Impact Parameter Distribution Research Articles

Classical trajectory calculations for H+-H collisions with the Wigner function as initial phase space distribution

For classical trajectory Monte-Carlo calculations the authors propose the use of the Wigner function for the initial distribution of the coordinates and momenta of the electron instead of the microcanonical phase space distribution. For physical reasons they restrict the Wigner function in energy space. The allowed energy interval is determined by a cut-off parameter Emax which is fitted to the experimental data. The calculated total cross sections for ionisation and charge transfer in H+-H collisions reproduce the experimental data in the range of incident proton energies Elab=25-200 keV. In addition they present the probabilities of ionisation and charge transfer as functions of the impact parameter and angular distributions of the ionised electron.

A molecular dynamics simulation study of the influence of the ion-atom potential function upon sputtering

A molecular dynamics simulation has been used to investigate the ion-atom potential function dependence of single-crystal sputtering. Atom yields from the (111) and (001) crystal orientations have been calculated at 5.0 keV for the Cu/Ar+ system for 28 Born-Mayer and Moliere potential functions. The multimer yield ratios, layer yield ratios, ejected atom energy distribution, and angular distribution of ejected atoms are essentially independent of the potential parameters chosen in a Born-Mayer function. The atom yields from a target orientation can be ordered, or estimated, from a knowledge of the magnitude of the potential function in a sensitive range of separations near the maximum of the impact parameter distribution function for that surface. This property of the potential functions has been reproduced at 600 eV. The Moliere function yields and other properties fall exactly into the positions predicted from a knowledge of magnitude of the potential function in the sensitive range. With the exception of the atom yields, yield ratios, and the atoms per single ion distribution the properties of the sputtered atom distribution can be computed from any of a large number of potential functions.

Wavefunction effects in inner-shell ionisation of light atoms by protons

An efficient computer code for calculating the impact parameter distribution of atomic ionisation probabilities caused by charge particle impact, has been developed. The program is based on the semiclassical approximation, and it allows the use of an arbitrary atomic central potential for deriving the one-electron orbitals that form the basis for the description of the atomic states. Extensive calculations are reported for proton induced K-shell ionisation in carbon and neon, covering energies in the range 0.1-10 MeV. Some calculations on proton-argon L-shell ionisation are also reported. Comparison between the results obtained using (screened) hydrogenic potentials and the recently reported energy optimised effective atomic central potentials demonstrates that wavefunction effects are generally important for inner-shell ionisation of light atoms. The agreement between theory and experiment in the K-shell case is improved for fast collisions upon using better wavefunctions.

Impact parameter interpretation of proton-proton scattering from a critical review of all ISR data

This paper contains a critical review of all the data produced at the ISR on proton-proton elastic scattering and total cross sections. This coherent and complete set of data is used to compute the impact parameter distribution of the proton-proton inelastic overlap integral. This impact parameter analysis has smaller errors than any other previously made, and confirms the good agreement with the geometrical scaling model while strongly disagreeing with models based on factorizing eikonals. For the first time we find indications of a second contribution to the peripheral rising of the proton-proton cross section in a region around 2.2 fm.

Classical relativistic trajectory method and application to the K-shell ionisation in U-U collisions

The classical trajectory method of Abrines and Percival (1966) has been extended to relativistic velocities of the charged particles. The method is based on the solution of the classical relativistic Hamilton equations of motion for a three-body system consisting of a target nucleus, a charged projectile and a single electron. The electromagnetic fields are taken to be of Lienard-Wiechert type with extended charges of the nuclei. The theory has been applied to calculate the cross section, mean impact parameter and angular distribution for the ionisation of a singly occupied K shell in U-U collisions at Elab=1 and 10 GeV/n.

Chemiluminescent ion–molecule reactions: Rotational–vibrational population distributions of CH+(A 1Π) and CD+(A 1Π) from C++H2 and C++D2 collisions

Optical emission in the 3000–5000 Å range was observed as a result of low-energy (3–8 eVCM) ion–molecule reactions of C+ ions with H2 and D2 molecules. Most of the emission is due to (A 1Π→X 1Σ+) transitions of CH+ and CD+, which were each studied with 6 Å and 20 Å FWHM optical resolution. Rotational lines of much higher J were observed than are present in spectra from conventional CH+ light sources. The observed spectra were simulated on a computer, taking into account the strong dependence of the electronic transition moment on the internuclear distance, as given by ab initio calculations. Relative band oscillator strengths fv′v″(J) were calculated for each rotational line J and were found to depend strongly on J as result of centrifugal stretching. These calculations were done using the R centroid approximation as well as the exact band oscillator strength formula. From the spectrum simulations, accurate rotational and vibrational population distributions were derived. A simple model is proposed which explains the isotope effect on the rotational distributions and gives directly the distribution of impact parameters leading to reaction.

