Glauber Theory Research Articles

Effect of in-medium NN cross section and finite range force on the reaction cross section for a deformed target nucleus

A method is described to calculate the heavy ion reaction cross section ${\ensuremath{\sigma}}_{R}$ when one of the two nuclei is deformed. This method is based on both the double folding model and the optical limit approximation to Glauber theory. Both finite range of the nucleon-nucleon $(NN)$ interaction and in-medium effects of $NN$ cross section are included. The interacting pair $^{12}\mathrm{C}\text{\ensuremath{-}}^{238}\mathrm{U}$ is taken to study orientation, energy, and deformation dependence of ${\ensuremath{\sigma}}_{R}$. The finite range force increases the value of ${\ensuremath{\sigma}}_{R}$ by about $5%$. In-medium $NN$ cross section decreases ${\ensuremath{\sigma}}_{R}$ by about $1.9%$ compared to the free $NN$ cross section.

Total Nuclear Reaction Cross Section Induced by Halo Nuclei and Stable Nuclei**The project supported by the State Key Basic Research Development Program of China under Contract No. G2000077400, the “100- Person Project” of the Chinese Academy of Sciences, National Natural Science Foundation of China under Grant Nos. 10175080, 10175082 10004012, and 19847002, and CAS Knowledge Innovation Project under Grant No.

We develop a method for calculation of the total reaction cross sections induced by the halo nuclei and stable nuclei. This approach is based on the Glauber theory, which is valid for nuclear reactions at high energies. It is extended for nuclear reactions at low energies and intermediate energies by including both the quantum correction and Coulomb correction under the assumption of the effective nuclear density distribution. The calculated results of the total reaction cross section induced by stable nuclei agree well with 30 experimental data within 10 percent accuracy.The comparison between the numerical results and 20 experimental data for the total nuclear reaction cross section induced by the neutron halo nuclei and the proton halo nuclei indicates a satisfactory agreement after considering the halo structure of these nuclei, which implies quite different mean fields for the nuclear reactions induced by halo nuclei and stable nuclei. The halo nucleon distributions and the root-mean-square radii of these nuclei can be extracted from the above comparison based on the improved Glauber model, which indicates clearly the halo structures of these nuclei. Especially, it is clear to see that the medium correction of the nucleon-nucleon collision has little effect on the total reaction cross sections induced by the halo nuclei due to the very weak binding and the very extended density distribution.

Mn2(saltmen)2Ni(pao)2(L)2](A)2 with L=pyridine, 4-picoline, 4-tert-butylpyridine, N-methylimidazole and A=ClO4-, BF4-, PF6-, ReO4-: a family of single-chain magnets.

A series of single-chain magnets, [Mn2(saltmen)2Ni(pao)2(L)2](A)2 (saltmen(2-)=N,N'-(1,1,2,2-tetramethylethylene) bis(salicylideneiminate), pao-=pyridine-2-aldoximate; A-=ClO4- with L=4-picoline; 2, 4-tert-butylpyridine; 3, N-methylimidazole; 4, and L=pyridine with A-=BF4-; 5, PF6-; 6, ReO4-; 7), was prepared by reactions between MnIII dimer units, i.e., [Mn2(saltmen)2(H2O)2](A)2 (A-=ClO4-, BF4-, PF6-) or Mn2(saltmen)2(ReO4)2, and NiII monomeric units, i.e., Ni(pao)2(L)2, in methanol/water media. The crystal structures of 4, 6, and 7 were established by single-crystal X-ray crystallography. These three compounds are isostructural with [Mn2(saltmen)2Ni(pao)2(py)2](ClO4)2 (1) (Clérac, R.; Miyasaka, H.; Yamashita, M.; Coulon, C. J. Am. Chem. Soc. 2002, 124, 12837) and crystallize in monoclinic space group C2/c. The linear arrangement of MnIII dimer units and NiII building blocks leads to an alternating chain having a repeating unit, [-(O)2-Mn-ON-Ni-NO-Mn-]. The chains are well separated with the nearest interchain intermetallic distance of 10.36 A for 4, 10.51 A for 6, and 10.30 A for 7, and there is no significant pi-pi interchain interaction between ligands. The void space between the chains is occupied by counteranions, which control the three-dimensional organization of the chains. The X-ray diffraction analysis (XRD) on a powder sample was also performed for all compounds. The XRD patterns for 1, 2, and 4-7 are very similar, emphasizing the isostructural nature of these materials although they have individually slight different interchain distances. Inversely, the XRD pattern for 3 reveals a completely different shape being indicative of the peculiar crystal packing compared to the others. Nevertheless, the one-dimensional nature of the structure is also kept in 3 as indicated by magnetic measurements. The whole family of compounds exhibits quasi-identical magnetic behavior compared to that described for 1. Above 30 K, the heterometallic chain can be described as an assembly of antiferromagnetically coupled Mn...Ni....Mn trimers (via oximate bridge, -24.2 K<J(Mn-Ni)/kB<-20.8 K) connected through Mn...Mn ferromagnetic interaction (via the phenolate bridge, J(Mn-Mn)/kB approximately +0.7 K), assuming a ferromagnetic chain with S=3 units. In the low temperature region, combined ac and dc magnetic measurements revealed for the whole series the systematic presence of a magnet behavior exhibiting coercivity and slow relaxation of magnetization below 3.5 K. This behavior was analyzed on the basis of Glauber's theory for an Ising one-dimensional system which predicts an activated dependence (Arrhenius law) of the relaxation time: tau=tau0 exp(Delta/kBT). Similar values of tau0 approximately 1x10(-10) s and Delta/kB approximately 70 K have been found along the series. The described compounds constitute the first example of the rational design of a single-chain magnet family and clearly demonstrate the unique behavior of these heterometallic chains independently of their interchain environments.

