Eur. Phys. J. D (2018) 72: 119 https://doi.org/10.1140/epjd/e2018-80307-9
THE EUROPEAN PHYSICAL JOURNAL D
Regular Article
Double ionization of water molecules by proton impact: the role of the direct ionization mechanism Dahbia Oubaziz 1 , Rachida Boulifa 1 , Zakia Aitelhadjali 1 , and Christophe Champion 2,a , b 1 2
Laboratoire de G´enie Electrique, LGE, Universit´e Mouloud Mammeri de Tizi-Ouzou, 15000 Tizi Ouzou, Algeria Centre d’Etudes Nucl´eaires de Bordeaux Gradignan, CENBG, CNRS/IN2P3, Universit´e de Bordeaux, 33170 Gradignan, France Received 5 May 2017 / Received in final form 1 December 2017 c EDP Sciences, Societ` Published online 21 June 2018 – a Italiana di Fisica, Springer-Verlag 2018 Abstract. The proton-induced double ionization process of water molecules is here theoretically investigated at high impact energy. The current 1st Born perturbative approach refers to an initial collisional state that both includes a target molecular wave function expressed as a single-center linear combination of atomic orbitals and a plane wave for describing the incident proton whereas the final collisional state is characterized by a scattered proton modeled by a plane wave and two ejected electrons modeled by Coulomb wave functions coupled via a Gamow factor. Total and partial cross sections are then provided and compared with existing experimental and theoretical data in considering the various dissociative channels. Additionally, a study of the energy dependence of the double ionization process is reported for impact energies ranging from 300 keV to 10 MeV.
1 Introduction Heavy charged particle-induced interactions on biomolecular targets are of great importance in many scientific fields such as physics, chemistry, and biology as well as in various interdisciplinary domains like radiobiology where applications may be numerous in particular in medicine [1–6] with a special emphasis in radiotherapy [7]. Indeed, it is nowadays well recognized that the rˆ ole played by the secondary electrons (delta rays) may be primordial in the cellular death as well as the chromosomal aberration induction. Thus, all along the history of any heavy charged particle in biological matter, a shower of secondaries may be observed, leading to high-energy electrons that may deposit their energy very far from the original particle track and then induce critical events responsible of cellular death as well as mutagenesis (see for example Ref. [8]). Besides, it is worthnoting that such electrons have a lethal contribution even after the irradiation time itself, namely, along the physico-chemical and purely chemical stages during which they may form stable molecules (e.g., H2 , O2 ), free radicals (e.g., H+ , OH+ ) as well as ion species. In this context, a detailed analysis of the original ionizing physical processes – and more particularly the multiple ionization – appears as crucial for predicting the radio-induced damages in biological targets like water. In this respect, the existing theoretical studies are mainly based on the independent electron model (IEM), a
e-mail:
[email protected] Present address: Centre Lasers Intenses et Applications (CELIA), Universit´e de Bordeaux, 33405 Talence, France. b
which consists in assuming that the passive (not ionized) electrons remain as frozen in their molecular orbitals during the collision. This approximation has been successfully used for numerous ionization reactions (see for example Ref. [9]). In this context, we have previously reported a series of theoretical approaches to describe the ionization process of targets of biological interest including water as well as DNA components. For more details, we refer the interested reader to references [10–13] where differential and total ionization cross sections were provided within a perturbative 1st Born approach and compared with experimental measurements for water and DNA/RNA components impacted by light and heavy ions as well as references [14–19] where a continuum distorted-wave– eikonal initial state (CDW-EIS) model was applied for describing the proton-induced ionization and electronic capture processes in biological matter. Finally, for sake of completeness, let us also mention the series of works provided by Illescas and co-workers within the classical trajectory Monte Carlo (CTMC) approach for describing collisions of protons, He2+ , and C6+ with water vapor (see for example the recent work of Illescas et al. [20]). Despite its great importance in radiobiology, in particular in terms of energy transfers – locally deposited (12–1300 eV) or converted into secondaries (∼ =200 eV) [21] – multiple ionizing processes induced by charged particles in biological matter have received less attention than single ionization. In this context, let us first cite the semi-classical approach proposed by Champion [21] based on the statistical energy-deposition model [22] within the local density approximation [23] to describe the experimental multiple ionization cross sections of water
Page 2 of 10
