Cross sections for bare and dressed carbon ions in water and neon

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Phys. Med. Biol. 58 (2013) 641–672 doi:10.1088/0031-9155/58/3/641

Cross sections for bare and dressed carbon ions inwater and neon

Thiansin Liamsuwan and Hooshang Nikjoo1

Radiation Biophysics Group, Department of Oncology-Pathology, Karolinska Institutet,SE-17176 Stockholm, Sweden

E-mail: hooshang.nikjoo@ki.se

Received 16 May 2012, in final form 13 September 2012Published 14 January 2013Online at stacks.iop.org/PMB/58/641

AbstractThe paper presents calculated cross sections for bare and dressed carbonprojectiles of charge states q (0 to 6) with energies 1–104 keV u–1 impactingon molecular water and atomic neon targets. The cross sections of water areof interest for radiobiological studies, but there are very few experimental datafor water in any phase, while those for liquid water are non-existent. Themore extensive experimental database for the neon target made it possible totest the reliability of the model calculations for the many-electron collisionsystem. The current calculations cover major single and double electronicinteractions of low and intermediate energy carbon projectiles. The three-bodyclassical trajectory Monte Carlo (CTMC) method was used for the calculationof one-electron transition probabilities for target ionization, electron captureand projectile electron loss. The many-electron problem was taken into accountusing statistical methods: a modified independent event model was used for pure(direct) and simultaneous target and projectile ionizations, and the independentparticle model for pure electron capture and electron capture accompanied bytarget ionization. Results are presented for double differential cross sections(DDCS) for total electron emission by carbon projectile impact on neon. Forthe water target, we present the following: single differential cross sections(SDCS) and DDCS for single target ionization; total cross sections (TCS) forelectron emission; TCS for the pure single electronic interactions; equilibriumcharge state fractions; and stopping cross sections. The results were foundto be in satisfactory agreement with the experimental data in many cases,including DDCS and SDCS for the single target ionization, TCS for the totalelectron emission and TCS for the pure single electron capture. The stoppingcross sections of this work are consistent with the other model calculationsfor projectile energies �800 keV u–1, but smaller than the other calculationsat lower energies. The discrepancy arises from the inclusion of all carboncharge states and coupling between electron capture and target ionizationchannels, while other models use an average projectile charge. The CTMC

1 Author to whom any correspondence should be addressed.

0031-9155/13/030641+32$33.00 © 2013 Institute of Physics and Engineering in Medicine Printed in the UK & the USA 641

642 T Liamsuwan and H Nikjoo

model presented here provides a tool for cross section calculations for lowand intermediate energy carbon projectiles. The calculated cross sections arerequired for Monte Carlo track structure simulations of full-slowing-downtracks of carbon ions. The work paves the way for biophysical studies anddosimetry at the cellular and subcellular levels in the Bragg peak area of atherapeutic carbon ion beam.

(Some figures may appear in colour only in the online journal)

1. Introduction

The aim of the work presented here is to calculate cross sections for carbon projectiles ofcharge states 0 to 6 in the energy range 1 keV u–1 to 10 MeV u–1 in water. The work is part ofthe development of the Monte Carlo track structure simulation for dosimetry and biophysicalstudies at the cellular level in the Bragg peak region of a typical therapeutic carbon ion beam.The description of the track structure code is given in the accompanying paper of this issue(Liamsuwan and Nikjoo 2013).

Development of Monte Carlo track structure codes simulating tracks of radiation,interaction by interaction, requires accurate cross sections (Nikjoo et al 2006). Track structuresimulations are commonly used in biophysical studies of dose response and radiation damageat atomic, molecular and cellular levels. For radiobiological studies, the medium of interest isusually liquid water, as this constitutes ∼80% of human cells. However, due to scarcity of theexperimental cross sections of liquid water, track calculations have been primarily based oncross sections of water vapour by scaling the path length of a particle to unit density of liquidwater (Paretzke 1987, Uehara et al 1993, Kramer and Kraft 1994, Nikjoo et al 2008).

In the recent past, use of ion beam in radiation therapy has raised major interest in trackstructure studies of ions from protons to carbon ions (Brahme 2004, Scholz and Elsasser2007, Nikjoo et al 2008). Event-by-event simulations of full-slowing-down tracks of protonsand heliums have been developed (Uehara et al 2001, Uehara and Nikjoo 2002, Nikjooet al 2008, Liamsuwan et al 2011b, Friedland et al 2011), while those for the entire low(1–25 keV u–1) and intermediate (25–104 keV u–1) energy range2 of ions heavier than heliumare not yet available, mainly due to scarcity of cross sections for heavy ion impact onwater in any phase. Development of full-slowing-down track simulations also requires aknowledge of charge state distributions, because projectile ions with energies below a fewMeV u–1 can be accompanied by bound electrons. Another approach constructing full-slowing-down simulations of ions has been made by Kramer and Kraft (1994), by scaling the crosssections for protons to represent those for heavier ions. As the scaling scheme fails to predictexperimental cross sections for low energy heavy ions (Toburen 2012), the latter approach islimited to the LET concept or macroscopic radial dose distributions.

From the physics point of view, the rationale of using an ion beam is the increased doseconformed in the Bragg peak region close to the end of the ion’s trajectory, and the relativelylow dose at the beam entrance. Low and intermediate energy carbon ions (�10 MeV u–1)contribute greatly to the dose maximum in the Bragg peak region of a carbon ion beam. Inparticular, stopping power for carbon ions exhibits a maximum below 1 MeV u–1. Carbonions with energies below 10 MeV u–1 have ranges of the order of hundreds of micrometres(Liamsuwan and Nikjoo 2013), corresponding to several layers of human cells. Due to the

2 The same notation as used in ICRU (2005).

Cross sections for carbon projectiles in water and neon 643

potentially high density of interactions along the tracks of low and intermediate energy carbonions, dosimetry at the microscopic scale and biological effectiveness of these ions in the Braggpeak area are of major interest.

The paper presents calculated total and differential cross sections for bare and dressedcarbon ions of the charge states q (from 0 to 6) and energies 1–104 keV u–1 impactingon molecular water and atomic neon targets. Experimental data for carbon ion impact onwater of any phase are very scarce, while those for liquid water are non-existent. Because ofthe similarity of electronic configurations between H2O and Ne, and since the experimentaldatabase for neon targets is much more extensive, the neon target was studied to test the modelcalculations for the many-electron collision system.

Theoretical studies of ion tracks in condensed phase of water are limited to protons(Emfietzoglou et al 2009, Dingfelder et al 2000) and to high energy bare heavier ions(Dingfelder et al 2006). The recent measurements of electron emission spectra from amorphousice (Toburen et al 2010, Dingfelder et al 2008) provide important data for benchmarkingcalculations of ion impact cross sections. However, the electron yield data contain thecontributions of electron emissions due to interactions of electrons and ions with amorphousice and the substrate. The inaccuracy of the cross sections for either of those contributionsgives rise to a discrepancy between calculations and experimental data. Therefore, a thoroughinvestigation of electron yield measurements in condensed phase is needed for both electronand ion impacts.

For the current calculations, the three-body classical trajectory Monte Carlo (CTMC)model was developed. The many-electron collision system was considered within theframework of multiple one-electron problems. The cross sections presented here cover themajor single and double electronic interactions of low and intermediate energy carbon ions.Such data are needed for event-by-event simulations of full-slowing-down tracks of carbonions. The calculated cross sections were also used to obtain equilibrium charge state fractionsand stopping cross sections for carbon ions in water.

1.1. Physics models for low and intermediate energy ion impact

In the low to intermediate energy region, interactions of carbon ions include not onlyindependent and simultaneous ionizations and excitations3 of the targets, but also chargeexchange between the target and projectile (electron capture from the target and electronloss from the projectile). Charge exchange is particularly important at relatively low impactenergies. It leads to energy loss (or gain) by the moving ion and a change in the charge stateof the ion. Consequently, the strength of subsequent interactions is affected. Moreover, duringimpact of low and intermediate energy ions both the target and projectile nuclei (two-centre)simultaneously influence emission spectra of the ejected electrons, giving rise to the so-calledtwo-centre effect (Fainstein et al 1991).

