Inelastic-collision cross sections of liquid water for interactions of energetic protons

Radiation Physics and Chemistry 59 (2000) 255±275

www.elsevier.com/locate/radphyschem

Inelastic-collision cross sections of liquid water for interactions of energetic protons p Michael Dingfelder*, Mitio Inokuti 1, Herwig G. Paretzke GSF ± National Research Center for Environment and Health, Institute of Radiation Protection, IngolstaÈdter Landstrabe 1, D-85764 Neuherberg, Germany Received 31 August 1999; accepted 28 January 2000

Abstract Cross-section data for inelastic interactions of energetic protons with liquid water, for use, e.g. as input in track structure analysis, are derived for an energy range from 0.1 keV to 10 GeV. At proton kinetic energies above about 500 keV, the ®rst Born approximation and the dielectric-response function determined earlier are used. At proton energies above several hundred MeV in particular, the Fermi-density e€ect is also incorporated. At energies below about 500 keV, which corresponds to a residual range of about 8.9  10ÿ6 m, cross-section values are derived semiempirically by an extensive and critical analysis of experimental and theoretical information concerning not only cross sections for individual processes such as ionisation, excitation, and charge transfer but also stopping power and other relevant quantities. Spectra of secondary electrons resulting from ionising collisions are also presented. The analysis also includes considerations of phase e€ects on cross sections. 7 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction Track structure calculations of charged particles (Paretzke, 1987; Ritchie et al., 1991) are useful for the

understanding of early physical and chemical stages of radiation actions on matter in general. They are used especially in research on the e€ects of radiation on the biological cell or on DNA (Friedland et al., 1999),

Soon after acceptance of this manuscript we became aware of a work on a similar topic entitled ``Calculations of electronic stopping cross sections for low-energy protons in water'' by S. Uehara, L.H. Toburen, W.E. Wilson, D.T. Goodhead and H. Nikjoo, also accepted for publication in Radiation Physics and Chemistry (this volume). The authors calculated the electronic stopping cross section for low-energy protons in water vapour using the charge state approach and taking into account excitation and ionisation by protons and hydrogen atoms as well as charge changing processes. Whenever possible pure numerical ®ts of experimental data were used to represent the interaction cross sections for certain processes, otherwise use of published semi-empirical models for water vapour were made. In this way the calculated stopping cross section overestimates tabulated and experimental values within about 20% for particle energies below 300 keV. On the other hand, this work concentrates on liquid water and uses also the tabulated and experimental stopping cross sections of water in di€erent phases (vapour, liquid and ice) to include phase e€ects and to adjust semi-empirical models for proton impact in liquid water. In this sense the two papers are complementary and agree in common results. * Corresponding author. Tel.: +49-89-3187-4113; fax: +49-89-3187-3363. E-mail address: [email protected] (M. Dingfelder). 1 Permanent address: Argonne National Laboratory, Physics Division, Argonne, IL 60439, USA. p

0969-806X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 0 0 ) 0 0 2 6 3 - 2

256

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

which has a highly inhomogeneous spatial and chemical structure. Track structure analysis based on computer simulations requires as input data cross sections for the interactions of electrons and other charged particles with molecules in matter under consideration. In the biological cell of soft tissues, water is the dominant component (ICRU, 1989) and is largely in the liquid state. However, because of the scarcity of experimental and theoretical knowledge, earlier track structure calculations for radiation biophysics used water in the vapour state as a model substance (Zaider et al., 1983; Paretzke, 1987). Interaction cross sections for protons and other charged particles with molecules have been studied for many decades. Recent data sources are an IAEAreport (IAEA, 1995), a review on cross sections in general (Inokuti, 1994) and a review article focusing on proton ionisation and charge transfer cross sections (Toburen, 1998). Rudd and co-workers reviewed experimental data and developed semi-empirical models for proton ionisation cross sections for various atoms and molecules (Rudd et al., 1985a, 1992). Other semiempirical and numerical data sets for proton interaction cross sections in water vapour are seen by Miller and Green (1973), Kaushik et al. (1981) and Zaider et al. (1983). Later on, cross sections for water clusters were also derived (Long and Paretzke, 1991) to study the in¯uence of phase e€ects in water, in detail. Following up the paper by Dingfelder et al. (1998) on cross sections of liquid water for electron inelastic scattering, the present article gives cross sections for proton interactions in an energy range from 0.1 keV to 10 GeV. The nature of the topic makes it appropriate to discuss it in two parts. First, we start with fast protons with kinetic energies exceeding about 500 keV. We use the Born- and Bethe-theories, which are applicable to such fast protons, to derive ionisation and excitation cross sections, including relativistic e€ects; also the Fermi-density correction is included. Second, we treat slow protons that have kinetic energies below about 500 keV. We use semi-empirical approaches to account for the deviation from the Born-theory and derive cross sections for ionisation, excitation, charge transfer, as well as for ionisation and electron loss of neutral hydrogen. Finally, theoretical frameworks together with pertinent experimental information about water vapour, liquid water and ice are used to account for phase e€ects in stopping power.

2.1. Elements of the Bethe-theory

2. Fast protons

Kmax ˆ

We call a proton fast, when the First Born Approximation (FBA) is valid and we can use the Born- and Bethe-theory to derive cross sections.

where t ˆ Mv 2 =2 is the proton kinetic energy. The integration limits in general depend on the mass M of the incoming particle. Here we have ignored the energy

We use the symbol s to express a (microscopic) cross section in the following sense: the mean number of collisions that occur during the passage of a particle through an in®nitesimal distance dx in a medium consisting of N molecules per unit volume is Nsdx: When considering a condensed material of ®xed N, it is convenient to use the product Ns as an index of the collision probability and to call it a (macroscopic) cross section S, which has the dimension of (length)ÿ1. For liquid water, the (mass) density is r ˆ 0:9982 g/ cm3 at 208C. One mole is A ˆ 18:0153 g, which consists of NA ˆ 6:02214  10 23 molecules of H2O. Therefore, the (number) density of molecules is N ˆ 3:343  10 22 molecules/cm3. The total number of electrons in the H2O molecule is Z ˆ 10: In the FBA, which applies to suciently fast protons, the most basic quantity is the (doubly) di€erential cross section d 2 S=dE dK, where E represents the energy transfer, and hÿ K the magnitude of the momentum transfer on a single collision. As Fano (1963) and Landau and Lifshitz (1970) explain in detail that this quantity is related to the energy-loss function Z2 …E, K † ˆ Im‰ÿ1=E…E, K †Š, where E…E, K † is the dielectric-response function, discussed in more detail in Section 2.2 for liquid water. For a proton of non-relativistic speed v, it is convenient to use the variable T ˆ mv 2 =2 ˆ …m=M †  (kinetic energy), where m is the electron mass …mc 2 ˆ 511:0 keV) and M is the proton mass …Mc 2 ˆ 938:27 MeV), and to express the relation as d 2S 1 Z2 …E, K† ˆ , dE dK pa0 T K

…1†

where a0 is the Bohr radius hÿ 2 =…me 2 † ˆ 5:2918  10 ÿ11 m. The di€erential cross section dS=dE for energy transfer E (to the electronic structure) is then given as dS 1 ˆ dE pa0 T

… Kmax Kmin

Z2 …E, K†

dK , K

where the integration limits are p 2M ÿp p
Kmin ˆ ÿ tÿ tÿE h

…2†

…3†

and p 2M ÿp p
t‡ tÿE , hÿ

…4†

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

257

 dS 1  ˆ A…E †x ‡ B  …E † , dE pa0 T

…9†

transfer to the translational, rotational, or vibrational motion of the molecule, which is generally much smaller than the energy transfer to the electronic structure (ICRU, 1993). For E  T (when the FBA is best justi®ed), it is possible to determine the T-dependence of the cross section analytically (Bethe, 1930; Inokuti, 1971). Note ®rst that E Kmin 1 ÿ hv

…5†

is small and decreases with speed v. In particular, it depends on v and E, but not explicitly on M. In contrast, p 2Mt Kmax 12 ÿ …6† h is very large and increases with v. Using the above properties of the integration limits and the fact that Z2 …E, K † approaches a limit Z2 …E, 0† as K 4 0, one ®nds that "  #   dS 1 T E ˆ ‡ B…E † ‡ O A…E † ln , …7† dE pa0 T Ry T where Ry represents the Rydberg energy me4 =…2hÿ 2 † ˆ 13:606 eV, 1 A…E † ˆ Z2 …E, 0†, 2

…8†

B(E ) is given in terms of two integrals, as discussed fully in Appendix B of Dingfelder et al. (1998), and O…E=T † accounts for higher order correction terms, which are neglected in this work. We call Eq. (7) the Bethe asymptotic cross section. Importantly, the coecients A(E ) and B(E ) are the same for protons as for electrons at the same speed, explicit dependence on M being absent. A weak dependence on M begins to appear in the remainder expressed by O…E=T †, which leads to a weak dependence of the stopping power on M, as discussed in detail by Bichsel and Inokuti (1998). The full Born-approximation cross section, Eq. (2), depends on M, and successively more strongly at lower speeds. At kinetic energies appreciable compared to the proton rest energy Mc 2 ˆ 938:27 MeV or higher, the Bethe di€erential cross section, Eq. (7), needs to be modi®ed in two respects. First, one has to use relativistic kinematics to relate momentum transfer with the scattering angle and energy transfer. Second, one has to account for the interactions transverse to the momentum transfer vector hÿ K~ in addition to the Cou~ as lomb interactions (which is longitudinal to hÿ K), fully discussed by Fano (1956, 1963). The result is

where T ˆ mv 2 =2 (not the kinetic energy), b ˆ v=c, and x is a variable de®ned by " # b2 x ˆ ln ÿ b 2: …10† 1 ÿ b2 The coecient A(E ) is the same as given by Eq. (8), and the new coecient B  …E † is given by B  …E † ˆ B…E † ‡ A…E † ln

ÿ 2 hc , e2

…11†

where the numerical coecient of A(E ) has the value 9.8405. The relativistic Bethe di€erential cross section, Eq. (9), is the same for electrons and protons, as well as for positrons and antiprotons, at the same speed. Its energy dependence is simplest in appearance, when b 2 dS=dE is plotted against the variable x. We call such a plot the Fano plot. At kinetic energies greatly exceeding the proton rest energy, the density e€ect ®rst pointed out by Fermi (1940) reduces the cross section by an amount depending on the material density and the particle speed. Further discussion is deferred to Section 2.3.

