A SIMPLIFICATION OF THE SECOND BORN APPROXIMATION
A simplification of the second born approximation
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Collisional excitation of the highly excited hydrogen atoms in the dipole form of the semiclassical impact parameter and Born approximations
Expressions for the excitation cross section of the highly excited states of the hydrogenlike atoms by fast charged particles have been derived in the dipole approximation of the semiclassical impact parameter and the Born approximations, making use of a formula for the asymptotic expansion of the oscillator strength of the hydrogenlike atoms given by Menzel. When only the leading term in the asymptotic expansion is retained, the expression for the cross section becomes identical to the expression obtained by the method of the classical collision and correspondence principle given by Percival and Richards. Comparisons are made between the Bethe coefficients obtained here and the Bethe coefficients of the Born approximation for transitions where the Born calculation is available. Satisfactory agreement is obtained only for n yields n + 1 transitions, with n the principal quantum number of the excited state.
Omidvar, K.↗
Charge-impact excitation of the highly excited hydrogen atoms in the dipole form of the semiclassical impact-parameter and Born approximations.
Derivation of expressions for the excitation cross section in the dipole approximation of the semiclassical impact parameter and the Born approximations, making use of a formula given by Menzel (1968, 1969) for the asymptotic expansion of the oscillator strength of the hydrogen-like atoms. When only the leading term in the asymptotic expansion is retained, the expression for the cross section becomes identical with the expression obtained by the method of the classical collision and correspondence principle given by Percival and Richards (1970). Comparisons are made between the Bethe coefficients obtained by the author and the Bethe coefficients of the Born approximation for transitions where the Born calculation is available. Satisfactory agreement is obtained only for n yields n + 1 transitions, where n is the principal quantum number of the excited state.
Omidvar, K.↗
Single-electron Born approximations for charge transfer from multielectron atoms to protons
The Born approximation, including the internuclear interaction, is used to compute cross sections for the transfer of one electron from a multielectron atom to an incident proton. When the full internuclear interaction is included, the results lie far above high-energy experimental K-shell data for protons and argon. However, when only enough internuclear interaction is included so that the total projectile-target interaction goes to zero asymptotically in accordance with plane-wave functions actually used, fair agreement at high energies is obtained. The latter form of the Born approximation is compared with data on K-shell capture from helium as well as argon and with the simpler approximation of Oppenheimer and of Brinkman and Kramers (OBK), where no internuclear interaction is included. The OBK results typically lie a factor of 2 to 8 above the data, while the present Born results are within a factor of 2 of most of the existing high-velocity K-shell data.
Omidvar, K.↗
Cosmic-ray streaming in the Born approximation
The present work invokes the Born approximation to derive a more accurate expression for the streaming of cosmic rays parallel to the mean magnetic field. While all prior results pertaining to the helicity dependence of the diffusion coefficient and convection speed can be recovered as special cases from this streaming equation, it is concluded that a new set of transport parameters presented here is more appropriate for the solar modulation of galactic cosmic rays. In addition, a new parameter related to time variability, which may be a dominant cause of charge sign-dependent transport of solar particles, is introduced.
Bieber, J. W.↗
Renormalized Born approximation.
