Three-cluster Systems Research Articles

Spectroscopic amplitudes in microscopic three-cluster systems

I investigate three-body spectroscopic amplitudes in a microscopic three-cluster model. The total wave functions are described in the resonating group method (RGM) formalism based on cluster wave functions and on relative functions defined in the hyperspherical method. Core excitations are included. I develop a method aimed to determine three-body spectroscopic amplitudes and spectroscopic factors. This technique also provides three-body wave functions where the antisymmetrization is treated approximately. In this way, the various channels are orthogonal to each other and can be used without the antisymmetrization operator. This method is well known in two-cluster systems, and is based on the eigenvalues and the eigenvectors of the antisymmetrization operator. I illustrate the formalism with the $^{6}\mathrm{He}$, $^{11}\mathrm{Li}$, and $^{14}\mathrm{Be}$ nuclei where $^{9}\mathrm{Li}$ and $^{12}\mathrm{Be}$ excited states are taken into account.

Two- and three-cluster decays of light nuclei within a hyperspherical harmonics approach

We consider a set of three-cluster systems ($^{4}$He, $^{7}$Li, $^{7}$Be, $^{8}$Be, $^{10}$Be) within a microscopic model which involves the hyperspherical harmonics to represent intercluster motion. We selected such three-cluster systems which have at least one binary channel. Our aim is to study whether the hyperspherical harmonics are able and under what conditions to describe two-body channel(s) (nondemocratic motion) or they are suitable for describing three-cluster continuum only (democratic motion). It is demonstrated that a rather restricted number of the hyperspherical harmonics allows us to describe bound states and scattering states in two-body continuum for a three-cluster system.

Advances in theab initiodescription of nuclear three-cluster systems

We introduce the extension of the ab initio no-core shell model with continuum to describe three-body cluster systems. We present results for the ground state of 6He and show improvements with respect to the description obtained within the no-core shell model and the no-core shell model/resonating group methods.

Thomas-Ehrman effect in a three-body model: TheNe16case

The dynamic mechanism of the Thomas-Ehrman shift is studied in three-cluster systems by example of $^{16}$Ne and $^{16}$C isobaric mirror partners. We predict configuration mixings for $0^+$ and $2^+$ states in $^{16}$Ne and $^{16}$C. Large isospin symmetry breaking on the level of wave function component weights is demonstrated for these states and discussed as three-body mechanism of Thomas-Ehrman shift. It is shown that the description of the Coulomb displacement energies requires a consistency among three parameters: the $^{16}$Ne decay energy $E_T$, the $^{15}$F ground state energy $E_r$, and the configuration mixing parameters for the $^{16}$Ne/$^{16}$C $0^+$ and $2^+$ states. Basing on this analysis we infer the $^{15}$F $1/2^+$ ground state energy to be $E_r=1.39-1.42$ MeV.

Continuous spectrum of three-cluster systems

For the example of the 5H = 3H + n + n nucleus, the effect of the Pauli exclusion principle on the asymptotic behavior of wave functions for the continuous spectrum of three-cluster systems is studied within a microscopic approach. It is shown that, in the asymptotic limit, the Schrodinger equation for the wave function that describes the relative motion of tritium and two neutrons reduces to the multichannel problem of valence-neutron scattering on the 4H subsystem. The single-channel approximation within which the phase shift for neutron scattering on the 4H subsystem occurring in the lowest energy state is the only feature that characterizes the continuous spectrum of the 5H nucleus is proposed and implemented.

The role of the Pauli principle in three-cluster systems composed of identical clusters

Within the microscopic model based on the algebraic version of the resonating group method the role of the Pauli principle in the formation of continuum wave function of nuclear systems composed of three identical s-clusters has been investigated. Emphasis is placed upon the study of the exchange effects contained in the genuine three-cluster norm kernel. Three-fermion, three-boson, three-dineutron ( 3 d ′ ) and 3 α systems are considered in detail. Simple analytical method of constructing the norm kernel for 3 α system is suggested. The Pauli-allowed basis functions for the 3 α and 3 d ′ systems are given in an explicit form and asymptotic behavior of these functions is established. Complete classification of the eigenfunctions and the eigenvalues of the 12C norm kernel by the Be 8 = α + α eigenvalues has been given for the first time. Spectrum of the 12C norm kernel is compared to that of the 5H system.

Faddeev type of equations for three-cluster systems with full treatment of the Pauli principle

We analyze peculiarities of the Pauli principle in thrre-cluster system. We demonstrated that the antisymmetrization operator, being a source of the three-body interaction, can be decomposed into components involving either permutations between all three clusters, or permutations between two clusters only. Introducing this into the Faddeev equations, one obtains three alternative but equivalent formulations. These sets of equations are presented in operator form, for practical applications we will make use full set of the oscillator functions.