The CTMC method (Abrines and Percival 1966) is particularly attractive for low andintermediate energy ion impact. The method is based on simultaneous trajectory simulationsof the collision partners (the projectile nucleus, target nucleus and active electrons), taking intoaccount the two-centre effect. Active electrons are initially bound electrons involved in theinteraction. In general, all interaction channels can be simultaneously included in the CTMCcalculations, while cross sections for dressed ions can be calculated by assigning a screeningfunction to the projectile’s nuclear charge (Reinhold et al 1990). Since the simulated active

3 By excitation we mean a process, in which a bound electron is excited to a higher energy level of the bound states,in contrast to ionization, by which the electron is emitted.

644 T Liamsuwan and H Nikjoo

electrons are those bound to the target nucleus, to the projectile nucleus or to both of them,target and projectile ionizations can be tracked.

Other published models for the intermediate energy range are the continuum distortedwave (CDW) calculations (Belkic 1978) and the eikonal initial state modification (CDW-EIS)(Crothers and McCann 1983). Both models take into account the two-centre effect, but withinthe framework of perturbative calculations. Therefore, they are expected to be valid for impactenergies above the maximum in the ionization cross section (>hundreds of keV u–1). For ionimpact on water and biomolecular targets, distorted wave calculations have been investigatedonly for bare ions (Olivera et al 1995, Bernal and Liendo 2007).

In the low energy region, the atomic and molecular orbital close coupling method (Kimuraand Lane 1990) is expected to give accurate cross sections for ions with impact energiesranging from a few tens of eV to 10 keV u–1. In the close coupling method, electronicstates of the collision system are described by a set of basis wave functions. The number ofrequired basis wave functions increases with impact energy, making the close coupling methodcomputationally impractical for impact energies above about 10 keV u–1.

1.2. The CTMC method

In the CTMC approach, interactions between the collision partners are described by theclassical Newtonian mechanics. The classical method is strictly valid in the energy range wheremolecular effects (at low energy limit) and dipole interactions (at high energy limit) are notimportant. Although the low energy limit has not been well studied in the literature, the CTMCapproach has been used successfully in the cross section calculation for ion impact with energyas low as 1 keV u–1 (Otranto and Olson 2008). Regarding the high energy limit, the classicalcalculation is expected to be valid within the Bohr’s kappa criterion, κ = 2Zpv0/vp > 1 orT < Z2

p · 100 keV u–1 (Sigmund 2004), where Zp is the projectile charge, vp the projectile

velocity (vp ≈ 0.2√

T (keVu−1) atomic units), v0 the Bohr velocity and T the specific energyof the projectile ion (kinetic energy per atomic mass unit u). According to this criterion, CTMCcalculations for bare carbon ions (C6+) are expected to be applicable for projectile energies<3.6 MeV u–1, and within a lower energy limit for lower-charge carbon ions. The region ofvalidity of the CTMC calculation may be extended up to several tens of MeV u–1 by takinginto account the equilibrium mean charge of the carbon projectile (see section 2.4) and theinitial orbital velocity of the active electron ve (ve � 1.9 atomic units for water (ICRU 1996)),using the analysis of Olson (2006). For higher energy impact, the Bethe theory is applicable;thus the cross sections for carbon ions can be scaled with the square of its charge from thecross sections for protons or electrons of the same speed.

CTMC calculations have been successfully performed for low and intermediate energyion impact on atomic and simple molecular targets such as H2 (Abrines and Percival 1966,Olson and Salop 1977). Recently, different CTMC models have been applied in cross sectioncalculations for ion impact on radiobiologically relevant targets. To our knowledge, thosecalculations have been made mainly for bare ions, or they are based on the assumption ofa one-active electron collision system for water and biomolecular targets. The latter worksinclude that of Otranto and Olson (2008) for bare ions of nuclear charges Z = 4–26 andenergies 10 eV u–1–100 keV u–1 incident on water; the model of Errea and colleagues (Illescaset al 2011, Errea et al 2007) for bare ions of hydrogen, helium and carbon with energies25 keV u–1–10 MeV u–1 impacting on water; and the CTMC coupled with the classicaloverbarrier (CTMC-COB) model of Champion’s group for bare ions (H+, He2+ and C6+)with energies 10 keV u–1–10 MeV u–1 impacting on water and nucleotide bases (Abbas et al2008, Lekadir et al 2009a, 2009b). We have also reported CTMC calculations based on the

Cross sections for carbon projectiles in water and neon 645

assumption of a single active electron for bare and dressed carbon ions with energies 1–104 keVu–1 incident on water (Liamsuwan et al 2011a, Liamsuwan and Nikjoo 2012).

1.3. The current approach for cross section calculation

The CTMC model presented in this work extends the calculations for radiobiologicallyrelevant targets, for the first time, to all charge states of carbon projectiles (q = 0–6) byconsidering the many-electron collision problem. The three-body CTMC method was used forcalculating the probabilities of one-electron transitions (interactions involving one activeelectron) for all orbital electrons of the target including inner shell electrons and for outermostshell electrons of the projectile ions. Statistical methods were applied for solving the many-electron problem: a modified independent event model (IEVM) (Crothers and McCarroll1987) was used for pure (direct) and simultaneous target and projectile ionizations; and theindependent particle model (IPM) (McGuire and Weaver 1977) was applied for pure electroncapture and electron capture accompanied by target ionization (transfer ionization). Energyand angular distributions of ejected electrons due to the target and projectile ionizationswere obtained with the three-body CTMC calculations. Acceleration of the active electronsbefore being emitted (the so-called Fermi-shuttle emission) was inherently taken into accountin the trajectory simulations.

In principle, excitation of the target and projectile, either independently or simultaneouslywith ionization, can be tracked in CTMC calculations by associating the final electron bindingenergies with the quantum states (Becker and MacKellar 1984). Such calculations have beenperformed for light targets, but have not been well studied for water molecules. Moreover,a much larger number of trajectories would be required to obtain good statistics for theprobability of excitation in a defined state. Also, antiscreening interactions (interactionsbetween the target electrons and projectile electrons) were not considered in the currentcalculations, because for these interactions at least four bodies need to be simulatedsimultaneously, making the calculations for ten-electron targets of water and neon far toocomplicated and time consuming.

The cross sections for carbon projectile impact on water and Ne were computed for thefollowing interaction channels:

(a) SI—pure single target ionization: Cq+ + H2O → Cq+ + H2O+ + e−,(b) SC—pure single electron capture: Cq+ + H2O → C(q–1)+ + H2O+,(c) SL—pure single projectile electron loss: Cq+ + H2O → C(q+1)+ + H2O + e−,(d) DI—pure double target ionization: Cq+ + H2O → Cq+ + H2O2+ + 2e−,(e) TI—transfer ionization: Cq+ + H2O → C(q–1)+ + H2O2+ + e−, and(f) LI—loss ionization: Cq+ + H2O → C(q+1)+ + H2O+ + 2e−,

where H2O in the above equations represents either the water or Ne targets. NomenclaturesSI, SC, SL and DI represent direct (pure) interactions, i.e. they are not accompanied byother interaction channels. TI is simultaneous target ionization and electron capture, and LIsimultaneous target ionization and projectile ionization. Projectile ionization is the same asprojectile electron loss, in this context.

In the following, we describe the CTMC model for the cross section calculations. Theresults are presented for double differential cross sections (DDCS) for total electron emissionby carbon ion impact on the neon target. For the water target, we present the calculatedDDCS and single differential cross sections (SDCS) for single target ionization, the total crosssections (TCS) for electron emission and the TCS for the pure single electronic interactions.The obtained equilibrium charge state fractions and stopping cross sections for carbon ions

646 T Liamsuwan and H Nikjoo

Figure 1. The collision scheme of the three bodies used in the current CTMC calculation: pdenotes the screened projectile nucleus, t the screened target nucleus, e the active electron, bthe impact parameter and vp the initial projectile velocity. The active electron is bound either tothe target core (ionization scheme) or to the projectile core (electron loss scheme). The CTMCcalculations were carried out using the relative coordinates (R1, R2, R3). In the ionization scheme,R1 = rp − rt, R2 = re − rt and R3 = R2 − R1, where r is the position vector of each collisionpartner in the Cartesian coordinate system (x, y, z). In the electron loss scheme, the index p isreplaced by t, and vice versa.

in water are also presented. The results were compared, where possible, with the availableexperimental data and other model calculations. Atomic units were used throughout unlessotherwise stated.