2.2. The dielectric-response function 2.2.1. The semi-empirical model The dielectric-response function E…E, K † is an important quantity related to the energy loss in the FBA. For liquid water E…E, K † is determined here in a semiempirical model (Dingfelder et al., 1998), which is mainly based on optical measurements by Heller et al. (1974). The imaginary part E2 …E, 0† is modelled in the optical limit, i.e., at momentum transfer hÿ K, by a superposition of Drude-like functions, while the real part E1 …E, 0† is calculated analytically using the Kramers±Kronig relation. The model uses experimental data as well as theoretical constraints like sum rules and asymptotic behaviour. The momentum-transfer dependence for the continuum is introduced by an impulse approximation, in which one views a target electron as nearly free at the moment of energy and momentum transfer and accounts for its binding through the instantaneous momentum of a stuck electron. 1 Our model considers ®ve excitation states, A~ B1 , 1 B~ A1 , Ryd A ‡ B, Ryd C ‡ D, and di€use bands, and ®ve ionisation shells, 1b1, 3a1, 1b2, 2a1, and the K-shell of oxygen. More details and documentation of this semi-empirical model can be found in our paper on

258

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

electron inelastic scattering cross sections (Dingfelder et al., 1998).

2.2.2. New experiments: The Bethe surface After the completion of the semi-empirical determination (Dingfelder et al., 1998) of the dielectric-response function, we became aware of the recent measurements by the Sendai group (Watanabe et al., 1997; Hayashi et al., 1998), who studied the Compton scattering of high-energy photons of synchrotron radiation in liquid water. The cross section for the Compton scattering of a photon by electrons in atoms, molecules, or a condensed matter is expressible as the product of the Klein±Nishina cross section (which applies to a free electron at rest) and a factor that accounts for the in¯uence of the binding of those electrons as long as their binding energies are much smaller than the photon energy and when the momentum transfer is not very small. This factor is in essence the generalised oscillator-strength spectrum, or the dynamic structure function. Therefore, it was sensible for those workers to present results in the form of the Bethe surface, viz., a plot of Z2 …E, K † over the plane indicating energy transfer E and momentum transfer ÿ hK The method of the Sendai group is particularly e€ective for the region near the Bethe ridge (Inokuti, 1971), viz., the peak around the line E ˆ (hÿ K † 2 =…2m†, or E=Ry ˆ …Ka0 † 2 , corresponding to the transfer of energy and momentum to a free electron. In this sense, the method is complementary to the optical measurements of Heller et al. (1974), which was the main source of data used previously by Dingfelder et al. (1998). A comparison of the semi-empirical determination with the recent Sendai measurements, discussed in detail elsewhere (Dingfelder and Inokuti, 1999a) indicates the following: the agreement found between the new experiment and our previous model is surprisingly good. No alarming discrepancy could be seen on a global view of the Bethe surface. This is due, in part, because the two sets of data have been constrained by the Bethe sum rule. The Sendai Bethe surface peaks at around 23 eV for low K, and the Bethe ridge is broad. In comparison, the semi-empirical model shows a sharply peaked Bethe ridge. The broader ridge of the Sendai group is likely to be closer to reality than the sharply peaked theoretical curve, while the Sendai data are probably unreliable near the optical limit, K 4 0, because of a low signal-to-noise ratio. However, it is unlikely that this di€erence would a€ect the dS=dE values obtained through the integration over K, at least for outer shells. Further experimental data, especially at low-K values, are desirable, for an improved determination of the Bethe surface.

2.2.3. K-shell of oxygen The semi-empirical approach as described in Section 2.2.1 needs to be improved for the K-shell ionisation in the case of proton or other light-ion impact. Recall that the K-dependence in the semi-empirical model was introduced using an impulse approximation, and therefore the whole momentum dependence is concentrated in a sharply peaked Bethe ridge. In reality, this ridge should be more broadened, as described in the discussion of the Bethe surface in the last section. This sharply peaked K-dependence leads to an overestimation of the total ionisation cross section for higher proton energies and to a breakdown at lower energies (see Fig. 1, dashed curve). The sharp peak is either completely in the integration area, or completely out, depending on the moving lower integration limit Kmin (Eq. (3)), which is much higher for the K-shell than for the outer shells, because of the higher energy transfer. (Recall that the binding energy of the K-shell is 539 eV). Therefore, we have to use a more realistic momentum dependence for the K-shell. This problem does not occur in case of electron impact, because of the di€erent kinematic regimes. For electrons, the peak is always completely inside the integration area, and therefore the peak shape does not appreciably a€ect the di€erential cross sections. We use the generalised-oscillator strength (GOS) of the scaled hydrogenic model (Bethe and Ashkin, 1956)

Fig. 1. Total ionisation cross section of the K-shell of oxygen vs. proton incident energy. Shown are Born approximations using the scaled hydrogenic generalised oscillator strength (present work, solid curve) and the Drude-like energy loss function with an impulse approximation for the momentum transfer dependence (present work, dashed curve), a semiempirical model (Rudd et al., 1992, dotted curve), a compilation and best ®t of experimental data (Paul and Sacher, 1989, diamonds) and an older experiment (Toburen and Wilson, 1977, squares).

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

to introduce the momentum dependence of the dielectric response function. The GOS, df …E, K †=dE, is, in general, related to the energy-loss function Z2 …E, K† ˆ

p E p2 1 df…E, K† , 2 Z E dE

…12†

where Ep is the nominal plasmon energy, de®ned by 1=2  Z Ep ˆ 28:816 r eV: A

…13†

For liquid water (Dingfelder et al., 1998), Ep ˆ 21:46 eV. In case of the K-shell, the energy-loss function is approximately equal to the imaginary part of the dielectric-response function (Dingfelder et al., 1997) -shell …E, K† ˆ ZK 2

-shell …E, K† EK 2 2 E1 …E, K† ‡ E22 …E,

K†

-shell …E, K† 1EK 2

…14†

within about 0.2%, because E1 …E, K †11 and E2 …E, K †  1: Therefore, we use -shell …E, K † ˆ ZK-shell …E, K †, neglecting contributions EK 2 2 of the K-shell to E1 : The GOS for the K-shell of oxygen in water is now approximated with the use of the GOS of the scaled hydrogenic model, df …H † …E, K †=dE (per electron), which will be described in detail below. In other words, we write df…E, K† df ˆ 2x dE

…H †

…E, K† , dE

…15†

where the factor 2 accounts for two electrons in the Kshell. The factor x arises from a renormalization. The known value of the total oscillator strength of the oxygen K-shell is 1.79, as stated in Section 4.1 of Dingfelder et al. (1998). However, the integral of df …H † …E, 0†=dE over the continuum is 0.88. Therefore, we renormalize the strength by the factor x ˆ 1:79=…2  0:88† ˆ 1:017: The GOS of the hydrogen atom is well known (Bethe, 1930; Inokuti, 1971). The GOS for transition from K-shells into the continuum for any atom with nuclear charge Z can be obtained by scaling the energy E/Ry and the momentum Ka0 with the charges (Bethe and Ashkin, 1956). Technically, this is done by replacing the Rydberg energy Ry by the scaled ``ground 2 state'' energy Ry Z eff , and the Bohr radius a0 by a0 =Zeff , where Zeff ˆ Z ÿ 0:3 is the e€ective nuclear charge, including screening e€ects from the other electrons, sometimes called Slater corrections (Slater, 1930). The binding energy of the scaled hydrogenic 2 model is then Ry Z eff ˆ 806:7 eV. The GOS for transition from the K-shell to the continuum, for energy 2 transfers E > Ry Z eff then reads

df

259

…H †

…E, K† dE   E 1 E …Ka0 † 2 ‡ ˆ 27 2 4 2 3Ry Ry Zeff Z eff

Z 12 eff h i3 h i3 2 2 2 …Ka0 ‡ ka0 † ‡Z eff …Ka0 ÿ ka0 † 2 ‡Z eff !) ( 2 2Z eff 2ka0 Zeff  exp ÿ arctan 2 ka0 …Ka0 † 2 ÿ…ka0 † 2 ‡Z eff    ÿ1 2pZeff  1 ÿ exp …16† ka0 p 2 : with ka0 ˆ E=Ry ÿ Z eff Notice that the experimental binding energy (Dingfelder et al., 1998) is IK-shell ˆ 539:0 eV. For energy 2 , ka0 becomes imagintransfers E with IK-shell < Ry Zeff ary. Therefore, we insert ka0 in Eq. (16) and transform the equation by analytical continuation. On the pure imaginary axis, the arctan function is a combination of logarithms and has a real value as it should. The last factor on the right-hand side of Eq. (16), sometimes called Sommerfeld factor, is then set at unity. Thus, 2 Eq. (16) is recast for IK-shell < E < Ry Z eff in the form …H †

…E, K† dE   E 1 E 2 ˆ 27 2 4 † ‡ …Ka 0 2 3Ry Ry Z eff Z eff

df

Z 12 eff i3
2 2 2 …Ka0 † ÿE=Ry ‡4…Ka0 † 2 Z eff 2 ÿE=Ry  Z =p Z eff 1 ÿ b eff 1‡b hÿ

…17†

with p 2 ÿ E=Ry 2Zeff Z eff bˆ 2 2 …Ka0 † ÿE=Ry ‡ 2Z eff

…18†

Using now this scaled hydrogenic GOS for calculating the ionisation cross section for the K-shell, we obtain a more realistic result. Fig. 1 shows the present result using the scaled hydrogenic GOS (solid curve) and the Drude-like energy-loss function with the impulse approximation for the momentum-transfer dependence as described in Section 2.2.1 (dashed curve), compared to data given by Paul and Sacher (1989) and by Toburen and Wilson (1977). The data presented by Paul and Sacher (1989) are in the form of a compilation and the best ®t to all data from available sources. Also shown is a semi-empirical result from Rudd et al. (1992) (dotted curve), described in Section 3.3. The present calculation using the scaled hydronic GOS certainly reproduces the overall trend of established experimen-

260

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

tal data. An apparent underestimate of the cross section around the maximum is in line with the usual behaviour of the FBA for proton collisions.

