Eikonal expansion

An eikonal expansion of the potential scattering T matrix is evaluated, without approximation, through third order in the inverse momentum. Based on the results, their correspondence with the WKB approximation and a new statement of the unitarity constraint, we propose a sequence of four approximations to the exact impact parameter (Fourier-Bessel) representation of the scattering matrix. The sequence consists of the Glauber approximation and three systematic corrections to the Glauber approximation. The corrections are analytic functions of the impact parameter for Yukawa and Gaussian potentials; they vanish for a Coulomb potential.The sequence of eikonal amplitudes is convergent at high energy and is clearly established for small momentum transfer. Validity for all momentum transfer is conjectured based on systematic cancellation, explicitly verified through third order in the expansion, of momentum transfer dependence in the eikonal impact parameter representation. Such cancellation is shown to occur in the explicit construction of the eikonal expansion of the second Born amplitude for a Yukawa potential.Numerical tests of the sequence of eikonal amplitudes show systematic increase of the angular range of validity by comparison with partial wave results for continuous potentials; the theory is not convergent for discontinuous potentials.The WKB phase shift formula is shown to produce a systematic connection with eikonal expansion results. From this we deduce a generating function for the eikonal phase corrections of arbitrary order and also conjecture a sum of the eikonal expansion valid in the limit of high energy and arbitrary potential strength.     

Inelastic atom-surface scattering: A comparison of classical and quantum treatments

The validity of the classical approximation is investigated for the calculation of the energy transfer and sticking coefficient of particles scattered by harmonic solids. If in zeroth order the static approximation for the force between particle and solid is valid, the energy transfer can be calculated classically, if the WKB approximation for the matrix element of the force holds. This is the case except at low impact energies. The characteristic energy below which quantum effects become important depends strongly on the long range behavior of the force.For the sticking coefficient one has to distinguish quantum effects of particle and solid. Quantum effects for the particle can again be neglected if the WKB approximation for the matrix elements is valid. The solid, however, can be treated classically only, if the number of excited phonons is large compared to unity. This requires energy transfers large compared to the Debye energy of the solid.Extract from doctoral thesis of R.S. submitted to Fakultät für Physik, Techn. Univ. München, 1979. Work supported in part by DFG, Sonderforschungsbereich 128     

Channel coupling and exchange of an alpha-particle cluster in deuteron scattering on 6Li nuclei

Existing experimental data on elastic and inelastic deuteron scattering on ⁶Li nuclei in the energy range from 8 to 50 MeV were analyzed within the approach of coupled reaction channels. The coupling of elastic scattering and inelastic scattering accompanied by the transition to the 3⁺ state at E _x = 2.186 MeV and the mechanism involving the exchange of an alpha-particle cluster were taken into account in respective calculations. The phenomenological potentials obtained from the present analysis describe well experimental angular distributions at all energies and in full angular ranges. The depths of the real and imaginary parts of the potentials in question depend smoothly on energy at fixed values of the remaining parameters. The energy dependence of relevant volume integrals agrees well with similar data for the p + ⁶Li, ?? + ⁶Li, and ¹²C + ¹²C systems and with the predictions of a microscopic theory.     

Sticking and displacement channels in the COg + O2/Pt(111) reaction

A molecular beam of CO, impinging on a Ft surface saturated with molecular oxygen, causes displacement of O₂ molecules into the gas phase. The kinetics of the displacement and associated CO sticking have been measured for CO kinetic energies in the range 0.06-1.83 eV. At low kinetic energies the main displacement channel is associated with the sticking of CO, which by dynamic energy and momentum transfer causes O₂ molecules to leave the surface, with a probability of 0.09 per stuck CO molecule. At the highest CO kinetic energies an additional displacement channel is appearing, namely inelastic (non-sticking) scattering of CO molecules, which deposit enough energy to displace adsorbed O₂ into the gas phase.     

On the low energy limit of reflection and sticking coefficients in atom surface scattering. I. Short range forces

The elastic scattering of atoms and molecules by solid surfaces is treated in a nonperturbative way using the single particle Green's function of the scattered particles. For sufficiently short ranged interactions the elastic reflection coefficient is found to approach unity in the limit of zero energy of the incident particles. Unitarity then implies zero total reaction coefficient including the sticking coefficient. The findings of the first order distorted wave Born approximation are thus generalized. The result is not in contradiction to the experimental observation of sticking coefficients increasing with decreasing energy, since quantitative approximate calculations show that the limit is reached only at extremely low energies.     

