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Simple stopping power formula for low and intermediate energy electrons
ARTICLE IN PRESS
Radiation Physics and Chemistry 72 (2005) 7–12
Simple stopping power formula for low and intermediate energy electrons Hasan Gum . u. s* Department of Physics, Faculty of Arts and Sciences, Ondokuz Mayis University, Samsun 55139, Turkey Received 29 November 2003; received in revised form 26 January 2004; accepted 4 March 2004
Abstract A simple stopping power formula, modified from that of Rohrlich and Carlson is presented. In this study analytical expressions for effective charge and effective mean excitation energies of target atoms in the modified Rohrlich and Carlson stopping power formula are used, while for effective charge of incoming electrons, Sugiyama’s semiempirical formula is used from Peterson and Green. The Lenz–Jensen statistical atomic density model has been used for calculations of effective charge and effective mean excitation energies. The calculated results of stopping power for electrons in the energy range from a few tens of electron volts to 10 MeV are found to be in good agreement to within 10% with the experimental data and a number of other calculations. r 2004 Elsevier Ltd. All rights reserved. Keywords: Electron stopping power; Effective charge; Effective mean excitation energy; Bragg rule; Bohr criterion
1. Introduction Stopping powers of matter for electrons are important in a wide variety of applications involving energy deposition. In radiation physics, chemistry, biology and medicine, it is often important to have simple but accurate information about the stopping power of various media for energetic electrons. Electron stopping power has been the subject of many studies (Pimblott and Siebbeles, 2002; Cengiz, 2002; Dingfelder et al., 1998; Sugiyama, 1985; La Verne and Mozumder, 1983; Berger and Seltzer, 1982; Seltzer and Berger, 1982; Ashley et al., 1979, 1982; Ashley, 1980; Tung et al., 1979, 1981; Spencer and Pal, 1978; Gupta et al., 1975; Dalgarno, 1960; Rohrlich and Carlson, 1954). While electron stopping powers at energies above 10 keV are theoretically well described and can be found in tables given in Pages et al. (1972), and ICRU 37 Report (1984), no practical calculation model of the stopping power exists for incoming electrons energies below 10 keV. *Corresponding author. Tel.: +90-362-4576020/5061; fax: +90-362-4576081. E-mail address: [email protected] (H. Gum . u. s-).
Sugiyama obtained stopping power values for intermediate electrons energies using a modified Rohrlich and Carlson model (Sugiyama, 1985; Rohrlich and Carlson, 1954). Sugiyama (1985) applied the Bohr stripping criterion (i.e v1 > bvF ðrB Þ ¼ ðb_=mÞ½3prðrB Þ1=3 ; vF ðrB Þ is the Thomas–Fermi velocity between r and r þ dr; b is the proportional constant of order of 1.26) to target atoms for effective charge and effective mean excitation energy. In applying this, he used the Thomas– Fermi atomic model and also evaluated the integrals numerically related to effective charge and effective mean excitation energy (Sugiyama, 1985). This method produced some uncertainties since in the model the upper limit of the integral has to be determined. The reason for this deviation originated from the effective charge and effective mean excitation energy calculation methods in which the upper limit of the integral has to be determined. Therefore, uncertainties arise both due to nature of the Thomas–Fermi atom and also to numerical calculations of integrals of Z and I (Sugiyama, 1985, Eqs. (4) and (5)). The purpose of this paper, is to obtain a stopping power formula for incident electrons, valid for the low and intermediate energy region (o10 keV). The stopping
0969-806X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2004.03.006
ARTICLE IN PRESS H. Gum . u. s / Radiation Physics and Chemistry 72 (2005) 7–12
8
power formula was based on a modification by Sugiyama (1985) of the Rohrlich and Carlson formulation (Rohrlich and Carlson, 1954). Analytical expres. sions obtained in previous work (Gum . u. s- and Koksal, 2002) on effective charge and effective mean excitation energies of target atoms were used, and also a semiempirical effective charge expression for incoming electrons, adjusted by Sugiyama (1985) to fit the Peterson and Green method (Peterson and Green, 1968). The calculating procedure thus described, was applied to incoming electrons on H2, N2, O2, H2O and CO2 targets which are tissue equivalent gas mixtures and are crucially important in microdosimetry, radiobiology and radiochemistry. Results from these were compared with the data and other theoretical results for wide range of incident electron energies.
