Eigenfunction expansions for stationary scattering theory in spaces with negative norm

The space with “negative” norm $\text{L}$_- with which the Hilbert space $\text{L}$ of wave-pockets is equipped is used as a natural carrier of non-square integrable functions which appear in scattering theory, such as the plane waves and distorted plane waves in potential scattering. The concept of a complete set of observables is formulated within this framework. The S-matrix formulae leading to the expression for the scattering cross-section are derived on a general level, and then are applied to the case of potential scattering.     

Annihilation of slow antiprotons in flight and in traps

The antiproton-nucleus annihilation in two possible experimental situations (low energy in-flight annihilation and annihilation in a trap) is investigated taking into account the Coulomb interaction. For these two cases, the formulas giving the relation between the antiproton-nucleus scattering lengtha _cs and observables (annihilation cross section and antiprotons time-of-life in a trap) are obtained. These relations are proposed to determinea _cs.     

Improved analytical formulae for correcting elastic-scattering effects in X-ray photoelectron spectroscopy

We present new predictive formulae for the correction parameters Q_x and β_eff that are used to account for the effects of elastic scattering in quantitative X-ray photoelectron spectroscopy. These formulae were derived from an analysis of published calculations of Q_x and β_eff from extensive Monte Carlo simulations for a group of elemental solids. Two formulae are given for each parameter. One formula is a function of the single-scattering albedo and the photoelectron emission angle, and is useful for emission angles between 0° and 80°. The other formula is a function only of the single-scattering albedo and is useful for emission angles between 0° and 50°. The single-scattering albedo is in turn a simple function of two material parameters, the inelastic mean free path and the transport mean free path. The latter parameters can be determined readily from available predictive formulae for any material or from databases. The root-mean-square and mean deviations of the Monte Carlo values of Q_x and β_eff from those found with the new formulae are comparable to or smaller than those found with formulae published by Seah and Gilmore.     

Analysis of elastic scattering of pi-mesons by nuclei within the microscopic optical potential

The microscopic pion-nucleus optical potential defined by the pion-nucleon scattering amplitude and the density distribution function of the target nucleus is applied to obtain the differential cross section of the elastic pion-nucleus scattering based on the solution to the relativistic wave equation. This allow one to account for effects of the relativization and distortion by the potential field. Data on π^±-meson scattering on ²⁸Si, ⁵⁸Ni, and ²⁰⁸Pb nuclei at T _lab = 291 MeV are analyzed and the parameters of the in-medium πN-amplitude are obtained. The parameters are compared with similar parameters for scattering on free nucleons.     

Deviation from Matthiessen's rule of dilute al-alloys, 3 group model

The deviation from Matthiessen's rule has been investigated. The calculations were based upon three group model, in which the Fermi surface is split up into three regions. The mean free paths over the three major regions are denoted byL ₁,L ₂ andL ₃. Besides the Fermi surface anisotropy, the anisotropy of electron scattering with phonons and impurities have also been employed in the calculations. Simple formulae for the residual resistivity, the resistivity associated with phonon scattering and the deviation from Matthiessen's rule Δ(C, T), in terms of the anisotropic parameters, were derived. The formulae were used to throw light on the deviation from Matthiessen's rule.     

Transport cross sections for charged particle scattering in (dense-) cesium plasma; electrical conductivity

Transport cross sections for the electron-ion and electron-electron scattering in (dense) cesium plasmas is determined with the partial wave method. The scattering phase shifts δ_l(k) are calculated numerically by means of the usual Numerov method and the advantageous amplitude — phase method for energies E = 10^-4··· 5 ryd and for Debye screening lengths R_D = (10···10³)a₀. Besides the usual Debye potential, also a Thomas-Fermi potential is applied for the cesium electron-ion interaction. The resulting transport cross sections are compared with the Born and quasi-classical approximations. Furthermore, the electrical conductivity of fully ionized cesium plasmas is determined and compared with respective Spitzer results.     

On the theory of elastic scattering of slow particles

A theory describing scattering of particles with low energy E from an arbitrary 3D potential is proposed. An exact expression is derived for the s scattering amplitude in the limit E → 0. Phenomenological constant κ appearing in the expression for the resonance scattering amplitude is expressed in terms of the parameters typical of the given potential.     

Semi-phenomenological bounds for the pion-pion coupling constant

Upper and lower bounds on the Chew-Mandelstam ππ coupling constant λ are derived as functions of the D-wave scattering length a₂. The only information used is the axiomatic analyticity domain of the ππ scattering amplitude, and the crossing and unitarity properties. For a₂ = 7 × 10^?4, one obtains ?0.164<λ<0.162. This is compared with previous bounds.     

