Dynamical Study of Tensor Observables in the Energy Range of 80 keV to 95 MeV: Tests of Effective Two-Body Models

 Realistic interactions are used to study tensor observables in the energy range of 80 keV to 95 MeV deuteron laboratory energy, as well as the differential cross section for the two-body photodisintegration of . The Siegert form of the E1 multipole operator in the long-wavelength limit is taken as the sole component of the electromagnetic interaction. The three-body Faddeev equations for the bound-state and continuum wave functions are solved using the Paris, Argonne V14, Bonn-A, and Bonn-B potentials. The corresponding nucleon-nucleon t-matrices are represented in a separable form using the Ernst-Shakin-Thaler representation. The Coulomb force between protons is neglected and no three-nucleon force is included. The contribution of nucleon-nucleon P-wave components to the observables is carefully studied, not only in the angular distribution of the observables, but also as a function of the deuteron laboratory energy for fixed centre-of-mass angle. Comparison with data is shown wherever it exists. Results with simple Yamaguchi-type interactions with variable %D-state in the deuteron are compared with realistic interactions and one of these model potentials is used to study the results in terms of contributions from specific wave-function components or terms in the electromagnetic operator. Effective two-body models are examined by means of a derivation that is consistent with the underlying three-body calculation and that leads to an effective two-body t-matrix for neutron-deuteron elastic scattering carrying the same on-shell amplitudes as the original three-body equations. Received September 21, 1999; revised December 23, 1999; accepted February 9, 2000     

Exact four-body calculation of the 4ΛHe-4ΛH binding energy difference

The coupled, two-variable integral equations that determine the ⁴_ΛHe and ⁴_ΛH bound states, when the NN and ΛN interactions are represented by separable potentials, are derived from the Schrödinger equation. The integral equations are solved numerically for simple s-wave potentials and for tensor potentials in the truncated t-matrix approximation without resort to separable expansion of the kernels. The Λ-separation energy difference ΔB_Λ resulting from the genuine four-body model is shown to be approximately twice as large as that coming from an “effective two-body” model calculation, when identical central potentials are used. The four-body model estimate of ΔB_Λ made with tensor forces is consistent with the experimental value, indicating that charge symmetry breaking implied by the low energy Λ N scattering parameters is compatible with that suggested by the known binding energy difference in the A = 4 hypernuclear isodoublet.     

Separable expansion of the t-matrix in the 3S1-3D1 channel

A recently proposed separable expansion for the t-matrix for local potentials is extended to the coupled channel of nucleon-nucleon interactions. The simple method yields a rather convergent separable expansion, with simple form factors, and the requirements of two particle unitarity and time reversal symmetry are maintained for approximations of any rank. The method is illustrated for the simplified Reid soft-core potential in the ³S₁-³D₁ channel, and the results compared with a recent calculation of Pieper.     

Relativistic effects on the tensor polarization in electron-deuteron scattering

The tensor analysing power P_e in electron deuteron scattering is calculated including relativistic corrections to the deuteron form factor as determined by Friar and Gross. These effects are found to be quite comparable to the differences between various potentials, and should therefore not be ignored.     

Measurements of the spin-correlation coefficients Cxx, Cyy and S for d-p scattering at Ed = 17.4, 19.5, 23.8 and 26.1 MeV

The spin-correlation coefficients C_xx, C_yy and S have been measured by scattering the vector and tensor polarized deuteron beam of the Grenoble cyclotron from a polarized proton target. Comparison is made with Faddeev equation calculations using a separable N-N interaction. The results are in good agreement only when P-waves and the tensor force are taken into account in the N-N interaction.     

