Four-body Nuclei Research Articles

Electromagnetic Breakup of Three- and Four-Body Nuclei

Recent progress in the non-relativistic description of the electromagnetic disintegration of three- and four-body nuclei is reported. At first, the relevant degrees of freedom are considered, paying particular attention to the role of the Δ-resonance. The consistency of electromagnetic currents with the strong interaction Hamiltonian is illustrated with examples from conventional models and chiral perturbation theory. The relevance of relativistic corrections for specific kinematics is pointed out. Furthermore, the Lorentz integral transform (LIT) method, the implementation of the Δ-resonance in the LIT approach, and the LIT inversion are briefly outlined. Various examples of specific processes in photo- and electrodisintegration are discussed in greater detail.

Electromagnetic reactions of few-body systems with the Lorentz integral transform method

Various electromagnetic few-body break-up reactions into the many-body continuum are calculated microscopically with the Lorentz integral transform (LIT) method. For three- and four-body nuclei the nuclear Hamiltonian includes two- and three-nucleon forces, while semirealistic interactions are used in case of six- and seven-body systems. Comparisons with experimental data are discussed. In addition various interesting aspects of the 4He photodisintegration are studied: investigation of a tetrahedrical symmetry of 4He and a test of non-local nuclear force models via the induced two-body currents.

State-dependent effective interaction for the hyperspherical formalism with noncentral forces

The recently developed effective interaction method for the hyperspherical harmonic formalism is extended to noncentral forces. Binding energies and radii of three- and four-body nuclei are calculated with AV6 and AV14 NN potentials. Excellent results for the convergence of the expansion are found, particularly for the three-nucleon system. Due to the higher density the convergence rate is a bit slower for the alpha particle. In comparison to central potential models there is only a very slight deterioration of the convergence due the tensor force, while other potential terms have no visible effect on the convergence. The obtained values for binding energy and radii also agree well with the results in the literature obtained with other few-body techniques.

Electron scattering from light nuclei

Elastic and inelastic scattering of electrons by three- and four-body nuclei is studied with the Monte Carlo method using wave functions based on realistic nuclear interactions. There is a reasonable agreement between the theoretical predictions and recent experimental data indicating the dominance of nucleon degrees of freedom in determining the structure of these nuclei.

Charge form factors of the three- and four-body nuclei.

The charge form factors of $^{3}\mathrm{H}$, $^{3}\mathrm{He}$, and $^{4}\mathrm{He}$ are calculated using the Monte Carlo method and variational ground-state wave functions obtained for the Argonne two-nucleon and Urbana-VII three-nucleon interactions. The model for the charge density operator contains the two-body exchange contributions of longest range. With some spread due to the uncertainty in the electromagnetic form factors of the nucleon the calculated charge form factors are in good agreement with the empirical values over the whole experimentally covered range of momentum transfer.

Coulomb sum rule of A=2, 3, and 4 nuclei.

The ${\ensuremath{\omega}}^{2}$-weighted sums ${W}_{L}^{(2)}$(k) of the longitudinal response of two-, three-, and four-body nuclei are calculated with realistic nuclear forces, and exact (A=2) and variational (A=3 and 4) wave functions. These and the \ensuremath{\omega}-weighted sums ${W}_{L}$(k) are used to estimate the unmeasured longitudinal response at \ensuremath{\omega}>${E}_{\mathrm{max}}$. The Coulomb sums obtained by integrating the measured response up to \ensuremath{\omega}=${E}_{\mathrm{max}}$ and the estimated response at \ensuremath{\omega}>${E}_{\mathrm{max}}$, are in good agreement with the theoretical predictions in light nuclei. The $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ data show the presence of proton-proton correlations, although they are not accurate enough to allow for a quantitative determination of the size of the repulsive core in the nucleon-nucleon interaction.

Sum rules of the transverse response functions of A =2, 3 and 4 nuclei

The non-energy- and energy-weighted sum rules of the transverse response functions of 2H, 3H, 3He and 4He are calculated using the realistic Argonne v 14 and Urbana model-VII three-nucleon interactions, exact ( A = 2) or variational ( A = 3 and 4) wave functions and one-body currents. The Monte Carlo method for the three- and four-body nuclei, and standard numerical techniques for the deuteron are employed to evaluate, without any approximation, the ground-state expectation values. The non-energy-weighted sum rule S T N ( k) is significantly affected by nucleon-nucleon correlations, particularly the long-range ones induced by the tensor component of the nuclear force. The strong spin-, isospin- and momentum-dependent terms of the interactions give large contributions to the energy-weighted sum rule W T N ( k), and enhance it by 25–300% in the momentum transfer region of interest. We analyse the transverse scattering data on 2H, 3H, 3He and 4He. S T N ( k) and W T N ( k) are used to estimate, in the quasi-elastic region of the transverse response, the contributions given by two- (or many-)body currents and the interference between one- and two(or many-)body currents. Although these “non-nucleonic” contributions are small in 2H, 3H and 3He, they are found to be quite large in 4He. Some of the implications of this fact are discussed.

