Nonrelativistic Amplitudes Research Articles

Bremsstrahlung radiation from slow electrons in a Coulomb field: Classical limit and quantum correction

Compact analytic expressions have been derived by a direct expansion in ħ → 0 for the nonrelativistic amplitude of Coulomb bremsstrahlung radiation (BR), the differential (in frequency and angles of the scattered electron) BR cross section, and the triply differential BR cross section that takes into account the bremsstrahlung photon direction and polarization and the scattered electron direction. They contain the classical limit and a quantum correction of the order of ħ at an arbitrary BR frequency ω. An explicit expression has been found for the quantum correction of the order of ħ to the classical BR spectrum.

Hilbert space theory for relativistic dynamics with reflection. Special cases

We present and study a novel class of one-dimensional Hilbert space eigenfunction transforms that diagonalize analytic difference operators encoding the (reduced) two-particle relativistic hyperbolic Calogero–Moser dynamics. The scattering is described by reflection and transmission amplitudes $t$ and $r$ with function-theoretic features that are quite different from nonrelativistic amplitudes. The axiomatic Hilbert space analysis in the appendices is inspired by and applied to the attractive two-particle relativistic Calogero–Moser dynamics for a sequence of special couplings. Together with the scattering function $u$ of the repulsive case, this leads to a triple of amplitudes $u, t$ and $r$ satisfying the Yang–Baxter equations.

(e, eγ)-coincidence studies to determine spin-resolved Stokes parameters of the 185 nm emission line in mercury

Direct measurements of the three Stokes parameters (polarization components) P1, P2 and P3 of the VUV Hg transition 6s6p1P1 → 6s21S0 (185 nm) have been carried out at electron impact energies of 15 eV, 50 eV and 100 eV. Within the experimental uncertainty, no influence of the electron spin was discovered for scattering angles θ ⩽ 30°. At 15 eV excitation energy and scattering angles θ ⩾ 80°, increasing spin effects become apparent. The experimental data are compared to theoretical predictions from a first-order full-relativistic distorted-wave model, a five-state Breit–Pauli R-matrix (close-coupling) approach, and a convergent close-coupling model, in which relativistic effects are accounted for by adding non-relativistic amplitudes using known intermediate-coupling coefficients. At scattering angles θ ⩽ 15°, all of the theories reproduce the experimental data well, whereas the CCC model exhibits the best overall agreement with experiment at large scattering angles.

Relativistic effects in proton-induced deuteron break-up at intermediate energies with forward emission of a fast proton pair

Recent data of the reaction $pd \to (pp) n$ with a fast forward pp pair with small excitation energy is analyzed within a covariant approach based on the Bethe-Salpeter formalism. Relativistic effects in cross-section are extracted and found to be large. It is demonstrated that the node of the non-relativistic amplitude is shifted and masked by relativistic effects from Lorentz boost and the negative-energy P components in the 1 S 0 Bethe-Salpeter amplitude of the pp pair.

Relativisitic longitudinal non-Abelian oscillations in quark-antiquark plasma

We study the relativistic version of the non-Abelian, longitudinal wave in quark-antiquark plasma reported earlier by Bhat et al [Phys. Rev. D39, 649 (1989)]. We have also relaxed various approximations they made in their analysis. Both the quark and antiquark dynamics are taken in our analysis. The non-linearity arising from non-Abelian field as well as from plasma are included. Hence it is an exact longitudinal mode in relativistic quark-antiquark plasma, relevant to the study of quark gluon plasma. We find that earlier results are reproduced for non-relativistic and low amplitude oscillations, but are modified for relativistic or large amplitude waves. Further more, the above results are based on just four first-order equations for gauge invariant quantities derived from gauge covariant twelve first-order equations.

Investigation of the Angle Dependence of the Photon–Atom Anomalous Scattering Factors

The angle dependence of the elastic photon-atom scattering amplitude is investigated using full relativistic S-matrix calculations for guidance. A simple quantitative approach is developed based on the full nonrelativistic amplitude for Rayleigh scattering from the ground state of a hydrogenic ion and it is successfully applied to neutral atoms of low and intermediate nuclear charge. This framework is useful for scattering of X-rays with energies above the K-shell ionization threshold but below energies where major relativistic corrections are necessary. The generalized anomalous scattering factors presented here go beyond the usual angle-independent anomalous-scattering-factor approximation, permitting a discussion of the full angular dependence of the anomalous scattering factors. Simple formulas, expressed in terms of elementary functions and a lookup table, are presented for these generalized anomalous scattering factors. Numerical examples illustrate that the angle dependence can be substantial. A fairly good representation of full numerical results is achieved.

