Challenges in Monte Carlo track structure modelling
Larry H. Toburen
Department of Physics, East Carolina University, Greenville, North Carolina, USA
Introduction
Track structure simulation has been the subject of extensive study since the discovery of the electron and the observa-tion of effects on the paths of moving charged particles as they pass through different materials; see, for example, the very early study by J. J. Thompson of energy loss by elec-trons (Thomson 1912) and the review of N. Bohr on energy
loss by heavy charged particles (Bohr 1948). In the calcula-tion of stopping power, energy loss per unit path length, the energy is considered to be lost uniformly along the path of the particle depending primarily on the charge and velocity of the moving particle and the mean excitation energy of the stopping medium. The art of accurately calculating stopping power has developed into a complex and demanding field, but still the end product is simply the energy lost per unit track length; information on the stochastic nature of energy deposition is lacking, except as it might affect energy and range straggling (Bohr 1948). Several reviews of stopping power have been published in the past few years; see for example, contributions by numerous authors in the recent book edited by Mozumder and Hatano (2004) and the review published by the International Council on Radiation Units and Measurement (ICRU 1992).
Chemists were the first to discover that stopping power was an inadequate description of charged particle tracks on which to predict chemical reactivity induced by energy deposited along the track. They found that one had to account for the statistical distribution of energy-loss events including events occurring off-axis to the track to predict the observed chemical yields. They generated track entities such as spurs, blobs, and short tracks to describe effects of the statistical variations in energy deposition on chemical yields (Mozumder and Magee 1966a, 1966b, 1966c). The inability of simple stopping power (energy loss), or linear energy transfer (energy absorbed), to describe chemical and biological response to irradiation by fast charged particles also led to the development of track structure models based on radial distributions of energy deposition along the track (Butts and Katz 1967, Chatterjee and Schaeder 1976). From these experiences came the first Monte Carlo (MC) codes to calculate the radial extent of regions of energy density that defined different chemical reaction regimes (Chatterjee et al. 1973), i.e., the track core versus the track penumbra; a pow-erful concept, if not a totally accurate picture of the physics. The development of these track models was accompanied by the development of a number of event-by-event Monte Carlo models of the structure on charged-particle tracks
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AbstractPurpose: Although great progress has been made, numerous challenges remain in the development of Monte Carlo (MC) charged-particle track structure simulation models. Such models have evolved from the simple gas phase target models to those using condensed phase interaction data coupled with complex targets representing cellular and molecular constituents of mammalian tissue. A wide choice of MC models is now available ranging from those based on the physics of continuous slowing down, to simulations following each interaction on an event-by-event basis. The choice of code depends largely on requirements for computational speed, and the degree of detail required; however, one must be continuously vigilant to recognise the inherent limitations of the model chosen. Conclusions: There remain numerous questions of the accuracy and completeness of the interaction physics that present challenges to MC modellers. Recent evidence suggests that the yields of electrons with energies less than a few hundred eV might be substantially overestimated by the elastic and inelastic cross-sections used in many codes. Densely ionising heavy ions present modelling challenges when the rate of energy loss is sufficient to ionise essentially ‘every’ atom along the ion path. Effects of electron capture and loss by moving heavy ions present significant challenges for modellers particularly for accurate simulation for ions heavier than protons and helium ions? The average effective-charge provides an inadequate description for estimating differential cross-sections for energy loss. These and other questions are considered.
Keywords: Monte Carlo, cross-sections, effective charge, electron transport, charge transfer, ionisation, differential cross-sections, ion and electron interactions
Correspondence: L. H. Toburen, Professor Emeritus, PhD, Department of Physics, East Carolina University, Greenville, NC 27858, USA. Tel: +1 252 328 1861. E-mail: [email protected]
(Received 27 December 2010; revised 8 March 2011; accepted 17 March 2011)
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Challenges in MC modelling 3
that attempted to incorporate the most up-to-date physics in more detailed models of the structure of heavy ion tracks; these many models are not reviewed here as they have been discussed in detail by Nikjoo et al. (1998). The take home message is simply that stopping power does not function well if one is looking for ways to predict chemical, and thereby biological, response to absorption of ionising radiation and numerous models of greater spatial detail have been devel-oped to describe energy deposition. It, of course, depends on the application intended as to which of the different types of models might be the best choice for that task.