2. Materials and methods

2.1. The three-body CTMC method

The coordinates of the projectile nucleus, target nucleus and active electron, as used in theCTMC calculations, are presented in figure 1. The active electron is bound either to the targetnucleus (ionization scheme) or to the projectile nucleus (electron loss scheme). We refer to thescreened nucleus to which the active electron is bound as a ‘core’. The target core is associatedwith the ionization scheme, and the projectile core with the electron loss scheme. For bothcollision schemes, the two nuclei are screened by their spectator electrons. The interactionbetween the three bodies (figure 1) was described using the Newtonian mechanics, given by

mid2ri

dt2= −

∑j �=i

∇iZi(ri j)Zj(ri j)

|ri − r j| (1)

anddri

dt= vi, (2)

where mi is the mass of the particle i, ri the position vector in the Cartesian coordinate system(x, y, z), vi the velocity vector and Zi(ri j) the effective charge of i with respect to distance ri j

to the other particle j. Zi(ri j) were calculated from a model potential formula, explained insection 2.2.

The CTMC calculations were carried out using the relative coordinates (R1, R2, R3). Inthe ionization scheme, R1 = rp − rt, R2 = re − rt and R3 = R2 − R1, where the indices p, tand e denote the projectile, target and active electron, respectively (cf figure 1). In the electronloss scheme, the index p is replaced by t, and vice versa.

In total, 12 differential equations (R1, R2 and their time derivatives) were integrated overtime using the Runge–Kutta method with an adaptive step size. The time integration (the

Cross sections for carbon projectiles in water and neon 647

so-called trajectory simulation) was performed until the probability of interaction betweenthe screened nuclei was negligible. For each projectile, the integration was tested for a smallnumber of trajectories using different time intervals. The shortest time period, for whichthe obtained interaction probabilities converged, was used in the final calculation. This timeinterval was at least 103 atomic units for the intermediate energy projectiles, and at least104 atomic units for the low energy projectiles.

Prior to the trajectory simulation, the initial radial distance of the active electron to thecore (rec or R2 in figure 1) was randomly sampled from a classical distribution (described insection 2.2). The momentum of the active electron relative to the core (pec) was calculatedusing the energy conservation law:

Eec = p2ec

2μec− Zc(rec)

rec, (3)

where Eec is the electron binding energy and μec the reduced mass of the active electron-core system. Directions of the position and momentum vectors were randomly selected, asdescribed in Olson and Salop (1977). The initial distance between both screened nuclei wasset large enough, to ensure that the starting position did not affect the resulting cross section.This distance varied from 103 to 104 atomic units for the highest and lowest projectile energiesstudied, respectively.

The trajectory simulations were performed for different impact parameters b, rangingfrom 0 to bmax, where at bmax the probability of interaction between the two screened nucleiwas negligibly small. In the ionization scheme, bmax ranged from 1 to 25 atomic units forthe innermost to outermost shell electrons, respectively. In the electron loss scheme, it was10–20 atomic units.

At the end of each trajectory simulation, the energies of the active electron relative to thetarget and projectile nuclei (Eet and Eep, respectively) were calculated according to equation (3).The interaction type was selected as target ionization or projectile ionization, if Eet and Eep

> 0; and as electron capture, if Eet � 0 and Eep > 0. For each impact parameter, morethan 104 trajectories were simulated. The one-electron transition probabilities of the targetionization Pi, electron capture Pc and projectile electron loss Pl were calculated according toPR = NR/Ntot, where NR is the number of the interaction type R recorded and Ntot the totalnumber of the simulated trajectories. The relative uncertainty of PR was estimated to 1/

√NR,

i.e. �1%, for 104 simulated trajectories.

2.2. Electron radial and momentum distributions

The initial distance of the active electron to the core was randomly selected from a classicalelectron radial distribution. Following the method of Reinhold and Falcon (1986), the initialelectronic state was described using the microcanonial distribution f (rec, pec), given by

f (rec, pec) ∝ δ

(−|Eb| − p2

ec

2μec+ Zc(rec)

rec

), (4)

where |Eb| is either the electron binding energy in the ionization scheme or the ionizationenergy of the carbon projectile in the electron loss scheme (given in table 1). rec is constrainedto 0 < rec < rmax, rmax is the positive root of − |Eb|+ Zc(rec )

rec, and Zc(rec) is the effective charge

of the core. Based on equation (4), the table of sampling for rec was generated, according to

w(rec) =∫ rec

0dr′μecr′2

√2μec

(−|Eb| + Zc(r′)

r′

), (5)

648 T Liamsuwan and H Nikjoo

Table 1. Electron binding energies of water (Zaider et al 1983) and neon (Cardona and Ley 1978),and the ionization energies of carbon projectiles (James and Lord 1992, Huheey et al 1993). NL

denotes the effective number of the active electrons for each orbital of the targets and for eachcharge state of the projectile.

Water Neon Carbon projectiles

Binding Binding IonizationOrbital energy (eV) NL Orbital energy (eV) NL Charge state energy (eV) NL

1b1 12.62 2 2p1/2, 3/2 21.65a 6 C0 11.26 23a1 14.75 2 2s 48.5 2 C1+ 24.39 11b2 18.51 2 1s 870.2 2 C2+ 47.90 22a1 32.4 2 C3+ 64.50 11a1 539.7 2 C4+ 392.14 2

C5+ 490.06 1a The binding energies for 2p1/2 and 2p3/2 electrons are 21.6 and 21.7 eV, respectively. In this work, we assumed theaverage binding energy for both orbitals.

Table 2. The screening parameters (ς0, ς1, η0 and η1) used in the effective charge formula(equations (6)–(9)), taken from Garvey et al (1975). N is the number of spectator electronsassociated with a given core.

Ionization scheme Electron loss scheme N ς0 ς1 η0 η1

C6+ C5+ 0 1 0 0 0C5+ C4+ 1 2.625 1.2996 1.770 1.1402C4+ C3+ 2 2.164 0.9764 1.750 0.6821C3+ C2+ 3 1.30 0.6465 1.880 0.5547C2+ C1+ 4 1.031 0.4924 2.0 0.4939C1+ C0 5 1.065 0.48 2.13 0.4434C0 – 6 1.179 0.4677 2.270 0.4143H2O/Ne – 9 1.792 0.4515 2.710 0.3671– H2O/Ne 10 1.712 0.3923 2.85 0.3469

where 0 < w(rec) < w(rmax). w(rec) is associated with the probability of finding an electronat a distance from 0 to rec from the core. In the trajectory simulation, a generated randomnumber (from 0 to 1) corresponds to the value of w(rec)/w(rmax).

In the early phase of this study, we tested the CTMC calculations with a constant effectivecharge of the screened target nucleus. It was found that electron backscatterings (mainly due tohard collisions) were largely suppressed, although the calculated spectra for forward electronemissions were consistent with the measured data reported in Dal Cappello et al (2009). Thisis because the constant screening strength became too large for close collision. Thus, in thecurrent calculations and in the previous work (Liamsuwan et al 2011a, Liamsuwan and Nikjoo2012), we made use of the model potential Vc(rec) of Garvey et al (1975) for the effectivecharge, namely

Vc(rec) = Zc(rec)

rec= Z − N[1 − �(rec)]

rec, (6)

�(rec) = 1

(η/ζ )[exp(recς ) − 1] + 1, (7)

Cross sections for carbon projectiles in water and neon 649

η = η0 + η1(Z − N − 1) (8)

and ς = ς0 + ς1(Z − N − 1), (9)

where Z is the bare nuclear charge of the core c, N the number of spectator electrons boundto the core and �(rec) the screening function parameterized with the screening parameters η

and ς . The screening parameters, as given in Garvey et al (1975), have been generalized forany atomic and ionic species with Z � 54. For the targets and projectiles under investigation,the screening parameters and the number of spectator electrons are listed in table 2. Here,the water molecule was described by a spherical symmetric, atomic picture with the nuclearcharge of 10. The electron binding energies for the molecular orbitals 1b1, 3a1, 1b2, 2a1 and1a1 are given in table 1. In the electron loss scheme, ten spectator electrons are associated withthe water molecule, or the neon atom, because the active electron is bound to the projectile’snucleus.

