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Stopping of swift ions in compounds
Nuclear Instruments and Methods in Physics Research B 218 (2004) 19–28 www.elsevier.com/locate/nimb
Stopping of swift ions in compounds A. Sharma a, A. Fettouhi b, A. Schinner c, P. Sigmund a
a
Physics Department, University of Southern Denmark, DK-5230 Odense M, Denmark b Physikalisches Institut, J.-Liebig Universit€at, D-35392 Giessen, Germany c Inst. f. Experimentalphysik, J.-Kepler-Universit€at, A-4040 Linz-Auhof, Austria
Abstract Stopping of light and heavy ions in compounds has been studied on the basis of the binary theory of electronic stopping with the focus on limitations of BraggÕs additivity rule. The case of LiF is analysed in detail because departures from Bragg additivity are expected to be particularly pronounced here, especially for penetrating antiprotons. A significant dependence on the atomic number and ionic charge of the projectile is predicted along with the more wellknown velocity dependence of departures from Bragg additivity. Such departures are found to be affected by shell correction and Barkas effect in addition to the more well-known dependence on the I-value. Comparisons with experimental results include H2 O vapor, CH4 , CO2 , mylar and polycarbonate. 2003 Elsevier B.V. All rights reserved. PACS: 34.50.Bw; 79.20.Nc Keywords: Stopping; Stopping power; Stopping force; Compound; BraggÕs rule; LiF
1. Introduction The stopping of charged particles in compounds is conventionally described by BraggÕs additivity rule which claims that the stopping cross section of an atom is independent of its chemical and physical environment. This implies that the stopping cross section of a molecule is equal to the sum of the stopping cross sections of its constituent atoms. Limitations of this rule have been known for at least half-a-century, not the least due to careful measurements with proton beams penetrating through molecular gases such as water vapor [1]. Experimental efforts in this area have focused on proton and helium beams and have been summarized in comprehensive reviews [2–4]. Typical results are similar to those found in [1], with a 10% departure from Bragg additivity around the
stopping maximum that decreases rapidly toward higher projectile speeds. Although a few theoretical studies have been made on specific systems [5,6], little systematic knowledge is available on departures from Bragg additivity as a function of the atomic number Z1 and the speed v of the projectile as well as its charge state. Recent work by three of us [7] indicates that dramatic departures from Bragg additivity may be found for specific materials like LiF where • the structure of the valence electrons differs markedly from those of the isolated atoms and • valence electrons make up a large fraction of all target electrons, in agreement with qualitative arguments by Thwaites [8]. We found particularly pronounced
0168-583X/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2003.12.009
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A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
departures from additivity in antiproton stopping. This was ascribed to the absence of projectile screening. The present note serves to follow up on those preliminary findings, while detailed documentation is in preparation [9].
(3) Projectile screening due to electrons accompanying a penetrating ion influence the range of interaction and hence the contribution of valence electrons to stopping. 2.2. Binary stopping theory
2. Theory 2.1. Preliminary considerations Theoretical considerations have frequently been based on BetheÕs formula which, in its simplest form, predicts the stopping cross section S to be given by [10] S¼
4pZ12 Ze4 2mv2 ; ln mv2 I
ð1Þ
where Z1 is the atomic number of the projectile and Z the number of electrons in a target molecule. Material properties enter through the I-value, called mean excitation energy or mean ionization potential [11]. The logarithmic dependence ensures that departures from Bragg additivity originating in the I-value decrease in significance with increasing projectile speed. It is not obvious whether changes in I-value are the dominating cause of departures from Bragg additivity, as has been frequently implied [4,8,12]: (1) In the energy range around and below the stopping maximum, classical stopping theory is known to be more reliable than the Bethe scheme. In the Bohr theory [13] the logarithm in Eq. (1) is replaced by ln
Cmv3 ; Z1 e2 x
ð2Þ
where x ¼ I= h and C ¼ 1:1229. The factor Z1 in the denominator must produce a projectile dependence which has not been studied to our knowledge. (2) Shell corrections are sensitive to the orbital motion of the target electrons, and the Barkas effect – i.e. contributions from higher-order-Z1 terms – hinges on their state of binding. These two effects must be influenced by changes in electron structure due to molecular binding.
A convenient theoretical scheme to account for these phenomena is the binary stopping theory developed recently by two of us. Its basic formalism has been fully described in [14], and numerous applications to stopping in elemental materials have been documented in [15]. Therefore we only characterize the main features here, as well as those additions that are necessary for stopping in compounds. • The physical model is close to BohrÕs [13], assuming harmonically-bound target electrons, but • no expansion in Z1 is necessary for either close or distant interactions, • this is possible due to an accurate modelling of electron binding via a screened Coulomb potential, • a smooth transition to the quantum regime at higher speed is achieved by use of BlochÕs theory [16,17]. • shell corrections are incorporated by kinetic theory [18], • static projectile screening is allowed for in accordance with the ionic charge state (frozen or equilibrium), • projectile excitation can be included and • electron capture and loss can be allowed for in charge equilibrium [19]. The scheme has been applied successfully to stopping of ions in metals, semiconductors, insulators and gases [15,20]. 2.3. Input Binary stopping theory needs the following numerical input: • Resonance frequencies and oscillator strengths, • orbital velocity spectra and • binding energies
A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
for all shells and/or subshells of the target and the projectile. Finding suitable data for resonance frequencies and oscillator strengths has been described in [15,22]. The same procedure can be applied to compound materials to the extent that their optical properties have been tabulated. Here we use data from [23,24]. In practice we leave the excitation spectrum for all but the outermost shell of each participating atom unchanged but collect all remaining electrons in one common valence shell.
