Our present understanding of the structure of the Hoyle state in $^{12}$C and other near-threshold states in $α$-conjugate nuclei is reviewed in the framework of the $α$-condensate model. The $^{12}$C Hoyle state, in particular, is a candidate for $α$-condensation, due to its large radius and $α$-cluster structure. The predicted features of nuclear $α$-particle condensates are reviewed along with a discussion of their experimental indicators, with a focus on precision break-up measurements. Two experiments are discussed in detail, firstly concerning the break-up of $^{12}$C and then the decays of heavier nuclei. With more theoretical input, and increasingly complex detector setups, precision break-up measurements can, in principle, provide insight into the structures of states in $α$-conjugate nuclei. However, the commonly-held belief that the decay of a condensate state will result in $N$ $α$-particles is challenged. We further conclude that unambiguously characterising excited states built on $α$-condensates is difficult, despite improvements in detector technology.
The Hoyle state and other resonances in the continuum above the 3 alpha threshold in 12C are studied in a microscopic cluster model. Whereas the Hoyle state is a very sharp resonance and can be treated reasonably well in bound state approximation, the other higher lying states require a proper treatment of the continuum. The model space consists of an internal region with 3 alpha particles on a triangular grid and an external region consisting of the 8Be ground state and excited (pseudo)-states of 8Be with an additional alpha. The microscopic R-matrix method is used to match the many-body wave…
$^3$He+$^{12}$C scattering is studied in a coupled channel method by using a double folding model with microscopic wave functions of $^{12}$C. Experimental angular distributions in elastic and inelastic scattering to the $2^+$ (4.44 MeV), 0$^+_2$ (7.65 MeV) and 3$^-$ (9.63 MeV) states of $^{12}$C are well reproduced. It is found that the Airy minimum of the prerainbow oscillations for the Hoyle state is considerably shifted to a larger angle due to its dilute density distribution compared with that of the normal ground state in agreement with the idea of $α$ particle condensation.
The excited states in ${^{12}{\rm C}}$ are investigated by using an extended version of the so-called Tohsaki-Horiuchi- Schuck-Röpke (THSR) wave function, where both the $3α$ condensate and ${^{8}{\rm Be}} + α$ cluster asymptotic configurations are included. A new method is also used to resolve spurious continuum coupling with physical states. We focus on the structures of the "Hoyle band" states, $2_2^+$, and $4_2^+$ states, which are recently observed above the Hoyle state and of the $0_3^+$ and $0_4^+$ states, which are also quite recently identified in experiment. Their resonance parameter…
Identification of alpha cluster states analogous to the 12C Hoyle state in heavier alpha- conjugate nuclei can provide tests of the existence of alpha condensates in nuclei. Such states are predicted for 16O, 20Ne, 24Mg, 28Si etc. at excitation energies slightly above the multi-alpha particle decay threshold, but have not yet been experimentally identified. The Thick Target Inverse Kinematics (TTIK) technique can be used to study the breakup of excited self-conjugate nuclei into many alpha particles. The reaction 20Ne+α was studied using a 20Ne beam at 12 MeV/nucleon from the K150 cyclotron at…
The excited state of the 12C nucleus known as the "Hoyle state" constitutes one of the most interesting, difficult and timely challenges in nuclear physics, as it plays a key role in the production of carbon via fusion of three alpha particles in red giant stars. In this letter, we present ab initio lattice calculations which unravel the structure of the Hoyle state, along with evidence for a low-lying spin-2 rotational excitation. For the 12C ground state and the first excited spin-2 state, we find a compact triangular configuration of alpha clusters. For the Hoyle state and the second excite…
The $α$-particle condensate in nuclei is a novel state described by a product state of $α$'s, all with their c.o.m. in the lowest 0S orbit. We demonstrate that a typical $α$-particle condensate is the Hoyle state ($E_{x}=7.65$ MeV, $0^+_2$ state in $^{12}$C), which plays a crucial role for the synthesis of $^{12}$C in the universe. The influence of antisymmentrization in the Hoyle state on the bosonic character of the $α$ particle is discussed in detail. It is shown to be weak. The bosonic aspects in the Hoyle state, therefore, are predominant. It is conjectured that $α$-particle condensate st…
We review the calculation of the Hoyle state of $^{12}$C in Nuclear Lattice Effective Field Theory (NLEFT) and its anthropic implications for the nucleosynthesis of $^{12}$C and $^{16}$O in red giant stars. We also review the extension of NLEFT to the regime of medium-mass nuclei, with emphasis on the determination of the ground-state energies of the alpha nuclei $^{16}$O, $^{20}$Ne, $^{24}$Mg and $^{28}$Si by means of Euclidean time projection. Finally, we review recent NLEFT results for the spectrum, electromagnetic properties, and alpha-cluster structure of $^{16}$O.