The radial distributions D(r) for the five molecular orbitals of water were calculated bysampling the radial distance rec from the data table (rec, w(rec)), equation (5). If a generatedrandom number had a value corresponding to w(rec)/w(rmax), the electron radial distance wasset equal to rec. The obtained radial distributions D(r) for the water molecule are presentedin figure 2(a) (averaged over 107 sampling trials), where

∫D(r) dr = 1. The corresponding

quantum mechanical (QM) distributions are given in figure 2(b).The QM electron radial distributions were calculated with the self-consistent field

molecular orbital (SCF-MO) radial wave functions �(r, θ, φ) of Moccia (1964), according to

D(r) = r2∫ 2π

0dφ

∫ π

0dθ sin θ |�(r, θ, φ)|2, (10)

where θ is the polar angle, φ the azimuthal angle and r the radial distance of the electronrelative to the molecular centre. As shown in figure 2, the classical radial distributions tend tovanish at smaller distances from the core than the QM distributions. Therefore, it is expectedthat glancing collision is less probable in the CTMC approach than in QM calculations.

In figure 3, the associated classical and QM electron momentum distributions D(p)

are presented for the water target. For the classical distributions, equation (3) was used tocalculate the electron momenta, which are associated with the scored radial distances(figure 2(a)). The QM distributions were calculated equivalently to equation (10), wherethe wave functions in the momentum space are the Fourier transforms of the radial wavefunctions.

In most cases, the classical and QM electron momentum distributions show similartrends, and both types of distributions have the maxima in the similar range of momenta(figure 3). The magnitudes of the classical and QM distributions are quite different for theouter orbitals, especially for the 1b2 and 2a1 orbitals, but they are in good agreement for theinnermost shell (1a1) (figure 3). Success of CTMC calculations for light targets (H and He)has been reasoned by consistency between classical and QM electron momentum distributionsof these targets (Otranto and Olson 2008). Based on the qualitative agreement between thecalculated classical and QM electron momentum distributions shown in figure 3, the atomicpicture used for describing the water molecule is reasonable to be implemented in the CTMCcalculations.

2.3. Statistical methods and cross section calculations

The interaction probability obtained using the three-body CTMC method is associated withthe collision involving one active electron (figure 1). In reality, several electrons can be active

650 T Liamsuwan and H Nikjoo

(a)

(b)

Figure 2. Electron radial distributions D(r) in the five molecular orbitals of water: (a) the classicaldistributions (equation (4)), and (b) the QM distributions calculated using the SCF-MO wavefunctions of Moccia (1964) (equation (10)). The distributions were normalized according to∫

D(r) dr = 1.

during the collision. The IEVM (Crothers and McCarroll 1987) and the IPM (McGuire andWeaver 1977) are statistical models, commonly used for calculating many-electron transitionprobabilities. In the IEVM active electrons are assumed to be correlated, while electron removalevents are consecutive and independent. If the electron correlation effect is negligibly small,the IPM can be applied.

Cross sections for carbon projectiles in water and neon 651

(a) (b)

(c) (d)

(e)

Figure 3. Electron momentum distributions D(p) in the five molecular orbitals of water: theclassical distributions (solid lines) and the QM distributions (dashed lines). The distributions werenormalized according to

∫D(p) dp = 1.

652 T Liamsuwan and H Nikjoo

By electron correlations we mean static correlations, i.e. changes in the two-centrepotential after an active electron is removed from the core. Dynamic electron correlations,arising from interactions between the active electrons and from final state rearrangements,are not taken into account in the IPM or the IEVM. Dynamic correlations are expected to beimportant for weakly coupling cases (Ford and Reading 1991), associated with high energyions, outside the scope of this work.

For the pure and simultaneous target and projectile ionizations, a modified form of theIEVM was used. These interactions occur mainly with the outer shell electrons, which aresensitive to changes in the two-centre potential. The IPM was used for the interactionsassociated with the electron capture (pure single electron capture and transfer ionization),due to strong forces between the electron and projectile nucleus during electron capturereactions.

In general, in the IEVM, the probability of single electron removal P(m)

single is given by

P(m)

single(b) =∑

L

NLPR(L, b)PNL−1n (L, b)

∏L′ �=L

PNL′n (L′, b), (11)

and the probability of double electron removal P(m)

double by

P(m)

double(b) = 1

c

∑L

NL(NL − 1)PR1 (L, b)PR2 (L, b)PNL−2n (L, b)

⎡⎣∏

L′ �=L

PNL′n (L′, b)

⎤⎦

+∑

L

NLPR1 (L, b)∑L′ �=L

NL′ PNL′R2

(L′, b)

×⎡⎣PNL−1

n (L, b)PNL′ −1n (L′, b)

∏L′′ �=L′ �=L

PNL′′n (L′′, b)

⎤⎦ , (12)

where the superscript (m) denotes the many-electron transition probability, NL is the numberof electrons in the orbital L, b is the impact parameter, and P and P are the one-electrontransition probabilities for removal of the first and second electrons, respectively. The factor1/c takes into account the indistinguishability between electrons of the same orbital during apure double electronic interaction, i.e. c = 2 for pure interactions, or otherwise c = 1. Thesubscripts R, R1 and R2 refer to electron removal reactions and the index n labels the probabilitythat none of the electrons is removed from the core (elastic scattering and excitation). For theionization scheme Pn = 1 − Pi − Pc, and for the electron loss scheme Pn = 1 − Pl, where theindices i, c and l refer to the target ionization, electron capture and projectile electron loss,respectively.

Table 1 shows the number of active electrons NL of the studied targets and projectiles. Forcarbon projectiles, only electrons from the outermost occupied shell were taken into account.For the Ne target, the same probabilities were used for electron removals from the 2p1/2 and2p3/2 orbitals, because the electron binding energies of both atomic orbitals are nearly thesame.

Considering the static correlation effect, P is associated with the potential of H2O+ (orNe+). In general, PR is smaller than PR, because the electron binding energy of a singlycharged atomic/molecular ion is larger than that of the neutral atom/molecule. Following themethod of Crothers and McCarroll (1987), it was assumed that the interacting electrons wereremoved from the same subshell L, and PR(L′) of the other subshells L′ were negligibly small(PR(L′) ≈ 0). We did not take into account the order of electron removals, meaning that theone-electron transition probability is the same for all active electrons of the same subshell, i.e.

Cross sections for carbon projectiles in water and neon 653

PR2 (L, b) ≈ PR2 (L, b) = PR1 (L, b) and Pn(L, b) ≈ Pn(L, b). In the modified version of IEVM,equations (11) and (12) were reduced to

P(m)

single(b) ≈∑

L

NLPR(L, b)PNL−1n (L, b) (13)

and

P(m)

double(b) ≈ 1

c

∑L

NL(NL − 1)PR1 (L, b)PR2 (L, b)PNL−2n (L, b). (14)

If all degrees of target ionization are taken into account, the many-electron transitionprobability of target ionization, P(m)

ion , is described by the modified IEVM as

P(m)

ion (b) =∑

L

NL∑k=1

NL!

(NL − k)!k!Pk

i (L, b)[1 − Pi(L, b)]NL−k

=∑

L

{1 − [1 − Pi(L, b)]NL}

≈∑

L

NLPi(L, b), (15)

where k is the degree of the target ionization and 1 − Pi the probability of non-ionizing events.In the ionization scheme, P(m)

ion corresponds to the many-electron transition probability ofthe direct target ionization and target ionization accompanied by another interaction. P(m)

ion isequivalent to the probability of the pure target ionization, if Pc Pi, thus Pn ≈ 1 − Pi. In theelectron loss scheme, the index ‘i’ is replaced by ‘l’ in equation (15). Since Pn = 1 − Pl, P(m)

ionis the many-electron transition probability of the pure projectile electron loss.

