Microscopic interpretation of 7Li+24Mg inelastic scattering at 34 MeV

Theoretical differential cross sections for ⁷Li+²⁴Mg inelastic scattering have been calculated in the DWA using a double folding model. The projectile and target transition densities are matched to electron scattering data and the nucleon-nucleon interaction is chosen to reproduce the real part of the optical potential at the distance $\text{D}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ where there is 50 % transmission in the elastic channel. There are no free parameters in the calculations. The agreement between the theoretical results and the available experimental data is good which serves to establish a consistency between elastic and inelastic scattering. The cross section for a mutual transition which leaves both the projectile and target in their first excited states is found to come mainly from relative L_r = 4 transfer. It is speculated that a quadrupole term in the optical potential might make important contributions at backward angles in the elastic cross section.     

Pion production and the higher resonances in pion-nucleon scattering

The reaction π + N → 2π + N has been studied in the vicinity of the higher resonances in the pion-nucleon cross section. The Low equation for the production amplitude is transformed into an integral equation by isolating the true one-meson intermediate states and discarding higher order contributions. The only part kept in the inhomogeneous term corresponds to the collision of a pion in the nucleon cloud with the incident pion in the resonant T = J = 1 state, which is simulated by an unstable vector Boson. Crossed terms are neglected and the 2π-N state is described by the static model. The terms kept in the sum over states describe the rescattering (off the nucleon) of one of the outgoing pions. The required off-the-energy-shell elastic scattering amplitude is approximated by the 3-3 resonance formula of Chew and Low. With these simplifications the Low equation for the production amplitude reduces to an easily soluble linear integral equation. The rescattering amplitude, which dominates the inhomogeneous term in the resonance region, is proportional to the 3-3 scattering amplitude of one of the outgoing pions. Although the result provides some support for the conventional isobar model, it is important to note that the largeness of the rescattering term arises from scattering far off the energy shell, rather than by “real” excitation as in the phenomenological isobar model. Quantitative calculations for the $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ channel leading to a p-wave ($\text{J =}\text{3}\text{2}$) and an s-wave pion produce a maximum in the cross section near 600 Mev incident pion lab energy. For a π-π resonance energy squared S = 10, agreement with experiment is obtained with a width about one third that suggested by nucleon electromagnetic structure. In our approximation, the well known 600 Mev $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ isospin $\text{1}\text{2}$ resonance occurs at the same energy as the 800 Mev $\text{D}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ isospin $\text{3}\text{2}$ resonance. It is assumed, but not proved, that the neglected terms are responsible for the splitting of the resonance energies. When this splitting is taken into account, the predicted charge state ratios near the second resonance agree well with existing data. The “third” resonance occurs for the state having two p-wave pions, according to the present theory, although no numerical calculations were made for this case. This point of view suggests that the $\text{F}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, $\text{P}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$, and $\text{P}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ incident channels contribute to the third resonance.     

Analyzing powers and cross sections in elastic p-d scattering at 65 MeV

Analyzing powers and differential cross sections for $\text{p}\text{-}\text{d}$ elastic scattering have been measured at 64.8 MeV. The angular distributions cover center-of-mass angles between 8° and 169°. The relative uncertainties of the analyzing power measurement are typically ± 0.005 at c.m. angles less than 80° and are in general ± 0.015 at the remaining angles. The absolute scale of the analyzing power measurement has an uncertainty of ± 0.013. The data are compared with three-body calculations based on the Faddeev theory. In contrast with the differential cross sections, the analyzing powers could not be reproduced without the D-wave nucleon-nucleon interactions.     

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     

On the left-hand cut in potential scattering

Partial wave $\text{N}\text{D}$ equations in potential scattering are solved for the exponential, Hulthén and Morse potentials. The driving terms are taken to be either the contributions of a finite number of Born terms or the total contributions of only the nearest singularities (first n poles). For repulsive potentials one observes ghosts, anomalous bound states or resonances if the order of approximation is small with respect to the potential strength. The origin and meaning of these unphysical phenomena are explained. For attractive potentials such anomalies occur only at very large potential strengths if at all. Input-equivalent Bargmann potentials are employed to determine the quality and nature of the approximate $\text{N}\text{D}$ solutions. Rough criteria for the validity of approximations within the $\text{N}\text{D}$ approach are formulated.     

Jet analysis in lepton-hadron scattering from QCD

For deep inelastic lepton-hadron scattering we deduce from QCD perturbation theory the cross sections $\text{d}\text{σ}\text{d}\text{T}$ and $\text{d}\text{σ}\text{d}\text{S}$ in terms of the collective jet variables thrust T and spherocity S. We find that the shape of these cross sections depends mainly on the total hadronic energy W. While present data are consistent with the cross sections calculated they do not yet prove or disprove the presence of three-jet contributions. We predict that these contributions will be clearly visible for W ? 12 to 15 GeV.     

