The first order of the hyperspherical harmonic expansion method

The hyperspherical harmonic expansion method is studied in this work. Our attention is focused on the properties of the L_m-approximation in which only the hyperspherical harmonics of minimal order are taken into account. Exact solutions of the Schrödinger equation for a few simple hyperspherical potentials are given. Recipes for constructing antisymmetric hyperspherical harmonics for fermions are investigated, and various procedures to derive the effective potential in the L_m-approximation are discussed. The method is applied to the calculation of ground state and hyperradial excited states (which are identified as the breathing modes) of doubly-magic nuclei. Finally, the energy per particle is derived in the L_m-approximation with Skyrme like forces for an infinitely heavy self-conjugate nucleus.     

Low-Lying S-States of Two-Electron Systems

The energies of the low-lying bound S-states of some two-electron systems (treating them as three-body systems) like negatively charged hydrogen, neutral helium, positively charged-lithium, beryllium, carbon, oxygen, neon, argon and negatively charged muonium and exotic positronium ions have been calculated employing hyperspherical harmonics expansion method. The matrix elements of two-body interactions involve Raynal–Revai coefficients which are particularly essential for the numerical solution of three-body Schr?dinger equation when the two-body potentials are other from Coulomb or harmonic. The technique has been applied for to two-electron ions ¹H^? (Z = 1) to ⁴⁰Ar¹⁶⁺ (Z = 18), negatively charged-muonium Mu^? and exotic positronium ion Ps^?(e ⁺ e ^? e ^?) considering purely Coulomb interaction. The available computer facility restricted reliable calculations up to 28 partial waves (i.e. K _m  = 28) and energies for higher K _m have been obtained by applying an extrapolation scheme suggested by Schneider.     

Variational Calculation on A = 3 and 4 Nuclei with Non-Local Potentials

The application of the hyperspherical harmonic approach to the case of non-local two-body 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 other ground-state properties of A = 3 and 4 nuclei are calculated using the CD Bonn 2000 and N3LO two-body potentials. The results are shown to be in excellent agreement with corresponding ones obtained by other accurate techniques.     

Five-dimensional SU(2)-monopole: Continuous spectrum

Hyperspherical and parabolic wave functions are calculated for the five-dimensional “charge-SU(2) monopole” system in the continuous spectrum. It is shown that the coefficients of parabolic-hyperspherical and hyperspherical-parabolic transformations are proportional to the generalized hyperge-ometric function ₃ F ₂{...|1}.     

Matrix elements of two-body operators between three-body hyperspherical basis states

Matrix elements of two-body operators between three-body spherically symmetric hyperspherical basis states are expressed through the Clebsch-Gordon coefficients and explicitly written functions of a radial variable.     

Recurrence relations between transformation coefficients of hyperspherical harmonics and their application to moshinsky coefficients

Closed formulae and recurrence relations for the transformation of a two-body harmonic oscillator wave function to the hyperspherical formalism are given. With them Moshinsky or Smirnov coefficients are obtained from the transformation coefficients of hyperspherical harmonics. For these coefficients the diagonalization method of Talman and Landéreduces to simple recurrence relations which can be used directly to compute them. New closed formulae for these coefficients are also derived : they are needed to compute the two simplest coefficients which determine the sign for the recurrence relation.     

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.     

An Integro-Differential Equation for Bose�CEinstein Condensates

We present an integro-differential equation describing systems with large number of bosons. The new equation includes the two-body correlations exactly into account and the kernel has a simple analytic form. The equation has been employed to obtain results for ${A\in\{10,100\}}$ ⁸⁷Rb atoms confined by an externally applied trapping potential V _trap(r). Our results are in excellent agreement with those of the Potential Harmonic Expansion Method (PHEM) and the Diffusion Monte Carlo (DMC) method.     

F L structure function and heavy quark contributions

In this paper we obtain the heavy-quark contribution to the longitudinal structure functions F _L (x, Q ²). Since F _L structure functions contains rather large heavy flavor contributions in the small x region, we need to use the massive operator matrix elements, which contribute to the heavy flavor Wilson coefficients in unpolarized deeply inelastic scattering in the region Q ²?>?>?m ². The method of QCD analysis, based on the Jacobi polynomials method, is also described. Our results for longitudinal structure function are in good agreement with the available experimental data.     

Bounds on fourth-generation fermion masses in the standard model

In the standard model, if the ratios of Yukawa couplings to gauge couplings are bounded functions of the renormalization point, then the fourth-generation fermion massesm _U , m _D , andm _L (in GeV) are constrained to lie within a domain approximately given by (m _U /213)^4.8+(m _D /209)^4.8+(m _L /161)^2.4<1.     

Electron scattering from some even fp- and N50-shell nuclei with implanting the influence of short-range correlation functions

The influence of two body short range correlations on elastic electron scattering charge form factors, charge densities as well as root mean square charge radii of some fp-shell nuclei (for example, ⁴⁸Ca, ⁵⁰Cr, ⁵⁴Fe, ⁵⁸Ni, ⁷⁰Ge and ⁷⁴Se) and some N50-shell nuclei (for example, ⁸⁸Sr and ⁹⁰Zr) is analyzed using the one- and two-body terms in the cluster expansion together with the single particle harmonic oscillator wave functions. The Jastrow-type correlation function is utilized to embed the effect of short range correlations into elastic charge form factors F(q) and charge densities ρ(r). Both F(q) and ρ(r) depend upon the harmonic oscillator parameter b and the correlation parameter β (which initiates from the Jastrow correlation function). Here, the parameters b and β are determined via the fitting to the measured charge form factors. The embedding of short range correlations imitates the measured charge form factors at the high momentum transfers (q ≥ 2 fm⁻¹). It is noticed that the implanting of short range correlations is required for obtaining a remarkable alteration in the computed elastic charge form factors which in turn leads to explain the data of electron scattering astonishingly throughout the entire range of considered momentum transfers.     

