Calculation of proton stopping power in the region of its maximum value for several organic materials and water

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1 28 December, 2000 Calculation of proton stopping power in the region of its maximum value for several organic materials and water A. Akkerman a, A. Breskin a, R. Chechik a and Y. Lifshitz b a The Weizmann Institute of Sciences, P.O. Box 26, 76100, Rehovot, Israel b Soreq Nuclear Research Center, 80100, Yavneh, Israel ABSTRACT The stopping powers for 10 solid organic materials and water have been calculated in the range of proton energies kev. Most of the presented results are new and are in good agreement with existing experimental data.the calculated data might be useful for applications in radiobiology and space research. Keywords: proton stopping power, organic materials, dielectric approach Author responsible for further correspondence: Dr. Rachel Chechik The Weizmann Institute of Sciences P.O.Box 26, 76100, Rehovot ISRAEL Tel: (+972) Fax: (+972) fnchecik@wis .weizmann.ac.il 1

2 INTRODUCTION AND DESCRIPTION OF THE MODEL Knowledge of proton stopping power (SP) in the region of its maximum value for a given medium is required in radiobiology for studies of radiation damage to tissue and for other applications. Therefore attempts have been made to estimate the SP both theoretically and experimentally. In the low energy range (Ep<50 kev), Lindhard s theory (Lindhard and Scharff, 1953) and its further modifications successfully account for the SP proportional relation to the proton velocity. At the high-energy range (Ep>300 kev) Bethe s theory (Bethe and Ashkin, 1953) adequately describes the SP versus Ep. In the peak region the mechanisms of proton-electron interactions change, which makes it difficult for self-consistent theoretical treatment. In addition, available experimental data in this energy interval, even for monatomic substances, fluctuate in value significantly. This prevents the use of fitting parameters for these data and in extrapolating them to complex organic molecules. A review of the subject is found in Semrand and Bauer (1985) and references therein and in ICRU Report-49 (1993). Many of the SP theories in the considered energy range are based on the dielectric response model (Ritchie, 1959). This approach was used successfully for calculating mean free paths (mfp) and SP for low energy electrons. As an instance, in Ashley (1991) this model was used to calculate the SP for protons. The model does not require experimentally determined mean ionization potentials and it includes, in a self-consistent way, the shell correction term. In this article we have extended Ashley's approach (Ashley, 1991) and calculated the proton SP in 10 organic compounds and in water, these being important materials for radiobiology and for other applications as insulating and resistive materials. For all of these materials experimentally derived Optical Energy Loss Functions (OELF) exist. The dielectric approach was recently used (Akkerman and Akkerman, 1999) to calculate the mfp and SPs of electrons at energies below 10 kev for the same organic materials, results being in good agreement with existing experimental values. For most of these materials the proton SP are presented here for the first time. We calculate the proton SP as a sum of terms SP=SP v +SP c +SP s that represent various processes: SP v is the valence electron excitation, SP c the core-electron ionization, and SP s the second order (in Z) correction term. According to the linear dielectric response theory the stopping power of a medium for a proton is expressed by the energy-loss function (ELF) Im(-1/ε(ω,q)) (Ritchie, 1959): E m SP v = dω ω dq 1/q Im(-1/ε(ω,q)), (1) 0 ω/v where E m =2V 2, V being the proton velocity and ħ = e = m 0 = 1. The ELF used in (1) is obtained from the optical energy-loss function (OELF) Im(-1/ε(ω,0)) by the quadratic extension for the momentum transfer q. For the core ionization SP c, induced by ions, we used Gryzinski's binary encounter theory (Gryzinski, 1965), which is a good approximation for light atomic mass materials. The dielectric response theory does not include ab initio the higher order in Z corrections to the SP, i.e. the Barkas- effect ( Z 3 ) (Ashley et al, 1972) and Bloch's term ( Z 4 ) 2

