Hyperspherical Harmonic Method Research Articles

Bound and scattering states with non-local potentials

The application of the hyperspherical harmonics method to the case of non-local potentials is described. Given the properties of the hyperspherical harmonic functions, there are no difficulties in considering the approach in both coordinate and momentum space. The binding energies and some scattering observables of A = 4 systems are calculated using modern two-body potentials. For the α -particle, the results are shown to be in excellent agreement with corresponding ones obtained by other techniques. A study of the n – 3 H scattering lengths is also reported.

VARIATIONAL DESCRIPTION OF FEW–NUCLEON SYSTEMS: BOUND AND SCATTERING STATES

Three– and four–nucleon systems are described using the hyperspherical harmonic (HH) method. Bound and scattering states are expanded in the HH basis and the corresponding binding energies and S-matrices are obtained using a variational principle. Modern nucleon-nucleon potentials plus three-nucleon interactions are considered. The calculated quantities as binding energies, cross section and polarization observables are accurate at the level of 1% or better.

Wavefunctions of helium-like systems in limiting regions

We find an approximate analytic form for the solution ψ(r 1, r 2, r 12) of the Schrödinger equation for a system of two electrons bound to a nucleus in the spatial regions r 1 = r 2 = 0 and r 12 = 0, which are of great importance for a number of physical processes. The forms are based on the well-known behavior of ψ(r 1, r 2, r 12) near the singular triple coalescence point. The approximate functions are compared to the locally exact ones obtained earlier by the correlation function hyperspherical harmonic (CFHH) method for the helium atom, light helium-like ions, and the negative ion of hydrogen H−. The functions are shown to determine a natural basis for the expansion of CFHH functions in the considered spatial region. We demonstrate how these approximate functions simplify calculations of high-energy ionization processes.

Numerical generation of hyperspherical harmonics for tetra-atomic systems

A numerical generation method of hyperspherical harmonics for tetra-atomic systems, in terms of row-orthonormal hyperspherical coordinates-a hyper-radius and eight angles-is presented. The nine-dimensional coordinate space is split into three three-dimensional spaces, the physical rotation, kinematic rotation, and kinematic invariant spaces. The eight-angle principal-axes-of-inertia hyperspherical harmonics are expanded in Wigner rotation matrices for the physical and kinematic rotation angles. The remaining two-angle harmonics defined in kinematic invariant space are expanded in a basis of trigonometric functions, and the diagonalization of the kinetic energy operator in this basis provides highly accurate harmonics. This trigonometric basis is chosen to provide a mathematically exact and finite expansion for the harmonics. Individually, each basis function does not satisfy appropriate boundary conditions at the poles of the kinetic energy operator; however, the numerically generated linear combination of these functions which constitutes the harmonic does. The size of this basis is minimized using the symmetries of the system, in particular, internal symmetries, involving different sets of coordinates in nine-dimensional space corresponding to the same physical configuration.

Continuum spectroscopy of Borromean two-neutron halo nuclei

Energy and angular correlation distributions of the three fragments in $^{6}\mathrm{He}$ breakup on $^{208}\mathrm{Pb}$ at a collision energy of 240 MeV/nucleon are discussed within the microscopic four-body distorted wave model and compared with experimental data. The nuclear structure of the ground state and low-energy three-body continuum of $^{6}\mathrm{He}$ is calculated by the method of hyperspherical harmonics within the three-body cluster model. Reflections of the fundamental permutation symmetry of the halo neutrons in angular and energy correlations are pointed out. The calculations describe the experimental data for fragment correlations near breakup threshold rather well, and the physics is contained in a few elementary modes; but with increasing excitation energy of $^{6}\mathrm{He}$, some striking deviations from experimental distributions are encountered. Possible reasons for this are discussed.

Ro-Vibrational States of Triplet H2D+

We present rotational term values for J < or = 3 of the vibrational states with up to twofold excitation of H2D+ in the lowest electronic triplet state (a3sigma(u)+). The calculations were performed using the method of hyperspherical harmonics and our recent accurate double many-body expansion potential energy surface.

