Binary-encounter Theory Research Articles

Behavior of Secondary Electrons: Estimation of the Ionization Yield in Poly(hydroxystyrene) under EUV Irradiation Based on Binary Encounter Theory and Continuous Slowing-Down Spectrum

Behavior of Secondary Electrons: Estimation of the Ionization Yield in Poly(hydroxystyrene) under EUV Irradiation Based on Binary Encounter Theory and Continuous Slowing-Down Spectrum

Theoretical partial ionization cross sections by electron impact for production of cations from CH3OH, CO2 and NH3

In the present article, the Plane Wave Born Approximation (PWBA) along with Generalised Oscillator Strength (GOS) proposed by Weizsacker-Williams has been employed to calculate the partial ionization cross sections due to electron impact for production of cations (CH3OH+, CH2OH+, CH2O+, CH3+, CH2+ CHO+, CH+, C+, CO+, OH+, H2+ and H+) from methanol, for production of cations (CO2+, CO+, O+ and C+) from carbon dioxide and for production of cations (NH3+, NH2+, NH+, N+ and H+) from ammonia from the ionization threshold to 1 keV. The use of GOS given by Weizsacker-Williams breaks the expression of cross section into two terms, Bethe cross section and Mott cross section. Both terms depend on Optical Oscillator Strength (OOS). Furthermore, binary encounter theory has also been used. By summing all the partial ionization cross sections, total ionization cross sections are also obtained. The calculated cross sections are compared with the previously available experimental data and theoretical cross sections. The present calculations are found in remarkable agreement with available experimental data and other previous theoretical results.

RBED cross sections for the ionization of atomic inner shells by electron-impact

The relativistic binary-encounter-dipole (RBED) model for electron-impact ionization of atoms combines classical binary-encounter theory and the asymptotic dipole interaction, which is based on the plane-wave Born approximation, with the only non-trivial ingredient being the optical oscillator strength (OOS). Due to the difficulty of obtaining accurate OOSs, the performance of the RBED model has so far not been fully assessed. In the present work we compare RBED inner-shell ionization cross sections (total and differential) of neutral atoms evaluated using three types of OOSs, namely an empirical power-law OOS, analytical hydrogenic OOSs and ab initio OOSs calculated numerically from self-consistent atomic potentials. We find that, compared to the distorted-wave Born approximation (DWBA), the RBED with either hydrogenic or numerical OOSs generally yields more accurate total cross sections (TCSs) than the RBED with the power-law OOS, especially for the most tightly bound shells. In the highly relativistic limit the RBED model does not recover the Bethe asymptotic behavior because of its different energy-dependent prefactor, hence we investigate an alternative prefactor which restores the correct Bethe asymptote. Finally, we suggest multiplying the RBED differential cross sections (DCSs) by the ratio of DWBA to RBED TCSs and verify that this renormalization improves the agreement with the DWBA DCSs.

Abstract ID: 165 Assessment of RBED electron-impact ionization cross sections for Monte Carlo electron transport

Electron-impact ionization cross sections for atoms and molecules are essential for the modelling of radiation effects in a wide range of applications. Current state-of-the-art theoretical calculations are based on the distorted-wave Born approximation (DWBA), which provides an improvement in accuracy and validity range over the plane-wave Born approximation (PWBA) by accounting consistently for exchange effects as well as the distortion of the projectile wave functions by the field of the target atom. However, numerical DWBA calculations are elaborate and slow, and tabulated data only exist for integrated cross sections, whereas differential cross sections are also needed in Monte Carlo simulations. The relativistic binary-encounter-dipole (RBED) model combines classical binary-encounter theory with Bethe’s PWBA-based dipole model without the need for any empirical adjustable parameters [1] . The analytical RBED model provides integrated as well as differential cross sections and is much simpler than the DWBA [2] , requiring as input only the binding energy, average kinetic energy, and optical oscillator strength (OOS) for each atomic subshell. Due to the difficulty of obtaining accurate OOSs, simplified models (known as RBEB and RBEQ) which assume a simple functional form for the OOS have also been proposed, and have been employed almost exclusively in studies since then. Here we calculate electron-impact ionization cross sections for a set of atoms spanning the periodic table using the more accurate RBED model, relying on OOSs obtained numerically with self-consistent potentials as described in [3] . Compared to RBEB, we find that the RBED results show better agreement with DWBA data for the K-shell of all atoms and over the entire energy range. The improvement is more pronounced for lower-Z elements, where the agreement between RBED and DWBA is near perfect. For higher shells, there is also better agreement at high energies for all elements, but near the threshold the improvement is less consistent and depends on the specific atomic shells. The disagreements between the RBED and DWBA cross sections near the threshold are due to the interplay between Coulomb and exchange effects, and highlight the limitations of using semi-empirical scaling factors. In the near future we will compile tables of RBED data for biologically-relevant elements, and ultimately implement them in Monte Carlo track-structure codes.