and Z2 eZb 3 cn I ¼ cnðg0 Þ þ ð Þ g1 ðZb Þ g2 ðZb Þ g3 ðZb Þ 2 Z2 2 ð3Þ where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pe2 3:675 ¼ gZ2 185:12 eV ; g0 ¼ g_Z2 ma30
ð4Þ
g1 ðZb Þ ¼ ð0:002647Z5b þ 0:043207Z4b þ 0:285923Z3b þ Z2b 1 þ 0:2647Zb ð5Þ þ 2Zb þ 2Þln Zb g2 ðZb Þ ¼ 0:0013237Z6b þ 0:0229271Z5b þ 0:171183Z4b þ 0:770847Z3b þ 2:5Z2b þ 5:91Zb þ 8:140677 ð6Þ and
2. Method The modified mass collision stopping power formula for incoming electrons can be written as (Sugiyama, 1985; Rohrlich and Carlson, 1954): 2 dE dE 4pe4 z N0 E ¼ Þ F ðtÞ=2 ; Z lnð Sx ¼ ¼ dx r dx I mv21 A 2 F ðtÞ ¼ 1 b2 þ ½ðt2 =8Þ ð2t þ 1Þln 2=ð1 þ t2 Þ;
ð1Þ
where m is the electron mass, v1 is the incident electron velocity, E is the electron energy, N ¼ N0 =A is the density of target atoms and denotes an effective quantity, i.e. the effective charge of incident electrons z, the effective number of target electrons Z2 ; A is the atomic weight of the target element, N0 Avogadro’s number, and b is the ratio of v1 =c; with c the velocity of light and I the effective mean excitation energy, r the charge density of the target. In Eq. (1) the factor in front 2 of the parenthesis can be written as ðk=Ab2 Þz Z2 ; since 2 4 2 k ¼ 4pe N0 =mc ¼ 0:307075 MeV cm (Sugiyama, 1981; . Gum . u. s- and Koksal, 2002). Z2 and I can be obtained from Bohr’s stripping criterion for the effective charge of heavy ions and target atoms, and effective mean excitation energies given by Sugiyama (Sugiyama, 1981, 1985) respectively, Z N Z ¼ 4pr2 nðrÞ dr; rb
cn I ¼
1 Z 2
Z
N
cn g_op ðrÞ 4pr2 nðrÞ r:
rb
Z2 and I can be obtained analytically from these expressions by using Lenz atomic model (Lenz, 1932; . Jensen, 1932; Gum . u. s- and Koksal, 2002), as follows: eZb Þð0:002647Z5b þ 0:043207Z4b þ 0:285923Z3b Z2 ¼ Z2 ð 2 þ Z2b þ 2Zb þ 2Þ ð2Þ
g3 ðZb Þ ¼ 3eZb Eið1; Zb Þ 4:2eZb Eið1; 3:7735849 þ Zb Þ: ð7Þ Here, Ei is the exponential integral (Abramowitz and Stegun, 1970). Zb is determined from the Bohr stripping b_ ½3p2 nðZb Þ1=3 ; with vLJ criterion and v1 ZbvLJ ðrb Þ ¼ m the orbital velocity of the projectile or target atoms. The latter can be rewritten as, vF ðZb Þ ¼ 0:034838cZ 2=3 ðeZb Þ1=3 ð1 þ 0:2647Zb Þ=Zb
ð8Þ
with b a proportional constant of value about 1.26 and gE21=2 as used by Lindhard and Scharff (1953). In this study, g values for each of the target atoms were adjusted in a way that normalized effective mean excitation ionization energy, while I, refers to empirically determined I values (Andersen and Ziegler, 1977; Ahlen, 1980; ICRU 37, 1984) at sufficiently high incident energies (EcI; i.e, large Zb). In addition, the Hartree–Fock–Slater orbital velocity (Herman and Skillman, 1963) is used to determine the Bohr criterion, giving better results than that of the Thomas–Fermi velocity vF ðrÞ: The present procedure is to multiply vF ðrÞ P P 1=2 ; where by a correction factor G ¼ eHF = eTF eHFS and eTF are the orbital electron energy of each shell in the Hartree–Fock–Slater and Thomas–Fermi approximation, respectively. The mean excitation ionization energy values and basic data for the calculations are given in Table 1. Calculated values of Z and I are shown in Fig. 1 as a function of incident electron energy. In this study, the semiempirical effective charge of incident electrons z is used with z being given by z ¼ 1 expð2200b1:78 Þ ð9Þ where b is again the ratio of v1 =c; with c the velocity of light (Sugiyama, 1985).
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Table 1 The mean excitation ionization energy values and basic data for calculation Z
A
I (eV)
M (g/mol)
NA =M 1021
g
D
1 6 7 8 18 36 47
1 12 14 16 40 84 107
19.2 81 88 98 182 358 478
1.0079 12.001 14.0067 15.9994 39.09480 83.80 107.868
597.7 50.16 43.02 37.66 15.08 7.190 5.585
2.529 1.59 1.656 1.614 1.332 1.310 1.340
1.09 1.165 1.13 1.106 1.06 1.02 1.02
100
1000 Z=47 36
Z=47 36 18
18 8
10
1 z*
1
1
10
z* 0.1
10
6
I* (eV)
* *
z, Z
1
8
100
6
100
1000
10000
100000
1000000
10
100
Electron Energy (eV)
Fig. 1. The effective atomic number of electrons Z of various target atoms as a function of incident electron energy.
