Scaling and Formulary cross sections for ion-atom impact ionization

The values of ion-atom ionization cross sections are frequently needed for many applications that utilize the propagation of fast ions through matter. When experimental data and theoretical calculations are not available, approximate formulas are frequently used. This paper briefly summarizes the most important theoretical results and approaches to cross section calculations in order to place the discussion in historical perspective and offer a concise introduction to the topic. Based on experimental data and theoretical predictions, a new fit for ionization cross sections is proposed. The range of validity and accuracy of several frequently used approximations (classical trajectory, the Born approximation, and so forth) are discussed using, as examples, the ionization cross sections of hydrogen and helium atoms by various fully stripped ions.


I. INTRODUCTION
Ion-atom ionizing collisions play an important role in many applications such as heavy ion inertial fusion [1], collisional and radiative processes in the Earth's upper atmosphere [2], ion-beam lifetimes in accelerators [3], atomic spectroscopy [4], and ion stopping in matter [5], and are of considerable interest in atomic physics [6]. The recent resurgence of interest in charged particle beam transport in background plasma is brought about by the recognition that plasma can be used as a magnetic lens. Applications of the plasma lens ranging from heavy ion fusion to high energy lepton colliders are discussed in Refs. [6][7][8][9][10]. In particular, both heavy ion fusion and high energy physics applications involve the transport of positive charges in plasma: partially stripped heavy elements for heavy ion fusion; positrons for electron-positrons colliders [9]; and high-density laser-produced proton beams for the fast ignition of inertial confinement fusion targets [11].
To estimate the ionization and stripping rates of fast ions propagating through gas or plasma, the values of ion-atom ionization cross sections are necessary. In contrast to the electron [12] and proton [13,14,15] ionization cross sections, where experimental data or theoretical calculations exist for practically any ion and atom, the knowledge of ionization cross sections by fast complex ions and atoms is far from complete [16,17,18,19]. When experimental data and theoretical calculations are not available, approximate formulas are frequently used.
The raison d'etre for this paper are the frequent requests that we have had from colleagues for a paper describing the regions of validity of different approximations and scaling laws in the calculation of ion-atom stripping cross sections. The experimental data on stripping cross sections at low projectile energy were collected in the late 1980s, while comprehensive quantum mechanical simulations were performed in the late 1990s. Having in hand both new experimental data and simulation results enabled us to identify regions of validity of different approximations and propose a new scaling law, which is the subject of the present paper.
The most popular formula for ionization cross sections was proposed by Gryzinski [20].
The web of science search engine [21] shows 457 citations of the paper, and most of the citing papers use Gryzinski's formula to evaluate the cross sections. In this approach, the cross section is specified by multiplication of a scaling factor and the unique function of the projectile velocity normalized to the orbital electron velocity. The popularity of Gryzinski's formula is based on the simplicity of the calculation, notwithstanding the fact that his formula is not accurate at small energies.
Another fit, proposed by Gillespie, gives results close to Gryzinski's formula at large energies, and makes corrections to Gryzinski's formula at small energies [22]. Although more accurate, Gillespie's fit is not frequently used in applications, because it requires a knowledge of fitting parameters not always known a priori.
In this paper, we propose a new fit formula for ionization cross section which has no fitting parameters. The formula is checked against available experimental data and theoretical predictions. Note that previous scaling laws either used fitting parameters or actually did not match experiments for a wide range of projectile velocities. We also briefly review the most important theoretical results and approaches to cross section calculations in order to place the discussion in historical perspective and offer nonspecialists a concise introduction to the topic.
The organization of this paper is as follows. In Sec.II we give a brief overview of key theoretical results and experimental data. Further details of the theoretical models are presented in Appendices A-C. The new proposed fit formula for ionization cross section is presented in Sec.III, including a detailed comparison with experimental data, and in Sec.IV the theoretical justification for the new fit formula is discussed.

II. BRIEF OVERVIEW OF THE THEORETICAL MODELS AND EXPERIMEN-TAL DATA FOR IONIZATION CROSS SECTIONS
There are several theoretical approaches to cross section calculations. These include: classical calculations that make use of a classical trajectory and the atomic electron velocity distribution functions given by quantum mechanics [this approach is frequently referred to as classical trajectory Monte Carlo (CTMC)]; quantum mechanical calculations based on Born, eikonal or quasiclassical approximations, and so forth [16,17,18,19]. All approaches are computationally intensive and the error and range of validity are difficult to estimate in most cases. Therefore, different fittings and scalings for cross sections are frequently used in practical applications.
Most scalings were developed using theories and simulations based on classical mechanics.
Classical trajectory calculations are easier to perform compared with quantum mechanical calculations. Moreover, in some cases the CTMC calculations yield results very close to the quantum-mechanical calculations [23,24,25,26]. The reason for similar results lies in the fact that the Rutherford scattering cross section is identical in both classical and quantum-mechanical derivations [27]. Therefore, when an ionizing collision is predominantly a consequence of the electron scattering at small impact parameters close to the nucleus, the quantum mechanical uncertainty in the scattering angle is small compared with the angle itself, and the classical calculation can yield an accurate description [28,29,30].
Whereas in the opposite limit, when an ionizing collision is predominantly a consequence of the electron scattering at large impact parameters far from the nucleus, the quantum mechanical uncertainty in the scattering angle is large compared with the angle itself, and the classical calculation can remarkably fail in computing the ionization cross section [31,32].
In the present analysis, we consider first the stripping or ionization cross section of the hydrogen-like electron orbits (for example one-electron ions), with nucleus of charge Z T , colliding with a fully stripped ion of charge Z p . Subsequently, we show that the approach can be generalized with reasonable accuracy for any electron orbital, making use of the ionization potential of the electron orbitals. Because different terminology is used in the literature, we call a stripping collision a collision in which the fast ion loses an electron in a collision with a stationary target ion or atom (in the laboratory frame); and we call an ionizing collision a collision in which a fast ion ionizes a stationary target ion or atom [16].
Both cases are physically equivalent to each other by changing the frame of reference, and further consideration can be given in the frame of the atom or ion being ionized.