This implies that the only change in the excitation spectrum compared to calculations assuming Bragg additivity is the resonance frequency characterizing the outermost shell. Orbital velocity spectra were found until now by Fourier transforming Hartree–Fock wave functions tabulated in [25], as described in [15], similar in principle to the procedure proposed in [26] but different in practice. We recently found that this procedure gives rise to significantly overestimated shell corrections and, therefore,
H ions in water vapor
30
2
eVcm ]
21
S [10
-1 5
20
10
Binary, hydrogenic Binary, Hartree-Fock Fainstein et al
Thin lines: Bragg 0 0.01
0.1
1
E [MeV]
H ions in water vapor
2
eVcm ]
30
S [10
-1 5
20
10
0 0.01
Bragg: Reynolds et al Water: Reynolds et al Water: Mitterschiffthaler et al Bragg: Binary, hydrogenic Water: Binary, hydrogenic
0.1
1
E [MeV] Fig. 1. Stopping of protons in water vapor. Upper graph: Three theoretical predictions. Binary theory for modified hydrogenic and Hartree–Fock velocity spectra, and Fainstein et al. [6]. Thick lines: molecular; thin lines: Bragg. Lower graph: Experimental data from [1,21] compared with binary theory, hydrogenic velocity spectra.
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A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
decided to apply modified hydrogenic wave functions instead. Here, the effective charge of each target shell has been fixed such as to match the ionization energy of that shell. As will be shown below, shell corrections contribute significantly to departures from Bragg additivity. As a logical consequence, velocity spectra calculated from molecular wave functions would have to be employed in quantitative estimates. This step has not yet been implemented. Unfortunately, the basic architecture of our PASS code [15] does not readily allow this feature to be incorporated. Orbital binding energies for atoms are available in numerous sources. In addition we use ionization potentials for the valence shell from [24].
3. Results 3.1. Water vapor Fig. 1 shows results for protons in water vapor. The upper graph demonstrates a significant influence of the adopted velocity profile both on the magnitude of the stopping force and the departure from Bragg additivity. We find reasonable agreement with quantal calculations [6] for hydrogenic velocity profiles. The latter predict a larger departure from Bragg additivity, which, however,
is not corroborated by experiment as is seen on the lower graph. With regard to our prediction, we note good agreement with the relative departure from Bragg additivity, while some discrepancy is observed in absolute stopping forces at projectile speeds well below the stopping maximum. This is a common feature also for noble-gas targets [15]. 3.2. Lithium fluoride Fig. 2 shows the predicted stopping force on antiprotons in LiF compared to the prediction of BraggÕs additivity rule. This graph differs substantially from a similar one published in [7]. Apart from an improvement in the underlying excitation spectrum, the main reason for the change is the use of hydrogenic wave functions that reduces substantially the departure from Bragg additivity. Fig. 3 shows calculations with and without shell correction. While the influence of the shell correction on the absolute magnitude of calculated stopping force is well known, it is seen that the departure from Bragg additivity is more pronounced in the shell-corrected curve, in particular around the stopping maximum. This is due to the fact that the jump of the 2s electron of a lithium atom to a 2p-like state in fluorine produces a change in orbital velocity in addition to changing the excitation frequency.
Stopping of antiprotons: Hydrogenic
20
2
eVcm ]
10
S [10
-1 5
5
2
1
0.5 0.01
Bragg Li F F Li
0.1
1
10
E [MeV] Fig. 2. Predicted stopping forces for LiF, metallic lithium and fluorine, including the prediction of BraggÕs rule.
A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
23
Antiprotons in Li F
2
5
S [10
eVcm ]
10
-15
20
2
Bragg: No shell correction LiF: No shell correction Bragg: Including shell correction LiF: Including shell correction
1 0.01
0.1
1
10
E [MeV] Fig. 3. Stopping of antiprotons in LiF. Influence of shell correction.
100
Relative departure [pct]
Protons and antiprotons in LiF
10
1
0.01
Antiproton Average Proton
0.1
1
10
E [MeV] Fig. 4. Relative departure from Bragg additivity for protons and antiprotons in LiF as well as their average.
Fig. 4 compares relative departures from Bragg additivity for protons and antiprotons in LiF. It is seen that the departure is greater for antiprotons than for protons. In other words, the Barkas effect – the difference between proton and antiproton stopping – opposes the effect of the shell correction for protons, in agreement with a general tendency regarding the stopping force in elemental materials.
Fig. 5 shows a comparison of stopping forces in LiF on several ions with calculations based on BraggÕs rule. It is seen that the departure from additivity decreases rapidly from antiprotons to heavier ions and is essentially vanishing from carbon on. The reason for this behavior emerges from Fig. 6 which demonstrates a pronounced dependence on ion charge: Bragg additivity is governed by excitation of valence electrons which
24
A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28 Ions in charge equilibrium in LiF Argon Neon Carbon Helium Hydrogen Antiproton
2
eVcm ]
1000
S [10
-1 5
100
10
Thin lines: Bragg Thick lines: LiF 1 0.01
0.1
1
10
E/A1 [MeV] Fig. 5. Stopping of ions in equilibrium charge state in LiF. Comparison with calculations based on BraggÕs rule.
