Hyperspherical Adiabatic Expansion Research Articles

Few-body structures in the mirror nuclei O11 and Li11

We investigate the dripline mirror nuclei, $^{11}$Li and $^{11}$O, located on the neutron and proton dripline, respectively. We calculate the lowest four states, $3/2^-$, $1/2^+$, $3/2^+$ and $5/2^+$, built on double occupancy in the nuclear $s_{1/2}$ and $p_{1/2}$ valence single-particle states. We use the hyperspherical adiabatic expansion method to solve the three-body problem for a frozen nuclear core surrounded by two identical nucleons. The four analogue states in $^{11}$O are obtained with precisely the same interactions as used for the four states in $^{11}$Li, except for addition of the Coulomb interaction from the charge of the substituted valence protons. Surprisingly the four energies deviate from each other only by less than a few hundred keV. Any of them could then turn out to be the ground state, due to the uncertainty related to the angular momentum and parity dependence of the three-body potential. Still, our calculations marginally favor the $1/2^+$ state. The structures of these four states in $^{11}$O deviate substantially from the analogue states in the mirror, $^{11}$Li.

Combining Few-Body Cluster Structures with Many-Body Mean-Field Methods

Nuclear cluster physics implicitly assumes a distinction between groups of degrees-of-freedom, that is the (frozen) intrinsic and (explicitly treated) relative cluster motion. We formulate a realistic and practical method to describe the coupled motion of these two sets of degrees-of-freedom. We derive a coupled set of differential equations for the system using the phenomenologically adjusted effective in-medium Skyrme type of nucleon-nucleon interaction. We select a two-nucleon plus core system where the mean-field approximation corresponding to the Skyrme interaction is used for the core. A hyperspherical adiabatic expansion of the Faddeev equations is used for the relative cluster motion. We shall specifically compare both the structure and the decay mechanism found from the traditional three-body calculations with the result using the new boundary condition provided by the full microscopic structure at small distance. The extended Hilbert space guaranties an improved wave function compared to both mean-field and three-body solutions. We shall investigate the structures and decay mechanism of $^{22}$C ($^{20}$C+n+n). In conclusion, we have developed a method combining nuclear few- and many-body techniques without losing the descriptive power of each approximation at medium-to-large distances and small distances respectively. The coupled set of equations are solved self-consistently, and both structure and dynamic evolution are studied.

Emergent universality in the two-neutron halo structure of C22

The structure of the two-neutron halo $^{22}\mathrm{C}$ is investigated by means of a renormalized zero-range three-body model, with interactions in the $s$-wave channel, and a finite-range model with two- and three-body forces provided by the hyperspherical adiabatic expansion method. In both models the halo wave function in configuration space is obtained by using as inputs the two-body scattering lengths and the two-neutron separation energy. The halo-matter density is computed for $^{22}\mathrm{C}$ with different three-body forces and low-energy parameters, with two-neutron separation energy within the range $50\phantom{\rule{4.pt}{0ex}}\text{keV}\ensuremath{\le}{S}_{2n}\ensuremath{\le}1000$ keV. The halo-neutron density depends weakly on the neutron-$^{20}\mathrm{C}$ scattering length as long as its absolute value is larger than the neutron-neutron one. The halo-neutron density is then analyzed by means of the root-mean-square radius, the probability density, and also the geometry, taking into account the angle between the two Jacobi coordinates. The results of finite-range and zero-range two-neutron-core models are compared. The effects in the halo structure of short-range and long-range three-body forces are studied, and the emergent universal behavior of the halo-neutron density and its geometry is pointed out.