3. Calculated results and discussion Figs. 1 and 2 show the effective number of electrons Z and effective mean excitation energy I as a function of incident electron energy. For high incident energies (i.e. for Zb -0), Z -Z; z -0; and I -I; while for low incident energies (i.e. for high Zb values), Z -0 and z -0: The values of the semiempirical effective charge of electrons z is affected for energies lower than 200 eV (Sugiyama, 1985). z has value of 0.97713 for 200 eV, of 0.86987 for 100 eV, and of 0.23108 for 10 eV. That has also some contribution in the improvement of the results which has significant contribution in the improvement of the results below 200 eV energy regions. In order to calculate the electronic stopping power of each of the compounds discussed herein, first the effective charge of the incident electron z was obtained, primarily using Eq. (9), and the effective charges of the target atoms Z2 and Z3 in the compound were then obtained using Eq. (2). Eq. (3) was used to calculate the effective mean excitation energy of the target atoms I2 and I3 : These values were later used in Eq. (1) to obtain the mass collision stoppin power,Sx. The mass collision
1000 10000 100000 Electron energy (eV)
1000000
Fig. 2. The effective mean excitation energy I of various target atoms as a function of incident electron energy.
stopping powers of the identified molecularly constituted compounds for incoming electrons were calculated from Bragg’s addition rule (Bragg and Kleeman, 1905) which can be written as Sx ¼ nSxðAÞ þ mSxðBÞ
ð10Þ
where n and m is the atomic fraction of element (A) and (B) respectively. The calculated results in this study are given in Figs. 3–6. Fig. 3 shows the mass stopping power values for electrons in liquid water. The results obtained in this study are in good agreement with the semiempirical formula of Kutcher and Green (1976), the recommendations of ICRU 16 (1970) for the stopping power of liquid water, the predictions of La Verne and Pimblott (1995) and of Paretzke (1988) and the NIST database (ESTAR, NIST database, 2003). The NIST database data is based on the calculations of ICRU 37 (1984) derived from Bethe theory. The mass stopping powers obtained by using the formalism described in this paper are in good agreement to within 25% with data predicted using Paretzke’s formulation for water vapour (Paretzke, 1988), 8% with recommendations of ICRU 16 (1970) except for data obtained at 20, 200 and 300 eV, 2% with data of predictions of La Verne and Pimblott (1995),
ARTICLE IN PRESS H. Gum . u. s / Radiation Physics and Chemistry 72 (2005) 7–12
10 1000
Hydrogen and Nitrogen 1000
10
1
Peterson 68 This study ICRU 37 ESTAR
H2
2
100
Se / (MeV.cm /g)
Kutcher 76 Verne 95 ICRU 16 Paretzke 88 ESTAR This study ICRU 37
2
Se / (MeV.cm /g)
Liquid Water
100
N2
10
1 1
10
100
1000
10000
100000 1000000
10
1E7
100
1000
10000
100000
Fig. 3. Comparison of mass stopping power Sx for electrons, incident on liquid water. Solid curves, —, are the present calculations. ’, semiempirical formula by Kutcher and Green (1976); o results of La Verne et al. (1995); results of ICRU 16 Report (1970); &, from Paretzke (1988); ., data from ICRU 37 (1984) and r, values obtained from, ESTAR package.
1E7
Fig. 5. Mass stopping power Sx, plotted against incident electron energy on H2 and N2. — present calculations; , semiempirical formula of Peterson and Green (1968);., data from ICRU 37 report (1984) and D, values obtained from the ESTAR package.
1000
Oxygen Carbon Dioxide 2 -1
Se / (MeV.cm g )
100
Peterson 68 This study Williart 03 ICRU 37 ESTAR
100
Waibel 91a Waibel 91b Waibel 91c Berger 82 This study ICRU 37 ESTAR
2
Se / (MeV cm /g)
1000000
Electron Energy (eV)
Electron Energy (eV)
10
10
1 1 1
1
10
100
1000
10000
100000 1000000
1E7
Electron Energy (eV)
Fig. 4. Mass stopping power Sx for electrons, in CO2 as a function of the electron energy. —, present present calculations; +, Waibel and Grosswendt (1991) data derived from ionization measurements with corrections for loss of ionization from secondary electron; Waibel and Grosswendt’s results from inelastic scattering cross sections (Waibel and Grosswendt, 1991) and , restricted energy transfer (Waibel and Grosswendt, 1991); &, theoretical data of Berger and Seltzer (1982); m, data from ICRU 37 report (1984) and r, values obtained from, ESTAR package.
25% with the semiempirical formula of Kutcher and Green (1976) for incident electron energies above 30 eV. The agreement with the NIST database (ESTAR, 2003) and ICRU 37 (1984) are better than 0.8% for the incident electron energies above 10 keV. Fig. 4 shows the stopping power values for electrons in carbon dioxide as a function of the electron energy.
10
100
1000
10000
100000 1000000
1E7
Electron Energy (eV)
Fig. 6. Mass stopping power Sx, plotted against incident electron energy on O2. —, present calculations; , semiempirical formula of Peterson and Green (1968); ., data from ICRU 37 report (1984) and D, values obtained from, the ESTAR package and experimental points o, from Williart et al. (2003).
