Faddeev Formalism Research Articles

Neutron-neutron quasifree scattering in neutron-deuteron breakup at 10 MeV

New measurements of the neutron-neutron quasifree scattering cross section in neutron-deuteron breakup at an incident neutron energy of 10.0 MeV are reported. The experiment setup was optimized to evaluate the technique for determining the integrated beam-target luminosity in neutron-neutron coincidence cross-section measurements in neutron-deuteron breakup. The measurements were carried out with a systematic uncertainty of $\pm 5.6 \%$. Our data are in agreement with theoretical calculations performed using the CD-Bonn nucleon-nucleon potential in the Faddeev formalism. The measured integrated cross section over the quasifree peak is $20.5 \pm 0.5 \text{(stat)} \pm 1.1 \text{(sys)}$ mb/sr$^2$ in comparison with the theory prediction of 20.1 mb/sr$^{2}$. These results validate our technique for determining the beam-target luminosity in neutron-deuteron breakup measurements.

Core-excitation effects in three-body breakup reactions studied using the Faddeev formalism

Previous studies of $(d,p)$ reactions in three-body (proton, neutron, nuclear core) systems revealed a nontrivial effect of the core excitation: the transfer cross section cannot be factorized into the spectroscopic factor and the single-particle cross section obtained neglecting the core excitation. This observable, up to a kinematic factor, is the angular distribution of the core nucleus in the $(p,d)$ reaction. The study of the core excitation effect for the most closely related observable in the $(p,pn)$ three-body breakup, i.e., the core angular distribution, is aimed in the present work. Breakup of the one-neutron halo nucleus in the collision with the proton is described using three-body Faddeev-type equations extended to include the excitation of the nuclear core. The integral equations for transition operators are solved in the momentum-space partial-wave representation. Breakup of 11Be nucleus as well as of model $A=11$ $p$-wave nuclei is studied at beam energies of 30, 60, and 200 MeV per nucleon. Angular and momentum distributions for the 10Be core in ground and excited states is calculated. In sharp contrast to $(p,d)$ reactions, the differential cross section in most cases factorizes quite well into the spectroscopic factor and the single-particle cross section. Due to different reaction mechanisms the core excitation effect in the breakup is very different from transfer reactions. A commonly accepted approach to evaluate the cross section, i.e., the rescaling of single-particle model results by the corresponding spectroscopic factor, appears to be reliable for breakup though it fails in general for transfer reactions.

Two-neutron halo state of 15B around 3.48 MeV by a three-body model**Supported by National Natural Science Foundation of China (Grant No. 11535004, 11761161001, 11375086, 11120101005, 11175085 and 11235001), by the National Major State Basic Research and Development of China, Grant No. 2016YFE0129300, and by the Science and Technology Development Fund of Macau under Grant No. 068/2011/A.

We investigate low-lying bound states of the neutron-rich nucleus 15B by assuming it is a three-body system made of an inert core 13B and two valence neutrons. The three-body wave functions are obtained using the Faddeev formalism. Special attention is paid to the excited state at 3.48(6) MeV observed in the 13C(14C,12N)15B reaction, whose properties are less clear theoretically. In our three-body model, besides the ground state , a second state is discovered at around 3.61 MeV, which might be identified with the excited state observed at 3.48(6) MeV. We study this state in detail. It turns out to be a two-neutron halo state with a large matter radius .

Propagation of Uncertainties of the Nucleon-Nucleon Potential to the Neutron-Deuteron Elastic Scattering Cross Section

Faddeev formalism is used to study the propagation of theoretical uncertainties from the two-nucleon force to three-nucleon scattering observables. Predictions are obtained with the One-Pion-Exchange Gaussian interaction, for which correlations between its parameters are known. Within the Monte Carlo approach we are able to estimate uncertainties of three-nucleon observables arising from the imprecise knowledge of the One-Pion-Exchange Gaussian potential parameters. We found that the uncertainties of this type are small for three-nucleon elastic scattering cross section at investigated here energies up to the nucleon laboratory energy of 200 MeV. They remain smaller than the dominant theoretical uncertainty arising from using various models of nucleon-nucleon interactions. We also compare the above-mentioned results with other types of theoretical uncertainties, that is with the ones stemming from order truncation errors and regulator dependencies, present in calculations based on the chiral interaction.

The neutron-deuteron scattering problem in the framework of the Faddeev formalism

The three-body Faddeev equations in the configuration space are solved numerically for study of the neutron-deuteron scattering process above the breakup threshold. The amplitudes for the binary scattering process and breakup are obtained by the developed asymptotic approach.

Scaling behavior of scattering observables for three-body systems near the unitary limit

This contribution reports recent investigations on low-energy scaling properties of three-body systems, by considering elastic s–wave collisions of a particle in a bound-state formed by the remaining two-body system. First, some previous results for the case of the halo nucleus 20C will be revised, for the neutron–19C scattering properties near the critical condition for the occurrence of an excited bound state in 20C, within a neutron–neutron–18C configuration. In our approach, we consider the Faddeev formalism with renormalized zero-range two-body interactions. Next, by considering the actual possibilities for verification of low-energy scaling properties in cold-atom laboratories, the approach is extended to atomic strongly-mass-imbalanced three-body systems, with two identical heavy particles and a light one. In this case, we consider that the heavy particle is being scattered by the light-heavy weakly-bound dimer. Our preliminary results for scattering observable are evidencing the universal scaling features.