Argon in LiF
2000
2
S [10 eVcm ]
1000
-15
500
18 + 12 + 6+ 0+
200
100
50 0.01
0.1
1
10
E/A1 [MeV] Fig. 6. Stopping of argon ions in LiF as a function of the ionic charge. Thick lines: molecular; thin lines: Bragg.
occurs in distant interactions. Projectile screening by accompanying electrons screens the potential and hence reduces the significance of valence excitations. Indeed, BraggÕs rule appears to hold very accurately for a neutral ion. The same feature was pointed out previously in connection with Z2 structure in heavy-ion stopping [7]. The tendency expressed by Fig. 5 goes strictly against the effect discussed in connection with the Z1 -dependence of the Bohr logarithm, Eq. (2). That effect, however, can be demonstrated if the
same plot is made for bare ions instead of ions in charge equilibrium. A comparison between the two cases at a fixed beam velocity is shown in Fig. 7 which demonstrates a clear difference. 3.3. Carbon compounds Carbon compounds have received considerable interest in the stopping literature [2,3]. The present survey presents our first explorative findings and is by no means meant to be exhaustive. It focuses on
A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
25
5
Stopping in LiF 1 MeV/u Relative departure [pct]
4
3
Bare ion Mean equilibrium charge
2
1
0 1
10
100
Z1 Fig. 7. Relative departure from Bragg additivity at 1 MeV/u for bare ions and ions in charge equilibrium.
100
2
-dE/dx [MeVcm /mg]
Stopping in methane gas
10
Krypton Argon Lithiun
1 0.01
0. 1
1
10
100
E/A1 [MeV] Fig. 8. Stopping of Li, Ar and Kr ions in methane gas. Solid lines: molecular; broken lines: BraggÕs rule. Experimental results from [27] (Li), [28,29] (Ar) and [29–31] (Kr).
ions heavier than helium because of the nearcomplete absence of systematic information – experimental or theoretical – for such systems. Fig. 8 shows predictions for stopping in methane, compared with experimental results. 1 It
1 Experimental data shown in Figs. 8–11 have been extracted from PaulÕs data base [33].
is important to note that our calculations assuming BraggÕs rule utilize the stopping cross section of amorphous carbon. With this cross section, predicted departures from Bragg additivity become rather small for Li and negligible for argon and krypton ions. While good agreement with experimental results is found for heavier ions, discrepancies are found for lithium which are not unlike those in elemental materials [15].
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A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
Similar results are shown in Fig. 9 for carbon dioxide. The remaining results refer to stopping in solid films. Fig. 10 shows predicted stopping forces in mylar. Since we do not have access to tabulated optical properties of this material, calculations have been performed on the basis of BraggÕs rule. Figs. 8 and 9 indicate that the magnitude of the expected error will not significantly affect our conclusions from this comparison. The agreement
Stopping in CO2 gas
2
-dE/dx [MeVcm /mg]
100
with experiment is seen to be quite satisfactory, the only exception being seen with the lightest ions at low energies. Finally, Fig. 11 shows a similar comparison for polycarbonate for a particularly large set of penetrating ions. Ion velocities are somewhat higher here than in previous examples, but excellent agreement is found for all ions even at velocities somewhat below the stopping maximum.
10
1
Krypton Argon
Lithium
0.001
0.01
0.1
1
10
100
E/A1 [MeV] Fig. 9. Same as Fig. 8 for carbon dioxide. Experimental data from [32] (Li), [28,29] (Ar) and [29–31] (Kr).
10
2
-dE/dx [MeVcm /mg]
Stopping in Mylar
Argon Chlorine Aluminium Sodium Oxygen Carbon Boron
1
0.1
1
10
100
E/A1 [MeV] Fig. 10. Stopping in mylar. Calculations based on BraggÕs rule. Experimental data from [34–41].
A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
27
50
20
2
-dE/dx [MeVcm /mg]
Stopping in polycarbonate
10
5
Silicon Magnesiu m Oxygen Carbon
2
1 0.1
1
10
E/A1 [MeV] 50
20
2
-dE/dx [MeVcm /mg]
Stopping in polycarbonate
10
Argon Aluminiu m Fluorin e Nitrogen Boron Lithiu m
5
2
1 0.1
1
10
E/A1 [MeV] Fig. 11. Stopping in polycarbonate. Calculations by BraggÕs rule. Experimental data from [41–44].
4. Conclusions In conclusion we find that • BraggÕs additivity rule is obeyed at projectile speeds well above the stopping maximum, within the accuracy of our model. • For ions heavier than boron and in charge equilibrium, departures from Bragg additivity are expected to be very small at all projectile speeds. • Conversely, for stripped ions, departures from Bragg additivity may increase with the atomic number of the ion.