Proton–nucleus elastic scattering and the equation of state of nuclear matter

We calculate differential cross sections for proton–nucleus elastic scattering by using a Glauber theory in the optical limit approximation and nucleon distributions that can be obtained in the framework of macroscopic nuclear models in a way dependent on the equation of state of uniform nuclear matter near the saturation density. We find that the peak angle calculated for unstable neutron-rich nuclei in the small momentum transfer regime increases as the parameter L characterizing the density dependence of the symmetry energy decreases. This is a feature associated with the L dependence of the predicted matter radii.

Intensity interferometry for a chaotic source with a collective flow and multiple scattering

We study the effects of a collective flow and multiple scattering on two-particle correlation measurements in Hanbury-Brown–Twiss (HBT) intensity interferometry. While the derivation in this paper is not rigorous, we propose a semi-quantitative derivation to shed further light on the interesting RHIC HBT puzzle. We find that under a collective flow the effective source distribution in a two-particle correlation measurement depends on the initial source distribution. In addition, it depends on a collective flow phase function which consists of terms that tend to cancel each other. As the detected particles traverse from the source point to the freeze-out point, they are subject to multiple scattering with medium particles. We examine the effects of multiple scattering on HBT correlations. Applying the Glauber theory to multiple scattering at high energies and the optical model at intermediate energies, we find that multiple scattering leads to an absorption and an effective density distribution that depends on the initial source distribution.

CHARGED PION PHOTOPRODUCTION FROM HYDROGEN AND DEUTERIUM AT JEFFERSON LAB

The γn → π-p and γp → π+n reactions are essential probes of the transition from meson-nucleon degrees of freedom to quark-gluon degrees of feedom in exclusive processes. The cross sections of these processes are also, advantageous, for the investigation of oscillatory behavior around the quark counting prediction, since they decrease relatively slower with energy compared with other photon-induced processes. Moreover, these photoreactions in nuclei can probe the QCD nuclear filtering and color transparency effects. In this talk, I discuss the preliminary results on the γp → π+n and γn → π-p processes at a center-of-mass angle of 90° from Jefferson Lab experiment E94-104. I also discuss a new experiment in which singles γp → π+n measurement from hydrogen, and coincidence γn → π-p measurements at the quasifree kinematics from deuterium and 12 C for photon energies between 2.25 GeV to 5.8 GeV in fine steps at a center-of-mass angle of 90° are planned. The proposed measurement will allow a detailed investigation of the oscillatory scaling behavior in photopion production processes and the study of the nuclear dependence of rather mysterious oscillations with energy that previous experiments have indicated. The various nuclear and perturbative QCD approaches, ranging from Glauber theory, to quark-counting, to Sudakov-corrected independent scattering, make dramatically different predictions for the experimental outcomes.

ORIENTATION AND DEFORMATION DEPENDENCE OF THE REACTION CROSS-SECTION FOR A DEFORMED TARGET NUCLEUS

The optical limit approximation to Glauber theory was used to calculate the reaction cross-section, σR, for a deformed target nucleus. A method is presented to include both density dependence of NN reaction cross-section and higher order deformations of the target nucleus. We studied orientation, energy and deformation dependence of σR for C 12– N 17 and C 12– U 238 interacting pairs. We found that the orientation dependence of σR for the heavy target U 238 depends on the value and sign of hexadecapole deformation and it is more than 2.2 times compared to the light deformed target nucleus N 17. The presence of hexadecapole deformation does not affect the value of σR averaged over all orientation of the target nucleus. A geometrical model was proposed to account for the orientation dependence of σR. We found that the error in this model is less than 10%.