Radiative capture reaction forNe17formation within a full three-body model

Background: The breakout from the hot Carbon-Nitrogen-Oxigen (CNO) cycles can trigger the rp-process in type I x-ray bursts. In this environment, a competition between $^{15}\text{O}(\alpha,\gamma){^{19}\text{Ne}}$ and the two-proton capture reaction $^{15}\text{O}(2p,\gamma){^{17}\text{Ne}}$ is expected. Purpose: Determine the three-body radiative capture reaction rate for ${^{17}\text{Ne}}$ formation including sequential and direct, resonant and non-resonant contributions on an equal footing. Method: Two different discretization methods have been applied to generate $^{17}$Ne states in a full three-body model: the analytical transformed harmonic oscillator method and the hyperspherical adiabatic expansion method. The binary $p$--$^{15}$O interaction has been adjusted to reproduce the known spectrum of the unbound $^{16}$F nucleus. The dominant $E1$ contributions to the $^{15}\text{O}(2p,\gamma){^{17}\text{Ne}}$ reaction rate have been calculated from the inverse photodissociation process. Results: Three-body calculations provide a reliable description of $^{17}$Ne states. The agreement with the available experimental data on $^{17}$Ne is discussed. It is shown that the $^{15}\text{O}(2p,\gamma){^{17}\text{Ne}}$ reaction rates computed within the two methods agree in a broad range of temperatures. The present calculations are compared with a previous theoretical estimation of the reaction rate. Conclusions: It is found that the full three-body model provides a reaction rate several orders of magnitude larger than the only previous estimation. The implications for the rp-process in type I x-ray bursts should be investigated.

Three-Body Coulomb Functions in the Hyperspherical Adiabatic Expansion Method

In this work we describe a numerical method devised to compute continuum three-body wave functions. The method is implemented using the hyperspherical adiabatic expansion for the three-body wave function imposing a box boundary condition. The continuum energy spectrum results discretized and, for specific quantum number values, all the possible incoming and outgoing channels are simultaneously computed. For a given energy, the hyperradial continuum functions form a matrix whose ij-term refers to specific incoming and outgoing channels. When applied to three-body systems interacting only through the Coulomb potential, this method provides the adiabatic representation of the regular three-body Coulomb wave function. The computation of the irregular Coulomb wave function representation is also discussed. These regular and irregular Coulomb functions can be used to extract the \({\mathcal {S}}\)-matrix for those reactions where, together with some short-range potential, the Coulomb interaction is also present. The method is illustrated in the case of the \(3\rightarrow 3\) process of three alpha particles.

Rotational character of the12Cspectrum investigated through inelastic cross sections via photon emission

In this work the bremsstrahlung and photon dissociation cross sections for transitions between 0+ , 2+ , and 4+ states in 12 C are computed. The nucleus is described within the three-alpha model, and the wave functions are computed by means of the hyperspherical adiabatic expansion method. The continuum states are discretized by imposing a box boundary condition. The transition strengths are obtained from the cross sections, and compared to schematic rotational model predictions. The computed results strongly suggest that the two lowest bands are made, respectively, by the states with angular momentum and parity {01 + , 21 + , 42 + } and {02 + , 22 + , 41 + }. The transitions between the states in the first band are consistent with the rotational pattern corresponding to three alphas in an equilateral triangular structure. For the second band, the transitions are also consistent with a rotational pattern, but with the three alphas in an aligned distribution.

Three-body bremsstrahlung and the rotational character of theC12spectrum

The electric quadrupole transitions between ${0}^{+},{2}^{+}$, and ${4}^{+}$ states in $^{12}\mathrm{C}$ are investigated in a $3\ensuremath{\alpha}$ model. The three-body wave functions are obtained by means of the hyperspherical adiabatic expansion method, and the continuum is discretized by imposing a box boundary condition. Corresponding expressions for the continuum three-body $(3\ensuremath{\alpha})$ bremsstrahlung and photon dissociation cross sections are derived and computed for two different $\ensuremath{\alpha}\text{\ensuremath{-}}\ensuremath{\alpha}$ potentials. The available experimental energy dependence is reproduced and a series of other cross sections are predicted. The transition strengths are defined and derived from the cross sections, and compared to schematic rotational model predictions. The computed properties of the $^{12}\mathrm{C}$ resonances suggest that the two lowest bands are made, respectively, by the states ${{0}_{1}^{+},{2}_{1}^{+},{4}_{2}^{+}}$ and ${{0}_{2}^{+},{2}_{2}^{+},{4}_{1}^{+}}$. The transitions between the states in the first band are consistent with the rotational pattern corresponding to three alphas in an equilateral triangular structure. For the second band, the transitions are also consistent with a rotational pattern, but with the three alphas in an aligned arrangement.