vapor impacted by 6.7 MeV/amu-Xe44+ ions [24]. Later on, inspired by the work of Zarour and Saalmann [25], Abbas et al. [26] investigated the double capture (DC), the double ionization (DI) and the simultaneous electron capture and electron ionization (usually called transfer ionization) of water molecules impacted by protons and α-particles. They employed a CTMC model where classical over-barrier (COB) criteria were applied for determining the electron emission conditions. Moreover, electron correlation was included in the calculation of the corresponding cross sections. However, only reactions involving electrons from the same target molecular orbital were considered. More recently, Murakami et al. [27–29] reported on a series of studies on the single and multiple ionization processes (capture and ionization). The authors used a non-perturbative basis generator model for calculating charge-state correlated cross sections for single- and multiple-electron removal processes (capture and ionization) in proton- and He+ –H2 O collisions for a wide range of impact energies spanning from 20 keV/amu to several MeV/amu. However, in this series of works, the study was essentially limited to multiple electron removal cross sections i.e. without differentiating with respect to capture and pure ionization, a difficult task essentially due to the intrinsic ambiguity about the transfer ionization contribution in particular when comparisons were performed with experiment. Details about the fragmentation model developed by the authors are nevertheless clearly exposed in reference [28] for extracting the pure ionization and capture respective contribution for He+ –H2 O collisions. Thus, they reported in reference [27] the double ionization cross sections for proton impact energies ranging from 20 keV to 1 MeV. More recently, Gulyas et al. [30] have reported total cross sections for the fragmentation of water molecule (OH+ + H+ and O+ + H+ ) by using the CDW-EIS model, an approach originally proposed by Gervais et al. [31] for modeling the multiple ionization and high LET effects in liquid water radiolysis. Finally, let us also cite the semi-empirical model reported by Meesungnoen et al. [32] where Rudd’s single ionization cross sections for water vapor [33] were adapted for studying the double ionization process in liquid water by assuming that the ratio between double and single ionization cross sections was known. On the experimental side, Werner et al. [34] have reported total cross sections (TCS) for water molecule fragmentation leading to H+ + H+ , O2+ , H+ , OH+ and O+ products for 100–350 keV protons and He+ ions by using a position- and time-sensitive multi-particle detector. Furthermore, Gobet et al. [35,36] determined a complete set of cross sections for water vapor ionization and electronic capture induced by protons in the energy range 20–150 keV, including total and partial cross sections for fragment production. The authors reported TCS in good agreement with the measurements of references [34–37]. Let us also mention the He2+ -ioninduced electron emission from water vapor studied by Ohsawa and co-workers [38] as well as the recent experience provided by Bhattacharjee and co-workers about the ionization of water molecules by fast bare oxygen ions [39]. Besides, it’s also worth to note the recent
Eur. Phys. J. D (2018) 72: 119
works of Montenegro and co-workers dedicated to water molecule fragmentation induced by proton and carbon impact [40,41] where multiple ionization and electron capture processes were scrutinized. More recently, Adoui and co-workers [42] reported on the ionization and fragmentation of water clusters by fast highly charged ions. Finally, let us cite the very recent experiment of Tavares et al. [43] devoted to the single and double ionization of water molecule by swift protons for impact energies ranging from 0.3 to 2 MeV. In their work, a multi-hit coincidence technique was used to measure the H+ + OH+ and H+ + O+ fragmentation channels. In the current work, we report on calculations of double ionization cross sections for H+ –H2 O collisions using the 1st Born approximation framework in which the initial state of the system includes a single-center molecular ground-state wave function [44] while the final channel is modeled by two independent Coulomb wave functions – for describing the two ejected electrons – coupled with a Gamov factor used for modeling the electron–electron repulsion. Besides, in order to go beyond the perturbative 1st Born approximation, the wave function of the scattered proton is described by a modified Coulomb wave function by considering the effective charge Z∗ seen by the scattered proton being in the field of the nucleus of the target together with the two ejected electrons. The manuscript is organized as follows: in Section 2, the theoretical model used to calculate the cross sections for the double ionization process is presented while a comparison in terms of TCS between our results and available experimental and theoretical data is given in Section 3. Finally, conclusions and prospective works are discussed in Section 4. Atomic units are used throughout unless otherwise indicated.
2 Theoretical model The direct double ionization of water molecule induced by proton impact may be schematized by − 2+ H+ + H2 O → H+ + e− . 1 + e2 + H2 O
(1)
We here consider a fast proton of mass Mp and initial momentum ki that double ionizes an isolated water molecule. The final state of the system is then characterized by a scattered proton of momentum ks and two ejected electrons of momentum k1 and k2 . The later are linked p to the electron kinetic energies √ √ via the relations ki = 2Mp Ei , k1 = 2E1 , k2 = 2E2 and ks = p 2Mp (Ei − E1 − E2 − I 2+ ), where I 2+ denotes the double ionization threshold and varies according to the two molecular orbitals involved in the collision (see Tab. 1). In the 1st Born approximation, the two ejected electrons are described by a product of two Coulomb wave functions while the incident and scattered proton are described by plane waves. In these conditions, the fivefold differential cross section for this process, i.e., differential in the energy of the two ejected electron k12 /2 and k22 /2, differential in the direction of the two ejected electrons dΩ1 and dΩ2 ,
Eur. Phys. J. D (2018) 72: 119
Page 3 of 10
( E E j1 j2 j j Ψi (ki ; r0 , r1 , r2 ; α1 , α2 ) = φ(ki ; r0 ) × ϕi 1 2 (r1 , r2 ) × |α1 , α2 i hΨf (ks , k1 , k2 ; r0 , r1 , r2 ; α1 , α2 )| = hφ(ks ; r0 ) × ϕf (k1 , k2 ; r1 , r2 )| × hα1 , α2 |.
