Non-eikonal corrections for the scattering of spin-one particles

The Wallace Fourier-Bessel expansion of the scattering amplitude is generalised to the case of the scattering of a spin-one particle from a potential with a single tensor coupling as well as central and spin-orbit terms. A generating function for the eikonal-phase (quantum) corrections is evaluated in closed form. For medium-energy deuteron-nucleus scattering, the first-order correction is dominant and is shown to be significant in the interpretation of analysing power measurements. This conclusion is supported by a numerical comparison of the eikonal observables, evaluated with and without corrections, with those obtained from a numerical resolution of the Schrödinger equation for d-⁵⁸ Ni scattering at incident deuteron energies of 400 and 700 MeV.Received: 4 December 2003, Published online: 10 August 2004PACS: 03.65.Nk Scattering theory - 24.70. + s Polarization phenomena in reactions - 25.10. + s Nuclear reactions involving few-nucleon systems - 25.40.Hs Transfer reactionsM.W. Gaber: Present address: Department of Radiological Sciences, St. Jude Childrens Research Hospital, 332 N. Lauderdale St., Memphis, TN 38105-2794, USA     

An interaction potential for heavy-ion scattering

The long-range part of the nucleus-nucleus interaction is taken to be given by folding the density distribution of one nucleus with the real part of the single-nucleon optical potential of the other. Analytic approximations are derived for the folded potential and its derivative in the case where the density distribution and single-nucleon optical potential have Saxon-Woods form factors of equal surface thickness. The approximations are generalised to the case of different surface thicknesses and are compared with a previous parametrisation due to Broglia and Winther. The variation with mass number of the central density of the Saxon-Woods matter distribution required to obtain the correct normalisation is shown to be large and an expression for the variation is given. Some calculations are performed on various elastic scattering data using the “quarter-point recipe” of Frahn's diffraction theory. The parameters required to fit the quarter-points of the elastic cross sections are shown to be consistent with their accepted values. It is shown, however, that the quarter-point recipe leads to a larger radius for ²⁰⁸Pb than for ²³²Th. The positions and heights of the pure Coulomb barrier (L = 0) are evaluated for various nuclei. The barrier radii are found to be sufficiently large to suggest that an interaction of the folded type should be reasonable in this region.     

A multichannel quasi-separable potential approach to nucleon-nucleus scattering

A quasi-separable potential model for two-body multichannel scattering is developed. Spin and Coulomb effects are taken into account. By a suitable choice of the separable nuclear interactions we arrive at simple analytic expressions for the transition amplitudes. Our model is applied to the study of the n-¹²C and p-¹²C scattering processes. Effects arising from the excitation of the target nucleus are well reproduced.     

Isobaric analogue resonances in proton scattering on Ba

Extensive experimental data for the resonance scatttering of protons to the ground state and first excited state in ¹³⁸Ba are presented. A Breit Wigner analysis with a background term of the form Σ_n(a_n/Eⁿ) is performed and the extracted background function is compared with optical-model calculations using the parameters given in a detailed investigation by Perey. Neutron spectroscopic factors calculated in different ways from this data are compared with the corresponding factors from (d, p). An accurate determination of resonance energies for the first two resonances reveals a deviation from a constant Coulomb shift of the order of 10 keV.     

Polarization effects on heavy-ion scattering in the distorted-wave method

On the basis of an exact formalism for DWBA methods we calculate the distorting potentials to be used in a standard distorted-wave Born approximation for systems with strongly coupled channels. We examine for a practically useful model some simple approximate treatments through comparisons of the polarization potentials in the case of elastic and inelastic ¹⁶O + ¹⁶O scattering. The adiabatic approximation omitting the radial kinetic energy in the propagator is found to lead to satisfactory agreement with the exact coupled-channel cross sections.     

The optical potential for 6Li-6Li elastic scattering at 156 MeV

Elastic scattering of ⁶Li from ⁶Li has been studied for a beam energy of 156 MeV. The experimental differential cross sections have been analysed on the basis of the optical model using various phenomenological forms. The spin-orbit interaction proves to be less significant. A semimicroscopic double-folding cluster model which generates the real part of the optical potential by an antisymmetrized dα cluster wave function of ⁶Li and αα, dd and dα interactions is well able to describe the experimental data.     

Deformation effects in heavy-ion scattering

Many experiments on the scattering of heavy ions confirm that the elastic cross sections display characteristics of Fresnel diffraction. Recent experiments on the scattering of very heavy ions, however, show marked deviations from the expected Fresnel shapes and an analysis of these experiments using the “quarter-point-recipe” of the Fresnel cross section yields a larger radius for ²⁰⁸Pb than for ²³²Th. It is shown that the deviations may be described in terms of the ground state deformations of the nuclei involved. Taking the deformations into account removes the above anomaly in the radii. The actual fit to the experimental cross sections is not very good and suggests that the real part of the nuclear potential may have an appreciable effect on the orbits of nuclei which are elastically scattered below the Coulomb barrier.     