A Monte Carlo track structure code for electrons(~10 eV-10 keV) and protons (~0.3-10 MeV) in water:partitioning of energy and collision events

An event-by-event Monte Carlo simulation code for track structure studies isdescribed. In the present form the code transports protons(~0.3-10 MeV) and electrons (~10 eV-10 keV) in a watermedium in the gas phase approximation. For the type of particles and energyrange considered, ionization, electronic excitation and electron elasticscattering are the most important collision events accounted for in thetransport simulation. Efforts were made to ensure that the analyticrepresentation of the various interaction cross sections rests on wellestablished experimental data and theory. For example, the secondary-electronspectrum as well as partial and total ionization cross sections arerepresented by a semitheoretical formulation combining Bethe's asymptoticexpansion and binary-encounter theory. Binding effects for five levels ofionization and eight levels of electronic excitation of the water molecule areexplicitly considered. The validity of the model cross sections is examinedagainst available experimental data and theoretical predictions from othersimilar studies. Results pertaining to the partitioning of energy loss andinteraction events for the first-collision probability and nanometre-sizetrack segments are presented.

Electron-impact total ionization cross sections of silicon and germanium hydrides

Electron-impact total ionization cross sections of some silicon and germanium compounds have been calculated by applying a new theoretical model that has been found to be reliable for a wide range of molecules. The new theory, the binary-encounter-Bethe (BEB) model, combines the binary-encounter theory and the Bethe theory for electron-impact ionization, and uses simple theoretical molecular orbital data—binding energies, average kinetic energies, and occupation numbers—which are readily available from molecular structure codes. Total ionization cross sections of SiH, SiH2, SiH3, SiH4, Si2H6, Si(CH3)4, GeH, GeH2, GeH3, GeH4, and Ge2H6 are presented for incident electron energies T from threshold to 1 keV, and compared to available experimental data. Theory and experiment agree well for SiHx, x=1–4, from thresholds to T<80 eV, while theoretical peaks occur at lower T than experimental peaks for SiHx, x=1–3. No experimental data are available for germanium hydrides for comparison. The theoretical cross sections are given by a compact analytic form suitable for applications in plasma processing.

CEMS and TEM investigations of iron implanted with nitrogen

It is well known that the depth distribution of nitrogen implanted into iron depends strongly on the implantation conditions. Parameters such as implanted dose or the temperature of the sample during implantation play an important role. At room temperature and low dose of implantation, the depth distribution of nitrogen is almost Gaussian and well predicted by the binary encounter theory. When the temperature during implantation is increasing (above 80°C) diffusion process takes place and the depth profile of nitrogen is split into two components: a bulk component which corresponds to nitrogen trapped in interstitial sites of the host crystal and a surface peak due to diffusion mechanism during the implantation. Resonant Nuclear Reaction Analysis (RNRA) has been used to depth profile nitrogen and Conversion Electron Mössbauer Spectroscopy (CEMS) as well as Transmission Electron Microscopy (TEM) have been used to characterize this surface component. Both techniques reveals the nature of the precipitation sites where nitrogen is trapped during diffusion process.