Probabilities for2s−2pexcitation ion-atom collisions

Expressions are derived using the semiclassical Coulomb approximation for probabilities for $2s\ensuremath{-}2p$ excitation during a collision between a target atom and a proton. Results of calculations are presented. For proton energies above 1 MeV the transition probabilities are less than \ensuremath{\sim}0.03. Unlike excitation from one atomic shell to another, our intrashell results are largely independent of the target atomic charge ${Z}_{2}$ and binding-energy difference $\ensuremath{\Delta}{E}_{2s2p}$. The shapes of our impact-parameter distributions, when scaled to the atomic radius, are similar for various targets. However, for a given target, the dependence of the $m=0$ and $m=\ifmmode\pm\else\textpm\fi{}1$ amplitudes on both impact parameter $B$ and $\ensuremath{\Delta}{E/}_{2s2p}$ is quite different, indicating that the polarization of light emitted in subsequent $2p\ensuremath{-}1s$ transitions may vary with both $B$ and $\ensuremath{\Delta}{E}_{2s2p}$. Furthermore, near $B=0$ the probabilities are sensitive to $\ensuremath{\Delta}{E}_{2s2p}$, which varies with the charge state of the target atom. Consequently, the authors suggest that dependences of collision cross sections and probabilities on the atomic charge state may be investigated by considering impact-parameter dependence or polarization fractions in $2s\ensuremath{-}2p$ atomic excitation by ion impact.

Azimuthal correlations and Bose Einstein effects in the reaction K +p → K +p2 π+2 π− AT 16 GeV/ c

Azimuthal distributions between all possible particle pairs are studied in the reaction K +p → K +p2( π + π −) at 16 GeV/ c. It is shown that the hypothesis of uncorrelated transverse momenta at fixed rapidities of the final-state particles, is sufficient to describe the main features of the data. An interesting exception occurs for identical particle pairs where a genuine short-range correlation, attributed to Bose-Einstein symmetrization, is observed. The correlation length of this effect is shown to be related to the impact parameter distribution of the produced pions.

The diffraction pattern of pp elastic scattering and the composite structure of the proton

The consequences of the quark-glue structure of hadrons for pp elastic scattering are examined in a simple model where the glue is taken to be the active constituent in normal (low p T) hadronic collisions. Earlier work based on the study of pp inelastic diffraction led to the conclusion that the glue-glue elastic amplitude t g is approximately Gaussian in the glue-glue impact parameter b g, with maximum opacity at b g = 0. A slight refinement of this approximation, in which t g is almost maximally opaque over a small b g interval and is Gaussian at larger b g, is now shown to account for the remarkable diffraction structure observed in pp elastic scattering at ISR energies. A very simple form is obtained for the form factor describing the impact parameter distribution of the glue inside a proton relative to the proton itself.

A simple geometrical approach to particle production in collisions with nuclei

It is argued that hadron collisions with nuclei are similar to hadron-hadron collisions, having similar properties for the impact parameter distributions and the leading particle spectra. The relevant existing high-energy data, including the universality of multiplicity distributions and the possibility of geometrical scaling in reactions with nuclei, are easily understood in the framework of geometrical models by extending to p-nucleus collisions what was learnt about impact parameter and leading particles in pp collisions. The questions of forward-backward correlations and photo- and electroproduction are briefly discussed.

Non-linear dependence of density of elastic stopping power

In the conventional way of calculating the energy loss of energetic particles in solid material, the target is treated as an extremely dense gas. This implies that for every collision the impact parameter distribution is given by πp 2. However, if the effective range of the interaction potential between a projectile and the target atoms is of the same order of magnitude as the average interatomic distance, this description fails, because it overestimates the influence of large impact parameters and thus overestimates the elastic stopping power. This overestimate is corrected if we describe the spatial distribution of the target atoms by the Poisson law and consider only successive interactions with target atoms in the forward directed half space, which have the smallest distance to the projectile. This results in an impact parameter distribution which favors small impact parameters and is negligibly small for impact parameters larger than the average interatomic distance, An expression for the elastic stopping power, based on this distribution is derived. Calculations for the elastic stopping power of noble-gas ions in copper are presented in the energy range of 0.1–100 keV.