Relativistic calculations for photonuclear reactions (II). nonrelativistic reductions and nuclear medium effects

The relativistic amplitude for the direct knock-out contribution to $\left( \gamma, p \right)$ reactions on nuclei is reduced to a nonrelativistic form using an effective Pauli reduction scheme. The reduction is carried out to second order in the inverse nucleon mass. It is found that the interaction Hamiltonian appearing in the nonrelativistic amplitude has significant dependence, starting at second order, on the vector and scalar mean nuclear potentials. These strong medium modifications are absent in traditional nonrelativistic calculations. Detailed comparisons show that these modifications are crucial to understanding the differences between relativistic and nonrelativistic models. These differences are also examined through reduction of the relativistic amplitude via the Foldy-Wouthuysen transformation. Similar medium modifications are obtained in this case as well. We discuss the implications of these medium modifications for the consistency of existing nonrelativistic calculations.

Nuclear medium effects in the relativistic treatment of quasifree electron scattering

Non-relativistic reduction of the S-matrix for the quasifree electron scattering process $A\left(~e, e'p~\right)A-1$ is studied in order to understand the source of differences between non-relativistic and relativistic models. We perform an effective Pauli reduction on the relativistic expression for the S-matrix in the one-photon exchange approximation. The reduction is applied to the nucleon current only; the electrons are treated fully relativistically. An expansion of the amplitude results in a power series in the nuclear potentials. The series is found to converge rapidly only if the nuclear potentials are included in the nuclear current operator. The results can be cast in a form which reproduces the non-relativistic amplitudes in the limit that the potentials are removed from the nuclear current operator. Large differences can be found between calculations which do and do not include the nuclear potentials in the different orders of the nuclear current operator. In the high missing momentum region we find that the non-relativistic calculations with potentials included in the nuclear current up to second order give results which are close to those of the fully relativistic calculation. This behavior is an indication of the importance of the medium modifications of the nuclear currents in this model, which are naturally built into the relativistic treatment of the reaction.

Comparison of relativistic and non-relativistic amplitudes for photonuclear reactions

The relativistic DWBA amplitude for direct knock-out in (γ, p) reactions on nuclei is subjected to an effective two-component Pauli reduction. The resulting interaction Hamiltonian exhibits strong dependence on the nuclear vector and scalar potentials. The leading term in a 1/( E+ M) expansion recovers the standard non-relativistic form of the amplitude. Large differences exist between the relativistic and non-relativistic calculations, particularly for polarization observables at intermediate energies. Calculations carried out to second order clearly indicate the crucial importance of taking into account the medium modifications, via the strong vector and scalar potentials, of the interaction Hamiltonian.

Multiphoton transitions in a strong field: Inclusion of the photon momentum.

Taking into account the photon momentum (i.e., without making the dipole approximation) we derive some different expressions for the nonrelativistic amplitude for a one-electron system to undergo a transition in a classical monochromatic field. As an application, we use the Keldysh approximation, generalized beyond the dipole approximation, to examine the effect of the photon momentum on the angular distribution for multiphoton detachment of a model ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ ion by a strong low-frequency field. We find that when the field is very strong the neglect of the transverse Doppler shift (but inclusion of the nonrelativistic Doppler shift) leads to a spuriously large rate for multiphoton detachment, even though the relativistic correction to the rate is very small.

The equal charge attraction in 2 + 1 topologically massive nonabelian gauge theories

The particle interaction in the 2 + 1 topologically massive nonabelian gauge theory is given. It turns out, that contrary to the usual case two equally charged topologically massive gauge bosons attract each other in the S-wave. The nonrelativistic amplitude for the SU (2) group is obtained, which can be generalized to the general case. Some applications are shortly discussed.

Gauge-invariant forms of the nonrelativistic Compton amplitude.

Various forms of the long-wavelength, nonrelativistic nuclear Compton amplitude are discussed. Simple manipulations allow the usual low-energy theorem for the amplitude to be cast into a manifestly gauge-invariant form, which facilitates interpretation.

Relativistic effects on parity-violating phenomena in the low-energy np to dγ process

The photon circular polarisation Pgamma and the asymmetry Agamma in the process n+p to d+ gamma are investigated in a Lorentz and gauge-invariant formalism in which the deuteron and initial np-distorted wavefunctions involve negative-energy components. In the low-energy limit, the formalism gives the conventional non-relativistic amplitude plus the relativistic correction terms associated with the negative-energy components of the deuteron and initial np-distorted wavefunctions. A large relativistic contribution to Pgamma , but a smaller one to Agamma , are suggested by estimates from certain models of the weak and strong interactions. The large relativistic contribution to Pgamma is not suppressed by terms required by gauge invariance.