Dispersion relations for elastic n-d scattering in the Amado model

The formulation of rigorous, dispersion relations for on-shell three-body amplitudes in a separable model is reported. The results of approximate $\text{N}\text{D}$ calculations of s-wave n-d elastic scattering are compared with exact, numerical solutions of the Faddeev equations.     

s- and p-wave neutron spectroscopy: Part V. Level Spacings in Heavy Nuclei

Total neutron cross sections were measured for natural rubidium and thallium and for the separated isotopes, Pb²⁰⁸, Pb²⁰⁷, Pb²⁰⁶, Tl²⁰³, and Rb⁸⁵. Approximate s-wave resonance spacings were estimated for Pb²⁰⁷, Y⁸⁹, Rb⁸⁷, Rb⁸⁵, Tl²⁰³, and Tl²⁰⁵. D₀ = D_J(2J + 1) = 2D_S(2I + 1) = 50, 24, 8, 4, and 30 kev, respectively, where D_s is the average (s-wave) level spacing for all channels. The spacings (D₀) of Pb²⁰⁶ and Pb²⁰⁸ were found to be of the order of hundreds of kew; there is also evidence that resonance spacings are very wide for Sr⁸⁶ and Ba¹³⁶. It is concluded that, in all compound nuclei with a neutron number silightly less than the magic numbers 50, 82, and 126, the resonance spacings are usually not much less than when the magic number is exceeded slightly. Since neutron excitation energies of these sub-magic nuclei are higher than the average, the observed wide level spacings below the magic numbers must be due to the properties of the nearly closed shells and can not possibly be caused only by low excitation energy of the compound nucleus. In considering these comparisons it is shown that, for s-wave resonances, the relation $\text{D}{}_{\mathrm{J}}\text{=}\text{D}{}_0\text{(2J + 1)}$ is a useful approximation in that a plot of D₀ is a much less erratic function of A than is the observed spacing. We also discuss the Bethe-Hurwitz effect, i.e., the influence on resonance spacing (apart from the 2J + 1 factor) of an unpaired nucleon in the target nucleus. We estimate that $\text{α ≧ 30}$ in the equation $\text{D}{}_0\text{(0)}\text{D}{}_0\text{(W)}\text{=}\text{exp}\text{(αW)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ where D₀(0) is the energy and spin independent spacing parameter, and W is the excitation energy of the compound nucleus.     

Statistical model cross sections for overlapping resonances

The cross section for a reaction dominated by numerous incoherent resonances is studied, as a function of the average resonance width $\text{Λ}$ divided by the spacing D. Both the Bohr and Wheeler formulae for the average cross sections, and the standard Ericson analysis of fluctuations, are found to be valid only for weakly overlapping resonances, $\text{Λ}\text{D}\text{?}\text{n}\text{2π}$, where n is the number of open scattering channels. Two different models are presented which respect unitarity whatever the resonance widths, namely a K-matrix or Weisskopf-Wigner model, and an eikonal type model. The cross sections are found to saturate either the elastic or the inelastic unitarity bound for strongly overlapping resonances, $\text{Λ}\text{D}\text{?}\text{n}\text{2π}$; and the fluctuations in the cross sections are found to have a coherence length of order nD, rather than $\text{Λ}$.     

Coherent π0 production in neutrino reactions

We have calculated the cross section and angular distribution of the neutral current process $\text{ν +}\text{N}\text{→ ν +}\text{N}\text{+ π}{}^0$ involving the coherent interaction of a neutrino with a complex nucleus. A contrast is made to incoherent production ν + n → ν + n + π⁰ on a single nucleon. The results are compared with observations from some recent experiments.     

Measurements of the Cross Sections and A y for D(p, n) Inclusive Breakup Reaction at 170 MeV

The effects of three-nucleon force (3NF) has been actively studied by using the nucleon–deuteron (Nd) scattering states. The differential cross sections of the elastic Nd scattering at the energy below 150 MeV can be well reproduced by incorporating 3NF in the Faddeev calculation based on modern nucleon–nucleon (NN) interactions. On the other hand, the differential cross sections of Nd elastic and inelastic scatterings at 250 MeV show large discrepancies between the data and the Faddeev calculations with 3NF. It indicates the presence of the missing features of the three nucleon system at this energy region. For the systematic study about the energy dependence of this large discrepancies, we measured the differential cross sections and the vector analyzing power A _y for the ²H(p, n) inclusive breakup reaction at 170 MeV. The experiment was carried out at RCNP by detecting scattered neutrons by using the neutron detector NPOL3. The data was compared with the results of the Faddeev calculations with and without the 3NF.     

Asymptotic method for determining the amplitude for three-particle breakup: Neutron-deuteron scattering

The process of neutron-deuteron scattering at energies above the deuteron-breakup threshold is described within the three-body formalism of Faddeev equations. Use is made of the method of solving Faddeev equations in configuration space on the basis of expanding wave-function components in the asymptotic region in bases of eigenfunctions of specially chosen operators. Asymptotically, wave-function components are represented in the form of an expansion in an orthonormalized basis of functions depending on the hyperangle. This basis makes it possible to orthogonalize the contributions of elastic-scattering and breakup channels. The proposed method permits determining scattering and breakup parameters from the asymptotic representation of the wave function without reconstructing it over the entire configuration space. The scattering and breakup amplitudes for states of total spin S = 1/2 and 3/2 were obtained for the s-wave Faddeev equation.     

