Sommerfeld-Maue Wave Function Research Articles

Analytic calculations of electron–nucleus bremsstrahlung cross sections above a few MeV including higher-order corrections and multiple scattering in the target

Calculations of the elementary process of electron–nucleus bremsstrahlung employing a partial–wave expansion of the initial and final electron wave functions are exact and offer results that match the experimental data, but they become impractical for increasingly higher electron energies, especially in some regions of the emitted photon phase space. Here we take the opposite approach and use analytic expressions based on the Olsen–Maximon–Wergeland additivity rule, valid above few MeV, to decouple the calculation of the screening and Coulomb corrections. The former is obtained from the first Born approximation, while the latter is evaluated within the formalism of the Furry–Sommerfeld–Maue wave functions using the leading–order expression by Elwert and Haug and the next–to–leading–order corrections by Roche, Ducos, and Proriol, which is presently the best way to calculate cross sections for electrons with energies in the range of hundreds of MeV. A detailed discussion of the approximations involved and an extensive comparison with experimental data have been published in previous works. Here we focus on the numerical procedures that have been developed and on the stringent cross checks that have been performed to ensure the accuracy of the results. Published benchmark calculations for bremsstrahlung in a pure Coulomb field, that stop at 100 keV, are extended here to 100 MeV by making available selected values that could be used by other independent researchers to validate their programs. We also discuss the numerical details of the procedure adopted to convolve the bremsstrahlung cross section with the impinging electron angular distribution due to multiple scattering in the target.

On higher–order corrections to nuclear bremsstrahlung cross sections

A quantitative understanding of bremsstrahlung by electrons in the screened field of atomic nuclei has been developed by several researchers over the last 80 years. However, some ranges of the possible parameters can still not be covered with good accuracy. Dirac partial-wave calculations are the best method developed to date, while analytic calculations are the only viable solution at high impinging electron energies. Recent progresses on both fronts will be discussed. In particular, state of the art partial wave results, including screening, allowed to test the Olsen–Maximon–Wergeland additivity rule discovered by analytic means. Closed expressions for the next-to-leading order corrections to the Furry–Sommerfeld–Maue wave functions are known. One next-to-next-to-leading order term is typically included in this type of approach. The effect of this extra term on the angular dependence of the double differential cross sections for photon emission by 2.5 MeV electrons is compared in detail with partial wave calculations which are exact at all orders.

Radiative-electron-capture-to-continuum cusp inU88++N2collisions and the high-energy endpoint of electron-nucleus bremsstrahlung

The radiative electron capture of a target electron into the projectile continuum has been studied for the collision system ${\text{U}}^{88+}+{\text{N}}_{2}\ensuremath{\rightarrow}{\text{U}}^{88+}+{[{\text{N}}_{2}^{+}]}^{*}+{e}^{\ensuremath{-}}+\ensuremath{\gamma}$ at 90 MeV/$\mathrm{u}$. Using a magnetic electron spectrometer, the energy distribution of cusp electrons emitted under an angle of ${0}^{\ensuremath{\circ}}$ with respect to the projectile beam and with a velocity close to the projectile velocity has been measured in coincidence with the emitted photons under various observation angles. The experimental results provide a stringent test for the corresponding process in inverse kinematics, namely, the theory of electron-nucleus bremsstrahlung at the high-energy endpoint. For comparison this process is calculated using fully relativistic Dirac wave functions and using semirelativistic Sommerfeld-Maue wave functions.

Relativistic antihydrogen production by pair production with positron capture

Antihydrogen atoms may rarely be produced by the collision of antiprotons with ions. At relativistic velocities, the antiproton may pass around the Coulomb field of the nucleus and the electron–positron pairs can be produced electromagnetically. After this pair production, not so often, positron can be captured by the antiproton and as a result, antihydrogen atoms may be produced. In this work, we have calculated the antihydrogen production cross section in the framework of perturbation theory, by applying Monte-Carlo integration techniques. In order to compute the lowest-order Feynman diagrams amplitudes, we used Darwin wave functions for the bound states of the positrons and Sommerfeld–Maue wave functions for the continuum states of the electrons.