Atomic units are used throughout this paper with e = = m e = 1, which corresponds to length normalized to a 0 = 2 /(m e e 2 ) = 0.529 · 10 −8 cm, velocity normalized to v 0 = e 2 / = 2.19 · 10 8 cm/s, energy normalized to E 0 = m e v 2 0 = 2Ry = 27.2eV , where Ry is the Rydberg energy. The normalizing coefficients are kept in all equations for robust application of the formulas. For efficient manipulation of the formulas it is worth noting that the normalized projectile ion velocity is v/v 0 = 0.2 E[keV /amu], where E is energy per nucleon in keV /amu. Therefore, 25keV /amu corresponds to the atomic velocity scale. Some papers express the normalized velocity v/v 0 as βα, where β = v/c, and v 0 /c = α = 1/137.
Here, c is the speed of light, and α is the fine structure constant.
For a one-electron ion, the typical scale for the electron orbital velocity is v nl = v 0 Z T .
Here, n, l is the standard notation for the main quantum number and the orbital angular momentum [27]. The collision dynamics is very different depending on whether v is smaller or larger than v nl .
A. Behavior of cross sections at large projectile velocities v > v nl If v >> v nl , the electron interaction with the projectile ion occurs for a very short time and the interaction time decreases as the velocity increases. Therefore, the ionization cross section also decreases as the velocity increases. In the opposite case v << v nl , the electron circulation around the target nucleus is much faster than the interaction time, and the momentum transfer from the projectile ion to the electron averages out due to the fast circulation. Thus, the cross section decreases as the projectile velocity decreases. This is why the cross section typically has a maximum at v = v max ∼ v nl , but as we shall see below, v max also depends on the charge of the projectile.

Thompson's treatment
In the first treatment, Thompson calculated the ionization cross section in the limit v >> v nl [33]. This treatment neglected completely the orbital motion of the target electrons and assumed a straight-line trajectory of the projectile. In this approximation, the velocity kick acquired by the electron during the collision is entirely in the direction perpendicular to the ion trajectory, because the final action of the force along the trajectory cancels out due to symmetry, i.e., the electron velocity change during the approaching phase is equal to minus the electron velocity change during the departing phase. The momentum acquired by the electron ( m e ∆v) from passing by the projectile moving with the speed v and impact parameter ρ is given by the integral over time of the force perpendicular to ion trajectory F ⊥ = e 2 Z p ρ/(ρ 2 + v 2 t 2 ) 3/2 , where t = 0 corresponds to the distance of the closest approach.
Time integration of the force yields From Eq.(1) it follows that only collisions with sufficiently small impact parameters result in ionization. The minimum impact parameter for ionization of an initially stationary electron (ρ min ) is m e ∆v(ρ min ) 2 /2 = I nl . During a collision with impact parameter ρ min the energy transfer from the projectile to the electron is equal to the ionization potential I nl = Z 2 T E 0 /2, or ∆v(ρ min ) = v nl . Substitution of Eq.(1) gives the total ionization cross section πρ 2 min [28,33] Similarly, Eq.(2) can be derived by averaging the Rutherford cross section over all scattering angles leading to ionization. Although the first derivation of Eq.(2) was done by Thompson [33] the formula is frequently referred to as the Bohr formula [16].

Gerjuoy's treatment
The following treatments account for the effect of finite electron orbital velocity. The most complete and accurate calculations were done by Gerjuoy [34]. He calculated the differential cross section dσ/d∆E(v e , v, ∆E) of energy transfer ∆E in the collision between the projectile ion and a free electron (the target atomic potential was neglected) with given initial speed v e (and arbitrary direction), by averaging the Rutherford cross section over all orientations of electron orbital velocity v e . The total cross section is then calculated by integration over the energy transfer for energies larger than the ionization potential, and weighted by the electron velocity distribution function f (v e ). This gives where A rather complicated analytical expression for dσ/d∆E(v e , v, ∆E) is given in Appendix A.
For large projectile ion velocities (v >> v nl ), the differential cross section can be expressed as [34] dσ high−energy Substituting Eq.(5) into Eq.(3) and Eq.(4) gives where σ Bohr is given by Eq.(2), and K nl ≡< m e v 2 e /2 > nl is the average orbital electron kinetic energy. For hydrogen-like electron orbitals, the average electron kinetic energy is equal to the ionization potential K nl = I nl [27], and B nl = 1. The B nl factors are introduced to account for the difference in the electron velocity distribution functions (EVDF) from the EVDF of the hydrogen-like electron orbitals. The data for K nl are calculated for many atoms in Ref. [35]. For example, the average kinetic energy for the helium atom is K nl ≡< m e v 2 e /2 >= 1.43E 0 , whereas I nl = 0.91E 0 , and therefore B He = 1.22. That is the reason that accounting for the finite orbital electron velocity gives a cross section which is 5/3 times larger than the Bohr formula in Eq.(2). This is a consequence of the fact that for an electron with nonzero velocity less energy transfer is required for ionization.
Classical mechanics gives the EVDF as a microcanonical ensemble, where Here, C is a normalization constant defined so that f (v e ) dv e = 1, and δ(...) denotes the Dirac delta-function. Interestingly, the EVDF for a one-electron ion is identical in both the quantum-mechanical and classical calculations [27,35] with where v nl is the scale of electron orbital velocity Although a microcanonical distribution provides the same velocity distribution as in quantum theory for hydrogen-like shells, this is not the case for other electron shells. Moreover, the spatial distribution of the charge density is poorly approximated even for hydrogen, vanishing identically for r > 2a 0 rather than decreasing exponentially [18]. Substituting the Here, the scaling function G GGV (x) is given by Eq.(C3) in Appendix A, using the tabulation of the function G(x) presented in Ref. [36] for x > 1, and in Ref. [37] for x < 1. The notation GGV stands for the classical trajectory calculation in Eq.(C3) due to Gerjuoy [34] using the fit of Garcia and Vriens [36].