• Significant departures from Bragg additivity are to be expected in the absence of projectile screening, at low projectile speed and, especially, for low-Z materials like LiF. We note that effective-charge scaling for heavyion stopping, proposed by Northcliffe [45] and employed in the SRIM [46] and MSTAR [47] codes, does not allow for any projectile dependence of departures from Bragg additivity. From a theoretical point of view we find it important to emphasize that the effects of projectile screening and shell correction are at least as
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A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
significant as the excitation spectrum via the Ivalue for the understanding of the overall departure from Bragg additivity. Acknowledgements This work was supported by grants from the Danish Natural Science Research Council (SNF). References [1] H.K. Reynolds, D.N.F. Dunbar, W.A. Wenzel, W. Whaling, Phys. Rev. 92 (1953) 742. [2] D.I. Thwaites, Rad. Res. 95 (1983) 495. [3] J.F. Ziegler, J.M. Manoyan, Nucl. Instr. and Meth. B 35 (1988) 215. [4] ICRU, Stopping Powers and Ranges for Protons and Alpha Particles, Vol. 49, ICRU Report, International Commission of Radiation Units and Measurements, Bethesda, MD, 1993. [5] J.R. Sabin, J. Oddershede, Nucl. Instr. and Meth. B 27 (1987) 280. [6] P.D. Fainstein, G.H. Olivera, R.D. Rivarola, Nucl. Instr. and Meth. B 107 (1996) 19. [7] P. Sigmund, A. Fettouhi, A. Schinner, Nucl. Instr. and Meth. B 209 (2003) 19. [8] D.I. Thwaites, Nucl. Instr. and Meth. B 69 (1992) 53. [9] A. Sharma, A. Fettouhi, A. Schinner, P. Sigmund, in preparation. [10] H. Bethe, Ann. Phys. 5 (1930) 324. [11] U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1. [12] P. Bauer, D. Semrad, Nucl. Instr. and Meth. B 182 (2001) 62. [13] N. Bohr, Philos. Mag. 25 (1913) 10. [14] P. Sigmund, A. Schinner, Eur. Phys. J. D 12 (2000) 425. [15] P. Sigmund, A. Schinner, Nucl. Instr. and Meth. B 195 (2002) 64. [16] F. Bloch, Ann. Phys. 16 (1933) 285. [17] J. Lindhard, A.H. Sørensen, Phys. Rev. A 53 (1996) 2443. [18] P. Sigmund, Phys. Rev. A 26 (1982) 2497. [19] P. Sigmund, L.G. Glazov, Eur. Phys. J. D 23 (2003) 211. [20] P. Sigmund, A. Schinner, Nucl. Instr. and Meth. B 193 (2002) 49. [21] C. Mitterschiffthaler, P. Bauer, Nucl. Instr. and Meth. B 48 (1990) 58. [22] P. Sigmund, A. Schinner, Eur. Phys. J. D 15 (2001) 165. [23] E.D. Palik, Electronic Handbook of Optical Constants of Solids, Version 1.0, SciVision, Academic Press, 2000.
[24] J. Berkowitz, Atomic and Molecular Photoabsorption. Absolute Total Cross Sections, Academic Press, San Diego, 2002. [25] E. Clementi, C. Roetti, Atomic Data Nucl. Data Tables 14 (1974) 177. [26] J.R. Sabin, J. Oddershede, Phys. Rev. A 26 (1982) 3209. [27] Y.A. Teplova, V.S. Nikolaev, I.S. Dimitriev, L.N. Fateeva, Sov. Phys. JETP 15 (1962) 31. [28] R. Bimbot, C. Cabot, D. Gardes, H. Gauvin, R. Hingmann, I. Orliange, L. De Reilhac, F. Hubert, Nucl. Instr. and Meth. B 44 (1989) 1. [29] J. Herault, R. Bimbot, H. Gauvin, B. Kubica, R. Anne, G. Bastin, F. Hubert, Nucl. Instr. and Meth. B 61 (1991) 156. [30] R. Bimbot, C. Cabot, D. Gardes, H. Gauvin, I. Orliange, L. DeReilhac, K. Subotic, F. Hubert, Nucl. Instr. and Meth. B 44 (1989) 19. [31] H. Geissel, Data measured at UNILAC 1977-81, Personal communication, 1998. [32] H.H. Andersen, F. Besenbacher, H. Knudsen, Nucl. Instr. and Meth. B 149 (1978) 121. [33] H. Paul, Stopping Power for Light Ions, 2003. Available from http://www.exphys.uni-linz.ac.at/stopping. [34] R. Bimbot, H. Gauvin, I. Orliange, R. Anne, G. Bastin, F. Hubert, Nucl. Instr. and Meth. B 17 (1986) 1. [35] H. Gauvin, R. Bimbot, J. Herault, R. Anne, G. Bastin, F. Hubert, Nucl. Instr. and Meth. B 28 (1987) 191. [36] E. Rauhala, J. R€ais€anen, Phys. Rev. B 37 (1988) 9249. [37] J. R€ais€anen, E. Rauhala, Phys. Rev. B 41 (1990) 3951. [38] C.W. Jin, X.T. Lu, X.J. Huang, Y.L. Ye, Z.H. Xia, H.T. Liu, D.X. Jiang, Phys. Rev. B 48 (1993) 6858. [39] J. R€ais€anen, E. Rauhala, Radiat. Eff. 128 (1994) 163. [40] A. Sharma, S. Kumar, S.K. Sharma, P.K. Diwan, N. Nath, V.K. Mittal, S. Ghosh, D.K. Avasthi, Nucl. Instr. and Meth. B 170 (2000) 323. [41] T. Alanko, J. Hyvonen, V. Kyllonen, P. Laitinen, A. Matilainen, J. R€ais€anen, A. Virtanen, J. Phys.: Cond. Matter 13 (2001) 10777. [42] E. Rauhala, J. R€ais€anen, Phys. Rev. B 42 (1990) 3877. [43] T. Alanko, W.H. Trzaska, V. Lyapin, J. R€ais€anen, G. Tiourine, A. Virtanen, Nucl. Instr. and Meth. B 190 (2002) 60. [44] A. Sharma, P.K. Diwan, S. Kumar, S.K. Sharma, V.K. Mittal, S.V.S.N. Rao, B. Sannakki, S. Ghosh, D.K. Avasthi, Nucl. Instr. and Meth. B 194 (2002) 7. [45] L.C. Northcliffe, Ann. Rev. Nucl. Sci. 13 (1963) 67. [46] J.F. Ziegler, The Stopping and Range of Ions in Matter, version SRIM-2003, 2003. Available from http://www. srim.org/. [47] H. Paul, A. Schinner, Nucl. Instr. and Meth. B 195 (2002) 166.
