Glauber Results Research Articles

Structure and reaction dynamics of zirconium in the relativistic mean field model

In the framework of relativistic mean field theory using NL3, NL3*, NLSH, DD-PC1 and DD-ME2 parameter sets, the ground state binding energy, charge radius, quadrupole deformation parameter of three key isotopes of Zr, i.e., 90,92,94Zr has been studied. The results are compared with the experimental data and are found in reasonable agreement. The two-neutron separation energy and β-decay half-life of these isotopes are also evaluated. Total reaction and elastic differential scattering cross-section are also studied for these isotopes as projectile with 12C as target, using above parameter sets in conjunction with Glauber model and results are compared with the experimental data. All these investigations support the abundance of these three isotopes of zirconium.

The role of projectile interactions in triply differential cross sections for excitation–ionization of helium

We report triply differential cross section (TDCS) for the simultaneous excitation–ionization of helium by electron impact for both coplanar and non-coplanar geometry. In the coplanar case, calculations have been performed for an incident energy of 500 eV and low ejection energies (3 and 10 eV), whereas in the noncoplanar case we have considered impact energies in the range 1240–4260 eV for a symmetric geometry. The present calculation is based on the eikonal approximation due to Glauber. We have incorporated the effect of post-collision interaction in the Glauber approximation. A comparison is made of the present calculations with the results of other theoretical methods and recent experiments. The Glauber results are in reasonably good agreement with the experiment for small scattering angles.

Fully differential cross section for single ionization of helium by 1 KeV electrons in the eikonal approximation

We report new results for fully differential cross sections for the single ionization of helium by 1 KeV electron impact at the ejection energy of 10 eV. Investigations have been made for both the scattering and perpendicular planes. The present calculation is based on the eikonal approximation due to Glauber. Here we have also incorporated the effect of classical post collision interaction (PCI) in the Glauber approximation (GA). A comparison is made of the present calculation with the results of other theoretical methods and the recent experiment [M. Dürr, C. Dimopoulou, B. Najjari, A. Dorn, K. Bartschat, I. Bray, D.V. Fursa, Z. Chen, D.H. Madison, J. Ullrich, Phys. Rev. A 77 (2008) 032717]. The Glauber results are in good agreement with the experiment in the scattering plane, but strong discrepancies are observed in the plane perpendicular to it. The effect of PCI is not substantial in the present kinematics.

Publishers’s Note: Incoherentρ0electroproduction off nuclei [Phys. Rev. C67, 054606 (2003)

In the present paper we investigate incoherent rho^0 electroproduction off complex nuclei. We derive a novel, simple expression for the incoherent electroproduction cross section in which one can clearly separate the final state interactions of the reaction products from the 'initial state interactions' of the photon that give rise to nuclear shadowing. In the special case of purely absorptive final state interactions we deduce from our expression the known Glauber result. A more realistic treatment of the final state interactions within a transport model is then used to compare our predictions with experimental data from the HERMES experiment.

Incoherentρ0electroproduction off nuclei

In the present paper we investigate incoherent ${\ensuremath{\rho}}^{0}$ electroproduction off complex nuclei. We derive a novel, simple expression for the incoherent electroproduction cross section in which one can clearly separate the final state interactions of the reaction products from the ``initial state interactions'' of the photon that give rise to nuclear shadowing. In the special case of purely absorptive final state interactions, we deduce from our expression the known Glauber result. A more realistic treatment of the final state interactions within a transport model is then used to compare our predictions with experimental data from the HERMES experiment.

Eikonal methods for multielectron ion-atom collisions

The Glauber and Born approximations were used to calculate cross sections for bare ions exciting hydrogen to its n = 2 level for an energy of 1.4 MeV/amu which corresponds to a speed of ν=7.5 a.u. results for the total cross sections for the incident bare ions H +, Be 4+, C 6+, O 8+, Ne 10+, Mg 12+, and P 15+ are presented. The Glauber approximation results make a significant correction to the Born approximation as the charge on the incident ion is increased. For P 15+ the Glauber result is over a factor of 2 larger than the Born result. Two eikonal approximations (G1 and G2) based on the Glauber approximation were carried out for an incident phosphorus ion in different ionization states for the same energy, 1.4 MeV/amu. It is argued that the G1 approximation, which has a structure similar to the Born approximation, is expected to be in better agreement with experimental data. However, there is no data with which to compare at this time.