Nucleus-nucleus scattering in the high-energy approximation and optical folding potential

A microscopic complex folding-model potential that reproduces the scattering amplitude of Glauber-Sitenko theory in its optical limit is obtained. The real and imaginary parts of this potential are dependent on energy and are determined by known data on the nuclear-density distributions and on the nucleon-nucleon scattering amplitude. For the real part, use is also made of a folding potential involing effective nucleon-nucleon forces and allowing for the nucleon-exchange term. Three forms of semimicroscopic optical potentials where the contributions of the template potentials—that is, the real and the imaginary folding-model potential—are controlled by adjusting two parameters are constructed on this basis. The efficiency of these microscopic and semimicroscopic potentials is tested by means of a comparison with the experimental differential cross sections for the elastic scattering of heavy ions ¹⁶O on nuclei at an energy of E ～ 100 MeV per nucleon.     

On the separation of the direct reaction contribution in fluctuating cross sections by the use of polarized particles

A method to determine the direct reaction contribution y_D to the differential cross section in the presence of Ericson fluctuations is proposed. For the scattering of polarized $\text{spin}\text{-}\text{1}\text{2}$ particles from spinless nuclei the statistical analysis of the differential cross section dσ/dΩ and of the differential analyzing power $\text{(}\text{d}\text{σ/}\text{d}\text{Ω)A}$ allows an unambiguous and model independent extraction of y_D. The method is applied to the elastic scattering of polarized protons from ²⁶Mg, ⁸⁸Sr and ⁹⁰Zr; the results are in agreement with Hauser-Feshbach calculations.     

Differential cross sections and analyzing powers for pd elastic scattering below 1.0 MeV

Differential cross sections for the elastic pd scattering were measured at seven energies between 0.4 and 1.0 MeV for scattering angles from θ_c.m. = 44.5° to 149.2°. A mixture of D₂ and Kr was used as target gas and the pd differential cross sections were determined relative to those of pKr scattering with a statistical error of $\text{Δσ}\text{σ}$ ～5 × 10^?3. Analyzing powers for $\text{p}$d scattering were measured at 0.8, 0.9 and 1.0 MeV with a statistical error of ΔA_y ～5 × 10^?4.     

Experimental cross section for dimuon production and the Drell-Yan model

We have measured the absolute cross section for dimuon production in hadron collisions at 200 GeV/c in the continuum region M = 4–8.5 GeV. In all the channels studied (pN, $\text{p}$N, π^±N and π^?H₂) the experimental cross section is significantly larger by a factor of 2.3 ± 0.5 than expected from the Drell-Yan model. Furthermore, our proton-nucleon data allow a determination of the nucleon valence structure function which agrees with the deep inelastic lepton scattering data.     

Complete set of deuteron analyzing powers for dp elastic scattering at intermediate energies

With the aim of clarifying roles of the 3NFs in nuclei experimental programs with polarized deuterons beams at intermediate energies are in progress at RIKEN RI Beam Factory. As the first step, we have measured a complete set of deuteron analyzing powers (iT ₁₁, T ₂₀, T ₂₁, T ₂₂) in deuteron-proton elastic scattering at 250 and 294 MeV/nucleon. The obtained data are compared with the Faddeev calculations based on the modern nucleon-nucleon forces together with the Tucson-Melbourne’99, and UrbanaIX three nucleon forces.     

The Faddeev equations for the Heisenberg ferromagnet

The Faddeev equations for the three-magnon T-matrix of the Heisenberg ferromagnet with nearest neighbour interactions are derived for the cubic lattice in arbitrary dimensions. The extreme case of spin $\text{1}\text{2}$ is considered and the kinematical restriction, that only one spin deviation per site is possible, has been taken into account rigorously. Hence the T-matrix is unitary and suited for the study of bound state as well as scattering state properties. The analytic solution of the homogeneous Faddeev equations in one dimension is given.     

Quantum chromodynamics and charm photoproduction

It is shown that charm photoproduction can be consistently described within an asymptotically free field theory. Quantum chromodynamics is used to derive sum rules for the total cross section σ_c^γ which includes both production of mesons with hidden charm (J/ψ, ψ′ and so on) and of charmed particles (pairs $\text{D}\text{D}$, $\text{F}\text{F}$ and so on). An estimate of σ_c^γ as a function of energy is given and fast growth is discovered up to energies ～ 1000 GeV. In this energy range σ_c^γ turns out to be equal to several μb. It is argued that measurements of charm photoproduction would give the most direct information on the gluon distribution within a nucleon. All the results are generalized to production of heavier particles containing new quarks. In particular, a simple rescaling law is derived connecting the cross sections for charm and beauty.