and differential in the direction of the scattered proton dΩs , is given by d5 σ ≡ σ (5) (Ω1 , Ω2 , Ωs , E1 , E2 ) dΩ1 dΩ2 dΩs dE1 dE2 5 5 X X (5) = σj1 j2 (Ω1 , Ω2 , Ωs , E1 , E2 ) j1 =1 j2 ≥j1 4
= (2π)
5 5 k1 k2 ks 2 X X 2 Mp |Tj1 j2 | . ki j =1 1
j2 ≥j1
(2) Consequently, the doubly differential cross sections (DDCS), namely, differential in ejection energy E1 and E2 are given by
Table 1. Binding energies of the various final states of the double ionized water molecule. The data are taken from [45]. Molecular final states
Binding energies (eV)
State multiplicity
1b−2 1 3a−2 1 1b−2 2 2a−2 1 −1 1b−1 1 3a1
39.7 44.4 52.2 83.3 41.3 38.6 44.9 43.0 63.9 57.1 47.1 44.9 65.2 58.8 70.3 63.9
Singlet Singlet Singlet Singlet Singlet Triplet Singlet Triplet Singlet Triplet Singlet Triplet Singlet Triplet Singlet Triplet
−1 1b−1 1 1b2 −1 1b−1 1 2a1 −1 3a−1 1 1b2 −1 3a−1 1 2a1
2
d σ 4 k1 k2 ks ≡ σ 2 (E1 , E2 ) = (2π) Mp2 dE1 dE2 ki ZZZ X 5 5 X 2 |Tj1 j2 | dkˆs dkˆ1 dkˆ2 , (3) × j1 =1 j2 ≥j1
where dkˆ = sinθdθdφ. The transition amplitude denoted Tj1 j2 refers to the simultaneous ejection of two electrons from two molecular orbitals labeled j 1 and j 2 , respectively. It may be written as Tj1 j2 = hΨf (ks , k1 , k2 ; r0 , r1 , r2 ; α1 , α2 ) |V (r0 , r1 , r2 )| E ×Ψij1 j2 (ki ; r0 , r1 , r2 ; α1 , α2 ) , (4)
(7)
−1 1b−1 2 2a1
In equation (4), the vectors |α1 ,α2 i indicate the spin of the two active electrons: four possibilities may be identified, namely, (u, u), (u, d ), (d, u) and (d, d ), where u (d ) refers to a spin up (down). Thus, we have 1 √ (|u di − |d ui) for a singlet state 2 |u ui |α1 ,α2 i = √1 (|u di + |d ui) for a triplet state. 2 |d di (6)
where V (r0 , r1 , r2 ) represents the interaction between the incoming proton and the water target. However, by using the well-known frozen core approximation we reduce here the N = 10 electron target problem to a two-electron target problem where the two active electrons are those which will be ejected after the collision. Moreover, we assume that the electrons in the doubly charged ion core are relatively unaffected by the ionization process. We assume here that the core is completely unrelaxed. Note that, this assumption is successfully used in the case of single ionization of water molecule [46–49]. Consequently, the potential V ≡ V (r0 , r1 , r2 ) involved in the transition matrix element may be written as
The initial state of the system {H+ , H2 O} is then described by the product of a plane wave, which represents the incident proton and the ground-state wave function of the water molecule, while the final state is described by the product of two Coulomb wave functions. Thus, we may write See equation (7) above.
(5)
The functions φ(ki ; r0 ) and φ(ks ; r0 ) refer to the plane wave functions associated to the incident and the scattered proton, respectively, while the functions ϕji 1 j2 (r1 , r2 ) and ϕf (k1 , k2 ; r1 , r2 ) refer to the initial and final wave functions, respectively
where ri refers to the position of the i th bound electron of the target and r0 that of the passing proton, both with respect to the oxygen nucleus.
ϕji 1 j2 (r1 , r2 ) ≡ [ϕji 1 j2 (r1 , r2 )]± υj (r1 ) × υj2 (r2 ) ± υj1 (r2 ) × υj2 (r1 ) √ , (8) = 1 2
V (r0 , r1 , r2 ) =
2 1 1 − − , r0 |r0 − r1 | |r0 − r2 |
Page 4 of 10
Eur. Phys. J. D (2018) 72: 119
ϕf (k1 , k2 ; r1 , r2 ) ≡ [ϕf (k1 , k2 ; r1 , r2 )]± ϕc (k1 ; r1 ) × ϕc (k2 ; r2 ) ± ϕc (k1 ; r2 ) × ϕc (k2 ; r1 ) √ = . (9) 2 Additionally, let us note that the ten bound electrons are distributed among the five one center molecular wave functions υj (r) (with j ranging from 1 to 5) corresponding to the five molecular orbitals 1b1 , 3a1 , 1b2 , 2a1 , and 1a1 , respectively. Each of them is expressed by linear combinations of Slater-type functions [44] and is written as Nat (j)
υj (r) =
X
ξ
ajk .Φnjk (r), jk ljk mjk
(10)
k=1
where Nat (j) is the number of Slater functions used in the development of the jth molecular orbital and ajk is the ξ weight of each real atomic component Φnjk (r), the jk ljk mjk latter being expressed as ξ
ξ
Φnjk (r) = Rnjk r), jk (r) Sljk ,mjk (ˆ jk ljk mjk
(11)
where the radial part is given by ξ
Rnjk jk (r) =
(2ξjk )njk +1/2 njk −1 −ξjk r p r e , (2njk )!
(12)
and where Sljk ,mjk (ˆ r) is the so-called real solid harmonic [50] expressed by 2 n mjk −|m | Yljk jk (ˆ Sljk ,mjk (ˆ r) = r) + (−1)mjk 2 |mjk | mjk |m | Yljk jk (ˆ r) if mjk 6= 0 × 2 |mjk |
Fig. 1. Total double ionization section for protons in water (in Mb). Comparison between the current results (solid line) and existing theoretical predictions in water (CDW-EIS calculations taken from [31], dashed line, and the results of Murakami et al. [27], dotted line) as well as experimental data in neon taken from various sources (circles [55], squares [56] and star [57]).