Electron-impact ionization cross sections of atmospheric molecules

A theoretical model for electron-impact total ionization cross sections, which has been found to be reliable for a wide range of molecules, is applied to molecules of interest to atmospheric science. The new theory, the binary-encounter-Bethe (BEB) model, combines the binary-encounter theory and the Bethe theory for electron-impact ionization, and uses simple theoretical data for the ground state of the target molecule, which are readily available from molecular structure codes. Total ionization cross sections of 11 molecules, CS, CS2, COS, CH4, H2S, NH3, NO2, N2O, O3, S2, and SO2, are presented for incident electron energies from threshold to 1 keV with an average accuracy of 15% or better at the cross section peak. We also found that the use of vertical ionization potentials (IPs) rather than adiabatic IPs for the lowest IPs significantly improves BEB cross sections between the threshold and cross section peak for molecules whose adiabatic and vertical IPs are different by ~1 eV or more (CH4 and NH3). The BEB cross sections are presented in a compact analytic form with a small number of constants, making the cross sections suitable for modeling applications.

Electron inelastic mean free path and stopping power modelling in alkali halides in the 50 eV–10 keV energy range

A model for calculating the electron inelastic mean free path and stopping power in insulators in the 50 eV–10 keV energy range is presented. Both valence and core electron contributions have been considered. The valence part has been estimated following the dielectric theory modified to include the energy gap; the core contribution has been evaluated on the basis of the classical binary encounter theory. Inelastic mean free path and stopping power calculations based on this model have been performed for several alkali halides: LiF, NaCl, KCl and CsI. They are compared to existing experimental data and Penn model’s predictions for the mean free path and to Bethe’s values for the stopping power; a fair agreement is found for incident electron energies higher than 100-200 eV.

New model for electron-impact ionization cross sections of molecules

A theoretical model for electron-impact ionization cross sections, which has been developed primarily for atoms and atomic ions, is applied to neutral molecules. The new model combines the binary-encounter theory and the Bethe theory for electron-impact ionization, and uses minimal theoretical data for the ground state of the target molecule, which are readily available from public-domain molecular structure codes such as GAMESS. The theory is called the binary-encounter Bethe (BEB) model, and does not, in principle, involve any adjustable parameters. Applications to 19 molecules, including H2, NO, CH2, C6H6, and SF6, are presented, demonstrating that the BEB model provides total ionization cross sections by electron impact from threshold to several keV with an average accuracy of 15% or better at the cross section peak, except for SiF3. The BEB model can be applied to stable molecules as well as to transient radicals.

Triton burnup in plasma focus plasmas

Pure deuterium plasma discharge from plasma focus breeds 1.01 MeV tritons via the D(d,p)T fusion branch, which has the same cross section as the D(d,n)3He (En=2.45 MeV) fusion branch. Tritons are trapped in and collide with the background deuterium plasma, producing 14.1 MeV neutrons via the D(t,n)4He reaction. The paper presents published in preliminary form as well as unpublished experimental data and theoretical studies of the neutron yield ratio R=Yn(14.1 MeV)/Yn(2.45 MeV). The experimental data were obtained from 1 MJ Frascati plasma focus operated at W=490 kJ with pure deuterium plasma (in the early 1980s). Neutrons were monitored using the nuclear activation method and nuclear emulsions. The present theoretical analysis of the experimental data is based on an exact adaptation of the binary encounter theory developed by Gryzinski. It is found that the experimentally defined value 1⋅10−3<R<3⋅10−3 can be explained theoretically only if one considers that tritium burnup occurs in the plasma domains of very high density (n≳1021 cm−3), high temperature (kT≳1 keV), and short trapping time (t0≤20 ns). These domains are known as efficient traps of MeV ions but are not the main source of D(d,n)3He fusion.

Binary-encounter-dipole model for electron-impact ionization.

A theoretical model, which is free of adjustable or fitted parameters, for calculating electron-impact ionization cross sections for atoms and molecules is presented. This model combines the binary-encounter theory with the dipole interaction of the Bethe theory for fast incident electrons. The ratios of the contributions from distant and close collisions and interference between the direct and exchange terms are determined by using the asymptotic behaviors predicted by the Bethe theory for ionization and for stopping cross sections. Our model prescribes procedures to calculate the singly differential cross section (energy distribution) for each subshell using the binding energy, average kinetic energy, and the differential dipole oscillator strengths for that subshell. Then the singly differential cross section is integrated over the ejected electron energy to obtain the total ionization cross section. The resulting total ionization cross section near the threshold is proportional to the excess energy of the projectile electron. We found that this model yields total ionization cross sections for a variety of atoms and molecules from threshold to several keV which are in good agreement (\ensuremath{\sim}10% or better on average) with known experimental results. The energy distributions also exhibit the expected shapes and magnitudes. We offer a simpler version of the model that can be used when differential oscillator strengths are not known. For the ionization of ions with an open-shell configuration, we found that a minor modification of our theory greatly improves agreement with experiment.

