Nucleon-nucleon Cross Section Research Articles

Heavy ion peripheral collisions at relativistic energies: Theory of giant quadrupole excitation.

The theory of excitation of the giant quadrupole resonance in E/Q=0.5--2.1 GeV heavy ion collisions is studied. Karol's soft spheres model based on nucleon-nucleon cross sections is generalized to deal with collective excitations in deformed nuclei. The giant quadrupole resonance cross sections are calculated in several cases. The results are shown to be in close agreement with the prediction of Blair's simple diffraction model except for excitation in the light nuclei $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$. Comparison with $^{16}\mathrm{O}$ fragmentation yield data is made.

Contribution to the Inelastic Proton-Nucleus Interaction Simulation in the Energy Range 50-300 MeV

Two simulation methods are presented: (1) A Monte Carlo method which describes the intranuclear cascade: the proton-nucleus interaction is regarded as a sequence of nucleon-nucleon and nucleon-cluster interactions and the nucleus is represented by a gas of non-interacting particles in a potential well. (2) A time dependent method based on the discretisation of classical equations of motion. The two-body interaction potential was chosen to insure (i) a correct fit of the first two moments of the elastic free nucleon-nucleon differential cross section: and (ii) a nucleus simulation such that the main nuclear characteristics (binding energy, density, average potential per nucleon) are in agreement with experimental data. Nucleon spatial distribution is performed with the nuclear isomorphic model. Results from these two methods for the emitted proton spectra are presented and compared with existing experimental data.

Transverse-momentum distribution of produced particles in ultrarelativistic nucleus-nucleus collisions.

In order to discern coherent or collective processes from incoherent processes in nucleus-nucleus reactions at high energies, we study the transverse-momentum distribution of the produced particles with an incoherent-multiple-collision model. In this model, the projectile nucleon makes successive inelastic collisions with nucleons in the target nucleus, the probability of such collisions being given by the thickness function and the nucleon-nucleon inelastic cross section. It is assumed that each baryon-baryon collision produces particles and degrades momenta just as a baryon-baryon collision in free space, and that there are no secondary collisions between the produced particles and the nucleons. We found that the average transverse momentum and the charged-multiplicity data at Fermilab and CERN ISR energies can be well explained by such a model. However, the average transverse momentum for some events observed by the Japanese-American cooperative emulsion experiment (JACEE) associated with large energy density in the central rapidity region differ markedly from the model results. Such a deviation indicates the presence of coherent or collective effects for these collisions and may indicate the possibility of a formation of quark-gluon plasma.

Nucleon-nucleus reactions at ultrarelativistic energies.

In order to discern coherent processes from incoherent and collective processes in nucleon-nucleus reactions at high energies, we study these reactions with an incoherent-multiple-collision model. In this model, the projectile nucleon makes successive inelastic collisions with nucleons in the target nucleus, the probability of such collisions being given by the thickness function and the nucleon-nucleon inelastic cross section. It is assumed that each baryon-baryon collision produces particles and degrades momenta just as a baryon-baryon collision in free space, and that there are no secondary collisions between the produced particles and the nucleons. We found that the inelastic proton data, the pseudorapidity distribution data ${\mathrm{dN}}^{\mathrm{pA}}$/d\ensuremath{\eta}, and the total nucleon-nucleus absorption data can be well explained by the incoherent-multiple-collision model. However, the single-particle fragmentation data for nonleading particles indicate that there is a reduction of the fragmentation cross section because of subsequent collisions of the leading baryon along the collision chain. Modification of the incoherent-multiple-collision model is suggested. The modified model gives cross sections for the reactions pA\ensuremath{\rightarrow}${\ensuremath{\pi}}^{+}$X, pA\ensuremath{\rightarrow}${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}}$X, pA\ensuremath{\rightarrow}${K}^{+}$X, pA\ensuremath{\rightarrow}${K}^{\mathrm{\ensuremath{-}}}$X, and pA\ensuremath{\rightarrow}p\ifmmode\bar\else\textasciimacron\fi{}X in the projectile fragmentation region in good agreement with experiment.

Spin-flip versus non-spin-flip interaction in forward (p, n) reactions

The ratio of the probabilities for triggering Gamow-Teller or Fermi transitions in forward (p, n) reactions is evaluated, using total and forward nucleon-nucleon cross sections. Our results indicate that the ratio extracted from the (p, n) reaction is not consistent with the free nucleon-nucleon data.

