=1/(150~{}{\rm MeV})=1.32$ fm. One important feature of the finite lattice spacing is the UV finiteness of the theory, as the largest possible momentum is given by $p_{\rm max}=\pi/a\simeq 314\,$MeV. Thus, the interaction is very soft and therefore most higher order corrections, including also the Coulomb effects, can be treated in perturbation theory. Another advantage of this approach is the fact that all possible configurations of nucleons are sampled, as depicted in Fig. 22.

This gives a first hint that the phenomenon of clustering indeed will arise quite naturally in this approach. In the actual calculations, the interactions between the nucleons are described in terms of auxiliary fields, which makes the approach particularly suited for highly parallel computation. In essence, each nucleon evolves in time from the starting at $t=t_{i}$ up to the final time $t_{f}$. The value of $t_{f}$ has to be large enough so that the asymptotic behavior of any observable for the $A$-nucleon state can be extracted. For further details, we refer to the detailed description of the LO chiral EFT interactions on the lattice in Ref. Borasoy et al. (2007a). See also the review in Ref. Lee (2009).

Another important fact is the approximate Wigner SU(4) symmetry of the nuclear interactions Wigner (1937). This is the observation that combined spin-isospin rotations of the nucleon four-vector $(p\uparrow,p\downarrow,n\uparrow,n\downarrow)$ leave the nuclear forces in the $S$-wave approximately invariant. This symmetry is broken by the OPE and the Coulomb interaction, but rather well respected by the four-nucleon short-range operators Mehen et al. (1999). Most importantly, in case of an exact Wigner symmetry, nuclei with spin and isospin zero do not show any sign oscillations Chen et al. (2004), which make finite density lattice simulations so difficult. This approximate symmetry can therefore be used as an inexpensive filter in the actual simulations. It can also be proven that for the case of attractive SU(4) interactions, the resulting nuclear binding energies must satisfy spectral convexity bounds that correspond to an alpha clustering phase Lee (2007). For further work on understanding Wigner symmetry within QCD and its consequences, see e.g. Refs. Lee (2004); Calle Cordon and Ruiz Arriola (2008); Beane et al. (2013). We will come back to this topic in subsection VII.2.

Before continuing, let us define what we mean by ab initio calculation in this context. The various parameters appearing in the lattice approach, like the LECs and the smearing parameters as defined below, are determined in fits to properties of few-particle systems like phase shifts and binding energies. Here, few means less or equal four. The properties of nuclei with larger atomic number can then be predicted without further parameter tuning to a precision that is given by the accuracy of the underlying chiral EFT Hamiltonian. Note that recently it has been found that fitting also to the low-energy $\alpha$-$\alpha$ $S$-wave phase shifts for determining the pertinent LECs provides some advantage in controlling higher-body interactions in larger systems Elhatisari et al. (2016b).

VII.2 Lattice formalism

To calculate the energy or any other static observable, we need an initial wave function for the nucleus under consideration, $|\Psi_{A}^{\mathrm{in}}\rangle$. Such a state can on one hand be chosen as a Slater-determinant states composed of delocalized standing waves in the periodic cube with $A$ nucleons, and on the other hand as localized $\alpha$-cluster trial states (or any other type of cluster state). Such localized states have been used in the investigations of 12C and 16O Epelbaum et al. (2011, 2012, 2014). These can be used to check the calculations with the delocalized initial states, but also allow to assess the spatial structure of the nuclei. It has to be understood that these states are always prepared with a given total angular momentum $J$ and parity $\pi$, that is a fixed $J^{\pi}$. Rotational symmetry breaking due to the lattice is an issue that will be discussed later. The central object of NLEFT is the Euclidean-time projection amplitude

[TABLE]

that allows to compute the “transient energy” $E_{A}(t)=-\partial[\ln Z_{A}(t)]/\partial t$. Here $H_{\mathrm{LO}}$ is the leading-order Hamiltonian. In the infinite time limit, this gives the ground-state energy, as all excited states have a larger energy and thus fall off faster. The initial and final states have been prepared using

[TABLE]

where $H_{\rm SU(4)}$ is a lattice Hamiltonian that approximates $H_{\mathrm{LO}}$ but maintains an exact Wigner SU(4) symmetry, as detailed in Ref. Borasoy et al. (2007a). This method has been considerably improved by the so-called “triangulation” procedure introduced in Ref. Lähde et al. (2014) using several initial states, which allows significant reduction in the error due to Euclidean time extrapolation. For example, using this method, the ground state energies of 12C and 16O can now be calculated with an absolute uncertainty of $\pm 200\,$keV. A detailed discussion of this method and the associated uncertainties is given in Ref. Lähde et al. (2015a).