Angular distributions for elastic and inelastic scattering of $^3$He from $^{12}$C were measured at energies 50.5 and 60 MeV. The Airy minimum of the prerainbow scattering was clearly observed in the angular distributions for the 0$_2^+$ (7.65 MeV) state of $^{12}$C (Hoyle state). The experimental results were analyzed with a coupled channels method with double folding potentials derived from the microscopic wave functions for the ground 0$_1^+$, 2$^+$ (4.44 MeV), 3$^-$ (9.64 MeV) and 0$_2^+$ states. The analysis supports the view that the Hoyle state is a three alpha particle condensate with …
The $^{12}$C nucleus with $N$=6 and $Z$=6 is a doubly closed-shell nucleus in a toroidal potential. In the description of the ground state and the Hoyle state of $^{12}$C in the resonating group method or the generator coordinate method, a superposition of the orientations of Wheeler's triangular cluster on the cluster plane would naturally generate an intrinsic toroidal density. A toroidal state also has a probability amplitude to overlap with a 3-alpha cluster, which is the dominant decay mode for the Hoyle state. For these reasons, we study a toroidal description of the states of $^{12}$C i…
Background: The structure of the Hoyle state, a highly $α$-clustered state at 7.65 MeV in $^{12}\mathrm{C}$, has long been the subject of debate. Understanding if the system comprises of three weakly-interacting $α$-particles in the 0s orbital, known as an $α$-condensate state, is possible by studying the decay branches of the Hoyle state. Purpose: The direct decay of the Hoyle state into three $α$-particles, rather than through the $^{8}\mathrm{Be}$ ground state, can be identified by studying the energy partition of the 3 $α$-particles arising from the decay. This paper provides details on th…
The second 2+ state of 12C, predicted over fifty years ago as an excitation of the Hoyle state, has been unambiguously identified using the 12C(g,a_0)8Be reaction. The alpha particles produced by the photodisintegration of 12C were detected using an Optical Time Projection Chamber (O-TPC). Data were collected at beam energies between 9.1 and 10.7 MeV using the intense nearly mono-energetic gamma-ray beams at the HIgS facility. The measured angular distributions determine the cross section and the E1-E2 relative phases as a function of energy leading to an unambiguous identification of the seco…
It is suggested that direct evidence of Bose-Einstein condensation of alpha particles is obtained by observing a phase mode (Nambu-Goldstone boson) with long wavelength even when characteristic features such as superfluidity is diffucult to observe. For the 7.65 MeV $0_2^+$ Hoyle state in $^{12}$C and 15.1 MeV $0^+$ state in $^{16}$O, which are candidates for an alpha particle condensate, it is suggested that the emergent band head $0^+$ state of the $K=0_2^+$ rotational band with a very large moment of inertia is considered to be a Nambu-Goldstone boson.
The root mean square radii of $^{20}$Ne in the short-lived excited states were experimentally deduced for the first time from the analyses of $α$+$^{20}$Ne diffraction scattering. Differential cross sections of the elastic and inelastic $α$+$^{20}$Ne scattering in the incident energy range from a few MeV/nucleon up to 100 MeV/nucleon were analyzed by the modified diffraction model. No significant radius enhancement for the members of K$^π$ = 0$_{1}^{+}$ and K$^π$ = 2$^{-}$ bands in comparison with the ground state was observed. At the same time 20 % radius enhancement was obtained for the K$^π…
Theoretical Studies of alpha Clustering in Nuclei and Beyond
University of Tokyo（东京大学）；RIKEN Nishina Center for Accelerator-Based Science（理化学研究所仁科加速器科学中心）；Florida State University（佛罗里达州立大学）；Osaka Metropolitan University（大阪公立大学）
The structure of $1/2^{\pm}$ states in $^{13}$C up to around the $3α+n$ threshold ($E_x = 12.3$~MeV) is investigated with a full four-body $3α+n$ orthogonality condition model (OCM) calculation, where the $3α$ OCM, the model space of which is the subspace of the $3α+n$ model, describes well the structure of the low-lying states of $^{12}$C including the $2^{+}_2$, $0^{+}_3$, and $0^{+}_4$ states, which have been recently observed above the Hoyle state ($0^+_2$). A full spectrum up to the $1/2^{-}_5$ ($1/2^{+}_3$) state is reproduced consistently with the lowest five $1/2^{-}$ (three $1/2^{+}$)…
The fragmentation of quasi-projectiles from the nuclear reaction $^{40}Ca$ + $^{12}C$ at 25 MeV/nucleon was used to produce excited states candidates to $α$-particle condensation. The experiment was performed at LNS-Catania using the CHIMERA multidetector. Accepting the emission simultaneity and equality among the $α$-particle kinetic energies as experimental criteria for deciding in favor of the condensate nature of an excited state, we analyze the $0_2^+$ and $2_2^+$ states of $^{12}$C and the $0_6^+$ state of $^{16}$O. A sub-class of events corresponding to the direct 3-$α$ decay of the Hoy…
The fragmentation of quasi-projectiles from the nuclear reaction $^{40}Ca$+$^{12}C$ at 25 MeV/nucleon was used to produce excited states candidates to $α$-particle condensation. The methodology relies on high granularity 4$π$ detection coupled to correlation function techniques. Under the assumption that the equality among the kinetic energies of the emitted $α$-particles and the emission simultaneity constitutes a reliable fingerprint of $α$ condensation, we identify several tens of events corresponding to the deexcitation of the Hoyle state of $^{12}$C which fulfill the condition.