Total cross sections are calculated by integration of the doubly differential cross sections reported in equation (3) over the ejection energies E1 and E2 , namely
Sljk ,0 (ˆ r) =
if mjk = 0.
Z
×
d2 σ dE1 dE2 , dE1 dE2
2
k2 1−e−2π/k2
0 1 − ks /k1 Z∗ = 1 − ks /k2 2 − ks /k1 − ks /k2
(16)
if if if if
ks k2 k1 ks
≥ k1 , k2 < ks < k1 < ks < k2 < k1 , k2 .
(17)
In equation (15), let us note that the upper limits of the integration E1 max and E2 max are defined by E1max = (Ei − I 2+ ) and E2max = E1max − E1 with Ei = I 2+ + Es + E1 + E2 [54]. Finally, let us note that the total double ionization cross section for each final state is given by σj1 j2 =
(14)
1 2πZ ∗ ∗ /k −2πZ s) ks (1 − e
with
Z when k1 > k2 when k1 = k2 when k2 > k1 .
(15)
where |f (ks )| =
2π e−2π/k1 k1 1−e−2π/k1 gG (k1 , k2 ) = 0 2π e−2π/k2
2
0
(13)
Besides, let us note that the target wave functions υj (r) correspond to a particular orientation of the molecular target, the latter being defined by the Euler angles (α, β, γ). Under these conditions, comparisons with experiments require to average the cross sections given in equation (2) over all the target orientations, that may be analytically carried out thanks to the reduction properties of the rotation matrix. Furthermore, integrations over the ejection directions dkˆ1 and dkˆ2 are analytically performed by using the closure relation of the spherical harmonics while the integration over the scattering direction dkˆs is numerically performed (see [51] for more details). Finally, let us add that we also introduced the wellknown Gamov factor gG (k1 , k2 ) for modeling the electronelectron repulsion and used the expression given by Defrance and co-workers [52,53], namely
E2max
gG (k1 , k2 ) ∗ |f (ks )| 0
Yl0jk (ˆ r)
E1max
Z σT =
E1 max
Z
E2 max
2
4
gG (k1 , k2 ) ∗ |f (ks )| (2π) ZZZ k1 k2 ks 2 2 × Mp |Tj1 j2 | dkˆs dkˆ1 dkˆ2 . (18) ki 0
0
Eur. Phys. J. D (2018) 72: 119
Page 5 of 10
Fig. 2. (a) Partial cross sections of water double ionization and (b) contribution of each final channel to the double ionization process.
3 Results The double ionization process is here investigated for isolated water molecules impacted by protons within the 0.3–10 MeV impact energy range. Partial and total cross sections as well as mean energy transfers are hereafter successively analyzed.
3.1 Double ionization cross sections We first report in Figure 1, the total double ionization section corresponding to the two-electron target removal process without differentiating the final channel of the system. Note that the vertical axis is linear. In absence of direct confrontation to experimental data in water, the current theoretical predictions are compared with an extensive series of measurements in the isoelectronic neon target [55–57]. The results are seen to agree well in shape with both experiment and theory. In this context, let us note that to the best of our knowledge, the sole existing theoretical approaches are that reported by Gervais et al. where multiple ionization cross sections were extracted from impact-parameter-dependent probabilities calculated within the CDW-EIS framework [31] as well as the recent study of Murakami et al. [27] where multipleelectron removal cross sections (capture and ionization) were calculated by using the non perturbative basis generator method. We observe that our predictions are of the same order of magnitude than those provided by the two other available theories for proton energies lower than about 1 MeV, showing an evident overestimation of the Ne experiment of a factor ≈2. For higher energies, our results asymptotically exhibit an overestimation of a factor ≈5 at 10 MeV, contrary to the CDW-EIS results, which show a very good agremeent. Double ionized water molecules are highly unstable and then rapidly dissociate resulting to various two-electron removal combinations. In this context, Figure 2a reports
on the absolute contribution to the total double ionization process for each final channel considered in this study, namely, those corresponding to the removal of two electrons from the same orbital, i.e., (1b1 )−2 , (3a1 )−2 , (1b2 )−2 , and (2a1 )−2 as well as those involving two electrons from two different orbitals, i.e. (1b1 )−1 (3a1 )−1 , (1b1 )−1 (1b2 )−1 , (1b1 )−1 (2a1 )−1 , (3a1 )−1 (1b2 )−1 , (3a1 )−1 (2a1 )−1 , and (1b2 )−1 (2a1 )−1 . Let us add that the double ionization from the innermost molecular orbital 1a1 is not here considered since negligible. We observe that all the molecular channels exhibit the same decreasing behavior with respect to the incident energy. Besides, Figure 2b demonstrates that the double ionizing processes corresponding to the ejection of electrons originating from two different molecular orbitals are predominant within the whole energy range. More precisely, we obtain a total contribution of the order of 85% (sum of all the contributions shown by the red lines) with a quasi-identical weight for all channels considered. Consequently, it clearly appears that the double electron removal from a given molecular orbital remains negligible whatever the orbital considered. 3.2 Fragmentation cross sections On the experimental side, it’s worthnoting that the multiple ionizing processes are usually analyzed in terms of fragmentation, leading difficult task for extracting the pure double ionization contribution. In this respect, the authors commonly define the two electron removal cross section σD as σD ≡ σ0,2 + σ ˜1,1 where the first term refers to the pure double ionization while the second one corresponds to the transfer ionization process (see for example Ref. [27]). However, in the high-energy regime, Murakami et al. reported that the corresponding cross section was very small, typically lower than 2 × 10−19 cm2 for Ei = 500 keV [27] i.e. very small in comparison to σ0,2 , that ioniz allows to assume that σD ∼ provided that = σ0,2 ≡ σD the proton energy is high enough, typically greater than about 500 keV.