Folding model potentials from realistic interactions for heavy-ion scattering

The double-folding model, with “realistic” nucleon-nucleon interactions based upon a G-matrix constructed from the Reid potential, is used to calculate the real part of the optical potential for heavy-ion scattering. The resulting potentials are shown to reproduce the observed elastic scattering for a large number of systems with bombarding energies from 5 to 20 MeV per nucleon. Some representative inelastic transitions are also reproduced. Exceptions are the elastic scattering of ⁶Li and ⁹Be for which the folded potentials must be reduced in strength by a factor of about two.The same effective interactions are shown to give a good account of two particular cases of alpha scattering as well as some cases of nucleon-nucleus scattering. Some typical examples of inelastic heavy-ion scattering are also predicted successfully.Some general properties of the folding model are reviewed and its theoretical basis is discussed. An explicit density-dependence is examined for one particular realistic interaction and found not to change the results. Single nucleon exchange is included in an approximate way and its importance is studied.In addition to being a study of the folding model, this work also provides a systematic and comprehensive optical model analysis of heavy-ion elastic scattering in this energy range.     

The 12C+24Mg elastic scattering: An example of anomalous transparency at coulomb barrier energies

Fifteen complete angular distributions of the elastic scattering of ¹²C+²⁴Mg were measured at energies around the Coulomb barrier (E_cm = 10.67–16 MeV). The angular distributions are strongly oscillating and could be well described by an optical potential family, whose real part was determined without continuous ambiguity. The imaginary part of this optical potential is very shallow. At four energies the inelastic scattering angular distributions leading to the 2⁺ state of the ²⁴Mg were also measured and analysed with coupled-channels calculations. The volume integrals of the optical potentials used in the coupled-channels calculations present the threshold anomaly in their energy dependence, with a clear Q-value dependence.     

Analisis of inelastic scattering of π-mesons from nuclei within the microscopic optical potential

Cross sections of inelastic scattering of π-mesons from Si, Ni, and Pb nuclei at energy T _lab = 291 MeV are calculated using the distorted wave approximation. The microscopic direct and transition optical potentials are determined by specifying the pion-nucleon scattering amplitude and the nuclear density distribution, where we use the in-medium πN amplitude parameters obtained earlier by analyzing the elastic scattering data for the same nuclei. The cross sections are calculated on the basis of the relativistic wave equation. The deformation parameters of the nuclei are obtained by comparing inelastic scattering cross sections with the appropriate experimental data.     

Inelastic proton scattering from deformed nuclei: (II). Results with a local approximation

We describe an approximate method of including exchange effects in proton inelastic scattering appropriate to situations where a coupled channel approach is essential, i.e. deformed nuclei. We demonstrate that in this approach the effects of exchange are not large, contrary to what is often said, provided consistence between elastic and inelastic scattering is maintained. The corrections may be roughly represented as a “shifted” zero range approximation which can be viewed as giving a rather smallL-dependent effect. The effect on the extraction of quadrupole moments should not exceed 4% and of hexadecapole moments 10% except at the lowest energies.     

Multi-step processes in high-energy elastic and inelastic nuclear scattering

Multi-step processes in elastic and inelastic nuclear scattering at intermediate and high energies are investigated using a formulation whereby a finite number of channels are explicitly treated while all the other channels are approximately accounted for through a “second-order potential matrix”. Within the framework of the eikonal approximation the problem reduces to a finite system of first-order coupled integro-differential equations with non-local potentials which depend on the two-body density matrix of the target nucleus. The relationship of the above formulation to the DWIA, the close-coupling method, and the Glauber multiple scattering model is examined. This approach is applied to the elastic and inelastic (2⁺, 4.43 MeV) scattering of 1 GeV nucleons by ¹²C. The corrections to the DWIA are sizeable, and the inelastic scattering appears to be very sensitive to the multi-step contributions and the nuclear structure.     

Temperature effect for an inelastic neutrino scattering cross section

The influence of temperature on inelastic neutrino scattering on hot nuclei is studied with the ⁵⁴Fe nucleus as an example. The strength distribution of charge-neutral Gamov-Teller transitions in ⁵⁴Fe at finite temperature is calculated within the framework of a random phase approximation using the thermo field dynamics formalism. It is shown that for neutrino energies lower than the energy of the Gamov-Teller resonance, the inelastic scattering cross section depends substantially on temperature.     

Scattering of <Emphasis Type="Italic">α</Emphasis> particles on <Superscript>11</Superscript>B nuclei at energies 40 and 50 MeV

The differential cross sections for elastic and inelastic scattering of α particles on ¹¹B nuclei at energies of 40 and 50 MeV were measured in the entire angular range. The measured angular distributions were analyzed in terms of the optical model, the distorted-wave Born approximation, and the coupled-channel-method. Optical model potentials and quadrupole (β₂) and hexadecapole (β₄) deformation parameters were found from this analysis. The rise in the cross sections at backward angles was shown to be associated with the transfer mechanism of the heavy ⁷Li cluster.     

Elastic and inelastic scattering of nucleons from 16O

Measurements of differential elastic and inelastic cross sections for neutron scattering from ¹⁶O at incident energies 18 to 26 MeV are presented. In addition to cross sections for neutron scattering differential cross sections for proton scattering up to 66 MeV are described in terms of phenomenological optical model potentials. At 24.5 MeV incident energy inelastic scattering up to 11.5 MeV excitation was measured. The elastic and inelastic compound nucleus contributions were examined. Direct inelastic scattering from the normal parity states was calculated using the DWBA and coupled-channel formalisms. The inelastic scattering cross section from non-normal parity state 2⁻ was calculated using the coupled-channel formalism via multi-step processes. Cross sections due to inelastic scattering from some of the states, which are thought to be members of an excited state rotational band were calculated using both vibrational and rotational approaches and were compared.     