First-order relativistic corrections to the theory of Coulomb excitation

It is suggested that relativistic effects may be significant in high-precision Coulomb-excitation experiments. First-order corrections to the non-relativistic theory of Coulomb excitation are determined with the restriction that all of the states involved in the excitation have the same parity and that the scattering angle is 180°. The results of some numerical calculations are presented to give an indication of the magnitude of the correction. A recent experiment to determine the B(E2;0⁺ → 2⁺) and excited-state quadrupole moment of ¹⁹⁸Hg is reanalyzed and the relativistic correction to the B(E2) is found to be several times the experimental uncertainty. Sizeable effects are also found in multiple Coulomb excitation.     

Elastic electron scattering on Zr and Mo

Elastic electron scattering on ⁹⁰Zr and ⁹²Mo has been measured at 200 and 300 MeV up to 2 fm⁻¹ momentum transfer. Data analysis led us to adopt the parabolic Gaussian shape as a valuable charge distribution for these nuclei. For ⁹⁰Zr, we have compared this model with charge distributions given by recent Hartree-Fock calculations.     

Elastic scattering of 119 MeV 3He particles and energy and mass-number dependence of optical potential parameters

The elastic scattering of 119 MeV ³He particles by ¹²C, ²⁷Al, ^58,60,62,64Ni, ⁵⁹Co, ^90,92Zr and ⁸⁹Y has been investigated over a wide angular range in order to study the mass-number dependence of the optical potential and odd-even differences in the elastic scattering. The elastic scattering cross sections have been analyzed in terms of the optimal model. The data, except for ¹²C, ²⁷Al and ⁵⁹Co, were fitted with r_R = 1.21 fm, a_R = 0.76 fm and r_I = 1.17 fm, fixed at the average values. The following formulae were obtained for the three remaining parameters by combining with the energy dependence of the parameters (90–120 MeV) obtained previously:

$\text{V=111.4?0.173E+}\text{Z}\text{A}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{+14.9}\text{N?Z}\text{A}\text{(}\text{MeV}\text{)}$

$\text{W}{}_{\mathrm{<mtext>D</mtext>}}\text{=24.8?0.028E (}\text{MeV}\text{)}$

,

$\text{a}{}_{\mathrm{I}}\text{=0.754+0.78}\text{N?Z}\text{A}\text{(}\text{fm}\text{)}$

The depths of the valleys of the angular distribution in the case of ${}^{59}\text{Co}\text{(I =}\text{7}\text{2}\text{)}$ are considerably shallower than those for Ni isotopes, while the depths in the case of ${}^{89}\text{Y}\text{(I =}\text{1}\text{2}\text{)}$ are nearly the same as those for Zr isotopes. Thus it may be concluded that the odd-even differences are attributed to the scattering from the quadrupole moment of the odd-A nucleus (if $\text{I >}\text{1}\text{2}$).     

Heavy-ion inelastic scattering

The ²⁴Mg(¹⁶O, ¹⁶O'γ)²⁴Mg(2⁺) reaction has been investigated at 42 MeV incident energy. In-plane (¹⁶O, γ) angular correlations were measured for outgoing ¹⁶O ions between 6° and 40°. The results are compared with DWBA and coupled channels calculations.     

Neutron elastic scattering on T = 0 nuclei

Neutron elastic scattering on Si, S and Ca has been measured at 11, 20 and 26 MeV using the Ohio University 11 MeV Tandem Van de Graaff. A time-of-flight technique was used and the angular distributions covered an angular range from 15° through 155°. The measured cross sections were corrected for dead time, source anisotropy, detector efficiency, finite geometry, neutron flux attenuation and multiple scattering. Individual as well as global fits to the data using an optical-model search code are presented. The comparison of the optical-model analysis to the neutron and proton elastic scattering data in the case of ⁴⁰Ca, allows an empirical determination of the Coulomb correction term which may be parametrized as $\text{0.46 Z/A}\text{1}\text{3}$. It is also shown that the elastic scattering and inelastic scattering to the first 2⁺ states in ²⁸Si and ³²S may be fitted using the same optical-model parameters obtained for ⁴⁰Ca using the coupled-channel formalism.     

Interaction potential between two 16O nuclei derived from the Skyrme interaction

The energy functional for the Skyrme density-dependent force is used to calculate the interaction potential between two ¹⁶O nuclei. A two-centre harmonic oscillator potential is employed to construct the density and kinetic energy density of the ground state of the combined system and of the separated nuclei. The antisymmetrization effects are investigated. The relative motion of the nuclei is taken approximately into account and the energy dependence of the potential derived from the Skyrme force is presented.     

