Nonrelativistic Dipole Approximation Research Articles

Non-relativistic non-dipole two-photon bremsstrahlung in a Coulomb field

We present analytical and numerical results on the cross section of the two-photon bremsstrahlung (2BrS) of a non-relativistic projectile scattered in a point Coulomb field. The effect of the radiation retardation is taken into account. The exact analytic evaluation of the point Coulomb two-photon non-dipole free - free matrix element, which characterizes the amplitude of the 2BrS process, is carried out following a method of momentum space integration developed by Gavrila. The obtained formulae are the generalization of the expressions derived earlier by several workers within the framework of the non-relativistic dipole approximation. Results of numerical calculations are compared with predictions of other theoretical approaches and with the experimental data on 2BrS.

K-shell radiative electron capture with bare 60-MeV/u Kr ions channeled in a Si crystal: Experiments and simulations.

We have performed K-shell radiative electron capture (K-REC) measurements with bare 60.1-MeV/u incident krypton ions, both in channeling conditions and for random orientation of a 37-\ensuremath{\mu}m silicon crystal. The sampled electron densities are quite different in each case, which has an influence both on the shape and on the amplitude of the K-REC photon peak. We have developed simulations of the K-REC photon lines: for this we have determined the impact parameter distribution at statistical equilibrium for various beam incidence conditions (direction and angular spread) using the continuum potential model for channeled ions. Multiple scattering effects were included. The K-REC photon peak was calculated within the nonrelativistic dipole approximation, K-REC being assumed to be a purely local process. Solid state electron densities were used, and impact parameter dependent electron momentum distributions (Compton profiles) were calculated for 2s and 2p silicon electrons. A remarkable agreement is found between the spectra measured with very high statistics, and the calculated ones, which leads to the following results: (i) The dependence of the K-REC yield on the beam incidence angle is obtained separately for silicon core and valence electrons, which was never observed before. We find that the core electron contribution to REC is still significant for axial alignment, whereas it is generally neglected in the literature. (ii) Electron Compton profiles are found to vary significantly with impact parameter. (iii) The free electron gas model represents a fair approximation for the description of valence electron Compton profiles. (iv) The K-REC cross section is measured with an absolute accuracy better than 20%, and found to be close to the value calculated within the nonrelativistic dipole approximation. \textcopyright{} 1996 The American Physical Society.

Retardation effects in nonrelativistic two-photon electron bremsstrahlung in the Coulomb field.

We investigate the deviations from the dipole approximation for the two-photon bremsstrahlung cross section ${\mathrm{\ensuremath{\sigma}}}_{4}$, corresponding to the observation of the emitted photons (but not the scattered electron) in coincidence, which has been the subject of recent experiments. Our analysis is for the Coulomb field, and it is done in the Born approximation. We find that retardation effects are important, even in the lower energy range (8--12 keV) of the Hippler experiments, but not in the particular configuration for which experiments were performed. Using the simple analytic expression that we derive, including retardation, in the nonrelativistic Born approximation, we show that, in the incident electron energy range 10--50 keV, the differences between the relativistic Born (Smirnov's approach) and nonrelativistic dipole approximation results are mainly retardation effects. \textcopyright{} 1996 The American Physical Society.

X-rays from recombination of bare ions with electrons in an electron cooler

Two basic mechanisms of X-ray production in recombination of heavy bare ions with free electrons in an electron cooler of a storage ring are discussed theoretically. In the first process, the radiative recombination of bare ions with electrons, the recombination rates for individual nl-states and the energy spectra of X-rays are described within the nonrelativistic dipole approximation in the low-energy limit. Next, the emission of characteristic X-rays due to the radiative deexcitation cascades in H-like ions are studied in terms of total branching ratios averaged over the initial population of l-states formed in radiative recombination. Strong enhancement of the population of selected low nl-states is found. A possibility to develop precise X-ray spectroscopy by using a 0°-Doppler-tuned spectrometer in a storage ring is also discussed.

Radiative electron capture in relativistic heavy-ion atom collisions

Based on systematic experimental studies and on exact relativistic calculations of the REC process in relativistic heavy-ion atom collisions, the limits of applicability of the nonrelativistic dipole approximation are discussed. In particular, direct comparison between a general REC scaling law and all available total electron-capture cross sections related to REC into bare and H-like ions is given. Similar to K-REC, a fortuitous agreement between the nonrelativistic approach and the experimental data is found. Moreover, the results of a dedicated L-REC experiment are presented, where this process was investigated for energy-resolved subshells. The measured angular distributions for REC into the j = 1 2 and j = 3 2 levels are in excellent agreement with exact relativistic calculations and manifest a complete breakdown of the nonrelativistic theory.

