Classical Deflection Function Research Articles

Orbiting phenomena in black hole scattering

Rainbow, glory and orbiting scattering are usually described by the properties of the classical deflection function related to the real part of the quantum mechanical scattering phase shift or by the diffractive pattern of the quantum mechanical cross sections. Here we show that the case of orbiting scattering of massless spin 0, 1 and 2 particles from Schwarzschild black holes can be characterized by a sudden rise in $|R_l|^2$ at a critical angular momentum $l_C$, which we show corresponds to the unstable circular orbits of these particles. For the cases, $s =0, 2$, we attempt a new interpretation of the Regge-Wheeler potential by identifying the quantum mechanical corrections to the effective potential of massless particles. We probe into the black hole scattering by using numerical and semi-analytical methods which give very good agreements with the exact numerical results. The limitations of previously used approximations as compared to the exact and semi-analytical results are discussed.

Kicking electrons

The concept of dominant interaction Hamiltonians is introduced. It is applied as a test case to classical planar electron–atom scattering. Each trajectory is governed in different time intervals by two variants of a separable approximate Hamiltonian. Switching between them results in exchange of energy between the two electrons. A second mechanism condenses the electron–electron interaction to instants in time and leads to an exchange of energy and angular momentum among the two electrons in the form of kicks. We calculate the approximate and full classical deflection functions and show that the latter can be interpreted in terms of the switching sequences of the approximate one. Finally, we demonstrate that the quantum results agree better with the approximate classical dynamical results than with the full ones.

The forward glory effect in the differential cross sections measured below Elab = 50 eV for the one-electron capture process in N5+–He collisions

A crossed-beam apparatus was used to measure the relative state-selective differential cross sections for the one-electron capture process in the N5+–He system from θlab = −3° to 21° at Elab = 27.5 and 47.5 eV. The main reaction channel was found to be N5+ (1s2 1S) + He → N4+ (1s23s 2S) + He+ + 16.8 eV. The differential cross section for this channel is pronounced at θcm = 0, and it shows another peak at a certain angle that depends on the collision energy. The O5+–He system at Elab = 47.5 eV was measured for comparison. The differential cross section for the main reaction channel O5+ (1s22s 2S) + He → O4+ (1s22s3p 1,3P) + He+ + 17 eV is zero at θcm = 0, and it shows only a single peak at θcm = 0.314 rad. The structures observed in the N5+–He system were analysed using the classical deflection function based on ab initio potentials. The peak observed at θcm = 0 is assigned to the forward glory effect and the second peak to the inelastic rainbow effect.

10.1007/s11448-008-2005-5

The quasi-channeling of positive and negative relativistic particles in a bent crystal is studied using the classical deflection function. It was shown that the potential scattering in a central field of bounded ring-like potentials may produce the ``reflected'' and ``refracted'' fractions of scattered particles. For particles with positive charge the ``reflected'' fraction is mainly presented; at the same time we predict that for particles with negative charge the ``refracted'' fraction should dominate. The effect of ``empty core'' for central scattering is also discussed. The average deflection angles for volume ``reflection'' and ``refraction'' are derived for accepted potential model of the crystal. The calculated average ``reflection'' angle is in satisfactory agreement with recent experimental data \cite{ivanov_2006}

Angular Dispersion and Deflection Function for Heavy Ion Elastic Scattering

The differential cross sections for elastic scattering products of 17F on 208Pb have been measured. The angular dispersion plots of ln(dσ/dθ) versus θ2 are obtained from the angular distribution of the elastic scattering differential cross sections. Systematical analysis on the angular dispersion for the available experimental data indicates that there is an angular dispersion turning angle at forward angular range within the grazing angle. This turning angle can be clarified as nuclear rainbow in classical deflection function. The exotic behaviour of the nuclear rainbow angle offers a new probe to investigate the halo and skin phenomena.

