The application of the hyperspherical harmonic approach to the case of the N-d scattering problem below deuteron breakup threshold is described. The nuclear Hamiltonian includes two- and three-nucleon interactions, in particular the Argonne v_{18}, the N3LO-Idaho, and the V_{low k} two-nucleon, and the Urbana IX and N2LO three-nucleon interactions. Some of these models are local, some are non-local. Also electromagnetic effects are included. Accurate calculations for many scattering observables at various center-of-mass energies are performed and the results are compared with the available experimental data. Furthermore, a χ^2 analysis of some of the Hamiltonian models has been performed to compare their capability to describe the scattering process.
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 allow…
With the goal of using chiral interactions at various orders to explore properties of the few-body nuclear systems, we write the recently developed local chiral interactions as spherical irreducible tensors and implement them in the hyperspherical harmonics expansion method. We devote particular attention to three-body forces at next-to-next-to leading order, which play an important role in reproducing experimental data. We check our implementation by benchmarking the ground-state properties of $^3$H, $^3$He and $^4$He against the available Monte Carlo calculations. We then confirm their order…
We have studied the solution of the six-nucleon bound state problem using the hyperspherical harmonic (HH) approach. For this study we have considered only two-body nuclear forces. In particular we have used a chiral nuclear potential evolved with the similarity renormalization group unitary transformation. A restricted basis has been selected by performing a careful analysis of the convergence of different HH classes. Finally, the binding energy and other properties of ${}^6$Li ground state are calculated and compared with the results obtained by other techniques. Then, we present a calculati…
A one-dimensional system of bosons interacting with contact and single-Gaussian forces is studied with an expansion in hyperspherical harmonics. The hyperradial potentials are calculated using the link between the hyperspherical harmonics and the single-particle harmonic-oscillator basis while the coupled hyperradial equations are solved with the Lagrange-mesh method. Extensions of this method are proposed to achieve good convergence with small numbers of mesh points for any truncation of hypermomentum. The convergence with hypermomentum strongly depends on the range of the two-body forces: it…
The non-symmetrized hyperspherical harmonics method for a three-body system, composed by two particles having equal masses, but different from the mass of the third particle, is reviewed and applied to the $^3$H, $^3$He nuclei and $^3_Λ$H hyper-nucleus, seen respectively as $nnp$, $ppn$ and $NNΛ$ three-body systems. The convergence of the method is first tested in order to estimate its accuracy. Then, the difference of binding energy between $^3$H and $^3$He due to the difference of the proton and the neutron masses is studied using several central spin-independent and spin-dependent potential…
The Hyperspherical Harmonics (HH) method is one of the most accurate techniques to solve the quantum mechanical problem for nuclear systems with $A\le 4$. In particular, by applying the Rayleigh-Ritz or Kohn variational principle, both bound and scattering states can be addressed, using either local or non-local interactions. Thanks to this versatility, the method can be used to test the two- and three-nucleon components of the nuclear interaction. In the present review we introduce the formalism of the HH method, both for bound and scattering states. In particular, we describe the implementat…
We demonstrate the ability to calculate electromagnetic sum rules with the \textit{ab initio} symmetry-adapted no-core shell model. By implementing the Lanczos algorithm, we compute non-energy weighted, energy weighted, and inverse energy weighted sum rules for electric monopole, dipole, and quadrupole transitions in $^4$He using realistic interactions. We benchmark the results with the hyperspherical harmonics method and show agreement within $2σ$, where the uncertainties are estimated from the use of the many-body technique. We investigate the dependence of the results on three different int…
A program to calculate the three-particle hyperspherical brackets is presented. Test results are listed and it is seen that the program is well applicable up to very high values of the hypermomentum and orbital momenta. The listed runs show that it is also very fast. Applications of the brackets to calculating interaction matrix elements and constructing hyperspherical bases for identical particles are described. Comparisons are done with the programs published previously.
Active-learning construction of hyperspherical-harmonics spaces for A = 3
European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Fondazione Bruno Kessler（欧洲核物理及相关领域理论研究中心 (ECT*)，布鲁诺·克勒斯勒基金会）；Data Science for Industry and Physics (DSIP), Fondazione Bruno Kessler（工业与物理学数据科学 (DSIP)，布鲁诺·克勒斯勒基金会）；Dipartimento di Fisica “E. Fermi”, Università di Pisa（比萨大学“E. 费米”物理系）；INFN-Sezione di Pisa（意大利国家核物理研究所比萨分部）
In this paper, we presented the ground state energies calculated for some single and double cascade hyperon($Ξ^{-}$) hypernuclei. For hypernuclei of the type $_Ξ^{A_c+1}$X we applied $Ξ+^A_c$X two-body model and for hypernuclei of the type $_{ΞΞ}^{A_c+2}$X we adopted $Ξ+Ξ+^A_c$X three-body model. A two-term Gaussian (Isle type) potential with adjustable parameters is chosen as the $Ξ$-nucleus interaction, while a two-term Yukawa-type potential has been chosen for the $ΞΞ$ pair. In the three-body model calculation, we adopted the hyperspherical harmonics expansion (HHE) formalism. The resulting…
Within the hyperspherical harmonics approach the three-body problem is reduced to a motion of one effective particle in a "strongly deformed" field, which is described in coupled-channel formalism. This method is especially suited to studies of phenomena characterized by genuine three-body dynamics, e.g. Borromean haloes and true three-body decays. The reduction of the hyperspherical equations set to a single-channel Schrödinger equation provides the basis for the use of the standard quasiclassical expression for calculations of widths for true three-body decays. We demonstrate that the quasic…
We study the 9 Be ground-state energy with non-local $α-$n and $α-α$ potentials derived from Cluster Effective Field Theory. The short-distance dependence of the interaction is regulated with a momentum cutoff. The potential parameters are fitted to reproduce the scattering length and effective range. We implement such potential models in a Non-Symmetrized Hyperspherical Harmonics (NSHH) code in momentum space. In addition we calculate ground state energies of various alpha nuclei. Work is in progress on a calculation of the photodisintegration of 9Be with the Lorentz Integral Transform (LIT) …