The last approximation in equation (15) is valid for Pi 1, where only single ionizationevents are to be considered. Due to the small magnitude of the differential probabilities Pi(l)(ε)

and Pi(l)(ε, θ ), equation (15) was applied for the calculations of single and double differentialcross sections in electron energy (ε) and emission angle (θ ) for the single target ionization andpure single projectile electron loss.

The calculations in the IPM were carried out by substituting PR (Pn) by PR (Pn) inequations (11) and (12).

The modified IEVM (equations (13), (14)) was used for P(m)

SI , P(m)

SL and P(m)DI , and the

IPM for P(m)

SC and P(m)TI . The many-electron transition probabilities of loss ionization P(m)

LI , totalelectron emission P(m)

emit, one-electron capture P(m)

q,q−1 and one-electron loss from the projectile

P(m)

q,q+1 were obtained with

P(m)LI (b) = P(m)

SI (b)P(m)

SL (b), (16)

P(m)

emit(b) = P(m)

SI (b) + 2P(m)DI (b) + 2P(m)

LI (b) + P(m)TI (b) + P(m)

SL (b), (17)

P(m)

q,q−1(b) = P(m)

SC (b) + P(m)TI (b) (18)

and

P(m)

q,q+1(b) = P(m)

SL (b) + P(m)LI (b), (19)

respectively.TCS, SDCS and DDCS were calculated using the many-electron transition probabilities,

P(m)R , according to

σR = 2π∑

i

biP(m)R (bi)�b, (20)

654 T Liamsuwan and H Nikjoo

dσR

dε= 2π

∑i

biP(m)R (bi, ε)�b

�ε(21)

and

d2σR

dεd�= 2π

∑i

biP(m)R (bi, ε, θ )�b

2π sin θ�θ�ε, (22)

respectively, where the index R denotes the interaction type (equations (11)–(19)), bi the ithimpact parameter simulated, �b the differential impact parameter and �ε the electron energyinterval used for scoring the ejected electrons.

2.4. Charge state distributions and stopping cross sections

TCS for the charge exchange (SC, SL, TI and LI) were used for the calculation of equilibriumcharge state fractions fq of the charge state q. In this calculation, we solved the system ofequations, given by

0 = fq−1σq−1,q − fq(σq,q−1 + σq,q+1) + fq+1σq+1,q, (23)

where the equilibrium mean charge (qequi) of carbon ions was calculated with

qequi =6∑

q=0

q fq. (24)

For each specific energy T , the stopping cross section S(T ) was calculated, according to

S(T ) =6∑

q=0

∑R

fq(T )Eq,R(T )σq,R(T ), (25)

where Eq,R is the energy loss of the projectile during the interaction R and σq,R theassociated TCS. For testing the CTMC model, average energy loss was used in equation(25). Full Monte Carlo simulations are presented in the accompanying paper (Liamsuwan andNikjoo 2013).

For the calculation of the stopping cross sections, target excitation was also taken intoaccount. The cross sections for excitation to the states A1B1, B1A1, Rydberg states and diffusebands of the water molecule were obtained by scaling the Miller and Green’s formula forproton impact (Miller and Green 1973) with the square of the projectile’s charge state. In thiscase, the target electrons were assumed to see the carbon projectile as completely screenedby all its bound electrons. The effective charge formula Zc(rec) (equation (6)) was not usedfor scaling the excitation cross sections, because Zc(rec) depends on the distance between theactive electron and the core c, while an interaction cross section is by definition averagedover all possible electron radial distances. Based on the similar calculation for proton impactreported by Uehara et al (2000), the target excitation contributes at maximum about 18% tothe total stopping cross sections at proton energy of about 15 keV u–1, and the contribution isabout 10% at asymptotic impact energies.

The energy loss by the pure single target ionization (ESI) was calculated from the sumof the mean potential energy of the target (BH2O,SI) and mean kinetic energy of the ionizedelectron (ε). Here BH2O,SI was obtained with

Cross sections for carbon projectiles in water and neon 655

BH2O,SI =∑L

|Eb(L)|σi(L)∑L

σi(L), (26)

where σi(L) is the total ionization cross section associated with the orbital L and |Eb(L)| theorbital binding energy (table 1). σi(L) was computed by inserting the one-electron transitionprobability of target ionization Pi(L, b) into equation (20), and ε was calculated from theSDCS for the single target ionization, dσion

/dε (equation (15)), according to

ε =∫

εdσion

dεdε. (27)

The energy loss by the pure single electron capture (ESC) was set equal to

ESC = T/λ + BH2O,SC − BC,q−1, (28)

where T/λ is accounted for the energy used in accelerating the captured electron to theprojectile’s speed, λ the proton-to-electron mass ratio (λ ≈ 1836), BC,q−1 the first ionizationpotential of Cq–1 (table 1) and BH2O,SC the mean electron binding energy of the target for theelectron capture reaction, i.e.

BH2O,SC =∑L

|Eb(L)|σc(L)∑L

σc(L), (29)

where σc(L) was calculated similar to σi(L) used in equation (26).For the pure single projectile electron loss we assumed the energy loss to be ESL =

BC,q, i.e. electrons ejected from the projectile were assumed to have the same kineticenergy as before the emissions. Therefore, there is no energy loss for acceleration of theelectrons.

For the double electronic interactions the energy losses were calculated from the sum ofthe energy expended during each of the involved interactions, namely EDI = 2ESI + BH2O,SI,ETI = ESC + ESI + (

0.5BH2O,SI + 0.5BH2O,SC)

and ELI = ESL + ESI. Due to the fact thatenergy required to overcome the binding energy of a singly ionized molecule is in generalhigher than that of the neutral molecule, we assumed that removal of the second electron fromH2O+ required double as much energy as the binding energy of the neutral H2O. An exceptionto this view is expected for K-shell ionization, discussed in Liamsuwan and Nikjoo (2013),but this was not considered here. Moreover, for the transfer ionization the order of the electroncapture and target ionization was not considered, and the associated mean electron bindingenergy was averaged between BH2O,SI and BH2O,SC.

3. Results and discussion

3.1. DDCS for Cq+ projectiles on the Ne target

Figures 4–6 show the CTMC calculations of DDCS for the total electron emission by C+ ofenergies 66.67–200 keV u–1 impacting on the Ne target, compared with the experimentaldata of Toburen and co-workers (Toburen et al 2006, Toburen 2011). The calculationstook into account the single target ionization and pure single projectile electron loss(equation (15)).

The CTMC results show fairly good agreement with the experimental data at theintermediate angles (figures 4–6, panels (b)), but at the small angles (panels (a)) the calculationoverestimates the measured electron yield. The overestimation is particularly observed at the

656 T Liamsuwan and H Nikjoo

(a) (b)

(c)

Figure 4. DDCS for the total electron emission (single target ionization and pure single projectileionizations, equation (15)) by the 66.67 keV u–1 C+ impact on Ne: solid lines—the current CTMCmodel; dotted lines—the experimental data of Toburen et al (2006) and Toburen (2011). For the130◦ electron emission, the dot-dashed line shows the contribution of target ionization (ioniz) andthe dashed line the contribution of projectile electron loss (eloss).

binary encounter peaks of the target ionization, i.e. at the electron energy ε ≈ (4T/λ) cos2 θ

(T : specific energy and θ : emission angle). The agreement between the calculation and themeasurement for the forward electron emissions becomes better with the increase in projectileenergy.

At small angles, the experimental data for the C+ impact with energies 100 and200 keV u–1 show structures at high electron energies. The structures seen arise from theDoppler shift Auger electron emissions from the projectile. Auger electron emissions werenot included in the CTMC calculations.

For the large angle scatterings, the calculated yields of low energy electrons emittedby the 66.67 and 100 keV u–1 C+ impact are smaller than those measured (panels (c) offigures 4, 5). Moreover, the electron loss peaks (at ε ≈ T/λ) at large angles tend to be smaller

Cross sections for carbon projectiles in water and neon 657

(a) (b)

(c)

Figure 5. DDCS for the total electron emission by the 100 keV u–1 C+ impact on Ne. The samedescription as for figure 4.

than the corresponding peaks of the measured spectra for the studied projectile energies (panels(c) of figures 4–6).