Polarization correlations in electron-nucleus bremsstrahlung: the short-wavelength limit

The short-wavelength limit of the electron-nucleus bremsstrahlung is re-investigated with special emphasis on the polarization correlations between incoming electron and emitted photons. A theoretical analysis of these correlations is performed within both the rigorous relativistic Dirac theory and the Dirac-Sommerfeld-Maue (DSM) approach which approximates the initial electronic state by a Sommerfeld-Maue wavefunction. Based on detailed calculations carried out for bremsstrahlung of electrons scattered from medium- and high-Z bare ions, we argue that the DSM approach is complementary to the exact partial-wave theory at sufficiently high collision energies. For such high-relativistic domains predictions are made for the polarization correlations relating to linearly polarized radiation.

Asymmetry of bremsstrahlung by moderately relativistic polarized electrons

The asymmetry of bremsstrahlung by transversely polarized electrons using Sommerfeld–Maue wave functions has been calculated. In order to simplify the final formula the relativistic approximation has been used. The results obtained show a linear dependence of the asymmetry on the atomic number Z for the range 4≤Z≤26. It is shown that the approach developed gives the zeroth asymmetry for ultrarelativistic energies as expected but for initial electron energies ∼5 MeV the asymmetry may achieve a magnitude ∼8% (for Z=26). Polarimetry of electrons with energy higher than 1 MeV (where conventional Mott polarimeters have large experimental difficulties) may be carried out using the bremsstrahlung process providing a reasonable asymmetry and high cross section.

Cross section of electron-nucleus bremsstrahlung near the short-wavelength limit

Using the partial-wave formalism the cross section of bremsstrahlung for a pure Coulomb field is calculated for various atomic numbers of the target nucleus and various incident electron energies. The results are compared to those of previous theories. For intermediate and large atomic numbers the latter cross sections undervalue significantly the present ones. Except for low atomic numbers the cross sections in Born approximation with the Coulomb correction by Elwert yield more accurate results than the cross sections obtained by using the approximate Sommerfeld-Maue wave function. The effect of screening is small for low atomic numbers. At nonrelativistic energies the number of required partial waves is rather large for target nuclei with high atomic numbers, contrary to the expectations. The pertinent results exceed the numerical values including screening by factors of up to five. They verify the assumption that the nonrelativistic cross section of Sommerfeld is fairly accurate even for high atomic numbers.

Bound-free electron-positron pair production in relativistic heavy-ion collisions

The bound-free electron-positron pair production is considered for relativistic heavy ion collisions. In particular, cross sections are calculated for the pair production with the simultaneous capture of the electron into the $1s$ ground state of one of the ions and for energies that are relevant for the relativistic heavy ion collider and the large hadron colliders. In the framework of perturbation theory, we applied Monte Carlo integration techniques to compute the lowest-order Feynman diagrams amplitudes by using Darwin wave functions for the bound states of the electrons and Sommerfeld-Maue wave functions for the continuum states of the positrons. Calculations were performed especially for the collision of $\text{Au}+\text{Au}$ at 100 GeV/nucleon and $\text{Pb}+\text{Pb}$ at 3400 GeV/nucleon.

Semirelativistic model for ionization of atomic hydrogen by electron impact

We present a semirelativistic model for the description of the ionization process of atomic hydrogen by electron impact in the first Born approximation by using the Darwin wave function to describe the bound state of atomic hydrogen and the Sommerfeld-Maue wave function to describe the ejected electron. This model, accurate to first order in $Z∕c$ in the relativistic correction, shows that, even at low kinetic energies of the incident electron, spin effects are small but not negligible. These effects become noticeable with increasing incident electron energies. All analytical calculations are exact and our semirelativistic results are compared with the results obtained in the nonrelativistic Coulomb Born approximation both for the coplanar asymmetric and the binary coplanar geometries.

