Effects of the Three-Nucleon Forces from Different Meson Exchanges on the Triton Binding Energy

Effects of the three-nucleon forces on the triton binding energy are investigated using different models for which the xN scattering amplitudes for the off-mass-shell pions are extrapolated from the on-mass-shell data. In the independent model potential, the threenucleon forces are constructed using the chiral symmetry through partial conservation of the axial current vectors and current algebra. On the other hand, the three-nucleon forces in the dependent model potential are derived from an effective Lagrangian which is constrained by chiral and gauge symmetry. The effect of the long range 2π and the repulsive ap-exchange three-nucleon forces on the triton binding energy are investigated. The bound state triton binding energies are then calculated by solving the Faddeev equations with a Hamiltonian including the three-nucleon forces, using different nucleon-nucleon interactions. The pionic form factors are considered with different sets of the form factor cutoff,parameter Λ. The 2π-exchange forces with S- and P-wave amplitudes are found to overbind the triton binding energy by about 0.4～Y 1.6 MeV. However, xp-exchange forces are found to reduce the 2π exchange forces by about 18% of its contribution, to the triton binding energy.     

Triton Photodisintegration with Effective Field Theory

Effective field theory (EFT) has been recently used for the calculation of neutron–deuteron radiative capture at very low energies. We present here the use of EFT to calculate the two-body photodisintegration of the triton, considering the three-body force. The calculated cross section shows sharp rising from threshold to maximum about 0.88 mb at ~13 MeV and decreasing slightly to about 0.81 mb at ~19 MeV, in agreement with the experimental data. Our results are in good agreement with the experimental data and the other calculations using modern realistic two- and three-nucleon forces, like AV18/UrbanaIX potential.     

Phenomenological Non-Local Nucleon-Nucleon Interactions and the Neutron-Deuteron Break-Up Process

 Phenomenological “inside non-local, outside Yukawa-tail”N-N interactions, which fit the N-N data and produce the correct triton binding energy, are tested in the n-d break-up process at bombarding energies of 10.5, 13, and 19 MeV. The sensitivity of measurable quantities to the type and to the components of the N-N interactions is investigated. In some kinematic regions the break-up nucleon vector-analyzing power shows a strong sensitivity to the details of the triplet P-wave interactions. Received October 20, 1998; accepted in final form January 30, 1999     

Polarization-Transfer Coefficients in and Reactions at MeV: Experiment Compared to Calculations with Recent Nuclear-Interaction Models

The polarization-transfer coefficients K , K and K , K have been measured in the elastic scattering reactions D(, )p and D(, )d at MeV, respectively. They are compared to solutions of the three-nucleon Faddeev equations obtained with the recent nucleon-nucleon interactions AV18, CD Bonn, NijmI and II. Effects of the Tucson-Melbourne three-nucleon force, adjusted separately to reproduce the triton binding energy for each of these potentials, are studied. Both and exhibit a scaling behaviour with the triton binding energy. For and the various predictions with two-nucleon forces only agree practically with each other but spread after inclusion of the three-nucleon force. The agreement of theory and data is fair but the neglect of the proton-proton Coulomb force precludes a final conclusion. Received July 28, 1997; revised February 3, 1998; accepted for publication March 11, 1998     

Can the SCA with relativistic Hartree-Fock wave functions describe the high energy ionisation probability

Recently experimental data of the L-shell ionisation probability of Au in the impact parameter region around the L-shell radius have been reported with α-particle energies in the range of 12–50 MeV. A very large discrepancy growing with increasing energy was found between the experimental and theoretical values. Refined calculations using the semiclassical approximation (SCA) with hyperbolic projectile trajectories and relativistic Hartree-Fock electron wave functions show, that this discrepancy can still be explained unsufficiently.     

Thomas-Fermi theory for atomic nuclei revisited

The recently developed semiclassical variational Wigner-Kirkwood (VWK) approach is applied to finite nuclei using external potentials and self-consistent mean fields derived from Skyrme interactions and from relativistic mean field theory. VWK consists of the Thomas-Fermi part plus a pure, perturbative ?² correction. In external potentials, VWK passes through the average of the quantal values of the accumulated level density and total energy as a function of the Fermi energy. However, there is a problem of overbinding when the energy per particle is displayed as a function of the particle number. The situation is analyzed comparing spherical and deformed harmonic oscillator potentials. In the self-consistent case, we show for Skyrme forces that VWK binding energies are very close to those obtained from extended Thomas-Fermi functionals of ?⁴ order, pointing to the rapid convergence of the VWK theory. This satisfying result, however, does not cure the overbinding problem, i.e., the semiclassical energies show more binding than they should. This feature is more pronounced in the case of Skyrme forces than with the relativistic mean field approach. However, even in the latter case the shell correction energy for e.g., ²⁰⁸Pb turns out to be only ∼−6 MeV what is about a factor two or three off the generally accepted value. As an ad hoc remedy, increasing the kinetic energy by 2.5%, leads to shell correction energies well acceptable throughout the periodic table. The general importance of the present studies for other finite Fermi systems, self-bound or in external potentials, is pointed out.     

