Effect of electron and γ polarization on two-photon bremsstrahlung of electrons in a nuclear field

The cross section for two-photon bremsstrahlung is integrated over the electron scattering angle and the (common) emission angle of the bremsstrahlung photons. Equations are found for the energetic, angular, and polarization distributions of the cross section. The cross section for two-photon bremsstrahlung for the case in which the momenta of the bremsstrahlung photons are parallel can be larger than the corresponding Bethe-Heitler cross section. The probabilities for certain polarization states of the electrons are also analyzed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 13–18, July, 1972.     

Large contribution of virtual Delbrück scattering to the emission of photons by relativistic nuclei in nucleus–nucleus and electron–nucleus collisions

Delbrück scattering is the elastic scattering of a photon in the Coulomb field of a nucleus via a virtual electron loop. The contribution of this virtual subprocess to the emission of a photon in the collision of ultra-relativistic nuclei, Z₁Z₂→Z₁Z₂γ, is considered. We identify the incoming virtual photon as being generated by one of the relativistic nuclei involved in the binary collision and the scattered photon as being emitted in the process. The energy and angular distributions of the photons are calculated. The discussed process has no infrared divergence. The total cross section obtained is 14 barn for Au–Au collisions at the RHIC collider and 50 barn for Pb–Pb collisions at the LHC collider. These cross sections are considerably larger than those for ordinary tree-level nuclear bremsstrahlung in the considered photon energy range, m_e⪡E_γ⪡m_eγ, where γ is the Lorentz factor of the nucleus. Finally, photon emission in electron–nucleus collisions, eZ→eZγ, is discussed in the context of the eRHIC option.     

An analytical expression of the amplitude of stripping reaction in the coulomb approximation

A method is derived for calculating the matrix element of stripping reaction (d,p) in the socalled distorted-wave Born approximation; only Coulomb scattering of incident and outgoing particles is taken into consideration. Coulomb wave functions are expressed by means of integral representation permitting integration over angular variables and, when the radial wave function of the trapped neutron is specially chosen, also calculation of the radial integral for non-zero orbital momentum of the trapped neutron. Relations are given for calculating the differential cross-section of reaction and the polarization of products. The special case, when the orbital momentum of the captured neutron is equal to one or two, is discussed in detail.     

Final-state summation in photoelectron diffraction

A formula was derived to perform the summation over the l_c±1 final states of the dipole transition matrix element and the scattering path operator. This formula shows that often it seems more convenient to speak about l_c+1 and l_c±1 channels instead of the l_c+1 and the l_c−1 channel. Using this formula and restricting the considerations on the first order scattering and the plane wave approximation, explicit formulas were given for the photoelectron intensity, the circular dichroism and the spin polarization in terms of products of the vectors involved in the problem: photon wavevector, photon polarization, magnetization, electron wave vector, and orientation of the spin detector. Additionally the scattering part contains the position vectors of the scatterers with respect to the emitter. These formulas are discussed as well for magnetic as for nonmagnetic samples and for s, p, d and f core states. General expressions are given for the peak rotation. Also it is possible to write down explicitly the dependence on absolute value and the phases of the complex radial matrix elements.     

Optical potential for 12C + 28Si scattering

¹²C + ²⁸Si elastic scattering angular distributions are smooth functions, relatively easily fit by optical potential predictions, at laboratory bombarding energies, E_1ab, within ≈ 10 MeV of the Coulomb barrier (i.e. up to E_1ab ≈ 27 MeV). Between E_1ab = 27 and 36 MeV the angular distributions show pronounced oscillatory structure which is not easily fit with an optical potential. No optical potential has been found to give a very good account of all the angular distributions simultaneously; the best simultaneous fit to all the data was achieved with a surface-transparent potential whose real and imaginary well depths are energy dependent.     

CompleteO(α) corrections to polarized Compton scattering

General expressions for Compton scattering and itsO(α) real and virtual corrections are given for electron and photon beams with arbitrary polarization. The expressions obtained are suitable for use in Monte Carlo simulations of Compton polarimeters planned for polarizede ⁺ e ^? colliders. The basis for the calculation is the spinor techniques recently applied to photon and gluon bremsstrahlung. The practical application of these techniques for massive fermions is discussed.     