Several Monte Carlo track structure codes have been developed for applications in Medical Physics. For the most part these codes are based on track segment linear energy transfer (LET) or stopping power (dE/dx) models that follow the dose profiles in volumes with dimensions of fractions of millimeters and larger. The strength of such models is the manner they can handle complex geometries involving interfaces between tissues of different types, e.g., tumour and muscle tissue, bone and bone marrow, air cavities, etc. The well known physics of stopping power provides a reliable description of the energy loss process in volumes of the sizes of interest to medical physics and the current focus of devel-opment of Medical Physics MC models is on developing better, more complete, descriptions of the geometric struc-tures of the targets of interest. The physics challenges of track-structure physics are much less critical to the track-average models of Medical Physics than to event-by-event applica-tions in microscopic targets, such as those representative of events occurring in the deoxyribonucleic acid (DNA) of cell nuclei.
On the other extreme of target size is that of quantum mechanical models of energy transport in condensed phase materials. These models address the interaction process predominantly at atomic dimensions. Such models of track structure are important for understanding effects of molecu-lar bonding and material structure on energy deposition, but tend to be sufficiently complex that it would seem to be some time before they can be applied in biologic media.
The most widely used track-structure models in mecha-nistic radiation biology are the event-by-event models using water as the absorbing media (see, for example, Nikjoo et al. 1998, Dingfelder et al. 2008a). Most of these models use essentially the same interaction physics for electron and ion impact; gas phase models use primarily experimental data and those based on liquid water use results of Born theory as formulated by Ritchie et al. (1991) and updated by numerous authors, see for example, Dingfelder et al. (1998). The theoretical liquid water cross-section calculations are based on the dielectric functions of liquid water and are evaluated via experimental optical oscillator strengths. As such they contain, in detail, the various energy loss channels for charged particles in water in the liquid state. Differences between event-by-event Monte Carlo codes using different input cross-sections have been presented by Paretzke et al. (1991), Nikjoo et al. (1998), Dingfelder et al. 2008a and others; it is not the object of this paper to provide detailed reviews of MC codes, but hopefully the articles quoted here will provide the interested reader a departure points for further reading.
Challenges to physics
Event-by-event Monte Carlo codes developed for application in biological sciences have had a great deal of success. With sophisticated models of DNA and chromosome structures (see for example structures presented by Bernhardt et al. 2003), calculations of DNA damage by ions and photons have become common. Among other endpoints, these models have been used to calculate DNA fragment yields (Campa et al. 2005, Friedland et al. 2005), DNA fragment size distribu-tions (Friedland et al. 2003) and DNA break points in genes (Friedland et al. 2001) produced by ionising X-rays and light ion irradiation. Perhaps one of the most exciting applications of charged particle track simulation was the ability to actu-ally confirm the correct chromosome structures between competing models (Rydberg et al. 1998).
