Energy and angular distributions in 250 eV electron and positron collisions with argon atom

We present energy and angular differential cross sections for single-ionization in collisions between electrons and positrons with argon atoms at 250 eV. We treat the collision classically using the three body approximation where the target atoms are described within the single active electron approximation using a Garvey model potential and only the outermost electron is involved in the collision dynamics. Our present classical trajectory Monte Carlo model is shown to describe the ionization cross sections reasonably well and agree with existing experimental data. We show that the energy distributions, both for electron and positron impact, have the same shape and structure. In contrast, the angular distributions for electron and positron impact behave completely different which it maybe be attributed to the projectile-target core interaction. We present also the ionization probabilities as a function of impact parameter. We found that for the case of positron impact the distribution is symmetric, while for the case of electron impact the distribution is asymmetric.


Introduction
The understanding of the ionization process in ion-atom collisions is of fundamental interest in fields ranging from atmospheric and interstellar physics to radiation damage of solids, surfaces and biological systems.One area of interest that received much attention in the 1970's and 1980's is the angular and energy distribution of the ejected electrons as this provides basic information about the collision dynamics and has direct application with respect to modeling radiation damage.Although the attention has shifted to triply (fully) differential studies which test theory more rigorously, experimental Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
doubly differential data are presented in an absolute scale rather than the relative scale used for triply differential data.Thus, being able to compare the theoretical and experimental results on an absolute scale is one important justification of the present work.Another is to compare results obtained for both electron and positron impact since various interactions can be studied separately or deduced from the comparisons.Examples of this include being able to separate projectile scattering from electron emission using positron impact and the importance of post-collision processes by comparing electron and positron impact data.
During the past years the cross sections with electron and positron impact have been studied extensively both experimentally [1][2][3][4][5] and using various models and methods such as applying the R-matrix approach [6], various version of the distorted-wave Born approximation (DWBA) [6][7][8][9][10][11] and the convergent close-coupling method [12].Here, not aiming for completeness, we mention only a few representative studies.DuBois et al [1] investigated experimentally how the 1st-and 2nd-order mechanisms influence the differential electron emission in positon and electron collisions with Ar atom.Ren et al presented the experimental tripledifferential cross sections of single ionization at 195 eV electron impact on Ar [2].In a combined experimental and theoretical work of Ren et al the low-energy electron-impact single ionization Ar(3p) has been studied [3].Babij et al measured the direct single-ionization cross section of Ar by positron impact just above the first ionization threshold [4].DuBois, and de Lucio presented experimental triply differential data for 200 eV positron and electron impact ionization of argon [5].Theoretically the R-matrix approach offers a powerful technique in the description of the cross sections.Bartschatt and Burke calculated the total and single differential ionization cross sections, including inner-shells, of argon atom using a two-state R-matrix approach in combination with the DWBA [6].The variation of DWBA was used by many authors to calculate the ionization cross sections of Ar by electron and positron impact [7][8][9][10][11].Recently the convergent close-coupling method was also used to calculate the ionization cross section of Ar by electron impact [12].
In addition to the various quantum mechanical calculations, many classical theoretical calculations have also been used for predicting the cross section in collision between electron and positron with atoms.One of the frequently used classical models is the classical trajectory Monte Carlo (CTMC) method.The CTMC method is used here because ( 1) it has been shown to be quite successful in dealing with a wide variety of processes in ion-atom collisions involving three or more particles [13][14][15][16][17][18][19], (2) it was shown that the method can also be successfully applied for the ionization studies at light particle impact like electrons and positrons [20][21][22][23][24][25][26][27].(3) It has the advantages of being non-perturbative and tracking the projectile and target particles independently, (4) individual interactions can easily be turned on or off in order to study their importance and (5) and it provides information generally not available in quantum mechanical codes such as the impact parameter and correlated information about the velocity components of the scattered projected and ejected electron.The key point of the calculations is the proper description of the collisions.
In this work, which is a prelude to follow up studies of the impact parameter associated with various scattering and ejection angles, we concentrate on the energy and angular differential cross sections for single-ionization in collisions between electron and positrons with argon atoms.The present work concentrates on 250 eV primary energy which was chosen because absolute cross sections for electron impact are available (see [28,29] which presents an overview of the data).We apply the CTMC method to separate and compare the projectile scattering and target emission components and go beyond previous studies by providing impact parameter information of the total ionization.

