Borromean Nuclei Research Articles

Structure and Decay at Rapid Proton Capture Waiting Points

We investigate the region of the nuclear chart around $A \simeq 70$ from a three-body perspective, where we compute reaction rates for the radiative capture of two protons. One key quantity is here the photon dissociation cross section for the inverse process where two protons are liberated from the borromean nucleus by photon bombardment. We find a number of peaks at low photon energy in this cross section where each peak is located at the energy corresponding to population of a three-body resonance. Thus, for these energies the decay or capture processes proceed through these resonances. However, the next step in the dissociation process still has the option of following several paths, that is either sequential decay by emission of one proton at a time with an intermediate two-body resonance as stepping stone, or direct decay into the continuum of both protons simultaneously. The astrophysical reaction rate is obtained by folding of the cross section as function of energy with the occupation probability for a Maxwell-Boltzmann temperature distribution. The reaction rate is then a function of temperature, and of course depending on the underlying three-body bound state and resonance structures. We show that a very simple formula at low temperature reproduces the elaborate numerically computed reaction rate.

Determination of a dineutron correlation in Borromean nuclei via a quasi-free knockout (p,pn) reaction

To discuss the dineutron correlation in the ground state, the quasi-free neutron knockout reaction on $^6$He is investigated.In the present work, the momentum distribution of the two emitted neutrons is calculated with the $\alpha$~+~$n$~+~$n$ three-body model and a simple reaction model to assess the effects of the knockout process via the $^5$He resonance and the target dependence in the momentum distribution. A clear signature of the dineutron correlation can be obtained by choosing the kinematical condition so the process via the $^5$He resonance is excluded because the $^5$He resonance drastically changes the momentum distribution. Using the proton target is important in the quantitative discussion on the dineutron correlation by the knockout reaction. In addition to theoretical arguments, a possible experimental setup to determine the dineutron correlation via the quasi-free knockout reaction is considered.

Coulomb breakup ofC22in a four-body model

Breakup cross sections are determined for the Borromean nucleus 22C by using a four-body eikonal model, including Coulomb corrections. Bound and continuum states are constructed within a 20C + n + n three-body model in hyperspherical coordinates. We compute continuum states with the correct asymptotic behavior through the R-matrix method. For the n+ n potential, we use the Minnesota interaction. As there is no precise experimental information on 21C, we define different parameter sets for the 20C + n potentials. These parameter sets provide different scattering lengths, and resonance energies of an expected 3/2+ excited state. Then we analyze the 22C ground-state energy and rms radius, as well as E1 strength distributions and breakup cross sections. The E1 strength distribution presents an enhancement at low energies. Its amplitude is associated with the low binding energy, rather than with a three-body resonance. We show that the shape of the cross section at low energies is sensitive to the ground-state properties. In addition, we suggest the existence of a low-energy 2+ resonance, which should be observable in breakup experiments.

Bound and unbound nuclear systems at the drip lines: a one-dimensional model

We construct a one-dimensional toy model to describe the main features of Borromean nuclei at the continuum threshold. The model consists of a core and two valence neutrons, unbound in the mean potential, that are bound by a residual point contact density-dependent interaction. Different discretization procedures are used (harmonic oscillator and transformed harmonic oscillator bases, or use of large rigid wall box). Resulting energies and wave functions, as well as inelastic transition intensities, are compared within the different discretization techniques, as well as with the exact results in the case of one particle and with the results of the di-neutron cluster model in the two particles case. Despite its simplicity, this model includes the main physical features of the structure of Borromean nuclei in an intuitive and computationally affordable framework, and will be extended to direct reaction calculations.

Exploring the Structure of the Most Neutron-rich Boron and Carbon Isotopes

The most neutron-rich isotopes of boron and carbon are of considerable interest in terms of their few-body character - most notably through the possible formation of twoneutron haloes in the Borromean nuclei 19 B and 22 C. Here a series of recent investigations are briefly reviewed, including preliminary results from reaction studies of 18 B and 21 C. It is concluded that 22 C exhibits a prominent halo arising from a significant ν 2S 1/2 2 valence neutron configuration. More generally, the structure of the dripline boron and carbon nuclei appears to be governed by the competition between the ν1d 5/2 and ν2s 1/2 single-particle orbitals, with a 5/2+ – 1/2+ level inversion very probably occuring in 21 C.