Method: To examine signatures of this alpha-condensation, a compound nucleus reaction using 160, 280, and 400 MeV 16O beams impinging on a carbon target was used to investigate the 12C(16O,7a) reaction. This permits a search for near-threshold states in the alpha-conjugate nuclei up to 24Mg. Results: Events up to an alpha-particle multiplicity of 7 were measured and the results were compared to both an Extended Hauser-Feshbach calculation and the Fermi break-up model. The measured multiplicity distribution exceeded that predicted from a sequential decay mechanism and had a better agreement wit…
The formation of alpha-clusters in nuclei close to the decay thresholds is discussed. These states can be considered to be boson-condensates, which are formed in a second order phase transition in a mixture of nucleons and alpha-particles. The de Broglie wavelength of the alpha-particles is larger than the nuclear diameter, therefore the coherent properties of the alpha-particles give particular effects for the study of such states. The states are above the thresholds thus the enhanced emission of multiple-alphas into the same direction is observed. The probability for the emission of multiple…
The spatial structure of the lowest 0$_1^+$, 0$_2^+$, 2$_1^+$ and 2$_2^+$ states of the $^{12}$C nucleus is studied within the 3$α$ model with the Buck, Friedrich, and Wheatley $αα$ potential with Pauli forbidden states in the $S$ and $D$ waves. The Pauli forbidden states in the three-body system are treated by the exact orthogonalization method. The largest contributions to the ground and excited 2$_1^+$ bound states energies come from the partial waves $(λ, \ell)=(2,2)$ and $(λ, \ell)=(4,4)$. As was found earlier, these bound states are created by the critical eigenstates of the three-body P…
When two heavy ions near the Fermi energy collide, a warm and low-density region can form in which fragments appear. This region is mainly dominated by proton (p) and alpha particles. In such an environment, the alphas interact with each other, and especially through strong resonances, form complex systems such as 8Be and 12C. Our experiments show that in the reactions 70(64)Zn(64Ni)+70(64)Zn(64Ni) at E/A=35 MeV/nucleon levels of 8Be appear around relative energies Eij=0.092 MeV, 3.03 MeV as well as above 10 MeV and 100 MeV. For the 3 alpha systems, multi resonance processes give rise to excit…
How do protons and neutrons bind to form nuclei? This is the central question of ab initio nuclear structure theory. While the answer may seem as simple as the fact that nuclear forces are attractive, the full story is more complex and interesting. In this work we present numerical evidence from ab initio lattice simulations showing that nature is near a quantum phase transition, a zero-temperature transition driven by quantum fluctuations. Using lattice effective field theory, we perform Monte Carlo simulations for systems with up to twenty nucleons. For even and equal numbers of protons and …
It is suggested that superfluid helium (condensate of 4-He atoms) may stimulate nuclear alpha-decay in a situation when an alpha-emitter moves through superfluid helium with fine-tuned velocity, so that the backward-emitted alpha-particle is at rest in the laboratory frame. It is shown that the probability of stimulated alpha-decay in this case may be sizable enough to be detected.
Nuclear clustering describes the appearance of structures resembling smaller nuclei such as alpha particles (4He nuclei) within the interior of a larger nucleus. While clustering is important for several well-known examples, much remains to be discovered about the general nature of clustering in nuclei. In this letter we present lattice Monte Carlo calculations based on chiral effective field theory for the ground states of helium, beryllium, carbon, and oxygen isotopes. By computing model-independent measures that probe three- and four-nucleon correlations at short distances, we determine the…
Since the pioneering discovery of molecular resonances in the 12C+12C reaction more than half a century ago a great deal of research work has been undertaken in alpha clustering. Our knowledge on physics of nuclear molecules has increased considerably and nuclear clustering remains one of the most fruitful domains of nuclear physics, facing some of the greatest challenges and opportunities in the years ahead. The occurrence of "exotic" shapes and Bose-Einstein alpha condensates in light N=Z alpha-conjugate nuclei is investigated. Various approaches of the superdeformed and hyperdeformed bands …
Knowledge on nuclear cluster physics has increased considerably since the pioneering discovery of 12C+12C resonances half a century ago and nuclear clustering remains one of the most fruitful domains of nuclear physics, facing some of the greatest challenges and opportunities in the years ahead. The occurrence of "exotic" shapes and/or Bose-Einstein alpha condensates in light N-Z alpha-conjugate nuclei is investigated. Evolution of clustering from stability to the drip-lines examined with clustering aspects persisting in light neutron-rich nuclei is consistent with the extension of the "Ikeda-…
Since the discovery of molecular resonances in $^{12}$C+$^{12}$C in the early sixties a great deal of research work has been undertaken to study alpha-clustering. Our knowledge on physics of nuclear molecules has increased considerably and nuclear clustering remains one of the most fruitful domains of nuclear physics, facing some of the greatest challenges and opportunities in the years ahead. Occurrence of "exotic" shapes and Bose-Einstein Condensates in light alpha-cluster nuclei are investigated. Various approaches of superdeformed/hyperdeformed shapes associated with quasimolecular resonan…