Production of 6He and 9Be by radiative capture and four-body recombination

In this work we estimate the production rates for the three-body radiative capture processes α + n + n → 6He + γ and α + α + n → 9Be + γ, as well as the four-body recombination reactions α + α + n + n → 6He + α, α + n + n + n → 6He + n, α + α + n + n → 9Be + n and α + α + α + n → BeBe + α. These processes compete as a source of 6He and 9Be. The hyperspherical adiabatic expansion method is used. With this method no assumption is made about the capture mechanism. Both sequential and direct capture are included. The production rates for the radiative and the four-body recombination processes are found to be comparable for a mass density of about 107g/cm3 ( ∼ 1030 neutrons/cm3) and temperatures of a few GK.

Recombination rates from potential models close to the unitary limit

We investigate universal behavior in the recombination rate of three bosons close to threshold. Using the He-He system as a reference, we solve the three-body Schr\"odinger equation above the dimer threshold for different potentials having large values of the two-body scattering length $a$. To this aim, we use the hyperspherical adiabatic expansion and we extract the $S$ matrix through the integral relations recently derived. The results are compared to the universal form, $\ensuremath{\alpha}\ensuremath{\approx}67.1{\mathrm{sin}}^{2}[{s}_{0}\mathrm{ln}({\ensuremath{\kappa}}_{*}a)+\ensuremath{\gamma}]$, for different values of $a$ and selected values of the three-body parameter ${\ensuremath{\kappa}}_{*}$. A good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections, which have been recently proposed [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. Furthermore, we analyze the validity of the above formula in the description of a very different system: neutron-neutron-proton recombination. Our analysis confirms the universal character of the process in systems of very different scales having a large two-body scattering length.

Three-particle decays of light-nuclei resonances

We have studied the three-particle decay of 12C, 9Be and 6Be resonances. These nuclei have been described as three-body systems by means of the complex scaled hyperspherical adiabatic expansion method. The short-distance part of the wave function is responsible for the energies, whereas the information related to the observable decay properties is contained at large distances, which must be computed accurately. As an illustration we show the results for the angular distribution of 9Be and 6Be resonances.

General integral relations for the description of scattering states using the hyperspherical adiabatic basis

In this work we investigate $1+2$ reactions within the framework of the hyperspherical adiabatic expansion method. With this aim two integral relations, derived from the Kohn variational principle, are used. A detailed derivation of these relations is shown. The expressions derived are general, not restricted to relative $s$ partial waves, and with applicability in multichannel reactions. The convergence of the $\mathcal{K}$ matrix in terms of the adiabatic potentials is investigated. Together with a simple model case used as a test for the method, we show results for the collision of a $^{4}\mathrm{He}$ atom on a $^{4}\mathrm{He}$${}_{2}$ dimer (only the elastic channel open), and for collisions involving a $^{6}\mathrm{Li}$ and two $^{4}\mathrm{He}$ atoms (two channels open).

Integral Relations for Multichannel Reactions

Multichannel scattering reactions can be described through two integral relations based on the Kohn variational principle. When used in combination with the hyperspherical adiabatic expansion method the pattern of convergence of the corresponding \({{\mathcal K}}\)-matrix is similar to the one found when applied to obtain bound states.

Three-body structure of low-lying 18Ne states

We investigate to what extent 18Ne can be descibed as a three-body system made of an inert 16O-core and two protons. We compare to experimental data and occasionally to shell model results. We obtain three-body wave functions with the hyperspherical adiabatic expansion method. We study the spectrum of 18Ne, the structure of the different states and the predominant transition strengths. Two 0+, two 2+, and one 4+ bound states are found where they are all known experimentally. Also one 3+ close to threshold is found and several negative parity states, 1-, 3-, 0-, 2-, most of them bound with respect to the 16O excited 3- state. The structures are extracted as partial wave components, as spatial sizes of matter and charge, and as probability distributions. Electromagnetic decay rates are calculated for these states. The dominating decay mode for the bound states is E2 and occasionally also M1.