On normalization of scattering matrices of polarized radiation

The condition of normalization of scattering matrix is derived when the polarized radiation is described by the Stokes parameters I, Q, U, V. The normalization of the matrix is based on energy conservation. It has a probabilistic meaning also. When the scattering particle is nonspherical, scattered radiation may depend not only on the angle between incident and scattered radiation but on orientation of the scattering plane also. In these cases, the known change of the Stokes parameters Q, U of the incident radiation with respect to various scattering planes influences the normalization. The derived normalization includes all elements of the first line of scattering matrix and the characteristics of polarization of the incident radiation. Dependence on this polarization is appeared because the polarization influences intensities of scattered radiation and, therefore, is included in energy conservation. The routine normalization includes the first element of the scattering matrix only. These two normalizations determine the different normalizing constants of the scattering matrix. The simple computational example of scattering by the particle that has the shape of a finite cylinder is considered. This example shows that the values of normalizing constants of the routine normalization may considerably differ from the ones of the obtained normalization. The results of the study may be useful in various investigations of radiation scattering, especially in the cases when the scattering particles are nonspherical.     

P-Matrix approach and parameters of low-energy scattering in the presence of a Coulomb potential

The scattering of two charged strongly interacting particles is described on the basis of the P-matrix approach. In the P matrix, it is proposed to isolate explicitly the background term corresponding to purely Coulomb interaction, whereby it becomes possible to improve convergence of the expansions used and to obtain a correct asymptotic behavior of observables at high energies. The expressions for the purely Coulomb background P matrix, its poles and residues, and purely Coulomb eigenfunctions in the P-matrix approach are obtained. The nuclear-Coulomb parameters of the low-energy scattering of two charged hadrons are investigated on the basis of this approach combined with the method for isolating the background P matrix. Simple explicit expressions for the nuclear-Coulomb scattering length and effective range in terms of the residual P matrix are derived. For models of short-range strong interaction, these expressions give a general form of nuclear-Coulomb parameters for low-energy scattering. Specific applications of the general expressions derived in this study are exemplified by considering, on the basis of these expressions, some exactly solvable models of strong interaction, including the hard-core model, and, for these models, the nuclear-Coulomb parameters for low-energy scattering at arbitrary values of the orbital angular momentum are found explicitly for the first time. In particular, the nuclear-Coulomb scattering length and effective range are obtained explicitly for the boundary-condition model, the model of a hard-core delta-shell potential, the Margenau model, and the model of square-well hard-core potential.     

Matrix elements in appearance potential spectroscopy of Al and its alloys

The Al K appearance potential spectrum (APS) of Al and some Al alloys, measured with an X-ray monochromator tuned to the photon energy of the AI K ← L_2,3 transition, are presented.Theoretical arguments and APS spectral simulations using convolutions of total densities of states (DOS), partial densities of states, and BIS data give evidence for the importance of the angular parts of the transition matrix elements, as well as the radial parts. It is found that theoretical estimates of plasmon effects in APS are not in accord with experimental observations.     

Lower and upper variational bounds in calculations of Coulomb and nuclear systems

All formulas that are necessary for deriving not only upper (E _U) but also lower (E _L) variational bounds on the energy of systems featuring a few nonrelativistic particles are obtained with trial functions in the form of expansions in multidimensional Gaussian functions or exponentials. For potentials that are used most widely, all matrix elements are expressed in terms of known functions, a circumstance that simplifies considerably relevant numerical calculations. This is so for systems featuring an arbitrary number of particles in the case of a Gaussian basis and for three-particle systems in the case of an exponential basis. Numerical results for E _U and E _L, which are characterized by record accuracies, are presented for some Coulomb and nuclear systems such as the He atom; the e ⁺ e ^? e ^?, ppμ^?, 3α, and 4α systems; and hypertritium (pnΛ). Lower bounds with exponential trial functions are obtained for the first time (the corresponding formulas are presented for the first time as well); for a Gaussian basis, lower bounds for Coulomb systems have not been known either. Given E _L and E _U, limits within which the exact value of energy, E ₀, lies can be indicated with confidence. Moreover, an analysis of the correlation between E _L and E _U with increasing number of terms in the expansion of the trial function makes it possible to improve the accuracy (at least by one order of magnitude) of the value E _∞ extrapolated to infinity. By considering specific examples, it is shown that the exponential basis is advantageous in relation to the Gaussian one.     