Renormalizing approximate wave functions so that amplitude is correct by means of matrix, with applications to Born approximation
BORN APPROXIMATION↗
Various facets of magnetic charge correlation: Micromagnetic and distorted-wave Born approximation simulations study
The emergent concept of the magnetic charge quasiparticle provides a new realm to study the evolution of magnetic properties in two-dimensional artificially frustrated magnets. Here we report on the exploration of magnetic phases due to various magnetic charge correlation using the complementary numerical techniques of micromagnetic and distorted-wave Born approximation simulations in artificial permalloy honeycomb lattice. The honeycomb element length varies between 10 nm and 100 nm, while the width and thickness are kept within the single domain limit. In addition to the charge ordered loop state, we observe disordered charge arrangement, characterized by the random distribution of ±Q charges, in single domain size honeycomb lattice. As the length of the honeycomb element increases, low multiplicity magnetic charges tend to form contiguous bands in thinner lattice. Thin honeycomb lattice with 100 nm element length exhibits a perfect spin ice pattern, which remains unaffected by the modest increase in the width of element size. We simulate scattering profiles under the pretext of distorted-wave Born approximation formalism for the micromagnetic phases. The results are expected to provide useful guidance in the experimental investigation of magnetic phases in an artificial honeycomb magnet.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
CMB weak-lensing beyond the Born approximation: a numerical approach
We perform a complete study of the gravitational lensing effect beyond the Born approximation on the Cosmic Microwave Background (CMB) anisotropies using a multiple-lens raytracing technique through cosmological N-body simulations of the DEMNUni suite. The impact of second-order effects accounting for the non-linear evolution of large-scale structures is evaluated propagating for the first time the full CMB lensing jacobian together with the light rays trajectories. We carefully investigate the robustness of our approach against several numerical effects in the raytracing procedure and in the N-body simulation itself, and find no evidence of large contaminations. We discuss the impact of beyond-Born corrections on lensed CMB observables, and compare our results with recent analytical predictions that appeared in the literature, finding a good agreement, and extend these results to smaller angular scales. We measure the gravitationally-induced CMB polarization rotation that appears in the geodesic equation at second order, and compare this result with the latest analytical predictions. We then present the detection prospect of beyond-Born effects with the future CMB-S4 experiment. We show that corrections to the temperature power spectrum can be measured only if a good control of the extragalactic foregrounds is achieved. Conversely, the beyond-Born corrections on E and B-modes power spectra will be much more difficult to detect.
79 ASTRONOMY AND ASTROPHYSICS↗
Total Born approximation cross sections for single electron loss by atoms and ions colliding with atoms
The first born approximation (FBA) is applied to the calculation of single electron loss cross sections for various ions and atoms containing from one to seven electrons. Screened hydrogenic wave functions were used for the states of the electron ejected from the projectile, and Hartree-Fock elastic and incoherent scattering factors were used to describe the target. The effect of the target atom on the scaling of projectile ionization cross sections with respect to the projectile nuclear charge was explored in the case of hydrogen-like ions. Scaling of the cross section with respect to the target nuclear charge for electron loss by Fe (+25) in collision with neutral atoms ranging from H to Fe is also examined. These results were compared to those of the binary encounter approximation and to the FBA for the case of ionization by completely stripped target ions.
Rule, D. W.↗
Total Born-approximation cross sections for single-electron loss by atoms and ions colliding with atoms
The first Born approximation (FBA) is applied to the calculation of single-electron-loss cross sections for various ions and atoms containing from one to seven electrons. Screened hydrogenic wave functions are used for the states of the electron ejected from the projectile, and Hartree-Fock elastic and incoherent scattering factors are used to describe the target. The effect of the target atom on the scaling of projectile ionization cross sections with respect to the projectile nuclear charge is explored in the case of hydrogenlike ions. Also examined is the scaling of the cross section with respect to the target nuclear charge for electron loss by Fe(25+) in collision with neutral atoms ranging from H to Fe. These results are compared with those of the binary-encounter approximation (BEA) and with the FBA for the case of ionization by completely stripped target ions. Electron-loss cross sections are also calculated for the ions O(i+) (i = 3-7) and N(i+) (i = 0-6) in collision with He targets in the energy range of approximately 0.1 to 100 MeV/nucleon. These results are found to be in excellent agreement with the available data near the peak of the ionization cross section.
Rule, D. W.↗
Ladder and cross terms in second-order distorted Born approximation
In the strong fluctuation theory for a bounded layer of random discrete scatterers, the second moments of the fields in the second-order distorted Born approximation are obtained for copolarized and cross-polarized fields. The backscattering cross sections per unit area are calculated by including the mutual coherence of the fields due to the coincidental ray paths, and that due to the opposite ray paths, corresponding to the ladder and cross terms in the Feynman diagramatic representation. It is proved that the contributions from ladder and cross terms for the copolarized backscattering cross sections are the same, while the contributions for the cross-polarized backscattering cross sections are of the same order. The bistatic scattering coefficients in the second-order approximation for both the ladder and cross terms are also obtained. The contributions from the cross terms explain the enhancement in the backscattering direction.