CALCULATION OF THE GROUND STATE HYPERFINE STRUCTURE FOR MUONIC HELIUM ATOMS WITH THE USE OF SOME LOCAL PROPERTIES OF THE WAVE FUNCTIONS

The ground-state hyperfine structure and other properties are calculated for muonic helium atoms (3 He +2μ-e- and 4 He +2μ-e-), with the use of some local properties of the wave functions in the domains where two particles are close to each other or far away. Simple wave functions incorporating these properties with one variational parameter is developed. Also, the electron-muon correlation function is considered in proposed wave functions. It has a correct behavior for r12 tending to zero and infinity. The calculated values for hyperfine structure, energy and expectation values of r2n in ground-state, are compared with the multibox variational approach and the correlation function hyperspherical harmonic method. Our obtained results are very close to the values calculated by mentioned methods and giving strong indications that the proposed wave functions in addition to being very simple, provide relatively accurate values for hyperfine structure, energy and expectation values of r2n, emphasizing on the importance of the local properties of the wave functions.

Calculation of the ground-state energy and average distance between particles for the nonsymmetric muonicHe3atom

A calculation of the ground-state energy and average distance between particles in the nonsymmetric muonic $^{3}\mathrm{He}$ atom is given. We have used a wave function with one free parameter, which satisfies boundary conditions such as the behavior of the wave function when two particles are close to each other or far away. In the proposed wave function, the electron-muon correlation function is also considered. It has a correct behavior for ${r}_{12}$ tending to zero and infinity. The calculated values for the energy and expectation values of ${r}^{2n}$ are compared with the multibox variational approach and the correlation function hyperspherical harmonic method. In addition, to show the importance and accuracy of approach used, the method is applied to evaluate the ground-state energy and average distance between the particles of nonsymmetric muonic $^{4}\mathrm{He}$ atom. Our obtained results are very close to the values calculated by the mentioned methods and giving strong indications that the proposed wave functions, in addition to being very simple, provide relatively accurate values for the energy and expectation values of ${r}^{2n}$, emphasizing the importance of the local properties of the wave function.

Electromagnetic structure of A=2 and 3 nuclei and the nuclear current operator

Different models for conserved two- and three-body electromagnetic currents are constructed from two- and three-nucleon interactions, using either meson-exchange mechanisms or minimal substitution in the momentum dependence of these interactions. The connection between these two different schemes is elucidated. A number of low-energy electronuclear observables, including (i) np radiative capture at thermal neutron energies and deuteron photodisintegration at low energies, (ii) nd and pd radiative capture reactions, and (iii) isoscalar and isovector magnetic form factors of $^{3}\mathrm{H}$ and $^{3}\mathrm{He}$, are calculated to make a comparative study of these models for the current operator. The realistic Argonne ${v}_{18}$ two-nucleon and Urbana IX or Tucson-Melbourne three-nucleon interactions are taken as a case study. For $A=3$ processes, the bound and continuum wave functions, both below and above deuteron breakup threshold, are obtained with the correlated hyperspherical harmonics method. Three-body currents give small but significant contributions to some of the polarization observables in the $^{2}\mathrm{H}$($p,\ensuremath{\gamma}$)$^{3}\mathrm{He}$ process and the $^{2}\mathrm{H}$($n,\ensuremath{\gamma}$)$^{3}\mathrm{H}$ cross section at thermal neutron energies. It is shown that the use of a current that did not exactly satisfy current conservation with the two- and three-nucleon interactions in the Hamiltonian was responsible for some of the discrepancies reported in previous studies between the experimental and theoretical polarization observables in pd radiative capture.

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.

Shape variation of the two-electron photoionization spectrum with photon energy growth

We trace the evolution of the shape of the two-electron photoionization spectrum with photon energy growth using quite a precise two-electron wave function, obtained by the correlation function hyperspherical harmonic method. We obtain the values of {omega}{sub 1} and {omega}{sub 2} at which the spectrum curve changes its shape. At {omega}={omega}{sub 1} the U shape changes to a W shape. At {omega}={omega}{sub 2} the central W peak splits into two. We consider ground states of the helium atom and of heliumlike ions with the nuclear charge Z, the negative ion of hydrogen H{sup -}, and the excited n{sup 1}S state of helium. The limiting laws for Z>>1 and n>>1 are obtained. The analysis is carried out without calculations of the particular energy distributions themselves.