With the degree of success that has been shown in the application of the modern Monte Carlo track simulation codes, one might ask, where are the remaining challenges? In reality, this becomes essentially a question as to what level one can average energy deposition without affecting results. Even with the studies of DNA damage, the statisti-cal nature of energy deposition is averaged, generally by determining the average energy required to produce strand breaks of a specific type in the volume occupied by the damage. However, when detailed knowledge of the initial spectrum of damage species is of interest, averaging must be limited to much smaller volumes and smaller energy deposition events. Can we reliably predict the frequency and nature of specific types of complex DNA damage with the level of physics used in current codes? We now know that repair fidelity is affected by the nature of these mul-tiply locally damaged sites (Keszenman and Sutherland 2010). We might speculate that very complex DNA damage simply kills cells (does not lead to mutation), and that sim-ple damage is repaired with high efficiency leading to no effects. But are there sub-classes of damage that might be only partially repaired, e.g., a single-strand break repaired in the presence of other local damage that are either not repaired or miss-repaired leading to a genomic instability and perhaps mutagenic behaviour? In addition, electrons with energies less than 10 electron volts (eV), the normal cut-off energy for event-by-event Monte Carlo codes used in radio-biology, have been found to also produce breaks in DNA (Huels et al. 2003). The number and energies of low-energy electron track-ends are particularly important parameters for predicting hazardous biologic endpoints. Thus, understanding the accuracy of our codes for dealing with low-energy electrons, whether produced directly by high energy ions or the result of ‘delta-rays’ slowing down, can be a challenge to physics. Recall that essentially all current liquid-phase transport codes use theoretical cross-sections for interactions in liquid water because detailed experimental studies of liquid water are either infeasible or extremely difficult. In addition, there have always been strong arguments against using water-vapour cross-sec-tions to represent interactions in tissue; the use of a water medium for electron transport studies has been justified because the cell is about 85% water.
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4 L. H. Toburen
Low-energy electron physicsLet us first consider the challenge of accurately modelling low-energy electron interactions in the condensed phase. A few years ago we initiated an experimental approach to develop data that might test the accuracy of electron production and transport accomplished in MC codes that use liquid water as the transport medium. We developed an experimental sys-tem using thin layers of amorphous solid water (water frozen on a copper substrate at ~40 Kelvin) as the transport medium because of its similarity to liquid water, and high-energy pro-tons to produce fast electrons that undergo transport within this ice. The electrons produced via interactions between fast protons and the electrons of the amorphous solid water (ASW) undergo elastic and inelastic scattering until they exit the sur-face of the thin ice layer where they are analysed as to emis-sion angle and exit energy (the details of these measurements were published by Toburen et al. 2010). A detailed discussion of the experiment is not provided here, but a comparison of the measured differential Yields g(e, q), where e is the ejected electron energy and q the emission angle, to those calculated by the MC code PARTRAC is shown in Figure 1; in this example the measured spectrum is for electrons ejected at 45° in respect to the outgoing proton beam, whereas calculations using the MC code were performed at 40 and 50°. In these calculations the MC code was extended so that one could follow transport of electrons with energies as low as 1 eV using inelastic electron scattering cross-sections for water from the work of Michaud et al. (2003) (see also Dingfelder et al. 2008b) for the discus-sion of modifications made to extend PARTRAC to 1 eV). The large differences observed in Figure 1 between the calculated and measured spectra for electron energies less than about 100 eV could have important consequences on the yields of complex damage to DNA estimated from such codes. A serious challenge remains for modellers to resolve the origins of these discrepancies if we are to be confident in the use of MC codes using theoretical interaction cross-sections for liquid water to accurately model detailed patterns of complex DNA damages.
