Investigation of the phase of the electromagnetic pion form factor in the asymptotic region

The phase of the electromagnetic pion form factor G(t) in the asymptotic region (t > 1 (GeV)²) is investigated by the dispersion sum-rule method. Using experimental data for G(t) at t < 1 (GeV)² we find that the mean weighted value of the asymptotic phase does not show strong deviations from the asymptotic value of the phase coming from an extrapolation of the data from the ?-resonance region with the help of the Gounaris-Sakurai formula. The contribution of the hypothetical ?-like resonances to the dispersion sum rules is roughly estimated. These estimates and the results of the calculation of the dispersion sum rules show that the existence of strong ?-like elastic resonances is forbidden in the 1–1.7 GeV $\text{t}$ range at least.     

Polarization of single particle states in heavy ion induced nuclear reactions within the two center shell model

In “slow” nucleus-nucleus collisions the single particle states will adjust continuously to an instantaneous two center potential. Transfer reactions will occur between polarized rather than between asymptotic states. In order to employ two center states in reaction calculations, an expansion in terms of asymptotic states with good angular momentum is required. This expansion is formulated, and equations for the expansion coefficients are derived. Model calculations for the system ⁴⁰Ca+¹⁶O indicate that the polarization of unoccupied proton states in ⁴⁰Ca is strong, while it is very weak for the tightly bound 1p_{$\text{1}\text{2}$} proton in ¹⁶O. The dependence of the polarization effect on several relevant parameters is discussed.     

Trilinear mode effects on transport coefficients

The effects of tri-linear modes on transport coefficients is illustrated by explicit calculations for the self-diffusion constant. In two dimensions the form of the asymptotic time behavior of the velocity autocorrelation function is changed from t^-1 (bilinear modes) to (t log t)^-1 (trilinear modes). Thus, no finite mode calculation in two dimensions can yield the correct asymptotic form. A heuristic self-consistent calculation including all mode orders yields the form [t log^{$\text{1}\text{2}$} t]^-1. In three dimensions, the form of the asymptotic time behavior is not changed when higher-order modes are included.     

Isotopic dependence of giant multipole resonances

A procedure is presented which allows the application of linear response theory and the random phase approximation to an open shell. The procedure is applied to Ca isotopes. The general features of giant multipole resonances are found to vary smoothly with the mass. The resonances exhibit more structure in the open 1f_{$\text{7}\text{2}$} shell nuclei. While the energy-weighted dipole sum is practically constant in all isotopes, the isoscalar quadrupole and octupole energy weighted sums increase continuously by ~ 30 % from ⁴⁰Ca to ⁴⁸Ca.     

Light-cone expansions,the parton model and virtual γγ scattering

It is shown that it is sufficient to use the light-cone algebra of currents and the algebra of bilocal operators to find the asymptotic behaviour of the γγ scattering amplitude when one (or two) of the photon masses q_1,2² is large, and for an arbitrary value of the energy squared s = (q₁+q₂)². A general form of this asymptotic behaviour is obtained. The box-diagram is dominant over the wide region in $\text{s(μ}{}^2\text{« s « q}{}_1{}^2\text{q}{}_2{}^2\text{/μ}{}^2\text{,μ ～ 1}\text{GeV}\text{)}$ and so the asymptotic amplitude is known completely. It is shown that the parton model of the type of ref.[8] gives the same predictions for the asymptotic behaviour of the γγ amplitude.     

D-state effects in (d, t) and (d, 3He) reactions

The influence of the deuteron and triton [³He] D-state on the form factor required in the DWBA analyses of the ($\text{d}$, t) [($\text{d}$, ³He)] reactions has been studied. The wave functions of Phillips are used for three-nucleon systems because they are designed to yield the correct asymptotic form for the (d, t) and (d, ³He) overlaps. The zero-range parameters D₀ and D₂ which enter into the calculation of the cross section and the tensor analysing power in the DWBA with the local energy approximation are calculated and found to yield values consistent with experimental data.     