Operator form of the three-nucleon scattering amplitude

To extend the applications of the so-called ``three-dimensional'' formalism to the description of three-nucleon scattering within the Faddeev formalism, we develop a general form of the three-nucleon scattering amplitude. This form significantly decreases the numerical complexity of the ``three-dimensional'' calculations by reducing the scattering amplitude to a linear combination of momentum-dependent spin operators and scalar functions of momenta. The number and structure of the spin operators is fixed and the scalar functions can be represented numerically using standard methods such as multidimensional arrays. In this paper, we show that all orders of the iterated Faddeev equation can be written in this general form. We argue that calculations utilizing the three-nucleon force will also conform to the same general form. Additionally, we show how the general form of the scattering amplitude can be used to transform the Faddeev equation to make it suitable for numerical calculations using iterative methods.

Reaction theory: Status and perspectives

The current status of the reaction theory of nuclear collisions involving weakly-bound exotic nuclei is presented. The problem is addressed within the Continuum Discretized Coupled Channel (CDCC) framework, recalling its foundations and applications, as well as its connection with the Faddeev formalism. Recent developments and improvements of the method, such as core and target excitations and the extension to three-body projectiles, are presented. The use of the CDCC wave function in the calculation of inclusive breakup reactions is also introduced.

The general configuration-space Faddeev formalism for studying pd scattering

The configuration-space Faddeev equations are derived for p-d scattering taking into account the difference in interaction between the participant particles. Appropriate modifications have been made in the well-known configuration-space equations for n-d scattering. To show the effect of these modifications, the s-wave calculations are performed for bound state and scattering problems. We model the charge symmetry breaking effect for 3 H and 3 He with a modified Malfliet-Tjon MT I-III potential. Results obtained for elastic n-d and p-d scattering at Elab =14.1 MeV are compared with our prediction (Ref. [1]) and those of the Los-Alamos/Iowa group (Ref. [2]) .

Scaling functions of two-neutron separation energies of20Cwith finite range potentials

The behaviour of an Efimov excited state is studied within a three-body Faddeev formalism for a general neutron-neutron-core system, where neutron-core is bound and neutron-neutron is unbound, by considering zero-ranged as well as finite-ranged two-body interactions. For the finite-ranged interactions we have considered a one-term separable Yamaguchi potential. The main objective is to study range corrections in a scaling approach, with focus in the exotic carbon halo nucleus $^{20}C$.

Configuration space Faddeev formalism for system of non-identical particles: three-body model for $^6_\Lambda$He

Configuration space Faddeev formalism for system of non-identical particles: three-body model for $^6_\Lambda$He

The double-Λ hypernucleus ΛΛ 11 Be

The ground-state energy of the double-Λ hypernucleus ΛΛ 11 Be is calculated within the frame-work of the five-body model ααnΛΛ. The five-body nucleus is described using the Faddeev formalism for the unequal mass clusters involved and the potential harmonics expansion of the corresponding amplitudes. The resulting coupled differential equations depend only on the inter-cluster interactions, ΛΛ, Λn, Λα, nα, and αα, which are fairly well known from previous investigations of hypernuclei. The results obtained are in excellent agreement with the recent KEK-E373 experiment as well as with the theoretical results obtained within the Gaussian expansionmethod by Hiyama et al. The ΛΛnn double hypernucleus is also discussed.

Low-Energy 12C States in Three-α Model

Cross sections for three-α photodisintegration process of the12C nucleus are calculated by a Faddeev formalism, from which 0+- and 2+ low-lying resonant 12C states are studied. The triple-alpha reaction rate is evaluated from the calculated cross sections of the disintegration process.

New treatment of breakup in few-body systems

A new approach to determining breakup amplitudes in few-body systems in the context of a Faddeev formalism based on lattice discretization of a continuum is described. Due to such discretization and use of finite-dimensional representations for all operators in the kernels of integral equations, breakup in few-body systems is interpreted as a partial case of multi-channel scattering and corresponds to transitions between the states of the discretized continuum of an asymptotic channel Hamiltonian. The case study is based on amplitudes of three-nucleon breakup n + d → n + n + p with semi-realistic NN interaction potentials.

Universal structure of the three-body system

A nonlocal energy-dependent two-body quasipotential (E2Q) is defined within the framework of the three-body Faddeev formalism. The Fourier transform of this E2Q generates an energy-dependent Yukawa-type local potential. After an appropriate average of the potential with respect to the energy, a variety of local potentials with different ranges are obtained. These include a Yukawa potential, a Van der Waals potential, and a $1/{r}^{2}$ potential. An interesting potential appears in the $\ensuremath{\pi}$$\mathrm{NN}$ system, which gives rise to the one-pion-exchange $\mathrm{NN}$ interaction. It is also found that the Yukawa potential is automatically accompanied by an additional longer-range interaction. This potential could give rise to an infinite number of bound states near the threshold above the deuteron bound state with a more interesting physics than the Efimov effect.