Collisional processes in muon catalyzed fusion as studied by the classical collision theory

Collisional processes involving a negative muon in the deuterium and tritium system were studied using the classical binary encounter theory. The time needed for slowing down of a 10 keV muon was found to be of the order of 10−8s to 10−12s, depending on the density of the system. The “Sticking Probabilities” for the d-t and d-d fusions were obtained to be 0.48% and 10.2%, respectively. The usefulness of the classical model for understanding fundamental processes in muon catalyzed fusion is suggested.

Fluctuation asymptotics of hopping quenching in disordered systems.

The impurity quenching of incoherent excitation accelerated by resonance migration in solid solutions is studied. It is shown that a lower bound for the survival probability N(t) is established by so-called encounter theory, which is linear in the quencher concentration and predicts an exponential decay at the end of the migration-accelerated stage following a short static decay. Another lower bound, which is the probability of not leaving the site where the excitation was initially created, decays much more slowly at the very end of a process. The crossover point of these lower bounds establishes an upper time boundary ${\mathit{t}}_{\mathit{f}}$ for binary approximation. The real quenching initially well described by binary-encounter theory is completed by the ``fluctuation asymptotics'' of quenching, which is multiparticle in principle. In the hopping quenching limit, the continuous-time random-walk equation is used to find the starting point ${\mathit{t}}_{\mathit{f}}$ of the fluctuation stage and the share of excitations that have survived at that time N(${\mathit{t}}_{\mathit{f}}$). When migration is fast enough the latter is shown to be small, and the binary description of preceding stages proves to be good. However, with a decrease of the donor concentration, an intermediate migration-accelerated stage is ousted by multiparticle static quenching from one side and the fluctuation asymptotics from another.

Theoretical Study of W Values in Hydrocarbon Gases

The recent measurements by D. Srdoc, B. Obelić, and I. Krajcar Bronić (J. Phys. B, 20, 4473-4484, 1987) and by I. Krajcar Bronić, D. Srdoc, and B. Obelić (Radiat. Res., 115, 213-222, 1988) revealed nonsmooth, oscillatory variation of the W value for low-energy electrons as a function of the number of carbon atoms in normal alkanes. This behavior is contrary to what has been expected generally and prompted us to undertake the present study. Calculations were carried out using the continuous-slowing-down approximation for the degradation spectrum, the binary-encounter theory for cross sections, and relevant molecular properties, to explain the behavior of the W value. The main contributor to the oscillatory variation is the branching ratio for neutral dissociation to ionization of super-excited states. We also present an interpretation of the trend of the W value in the series C2H2, C2H4, and C2H6.

Erratum: Bethe stopping theory for a harmonic oscillator and Bohr's oscillator model of atomic stopping

Bethe's expressions for the stopping cross section and the straggling parameter of a penetrating point charge have been evaluated for a spherical harmonic oscillator as a target. The results, which are rigorous except for the neglect of relativistic corrections and higher-order Born terms, are given in tabulated form as well as in the form of asymptotic shell-correction expansions to arbitrary order. Existing general expressions for the stopping cross section and straggling are confirmed as far as they go. The range of validity of various model theories of atomic stopping is tested. At velocities down to about the stopping maximum, the agreement is very good for the kinetic theory, while a Fermi gas with a properly chosen electron density yields a significantly different stopping maximum. The binary encounter theory underestimates the stopping power at all speeds. The dielectric or local-density approach does not reproduce the correct scaling behavior. None of the model theories reproduces the threshold behavior. For light projectiles, i.e., positrons and electrons, a Franck-Hertz-type structure is observed in the velocity dependence of the stopping cross section which is particularly pronounced near threshold. An equipartition rule is derived for contributions to the stopping cross section from low excitations, i.e., distant collisions, and close encounters. A straight extension of Bohr's oscillator model of atomic stopping is proposed which is shown to reproduce leading shell corrections in both stopping power and straggling and which, when applied to hydrogen, accurately describes the stopping power at velocities around and above the stopping maximum.