Charge transferAnother challenge for MC modellers who are interested in applications of MC codes to the slowing of heavy ions (and all ‘fast’ ions must ultimately become sufficiently slow) is the inclusion of effects of charge transfer. When fast ions slow to energies less than a few million electron volts per atomic mass unit (MeV/u) they begin to capture bound electrons from the transport medium. The probability, or cross-section, for electron capture or loss is commonly des-ignated as sif, where i designates the initial and f the final projectile charge state, e.g., s10 refers to the probability an ion with net charge +1 will capture an electron from a tar-get atom to become neutral (charge state zero). The capture process leads to additional energy loss channels that include adding the necessary kinetic energy to the previously bound target electron to bring it to the velocity of the ion, and the transfer of ‘potential’ energy Et = BI − Bt between target and projectile, where BI is the binding energy of the state of the ion to which the electron is transferred and Bt is the binding energy of the initial state of the electron in the target. Note that Et can be either positive or negative depending on the relative potential energies of the states involved; with highly charged ions this aspect of charge transfer generally leads to positive energy, i.e., the projectile actually gains energy. Of course, once an electron (or several electrons) has been cap-tured by the moving ion, these electrons can be stripped in a subsequent collision, causing an additional loss of energy by the ion and leaving an energetic fee electron moving in the interaction medium (one having velocity near that of the ion from which it was stripped). Interactions involving electron capture lead to energy loss by the ion and a target ion being formed although no free electron is produced. On the other hand, with projectile-electron loss an energetic free electron is generated with no target ion being produced. Generally, however, these processes (capture and loss) lead to simulta-neous excitation of the collision partner, i.e., electron capture with projectile excitation or electron loss with target excita-tion (Manson and Toburen 1981). Perhaps the most impor-tant aspect of electron capture and loss, in addition to a change in number of free electrons produced beyond simple ionisation, is the change in the ‘effective’ charge of the mov-ing ion that leads to a change in the interaction potential with succeeding interactions.
A complete experimental or theoretical understanding of the charge-transfer process is currently lacking. Little has been published regarding charge-transfer cross-sections with the exception of ions and targets of importance to fusion energy. Protons and helium ions with energies less than a few MeV and low-energy carbon, oxygen, and nitrogen ions have been studied under sponsorship of the fusion energy com-munity as important plasma constituents and contaminants. The data in Figure 2 suggest that for singly-charged ions, electron capture and loss cross-sections might be relatively independent of the ion species. The data shown here for singly charged ions of Hydrogen (H), Helium (He), Carbon (C), and Chlorine (Cl) interacting with neon and methane target gases would suggest a common ‘average’ line could be drawn representing the single electron capture process and a second line could be fitted to the single electron loss
Figure 1. Comparison of the measured energy spectrum of electrons ejected from a thin layer of amorphous solid water at 45° to the for-ward direction of the exiting 6-MeV proton beam to spectra simulated using the Monte Carlo code PARTRAC where liquid water is used as the transport medium. The experimental data are described in detail by Toburen et al. (2010), and the PARTRAC code as used here has been described by Dingfelder et al. 2008b.
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Challenges in MC modelling 5
the position of ion production to the interaction region lead-ing to varying amounts of decay of excited states that might be produced. Thus MC modelling of intermediate to low-energy ions presents a strong challenge to modelers who want to include cross-sections for electron capture and loss in track structure simulations. And this challenge only gets worse for describing heavier ions interacting with biologically relevant targets where even less data are available to guide modelers.
Effective chargeThe effects of electron capture and loss lead to changes in energy loss as noted above, but perhaps more importantly, these processes lead to changes in the interaction potential between the moving ion and the target electrons. For stop-ping power these changes are incorporated via an effective charge (Zeff) that can be estimated by theory, i.e., using the empirical formula of Barkus (1963) or other approaches. In Figure 4, the effective charge for hydrogen ions, based on the Barkus formula, is found in relatively good agreement with the measurements of Yarlagadda et al. (1978) derived from experimental stopping power data. However, it should be emphasised that the effective charge as defined for use in stopping power is not the same quantity as the average charge of an ion (Zave, or sometimes designated as qave) as deter-mined from projectile charge measurements. In Figure 4 the effective charge is also compared with the average charge of an ion (Zave) derived from the experimental data of McDaniel et al. (1979); note that only for the largest velocities are these two quantities similar, and in that case the average charge is essentially that of a bare proton. The difference at smaller relative velocities (vo in Figure 4 is the velocity of the bound electron prior to capture and v1 is the ion velocity) is largely related to the fact that stopping power implies an energy loss has taken place and this occurs at small impact parameters where the bound projectile electrons are somewhat less