Asymmetry in polarized e-d inelastic scattering

The parity violating asymmetry is calculated for polarized electron-deuteron inelastic scattering in the closure approximation for the outgoing nucleons. The results are applicable in the intermediate-energy region of the incident electron where asymmetries are found to be in the range accessible to the present experiments.     

Polarization in neutron-proton scattering at 16.2 MeV

The polarization in the scattering of neutrons from protons has been measured at a lab neutron energy of 16.2 MeV for the three c.m. angles 70°, 100° and 130°, at which the measured polarizations were (+2.95 ± 0.86) %, (+1.84 ± 0.91)% and (+1.62 ± 0.99) % respectively.     

High-resolution proton scattering on 50Ti

Differential cross sections were measured for ⁵⁰Ti(p, p) at four angles for E_p = 1.83 to 2.97 MeV, with an overall energy resolution of about 350 eV. Spins, parities and total widths were extracted for 212 levels. An energy region near E_p = 1.37 MeV was also examined in order to study the analogue of the ground state of ⁵Ti. Coulomb energies and spectroscopic factors were determined for the analogues of the ground and first excited states of ⁵¹Ti. The latter analogue was highly fragmented. The s-wave spacing and width distributions were analyzed and the number of missing levels estimated. The $\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and $\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ proton strength functions were determined.     

On the evaluation of the quantum corrections to periodic TDHF orbits

The evaluation of the leading order quantum correction to periodic mean-fields within the path integral approach is reinvestigated. The corresponding gaussian functional integral is well defined only after restoring the time-translation invariance broken by the time-dependent meanfield approximation. The particular structure of the action function permits one to restore the invariance in two different ways, that seem to exhibit an ambiguity in the evaluation of the leading order quantum correction. We prove, however, that both ways of restoring the time-translation invariance yield the same result, showing that the leading order quantum correction is uniquely defined within the path integral approach.     

Inelastic electron scattering from He

The spectra of electrons scattered inelastically from ⁴He have been measured at incident energies from 150 MeV to 400 MeV for scattering angles from 38° to 90°. Through the use of a liquid ⁴He target, a high momentum resolution (≈0.25%) was obtained, and the 20 MeV 0⁺ state of the -particle was observed for the first time in electron scattering. The excitation energy and the total width of this state were determined and are in good agreement with the results from other experimental methods. It was found that the total disintegration cross section appears be smaller than 2(σ_p+σ_n) by more than an order of magnitude at the lowest q²(≈ 0.33 fm⁻²). In earlier work the assumption of a total cross section of 2(σ_p+σ_n) was found to be quite accurate for higher q². As a new result, the partial radiative width for the 0⁺ state is determined to be Γ_rad = 1.1±0.3 meV.     

Neutron scattering from 208Pb

Elastic scattering of 7, 9, 11, 20 and 26 MeV neutrons from ²⁰⁸Pb has been measured with the Ohio University Tandem Van de Graaff accelerator. Standard pulsed beam time-of-flight techniques were employed. Measurements of the incident flux at 0° were used to normalize the differential cross sections. The measured cross sections were corrected for dead time, detector efficiency, flux attenuation, multiple scattering, finite geometry, neutron source anisotropy and compound elastic contribution. Relative uncertainties are estimated to be between 5%–10% and the uncertainty in the normalization is estimated to be less than 5 %. The data were used to obtain neutron optical potential parameters. A comparison with proton optical parameters is presented, and the (p, n) quasi-elastic cross section is calculated and compared with available data. Deformation parameters for the 3^? state (Q = ?2.615 MeV) and 5ā (Q = ?3.198 MeV) in ²⁰⁸Pb were obtained at incident energies of 11 and 26 MeV.     

Deep-inelastic electron scattering from carbon

We have measured the deep-inelastic electron scattering from carbon up to and including the Δ-region at 36°, 60°, 90° and 145°. The systematic decomposition of the transverse and longitudinal response functions has been obtained by means of a Rosenbluth-type analysis of the data in the momentum transfer interval 200 MeV/c /s |q| /s 600 MeV/c. A comparison with theoretical calculations which extend over the quasielastic and Δ-peak regions is presented. A reduction of the differences between our data and theory seems obtainable through the introduction of meson-exchange currents, resonant and non-resonant meson production, and the use of the shell model. Our experimental Coulomb sum-rule estimates at higher $\text{|}\text{q}\text{|}$ agree with independent particle model predictions. We compare our results in the vicinity of the Δ-peak with the total absorption cross section for real photons.