Elastic Charge Form Factors Research Articles

Bethe-Salpeter calculation of three-nucleon observables with multirank separable interactions.

Bethe-Salpeter equations for two and three particles are employed to study relativistic effects in the three-nucleon bound-state system. The used covariant, multirank separable interactions are constructed from one phenomenological (Graz II) and two different meson-exchange potentials (Paris and Bonn). The obtained increase in the triton binding energy due to relativity ranges from 0.29 to 0.38 MeV. In an approximation where boost effects are neglected and an angular-averaging procedure is applied in the construction of the total trinucleon wave function, the elastic charge form factors are calculated.

Effects of pion exchange currents and short-range correlation on charge form factor of4He

Effects of short-range correlation (SRC) and meson exchange currents (MEC) on elastic charge form factor of4He have been investigated. The one-pion exchange approximation is employed in the nonrelativistic limit. Form factors are calculated with SRC and MEC for a wide range of momentum transfer (up toq2=100 fm−2). It is noticed that SRC produces the first minimum and second maximum in good agreement with the experimental data. The inclusion of MEC does not improve the fit if not appreciably worsens it. However, the MEC does produce the second minimum and third maximum which SRC alone does not.

Simple procedure for solving few-body equations

In this work we propose a procedure for iterative improvement of the adiabatic solution of the hyperspherical-harmonics technique. The usefulness of the method is demonstrated by calculating the binding energies of the ground and first excited state of the4He nucleus using several nucleon-nucleon interactions. The procedure is found to be simple and very stable. We compare the elastic charge form factor of the4He nucleus obtained by the usual adiabatic solutions and the improved version of this work. This comparison gives valuable physical insight into the nature of the adiabatic approximations.

Electromagnetic properties of 6Li in a cluster model with breathing clusters.

Electromagnetic properties of $^{6}\mathrm{Li}$ are studied using a microscopic (\ensuremath{\alpha}+d) cluster model. In addition to the ground state of the clusters, their breathing excited states are included in the wave function in order to take into account the distortion of the clusters. The generator coordinate calculations are free from the spurious center-of-mass motion and arbitrary parameters. The cluster stability condition is satisfied. The elastic charge form factor ${F}_{C0}$ is in good agreement with experiment up to momentum transfer 8 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}2}$. The discrepancy appearing at momentum transfer ${q}^{2}$>8 ${\mathrm{fm}}^{\mathrm{\ensuremath{-}}2}$ must be attributed to the omission of the short range nucleon correlation. The ground state magnetic form factor ${F}_{M1}$ and the inelastic charge form factor ${F}_{C2}^{\mathrm{*}}$ are also well described. The breathing excited states of d influence the behavior of ${F}_{C0}$ at high momentum transfer only, but they have an effect on ${F}_{M1}$ and ${F}_{C2}^{\mathrm{*}}$ even at low momentum transfer. The effect of the breathing states of \ensuremath{\alpha} on the form factors proves to be negligible except at high momentum transfer. The ground-state charge density, rms charge radius, the magnetic dipole moment and a reduced transition strength are also obtained in fair agreement with experiment.

Electron-nucleus scattering at high Q

Elastic charge and magnetic form factors of 3He, 17O, and 209Bi are calculated using a consistent relativistic formalism. The wavefunctions are generated by the self-consistent relativistic Hartree formalism, and a relativistically covariant effective current operator is used. Results are obtained for momentum transfers up to 7.5 fm −1.

Six-quark bags and the charge density difference between Pb and Tl

Six-quark bags (even if present in copious numbers) can have little qualitative influence on computations of the elastic charge form factors of heavy nuclei.

Relativistic treatment of the few body system

Some specific examples of effects of mesonic degrees of freedom and special relativity in the two- and three-nucleon system are reviewed. For the twonucleon system a relativistic covariant formulation of the one boson exchange and isobar model is described based on the Bethe-Salpeter equation. Its extension to dynamical equations satisfying three-particle unitarity above one pion production threshold is presented. Moreover, the meson theoretical basis of the Dirac approach for elastic proton-nucleus scattering is discussed using the above mentioned relativistic one boson exchange model. In the past two decades strong evidence has been presented that a nonrelativistic description of the few nucleon system in terms of pair-wise interaction is not in accordance with the various experimental results. Well known examples are radiative neutron capture and the electro disintegration of the deuteron'. We review briefly two other cases in view of some recent interesting experimental and theoretical work. NN interactions which are believed to describe the nuclear force in an accurate way at least up to the one pion production threshold, leads to the underbinding of the triton typically by l-11 MeV whereas the secondary maximum in the elastic charge form factor of 3 He is much too low in the theoretical predictions. In table I some recent results are summarized for the case of the Paris and Reid-soft-core (RSC) potentials. The 5-channel calculations contain only the 'so, 3s1-3 D1 two-nucleon partial waves potentials while the complete 18-channel analysis has all the components up to angular momentum a=2 included. The results of the triton binding calculations of refs. 2 and 3 agree very well for the RSC potential. A puzzling point however is that in the Paris case they differ by 0.2 MeV which is greater than the suggested numerical inaccuracy. Possible explanations for the failure of the nonrelativistic description are the presence of the non-nucleonic degrees of freedom such as mesons and quarks and effects of special relativity. Attempts to explain the differences by the presence of mesonic degrees of freedom have been rather successful. They give rise to effective three-nucleon (3N) forces and so-called mesonic exchange

Tritium form factors at lowq

The elastic charge and magnetic form factors of $^{3}\mathrm{H}$ have been measured in the region $0.0477<{q}^{2}<2.96$ ${\mathrm{fm}}^{\ensuremath{-}2}$. Throughout this range, the charge form factor is found to be larger than previous measurements, whereas the magnetic form factor agrees with the earlier work. The change in the charge form factor slightly increases the discrepancy between the calculated and observed binding energy difference between $^{3}\mathrm{H}$ and $^{3}\mathrm{He}$.