Current Status of Nuclear Physics Research

In this review we discuss the current status of research in nuclear physics which is being carried out in different centers in the World. For this purpose we supply a short account of the development in the area which evolved over the last 9 decades, since the discovery of the neutron. The evolution of the physics of the atomic nucleus went through many stages as more data become available. We briefly discuss models introduced to discern the physics behind the experimental discoveries, such as the shell model, the collective model, the statistical model, the interacting boson model, etc., some of these models may be seemingly in conflict with each other, but this was shown to be only apparent. The richness of the ideas and abundance of theoretical models attests to the important fact that the nucleus is a really singular system in the sense that it evolves from two-body bound states such as the deuteron, to few-body bound states, such as $^4$He, $^7$Li, $^9$Be etc. and up the ladder to heavier bound nuclei containing up to more than 200 nucleons. Clearly statistical mechanics does not work for such finite system, neither does other theories applicable to condensed matter systems. The richness of nuclear physics stems from these restrictions. New theories and models are presently being developed. Theories of the structure and reactions of neutron-rich and proton-rich nuclei, called exotic nuclei, halo nuclei, or Borromean nuclei deal with the wealth of experimental data available in the last 35 years. Further, nuclear astrophysics and stellar and Big Bang nucleosynthesis have become a more mature subject. Due to limited space, this review only covers a few selected topics, mainly those with which the authors have worked with.

Walking at the drip line

Among exotic nuclei those at the drip line which are unstable against neutron emission are particularly interesting because they convey information on the nuclear force in the most extreme situations. Strictly speaking they are not ''nuclei" but they exist thanks to long living resonances between a neutron and a bound ''core" nucleus. Adding one more neutron they become bound and are called "borromean". Being particularly exotic they have attracted much attention in past years, see for example Refs.[1, 2, 3]. One very challenging example is 13Be whose level ordering has been discussed in a large number of papers in which it has been studied by transfer [4] and fragmentation experiments [5]-[11], or it has been discussed theoretically[12]-[19]. Although projectile fragmentation spectra show evident similarities, the interpretations of data all differ from each other. In this paper we argue that a way trough the problem could be to try to establish first, or at the same time, the quite elusive "nature" of the second s-state in the Beryllium isotopes with A=9-14. On the other hand there are other recent neutron removal experiments leading to nuclei unstable by one or more proton emissions [20], and thus somewhat mirror to borromean nuclei, performed with nuclei close to the proton drip line. It has been shown that by taking in coincidence all (charged) particles but the removed neutron, reconstructing the invariant mass and gating on the ground state peak, it is possible to obtain the longitudinal momentum distribution of the unbound "core". One can link it to the original wave function of the bound orbital and thus determine the initial neutron angular momentum from the shape of the distribution and the initial occupation probability from the absolute removal cross section. Then it is clear that modern experiments and theories are able to study unstable nuclei with the same degree of accuracy as stable nuclei. Such a line of research offers a great potential for numerous further studies beyond the drip line.

Astrophysical reaction rate forBe9formation within a three-body approach

The structure of the Borromean nucleus $^9$Be ($\alpha+\alpha+n$) is addressed within a three-body approach using the analytical transformed harmonic oscillator method. The three-body formalism provides an accurate description of the radiative capture reaction rate for the entire temperature range relevant in Astrophysics. At high temperatures, results match the calculations based on two-step sequential processes. At low temperatures, where the particles have no access to intermediate two-body resonances, the three-body direct capture leads to reaction rates larger than the sequential processes. These results support the reliability of the method for systems with several charged particles.

Spatial periphery of lithium isotopes

The spatial structure of lithium isotopes is studied with the aid of the charge-exchange and (t, p) reactions on lithium nuclei. It is shown that an excited isobaric-analog state of 6Li (0+, 3.56MeV) has a halo structure formed by a proton and a neutron, that, in the 9Li nucleus, there is virtually no neutron halo, and that 11Li is a Borromean nucleus formed by a 9Li core and a two-neutron halo manifesting itself in cigar-like and dineutron configurations.

