We have analyzed $^{3}$He scattering from $^{12}$C at 34.7 and 72 MeV in a coupled channel method with a double folding potential derived from the precise wave functions for the ground 0$^+$ state and $0_2^+$ (7.65 MeV)
Hoyle state, which has been suggested to be an $α$ particle condensate. It is found that strong refraction of $^3$He in the Hoyle state can be clearly seen in the experimental angular distribution at {\it low} incident energy region as an Airy minimum of the {\it pre-rainbow oscillations}.
At low densities, with decreasing temperatures, in symmetric nuclear matter alpha-particles are formed, which eventually give raise to a quantum condensate with four-nucleon alpha-like correlations (quartetting). Starting with a model of alpha-matter, where undistorted alpha particles interact via an effective interaction such as the Ali-Bodmer potential, the suppression of the condensate fraction at zero temperature with increasing density is considered. Using a Jastrow-Feenberg approach, it is found that the condensate fraction vanishes near saturation density. Additionally, the modification…
It is proved that alpha-particle states are well described by the Elliott SU(3) model. This result is used to analyze the alpha-particle condensation effect. It is shown that these states possess the basic attributes of the alpha-condensate and, also, the normal nuclear density on frequent occasions. The statistics of alpha-particles (and of arbitrary composite bosons) turns out to be something other than the Bose-Einstein, Fermi-Dirac statistics, and parastatistics.
The present understanding of the structure of the Hoyle state in 12C is reviewed. Most of the theoretical approaches to the Hoyle state are shortly summarized. The corresponding results are analyzed with respect to whether they give evidence to the alpha particle condensation structure of the Hoyle state (and other Hoyle-like states in heavier self-conjugate nuclei) or not.
The phase diagram of isospin-symmetric chemically equilibrated mixture of alpha particles and nucleons is studied in the mean-field approximation. Skyrme-like parametrization is used for the mean-field potentials as functions of partial densities of nucleons and alphas. We find that there is a threshold value a* of the parameter a(N-alpha) which describes the attractive interaction between alpha-particles and nucleons. At a(N-alpha)<a* the ground state of nuclear matter at zero temperature behaves as a pure system of interacting nucleons, whereas at a(N-alpha)>a* the nuclear ground state inclu…
A round up of the present status of the conjecture that n alpha nuclei form an alpha-particle condensate in excited states close to the n alpha threshold is given. Experiments which could demonstrate the condensate character are proposed. Possible lines of further theoretical developments are discussed.
The 4-alpha condensate state for ^{16}O is discussed with the THSR (Tohsaki-Horiuchi-Schuck-Roepke) wave function which has alpha-particle condensate character. Taking into account a proper treatment of resonances, it is found that the 4-alpha THSR wave function yields a fourth 0^+ state in the continuum above the 4-alpha-breakup threshold in addition to the three 0^+ states obtained in a previous analysis. It is shown that this fourth 0^+ ((0_4^+)_{THSR}) state has an analogous structure to the Hoyle state, since it has a very dilute density and a large component of alpha+^{12}C(0_2^+) config…
The onset of quartetting, i.e. alpha-particle condensation, in symmetric nuclear matter is studied with the help of an in-medium modified four nucleon equation. It is found that at very low density quartetting wins over pairing, because of the strong binding of the alpha-particles. The critical temperature can reach values up to around 6 MeV. Also the disappearance of alpha-particles with increasing density, i.e. the Mott transition, is investigated. In finite nuclei the Hoyle state, that is the 0_2^+ of 12C, is identified as an "alpha-particle condensate" state. It is conjectured that such st…
In order to discuss the spatial extention of the second 0+ state of 12C (Hoyle state), we analyze the inelastic form factor of electron scattering to the Hoyle state, which our 3 alpha condensate wave function reproduces very well like previous 3 alpha RGM/GCM models. The analysis is made by varying the size of the Hoyle state artificially. As a result, we find that only the maximum value of the form factor sensitively depends on its size, while the positions of maximum and minimum are almost unchanged. This size dependence is found to come from a characteristic feature of the transition densi…
We discuss the treatment of isovector pairing by an alpha-like quartet condensate which conserves exactly the particle number, the spin and the isospin. The results show that the quartet condensate describes accurately the isovector pairing correlations in the ground state of systems with an equal number of protons and neutrons
The ground state energy of ideal alpha-matter at T=0 is analyzed within the framework of variational theory of Bose quantum liquids. Calculations are done for three local alpha-alpha potentials with positive volume integrals and two-body correlation functions obtained from the Pandharipande-Bethe equation. The energy per particle of $\alp$ matter is evaluated in the cluster expansion formalism up to four-body diagrams, and using the HNC/0 and HNC/4 approximation for a Bose liquid. Inclusion of higher order terms in the cluster expansion of the condensate fraction are leading to a stronger depl…
There has been a recent flurry of interest in the possibility of condensates of $α$-particles in nuclei. In this letter we discuss occurrence conditions for such states. Using the quantality condition of Mottelson we show that condensates are only marginally expected in $α$-particle states. We proceed to demonstrate that few-body nuclear condensates are ill-defined, and emphasize the conflict between $α$-localization and $α$-condensate formation. We also explore the connection between Ikeda diagrams, linear chains, and Tonks-Girardeau gases. Our findings show that no new information is contain…
We construct a BCS-like model that combines nucleonic pairing correlations and possible quartic correlations of alpha-type in a single variational wave function and derive corresponding gap equations. In the approximation of large logarithms typical for the BCS approach, we show that the system reveals two possible types of a condensate which cannot coexist. If the alpha-type condensate prevails at N=Z, the growth of the neutron excess will naturally lead to the first order phase transition to the nucleon condensates.