Total inclusive neutron cross sections and multiplicities in nucleus-nucleus collisions at intermediate energies

The total integrated inclusive cross section, $\ensuremath{\sigma}(T<{T}_{0})$, for the emission of neutrons above an energy ${T}_{0}$ by neon ions with an average energy of 337 MeV per nucleon reacting in targets of uranium, copper, aluminum, and carbon is described by ${\overline{\ensuremath{\sigma}}}_{\mathrm{NN}}{(\frac{{\overline{R}}_{G}}{{r}_{0}})}^{\ensuremath{\alpha}({T}_{0})}$. Here ${\overline{\ensuremath{\sigma}}}_{\mathrm{NN}}$ is the isospin-averaged nucleon-nucleon cross section evaluated at an energy equal to the bombarding energy per nucleon, and ${\overline{R}}_{G}$ is the arithmetic mean value of the radii of the projectile and the target measured in units of the radius parameter ${r}_{0}(=1.2 \mathrm{fm})$. In the limit ${T}_{0}=0$, the exponent $\ensuremath{\alpha}({T}_{0})=5$. A useful formula is derived for calculating mean neutron multiplicities in nucleus-nucleus collisions.NUCLEAR REACTIONS C, Al, Cu, $\mathrm{U}(\mathrm{N}\mathrm{e},\mathrm{n})X$, $E=337$ MeV/nucleon. Total inclusive neutron cross sections and mean multiplicities represented by simple one-parameter formulae. Compared results with predictions of the participantspectator model.

Elastic proton-nucleus scattering at 800 MeV: a comparison between the optical model and the Glauber model

Optical-model analyses of 800 MeV elastic proton scattering by eleven nuclei are performed. Relations between the optical-model parameters and the nucleon-nucleon parameters of the Glauber model are derived. The results indicate that the nucleon-nucleon cross section in the Glauber calculations should be somewhat larger than the free one and that the slope parameters for the spin-dependent amplitude should be considerably larger than assumed in previous analyses.

Do nuclei flow at high energies?

Nuclear flow patterns for E lab = 250 MeV/nucleon are generated using the cascade code of Cugnon et al. and analyzed in terms of a kinetic flow tensor. The effect of increasing the nucleon-nucleon cross section and the difference between repulsive and stochastic scattering styles are studied. A flow diagram is proposed to aid the experimental search for such flow patterns.

Viscous heating of expanding fireballs

The effect of a finite mean free path on the expansion stage of central collisions between heavy nuclei is studied in a nonrelativistic fluid dynamical approach. A gas of point particles with localized interactions is used. Kinetic theory then provides the thermal conductivity and viscosity coefficients in terms of the nucleon-nucleon cross section. About 10% additional entropy is generated during the expansion stage of central collisions at a beam energy of 800 MeV/A due to viscous heating of the system. Surprisingly there is only a weak dependence on the mass of the colliding nuclei. These results are in substantial agreement with recent intranuclear cascade calculations, although at present the fluid dynamical results are slightly ambiguous due to an uncertainty in the breakup criterion.

Pauli blocking and Fermi motion effects in ion-ion collisions

A geometrical model for Pauli blocking and Fermi motion effects in ion-ion collisions is presented. The results, given as effective nucleon-nucleon total cross sections, incicate that the Pauli blocking reduces the nucleon-nucleon cross section in the ion-ion environment by a larger amount than previously estimated.

Mean free path of nucleons in a Fermi gas at finite temperature

The mean free path of a nucleon in a nuclear Fermi gas at finite temperature is calculated by utilizing the free nucleon-nucleon cross section modified to suppress final states excluded by the Pauli principle. The results agree with an earlier zero-temperature calculation but yield substantially smaller values than a previous finite-temperature analysis. The Fermi gas mean free paths are some two to four times shorter than those implied by phenomenological imaginary optical potentials, suggesting that the present Fermi gas model fails to adequately describe the physical processes determining the mean free path. Even so, the present results, taken as lower bounds on the mean free path, require temperatures of some 4.5 MeV before the mean free path of bound nucleons becomes as short as the nuclear diameter. It follows that very high excitation energies are prerequisite to any short mean free path assumption in nuclear heavy-ion collisions.

Microscopic Description of Nucleon-Nucleus Total Reaction Cross Sections

Microscopic calculations of the total reaction cross sections for protons on $^{12}\mathrm{C}$, $^{27}\mathrm{Al}$, $^{40}\mathrm{Ca}$, and $^{208}\mathrm{Pb}$, and neutrons on $^{27}\mathrm{Al}$ and $^{208}\mathrm{Pb}$ have been made, which provide for the first time an excellent description of the data for projectile energies from 15 MeV through 1 GeV. The calculations are based on the experimental nucleon-nucleon total cross sections and explicitly include the effects of the real nuclear potential, the Coulomb potential, Pauli blocking, and Fermi motion.