Bethe's treatment
The classical calculations underestimate the cross sections for very high projectile velocities v >> v nl . The scattering angle of the projectile due to collision with the target atom is of order θ c = ∆p/Mv, where ∆p is the momentum transfer in the collision, and M is the mass of the projectile particle. The minimum energy transfer from the projectile is determined by the ionization potential, with ∆E = v∆p > I nl , and ∆p > ∆p min ≡ I nl /v.
Here, we use the fact that the momentum transfer ∆p is predominantly in the direction perpendicular to the projectile velocity. The projectile particle with wave vector k = Mv/ undergoes diffraction on the object of the target atomic size a nl with the diffraction angle of order θ d = 1/(ka nl ) = /(Mva nl ) [28]. At large projectile velocities v >> v nl , it follows that ∆p min ≡ I nl /v << /a nl , because v nl = I nl a nl / for hydrogen-like electron orbitals.
And for small ∆p ∼ ∆p min , it follows that θ c = ∆p/Mv << θ d = /(Mva nl ). Therefore, the collision can not be described by classical mechanics.
Bethe made use of the Born approximation of quantum mechanics to calculate cross sections [38] (see Appendix B for details). This yields for v >> v nl If the projectile speed is much larger than the electron orbital velocity v >> v nl , the logarithmic term on the right-hand side of Eq.(11) contributes substantially to the cross section, and as a result the quantum mechanical calculation in Eq.(11) gives a larger cross section than the classical trajectory treatment in Eq. (6). The quantum mechanical cross section is larger than the classical trajectory cross section due to the contribution of large impact parameters (ρ) to the quantum-mechanical cross section, where the ionization is forbidden in classical mechanics because the energy transfer calculated by classical mechanics is less than the ionization potential [∆E = v∆p c (ρ) < I nl , where ∆p c is the momentum transfer given by classical mechanics in Eq.(1)]. However, ionization is possible due to diffraction in quantum mechanics [39]. Moreover, integration over these large impact parameters where the ionization is forbidden in classical mechanics, contributes considerably to the total ionization cross section (see Appendix B for further details).

Gryzinski's treatment
Gryzinski attempted to obtain the ionization cross sections using only classical mechanics similarly to Gerjuoy. But, in order to match the asymptotic behavior of the Bethe formula in Eq.(11) at large projectile velocities, Gryzinski assumed an artificial electron velocity distribution function (EVDF) instead of the correct EVDF in Eq.(8) [20], i.e., The ionization cross section was calculated by averaging the Rutherford cross section over all possible electron velocities, similar to the Gerjuoy calculation in Eq.(3), but was less accurate for small velocities v < v nl . The effect of using the EVDF in Eq. (12) is to populate the EVDF tail with a much larger fraction of high-energy electrons with v e >> v nl , for the correct EVDF in Eq. (8).
As a result, the average electron kinetic energy < m e v 2 e /2 > diverges, which leads to a considerable enhancement of the ionization cross section at high projectile velocities. For v >> v nl , Gerjuoy's calculation of the differential cross section dσ/d∆E(v e , v, ∆E) of energy transfer ∆E is similar to Gryzinski's. Therefore, we can substitute Eq.(12) into Eqs. (5) and (4). Because in the limit v >> v nl the ionization cross section is proportional to the average electron kinetic energy < m e v 2 e /2 > [Eq. (6)], and the average kinetic energy diverges, it follows that a small population of high-speed electrons contributes considerably to the cross section. Using the general expression for dσ/d∆E(v e , v, ∆E) avoids singularity and yields the logarithmic term in the ionization cross section similar to the Bethe formula in Eq. (11). After a number of additional simplifications and assumptions, Gryzinski suggested an approximation for the cross section in the form given by Eq.(10) with [20] Here, the function G Gryz (x) is specified by Eq.(C6) of Appendix C. In Eq.(13), the function G Gryz (x) has the following limit which is close to Bethe's result in Eq.(11), For 10 < x < 40, it follows that Therefore, the Gryzinski formula can be viewed as a fit to the Bethe formula at large velocities v >> v nl with some rather arbitrary continuation to small velocities v << v nl . Figure 1 shows the experimental data for the cross section for ionizing collisions of fully stripped ions colliding with a hydrogen atom, where X q+ denotes fully stripped ions of H, He, Li, C atoms, and (1s) symbolizes the ground state of a hydrogen atom. The experimental data for H + ions were taken from [40] (note that authors of this reference concluded that the previous measurements of the cross sections were inaccurate); from [41] for He +2 , C +6 ions ; and from [42] for Li +3 ions.
and the interaction time is of order ρ ioniz /v. The electron circulation time is where v nl is the electron orbital velocity, which scales as v nl = Z T v 0 , and a nl is the ion radius a nl = a 0 /Z T [39]. Therefore the condition Here, Z p is the charge of the fully stripped projectile and Z T is the nuclear charge of the target atom or ion. For velocities larger than v max , the ionization cross section decreases as the velocity increases [see Eq.(11)] due to the decreasing interaction time with an increase in velocity. On the other hand, for velocities less than v max , the collision becomes more adiabatic. The influence of the projectile is averaged out due to the slower motion of the projectile compared with the electron orbital velocity, and the ionization cross section decreases with decreasing projectile velocity. Thus, the cross section has a maximum at v ≃ v max [Eq. (19)].
Note that if the projectile speed is comparable with or smaller than the electron orbital velocity v < v nl , the Born approximation of quantum mechanical theory is not valid. Cumbersome quantum mechanical simulations are necessary for an exact calculation of the cross sections, as for example in Ref. [43]. Nevertheless for the case 2Z p ∼ Z T the maximum of the cross section calculated from the Born approximation is similar to the experimental results. To describe the behavior of the cross section near the maximum, the second-order correction in the parameter v nl /v has been calculated in [44], yielding the cross section in the form where (20) Equation (21) agrees with the exact calculation [Eq.(B1)] within 2% for v > 1, and within 20% for 0.2 < v < 1.
Equation (21) was derived making use of the unperturbed atomic electron wave functions, which implicitly assumes that the projectile particle transfers momentum to the electron and departs to large distances, where it does not affect the electron to be ionized. The wave function can therefore be described as a continuous spectrum of the atomic electron, not affected by the projectile.