Elastic-scattering cross-section for electrons and positrons from atomic hydrogen

The differential cross-sections of atomic hydrogen for elastic scattering of electrons and positrons have been rederived with the help of a method using a single parameter-dependent unitary shift operator for the calculation of the direct contribution. When the parameter approaches zerc the new method leads to the well-known conventional Glauber results. The numerical calculations include polarization effects and the exchange corrections obtained according to alternative approximation methods. Results calculated with Franco’s exchange show a definite improvement over the earlier results for medium energy electrons at large angles of scattering. Total elastic cross-sections have been calculated for 50 and 100eV electrons and positrons.

Multiple scattering in dilute systems

Multiple scattering in the limit in which the constituent scatterers are far removed from each other is studied. It is seen how this situation leads, in the Watson multiple-scattering series, to the dominance of on-shell propagation between the constituents. It also yields a version of the Glauber multiple-scattering theory in which no assumption is made to the effect that the individual projectile-constituent scattering amplitudes are eikonal in form. This approach allows for the immediate extraction of Fresnel corrections to the Glauber result and explains several apparent contradictions as to the relationship between the Watson multiple-scattering series approximated with on-shell propagation only and the Glauber theory.

Isobaric intermediate states in proton-nucleus elastic scattering

The effects of propagation of isobaric nucleon states in the intermediate steps of multiple scattering processes have been studied and applied to proton-4He elastic scattering at 1 GeV. The calculations are performed within the Glauber model and results are given for the differential cross section, the polarisation and the spin rotation parameter. In the conclusions the authors stress the large cancellations observed between terms of various orders and the great sensitivity of the effects to the nucleon-nucleon amplitudes.

Calculation of arbitrary 1s-nd excitations of atomic hydrogen by electron impact using the Glauber approximation

The 1s to nd excitations in e--H collisions are studied within the framework of the Glauber theory using a recently proposed method of reduction of the Glauber scattering amplitude. Calculations are performed in the incident electron energy range 15-400 eV for the state n=4, and for the asymptotic limit n to infinity and tabular results of properly scaled differential and integrated Glauber cross sections are presented. The Glauber results are compared with the existing theoretical predictions. A unified graphical representation of the integrated 1s to nd (n>or=4) Glauber cross sections above 50 eV normalised to the scaled asymptotic value at 100 eV presented.

Differential cross sections for excitation of atomic hydrogen by proton impact

The differential cross section for the excitation of the hydrogen atom to the $2s$ and $2p$ states are calculated by applying the Coulomb-projected-Born (CPB) method as proposed by Geltman. The results for the excitations of the hydrogen atom for the incident proton energies of 25, 50, 100, and 150 keV and for the scattering angles $\ensuremath{\theta}$ varying from 0 to 1.5 (in ${10}^{\ensuremath{-}3}$ rad c.m.) obtained by the present CPB approach are compared with the experimental observations and other existing theoretical calculations. At the incident energy 100 keV the present CPB results, except at large scattering angles, are found to be in good agreement with the observed results and are as reliable as the calculated results obtained by applying the many-state close-coupling (CC) approximation. For 50 keV and for scattering angles varying from 0.4 to 1.0 (in ${10}^{\ensuremath{-}3}$ rad c.m.) the present CPB results agree with the observed data and are as reliable as the CC and the Glauber results, however with the decrease of scattering angle ($\ensuremath{\theta}\ensuremath{\le}0.4$ in ${10}^{\ensuremath{-}3}$ rad c.m.) the results obtained by the CPB method become less reliable compared to the CC or the Glauber results. At 25 keV the present CPB results for the differential cross sections underestimate the observed values throughout the angular region considered except at the large angles, where $\ensuremath{\theta}\ensuremath{\ge}0.5$ in ${10}^{\ensuremath{-}3}$ rad c.m. On the other hand, the Glauber results at this energy give quite good agreement throughout the angular region, whereas the Born results grossly overestimate the observed differential cross sections.