Operator approach to the calculation of nonrelativistic amplitudes in a Coulomb field

A new method for the calculation of the nonrelativistic amplitudes in a Coulomb field is proposed. The method is based on an operator representation of the Green function for a charged particle with a subsequent disentanglement of operator expressions. The dynamical algebra O (4,2) of the hydrogen atom is used. The calculation of the bremsstrahlung cross section is performed.

Alternative partial wave representations for ultrarelativistic Dirac potential scattering

After reviewing briefly the point Coulomb scattering problem in the nonrelativistic, Klein-Gordon, and Dirac cases, the form of the ultrarelativistic vector (Coulomb-like) potential scattering amplitude f/sub 0/ is developed in partial wave form for the ''spinor'' representation. The latter is one of a variety of different Dirac scattering amplitudes, and arises naturally in perturbation theory as the coefficient of a spinor matrix element. Unlike the traditional amplitudes, f/sub 0/ is not required to vanish at theta = ..pi.., which makes it the natural choice for eikonal treatments. This representation is also the one closest to the usual nonrelativistic and Klein-Gordon amplitudes. Summation by parts, a potentially powerful rearrangement technique, is used to verify several previously known relationships among the various amplitudes, and illustrates the problems associated with summing poorly convergent series.

Fresnel and Recoil Corrections to the Glauber Theory of Elastic Electron Scattering by Hydrogen Atoms and Hydrogen-like Ions

The influence of Fresnel and recoil corrections on the non-relativistic amplitude for elastic scattering of electrons and positrons by hydrogen atoms and hydrogen-like ions is investigated in the straight-line Glauber approximation without exchange terms. Numerical results are presented for scattering on hydrogen.

Saturation of confining forces in a relativistic quark model for the baryons

A relativistic quark model for the baryons with saturating three-particle forces is investigated. The properties of relativistic three-fermion amplitudes are analyzed with respect to Lorentz transformations and permutations. Six different classes of possible structures in spin space are found. They serve as an appropriate basis for the classification and calculation of spin-dependent interactions. The quantum numbers and amplitudes for the orbital part are determined for Euclidean relative vectors with help of the irreducible representations of the groups SU(4) ⊃ SO(4) ⊃ SO(3). These kinematical results together with the Green's function techniques of relativistic quantum field theory are applied to a Bethe-Salpeter model for the binding of three heavy quarks inside a baryon. We give as an example a confining saturating interaction which yields baryon quantum numbers similar to those of the non-relativistic harmonic oscillator model. However, the spin structure of the amplitudes obtained in this way differs from the boosted non-relativistic ones. This feature is important, since the phenomenological discussion of photoproduction and strong decays of the baryon resonances shows that at least sizable corrections to the non-relativistic amplitudes are necessary.

Covariant harmonic oscillators and diffractive excitations

We use convariant harmonic-oscillator wave functions to describe quark-model hadrons in Glauber's model of diffractive scattering. It is shown that the Glauber model can be constructed in the center-of-mass system in terms of fully covariant quantities. For elastic scattering, the covariant model gives the same result as that using nonrelativistic harmonic-oscillator wave functions. For the transition from the $n=0$ to $n=2$ states, which includes the diffractive excitations to the $N(1470)$ and $N(1690)$ resonances, the relativistic effect is simply a multiplication of the nonrelativistic amplitude by the factor ($1\ensuremath{-}{\ensuremath{\alpha}}^{2}$), $\ensuremath{\alpha}$ being the velocity difference between the incoming nucleon and the final-state resonance. We discuss the effects of these results on the existing nonrelativistic calculations.

Two-photon absorption spectroscopy of metastable hydrogen-like atoms

Recent experimental technology points to the possibility of measuring the 2S–3S transition in hydrogen-like atoms via two-photon absorption. The non-relativistic amplitude for this process is evaluated exactly and numerical values are given.

Two-Variable Galilei-Group Expansions of Nonrelativistic Scattering Amplitudes

Two-variable Galilei-group expansions are derived for the two-particle nonrelativistic scattering amplitude. These expansions contain the usual partial-wave and eikonal expansions and supplement them by further expansions in the remaining kinematic variable. The expansions are written both for square-integrable and for asymptotically increasing amplitudes and are shown to correspond to the nonrelativistic limit of previously considered relativistic two-variable expansions. Dynamical singularities (Breit-Wigner resonances, bound states, poles in the impact-parameter plane, etc.) are investigated and related to the asymptotic behavior of the expansion coefficients (or Galilei amplitudes). The threshold and high-energy limits of the expansions are discussed. As a mathematical by-product we give a classification of the subgroups of E(3) and also some results on the representation theory of this group; in particular we study the Clebsch-Gordan coefficients.