Large basis shell-model study of nucleon-transfer overlaps

A new approach to two-nucleon transfer overlap calculations is presented. It is shown that the usual well-depth method underestimates the ¹⁶O (t,p) ¹⁸O two-nucleon transfer form factor in the tail region by as much as 50%.     

Nucleon charge form factors and chiral quark models

Calculations of the proton and neutron charge form factors G_E^p,n(q²) are presented, based on chiral bag as well as confining Dirac potential models with chiral pion-quark coupling. Special emphasis is placed on a detailed treatment of the charged pion cloud contribution to the nucleon current. The role of a finite extension of the pion-quark vertex in truncating the summation over intermediate quark bag states is studied. Quark core radii (including recoil corrections) are constrained by a simultaneous calculation of the nucleon axial form factor. The proton charge form factor is well reproduced for $\text{|q}{}^2\text{|}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? 0.7}\text{GeV}$ with quark core rms radii between 0.5–0.6 fm. About $\text{1}\text{3}$ of the proton charge is carried by the pion cloud in this model. The neutron charge form factor is obtained with the correct sign and overall q² dependence but needs further refinements, probably at the level of the isoscalar form factor.     

On the predictions of the coexistence model for the reaction 42Ca(p,t)40Ca

The coexistence model as formulated by Gerace and Green for ⁴⁰Ca has been shown to predict cross sections for the reaction ⁴²Ca(p,t)⁴⁰Ca in strong disagreement with experiment. In the present letter we demonstrate how a much improved agreement can be obtained by introducing additional kinds of basis states. Especially important are triaxially deformed components, which contain admixtures of the $\text{1}\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ states.     

Three-body calculation of the 2H(α, αp)n reaction

The ²H(α, αp)n reaction at 15–42 MeV is analysed by solving the Amado-Lovelace equations. Rank-one separable potentials with the Yamaguchi-type form factor are assumed in ${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$s, ${}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$$\text{p and}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{p}$ waves for the N-α subsystem and in the triplet s-wave for the n-p subsystem. The agreement with experiment is quite good in this energy region. The role of multiple-scattering terms is discussed. Interesting features of the reaction mechanism, such as the decay process through the 1⁺ resonance of ⁶Li, are found in this energy region. The triton transfer process at 15 MeV predicted by the modified impulse approximation is not found. The numerical method is discussed. It is shown that the rotated contour method causes no difficulties even when it is applied to breakup amplitudes.     

f72 neutron occupancies in the Ca isotopes

Methods of obtaining nucleon occupancies from total energy-weighted sum rules for spectroscopic factors are described and applied to $\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$ neutron transfer data for Ca isotopes. The $\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$ neutron occupancies obtained are consistent with shell model expectations and, for ⁴¹Ca and ⁴³Ca, with values previously obtained from an analysis using non-energy-weighted sum rules.     

Spin diffusion in Heisenberg Ferromagnets in the high temperature limit

A study is made of the time dependence of the memory function given by the mode-mode theory for the Heisenberg Ferromagnet in the high temperature limit. For very long times this function is found to go as $\text{t}{}^{\mathrm{<mtext>?5</mtext><mtext>2</mtext>}}$ exp (-Dq²t/2). This leads to the conclusion that spin diffusion is not predicted in the asymptotic time region, in contrast to what has been stated before.     