Radiative electron capture studied in relativistic heavy-ion-atom collisions.

The process of radiative electron capture (REC) in relativistic collisions of high-Z ions with low-Z gaseous and solid targets is studied experimentally and theoretically. The observed x-ray spectra are analyzed with respect to photon angular distributions as well as to total K-REC cross sections. The experimental results for angle-differential cross sections are well reproduced by exact relativistic calculations which yield significant deviations from standard ${\mathrm{sin}}^{2}$\ensuremath{\theta} distributions. Total cross sections for K-REC are shown to follow a simple scaling rule obtained from exact relativistic calculations as well as from a nonrelativistic dipole approximation. The agreement between these different theoretical approaches must be regarded as fortuitous, but it lends support to the use of the nonrelativistic approach for practical purposes.

Recombination of D+ and He+ ions with low-energy free electrons.

We have measured the rate coefficients for the recombination of ${\mathrm{D}}^{+}$ ions with free electrons in the electron cooler of the storage ring CRYRING, as a function of the relative energy in the interval from 0 to \ensuremath{\sim}10 eV. Also measured is the rate coefficient for ${\mathrm{He}}^{+}$-e recombination at zero relative energy. The experimental data are reported at two different transverse temperatures of the electron beam, namely, 0.1 and 0.01 eV/k. The present data are compared with calculations for radiative recombination, which is expected to be the dominant recombination process for our experimental conditions. The calculations of the rate coefficients are performed using the nonrelativistic dipole approximation represented by the semiclassical Kramers approximation, corrected by the Gaunt factor. The measured rate coefficients are systematically higher than the theoretical predictions for radiative recombination, corrected for field-ionization effects in the dipole magnets of the ring. Other contributions to the recombination rate, i.e., collisional-radiative recombination and radiative stabilization, are discussed.

Light-induced radiative correction to the Rutherford formula for Coulomb scattering

The light-induced correction to the Rutherford cross section, linear in the intensity, is derived, with an accurate account for the Coulomb potential in the nonrelativistic dipole approximation. Various limits are investigated. A specific asymmetry is discovered, which consists in a dependence of the cross section on the sign of the photon circular polarization.

Relativistic, retardation, and multipole effects in photoionization cross sections: Z, n, and l dependence.

The limits of validity of the nonrelativistic dipole formulation of the total photoionization cross section are explored. Relativistic and nonrelativistic independent particle model calculations of all subshells of C, Sn, and U are performed from 0 to 200 keV above threshold in each case. The comparison shows a remarkable agreement between nonrelativistic dipole and full relativistic multipole results for s subshells, even at the highest energies investigated where nonrelativistic dipole approximation should show significant breakdown; this phenomenon does not occur for l\ensuremath{\ne}0 subshells or in angular distributions. The phenomenon is explained in terms of an approximate cancellation among relativistic, retardation, and multipole effects which occurs only for s subshells.

Results from the nonrelativistic dipole-approximation theory of two-photon electron bremsstrahlung in the Coulomb field.

We extend previous nonrelativistic dipole-approximation theoretical studies of two-photon electron bremsstrahlung in the Coulomb field by calculations exploring the cross section that describes the emitted photons, irrespective of the direction of the scattered electron, and its dependence on the charge Z of the target, for different detection geometries. A Born-approximation equation, valid for any emitted-photon configuration, is derived. Comparisons with the most recent experimental data are presented.

Direct radiative recombination and photoeffect cross sections for arbitrary atomic Coulomb bound states

Simple exact analytical cross sections for direct radiative recombination (DRR) and atomic photoeffect for arbitrary (n, l, m) states are obtained in the nonrelativistic Coulomb dipole approximation. These may also be used in the case of DRR or photoeffect on the Rydberg states of highly ionized atoms with N electrons by considering an effective ionic charge Z eff =Z-N. Asymptotic formulae in the energy of the incoming particle are derived. The numerical values of the DRR cross sections obtained with our formulae are in very good agreement with the existing numerical relativistic results. This is true for high Z values and large energies of the incoming electron

Double photoionization of molecular hydrogen.

The cross section for two-electron photoionization of the hydrogen molecule by high-energy photons is predicted in the nonrelativistic dipole approximation to be 2.25% of the cross section for single photoionization.

Photon polarization in radiative recombination of bare ions with low-energy free electrons.