2.3. The Fermi-density e€ect This e€ect is most often discussed in connection with the stopping power (Fermi, 1940; Fano, 1963; Landau and Lifshitz, 1970; Jackson, 1998). More recently it has been discussed for the inner-shell ionisation cross sections theoretically (Sùrensen and Uggerhùj, 1986; Sùrensen, 1987) and experimentally (Bak et al., 1983, 1986; Meyerhof et al., 1992; Spooner et al., 1994). Considerations on the Fermi-density e€ect on the cross section di€erential with respect to energy transfer dS=dE are in progress, as seen in a recent initial report (Dingfelder and Inokuti, 1999b). The origin of the e€ect is seen in the minimum momentum transfer hÿ K, de®ned by Eq. (5). This may be interpreted to mean that any of the molecules within the cylinder around the particle path with the radius bmax ˆ 1=Kmin ˆ hÿ v=E (according to Eq. (5)) may receive energy transfer from the particle, but none of the molecules farther away from the track. This distance bmax may be called the maximum impact parameter e€ective for energy transfer. Consideration of relativistic kinematics leads to a more accurate result (Jackson, 1998)   ÿ  ghÿ v Ry hc bmax ˆ ˆ 2 2 a0 bg , …19† E e E where g ˆ …1 ÿ b 2 † ÿ1=2 : Notice that 2…hÿ c=e 2 †a0 ˆ 1:4503  10 ÿ6 cm. Therefore, for lower excitations with E  Ry, bmax far exceeds intermolecular distances in liquid water. Many molecules within bmax are electrically polarised by the particle and thereby screen a part of the electromagnetic interactions between the particle and a molecule that receives energy. For higher excitation, bmax is smaller and so is the shielding e€ect. 2.3.1. Stopping power The stopping power S is the mean energy loss per unit path length, and is evaluated as …   dS Sˆ E dE, …20† dE where the integral covers all possible energy transfers E including discrete and continuous spectra. One may also write S ˆ Nsst , where

…21†

…   ds dE sst ˆ E dE

…22†

is called the stopping cross section and has the dimension of (energy)  (area). For a particle of charge ze and high enough speed v ˆ bc, S is given as (Fano, 1963)    
ÿ 4pz 2 e4 2mv 2 1 2 2 Sˆ d , ÿ b ÿ ln 1 ÿ b NZ ln ÿ 2 mv 2 I …23† where the mean excitation energy I and the Fermi-density correction d are both determined by Z2 …E, 0†: First, the mean excitation energy I is de®ned by   … 2 1 1 ln I ˆ E…ln E †Z2 …E, 0† dE, …24† p E p2 0 and I ˆ 81:8 eV for liquid water (Dingfelder et al., 1998) in the present approach. Next, the Fermi-density correction d is calculated as (Fano, 1963; Inokuti and Smith, 1982)     … 2 1 1 L2 dˆ E ln 1 ‡ 2 Z2 …E, 0† dE p E p2 0 E !
L2 ÿ ÿ 1 ÿ b2 , …25† 2 Ep where L is a real-valued function of b 2 de®ned as the positive root of the equation 1 ÿ b 2 E…iL, 0† ˆ 0:

…26†

The quantity E…iL, 0† here means the analytic continuation of E…E, 0† from real values of E (or frequency E=hÿ † to pure imaginary values iL, which physically describe a damping response instead of an oscillatory response of the medium. To determine the function L ˆ L…b 2 †, it is appropriate to cast Eq. (26) in the form (Inokuti and Smith, 1982)   …1 2 1 1 E…iL, 0† ÿ 1 ˆ EE2 …E, 0† 2 dE ˆ 2 ÿ 1, p 0 E ‡ L2 b …27† which is suitable for an insulator, for which the integrand is always positive and E…0, 0† > 1: There is a positive root L…b 2 † of Eq. (26) only when 1=E…0, 0† < b 2 : For liquid water (Dingfelder et al., 1998), E…0, 0† ˆ 1:872, and a positive root occurs for b 2 > 0:534, viz., for the proton kinetic energy greater than 436.5 MeV, or for the electron kinetic energy greater than 237.7 keV. In practice, the relation between L and b 2 is deter-

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

261

on no account more sensitive to details of the dielectric-response function than the mean excitation energy I, as Inokuti and Smith (1982) also concluded in their work on metallic aluminium. 2.3.2. Cross section for a ®xed energy transfer To consider the Fermi-density e€ect on the cross section dS=dE for a ®xed energy transfer E, we recall the sum rule (Fano, 1963)   … 2 1 1 EZ2 …E, 0† dE ˆ 1, …28† p E p2 0

2

2

Fig. 2. Real-valued function L…b † for liquid water vs. b , de®ned in Eq. (25), calculated using Eq. (26). The inlet shows an enlargement of the curve for b 2 R0:7:

and rewrite Eq. (25) as   … 2 1 1 dˆ EZ2 …E, 0†D…E, b† dE, p E p2 0 where we de®ne 

L2 D…E, b† ˆ ln 1 ‡ 2 E

 ÿ

!
L2 ÿ 1 ÿ b2 : 2 Ep

…29†

…30†

mined by evaluating the integral of Eq. (27) on a mesh of chosen L values and relating them to b 2 : The desired function L…b 2 † is then obtained by interpolation of these calculated data, as shown in Fig. 2. Once the function L…b 2 † is determined, d can easily be calculated. Note that d depends on b 2 only and its value applies to protons and electrons at the same speed. Fig. 3 shows the Fermi-density correction d as a function of the proton kinetic energy and an earlier calculation by Ashley (1982a, 1982b). Our result for d is close to that of Ashley (1982b), who used a di€erent data set for E…E, 0†: The close agreement con®rms the view of Fano (1963) that d is

Close examination of the derivation by Fano (1963) indicates that d appears in his ``low-Q'' contribution to S of Eq. (23). Note that his variable Q is equal to …Ka0 † 2 Ry in our notation. The low-Q contribution may be expressed as   …
2pz 2 e4 2 1  ÿ Slow-Q ˆ L E, b 2 NZ p E p2 mv 2 ÿ
 ÿ D E, b 2 EZ2 …E, 0† dE, …31†

Fig. 3. Fermi-density correction d to the stopping power S for liquid water vs. proton incident energy t compared with the previous calculation of Ashley (1982a, 1982b) for electrons at the same speed (circles).

Fig. 4. Fermi-density e€ect on di€erential cross sections: Modi®cation factor f…E, b 2 † vs. energy transfer E for several incident proton energies …0:55Rb 2 R0:95), as de®ned in Eq. (33).

262

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

where 

ÿ
2mv 2 Q1 L E, b 2 ˆ ln I2




ÿ ÿ ln 1 ÿ b 2 ÿ b 2 ,

…32†

and Q1 is the upper limit of Q, which is set to Q1 ˆ 1 Ry for evaluation. As we expect, the resulting values of L…E, b 2 † are insensitive to Q1. In other words, the use of L…E, b 2 † alone leads to the Bethe stopping power, and the presence of D…E, b 2 † leads to S including the Fermi-density e€ect. We interpret Eq. (31) as follows. The Fermi-density e€ect reduces the probability of energy transfer E from L…E, b 2 †Z2 …E, 0† to ‰L…E, b 2 † ÿ D…E, b 2 †ŠZ2 …E, 0†, apart from the factor in front of the integral of Eq. (31). In other words, the cross section dS=dE is modi®ed by the factor
ÿ ÿ
D E, b 2 2 ÿ
: f E, b ˆ 1 ÿ …33† L E, b 2 2

Fig. 4 shows f…E, b † for liquid water for several ®xed incident proton energies …0:55Rb 2 R0:95† as a function of energy transfer E. f is a monotonously increasing function starting with values below unity for small energy transfers and reaching a constant value slightly above unity for high energy transfers. For lower proton energies …b 2 ˆ 0:55† the factor is almost constant and close to unity, f11, for all energy transfers. The Fermi-density e€ect does not in¯uence the cross section visibly. With increasing proton energies …b 2 > 0:55† f decreases for smaller energy transfers while the constant value for higher energy transfers slightly increases. Now, the Fermi-density e€ect modi®es the di€erential cross section signi®cantly and reduces the total cross section for higher velocities b 2 : 2.4. The applicability of the ®rst born approximation (FBA) An important di€erence between an electron and a proton (or a heavier charged particle) concerns the justi®cation of the FBA. The universal criterion, applicable to all particles, is that the speed v of an incident particle far exceeds the mean orbital speed vm of a molecular electron that receives energy. Then, the (doubly) di€erential cross section, Eq. (1), is applicable (except at extremely high values of the momentum transfer, at which the cross section is very small). However, the cross section di€erential in energy transfer, Eq. (2), can be derived also from the ®rst-order treatment in the semi-classical approximation as explained by Bethe and Jackiw (1997). This derivation rests on the condition that the momentum Mv of the incident particle far exceeds the mean orbital momentum mvm of the molecular electron, which holds down

to much lower speeds for a proton or a heavier particle. Furthermore, the electron-exchange e€ect, as discussed in Section 3.4 of Dingfelder et al. (1998), a€ects the ionisation cross section by electron collisions at moderately high speeds. At lower speeds, e€ects not included in the FBA need to be accounted for. They include e€ects of the distortion of incident waves, and coupled-channel e€ects. Inspection of coupled-channel calculations on proton collisions with hydrogen, helium, and other cases (Kimura and Lane, 1990; Schiwietz, 1990; Grande and Schiwietz, 1992) indicate how these e€ects can in¯uence cross sections, although we are aware of no similar calculations speci®cally on liquid water. A lower limit of the validity of the FBA might be 200 keV, and more conservatively in the range from 300 to 500 keV. At lower energies we use a semi-empirical approach fully described in Section 3. 2.5. Results Using the semi-empirical determination (Dingfelder et al., 1998) of the dielectric-response function E…E, K †, as described in Section 2.2, we calculated the di€erential cross sections using Eq. (2). The integration over momentum-transfer needs a careful treatment, because of the sharply peaked energy-loss function. Furthermore, the main contributions to the integrand of Eq. (2) are often concentrated in a narrow interval around the Bethe ridge. Thus we separated the integral in several integrals using linear and logarithmic integration schemes based on a Romberg method (Press et al., 1992) with a trapezoidal formula. For relativistic energies …t > 10 MeV) we used the relativistic Bethe formula (Eq. (9)). The determination of the Bethe coecients A(E ) and B(E ) is discussed in detail by Dingfelder et al. (1998).

Fig. 5. Energy di€erential cross sections ds=dE, summed over all ionisation shells, vs. energy transfer E for di€erent incident proton energies t (316 keV, 1 and 10 MeV).

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

Fig. 5 shows the resulting energy di€erential cross sections, ds=dE ˆ N ÿ1 dS=dE, summed over all ®ve molecular ionisation shells of water for di€erent incident proton energies (from top to bottom t ˆ 316 keV, 1 and 10 MeV) versus the energy transfer E. The presence of the K-shell is clearly seen in the lower curves, while it is hidden in the sharp decrease of the cross section in the upper curve. This decrease of the cross section is related to the maximum energy transfer from the proton to a free electron, which is (non-relativistically) four times the kinetic energy of an electron of the same speed, Emax ˆ 4…m=M †t: The shown range of incident proton energies corresponds to the range of validity of the non-relativistic FBA. „ The total ionisation cross section s ˆ …ds=dE † dE is shown in Fig. 6 for proton kinetic energies from 1 keV to 10 GeV. It has been calculated in three parts: (1) the semi-empirical approach below 500 keV, (2) the FBA between 500 keV and 10 MeV, and (3) the relativistic Bethe formula above 10 MeV. The Fermi-density correction is not included in the data shown in this ®gure. The dotted curve shows the failure of the FBA for incident proton energies below 500 keV and for relativistic energies above 10 MeV. For comparison, the electron-impact ionisation cross section, calculated earlier (Dingfelder et al., 1998), is plotted against the energies of protons at the same speed (dashed curve). For electrons the FBA fails at speeds much higher than for protons. One sees also the very weak mass

Fig. 6. Total ionisation cross section for protons in liquid water. The solid curve consists of three parts: the semi-empirical formula, as described in Section 3.3 for proton energies below 500 keV, the Born formula for proton energies in the range from 500 keV to 10 MeV, and the relativistic Bethe formula for proton energies exceeding 10 MeV. The dashed curve shows the results of the non-relativistic Born formula which are inadequate at energies below 500 keV and above 10 MeV. Also shown is the electron impact ionisation cross section for electrons (Dingfelder et al., 1998, dashed curve) with the same speed as the protons.