In order to compute the low-lying excited states of a given nucleus, the Euclidean time projection method is extended to a multi-channel calculation. Take the 12C nucleus as an example Epelbaum et al. (2011). Using the auxiliary-field formalism, one applies the exponential operator $\exp({-Ht})$ to a set of different single-nucleon standing waves in the periodic cube. From these standing waves one then builds initial states consisting of Slater determinants of $6$ protons and $6$ neutrons each and extracts orthogonalized energy eigenstates with the desired quantum properties. In Ref. Epelbaum et al. (2011) four states were found with even parity and total momentum equal to zero. As is well known, the lattice discretization of space and periodic boundaries reduce the full rotational group to a cubic subgroup. This complicates the identification of spin states. However, the number of energy levels seen for each value of $J_{z}$ allows one to identify different spin states. This method can be refined by not only using delocalized standing waves but also initial cluster states with nucleons grouped into Gaussian wave packets arranged in certain geometrical configurations, see Ref. Epelbaum et al. (2012). As shown in Fig. 23, various configurations that corresponds to $J^{\pi}=0^{+}$ indeed leads to the ground state (left panel) or the first excited $0^{+}$ Hoyle state about 7 MeV above the ground state.

We have already noted that the proximity of the Hoyle state energy to the triple-$\alpha$ threshold is important for the production of carbon in the universe. As with any near-threshold state, this very low-energy scale is well-separated from other energy scales, and this separation of scales forms the basis for halo effective field theory Bertulani et al. (2002); Bedaque et al. (2003); Higa et al. (2008). One interesting theoretical question is whether this proximity of the Hoyle state to the triple-$\alpha$ threshold is a generic feature of quantum chromodynamics or is something needs to be fine-tuned. The quark mass dependence of the Hoyle state energy has been studied using lattice simulations Epelbaum et al. (2013a, b) in connection with the anthropic principle, the production of carbon and oxygen, and the fine-tuning of the parameters of nature Beane and Savage (2003b); Epelbaum et al. (2003); Beane and Savage (2003a); Epelbaum et al. (2002); Berengut et al. (2013); Bedaque et al. (2011); Meißner (2015).

We now discuss two-body scattering on the lattice. This is not only an important ingredient to fix the LECs of the effective Lagrangian but can also be used to investigate the nuclear dynamics encoded in nuclear reactions. The best known and most frequently used method is due to Lüscher, who showed that the energy of an interacting two-particle system in a finite volume can be related to the infinite-volume phase shift at the same energy Lüscher (1986, 1991). This method has by now been extended to cope with higher partial waves, partial-wave mixing, multi-channel scattering, boosted frames and all possible types of boundary conditions, see e.g. Refs. He et al. (2005); Liu et al. (2006); Bernard et al. (2008); Lage et al. (2009); Luu and Savage (2011); König et al. (2011); Bour et al. (2011); König et al. (2012); Göckeler et al. (2012); Li and Liu (2013); Briceno et al. (2013); Li et al. (2014); Li and Wu (2015). However, for the case of nucleon-nucleon scattering, which involves higher energies, spin-orbit coupling and partial-wave mixing, the method has more limited accuracy. A more robust approach that makes use of the non-relativistic character of the nuclear problem is the so-called spherical wall approach, used for numerical calculations as early as in Ref. Carlson et al. (1984), but reinvented for the lattice formulation used here in Ref. Borasoy et al. (2007b). Here, one imposes a hard spherical wall boundary on the relative separation between the two particles at some radius $R_{\rm W}$. In that way, copies of the interactions produced by the periodic lattice are removed and, from the solution of the Schrödinger equation for spherical standing waves at $r=R_{\rm W}$, one can easily recover the phase shift for a given partial wave. Mixing of the partial waves caused by spin-orbit coupling is also easily dealt with. This method has been improved significantly in Ref. Lu et al. (2016). First, so-called radial position states for a given partial wave are constructed according to

[TABLE]

with $Y_{\ell,\ell_{z}}$ spherical harmonics with angular momentum quantum numbers $\ell,\ell_{z}$, and $r$ is to be restricted to be less than half the box size $L/2$. Angular momentum is not conserved on the lattice. However the amount rotational invariance breaking decreases with increasing radial distance, and we can use spherical harmonics to dial the corresponding partial waves. This projection allows one to construct the so-called radial lattice Hamiltonian. Second, one introduces auxiliary potentials in the region immediately in front of the spherical wall. By tuning the depth of this potential, one can dial the scattering energy. In case of partial-wave mixing, this potential is chosen such that time-reversal symmetry is broken. This allows to extract phase shifts and scattering angles from the real and imaginary parts of the wave function. For details, see Ref. Lu et al. (2016).