On the Regge-pole description of nucleus-nucleus scattering

The properties of the Regge poles of theS-matrix for scattering of strongly-absorbed nuclear particles are considered. Simple formulae are obtained for describing the Regge trajectories in terms of the nuclear radius, the quasi stationary levels in the combined nuclear-Coulomb-potential and the widths of these levels. The predictions of these formulae are compared with the Regge trajectories obtained previously, for a Woods-Saxon potential, and with those required to fit¹⁶O-¹²C backward scattering.     

Perturbative Handling of the Renner-Teller Effect and Spin-Orbit Coupling in Π Electronic States of Triatomic and Tetra-atomic Molecules

Second-order perturbative formulae for handling the Renner-Teller effect combined with the spin-orbit coupling in Π electronic states of triatomic and symmetric (ABBA-type) tetra-atomic molecules with linear equilibrium geometry are derived. Two schemes for partition of the model Hamiltonian are employed: In the first the spin-orbit coupling term is treated as a perturbation, in the second it is included in the zeroth-order Hamiltonian. It is demonstrated that both approaches lead to the same results when the spin-orbit coupling constant is small compared to the bending frequency, but much larger than the splitting of potential surfaces upon bending. The perturbative formulae derived for tetra-atomic molecules are used to compute the spectrum of the X²Π_u state of the acetylene ion, employing the parameters obtained in ab initio calculations. The results are compared with those generated in corresponding variational computations.     

Chemical potentials in gauge theories

One-loop calculations of the thermodynamic potential Ω are presented for temperature gauge and non-gauge theories. Prototypical formulae are derived which give Ω as a function of both (i) boson and/or fermion chemical potential, and in the case of gauge theories (ii) the thermal vacuum parameter A₀=const (A_μ is the euclidean gauge potential). From these basic abelian gauge theory formulae, the one-loop contribution to Ω can readily be constructed for Yang-Mills theories, and also for non-gauge theories.     

Path integral methods for inelastic scattering

Corrections to the primitive semi-classical amplitude for multiple inelastic scattering are obtained from a path integral formulation of scattering theory. The path integrals are calculated by making an expansion about a classical orbit describing elastic scattering. Terms are collected to give a series in inverse powers of the reduced mass m of relative motion of the target and projectile. The leading term is the primitive semi-classical amplitude for multiple excitation and explicit formulae are given for the corrections of order $\text{1}\text{m}$. These are calculated in detail for a one-dimensional model. It is shown that some, but not all, of the corrections can be included by evaluating the primitive amplitude with a symmetrized orbit.     

Non-coherent scattering in subordinate lines: III. Generalized redistribution functions

The concept of the redistribution function, which was originally introduced in order to describe a correlation between frequencies and directions of the absorbed and emitted photon in resonance scattering, has been extended to other resonance two-photon processes including resonance Raman scattering, resonance two-photon absorption and emission, and inverse Raman scattering. We have derived, within the frame of the impact approximation, the appropriate form of the generalized redistribution function. Using a suitable formalism, the generalized redistribution function takes the same form for all types of two-photon processes and contains all the redistribution functions, considered previously, as various limiting cases. In analogy to Hummer's original scheme of redistribution functions, we have derived a similar set of generalized redistribution functions, denoted as p_i(i = I, II, III, IV, V), and we have shown that the most general case is described by a linear combination of p_III and p_V, analogously to the previous results. Explicit formulae for the velocity-averaged (laboratory-frame) generalized redistribution functions p_i(i = I, II, III, IV, V) are given and possible numerical methods for their evaluation are briefly indicated.     

Curve-crossing and the WKB approximation

The transition probabilities and phase changes associated with passage through a potential curve crossing are derived in the form of connection formulae between WKB solutions on either side of the crossing point. The formulae are expressed in terms of three integrals v and ε_± which may be evaluated for arbitrary potential curves and interaction function without knowledge of the associated wavefunctions. The theory, which is fully developed in the chemical energy region (< 100 ev), is applied to a model for the covalent-ionic crossing responsible for inelastic scattering of alkali atoms from neutral targets and its extension to the higher energy region for this model indicated.     