How the Pauli principle governs the decay of three-cluster systems

New approach to the problem of multichannel continuum spectrum of three-cluster systems composed of an s-cluster and two neutrons is suggested based on the discrete representation of a complete basis of allowed states of the multiparticle harmonic oscillator. The structure of the eigenfunctions and behavior of the eigenvalues of the three-cluster norm kernel are analyzed. Classification of the eigenvalues of the three-cluster systems with the help of eigenvalues of the two-body subsystem is suggested. Asymptotic boundary conditions for a three-cluster wave function in the continuum consistent with the requirements of the Pauli principle are established. Such asymptotic behavior corresponds rather to subsequent decay of the three-cluster system than to the so-called “democratic decay” associated with the hyperspherical harmonics. The 3H + n + n configuration of the 5H nucleus is considered in detail.

The Complex Scaling Method for Many-Body Resonances and Its Applications to Three-Body Resonances

Resonance phenomena in quantum physics are very familiar in many fields of physics, but we have not yet obtained a complete physical understanding, mathematical description or computational treatment, especially in the case of many-body resonances. Recently, in experimental developments concerning unstable nuclear physics and heavy-ion nuclear reactions, much interest has been concentrated on many-body resonance problems. In the last quarter century, theoretical and mathematical treatments of many-body resonances have experienced great development through application of the complex scaling method (CSM). We can now treat resonant states of three-body systems in the same way as those of twobody systems. In this article, starting from the definition of a resonant state and discussion of its norm, we present a summary of recent studies of CSM to treat many-body resonances and applications to three-body resonant states in two-neutron halo nuclei and three-cluster systems.

A microscopic three-cluster model in the hyperspherical formalism

The hyperspherical formalism is applied in the framework of the microscopic generator-coordinate method. Three-cluster systems are described with a single generator coordinate, the hyperradius. This model is a natural extension of previous microscopic theories, and can be applied to bound states and resonances in a range of nuclei exhibiting a cluster structure. As an illustrative example, the model is applied to 6He and 6Li. Density distributions and electric dipole transition strengths are calculated, including B( E2;0 +→2 +) in 6He. The results are consistent with a neutron halo structure for the ground state of 6He, and with a proton–neutron halo structure of its isobaric analog state in 6Li. No 1 − “soft dipole” resonance is found in 6He.

Polarization Effects on the $^3\overrightarrow{\rm He}(\overrightarrow{d},p)$4He Fusion Reaction in the 3/2+Resonance Region

A cross section enhancement of the 3 HE(d,p) 4 He fusion reaction in the 3/2 + resonance region is presented. It is pointed out that a residual interaction in a nuclear three-body system is generated by an atomic target with an electron-cloud, as is the case for atomic 3 He and the deuteron. A complete nuclear three-charged-particle Faddeev-type equation is derived for the first time in momentum space. Our formulation could be useful not only to investigate three-cluster systems with such a residual potential but also to obtain the astrophysical S-factor in the ultra low-energy region. The importance of a new renormalization amplitude is emphasized, which is missing in the previous renormalization method.

Three-Cluster Equation Using the Two-Cluster RGM Kernel

We propose a new type of three-cluster equation which uses two-cluster resonating-group-method (RGM) kernels. In this equation, the orthogonality of the total wave-function to two-cluster Pauli-forbidden states is essential to eliminate redundant components admixed in the three-cluster systems. The explicit energy-dependence inherent in the exchange RGM kernel is self-consistently determined. For bound-state problems, this equation is straightforwardly transformed to the Faddeev equation which uses a modified singularity-free T-matrix constructed from the two-cluster RGM kernel. The approximation of the present three-cluster formalism can be examined with more complete calculation using the three-cluster RGM. As a simple example, we discuss three di-neutron (3d') and 3 alpha systems in the harmonic-oscillator variational calculation. The result of the Faddeev calculation is also presented for the 3' system.

Nature of halo in light nuclei

A feature peculiar to light neutron-rich nuclei is that their lowest decay thresholds are only slightly above their ground states. Among them, 6He and 11Li are two most striking examples. The energy needed to break 6He (11Li) into an alpha particle (9Li) and two neutrons is about 1 MeV (300 keV). So small a value prompts one to construct their theory by analogy with the zero-range-nuclear-force approximation previously applied to the deuteron. A more detailed analysis shows, however, that the simple version of this approximation applied to systems that decay through a three-particle channel does not take into account some important features of these systems and requires significant improvements. First, with increasing distance between three particles, the potential energy decreases, in contrast to what is observed for binary systems, in inverse proportion to the hyperradius cubed. Second, the Pauli exclusion principle adds complexity even in the asymptotic domain, and we meet its demands in constructing the 6He and 11Li wave functions in the continuum. An approach is proposed to analyze weakly bound three-cluster systems that takes into account the aforementioned features and which describes correctly the experimentally observed structure of bound and unbound states above the threshold for three-particle decay.