Kernradien von12C,13C,14N und16O aus Elektronenstreuung zwischen 30 und 60 MeV

Elastic electron scattering cross sections of¹²C,¹⁴N and¹⁶O relative to the proton, and of¹³C relative to¹²C have been measured. Using harmonic oscillator wave functions the followingrms charge radiiR _m were deduced by phaseshift calculations: 2.395 (28) fm for¹²C, 2.384 (47) fm for¹³C, 2.492 (33) fm for¹⁵N, 2.666 (33) fm for¹⁶O. The ratioR _m (¹³C)/R _m (¹²C) is 0.995±0.008. The errors given do not include uncertainties from the model dependence of the evaluation which may be of the same magnitude.     

Nuclear charge radii of Al and Si from elastic electron scattering between 25 and 60 MeV

Elastic electron scattering cross sections of²⁷Al and Si (natural isotopic mixture) have been measured relative to carbon. The rms charge radiiR _m , deduced with partial wave calculations, are (3.01±0.05) fm for²⁷Al and (3.06±0.05) fm for Si, in good agreement with results from muonic X-ray energies. The values given are those for a Fermi charge distribution with skin thickness 2.5 fm; harmonic oscillator shell model distributions yield radii smaller by 0.03 fm. The ratioR _m (²⁷Al)/R_m(Si) is 0.984±0.016.     

Hyperspherical harmonics and the nuclear four-body problem

A method proposed earlier by Aguilera, Moshinsky, and Kramer, for adapting a system of translationally invariant four-particle harmonic oscillator functions to the symmetry of the permutation group S(4), is applied to hyperspherical harmonic functions depending on three relative vectors. Except for a few cases in which diagonalization of matrices is required, the method gives closed formulas for orthonormal sets of harmonic functions with good permutational symmetry. The matrix elements of S(4) permutations with respect to the harmonic functions are obtained.     

Relativistically corrected masses of ground-state baryons in hyperspherical harmonic quark model

The masses of the ground-state light baryons are calculated in the quark model. The unperturbed wave functions correspond to a hyperspherical harmonic confinement. The perturbation includes a short-range potential and all the relevant relativistic corrections of the order ofv ²/c ². Results are compared with the experimental values and found to be in good agreement. This may be a test not of the hyperspherical harmonic model so much as of the feasibility of a simple (but consistent) relativistically corrected fit of the light baryon masses.     

Meson-exchange corrections to the nuclear weak axial charge density in the hard pion model and 0+ 3 0− transitions in A = 16 nuclei

Starting with the hard-pion model based on a minimal chiral invariant phenomenological lagrangian, the two-body part of the time component of the weak axial-vector current is constructed in the tree approximation. Pion, rho- and A₁-meson exchanges are considered. The mesonic exchange operator obtained is applied to describe the purely weak axial 0⁺?0^?, ΔT = 1 transition in the nuclear A = 16 system. In order to treat nuclear structure correlation effects, explicit use of shell-model wave functions with configuration mixing is made. We confirm the large enhancement of the nuclear weak axial charge density with respect to impulse approximation.     

The effect of density dependent Av18 effective interaction on the ground state properties of heavy closed shell nuclei

The channel-dependent Argonne Av₁₈ effective two-body interactions (CDEI) which are generated through the lowest order constrained variational (LOCV) calculation for asymmetric nuclear matter with the charge-dependent Av₁₈ bare nucleon–nucleon potential are used to calculate the ground state properties of heavy closed shell nuclei such as ⁴⁸Ca, ⁹⁰Zr, ¹²⁰Sn and ²⁰⁸Pb. The harmonic oscillator basis, and the local density approximation (LDA) are applied to create the relative and the center of mass dependent effective two-body potential. We get more binding with respect to the similar calculation with the Reid types potentials. It is tried to omit the LDA and perform full calculation with the Av₁₈CDEI for light nuclei. The results indicate that the LDA works quite well. It is also shown that in case of heavy closed shell nuclei and unlike our previous report with Reid68Day   interaction, the contributions of higher partial waves (J>2$J>2$) are very important for the calculations with Av₁₈ potential and we get reasonable agreement between our calculated binding energies and RMS radii, with those predicated by the others methods, and the experimental data. Finally, the various aspects of channel and density dependent two-body effective interactions are discussed.     

Effect of the three-body force on trinucleon bound systems consideringS andS′-state admixture

We report the calculation of binding energy, charge form factor and point-like proton density of both³H and³He by the hyperspherical harmonics method with the inclusion of two-pion exchange three-nucleon force (Fujita-Miyazawa type). For the two-body force theN-N Afnan-Tang S-3 potential is taken. Coulomb and three-body forces are treated nonperturbatively. In this calculation the mixed symmetryS′-state of the trinucleon ground state is considered along with the space totally symmetricS-state.