Energy weighted sums of longitudinal response

The energy weighted sums W L( k) of longitudinal response of three- and four-body nuclei and nuclear matter are calculated with realistic nuclear forces and variational wave functions. Results with Faddeev wave functions are also reported for three-body nuclei. The isospin exchange interactions give large contributions to W L( k) and enhance it by 10–80% in the region of interest. We analyze the longitudinal scattering data on 3He, 12C, 40Ca, 48Ca and 56Fe. The W L( k) of 12C, 40Ca, 48Ca and 56Fe are estimated from that of nuclear matter by taking into account its dependence on neutron excess ( N− Z)/ A. The experimental data provide S L( k, ω) at ω < E max, and in the best cases, 3He ( 12C) at k = 350 MeV/ c, it gives only 86% (68%) of the predicted sum. The response at ω > E max is assumed to be given by an exponential or square tail that joins the data at ω < E max, and gives the remainder of W L( k). The contribution of the tail to the longitudinal structure function S L( k) is not too sensitive to its assumed (exponential or square) shape, and greatly improves the agreement between experiment and theory of S L( k).

Exact ground states of few-body nuclei with and without three-body forces

Approximation-free solutions of the Schr\odinger equation are given for the ground states of three- and four-body nuclei, employing spin-independent central two- and three-body forces. Comparison with the results of more approximate few-body methods shows their relative accuracy.

A study of three-nucleon interaction in three- and four-body nuclei

The variational Monte Carlo method is used to study the effect of model three-nucleon interactions on the binding energy and structure of 3He and 4He nuclei. Realistic two-nucleon intertions, such as Reid v 8 and Urbana v 14, as well as simple spin-isospin independent two-nucleon interactions such as the Malfliet-Tjon model V are used in this study. The three-nucleon interaction V ijk is assumed to depend on r ij , r ik and the angle θ i , of the triangle r i, r j and r k . It is shown that V ijk having P 2 (cos θ i ) angle dependence can explain the binding energy and the dip in the proton distribution of 3He nucleus. Results of limited studies of the effect of spin-isospin dependence of V ijk , are also presented. Simple models of V ijk , having only P 0 (cos θ i ) and P 2 (cos θ i ) terms can bring the theory of these nuclei closer to experiment, however they seem to give a much smaller dip in the proton distribution of 4He.

Independent-pair wave functions for the three- and four-body nuclei

A method which was described previously for generating independent-pair wave functions for three-nucleon systems is generalized so that the S′ state with mixed symmetry can be included in three-body calculations and so that the method can be used to generate independent pair wave functions for the alpha particle. This procedure is used to find approximate ground-state functions for several three- and four-body Hamiltonians. The functions so obtained are used to calculate an upper and, in some cases, a lower bound to the ground-state energy and to calculate the charge form factors for these three- and four-body systems. When the interactions are chosen so that both low- and high-energy two-nucleon data are correctly predicted, then the calculated binding energies are slightly too large, the ratio of the form factors for 3H to that of 3He is consistent with the experimental ratio, and the form factors for all three nuclei are too large at large momentum transfers. By increasing the radius of the hard core, the calculated form factor functions are decreased closer to the experimental ones.

Polarized wave functions for a few-body model nucleus

An approximation scheme for the wave function of a few-body system is developed, in which intrinsic polarization is introduced in a natural way. The polarized wave function is tested with model three- and four-body nuclei, and it is found that two different polarizations are nearly degenerate. The resulting resonant wave function yields accurate binding energies. Furthermore, the accuracy of the spatial density is checked by means of an error-estimating principle. The result is that the advantages of simplicity and generality of the polarized wave function are coupled with very good simulation of the correct wave function.

Monte Carlo Calculations of the Ground State of Three- and Four-Body Nuclei

By means of a suitable Green's function, the Schr\"odinger equation for the few-body nuclear problem is written as an integral equation. In this equation, the binding energy of the ground state is assumed known and the strength of potential required to give this energy is an eigenvalue to be determined. A random walk can be devised whose collision density satisfies the same integral equation. The simulation of this random walk therefore permits an exact numerical solution by Monte Carlo methods. The calculation has been carried out with pairwise potentials of square, Gauss, and exponential shape.