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.

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.

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.

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.

Ro-vibrational states of triplet H 3+ ( a3Σ u+): The lowest 19 bands

We have performed extensive calculations of the ro-vibrational states of triplet H 3 +, using the method of hyperspherical harmonics and our recently reported double many-body expansion potential energy surface. The rotational term values of the lowest 19 states are presented here for a total angular momentum of J⩽10.

Analytical Structure Matching and Very Precise Approach to the Coulombic Quantum Three-Body Problem

A powerful approach to solve the Coulombic quantum three-body problem is proposed. The approach is exponentially convergent and more efficient than the hyperspherical coordinate method and the correlation-function hyperspherical harmonic method. This approach is numerically competitive with the variational methods, such as that using the Hylleraas-type basis functions. Numerical comparisons are made to demonstrate the efficiency of this approach, by calculating the nonrelativistic and infinite-nuclear-mass limit of the ground state energy of the helium atom. The exponential convergency of this approach is due to the full matching between the analytical structure of the basis functions that are used in this paper and the true wavefunction. This full matching was not reached by most other methods. For example, the variational method using the Hylleraas-type basis does not reflects the logarithmic singularity of the true wavefunction at the origin as predicted by Bartlett and Fock. Two important approaches are proposed in this work to reach this full matching: the coordinate transformation method and the asymptotic series method. Besides these, this work makes use of the least square method to substitute complicated numerical integrations in solving the Schrödinger equation without much loss of accuracy, which is routinely used by people to fit a theoretical curve with discrete experimental data, but here is used to simplify the computation.

Local properties of three-body atomic wave functions

The local properties and accuracy of the positronium negative-ion $({\mathrm{Ps}}^{\ensuremath{-}})$ ground-state wave functions obtained by the stochastic variational method (SVM) and by direct solution of the Schr\odinger equation with the help of the correlation-function hyperspherical-harmonic method (CFHHM) are studied and compared. Though the energy, calculated by both methods, agrees to up to ten digits, the amplitudes of the values of the operator $D=H\ensuremath{\Psi}/E\ensuremath{\Psi}\ensuremath{-}1,$ characterizing local deviation of the wave function from its true value, in all of the coordinate space in the SVM are consistently larger (by up to five orders of magnitude) than in the CFHHM, despite the fact that the SVM observables except $〈\ensuremath{\delta}({\mathbf{r}}_{k})〉$ converge to significantly more digits than the CFHHM observables for their respective selected bases.

Three-cluster states in the spectrum of [formula omitted]: (1) Structure and β-decay of [formula omitted] to 16.6 MeV state

The structure and spectrum of the three-cluster states in 8 Be around 16–20 MeV was studied in the framework of the hyperspherical harmonic method. For that purpose we have developed a potential formalism for T– p and 3 He – n (3+1) scattering and mechanism for the three-body mixed isospin calculations. We have studied the mechanism of the isospin mixing for 1 +, 2 + doublets in 8 Be and found a sharp sensitivity of isospin mixing to the experimentally not well defined parameters of the 3+1 intercluster interaction. The matrix elements of the β-decay of 8 B to 2 + doublet 16.6, 16.9 MeV were calculated and estimates were made for the probability of electron capture to 17.64 MeV state. It is found that about 90% of the Gamow–Teller (GT) sum rule is concentrated in the three-cluster 2 + and 1 + states, situated below the three-body thresholds, within narrow boundaries (16.5–19.0 MeV).

Exact solutions of the Schr�dinger equation for1,3S states of the atom with Fues-Kratzer-type potential

In this article exact solutions of a two-electron Schroedinger equation for the Coulomb potential were extended to the Fues-Kratzer-type potential: ({cflx Z}({Omega})/r) + ({cflx A}/r{sup 2}). The wave function {Psi}(r, {Omega}) is expanded into generalized Laguerre polynomials and hyperspherical harmonics. An analytical expression of two-electron systems is given for matrix elements and accurate energy eigenvalues of the excited state of {sup 1,3}S helium are calculated by using the hyperspherical harmonics method. The present results are compared with previous theoretical calculations and it is concluded that the convergence of energy eigenvalues is faster.