Eikonal description of quantal scattering

Elastic and inelastic quantal scattering is described by a theory in which the contribution of a range of impact parameters to the scattering amplitude is determined by a phase integral (“eikonal”) which is integrated along a real curved “quantal” trajectory. This amplitude reduces to the Glauber expression in the high-energy, forward-angle limit, and to the usual semiclassical amplitude in the classical limit. The formulation can be applied to the study of heavy-ion scattering. The quantal trajectories are investigated analytically for the case of Coulomb scattering. A numerical analysis of elastic ¹⁶O¹⁶O scattering is carried out. The results show appreciable improvement as compared with the Glauber approximation.     

The Wigner function for two dimensional tori: Uniform approximation and projections

The Wigner representation of a quantum state, corresponding to a classically integrable Hamiltonian, has been shown to be intimately tied to a classical phase space torus of the same energy. The fact that the semiclassical approximation of the Wigner function there derived turns out to be singular on the torus, as well as on the “Wigner caustic” which contains it, is due to well known limitations of the stationary phase method. The uniform approximation, here derived, does indeed ascribe to the Wigner function a high amplitude along the Wigner caustic, but this is modulated by rapid oscillations except at the torus itself. Asymptotic expansion away from the torus leads back to the semiclassical approximation. Close to the torus the Wigner function is described by a simple transitional approximation which can be resolved into a product of Wigner functions corresponding to one dimensional tori. These results permit one to explicitly project the Wigner function onto any (Lagrangian) coordinate plane so as to obtain the corresponding wave intensity.     

Quantenfeldtheorie wechselwirkender Vielteilchensysteme zur semiklassischen Beschreibung von kollektiver Bewegung mit großen Amplituden

By using path integral methods a collective quantum field theory of interacting many-body systems is developed, the classical limit of which is given by the time-dependent mean-field approximation. In this way the mean-field approximation is embedded into the full quantum mechanics and the quantum corrections to the “classical” mean-field approximation can be systematically evaluated. By including the dominant quantum corrections to the mean-field approximation a semiclassical theory of large amplitude collective motions in many-body-systems, which show a highly nonlinear dynamic and are not accessible to perturbation theoretical methods, is derived. The semiclassical theory is developed explicitly for bound states and decay processes like nuclear fission. In the case of bound states this leads to the quantization of the time-dependent Hartree-Fock-Theory, which is demonstrated for a uniform nuclear rotation.     

Semiclassical scattering theory with complex trajectories. I. Elastic waves

The WKB approximation to the one-particle Schrödinger equation is used to obtain the wave function at a given point as a sum of semiclassical terms, each of them corresponding to a different classical trajectory ending up at the same point. Besides the usual, real trajectories, also possible complex solutions of the classical equations of motion are considered. The simplicity of the method makes its use easy in practical cases and allows realistic calculations. The general solution of the one-dimensional WKB equations for an arbitrary number of complex turning points is given, and the solution is applied to calculate the position of the Regge poles of the scattering amplitude. The solution of the WKB equations in three dimensions for a central analytical potential is also obtained in a way that can be easily generalized to N-dimensions, provided the problem is separable. A multiple reflection series is derived, leading to a separation of the scattering amplitude into a smooth “background” term (single reflection approximation) that can be treated using classical but complex trajectories and a second resonating term that can be treated using the Sommerfeld-Watson transformation. The physical interpretation of the complex solutions of the classical equations of motion is given: they describe diffractive effects such as Fresnel, Fraunhofer diffraction, or the penetration of the quantal wave into shadow regions of caustics. They arise also in the scattering by a complex potential in an absorptive medium. The comparison with exact quantal calculations shows an astonishingly good agreement, and establishes the complex semiclassical approximation as a quantitative tool even in cases where the potential varies rapidly within a fraction of a wavelength. An approximate property of classical paths is discussed. The general pattern of the trajectories depends only on the product ? = EΘ, and not on energy and angle separately. This property is confirmed by experiments and besides the signature it gives for the semiclassical behavior, it simplifies considerably the search for all trajectories scattering through the same angle. Finally, a general classification of the different types of elastic heavy ion cross sections is given.     

Collisions of halo nuclei within a dynamical eikonal approximation

The dynamical eikonal approximation unifies the semiclassical time-dependent and eikonal methods. It allows calculating differential cross sections for elastic scattering and breakup in a quantal way by taking into account interference effects. Good agreement is obtained with experiment for 11Be breakup on 208Pb. Dynamical effects are weak for elastic scattering.     