Analytical transformed harmonic oscillator basis for three-body nuclei of astrophysical interest: Application to6He

Recently, a square-integrable discrete basis, obtained performing a simple analytical local scale transformation to the harmonic oscillator basis, has been proposed and successfully applied to study the properties of two-body systems. Here, the method is generalized to study three-body systems. To test the goodness of the formalism and establish its applicability and limitations, the capture reaction rate for the nucleosynthesis of the Borromean nucleus ${}^{6}$He (${}^{4}\text{He}+n+n$) is addressed. Results are compared with previous publications and with calculations based on actual three-body continuum wave functions, which can be generated for this simple case. The obtained results encourage the application to other Borromean nuclei of astrophysical interest such as ${}^{9}$Be and ${}^{12}$C, for which actual three-body continuum calculations are very involved.

Momentum profile analysis in one-neutron knockout from Borromean nuclei

One-neutron knockout reactions from Borromean nuclei are analyzed using a profile function analysis technique. The profile function, which is derived as the square root of the variance of the measured fragment + neutron momenta as a function of their relative energy, is shown to be very sensitive to the angular momentum of the knocked out neutron. Three cases are studied here: 7He, where the profile function analysis shows a presence of (s1/2)2 component in the 8He ground-state wave-function, 10Li, where the presence of a 11(2)% d-wave contribution to the relative energy spectrum above 1.5 MeV is found and, finally, the presence of a major s contribution around 0.5 MeV relative energy in the 13Be case and that the observed decay to the 12Be 2+ state originates in a d state in 13Be.

Unbound nuclei studied by projectile fragmentation

The level ordering in the unbound nucleus 13Be is firmly established by using a time-dependent projectile fragmentation model, comparing to experimental data and using structure inputs obtained from a semi-phenomenological core-vibration coupling model of two-neutron halo nuclei. Thus, from the shell ordering, the neutron–core interaction is determined, which is needed by any three-body model of Borromean nuclei. As a consistency test, we show that the energy spectra of some beryllium isotopes obtained using the particle–particle random phase approximation predict the same shell ordering as those extracted from the reaction model versus data analysis.

New Measurements for 8He Excited States

Breakup reactions of the double borromean nucleus 8He were measured at 82.3 MeV/u on CH2 and C targets. The coincident detection of two decaying neutrons and the high performance for neutron cross talk rejection are realized in this experiment. The relative energy spectrum for 8He was reconstructed with good statistics. The spectrum exhibits a structure of two resonant peaks, one at an excitation energy of about 3.0 and the other at about 4.14 MeV, respectively. Substantially larger cross section for the first resonance is observed in comparison to the results reported previously.

Effect of proton-proton Coulomb repulsion on soft dipole excitations of light proton-rich nuclei

We perform three-body model calculations for soft dipole excitations of the proton-rich Borromean nucleus ${}^{17}$Ne. To this end, we assume that ${}^{17}$Ne takes the ${}^{15}\mathrm{O}+p+p$ structure, in which the two valence protons are excited from the ${0}^{+}$ ground state configuration to ${1}^{\ensuremath{-}}$ continuum states. We employ a density-dependent contact force for the nuclear part of the pairing interaction, and discretize the continuum states with the box boundary condition. We show by explicitly including the Coulomb interaction between the valence protons that the Coulomb repulsion does not significantly alter the $E1$ strength distribution. We point out that the effect of the Coulomb repulsion in fact can be well simulated by renormalizing the nuclear pairing interaction.

Two-particle correlations in continuum dipole transitions in Borromean nuclei

We use a three-body model to discuss the role of dineutron correlation in the ground state and the dipole excitations of typical weakly-bound Borromean nuclei, 11Li and 6He. We first show that the wave function for the neutron pair in the ground state is strongly concentrated on the nuclear surface for both nuclei. We then show that the energy distributions for the two emitted neutrons from the dipole excitation, on the other hand, are considerably different between the two nuclei, even though they show similar strong dineutron correlations in the ground state to each other. Our calculation indicates that this different behaviour of the energy distribution primarily reflects the interaction between the neutron and the core nucleus, rather than the interaction between the valence neutrons.