For the electron backscatterings at 130◦, the contributions of the target ionization andprojectile ionization are given in the panels (c) of figures 4–6. The yields of low and high energyelectrons mainly arise from the target ionization. In this case, agreement between the calculatedand measured electron spectra is improved with increasing projectile energy. In contrast, theCTMC calculation underestimates the projectile electron yields in the backward directionfor all C+ impacts investigated. The comparison implies limitation of the CTMC model forprojectile electron loss at large angles. The disagreement with the experimental data mayalso imply the shortcomings of the screening function (equations (6)–(9)) for close collisions(leading to backscatterings). Nevertheless, backscatterings make only a small contribution tototal electron yield summed over all ejection angles, associated with SDCS.

In figure 7, the calculated yields of projectile electrons at 15◦, 90◦ and 130◦ are presented.As expected, the peaks at 15◦ (forward direction) are most pronounced compared with the

658 T Liamsuwan and H Nikjoo

(a) (b)

(c)

Figure 6. DDCS for the total electron emission by the 200 keV u–1 C+ impact on Ne. The samedescription as for figure 4.

peaks at 90◦ and 130◦. In the forward direction, both projectile ionization and target ionizationare important. Thus, the projectile electron loss peaks in the calculated electron yields are notwell observed in the panels (a) of figures 4–6.

As shown in figure 7(c), at 130◦ the electron loss spectra are relatively flat at low electronenergies, and begin to decrease continuously above a threshold energy. The threshold energyis close to T/λ (when electron’s velocity is similar to that of the projectile) for the 100 and200 keV u–1 C+, but for the 66.67 keV u–1 C+ the shoulder of the electron loss spectrum shiftstowards electron energy higher than T/λ.

The spectra of ejected projectile electrons, shown in figure 7, arise from the momentumdistributions of the projectile electrons before the collisions (equation (4)). If the projectileelectrons are assumed to travel initially with the carbon projectile’s speed (specific energy T )before the emissions, the ejected projectile electrons would have the constant energy of ∼T/λ

in both forward and backward directions, as described by binary encounter collisions betweenthe projectile electrons and screened target nucleus (Drepper and Briggs 1976).

Cross sections for carbon projectiles in water and neon 659

(a) (b)

(c)

Figure 7. DDCS for the pure single projectile electron loss (equation (15)) by the 66.67, 100 and200 keV u–1 C+ impact on Ne. The emission angles are (a) 15◦, (b) 90◦ and (c) 130◦.

3.2. Cross sections for Cq+ projectiles on the water target

3.2.1. Single and double differential cross sections. Figure 8 shows DDCS for the singletarget ionization (equation (15)) by the 6 MeV u–1 C6+ impact on water. The experimentaldata and the first Born approximation (FBA) model calculations reported in Dal Cappello et al(2009) are presented for comparison.

The CTMC results agree reasonably well with the measured DDCS at the intermediateand large angles (panels (b) and (c)), but at angles smaller than 60◦ (panel (a)) theyields of high energy electrons are overestimated by the CTMC calculations. At the smallangles, the FBA model gives better agreement to the measurement than the CTMC model(panel (a)).

The corresponding SDCS are presented in figure 9, together with the experimental dataof Ohsawa et al (2009) and the FBA model calculations of Dal Cappello et al (2009). Forthe SDCS, the CTMC calculations are in good agreement with the experimental data, while

660 T Liamsuwan and H Nikjoo

(a) (b)

(c)

Figure 8. DDCS for the single target ionization (equation (15)) by the 6 MeV u–1 C6+ impact onwater at emission angles: (a) 20◦–60◦; (b) 70◦–110◦; and (c) 120◦–160◦. This work (solid curves),the FBA model calculations of Dal Cappello et al (2009) (dashed curves) and the experimentaldata reported in Dal Cappello et al (2009) (symbols).

the FBA model overestimates the measured data over a broad range of electron energies. Thecomparison implies that most of the ionized electrons were ejected into intermediate angles,where good agreement was obtained between the DDCS calculated using the CTMC modeland the experimental data (cf figure 8(b)).

3.2.2. Total cross sections.Total electron emissionFigure 10 shows the TCS for the total electron emission by the C6+ impact on water. Thecalculations took into account the contributions of the direct target ionizations (SI and DI)and transfer ionization (TI), according to equation (17). Also presented are the contributionsof SI alone, and the TCS for the single target ionization based on one-electron transition

Cross sections for carbon projectiles in water and neon 661

Figure 9. SDCS for the single target ionization (equation (15)) by the 6 MeV u–1 C6+ impact onwater: this work (solid line); the FBA model calculation of Dal Cappello et al (2009) (dashed line);and the experimental data of Ohsawa et al (2009) (symbols).

probabilities, as described in the previous work (Liamsuwan and Nikjoo 2012). Comparisonswere made with the three-centre CTMC model calculations of Illescas et al (2011), theCDW-EIS model results of Bernal and Liendo (2007), the FBA model calculations of DalCappello et al (2009) and the empirical point at 6 MeV u–1 obtained by integrating theexperimental SDCS of Ohsawa et al (2009) over electron energies �1 eV (cf figure 9). Anumber of comments and questions arise from the comparison in figure 10 as discussedbelow.

Based on the current calculations, SI made a contribution of 65–80% to the total electronemission (figure 10). The contribution of TI maximized with 14% at ∼50 keV u–1, and it wasnegligibly small above a few hundreds of keV u–1 (data not shown). The rest of the ejectedelectrons were emitted by the direct double target ionizations (up to 35% contribution).

The current calculations largely reduced the TCS for the total electron emission below100 keV u–1, compared with the TCS for the single target ionization based on our previousmodel (Liamsuwan and Nikjoo 2012). In the previous work, we used the one-electron transitionprobability Pi to calculate the TCS for the single target ionization (similar to equation (20)) anddid not take into account the contribution of the double target ionization. The probability ofthe single target ionization used in the previous model is equivalent to the last approximationin equation (15), where both SI and TI are considered within the modified IEVM. In thiswork, the cross sections for TI were calculated based on the IPM. The reduction of thecross sections for the total electron emission obtained here is due to the decrease in thecross sections for SI. By considering the probability of having none of the electrons removedfrom the core (Pn � 1), P(m)

SI � Pi as shown in equation (13). The agreement between thecurrent and previous CTMC model calculations for the total electron emission at energy �500 keV u–1 (figure 10) arises from the fact that the direct target ionization is the dominantinteraction channel, and TI is negligible, in this energy range.

662 T Liamsuwan and H Nikjoo

Figure 10. TCS for the total electron emission (SI, DI and TI) by C6+ impact on water: the solidline with filled squares is the present work based on the many-electron transition probabilities(equation (17)); the dotted line with circles is the present calculation for SI; the short dashedline with diamonds is the TCS for single target ionization based on the one-electron transitionprobability described in Liamsuwan and Nikjoo (2012). Comparisons are made with the empiricalpoint at 6 MeV u–1 obtained by integrating the experimental SDCS of Ohsawa et al (2009) overelectron energy range �1 eV (cf figure 9); the CTMC model of Illescas et al (2011) based on thethree-centre model potential of the water target (dot-dashed line); the CDW-EIS model of Bernaland Liendo (2007) (double-dot-dashed line); and the FBA model of Dal Cappello et al (2009) (longdashed line).

The current calculation gives about 11% increase in the TCS compared with the empiricaldata point at 6 MeV u–1. The three-centre CTMC model of Illescas et al (2011) underestimatesthe empirical result, while the FBA and CDW-EIS model calculations are in good agreementwith the single point measurement.

Different energy dependences of the TCS were observed between our calculationsand those of the FBA and CDW-EIS models. As the impact energy decreases to below500 keV u–1, both of the CTMC models (this work and Illescas et al (2011)) and the CDW-EISmodel tend to produce the cross sections in a similar order of magnitudes, while the FBAmodel largely overestimates the other model calculations shown in figure 10.