The aforementioned SU(4) symmetry can be further utilized to suppress the sign oscillations in auxiliary-field Monte Carlo calculations. The underlying idea is to smoothly connect the LO lattice Hamiltonian with an SU(4)-symmetric counterpart that does not suffer from any sign oscillations. In that way, one can construct a one-parameter family of Hamiltonians, $H(d)=dH_{\rm LO}+(1-d)H_{\rm SU(4)}$. For $d=1$, one obviously recovers the microscopic chiral Hamiltonian. One can then perform simulations for various values of $d$ and extrapolate to the limit $d=1$. This method is called symmetry-sign extrapolation (SSE) Lähde et al. (2015b). In contrast to techniques introduced earlier in shell model Monte Carlo calculations Alhassid et al. (1994); Koonin et al. (1997), the magnitude of the sign oscillations in this approach are typically quadratic in $d$ and are thus milder.

Another important issue in the lattice simulations is interaction smearing. The interactions are not strictly point-like but distributed over neighboring lattice sites. This method is very common in lattice QCD to enhance the strength of a given quark source, see e.g. Refs. Güsken (1990); Daniel et al. (1992); Allton et al. (1993); Morningstar and Peardon (2004); Hasenfratz et al. (2007); Edwards et al. (2008) for some groundbreaking work. In NLEFT, smearing is done for various reasons. First, the nucleon-nucleon interaction terms are smeared with a Gaussian-type function, whose depth and width is fixed from the averaged nucleon-nucleon $S$-wave scattering lengths and effective ranges. As discussed in detail in Ref. Borasoy et al. (2007a), this type of smearing is required to avoid overbinding due to the configurations with four nucleons on one lattice site, cf. Fig. 22. This has the added value that important effective range corrections are treated non-perturbatively rather than perturbatively, i.e. some important higher-order corrections are being resummed. Second, a novel type of non-local smearing was introduced in Ref. Elhatisari et al. (2016b). For that, one considers non-local nucleon annihilation (creation) operators and two-nucleon densities, such as

[TABLE]

where the three-vector ${\bf n}$ denotes a lattice site, and $\sum_{\langle{\bf n^{\prime}\,n}\rangle}$ denotes the sum over nearest-neighbor lattice sites of ${\bf n}$, so that $|{\bf n^{\prime}}-{\bf n}|=a$. The smearing parameter $s_{\rm NL}$ is to be determined together with the other parameters and LECs as discussed later. This non-local smearing offers the possibility of considering non-local short-range interactions on the lattice, as discussed in subsection VII.4.3.

Another issue to be addressed is the lattice spacing dependence. In contrast to lattice QCD, in NLEFT one does not perform the continuum limit $a\to 0$ as we are dealing with an effective field theory that only makes sense below some hard (breakdown) momentum scale $\Lambda$. Physically, one can understand this very intuitively, the EFT is not appropriate to resolve the inner structure of the nucleon at distances less than the proton charge radius of about 0.85 fm. Therefore, one expects that the calculations within NLEFT are invariant under variations of $a$ between 1 and 2 fm, provided that the LECs are properly readjusted. This expectation is indeed borne out by explicit calculations. In Ref. Klein et al. (2015) it was shown within the pionless as well as the pionful LO EFT that the $S$-wave phase shifts and the deuteron binding energy can be reproduced for $0.5\lesssim a\lesssim 2.0\,$fm. This has recently been sharpened by studying the neutron-proton interactions to NNLO for lattice spacings from $1\ldots 2$ fm Alarcón et al. (2017). Presently, larger systems are being systematically investigated to establish this $a$-independence in general.

Finally, we mention that simple $\alpha$-cluster models have been used in Refs. Lu et al. (2014, 2015) to gain a deeper understanding of the effects of the rotational symmetry breaking on the lattice and to develop methods to overcome this. It was demonstrated in Ref. Lu et al. (2014) that lattice spacing errors are closely related to the commensurability of the lattice with the intrinsic length scales of the system and that rotational symmetry breaking effects can be significantly reduced by using improved lattice actions. In particular, the physical energy levels are accurately reproduced by the weighted average of a given spin multiplets. Further, in Ref. Lu et al. (2015) the matrix elements of multipole moment operators were studied. It could be shown that the physical reduced matrix element is well reproduced by averaging over all possible orientations of the quantum state, and this is expressed as a sum of matrix elements weighted by the corresponding Clebsch-Gordan coefficients. These methods will become important when further investigations of the electromagnetic structure of nuclei using NLEFT are performed.