3 (Bloch, 1933), though these terms may play a significant role at relatively low proton energies, in the vicinity of the SP maximum and below. Ashley (Ashley, 1991) has given analytical expressions for these terms. The calculation of SP s requires the minimal impact parameter a, which distinguishes distant collisions from close impacts. Two different choices of this parameter are available, from Jackson and McCarthy (1972) and Lindhard (1976). RESULTS AND DISCUSSION First we fit the OELF by a sum of Drude-like functions with quadratic extension for the momentum transfer. When extrapolated to the first core-ionization level, this sum shows reasonable agreement with the OELF calculated from the photoionization cross sections (Biggs and Lighthill (1990). The major sum-rules, including the mean ionization potentials as in Akkerman and Akkerman (1999), were found to be in close agreement with experimental values. We have also obtained a good agreement between our calculated SP s for C, Al and Cu, the experimental SP's of Semrad and Bauer (1985) and calculated values of Ashley (1991). Minor differences, still observed in the vicinity of the SP maximum and above, could be due to differences in the ELF tails. At energies below the SP maximum (Ep<50 kev) the deviation of our calculated SP's from experimental data become significantly larger: our data decrease much faster than expected from proportionality of the SP to the proton velocity. Hence for these energies Lindhard's SP theory and its further improvements are preferable. Using the scheme described above we have calculated the valence and core parts of the SP and the high order correction terms for all materials listed in Table 1. It follows that the treatment of Jackson and McCarthy (1972) for the Barkas-effect yields better agreement than Lindhard's. For Polyethylene and PMMA excellent agreement is found with ICRU-49 (1993) in the vicinity of the SP maximum (see figure 1). For water (figure 2) the values for our data are somewhat below those of ICRU-49 (1993), being possibly explained by differences in the OELF data. In our SP calculations for all the organic materials listed in Table 1, the a's were taken from Jackson and McCarthy (1972). We note that the model may seem fragmentary, as it treats separately contributions to the SP from different excitation terms. Such division could be eliminated if the OELF was known up to very large energy losses, including the core-excitation part. However, such division is unavoidable for the high-order in Z correction terms, which do not follow directly from the dielectric response function. We conclude that the calculated SP's are practically uninfluenced by the method chosen for the momentum transfer extension (q>0), as opposed to that adopted by Planes et al, (1996). The same is concluded from comparison of our electron mfp calculation (Akkerman et al 1999) with those obtained by Tanuma et al (1993), using Lindhard's function for the q-extension of the OELF. 3

4 REFERENCES Akkerman A. and Akkerman E., (1999) Characteristics of electron inelastic interactions in organic compounds and water over the energy range ev, J. Appl. Phys. 86, Ashley J.C., Ritchie R.H. and Brandt W., (1972) Z 3 effect in stopping power of matter for charged particles, Phys. Rev. B 5, Ashley J.C., (1991) Optical data model for stopping power of condensed matter for protons and antiprotons, J. Phys. Condens. Matter 3, Bethe H.A. and Ashkin J., (1953) Passage of radiation through Matter in: Experimental Nuclear Physics, Vol. I, ed. E. Segre (Wiley, New York), Bloch F., (1933) Zur bremsung rasch bewegter teilchen beim durgang durch die materie, Ann. Phys. 16, Biggs F. and Lighthill R., (1990) Analytical approximation for X-ray cross sections III, Sandia National Lab. Report SAND Gryzinski M., (1965) Classical theory of atomic collisions I: Theory of inelastic collisions, Phys. Rev. A 138, ICRU (International Commission on Radiation Units and Measures), (1993) Stopping powers and ranges for protons and alpha particles, Report No 49 ICRU, Bethesda MD. Jackson J.D. and McCarthy R.L., (1972) Z 3 Corrections to energy loss and range, Phys. Rev. B 6, Lindhard J. and Scharff M., (1953) Energy loss in matter by fast particles of low charge, Kgl. Danske Videnscab. Selskab. Mat-Fys. Medd. 27 (15) (full issue). Lindhard J., (1976) The Barkas effect - or Z 3, Z 4 corrections to stopping of swift charged particles, Nucl. Instrum. Meth. B 2, 1-5. Planes D.J., Garcia-Molina R., Abril I. and Arista N.R., (1996) Wavenumber dependence of the energy loss function of graphite and aluminum, J. Electr. Spectrosc. Relat. Phenomena 82, Ritchie R.H., (1959) Interaction of charged particles with a degenerate Fermi-Dirac electron gas, Phys. Rev. 114, Semrad D. and Bauer P., (1985) Formulae and theories to predict proton stopping powers around the maximum, Nucl. Instrum. Meth. B 12, Tanuma S, Powell C.J, and Penn D.R., (1993) Calculations of the electron inelastic mean free paths V: Data for 14 organic compounds over the energy range ev, Surf. Interface Anal. 21,

5 Table 1. The calculated SP (in ev/å) for: 1-Polyethylene, 2-Guanine, 3-Poly(2-vinylpyridine), 4-Diphenyl-hexatriene, 5- β-carotene, 6-Polystyrene, 7-PMMA, 8-Paraffin, 9-Polyacetelene, 10-PBS, 11-Water. E[keV] Polyethylene SP, ev/a o Proton energy, kev Figure 1. Proton stopping power in Polyethylene. Open squares are our calculations, solid circles are ICRU-49 data. 5

6 8 Water SP, ev/a o Proton energy, kev Figure 2. Proton stopping power for water. Open squares are our calculations, solid squares are ICRU-49 data. 6

Calculation of the Stopping Power for Intermediate Energy Positrons. Önder Kabaday, and M. Çaǧatay Tufan

CHINESE JOURNAL OF PHYSICS VOL. 44, NO. 4 AUGUST 006 Calculation of the Stopping Power for Intermediate Energy Positrons Hasan Gümüş, Önder Kabaday, and M. Çaǧatay Tufan Department of Physics, Faculty