Hyperspherical Harmonics Research Articles

Low energyn−H3scattering: A novel testground for nuclear interactions

Low energy $n\text{\ensuremath{-}}^{3}\mathrm{H}$ elastic cross sections near the resonance peak are calculated by solving the four-nucleon problem with realistic NN interactions. Three different methods---Alt, Grassberger, and Shandas; hyperspherical harmonics; and Faddeev-Yakubovsky---have been used and their respective results are compared. We conclude on a failure of the existing NN forces to reproduce the $n\text{\ensuremath{-}}^{3}\mathrm{H}$ total cross section.

Calculation of theα-particle ground state within the hyperspherical harmonic basis

The problem of calculating the four--nucleon bound state properties for the case of realistic two- and three-body nuclear potentials is studied using the hyperspherical harmonic (HH) approach. A careful analysis of the convergence of different classes of HH functions has been performed. A restricted basis is chosen to allow for accurate estimates of the binding energy and other properties of the 4He ground state. Results for various modern two-nucleon and two- plus three-nucleon interactions are presented. The 4He asymptotic normalization constants for separation in 2+2 and 1+3 clusters are also computed.

Quantum properties of many boson system: Path integral approach

We report for the first time a formulation for finding the path integral of many boson systems using hyperspherical harmonics. The accuracy of the method is established by comparison with analytical results of simple systems. The partition function, and hence several thermodynamic properties are also calculated using interactions that are relevant to Bose–Einstein condensation.

Electrodisintegration of 3He below and above deuteron breakup threshold

Recent advances in the study of electrodisintegration of 3He are presented and discussed. The pair-correlated hyperspherical harmonics method is used to calculate the initial and final state wave functions, with a realistic Hamiltonian consisting of the Argonne v18 two-nucleon and Urbana IX three-nucleon interactions. The model for the nuclear current and charge operators retains one- and many-body contributions. Particular attention is made in the construction of the two-body current operators arising from the momentum-dependent part of the two-nucleon interaction. Three-body current operators are also included so that the full current operator is strictly conserved. The present model for the nuclear current operator is tested comparing theoretical predictions and experimental data of pd radiative capture cross section and spin observables.

Method for solving the many-body bound state nuclear problem

We present a method based on hyperspherical harmonics to solve the nuclear many-body problem. It is an extension of accurate methods used for studying few-body systems to many bodies and is based on the assumption that nucleons in nuclei interact mainly via pairwise forces. This leads to a two-variable integro-differential equation which is easy to solve. Unlike methods that utilize effective interactions, the present one employs directly nucleon–nucleon potentials and therefore nuclear correlations are included in an unambiguous way. Three body forces can also be included in the formalism. Details on how to obtain the various ingredients entering into the equation for the A-body system are given. Employing our formalism we calculated the binding energies for closed and open shell nuclei with central forces where the bound states are defined by a single hyperspherical harmonic. The results found are in agreement with those obtained by other methods.

Potential harmonics expansion method for trapped interacting bosons: Inclusion of two-body correlation

We study a system of $A$ identical interacting bosons trapped by an external field by solving ab initio the many-body Schroedinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious problems due to the large degeneracy of HH basis, symmetrization of the wave function, calculation of the matrix elements, etc. for large $A$. Instead of the HH basis, here we use the "potential harmonics" (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in $3(A-1)$-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is extremely dilute. Unlike the $(3A-4)$ hyperspherical variables in HH basis, the PH basis involves only three {\it{active}} variables. It drastically reduces the number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any $A$. One can easily incorporate realistic atom-atom interactions in a straight forward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number.

Incorporation of Three-Nucleon Force in the Effective-Interaction Hyperspherical-Harmonic Approach

It is shown how a bare three-nucleon force is incorporated into the formalism of the effective interaction approach for hyperspherical harmonics. As a practical example we calculate the ground state properties of 3H and 3He using the Argonne V18 nucleon-nucleon potential and the Urbana IX three-nucleon force. A very good convergence of binding energies and matter radii is obtained. We also find a very good agreement of our results compared to other high precision calculations.

Structure of theLi11continuum from breakup on proton target

Inelastic scattering of $^{11}\mathrm{Li}$ on proton targets at energy of $68\phantom{\rule{0.3em}{0ex}}\mathrm{MeV}∕\text{nucleon}$ has been studied in the microscopic four-body distorted-wave model. The ground state and three-body continuum excitations of $^{11}\mathrm{Li}$ were calculated in the $^{9}\mathrm{Li}+n+n$ cluster model using the method of hyperspherical harmonics. The theory describes the experimental data well and clarifies contributions from the three-body monopole resonance in $^{11}\mathrm{Li}$ and from its soft dipole excitation. Various aspects of the $^{11}\mathrm{Li}$ excitations and their nature are discussed in terms of correlation plots.

Ab initio calculation of 7Li photodisintegration

The 7Li total photoabsorption cross section is calculated microscopically, using the semi-realistic AV4' nucleon–nucleon potential. The interaction of the final 7-nucleon system is fully taken into account via the Lorentz integral transform (LIT) method. For the calculation of the LIT we use expansions in hyperspherical harmonics (HH) in conjunction with the HH effective interaction (EIHH) approach. The convergence of the LIT expansion is discussed in detail. The calculated cross section agrees quite well with the available experimental data, which cover an energy range from threshold up to 100 MeV.

Modern theories of low-energy astrophysical reactions

We summarize recent ab initio studies of low-energy electroweak reactions of astrophysical interest, relevant for both big bang nucleosynthesis and solar neutrino production. The calculational methods include direct integration for np radiative and pp weak capture, correlated hyperspherical harmonics for reactions of A=3,4 nuclei, and variational Monte Carlo for A=6,7 nuclei. Realistic nucleon–nucleon and three-nucleon interactions and consistent current operators are used as input.