This assumption breaks down at low projectile velocities when the projectile velocity is comparable with the electron orbital velocity. Indeed, the electron kinetic energy in the frame of the projectile is of order m e v 2 /2 and the potential energy Z p e 2 /ρ ioniz , where ρ ioniz is the impact parameter leading to ionization, given by Eq. (18). Substituting ρ ioniz from Eq.(18) into electron potential energy Z p e 2 /ρ ioniz gives that potential energy is larger than Therefore, under the condition in Eq. (22), an electron can be effectively captured by the projectile after the collision instead of leading to ionization. As a result, the ionization cross section is small compared with the charge exchange cross section at low projectile velocities.
The assumption of the unperturbed electron wave function results in grossly overestimated ionization cross sections as can be seen in Fig.1.
The ionization cross sections are also difficult to measure at small projectile energies, because careful separation between the large charge exchange cross section and the small ionization cross section is necessary for the correct measurement [40]. Therefore, early measurements of the ionization cross section at small velocities were not always accurate [16,40].

Gillespie's treatment
To account for the difference between the Born approximation results and the experimental data for v < v max , Gillespie proposed to fit the cross sections to the following function [22], Here, λ nl is a constant, which characterizes the ionized atom or ion (for example, for the ground state of H, λ nl = 0.76), and σ Bethe mod is the cross section in the Born approximation in the form of Eq.(20). Gillespie's Eq.(23) proved to fit very well existing experimental cross sections for hydrogen atom ionization by H + , He +2 , Li +2 ,Li +3 , C +4 , N +5 , N +4 , O +5 ions, and less well for He and H molecules with the same ions [22]. Because σ Bethe mod (v) becomes negative for v < 0.7, Gillespie's Eq. (23) can not be applied to these low projectile velocities.
In principle, the general fit σ BA f it in Eq.(21) can be used instead of σ Bethe mod in Eq. (20). However, because the two formulas differ considerably in the range of interest, 0.7 < v < 1, the fitting coefficients λ nl have to be updated for use with σ BA f it . Although Gillespie's fit proved to be very useful, there are a number of reasons to look for another fit. Gryzinski's Eq.(C6) is frequently used, because it requires only knowledge of one function for calculations of cross sections, notwithstanding the fact that it overestimates the cross sections at low energies.

Bohr and Linhard's treatment
For v v nl , a universal curve is expected if both the cross sections and the square of impact velocity are divided by Z p [48]. This scaling was established for the total electron loss cross section σ el , which includes both the charge exchange cross section σ ce and the ionization cross section. Based on the results of classical trajectory Monte Carlo (CTMC) calculations, Olson proposed the following fit [49], where f (x) describes the scaled cross sections Here, γ nl and A nl are constants, for example, γ H = 5/4 = 1.12 and A H = 16/3 for atomic hydrogen, and γ He = 1.44 and A he = 3.57 for helium. The scaling in Eq. (24) was also demonstrated analytically by Janev [50]. For v << v 0 Z p , σ el is dominated by charge exchange, σ ce ≈ σ el , and Eq.(24) gives a constant cross section for charge exchange, where Here, N nl is the number of electrons in the orbital nl, and the B nl factors Eq. (7) [42,50] confirm the scaling in Eq.(24) for 1. developed, which use up to ten fitting parameters to describe the ionization cross sections over the entire projectile energy range [19].

III. NEW FIT FORMULA FOR THE IONIZATION CROSS SECTION
Analysis of the experimental data in Fig.1 shows that the maxima of the experimentally measured cross sections occur at Z p + 1, not at Z p as would be the case according to Olson's scaling in Eq. (24). Therefore, it is natural to plot cross sections as a function of the normalized velocity v/(v nl Z p + 1). Note that at large velocities, according to Eq.(6) Therefore, making use of the normalized velocity v/(v nl Z p + 1) requires normalization of the cross sections according to σ/ Z 2 p /(Z p + 1) . As a consequence, instead of Eq.(25), we propose the following scaling The resulting universal function can be fitted with various functions, but the simplest fit was proposed by Rost and Pattard [52]. They showed that if both the cross section and the projectile velocity are normalized to the values of cross section and projectile velocity at the cross section maximum, then the scaled cross section σ/σ max is well described by the fitting function Here, σ max is the maximum of the cross section, which occurs at velocity v max . For the present study (the case of the ionization cross section by the bare projectile), we predict that where the coefficients B nl depend only weakly on the projectile charge. From Fig.3 one can At large x >> 1, Eq.(30) approaches the Bethe formula in Eq. (15), and at small x < 1, Eq. (30). This gives it a general form and introduces small errors of less than 8%.
We have applied the new fit in Eqs. (26) and (30) to the ionization cross section of helium, shown in Fig.4a. The symbols in Fig.4a denote the experimental data for H + , He +2 , Li +3 [53,54], for C +6 [55], for I +Zp and U +Zp [56], and for Au +Zp [57], where Z p = 10 − 40. The solid curves correspond to the continuum-distorted-wave-eikonal initial state (CDW-EIS) theoretical calculation from Ref. [58], which is a generalization of the Born approximation.
The CDW-EIS theory accounts for the distortion of the electron wave function by the projectile. From Fig.4a it is evident that the CDW-EIS theory overestimates the cross section near the maximum, and underestimates the cross section at small energies. Note that one experimental point in Fig.4 for C +6 projectiles is located far away from the fit. The error bar for this point is about 30% [55]. This data may be inaccurate, as the experimental point is higher than the predictions of CDW-EIS theory, which overestimates the cross section near the maxima of the cross sections for all other ions. The reason for the large scatter in the uranium data on the scaled plot at small energies is not clear, because the experimental data for all other projectiles are located much closer to the fit line. In this section we discuss the theoretical foundations for the new fit to the ionization cross section given by Eq. (26) and Eq. (30). We start with an analysis of high projectile velocities.