Inelastic processes and form factor effects in the 162, 164Dy(3He,d) reactions at 46.5 MeV

The 162, 164Dy(³He, d) reactions at E_3He = 46.5 MeV are analyzed using the coupled channels Born approximation (CCBA) and improved form factors derived from a deformed Woods-Saxon potential. The latter are generated using the coupled channels procedure of Rost. The transitions considered populate the $\text{7}\text{2}$^?[523], $\text{1}\text{2}$⁺[411], $\text{3}\text{2}$⁺[411], $\text{1}\text{2}$^?[541] and $\text{5}\text{2}$⁺ orbitals in ^{163, 165}Ho. Indirect processes induced by inelastic scattering are found to have an influence on the cross sections comparable to that deduced for neutron transfer reactions on rare earth nuclei at lower energies. Considered alone, these can alter the cross sections even of strong transitions by a factor of two and of weaker ones by an order of magnitude. For the weaker transitions equally large changes can result when the improved form factors, rather than conventional spherical Woods-Saxon functions, are used in the calculations. In the examples considered these two effects tend to cancel, often, but not always, resulting in predicted cross sections similar in magnitude to the results of conventional DWBA calculations made with spherical Woods-Saxon form factors. The CCBA angular distributions are generally similar in shape to DWBA predictions, which usually give good fits to the experimental angular distributions over the 0–35° range of the data. Compared with DWBA predictions which use (he same optical parameters, but spherical Woods-Saxon form factors, the CCBA with deformed Woods-Saxon form factors is in better overall agreement with the experimental cross-section magnitudes. However there are a number of cases in which the CCBA, although usually predicting larger cross sections than the DWBA, still underestimates the experimental cross sections by nearly factor of two. These cases all occur in the $\text{71}\text{2}$^?[541] band or in the strongly Coriolis mixed $\text{1}\text{2}$⁺[411] and $\text{3}\text{2}$⁺[411] bands, and include the majority of transitions populating these orbitals. Since both nuclear structure and reaction mechanism effects are interwoven m the calculations, further data would be most useful in probing the origin of the discrepancy.     

Elastic and inelastic scattering of 15 MeV polarized deuterons from isotopes of Ca,Cu, Sr,Zr, and Mo

The elastic and inelastic scattering of 15 MeV polarized deuterons from ⁴⁸Ca, ⁶³Cu, ⁸⁸Sr, ⁹⁰Zr, ⁹²Zr, and ⁹²Mo has been investigated. Angular distributions of the cross section and vector analyzing power iT₁₁ have been measured for all these nuclei; the tensor analyzing powers T₂₀ and T₂₂ have been studied for ⁹²Zr. Cross sections and vector analyzing powers are generally well explained by the optical model for elastic scattering and by the DWBA with a macroscopic form factor for the inelastic scattering; this is consistent with previous work. Distributions for ⁴⁸Ca, however, are poorly fitted. Anomalous behavior of the N = 50 nuclei found in the inelastic scattering of polarized protons is not present for deuterons. Tensor analyzing powers are not well explained by standard procedures: use of approximate folding model optical parameters did not improve the fits. The distribution of iT₁₁ for the $\text{1}\text{2}$^? state in ⁶³Cu is significantly different from the distributions for the $\text{5}\text{2}$^? and $\text{7}\text{2}$^? states.     

Accurate determination of the 13C-12C+n spectroscopic factor

We have measured ¹²C-¹³C elastic cross sections at 12 MeV between 40°–140° in 1° steps to ±1%. The observed oscillatory interference between Coulomb scattering and the neutron transfer process is analyzed using exact finite-range DWBA, and a model-independent value of C² = 2.55±0.10 for the asymptotic normalization of the $\text{1}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ neutron wave function in ¹³C is obtained. Using radial wave functions determined by elastic electron scattering the spectroscopic factor is found to be S = 0.81±0.04.     

