Three-Body Bound State Calculations Without Angular Momentum Decomposition
C. Elster and W. Schadow, Ohio University
Research Objectives
The objective is to carry out few-nucleon calculations without the traditionally employed angular momentum decomposition. At a few hundred MeV projectile energy, quite a few angular momentum states are required to achieve convergence of scattering observables. Before treating the scattering problem, we gained experience in the three-dimensional approach in few-nucleon physics by treating the bound state equations for three identical particles in momentum space.
We formulated the Faddeev equations as function of vector Jacobi momenta, specifically the magnitudes of the momenta and the angle between them, and demonstrated their numerical feasibility and the accuracy of their solutions. For the two-body force, we concentrated on a superposition of an attractive and repulsive Yukawa interaction, which is typical for nuclear physics. The two-body t-matrix, which enters the Faddeev equations, is also calculated directly without partial wave decomposition.
Computational Approach
The discretized Faddeev equation for a bound state (neglecting spin degrees of freedom) is an integral equation in 3 variables on a typical grid of 90 x 109 x 42 (momentum magnitudes p,q, and angle between the momentum vectors). The eigenvalue equation for the bound state is solved iteratively by using Lanczos-type techniques, here the power method.
The numerical treatment can be divided into two steps, namely calculation of the kernel, i.e., setting up the integral equation, and the iteration (on average 5) of the equation to obtain the eigenvalue. For the kernel, a two-body t-matrix (with the two-nucleon interaction as driving term) is obtained by solving a system of linear equations of the form A x x = b,where A is typically a 4000 x 4000 matrix. This system is solved for about 100 different vectors b.
The calculated t-matrix is then interpolated to the variables needed in the Faddeev equations. The interpolations are performed using cubic Hermite splines. The number of required interpolations is typically 1.8 x 108.
Accomplishments
As a first test for the numerical accuracy of the solution of the Faddeev equation as a function of vector variables, we determined the energy eigenvalue of the bound system and compared our result with the one obtained in a traditional Faddeev calculation carried out on a partial wave truncated basis. We achieved excellent agreement (5 significant figures) between the two approaches, as well as excellent agreement with calculations in the literature.
For a stringent test of the three-dimensional wave function obtained from the Faddeev amplitude, we inserted it into the 3N Schrödinger equation and evaluated the accuracy with which the eigenvalue equation is fulfilled throughout the entire space where our solution is defined. We found that within the physical relevant momentum region, namely the magnitudes of the Jacobi momenta less than 10 fm-1, the 3N Schrödinger equation is fulfilled with high accuracy by our numerical solution.
The real part of the half-shell two-nucleon t-matrix, T(q, q0, x, E), as function of q and the angle x = cos(theta) between the momentum vectors q and q0 at center-of-mass energies E = 25 MeV (green) and E = -25 MeV (purple). The vector q0 describes the momentum of the incident nucleon. The employed nucleon-nucleon interaction is of Malfliet-Tjon type.
Significance
Nuclear scattering at intermediate energies of a few hundred MeV requires quite a few angular momentum states in order to achieve convergence of scattering observables. Presently employed computational methods for 3N scattering at higher energies, using conventional partial-wave expansions, have intrinsic limitations, since with increasing energy the number of channel quantum numbers strongly proliferates, leading to increasing numerical difficulties with respect to accuracy and storage requirements.
This work represents an alternative computational approach by solving the Faddeev equations directly in a three-dimensional (3D) form in momentum space. The incorporation of the boundary conditions for three-body scattering does not change for 3D solutions of the Faddeev equations. In the integral from in momentum space, which we are using, they are automatically included. This approach will allow us to extend the investigations concerning the importance of three-nucleon forces in a computationally sound way to the energy regime up to 300 MeV, a regime of current experimental efforts at the Indiana University Cyclotron Facility (IUCF), the Kernfysisch Versneller Instituut (KVI), and the Research Center for Nuclear Physics (RCNP).
Publications
C. Elster, W. Schadow, A. Nogga, and W. Glöckle, "Three body bound state calculations without angular momentum decomposition," Few-Body Systems (submitted, 1998).
C. Elster, W. Schadow, H. Kamada, and W. Glöckle, "Shadowing and antishadowing effects in a model for the n+d total cross section," Phys. Rev. C (in press, 1998).