263

dependence of the Born formula. The small di€erences between the dashed curve and the solid curve at higher speeds arise from the present treatment of the K-shell di€erent from the calculation for electrons (Dingfelder et al., 1998). The electron and proton cross sections should agree at the same speed in the Bethe-formula when the same dielectric-response function is used; mass-dependent terms only occur in higher order correction terms, which are not included in Eq. (9). All the cross sections of Fig. 6 are shown again in Fig. 7 in a relativistic Fano plot. The Fano plot is a more sensitive test of the consistency of the cross sections. There, according to Eq. (9), the total ionisation cross section multiplied by the speed squared is plotted versus x (Eq. (10)). For higher speeds the cross section shows the asymptotic straight line, as expected. The deviation from the straight line for lower speeds are clearly seen as well as the failure of the Born-approximation (dotted curve) for low speeds, and also in the relativistic region. The dashed curve, again, shows the recently derived electron cross section at the same speed. All the features of the cross sections discussed in the preceding paragraph are more clearly seen in the Fano plot. Finally, the di€erential ionisation cross sections are displayed in Fig. 8 in the form of a Platzman plot, which is a simple, but powerful way graphically to present and evaluate such data (ICRU, 1996). Here, the ratio Y of the energy di€erential cross section ds=dE to the Rutherford cross section for a single target electron is plotted as a function of Ry=E, where E ˆ W ‡ I1 is the sum of the secondary electron kinetic energy W and the lowest binding energy for liquid water (Dingfelder et al., 1998) I1 ˆ 10:79 eV. Each curve corresponds to a ®xed proton energy t: By choosing Ry=E as a variable on the horizontal axis instead of E, the area under the curve is proportional

Fig. 7. Relativistic Fano plot of the total ionisation cross section. Shown are the same data as in Fig. 6.

264

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

Fig. 8. Platzman plot of the secondary electron spectra for the Born approach. Symbol E ˆ W ‡ I1 represents the secondary electron energy plus the binding energy of the outermost shell, I1 ˆ 10:79 eV. The curves are for di€erent proton energies, from top to bottom for t ˆ 10, 5, 2, 1 MeV, 500 and 316 keV.

to the total ionisation cross section multiplied by the proton energy t: The curves show a maximum at around Ry=E ˆ 0:55, which corresponds to a secondary electron energy of W114 eV, and reaches a value of Z ˆ 10 for high energy transfers …Ry=E 4 0), which is the number of electrons in the water molecule. The shape is similar to the case of electron impact (Dingfelder et al., 1998, Fig. 4), for which the maximum is at the same energy but narrower and less smooth. The peak at high energy transfers …Ry=E 4 0), seen for electron impact and resulting from the electron exchange contributions, i.e., the indistinguishability of electrons, is missing for proton impact. 3. Slow protons 3.1. Background As indicated in the last section, the FBA is not valid for describing slow-proton interactions. We call a proton slow, when its kinetic energy is below 500 keV. Table 1 List of all inelastic processes for slow protons considered in this worka (1) Excitation (2) Ionisation (3a) Charge transfer (3b) . . . into continuum (4) Stripping (5) Ionisation a

More explication in the text.

p ‡ H 2 O 4 p ‡ H2 O p ‡ H 2 O 4 p ‡ H2 O‡ ‡ e ÿ p ‡ H2 O 4 H1 ‡ H2 O‡ p ‡ H2 O 4 …p ‡ e ÿ † ‡ H2 O‡ H ‡ H2 O 4 p ‡ e ÿ ‡ H2 O H ‡ H 2 O 4 H ‡ H2 O‡ ‡ e ÿ

For slower and slower protons other inelastic processes, which are suppressed for higher energies, become more and more important. All the inelastic processes considered in this study are listed in Table 1. Aside from excitation (Table 1, (1)) and ionisation (2), which decrease in importance below 100 keV, charge transfer (3) becomes dominant. In this process an electron from the water molecule is transferred to the moving slow proton, to form a neutral hydrogen atom H, either in any bound state (ground or excited state) (3a) or a continuum state (3b), where the electron is travelling together with the proton at the same speed. After the charge transfer process the neutral hydrogen atom becomes either stripped (4) or ionises (5) a water molecule. In the ®rst case, a proton, and a (possibly excited) water molecule remains, an electron is ejected in forward direction with nearly the same velocity as the proton. In the latter case, the neutral hydrogen, an ionised water molecule and an ejected electron with a similar distribution of electron energy and ejection angle as for protons result. Excitations of the water molecule by the neutral hydrogen atom should occur rarely and are neglected in this study. At lower incident energies, say below 1 keV, also elastic scattering of the proton by the water molecules becomes important. This process is also called nuclear scattering. In the following subsections we review all the experiments which we are aware of and which refer largely to water vapour, and present cross section sets suitable to use for liquid water. These sets are obtained taking into account medium phase e€ects, which have been experimentally seen in stopping cross sections, for which data on the vapour, liquid and ice phase of water exist. The derivation and calculation of the stopping cross sections are described in Section 3.8. Most of the experiments refer to electron capture and loss processes, while experimental data on ionisation cross sections are scarce. The total cross sections for the di€erent processes are characterised by the initial and the ®nal charge state of the incident particle, and the total cross section for a certain process is written as sij , where i indicates the incident charge state and j the ®nal charge state. Therefore, s11 refers to the total ionisation cross section of the proton, s10 to the charge transfer to the proton (also called capture process), and s01 to the stripping of hydrogen (also called electron-loss process). Note, that for practical use of the following semiempirical equations of this chapter all variables and parameters of dimension energy are used in units of eV (e.g. primary particle energy t, binding energies) and all variables and parameters of dimension length in units of meter (e.g. Bohr radius a0).

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

3.2. Excitations To our knowledge, there are no experiments on excitation cross sections for proton impact, neither for water vapour nor for liquid water, apart from the work of Yousif and co-workers (Yousif et al., 1986), who present cross sections for Balmer Ha emission from H2O through dissociative excitations of the target in the energy range from 5 to 100 keV. Semi-empirical treatments relate proton excitation cross sections to electron excitation cross sections and use systematics of the cross sections for their analytical representation. We know the speed dependence and the very weak mass dependence of the FBA cross sections. The electron excitation cross sections calculated previously for liquid water (Dingfelder et al., 1998) also include lowenergy corrections, which account for deviations from the FBA. After careful deliberations we adopt the following semi-empirical model which uses a speed scaling on the basis of the electron excitation cross sections, together with extensions towards lower proton energies. This idea was developed by Miller and Green (1973), who also presented proton excitation cross sections for water vapour. The cross section for a single excitation level k is represented by the analytic form sproton exc, k …t† ˆ

O

s0 …Za † …t ÿ Ek †n , J O‡n ‡ tO‡n

…34†

where s0 is a constant …s0 ˆ 10 ÿ20 m2), Z the number of electrons in the target material, and Ek the excitation energy. Symbols n and O are dimensionless parameters and a and J parameters of the dimension of energy (eV). The parameters a and O represent roughly the highenergy limit and are inferred from the speed scaling of electron and proton cross sections in the Born-theory, while J and n describe the low-energy behaviour and are inferred from the systematics of such parameters from a variety of substances (Green and McNeal, 1972). The parameters J and a are related to the maximum semax of the electron excitation cross section, at an electron energy Tmax. Thus, J and a are given by J ˆ C2

 1=…O‡n† M O Tmax m n

…35†

a ˆ C2

 1=O M Tmax C1 semax …O ‡ n † , m Z s0 v

…36†

where C1 and C2 are parameters connected with O and n: The derivation of Eqs. (35) and (36) is complicated and not unique, as described in detail by Miller and Green (1973). They also made recommendations for

265

the other parameters: n ˆ 1, C1 ˆ 4, C2 ˆ 0:25, if O11 and n ˆ 2, C1 ˆ 4, C2 ˆ 1, if O  1: We choose the parameters a and O so that the semi-empirical approach agrees in the high-energy limit with the results obtained from the FBA. In detail, O represents the slope of the cross section, and C1 is used to adjust the magnitude of the cross section for incident proton energies exceeding 500 keV. The parameters n and C2 are taken from the recommendations of Miller and Green, discussed above. The set of parameter values thus determined is displayed in Table 2 for all ®ve excited states considered. The table includes also the values of the cross section maximum semax for electron impact (Dingfelder et al., 1998) and the electron kinetic energy Tmax there. Fig. 9 shows the excitation cross section for proton impact, as calculated from Eq. (34) together with the contributions from the di€erent excited states for proton incident energies from a few 10 eV to 10 MeV. Above 500 keV incident proton energy the semi-empirical cross sections agree with the results obtained from the FBA. Fig. 9 includes the result, shown by the dashed curve, of the straight forward scaling, de®ned by electron … † sproton T, exc …t† ˆ sexc

Tˆ

m t: M

…37†

This cross section has been sometimes used in earlier track-structure simulations (see e.g. Zaider et al., 1983), which only considered fast protons. However, the use of Eq. (37) for slow protons leads to the unrealistic drop of the proton excitation cross section,

Fig. 9. Total excitation cross section sexc for liquid water vs. proton incident energy. Shown are the total excitation cross section (top curve) and the contributions of the di€erent excited states, as calculated in the Miller model (proton energies below 500 keV) and the Born formula (proton energies above 500 keV). Also shown is the total electron excitation cross section (dashed curve) for impact of electrons at the same speed.