The experimental data are taken from Waibel and Grosswendt (Waibel and Grosswendt, 1991). In this figure, the data identified as ‘Waibel 91 a’ shows the derived ionization measurement for the correction of loss of ionization from secondary electrons, while the data ‘Waibel 91 b’ gives the restricted energy transfer without correction; ‘Waibel 91 c’ illustrates results from inelastic scattering cross sections (Waibel and Grosswendt, 1991). The solid curve indicate the present calculational results. Agreement with the data ‘Waibel 91 a’ and ‘Waibel 91 b’ is good (better than 10%) around stopping power maxima. Various stopping power data for carbon dioxide for energetic
ARTICLE IN PRESS H. Gum . u. s / Radiation Physics and Chemistry 72 (2005) 7–12
electrons (Berger and Seltzer, 1982; ESTAR, NIST database, 2003; ICRU 37, 1984) are also shown in Fig. 4. The stopping powers obtained using the formalism described here are in good agreement with experimental data and results of ICRU 37 (1984) and ESTAR in high energy regions and with the theoretical data of Berger and Seltzer (1982) at intermediate energies. Fig. 5 indicates the stopping power values for hydrogen and nitrogen for incoming electrons. The calculated mass stopping power values are compared with the semiempirical formula of Peterson and Green (1968) and the values of the NIST database (ESTAR, NIST database, 2003) and ICRU 37 (1984). Agreement with the semiempirical results are satisfactory, within 8%, except for data obtained at lower energies for H2 as has similarly been seen by Sugiyama (1985) in his calculated results. It would be expected that if higher order z2 and z3 corrections were included in the stopping power, then the agreement with experiment would be better. Fig. 6 indicates the stopping power values of oxygen for incident electrons. The calculated mass stopping power values are compared with the semiempirical formula of Peterson and Green (1968) and the values of NIST database (ESTAR, NIST database, 2003) and ICRU 37 (1984) and with experimental data taken from Williart et al. (2003). Agreement with the semiempirical results and other calculation results together with the experimental data are satisfactory, i.e to within 10% with semiempirical formula Peterson and Green (1968) except for data obtained at 10 eV and Williart et al. (2003), 1% with the values of NIST database (ESTAR, 2003) and ICRU 37 (1984).
4. Concluding remarks In this work, collision stopping power values for electrons on several target materials have been evaluated by the modified version of the Rohrlich and Carlson formula. The effective number of electrons and the effective mean excitation energy are introduced, determined from analytical expressions using the Bohr stripping criterion for heavy ions. The calculational results of stopping powers for the targets H2, N2, O2 and H2O are found to be in good agreement with semiempirical and a number of other theoretical predictions, together with experimental results. The collision stopping power depends sensitively on the charge-state or effective charge of projectile electrons and target atoms. The simple effective charge expressions is remarkably successful, and is highly useful for practical computation. The present electron stopping power calculation method should be especially useful for biomedical dosimetry.
11
References Abramowitz, M., Stegun, I.A., 1970. Handbook of Mathematical Functions. National Bureau of Standards, Appl. Math. Ser. No. 55. US GPO, Washington, DC. Andersen, H.H., Ziegler, V.P., 1977. Hydrogen: Stopping Power and Ranges in All Elements. The Stopping and Ranges of Ions in Matter, vol 3. Pergamon Press, New York. Ahlen, S.P., 1980. Theoretical and experimental aspects of the energy loss of relativistic heavily ionizing particles. Rev. Mod. Phys. 52, 121–172. Ashley, J.C., 1980. Inelastic interactions of low-energy electrons with organic solids: simple formulae for mean free paths and stopping powers. IEEE Trans. Nucl. Sci. NS-27, 1454–1458. Ashley, J.C., Tung, C.J., Ritchie, R.H., 1979. Electron inelastic mean free paths and energy losses in solids, 1. Aluminum metal. Surf. Sci. 81, 409–425. Ashley, J.C, Tung, C.J., Ritchie, R.H., 1982. Stopping power of liquid water for low energy electrons. Radiol. Res. 89, 25–31. Berger, M.J., Seltzer, S.M., 1982. Stopping powers and ranges of electrons and positrons. National Bureau of Standarts Report, NBSIR 82–2550 A. Bragg, W.H., Kleeman, R., 1905. On the alpha particles of radium and their loss of range in passing through various atoms and molecules. Philos. Mag. 10, 318–340. Cengiz, A., 2002. Approximation inelastic scattering cross sections of electrons. Radiat. Phys. Chem. 65, 1–18. Dalgarno, A., 1960. The stopping powers of atoms. Proc. Phys. Soc. (London) 76, 422–424. Dingfelder, M., Hantke, D., Inokuti, M., Paretzke, H.G., 1998. Electron inelastic-scattering cross sections in liquid water. Radiat. Phys. Chem. 53, 1–18. ESTAR: 2003. Stopping Power and Range Tables for Electron http://physcs.nist.gov/PhysRefData/Star/Text/ ESTAR.html. Gupta, P., Jhanwar, B.I., Khare, S.P., 1975. Stopping power of atmospheric gases for electrons. Physica B79, 311–321. . Gum . u. s-, H., Koksal, F., 2002. Effective stopping charges and stopping power calculations for heavy ions. Radiat. Eff. Def. Solids 157, 445–458. Herman, F., Skillman, S., 1963. Atomic Structure Calculations. Prentice-Hall, Englewood Cliffs, NJ. ICRU, Report No. 16, 1970. Linear Energy Transfer. International Commission on Radiation Units and Measurements, Washington, DC. ICRU, Report No. 37, 1984. Stopping powers for electrons and positrons. International Commission on Radiation Units and Measurements, Bethesda, MD. . Jensen, H., 1932. Uber die Anwendbarkeit der Statistischen Methode auf Ionengitter. Zeitung. F. Phys 77, 713. Kutcher, G.J., Green, A.E.S., 1976. A model for energy deposition in liquid water. Radiat. Res. 67, 408–425. La Verne, J.A., Mozumder, A., 1983. Penetration of low-energy electrons in water. Radiat. Res. 96, 219–234. La Verne, J.A., Pimblott, S.M., 1995. Electron energy loss distributions in solid, dry dna. Radiat. Res. 14, 208–215. Lenz, W., 1932. Die ladungsverteilung in Ionen und die Gitterkonstante des Rubidiumbromids nach der Statistischen Methode. Zeitung. F. Physik 77, 722.