Theory
In the present work the CTMC simulations were made using a three-body approximation where the many-electron argon atom is replaced by a one-electron atom [23,24].Therefore, in our CTMC model the three particles are the projectile (P), one atomic active target electron (e), and the remaining target ion (T) consisting of the target nucleus and the remainder target electrons.Figure 1 shows the relative position vectors of the three-body collision system.
The three particles are characterized by their masses and charges.We note that this model is the classical analogue of the quantum-mechanical effective single-electron treatment of the collisions in which the electrons are treated equivalently.For the description of the interaction among the particles a central model potential developed by Green [30], which is based on Hartree-Fock calculations, is used.The potential can be written as: where Z is the nuclear charge, N is the total number of electrons in the atom or ion, r is the distance between the nucleus and the test charge q, and The potential parameters ξ and η can be obtained in such a way that they minimize the energy for a given atom or ion.Using the energy minimization, Garvey et al obtained the following parameters for Ar: η = 3.50 and ξ = 0.957 (in atomic units (au) [31].We note that this type of potential has further advantages, because it has a correct asymptotic form for both small (equation ( 3)) and large (equation ( 4)) values of r The Lagrange equation for the three particles can be written as: where and − → r , Z and m are the position vector, the charge and the mass of the noted particle, respectively, and ⃗ r and similar quantities in following equations are velocity vectors.Then the equations of motion can be calculated as: Introducing the relative position vectors ⃗ A =⃗ r e −⃗ r T , ⃗ B = ⃗ r T −⃗ r P , and ⃗ C =⃗ r P −⃗ r e , in such a way that ⃗ A + ⃗ B + ⃗ C = ⃗ 0, after some elementary calculus we can write: ⃗ A and ⃗ B and similar quantities in following equations are acceleration vectors.
These differential equations were integrated with respect to the time as independent variable by the standard Runge-Kutta method for a given set of initial conditions.Equations ( 9) and ( 10) contain 12 coupled first-order differential equations.Therefore, we need to consider and specify 12 initial values of initial conditions.These are the coordinates and the velocities of an internal motion of (T,e) atomic system and the relative projectile ion-atomic center-of-mass motion.The origin of our coordinate-system in the laboratory frame is the centerof-mass of the target atom and the z axis is parallel to the velocity vector of the projectile (see figure 1).The initial relative motion is specified by the velocity of the projectile and the distance between the projectile and the atomic center-of-mass: During our CTMC simulations v p is fixed.The impact parameter must be chosen so that it reproduces a uniform flux of incident particles.Except for elastic collisions we determine a maximum value of impact parameter, b max , such that for impact parameters above b max , the probabilities of the investigated processes are zero or negligible.The initial distance, R, between the projectile ion and target atom is chosen at sufficiently large internuclear separations, where the projectile ion and target atom interactions are negligible.In practice we have used R = (4,5) b max Z P .
The initial electronic state of the target atom is obtained from the microcanonical distribution.These are selected in a similar fashion as described by Reinhold and Falcon [32] for non-Coulombic systems.A microcanonical ensemble characterizes the initial state of the target constrained to an initial binding energy of the given shell: where K 1 is a normalization constant, E 0 is the ionization energy of the active electron, V(A) is the electron and targetcore potential, A is the length of the vector ⃗ A, and µ Te is the reduced mass of particles 'T' and 'e'.According to the equation ( 13), the electronic coordinate is confined to the intervals where the relation is verified.In the following we assume that equation ( 14) has only one root, A 0 .Therefore the values of A are then confined to the single interval 0 < A < A 0 .Potentials satisfying this condition represent the electron-core interaction.In order to generate an initial condition for the active electron, we must perform a transformation from the variables ( ⃗ A, ⃗ A) to a set of uniformly distributed variables completely specifying the initial state of the system given by equation ( 13).This transformation is a combination of two successive changes of coordinates (see [32]), and finally the required distribution can be written as: where and the independent variables are w, ϑ r , ϑ v , ϕ r , ϕ v .We note, that for or A < A 0 , w is always within the interval An initial condition for the active electron now can be easily generated.The random electronic state specified by the binding energy of the electron in the target atom, E 0 , can be selected by five random numbers distributed in the following ranges: The corresponding initial conditions for ⃗ A and ⃗ A are then obtained from the following relations: For practical reasons, at the beginning of our CTMC calculations the values of A and the corresponding values of w, computed numerically from equation ( 16) are tabulated.During the Monte Carlo simulations the particular values of A are selected from this table using interpolation.For a given set of initial conditions the three-body, three-dimensional CTMC calculation is performed as described by Tőkési and Kövér [23].
The energy and angular differential cross-sections were computed with the following formulas: The statistical uncertainty of the cross section is given by In equations ( 21)-( 23) T N is the total number of trajectories calculated for impact parameters less than b max , T (i ) N is the number of trajectories that satisfy the criteria for ionization, and b j (i) is the actual impact parameter for the trajectory corresponding to the ionization process under consideration in the energy interval ∆E and the emission angle interval ∆Ω of the electron.Note that here, unlike previous studies, one of the goals is to investigate individual processes as a function of impact parameter and see how this differs when the various Coulomb forces are reversed by changing the sign of the projectile charge.Our CTMC results are compared on the singly and doubly differential level by comparing with the absolute electron impact measurements of DuBois [28,29].
We performed a large number of classical trajectory simulations based on our 3-body code.While for the electron and Ar(3p) simulations, 3.1 × 10 7 trajectories were followed; for the positron and Ar(3p) collisions, 1.0 × 10 7 individual trajectories were calculated.The multi-electronic Ar atom was modeled by the model potential in equation (1).During the simulation we only account for ionization from the Ar 3p shell.The initial state of the target is characterized by a microcanonical ensemble, which is constrained to an initial binding energy of 0.581 a.u., at a relatively large distance from the collision center, choosing the initial parameters randomly.The differential cross sections were calculated at large separation of the particles after the ionization occurs.The convergence of our calculations were tested at two values of the separation, e.g., the integration of the classical equation of motions were stopped at 1000 a.u. and 100 000 a.u.from the collision center.We found no visible difference in the differential cross sections evaluated at these two distances.Therefore, to minimize computation time the calculations are stopped at 1000 a.u.