Imaginary part of the ion-ion interaction potential

The energy density formalism of Brueckner is extended to the complex domain with the use of a complex two-body effective interaction, so as to generate simultaneously the real and imaginary part of the ion-ion interaction potential. A simple effective interaction consisting of a density dependent delta function repulsion and a Gaussian attraction is chosen for the real part of the complex effective interaction, whereas the imaginary component is derived by using the forward scattering amplitude approximation. In using this approximation, emphasis has been given to use the complex two-body matrix $\mathcal{T}$, where the propagator has the nuclear many-body Hamiltonian rather than the free two-body matrix $t$. For the average nucleon-nucleon cross section $〈\ensuremath{\sigma}〉$, required in constructing the complex effective interaction, a modified version of the relation suggested by Bohr and Mottelson is used which compares quite well with that of Clemental and Villi. The real part of the ion-ion interaction potential calculated in this model has already been reported in an earlier work and the present work deals with the calculation of the imaginary part of the potential for five pairs of spherical nuclei. The region of the potential beyond the surface is parametrized into a Woods-Saxon shape. The calculated imaginary potential for the system $^{40}\mathrm{Ca}$-$^{16}\mathrm{O}$ is compared with the phenomenological one of Becchetti et al. and the agreement between the two sets of potentials is found to be quite good in the region beyond the strong absorption radius.NUCLEAR REACTIONS Complex two-body effective interaction, forward scattering amplitude approximation, imaginary part of ion-ion interaction potential.

Geometrical aspects of high-energy peripheral nucleus-nucleus collisions

The abrasion-ablation model of relativistic nucleus-nucleus (heavy ion) fragmentation reactions is extended to mass number dependence and factorization properties of the cross sections. Weak (fragment-target) factorization is approximately valid while strong (projectile-target) factorization is violated. Calculated projectile-target factors γ PT for the fragmentation of 12C or 16O agree with the Berkeley and Orsay experiments. We propose using the fitted exponent y of the expression σ exp ∼(A P 1 3 + A T 1 3 ) y as a geometrical signature for discriminating peripheral and non-peripheral collisions for medium and heavy nuclei. The weak, logarithmic dependence of n-nucleon processes on the total nucleon-nucleon cross section σ NN implies weak beam energy dependence of fragmentation cross sections. A number of definite experimental predictions are made.

The imaginary components of T=1 nucleon-nucleon phase-shifts and their effect on spin-dependent inelastic cross sections

By means of a model involving the Delta (1236), an improved set of imaginary components for the T=1 nucleon-nucleon phase-shifts is obtained. These phase-shifts are then used to calculate inelastic nucleon-nucleon cross sections below 720 MeV, including the spin-dependent inelastic cross sections Delta sigma Ti and Delta sigma Li.

High momentum nucleons from relativistic nuclear collisions

A hard-scattering mechanism is proposed to describe proton and pion spectra from high-energy nuclear collisions. Results of a parameter-free model using free nucleon-nucleon cross sections and the empirical momentum distribution for nucleons within the nuclei are compared with data from C + C, Ne + NaF, and C + Pb collisions at 800 MeV/nucleon.

Collisions between composite particles at medium and high energies

Collisions between nuclei are studied by means of a simple extension of the Glauber high energy approximation. An expression for the optical phase shift function, exact within the framework of the Glauber approximation, is expanded in an infinite series and includes the effects of nuclear correlations. The first term corresponds to the standard optical limit result of the Glauber theory, and the higher order corrections arise from the processes in which one or more nucleons of either nucleus can undergo multiple collisions. The center-of-mass correlation is treated consistently so that our results do not exhibit the large-q divergence which characterizes the usual optical limit. It is shown that with realistic constraints on nucleon-nucleon total cross sections, the optical phase shift function does not approach the usual optical limit result when the mass numbers of the colliding nuclei become very large. With a proper treatment of center-of-mass correlations, the optical phase shift series converges rapidly for light nuclei and allows one to perform realistic calculations. The effects of higher order corrections on total and inelastic cross sections and on elastic scattering intensities are examined. The effects of the Coulomb field are included in an average phase approximation and results are comparedmore » with measurements.« less

Dynamics ofN−NTotal Cross Sections at Medium Energies

We show that a relativistic, one-pion-exchange, three-body theory including the (3,3) isobar and full spin dependence adequately describes the inelastic nucleon-nucleon cross section at energies where only single-pion production is important. Unitarization of the amplitude plays a significant role in obtaining the correct energy dependence. To fit elastic cross sections, however, will require inclusion of short-range potentials.

Rows on rows — A theory for collisions between heavy ions at high energy

The double differential cross section dσ/ dΩ dE is computed for protons and pions from collisions between nuclei at high energies (250 to 2100 MeV/nucleon). We derive the following within Glauber theory: At high energies the interaction between two nuclei reduces to a series of collisions between rows of target nucleons and rows of projectile nucleons. The scattering “rows on rows” can be obtained from a one-dimensional cascade calculation. We perform it in a simple approximation, using experimental nucleon-nucleon cross sections as the only input. The introduction of the Δ-degree of freedom leads to a satisfactory description of the proton and pion data at 2.1 GeV/nucleon. At 250 MeV/nucleon our agreement with experiment compares favourably with the one achieved in other approaches. A scaling law may be the reason why many simple approaches come already close to experiment. We discuss the question of thermalization.

Nucleon mean free path in nuclear matter

In calculations of nuclear reaction yields at incident energies of some tens of MeV consistently better agreement with experiments is obtained by assuming a nucleon mean free path in nuclear matter longer than that deduced from the Fermi gas model and free nucleon-nucleon cross sections.