266

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

the secondary-electron spectra from charged particle interactions is published by the ICRU (1996). There, di€erent theoretical and semi-empirical approaches are discussed and parameters and experimental information are given for a wider variety of atoms and molecules. After thorough considerations we decided to use the following semi-empirical approach based on ideas suggested by Rudd and co-workers (Rudd et al., 1985a, 1992). In detail, the single di€erential cross section ds=dE, di€erential in energy transfer E is given by

towards its vanishing at t ˆ 15 keV, which corresponds to the excitation threshold Ek ˆ 8:17 eV for electrons (Dingfelder et al., 1998). The excitation cross sections for neutral hydrogen impact are neglected in view of lack of experimental or theoretical information and the probable unimportance for our purpose. 3.3. Ionisation by protons Experiments on proton ionisation cross sections in liquid water do not seem to have been carried out, while measurements in water vapour are scarce. Toburen and Wilson (1977) measured energy and angular distributions of electrons ejected from water vapour by protons in the energy range from 300 keV to 1.5 MeV. They present also single di€erential cross sections, but not the total ionisation cross section. A calculation of total ionisation cross sections, based on these data, can be found in Kaushik et al. (1981). Additional measurements for higher proton energies are reported by Wilson et al. (1984); a numerical representation is given by Wilson and Nikjoo (1999). Rudd et al. (1985b) presented total ionisation and charge transfer cross sections for water vapour by measuring the gross cross sections for the production of positive and negative charge and calculated the total cross sections (Rudd et al., 1983). Bolorizadeh and Rudd (1986a) presented di€erential cross sections, di€erential in scattering angle and secondary-electron ejection energy for slow proton energies (15±150 keV). Gibson and Reid (1987) also measured these cross sections for the single incident energy of 50 keV. Rudd and co-workers reviewed all experimental and theoretical approaches to total cross sections (Rudd et al., 1985a) and di€erential cross sections (Rudd et al., 1992). They developed semi-empirical models for the ionisation cross sections and obtained necessary parameters by best ®ts to all known data. These semiempirical models reproduce experimental data on di€erential and total cross sections reasonably well but do not supply realistic information on the partitioning into subshells. Also, they show diculties in reproducing stopping cross sections. A more recent survey on

X ds j ds Gj , ˆ dE dWj all j

…38†

where Wj ˆ E ÿ Ij is the secondary-electron kinetic energy, Ij the ionisation energy of sub-shell j in liquid water and Gj the partitioning factor to adjust the contributions of the di€erent subshells to the results obtained from the Born approximation. In the Rudd model the secondary-electron energy spectrum dsj =dWj is represented by a combination of low and high energy parts. Thus, the secondary-electron energy distribution for single ionisation of sub-shell j is given by ds j S F1 …v† ‡ wF2 …v† ˆ  :   Bj …1 ‡ w †3 1 ‡ exp a…w ÿ wc †=v dw

…39†

where the two functions F1 and F2, which consist of di€erent low and high energy parts, are given by F1 …v† ˆ L1 …v† ‡ H1 …v†, F2 …v† ˆ

…40†

L2 …v†H2 …v† , L2 …v† ‡ H2 …v†

with L1 …v† ˆ

C1 vD1 , 1 ‡ E1 v…D1 ‡4†

L2 …v† ˆ C2 vD2 ,

H1 ˆ

H2 …v† ˆ


ÿ A1 ln 1 ‡ v 2 , v 2 ‡ B1 =v 2

A2 B2 ‡ 4 v2 v

…41†

The dimensionless variables used here are a scaled sec-

Table 2 Parameters for the excitation cross section (Eq. (34)) for liquid water k

Excited state

Ek (eV)

a (eV)

J (eV)

O

n

C1

C2

Tmax (eV)

semax (m2)

1 2 3 4 5

1 A~ B1 1 B~ A1 Ryd A ‡ B Ryd C ‡ D Di€use bands

8.17 10.13 11.31 12.91 14.50

876 2084 1373 692 900

19820 23490 27770 30830 33080

0.85 0.88 0.88 0.78 0.78

1 1 1 1 1

3.132 3.468 3.324 3.028 3.028

0.25 0.25 0.25 0.25 0.25

47.15 54.78 65.75 77.22 82.86

8.00  10ÿ22 1.30  10ÿ22 8.11  10ÿ22 5.19  10ÿ22 6.03  10ÿ22

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275 Table 3 Parameter sets for the di€erential ionisation cross sections (Eq. (39)) on proton impact for liquid water (this work), water vapour, and K-shell ionisation (Rudd et al., 1992) Parameter

Liquid

Vapour

K-Shell

A1 B1 C1 D1 E1 A2 B2 C2 D2 a

1.02 82.0 0.45 ÿ0.80 0.38 1.07 14.6 0.60 0.04 0.64

0.97 82.0 0.40 ÿ0.30 0.38 1.04 17.3 0.76 0.04 0.64

1.25 0.50 1.00 1.00 3.00 1.10 1.30 1.00 0.00 0.66

ondary-electron energy w ˆ W=Bj , and a scaled speed squared of the particle v 2 ˆ T=Bj , where Bj is the binding energy of the sub-shell j. Furthermore, S ˆ 4pa02 Nj …Ry=Bj † 2 , with Nj the number of electrons in the sub-shell j, wc ˆ 4v 2 ÿ 2v ÿ Ry=…4Bj † and a, a numerical parameter related to the relative size of the target molecule. Note that the symbols in Eqs. (38)±(41) have been taken from Rudd and co-workers (Rudd et al., 1985a, 1992). In particular, C1, C2, v and S are not to be confused with the same symbols used earlier in the present paper. Parameters for many atoms and molecules, including water vapour, can be found in the work by Rudd et al. (1992) and in the ICRU report 55 (ICRU, 1996). The parameter set for liquid water is displayed in Table 3 together with the parameter set for water vapour. Also given there is a special parameter set for the K-shell ionisation (Rudd et al., 1992), common to both phases. The Rudd model gives values of the K-shell ionisation cross section close to the numerical compilation by Paul and Sacher (1989), as seen in Fig. 1, and should be adequate for practical applications. The parameter set for liquid water has been

267

obtained in the following way: we started with the parameter set for water vapour, and modi®ed it in order to reproduce the recommended ICRU stopping cross section (ICRU, 1993) for liquid water, which is the only relevant experimental information on the liquid phase, taking into account a realistic partitioning into subshells. The partitioning factor Gj is determined by adjusting the shell contributions of the semi-empirical model at higher incident particle energies …t > 500 keV) to the results of the Born approximation. The obtained values of Gj are displayed in Table 4. The parameter set for water vapour itself does not reproduce the recommended ICRU stopping cross sections for water vapour (ICRU, 1993) very well. Note that the binding energies Bj of the vapour phase are used as parameters in Eq. (39) and cannot be changed or even replaced with those …Ij † of the liquid phase. For completeness the binding energies of both phases are also given in Table 4. In the adoption of the parameter set for liquid we chose to modify the secondary-electron spectra minimally on going from vapour to liquid. The adopted spectra for liquid are slightly less intense (as seen in Fig. 10). This trend is consistent with a general expectation (Inokuti, 1991) that secondary electrons, especially of lower energies, should be suppressed owing to scattering by surrounding molecules in liquid. Fig. 10 shows the ratio of the semi-empirical single di€erential energy transfer cross section ds=dE to the Rutherford cross section …ds=dE †Rutherford ˆ 4pa02 Ry 2 =…TE 2 † for liquid water (solid curves) and water vapour (dotted curves) for di€erent incident proton energies (from bottom to top t ˆ 15, 30, 100, 300 keV, and 1 MeV) as a function of the secondary-elec-

Table 4 Ionisation thresholds and number of electrons N for di€erent ionisation sub-shells in water vapour …Bj , Rudd et al., 1992) and liquid water …Ij , Dingfelder et al., 1998). Also given the partitioning factor Gj Shell j

Ij (eV)

Bj (eV)

Nj

Gj

1a1 2a1 1b2 3a1 1b1

539.00 32.30 16.05 13.39 10.79

539.70 32.20 18.55 14.73 12.61

2 2 2 2 2

1.00 0.52 1.11 1.11 0.99

Fig. 10. Ratios of the semi-empirical di€erential cross sections for liquid water (solid curves) and water vapour (dotted curves) to the Rutherford cross section vs. energy transfer for di€erent incident proton energies (from bottom to top t ˆ 15, 30, 100, 300 keV, and 1 MeV), compared to experimental data of Bolorizadeh and Rudd (1986a) (circles), and of Toburen and Wilson (1977) (diamonds).

268

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

tron energy W. Also shown are experimental data for proton impact on water vapour by Bolorizadeh and Rudd (1986a) (circles) and by Toburen and Wilson (1977) (diamonds). The total ionisation cross section is obtained by numerically integrating ds=dE over E. Alternatively, a semi-empirical formula for the total ionisation cross section is given by Rudd et al. (1985a), and it leads to values close to those obtained by numerical integration. This formula is also used to represent other total cross sections such as the stripping cross section of hydrogen. The total cross section is represented by the harmonic mean of a high-energy part shigh and a low-energy part slow  s…t† ˆ

1 slow

‡

1 shigh

 ÿ1

:

…42†

The low-energy part is given as  D ! T 2 slow …t† ˆ 4pa0 C ‡F , Ry

…43†

where C, D and F are parameters, and T ˆ …m=M †t represents the kinetic energy of an electron travelling with the same speed as the proton. The high-energy part is given as      Ry Ry shigh …t† ˆ 4pa02 A ln 1 ‡ ‡B , …44† T T where A and B are parameters. (Note that the symbols in Eqs. (42)±(44) follow Rudd et al. (1985a). In particular, A, B, C, D and F are all parameters, and are not to be confused with our A(E ), B(E ), etc.) Parameters for total ionisation cross sections can be found for various atoms and molecules in the review article of Rudd et al. (1985a), and for water vapour in Rudd et al. (1985b). Parameter sets used in the present work are displayed in Table 5. Fig. 11 presents total ionisation cross sections obtained by numerical integration of the di€erential cross sections. Shown are results of the present calcu-

lation for proton impact (solid curve) and hydrogen impact (dashed curve), to be further discussed in Section 3.6, on liquid water, the recommended cross section for proton impact on water vapour (Rudd et al., 1992) (dash±dotted curve) and various experimental data from Rudd et al. (1985b) (symbols). The liquid water cross section is somewhat smaller than the vapour cross section at high incident energies and around the maximum, which is shifted to a little lower incident energies, but is comparable or even larger at lower incident energies. Indeed, the maximum of the liquid water stopping cross section (to be discussed further in Section 3.7) is smaller in its magnitude and width, which indicates that the cross section receives less contributions around the maximum, and more contributions at lower energies again. 3.4. Charge transfer In contrast to the very few measurements of ionisation cross sections there is a good data base for charge-changing events. A recent review article on newer data for various molecules is published by Toburen (1998); an older one by Tawara (1978) is a good data source on cross sections of various atoms and molecules, too. There are many measurements of the electron-capture cross section s10 in water vapour, but there is no experimental corresponding information on the liquid or solid phase. Unfortunately, the absolute values of the cross sections measured by di€erent workers di€er by up to one order of magnitude. Therefore, in a most recent measurement, Lindsay et al. (1997) tried to con®rm or reject earlier measurements.