Page 6 of 10
Eur. Phys. J. D (2018) 72: 119
Fig. 3. Cross sections for the fragmentation channels of doubly ionized H2 O molecules impacted by protons. Current predictions (solid line) compared with calculations from Gulyas et al. [30] (dotted line), Murakami et al. [27–29] (dash-dotted line) and semi-empirical predictions taken from Tavares et al. [43] (dash-dot-dotted line). Experiments are taken from Tavares et al. [43] (stars) and Werner et al. [34] (diamond).
Furthermore, to express the partial cross sections for fragment-ion production in the double ionization process, namely, the OH+ + H+ , O+ + H+ and H+ + H+ channels, we used the branching ratios suggested by Murakami et al. [27], namely + 2+ 60% + H2 O −→ H +OH 20% H2 O2+ −→ H+ +H+ +O 20% H2 O2+ −→ H+ +H + O+
(19)
that leads to the following definitions ioniz ≡ 0.6σ0,2 σH+ −OH+ = 0.6σD ∼ = 0.6σD ioniz ∼ σH+ −O+ = 0.2σD = 0.2σD ≡ 0.2σ0,2 ioniz σH+ −H+ = 0.2σD ∼ ≡ 0.2σ0,2 . = 0.2σD
(20)
Therefore, our calculations are compared with the experimental data reported by Tavares et al. [43] for the OH+ + H+ and O+ + H+ channels as well as that
Eur. Phys. J. D (2018) 72: 119
taken from Werner et al. [34] for the H+ + H+ fragmentation pathway. Additionally, the current fragmentation descriptions of the OH+ + H+ and O+ + H+ channels are compared with the calculations provided by Gulyas et al. [30] (dotted line), Murakami et al. [27–29] (dash-dotted line) as well as the semi-empirical predictions taken from Tavares et al. [43] (dash-dot-dotted line). In this respect, let us first remind that speaking of double ionization implies the competitive occurrence of two different mechanisms: i ) a direct ionization process, which corresponds to the direct interaction of the projectile with the target electrons that leads to the ejection to the continuum of one or more of them, leaving the residual target in a highly excited state with vacancies in inner shells, ii ) a second mechanism - assumed to be produced once the projectile is far away - during which the above-created vacancies are filled by spectator electrons of the target, the energy generated during these cascades being employed to release photons or to produce auto-ionization of the remaining bound electrons through Auger-type emission. Furthermore, let us add that such a post-collisional reaction may either correspond to inter-shell Auger and/or intra-shell Coster-Kronig electron ionization, depending of the shell of the ejected electron relatively to that occupied by the electron filling the vacancy. However, it is worthnoting that direct two-electron removal or the Auger process will lead to very similar two-valence hole configuration as underlined by Adoui et al. [42]. In this context, it was theoretically demonstrated that at high collision velocities, the post-collisional dominate the cross sections in comparison with direct ionization contributions [58]. More precisely, the authors reported that the Bohr parameter Z/vi (where Z and vi refer to the projectile charge and velocity, respectively) was adequate to estimate the physical conditions under which Auger-type emission mechanisms start to give the main contribution to the double ionization cross sections. Furthermore, there is an overall consensus to consider that the direct mechanism between the projectile and the target may exclusively be described via three scenarios, namely, the so-called Shake-Off (SO), the Two-Step 1 (TS1), and the Two-Step 2 (TS2) mechanisms. The SO process is a single interaction between the incident projectile and one-target electron, which leads to the ejection of a first electron, the second ejected one resulting from the relaxation process from the sudden change of the interaction potential. Concerning the two-step mechanisms, we distinguish the TS1 process constituted by a first interaction between the incoming proton and the target leading to a first ejected electron that interacts, in a second step, with another target electron that results in the ejection of a second electron. The TS2 process takes into account two interactions between the incoming projectile and the target i.e. the incident proton first collides with a target electron and then interacts with another one (for more details we refer the reader to Ref. [59]). This being said, it is clear that in our 1st Born approach the double ionization process is seen as a direct single step interaction, namely, the SO mechanism while in the other theories available in the literature dedicated to water double ionization, the description only considers the TS2
Page 7 of 10
Fig. 4. 1st Born total cross sections for single ionization (black line) and double ionization (red line) of isolated water molecules by proton impact. Current data provided by 1st Born calculations (solid line) compared with the CDW-EIS results taken from Gervais et al. [31] (circles).
