Relativistic Born Approximation Research Articles

Multiple ionization of atoms by highly charged ions

A method is suggested for quickly and easily estimating multiple ionization (MI) cross sections of heavy atoms colliding with highly charged ions, using the independent-particle model (IPM). One-electron ionization probabilities p(b) are calculated using the geometrical model for values at zero impact parameter b and the relativistic Born approximation for one-electron ionization cross sections. Numerical results of MI cross sections are presented for Ne and Ar atoms colliding with Ar8+, Fe20+, Au24+, Bi67+ and U90+ ions at energies 1 MeV u–10 GeV uand compared with available experimental data and CTMC (classical trajectory Monte Carlo) calculations. The present method of calculation describes experimental dependencies of MI cross sections on the number of ejected electrons m within a factor of two to three. Numerical calculations show that at intermediate ion energies E = 1 − 10 MeV u, the contribution of MI cross sections to the total, i.e. summed over all m values, is quite large ∼35% and decreases with increasing energy.

Extension of PENELOPE to protons: Simulation of nuclear reactions and benchmark with Geant4

Describing the implementation of nuclear reactions in the extension of the Monte Carlo code (MC) PENELOPE to protons (PENH) and benchmarking with Geant4. PENH is based on mixed-simulation mechanics for both elastic and inelastic electromagnetic collisions (EM). The adopted differential cross sections for EM elastic collisions are calculated using the eikonal approximation with the Dirac-Hartree-Fock-Slater atomic potential. Cross sections for EM inelastic collisions are computed within the relativistic Born approximation, using the Sternheimer-Liljequist model of the generalized oscillator strength. Nuclear elastic and inelastic collisions were simulated using explicitly the scattering analysis interactive dialin database for (1)H and ICRU 63 data for (12)C, (14)N, (16)O, (31)P, and (40)Ca. Secondary protons, alphas, and deuterons were all simulated as protons, with the energy adapted to ensure consistent range. Prompt gamma emission can also be simulated upon user request. Simulations were performed in a water phantom with nuclear interactions switched off or on and integral depth-dose distributions were compared. Binary-cascade and precompound models were used for Geant4. Initial energies of 100 and 250 MeV were considered. For cases with no nuclear interactions simulated, additional simulations in a water phantom with tight resolution (1 mm in all directions) were performed with FLUKA. Finally, integral depth-dose distributions for a 250 MeV energy were computed with Geant4 and PENH in a homogeneous phantom with, first, ICRU striated muscle and, second, ICRU compact bone. For simulations with EM collisions only, integral depth-dose distributions were within 1%/1 mm for doses higher than 10% of the Bragg-peak dose. For central-axis depth-dose and lateral profiles in a phantom with tight resolution, there are significant deviations between Geant4 and PENH (up to 60%/1 cm for depth-dose distributions). The agreement is much better with FLUKA, with deviations within 3%/3 mm. When nuclear interactions were turned on, agreement (within 6% before the Bragg-peak) between PENH and Geant4 was consistent with uncertainties on nuclear models and cross sections, whatever the material simulated (water, muscle, or bone). A detailed and flexible description of nuclear reactions has been implemented in the PENH extension of PENELOPE to protons, which utilizes a mixed-simulation scheme for both elastic and inelastic EM collisions, analogous to the well-established algorithm for electrons/positrons. PENH is compatible with all current main programs that use PENELOPE as the MC engine. The nuclear model of PENH is realistic enough to give dose distributions in fair agreement with those computed by Geant4.

Charge-changing processes in collisions of heavy many-electron ions with neutral atoms

Some features of charge-changing processes, namely, electron capture (EC) and electron loss (EL), are considered for heavy many-electron ions colliding with neutral atoms in a wide range of ion energy E = 10 keV/u–100 GeV/u. The discussion is based on cross-section calculations performed by available computer codes, namely, CAPTURE, DEPOSIT and RICODE. The RICODE (Relativistic Ionization CODE), which provides calculation of single-electron loss cross sections in the relativistic energy regime, was recently created on the basis of the relativistic Born approximation and is described in the Appendix A. In addition, a semi-empirical formula for single-electron loss cross sections is suggested based on properties of the Born approximation and numerical calculations by the RICODE program. To cover also the low and intermediate collision energies, EL cross sections are obtained by the recently created DEPOSIT code which provides calculation of single- and multiple-electron as well as the total cross sections. Based on the results obtained by these codes, recommended capture and loss cross sections for heavy ions like xenon, uranium and lead ions colliding with neutral atoms are presented over a wide energy range.