Cross section calculations in Glauber model: I. Core plus one-nucleon case

This paper presents a Fortran program to calculate cross sections for various reactions between composite particles in the framework of Glauber theory. The projectile nucleus is described as a core plus one valence-nucleon system. The input data needed for the calculation are the core and target densities and the nucleon–nucleon profile function. The single-particle wave function of the valence nucleon may be provided by a user or generated by the code if appropriate quantum numbers of the wave function are specified. Multiple integrations involving the valence-nucleon wave function are performed by a Monte Carlo technique with Metropolis algorithm. The program gives the total reaction cross section, the one-nucleon removal cross section, the elastic differential cross section, and the longitudinal momentum distribution of the core fragment. Sample calculations include the reactions of 13,19C+ 12C, 11Be+ 12C, 11Be+ 9Be, and 8B+ 12C systems.

Monte Carlo integration in Glauber model analysis of reactions of halo nuclei

Reaction and elastic differential cross sections are calculated for light nuclei in the framework of the Glauber theory. The optical phase-shift function is evaluated by Monte Carlo integration. This enables us to use the most accurate wave functions and calculate the phase-shift functions without approximation. Examples of proton-nucleus (e.g., ${p\ensuremath{-}}^{6}\mathrm{He},$ ${p\ensuremath{-}}^{6}\mathrm{Li})$ and nucleus-nucleus ${(\mathrm{e}.\mathrm{g}.,}^{6}\mathrm{He}{\ensuremath{-}}^{12}\mathrm{C})$ scatterings illustrate the effectiveness of the method. This approach gives us a possibility of a more stringent analysis of the high-energy reactions of halo nuclei.

Hadron formation in high energy photonuclear reactions

We present a new method to account for coherence length effects in a semiclassical transport model. This allows us to describe photoproduction and electroproduction at large nuclei $(A&gt;12)$ and high energies using a realistic coupled channel description of the final state interactions that goes beyond simple Glauber theory. We show that the purely absorptive treatment of the final state interactions can lead to wrong estimates of color transparency and formation time effects in particle production. As an example, we discuss exclusive ${\ensuremath{\rho}}^{0}$ photoproduction on Pb at a photon energy of 7 GeV as well as ${K}^{+}$ production in the photon energy range 1--7 GeV.

Proton Reaction Cross Sections Measured in the BNL/AGS E943 Experiment

We have measured proton reaction cross sections over a wide mass and energy range at the Brookhaven AGS accelerator. The samples were elemental Be, C, Al, Cu, W, and Pb; the measurements were carried out at ten incident proton kinetic energies in the range 0.54 to 7.8 GeV. The experiment was similar to an earlier experiment in the 200- 550 MeV range by Renberg et al.1) The new results are in good agreement with those of Renberg et al. at the overlap point near 550 MeV. The combined results of the two experiments show an energy dependence expected from the behavior of the nucleon-nucleon cross sections. The results are reproduced by calculations based on variants of the impulse approximation and Glauber theory.

Calculation of nucleus–nucleus cross sections at intermediate energies using Glauber theory

Nucleus–nucleus cross sections are calculated using nucleon–nucleus optical potentials in the framework of Glauber theory. The resulting cross sections are compared with experimental data. The reaction cross sections for various light nuclei including loosely bound nuclei are calculated from the nucleus–nucleus phase shift functions which need the nuclear density and the optical potential as an input. The calculated reaction cross sections in the energy range of 30 to 800 MeV/nucleon are in agreement with experimental data within error bars. The elastic differential cross sections of 4,6He+ 12C and 12C+ 12C are calculated at 41 MeV/nucleon and 30, 200 MeV/nucleon, respectively. At lower energy, a correction beyond the eikonal approximation improves the agreement between measured and calculated angular distributions significantly.

Modified Glauber model for the total reaction cross-section of 12C + 12C collisions

Glauber's theory has been adopted to calculate the total heavy-ion reaction cross-sections at high energies. At relatively low energies, Glauber's total reaction cross-section has been modified in order to take into account the Coulomb field effect and is called modified Glauber model I. In addition to the Coulomb field effect, the nuclear effect has also been taken into account in the Glauber model and is called modified Glauber model II. An analytical expression for the transparency function for heavy-ion reactions, involving the nuclear densities of the colliding ions and the nucleon-nucleon cross- section, has been obtained within the framework of the modified Glauber models I and II. The transparency and the total reaction cross-sections of the 12C + 12C collisions are calculated at different bombarding energies. The obtained results are in good agreement with the experimental data and with previous theoretical calculations.