The polarization of photons emitted from radiative recombination of bare ions with free electrons is discussed in the nonrelativistic dipole approximation in the low-relative-energy limit. Within this approach an analytical expression is derived for the degree of polarization of photons from radiative recombination of ions with fixed-energy electrons for an arbitrary (n, l) state. For a relative-velocity distribution of the electron beam, characterized by the longitudinal kT ∥ and transverse kT ⊥ electron-beam temperatures, we found also analytical forms for the polarization rates

Radiative recombination in the low-energy regime

We give analytical formulas for the cross section and rate coefficient for radiative recombination between bare ions and free electrons into a fixed n state. Those were derived in the low-energy regime within the nonrelativistic dipole approximation by using the sum rule given by Burgess. An analytical expression for the low-energy Gaunt factor is also derived. The results are briefly discussed in the context of current atomic physics experiments using the electron cooler in heavy ion storage rings.

Radiative recombination of bare ions with low-energy free electrons.

Radiative recombination of bare ions with electrons is discussed as a time-reversed photoionization process in the nonrelativistic dipole approximation. Analytical expressions for the radiative-recombination cross section and the angular distribution of the emitted photons are derived for an arbitrary (n,l) state in the low-energy limit. In this approximation, the anisotropy of the emitted photons is described by an energy-independent parameter ${\mathrm{\ensuremath{\beta}}}_{\mathit{n}\mathit{l}}$. The radiative-recombination rate coefficients, both differential and angle integrated, for any (n,l) state can, in this limit, be expressed in analytical form. This result holds also for an anisotropic electron-velocity distribution characterized by a longitudinal ${\mathit{kT}}_{\mathrm{\ensuremath{\parallel}}}$ and transverse ${\mathit{kT}}_{\mathrm{\ensuremath{\perp}}}$ electron-beam temperatures. The results are discussed in context of the state-selective radiative-recombination experiments planned to be studied in electron coolers of heavy-ion storage rings.

Bound states of atomic hydrogen in a laser field beyond the nonrelativistic dipole approximation.

Recent calculations on the bound states of atomic hydrogen in a laser field in the nonrelativistic dipole approximation have produced metastable states with decreasing binding energy as a function of ${\mathrm{\ensuremath{\alpha}}}_{0}$\ensuremath{\sim}${\mathit{I}}^{1/2}$${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}2}$ (where I is the laser intensity and \ensuremath{\omega} is its frequency). We extend the calculations by removing these restrictions. We find the limit of validity of the previous calculations and find that beyond this limit of ${\mathrm{\ensuremath{\alpha}}}_{0}$ the lowest bound state persists but that its character is completely changed.

Stimulated two-photon free-free transitions in a Coulomb potential: Formalism.

We present a calculation of the cross sections for two-photon free-free transitions of an electron colliding with a Coulomb center of force. The calculation is based on second-order perturbation theory, in the nonrelativistic dipole approximation. The matrix elements for absorption and emission were integrated analytically in momentum space, following a method developed earlier by one of us. This makes use of the Schwinger integral representation for the Coulomb Green's function. The result was expressed in terms of integrals over hypergeometric functions of the Gauss type. Simple limiting forms of these complicated expressions were found in the first Born approximation, and at low and high photon energies. The results derived agree with those obtained by direct calculations done for these limits. Finally, concluding remarks on the analytic part of our work are made. The numerical computation of the two-photon free-free transition cross sections for absorption and emission in various geometries is planned to be presented at a later time.

Two-photon ionization of atomic hydrogen with elliptically polarized light.

The theory of two-photon ionization of a hydrogenic state in the nonrelativistic dipole approximation is generalized for elliptically polarized light. An application to the metastable 2S state of atomic hydrogen is made. Significant differences in the angular distribution of the outgoing electrons are found depending upon the polarization of the photons. It is claimed that two-photon ionization employing elliptically polarized photons from lasers may provide an additional test for the theories of multiphoton ionization.

Direct radiative recombination cross sections for arbitrary nS, nP and nD subshells

Exact analytical cross sections for direct radiative recombination (DRR) on nS, nP and nD states with arbitrary principal quantum number n are obtained in the non-relativistic Coulomb dipole approximation. They may be used in the case of DRR on the Rydberg states of N-electrons ions by considering the ionic charge Zeff=Z-N instead of the atomic charge Z. Numerical values of the cross sections are given for a large range of incoming electron energies and ionic changes, involving nuclear fusion plasma.

Systematic study of 1s-ns and 1s-nd two-photon transitions of hydrogenlike atoms.

We present the results of a numerical study of the invariant amplitudes of 1s-ns and 1s-nd two-photon transitions in hydrogenlike atoms in the nonrelativistic dipole approximation, based on compact analytic formulas. We report also results valid in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. The 1s-nd transitions with ng6 are studied for the first time. For other particular cases our numerical results are in agreement with previous calculations in the literature. The investigated energy region of the two-photon spectrum is extended. The position of the transparencies in the second half of the spectrum is found to be almost n independent.