The energy-deposition model: electron loss of heavy ions in collisions with neutral atoms at low and intermediate energies

Single- and multiple-electron loss processes of heavy many-electron ions (positive and negative) in collisions with neutral atoms at low and intermediate energies are considered using the energy-deposition model. The DEPOSIT computer code, created earlier to calculate electron-loss cross sections at high projectile energies, is extended for low and intermediate energies. A description of a new version of the DEPOSIT code is given, and the limits of validity for collision velocity in the model are discussed. Calculated electron-loss cross sections for heavy ions and atoms (N+, Ar+, Xe+, U+, U28+, W, W+, Ge−, Au−), colliding with neutral atoms (He, Ne, Ar, W), are compared with the available experimental and theoretical data at energies E > 10 keV/u. It is found that in most cases the agreement between the experimental data and the present model is within a factor of 2. Combining results obtained by the DEPOSIT code at low and intermediate energies with those by the LOSS-R code at high energies (relativistic Born approximation), recommended electron-loss cross sections in a wide range of collision energy are presented.

Modified Kolbensvedt model for the electron impact K-shell ionization cross-sections of atoms and ions

A modification of the Kolbensvedt model, MKLV, in terms of ionic, correlation (between the incident and target electrons) and relativistic corrections, is proposed to calculate the K-shell electron impact ionization cross-sections of neutral and ionic targets. The modified model, with a single set of parameters, is found to reproduce satisfactorily the experimental data for the neutral H, Al, Ar, Mn, Ge, Mo, Ag, Sn, Au, Bi and U and ionic Li+, B4+ and O7+ targets better than the existing empirical models from threshold energies to as high as 104 MeV. For the Ag target, the calculations from MKLV follow closely the results of Scofield, and for Au, those of Scofield, and Ndefru and Malik, both done in the relativistic Born approximation with the Moller interaction, at energies higher than the peak region.

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.

Excitationof the lowest autoionizingnp5(n+1)s2, 2P3/2,1/2states of Na(n = 2), K(n = 3), Rb(n = 4) andCs(n = 5) byelectron impact

Relativistic distorted-wave (RDW) calculations are carried out for the electron impact excitation of the lowest autoionizing states np5(n+1)s2, 2P3/2,1/2 in Na(n = 2), K(n = 3), Rb(n = 4) and Cs(n = 5) alkali atoms from the ground np6(n+1)s, 2S1/2 state. Detailed results in the range of near threshold to 1.5 keV incident electron energies are obtained for total cross sections of the magnetic substates of the individual 2P3/2 and 2P1/2 states. Utilizing these, the results are presented for their total cross sections, alignment parameter 20 of the 2P3/2 state and the intensity of the ejected electrons from the 2P3/2 state. RDW calculations excluding the exchange effect and relativistic Born approximation calculations are also performed for comparison. The results are compared and discussed in the light of earlier available results and the recently reported experimental data from the Freiburg group (Germany).

RELATIVISTIC EFFECT IN INELASTIC COLLISION OF ELECTRON WITH ATOM OR ION

Based on relativistic Born approximation theory, we have calculated the relati-vistic and non-relativistic generalized oscillator strengthes (GOS) of the Li-liks iso-electron sequence. The difference between the two calculations shows the relativistic effect of both the target and impact electron in the inelastic collision process. For low Z target (e.g. Li), the relativistic effect is negligible. For the 2s?p transition of Au+76, the relativistic GOS is 27.1% smaller than the non-relativistic one. At high impact energy, the contribution of transverse term of GOS to the differential cross section is important at certain angle. At extremely high impact energy (e.g. 106, 107eV), the transverse term of GOS becomes dominant at forward small angle and such forward peak makes overwhelming contribution to the totalinelastic cross section.