Quarks in a constant self-dual Euclidean background field: (II). Influence on the stability of the background field

The fermionic determinant is expanded in powers of the oscillations around a semiclassical background gluon field. In the quadratic approximation, with a translation invariant (anti)self-dual background SU(2) gluon field, the changes of the zero eigenvalues are investigated in a perturbative approximation. Subtracting the field-idependent contributions, the eigenvalue of the “color-longitudinal” zero mode is unchanged while those of the “color-transverse” zero modes get positive shifts, suggesting a further stabilization of the gluonic oscillations.     

Semiclassical description of reactions at energies near the Coulomb barrier

The modified semiclassical approximation of Coulomb matrix elements is extended to include effects of distorting nuclear potentials in the scattering wave functions. The applicability and efficiency of the proposed semiclassical method are discussed. The advantages of this approximation are shown for a typical heavy-ion transfer reaction.     

Semiclassical treatment of quadrupole shape effects in the elastic scattering of heavy ions below the Coulomb barrier

A perturbative semiclassical approximation for the elastics-matrix is used to derive simple and accurate formulae describing the effects of a nuclear quadrupole deformation on the elastic scattering of aligned nuclei. Expressions are derived for the second rank tensor components of the analysing power, the ratios of which turn out to be simple trigonometric functions of the scattering angle in agreement with experimental observations. The approximations are discussed in some detail and higher order corrections are derived. In an appendix we derive a semiclassical first-order perturbation formula describing the effect of a non-central complex perturbation in the presence of a non-perturbatively treated central term. Our formula is in disagreement with some earlier published formulae which fail to treat the real part of the perturbation correctly.     

Excitation of collective nuclear states by fast particles

The elastic and inelastic scattering of a fast particle by a vibrating nucleus is discussed in the semicalssical approximation. For elastic scattering it is shown that the effect of the vibrations can be described by an effective deformed optical potential which is axially symmetric about the incident direction. Explicit results are obtained for inelastic scattering. In an appendix, the validity of the semiclassical approximation for potential scattering is discussed and numerical tests made.     

The electric conductivity of a laminated metal system (alternating magnetic and nonmagnetic layers)

A set of kinetic equations for the distribution functions of carriers differing both by the energy spectrum and by the spin projection is used to investigate the conductivity of a multilayer sample (alternating layers of magnetic (m) and nonmagnetic (_n) metals). The boundary conditions on the interlayer surfaces are derived in an approximation in which the surface scattering is divided into “specular” and “diffuse” scattering and is characterized by scattering parameters (reflection and transmission) which are related to each other by relations dependent on spin projections and on the type of spectrum. The problem on the longitudinal (with respect to the layers) current is treated; situations are analyzed in which the variation in conductivity due to the change of mutual orientation of magnetization in successive m layers from antiparallel to parallel may be of the order of the values of the conductivity proper (the so-called giant magnetoresistance effect). This is possible only in the case of thin (compared with the free path) n layers (in m layers, the ratios of the characteristic dimensions may be arbitrary) and in the mandatory presence of specular surface scattering. Results are given for different possible ratios of Fermi momenta of electron groups and for different fractions of specular and diffuse scattering. The possibility of realizing the effects of both signs is demonstrated.     

On the microscopic description of nuclei at high spins

The behaviour of nuclei at high angular momenta is studied within the framework of many-body Greenfunctions. A theory is developed which can be applied to interacting Fermi-systems in strong external fields. In a qualitative calculation, using the semiclassical approximation, it is shown that at high angular momenta the groundstate band can nearly coincide with the first vibrational band, which corresponds to a breakdown of the usual RPA approximation. We also obtain the socalled “backbending” effect.     

“Optimal” approximation for proton elastic scattering on deuterium at large momentum transfers

The accuracy of the “optimal” approximation for the proton-deuteron elastic scattering at intermediate energies is studied in a wide range of momentum transfers. The s- and d-state deuteron wave functions corresponding to the Paris potential are used in the numerical calculations. It is found that for the backward scattering the accuracy of the “optimal” approximation is not completely controlled by the ratio of the internal nucleon velocity to the projectile velocity. The calculated corrections to this approximation achieve values of 10–20% for the pd backward scattering at 0.5 GeV. The moduli of the relative corrections decrease in general with increasing energy at fixed scattering angle.     