Results and discussions
Figure 2 shows the energy differential cross sections of single target ionization in collisions between 250 eV electrons and Ar atoms.Due to the fact that within the CTMC calculations the electrons are distinguishable, we show results in separation, i.e contributions from the target electron and from the scattered projectile electron.
The sum of the electron yields is in reasonably good agreement with the experimental data with the biggest deviation for energies between 30 and 130 eV.For the lower portion of this energy region, the blue curve shows the discrepancy between experiment and theory is most likely due to the ejected target electron contribution which is monotonically decreasing with increasing energy but is an order of magnitude larger than the scattered projectile electron contribution, the dashed red curve, which increases with energy.However, for the higher end of the region showing discrepancies, the ejected and scattered electron contributions are comparable so it is unclear which is responsible for the discrepancy.Overall, figure 2 shows that at low energies the measured electrons can be attributed to the target electrons and for higher energies to the scattered projectile electrons.As a final note, the small oscillations seen in the blue curve and also in some of the following figures is due to the statistical nature of the Monte Carlo simulations which becomes apparent for rare events, i.e. relatively small cross sections.
Figure 3 shows the angular differential cross sections of target single ionization in collisions between 250 eV electron and Ar atom.In this case, the sum of the target electron and the scattered projectile electron distributions agree well with the measured data.The target electron contribution has  no strong angular dependence; it is almost uniform.In contrast, the scattered electrons have a strong angular dependence.Preliminary studies imply that the bump at 160 • is associated with the projectile-target core interaction.We note further that the position of the bump is changed with the incident electron energy.At present we do not understand what is occurring but plan to address this in detail later.
To test the effect of the sign of the projectile we also performed simulations for 250 eV positron impact.Figure 4 shows the energy differential cross sections which have similar shapes and structures both for the target electron contributions and for the scattered projectile as we obtained for electron impact.Figure 5 shows the angular differential cross sections by 250 eV positron impact on argon target.The angular distributions are completely different from what we obtained for the case of electron impact.Here, the scattered positrons decrease monotonically with angle, whereas for electron impact there was an increase at large angles.A scenario that could possibly explain this is that for electron impact the scattered electron orbits around the positively charged ion core and exits in the backward direction whereas, for positron impact, there is no attractive force by the ion core for this to occur.We also note that we do not observe any bump in the scattered positron distribution which also supports the idea of a scattered projectile-target core interaction.In contrast to the nearly isotropic distribution observed for electron impact, the ejected target electron distributions for positron impact show more change as a function of the scattering angles.Figure 6 shows bP(b) for single ionization of Ar(3p) at 250 eV electron and positron impact energy.For comparison, the black dotted curve shows the radial distribution of the argon electrons.For the case of positron impact, the ionization probability has a symmetric, almost Gaussian, shape with a maximum around 1.4 a.u.Whereas for electron impact, it is asymmetric with a maximum around 0.93 a.u.Thus, for electron impact, ionization occurs at the inner portion of the 3p lobe whereas for positron impact it mostly takes place slightly outside the center of the lobe.But the integrals of these curves which are proportional to the total cross sections are almost the same for electron and positron impact.