A. Behavior of cross sections at large projectile velocities v > v nl
In the region of high projectile velocities the new fit predicts the ionization cross section which differs from the Bethe formula in Eq. (11). [The factor (Z p + 1)/2 appears in the denominator under the logarithm in the first term on the right hand side of Eq. (31).] We claim that incorporating this factor gives a better cross section estimate than the Bethe formula. A comparison of the existing experimental data with the Bethe formula in Eq. (11) and the fit formula in Eq. (31) is shown in Fig.5. The experimentally estimated uncertainty of 5.5% [42] is shown by the error bar. The region of validity of the Born approximation and, hence, the Bethe formula is [27,28] v > max(2Z The first condition in Eq. (32) assures that the projectile potential is taken into account in the Born approximation; the second condition allows use of the unperturbed atomic wave function. Unfortunately the experimental data exists in the region in Eq.(32) only for the ionization of hydrogen by protons. Figure 5 shows that the Bethe formula describes the experimental data for ionization of hydrogen by protons within the error bar only for v > 6v 0 . Application of the fit formula instead of the Bethe formula reduces discrepancy with the data.
The applicability of the Born theory and the Bethe formula in Eq.(11) was studied experimentally in Refs. [32,55,57,59]. It was confirmed that the necessary condition for the validity of the Bethe formula is given by the condition in Eq. (32). The failure of the Bethe formula for large Z p is apparent from the experimental data for gold ions shown in Fig.4(a). The ion velocity corresponds to v = 12v 0 or v = 8.9v nl , whereas Z p = 24, 43, 54, and does not satisfy the condition in Eq. (32). As a result, the cross sections are much smaller than given by the Bethe formula, as evident from Fig.4(a). (At large projectile energies, all data merge to the Bethe formula, which corresponds to a straight line in a logarithmic plot, similar to Fig.1.) The applicability of the Bethe formula is limited by the validity of the Born approximation. One of the easiest ways to correct it was suggested in Ref. [64]. Firstly, the Born approximation is considered, making use of a classical trajectory for the projectile and a quantum mechanical description in the Born approximation for the electron. In this approximation, the probability of ionization or excitation is a function of the impact parameter ρ.
Here, for brevity, we shall consider only ionization of the hydrogen atom. The projectile particle interacts with the atomic electron with a potential energy V (R, r e ) = −Z p e 2 /|R − r e |, where R(t) = ρ + vt is the classical trajectory of the projectile particle, and r e describes the position of the electron relative to the nucleus of the atom. For any impact parameter ρ, the probability of ionization is given by the square of the transition amplitude Here, ∆E is the transferred energy in the transition, and Ψ i and Ψ f are the initial and final electron wave functions, respectively. It can be shown that the calculations of ionatom ionization cross sections using the conventional Born approximation describing the collision making use of momentum transfer (outlined in Appendix B) and the semiclassical Born approximation making use of the assumption of the straight line classical projectile trajectory [Eq. (33)] are equivalent [39].
For large impact parameters ρ >> a 0 , we can expand V (R, r e ) in powers of r enl /R according to The first term does not contribute to the matrix element in Eq.(33) due to the orthogonality of the final and initial states. Substituting Eq.(34) into Eq.(33) and integrating in time where ω = ∆E/ , and K n is the modified Bessel function. Expanding the Bessel functions for small and large arguments, or simply evaluating the integrand in Eq.(35) approximately, we can approximate and neglect the second term on the right hand side in Eq. (35), which is small compared with the first term. The probability of ionization vanishes for ρ > ρ max ≃ v/ω = 2a 0 v/v 0 , corresponding to the adiabatic limit. For ρ > ρ max , the collision time ρ max /v > a 0 /v 0 is much longer than the electron circulation time around the nucleus, and the collision is adiabatic.
The square of electron dipole matrix element averaged over all possible momenta of the ionized electron is [38] f dr e Ψ i (r e )x e Ψ * f (r e ) 2 = 0.283a 2 0 .
Note that the sum over all final states including both ionization and excitation gives In this sum, 0.717 corresponds to excitation, and 0.283 corresponds to ionization [38].
For large impact parameters the momentum transfer to the electron is small and we can neglect the electron kinetic energy of the ejected electron compared with the ionization potential. As a result, ∆E ≈ I H = E 0 /2 and ω = v 0 /2a 0 (in atomic units). Finally for ρ > a 0 , the ionization probability is The ionization cross section is given by the integral For ρ > a 0 , we can use Eq.(39) to estimate P BA (ρ). For ρ < a 0 , the dipole approximation in Eq. (34) is not valid. To evaluate P BA (ρ) approximately for ρ < a 0 , we can utilize the fact that dte i∆Et/ V (R, r e ) is a weak function of ρ for ρ < a 0 , and therefore P BA (ρ) ≈ P BA (a 0 ).
Substituting P BA (ρ) ≈ P BA (a 0 ) for ρ < a 0 , and P BA (ρ) from Eq.(39) for ρ > a 0 , into Eq. (40) gives The first term in Eq.(41) comes from contributions of impact parameters ρ < a 0 , and the second term originates from contributions of large impact parameters ρ > a 0 , respectively.
Comparison with the exact result in the Born approximation in Eq. (11) shows that the contribution of impact parameters ρ < a 0 is underestimated, and 1/2 should be replaced by 1.52. The above considerations are valid if the total probability of ionization and excitation [P tot BA (ρ) = (2Z p a 0 v 0 /ρv) 2 , for ρ > a 0 ] for the entire region of impact parameters is less than unity, which requires 2Z p v 0 /v < 1. (Note that the total probability of ionization and excitation is about 4 times larger for ionization only.) For 2Z p v 0 /v > 1, the total probability of the ionization and excitation P tot BA (ρ) calculated using the Born approximation is more than unity, P tot BA (ρ) > 1, for impact parameters ρ < ρ break = 2Z p a 0 v 0 /v, indicating the breakdown of the Born approximation [64]. Similar to the previous case, we can estimate the ionization probability P BA (ρ) from Eq.(39) for ρ > ρ break > a 0 and assume P BA (ρ) ≈ P BA (ρ break ) = 0.283 for ρ < ρ break . These considerations result in a cross section estimate similar to the Bethe formula but with the logarithmic term in the form ln(ρ max /ρ min ) = ln(v 2 /v 2 0 Z p ), which gives This calculation results in a smaller cross section than the Bethe formula for 2Z p v 0 /v > 1.