Photoelectron spectroscopy of localized levels near surfaces: Scattering effects and relaxation shifts

We calculate the spectrum of photoelectrons excited from localized levels in solids far into the continuum (XPS situation), in a model that takes into account exactly their interaction with a boson-like spatially non-uniform field. The general results are applied to discuss the spectra of photoelectrons in metals, where the dominant interaction is the coupling to bulk and surface plasmons. In particular, we study the modification of spectral sum rules connecting the strength of inelastic processes with the energy shifts in the spectrum, showing that the electron (“extrinsic”) scattering changes the average position of the spectrum with respect to the pure “intrinsic” result in the “sudden approximation”, which leads also to its dependence on the excitation energy and position of localized level. We estimate these deviations in a simple model which nevertheless reproduces correctly the dynamical aspects of the problem.     

Reflection and interference structure in diatomic Franck-Condon distributions

The Franck-Condon distributions for diatomic radiative transitions from a single vibrational level of a given electronic state to all possible levels (bound and free) of a second electronic state exhibit either “reflection” or “interference” structure. In reflection structure there is a one-to-one mapping of peaks in the initial state probability distribution into peaks in the spectrum. No such simple relationship is known for interference structure, originally termed “internal diffraction” by Condon [Phys. Rev.32, 858–872 (1928)]. A semiclassical treatment of the quantum mechanical overlap integrals shows that the condition for reflection structure is a monotonic difference potential in the range of internuclear distance sampled by the initial wavefunction, whereas interference structure occurs when a “polytonic” difference potential is sampled. The qualitative validity of the semiclassical treatment is illustrated through quantum calculations which show that the Franck-Condon integrals accumulate near the points of stationary phase.     

Non-eikonal corrections in potential scattering on two centres

Scattering of a spinless projectile from a system of two (over-lapping or non-overlapping) local potentials is studied in the first-order eikonal expansion. Corrections to the Glauber formula are found to come mainly from the non-eikonal propagation in the “direct” double scattering, and not from multiple scattering (“reflections”).A comparison is made between the exact (to the leading order) and two approximate results: (i) based on the theory of scattering from non-overlapping potentials, and (ii) neglecting the reflection terms and adopting the local “Born-like” off-shell extrapolation of the two-body amplitudes. The second approximation is found to be more accurate and technically simpler in calculation of multiple scattering from many-body systems.     

Disordered clean surfaces and adsorbates investigated with atom-surface scattering

In this paper two different physical situations are considered which can be treated with the same method: a fluid adsorbate (disordered in the x, y plane) and a clean surface with random steps (disordered in the z direction). The hard corrugated wall model is used in the eikonal approximation; the differences between the two cases arise only from the different statistical properties of the two physical situations. The differential scattering probability is evaluated. For the fluid adsorbate the latter splits into a coherent (purely specular) contribution and an incoherent one (which is, in fact, weakly inelastic and related to classical diffusion on the surface). For stepped “rough” surfaces only incoherent scattering is present and the differential scattering probability for hexagonal lattices is given.     

Attosecond physics with neutrons and electrons: Ultrafast entanglement and decoherence phenomena involving protons in condensed matter and molecules

Nuclei and electrons in condensed matter and/or molecules are usually entangled, due to the prevailing electromagnetic interactions. Usually, the “environment” of a microscopic scattering system (e.g., a proton) causes an ultrafast decoherence, thus making atomic and/or nuclear entanglement effects not directly accessible to experiments. However, neutron Compton scattering (NCS) and electron Compton scattering represent ultrafast techniques operating in the sub-femtosecond timescale, thus opening a way for investigation of such dehoherence and short-lived entanglement phenomena of atoms in molecules and condensed matter. The experimental context of NCS and a new striking scattering effect from protons (H-atoms) in several condensed systems and molecules are described. In short, one observes an “anomalous” decrease of scattering intensity from protons, which seem to become partially “invisible” to the neutrons. The experiments apply large energy (several electronvolts) and momentum (10–200 Å^?1 transfers, and the collisional (or scattering) time between the neutron and a struck proton is only 100–1000 attoseconds long. Similar results are also obtained with electron-atom Compton scattering at large momentum transfers. As an example, we present new NCS experimental results from a single crystal, which also provide new physical insights into the attosecond quantum dynamics of protons in molecules and condensed matter. Theoretical discussions and models are presented which show that the effect under consideration is caused by the non-unitary time evolution (due to decoherence) of open quantum systems during the ultrashort, but finite, time-window of the neutron-proton scattering process. The conceptual connection with the well known Quantum Zeno Effect is pointed out. The experimental results, together with their qualitative interpretation “from first principles,” show that epithermal neutrons being available at spallation sources, and electron spectrometers providing large momentum transfers, may represent novel tools for investigation of thus far unknown physical and chemical attosecond phenomena.