Off-shell effects in the three-nucleon system

The triton binding energy and the neutron-deuteron doublet scattering length have been studied with phase-equivalent rank-2 separable potentials in S-waves. The potentials have been constructed from the phase shifts using the solution of the inverse scattering problem. Large off-shell effects are connected with the appearance of additional nodes in the deuteron or the low-energy singlet wave functions.     

Binding energy of triton and scattering lengths in neutron-deuteron collision in Faddeev formalism with local potential

A separable representation for the off-shell two-body t-matrix for a local Hulthén potential is presented, in which deuteron states are chosen as the expansion bases. Using the Faddeev equations with these t-matrices as input, the ground state energy of the triton and doublet and quartet scattering lengths in neutron-deuteron scattering, have been computed. The results have been compared with the experimental findings and the theoretical results of Sitenko et al. obtained in the Sturmian function representation with the same Hulthén potential.     

On the unitary pole expansion of the potentials I and III of Malfliet and Tjon

The unitary pole expansion for the local soft-core potentials I and III of Malfliet and Tjon is obtained, and the Faddeev-Lovelace equations are solved for the separable potentials. The energyE _T of the triton and the doublet scattering length² a have been obtained. The results are ?8.59 MeV ≦E_T≦ ?8.57 MeV and 0.89 fm≦² a≦0.91 fm. For the quartet scattering length a value₄ a = 6.39 fm has been found.     

Effective-range function for doublet nd scattering from an analysis of modern data

The parameters of the generalized effective-range function K(k ²) having a pole are found by using the results that were obtained by calculating the S-wave phase shift δ(E) for doublet nd scattering and the triton binding energy on the basis of Faddeev equations and within the N/D method and which were presented in the literature. The convergence of the expansion of K(k ²) in powers of momentum is studied. The binding energy of the virtual triton and the residues of the partial-wave scattering amplitudes at the poles corresponding to the bound and virtual states are calculated. Correlations between the binding energies of the bound and virtual states of the triton, on one hand, and the doublet scattering length for nd interaction, on the other hand, are considered. The function K(k ²) is also calculated within a two-body model featuring various potentials.     

The unitary pole expansion of the off-shell t-matrix elements for local potentials with hard cores

A convenient separable expansion of the t-matrix, similar to the one developed recently by Harms for local potentials with soft cores, has been obtained in the case of a hard-core potential as well. The unitary pole expansion (UPE), as it is called, is found to be strongly convergent. The UPE values of the off-shell t-matrix elements for a hard-core potential with an exponential shape agree fairly well with those obtained by Laughlin and Scott from the direct solution of the inhomogeneous Lippman-Schwinger equation.     

Off-shell effects in the three-nucleon problem

Using a previously reported solution of the inverse scattering problem for rank two separable potentials, off-shell effects on the binding energy of the model triton are investigated. Much larger effects are found than previously reported in the literature.     

3H at Next-to-Next-to-Next-to Leading Order of the Chiral Expansion

The chiral three-nucleon force (3NF) at next-to-next-to-next-to leading order (N³LO) is used to calculate the triton wave function and the doublet nucleon–deuteron scattering length. This allows us to fix the values of the low-energy constants which are free parameters of the theory. The obtained values of these parameters, the expectation values of the kinetic energy, two- and three-body potentials and individual contributions of different parts of 3NF are given.     

Doublet coulomb-nuclear scattering length and other parameters of the effective-range function for proton-deuteron scattering from an analysis of present-day data

Parameters of the effective-range function K(k ²) having a pole are found from the results that were obtained by calculating the S-wave phase shifts for doublet proton-deuteron scattering and the binding energies of three-nucleon nuclei on the basis of Faddeev equations and within the N/D method and which are known from the literature. The convergence of the expansion of K(k ²) in powers of the momentum is studied. The energy of the proton-deuteron resonance corresponding to the virtual triton state is calculated.     

可分的非定域中心力和轻核结合能

本文考虑了用可分的非定域中心力同时符合低能核子散射和轻核结合能的可能性。用低能散射数据定得了两种势阱的参量。用我们的参量和Yamaguchi的参量计算了氚和α粒子的结合能,结果都过大。因此,可分的非定域力和普通定域力类似:符合低能散射的中心力给出过大的轻核结合能。改变位阱形状并不能改进结果。     

The tensor force in nuclear saturation

One-term separable potentials in the ³S-³D channel are constructed which fit the following low-energy nucleon-nucleon data: the triplet effective range and scattering length, deuteron binding energy and quadrupole moment. They also yield ³D₁ phase shifts which have the correct sign. These potentials differ, however, in the amount of deuteron D-state probability, P_D, which they predict, where P_D ranges from 1 % to 9 %. Binding energy calculations of infinite nuclear matter and ⁴He are performed in order to test the effect of the tensor force on nuclear saturation properties. It is found that the larger the D-state probability, the smaller the energy per particle and saturation density. Detailed comparisons with local potentials in nuclear matter are also presented.