Eikonal expansion in electron scattering: I. Elastic scattering

A systematic eikonal expansion for the scattering of high-energy electrons from nuclei is derived which starts from the iterated Dirac equation. The resulting scattering amplitude is written in an impact parameter representation depending on eikonal phases which are proportional to inverse powers of the energy. The first two correction terms to the leading Glauber-Baker amplitude are calculated. For a Coulomb potential they agree with a sinθ-expansion of the relativistic Coulomb scattering amplitude. In the case of scattering from an extended charge distribution at sufficiently high energies numerical partial wave calculations are accurately reproduced.     

Nonlinear and nonlocal transport effects in simple fluids

Correlation function formulae for transport coefficients of 2nd order in arbitrarily dense fluids are derived, using a modified Chapman-Enskog solution of the Liouville equation. Some static correlations are neglected. Approximate evaluation for dilute gases gives essentially the same results as the solution of Boltzmann's equation. As an application higher order transport effects in the critical region are estimated. It is conjectured that they are apparent in sound absorption and the line width of Rayleigh scattering if (T?T_c)/T _c?10^?3.     

Tensor interaction in heavy-ion scattering: (II). Partial-wave expansions

The turning-point-model relations between components of rank-2 tensor analysing power in polarized heavy-ion scattering, which were derived semi-classically in our previous work, are re-derived quantum-mechanically without resorting to any classical concept. Using partial-wave expansions of the scattering amplitudes, the turning-point-model relation is reduced to a relation between the diagonal and off-diagonal components of the tensor scattering amplitude for each type of tensor interaction (T_R, T_P and T_L). Numerical investigations and analytical investigations using a plane-wave expansion method show that these relations arise from universal features of heavy-ion scattering: large angular momenta and the short-range nature of the tensor potentials. An interrelation between the three types of tensor interaction is also obtained for each partial wave.     

Two-dimensional friction oscillator: group-preserving scheme and handy formulae

With vibration isolation of buildings and turbomachinery blades in mind, we study the dynamic behaviour of a single-mass two-degree-of-freedom oscillator with dry friction damper, viscous damper and elastic spring connected in parallel. The mass is mounted on an elastic supporting plate allowing movement in two directions on a plane. We formulate a multi-dimensional friction model, from which the sliding conditions and the sticking conditions of the mass are derived. For calculations we develop a group-preserving scheme, which preserves the projective proper orthochronous Lorentz group PSO_o(2,1) symmetry of the model in the sliding phase so as to satisfy automatically the sliding conditions at each time step without iteration at all. The oscillator is then subjected to simple harmonic excitations, and the responses are displayed. According to the simple harmonic balance method together with a circular orbit assumption on displacements, we derive closed-form formulae for handily estimating the steady state responses, which are then compared with the results calculated by the group-preserving scheme to confirm the applicability of the formulae. We also derive formulae specifically for a two-dimensional friction oscillator with rigid base support, which include an exact formula of the magnification factor and a simple formula for estimating the minimum driving force amplitude (or the maximum friction force bound) to avoid sticking.     

Isotope effect in resonant Raman scattering and induced IR spectra of trans-polyacetylene

We present a full explanation of the isotope effect in resonant Raman scattering, doping induced, and photo-induced IR-active vibrations of previously reported data in trans-(CH)_x, (¹³CH)_x and (CD)_x. The interpretation is given in terms of Raman scattering from amplitude modes of the dimerized chains and absorption by charged induced IR-active modes. It is shown that the bare phonon frequencies are properly modified by the isotope mass whereas the observed frequencies are derived by the dressed phonon propagator, and therefore do not follow the usual mass rule.     

Solving the relativistic inverse scattering problem with allowance for inelasticity effects on the basis of <Emphasis Type="Italic">N/D</Emphasis> equations and application of the resulting solution to an analysis of nucleon-nucleon interaction

A manifestly Poincaré-invariant approach to solving the inverse scattering problem is developed with allowance for inelasticity effects. The equations of the N/D method are used as dynamical equations in this approach. Two versions of the approach are considered. In the first version (method A), the required equations are constructed on the basis of the maximal-analyticity principle, which constitutes the basis of dynamical S-matrix theory. In formulating the second version of equations (method B), it is assumed that a partial-wave scattering amplitude may develop dynamical singularities that violate the requirement of maximal analyticity. The dynamics of interaction components that violate maximal analyticity is described within the model of a nonlocal separable potential. The method is used to analyze nucleon-nucleon interaction in the ¹S₀ and ³S₁ states. The results obtained by solving the inverse scattering problem for potential functions are compared with the predictions of the one-boson-exchange model.