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Lindhard, J., Scharff, M., 1953. Energy loss in matter by fast particles of low charge. Kgl. Danske Vidensk. Selsk. Mat.Fys. Medd. 27 (15), 1–31. Paretzke, H.G., 1988. Simulation von elektronenspuren im energiebereich 0.01–10 keV in wasserdampf. GSF-Bericht, 24/88. GSF, Neuherberg. Pages, L., Bertel, E., Joffre, H., Sklavenitis, L., 1972. Energy loss, range and bremsstrahlung yield for 10 to 10 MeV electrons in various elements and chemical compounds. At. Data 4, 1–127. Peterson, L.R., Green, A.E.S., 1968. The relation between ionization yields, cross sections and loss functions. J. Phys. B 1, 1131–1140. Pimblott, S.M., Siebbeles, L.D.A., 2002. Energy loss by nonrelativistic electrons and positrons in liquid water. Nucl. Instr. Methods B 194, 237–250. Rohrlich, F., Carlson, B.C., 1954. Positron–electron differences in energy loss and multiple scattering. Phys Rev 93, 38–44. Seltzer, S.M., Berger, M.J., 1982. Evaluation of the collision stopping power of elements and compounds for electrons and positrons. Int. J. Appl. Radiat. Isot 33, 1189–1218.
Spencer, L.V., Pal, R., 1978. Numerical solution of the electron slowing down equation for molecular hydrogen. National Bureau of Standarts Report NBSIR 78–1523. Sugiyama, H., 1981. Electronic stopping power formula for intermediate energies. Radiat. Eff. 56, 205–212. Sugiyama, H., 1985. Stopping power formula for intermediate energy electrons. Phys. Med. Biol. 30, 4331–4335. Tung, C.J., Ashley, J.C., Ritchie, R.H., 1979. Electron inelastic mean free paths and energy losses in solids II. Surf. Sci. 81, 427–439. Tung, C.J., Chau, T.L., Kwei, C.M., 1981. Stopping power of diamond for low energy electrons. Radiat. Eff. 59, 7–11. Waibel, E., Grosswendt, B., 1991. Degradation of low-energy electrons in carbon dioxide: energy loss and ionization. Nucl. Instrum. Methods. B 53, 239–250. Williart, A., Kendall, P.A., Blanco, F., Tegeder, P., Garc!ıa, G., Mason, N.J., 2003. Inelastic scattering and stopping power for electrons in O2 and O3 at intermediate and high energies, 0.3–5 kev. Chem. Phys. Lett. 375, 39–44.
Radiation Physics and Chemistry 72 (2005) 7–12
Simple stopping power formula for low and intermediate energy electrons Hasan Gum . u. s* Department of Physics, Faculty of Arts and Sciences, Ondokuz Mayis University, Samsun 55139, Turkey Received 29 November 2003; received in revised form 26 January 2004; accepted 4 March 2004
Abstract A simple stopping power formula, modified from that of Rohrlich and Carlson is presented. In this study analytical expressions for effective charge and effective mean excitation energies of target atoms in the modified Rohrlich and Carlson stopping power formula are used, while for effective charge of incoming electrons, Sugiyama’s semiempirical formula is used from Peterson and Green. The Lenz–Jensen statistical atomic density model has been used for calculations of effective charge and effective mean excitation energies. The calculated results of stopping power for electrons in the energy range from a few tens of electron volts to 10 MeV are found to be in good agreement to within 10% with the experimental data and a number of other calculations. r 2004 Elsevier Ltd. All rights reserved. Keywords: Electron stopping power; Effective charge; Effective mean excitation energy; Bragg rule; Bohr criterion
1. Introduction Stopping powers of matter for electrons are important in a wide variety of applications involving energy deposition. In radiation physics, chemistry, biology and medicine, it is often important to have simple but accurate information about the stopping power of various media for energetic electrons. Electron stopping power has been the subject of many studies (Pimblott and Siebbeles, 2002; Cengiz, 2002; Dingfelder et al., 1998; Sugiyama, 1985; La Verne and Mozumder, 1983; Berger and Seltzer, 1982; Seltzer and Berger, 1982; Ashley et al., 1979, 1982; Ashley, 1980; Tung et al., 1979, 1981; Spencer and Pal, 1978; Gupta et al., 1975; Dalgarno, 1960; Rohrlich and Carlson, 1954). While electron stopping powers at energies above 10 keV are theoretically well described and can be found in tables given in Pages et al. (1972), and ICRU 37 Report (1984), no practical calculation model of the stopping power exists for incoming electrons energies below 10 keV. *Corresponding author. Tel.: +90-362-4576020/5061; fax: +90-362-4576081. E-mail address: [email protected] (H. Gum . u. s-).