cross-sections regardless of ion, with relatively small uncer-tainties. This simple model certainly would be expected to break down if the binding energies of the ion species were markedly different, but the similarity is hopeful for predict-ing cross-section where data are lacking. The data in Figure 3 extend the electron capture cross-sections to additional light ions and to higher charge states; the lack of continuity of data over a large energy range now becomes apparent, even for the simple helium target shown here, and for relatively light ions. Also, the oxygen ion data (On+), where n is the number of missing bound electrons, clearly exhibit wide variation in the cross-sections presented by different authors. These variations are generally attributed to different amounts of meta-stable excited states being carried by the incoming ions used in these studies; differences can occur when different methods are used to produce the specific charge state of the incident ion, or when there are differences in the time from
Figure 2. Single electron capture and loss by singly charged ions in neon and methane. Data shown are: -s10 C -Ne, -s12 C -Ne, -s10 C1 -Ne, -s12 C1 -Ne (Toburen et al. 2006); -°-s10 C -Ne (Rottmann et al. 1992); ∙∙∙∙ s10 H -Ne (Rudd et al. 1983); — — s01 H°-CH4 and ∆ s10 H -CH4 (Toburen et al. 1968); s10 C -CH4 (Itoh et al. 1995); - - - s10 He -Ne and s12 He -Ne (Rudd et al. 1985); and —— s10 H -Ne (Barnett et al. 1977).
Figure 3. Single electron capture from helium. The data for O , —, C ——, and N - - - are from Kusakabe et al. (1990); C - (Ishii et al. 2004); C - (Unterreiter et al. 1991); C3 - and C5 - (Melo et al. 1999); O (3P) — - — (Kimura et al. 1996); C - (Rottman et al. 1992); C3 - (Montenegro et al. 1992); C2 –o– (Janev et al. 1988); C6 -∆ (Shinpaugh et al. 1992); and C -x (Evans 2005).
Figure 4. Comparison of the effective charge of a proton, as a function of the ratio of its velocity v1 to that of the bound target electron vo. The solid line is derived from experimental stopping power (Yarlagadda et al. 1978), the dotted line is the effective charge determined from the empirical formula of Barkus (1963), and the dashed line is the average charge of the proton qave determined from data tabulated by McDaniel et al. (1979).
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6 L. H. Toburen
effective at screening the nuclear charge. Thus, the average effective charge is always somewhat larger than the overall average charge, except in the trivial case where the ion is bare. Because the effective charge of any ion carrying bound electrons depends on the impact parameter from which it is viewed it is strongly dependent on the energy loss; the larger the energy loss, the smaller the associated impact parameter, and the larger the effective charge (Toburen et al. 1981). This is particularly important for estimating the differential cross-sections for ionisation by ions that posses bound electrons and this is discussed below.
Because projectile electrons screen the projectile charge in an impact parameter-dependent manner one cannot use a fixed value for the average effective charge to scale differ-ential ionisation cross-sections from bare projectile data; an impact dependent and thereby energy loss dependent effective charge must be used. We can explore this by first considering mechanisms responsible for the spectra of elec-trons that are produced when water vapour is ionised by a moving neutral hydrogen atom (H°) with kinetic energy of a few hundred kilo-electron volts (keV). Note that in ion beam studies, the ion is generally defined by its chemical symbol and a superscript defining the charge of the ion, e.g., H+1 or simply H+ would designate a proton. The mechanisms lead-ing to the spectra of electrons produced in collisions of H° with water vapour (H2O) are illustrated in Figure 5 where the data of Bolorizadeh and Rudd (1986a) for 150 keV impact are shown along with the mechanisms that contribute to the electron spectrum. The spectrum produced by H° impact is dominated by a maximum for electrons of the lowest ejected energies observed (a few eV). A secondary, but smaller maxi-mum occurs for ejected electron energies of about 81 eV; in this case this is an energy at which the velocity of the electron is equal to the moving H° particle and the peak is formed by electrons stripped from the moving H° particle. For ejected electron energies above the stripped electron peak energy, the H° produced continuum follows closely the proton spec-trum that was obtained in this case from the model of Rudd et al. (Rudd et al. 1992, ICRU 1995). The dotted line shows the spectrum expected for target ionisation by the screened H° particle; as expected the cross-sections for target ionisation decrease as the screening becomes more efficient at large impact parameters (small energy loss). The dashed lines shown in Figure 5 for low-energy electron emission from water vapour were taken from the work of Bolorizadeh and Rudd (1986b) for ionisation of water vapour by fast electrons. It is apparent that the low-energy portion of the spectrum is dominated by inelastic scattering of electrons initially bound to the incident H° particle and the peak in the spectrum at about 81 eV results from elastic scattering of these electrons – the peak is broadened by the Doppler shifted initial velocity distribution, or the Compton profile, of electrons bound to H°.