mechanism (Murakami et al. [29] as well as Gulyas et al. [30]), the semi-empirical calculation of Tavares et al. [43] being on the other hand exclusively based on the Augerlike process. Let us first turn our attention to the direct double ionization process leading to the OH+ + H+ fragment channel (Fig. 3a). Surprisingly, we observe that our results are in very good agreement with the semi-empirical Auger-like predictions taken from Tavares et al. [43] (dash-dot-dotted line) in the whole energy range whereas evident discrepancies are shown with the TS2 calculations provided by Murakami et al. (dash-dotted line) and Gulyas et al. (dotted line), that could be interpreted as an indication that the TS2 mechanism is dominant in the low-energy regime, typically below 750 keV. The contribution of the Auger-like deexcitation is less evident in the energy range covered by the current theory/experiment confrontation (Ei < 3 MeV), that is coherent with the conclusions of Tachino et al. [58] who reported that the Auger-type emission mechanisms essentially contribute to the double ionization cross sections for Z/vi ≤ 0.3 to become dominant for Z/vi ≤ 1 i.e. for (Ei ≥ 2.5 MeV). Unfortunately, the upper limit of the energy range covered by the available experiment is too low to justify this statement. Interestingly, Figure 3a shows that our predictions match very well with the experiment showing both in shape and magnitude (in absolute linear scale) for impact energies greater than about 750 keV, that clearly points out the ability of the direct double ionization mechanism (SO) to reproduce the experimental OH+ + H+ fragment channel observations at intermediate impact energies. Similarly, for the O+ + H+ fragmentation channel (Fig. 3b), our results exhibit a fairly good agreement with both experiment and the TS2-based theories provided by Murakami et al. (dash-dotted line) and Gulyas et al. (dotted line) for proton intermediate energies.
Page 8 of 10
Eur. Phys. J. D (2018) 72: 119
Fig. 5. Mean energy transfers during proton-induced single [60] and double ionization of water molecules (black and red line, respectively): (a) mean kinetic energy transfer and (b) average deposited energy.
Regarding the H+ + H+ dissociation channel, the only existing experimental cross sections are to the best of our knowledge those reported by Werner et al. [34] for 100–350 keV protons. Thus, Figure 3c shows that at Ei = 300 keV, our calculated total cross section is of the order of ≈1.5.10−18 cm2 i.e. three times smaller than the experimental observations. 3.3 Double versus single proton-induced ionization of water molecule In Figure 4, we first compare the total cross section of the direct double ionization to its homologous for single ionization, the latter being taken from one of our previous works [10] where ab initio differential and total cross sections for the direct ionization of water vapor by protons in the incident energy range 0.1–100 MeV were provided within the 1st Born approximation. We clearly observe that the two collisional processes exhibit a similar behavior versus the proton impact energies with in particular a constant proportionality ratio R2 = σ2 /σ1 of about 2-3% all along the whole energy range. To the best of our knowledge, only rare cross section values are available in the literature for water molecules impacted by protons. Thus, for comparison purpose, we report in Figure 4 the single and double ionization cross sections provided by Gervais et al. [31] for protons in liquid water. Let us remind that in their study, the authors used the CDW-EIS approach to evaluate the multiple ionization cross sections from impact-parameter-dependent probabilities that implies that their approach only takes into account the TS2 mechanism and consequently should provide an accurate picture of the double ionization processes in the low-energy regime, where the TS2 scenario seems to be dominant, typically for proton energies lower than 750 keV. In this respect, Figure 4 shows an overall agreement between the two theories for the case of the single ionization. However, strong discrepancies are pointed out
for double ionization case, all the more as the incident proton energy increases with in particular a ratio of about 15 at 10 MeV. In absence of available measurement, it is a difficult task to check the accuracy of both theories and further experiment would obviously be welcomed to check the accuracy of both theories. Correlatively, it seemed us pertinent to study the variation of the mean energy transferred during the double ionization process as a function of the incident proton energy. Thus, we first report in Figure 5a the mean kinetic energy transferred, this latter being simply deduced from the singly differential cross sections. The results also obtained clearly indicate that for incident proton energies lower than ≈2 MeV the mean kinetic energy transfer increases linearly with respect to the incident energy to asymptotically tend to a constant value of about 145 eV. In absence of experimental data, we compared the obtained values with those reported by Champion in a theoretical study dedicated to the direct single ionization of water molecules induced by proton impact [60]. Then, we observe a similar tendency with in particular an asymptotic value of about 60 eV reached at high impact energies. More precisely, the corresponding double ionization vs. single ionization ratio exhibits a constant value of about 1.9 for Ei ≤ 2 MeV and 2.3 beyond. Similarly, we also report in Figure 5b the evolution of the mean deposited energy with respect to the proton energy for both double and single ionization of water molecule. This potential energy is calculated from the sum of all the corresponding partial cross section σj1 ,j2 weighted by the associated double ionization threshold energy (Ij2+ ), namely, 1 ,j2 P5 hEdep i =
P5 2+ j1 =1 j2 ≥j1 σj1 ,j2 Ij1 ,j2 P5 P5 j1 =1 j2 ≥j1 σj1 ,j2
P5 =
j1 =1
P5
j2 ≥j1
σj1 ,j2 Ij2+ 1 ,j2
T CS
.
(21)
Eur. Phys. J. D (2018) 72: 119
Thus, for the double ionization we obtain a mean deposited energy of about 50.5 eV all along the whole energy range vs. about 16.5 eV for single ionization [60], leading to a constant ratio of about 3 that clearly demonstrates the key rˆole played by the double ionization process in the total energy deposit pattern induced by protons in water and then in biological medium by extension. Let us note that this ratio is roughly of the same order of magnitude that the double versus single ionization threshold energy (see [61]).