Analysis of 16O + 28Si elastic scattering in the laboratory energy range 50.0 MeV to 142.5 MeV

We have successfully carried out an optical model analysis of 16O + 28Si differential scattering cross section σ(θ) in the laboratory energy range EL from 50.0 MeV to 142.5 MeV and angular range θ from 0° to 100° using the LC potential suggested by Lee and Chan. This potential is very successful in explaining the differential cross section σ(θ) at EL = 50.0 and 55.0 MeV in the entire centre of mass angle range reproducing the enhanced back angle oscillations correctly. The LC potential has only seven parameters, including the Coulomb radius parameter, which is much smaller than the number of parameters used in recent calculations on the system. In the light of an excellent fit to the data, we make a detailed analysis of different features of the LC potential and compare it with the typical heavy-ion potential having a Woods–Saxon (WS) form factor used in the analysis of 18O + 58Ni at EL = 60.0 MeV. These features include the behaviour of the real effective potential around grazing partial wave (ℓg), angular momentum (ℓ) dependence of classical deflection function (Θ), reflection function |Sℓ| and the term |1 − Sℓ|2 governing the total partial wave elastic cross section. We also calculate the barrier region resonance positions generated by the partial waves around ℓg. We find that the orbiting phenomenon generated by the flatter, surface transparent effective potential and the resulting coherent behaviour of only a small number of partial waves in the surface region, resulting in the interference of absorptive region and Coulombic region amplitudes, are primarily responsible for back angle oscillations. As a quantum-mechanical consequence of orbiting behaviour, we find close clustering of resonances corresponding to a set of partial waves around ℓg which is the special feature of the LC potential. On the basis of this, we envisage that 16O + 28Si is likely to show large enhanced back angle oscillations at EL = 66.0 and 72.0 MeV. It will be interesting if this is experimentally investigated.

Intermolecular motion in strong infrared laser fields

Abstract The effect of a strong infrared laser field on the collision between two rare gas atoms is examined as a model of intermolecular motion under such conditions. After examination of the classical collision dynamics, three novel classes of collisions are identified, all of which result from interaction with the angular potential well. The signatures of these “collision mechanisms” in the classical deflection function and differential cross-section are determined. The generality of the results and the feasibility of using a modified crossed molecular beam experiment to observe these laser-induced effects are discussed.

COULOMB-MODIFIED EIKONAL PHASE SHIFT ANALYSIS BASED ON HYPERBOLIC TRAJECTORY FOR 12C + 12C ELASTIC SCATTERINGS

We present a Coulomb-modified eikonal model formalism based on hyperbolic trajectory for heavy-ion elastic scattering. This formalism has been applied satisfactorily to elastic scatterings of the 12 C + 12 C system at E lab =240, 360 and 1016 MeV. The presence of a nuclear rainbow in this system is evidenced through a classical deflection function. The Fraunhöfer oscillations observed in the elastic angular distributions can be explained due to interference between the near- and far-side amplitudes. We have found that the hyperbolic trajectory effect on the eikonal model is important when the absorptive potential is weak and the real potential is strong.

SECOND-ORDER EIKONAL MODEL ANALYSIS OF16O+16OELASTIC SCATTERING

We analyze the elastic scattering angular distributions of the16O +16O system at Elab=480 MeV and 704 MeV within the framework of the second-order eikonal model based on Coulomb trajectories of colliding nuclei. The diffractive oscillatory structure observed in the elastic angular distribution could be explained due to the interference between the near- and far-side scattering amplitudes. The presence of a nuclear rainbow in this system is evidenced through a classical deflection function. The effective optical potential is developed from the second-order non-eikonal phase shifts.

Investigations of the Scattering of UF5+ Ions by N2 Molecules

The orientation-averaged classical deflection function for scattering of UF5+ ions by N2 molecules is calculated in the collision energy range 2–10 keV, using additive interatomic potentials to calculate the anisotropic interaction of the molecular particles. The elastic scattering cross sections are calculated for ions and the interaction potential of the colliding particles is reconstructed using the Firsov method based on classical mechanics.