Jin, Y. Q.↗
A simple correction for the Born approximation for electron impact excitation of hydrogenic ions
An analytic procedure is developed for obtaining the electron impact excitation cross sections for hydrogenic ions. A simple modification of the Born approximation, in order to include Coulomb focusing of the continuum wave functions, produces the correct energy dependence of the cross section near excitation threshold. This procedure is used to compute the 1s yields 2s and 1s yields 2p excitation cross sections for hydrogenic ions (Z approaches infinity), and compare these modified-Born cross sections with numerical Colomb-Born calculations. The modified-Born cross sections agree well with the Coulomb-Born cross sections and thus provide a reliable generalized analytic expression for electron impact excitation of hydrogenic ions. By treating effective charges, the results in this work can be extended to nonhydrogenic target systems.
Jung, Young-Dae↗
Diffuse X-ray scattering from polished silicon: application of the distorted wave Born approximation
Measured diffuse X-ray scattering data for a `smooth' as well as for a `rough' silicon sample were fit to theoretical expressions within the distorted wave Born approximation (DWBA). Data for the power spectral density (PSD) for both samples were also obtained by means of atomic force microscopy and optical interferometry. The Fourier transforms of trial correlation functions were fit to the PSD data and then applied to the DWBA formalism. The net correlation functions needed to fit the PSD data for each sample comprised the sum of two terms with different cutoff lengths and different self-affine fractal exponents. At zero distance these correlation functions added up to yield net values of σ 2 = (2) 2 and (71) 2 Å 2 for the smooth and rough samples, respectively. X-ray scattering data were obtained at beamline 1-BM of the Advanced Photon Source. Data and fits at values of q z = 0.05 and 0.10 Å −1 for the smooth sample are reported. Good fits for the smooth sample were obtained at both q z values simultaneously, that is, identical fitting parameters were applied at both values of q z . The smooth sample also exhibited weak Yoneda wings and a clear distinction between the strong specular scattering and the weak diffuse scattering. Data for the rough sample were qualitatively different and exhibited very weak scattering at the specular condition in contrast to extremely large Yoneda wings. Fits for the rough sample are reported for q z = 0.04, 0.05, and 0.06 Å −1 . Although the large Yoneda wings could be fit quite well in both position and amplitude, scattering near the specular condition could not be equally well fit by applying the same fitting parameters at all values of q z . Albeit imperfect, best-fitting results at the specular condition were obtained by invoking only diffuse scattering, that is, without including a separate theoretical expression for specular scattering.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Three-body Faddeev equations in two-particle Alt-Grassberger-Sandhas form with Distorted-Wave-Born-approximation amplitudes as effective potentials
A(d, p)B reactions on heavier nuclei are peripheral at sub-Coulomb energies and can be peripheral even at energies above the Coulomb barrier due to the presence of the distorted waves in the initial and final channels. Usually, to analyze such reactions the distorted-wave-Born-approximation (DWBA) is used. The DWBA amplitude for peripheral reactions is parametrized in terms of the asymptotic normalization coefficient (ANC) of the bound state B = (n A). However, in the DWBA, the coupling of the different channels is not taken into account explicitly. The three-body Faddeev equations written as the two-particle Alt-Grassberger-Sandhas (AGS) equations are very suitable for the analysis of the A(d, p)B reactions because they take into account explicitly the coupling of the different channels. It is well known that the overall normalization of the DWBA amplitude or even of the more advanced continuum-discretized coupled channels (CDCC) amplitude for peripheral reactions (both sub-Coulomb and above the Coulomb barrier) is determined by the ANC. However, it is not apparent that it is the case for the AGS solution due to the coupling of the (d, p) channel to other channels. In this paper, it is proved that the sub-Coulomb A(d, p)B reaction amplitude, which is a solution of the two-body AGS equations, is peripheral and is parametrized in terms of the ANC of the bound state B = (nA) if the corresponding DWBA amplitude is peripheral. Both nonlocal separable and local nuclear interaction potentials between the constituent particles are considered. To prove the peripheral character of the AGS amplitude for the sub-Coulomb A(d, p)B reactions the effective potentials are