SELECTED TOPICS IN HIGH ENERGY SEMI-EXCLUSIVE ELECTRO-NUCLEAR REACTIONS

We review the present status of the theory of high energy reactions with semi-exclusive nucleon electro-production from nuclear targets. We demonstrate how the increase of transferred energies in these reactions opens a completely new window for study of the microscopic nuclear structure at small distances. The simplifications in theoretical descriptions associated with the increase in the energies are discussed. The theoretical framework for calculation of high energy nuclear reactions based on the effective Feynman diagram rules is described in detail. The result of this approach is the generalized eikonal approximation (GEA), which is reduced to the Glauber approximation when nucleon recoil is neglected. The method of GEA is demonstrated in the calculation of high energy electro-disintegration of the deuteron and A=3 targets. Subsequently, we generalize the obtained formulae for A&gt;3 nuclei. The relation of GEA to the Glauber theory is analyzed. Then, based on the GEA framework we discuss some of the phenomena which can be studied in exclusive reactions: nuclear transparency and short-range correlations in nuclei. We illustrate how light-cone dynamics of high-energy scattering emerge naturally in high energy electro-nuclear reactions.

Time-ordering of quark multi-scattering processes in hadron–proton scattering at high energy

In the framework of non-relativistic Glauber theory with the quark model, the time-ordering of quark multi-scattering processes in hadron–proton elastic scattering at FNAL and CERN-ISR ranges of energy is considered. The time-ordering effect in the Glauber approximation with the geometrical scaling property of hadron reactions is presented. The phase variation effect, as suggested by Franco, is also discussed. A cloud effect is taken into account to obtain a good fit with the experimental data of elastic scattering differential cross section in the forward direction. The results obtained for proton–proton scattering at 200, 290, 500, 1070 and 1500 GeV/c lead us to neglect the phase variation effect in this range of energy. At the same time, the time-ordering of quark multi-scattering processes plays an important role in obtaining a good fit in the range of momentum transferred squared where q2 > 3.5 (GeV/c)2. In the case of π−-proton elastic scattering at 200 GeV/c, where the phase variation leads to a good fit with the experimental data, the sensitivity with respect to the time-ordering effect can be observed for q2 > 6 (GeV/c)2.

Study of 1s internal bremsstrahlung spectrum from 57Co

The internal bremsstrahlung contribution from the electron capture of 57Co has been measured in coincidence with K-X-ray of the residual atom. The end-point energy (EPE) is extracted from the data using the linearised Jauch plot. The transition energy obtained using the (EPE) is 842.7 keV, which is close to the value given by Audi and Wapstra. The measured intensity and shape factor from 300 to 600 keV are found to be in good agreement with the Glauber and Martin theory.

Nucleus-Nucleus Scattering Based on a Modified Glauber Theory**The project supported by National Natural Science Foundation of China under Grant Nos 10075080, 19847002, 19835010 and Major State Basic Research Development Program under Contract No. G200077407

A modified microscopic Glauber theory has been extended to investigation of the reaction and elastic differential cross sections of various projectile-target collisions at low and intermediate energies. Through a systematic study, we find that the inclusion of the finite range interaction and Coulomb modifications plays very important role in the Glauber theory to reproduce the experimental data at these energy ranges. Usually the effect of the Coulomb modification is to decrease the reaction cross sections, on the contrary that of the finite range interaction modification increases them. The angular distributions calculated by the Glauber theory including these two corrections are in good agreement with the experimental data.

Nuclear transparency in heavy-ion collisions at 14.6 GeV [formula omitted]nucleon

The probability of a projectile nucleon to traverse a target nucleus without interaction is calculated for central Si–Pb collisions and compared to the data of E814. The calculations are performed in two independent ways, via Glauber theory and using the transport code UrQMD. For central collisions Glauber predictions are about 30 to 50% higher than experiment, while the output of UrQMD does not show the experimental peak of beam rapidity particles.

Nuclear shadowing at low photon energies

We calculate the shadowing effect in nuclear photoabsorption at low photon energies (1-3 GeV) within a multiple scattering approach. We avoid some of the high energy approximations that are usually made in simple Glauber theory like the narrow width and the eikonal approximation. We find that the main contribution to nuclear shadowing at low energies stems from $\rho^0$ mesons with masses well below their pole mass. We also show that the possibility of scattering in non forward directions allows for a new contribution to shadowing at low energies: the production of neutral pions as intermediate hadronic states enhances the shadowing effect in the onset region. For light nuclei and small photon energies they give rise to about 30% of the total shadowing effect.

Decay energy of 55Fe from its inner bremsstrahlung spectrum

Several measurements of decay energy using the inner Bremsstrahlung spectrum (IB) due to radiative electron capture in 55Fe has been made. But the results are not uniform. Hence another attempt has been made at the same. Experimental data was obtained with a 4.445 cm. dia × 5.08 cm thick Nal (Tl) detector. It was subjected to suitable statistical treatment and various corrections using Liden and Starfelt procedure. The corrected spectrum agrees well with the Glauber and Martin theory for 1s electron capture beyond 100 keV. From the Jauch plot, the decay energy of 232.36±0.64 keV was obtained.