Numerical study of a selfconsistent two-body theory for colliding ions in a one-dimensional model

A selfconsistent quantal two-body theory as obtained from the density-matrix hierarchy is solved numerically for the first time for a one-dimensional system modelling α+ α collisions at 80 MeV/u. Various truncation schemes for the two-body correlation function are investigated in order to explore the convergence properties of the theory. We find that perturbative treatments with respect to two-body processes do not yield reliable results in the energy regime investigated and that the nuclear stopping power sensitively depends on the order of the two-body correlations considered.     

The central and tensor force matrix elements for arbitrary average and two-body potentials

A general method is described for computing central as well as tensor force matrix elements for shell-model states. The method consists of expanding the two-body potential into free multipole fields. The advantages of the method are that

(i) arbitrary forms of the average and two-body potential can be chosen; (ii) even in the case of a finite-range force, double integrals are eliminated, and the radial integrals are independent of the shape and parameters of the two-body force; and (iii) the central and the tensor forces are treated on the same footing.

FORTRAN codes employing this method are available for the computation of the two-body force matrix elements for an arbitrary nucleus.     

The potential harmonic expansion method

Various properties of the hyperspherical potential basis are investigated. The expansion of any two-body function, in particular the two-body potential, is given. The matrix elements with two and three potential harmonics needed for the construction of the potential matrix are calculated. Useful recurrence formulae are derived. The concept of potential basis is extended to systems with any number of fermions. A method for improving the accuracy of the expansion of the wavefunction by taking into account more than the two-body correlations is suggested.     

A unitary model for three-body collisions

A connected 3 → 3 formalism for three-body collision processes is reduced to a hierarchy of three on-energy-shell integral equations and one off-energy-shell integral equation. Only the on-energy-shell equations, which involve only on-energy-shell three-body and two-body amplitudes, need be solved exactly in order to obtain elastic and break-up amplitudes satisfying the unitarity constraints exactly. Applied to n-d break-up, the on-energy-shell equations ensure that the n-d initial-state interaction, the nucleon-nucleon final-state interactions, and more complicated 3 → 3 processes are correctly described. After angular momentum analysis the on-energy-shell equations are one-dimensional integral equations, even in the case of local two-body potentials. This unitary model provides a practical scheme for calculating approximate three-body elastic and break-up amplitudes when two-body local potentials are used to describe the two-body subsystems.     

One- and two-body dissipation in heavy-ion collisions

A microscopic theory has been formulated for one- and two-body dissipation in collisions between two heavy nuclei. With a nucleon-nucleon interaction as the basic perturbation in a density matrix approach with “linear response” approximations, the one- and two-body nuclear friction coefficients for the ⁴⁰Ca + ⁴⁰Ca system have been calculated and their dependence on relative kinetic energy and smearing of nuclear single-particle states was obtained. The results of our calculation show that: (a) the combined one- and two-body friction coefficients compare favourably with phenomenological values, (b) the one-body dissipation is more effective than two-body in kinetic energy damping, while both the mechanisms are comparable for the damping of relative angular momentum, (c) the importance of the two-body friction compared to one-body increases at higher relative kinetic energies and (d) the effect of introducing a smearing in nuclear levels appears as a lowering of nuclear friction.     

Interplay between two- and three-body interaction in light nuclei and nuclear matter

A recent study of different models of three-nucleon interaction (TNI) in ³He, ³H, ⁴He and nuclear matter is extended to study the influence of different choices of the accompanying two-body interaction. A new two-body potential, Argonne υ₁₄, is coupled with both the Tucson and isobar intermediate-state models of two-pion-exchange TNI, with a phenomenological intermediate-range repulsive TNI added to the latter. Variational calculations are carried out for these systems, and compared to the earlier work. We find that a stronger tensor component in the two-body potential, as typified by a larger deuteron D-state percentage, gives more attraction for the TNI, counteracting the saturation effect obtained when only two-body forces are considered.     

Differential equations in momentum space for the three-body problem in the case of pointlike pair interactions

Corrects schemes for solving equations of three-body dynamics for systems governed by a zerorange two-body interaction are considered. Correlations between spectral features of a three-boson system are obtained. The results are compared with the results obtained by calculating the spectra and scattering lengths in the system of three helium atoms with realistic two-body interaction potentials.     

Lorentz Boosted Potential for a Two-Body System with Unequal Masses

We produce a Lorentz boosted two-body potential for particles of different mass that is phase equivalent to a given realistic non-relativistic two-body potential. The relativistic potential is related to the nonrelativistic potential using the Coester–Pieper–Serduke scheme, which ensures that the same scattering wave functions are obtained from the relativistic and non-relativistic potentials. This implies that the phase shifts are identical functions of the relative momentum. To construct the potential we use an iterative scheme that generalizes one that has been applied successfully to two-body systems with equal masses.     