Note that in the above analysis we have used unperturbed electron wave functions, which is valid only for v >> v 0 .
While a number of smart semi-empirical ways to improve the first Born approximation were developed [60,61,62], the rigorous approaches to improve the Bethe formula are based on the eikonal approximation instead of the Born approximation [63]. The eikonal approximation is justified if ka nl > 1, where k is the projectile particle wave vector k = Mv/ , and the projectile kinetic energy is large compared to the potential energy interaction with the target. For heavy projectile particles with mass much larger than the electron mass, these conditions are well satisfied. The ionization cross section in the eikonal approximation is given by [27] where f (q) is the amplitude of ionization with momentum transfer q The eikonal approximation in Eqs. (43) and (44)  v v 0 . This is typically performed in the distorted wave approximation [16].
Therefore, the correction to the Born approximation in Eq.(42) and the eikonal approximation give a formula similar to Eq.(31) but with a factor α Z p (α is a coefficient of order unity), instead of (Z p + 1)/2. At large velocities, both formulas give similar results.

B. Behavior of cross sections at small projectile velocities v < v nl
If the projectile velocity is small compared with the orbital velocity, the collision is adiabatic and the electron circulates many times around both nuclei. The electronic energy states need to be determined in such a quasimolecule as a function of the positions of both nuclei at a particular time. In both the quantum mechanical and the classical approaches, ionization is only possible if during the collision the initial and final electronic terms cross at some instant. In classical mechanics this corresponds to the so-called "v/2 mechanism". In a collisional system comprised of two nuclei of equal charges (say ionization of hydrogen by a proton), an electron which is exactly in between the two nuclei experiences a very small electric field because the electric fields from both nuclei exactly cancel for all times at this point. The electron can "ride" this saddle point of the potential if its velocity is equal to one-half the velocity of the projectile. The collision dynamics is illustrated in Fig.6. From Fig.6 one can see that the electron is stranded in between the protons at t = 15a 0 /v 0 and its velocity projection on the x-axis is one-half of the projectile velocity. A small variation of the initial condition from z = −1.606756a 0 (solid line) to z = −1.606751a 0 (dotted line) completely changes the result of the collision. After the collision the electron stays near the first nucleus and does not become ionized. As a result, the probability of ionization is extremely small even though the projectile velocity is not small (for the conditions in Fig.6, v = 1/2 in atomic units). The mechanism for ionization described above is also so-called T-promotion in quantum mechanical descriptions [67].
Another mechanism for ionization is attributed to the so-called S-promotion mechanism [67]. It is associated with the special type of trajectory of the electron in the field of two positive charges, shown in Fig.6(c). Figure 6(c) shows that an electron with particular initial conditions tends to spiral with a large number of turns enclosing a segment of the straight line joining the nuclei Fig.6(c) [68]. Such a trajectory is unstable -a small variation of initial conditions results in a completely different trajectory as shown in Fig.6(c). Analysis of the electron motion in the field of two positive charges, Z T and Z P , which are separated by a distance R is best described in elliptical coordinates where r p and r T are the distances from the electron to the projectile and target nuclei, respectively. Making use of atomic units, the classical trajectory in terms of the variables ξ and η can be expressed as [68] dξ dt where the canonical momentums P ξ and P η are Here E < 0 is the total energy of the electron, P φ = ξηdφ/dt is the rotational momentum around the straight line joining the nuclei, and λ is the integral of motion (for stationary nuclei) Here, ζ is the closest distance from the electron to the straight line joining the nuclei; P ζ is the vector dot product of the electron momentum with the ζ-axis; M 2 = (r × p) 2 is the total rotational momentum; and θ P and θ T are the angles between r p and R, and r T and −R, respectively. Moreover, r p is the radius vector from the projectile to the electron; r T is the radius vector from the target nucleus to the electron; and R is the radius vector from the projectile to the target nucleus. The canonical momentum P ξ in Eq. (47) tends to infinity if ξ → 1, preventing the electron from approaching a segment of the straight line joining the nuclei, ξ = 1. In the special case The probability of ionization is greatly enhanced in quantum mechanics due to tunnelling into classically forbidden regions of phase space. The cross sections can be calculated using the quasiclassical method, where the probability of transition is given by where S(ρ, ǫ) = n c pdR.
Here, S(ρ, ǫ) is the classical action of the projectile ion, and p = 2M(ǫ − U(R, ρ) − E i ) is the projectile momentum, generalized to classically forbidden regions of phase space where p is complex [27]. The integration contour in Eq. (51) is in the complex R plane around the branch points (R c n ) where the initial and final electronic terms cross [E f (R c n ) = E i (R c n )]. Moreover, n numerates different branch points or channels of ionization for S and Tpromotions. The resulting cross section for hydrogen ionization by collision with a proton is [67] where n labels many different channels, and the coefficients ∆ n and R n are of order unity in atomic units (R n is determined by the branch points R cn ). In the range of projectile velocities v = 0.4 − 1, we find that Eq.(53) can be approximated to within 10% accuracy by only two exponents with R 1 = 1.9 , ∆ 1 = 0.53 (corresponding to S-promotion) and Eq. (53) for v in the range v = 0.5 − 1. While the data for hydrogen at very low projectile velocity is absent, and the fit agrees well for the entire dataset in Fig.3, the disagreement is clearly seen when the fit is compared with the experimental data for the ionization of He shown in Fig.4(d). Adiabatic theory results are absent for helium, but the experimental ionization cross section of He by protons can be described by Eq.(53) with different coefficients ∆ n and R n . The behavior of the experimental ionization cross section of He by He +2 is somewhat puzzling because of the very slow decrease of the cross section for small projectile velocity.