Sugiyama obtained stopping power values for intermediate electrons energies using a modified Rohrlich and Carlson model (Sugiyama, 1985; Rohrlich and Carlson, 1954). Sugiyama (1985) applied the Bohr stripping criterion (i.e v1 > bvF ðrB Þ ¼ ðb_=mÞ½3prðrB Þ1=3 ; vF ðrB Þ is the Thomas–Fermi velocity between r and r þ dr; b is the proportional constant of order of 1.26) to target atoms for effective charge and effective mean excitation energy. In applying this, he used the Thomas– Fermi atomic model and also evaluated the integrals numerically related to effective charge and effective mean excitation energy (Sugiyama, 1985). This method produced some uncertainties since in the model the upper limit of the integral has to be determined. The reason for this deviation originated from the effective charge and effective mean excitation energy calculation methods in which the upper limit of the integral has to be determined. Therefore, uncertainties arise both due to nature of the Thomas–Fermi atom and also to numerical calculations of integrals of Z and I (Sugiyama, 1985, Eqs. (4) and (5)). The purpose of this paper, is to obtain a stopping power formula for incident electrons, valid for the low and intermediate energy region (o10 keV). The stopping
0969-806X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2004.03.006
ARTICLE IN PRESS H. Gum . u. s / Radiation Physics and Chemistry 72 (2005) 7–12
8
power formula was based on a modification by Sugiyama (1985) of the Rohrlich and Carlson formulation (Rohrlich and Carlson, 1954). Analytical expres. sions obtained in previous work (Gum . u. s- and Koksal, 2002) on effective charge and effective mean excitation energies of target atoms were used, and also a semiempirical effective charge expression for incoming electrons, adjusted by Sugiyama (1985) to fit the Peterson and Green method (Peterson and Green, 1968). The calculating procedure thus described, was applied to incoming electrons on H2, N2, O2, H2O and CO2 targets which are tissue equivalent gas mixtures and are crucially important in microdosimetry, radiobiology and radiochemistry. Results from these were compared with the data and other theoretical results for wide range of incident electron energies.
and Z2 eZb 3 cn I ¼ cnðg0 Þ þ ð Þ g1 ðZb Þ g2 ðZb Þ g3 ðZb Þ 2 Z2 2 ð3Þ where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pe2 3:675 ¼ gZ2 185:12 eV ; g0 ¼ g_Z2 ma30
ð4Þ
g1 ðZb Þ ¼ ð0:002647Z5b þ 0:043207Z4b þ 0:285923Z3b þ Z2b 1 þ 0:2647Zb ð5Þ þ 2Zb þ 2Þln Zb g2 ðZb Þ ¼ 0:0013237Z6b þ 0:0229271Z5b þ 0:171183Z4b þ 0:770847Z3b þ 2:5Z2b þ 5:91Zb þ 8:140677 ð6Þ and
2. Method The modified mass collision stopping power formula for incoming electrons can be written as (Sugiyama, 1985; Rohrlich and Carlson, 1954): 2 dE dE 4pe4 z N0 E ¼ Þ F ðtÞ=2 ; Z lnð Sx ¼ ¼ dx r dx I mv21 A 2 F ðtÞ ¼ 1 b2 þ ½ðt2 =8Þ ð2t þ 1Þln 2=ð1 þ t2 Þ;
ð1Þ
where m is the electron mass, v1 is the incident electron velocity, E is the electron energy, N ¼ N0 =A is the density of target atoms and denotes an effective quantity, i.e. the effective charge of incident electrons z, the effective number of target electrons Z2 ; A is the atomic weight of the target element, N0 Avogadro’s number, and b is the ratio of v1 =c; with c the velocity of light and I the effective mean excitation energy, r the charge density of the target. In Eq. (1) the factor in front 2 of the parenthesis can be written as ðk=Ab2 Þz Z2 ; since 2 4 2 k ¼ 4pe N0 =mc ¼ 0:307075 MeV cm (Sugiyama, 1981; . Gum . u. s- and Koksal, 2002). Z2 and I can be obtained from Bohr’s stripping criterion for the effective charge of heavy ions and target atoms, and effective mean excitation energies given by Sugiyama (Sugiyama, 1981, 1985) respectively, Z N Z ¼ 4pr2 nðrÞ dr; rb
cn I ¼
1 Z 2
Z
N
cn g_op ðrÞ 4pr2 nðrÞ r:
rb
Z2 and I can be obtained analytically from these expressions by using Lenz atomic model (Lenz, 1932; . Jensen, 1932; Gum . u. s- and Koksal, 2002), as follows: eZb Þð0:002647Z5b þ 0:043207Z4b þ 0:285923Z3b Z2 ¼ Z2 ð 2 þ Z2b þ 2Zb þ 2Þ ð2Þ
g3 ðZb Þ ¼ 3eZb Eið1; Zb Þ 4:2eZb Eið1; 3:7735849 þ Zb Þ: ð7Þ Here, Ei is the exponential integral (Abramowitz and Stegun, 1970). Zb is determined from the Bohr stripping b_ ½3p2 nðZb Þ1=3 ; with vLJ criterion and v1 ZbvLJ ðrb Þ ¼ m the orbital velocity of the projectile or target atoms. The latter can be rewritten as, vF ðZb Þ ¼ 0:034838cZ 2=3 ðeZb Þ1=3 ð1 þ 0:2647Zb Þ=Zb
ð8Þ