4 Conclusions Partial and total cross sections for the double-ionization of water were here calculated for 300–10 000 keV proton impact by using a first Born model. In absence of experimental data, the current results have been compared to previous calculations reported on the double ionization of liquid water radiolysis as well as experimental data for isoelectronic neon target. From a general point of view, a reasonable agreement is observed at intermediate impact energy, namely, for 750 keV ≤ Ei ≤ 2.5 MeV, an overestimation of a factor ≈5 being shown at higher energies. More precisely, it was found that the direct double ionization mechanism (SO) is able to reproduce the experimental OH+ + H+ fragment channel observations at intermediate impact energies. Furthermore, our results show a very good agreement in the whole energy range with the semi empirical Auger-like predictions while evident discrepancies are shown with the TS2 mechanism. Consequently, it clearly appears that the TS2 mechanism is dominating in the low-energy regime, typically below 750 keV. For the double ionization O+ + H+ channel, the evaluated cross sections show fairly good agreement with both experiment and TS2 calculations at medium and high impact energies. Finally, energetic considerations have been analyzed in order to assess the role played by the double ionizing process in the water molecule irradiation. In this context, we have demonstrated its prime importance in the total energy deposit pattern. However, the current study has to be seen as prospective and should serve to inspire further experimental and theoretical study of water double ionization.
Author contribution statement All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.
References 1. B. Boudaiffa, P. Cloutier, D. Hunting, M.A. Huels, L. Sanche, Science 287, 1658 (2000) 2. F.A. Cucinotta, H. Nikjoo, D.T. Goodhead, Radiat. Res. 153, 459 (2000)
Page 9 of 10 3. S. Uehara, H. Nikjoo, D.T. Goodhead, Phys. Med. Biol. 38, 1841 (1993) 4. D. Belkic, in Advances in quantum chemistry, edited by D. Belkic (Elsevier, Amsterdam, 2013), Vol. 65, pp. 1–449 5. B.C. Garett et al., Chem. Rev. 105, 355 (2005) 6. A. Brahme, Int. J. Radiat. Oncol. Biol. Phys. 58, 603 (2004) 7. U. Amaldi, G. Kraft, Rep. Prog. Phys. 68, 1861 (2005) 8. A. Boissi´ere, C. Champion, A. Touati, M.-A. Herv´e du Penhoat, L. Sabatier, A. Chatterjee, A. Chetioui, Radiat. Res. 167, 493 (2007) 9. P.D. Fainstein, V.H. Ponce, R.D. Rivarola, J. Phys. B: At. Mol. Opt. Phys. 24, 3091 (1991) 10. O. Boudrioua, C. Champion, C. Dal Cappello, Y.V. Popov, Phys. Rev. A. 75, 022720 (2007) 11. C. Champion, H. Lekadir, M.E. Galassi, O. F´ ojon, R.D. Rivarola, J. Hanssen, Phys. Med. Biol. 55, 6053 (2010) 12. A. Itoh, Y. Iriki, M. Imai, C. Champion, R.D. Rivarola, Phys. Rev. A 88, 052711 (2013) 13. A.N. Agnihotri, S. Nandi, S. Kasthurirangan, A. Kumar, M.E. Galassi, R.D. Rivarola, C. Champion, L.C. Tribedi, Phys. Rev. A 87, 032716 (2013) 14. M.E. Galassi, C. Champion, P.F. Weck, R.D. Rivarola, O. Foj´ on, J. Hanssen, Phys. Med. Biol. 57, 2081 (2012) 15. C. Champion, P.F. Weck, H. Lekadir, M.E. Galassi, O. Foj´ on, P. Abufager, R.D. Rivarola, J. Hanssen, Phys. Med. Biol. 57, 3039 (2012) 16. Y. Iriki, Y. Kikuchi, M. Imai, A. Itoh, Phys. Rev. A 84, 032704 (2011) 17. A.N. Agnihotri, S. Kasthurirangan, S. Nandi, A. Kumar, M.E. Galassi, R.D. Rivarola, O. Foj´ on, C. Champion, J. Hanssen, H. Lekadir, P.F. Weck, L.C. Tribedi, Phys. Rev. A 85, 032711 (2012) 18. L.C. Tribedi, A.n. Agnihotri, M.E. Galassi, R.D. Rivarola, C. Champion, Eur. Phys. J. D 66, 303 (2012) 19. A.N. Agnihotri, S. Kasthurirangan, S. Nandi, A. Kumar, C. Champion, H. Lekadir, J. Hanssen, P.F. Weck, M.E. Galassi, R.D. Rivarola, O. Foj´ on, L.C. Tribedi, J. Phys. B 46, 185201 (2013) 20. C. Illescas, L.F. Errea, L. Mendez, B. Pons, I. Rabaden, A. Riera, Phys. Rev. A 83, 052704 (2011) 21. C. Champion, Nucl. Instr. Meth. B 205, 671 (2003) 22. A. Russek, J. Meli, Physica 46, 222 (1970) 23. J. Lindhard, M. Scharff, Mat. Fys. Medd. Dan. Vid. Selsk. 