Polarization bremsstrahlung in α decay

A mechanism of formation of electromagnetic radiation that accompanies α decay and is associated with the emission of photons by electrons of atomic shells due to the scattering of α particles by these atoms (polarization bremsstrahlung) is proposed. It is shown that, when the photon energy is no higher than the energy of K electrons of an atom, polarization bremsstrahlung makes a significant contribution to the bremsstrahlung in α decay.     

Concept of formation length in radiation theory

The features of electromagnetic processes are considered which connected with finite size of space region in which final particles (photon, electron–positron pair) are formed. The longitudinal dimension of the region is known as the formation length. If some external agent is acting on an electron while traveling this distance the emission process can be disrupted. There are different agents: multiple scattering of projectile, polarization of a medium, action of external fields, etc. The theory of radiation under influence of the multiple scattering, the Landau–Pomeranchuk–Migdal (LPM) effect, is presented. The probability of radiation is calculated with an accuracy up to “next to leading logarithm” and with the Coulomb corrections taken into account. The integral characteristics of bremsstrahlung are given, it is shown that the effective radiation length increases due to the LPM effect at high energy. The LPM effect for pair creation is also presented. The multiple scattering influences also on radiative corrections in a medium (and an external field too) including the anomalous magnetic moment of an electron and the polarization tensor as well as coherent scattering of a photon in a Coulomb field. The polarization of a medium alters the radiation probability in soft part of spectrum. Specific features of radiation from a target of finite thickness include: the boundary photon emission, interference effects for thin target, multi-photon radiation. The theory predictions are compared with experimental data obtained at SLAC and CERN SPS. For electron–positron colliding beams following items are discussed: the separation of coherent and incoherent mechanisms of radiation, the beam-size effect in bremsstrahlung, coherent radiation and mechanisms of electron–positron creation.     

The 12C+24Mg elastic scattering: An example of anomalous transparency at coulomb barrier energies

Fifteen complete angular distributions of the elastic scattering of ¹²C+²⁴Mg were measured at energies around the Coulomb barrier (E_cm = 10.67–16 MeV). The angular distributions are strongly oscillating and could be well described by an optical potential family, whose real part was determined without continuous ambiguity. The imaginary part of this optical potential is very shallow. At four energies the inelastic scattering angular distributions leading to the 2⁺ state of the ²⁴Mg were also measured and analysed with coupled-channels calculations. The volume integrals of the optical potentials used in the coupled-channels calculations present the threshold anomaly in their energy dependence, with a clear Q-value dependence.     

Study of nuclear-coulomb interference effects in inelastic deuteron scattering on 238U

The angular distributions of elastically and inelastically scattered deuterons from ²³⁸U at E = 17 MeV are compared to coupled-channel calculations. The cross sections at small scattering angles are strongly influenced by nuclear-Coulomb interference effects and allow a simultaneous extraction of nuclear (optical potential) and charge quadrupole deformation parameters. Two different deformed Coulomb potentials and the parameters of the optical model are discussed.     

A comparative study of a Heusler alloy Co2FeGe using LSDA and LSDA+U

We have calculated the on-site Coulomb repulsion (U) for the transition elements Co and Fe. To study the impact of Hubbard potential or on-site Coulomb repulsion (U) on structural and electronic properties the calculated values of U were added on GGA and LSDA. We performed the structure optimization of Co₂FeGe based on the generalized gradient approximation (GGA and GGA+U). The calculation of electronic structure was based on the full potential linear augmented plane wave (FP-LAPW) method and local spin density approximation (LSDA) as well as exchange correlation LSDA+U. The Heusler alloy Co₂FeGe fails to give the half-metallic ferromagnetism (HMF) when treated with LSDA. The LSDA+U gives a good result to prove that Co₂FeGe is a HMF with a large gap of 1.10 eV and the Fermi energy (E_F) lies at the middle of the gap of minority spin. The calculated density of states (DOS) and band structure show that Co₂FeGe is a HMF when treated with LSDA+U.     