On the Accuracy of Wave Functions and Charge Form Factors Obtained from ATMS Calculations

A function B(E) is introduced which measures the impurity of a trial wave function coming form the admixture of eigenstates with energies equal to or larger than E. An upper bound of B(E) is derived from the gap between the Ritz and Temple bounds of the energy. A small energy gap means a very small impurity at large values of E. Implications on form factor calculations based on the ATMS method are discussed. Around and above the region of the second maximum of the elastic charge form factor, in particular, the ATMS results are very reliable.

Elastic and inelastic charge form factors forLi36from electron scattering

A harmonic oscillator hard core plus residual two-body potential has been used to obtain the independent-particle shell-model wave functions for $_{3}^{6}\mathrm{Li}$. These wave functions have been utilized to calculate the theoretical elastic and inelastic electron scattering charge form factors. These are then compared with the experimental data.NUCLEAR REACTIONS Elastic and inelastic charge form factors for $_{3}^{6}\mathrm{Li}$ from electron scattering using harmonic hard core plus residual two-body potential.

Bound-State Calculations ofHe4with the Reid Soft-Core Interaction

The binding energy of $^{4}\mathrm{He}$ is calculated within the framework of the Yakubovsky equations for the Reid soft-core potential. Use is made of separable expansions of the subamplitudes. With neglect of the Coulomb effect and the non-$s$-wave parts of the two-nucleon $t$ matrix, the binding is found to be 19.5 MeV. With use of the solutions of the integral equations the elastic charge form factor is calculated. In addition to the underbinding of $^{4}\mathrm{He}$ it is found that the secondary maximum in the form factor is much too low as compared to experiment.

Some calculations for 12C in the α-particle model

In the representation of the 12C nucleus as three finite structureless α particles interacting through phenomenological potentials, we calculate the lower 0 + and 2 + energy levels, elastic and inelastic charge form factor, and transition widths. We use a variational method taking as a trial function a linear combination of harmonic-oscillator eigenfunctions in an approximation of 10 quanta.

Li6cluster-model phenomenology

A phenomenological $\ensuremath{\alpha}+d$ cluster-model wave function for the $^{6}\mathrm{Li}$ ground state is suggested and its properties fixed using data on the static $M1$ and $E2$ moments and elastic charge form factors of $^{6}\mathrm{Li}$. The resulting $\ensuremath{\alpha}+d$ momentum distribution is then compared with that deduced from knockout and transfer reactions.[NUCLEAR STRUCTURE $^{6}\mathrm{Li}$, phenom. $\ensuremath{\alpha}+d$ cluster-model wave function; parameters determined from elastic charge form factor and static $M1$ and $E2$ moments. Calculated $\ensuremath{\alpha}+d$ momentum distribution compared reactions. $^{6}\mathrm{Li}(\ensuremath{\alpha},2\ensuremath{\alpha})$, $^{6}\mathrm{Li}({\ensuremath{\pi}}^{\ensuremath{-}},2n)$, $^{12}\mathrm{C}(^{6}\mathrm{Li},d)$.]

Phenomenological Wave Function ofLi6

In the framework of the $\ensuremath{\alpha}$-deuteron cluster model, the $J=1$ deformation is considered in the phenomenological wave function of $^{6}\mathrm{Li}$ so as to produce the charge form factor and the quadrupole moment at the same time. In our model the deviation from the $S$ state of the relative motion between the $\ensuremath{\alpha}$ particle and the deuteron compensates the quadrupole moment of the free deuteron, ${Q}_{d}^{\mathrm{exp}}=+0.278\ifmmode\pm\else\textpm\fi{}0.008$ ${\mathrm{fm}}^{2}$, to give that of $^{6}\mathrm{Li}$, ${Q}_{\mathrm{Li}}^{\mathrm{exp}}=\ensuremath{-}0.08\ifmmode\pm\else\textpm\fi{}0.008$ ${\mathrm{fm}}^{2}$. It is shown that the quadrupole contribution plays an important role in filling up the diffraction minimum around ${q}^{2}=8$ ${\mathrm{fm}}^{\ensuremath{-}2}$ in the elastic charge form factor.

Investigation of the low-lying states in 6Li by inelastic electron scattering

The inelastic electron scattering cross sections for the excitation of the low-lying states of 6Li have been measured in the momentum transfer region q = 0.7–1.8 fm −, with emphasis on the transverse transitions to the 3.56 and 5.36 MeV T = 1 states. The intermediate coupling shell-model wave functions both in these states and in the ground state have been determined using radial wave functions generated in finite potentials. The form factors, especially the form factor for the M1 transition to the 3.56 MeV state, are sensitive to the radial distribution of the outer nucleons in 6Li. The form factor and nucleon distribution of the four 6Li core nucleons relative to the nuclear center of mass have been deduced from the well-known elastic charge form factor and fitted to a Gaussian shape.

Elastic form factor of the 6Li nucleus using improved radial wave functions

A variational calculation is performed for the ground state of the 6Li nucleus using trial radial wave functions, which are linear combinations of the harmonic oscillator and the exponential type. Two sets of variational parameters are obtained using Volkov and Brink interactions. The elastic charge form factor of the 6Li nucleus with the Brink interaction is found to be in good agreement with the experimental result and the result obtained by Bouten et al. using the projected Hartree-Fock technique. It is shown that the charge density obtained with the improved radial wave functions is in agreement with the charge density given by the phenomenological model of Hofstadter.