Conclusions
We have presented studies of the single differential ionization cross sections in collisions between 250 eV electron and positron impact with Ar(3p) target.The calculations were performed classically using the three body CTMC approximation.We found that our present CTMC model, where the target atoms were described within the single active electron approximation, describes reasonably well the ionization cross sections and agrees with existing experimental data.We have shown that the energy distributions, both for electron and positron impact, have the same shape and structure.At the same time, the angular distributions behave completely different which we suggest is associated with a projectiletarget core interaction.Preliminary work suggests that an observed bump in the scattered electron distributions for electron impact is also due to a projectile-target core interaction.Further investigations for a range of impact energies are in progress to investigate and clarify this.The ionization probabilities as a function of impact parameter were also presented.We found different probability distributions for electron and positron impact.For the case of positron the distribution is symmetric, for the case of electron impact the distribution is asymmetric.Further works, using different incident energies, are in progress to clarify and identify the source of the bump in the angular differential cross sections.

Figure 1 .
Figure 1.The relative position vectors of the particles involved in three-body collisions.⃗ A =⃗ re −⃗ r T , ⃗ B =⃗ r T −⃗ r P , ⃗ C =⃗ r P −⃗ re, ⃗ r Te is the position vector of the center-of-mass of the target system, and b is the impact parameter.

Figure 2 .
Figure 2. Energy differential cross sections by 250 eV electron impact on argon target.Solid red circle: experimental data [12, 13], red dashed line: present CTMC results, scattered projectile electron contribution, blue line: present CTMC results, ejected target electron contribution, green line: present CTMC results, sum of the projectile and target electron contribution.

Figure 3 .
Figure 3. Angular differential cross sections by 250 eV electron on argon target.Solid red circle: experimental data [12, 13], red dashed line: present CTMC results, projectile electron contribution, blue line: present CTMC results, target electron contribution, green line: present CTMC results, sum of the projectile and target electron contribution.

4 .
Energy differential cross sections by 250 eV positron impact on argon target.Red line: present CTMC results, positron contribution, blue line: present CTMC results, target electron contribution.

Figure 5 .
Figure 5. Angular differential cross sections by 250 eV positron impact on argon target.Red line: present CTMC results, scattered positron contribution, blue line: present CTMC results, target electron contribution.

Figure 6 .
Figure 6.bP(b) versus b for single ionization of Ar(3p) at 250 eV projectile impact energy.Blue line: electron projectile, red line: positron projectile.The dashed black curve shows the radial distribution of the argon electrons.