with b a proportional constant of value about 1.26 and gE21=2 as used by Lindhard and Scharff (1953). In this study, g values for each of the target atoms were adjusted in a way that normalized effective mean excitation ionization energy, while I, refers to empirically determined I values (Andersen and Ziegler, 1977; Ahlen, 1980; ICRU 37, 1984) at sufficiently high incident energies (EcI; i.e, large Zb). In addition, the Hartree–Fock–Slater orbital velocity (Herman and Skillman, 1963) is used to determine the Bohr criterion, giving better results than that of the Thomas–Fermi velocity vF ðrÞ: The present procedure is to multiply vF ðrÞ P P 1=2 ; where by a correction factor G ¼ eHF = eTF eHFS and eTF are the orbital electron energy of each shell in the Hartree–Fock–Slater and Thomas–Fermi approximation, respectively. The mean excitation ionization energy values and basic data for the calculations are given in Table 1. Calculated values of Z and I are shown in Fig. 1 as a function of incident electron energy. In this study, the semiempirical effective charge of incident electrons z is used with z being given by z ¼ 1 expð2200b1:78 Þ ð9Þ where b is again the ratio of v1 =c; with c the velocity of light (Sugiyama, 1985).
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Table 1 The mean excitation ionization energy values and basic data for calculation Z
A
I (eV)
M (g/mol)
NA =M 1021
g
D
1 6 7 8 18 36 47
1 12 14 16 40 84 107
19.2 81 88 98 182 358 478
1.0079 12.001 14.0067 15.9994 39.09480 83.80 107.868
597.7 50.16 43.02 37.66 15.08 7.190 5.585
2.529 1.59 1.656 1.614 1.332 1.310 1.340
1.09 1.165 1.13 1.106 1.06 1.02 1.02
100
1000 Z=47 36
Z=47 36 18
18 8
10
1 z*
1
1
10
z* 0.1
10
6
I* (eV)
* *
z, Z
1
8
100
6
100
1000
10000
100000
1000000
10
100
Electron Energy (eV)
Fig. 1. The effective atomic number of electrons Z of various target atoms as a function of incident electron energy.
3. Calculated results and discussion Figs. 1 and 2 show the effective number of electrons Z and effective mean excitation energy I as a function of incident electron energy. For high incident energies (i.e. for Zb -0), Z -Z; z -0; and I -I; while for low incident energies (i.e. for high Zb values), Z -0 and z -0: The values of the semiempirical effective charge of electrons z is affected for energies lower than 200 eV (Sugiyama, 1985). z has value of 0.97713 for 200 eV, of 0.86987 for 100 eV, and of 0.23108 for 10 eV. That has also some contribution in the improvement of the results which has significant contribution in the improvement of the results below 200 eV energy regions. In order to calculate the electronic stopping power of each of the compounds discussed herein, first the effective charge of the incident electron z was obtained, primarily using Eq. (9), and the effective charges of the target atoms Z2 and Z3 in the compound were then obtained using Eq. (2). Eq. (3) was used to calculate the effective mean excitation energy of the target atoms I2 and I3 : These values were later used in Eq. (1) to obtain the mass collision stoppin power,Sx. The mass collision
1000 10000 100000 Electron energy (eV)
1000000
Fig. 2. The effective mean excitation energy I of various target atoms as a function of incident electron energy.
stopping powers of the identified molecularly constituted compounds for incoming electrons were calculated from Bragg’s addition rule (Bragg and Kleeman, 1905) which can be written as Sx ¼ nSxðAÞ þ mSxðBÞ
ð10Þ
where n and m is the atomic fraction of element (A) and (B) respectively. The calculated results in this study are given in Figs. 3–6. Fig. 3 shows the mass stopping power values for electrons in liquid water. The results obtained in this study are in good agreement with the semiempirical formula of Kutcher and Green (1976), the recommendations of ICRU 16 (1970) for the stopping power of liquid water, the predictions of La Verne and Pimblott (1995) and of Paretzke (1988) and the NIST database (ESTAR, NIST database, 2003). The NIST database data is based on the calculations of ICRU 37 (1984) derived from Bethe theory. The mass stopping powers obtained by using the formalism described in this paper are in good agreement to within 25% with data predicted using Paretzke’s formulation for water vapour (Paretzke, 1988), 8% with recommendations of ICRU 16 (1970) except for data obtained at 20, 200 and 300 eV, 2% with data of predictions of La Verne and Pimblott (1995),
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10 1000
Hydrogen and Nitrogen 1000
10
1
Peterson 68 This study ICRU 37 ESTAR
H2
2
100
Se / (MeV.cm /g)
Kutcher 76 Verne 95 ICRU 16 Paretzke 88 ESTAR This study ICRU 37
2
Se / (MeV.cm /g)
Liquid Water
100
N2
10
1 1
10
100
1000
10000
100000 1000000
10
1E7
100
1000
10000
100000
Fig. 3. Comparison of mass stopping power Sx for electrons, incident on liquid water. Solid curves, —, are the present calculations. ’, semiempirical formula by Kutcher and Green (1976); o results of La Verne et al. (1995); results of ICRU 16 Report (1970); &, from Paretzke (1988); ., data from ICRU 37 (1984) and r, values obtained from, ESTAR package.