FOUR-BODY CONTINUUM-DISCRETIZED COUPLED-CHANNEL CALCULATIONS APPLIED TO 6He REACTIONS AROUND THE COULOMB BARRIER

The scattering of a weakly bound three-body system by a target is studied within the four-body continuum-discretized coupled-channels (4b-CDCC) framework. Two different methods, the transformed harmonic oscillator (THO) method and the binning procedure, are used for discretizing the three-body continuum. The formalism is applied to different reactions induced by the Borromean nucleus 6 He at energies around the Coulomb barrier: 6 He +64 Zn at 13.6 MeV, 6 He +120 Sn at 17.4 MeV, and 6 He +208 Pb at 22 MeV. Elastic cross section distributions are presented for these reactions comparing both discretization methods, THO and binning, as the mass of the target increases.

Pair and single neutron transfer with Borromean 8He

Direct observation of the survival of 199Au residues after 2n transfer in the He8+Au197 system and the absence of the corresponding 67Cu in the He8+Cu65 system at various energies are reported. The measurements of the surprisingly large cross sections for 199Au, coupled with the integral cross sections for the various Au residues, is used to obtain the first model-independent lower limits on the ratio of 2n to 1n transfer cross sections from 8He to a heavy target. A comparison of the transfer cross sections for 6,8He on these targets highlights the differences in the interactions of these Borromean nuclei. These measurements for the most neutron-rich nuclei on different targets highlight the need to probe the reaction mechanism with various targets and represent an experimental advance towards understanding specific features of pairing in the dynamics of dilute nuclear systems.

Constraints on two-neutron separation energy in the Borromean 22C nucleus

The recently extracted matter radius of carbon isotope 22C allows us to estimate the mean-square distance of a halo neutron with respect to the center-of-mass of this nucleus. By considering this information, we suggest an energy region for an experimental investigation of the unbound 21C virtual state. Our analysis, in a renormalized zero-ranged three-body model, also indicates that the two-neutron separation energy in 22C is expected to be found below ∼0.4 MeV, where the 22C is approximated by a Borromean configuration with a pointlike 20C and two s-wave halo neutrons. A virtual-state energy of 21C close to zero, would make the 22C, within Borromean nuclei configurations, the most promising candidate to present an excited bound Efimov state or a continuum three-body resonance.

Simultaneous analysis of elastic, breakup, and fusion channels for6He induced reactions at energies near the Coulomb barrier

We present some recent applications of the Continuum-Discretized Coupled-Channels (CDCC) and Coupled-Reaction-Channels (CRC) methods to the study of reactions induced by weakly-bound nuclei on stable targets. The methods are applied to the scattering of the Borromean nucleus 6 He at energies around the Coulomb barrier. Elastic scattering data are very well described by four-body CDCC calculations (based on a three-body description of the 6 He nucleus) but also by simpler three-body CDCC calculations, using a suitable two-body model for 6 He. By contrast, inclusivedata are better described using the CRC approach, which treats the breakup within the transfer to the continuum picture and, unlike the CDCC method, allows the inclusion of transfer to bound states of the target. We explore also the possibility of calculating the fusion cross sections using the CRC framework.

Erratum: Diproton correlation in the proton-rich Borromean nucleusNe17[Phys. Rev. C82, 024315 (2010)

We carry out three-body-model calculations for a proton-rich Borromean nucleus $^{17}$Ne by assuming a $^{15}$O + $p$ + $p$ structure. To this end, we use a density-dependent contact interaction between the valence protons, explicitly treating also the Coulomb interaction. We find that the two-particle density distribution for $^{17}$Ne is similar to that for $^{16}$C, which has two valence neutrons outside the $N$ = 8 core. That is, the two protons take a spatially compact configuration, while the Coulomb repulsion plays a minor role. This indicates that there is a strong diproton correlation in the ground state of the $^{17}$Ne nucleus. We also show that the Coulomb interaction reduces the expectation value of the proton-proton interaction by about 14\%.