Stopping of swift ions in compounds A. Sharma a, A. Fettouhi b, A. Schinner c, P. Sigmund a
a
Physics Department, University of Southern Denmark, DK-5230 Odense M, Denmark b Physikalisches Institut, J.-Liebig Universit€at, D-35392 Giessen, Germany c Inst. f. Experimentalphysik, J.-Kepler-Universit€at, A-4040 Linz-Auhof, Austria
Abstract Stopping of light and heavy ions in compounds has been studied on the basis of the binary theory of electronic stopping with the focus on limitations of BraggÕs additivity rule. The case of LiF is analysed in detail because departures from Bragg additivity are expected to be particularly pronounced here, especially for penetrating antiprotons. A significant dependence on the atomic number and ionic charge of the projectile is predicted along with the more wellknown velocity dependence of departures from Bragg additivity. Such departures are found to be affected by shell correction and Barkas effect in addition to the more well-known dependence on the I-value. Comparisons with experimental results include H2 O vapor, CH4 , CO2 , mylar and polycarbonate. 2003 Elsevier B.V. All rights reserved. PACS: 34.50.Bw; 79.20.Nc Keywords: Stopping; Stopping power; Stopping force; Compound; BraggÕs rule; LiF
1. Introduction The stopping of charged particles in compounds is conventionally described by BraggÕs additivity rule which claims that the stopping cross section of an atom is independent of its chemical and physical environment. This implies that the stopping cross section of a molecule is equal to the sum of the stopping cross sections of its constituent atoms. Limitations of this rule have been known for at least half-a-century, not the least due to careful measurements with proton beams penetrating through molecular gases such as water vapor [1]. Experimental efforts in this area have focused on proton and helium beams and have been summarized in comprehensive reviews [2–4]. Typical results are similar to those found in [1], with a 10% departure from Bragg additivity around the
stopping maximum that decreases rapidly toward higher projectile speeds. Although a few theoretical studies have been made on specific systems [5,6], little systematic knowledge is available on departures from Bragg additivity as a function of the atomic number Z1 and the speed v of the projectile as well as its charge state. Recent work by three of us [7] indicates that dramatic departures from Bragg additivity may be found for specific materials like LiF where • the structure of the valence electrons differs markedly from those of the isolated atoms and • valence electrons make up a large fraction of all target electrons, in agreement with qualitative arguments by Thwaites [8]. We found particularly pronounced
0168-583X/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2003.12.009
20
A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
departures from additivity in antiproton stopping. This was ascribed to the absence of projectile screening. The present note serves to follow up on those preliminary findings, while detailed documentation is in preparation [9].
(3) Projectile screening due to electrons accompanying a penetrating ion influence the range of interaction and hence the contribution of valence electrons to stopping. 2.2. Binary stopping theory
2. Theory 2.1. Preliminary considerations Theoretical considerations have frequently been based on BetheÕs formula which, in its simplest form, predicts the stopping cross section S to be given by [10] S¼
4pZ12 Ze4 2mv2 ; ln mv2 I
ð1Þ
where Z1 is the atomic number of the projectile and Z the number of electrons in a target molecule. Material properties enter through the I-value, called mean excitation energy or mean ionization potential [11]. The logarithmic dependence ensures that departures from Bragg additivity originating in the I-value decrease in significance with increasing projectile speed. It is not obvious whether changes in I-value are the dominating cause of departures from Bragg additivity, as has been frequently implied [4,8,12]: (1) In the energy range around and below the stopping maximum, classical stopping theory is known to be more reliable than the Bethe scheme. In the Bohr theory [13] the logarithm in Eq. (1) is replaced by ln
Cmv3 ; Z1 e2 x
ð2Þ
where x ¼ I= h and C ¼ 1:1229. The factor Z1 in the denominator must produce a projectile dependence which has not been studied to our knowledge. (2) Shell corrections are sensitive to the orbital motion of the target electrons, and the Barkas effect – i.e. contributions from higher-order-Z1 terms – hinges on their state of binding. These two effects must be influenced by changes in electron structure due to molecular binding.
A convenient theoretical scheme to account for these phenomena is the binary stopping theory developed recently by two of us. Its basic formalism has been fully described in [14], and numerous applications to stopping in elemental materials have been documented in [15]. Therefore we only characterize the main features here, as well as those additions that are necessary for stopping in compounds. • The physical model is close to BohrÕs [13], assuming harmonically-bound target electrons, but • no expansion in Z1 is necessary for either close or distant interactions, • this is possible due to an accurate modelling of electron binding via a screened Coulomb potential, • a smooth transition to the quantum regime at higher speed is achieved by use of BlochÕs theory [16,17]. • shell corrections are incorporated by kinetic theory [18], • static projectile screening is allowed for in accordance with the ionic charge state (frozen or equilibrium), • projectile excitation can be included and • electron capture and loss can be allowed for in charge equilibrium [19]. The scheme has been applied successfully to stopping of ions in metals, semiconductors, insulators and gases [15,20]. 2.3. Input Binary stopping theory needs the following numerical input: • Resonance frequencies and oscillator strengths, • orbital velocity spectra and • binding energies
A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
for all shells and/or subshells of the target and the projectile. Finding suitable data for resonance frequencies and oscillator strengths has been described in [15,22]. The same procedure can be applied to compound materials to the extent that their optical properties have been tabulated. Here we use data from [23,24]. In practice we leave the excitation spectrum for all but the outermost shell of each participating atom unchanged but collect all remaining electrons in one common valence shell.