The two sets of the CTMC model calculations (this work and Illescas et al (2011)) diverge.Differences between both CTMC models arise from the calculations of the transfer ionizationprobability (equation (12)), and the target description. Since the transfer ionization makesonly a small contribution to the total electron emission above 1 MeV u–1, the discrepancyin this energy range is expected to mainly arise from the implementation of different targetdescriptions. Specifically, we used the atomic picture for the water molecule and assigned themodel potential (equation (6)) and realistic electron binding energies to the water molecule(table 1). In the CTMC model of Illescas et al (2011) the water molecule was described withthree-centre potential composed of two hydrogens and one oxygen, and trajectory simulationswere averaged over ten directions relative to the configuration of the three nuclei. DifferentCTMC models have used different target descriptions, e.g. Otranto and Olson (2008) used

Cross sections for carbon projectiles in water and neon 663

Figure 11. TCS for the direct single target ionization (equation (13)) by the C0–C6+ impacts onwater. The symbols are calculated points connected with lines. The numbers denote the chargestate q. The experimental data of Montenegro et al (2007) for production of positive water ionsdue to direct target ionization by C0 impact are given (symbols without connecting lines). In theexperimental data, the contributions of the single target ionization and higher degrees of targetionization are not discriminated.

the constant effective charge of 1.6 for the water molecule, while the CTMC-COB modelof Abbas et al (2008) described the water molecule as a point charge with the effectivecharge of 1.

As shown in figures 8 and 9, the relatively simple atomic description used in the currentCTMC model reproduces the electron yields for the single target ionization reasonably well.Moreover, the cross section for the total electron emission at 6 MeV u–1 calculated in this workis in closer agreement to the empirical data point, than the result of the three-centre CTMCapproach (cf figure 10).Pure single target ionization (SI)Figure 11 shows the calculated TCS for the pure single target ionization (σSI) by the C0–C6+ impact on water (equation (13)). Comparison is made with the experimental data ofMontenegro et al (2007) for production of positive water ions due to pure target ionizationby C0 impact. Differences are seen between the calculations and experimental data. Themeasurements did not discriminate between the contributions of single ionization andhigher degrees of ionization. Also, the ‘peak’ of the experimental cross sections appearsat unexpectedly low energy (at ∼50 keV u–1). This peak is not observed in the calculations.Compared with our previous CTMC calculations (Liamsuwan and Nikjoo 2012, Liamsuwanet al 2011a), in this work we recalculated the one-electron transition probabilities for the1 keV u–1 impact of all carbon charge states to obtain better statistics. Thus, differences in Pi

between the present and previous calculations are seen in the range of 1–10 keV u–1 (data notshown).

As shown in figure 11, two different charge-state dependences of the direct single targetionization cross sections can be seen above and below the projectile energy of ∼100 keV u–1.

664 T Liamsuwan and H Nikjoo

Figure 12. TCS for the pure single electron capture by the C+–C6+ impacts on water: thiswork (crosses with connecting lines); and the experimental data for C+ (Montenegro et al 2007)(triangles), C3+ (Greenwood et al 2001) (diamond), C5+ (Mawhorter et al 2007) (circle) andC6+ (Greenwood et al 2001) (square). The dashed line is our previous CTMC calculation withouttaking into account the many-electron collision problem (Liamsuwan and Nikjoo 2012).

The decrease of the ionization cross section below 100 keV u–1 is associated with the increasedprobability of the electron capture at low impact energies (see figure 12). The reduction ofthe ionization cross sections is more significant for a higher-charge carbon ion. In particular,below 10 keV u–1 energy, the one-electron transition probabilities of the target ionization bythe C2+–C6+ impact were very small. Therefore, the target ionization cross sections for theseions below 10 keV u–1 energy were not considered.Pure single electron capture (SC)The calculated TCS for the pure single electron capture (σSC) for the C+–C6+ impacton water are presented in figure 12, compared with the experimental data reported inGreenwood et al (2001), Montenegro et al (2007) and Mawhorter et al (2007). For the highlycharged carbon ions with q = 5 and 6, agreement between the CTMC results and the singlemeasured data points is good. However, the energy dependence of the calculated cross sectionscannot be validated, due to lack of the experimental data for other projectile energies. For theC3+ impact, the calculated TCS is about 1.6 times larger than the measured data point ofGreenwood et al (2001), and for the C+ impact the CTMC model calculations are about1.6 times smaller than the measured data of Montenegro et al (2007). The probabilities of theelectron capture for projectile energies larger than 1 MeV u–1 are very low and therefore notincluded in the figure.

In figure 12, the results of the previous work (Liamsuwan and Nikjoo 2012) are also givenfor the C6+ impact. As mentioned earlier, the previous calculations did not take into accountthe many-electron collision problem. The current calculation, applying the IPM (section 2.3)for the electron capture cross sections, has significantly improved the agreement between theCTMC model calculations and the experimental data, as seen in figure 12.

Cross sections for carbon projectiles in water and neon 665

Figure 13. TCS for the pure single projectile electron loss (equation (13)) by the C0–C5+ impactson water. The symbols are calculated points and connected with the lines.

Pure single projectile electron lossFigure 13 shows the calculated TCS for the pure single projectile electron loss (σSL) for theC0–C5+ impact on water (equation (13)). In general, σSL increases with decreasing chargestate, because the projectile’s ionization potential is lower for a lower-charge ion (table 1).Exception is seen at projectile energies >1 MeV u–1, where σSL for the C2+ impact are slightlylarger than those for the C+ impact. The reason is that the effective number of the activeprojectile electrons for C2+ (two 2s-electrons) was assumed to be larger than that for C+ (one2p-electron) (table 1).

Threshold behaviour was observed for the one-electron transition probabilities forprojectile electron loss. The probabilities, and therefore the cross sections, are negligiblysmall for the C2+ and C3+ impact with energies below 10 keV u–1, and for the C4+ andC5+ impact with energies lower than 100 keV u–1, as shown in figure 13.

3.3. Equilibrium charge state distributions

The calculated interaction cross sections were interpolated with cubic B-splines, and used forthe calculation of equilibrium charge state fractions and stopping cross sections.

The obtained equilibrium charge state distributions (equation (23)) are presented infigure 14, along with the experimental evaluations for C0 and C+ (Montenegro et al 2007). Asexpected, lower charged carbon ions are mainly present at lower impact energies, and highlystripped carbon ions at relatively high impact energies.

Based on the calculation, carbon ions are fully stripped (q = 6) at projectile energies above2 MeV u–1 (figure 14). The obtained threshold energy is lower than about 3 MeV u–1, expectedfrom measurements with atomic targets (mostly carried out with carbon foils). In general,the measured equilibrium mean charge states tend to be marginally sensitive to the nature ofthe target. However, Martin (1965) has reported an anomalously high mean charge of carbon

666 T Liamsuwan and H Nikjoo

Figure 14. Equilibrium charge state distributions of carbon ions in water (equation (23)). Thenumber given next to each curve labels the corresponding charge state q. The symbols are theempirical results for C0 (solid circles) and C+ (open circles) reported in Montenegro et al (2007).

ions in H2 gas, compared to the measurements with N2, Ar and Ni targets. To our knowledge,equilibrium mean charges of carbon ions in water have not been determined experimentally,and it is uncertain if the values for water molecules are similar to those for atomic and othermolecular targets. Nevertheless, we found that the calculated fractions of C0 and C+ are insimilar magnitude to those evaluated from the measurements of water molecule fragments(Montenegro et al 2007). From the latter experimental study, the projectile energy, at whichthe fractions of C0 and C+ are equal, is somewhat lower than the energy value obtained in thiswork (see figure 14). The difference may arise from the fact that the experimental evaluation(Montenegro et al 2007) considered only the C0 and C+ fractions, while all charge statefractions were taken into account in this work.