Table 5 Parameter sets for total cross sections (Eq. (42)). s‡ and sÿ are taken from Rudd et al. (1985b)

A B C D F

sÿ

s‡

s01

2.98 4.42 1.48 0.75 ±

2.98 4.42 1.48 0.75 4.80

2.835 0.310 2.100 0.760 ±

Fig. 11. Total ionisation cross sections vs. incident particle energies. Shown are the proton ionisation cross sections (solid curve) and the hydrogen ionisation cross section (dashed curve) for liquid water and the proton ionisation cross section (dashed±dotted curve) for water vapour, compared to various measurements (di€erent symbols) for proton impact on water vapour (Rudd et al., 1985b).

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

They measured the angular di€erential and total charge-transfer cross sections at proton energies of 500 eV, 1.5 and 5 keV and found good agreement with the measurements by Dagnac et al. (1970) (concerning charge-transfer and electron-loss cross sections for proton energies in the range from 2 to 60 keV) and by Berkner et al. (1970) (concerning charge-transfer cross sections for proton energies in the range from 150 eV to 5 keV), while the data of Chambers (cited by Lindsay et al., 1997) (in the energy range from 1 to 10 keV) are somewhat higher than the new measurements. They discarded the data by Coplan and Ogilvie (1970) for the proton energy range from 100 to 500 eV, which are higher by a factor of 3, and the data of Koopman (1968) for the proton energy range from 40 eV to 1.5 keV, which are about an order of lower magnitude. Other data are due to measurements by Cable (cited by Koopman, 1968) for protons between 150 and 500 eV and by Toburen et al. (1968) (concerning chargetransfer and electron-loss cross sections for an energy range from 100 keV to 2.5 MeV), which is the only data source for higher proton energies. Rudd and co-workers (Rudd et al., 1985b) measured the production cross section of positive …s‡ † and negative …sÿ † charges, performing separate measurements with di€erent accelerators in the total energy range from 7 keV up to 4 MeV, and ®tted them to their semi-empirical model Eqs. (42)±(44) for total cross sections. The charge-transfer cross section is then s10 ˆ s‡ ÿ sÿ ; the parameter set is displayed in Table 5. The resulting cross section agrees well with the experimental data for proton energies of 1±100 keV, but is clearly higher than the experimental data of Toburen et al. (1968) for energies above 100 keV on one hand, and leads to a constant value for energies below 1 keV on the other hand, while most of the other experimental values increase with decreasing proton energies. This probably means that the simple semi-empirical model is not ¯exible enough to account for details. In particular, the determination of s10 by the subtraction is not very precise when s‡ and sÿ are comparable. Consequently we decided to represent the chargetransfer cross section by an analytic formula, consisting of straight lines for low and high proton energies on a doubly logarithmic scale, both connected by a power law. In detail, the charge transfer cross section is expressed by s10 …t† ˆ 10Y…X † ,

…45†

where X ˆ log…t†, and

t in eV,

…46†

269

Table 6 Parameter set for the charge transfer cross section (Eq. (45)) a0 b0 c0 d0 a1 b1 x0 x1

ÿ0.180 ÿ18.22 0.215 3.550 ÿ3.600 ÿ1.997 3.450 5.251

  Y…X † ˆ a0 X ‡ b0 ÿ c0 …X ÿ x0 †d0 Y…X ÿ x 0 † Y…x 1 ÿ X† ‡ …a1 X ‡ b1 †Y…X ÿ x 1 †, …47† where Y…x† represents the Heaviside step function. The parameters a0 and b0 determine the low-energy straight line, a1 and b1 the high-energy one, c0 and d0 the power law in between, connected to the low-energy straight line at x0. The connection point x1 to the high-energy line and the parameter b1 are calculated as x1 ˆ



a0 ÿ a1 c0 d0

1=…d0 ÿ1†

‡x 0 ,

b1 ˆ …a0 ÿ a1 †x 1 ‡ b0 ÿ c0 …x 1 ÿ x 0 †d0 ,

…48†

…49†

using the ®rst derivative. The parameters were chosen by considering the experimental data on water vapour as a basis, i.e., so that s10 is close to the data of Lindsay et al. (1997) and Dagnac et al. (1970) for low and medium energies and of Toburen et al. (1968) for higher energies, and by adjusting the contributions of the cross section to the total stopping cross section in respect to the recommended values for the liquid phase. The charge transfer cross section mainly in¯uences the probability of ®nding the projectile in a de®nite charge state, but also the contribution of the charge changing process. The parameter set is displayed in Table 6; the charge transfer cross section (solid curve) is plotted in Fig. 12 and compared to various experimental data. The ®gure also shows the semi-empirical model of Rudd et al. (1985b) for water vapour (dash±dotted curve) as described above. 3.5. Stripping of neutral hydrogen Dagnac et al. (1970) measured the stripping cross section for lower energies, while Toburen et al. (1968) presented data for higher energies. Both groups also review theoretical treatments mostly concerning the hydrogen target. Because there is no satisfactory theoretical treatment of water we decided to use again the

270

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

Fig. 12. Charge-changing cross sections for liquid water vs. incident particle energy. Shown are the charge transfer cross section s10 (solid curve) and the electron loss cross section s01 (dashed curve), compared to various experiments for water vapour (symbols). Also shown is the charge transfer cross section for water vapour (Rudd et al., 1985b), dash±dotted curve.

semi-empirical formulas, Eqs. (42)±(44), and ®t the parameters to these existing experiments. We again use the freedom of adjusting the parameters to reproduce the stopping cross section. The charge-transfer cross section and the stripping (or electron-loss) cross section together determine the probability of ®nding the projectile either in the ``proton'' state or in the ``hydrogen'' state. Both cross sections together determine the contribution of the charge-changing process and in¯uence the total stopping cross section. The parameters obtained are shown in Table 5; the stripping cross section itself is shown as the dashed curve in Fig. 12. 3.6. Ionisation by neutral hydrogen For fast collisions, hydrogen-impact cross sections di€er from proton-impact cross sections by the e€ects of screening of the nuclear charge by the bound hydrogen electron, by the contributions to the electron spectra from the stripping of the hydrogen electron and by contributions from electron±electron interactions of the hydrogen electron with the target electrons. The relative importance of these three processes depends on the incident particle energy, the ejected electron energy and the ejected electron angle. Unfortunately, there seems to be no experimental information on these processes for the liquid or solid phase. Fortunately, there are data for proton- and hydrogen-impact in gas targets (IAEA, 1995; Toburen, 1998), including water

vapour (Bolorizadeh and Rudd, 1986a, 1986b). Bolorizadeh and Rudd measured angular and energy di€erential cross sections for proton (Bolorizadeh and Rudd, 1986a) and hydrogen (Bolorizadeh and Rudd, 1986b) impact on water vapour and presented the ratio of the single di€erential cross section for hydrogen impact to that for proton-impact as a function of the secondary-electron energy (cf. Fig. 5 of Bolorizadeh and Rudd, 1986b). This ratio is almost independent of the secondary-electron energy for a ®xed incident particle energy, except for the contributions from the stripping of the hydrogen electron, i.e., the broad peak at the electron velocity nearly equal to the projectile velocity. The ratio is about 2.0 for 20-keV incident energy and decreases down to 0.9 for 100-keV incident energy. Therefore, after thorough considerations we decided to use the same secondary-electron spectrum for proton impact as a starting point, but modi®ed by a function g…t† that depends only on the incident particle energy t and not on the energy transfer E. On the basis of data (Bolorizadeh and Rudd, 1986b; Toburen, 1998) we adopt the function g…t† in the following way. At low incident particle energies, g…t† is a constant higher than unity. At higher incident energies, g…t† is slightly lower than unity, in view of the screening of the nuclear charge by the bound electron in hydrogen. Thus, the energy-transfer di€erential cross section for ionisation by hydrogen impact is given by     ds ds ˆ g…t† , …50† dE hydrogen dE proton with    ÿ1 log…t† ÿ 4:2 g…t† ˆ 0:8 1 ‡ exp ‡0:9: 0:5

…51†

The parameters are determined by considering the experimental data on water vapour as a basis and by adjusting the contribution of the cross section to the recommended total stopping cross section in liquid water by ICRU (1993). Consequently, the total ionisation cross section for hydrogen impact, shown by the dashed curve in Fig. 11, is larger by a factor of 1.7 than the proton ionisation cross section for low incident particle energies and by a factor of 0.9 smaller for high incident particle energies. 3.7. Phase e€ects Some information of the in¯uence of the state of aggregation (phase e€ect) to the cross sections can be obtained from electronic stopping cross sections. The stopping cross sections sst are of interest in many ®elds of science dealing with radiation. Therefore, the ICRU

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

published recommended stopping cross sections for most of the molecules and materials of interest to radiation science (ICRU, 1993), which are based on both theoretical and experimental information. Because of the relevance of water in medical applications of ionising radiation (i.e. radiotherapy) already in 1952 measurements with protons as incident particles were performed (Wenzel and Whaling, 1952; Reynolds et al., 1953). Also some theoretical work has been done to calculate stopping cross sections for water explicitly in di€erent phases (Xu et al., 1985; Fainstein et al., 1996). Recent years have seen extensive studies on phase e€ects in several materials, for instance, on the energy loss of protons in gaseous and solid zinc targets (Bauer et al., 1992; Arnau et al., 1994) as well as in water vapour and ice (Mitterschi€thaler and Bauer, 1990; Bauer et al., 1994; Bauer et al., 1997). A notable phase e€ect was found in these studies between water vapour and ice in the energy range around the stopping cross section maximum. The maximum of the stopping cross section of ice is smaller in the absolute value and narrower than the maximum of water vapour, and it is shifted to somewhat higher proton energies compared to the vapour phase. At lower particle energies, the phase e€ect amounts to as much as to 15% at 20 keV. † We see that s…vapour† > s…ice† > s…liquid in the maximum st st st …liquid † using the ICRU data for sst (ICRU, 1993). At higher proton energies the phase e€ect is small. Here, the in¯uence of charge-changing processes is negligible. Therefore, we can use the Bethe theory to estimate the phase e€ect (Bauer et al., 1994) as 1† 2† s…phase ÿ s…phase ˆ st st

  Iphase 2 4pZ e4 , ln 2 mv Iphase 1

…52†

271

state approach (Bates and Gring, 1953; Allison and Warshaw, 1953), where sst is a sum over all charged states i of the projectile sst ˆ

X

ÿ
Fi sst,i ‡ sij Tij :