The data in Figure 6 provide an extension to the dis-cussion of the effects of projectile bound electrons for a somewhat heavier ion, in this case C+ at 100 keV/u (Toburen et al. 2006), shown as the open circles, and compared to equal velocity proton impact (Crooks and Rudd 1971) shown as the solid circles. These data are shown in order
to emphasise the errors contributed by scaling differential cross-section data for protons to heavy ions by use of simple (Zeff)2 scaling. The proton data for ionisation of neon scaled by the square of the carbon projectile’s nuclear charge (ZC)2 are shown as the dot-dot-dashed line; note that only for energy loss (ejected electron energies) larger than a few hundred electron volts is there agreement between proton data scaled in this manner and C+ results; C+ looks like bare carbon for large energy loss, the dot-dot-dashed line representing scaled bare carbon projectiles is equal to and hidden behind the C+ data at the highest energies in Figure 6. For low-energy electrons there are large differences, e.g., at 10 eV the proton cross-sections scaled to bare carbon cross-sections are nearly an order of magnitude larger than the measured singly charged carbon ion data. Obviously, if a single value for an effective charge were used, say equal to the equilibrium charge of Zeff = 2.2 (Ferguson 1974), the high-energy electron cross-sections would be underesti-mated by about a factor of 7 and the low-energy data would be overestimated by about a factor of 4. It is clear that one must use an energy-loss-dependent effective charge to improve the scaling. In Figure 6 we also show the use of a scaling technique developed by Toburen et al. (1981), where the screening depends on the impact parameter R and is related to energy loss by the distance of closest approach via
Figure 5. Mechanisms for ionisation of water in the vapour phase by 150 keV neutral hydrogen particles; s(e) is the differential ionisation cross section, differential in the ejected electron energy e and the units are barns (1 barn 10−28 m2). The dashed line for neutral hydrogen data is from the work of Bolarizadeh and Rudd (1986a); the electron impact data (dot-dash and dot-dot-dash lines) are from Bolarizadeh and Rudd (1986b). The solid line represents ionisation of water vapour by 150 keV proton impact determined by the model of Rudd (Rudd et al. 1992, see ICRU 1995 for a discussion of the model and parameters for water vapour) and the dotted line represents target ionisation by 150 keV H° particles obtained by multiplying the results for proton impact by an impact parameter-dependent effective charge estimated from the work of Toburen et al. (1981).