Another remarkable characteristic of the equilibrium charge state distributions (figure 14)is the enhancement of the C4+ fraction up to about 78% at its maximum. For comparison, themaxima of the C2+, C3+ and C5+ fractions are in the range of 37–48%. Similar enhancement ofthe C4+ fraction has been reported for carbon foil targets (Shima et al 1992). The enhancementof the C4+ fraction can be explained by the shell effect: the ionization energy of the 1s electronof carbon (the active electron of C4+) is much higher than the ionization energy of the 2selectron (the active electron of C3+) (table 1). Thus, the cross section for the one-electronremoval from C4+ (σ4,5) is much smaller than the cross section for the one-electron removalfrom C3+ (σ3,4) (cf figure 13). In comparison, differences between σq,q+1 for the other pairsof adjacent charge states q are relatively similar to each other, and they are much smaller thanthe difference between σ4,5 and σ3,4 (cf figure 13). In addition to this, the one-electron capturecross section σq,q−1 (equation (18)) for all charge states q increases at a relatively similar ratewith the increasing charge, from q to q + 1, similar to the charge state dependence of the crosssections for the pure single electron capture (figure 12).

Cross sections for carbon projectiles in water and neon 667

Figure 15. Stopping cross sections for carbon projectiles in water (equation (25)). The results ofthis work are given for the total stopping cross sections (solid line, labelled ‘This work’), and thestopping cross sections for C6+ without taking into account energy loss by the charge exchangereactions (dot-dashed line, labelled ‘C6+ without CE’). The dashed line shows the results of ourprevious work (Liamsuwan and Nikjoo 2012), without taking into account the many-electroncollision problem or double electronic processes. The other model calculations were taken fromthe ICRU report (Sigmund et al 2005) (short-dashed line), the code SRIM-2010 (Ziegler et al 2010)(dotted line), Watt (1996) (long-dashed line) and the CasP4.1 code (Grande and Schiwietz 2009)(double-dot-dashed line). The inset shows the contributions of each charge state to the total stoppingcross sections. The numbers denote the charge state q.

3.4. Stopping cross sections

The calculated stopping cross sections (equation (25)) are presented in figure 15. The insetshows the contribution of each charge state. For comparison, the model calculation resultsof the ICRU Report (Sigmund et al 2005), Watt (1996), the SRIM-2010 code4 (Ziegler et al2010) and the CasP4.1 code5 (Grande and Schiwietz 2009) are presented. The current resultsare consistent with the other published models at projectile energies �800 keV u–1, but deviatefrom those models at lower energies. In particular, below 500 keV u–1, our results are smallerthan the other model calculations. Exception is seen at impact energies below 40 keV u–1,where our results are slightly larger than the CasP4.1 code calculations.

In the current calculations, the coupling between the target ionization and electron capturechannels within the three-body CTMC method was taken into account. Also, carbon projectilesof all charge states were considered according to equation (25). The other published modelsconsidered only the screening of the projectile’s nuclear charge, while the coupling betweentarget ionization and electron capture and the energy loss by the charge exchange reactionswere not included (Ziegler et al 2010, Grande and Schiwietz 2009, Sigmund et al 2005,Watt 1996).

Using a different approach, we calculated the stopping cross sections for C6+ by taking intoaccount the coupling between the target ionization and electron capture channels, but included

4 Stopping and range of ions in matter (SRIM).5 Convolution approximation for swift particles (CasP).

668 T Liamsuwan and H Nikjoo

only the energy loss due to the direct target ionization (SI and DI) and target excitation (thecurve labelled with ‘C6+ without CE’ in figure 15). The resulting stopping cross sectionsagreed with the other published model calculations for total stopping cross sections withinthe discrepancies between those model results (figure 15). The equivalent calculations havealso been performed for the other charge states (not shown here). We found that the obtainedstopping cross sections (without CE) decreased with a lower charge state of carbon ions. Inmost cases, the interaction strength is smaller for a lower-charge ion (except for projectileelectron loss). By incorporating all charge states in the calculation of total stopping crosssections (as given in equation (25)), the obtained results are expected to be lower than thecalculations without dressed ions. This is essentially the case shown in figure 15, whereour results (labelled ‘This work’) are smaller than other model calculations at relatively lowenergies.

It is noteworthy that below 500 keV u–1 all model calculations given in figure 15 havenotable discrepancies with each other. Future experiments on stopping of carbon ions in water(and on other biologically relevant targets) are needed for validation of model calculations forcross sections and stopping cross sections.

A comparison of the stopping cross sections is also made with our previous work(Liamsuwan and Nikjoo 2012), which considered neither the many-electron problem northe double electronic interactions (figure 15). In the energy range above 300 keV u–1, thecurrent approach significantly increased the stopping cross sections compared with the previouscalculation. This leads to better agreement between the CTMC results and the other modelcalculations in the MeV u–1 energy range, where all models are expected to be consistent.This improvement is due to the consideration of many-electron transition probabilities(section 2.3), and, to some extent, due to the inclusion of the double electronic interactions(section 2.4).

4. Conclusions

The paper presents calculated cross sections based on the three-body CTMC method for C0–C6+ in the energy range of 1–104 keV u–1 impacting on molecular water (similar situationas in water vapour) and atomic neon targets. The calculations for water are of major interestfor biophysical studies in the Bragg peak region of a carbon ion beam, while those for neonare of interest because of the similarity of its electronic structure to the water molecule.The calculations for neon were carried out to test the CTMC model for the many-electrontarget, because the experimental data for neon are much more readily available than those forwater.

In this work, the many-electron collision problem was taken into account using the IPMfor pure electron capture and transfer ionization and using the modified IEVM for pure targetand projectile ionizations and for loss ionization. The cross sections were computed for totalelectron emission (figures 4–10), pure single and double target ionizations (used for figures 10,11), pure single electron capture (figure 12), pure single projectile electron loss (figure 13),transfer ionization (used for figures 10 and 14) and loss ionization (used for figure 14). Thecross section data set covers major electronic interactions for low and intermediate energycarbon ions of all charge states. The paper presents, for the first time, a set of cross sectionsrequired for event-by-event track simulations of full-slowing-down carbon ions in the Braggpeak region, including all carbon charge states.

The results were compared with the available experimental data and the othermodel calculations. The calculated electron emission spectra show satisfactory agreementwith the experimental data for both neon and water targets, especially in the case of

Cross sections for carbon projectiles in water and neon 669

the target ionization. For the pure projectile electron loss, the CTMC model tends tounderestimate the electron loss peaks at large angles, implying the limitation of the modelfor backscatterings of projectile electrons. Nevertheless, backscatterings are expected to makea small contribution to the projectile electron emission spectrum summed over all emissionangles.

The calculated TCS for the water target are also in reasonable agreement with the existingmeasurements for the total electron emission and pure single electron capture. Using thecalculated charge exchange cross sections, we determined the equilibrium charge state fractionsof carbon ions in water. The equilibrium mean charge tends to be larger than those obtainedfrom measurements with atomic targets. However, the fractions of C0 and C+ are similar tothe empirical data, and the characteristic enhancement of the C4+ fraction observed in themeasurement was reproduced by the calculation.

The results for the stopping cross sections show consistency with the other modelcalculations at projectile energies �800 keV u–1, but deviate from those theoretical resultsin the lower energy region. The discrepancy arises from the fact that the present calculationsconsidered the contributions of the dressed ions, and the coupling between the electron captureand target ionization channels. In contrast, the other model calculations only took into accountthe average screened nuclear charge of carbon projectiles.

The CTMC model as presented in this work provides a tool for cross section calculationfor bare and dressed carbon ions in the low and intermediate energy ranges. The inclusionof many active electrons in the calculations has significantly improved the results forthe cross sections and stopping cross sections, compared with the previous calculationthat only considered the one-electron problem. The cross sections of the water target, ascomputed, are required inputs for event-by-event simulations of full-slowing-down tracksof carbon ions. The track simulations at a molecular level can be used for biophysicalstudy of radiation damage produced in the Bragg peak area of a therapeutic carbonion beam. The development described in this paper paves the way to investigate doseinhomogeneity in the Bragg peak area at the cellular level, a long standing problem in radiationtherapy.

Acknowledgments

We thank Professor Larry H Toburen for valuable comments and discussion of this workand for providing the experimental data for the neon target. We thank Professors ShuzoUehara (Kyushu), David Liljequist (Stockholm) and Dimitris Emfietzoglou (Ioannina)for their scientific support and useful discussions. The work was partially supported byStralsakerhetsmyndigheten (SSM) Swedish Radiation Safety Authority, Karolinska Institutet,and Department of Physics at Stockholm University.

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