27, 15 (1953) 24. L. Adoui, C. Caraby, A. Cassimi, D. Leli´evre, J.P. Grandin, A. Dubois, J. Phys. B 32, 631 (1999) 25. B. Zarour, U. Saalmann, Nucl. Instr. Meth. B 205, 610 (2003) 26. I. Abbas, C. Champion, B. Zarour, B. Lasri, J. Hanssen, Phys. Med. Biol. 53, N41 (2008) 27. M. Murakami, T. Kirchner, M. Horbatsch, H.J. L¨ udde, Phys. Rev. A 85, 052704 (2012) 28. M. Murakami, T. Kirchner, M. Horbatsch, H.J. L¨ udde, Phys. Rev. A 86, 022719 (2012) 29. M. Murakami, T. Kirchner, M. Horbatsch, H.J. L¨ udde, Phys. Rev. A 85, 022713 (2012) 30. L. Gulyas, S. Egri, H. Ghavaminia, A. Igarashi, Phys. Rev. A 93, 032704 (2016) 31. B. Gervais, M. Beuve, G.H. Olivera, M.E. Galassi, Radiat. Phys. Chem. 75, 493 (2006) 32. J. Meesungnoen, A. Filali-Mouhim, N. Snitwongse, N. Ayudhya, S. Mankhetkorn, J.P. Jay-Guerin, Chem. Phys. Lett. 377, 419 (2003)
Page 10 of 10 33. M.E. Rudd, Rad. Prot. Dosim. 31, 17 (1990) 34. U. Werner, K. Beckord, J. Becker, H.O. Lutz, Phys. Rev. Lett. 74, 1962 (1995) 35. F. Gobet, B. Farizon, M. Farizon, J.M. Gaillard, M. Carr´e, M. Lezius, P. Scheier, and T.D. M¨ ark, Phys. Rev. Lett. 86, 3751 (2001) 36. F. Gobet, S. Eden, B. Coupier, J. Tabet, B. Farizon, M. Farizon, M.J. Gaillard, M. Carr´e, S. Ouaskit, T.D. M¨ ark, P. Scheier, Phys. Rev. A 70, 062716 (2004) 37. M.E. Rudd, T.V. Goffe, R.D. DuBois, L.H. Toburen, Phys. Rev. A 31, 492 (1985) 38. D. Ohsawa et al., Phys. Rev. A 72, 062710 (2005) 39. S. Bhattacharjee et al., J. Phys. B: At. Mol. Opt. Phys. 49, 065202 (2016) 40. E.C. Montenegro, M.B. Shah, H. Luna, S.W.J. Scully, A.L.F. de Barros, J.A. Wyer, J. Lecointre, Phys. Rev. Lett. 99, 213201 (2007) 41. H. Luna, A.L.F. de Barros, J.A. Wyer, S.W.J. Scully, J. Lecointre, P.M.Y. Garcia, G.M. Sigaud, A.C.F. Santos, V. Senthil, M.B. Shah, C.J. Latimer, E.C. Montenegro, Phys. Rev. A 75, 042711 (2007) 42. L. Adoui et al., J. Phys. B: At. Mol. Opt. Phys. 42, 075101 (2009) 43. A.C. Tavares, H. Luna, W. Wolff, E.C. Montenegro, Phys. Rev. A 92, 032714 (2015) 44. R. Moccia, J. Chem. Phys. 40, 2186 (1964) 45. F. Tarentelli, A. Tarentelli, A. Sgamelloti, J. Chem. Phys. 83, 4683 (1985) 46. C. Champion, J. Hanssen, P.-A. Hervieux, Phys. Rev. A 63, 052720 (2001) 47. C. Champion, J. Hanssen, P.-A. Hervieux, J. Chem. Phys. 121, 9423 (2004) 48. C. Champion, C. Dal Cappello, S. Houamer, A. Mansouri, Phys. Rev. A 73, 012717 (2006)
Eur. Phys. J. D (2018) 72: 119 49. C. Champion, O. Boudrioua, C. Dal Cappello, Y. Sato, and D. Ohsawa, Phys. Rev. A 75, 032724 (2007) 50. H. Trygve, P. Jorgensen, J. Olsen, Molecular electronic – structure theory (Wiley, Chichester, England, 2000) 51. D. Oubaziz, C. Champion, H. Aouchiche, Phys. Rev. A 88, 042709 (2013) 52. P. Defrance, T.M. Kereselidze, I.L. Noselidze, M.F. Tzulukidze, J. Phys. B: At. Mol. Opt. Phys. 34, 4957 (2001) 53. P. Defrance, J.J. Jureta, T. Kereselidze, J. Lecointre, Z.S. Machavariani, J. Phys. B: At. Mol. Opt. Phys. 42, 025202 (2009) 54. E. Bahati, H. Cherkani-Hassani, P. Defrance, J.J. Jureta, T. Kereselidze, Z. Machavariani, I. Noselidze, J. Phys. B: At. Mol. Opt. Phys. 38, 1261 (2005) 55. R.D. DuBois, L.H. Toburen, M.E. Rudd, Phys. Rev. A 29, 70 (1984) 56. E.G. Cavalkanti, G.M. Sigaud, E.C. Montenegro, M.M. Sant’Anna, H. Schmidt-B¨ ocking, J. Phys. B: At. Mol. Opt. Phys. 35, 3937 (2002) 57. W.S. Melo, A.C.F. Santos, G.M. Sigaud, E.C. Montenegro, J. Phys. B: At. Mol. Opt. Phys. 35, L187 (2002) 58. C.A. Tachino, M.E. Galassi, R.D. Rivarola, Phys. Rev. A 80, 014701 (2009) 59. J H McGuire, Electron correlation dynamics in atomic collisions (Cambridge University Press, Cambridge, England, 1997) 60. C. Champion, M.A. Quinto, J.M. Monti, M.E. Galassi, P.F. Weck, O.A. Fojon, J. Hanssen, R.D. Rivarola, Phys. Med. Biol. 60, 7805 (2015) 61. D. Oubaziz, M.A. Quinto, C. Champion, Phys. Rev. A 91, 022703 (2015)