Nucleon-nucleus optical model potential: (1). Nuclear matter approach

We present a new method for the approximate solution of the Bethe-Goldstone equation when it has a singular kernel. The method reduces the integral equation to a set of coupled differential equations which are easily solved. In the special case of binding energy calculations, non-singular kernel, our method is equivalent to the reference spectrum method. A particular advantage is that there is no ambiguity in the treatment of hard-core N-N interactions. We perform calculations both for the Hamada-Johnston and Reid hard-core internucleon potentials and in intermediate states always use self-consistent single-particle energies. We apply the method to calculate in nuclear matter the binding energy/nucleon and the nucleon optical potential. Our results for the binding energy differ by about 2 MeV from those published for similar calculations. The difference is a consequence of our use of self-consistent energies and a greater number of partial waves, L ≦ 4. For the optical potential we obtain a logarithmic variation with incident energy E for E > 100 MeV, in agreement with experimental data. We also obtain better agreement with experiment than other authors for the energy variation in the the range 40 MeV < E < 100 MeV. This improvement is a consequence of our use of a higher number of partial waves.     

Non-equilibrium dynamics in high-energy nucleus-nucleus collisions: (II). Calculation of inclusive particle spectra

We present theoretical results for proton, deuteron, triton, ³He and ⁴He inclusive particle spectra measured in 250 MeV/amu and 400 MeV/amu nucleus-nucleus reactions. The calculations are based on a multiple collision model which is derived from the relativistic Boltzmann equation and which includes composites formation through thermodynamics. The agreement with the data is reasonable to good and the remaining discrepancies are discussed.     

OBEP and eikonal form factor: (II). Nuclear matter results

Nuclear matter properties are calculated in a first-order Brueckner-Bethe calculation using a one-boson exchange potential recently proposed by the authors, in which the phenomenological cutoff of dipole type used so far has been replaced by a form factor obtained from an eikonal approximation to multiple vector meson exchange processes. We find ?23.5 MeV saturation energy at a Fermi momentum k_F = 1.77 fm^?1, i.e. about 12 MeV more binding than realistic OBEP using dipole-type cutoffs and about 8 MeV overbinding compared to the empirical value of 16 MeV. On the other hand, estimates suggest that, compared to the Reid soft-core potential, this new OBEP predicts about 1.5 MeV more binding in the case of the triton and about 4 MeV more binding in the case of ¹⁶O, i.e. gives nearly the correct empirical result. The additional binding is traced back to the small deuteron D-state probability of 4.32% predicted by this OBEP, which is a consequence of the special structure of the eikonal form factor. However, taking the effect of the Δ-resonance into account recently given by Green and Niskanen, one arrives at ?14 MeV saturation energy for nuclear matter at k_F = 1.36 fm^?1, whereas the results for the triton and ¹⁶O are changed to a negligible extent only.     

Fission of highly excited fragments from collisions of 750 A.MeV 238U-ions on 208Pb

Fragments of relativistic 750 A.MeV U-projectiles were investigated by using the fragment separator FRS for magnetic selection of reaction products including ray-tracing and ΔE-ToF techniques. For elements between Ge and Sb, measurements of isotopic yield distributions and velocities revealed three processes: fragmentation, low-energy fission, and high-energy fission. The last of these regimes is presently reported. First and second moments of distributions of mass numbers, atomic numbers and velocities of the corresponding fragments allowed us to identify ¹⁰¹ ₄₃Tc₅₆ as the most probable fragment of a high energy symmetric fission reaction. Moreover, we could deduce a hypothetical mean fissioning fragmentation product ²⁰⁸Rn and its highly excited pre-fragmentation parent ²²⁷Ra produced in a primary abrasion reaction at an excitation energy of about 290 MeV. Received: 26 January 1998 / Revised version: 16 March 1998     

Study of the Superdeformed State of 196Pb in the Relativistic Mean Field Theory

Based on the constrained relativistic mean field (RMF) theory, the superdeformed states of ¹⁹⁶Pb are systematically investigated with four different interactions, TMA, PK1, NL3 and NL-SH. The potential surface, the quadruple deformation of ground and superdeformed states, and the excitation energies of superdeformed states are calculated. The results show that the shape of ¹⁹⁶Pb is oblate for the ground state with deformation β₂≈－0.15, and prolate for the superdeformed states with deformation β₂≈0.60. The calculated excitation energy and the depth of the potential well of the superdeformed state are approximately equal to 4.5MeV and 1.6MeV, respectively. These results are in good agreement with the current experimental data. It indicates that RMF theory can well describe the energy of the band head of superdeformed rotational band in ¹⁹⁶Pb.     

An application of a new harmonic-oscillator basis to the calculation of trinucleon ground-state observables

A new harmonic-oscillator basis for trinucleon ground-state calculations is introduced, featuring different oscillator radii for the two intrinsic variables. This basis allows charge-dependent interactions to be handled and seems to yield a better convergence with respect to the previously used oscillator bases. A test calculation with the Reid soft-core interaction is presented. The resulting (extrapolated) triton binding energy is 7.3 ± 0.2 MeV, and the first minimum in the ³He charge form factor occurs at q² = 13.1 fm^?2.     