Bethe stopping theory for a harmonic oscillator and Bohr’s oscillator model of atomic stopping

Bethe's expressions for the stopping cross section and the straggling parameter of a penetrating point charge have been evaluated for a spherical harmonic oscillator as a target. The results, which are rigorous except for the neglect of relativistic corrections and higher-order Born terms, are given in tabulated form as well as in the form of asymptotic shell-correction expansions to arbitrary order. Existing general expressions for the stopping cross section and straggling are confirmed as far as they go. The range of validity of various model theories of atomic stopping is tested. At velocities down to about the stopping maximum, the agreement is very good for the kinetic theory, while a Fermi gas with a properly chosen electron density yields a significantly different stopping maximum. The binary encounter theory underestimates the stopping power at all speeds. The dielectric or local-density approach does not reproduce the correct scaling behavior. None of the model theories reproduces the threshold behavior. For light projectiles, i.e., positrons and electrons, a Franck-Hertz-type structure is observed in the velocity dependence of the stopping cross section which is particularly pronounced near threshold. An equipartition rule is derived for contributions to the stopping cross section from low excitations, i.e., distant collisions, and close encounters. A straight extension of Bohr's oscillator model of atomic stopping is proposed which is shown to reproduce leading shell corrections in both stopping power and straggling and which, when applied to hydrogen, accurately describes the stopping power at velocities around and above the stopping maximum.

Investigation of hydrogen stopping in noble metals around the stopping power maximum

The stopping cross sections of copper, silver and gold for protons and deuterons were measured in the energy range from 50 keV/amu to 500 keV/amu. All measurements were performed on supported films evaporated onto polished backings employing both transmission and backscattering type techniques. By use of different hydrogen isotopes — assuming velocity scaling — and of different target thicknesses we showed that plural and multiple scattering do not influence our results within the experimental uncertainties of ± 3%. As a cross check the stopping ratios were also measured by evaluation of the heights of the backscattering spectra and were found to be consistent with the absolute values. Our results are compared to the theory of Lindhard-Winther using different solid state charge distributions and to the binary encounter theory of Kührt and Wedell. Best agreement is achieved for those calculations which take the correct density of the valence electrons into account. Furthermore, our results are compared to the fitted data by Andersen and Ziegler (1977), by Janni (1982) and by Ziegler (to be published). In general, better agreement with the more recent fit-data was found.

Single and double ionization of neon 2s- and 2p-subshells by proton and electron impact

Absolute single and double ionization cross sections of neon 2s- and 2p-subshells for proton (40–900 keV) and electron impact (0.2–10 keV) have been measured using photon spectroscopy in the spectral range of the vacuum ultraviolet. Cross sections for double ionization decrease more rapidly with increasing impact energy than cross sections for single ionization. No definite asymptotic energy dependence of a Bethe-Fano-plot could be found for double ionization in contrast to single ionization. The experimental results are compared with theoretical predictions of the shake-off model and Gryzinski's classical binary encounter theory. Better agreement is found with the latter, indicating that successive binary collisions have to be considered as a strong mechanism for double ionization by protons or electrons of the investigated energy range. Comparison is made with other experimental results for double ionization by photon impact or capture ionization by proton impact.

Relativistic binary-encounter collision and stopping theory: general expressions

An expression for the energy transfer between two particles moving with relativistic velocities has been given on the basis of a binary-encounter description of elastic scattering. By means of a Lorentz transformation of the differential scattering cross section in the center-of-mass system the differential cross section and the total cross section have been established for relativistic particle velocities. Expressions have been derived that relate stopping power and straggling for a projectile penetrating a medium in internal motion to the stopping parameters of a medium at rest. These expressions contain the velocity distribution of target particles, kinematic factors and the Moller speed.