D-state Wave Function Research Articles

Machine learning the deuteron

We use machine learning techniques to solve the nuclear two-body bound state problem, the deuteron. We use a minimal one-layer, feed-forward neural network to represent the deuteron S- and D-state wavefunction in momentum space, and solve the problem variationally using ready-made machine learning tools. We benchmark our results with exact diagonalisation solutions. We find that a network with 6 hidden nodes (or 24 parameters) can provide a faithful representation of the ground state wavefunction, with a binding energy that is within 0.1% of exact results. This exploratory proof-of-principle simulation may provide insight for future potential solutions of the nuclear many-body problem using variational artificial neural network techniques.

Graph-theoretical study of the nodal structure of the degenerate-state wave functions

An invariant graph-theoretical approach is proposed which allows to be determined the nodal structure of degenerate-state wave functions up to some equivalence. Nodal structures of p- and d-state wave functions for one-electron systems in a spherically symmetrical field are analyzed.

Polarization correlation coefficient for the [formula omitted]( [formula omitted]) 4He reaction

A new approach to study the high momentum component of deuteron D-state wave function by measuring the polarization correlation coefficient, C // [ =1+ 1 4 A yy+A xx + 3 4 C y,y+C x,x ], for the 3 He → ( d → ,p ) 4He reaction is proposed. The first measurement of C // has been carried out at E d =270 MeV by using a newly constructed spin-exchange-type polarized 3He target. The obtained value is 0.223±0.044(statistical)±0.037(systematic). DWBA calculations are unable to reproduce the present results. Each component of C // is compared with data for the d+ p backward scattering and the deuteron inclusive breakup. A similarity in polarization observables is found among these reactions.

Polarized 3He project at RIKEN

A new approach to study the high momentum component of deuteron D-state wave function by measuring the polarization correlation coefficient, C ∥, for the 3 H e( d , p) 4He reaction has been proposed. In order to carry out the experiment, a spin exchange type polarized 3He target was constructed at RIKEN Accelerator Research Facility (RARF). The density of the target was 2.2 × 10 20 cm −3 (8.2 amagats), and the polarization was about 10%. The spin flip system worked well. The experiment has been carried out in November, 1996, at RARF, using a polarized deuteron beam of 270 MeV.

A positive short range tensor model for the deuteron

A phenomenological model for the neutron-proton system in the 3S 1− 3D 1 coupled states is introduced. This model, unlike most phenomenological models, has a tensor potential which is positive at short distances. This produces a node in the D-state wavefunction.

P- [formula omitted] Radiative capture and the 3He D-state

The tensor analyzing power A yy ( θ) of the reaction 1H( d , γ) 3He has been measured at θ lab = 90° and projectile energies of 29.2 and 45.3 MeV. The reaction proceeds mainly through an E1 transition from the unbound 2 P 1 2 and 4 P 3 2 proton-deuteron states to the bound 3He ground state. The tensor analyzing power arises from the interference of the transitions to the 3He S- and D-states and offers an ideal way to check the D-state wave functions predicted by Faddeev calculations. Effects of meson-exchange currents are small. An exact Faddeev calculation has been performed for both continuum and bound states using the RSC interaction. The comparison with experiment shows that in the region 2–5 fm the calculated D-state wave function is too large by ∼20%.

1H + 2[formula omitted] radiative capture and the 3He d-state

The tensor analyzing power of the reaction 1 H( d, γ) 3 He has been measured at θ cm = 96° and deuteron energies of 29.2 and 45.3 MeV. It arises from the interference of the transitions to the 3He S- and D-states and allows a selective check of the D-state wave function. The experimental result is compared to a Faddeev calculation treating both continuum and bound states exactly.

Optimal basis sets for deep levels. II. Defect-molecule approximation.

We test the ``defect-molecule'' model of Coulson and Kearsley by calculating s- and p-type bound-state energies in silicon for attractive and repulsive Gaussian substitutional impurity potentials with exponential coefficients \ensuremath{\alpha}=0.2 and 1 a.u. The model is shown to work surprisingly well, which justifies its widespread use in the interpretation of experimental and computational results. The leading correction is found to be a d-state wave function on the impurity site. This enters for p states only and is much more important for repulsive than for attractive potentials. The basis set of the wave functions is much smaller than the nonorthogonal basis of the Green's function. In order to treat this problem properly we have developed a least-squares method of approximating the Green's-function equation. The method should be applicable to other problems also.