VII.3 Adiabatic projection method

To study reactions and inelastic processes on the lattice, one makes use of the adiabatic projection method (APM). The APM has been developed in Refs. Rupak and Lee (2013); Pine et al. (2013) and further refined in Refs. Elhatisari and Lee (2014); Rokash et al. (2015); Elhatisari et al. (2016a). From the set-up, it is similar to the recent studies combining the resonating group method with the no-core-shell-model Quaglioni and Navrátil (2008); Navrátil et al. (2010); Navrátil and Quaglioni (2012); Romero-Redondo et al. (2014). Within the APM, the cluster-cluster scattering problem on the lattice is evaluated in a two-step procedure. First, one uses Euclidean time projection to determine an adiabatic Hamiltonian for the participating clusters. Strictly speaking, for finite temporal lattice spacing, an adiabatic transfer matrix rather than the Hamiltonian is constructed, but the method is essentially the same, and for simplicity, the Hamiltonian formulation will be discussed here. In the second step, this adiabatic Hamiltonian is then used to calculate the pertinent phase shifts. The biggest advantage of the APM is that the computational time appears to scale with the number of interacting constituents, $t_{\rm CPU}\sim(A_{1}+A_{2})^{2}$, with $A_{i}$ the number of nucleons in cluster $i$, while more conventional approaches exhibit factorial scaling with increasing atomic number.

Consider an $L^{3}$ periodic lattice and a set of two-cluster states $|\vec{R}\rangle$ labeled by their separation vector $\vec{R}$, as illustrated in Fig. 24. In general, there are spin and flavor indices for these states, but we suppress writing the indices for notational simplicity. Also, it is favorable to perform a radial projection as given in Eq. (7.63). However, the exact form of these two-cluster states is not important except that they are localized so that for large separations they factorize as a tensor product of two individual clusters, $|\vec{R}\rangle=\sum_{\vec{r}}|\vec{r}+\vec{R}\rangle_{1}\otimes|\vec{r}\rangle_{2}.$ These states are propagated in Euclidean time to form dressed cluster states, $|\vec{R}\rangle_{\tau}=\exp(-H\tau)|\vec{R}\rangle$. An important consequence of this evolution in Euclidean time with the microscopic Hamiltonian is the fact that deformations and polarizations of the interacting clusters are incorporated automatically. Also, in this way one projects onto the space of low-energy scattering states in the finite volume, so that in the limit of large Euclidean time, these dressed cluster states span the low-energy subspace of two-cluster continuum states. Next, matrix elements of the microscopic Hamiltonian with respect to the dressed cluster states are formed, $\left[H_{\tau}\right]_{\vec{R},\vec{R}^{\prime}}=\ _{\tau}\langle\vec{R}|H|\vec{R}^{\prime}\rangle_{\tau}$. However, since the dressed cluster states $|\vec{R}\rangle_{\tau}$ are, in general, not orthogonal, one needs to construct the norm matrix $N_{\tau}$, $\left[N_{\tau}\right]_{\vec{R},\vec{R}^{\prime}}=\ _{\tau}\langle\vec{R}|\vec{R}^{\prime}\rangle_{\tau}$, so that the Hermitian adiabatic Hamiltonian can be readily calculated

[TABLE]

In the limit of large $\tau$, the spectrum of $H^{a}_{\tau}$ exactly reproduces the low-energy finite volume spectrum of the microscopic Hamiltonian $H$. From this adiabatic Hamiltonian, elastic phase shifts can be calculated using the methods discussed above. Inelastic processes can also be dealt within this scheme by including additional channels, see Ref. Pine et al. (2013) for details. One remark is in order. Since one is working in Euclidean time, the time evolution operator acts indeed as a diffusion operator. The precise definition of the asymptotic states must therefore account for this, and in fact one can define an asymptotic radius $R_{\epsilon}$ as the radius such that for $|\vec{R}|>R_{\epsilon}$ the amount of overlap between the cluster wave packets is less than $\epsilon$ Rokash et al. (2015). Consequently, in the asymptotic region $|\vec{R}|>R_{\epsilon}$, the dressed clusters are widely separated and interact only through long range forces such as the Coulomb interaction. For cases where there are no long range interactions, the scattering states of the adiabatic Hamiltonian are given by a superposition of Bessel functions in the asymptotic region. For the case with Coulomb interactions, the scattering states of the adiabatic Hamiltonian in the asymptotic region correspond to a superposition of Coulomb wave functions. The latter case is schematically shown in Fig. 25. A much refined version of the adiabatic Hamiltonian based on an improved radial “binning” was given in Ref. Elhatisari et al. (2016a). Thus, large-scale numerical computations of nucleus-nucleus scattering and reactions using Monte Carlo methods are possible. We discuss the archetypical process of elastic $\alpha$-$\alpha$ scattering in subsection VII.4.2.