ΛΛ 6 He and Λ 9 Be systems in the three-body cluster model treated on the basis of differential Faddeev equations

The ΛΛ 6 He and Λ 9 Be hypernuclei are treated as the S=0, T=0 (for the former) and S=1/2, T=0 (for the latter) bound states of the three-cluster systems ΛΛα and Λαα, respectively. The cluster-reduction method is used to solve the s-wave differential Faddeev equations for these systems. On the basis of the MT I–III model, the ΛΛ interaction potential is specified in the form \(V_{\Lambda \Lambda } = \frac{2}{3}V_{NN} \). Phenomenological potentials are used to describe Λα and αα interactions. The binding energies of the ΛΛ 6 He and Λ 9 Be hypernuclei and the parameters of low-energy Λ-hyperon and α-particle scattering on a Λ 5 He hypernucleus are calculated. It is shown that the proposed ΛΛ interaction potential makes it possible to reproduce faithfully the binding energy of the ΛΛ 6 He hypernucleus and that scattering in the Λ Λ 5 He system is similar to neutron scattering on a deuteron.

On the complete basis of Pauli-allowed states of three-cluster systems in the Fock–Bargmann space

In the Fock–Bargmann space, the complete harmonic-oscillator basis for the three-cluster system is constructed. The indices of the reduction U(6)⊇U(2)×U(3) are chosen as the quantum numbers. The Pauli-forbidden states are eliminated by the orthogonal transformation of basis functions. The basis states are obtained in terms of hypergeometric functions and the spherical Wigner functions. Their simple form allows one to solve the problem of calculating the matrix elements of the microscopic Hamiltonian needed for the study of three-cluster systems within the algebraic version of RGM.

Off-shell ambiguity and three-body force in the resonating group model

The off-shell ambiguity of the relative motion function in the resonating group model is discussed. It is shown that off-shell transformations can be used to enhance and reduce three-body force effects in three-cluster systems. The superposition principle of forces is discussed as a criterion for the selection of a physical off-shell behaviour of the relative motion function. As an example the three-cluster system npe − is treated in the framework of a six-quark plus electron resonating group model. It is shown that, at the energy of the deuteron, the three-body force is very small in the version of the resonating group model where the relative motion function is transformed with the square root of the norm operator whereas other types of relative motion functions which are discussed in literature, especially those with a short-distance node, give uncomfortably strong three-body forces.

Pauli-corrected Breit-Wigner formulae for compound resonances in two- and three-cluster systems

Breit-Wigner formulae for compound resonances in two- and three-cluster systems are derived. Analytic corrections due to the Pauli principle with regard to corresponding formulae for partial resonance widths and resonance pole shifts are given. In the case of a system of three composite particles a model is formulated in which sudden decay into three clusters is determined by the knowledge of the microscopical compound state of the metastable decaying nucleus. The derivation of the Breit-Wigner formulae in the case of a three-cluster system is based on the study of the asymptotic behavior of the full three-body Green's function. In the case of the two-body channel situation the formalism presented here is a modification of the Wildermuth-Benöhr reaction theory.

Effective interactions in two- and three-cluster systems

Effective interactions in two- and three-cluster systems

Importance of the three-body Pauli potential in three-cluster systems

The ground state energies of the 3α system and of theA-α-α system are calculated in the three-cluster fish bone optical model [1], It is found that the three-cluster Pauli potential is very strong in theM-version of the model and weak in the (off-shell transformed)¯M-version. Its influence on theΛ-α-α binding energy and ground state wave function is almost negligible in the¯M-version. The present results indicate that phenomenological three-cluster models with energy-independent two-cluster interactions are not necessarily in contradiction with more microscopic models. It seems to be important, however, to use potentials which contain in their off-shell behaviour both Pauli-principle and saturation character of the nuclear forces.

Chapter V. A Coupled Channel Variational Method for Microscopic Study of Reactions between Complex Nuclei

A precise generalization of the Kohn-Hulthen-Kato variational method for scattering matrix is made for the purpose of microscopic study of reactions between complex nuclei with many channels coupled. The frameworks is given by the resonating group method (RGM) both for two-cluster systems and for three-cluster systems. The scattering boundary condition is imposed on the trial functions of the dynamical coordinate space, while the requires matrix elements between the RGM trial basis functions for the internal region are described in terms of the generator coordinate (GC) kernels and the corresponding GC trial basis functions. Detailed numerical tests are exhibited with showing very good accuracy and sufficiently short computational time in the result of the present method, which is encouraging enough to apply the method to coupled channel studies of various-kind reactions based both on the microscopic models and on certain phenomenological models.