D. R. Lehman

Quality of the three-nucleon bound-state wave function from a two-nucleon separable expansion method

The numerical quality of the3H wave function obtained by the separable expansion method of Ernst, Shakin, and Thaler is examined. Separable approximations to the Paris potential with increasing accuracy are used in the1S0 and3S1-3D1 partial waves to calculate the binding energy, wave function, wave-function component percentages, and theS- andD-wave asymptotic normalization constants of3H. The results are compared with existing five-channel calculations obtained directly (without expansion) from the Paris potential to determine convergence. It is found that the results converge rapidly to the right values, indicating that the3H wave function thus obtained is of high quality and essentially indistinguishable from that obtained directly from the Paris interaction.

Numerical evaluation of integrals containing a spherical Bessel function by product integration

A method is developed for numerical evaluation of integrals with k‐integration range from 0 to ∞ that contain a spherical Bessel function jl(kr) explicitly. The required quadrature weights are easily calculated and the rate of convergence is rapid—only a relatively small number of quadrature points is needed—for an accurate evaluation even when r is large. The quadrature rule is obtained by the method of product integration. With the abscissas chosen to be those of Clenshaw–Curtis and the Chebyshev polynomials as the interpolating polynomials, quadrature weights are obtained that depend on the spherical Bessel function. An inhomogenous recurrence relation is derived from which the weights can be calculated without accumulation of roundoff error. The procedure is summarized as an easily implementable algorithm. Questions of convergence are discussed and the rate of convergence demonstrated for several test integrals. Alternative procedures are given for generating the integration weights and an error analy...

L-S coupling and the magnetic moment of6Li from three-body models

Recently, the shell structure of6Li was calculated from three-body models (αpn) where the only input is the required nucleon-nucleon and alpha-nucleon interactions. There it was learned, within the framework ofj-j coupling for the two valence nucleons with their coordinates origin on the alpha particle, that orbitals beyond thep-shell, e.g. thes-d shell, play a significant role in the structure of6Li. In the present work, we extend these calculations to thef-shell. Thef-shell orbital probabilities add ∼4% to the normalization, thus bring all the models to within 3–5% of complete convergence. We then use thej-j coupling orbital amplitudes up to thef-shell to construct the corresponding amplitudes forL-S coupling. We find theL-S orbital probabilities, and compare them with theL-S component probabilities calculated directly by recoupling the three-body wave function from its “natural” Jacobi-coordinate form. The6Li magnetic moment is determined from the directL-S probabilities. The most realistic models yield magnetic moments about 2.5% higher than experiment.

The Λ4HeΛ4H binding energy difference

Abstract The Λ 4 He Λ 4 H binding energy difference is studied using separable Λ-p and Λ-n interactions. The result of the exact 4-body calculation is found to be more than twice the result of an effective 2-body model calculation.

6He β-decay from a three-body model of the A=6 system

Abstract The rate of 6 He β-decay is computed from 6 He and 6 Li separable-potential three-body wave functions. The value of ft predicted with the wave functions derived from the most complete set of αN and NN interactions is 807±12 s compared to the experimental value of 807±2 s.

Exact three-body calculation of polarization observables in 1H (d, γ)3He☆

Abstract An exact three-body calculation of the radiative capture of polarized deuterons on protons is carried out assuming the transition is pure E1. It is found that exact three-body dynamics in the initial state is essential for interpreting the magnitude of the polarization observables. Furthermore, of all the components making up the 3He wave function, it is the D-state that is totally responsible for the character of the tensor analyzing powers, even though there is little sensitivity to the percentage D-state in the deuteron generated by the nucleon-nucleon interaction models considered.

Angular reduction in multiparticle matrix elements

A general method for reduction of coupled spherical harmonic products is presented. When the total angular coupling is zero, the reduction leads to an explicitly real expression in the scalar products of the unit vector arguments of the spherical harmonics. For nonscalar couplings, the reduction gives Cartesian tensor forms for the spherical harmonic products; tensors built from the physical vectors in the original expression. The reduction for arbitrary couplings is given in closed form, making it amenable to symbolic manipulation on a computer. The final expressions do not depend on a special choice of coordinate axes, nor do they contain azimuthal quantum number summations, or do they have complex tensor terms for couplings to a scalar; consequently, they are easily interpretable from the properties of the physical vectors they contain.

Accurate triton calculation with a local realistic N-N interaction with a separable expansion method

It is demonstrated that a clear convergent series for the three‐body observables is obtained with a new separable expansion method where several techniques are unified. In the method, we utilize both the separable expansions of EST and of Adhikari‐Sloan, splitting the original potential into two potentials with different ranges. The Adhikari‐Sloan method is re‐formulated in a simple manner to show the possibility of a separable expansion more clearly. The method is applied to the AV14 potential as an example. The binding energy with some other observables of triton are calculated. This work definitely shows that a separable expansion is both reliable and a very efficient method among all possible methods of ‘‘rigorous’’ three‐body calculations.