Applying Faddeev equations to the n-d scattering problem

The problem of n-d scattering above the breakup threshold is investigated using the three-body Faddeev formalism. The method for solving the configuration-space Faddeev equations is based on a finite-basis set expansion of the Faddeev components. The finite basis appears from a hyperangle-dependent operator. The method is applied to solve a boundary problem describing the scattering process, allowing us to obtain scattering parameters from the asymptotic representation of the wave function without its reconstructing over the entire configuration space. For the Faddeev s-wave equation, the values of the amplitudes of elastic scattering and breakup are calculated for the states with full spin: S = 1/2, 3/2.

Testing the continuum-discretized coupled channels method for deuteron-induced reactions

The continuum-discretized coupled channels (CDCC) method is a well established theory for direct nuclear reactions which includes breakup to all orders. Alternatively, the three-body problem can be solved exactly within the Faddeev formalism which explicitly includes breakup and transfer channels to all orders. With the aim to understand how CDCC compares with the exact three-body Faddeev formulation, we study deuteron-induced reactions on (i) ${}^{10}$Be at ${E}_{\mathrm{d}}=21.4,40.9\phantom{\rule{0.28em}{0ex}}\mathrm{and}\phantom{\rule{0.28em}{0ex}}71$ MeV; (ii) ${}^{12}$C at ${E}_{\mathrm{d}}=12\phantom{\rule{0.28em}{0ex}}\mathrm{and}\phantom{\rule{0.28em}{0ex}}56$ MeV; and (iii) ${}^{48}$Ca at ${E}_{\mathrm{d}}=56$ MeV. We calculate elastic, transfer, and breakup cross sections. Overall, the discrepancies found for elastic scattering are small with the exception of very backward angles. For transfer cross sections at low energy, $\ensuremath{\sim}$10 MeV/u, CDCC is in good agreement with the Faddeev-type results and the discrepancy increases with beam energy. In contrast, breakup observables obtained with CDCC are in good agreement with Faddeev-type results for all but the lower energies considered here.

Towards three-body unitarity inD+→K−π+π+

We assess the importance of final state interactions in $D^+ \rar K^- \p^+ \p^+$, stressing the consistency between two- and three-body interactions. The basic building block in the calculation is a $K\pi$ amplitude based on unitarized chiral perturbation theory and with parameters determined by a fit to elastic LASS data. Its analytic extension to the second sheet allows the determination of two poles, associated with the $\k$ and the $K^*(1430)$, and a representation of the amplitude based on them is constructed. The problem of unitarity in the three-body system is formulated in terms of an integral equation, inspired in the Faddeev formalism, which implements a convolution between the weak vertex and the final state hadronic interaction. Three different topologies are considered for the former and, subsequently, the decay amplitude is expressed as a perturbation series. Each term in this series is systematically related to the previous one and a re-summation was performed. Remaining effects owing to single and double rescattering processes were then added and results compared to FOCUS data. We found that proper three-body effects are important at threshold and fade away rapidly at higher energies. Our model, based on a vector weak vertex, can describe qualitative features of the modulus of the decay amplitude and agrees well with its phase in the elastic region.

Quark-Model Baryon-Baryon Interaction Applied to Neutron-Deuteron Scattering. II

The neutron-deuteron (nd) scattering is solved in the Faddeev formalism, employing the energy-independent version of the quark-model baryon-baryon interaction fss2. The differential cross sections and the spin polarization of the elastic scattering up to the neutron incident energy $E_n=65$ MeV are well reproduced without reinforcing fss2 with the three-body force. The vector analyzing-power of the neutron, $A_y(\theta)$, in the energy region $E_n$ < 25 MeV is largely improved in comparison with the predictions by the meson-exchange potentials, thus yielding a partial solution of the long-standing $A_y$-puzzle owing to the nonlocality of the short-range repulsion produced by the quark-model baryon-baryon interaction. The large Coulomb effect in the vector and tensor analyzing-powers in $E_n$ < 10 MeV is also analyzed based on the Vincent and Phatak method and recent detailed studies by other authors.

EFFECT OF AN OFF-SHELL TRANSFORMATION OF THE QUARK-MODEL NN INTERACTION IN THE NEUTRON-DEUTERON SCATTERING

The neutron-deuteron (nd) scattering is studied in the Faddeev formalism, employing the NN sector of the quark-model baryon-baryon interaction fss2. The energy-dependence of the NN interaction, inherent to the resonating-group formulation of two three-quark clusters, is eliminated by the standard off-shell transformation utilizing the square root of the normalization kernel. This procedure yields an extra nonlocality to the quark-model exchange kernel, which is very important to reproduce all the observables for the bound state and the elastic scattering below En ≤ 65 MeV . The deuteron breakup differential cross sections are also examined.