Antiproton-nucleus scattering with self-consistent model for medium corrections

The antiproton-nucleon t-matrix with propagation in the nuclear medium is calculated selfconsistently and applied to the construction of optical potentials for the elastic scattering of antiprotons from nuclei. We find that this treatment gives results that are sensitive to medium corrections even though the strong absorption acts to mask these corrections partially. The agreement with scattering experiments at 46.8 MeV on ¹²C is very good. We compare potentials containing medium corrections to those based on free $\text{p}\text{N}$ amplitudes for ¹²C and ⁴⁰Ca. The local approximation to the optical potential is found attractive at low energies, becoming shallower with increasing bombardment energy in the range considered here (up to about E_L = 120 MeV).     

Global optical model potentials for symmetrical lithium systems: 6Li+6Li, 7Li+7Li at Elab = 5–40 MeV

Angular distributions of ⁶Li+⁶Li elastic scattering were measured for E_lab = 5–40 MeV. An optical model analysis of these data together with older data of ⁷Li+⁷Li elastic scattering taken at E_lab = 8–17 MeV was performed with the aim to search for a “global” OM potential which describes elastic scattering in both LiLi systems in a broad energy range. Both surface and volume absorbing potentials can be found which fulfill this requirement if a linear energy dependence is assumed of the depths of the real as well as the imaginary potential. These depths, if fitted to individual angular distributions, are found to vary in a correlated manner with the beam energy. This is taken as indication of strong coupling between elastic, inelastic, and reaction channels. This is corroborated by the existence of resonances in reaction channels at these energies where the potential depths are most pronouncedly changing.     

On the reconstruction of an optical potential on the basis of experimental data in the WKB approximation

The inverse scattering problem at a fixed energy for a complex-valued potential is solved in the WKB approximation. The method is used to reconstruct the optical potential for elastic ¹⁶O + ¹⁶O scattering at E _lab = 350 MeV. The stability of the solution against small changes in the scattering matrix is studied.     

Coupling to the elastic channel in electron impact spectroscopy: A simple second born model and its application to excitation of the 61P1 state of mercury and to the excitation of molecules

The matrix element in the infinite channel close coupling approximation responsible for coupling to the elastic channel in electron impact inelastic encounters is investigated. The contribution from the imaginary part of the energy denominator in the elastic coupling matrix element for dipole allowed transitions is shown to yield large angle differential cross sections in good agreement with experiment. This coupling mechanism predicts that the shape of the inelastic differential cross section will be dominated by the shape of the elastic cross section in the large angle high energy limit. In fact the coupling matrix element exhibits a dependence on incident energy, k², and momentum transfer, K, of the form 1/kK² which is in agreement with the theoretical predictions of Huo and means that in the limit of high incident energy the non-first-Born elastic coupling will dominate the angular dependence of the inelastic differential cross section at large scattering angles. In the case of molecular electron impact spectra it is shown that the analog of the Massey—Moore coherence features depending on the symmetry of the states involved in the excitation process will also occur in the coupling contribution. It is suggested that this new mechanism for producing coherent features in inelastic differential cross sections may be the explanation of the coherent features observed experimentally by Karle and Swick.It can be concluded on the basis of the results obtained here that the coupling to the elastic cross section provided by the imaginary contribution from the second Born energy denominator is sufficient to explain presently available experimental data on the large angle differential cross section and spin polarization. The simple coupling model was found to be inadequate to explain the small angle differential cross section in the range 10° < θ < 30° even at incident energies as high as 400 eV. The calculations also showed significant differences between first and second Born calculations at zero scattering angle. No conclusion can be drawn about this observation as all the omitted terms should make significant contributions in the small angle region. It is important to again emphasize that the large angle scattering even in the limit of high incident electron energy will be completely dominated by the coupling to the elastic channel^{7, 11}. On the basis of this work it appears that the coherent structure in the large angle inelastic differential cross section observed by Swick and Karle^{12, 13} at incident electron energies in the keV region may well be due to coupling to the elastic channel.     

On the approximate solution through continued fractions of the schrödinger equation with central potentials for positive energies

We present an extension to positive energies of an approximation method based on continued fractions already used to provide approximate energies and wave functions for bound states. A formalism particularly adapted for numerical computation is proposed and its applications are shown for the free particle case and short and long range central potentials, for the lowest angular momenta. The domain of convergence is displayed; especially for low and medium energies, the results are quite good. Some care is devoted to the successful determination of theS-wave scattering length for the Yukawa potential. The mathematical relationship with Calogero's formulation of the problem is established.