Glauber cross sections for arbitrary excitation of ground-state hydrogen by electron and proton impact

The Glauber scattering amplitude for $1s\ensuremath{-}nlm$ excitations of hydrogen by charged particle impact is reduced to a form numerically computable for arbitrary values of $n$, $l$, and $m$. The increasing number of repeated parametric differentiations appearing in earlier methods with the increase of $n$ is replaced in our method by a single contour integration, which is especially useful for large-$n$ states. In addition, whereas the available closed-form expressions would involve the calculation of a large number of hypergeometric functions as $l$ and $m$ increase for a given $n$, the present integral form involves only one hypergeometric function for any $n$, $l$, and $m$. The asymptotic form of the $1s\ensuremath{-}nlm$ amplitude as $n\ensuremath{\rightarrow}\ensuremath{\infty}$ is also presented. The procedure is applied for studying the $1s\ensuremath{\rightarrow}\mathrm{ns}(n=3, 4, 10, \mathrm{and} \ensuremath{\infty})$ and $1s\ensuremath{\rightarrow}\mathrm{np}(n=3, 4, \mathrm{and} \ensuremath{\infty})$ excitations of hydrogen by electron and proton impact. Tabular results of the cross sections are presented for differential and integrated cross sections in ${e}^{\ensuremath{-}}$-H collisions and for integrated cross sections in ${\mathrm{H}}^{+}$-H collisions. The qualitative features observed earlier for low $\mathrm{ns}$ and $\mathrm{np}$ excitations are found to exist also for $n\ensuremath{\ge}3$ excitations. The $1s\ensuremath{\rightarrow}\mathrm{ns}$ and $1s\ensuremath{\rightarrow}\mathrm{np}$ cross sections obey an ${n}^{\ensuremath{-}3}$ law asymptotically as ${n}^{\ensuremath{-}2}\ensuremath{\rightarrow}0$, which is in conformity with the analytical results obtained quite generally for $1s\ensuremath{\rightarrow}\mathrm{nlm}$ excitations. For proton impact $1s\ensuremath{\rightarrow}3p$ excitations, the Glauber results are in excellent agreement with a recent 34-state calculation. A unified graphical representation of various normalized cross sections above some specific energy is presented, whence a knowledge of the high-energy Born cross sections for any $1s\ensuremath{\rightarrow}\mathrm{ns}$ or $1s\ensuremath{\rightarrow}\mathrm{np}$ excitation suffices to determine the absolute Glauber cross sections for the process.

Eikonal Approach to Break-Up Processes of Hydrogenic Ions

The Glauber approximation for an inelastic Coulomb scattering is first applied to the ionization process of hydrogenic ions by taking account of the asymptotic Coulomb interaction. The result of integrated ionization cross sections in the non-exchange approximation is compared with the corresponding Coulomb-Born result and it is shown that the Glauber result is in reasonable agreement with experimental data. The reduced cross section in the limit of infinitely large nuclear charge is also evaluated for hydrogenic heavy ions from the general and practical point of view.

Calculation of proton-impact excitation of helium using the Glauber approximation

Cross sections for the $1^{1}S\ensuremath{\rightarrow}n^{1}S$ ($n=2,3,4$) excitations of atomic helium under proton impact are calculated with incident energy ranging from 25 to 1000 keV employing the Glauber approximation. A properly orthogonalized set of He wave functions are used. The generalized oscillator strengths for $1^{1}S\ensuremath{\rightarrow}2^{1}S$ and $1^{1}S\ensuremath{\rightarrow}3^{1}S$ excitations are also calculated with these wave functions, and the results are compared with the existing theoretical and experimental data. The full Glauber scattering amplitude is separated into single-and double-scattering parts. The single-scattering amplitude is evaluated from a closed-form expression. A numerical method is employed to calculate some integrals occuring in the double-scattering amplitude. At incident energies above 500 keV, the single-scattering contribution is found to dominate the Glauber result. The effect of the double-scattering term is appreciable at intermediate energies, where it substantially lowers the cross sections from the corresponding single-scattering result. The Glauber result at intermediate energies also underestimates most of the existing theoretical and experimental cross sections, which already show a wide variation among themselves in absolute values. However, the functional dependence of the Glauber cross sections on energy is similar to that depicted by other calculations and measurements. Furthermore, the average ratio of the Glauber cross sections for $3^{1}S$ and $4^{1}S$ excitations at high energies show reasonable agreement with those obtained from other theories, as also from the ${n}^{\ensuremath{-}3}$ law of cross sections.

E-H exchange scattering in the generalised Coulomb-projected Ochkur-Born approximation

Exchange amplitudes have been obtained by carrying out Ochkur reduction on the modified Born model recently proposed by Junker (1975). In the case of 1s to 1s and 1s to 2s e-H exchange scattering at intermediate energies present results are found to be much better than the usual Ochkur results and are quite close to the Glauber and close-coupling results. Another advantage is that the present method is very economical on computer time.