In view of these observations, the applicability of the new fit is limited to v/[v nl (Z p + 1)] > 0.5. Note that for small projectile velocity the ionization cross section is ten times smaller than the maximum of the cross section, σ max , and the ionization cross section is completely dominated by charge exchange, whose cross section is comparable to σ max . Consequently both experimental measurements and theoretical simulations are very difficult for very small projectile velocity. where and dσ/d∆E(v e , v, ∆E) is defined by [34] dσ d∆E where Here, v up and v low are defined by For very large projectile velocities v >> v e , it follows that S ≈ 8v e (2v 2 e /3 + ∆E/m e ), and Eq.(A3) yields (v, I nl , Z p ) = 10 3 In the general case with v ∼ v e , substituting the EVDF Eq.(8) into Eqs.(A2) and (A1) yields where The approximate formula for G classical (x) is given below in Eq.(C3).

APPENDIX B: THE BORN APPROXIMATION
Although the Born approximation is valid only for large projectile velocities v >> Z p v 0 [27], the Born approximation does give results close to the experimental data even outside its validity range [47]. Therefore, we have studied cross sections in the Born approximation for the entire velocity range.
In the Born approximation, the ionization cross section for hydrogen atoms by impact of fully stripped projectile atoms with charge Z p is given by [16,38,39], where P I nl (q, v) is the probability of ionization, and qm e v 0 is the momentum transfer during the collision. We introducing the velocity in atomic units v ≡ v/v 0 , and P I nl (q, v)is determined by [38] Here, Θ(x) is the Heaviside function, and dP (q, κ)/dκ is the differential probability of ejecting an electron with momentum κm e v 0 when the momentum transfer from the projectile is In Eq.(B3), Ψ * κ (p) and Ψ * κ (r) are the wave functions of the continuous spectrum (ionized electron) in momentum space and coordinate space, respectively; Ψ 0 (p) and Ψ 0 (r) are the wave functions of the ground state, and star ( * ) denotes complex conjugate. According to [38], dP (q, κ) dκ For q >> 1, the function dP (q, κ)/dκ has a sharp maximum at κ = q [27] dP (q, κ) dκ which simply means that the entire momentum q is transferred to the ionized electron momentum κ. At small q < 1, dP (q, κ)/dκ ∼ κq 2 and the width of the function P (q, κ) as a function of κ is of order unity in atomic units.
For large projectile velocity v >> v 0 , considerable simplification can be made by neglecting the electron kinetic energy 1 2 κ 2 in the argument of the Heaviside function in Eq.(B2). The approximation is referred to as the close-coupling approximation. In this case, P (q, v) can be characterized by a function of one argument, S inh (q), with The function S inh (q) is refereed to as the total ionization transition strength [46]. . Therefore, we split the integration in Eq.(B1) into the two regions q < q up and q > q up , where q up = 1/2. In the first region q < q up , it follows that where q min = v 0 I nl /vE 0 . In the second region, only the range of q up < q < 2 contributes to the integral, because at large q >> 1, P I nl (q, v)/q 3 ≈ 1/q 3 and the contribution to the integral for large q quickly decreases to zero. At very large q > 2v, P I nl (q, v) became smaller than unity, but this region does not contribute to the integral and can be neglected.
As a result, the integral ∞ qup dqP I nl (q, v)/q 3 does not depend on v (for the large v under consideration). The integration from q up to infinity gives ∞ qup dqP I nl (q, v)/q 3 ≈ 0.666, and finally the result is similar to the Bethe formula in Eq. (11) with The small differences from the Bethe formula are due to utilization of the close coupled approximation in Eq.(B10), which overestimates P I nl (q, v) at small q [see Fig.7].
Comparison with the exact calculation (Fig.1) shows that the Bethe asymptotic result is close to the exact calculation in Eq.(B1) for v > 2. To extend the Bethe formula to lower velocities, the second-order correction in the parameter v 0 /v has been calculated in [44], yielding the cross section in the form where v = v/v 0 . Equation (B12) agrees with the exact calculation in Eq.(B1) to within 10% for v > 1.1. We have developed the following fit for the cross section in the Born approximation, which agrees with the exact calculation in Eq.(B1) to within 2% for v > 1, and to within 20% for 0.2 < v < 1.
The previous analysis was performed for the hydrogen atom. In the case of hydrogen-like electron orbitals, the similarity principle can be used. The quantity dP (q, κ)/dκ is identical for different electron orbitals if q, κ are scaled with the factor 1/Z T = v 0 /v nl [27]. Therefore, where H denotes hydrogen atom, and where As we have noted for helium, most scalings can be used even for non-hydrogen-like electron orbitals, provided the relationship in Eq.(B15) is used. calculated in the classical trajectory and Born approximations, we present both cross sections in the form of Eq.(B1). In the limit v >> v nl , the momentum transferred to the electron during a collision with impact parameter ρ is given by Eq.
(1), i.e., where x−axis is chosen in the direction perpendicular to the projectile ion trajectory along the momentum transfer. Because v >> v nl , the electron velocity is neglected in Eq.(B16).
In classical mechanics, ionization occurs if the energy transfer to the electron is more than the ionization potential, [(m e v e + q) 2 − m 2 e v 2 ]/2m e > I nl . A small momentum transfer to the electron along the projectile trajectory q z (ρ) can be determined making use of the energy conservation. Due to conservation of the momentum, the momentum transferred from the projectile particle is −q z (ρ). The projectile energy In the limit v >> v e , it follows that q z << q x , and consequently the total transferred momentum to the electron is q = q 2 x + q 2 z ≃ q x . The momentum of the ejected electron can be determined from the energy conservation relation In classical mechanics, the ionization probability of the ejected electron with momentum κ in a collision with total momentum transfer q is given by the integral over the electron distribution function, Introducing the one-dimensional electron distribution function and substituting q ≃ q x , Eq.(B19) simplifies to become For hydrogen-like electron orbitals given by Eq. (8), f x (v ex ) can be readily calculated to be Substituting the hydrogen-like electron distribution function Eq.(B22) into Eq.(B29) gives in atomic units Let us compare Eq.(B23) with the quantum mechanical result Eq.(B5). In the limit q >> 1, κ ≈ q and the two functions are equivalent. Both functions dP (q, κ)/dκ have a maximum at κ = q, and the width of the maximum is of order 1, which simply means that the entire momentum q is transferred to the ionized electron momentum κ.