RELATIVISTIC CALCULATION OF THE TWO-PHOTON BREMSSTRAHLUNG PROCESS

We present analytical and numerical results on the relativistic calculation of the two-photon bremsstrahlung process. For the approximate calculation of relativistic free–free two-photon matrix elements in the central field, we applied the method based on the accurate treatment of the diagonal singularities in the relativistic free–free single-photon matrix elements. Numerical calculations are carried out with the Furry–Sommerfeld–Maue wave functions. The obtained results are compared with the results of the relativistic Born approximation.

Asymmetry of bremsstrahlung from polarized electrons

The triply differential cross section of unpolarized bremsstrahlung produced by polarized electrons in the Coulomb field of a nucleus is calculated with the aid of Sommerfeld-Maue wave functions. The photon asymmetry originating from transversely polarized electrons is calculated for various values of energies and angles, in particular for the parameters of recent experiments. As a function of the atomic number Z of the target nucleus, the asymmetry shows a distinct deviation from a linear dependence, in contrast to a previous calculation exact to lowest order in the Coulomb parameter Z/137. The cross-section formula is integrated numerically over the angles of the outgoing electrons to obtain the photon asymmetry observed in non-coincidence experiments.

Angular distribution of electron-positron pairs produced in relativistic heavy-ion collisions.

Angular distributions of free electron-positron pairs produced in relativistic ion-atom collisions are calculated within a first-order Born approximation. The electron and positron continuum states in the field of the high-Z target nucleus are described by relativistic Sommerfeld-Maue wave functions. The cross-section dependence on the polar angles and on the lepton energies is calculated for several experimentally relevant cases. The azimuthal anisotropy for given polar angles is also presented.

Electron-positron pair production in relativistic heavy-ion collisions.

First-order Born cross sections for electron-positron pair production in relativistic heavy-ion collisions are calculated. Sommerfeld-Maue wave functions are employed to describe the continuum state of the emitted electron and positron in the field of the heavy target nucleus. Double-differential, single-differential, and total cross sections are presented in a wide energy regime from 1 GeV/u up to 20 TeV/u. We compare our results with first-Born calculations using a partial-wave expansion and with lowest-order QED calculations. The target-charge dependence is investigated.

Creation of e--e+ pairs in the target field followed by e- capture in the target

Electron capture in a bound state of the target following e--e+ production in the same centre is studied for collision between bare heavy ions at relativistic impact energies. The process is described by using a first-order Born approximation where Darwin and Sommerfeld-Maue wavefunctions are introduced to represent the electron and positron states. Comparisons are made for single differential cross sections, as a function of the final positron energy, with other existing theoretical calculations.

High-energy wave functions for the screened Coulomb field

An eikonal formalism is developed to calculate corrections due to screening to the exactly solvable case of the unscreened Coulomb field. The high-energy Sommerfeld-Maue wave functions are used, and, by construction, the unscreened case is reproduced exactly for all coordinate space. The modification caused by screening is found to be multiplicative, and it appears as a phase factor $\mathrm{exp}{\ensuremath{\mp}\mathrm{iZ}{e}^{2}(\frac{E}{p})\mathrm{ln}[p(r\ifmmode\pm\else\textpm\fi{}z)]\ensuremath{-}i(\frac{E}{p})\ifmmode\times\else\texttimes\fi{}\ensuremath{\int}{\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}}^{z}V(\ensuremath{\rho}, \ensuremath{\xi})d\ensuremath{\xi}}$, where $p$ is a large momentum, $E$ is the corresponding relativistic energy, and $V$ is the screened Coulomb field. Our approximate high-energy wave function is given by one formula for all coordinate values, without any need to introduce partial waves, is easy to calculate numerically, and is shown to satisfy a second-order differential equation to a very good accuracy practically in the whole space. For example, the outgoing solution satisfies an appropriate equation to an accuracy better then 3% in a very large region of cylindrical variables $\ensuremath{\rho}\ensuremath{\ge}0.1$, $z\ensuremath{\le}2p$, which expands with increasing $p$.