Modern theories of low-energy astrophysical reactions

We summarize recent ab initio studies of low-energy electroweak reactions of astrophysical interest, relevant for both big bang nucleosynthesis and solar neutrino production. The calculational methods include direct integration for np radiative and pp weak capture, correlated hyperspherical harmonics for reactions of A = 3 , 4 nuclei, and variational Monte Carlo for A = 6 , 7 nuclei. Realistic nucleon–nucleon and three-nucleon interactions and consistent current operators are used as input.

Hyperspherical Harmonics for Tetraatomic Systems. 2. The Weak Interaction Region

Hyperspherical harmonics as functions of hyperspherical coordinates appropriate for describing the weak interaction region of nuclear configuration space of tetraatomic systems are derived. They constitute an efficient basis set for calculating the corresponding local hyperspherical surface functions that avoid overcompleteness problems. Together with the hyperspherical harmonics for the strong interaction region of that space obtained previously, they should permit efficient benchmark-quality calculations of state-to-state differential and integral cross sections for those systems.

Effective interaction method for hyperspherical harmonics

The effective interaction hyperspherical harmonics (EIHH) method [1] is outlined. Recent extensions of the approach are discussed. Results for binding energies and radii of various p-shell nuclei are shown.

Exact solutions of the spherically symmetric multidimensional isotropic harmonic oscillator

The complete orthonormalised energy eigenfunctions and the energy eigenvalues of the spherically symmetric isotropic harmonic oscillator in N dimensions, are obtained through the methods of separation of variables. Also, the degeneracy of the energy levels are examined. KEY WORDS: - Schrödinger Equation, Isotropic harmonic oscillator, Hyperspherical harmonics, Multidimensional space. Global Jnl Pure & Applied Sciences Vol.10(2) 2004: 343-349

Effective interaction method for hyperspherical harmonics

The effective interaction hyperspherical harmonics (EIHH) method [1] is outlined. Recent extensions of the approach are discussed. Results for binding energies and radii of various p-shell nuclei are shown.

Laplacian eigenmodes for the three-sphere

The vector space of the eigenfunctions of the Laplacian on the three-sphere S3, corresponding to the same eigenvalue ?k = ?k(k + 2), has dimension (k + 1)2. After recalling the standard bases for , we introduce a new basis B3, constructed from the reductions to S3 of a peculiar homogeneous harmonic polynomial involving null vectors. We give the transformation laws between this basis and the usual hyper-spherical harmonics. Thanks to the quaternionic representations of S3 and SO(4), we are able to write explicitly the transformation properties of B3, and thus of any eigenmode, under an arbitrary rotation of SO(4). This offers the possibility of selecting those functions of which remain invariant under a chosen rotation of SO(4). When the rotation is a holonomy transformation of a spherical space S3/?, this gives a method for calculating the eigenmodes of S3/?, which remains an open problem in general. We illustrate our method by (re-)deriving the eigenmodes of lens and prism space. In a companion paper, we present the derivation for dodecahedral space.

Photodisintegration of Light Nuclei

The first microscopic calculation of inelastic reactions on nuclei with mass number A = 6 is presented. In particular, we consider the total photoabsorption cross section of 6He and 6Li. It is shown that the halo nucleus 6He exhibits two types of dipole resonances, the normal giant dipole resonance and a low-energy soft dipole resonance associated with the excitation of the outer neutrons. Comparing our results with available experimental data we note the role of P-wave interaction. Calculations are performed using the Lorentz integral transform (LIT) method and the effective interaction in the hyperspherical harmonics formalism (EIHH).

Selected Topics in Correlated Hyperspherical Harmonics

Correlated hyperspherical-harmonic basis functions are used to expand the three- and four-nucleon wave functions. Bound and scattering states are considered. Results for the binding energies of 3H, 3He, and 4He calculated using modern two- and three-nucleon forces are given and discussed. For scattering states, results for N-d differential cross section and vector analyzing powers are shown. The importance of the Coulomb and magnetic moment effects are discussed.

Hyperspherical harmonics with orthogonal symmetry in the shell model approach and its application to light nuclei

A method is proposed for constructing fully antisymmetric $A$-body hyperspherical harmonics with well-defined orthogonal symmetry. These hyperspherical harmonics are obtained by diagonalizing the matrix of the second Casimir operator for the orthogonal group, calculated in the hyperspherical basis constructed with the help of the Slater determinants of the oscillator translation-invariant shell model. The introduction of orthogonal symmetry has computational benefits and also provides a physical insight into collective modes of motion of atomic nuclei. Numerical applications have been performed for $^{3\char21{}5,7}\mathrm{H}$ and $^{4\char21{}10}\mathrm{He}$.

Three-body continuum spatial correlations in Borromean halo nuclei

Spatial correlations in the three-body continuum of Borromean (having no bound binary subsystems) three-body systems are discussed. The hyperspherical harmonics method is used to investigate low-lying resonances and the soft dipole mode in the two-neutron halo nucleus $^{6}\mathrm{He}$, which has only the $\ensuremath{\alpha}+n+n$ continuum for excitation energies below $13\phantom{\rule{0.3em}{0ex}}\text{MeV}$. The spatial correlations reveal characteristic structures for true three-body resonances, a moderate amplification in the interior region for above-barrier resonances and long-range correlations in the cases of three-body ${1}^{\ensuremath{-}}$ virtual and ${0}^{+}$ continuum states.