VII.4 Results

VII.4.1 Alpha-cluster nuclei

In Refs. Epelbaum et al. (2011, 2012) the even-parity spectrum and structure of 12C was calculated. The underlying Hamiltonian was given at NNLO precision, which includes the first contributions of the three-nucleon force (3NF). The 11 LECs related to the nucleon-nucleon interactions were fixed from the $S$- and $P$-wave np phase shifts as well from the pp and nn scattering lengths. The two LECs related to the 3NF were fixed from the triton binding energy and the weak axial-vector current. With that, the binding energy of 4He is $-28.3(6)\,$MeV, in agreement with the empirical value. The next $\alpha$-type nucleus, 8Be, is bound with $-55(2)\,$MeV, compared to the empirical value of $-56.5\,$MeV, which is above the $2\alpha$ threshold, i.e. 8Be is unbound in nature. Nevertheless, 8Be is long-lived, so given the accuracy of the NNLO calculation, this agreement is satisfactory. The resulting even-parity spectrum of 12C is shown in the NLEFT results presented in Fig. 4. The uncertainties on the energy levels have been considerably reduced to what was quoted in the original papers Epelbaum et al. (2011, 2012), the ground state can now be calculated with an uncertainty of about 200 keV, and similar errors are expected for the excited states. Most importantly, the clustering arises very naturally, as already discussed in subsection VII.2. Also, by using initial cluster-type states, one can map out the most important contributions for a state of given energy, spin and parity. We find that the ground and the first excited $2^{+}$ state of 12C are mostly given by a compact triangular configuration of three alphas, while the Hoyle state and the second $2^{+}$ receive a large contribution from the so-called “bent-arm” configuration (obtuse triangle). This is an indication that the second $2^{+}$ state is indeed a rotational excitation of the Hoyle state. However, one has to be aware that such “pictorials” of the wave function are resolution-dependent, that means for a finer lattice spacing one will be able to resolve these structures in more detail. The charge radii, quadrupole moments and electromagnetic transitions among the low-lying even-parity states of 12C have also been calculated at LO. These results tend to be on the low side of the experimental values. This can be traced back to the fact that at LO, the charge radius comes out about 10% too small. If one scales the corresponding moments and transition elements with appropriate powers of $r(0^{+}_{1})^{\rm exp}/r(0^{+}_{1})^{\rm LO}$, the agreement is quite satisfactory. Of course, this needs to be backed up in the future by higher order calculations of these observables.

Before elaborating on the structure of heavier nuclei, it is important to scrutinize the NNLO forces. This was done in Ref. Lähde et al. (2014), where it was shown that for $\alpha$-cluster nuclei beyond $A=12$ an overbinding appears, that grows with atomic number, as shown in Table 4. This has also been observed in other ab initio approaches using soft interactions, see e.g. Refs. Hagen et al. (2012); Jurgenson et al. (2013); Roth et al. (2011)555We note that the NNLOsat interaction in Ref. Ekström et al. (2015) is a soft interaction that does not overbind medium-mass nuclei, and thus there are other aspects of the interactions that also come into play.. In Ref. Lähde et al. (2014) this problem was overcome by adding an effective repulsive four-nucleon force, whose strength was determined from the ground state energy of 24Mg. As one can see from Table 4, including this, one achieves a very good description of the ground state energies of all $\alpha$-cluster nuclei up to 28Si. Another method to overcome this deficiency will be discussed in subsection VII.4.3.

The even-parity spectrum and structure of 16O has discussed in Ref. Epelbaum et al. (2014). The ground state has $J^{\pi}=0^{+}$ and its energy is within 3% of the empirical value, cf. Table 4. One finds a second $0^{+}$ state at $-123(2)\,$MeV and the first $2^{+}$ state at the same energy. This is consistent with the empirical values, $E(0^{+}_{2})=-121.57\,$MeV and $E(2^{+}_{1})=-120.70\,$MeV. By measuring four-nucleon correlations, one finds that the dominant cluster configuration on the lattice is the tetrahedron, see Fig. 26 (left), while the excited states have a strong overlap with the planar-type configurations also shown in Fig. 26 (right).

This implies that the first $2^{+}$ state is a rotational excitation of the first excited $0^{+}$. As in the case of 12C, the charge radius of the ground state comes out too small, we get $r(0^{+}_{1})^{\rm LO}=2.3(1)\,$fm, while the empirical value is $2.710(15)\,$fm. This again is due to the overbinding at LO. If one rescales as described above, one find that the predictions for the $E2$ and $E0$ transitions are in good agreement with the experimental values. In particular, NLEFT is able to explain the empirical value of $B(E2,2^{+}_{1}\to 0^{+}_{2})$, which is $\simeq 30$ times larger than the Weisskopf single-particle shell model estimate. This provides further confirmation of the interpretation of the $2^{+}_{1}$ state as a rotational excitation of the $0^{+}_{2}$ state. Again, more detailed higher calculations of the electromagnetic response of 16O within NLEFT are needed.