expressed in terms of the corresponding sub-Coulomb DWBA amplitudes of the different channels. The analysis of the A(d, p)B reactions above the Coulomb barrier requires the inclusion of the optical potentials. Hence, to analyze such reactions, the AGS equations are generalized by including the optical nuclear potentials in the same manner as it is done in the DWBA. The obtained AGS equations contain the DWBA effective potentials with distorted waves generated by the sum of the nuclear optical and the channel Coulomb potentials. It is shown that if the DWBA amplitude is peripheral than it is also the case for the AGS amplitude, which is also parametrized in terms of the ANC of the bound state B = (nA). Furthermore, the inclusion of the coupling of the different channels in the AGS formalism allows one to improve the treatment of the peripheral sub-Coulomb and above the Coulomb (d, p) reactions compared to the DWBA and CDCC method.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Calculation of the total electron excitation cross section in the Born approximation using Slater wave functions for the Li (2s yields 2p), Li (2s yields 3p), Na (3s yields 4p), Mg (3p yields 4s), Ca (4s yields 4p) and K (4s yields 4p) excitations
Excitation of neutral atoms by inelastic scattering of incident electrons in gaseous nebulae were investigated using Slater Wave functions to describe the initial and final states of the atom. Total cross sections using the Born Approximation are calculated for: Li(2s yields 2p), Na(3s yields 4p), k(4s yields 4p). The intensity of emitted radiation from gaseous nebulae is also calculated, and Maxwell distribution is employed to average the kinetic energy of electrons.
Simsic, P. L.↗
Distorted-wave born approximation calculations for turbulence scattering in an upward-refracting atmosphere
Weiner and Keast observed that in an upward-refracting atmosphere, the relative sound pressure level versus range follows a characteristic 'step' function. The observed step function has recently been predicted qualitatively and quantitatively by including the effects of small-scale turbulence in a parabolic equation (PE) calculation. (Gilbert et al., J. Acoust. Soc. Am. 87, 2428-2437 (1990)). The PE results to single-scattering calculations based on the distorted-wave Born approximation (DWBA) are compared. The purpose is to obtain a better understanding of the physical mechanisms that produce the step-function. The PE calculations and DWBA calculations are compared to each other and to the data of Weiner and Keast for upwind propagation (strong upward refraction) and crosswind propagation (weak upward refraction) at frequencies of 424 Hz and 848 Hz. The DWBA calculations, which include only single scattering from turbulence, agree with the PE calculations and with the data in all cases except for upwind propagation at 848 Hz. Consequently, it appears that in all cases except one, the observed step function can be understood in terms of single scattering from an upward-refracted 'skywave' into the refractive shadow zone. For upwind propagation at 848 Hz, the DWBA calculation gives levels in the shadow zone that are much below both the PE and the data.
Gilbert, Kenneth E.↗
Electron Capture into Rydberg States According to the Born Approximation: Correction of an Error
A general expression is derived for the Born amplitude for electron capture into high principal, and arbitrary angular momentum, quantum numbers, n and l, by bare nuclei from hydrogenic atom targets. Cross sections for capture into high n and l = 0,1 are given in analytic forms. The limiting values of the ratios of the Born over OBK cross sections for n >> 1 and l = 0,1 are found, and shown that they confirm the empirical law of Jackson and Schiff only approximately. For high angular momenta, it is shown that the Born cross section approaches the OBK cross section. An error in a previous publication by the author has been corrected.
Omidvar, Kazem↗
Born-approximation and radiative corrections to pair production in photon-photon collisions
Aspects of pair production in photon-photon collisions, which can be of great importance in a variety of astrophysical settings, are examined. In particular, the correction associated with the use of Coulomb rather than plane-wave functions to describe the outgoing e(+) e(-) is evaluated. This is important only in the energy domain near threshold where e(+) e(-) are nonrelativistic, and the effect is evaluated. Because of the extreme simplicity of the nonrelativistic Born limit, the corresponding derivation is briefly outlined using noncovariant perturbation theory. The relative magnitude of the Born correction is comparable to radiative-correction effects. These effects can be evaluated easily in the nonrelativistic limit, and an approximate calculation is outlined.
Gould, Robert J.↗