Semiclassical Description of Scattering

The quantum-mechanical scattering amplitude can be simply related to the classical deflection function when the conditions for a semiclassical analysis of the quantum-mechanical scattering are met. Various interesting characteristic features of the scattering are related to special features of the classical deflection function. The characteristic types of scattering discussed are: interference, when the deflection function possesses more than one branch at a given angle; rainbow scattering, when the deflection function has a relative maximum or minimum; glory scattering, when the deflection function passes smoothly through 0° or through an integral multiple of ±π; and orbiting, when the deflection function possesses a singularity. The consideration of the characteristic features of semiclassical scattering makes possible the analysis of an observed differential cross section to yield the classical deflection function, which in turn may be used to construct the potential.

NON-EIKONAL PHASE SHIFT ANALYSIS FOR 12C + 12C ELASTIC SCATTERING

We present first-order non-eikonal correction to the eikonal phase shifts for heavy ion elastic scattering based on Coulomb trajectories of colliding nuclei. It has been applied satisfactorily to elastic angular distributions of the 12 C + 12 C system at E lab = 240, 360 and 1016 MeV. The refractive oscillations observed in the elastic scattering angular distributions could be explained due to interference between the near- and far-side amplitudes. The presence of a nuclear rainbow is evidenced through classical deflection function. We have found that the first-order non-eikonal effect on the imaginary potential is important when the absorptive potential is weak and the real potential is strong.

Quantum-mechanical deflection function

The classical deflection function for scattering by a radially symmetric potential corresponds to twice the angular momentum derivative of the scattering phase shifts calculated in the conventional WKB approximation. A ``quantum-mechanical deflection function'' defined as twice the angular momentum derivative of the quantum-mechanical phase shifts (without WKB approximation) is able to incorporate into the simple trajectory picture certain quantum aspects of the scattering process, e.g., the influence of the Goos-H\"anchen shift. We demonstrate the performance of this quantum-mechanical deflection function for sharp-edged and smooth-edged two-dimensional scattering potentials.

Angular Distributions of Products in the Photodissociation of Diatomic Molecules

The issue of angular distributions of the products of photodissociation of diatomic molecules is reexamined on two aspects: (1) the nature of the state that is prepared by excitation from a bound level into a continuum, and (2) a demonstration that a semiclassical connection between the J-derivative of the scattering phase shift and the classical deflection function leads to a recovery of the classical expression for correction of the angular distribution owing to finite rotation of the molecule during photodissociation. Higher-order quantum contributions to this correction also result from these observations.

General Rydberg Atoms in Crossed Electric and Magnetic Fields: Calculation of Semiclassical Recurrence Spectra with Special Emphasis on Core Effects

The photoabsorption spectra of highly excited Rydberg atoms in external fields are known to be related to closed classical orbits of the excited electron. For atoms in a pure magnetic field, scaled recurrence spectra have successfully been calculated from classical orbits using the semiclassical closed-orbit theory. Rydberg atoms in crossed fields are more complicated as regards their theoretical and numerical treatment, as the Hamiltonian is nonseparable in three degrees of freedom, in contrast to only two nonseparable degrees of freedom in a pure magnetic field. In this paper, we present an extension of the closed-orbit theory to three degrees of freedom, taking into account arbitrary quantum defects. Motivated by nonhydrogenic resonances discovered in experimental scaled recurrence spectra of rubidium atoms, we investigate the influence of the ionic core on the three-dimensional closed orbits, using a simple model potential. We find that the introduction of a core potential results in an extreme increase of the number of closed orbits as compared to hydrogen. The novel orbits appear to be composed of hydrogenic orbits and are created through scattering by the core potential. Investigating the classical deflection function of the model potential, general properties of the new orbits can be explained. With the closed-orbit theory extended to three nonseparable degrees of freedom, we are able to calculate scaled recurrence spectra for Rydberg atoms in crossed fields with arbitrary quantum defects. Our results are in good agreement with the experimental spectra. In particular, the nonhydrogenic resonances can be explained in terms of the new orbits created by classical core scattering.

