The objective is to carry out few-nucleon calculations without the traditionally employed angular momentum decomposition. It is well known that nuclear scattering at intermediate energies of a few hundred MeV requires quite a few angular momentum states in order to achieve convergence of e.g. scattering observables. Presently employed computational methods for three nucleon (3N) scattering at higher energies using conventional partial wave expansions have intrinsic limitations, since with increasing energy the amount of channel quantum numbers strongly proliferates, leading to increasing numerical difficulties with respect to accuracy as well as storage requirements.
Three nucleon (3N) bound state calculations are traditionally carried
out by solving Faddeev equations on a partial wave basis. After truncation
this leads to a set of a finite number of coupled equations in two
variables for the amplitude. The calculations are performed either
space , in configuration space, or in a hybrid fashion using
both spaces. Though a few partial waves often provide qualitative
insight, modern three nucleon bound state calculations need 34 or more
different isospin, spin and orbital angular momentum combinations.
It appears therefore natural to avoid a partial wave representation
completely and work directly with vector variables.
We formulated the Faddeev equation including 3N forces as function of vector Jacobi momenta, specifically the magnitudes of the momenta and the angles between them, and demonstrate their numerical feasibility and the accuracy of their solution. As two-body force we concentrate 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 equation is also calculated directly without partial wave decomposition. As 3N force we consider an attractive force mediated by two scalar mesons.
Because of the ease in solving the Faddeev equation including three-body forces in this fashion, we allowed ourselves to explore various combinations of two-and three-body forces and investigated whether wave function properties change qualitatively when looking at scenarios where the three-body system is dominated by two-body or three-body forces.
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 65*65*42 (momentum magnitudes p,q, and angle between the momentum
vectors). A priori the multidimensionality of the integral equation to
be solved requires more memory. However, on an MPP system the number of
variables and thus the memory do not pose a computational problem, since
a variable defining
a specific dimension of the grid can be distributed over a number of
processors, leaving a lower dimensional grid on each processor. As such,
our three-dimensional approach is ideally suited as MPP application, and
we can achieve an almost perfect load balance in our runs.
The eigenvalue equation for the bound state is solved iteratively by using Lanczo's type techniques, here the method of iterated orthogonal eigenvectors. For a typical run ten orthogonal eigenvectors are calculated per energy. We need about 5 to 7 energy iterations to find the ground state energy.
The calculation of the kernel of the integral equation means evaluating the matrix elements on a fixed grid p,q, and angle x=cos(p.q). The two-body t-matrix (with the two-nucleon pair interaction as the driving term) is obtained by solving a system of linear equations of the form A*x = b, where A is typically a 2500*2500 matrix. This system is solved for about 60 different vectors b, distributed over correspondingly many processors. The integrations over the 3NF terms are also distributed over the same number of processors, which depends on the size of the q-grid.
As first test for the numerical accuracy of the solution of the Faddeev
equation as function of vector variables we determined the energy eigenvalue
of the bound system with 2N forces alone, and compared out result with
the one obtained in a traditional Faddeev calculation carried out in a
partial wave truncated basis. We achieved excellent agreement (5 significant
figures) between the two approaches as well as with calculations in the
Then we incorporated scalar 3N forces of Fujita-Miyazawa type into the calculation and calculated the binding energy E of the three-body system, the wave function, and with the wave function the expectation value of the Hamiltonian <H>. Both values agree within 0.1%.
We studied the influence of the strength and range of the three-body force on the wave function, single particle distributions and two-body correlation functions. We also investigated the extreme case of pure three-body forces acting in the system.
Nuclear scattering at intermediate energies of a few hundred MeV requires quite a few angular momentum states in order to achieve convergence of e.g. 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 as well as 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 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 where currently experimental efforts at the Indiana University Cyclotron Facility (IUCF), the Kernfysisch Versneller Instituut (KVI), as well as the Research Center for Nuclear Physics (RCNP) are under way.
H. Liu, Ch. Elster, W. Glöckle, Model Study of Three-Body Forces in the Three-Body Bound State, http://xxx.lanl.gov/ps/nucl-th/0207062, submitted to Few-Body Systems.
H. Liu, Ch. Elster, W. Glöckle, Numerical Implementation of Three-Body
Forces into Bound State Faddeev Calculations in Three Dimensions,
to appear in Computer Physics Communication, and http://xxx.lanl.gov/ps/nucl-th/0204027.
I. Fachruddin, Ch. Elster, and W. Glöckle, New Forms of Deuteron Equations and Wave Functions Representations, Phys. Rev. C63, 054003-1 (2001).
I. Fachruddin, Ch. Elster, and W. Glöckle, Nucleon-Nucleon Scattering in a Three Dimensional Approach, Phys. Rev. C62, 044002-1 (2000).
W. Schadow, Ch. Elster, and W. Glöckle, Three-Body Scattering Below Breakup Threshold: An Approach without using Partial Waves, Few-Body Systems 28, 15 (2000).
Ch. Elster, W. Schadow, A. Nogga, W. Glöckle, Three Body Bound
State Calculations without Angular Momentum Decomposition, Few-Body
Systems 27, 83 (1999).
The magnitude of the 3N bound state wave function Psi(p,q,x)
for x=cos(theta)=1 in units fm3 calculated with a Malfliet-Tjon
like two-body potential, containing a short range repulsion and an intermediate
The magnitude of the 3N bound state wave function Psi(p,q,x)
for x=cos(theta)=1 in units fm3 calculated with a scalar three-body
force of Fujita-Miyazawa type, containing a short-range repulsion and an
intermediate range attraction.