The 16O(γ, po) reaction at intermediate photon energies

The ¹⁶O(γ, p_o) reaction has been investigated between photon energies E_γ = 40 and 105 MeV using a bremsstrahlung beam, full angular distributions being obtained at E_γ = 60, 80 and 100 MeV. While cross-section calculations which introduce a residual interaction of Yukawa form agree well with the data, the need for further work is indicated.     

Absorption of electromagnetic energy by slow electrons under scattering from Coulomb centers

Simple analytical expressions are obtained for the rate of the inverse stimulated bremsstrahlung absorption under electron scattering from a Coulomb center with charge Z in the presence of the electromagnetic field. The initial and final values of electron energy are assumed to be small compared to the Rydberg energy Z ² (atomic units are used throughout). Single-photon processes of absorption and induced radiation of photon by electron are treated. It is assumed that the electromagnetic field frequency ω is rather low, so that the condition Zω/p ³ ? 1, where p is the electron momentum, and the condition ?ω ? p ² are valid. However, this frequency is assumed to be fairly high compared to the electron-Coulomb center collision frequency: ω ? v _nei. The dependences of the rates of photon absorption and induced radiation on the angle θ between the direction of incident electron and the electromagnetic field polarization vector (assumed to be linearly polarized) are obtained. It is demonstrated that, for any angles θ, the rate of photon absorption is higher than the rate of induced radiation and, therefore, the Marcuse effect for slow electrons (electromagnetic field amplification) is absent. It is further demonstrated that a slow electron on the average absorbs double ponderomotive energy per collision with an ion (Coulomb center) in Maxwellian plasma. This agrees both with the known results calculation for fast electrons and with the known results of the calculation based on the classical Boltzmann kinetic equation for plasma.     

Coulomb corrections to the parameters of the Landau-Pomeranchuk-Migdal effect theory

Using the Coulomb correction to the screening angular parameter of the Molière multiple scattering theory we obtained analytically and numerically the Coulomb corrections to the quantities of the Migdal LPM effect theory. We showed that the Coulomb correction to the spectral bremsstrahlung rate allows one to eliminate the discrepancy between the predictions of the LPM effect theory and its measurement at least for high-Z targets and also to improve additionally the agreement between the predictions of the LPM effect theory analogue for a thin layer of matter and experimental data.     

Hard photons and neutral pions as probes of hot and dense nuclear matter

The dynamics of heavy-ion collisions is studied in an energy domain in the vicinity of the Fermi energy. The early history of the collision is analyzed from the theoretical and experimental point of view in which the message conveyed by bremsstrahlung photons and neutral pions is exploited. The Boltzmann-Uehling-Uhlenbeck model and the Dubna Cascade Model, both based on similar principles but each adopting different computation technics, are briefly described and their respective predictions are discussed. In particular the emission pattern of bremsstrahlung photons is discussed. The photon production has been measured in the systems ⁸⁶Kr+⁵⁸Ni at 60A MeV, ¹⁸¹Ta+¹⁹⁷Au at 40A MeV and ²⁰⁸Pb+¹⁹⁷Au at 30A MeV and energy spectra, angular distributions and two-photon correlations have been analyzed. We find that bremsstrahlung photons are emitted from two distinct sources that can be correlated with nuclear-matter density oscillations. The properties of photon emission are discussed in terms of collective properties of nuclear matter. The high energy tail of the photon spectrum is interpreted by π⁰ and Δ decay but predominantly by radiative capture of pions. The π⁰ absorption in the nuclear medium is further analyzed by examining their emission pattern.