A calculation of some parameters in Migdal's theory of nuclei

Some of the parameters in Migdal's theory of nuclei have been calculated. We used principally Landau's phenomenological approach to obtain the quasi-particle interaction. The energy was calculated by Brueckner's theory. We have used effective interactions which in infinite nuclear matter give the same potential energies in (¹S₀+¹D₂ and (³S₁+³D₁) states as the Hamada-Johnston potential. Other contributions to the binding energy were neglected. The nuclear compressibility K was calculated both from the energy-versus-density curve and from the calculated parameters of Migdal's theory. The results were not in conflict with each other. The values of K obtained were 100–150 MeV, which agrees reasonably well with other calculations in infinite nuclear matter, but they are only about of the values bsed by Migdal and collaborators. We obtained 0.7 for the effective mass and 22–25 MeV for the symmetry energy. The latter result agrees reasonably well with empirical values. An attempt to use the Green function approach in the calculations of the parameters was not successful because of poor convergence, if any.     

Relativistic Calculations for Trapped Ions

We present recent results in the field of total binding energy calculations, Landщ factors, quantum electrodynamics corrections and lifetime that are of interest for ion traps and ion sources. We describe in detail MCDF and RMBPT calculation of ionic binding energies, which are needed for the determination of atomic masses from highly charged ion measurements. We also show new results concerning Landщ factor in 3-electron ions. Finally we describe how relativistic calculations can help understand the physics of heavy ion production ion sources. This revised version was published online in July 2006 with corrections to the Cover Date.     

Universal correlations in pion-less EFT with the resonating group method: Three and four nucleons

The Effective Field Theory “without pions” at next-to-leading order is used to analyze universal bound-state and scattering properties of the 3- and 4-nucleon system. Results of a variety of phase shift equivalent nuclear potentials are presented for bound-state properties of ³H and ⁴He , and for the singlet S -wave ³He -neutron scattering length a ₀(³He-n) . The calculations are performed with the Refined Resonating Group Method and include a full treatment of the Coulomb interaction and the leading-order 3-nucleon interaction. The results compare favorably with data and values from AV18(+UIX) model calculations. A new correlation between a ₀(³He-n) and the ³H binding energy is found. Furthermore, we confirm at next-to-leading order the correlations, already found at leading order, between the ³H binding energy and the ³H charge radius, and the Tjon line. With the ³H binding energy as input, we get predictions of the effective field theory “without pions” at next-to-leading order for the root mean square charge radius of ³H of (1.6±0.2) fm, for the ⁴He binding energy of (28±2.5) MeV, and for Re{a ₀(³He-n)} of (7.5±0.6) fm. Including the Coulomb interaction, the splitting in binding energy between ³H and ³He is found to be (0.66±0.03) MeV. The discrepancy to data of (0.10±0.03) MeV is model independently attributed to higher-order charge independence breaking interactions. We also demonstrate that different results for the same observable stem from higher-order effects, and carefully assess that numerical uncertainties are negligible. Our results demonstrate the convergence and usefulness of the pion-less theory at next-to-leading order in the ⁴He channel. We conclude that no 4-nucleon interaction is needed to renormalize the theory at next-to-leading order in the 4-nucleon sector.     

LOCV calculation of nuclear matter with relativistic Hamiltonian

The equation of state of symmetric nuclear matter is calculated using the relativistic Hamiltonian (H_R) with potentials which have been fitted with the N -N scattering data using the relativistic two-body Hamiltonian ( [(v)\tilde]₁₄ \tilde{{v}}_{{14}}^{} and the non-relativistic two-body Hamiltonian, i.e. the Argonne V₁₄ interaction. The boost interaction corrections as well as the relativistic one-body and two-body kinetic energy corrections in cluster expansion energy within the lowest-order-constrained variational method are calculated. It is shown that the relativistic corrections reduce the binding energy by 1.5MeV for [(v)\tilde]₁₄ \tilde{{v}}_{{14}}^{} and AV₁₄ interactions. The symmetric nuclear-matter saturation energy is about -16.43 MeV at r \rho = 0.253 (fm^-3) with [(v)\tilde]₁₄ \tilde{{v}}_{{14}}^{} interaction plus relativistic corrections. Finally, various properties of the symmetric nuclear matter are given and a comparison is made with the other many-body calculations.     

High-precision covariant one-boson-exchange potentials for np scattering below 350 MeV

Using the covariant spectator theory (CST), we have found one-boson-exchange (OBE) potentials that fit the 2006 world np data below 350?MeV with a χ²/N _data very close to 1, for a total of 3788 data. Our potentials have significantly fewer adjustable parameters than previous high-precision potentials, and they also reproduce the experimental triton binding energy without introducing additional irreducible three-nucleon forces.