Simple local NN potentials involving forbidden states and polarization in ed scattering

A simple NN potential is proposed in an analytic form. The parameters of this potential are fixed by fitting the effective range, the scattering length, and the deuteron binding energy. The phase shifts for np scattering at energies ranging up to 500 MeV and the properties of the deuteron are calculated with the resulting parameters. The effect of the πNN coupling constant on the potential parameters and on the accuracy in describing various properties of NN interaction is explored. The results of the present calculations are found to be in good agreement with experimental data, with available results from NN partial-wave analyses, and with the results of calculations with Nijmegen potentials. The tensor polarization t ₂₀ in elastic electron-deuteron scattering is analyzed by using some NN interactions.     

Deuteron-proton elastic scattering at intermediate energies

The deuteron-proton elastic scattering has been studied in the multiple-scattering expansion formalism. Primary attention has been given to such relativistic problems as a deuteron wave function in a moving frame and transformation of spin states due to Wigner rotation. Parametrization of the nucleon—nucleon t matrix has been used to take the off-energy-shell effects into account. The vector, A _y , and tensor, A _yy , analyzing powers of the deuteron have been calculated at two deuteron kinetic energies: 395 and 1200 MeV. The obtained results are compared with the experimental data. The text was submitted by the authors in English.     

Sensitivity of polarization observables in elastic ed scattering to the neutron form factors

The deuteron elastic form factors are calculated within the Bethe-Salpeter approach with separable interaction. The charge, quadrupole, and magnetic form factors [F _C(q ²), F _Q(q ²), and F _M(q ²), respectively]; the structure functions A(q ²) and B(q ²); and the tensor polarization components T ₂₀(q ², T ₂₁(q ²), and T ₂₂(q ²) are investigated up to ?q ²=50 fm^?2. The role of relativistic effects is discussed, and a comparison with nonrelativistic calculations is performed. The effect of the neutron form factors on the deuteron form factors and especially on tensor polarization components is discussed too.     

Deuteron form factors for the Nijmegen potentials

The deuteron form factors and tensor polarizations in elastic ed scattering are considered for four versions of the Nijmegen nucleon-nucleon potentials. The numerical deuteron wave functions in these potentials are approximated by a series of Gaussian functions with the result that it can be used in any computations of integrated characteristics. The quality of this approximation of the wave function is exemplified by comparing the results that it produces for the momentum distributions, the quadrupole moment, the D-state probability, and the deuteron radius with the results of the corresponding precise calculations.     

A separable approximation to the two-body t-matrix for short-range central local potentials

The S-wave potential in momentum space, V₀(p, p′), corresponding to a local two-body central short-range potential may be expanded in separable form with the help of suitable quadrature formulae. With a two-term separable expansion for a variety of nuclear potentials, the resulting t-matrix is found to be in close agreement with the corresponding result with a six-term expansion for small values of p and p′ which are important in the calculation of the trinucleon binding energy.     

Three-body model for stripping nuclear reactions

A three-body model for the deuteron stripping nuclear reactions is presented. A set of three integral equations is obtained for the wave functions of the three-body problem by introducing a decomposition into angular momentum states into the Lippmann-Schwinger equation. Simple two-particle interactions with separable potentials are used. These separable potentials reduce the three-body problem to the solution of coupled sets of one-dimensional Fredholm integral equations. The angular distributions for²⁸Si(d,p)²⁹Si and⁴⁰Ca(d, p)⁴¹Ca stripping reactions are calculated. From the extracted spectroscopic factors, good agreement with the experimental measurements is obtained.     

Elastic electron deuteron scattering on a tensor polarized solid ND3 target

Elastic electron deuteron scattering has been measured at the Bonn 2.5 GeV electron-synchrotron using a tensor polarized ND₃ target. The scattered electrons and deuterons were detected in coincidence by two magnetic spectrometers. Data were taken at an average value of four momentum transfer ofQ ²=0.495 (GeV/c)² in order to separate the electric monopole and quadrupole form factors of the deuteron. To have a sufficient control of systematic effects, both polarized and unpolarized cross sections were measured with an identical experimental set up. In the ratioR _EXP=dσ_pol/dσ_unp systematic effects are expected to cancel. This ratioR _EXP is related to the theoretical polarized cross section d dσ _T where the spin quantization axis for the deuterons is orientated in the ed-scattering plane, perpendicular to the virtual photon direction. Our final result for the corresponding recoil tensor polarization ist ₂₀=?2.01±1.13.     