Modified Sommerfeld-Maue wave function: Application to the pair-production process

A solution of the Dirac equation for an electron in a point-Coulomb potential is given in the form of an expansion in the parameter $\frac{1}{r}$. The main part of the solution is the Sommerfeld-Maue wave function. We introduce a correction term. All orders in $\ensuremath{\alpha}Z$ are kept. The given solution (Sommerfeld-Maue plus correction term) contains the exact contributions of orders 0 and 1 in the expansion in $\frac{1}{r}$. This modified Sommerfeld-Maue wave function is used to get the cross section for the pair-production process by $\ensuremath{\gamma}$ rays in the field of a nucleus. The cross section is written in the form of the Fink-Pratt cross section plus a correction term. The formula is valid for all $\ensuremath{\alpha}Z$. The energetic relative accuracy should be of order $\frac{1}{{E}^{2}}$. The small-angle approximation is not introduced.

Comment on the high-frequency limit of the bremsstrahlung cross section

Referring to recent papers the validity of the bremsstrahlung cross section at the high-frequency limit, as obtained by use of the Sommerfeld-Maue wave function, is discussed.

Second-Order Calculations of the Single-Quantum Annihilation Cross Sections in Various Atomic Shells

A general method for calculating the cross sections for single-quantum annihilation (SQA) of positrons in various atomic shells, valid to second order in the $\ensuremath{\alpha}Z$ expansion (one order beyond the leading term), is presented with the help of the modified Sommerfeld-Maue wave function for incoming positrons and the relativistic wave functions for the bound electrons in any atomic shells. It is shown in this approximation that the differential cross sections are of the form ${\ensuremath{\sigma}}_{\mathrm{njl}}(k, q)={C}_{\mathrm{nl}}{\ensuremath{\sigma}}_{{n}_{0}\mathrm{jl}}(k, q)$, where ${n}_{0}=l+1$, and $k$ and $q$ are the photon energy and the magnitude of momentum transfer to a nucleus, respectively. The factor ${C}_{\mathrm{nl}}$ is evaluated in general. The SQA cross sections are calculated for the $1{s}_{\frac{1}{2}}$, $2{p}_{\frac{1}{2}}$, and $2{p}_{\frac{3}{2}}$ states to 2nd order in the $\ensuremath{\alpha}Z$ expansion and for the $3{d}_{\frac{3}{2}}$ and $3{d}_{\frac{5}{2}}$ states to lowest order in the $\ensuremath{\alpha}Z$ expansion, and are in closed form. The relationship between the SQA cross section for any state and the corresponding cross section for the atomic photoeffect is discussed. The leading term for the SQA cross section is closely related to that for the photoeffect, but there is a distinct difference between the correction terms for the former and the latter. The comparison of the results of the present work for the $K$ and ${L}_{\mathrm{I}}$ shells with a recent numerical calculation done by others shows that for low and intermediate energies, the above factorization rule is valid within 1.4% for $Z=47$ and 3.8% for $Z=90$. However, the $1{s}_{\frac{1}{2}}$ cross section for $Z\ensuremath{\geqq}82$ is negative for any energy below 2.264 Mev.

Approximate scattering solutions of the dirac equation for electron-nucleus processes in light nuclei

Scattering solutions of the second-order Dirac equation for the case of the Coulomb potential and which are correct to first order in the coupling constantZe2/hc are investigated and found to describe pure Coulomb scattering equally well as the Sommerfeld-Maue wave functions. Errors introduced by the use of these solutions are studied in a numerical calculation of cross sections for nuclear electric-quadrupole excitation by high-energy electrons. The use of these wave functions is suggested for simplified calculations of lowest-order Coulomb corrections to Born approximation results for various electron-nucleus processes.

Coulomb Scattering of Polarized Electrons

An approximate analytical expression for the spin dependent cross section for Coulomb scattering, valid to order ( alpha Z)/sup 4/, is developed. The development is based on a modification of the Sommerfeld-Maue wave function. The ( alpha Z)/sup 2/ and ( alpha Z)/sup 3/ terms in the resulting cross section are identical with the corresponding terms found by the second Born approximation. The ( alpha Z)/sup 4/ term is verified by analytically summing the corresponding term in the Mott series. Graphs are presented which compare the approximate ana lytical expressions for both the cross section and asymmetry function with exact numerical results. (auth)