Semiclassical S-matrix theory for atomic fragmentation

A semiclassical scattering approach is developed which can handle long-range (Coulomb) forces without the knowledge of the asymptotic wave function for multiple charged fragments in the continuum. The classical cross section for potential and inelastic scattering including fragmentation (ionization) is derived from first principles in a form which allows for a simple extension to semiclassical scattering amplitudes as a sum over classical orbits and their associated actions. The object of primary importance is the classical deflection function which can show regular and chaotic behavior. Applications to electron impact ionization of hydrogen and electron–atom scattering in general are discussed in a reduced phase space, motivated by partial fixed points of the respective scattering systems. Special emphasis, also in connection with chaotic scattering, is put on threshold ionization. Finally, motivated by the reflection principle for molecules, a semiclassical hybrid approach is introduced for photoabsorption cross sections of atoms where the time-dependent propagator is approximated semiclassically in a short-time limit with the Baker–Hausdorff formula. Applications to one- and two-electron atoms are followed by a presentation of double photoionization of helium, treated in combination with the semiclassical S-matrix for scattering.

Fusion and transfer reactions in the 19F+165Ho system at energies near the Coulomb barrier.

Results of the measurements of fusion cross sections and average angular momenta in the {sup 19}F + {sup 165}Ho system, for center-of-mass energies ranging from 7 MeV below to 13 MeV above the Coulomb barrier are presented. The average angular momenta of the compound system were obtained using three independent methods, namely, the {gamma}-ray multiplicity, the ratios of the evaporation residues, and the fusion excitation function. The transfer probabilities for various one- and two-particle channels have also been measured at a beam energy of 110 MeV, an energy much higher than the Coulomb barrier. The measured {ital Q}-integrated one- and two-proton transfer probabilities have been analyzed in terms of an earlier suggested semiclassical calculation taking into account the nuclear branch in addition to the usual Coulomb branch of the classical deflection function. It is seen that the {open_quote}{open_quote}slope anomaly{close_quote}{close_quote} in the measured one- and two-proton transfer probabilities as a function of the distance of closest approach can be explained within this framework. The strengths of the form factors, required in a coupled channel calculation of fusion, for the important transfer channels were obtained from the data within such a semiclassical analysis. The observed enhancement of the fusion cross sections andmore » average angular momenta is discussed in terms of the coupled channel model using static deformation for the target, inelastic excitations in the projectile, and couplings to the transfer channels. The data are also compared with the neutron flow model suggested by Stelson. {copyright} {ital 1996 The American Physical Society.}« less

The Runge–Lenz vector and perturbed Rutherford scattering

The scattering of a particle in a perturbed Coulomb field is studied by means of the Runge–Lenz vector. We show that this leads to a simple method for calculating the classical deflection function, valid for quite general perturbing forces. With the same techniques we obtain the relativistic correction of order c−2 to Rutherford’s scattering formula.

Semiclassical description of one- and two-nucleon transder reactions at large distances in heavy ion collisions

It is shown that the variation of the one- and two-nucleon transfer probability with distance can be understood on the basis of semiclassical models if the contributions from the nuclear and Coulomb branches of the classical deflection function are considered. A shallow potential, with a diffuseness parameter a = 0.8–0.9 fm for the real part of the heavy ion optical potential, was found necessary to understand the data.

Angular distribution of ions axially channeled in a very thin crystal: Experimental and theoretical results.

We have studied the angular distributions for 6--30-MeV C[sup [ital q]+] ([ital q]=4--6) and 2--9-MeV H[sup +] ions axially channeled in the [001] direction of a thin silicon crystal (1792 and 1900 A). We report highly structured two-dimensional angular distributions that depend sensitively on the projectile's velocity and incident charge state and the target's thickness and azimuthal orientation. Some structure in the angular contour maps is the result of a rainbow effect in axial channeling (i.e., extrema in the classical deflection function). State-to-state charge-state distributions, which are required to interpret the data accurately, have also been measured. All measured angular distributions have been explained via Monte Carlo trajectory calculations using Moliere's approximation to the Thomas-Fermi screening function and a screening length given by target electrons alone. The calculations indicate that all projectile velocity and charge-state effects and the target-thickness effects observed are the result of the projectile's transverse oscillatory motion in the channel. Using this information, we show that swift heavy-ion and proton angular distributions are simply related using a scaling law that depends only on the projectile's velocity and charge-to-mass ratio and on the crystal thickness.