On mass corrections and the axial coupling constant in the chiral quark model

We investigate the higher order corrections to the baryon masses and the baryon axial coupling constant g A in a perturbative treatment of the Goldstone pion field. The zeroth-order quark equations of motion in the chiral quark bag model gives the MIT bag results. The next order gives the dominant corrections to the masses and g A . We show that the perturbation expansion of the pion field is justified for large quark bag radii R, but even for R ∼ 1 fm these higher order corrections calculated here are not negligible. A new spacial configuration for the quarks is introduced by the pion tensor force between different up and/or down quarks. The result is that the quarks in baryons have a mixture of S- and D-state wave functions which have important implications on, e.g., g A and the magnetic moments for different baryons as discussed by Glashow. Only baryons with two or more u/d quarks can have D-states. We neglect the tensor forces (and D-states) implied by the gluon exchange between quarks in this discussion.

On the total pion cross section of aligned deuterons

The total pion-deuteron cross section depends upon the tensor polarization of the target nucleus. Estimates of the contributions of different mechanisms to this effect are presented and they all depend sensitively on the deuteron D-state wavefunction. At high energies the multiple scattering seems to be the most important, while in the resonance region the Fermi motion takes over. Meson-exchange current phenomena may be significant above 1 GeV/c.

The magnetic form factors of 3He and 3H with mesonic-exchange corrections : Effects of the D-state and the role of short-range correlations

We show that the D-state of the trinucleon and the short-range behavior of the S- and D-state wave functions are extremely important in determining the magnetic form factors of the three-body system. We present a complete calculation of the form factors taking into account the conventional impulse approximation for the nuclear current and contributions from mesonic exchange currents. The D-state wave function has been determined by perturbation theory.

Approximate d-states and meson exchange currents

A simple approximate form for the [21] D-state component of the 3-nucleon ground state wave function, which has correct asymptotic behaviour is introduced and compared to a previously used form. We discuss the implications of approximate D-state wave functions in evaluation of meson exchange current effects.

Final-state interactions in the break-up reaction of deuterons by nucleons

The nucleon-nucleon correlation spectra of the cross sections for the break-up reaction of douterons by nucleons are calculated taking account of final-state interactions between nucleons in pairs. The higher partial waves in the interactions and the D-state wave function of deuterons are included. The partial waves are obtained by solving the Schrödinger equations with the Hamada-Johnston potential for two-body scattering states. The results are compared with experimental data for 198, 156, 100, 46, 45, 23 and 17 MeV incident protons.

Final-state interactions in the reaction 2 H (p, 2p) n

The cross sections for the reaction 2H(p, 2p)n are calculated by taking into account final state interactions including S, P and D waves for all pairs of outgoing nucleons. The D-state wave functions of the deuteron are also included in the calculation. The angular correlation distributions of the cross sections are compared with data at 156 MeV incident protons.

The 3He 3H charge form factor and the 3He coulomb energy

A formula for the Coulomb energy difference of 3He and 3H is derived on the assumption that the trinucleon S-state wave function and the symmetric part of the S'- and D-state wave functions can be represented as a linear superposition of Gaussians, The formula contains only experimentally measured quantities, since the nuclear physics can be parameterized directly in terms of the observed 3He and 3H charge form factors. The effect of two-body correlations is also discussed. The result for the Coulomb energy difference is 0.64±0.01 MeV.

Electrodynamics of direct interparticle action. II. Relativistic treatment of radiative processes: F. Hoyle and J. V. Narlikar. Institute of Theoretical Astronomy, University of Cambridge, Cambridge, England

This chapter discusses the three reactions where theory and experiment interact intermittently, and those are ND radiative capture, quasifree electron proton scattering in three-particle nuclei, and pion double charge exchange. The radiative ND capture reaction at low energies is fairly well understood. There are El transitions that depend on the D-states of the two- nucleon and three-nucleon bound states. However, the most important property is the long range parts of the D-state wave functions. The contribution to the isotropic term arises from Ml transitions that depend on the S1-state of the 3N bound state and are sensitive to meson exchange currents. The radiative capture of thermal neutrons in deuterium is a Ml transition. When the cross-section is calculated using the Ml operator given by only the nucleon spin and orbital magnetic moments, one finds a value of 0.15 mb, a discrepancy of a factor of 4. The experimental cross-section is obtained by including a phenomenological exchange operator with a magnitude consistent with the He3 and H3 magnetic moments. A more realistic representation of the meson exchange effects is possible and desirable. The chapter also describes an experiment on quasifree electron proton scattering in He3 and discusses an investigations on the pion double charge exchange reactions.

On the choice of D-state wave functions in triton calculations

A set of D-state wave functions with the correct asymptotic properties is suggested for use in variation calculations of the triton energy. The matrix elements of the kinetic and potential energy operators are evaluated for these functions.