VII.4.2 Ab initio alpha-alpha scattering

Although there has been impressive progress in ab initio calculations of nuclear scattering and reactions in the recent years, see e.g. Refs. Nollett et al. (2007); Navrátil et al. (2010); Navrátil and Quaglioni (2012); Hagen and Michel (2012); Orlandini et al. (2014), the aforementioned computational limits did so far not allow to consider astrophysically relevant reactions like elastic $\alpha$-$\alpha$, $\alpha$-12C or 12C-12C scattering. A major step forward done in these directions was reported in Ref. Elhatisari et al. (2015). There, the first ab initio calculation of $\alpha$-$\alpha$ scattering based on chiral EFT and using the lattice formulation was discussed. It is based on the same NNLO chiral Hamiltonian that was used for the analysis of 12C and 16O, and so all parameters had been determined before. Using the APM, the $S$- and $D$-wave scattering phase shifts could be calculated as shown in Fig. 27. For more details on the actual computations, see Ref. Elhatisari et al. (2015). In the chiral counting employed, the Coulomb interactions only appear at NLO, therefore the LO curves deviate significantly from the data. However, already at NLO one finds a good description of the $S$-wave and a fair description for the $D$-wave. While the NNLO corrections in the $S$-wave are very small, these corrections bring the $D$-wave close to the data, although there is still some room for improvement. The observed energy of the $S$-wave resonance is 0.09184 MeV above threshold. For the lattice results, the ground state is found at 0.79(9) MeV below threshold at LO, and 0.11(1) MeV below threshold at both NLO and NNLO. The $D$-wave resonance is located at $E_{R}=2.92(18)\,$MeV and $\Gamma=1.34(50)\,$MeV  Afzal et al. (1969), but there is some model-dependence as discussed in Ref. Elhatisari et al. (2015). In NLEFT, one finds at NNLO $E_{R}=3.27(12)\,$MeV and $\Gamma=2.09(16)\,$MeV. This calculation can be considered a benchmark for ab initio calculations of nuclear scattering processes. Clearly, it needs to be refined by going to higher orders and also working with finer lattices. However, arguably the most significant finding of this investigation is the fact that the computing time scales approximately quadratically with the number of nucleons involved. Therefore, the computation of the “holy grail” of nuclear astrophysics Fowler (1984), namely the reaction $\alpha+^{12}{\rm C}\to^{16}{\rm O}+\gamma$ at stellar energies, is in reach.

VII.4.3 Nuclear binding near a quantum phase transition

We had already seen that the NNLO forces overbind in larger nuclei, so higher-order calculations will be needed and eventually higher-body forces might be required. In Ref. Elhatisari et al. (2016b) two ideas were combined to give further insight into how nuclei are formed and what role $\alpha$-clustering plays. First, the non-local smearing already discussed in subsection VII.2 was utilized to construct two new LO interactions, motivated by the hope that the smearing would further suppress the sign oscillations. Second, it was speculated that determining the LECs from fitting also to data from nucleus-nucleus scattering might make the troublesome higher order corrections small. To quantify these ideas, two different LO interactions were constructed. More precisely, interaction A consists of non-local short-range interactions and one-pion exchange, supplemented by the Coulomb interaction. Interaction B has in addition local short-distance interactions. Second, while interaction A was entirely determined by a fit to np scattering data and the deuteron binding energy, interaction B was in addition tuned to the $S$-wave $\alpha$-$\alpha$ phase shifts. The resulting ground state energies for 3H, 3He, 4He and $\alpha$-cluster nuclei are given in Table 5. While the results up to 8Be are similar, interaction A fails to describe the heavier nuclei, quite in contrast to interaction B, which gives an amazingly good description. From this one concludes that $\alpha$-$\alpha$ scattering is quite sensitive to the degree of locality of the nucleon-nucleon lattice interactions. This can be understood from the compactness of the $\alpha$-particle wave function, as explained in more detail in Ref. Elhatisari et al. (2016b).