…53†

i, j, i6ˆj

Here, Fi denotes the probability to ®nd the ion in the charge state i, sst,i the (electronic) stopping cross section of charge state i, sij the charge-changing cross section from charge state i to charge state j (capture and loss of electrons, respectively) and Tij the energy loss connected with the charge changing event. We consider only one electron capture and loss processes and neglect multi-electron processes, in the present work. Therefore, our system contains two di€erent charge states of the projectile: i ˆ 0: neutral hydrogen H and i ˆ 1: proton p. Therefore, Eq. (53) reduces to ÿ
ÿ
sst ˆ F0 sst, 0 ‡ s01 T01 ‡ F1 sst, 1 ‡ s10 T10 : …54† A lower bound of the sum T01 ‡ T10 , i.e., the energy loss per charge-changing cycle, is given by the sum of the ionisation energy of the water molecule I0 and the kinetic energy of the electron, when travelling with the speed v of the proton, T ˆ mv 2 =2, T01 ‡ T10 ˆ I0 ‡

mv 2 : 2

…55†

The probabilities F0 and F1 are F0 ˆ

s10 , s01 ‡ s10

and F1 ˆ

s01 : s01 ‡ s10

…56†

Therefore, we can write

where I is the mean excitation energy of each phase. The recommended values (ICRU, 1984) are Ivapour ˆ 71:6 eV, Iice ˆ 75:0 eV and Iliquid ˆ 75:0 eV, newer experimental values are Iice ˆ 80:9 eV (Bauer et al., 1994) and Iliquid ˆ 79:7 eV (Bichsel and Hiraoka, 1992). Our model (Dingfelder et al., 1998) gives Iliquid ˆ 81:8 eV. Using the recommended values the di€erence at 300 keV is s…vapour† ÿ s…liquid=ice† ˆ 3:7  10 ÿ20 eV m2, correst st sponding to about 2% (Bauer et al., 1994). 3.8. Stopping cross sections Ions moving in a medium can capture and lose electrons and thus are in a dynamic charge equilibrium. The electron loss and capture processes will contribute to the stopping cross section considerably and the interaction of the ion in di€erent charge states will in¯uence the total stopping cross section. Theoretically, the stopping cross section sst for a charged ion in a medium is described by the charge

Fig. 13. Total cross sections for the di€erent processes in liquid water: proton ionisation s11 (solid curve), hydrogen ionisation s00 (long dashed curve), charge transfer s10 (dash± dotted curve), electron loss s01 (short dashed curve), and proton excitation cross section (dotted curve).

272

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

Fig. 14. Stopping cross section sst for liquid water (solid curve) with contributions from proton ionisation (short dashed curve), hydrogen ionisation (dash±dotted curve), charge changing processes (long dashed curve) and proton excitation (dotted curve), compared with experimental and recommended data for the di€erent water phases: liquid water (ICRU, 1993, circles), water vapour (ICRU, 1993, diamonds), and ice (Bauer et al., 1994, squares).

sst ˆ F0 sst,0 ‡ F1 sst,1 ‡ sst,CC

…57†

with the stopping cross sections sst, i of charge state i … dsii sst,i ˆ E dE …58† dE all E and the charge-changing stopping cross section sst,CC sst,CC ˆ

s01 s10 …I0 ‡ T†: s01 ‡ s10

…59†

while the ionisation cross sections and the energy-loss cross sections, all similar in shape and order of magnitude, decrease in this energy range. The hydrogenimpact ionisation cross section is larger than the proton-impact ionisation cross section for energies below 100 keV. The electron-loss cross section is always smaller than the ionisation cross sections. The excitation cross section follows more or less the ionisation cross section in shape and is one order of magnitude smaller. The stopping cross section calculated from these cross sections is shown in Fig. 14 (solid curve) together with the contributions from proton-impact ionisation F1 sst,1 (short dashed curve), hydrogen-impact ionisation F0 sst,0 (long dashed curve), charge-changing processes sst,CC (dash±dotted curve), and from protonimpact excitations (dotted curve) compared to the recommended ICRU stopping cross section for liquid water (ICRU, 1993) (circles). Also shown are the recommended ICRU stopping cross sections for water vapour (ICRU, 1993) (diamonds) and the experimental stopping cross sections for ice (Bauer et al., 1994) (squares). Since we used the stopping cross sections to normalize and to adjust parameter sets for di€erent processes, our model calculation shows an excellent agreement with the recommended values. At high energies only the proton ionisation cross section contributes appreciably to the stopping cross section; below 300 keV contributions from all other processes appear. The stopping cross section near the maximum depends on all contributions. Below 20 keV it is dominated by the hydrogen ionisation contribution. The contribution from excitations is less than a few percent. 4. Concluding remarks

3.9. Results The total stopping cross sections of the vapour and the liquid phase are used to adjust parameter sets for the di€erent processes and to adopt them for the liquid phase. Fig. 13 shows all total cross sections together: the proton excitation cross section sexc (dotted curve), the 1 proton ionisation cross section s11 (solid curve), the hydrogen ionisation cross section s00 (long dashed curve), the charge-transfer cross section s10 (dash± dotted curve) and the electron-loss cross section s01 (short dashed curve). One recognises that the chargetransfer cross section dominates for proton energies below, say 30 keV, increasing with decreasing energies, 2 A copy of our computer code (in C) generating the present cross sections can be transmitted in an electronically readable form upon request to [email protected]

The present paper reports the results of our work on a self-consistent set of proton inelastic scattering cross sections for liquid water for a wide energy range from very slow protons (100 eV) to highly relativistic ones (10 GeV). This set includes excitation and ionisation cross sections of protons, but also charge transfer, and ionisation and electron loss of moving neutral hydrogen atoms. The data can be used, for example, in track-structure analysis and other quantitative modelling studies in radiation physics, chemistry and biology2. The present article is the third report on our continuing e€orts toward establishing the best possible set of cross-section data for liquid water, following up Dingfelder et al. (1998) and Dingfelder and Inokuti (1999a), which primarily concern electron interactions. We now plan to report soon on those aspects of proton interactions, such as elastic scattering by a water molecule as a whole and the angular distribution of

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

secondary electrons, which are crucial to the determination of the spatial structure of tracks. Eventually we will extend our work to treat heavier ions of importance to radiation biology and medicine, e. g., a-particles as well as carbon, nitrogen and oxygen ions.

Acknowledgements We thank L.H. Toburen, P. Bauer and P. Jacob for helpful discussions including valuable ideas to di€erent parts of this paper. The present work is supported by the European Community under Contract No. FI4PCT95-0011 ``Biophysical Models for the Induction of Cancer by Radiation'' and in part by the Department of Energy, Oce of Science, Nuclear Physics Division, under Contract No. W-31-109-Eng-38.

References Allison, S.K., Warshaw, S.D., 1953. Passage of heavy particles through matter. Rev. Mod. Phys. 25, 779±817. Arnau, A., Bauer, P., Kastner, F., Salin, A., Ponce, V.H., Fainstein, P.D., Echenique, P.M., 1994. Phase e€ect in the energy loss of hydrogen projectiles in zinc targets. Phys. Rev. B 49, 6470±6480. Ashley, J.C., 1982a. Stopping power of liquid water for lowenergy electrons. Radiat. Res. 89, 25±31. Ashley, J.C., 1982b. Density e€ect in liquid water. Radiat. Res. 89, 32±37. Bak, J.F., Meyer, F.E., Petersen, J.B.B., Uggerhùj, E., éstergaard, K., Mùller, S.P., Sùrensen, A.H., Si€ert, P., 1983. New experimental investigations of the density e€ect on inner shell excitation and energy loss. Phys. Rev. Lett. 51, 1163±1166. Bak, J.F., Petersen, J.B.B., Uggerhùj, E., éstergaard, K., Mùller, S.P., Sùrensen, A.H., 1986. In¯uence of transition radiation and density e€ect on atomic K-shell excitation. Physica Scripta 33, 147±155. Bates, D.R., Gring, G.W., 1953. Inelastic collisions between heavy particles I: Excitation and ionisation of hydrogen atoms in fast encounters with protons and with other hydrogen atoms. Proc. Phys. Soc. London, Sect. A 66, 961±971. Bauer, P., Kastner, F., Arnau, A., Salin, A., Fainstein, P.D., Ponce, V.H., Echenique, P.M., 1992. Phase e€ect in the energy loss of H projectiles in Zn targets: experimental evidence and theoretical explanation. Phys. Rev. Lett. 69, 1137±1139. Bauer, P., KaÈferboÈck, W., NecÏas, V., 1994. Investigation of the electronic loss of hydrogen ions in H2O: in¯uence of the state of aggregation. Nucl. Instr. Meth. 93, 132±136. Bauer, P., Gosler, R., Aumayr, F., Semrad, D., Arnau, A., Zarate, E., Diez±MuinÄo, R., 1997. Contribution of valence electrons to the electronic energy loss of hydrogen ions in oxides. Nucl. Instr. Meth. B 125, 102±105. Berkner, K.H., Pyle, R.V., Stearns, J.W., 1970. Cross sections for electron capture by 0.3 to 70 keV deuterons in H2,

273

H2O, CO, CH4, and C8F16 gases. Nucl. Fus. 10 (1970), 145±149. Bethe, H., 1930. Zur Theoorie des Durchgangs schneller Korpuskularstrahlen durch Materie. Ann. Phys. (Leipzig) 5, 325±400. Bethe, H., Ashkin, J., 1956. SegreÁ, E. (Ed.), Experimental Nuclear Physics, vol I. Wiley, New York, p. 166. Bethe, H.A., Jackiw, R.W., 1997. Intermediate Quantum Mechanics Addison. Wesley, Reading, MA Reprint Series. Bichsel, H., Hiraoka, T., 1992. Energy loss of 70 MeV protons in elements. Nucl. Instr. Meth. 66, 345±351. Bichsel, H., Inokuti, M., 1998. Di€erence in stopping power for protons and deuterons of a given speed. Nucl. Instr. Meth. B 134, 161±164. Bolorizadeh, M.A., Rudd, M.E., 1986a. Angular and energy dependence of cross sections for ejection of electrons from water vapor. II. 15±150 keV proton impact. Phys. Rev. A 33, 888±892. Bolorizadeh, M.A., Rudd, M.E., 1986b. Angular and energy dependence of cross sections for ejection of electrons for water vapor. III. 20±150 keV neutral-hydrogen impact. Phys. Rev. A 33, 893±896. Coplan, M.A., Ogilvie, K.W., 1970. Charge exchange for H+ and H+ 2 in H2O, CO2, and NH3. J. Chem. Phys. 52, 4154± 4160. Dagnac, R., Blanc, D., Molina, D., 1970. A study on the col‡ ‡ lision of hydrogen ions H‡ 1 , H2 and H3 with a water± vapour target. J. Phys. B 3, 1239±1251. Dingfelder, M., Hantke, D., Inokuti, M., 1997. Interaction cross sections for electron inelastic scattering in liquid water. In: Goodhead, D.T., Neill, P.O, Menzel, H. (Eds.), Microdosimetry: An Interdisciplinary Approach. The Royal Society of Chemistry, Cambridge, pp. 23±26. Dingfelder, M., Hantke, D., Inokuti, M., Paretzke, H.G., 1998. Electron inelastic-scattering cross sections in liquid water. Radiat. Phys. Chem. 53, 1±18. Dingfelder, M., Inokuti, M., 1999a. The Bethe surface of liquid water. Radiat. Environ. Biophys. 38, 93±96. Dingfelder, M., Inokuti, M., 1999b. Fermi density e€ect on the cross section for a ®xed energy transfer, with application to liquid water. In: International Conference on Atomic Collisions in Solids ICACS±18, Book of Abstracts, Odense, Denmark, August 3±8, 48. Fainstein, P.D., Olivera, G.H., Rivarola, R.D., 1996. Theoretical calculations of the stopping power for protons traversing H, He and simple molecular targets. Nucl. Instr. Meth. B 107, 19±26. Fano, U., 1956. Di€erential inelastic scattering of relativistic charged particles. Phys. Rev. 102, 385±387. Fano, U., 1963. Penetration of protons, alpha particles, and mesons. Ann. Rev. Nucl. Sci. 13, 1±66. Fermi, E., 1940. The ionisation loss of energy in gases and in condensed materials. Phys. Rev. 57, 485±493. Friedland, W., Jacob, P., Paretzke, H.G., Merzagora, M., Ottolenghi, A., 1999. Simulation of DNA fragment distributions after irradiation with photons. Radiat. Environ. Biophys. 38, 39±47. Gibson, D.K., Reid, I.D., 1987. Energy and angular distributions of electrons ejected from various gases by 50 keV protons. Radiat. Res. 112, 418±425. Grande, P.L., Schiwietz, G., 1992. Energy loss of slow ions:

274

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275

one-band calculation for alkaline metals. Phys. Lett. A 163, 439±446. Green, A.E.S., McNeal, R.J., 1972. Analytic cross sections for inelastic collisions of protons and hydrogen atoms with atomic and molecular gases. J. Geophys. Res. 76, 133±144. Hayashi, H., Watanabe, N., Udagawa, Y., Kao, C.-C., 1998. Optical spectra of liquid water in vacuum uv region by means of inelastic X-ray scattering spectroscopy. J. Chem. Phys. 108, 823±825. Heller, J.M., Hamm Jr, R.N., Birkho€, R.D., Painter, L.R., 1974. Collective oscillation in liquid water. J. Chem. Phys. 60, 3483±3486. Inokuti, M., 1971. Inelastic collisions of fast charged particles with atoms and molecules Ð the Bethe theory revisited. Rev. Mod. Phys. 43, 297±347 (See also addenda: Inokuti, M., Itikawa, Y., Turner, J.E. 1978. Rev. Mod. Phys. 50, 23±35. Inokuti, M., Smith, D.Y., 1982. Fermi density e€ect on the stopping power of metallic aluminium. Phys. Rev. B 25, 61±66. Inokuti, M., 1991. How is radiation energy absorption di€erent between the condensed phase and the gas phase? Radiat. E€ects Defects Solids 117, 143±162. Inokuti, M. (Ed.), 1994. Advances in Atomic, Molecular and Optical Physics, Vol. 33: Cross-Section Data. Academic Press, Boston. International Atomic Energy Agency, 1995. Atomic and Molecular Data for Radiation Therapy and Related Research. IAEA±TECDOC±799, IAEA, Vienna. International Commission on Radiation Units and Measurements, 1984. Stopping Powers for Electrons and Positrons. ICRU Report 37. International Commission on Radiation Units and Measurements, Bethesda, Maryland. International Commission on Radiation Units and Measurements, 1989. Tissue Substitutes in Radiation Dosimetry and Measurements. ICRU Report 44. International Commission on Radiation Units and Measurements, Bethesda, Maryland. International Commission on Radiation Units and Measurements, 1993. Stopping Powers and Ranges for Protons and Alpha Particles. ICRU Report 49. International Commission on Radiation Units and Measurements, Bethesda, Maryland. International Commission on Radiation Units and Measurements, 1996. Secondary Electron Spectra from Charged Particle Interactions. ICRU Report 55. International Commission on Radiation Units and Measurements, Bethesda, Maryland. Jackson, J.D., 1998. Classical Electrodynamics, 3rd ed. Wiley, New York (Chapter 13). Kaushik, Y.D., Khare, S.P., Kumar, A., 1981. Inelastic collisions of protons with water vapour. Physica 106 C, 128± 134. Kimura, M., Lane, N.F., 1990. The low-energy, heavy-particle collisions Ð a close-coupling treatment. In: Bates, D., Bederson, B. (Eds.), Advances in Atomic, Molecular and Optical Physics, vol. 26. Academic Press, Boston, pp. 80± 160. Koopman, D.W., 1968. Light-ion charge exchange in atmospheric gases. Phys. Rev. 166, 57±62. Landau, L.D., Lifshitz, E.M., 1970. In: Sykes, J.B., Bell, J.S.,

Kearsley, M.J. (Eds.), Electrodynamics of Continuous Media, 2nd ed. Pergamon Press, London. Lindsay, B.G., Siegla€, D.R., Smith, K.A., Stebbings, R.F., 1997. Charge transfer of 0.5, 1.5, and 5 keV protons with H2O: Absolute di€erential and integral cross sections. Phys. Rev. A 55, 3945±3946. Long, K.A., Paretzke, H.G., 1991. Comparison of calculated cross sections for secondary electron emission from a water molecule and clusters of water molecules by protons. J. Chem. Phys. 95, 1049±1056. Meyerhof, W.E., Jensen, D.G., Kawall, D.M., Kuhn, S.E., Spooner, D.W., Meziani, Z.E., Faust, D.N., 1992. Density e€ect in Cu K-shell ionisation by 5.1-GeV electrons. Phys. Rev. Lett. 68, 2293±2296. Miller, J.H., Green, A.E.S., 1973. Proton energy degradation in water vapor. Radiat. Res. 54, 343±363. Mitterschi€thaler, C., Bauer, P., 1990. Stopping cross section of water vapor for hydrogen ions. Nucl. Instr. Meth. B 48, 58±60. Paretzke, H.G., 1987. Radiation track structure theory. In: Freeman, G.R. (Ed.), Kinetics of Non-Homogeneous Processes. Wiley, New York, pp. 89±170. Paul, H., Sacher, J., 1989. Fitted empirical reference cross sections for K±shell ionisation by protons. At. Data Nucl. Data Tables 42, 105±156. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in Fortran, 2nd ed. Cambridge University Press, Cambridge. Reynolds, H.K., Dunbar, D.N.F., Wenzel, W.A., Whaling, W., 1953. The stopping cross section of gases for protons, 30±600 keV. Phys. Rev. 92, 742±748. Ritchie, R.H., Hamm, R.N., Turner, J.E., Wright, H.A., Bolch, W.E., 1991. Radiation interactions and energy transport in the condensed phase. In: Glass, W.A., Varma, M.N. (Eds.), Physical and Chemical Mechanisms in Molecular Radiation Biology. Plenum Press, New York, pp. 99±133. Rudd, M.E., DuBois, R.D., Toburen, L.H., Ratcli€e, C.A., Go€e, T.V., 1983. Cross sections for ionisation of gases by 5±4000 keV protons and for electron capture by 5±150 keV protons. Phys. Rev. A 28, 3244. Rudd, M.E., Kim, Y.-K., Madison, D.H., Gallagher, J.W., 1985a. Electron production in proton collisions with atoms and molecules: total cross sections. Rev. Mod. Phys. 57, 965. Rudd, M.E., Go€e, T.V., DuBois, R.D., Toburen, L.H., 1985b. Cross sections for ionisation of water vapor by 7± 4000 keV protons. Phys. Rev. A 31, 492±494. Rudd, M.E., Kim, Y.-K., Madison, D.H., Gay, T.J., 1992. Electron production in proton collisions with atoms and molecules: energy distributions. Rev. Mod. Phys. 64, 441± 490. Schiwietz, G., 1990. Coupled-channel calculation of stopping powers for intermediate-energy light ions penetrating atomic H and He targets. Phys. Rev. A 42, 296±306. Slater, J.C., 1930. Atomic shielding constant. Phys. Rev. 36, 57±64. Sùrensen, A.H., 1987. Atomic K-shell excitation at ultrarelativistic impact energies. Phys. Rev. A 36, 3125±3137. Sùrensen, A.H., Uggerhùj, E., 1986. The hunt for the density

M. Dingfelder et al. / Radiation Physics and Chemistry 59 (2000) 255±275 e€ect in inner shell excitations. Comments At. Mol. Phys. 17, 285±309. Spooner, D.W., Meyerhof, W.E., Ku€ners, J.J., Montenegro, E.C., Ishii, K., Kuhn, S.E., Kawall, D.M., Jensen, D.G., Meziani, Z.-E., 1994. Density e€ect in relativistic K-shell ionisation. Z. Phys. D 29, 265±268. Tawara, H., 1978. Cross sections for charge transfer of hydrogen beams in gases and vapors in the energy range of 10 eV to 10 keV. At. Data Nucl. Data Tables 22, 491±525. Toburen, L.H., Nakai, M.Y., Langley, R.A., 1968. Measurement of high-energy charge-transfer cross sections for incident protons and atomic hydrogen in various gases. Phys. Rev. 171, 114±122. Toburen, L.H., Wilson, W.E., 1977. Energy and angular distributions of electrons ejected from water vapor by 0.3±1.5 MeV. J. Chem. Phys. 66, 5202±5213. Toburen, L.H., 1998. Ionisation and charge-transfer: basic data for track structure calculations. Radiat. Environ. Biophys. 37, 221±233. Watanabe, N., Hayashi, H., Udagawa, Y., 1997. Bethe surface of liquid water determined by inelastic X-ray scatter-

275

ing spectroscopy and electron correlation e€ects. Bull. Chem. Soc. Jpn. 70, 719±726. Wenzel, W.A., Whaling, W., 1952. The stopping cross section in D2O ice. Phys. Rev. 87, 499±503. Wilson, W.E., Miller, J.H., Toburen, L.H., Manson, S.T., 1984. Di€erential cross sections for ionisation of methane, ammonia and water vapor by high velocity ions. J. Chem. Phys. 80, 5631±5638. Wilson, W.E., Nikjoo, H., 1999. A Monte Carlo code for positive ion track simulations. Radiat. Environ. Biophys. 38, 97±104. Xu, Y.J., Khandelwal, G.S., Wilson, J.W., 1985. Proton stopping cross sections of liquid water. Phys. Rev. A 32, 629± 632. Yousif, F.B., Geddes, J., Gilbody, H.B., 1986. Balmer a emis‡ sion in collisions of H, H+, H‡ 2 and H3 with N2, O2 and H2O. J. Phys. B 19, 217±231. Zaider, M., Brenner, D.J., Wilson, W.E., 1983. The applications of track calculations to radiobiology I. Monte Carlo simulation of proton tracks. Radiat. Res. 95, 231± 247.