This implies that the only change in the excitation spectrum compared to calculations assuming Bragg additivity is the resonance frequency characterizing the outermost shell. Orbital velocity spectra were found until now by Fourier transforming Hartree–Fock wave functions tabulated in [25], as described in [15], similar in principle to the procedure proposed in [26] but different in practice. We recently found that this procedure gives rise to significantly overestimated shell corrections and, therefore,
H ions in water vapor
30
2
eVcm ]
21
S [10
-1 5
20
10
Binary, hydrogenic Binary, Hartree-Fock Fainstein et al
Thin lines: Bragg 0 0.01
0.1
1
E [MeV]
H ions in water vapor
2
eVcm ]
30
S [10
-1 5
20
10
0 0.01
Bragg: Reynolds et al Water: Reynolds et al Water: Mitterschiffthaler et al Bragg: Binary, hydrogenic Water: Binary, hydrogenic
0.1
1
E [MeV] Fig. 1. Stopping of protons in water vapor. Upper graph: Three theoretical predictions. Binary theory for modified hydrogenic and Hartree–Fock velocity spectra, and Fainstein et al. [6]. Thick lines: molecular; thin lines: Bragg. Lower graph: Experimental data from [1,21] compared with binary theory, hydrogenic velocity spectra.
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decided to apply modified hydrogenic wave functions instead. Here, the effective charge of each target shell has been fixed such as to match the ionization energy of that shell. As will be shown below, shell corrections contribute significantly to departures from Bragg additivity. As a logical consequence, velocity spectra calculated from molecular wave functions would have to be employed in quantitative estimates. This step has not yet been implemented. Unfortunately, the basic architecture of our PASS code [15] does not readily allow this feature to be incorporated. Orbital binding energies for atoms are available in numerous sources. In addition we use ionization potentials for the valence shell from [24].
3. Results 3.1. Water vapor Fig. 1 shows results for protons in water vapor. The upper graph demonstrates a significant influence of the adopted velocity profile both on the magnitude of the stopping force and the departure from Bragg additivity. We find reasonable agreement with quantal calculations [6] for hydrogenic velocity profiles. The latter predict a larger departure from Bragg additivity, which, however,
is not corroborated by experiment as is seen on the lower graph. With regard to our prediction, we note good agreement with the relative departure from Bragg additivity, while some discrepancy is observed in absolute stopping forces at projectile speeds well below the stopping maximum. This is a common feature also for noble-gas targets [15]. 3.2. Lithium fluoride Fig. 2 shows the predicted stopping force on antiprotons in LiF compared to the prediction of BraggÕs additivity rule. This graph differs substantially from a similar one published in [7]. Apart from an improvement in the underlying excitation spectrum, the main reason for the change is the use of hydrogenic wave functions that reduces substantially the departure from Bragg additivity. Fig. 3 shows calculations with and without shell correction. While the influence of the shell correction on the absolute magnitude of calculated stopping force is well known, it is seen that the departure from Bragg additivity is more pronounced in the shell-corrected curve, in particular around the stopping maximum. This is due to the fact that the jump of the 2s electron of a lithium atom to a 2p-like state in fluorine produces a change in orbital velocity in addition to changing the excitation frequency.
Stopping of antiprotons: Hydrogenic
20
2
eVcm ]
10
S [10
-1 5
5
2
1
0.5 0.01
Bragg Li F F Li
0.1
1
10
E [MeV] Fig. 2. Predicted stopping forces for LiF, metallic lithium and fluorine, including the prediction of BraggÕs rule.
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Antiprotons in Li F
2
5
S [10
eVcm ]
10
-15
20
2
Bragg: No shell correction LiF: No shell correction Bragg: Including shell correction LiF: Including shell correction
1 0.01
0.1
1
10
E [MeV] Fig. 3. Stopping of antiprotons in LiF. Influence of shell correction.
100
Relative departure [pct]
Protons and antiprotons in LiF
10
1
0.01
Antiproton Average Proton
0.1
1
10
E [MeV] Fig. 4. Relative departure from Bragg additivity for protons and antiprotons in LiF as well as their average.
Fig. 4 compares relative departures from Bragg additivity for protons and antiprotons in LiF. It is seen that the departure is greater for antiprotons than for protons. In other words, the Barkas effect – the difference between proton and antiproton stopping – opposes the effect of the shell correction for protons, in agreement with a general tendency regarding the stopping force in elemental materials.
Fig. 5 shows a comparison of stopping forces in LiF on several ions with calculations based on BraggÕs rule. It is seen that the departure from additivity decreases rapidly from antiprotons to heavier ions and is essentially vanishing from carbon on. The reason for this behavior emerges from Fig. 6 which demonstrates a pronounced dependence on ion charge: Bragg additivity is governed by excitation of valence electrons which
24
A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28 Ions in charge equilibrium in LiF Argon Neon Carbon Helium Hydrogen Antiproton
2
eVcm ]
1000
S [10
-1 5
100
10
Thin lines: Bragg Thick lines: LiF 1 0.01
0.1
1
10
E/A1 [MeV] Fig. 5. Stopping of ions in equilibrium charge state in LiF. Comparison with calculations based on BraggÕs rule.