1E7
Fig. 5. Mass stopping power Sx, plotted against incident electron energy on H2 and N2. — present calculations; , semiempirical formula of Peterson and Green (1968);., data from ICRU 37 report (1984) and D, values obtained from the ESTAR package.
1000
Oxygen Carbon Dioxide 2 -1
Se / (MeV.cm g )
100
Peterson 68 This study Williart 03 ICRU 37 ESTAR
100
Waibel 91a Waibel 91b Waibel 91c Berger 82 This study ICRU 37 ESTAR
2
Se / (MeV cm /g)
1000000
Electron Energy (eV)
Electron Energy (eV)
10
10
1 1 1
1
10
100
1000
10000
100000 1000000
1E7
Electron Energy (eV)
Fig. 4. Mass stopping power Sx for electrons, in CO2 as a function of the electron energy. —, present present calculations; +, Waibel and Grosswendt (1991) data derived from ionization measurements with corrections for loss of ionization from secondary electron; Waibel and Grosswendt’s results from inelastic scattering cross sections (Waibel and Grosswendt, 1991) and , restricted energy transfer (Waibel and Grosswendt, 1991); &, theoretical data of Berger and Seltzer (1982); m, data from ICRU 37 report (1984) and r, values obtained from, ESTAR package.
25% with the semiempirical formula of Kutcher and Green (1976) for incident electron energies above 30 eV. The agreement with the NIST database (ESTAR, 2003) and ICRU 37 (1984) are better than 0.8% for the incident electron energies above 10 keV. Fig. 4 shows the stopping power values for electrons in carbon dioxide as a function of the electron energy.
10
100
1000
10000
100000 1000000
1E7
Electron Energy (eV)
Fig. 6. Mass stopping power Sx, plotted against incident electron energy on O2. —, present calculations; , semiempirical formula of Peterson and Green (1968); ., data from ICRU 37 report (1984) and D, values obtained from, the ESTAR package and experimental points o, from Williart et al. (2003).
The experimental data are taken from Waibel and Grosswendt (Waibel and Grosswendt, 1991). In this figure, the data identified as ‘Waibel 91 a’ shows the derived ionization measurement for the correction of loss of ionization from secondary electrons, while the data ‘Waibel 91 b’ gives the restricted energy transfer without correction; ‘Waibel 91 c’ illustrates results from inelastic scattering cross sections (Waibel and Grosswendt, 1991). The solid curve indicate the present calculational results. Agreement with the data ‘Waibel 91 a’ and ‘Waibel 91 b’ is good (better than 10%) around stopping power maxima. Various stopping power data for carbon dioxide for energetic
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electrons (Berger and Seltzer, 1982; ESTAR, NIST database, 2003; ICRU 37, 1984) are also shown in Fig. 4. The stopping powers obtained using the formalism described here are in good agreement with experimental data and results of ICRU 37 (1984) and ESTAR in high energy regions and with the theoretical data of Berger and Seltzer (1982) at intermediate energies. Fig. 5 indicates the stopping power values for hydrogen and nitrogen for incoming electrons. The calculated mass stopping power values are compared with the semiempirical formula of Peterson and Green (1968) and the values of the NIST database (ESTAR, NIST database, 2003) and ICRU 37 (1984). Agreement with the semiempirical results are satisfactory, within 8%, except for data obtained at lower energies for H2 as has similarly been seen by Sugiyama (1985) in his calculated results. It would be expected that if higher order z2 and z3 corrections were included in the stopping power, then the agreement with experiment would be better. Fig. 6 indicates the stopping power values of oxygen for incident electrons. The calculated mass stopping power values are compared with the semiempirical formula of Peterson and Green (1968) and the values of NIST database (ESTAR, NIST database, 2003) and ICRU 37 (1984) and with experimental data taken from Williart et al. (2003). Agreement with the semiempirical results and other calculation results together with the experimental data are satisfactory, i.e to within 10% with semiempirical formula Peterson and Green (1968) except for data obtained at 10 eV and Williart et al. (2003), 1% with the values of NIST database (ESTAR, 2003) and ICRU 37 (1984).
4. Concluding remarks In this work, collision stopping power values for electrons on several target materials have been evaluated by the modified version of the Rohrlich and Carlson formula. The effective number of electrons and the effective mean excitation energy are introduced, determined from analytical expressions using the Bohr stripping criterion for heavy ions. The calculational results of stopping powers for the targets H2, N2, O2 and H2O are found to be in good agreement with semiempirical and a number of other theoretical predictions, together with experimental results. The collision stopping power depends sensitively on the charge-state or effective charge of projectile electrons and target atoms. The simple effective charge expressions is remarkably successful, and is highly useful for practical computation. The present electron stopping power calculation method should be especially useful for biomedical dosimetry.
11
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