Charge equilibrium of relativistic uranium ions in matter

Calculations are made for the electron capture and loss cross sections and the charge state distributions of uranium ions at energies of 200 MeV/nucleon, 437 MeV/nucleon and 962 MeV/nucleon colliding with low- Z and high- Z targets. The charge states of the projectile ions are determined from the equilibrium condition between electron loss from the ion and electron capture from a target atom into the projectile ion. The electron loss cross sections are formulated using the relativistic Born approximation. The electron capture process is considered to be non-radiative electron capture (non-REC) and radiative electron capture (REC) in the relativistic impact velocity range. The non-REC cross sections are calculated by Eichler's formula based on the eikonal theory, while for the evaluation of the relativistic REC process the contributions of internal e +e − pair creation and successive e +e − pair annihilation processes are taken into account in addition to the usual REC photon emission process.

Radiative corrections to the atomic photoeffect

The radiative corrections to the photoeffect are evaluated for the K-shell of hydrogenlike atoms to lowest order in the radiation field. The corrective matrix element is first discussed within the framework of the bound-state interaction (Furry) picture. We then employ the relativistic Born approximation for the Coulomb Dirac propagator and the final continuum wave function of the electron to obtain an expression which is correct to lowest order in ..cap alpha..Z, provided ..cap alpha..Z/..beta.. very-much-less-than 1, where ..beta.. is the velocity of the ejected electron. The renormalization program is carried out completely, and the lowest-order radiative corrections are given explicitly in terms of the familiar first-order invariant functions of QED. The matrix element which results is further analyzed in terms of invariant amplitudes which are expressed as sums of Feynman parameter integrals. Finally, we evaluate the differential cross section assuming the polarizations of the electron and photon are not observed. Infrared divergences are eliminated from this cross section by allowing for the possibility that an unobserved soft photon is emitted along with the photoelectron. Although in general a numerical evaluation is necessary, analytic expressions are given for the low- and high-energy limits of our final expression for the radiative correctionsmore » to the photoeffect. We find that, while the corrections are small at low energy, for incident photons in the range 1--10 MeV and for electrons emitted near the forward direction (finite momentum transfer), the radiative corrections tend to reduce the photoeffect differential cross section by 1.0 to 7.0%.« less

Dependence of atomic photoeffect on a bound-electron magnetic substate

Fl\"ugge, Mehlhorn, and Schmidt have suggested that the dependence of the atomic photoeffect on a bound-electron magnetic substate will have observable consequences for the angular distribution of subsequent Auger electrons. Because of the constraints of parity and time-reversal invariance, this possibility first arises for ejection from ${L}_{\mathrm{III}}$ and ${M}_{\mathrm{III}}$ subshells. We have calculated the relative probabilities of $|{j}_{z}|=\frac{3}{2}$ and $|{j}_{z}|=\frac{1}{2}$ ejection from these subshells, both with the relativistic Born approximation (Gavrila's formulation) and in the nonrelativistic dipole approximation. The ratio of these cross sections can have a dramatic energy dependence, ranging from 7 at high energies to 9/11 in the low-energy domain, and dropping to 1/3 near threshold if there is a Cooper minimum. Screening effects are discussed.

On the properties of spin 2 particles

A theory of spin 2 particles has been worked out. The wave equations are of first order and a symbolic approach which is developed permits easy computation of quantities of physical interest. An interesting application of the sum rules for spin 2 particles is the calculation of the Coulomb scattering in relativistic Born approximation. The theory is also applied to calculate the decay of a spin 2+ particle into three pseudoscalar particles and into two Dirac particles. The bearing of the decay calculations on the spin 2+ hypothesis for the K-meson is discussed.

The Relativistic Theory of the Specific Primary Ionization and Stopping Power and its Application in Helium

Using a relativistic Born approximation due to Bethe, an expression is derived for the collision cross section of relativistic electrons with an atomic shell in terms of Schrödinger wave functions. Then, by comparison with Møller's work for hydrogen, the stopping power and specific primary ionization are obtained as functions of a number of atomic parameters. These parameters are calculated for the case of helium and the results compared with experiment.