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Challenges in MC modelling 7
a water molecule, with dimensions of 1 Å between its constitu-ent hydrogen and oxygen, one might ask what the meaning of incurring ionisations located so closely spaced from the pas-sage of a heavy charged particle. Either the mean energy of the ejected electrons is much greater than the 50 eV estimate used here (based on electrons ejected by MeV protons), or the initial energy loss process includes large contributions of mul-tiple ionisation events! Can we find evidence that heavy ions produce extensive amounts of multiply ionising events along their path? The literature on multiple ionisation of atomic and molecular targets by fast heavy ions is not extensive, but there have been some data presented for selected ions interacting with generally light atoms. As might be expected, we find the highest probability is for single ionisation even for Bragg peak ions. Cocke (1979) found that multiple ionisation of argon atoms by 34 MeV Cl9+ ions (Bragg peak energy) led to the domi-nant charge state being Ar+1, although charge states up to Ar+11 were seen, and the probability decreased reasonably quickly with ionisation state, i.e., Ar2+ was produced with a probability of about 1/3 that of Ar1+ and Ar3+ was about ½ of the Ar2+ yield. From this data one could estimate the average number of elec-trons ejected per collision as about 1.7 (in gaseous argon); and based on an average energy loss of 50 eV/ion pair and the stop-ping power of 15 MeV/(mg/cm2) suggests somewhat more than 2 ionisations per Å. Maybe this back-of-the-envelope estimate is close enough to confirm that multiple ionisation is indeed the mechanism for the large energy losses by Bragg peak heavy ions; no doubt the 50 eV per ionisation is small when multiple ionisa-tions are considered. Still, energy losses accounting for as much as 8 ionisations per Å, as we estimated for Bragg peak iron ions, are a little hard to image even when multiple ionisa-tion is included. But, if these large ionisation densities are true, the next challenge is to associate the multiple electrons ejected and the target ion potentials produced along the charged particle track with subsequent chemical reactions if one is to understand the damage that might result at the molecular level in a biologic target like DNA.
Conclusions
This discussion has only touched on a few of the challenges that MC modellers must face as we try to better understand the chemical and biological effects of heavy ions at the molecular level and solutions to these challenges are not clear. It is perhaps premature to worry about such effects in light of the successes compiled by current models, but as molecular biology becomes more skillful at identifying molecular damage produced at early times following irra-diation, and as the consequences of different patterns on damage on the fidelity of biologic repair are discovered, these uncertainties in the physics of modelling track simu-lation are likely to become increasing important. Hope-fully physicists will continue to provide the data needed to address model weaknesses and modellers will incorporate these new data as we seek to provide better tools for under-standing the mechanism responsible for biologic response to ionising radiation and to reduce the uncertainties in estimating risks from new and different radiation source characteristics.
the Massey Criterion, i.e., R = v/DE, where R is the adia-batic interaction distance, v is the projectile velocity, and ∆E is the energy transfer in the collision. This impact-dependent effective charge Zeff(R) used in the Z2 scaling provides a marked improvement over the use of the traditional Zeff; perhaps this provides a simple approximate method for future work in meeting the challenge of dealing with the effects of electron capture and loss on differential ionisa-tion cross-sections used in track structure simulation.
Ionisation densityAnother challenge awaiting modellers, who attempt to describe the detailed chemistry following interactions of heavy charged particles with water, or other descriptions of the biologic medium, is that of dealing with the extremely high densities of ionisa-tion that can occur along the track of such ions. Simple estimates of the ionisation density along a proton track at energies near the Bragg peak (maxi-mum in the stopping power) leads to about one ionisation per 10 Angstroms (Å), or about an ionisation for every 2–3 molecules traversed along the path of the proton; thus ionisations are separated sufficiently that one is generally independent of another. This estimate is based on the stop-ping power in water and an estimate of about 50 eV per ionisation; the average value of 50 eV/ionisation might be a bit high, but it provides a rough estimate. With the same analysis, a fluorine ion with velocity near the Bragg peak can produce perhaps 2 ionisa-tions per Å, and under the same conditions an iron ion would produce 8 ionisations per Å. For
Figure 6. The electron spectrum for ionisation of neon by 100 keV/ u C+ ions (o) compared to data for 100 keV/u protons (•); s(e) is the differential ionisation cross section, differential in the ejected electron energy e and the units are barns (1 barn 10–28 m2). The proton data are from Crooks and Rudd (1971); the proton impact model is from Rudd et al. (1992); and the impact parameter-dependent effect charge of the projectile Zeff(R), where R is the distance between the moving charged particle and the active electron at closest approach, is obtained from the method described in Toburen et al. (1981).
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8 L. H. Toburen
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Declaration of interest
The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper.
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