From Table 5 one further reads off that in the absence of Coulomb interactions, the binding energy for a nucleus made of $N$ $\alpha$-particles is exactly $N$ times the $\alpha$ energy for interaction A, that is it describes a Bose-condensed gas of particles. These observations allows one to draw interesting conclusions about the many-body limit. As usual, the Coulomb interactions are switched off in order to take the many-body limit. One can then define a one-parameter family of interactions via $V_{\lambda}=(1-\lambda)V_{\rm A}+\lambda V_{\rm B}$. While the properties of the two, three, and four nucleon systems vary only slightly with $\lambda$, the many-body ground state of $V_{\lambda}$ undergoes a quantum phase transition from a Bose-condensed gas to a nuclear liquid. The corresponding zero temperature phase diagram is sketched in Fig. 28. The phase transition occurs when the $\alpha$-$\alpha$ $S$-wave scattering length $a_{\alpha\alpha}$ crosses zero, and the Bose gas collapses due to the attractive interactions Stoof (1994); Kagan et al. (1998). At slightly larger $\lambda$, finite $\alpha$-type nuclei also become bound, starting with the largest nuclei first. The last $\alpha$-like nucleus to be bound is 8Be in the so-called unitarity limit $|a_{\alpha\alpha}|=\infty$. Superimposed on the phase diagram, the $\alpha$-like nuclear ground state energies $E_{A}$ for $A$ nucleons up to $A=20$ relative to the corresponding multi-alpha threshold $E_{\alpha}A/4$ are also depicted. This shows that by varying $\lambda$, one can move any $\alpha$-cluster state up or down with respect to the $\alpha$ separation thresholds. This can be used as a new window to view the structure of these exotic nuclear states. In particular, this allows one to continuously connect the Hoyle state wave function without Coulomb interactions to a universal Efimov trimer Efimov (1971); Braaten and Hammer (2006); Kraemer et al. (2006).

Another interesting system is the second $0^{+}$ state of 16O , which should be continuously connected to a universal Efimov tetramer Hammer and Platter (2007); Kraemer et al. (2006); von Stecher et al. (2009). In summary, the main findings of this work are that the $\alpha$-$\alpha$ interaction is a key control parameter which determines whether the ground state of a many-nucleon system is a Bose-condensed gas of $\alpha$-particles or a nuclear liquid. The proximity of this first-order quantum phase transition may explain why seemingly similar nuclear interactions can produce very different results in ab initio nuclear structure calculations. These conclusions need to be solidified by more detailed higher order calculations. Similar results have been found in Ref. Ebran et al. (2012, 2013, 2014b, 2014a, 2015) using density functional methods.

One might ask what the dependence on $\lambda$ means for future nuclear structure calculations for heavier systems using chiral effective field theory. It suggests that the order-by-order convergence of chiral effective field theory might benefit from some optimization of the forces and regulators used in the chiral interactions. This need for optimization may not be visible in few-nucleon observables until very high-orders in chiral effective field theory. But the dependence on $\lambda$ appears as a leading-order effect in the framework of cluster effective field theory for two low-energy $\alpha$-particles. This suggests that some acceleration of the convergence of chiral effective field theory in heavier systems might be possible by making links to cluster effective field theory.

VII.4.4 Clustering in neutron-rich nuclei

In addition to the discussion of the quantum phase transition, another development in Ref. Elhatisari et al. (2016b) was the use of non-local interactions to reduce sign oscillations in the lattice Monte Carlo simulations. This idea was utilized in Ref. Elhatisari et al. (2017) to perform lattice simulations of neutron-rich nuclei. While this work only considered interactions at leading order in chiral effective field theory, the ground state energies of the hydrogen, helium, beryllium, carbon, and oxygen isotopes could be reproduced with an error of 0.7 MeV per nucleon or less with only three adjustable parameters.

In Ref. Elhatisari et al. (2017) a new model-independent method was also introduced for measuring clustering in nuclei using localized three- and four-nucleon operators. Let $\rho(\bf n)$ be the total nucleon density operator on lattice site $\bf n$. $\rho_{3}$ is defined as the expectation value of $:\rho^{3}({\bf n})/3!:$ summed over $\bf n$, where the $::$ symbols denote normal-ordering where all annihilation operators are moved to the right and all creation operators are moved to the left. Similarly $\rho_{4}$ is defined as the expectation value of $:\rho^{4}({\bf n})/4!:$ summed over $\bf n$.

Although the expectation values $\rho_{3}$ and $\rho_{4}$ depend on the manner in which short-distance physics is regularized, the leading part of this dependence is an overall factor which does not depend on the nucleus being considered. So if $\rho_{3,\alpha}$ and $\rho_{4,\alpha}$ are the corresponding values for the $\alpha$-particle, then the ratios $\rho_{3}/\rho_{3,\alpha}$ and $\rho_{4}/\rho_{4,\alpha}$ are free from short-distance divergences and are model-independent quantities up to contributions from higher-dimensional operators in an operator product expansion. In Ref. Elhatisari et al. (2017) the quantities $\rho_{3}/\rho_{3,\alpha}$ and $\rho_{4}/\rho_{4,\alpha}$ were computed and used the quantify the amount of $\alpha$-clustering in the helium, beryllium, carbon, and oxygen isotopes in a model-independent manner. It was observed that these ratios $\rho_{3}/\rho_{3,\alpha}$ and $\rho_{4}/\rho_{4,\alpha}$ could be used to probe the shape of the $\alpha$-clusters as well as the amount of quantum entanglement of nucleons from different $\alpha$-clusters.