Binding He atoms to hydrogen moieties: quantum features from ultraweak interactions

The possible existence of bound states in small helium clusters containing atomic and molecular hydrogen has been investigated by means of three completely independent sets of calculations. The first set employs the Jacobi–Davidson filtering procedure recently implemented in the Distributed Gaussian Functions (DGF) approach, the second one is based on the solution of the Faddeev equation by means of the hyperspherical adiabatic expansion method and the third follows the Quantum Monte Carlo approach. The partners within each complex are taken to interact via two-body potentials chosen among the most accurate existing in the literature. All the present quantum treatments show that no bound states are found for H(4He)2 and for D(4He)2 while only one very diffuse bound state is present in T(4He)2, where H, D and T are the three isotopes of hydrogen. The substitution of the atomic dopant with its molecular counterpart, H2, also gives rise to only one bound state whose spatial attributes are reported and analysed, underlining again the ultraweak nature of the interaction forces and the need to implement highly specific tools for the investigation of the ensuing bound states.

Few-Body Reactions in Nuclear Astrophysics: application to6He and9Be production

In this work we obtain the astrophysical reaction and production rates for the two-particle radiative capture processes α + n + n → 6 He + γ and α + α + n → 9 Be + γ . The hyperspherical adiabatic expansion method is used. The four-body recombination reactions α + α + n + n → 6 He + α, α + n + n + n → 6 He + n, α + α + n + n → 9 Be + n and α + α + α + n → 9 Be + α are also investigated.

Astrophysical reaction rates for 6He and 9Be production by electromagnetic radiative capture and four-body recombination

The triple alpha reaction α+α+α → 12C + γ is considered to be the most relevant one in stars at the helium burning stage, since it permits to bridge the A=5 and A=8 instability gaps. However, at higher temperatures and neutron rich environments, other reactions can also play a relevant role. In this work we investigate the astrophysical reaction and production rates for the two-particle radiative capture processes α+n+n → 6He+γ and α+α+n → 9Be+γ. The hyperspherical adiabatic expansion method is used. With this method no assumption is made about the capture mechanism. The four-body recombination reactions α+α+n+n → 6He + α, α + n + n + n → 6He + n, α+α+n+n → 9Be + n and α + α + α + n → 9Be + α are also investigated as a possible alternative as a source of 6He and 9Be.

Integral Relations for Three-Body Continuum States with the Adiabatic Expansion

The application of the hyperspherical adiabatic expansion to describe three-body scattering states suffers from the problem of very slow convergence. Contrary to what happens for bound states, a huge number of hyper-radial equations has to be solved, and even if done, the extraction of the scattering amplitude is problematic. In this Letter we show how to obtain accurate scattering phase shifts using the hyperspherical adiabatic expansion. To this aim two integral relations, derived from the Kohn variational principle, are used. The convergence of this procedure is as fast as for bound states.

Continuum three-body states using the hyperspherical adiabatic basis set

In this paper we investigate the feasibility of employing the hyperspherical adiabatic (HA) basis set to describe continuum three-nucleon states. In particular, the HA expansion is compared with the simpler expansion on hyperspherical harmonics (HH). A practical numerical application is presented using the MT-III potential.

ALPHA-PARTICLE MOMENTUM DISTRIBUTIONS FROM 12C DECAYING RESONANCES

The computed α particle momentum distributions from the decay of low-lying 12 C resonances are shown. The wave function of the decaying fragments is computed by means of the complex scaled hyperspherical adiabatic expansion method. The large-distance part of the wave functions is crucial and has to be accurately calculated. We discuss energy distributions, angular distributions and Dalitz plots for the 4+, 1+ and 4- states of 12 C .

Momentum distributions of α particles from decaying low-lyingC12resonances

The complex scaled hyperspherical adiabatic expansion method is used to compute momentum and energy distributions of the three \ensuremath{\alpha} particles emerging from the decay of low-lying $^{12}\mathrm{C}$ resonances. The large distance continuum properties of the wave functions are crucial and must be accurately calculated. We discuss separately decays of natural-parity states: two ${0}^{+}$, one ${1}^{\ensuremath{-}}$, three ${2}^{+}$, one ${3}^{\ensuremath{-}}$, two ${4}^{+}$, one ${6}^{+}$, and one of each of unnatural parity ${1}^{+},{2}^{\ensuremath{-}},{3}^{+},{4}^{\ensuremath{-}}$. The lowest natural-parity state of each ${J}^{\ensuremath{\pi}}$ decays predominantly sequentially via the $^{8}\mathrm{Be}$ ground state, whereas other states, including unnatural-parity states, predominantly decay directly to the continuum. We present Dalitz plots and systematic detailed momentum correlations of the emerging \ensuremath{\alpha} particles.