Non-symmetrized Hyperspherical Harmonics Method for Non-equal Mass Three-Body Systems

Nannini, A.; Marcucci, L. E.

doi:10.3389/fphy.2018.00122

Cited by 9 publications

(9 citation statements)

References 23 publications

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“…Using this relation, it is found that B Λ ( 3 Λ H) = 0.237 MeV. The convergence behaviour shown in these results is identical to that in [49], where the Non-Symmetrized Hyperspherical Harmonics method was used in computing the binding energy of a triton. Using lambda-nucleon Gaussian potentials from [50], the computed hypertriton binding energy in [49] also showed a comparable convergence behaviour.…”

Section: Resultssupporting

confidence: 64%

“…The convergence behaviour shown in these results is identical to that in [49], where the Non-Symmetrized Hyperspherical Harmonics method was used in computing the binding energy of a triton. Using lambda-nucleon Gaussian potentials from [50], the computed hypertriton binding energy in [49] also showed a comparable convergence behaviour. In some applications of the hyperspherical harmonics method, convergence is usually accelerated by including either a pair correlation factor or a Jastrow correlation factor in Equation (17).…”

Section: Resultssupporting

confidence: 63%

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Faddeev calculations on lambda hypertriton with potentials from Gel'fand-Levitan-Marchenko theory

Meoto,

Lekala

2019

Preprint

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Binding energy and root-mean-square radius are computed for the ground state of the lambda hypertriton (T = 0, J π = 1/2 + ). The computations are carried out using spin-averaged lambdaproton and lambda-neutron potentials restored from theoretical scattering phases through Gel'fand-Levitan-Marchenko theory. The lambda hypertriton is treated as a three-body system consisting of lambda-proton, lambda-neutron and proton-neutron subsystems. In coordinate space, the dynamics of the system is described using the Differential Faddeev Equations in hyperspherical variables. Faddeev amplitudes are represented as hyperradial wavefunctions on a basis of hyperspherical polynomials. These hyperradial wavefunctions are further expanded on a basis of associated Laguerre polynomials, resulting in a system of coupled hyperradial equations. By solving the eigenvalue problem derived from this system of coupled hyperradial equations, the binding energy and rootmean-square matter radius computed are -2.462 MeV and 7.00 fm respectively.

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Section: Resultssupporting

confidence: 64%

Section: Resultssupporting

confidence: 63%

Faddeev calculations on lambda hypertriton with potentials from Gel'fand-Levitan-Marchenko theory

Meoto,

Lekala

2019

Preprint

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“…[5] for the mass difference effect for 3 H is 2 keV, and for 3 He is 7 keV calculated using a semi-realistic potential. Similar calculations [7] performed with different N N potentials resulted in the differences from 4 keV to 7 keV for the 3 H. Comparing both models, AAA and AAB, we assume that the mass difference adds uncertainty of several keV to AAA calculations for the 3 H binding energy when a comparison with experimental data is proposed. This uncertainty is above the numerical accuracy of ≤ 1 keV evaluated for the realistic AAA calculations in Refs.…”

Section: Introductionmentioning

confidence: 79%

Mass-Energy Equivalence in Bound Three-Nucleon Systems

Filikhin¹,

Suslov²,

Vlahovic³

2021

Preprint

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The mass defect formula reflects the equivalence of mass and energy for bound nuclear systems. We consider the three-nucleon systems 3 H and 3 He assuming the neutron and proton to be indistinguishable particles (AAA model) or taking into account realistic masses for neutrons and protons (AAB model). We discuss the disagreement between the AAA model and the mass defect formula, which is related naturally to the AAB model. Also, we demonstrate that ignoring the neutronproton mass difference leads to the uncertainty for AAA calculations of several keV that overlap the realistic calculations accuracy estimated before in the literature. To show another manifestation of the mass-energy equivalence, we perform AAA realistic calculations varying nucleon mass and scaling the depth of pair potential. The mass-energy compensation effect is demonstrated, and an approach based on the effective nucleon mass is proposed. Comparing the results of different authors, we show that the contribution of attractive three-body potential to the Hamiltonian can be compensated by increasing the nucleon mass. For the binding energy calculations, we apply the differential Faddeev equations.

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“…Different applications followed this procedure for bosons as well as for fermions (see Refs. [31,[50][51][52][53]). The advantage of eliminating the orthonormalization of the states has to be balanced by the fact that in this case one has to work with the full basis of HH functions, whose degenerancy rapidly increases with K and the number of particles A.…”

Section: [K]mentioning

confidence: 99%

“…( 48) the first order results of Eqs. ( 50) and (53). Such second-order calculation provides then a symmetric Kmatrix.…”

Section: The a = 3 And 4 Scattering Statesmentioning

confidence: 99%

The Hyperspherical Harmonics Method: A Tool for Testing and Improving Nuclear Interaction Models

et al. 2020

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The Hyperspherical Harmonics (HH) method is one of the most accurate techniques to solve the quantum mechanical problem for nuclear systems with A ≤ 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 implementation of the method to study the A = 3 and 4 scattering problem. Second, we present a selected choice of results of the last decade, most representative of the latest achievements. Finally, we conclude with a discussion of what we believe will be the most significant developments within the HH method for the next five-to-ten years.on Monte Carlo techniques, as the variational Monte Carlo (VMC) or the Green's function Monte Carlo (GFMC) methods (see Ref. [7], and references therein). There are then the methods linked to the shell model, as the no-core shell model (NCSM) or the realistic shell model (RSM), see Refs. [8] and [9,10], respectively. All these methods are quite powerful to study medium-mass nuclear bound states, but less accurate, apart from the GFMC and NCSM, for very light nuclei, as those with A = 3, 4. Furthermore, their extension to the scattering systems is not so trivial, and, in some cases, still not at reach.Restricting ourselves to the A = 3, 4 nuclear systems, both bound and scattering states, we have at hand very few accurate ab-initio methods, i.e. the Faddeev (Faddeev-Yakubovsky for A = 4) equations (FE) technique, solved in coordinate-or in momentum-space, the method based on the Alt-Grassberger-Sandhas (AGS) equations solved in momentum space, and the Hyperspherical Harmonics (HH) method presented here. We refer the reader to Refs. [11, 12] for the FE method in coordinate space, to Refs. [13,14] for the FE method in momentum space, to Refs. [15,16] for recent reviews on the AGS method. Clearly, each method has advantages and drawbacks. For instance, the FE method in momentum space can be applied to A = 3, 4 bound and scattering states in a wide energy range. However, the inclusion of the Coulomb interaction for charged particle scattering states is quite problematic. The FE method in coordinate space can handle the Coulomb interaction, but it has not yet been applied to scattering problems at very low-energy, and it has been applied only recently to study systems with larger A values [17]. It is though a method with in principle great possibilities of extension [12]. The AGS method, although working in momentum space, can handle the Coulomb interaction, and can be applied to a large variety of A = 3, 4 scattering states, in a wide energy range. However, the very low energy range, that of interest for nuclear astrophysics, i.e. below ∼ 100 keV, is still not accessible with the AGS method. The method has ...

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Non-symmetrized Hyperspherical Harmonics Method for Non-equal Mass Three-Body Systems

Cited by 9 publications

References 23 publications

Faddeev calculations on lambda hypertriton with potentials from Gel'fand-Levitan-Marchenko theory

Faddeev calculations on lambda hypertriton with potentials from Gel'fand-Levitan-Marchenko theory

Mass-Energy Equivalence in Bound Three-Nucleon Systems

The Hyperspherical Harmonics Method: A Tool for Testing and Improving Nuclear Interaction Models

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