A comparison of two‐dimensional and three‐dimensional trajectory calculations in thermal HBr2 and H2Br systems

AbstractTwo‐dimensional (2D) and three‐dimensional (3D) quasiclassical trajectory calculations on H + Br2 at 300°K and H + HBr at 1000°K are reported. Angular scattering, energy disposal, and impact parameter distributions for reactive collisions are compared after removal of phase‐space factors (dimensionality bias) as a means of examining the similarities and differences in the dynamic bias in 2D and 3D. Qualitatively, for all reactive processes studied, the 3D trajectory calculated distributions are reproduced by the phase‐space adjusted 2D trajectory data. Thus the surprisal of these angular scattering, energy disposal, and impact parameter distributions is dimensionally invariant, and the same dynamic bias appears in 2D and 3D. A systematic method for converting 2D reaction probabilities and maximum reactive impact parameters into 3D rate coefficients is presented. We find that trajectory calculated 3D rate coefficients may in general differ markedly from those derived from 2D trajectory data. In particular, the surprisal associated with rate coefficients depends on dimensionality for the H + HBr → H2 + Br reaction, but is invariant for the H′ + HBr → H′Br + Br and H + Br2 → HBr + Br reactions.

The impact parameter structure of multiparticle production

High-energy exclusive and inclusive cross sections are discussed in terms of their dependence on the transverse-position variables (impact parameters) of incident and outgoing particles. The objective is to clarify the points of conflict and agreement of various models with each other and with intuitive ideas based on macroscopic collisions. First the impact parameter representation of states and amplitudes is reviewed. New impact parameter conservation laws and sum rules are derived from Lorentz invariance. The generalized optical theorem of Mueller is extended to give the impact parameter distribution of produced particles. Mueller-Regge behaviour is shown to imply three-dimensional limiting fragmentation in (impact parameter, rapidity)- space and a specific linear structure of the final-state particle density in the central region of this space. The predictions of weak- and strong-coupling multiperipheral models, the φ 3 ladder eikonal model and the dual resonance model are presented, with emphasis on the dependence of average multiplicity on impact parameter and the dependence of mean square impact parameter on multiplicity. Special techniques for strong-coupling multiperipheral models are used to study the breakdown of the random-walk impact-parameter structure of the weak-coupling case, and to show explicitly that the average multiplicity is a decreasing function of impact parameter in the strongly coupled ABFST model.

On the nature of absorption

The imaginary helicity non-flip ω exchange amplitudes in KN and NN scattering are determined versus t and s from data on elastic differential cross section differences. The ω-universality prediction Im ω N ( t) = 3 Im ω K ( t), is found to be satisfied for 0 ⩽ | t| ⩽ 0.5, indicating that the absorptive cuts at small t preserve universality: The impact parameter distributions of the ω amplitudes show: 1. (i) a peripheral peak centered around b ≈ 1 fm, with Δb ≈ ± 0.5 fm in KN and Δb ≈ ± 0.75 fm in NN; 2. (ii) a sizeable negative contribution at small b, more negative in NN than in KN, 3. (iii) a smaller tail of negative contributions around b ≈ 2 fm. Strong absorption of the ω-pole imaginary amplitude with λ values around two is needed to reproduce this structure. A new version of the absorption model, suggested by πN amplitude analysis, is confirmed by our results.

Nonresonant Vibration-to-Vibration Energy Transfer Due to Dipole–Dipole Interactions

A simple model for nonresonant vibration-to-vibration energy transfer due to dipole–dipole interactions is used to yield analytic expressions for the energy-transfer probability as a function of impact parameter, relative velocity, and deviation from resonance. Where the deviation from resonance is sufficiently large, the probabilities may be integrated over the impact parameter distribution, and in some cases over the velocity distribution, to obtain cross sections for V → V energy transfer.

Kinetic Emission of Electrons from Monocrystalline Targets

This paper develops a single-collision theory of the emission of electrons from a metal surface subjected to high-energy ion bombardment, which gives reasonable agreement with the data of Carlston, Magnuson, Mahadevan, and Harrison, in the 1 to 10-keV energy range. The model is based upon a Thomas-Fermi-Firsov energy-transfer calculation which has been modified to include an explicit dependence upon lattice orientation. Orientation effects appear naturally, and orientation-dependent cross sections are not required, but the distribution of possible impact parameters for a particular crystal orientation is of central importance. The theory has been used to determine semiempirical interaction potentials between the moving particle and a lattice atom. These potentials are more similar to Abrahamson's atom-atom potentials than to the Gibson potentials used in radiation-damage studies.