Another development in Ref. Elhatisari et al. (2017) was the determination of $\alpha$-cluster correlations in the carbon isotopes 12C, 14C, and 16C by measuring density correlations among the three spin-up protons. This approach relies on the fact that, on average, there is only one spin-up proton within each $\alpha$-cluster. The similarities among the 12C, 14C, and 16C $\alpha$-cluster geometries suggest that there should be $\alpha$-cluster states in 14C and 16C that are analogs of the $\alpha$-cluster states in 12C. For example, the bound $0^{+}_{2}$ state at 6.59 MeV above the ground state of 14C could be a bound-state analog to the Hoyle state resonance in 12C.

VIII Summary and outlook

We have presented a review on the current status and understanding of microscopic clustering in nuclei. We began with a history of the field and then discussed recent experimental results on $\alpha$-conjugate nuclei, molecular structures in neutron-rich nuclei, and constraints for ab initio theory. There has been impressive progress in recent years clarifying clustering phenomena in 8,9,10Be, 10,12,13,14C, 16O, and several other nuclei. However, many more precision measurements are needed, and these will provide vital benchmarks for first principles calculations. In addition to rotational bands, form factors, electromagnetic transition strengths, decays, and reaction cross sections, model-independent assessments of clustering such as ANCs are also very useful in making connections to ab initio theory.

There are also new opportunities for discovery in exploring clustering phenomena over a wide range of nuclear systems, from light to heavy nuclei and from the proton drip line to the neutron drip line. One of the fundamental questions of the field is understanding how prevalent nuclear clustering is across the nuclear chart. This includes systems where clustering is more subtly expressed and mixed with other effects such as particle-hole excitations. Having a large empirical database of nuclear phenomena will shed light on the control parameters for nuclear cluster formation and stability.

On the theoretical side we have discussed methods used to study microscopic clustering. We reviewed the resonating group and generator coordinate methods, antisymmetrized molecular dynamics, Tohsaki-Horiuchi-Schuck-Röpke wave function and container model, no-core shell model, continuum quantum Monte Carlo, and lattice effective field theory.

While there have been many significant advances in the past decade, the field of microscopic nuclear clustering theory is now just entering the era of precision calculations. The future holds many opportunities for improvement in theory, methods or algorithms, and analysis. With the rapid growth of ab initio nuclear theory in the past few years, one great challenge for the field is to describe nuclear clustering from first principles with controlled systematic errors. This is no easy task as recent studies have found that the interactions between nuclear clusters are very sensitive to details of the nuclear forces.

One area where all theoretical groups may choose to invest time and effort is on error quantification and the systematic reduction of errors. One question relevant to all groups is how results on nuclear clustering depend on the microscopic nuclear forces utilized. The follow-up question is how this difference can be systematically reduced by including the relevant missing physics. For lattice calculations another important question is to estimate and reduce the size of lattice discretization errors. For methods based on finite basis truncation or variational parameter optimization, a key question is the residual dependence on the choice of truncated space or variational ansatz. For continuum quantum Monte Carlo, the analogous question would be the dependence on wave function constraints and the trial wave function.

In addition to reproducing observed experimental data, another challenge for theoretical calculations is to compute model-independent observables that provide a quantitative measure of clustering and also serve as standard benchmarks for all different theoretical approaches. We have already mentioned ANCs for shallow bound states, but other model-independent observables could also be computed and defined for resonances as well.

We hope that our review captures some of the excitement of the growing and vibrant field of nuclear clustering. With many open questions and challenges still remaining, we anticipate fascinating new chapters to be written in the coming years.

Acknowledgments

We acknowledge the work of our collaborators J. Alarcón, D. Du, S. Elhatisari, E. Epelbaum, Y. Funaki, N. Klein, H. Krebs, T. Lähde, N. Li, B. Lu, T. Luu, M. Lyu, Z. Ren, A. Rokash, G. Röpke, P. Schuck, A. Tohsaki, C. Xu, T. Yamada, and B. Zhou. We also acknowledge helpful discussions with K. Launey, P. Navrátil, W. Nazarewicz, T. Neff, and S. Pieper. Partial financial support was provided by JSPS KAKENHI Grant Number 26400270, JSPS KAKENHI Grant Number JP16K05351, Deutsche Forschungsgemeinschaft (Sino-German CRC 110), the Helmholtz Association (Contract No. VH-VI-417), BMBF (Grant No. 05P15PCFN1), the U.S. Department of Energy (DE-FG02-03ER41260), U.S. National Science Foundation grant No. PHY-1307453 and by The Chinese Academy of Sciences (CAS) President’s International Fellowship Initiative (PIFI) grant no. 2017VMA0025. Computing resources provided by the Jülich Supercomputing Centre at Forschungszentrum Jülich and RWTH Aachen.