Argon in LiF
2000
2
S [10 eVcm ]
1000
-15
500
18 + 12 + 6+ 0+
200
100
50 0.01
0.1
1
10
E/A1 [MeV] Fig. 6. Stopping of argon ions in LiF as a function of the ionic charge. Thick lines: molecular; thin lines: Bragg.
occurs in distant interactions. Projectile screening by accompanying electrons screens the potential and hence reduces the significance of valence excitations. Indeed, BraggÕs rule appears to hold very accurately for a neutral ion. The same feature was pointed out previously in connection with Z2 structure in heavy-ion stopping [7]. The tendency expressed by Fig. 5 goes strictly against the effect discussed in connection with the Z1 -dependence of the Bohr logarithm, Eq. (2). That effect, however, can be demonstrated if the
same plot is made for bare ions instead of ions in charge equilibrium. A comparison between the two cases at a fixed beam velocity is shown in Fig. 7 which demonstrates a clear difference. 3.3. Carbon compounds Carbon compounds have received considerable interest in the stopping literature [2,3]. The present survey presents our first explorative findings and is by no means meant to be exhaustive. It focuses on
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25
5
Stopping in LiF 1 MeV/u Relative departure [pct]
4
3
Bare ion Mean equilibrium charge
2
1
0 1
10
100
Z1 Fig. 7. Relative departure from Bragg additivity at 1 MeV/u for bare ions and ions in charge equilibrium.
100
2
-dE/dx [MeVcm /mg]
Stopping in methane gas
10
Krypton Argon Lithiun
1 0.01
0. 1
1
10
100
E/A1 [MeV] Fig. 8. Stopping of Li, Ar and Kr ions in methane gas. Solid lines: molecular; broken lines: BraggÕs rule. Experimental results from [27] (Li), [28,29] (Ar) and [29–31] (Kr).
ions heavier than helium because of the nearcomplete absence of systematic information – experimental or theoretical – for such systems. Fig. 8 shows predictions for stopping in methane, compared with experimental results. 1 It
1 Experimental data shown in Figs. 8–11 have been extracted from PaulÕs data base [33].
is important to note that our calculations assuming BraggÕs rule utilize the stopping cross section of amorphous carbon. With this cross section, predicted departures from Bragg additivity become rather small for Li and negligible for argon and krypton ions. While good agreement with experimental results is found for heavier ions, discrepancies are found for lithium which are not unlike those in elemental materials [15].
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Similar results are shown in Fig. 9 for carbon dioxide. The remaining results refer to stopping in solid films. Fig. 10 shows predicted stopping forces in mylar. Since we do not have access to tabulated optical properties of this material, calculations have been performed on the basis of BraggÕs rule. Figs. 8 and 9 indicate that the magnitude of the expected error will not significantly affect our conclusions from this comparison. The agreement
Stopping in CO2 gas
2
-dE/dx [MeVcm /mg]
100
with experiment is seen to be quite satisfactory, the only exception being seen with the lightest ions at low energies. Finally, Fig. 11 shows a similar comparison for polycarbonate for a particularly large set of penetrating ions. Ion velocities are somewhat higher here than in previous examples, but excellent agreement is found for all ions even at velocities somewhat below the stopping maximum.
10
1
Krypton Argon
Lithium
0.001
0.01
0.1
1
10
100
E/A1 [MeV] Fig. 9. Same as Fig. 8 for carbon dioxide. Experimental data from [32] (Li), [28,29] (Ar) and [29–31] (Kr).
10
2
-dE/dx [MeVcm /mg]
Stopping in Mylar
Argon Chlorine Aluminium Sodium Oxygen Carbon Boron
1
0.1
1
10
100
E/A1 [MeV] Fig. 10. Stopping in mylar. Calculations based on BraggÕs rule. Experimental data from [34–41].
A. Sharma et al. / Nucl. Instr. and Meth. in Phys. Res. B 218 (2004) 19–28
27
50
20
2
-dE/dx [MeVcm /mg]
Stopping in polycarbonate
10
5
Silicon Magnesiu m Oxygen Carbon
2
1 0.1
1
10
E/A1 [MeV] 50
20
2
-dE/dx [MeVcm /mg]
Stopping in polycarbonate
10
Argon Aluminiu m Fluorin e Nitrogen Boron Lithiu m
5
2
1 0.1
1
10
E/A1 [MeV] Fig. 11. Stopping in polycarbonate. Calculations by BraggÕs rule. Experimental data from [41–44].
4. Conclusions In conclusion we find that • BraggÕs additivity rule is obeyed at projectile speeds well above the stopping maximum, within the accuracy of our model. • For ions heavier than boron and in charge equilibrium, departures from Bragg additivity are expected to be very small at all projectile speeds. • Conversely, for stripped ions, departures from Bragg additivity may increase with the atomic number of the ion.