Pseudospin formulation of kinetic Ising models

It is shown that Glauber's kinetic Ising model is equivalent to an Ising model with multispin interactions in a transverse field. In one dimension, certain of Glauber's results are recovered by using the well-known fermion representation for spin-$\frac{1}{2}$.

High-energy heavy-ion scattering and the optical phase shift function

We derive, within the framework of the Glauber approximation, an exact expression for the optical phase shift function $\overline{\ensuremath{\chi}}(b)$ which takes into account the center-of-mass correlation function. We calculate total and differential cross sections using the first-order term for $\overline{\ensuremath{\chi}}(b)$ and compare them with the usual (first-order) optical limit results and, for the lighter nuclei, with the exact Glauber results. The nucleus-nucleus elastic scattering amplitude does not exhibit the large-$q$ divergence which characterizes the usual optical limit. Our results improve the calculated total and differential cross sections dramatically for light nuclei, and significantly for medium and heavy nuclei.NUCLEAR REACTIONS Glauber approximation, high-energy scattering theory, center-of-mass effects $^{2}\mathrm{H}$, $^{4}\mathrm{He}$, $^{12}\mathrm{C}$, $^{16}\mathrm{O}$, $^{24}\mathrm{Mg}$, $^{40}\mathrm{Ca}$ projectiles and targets, $E=2.1$ GeV/nucleon; calculated $\ensuremath{\sigma}$, $^{4}\mathrm{He}$ ($^{4}\mathrm{He}$, $^{4}\mathrm{He}$), $^{12}\mathrm{C}$ ($^{12}\mathrm{C}$, $^{12}\mathrm{C}$), $E=2.1$ GeV/nucleon; calculated $\ensuremath{\sigma}(\ensuremath{\theta})$.

Cross sections for atomicK-shell ionization by ion impact in the single-particle Glauber approximation

Total cross sections have been computed for $K$-shell ionization in selected multielectron atoms by proton impact using the single-particle Glauber approximation with hydrogenic wave functions. Our single-particle Glauber approximation corresponds to an independent-particle model where explicit coupling with target electrons not directly involved in the interaction is omitted. In principle, the Glauber approximation is valid over a wider region of scattering parameters than is the Born approximation. However, our Glauber results are not in significantly better agreement with observation than simpler Born predictions. For all targets considered at projectile velocities near the peak of the cross section, single-particle Glauber results tend to lie beneath observed cross sections by proton impact, in contrast to previous results for electron impact where, as expected, Glauber results are in better agreement with observations than Born results for low-${Z}_{2}$ targets. Some explanations of the discrepancy and possible extensions of our methods are considered. It is also found that, as the atomic number of the target, ${Z}_{2}$, increases, the relative contribution of Glauber corrections to corresponding Born calculations decreases at projectile velocities near and above the peak of the $K$-shell ionization cross section. For ${Z}_{2}\ensuremath{\gtrsim}5$, the Glauber scattering corrections to Born results for $K$-shell ionization at projectile velocities near and above the peak of the cross section are less than a few percent, e.g., less than errors due to our use of hydrogenic target wave functions. For ${\mathrm{O}}^{+8}$ + Ne, unlike ${p}^{+}$ +Ne, significant differences are apparent between Glauber and Born predictions for direct Coulomb ionization.

Glauber cross sections for the2s→2presonance excitation in lithium by electron impact

The cross sections for the electron-impact-induced resonance transition ($2s\ensuremath{\rightarrow}2p$) in lithium are calculated in the Glauber approximation using Franco's method of explicitly involving all the electrons of the many-electron target. Our results are always a bit smaller than the frozen Glauber results, but approach them as the energy increases. However, these differences are not very significant, indicating that the inner electrons (those in $1s$ state) are rather inert. Both of the Glauber results are found to be in good agreement above 15 eV with the recent experimental results of Leep and Gallagher.

Electron-hydrogen elastic scattering in the eikonal approximation

Differential and total cross sections for elastic scattering of electrons from atomic hydrogen have been calculated using the full eikonal approximation. The eikonal results for 50-, 100-, and 200-eV incident electrons are compared with other theoretical calculations and with recent experimental data. It is found that the full eikonal approximation predicts both small-angle differential cross sections and total cross sections, which are in better agreement with experiment than either the Born or Glauber results.