In nuclear matter no single-particle potential in intermediate states is used; in ⁴He, , where f is varied such that the absolute value of the diagram with a single potential insertion in a particle line is minimized. It is found in ⁴He that f= 0.75 and that this result is almost independent of both the potential employed and of ω. Furthermore, for 0 f 1.5, the total energy is independent of f.     

Phenomenology of the nucleon-nucleon potential

A local nucleon-nucleon potential expansion is developed in terms of orthogonal projectors. Considering the nucleon-nucleon (NN) potential as a completely phenomenological structure, the expansion provides an opportunity to obtain the NN scattering phase shifts that can be described by applying a restricted set of operators, dependent on angular and spin-isospin degrees of freedom of the interacting nucleons. The results obtained with an approximation for eight basic operators (central, spin-orbit and tensorial) are consistent with experience in the field, and provide directions for further modifications of realistic NN potentials.     

Coupled channel approach to the off-shell transition matrix

We propose a coupled-channel separable potential model with the feature that the off-shell elastic transition matrix can be generated directly from the scattering phase shifts. This should be useful in constructing pion-nucleus optical potentials.     

Equivalent local potential for the fish bone optical potential by inversion of its phase shifts

By inversion of the fish bone optical model phase shifts for α -¹⁶O scattering we confirm the presence of a Pauli barrier in the surface region of the corresponding equivalent local potential. In the interior the potential depth increases with decreasing energy, in agreement with the semiclassical results of Horiuchi. The advantages of the inversion method for the determination of equivalent local potentials are discussed.     

A density expansion of the pion-nucleus optical potential. II. Second order terms

The second order terms in a density expansion of the pion optical potential V_opt are evaluated quantitatively. The coefficients of these terms are proportional to various combinations of on- and off-shell nucleon-nucleon T-matrices, averaged over the distribution of two nucleon relative momenta in the Fermi sea. The on-shell contributions can be obtained directly from experimental phase shifts, but the calculation of the off-shell-parts requires a model for the nucleon-nucleon potential. We consider a number of realistic local and nonlocal, separable potentials which fit nucleon-nucleon phase shifts, in order to study the variations in V_opt which arise from differences in the off-shell behavior of T. We find absorptive (imaginary) and dispersive (real) contributions to V_opt which are of comparable magnitude. The dispersive part, which leads to a real energy shift associated with the two nucleon absorptive process, has not been previously estimated quantitatively. We compare our results to empirical potentials obtained by fitting energy level shifts and widths in pi-mesic atoms, as well as theoretical estimates based on threshold cross sections for the processes π + N + N ? N + N.     

Precision neutron interferometric measurement of the nd coherent neutron scattering length and consequences for models of three-nucleon forces

We have performed the first high precision measurement of the coherent neutron scattering length of deuterium in a pure sample using neutron interferometry. We find b(nd)=(6.665+/-0.004) fm in agreement with the world average of previous measurements using different techniques, b(nd)=(6.6730+/-0.0045) fm. We compare the new world average for the nd coherent scattering length b(nd)=(6.669+/-0.003) fm to calculations of the doublet and quartet scattering lengths from several modern nucleon-nucleon potential models with three-nucleon force (3NF) additions and show that almost all theories are in serious disagreement with experiment. This comparison is a more stringent test of the models than past comparisons with the less precisely determined doublet scattering length of (2)a(nd)=(0.65+/-0.04) fm.     

Elastic neutron-deuteron scattering with local potentials

Neutron-deuteron scattering is studied within the Faddeev formalism using local two-body S-wave potentials. The equations are solved by applying Padé technique to the multiple scattering series. The rate of convergence is found to be rapid enough for practical purposes. With this method the differential and total cross sections are calculated up to 50 MeV neutron laboratory energy and the results are compared with those of the separable potential model. In particular large differences are found in the forward direction of the differential cross-sections. Up to 22 MeV there is good agreement with experimental data. The differential cross-sections for local potentials up to about 14 MeV depend very little on the existence of a soft core in the two-body spin-triplet potential, but its effect becomes more important at higher energies.     

Pion-Nucleon Interaction and Two-Pion-Exchange Three-Nucleon Forces

We compare the πN scattering amplitudes that underlie 2π-exchange three-nucleon forces (TBFs) with the experimental πN amplitudes in the form of partial-wave phase shifts and subthreshold invariant amplitudes. The amplitudes of the Tucson-Melbourne and Brazil TBFs when taken on-pion-mass-shell predict scattering lengths at threshold and phase shifts (slightly) above threshold which are in good agreement with the experimental amplitudes, except for the S-waves. Partial wave amplitudes from separable potentials, recently employed in a 2π-TBF calculation, were continued below threshold, summed into invariant amplitudes, and compared with the experimental amplitudes in this kinematic region, which is most relevant to the kinematics of TBFs. The separable-potential invariant amplitudes, in contrast to those of TM and Brazil TBFs, do not compare well quantitatively with the experimental amplitudes in this region but have a similar qualitative behaviour. The very small TBF effect in the triton of the separable-potential amplitude appears to be due to the πNN vertex function rather than the πN amplitude itself. Received April 27, 1994; revised August 2, 1994; accepted for publication August 31, 1994