Center-of-mass corrections to overlap factors for two-nucleon transfer

The following simple recipe is presented for making c.m. corrections to the overlap factors involved in two-nucleon transfer: Expand the shell-model overlap in products of harmonic oscillator wave functions of the relative and c.m. coordinates of the two nucleons. Change the oscillator ν-value to $\text{Aν}\text{(A + 2)}$ in the c.m. functions, and multiply each term in the sum by [$\text{(A + 2)}\text{A}\text{]}\text{1}\text{2}{}^{\mathrm{N}}$, where N is the energy quantum number associated with the c.m. state of that term. The recipe depends on the value of ν used in the expansion. The proper choice of ν is discussed, and numerical examples are presented which indicate the magnitude of the correction. The zero-range form factor for two-neutron stripping on ¹⁶O is increased by a factor of about two, or more, in the surface region, resulting in a factor of about five in the cross section. The correction is greater at larger distances, implying that heavy-ion reactions will be effected more than light ion reactions.     

The radial distribution of the 1f72 neutron in 41Ca from the 40Ca(13C, 12C)41Ca reaction

The reaction ⁴⁰Ca(¹³C, ¹²C)⁴¹Ca leading to the ground and low-lying $\text{3}\text{2}$^? levels has been studied at bombarding energies of 18.5 and 19.0 MeV, close to the Coulomb barrier. The cross sections have been analysed using the DWBA with previously measured calibrations to obtain the rms radii of 1f_{$\text{7}\text{2}$} and 2p_{$\text{3}\text{2}$} orbits in the ⁴¹Ca levels. The rms radius of the 1f_{$\text{7}\text{2}$} point neutron orbit in the ground state relative to the ⁴⁰Ca core was determined to be 3.89 ± 0.12 fm. This is compared with various theoretical predictions based on Hartree-Fock theory. The 1f_{$\text{7}\text{2}$} orbit radius gives the point neutron excess size to a good approximation for comparison with the value of 3.45 fm deduced from Coulomb energy differences. The consistency of the results with ⁴⁰Ca(t, d)⁴¹Ca experiments using a (t/d), zero-range normalization of 3.26 × 10⁴ MeV² · fm³ is demonstrated, and trends in neutron size parameters through the Ca isotopes are discussed.     

Electron scattering from the 3.65 MeV (0+, T = 1) state in 6Li at high momentum transfer

The form factor ofthe 3.56MeV(0⁺, T = 1) state of ⁶Li has been measured for momentum transfers q = 1.0–3.0 fm^?1, and the $\text{2.18}\text{MeV}\text{(3}{}^{\mathrm{+}}\text{, T = 0)}$ and $\text{5.37}\text{MeV}\text{(2}{}^{\mathrm{+}}\text{, T = 1)}$ states have been measured up to q = 2.5 fm^?1. The 3.56 MeV form factor is analysed in terms of a phenomenological shell model with l = i valence nucleons. The radial wave functions are found to have a greater radial distribution than given by the harmonic oscillator, more closely resembling Woods-Saxon functions. The M1 form factor is found to decrease at high momentum transfer somewhat more slowly than the models predict. A technique for determining the Mλ transition current density based on the Fourier-Bessel analysis is developed and applied to the M1 transition. The M1 transition current density is obtained within a moderate error band and compared with the harmonic oscillator and Woods-Saxon densities. The M1 radiative width is 8.18 ± 0.25 eV, in agreement with previous measurements.     

E2 core polarization for sd-shell single-particle states calculated with a skyrme-type interaction

We have studied the core-polarization effect on the 1s0d-shell single-particle electromagnetic quadrupole transitions due to coupling with the quadrupole giant resonances. The self-consistent Hartree-Fock + RPA method is applied for the calculations of the single-particle wave functions and the response functions of the giant resonances. The particle-vibration-coupling model is used to calculate the core-polarization effect in the vicinity of ¹⁶O and ⁴⁰Ca. The effective coupling hamiltonian is determined by the SGII Skyrme-type interaction which is used in the HF + RPA and particle-vibration-coupling calculation. The results are discussed for the proton and neutron effective charges and for the longitudinal and transverse form factor for the $\text{0}\text{d}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}\text{→ 1}\text{s}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}{}^{\mathrm{?1}}$ pro single-particle transition in ³⁹K. Good agreement with recent longitudinal data for this transition is obtained.