Moreover it is possible to prove that the classical mechanical dP c (q, κ)/dκ is equivalent to the quantum mechanical function dP q (q, κ)/dκ for any s−electron orbital (spherically symmetrical wave function). Indeed, for large k >> 1, the ejected electron can be described as a sum over plane waves Ψ * κ (r) ≈ e ikr , and substituting Ψ * κ (r) into Eq.(B3) gives dP q (q, κ) dκ where integral over do k = 2π sin ϑdϑ designates averaging over all directions of the k-vector, ϑ is the angle between q and k, and f (v e ) is the electron distribution function in velocity space. Note that |q − k| 2 = q 2 +k 2 − 2q · k = (q − k) 2 + 4qk sin ϑ/2 2 . In the limit q >> 1, k ≈ q and only small ϑ contribute to the integral in Eq.(B24). Therefore, averaging over all directions of the k-vector gives Introducing v ⊥ = kϑ/m e , the integral in Eq.(B25) takes form where f x is the one-dimensional electron velocity distribution function. Substituting Eqs.(B26) and (B25) into Eq.(B24) yields Note that in the limit q >> m e v nl , it follows that κ ≈ q, and Eq.(B21) becomes Finally, comparing Eqs.(B27) and (B28) we arrive at the equivalence of functions dP (q, κ)/dκ in quantum mechanics and classical mechanics in the limit q >> m e v nl .
The situation is completely different for small q << m e v nl . From Eq.(B23) it follows that dP c (q, κ)/dκ ∼ κq 5 , and dP c (q, κ)/dκ is much smaller than dP q (q, κ)/dκ ∼ κq 2 . Therefore, classical mechanics strongly underestimates the probability of ionization for small transferred momentum q < m e v nl .
The total probability of ionization in classical mechanics is Equation (B29) simplifies to become The differential cross section for momentum transfer q is given by where ρ(q) is given by Eq.(B16). Substituting ρ(q) from Eq.(B16) into Eq.(B31) gives which is the Rutherford differential cross section for scattering at small angles. Finally, the total ionization cross section is In Eq. (B33), we accounted for the fact that the minimum q is q = I nl /v. Note that in the region q = [1 − 3]I nl /v ionization occurs due the collisions with very fast electrons v e ∼ v >> v nl , and q x ∼ q z . The previous analysis which assumed v e << v and q x >> q z is not valid in this region of extremely small q. However, because P c (q)/q 3 → 0 as  Fig.8. Figure 8 shows that the functions P I nl (q, v) and P c (q) are nearly identical for q > 0.6. The classical probability of ionization P c (q) rapidly tends to zero for q < 0.6, while the quantum probability of ionization, P q (q) ≈ 0.283q 2 , is much larger than P c (q) at small q. The cross section is determined by P q (q)/q 3 . Therefore the region of small q contributes considerably to the quantum mechanical cross section. Note that P q (q)/q 3 → 0 as q → I nl / v . It follows that the region of small q contributes most to the cross section [compare Fig.8(a) for v = 5, and Fig.8(b) for v = 15]. For v = 5, the classical mechanical ionization cross section in atomic units is σ c = 0.23, and the quantum mechanical ionization cross section is σ q = 0.30, which is 30% larger than the classical mechanical cross section. For v = 15, σ c = 0.025 and σ q = 0.043, which is 70% larger. In the general case with v ∼ v nl , the classical mechanical calculation accounting for the finite electron velocity in the atom, but neglecting the influence of the target nucleus on the electron has been performed by Gerjuoy [34] [see Appendix A]. This gives The tabulation of the function G GGV (x) is presented in Ref. [36] for x > 1, and in Ref. [37] for x < 1, which gives f or x > 1, Gryzinski's approximation for the ionization cross section [20] expressed in the form of Eq.(C2) is given by Bethe's asymptotic quantum mechanical calculation in the Born approximation [38] is valid for v/v 0 > 2Z p and v >> v nl [27], and can be expressed as The region of validity of the Born approximation and, hence, the Bethe formula is [27,28] v > max(2Z p v 0 , v nl ).
The first condition in Eq.(C8a) assures that the projectile potential is taken into account in the Born approximation; the second condition allows use of the unperturbed atomic wave function.
To describe the behavior of the cross section near the maximum, the second-order correction in the parameter v nl /v has been calculated in Ref. [44], yielding the cross section in the form where In the general case with v ∼ v nl , the ionization cross section in the Born approximation was first calculated in Ref. [47]. We have developed the following fit for the Bates and Griffing . (C10) The Bethe cross section valid for relativistic particles [39] is given by where β 2 p = v p /c, c is the speed of light, γ p = 1/ 1 − β 2 p , and M 2 ion and C ion are characteristic constants depending on the ionized atom or ion. For the hydrogen atom, M 2 ion = 0.283 and C ion = 4.04.
Gillespie's fit for the ionization cross sections [46] is given by where λ nl is a characteristic constant of the ionized atom or ion (for example, for the ground state of atomic hydrogen, λ nl = 0.76), and σ Bethe mod is the modified Bethe cross section defined in Eq.(C9).
The Olson scaling [49] for the total electron loss cross section σ el , which includes both the charge exchange cross section σ ce and the ionization cross section, is given by where f (x) describes the scaled cross sections Rost and Pattard [52] proposed a fit for the ionization cross section, which utilizes two fitting parameters, namely the maximum value of the cross section and projectile energy corresponding to the maximum value of the cross section. They showed that if both the cross section and the projectile velocity are normalized to the values of the cross section and the projectile velocity at the cross section maximum, then the scaled cross section σ/σ max is well described by the fitting function [52] σ where σ max is the maximum cross section, which occurs at the velocity v max .
We have shown that for ionization by a bare projectile, the values σ max and v max are well defined by the projectile charge Z p , with where the coefficient B nl depends weakly on the projectile charge.
In all previous equations cross section are given per electron in the orbital. If N nl is the number of electrons in the orbital, the ionization cross section of any electron in the orbital should be increased by the factor N nl .
Finally, it should be noted that a number of other semi-empirical models have been developed, which use up to ten fitting parameters to describe the ionization cross sections over the entire projectile energy range [19].