• Significant departures from Bragg additivity are to be expected in the absence of projectile screening, at low projectile speed and, especially, for low-Z materials like LiF. We note that effective-charge scaling for heavyion stopping, proposed by Northcliffe [45] and employed in the SRIM [46] and MSTAR [47] codes, does not allow for any projectile dependence of departures from Bragg additivity. From a theoretical point of view we find it important to emphasize that the effects of projectile screening and shell correction are at least as
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significant as the excitation spectrum via the Ivalue for the understanding of the overall departure from Bragg additivity. Acknowledgements This work was supported by grants from the Danish Natural Science Research Council (SNF). References [1] H.K. Reynolds, D.N.F. Dunbar, W.A. Wenzel, W. Whaling, Phys. Rev. 92 (1953) 742. [2] D.I. Thwaites, Rad. Res. 95 (1983) 495. [3] J.F. Ziegler, J.M. Manoyan, Nucl. Instr. and Meth. B 35 (1988) 215. [4] ICRU, Stopping Powers and Ranges for Protons and Alpha Particles, Vol. 49, ICRU Report, International Commission of Radiation Units and Measurements, Bethesda, MD, 1993. [5] J.R. Sabin, J. Oddershede, Nucl. Instr. and Meth. B 27 (1987) 280. [6] P.D. Fainstein, G.H. Olivera, R.D. Rivarola, Nucl. Instr. and Meth. B 107 (1996) 19. [7] P. Sigmund, A. Fettouhi, A. Schinner, Nucl. Instr. and Meth. B 209 (2003) 19. [8] D.I. Thwaites, Nucl. Instr. and Meth. B 69 (1992) 53. [9] A. Sharma, A. Fettouhi, A. Schinner, P. Sigmund, in preparation. [10] H. Bethe, Ann. Phys. 5 (1930) 324. [11] U. Fano, Ann. Rev. Nucl. Sci. 13 (1963) 1. [12] P. Bauer, D. Semrad, Nucl. Instr. and Meth. B 182 (2001) 62. [13] N. Bohr, Philos. Mag. 25 (1913) 10. [14] P. Sigmund, A. Schinner, Eur. Phys. J. D 12 (2000) 425. [15] P. Sigmund, A. Schinner, Nucl. Instr. and Meth. B 195 (2002) 64. [16] F. Bloch, Ann. Phys. 16 (1933) 285. [17] J. Lindhard, A.H. Sørensen, Phys. Rev. A 53 (1996) 2443. [18] P. Sigmund, Phys. Rev. A 26 (1982) 2497. [19] P. Sigmund, L.G. Glazov, Eur. Phys. J. D 23 (2003) 211. [20] P. Sigmund, A. Schinner, Nucl. Instr. and Meth. B 193 (2002) 49. [21] C. Mitterschiffthaler, P. Bauer, Nucl. Instr. and Meth. B 48 (1990) 58. [22] P. Sigmund, A. Schinner, Eur. Phys. J. D 15 (2001) 165. [23] E.D. Palik, Electronic Handbook of Optical Constants of Solids, Version 1.0, SciVision, Academic Press, 2000.
[24] J. Berkowitz, Atomic and Molecular Photoabsorption. Absolute Total Cross Sections, Academic Press, San Diego, 2002. [25] E. Clementi, C. Roetti, Atomic Data Nucl. Data Tables 14 (1974) 177. [26] J.R. Sabin, J. Oddershede, Phys. Rev. A 26 (1982) 3209. [27] Y.A. Teplova, V.S. Nikolaev, I.S. Dimitriev, L.N. Fateeva, Sov. Phys. JETP 15 (1962) 31. [28] R. Bimbot, C. Cabot, D. Gardes, H. Gauvin, R. Hingmann, I. Orliange, L. De Reilhac, F. Hubert, Nucl. Instr. and Meth. B 44 (1989) 1. [29] J. Herault, R. Bimbot, H. Gauvin, B. Kubica, R. Anne, G. Bastin, F. Hubert, Nucl. Instr. and Meth. B 61 (1991) 156. [30] R. Bimbot, C. Cabot, D. Gardes, H. Gauvin, I. Orliange, L. DeReilhac, K. Subotic, F. Hubert, Nucl. Instr. and Meth. B 44 (1989) 19. [31] H. Geissel, Data measured at UNILAC 1977-81, Personal communication, 1998. [32] H.H. Andersen, F. Besenbacher, H. Knudsen, Nucl. Instr. and Meth. B 149 (1978) 121. [33] H. Paul, Stopping Power for Light Ions, 2003. Available from http://www.exphys.uni-linz.ac.at/stopping. [34] R. Bimbot, H. Gauvin, I. Orliange, R. Anne, G. Bastin, F. Hubert, Nucl. Instr. and Meth. B 17 (1986) 1. [35] H. Gauvin, R. Bimbot, J. Herault, R. Anne, G. Bastin, F. Hubert, Nucl. Instr. and Meth. B 28 (1987) 191. [36] E. Rauhala, J. R€ais€anen, Phys. Rev. B 37 (1988) 9249. [37] J. R€ais€anen, E. Rauhala, Phys. Rev. B 41 (1990) 3951. [38] C.W. Jin, X.T. Lu, X.J. Huang, Y.L. Ye, Z.H. Xia, H.T. Liu, D.X. Jiang, Phys. Rev. B 48 (1993) 6858. [39] J. R€ais€anen, E. Rauhala, Radiat. Eff. 128 (1994) 163. [40] A. Sharma, S. Kumar, S.K. Sharma, P.K. Diwan, N. Nath, V.K. Mittal, S. Ghosh, D.K. Avasthi, Nucl. Instr. and Meth. B 170 (2000) 323. [41] T. Alanko, J. Hyvonen, V. Kyllonen, P. Laitinen, A. Matilainen, J. R€ais€anen, A. Virtanen, J. Phys.: Cond. Matter 13 (2001) 10777. [42] E. Rauhala, J. R€ais€anen, Phys. Rev. B 42 (1990) 3877. [43] T. Alanko, W.H. Trzaska, V. Lyapin, J. R€ais€anen, G. Tiourine, A. Virtanen, Nucl. Instr. and Meth. B 190 (2002) 60. [44] A. Sharma, P.K. Diwan, S. Kumar, S.K. Sharma, V.K. Mittal, S.V.S.N. Rao, B. Sannakki, S. Ghosh, D.K. Avasthi, Nucl. Instr. and Meth. B 194 (2002) 7. [45] L.C. Northcliffe, Ann. Rev. Nucl. Sci. 13 (1963) 67. [46] J.F. Ziegler, The Stopping and Range of Ions in Matter, version SRIM-2003, 2003. Available from http://www. srim.org/. [47] H. Paul, A. Schinner, Nucl. Instr. and Meth. B 195 (2002) 166.