Three-body calculation of neutron-deuteron elastic scattering in the energy region from 2.5 MeV up to 50 MeV

The three-body Faddeev equations for neutron-deuteron scattering are solved in the energy region from 2.5 MeV to 50 MeV of the incident neutron energy with small energy steps. Higher-rank separable potentials are used in the¹ s ₀ wave and in the³ s ₁—³ d ₁ waves, while rank-1 separable potentials are used in¹ p ₁,³ p _0,1,2 ¹ d ₂ ³ d _2,3 waves. The calculation is compared with experiments for the total cross section, the total break-up cross section, the differential cross section and the analyzing power of neutron-deuteron scattering. The improvements in the agreement as compared to previous calculations are impressive in many cases. Especially, the calculated total cross section agrees with the experiment below 30 MeV within the error bars, which are as small as 1%. A discussion on the numerical accuracy is given. General aspects of the calculated cross section are discussed. It is pointed out that thes-wave asymptotic normalization of the deuteron wave function (A _s ) is important.     

Effect of the deuteron anisotropy: Longitudinal and transverse components of the electric dipole polarizability

The anisotropy effect of the electric polarization (stretching) of the deuteron in the Coulomb field, caused by the tensor character of the nuclear force, is investigated. The values of the longitudinal (with the major axis, or the spin of the deuteron, directed along the electric field), and transverse components of the deuteron electric dipole polarizability that correspond to the low-energy n-p data, are predicted to be α₆ = 0.669 fm³ and α_⊥ = 0.555 fm³ (the potential YYm). The values of the major and minor semi-axes of the deuteron are calculated.     

Precise deuteron binding energy calculations

A new minimization approach to two-body binding energy calculation is suggested and applied to deuteron. The results of calculations with very high accuracy (at least 10 eV) for several nucleon--nucleon potentials are given. The suggestion to use new precise experimental values of deuteron binding energyE _D and calculated one as a constraint on N-N potentials follows from a study ofE _D sensitivity on the potential parameters.The authors are indebted to dr. E. Truhlik for a loan of initial version of DEUT code and useful discussions. The advices of dr. J. Piút are acknowledged. The presented work is based on thesis of P.K.     

Nucleon form factors for elastic electron-deuteron scattering at high momentum transfer

Elastic electron-deuteron scattering at high momentum transfer is investigated within the Bethe-Salpeter approach. The relativistic covariant Graz II separable kernel of nucleon-nucleon interactions is used to analyze the deuteron structure functions, form factors, and tensor of polarization components. The modern data for the electromagnetic structure of nucleons from the double polarization experiments, as well as some other models of the nucleon form factors, are considered.     

The tensor analyzing power Azz AT θ = 0° for the He( , p)He reaction

The tensor analyzing power f_zz has been measured for the ³He( , p)⁴He reaction at 0 = 0° over an incident deuteron energy range E_d = 6.6–15.8 MeV in steps of 0.5 MeV. The present results agree with and extend the previous measurements of Grüebler et al. The present results indicate that this reaction is a very good tensor analyzer for polarized deuteron beams with energies up to 15.8 MeV.     

The determination of the D-state asymptotic normalization of the deuteron and triton wavefunctions by the continuation of the 2H(b, p)3H tensor polarizations

By continuation of 14 tensor polarizations to the stripping and pick-up poles of the ²H($\text{d}$,p)³H reaction we have determined ?_D values for the deuteron and triton wavefunctions. The new precision value for the deuteron 0.0272 ± 0.004 suggests an increase of the applied strength of the OPE potentials. For the triton a value of 0.051±0.005 was found.