Two-body density matrix for closed s-d shell nuclei

The two-body density matrix for ⁴He,¹⁶O and ⁴⁰Ca within the Low-order approximation of the Jastrow correlation method is considered. Closed analytical expressions for the two-body density matrix, the center of mass and relative local densities and momentum distributions are presented. The effects of the short-range correlations on the two-body nuclear characteristics are investigated. Received: 11 September 1999 / Revised version: 20 December 1999     

Relativistic wave equation for a boson-fermion system

We present a two-body relativistic wave equation for a system composed of a boson and a fermion. One-body equations such as the Dirac and the Klein-Gordon equations are often used as an approximate equation for relativistic two-body systems. However, when the masses of two particles are not very different, the use of one-body equations comes into question. We use the Feshbach-Villars formalism for the boson so that the wave equation can be given in the form of an eigenvalue equation for the Hamiltonian. Differences between our equation and the one-body equations are examined and illustrated in a numerical example of a two-body system with scalar and vector potentials.Communicated by: W. Weise     

ON THE SHORT RANGE INTERACTIONS AMONG SKYRMIONS

A determination of the short range characteristics for the two-body and many-body interac-tions among Skyrmions is given, The effects of quantization is shown for the two-body interaction and the results are compared with those in the sixquark model.The sensitivity with respect to the pion mass is shown.     

Low-energy elastic scattemng of pions by the deuteron

The problem of the elastic scattering of pions by a deuteron is considered using the separable representation of the two-body t-matrix. The Faddeev equations are reduced to a set of one-dimensional integral equations by separating the angular variables. The dependence of the π-d scattering length on the form of two-body interaction and on the values of the π-N scattering lengths is studied in the case of a one-term nonlocal potential with separable variables. The π-d scattering length proves to be practically independent of the two-body interaction form, and is essentially dependent on the values of the π-N scattering lengths.     

Exactness of two-body cluster expansions in many-body quantum theory

The Horn-Weinstein formula and the variational principle, combined with numerical results for a few many-electron systems, are used to provide support for a conjecture that the exact ground-state wave function for a Hamiltonian system containing up to two-body terms may be represented by an exponential cluster expansion employing a finite two-body operator.     

Two-Body Correlation Transport Theory for Heavy Ion Collisions Ⅱ.The Time-Evolution Equations of Single Particle States Given by Means of the Time Dependent Variational Principle and Antisymmetrized Molecular Dynamics

Using the two-body correlation dynamics and the time dependent variational principle the equations of nonrelativistic two-body correlation transport theory(TBCTT) for heavy ion collisions are established.This theory is used to discuss Fermionic Molecular Dynamics (FMD) and Antisymmetrized Molecular Dynamics (AMD).The collision effects can naturely appear in the time-evolution equations of its single particle states.     

Can the eigenstates of a many-body Hamiltonian be represented exactly using a general two-body cluster expansion?

It is proved that a recent conjecture that the exact ground-state wave function of an arbitrary many-fermion system with one- and two-body interactions may be represented by an exponential cluster expansion employing finite two-body operators, starting from any reference function sufficiently close to the exact eigenfunction, is not valid. We show that the space of initial reference functions which lead to the exact ground state is of dimension equal to the number of two-body operators. If the dimension of the multiparticle space is greater than the number of two-body operators, then the space of good reference functions is of measure zero in it.     

On statistical mechanics of systems with highly singular two-body potentials

A finite volume statistical mechanics for highly singular two-body potentials is studied. The case of “point” hard-core particles (Lennard-Jones type two-body potentials) is discussed and the mathematical background for this physical idealization is given. Firstly the definition of a self-adjoint Hamiltonian for such a system is considered. Then a natural cutoff procedure for two-body highly singular potentials is proposed. The main result is the proof of a convergence theorem for partition sums (or free energies) when the cutoff parameter is removed to infinity. The question of stability of the cutoff interactions is also discussed. These results are illustrated by a consideration of the Lennard-Jones potential (12-6). Our results are valid in all dimensions greater than two and for potentials being not inevitably spherically symmetric.     

Four-Body Bound-State Calculations in Three-Dimensional Approach

The four-body bound state with two-body interactions is formulated in three-dimensional approach, a recently developed momentum-space representation which greatly simplifies the numerical calculations of few-body systems without performing the partial wave decomposition. The obtained three-dimensional Faddeev-Yakubovsky integral equations are solved with two-body spin-independent and spin-averaged potentials. This is the first step toward the calculations of the four-nucleon bound-state problem in three-dimensional approach. Results for four-body binding energies are in good agreement with achievements of the other methods.     

Self-consistent truncation of the BBGKY hierarchy on the two-body level

We present a coupled set of equations for the one-body density matrix and the two-body correlation function consistent with trace relations and conservation laws which provide an extension of correlation dynamics on the two-body level. The additional interaction terms are discussed in the context of time-dependent G-matrix theory and NQCD and studied numerically by one-dimensional calculations for colliding finite fermion systems.Supported by BMFT and GSI Darmstadt