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…
In order to study non-zero spin excitations of the recently proposed alpha-cluster condensation in the self-conjugate 4n nuclei, spatial deformation is introduced into the model wave function of the alpha-cluster condensate. The rotational band states of 8Be are investigated as a first step of a test case for the study of the deformation of the alpha-cluster condensate. Calculations reproduce well the binding energy of the 0+ ground state and also the excitation energy of the 2+ state. Our 0+ wave function is found to be almost exactly equal to the 0+ wave function obtained by the generator co…
It is shown that a dilute property of an $α$ particle condensate can be seen in the Airy structure of nuclear rainbow and prerainbow scattering. The dilute property of the Hoyle state of $^{12}$C with a developed $α$ cluster structure is discussed by studying refractive $^3$He+$^{12}$C and $α$+$^{12}$C scattering.
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…
The states with $J^π=0^+$, $2^+$, and $4^+$ of $^{12}$C with excitation energies less than about 15 MeV are investigated with the alpha condensate wave function with spatial deformation and by using the method of ACCC (analytic continuation in the coupling constant) which is necessary for a proper treatment of resonance states. The calculated energy and width of the recently observed $2_2^+$ state are found to be well reproduced. The obtained $2_2^+$ wave function has a large overlap with a single condensate wave function of $3α$ gas-like structure. The density distribution is shown to be almo…
In a recent paper, arXiv0712.1191 [nucl-th], Zinner and Jensen (ZJ) expressed strong doubts about the concept of alpha-particle condensation in finite nuclei. In this article we give a reply which, essentially, is point by point (but not in the order).
A new $α$-cluster wave function is proposed which is of the $α$-particle condensate type. Applications to $^{12}$C and $^{16}$O show that states of low density close to the 3 resp. 4 $α$-particle threshold in both nuclei are possibly of this kind. It is conjectured that all self-conjugate 4$n$ nuclei may show similar features.
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…
In this article we review the present status of alpha clustering in nuclear systems. First of all, an important aspect is condensation in nuclear matter. Second, the alpha container model, recently been proposed by Tohsaki-Horiuchi-Schuck-Roepke (THSR), will be outlined and the ensuing condensate aspect of the Hoyle state at 7.65 MeV in 12C will be investigated in some detail. After 15 years since the proposal of the alpha condensation concept a critical assessment of this idea will be given. Alpha gas states in other nuclei like 16O and 13C will be considered. The THSR wave function can also …
The theory of quartet condensation is further developed. The onset of quartetting in homgeneous fermionic matter is studied with the help of an in-medium modified four fermion equation. It is found that at very low density quartetting wins over pairing. At zero temperature, in analogy to pairing, a set of equations for the quartet order parameter is given. Contrary to pairing, quartetting only exists for strong coupling and breaks down for weak coupling. Reasons for this finding are detailed. In an application to nuclear matter, the critical temperature for alpha particle condensation can reac…
The equation of state and phase diagram of isospin-symmetric chemically equilibrated mixture of alpha particles and nucleons are studied in the mean-field approximation. The model takes into account the effects of Fermi and Bose statistics for nucleons and alphas, respectively. We use Skyrme-like parametrization of the mean-field potentials as functions of partial densities, which contain both attractive and repulsive terms. Parameters of these potentials are chosen by fitting known properties of pure nucleon- and pure alpha matter at zero temperature. The sensitivity of results to the choice …
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…
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.
In order to explore the 4 α-particle condensate state in 16O, we solve a full four-body equation of motion based on the 4 αOCM (Orthogonality Condition Model) in a large 4 αmodel space spanned by Gaussian basis functions. A full spectrum up to the 0_6^+ state is reproduced consistently with the lowest six 0^+ states of the experimental spectrum. The 0^+_6 state is obtained at about 2 MeV above the 4 αbreakup threshold and has a dilute density structure, with a radius of about 5 fm. The state has an appreciably large αcondensate fraction of 61 %, and a large component of α+12C(0_2^+) configurat…
Certain aspects of the recently proposed antisymmetrised alpha particle product state wave function, or THSR alpha cluster wave function, for the description of the ground state in 8Be, the Hoyle state in 12C, and analogous states in heavier nuclei, are elaborated in detail. For instance, the influence of antisymmetrisation in the Hoyle state on the bosonic character of the alpha particles is studied carefully. It is shown to be weak, so that bosonic aspects are predominant. The de Broglie wave length of alpha particles in the Hoyle state is shown to be much larger than the inter-alpha distanc…
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.
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 …
Four-particle correlations in fermion systems at finite temperatures are investigated with special attention to the formation of a condensate. Instead of the instability of the normal state with respect to the onset of pairing described by the Gorkov equation, a new equation is obtained which describes the onset of quartetting. Within a model calculation for symmetric nuclear matter, we find that below a critical density, the four-particle condensation (alpha-like quartetting) is favored over deuteron condensation (triplet pairing). This pairing-quartetting competition is expected to be a gene…
We briefly report the modern status of heavy quark sum rules (HQSR) based on stability criteria by emphasizing the recent progresses for determining the QCD parameters (alpha_s, m_{c,b} and gluon condensates)where their correlations have been taken into account. The results: alpha_s(M_Z)=0.1181(16)(3), m_c(m_c)=1286(16) MeV, m_b(m_b)=4202(7) MeV,<alpha_s G^2> = (6.49+-0.35)10^-2 GeV^4, < g^3 G^3 >= (8.2+-1.0) GeV^2 <alpha_s G^2> and the ones from recent light quark sum rules are summarized in Table 2. One can notice that the SVZ value of <alpha_s G^2> has been underestimated by a factor 1.6, <…
