J . Phys. Chem. 1991, 95, 3907-3914
3907
ARTICLES Energy Loss by Electrons in Gaseous Saturated Hydrocarbons Simon M. Pimblott* and Jay A. Laverne Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: October 22, 1990)
The stopping powers, the inelastic mean free paths, the ranges, and the distributions of energy loss events for electrons in some gaseous alkanes have been calculated from experimentally based dipole oscillator strength distributions. The results show very little difference between the members of the homologous series of straight-chain hydrocarbons, CnH1,,+]where n = 2, 4, 6, and 8. For instance, the integrated path length of a 1-MeV electron in ethane of unit density is 0.38 cm and increases to 0.40 cm in octane of the same density. The stopping power of a 1-MeV electron in unit density hexane is 0.2 eV/nm, and the inelastic mean free path is 220 nm. The corresponding values for the other alkanes differ by less than 10%. The distribution of energy lass events along the track of a high-energy electron in these gaseous hydrocarbons is not significantly affected by the value of n, and it is insensitive to the incident electron energy from 10 keV to 1 MeV. The most probable energy loss along the track of a 1-MeV electron is 14 eV in ethane, 15 eV in butane and hexane, and 16 eV in octane. The mean energy loss is 25 eV in butane, hexane, and Octane and is 1 eV less in ethane.
1. Introduction When ionizing radiation passes through a medium, it is attenuated by a series of energy loss events that involve the transfer of energy from the radiation to the molecular electrons of the medium. The energy deposition results in the excitation of the molecules and can lead to ionization or to dissociation.lv2, Consequently the path of an ionizing particle is marked by a track of highly reactive radicals and ions, the chemistry of which is strongly influenced by the energy transfer from the radiation to the medium." The local energy deposition density depends upon the nature of the radiation, and the important parameters that influence the radiation chemistry are the proximity of the energy loss events, the amount of energy lost in each event, and the types of molecular fragmentation that result. We have discussed the deposition of energy by electrons in the radiolysis of water in previous papers,'-'O and in this paper we extend this work to include gaseous alkanes. The radiolysis of hydrocarbons has been a field of continuing study since the pioneering work of Lind in the 192Os.Il This interest has been both theoretical and experimental and has encompassed all three phases. A wide variety of topics have been (1 ) Chatterjee, A. In Radiation Chemistry. Principles and Applications; Farhataziz, Rodgers, M . A. J., Eds.; VCH Publishers: New York, 1987. (2) Paretzke. H. G. In Kinetics of Nonhomogeneous Processes. A Practical Inrroducrionf w Chemisis, Biologists, Physicists and Material Scieniisrs; Freeman, G. R.. Ed.; Wiley-lnterscience: New York, 1987. (3) Ausloos, P. Fundamental Processes in Radiation Chemistry; Interscience: New York, 1968. (4) Foldiak, G. Radiation Chemistry of Hydrocarbons; Elsevier: Amsterdam, I98 I . (5) Farhataziz; Rodgers, M. A. J. Radiation Chemistry. Principles and Applications; VCH Publishers: New York, 1987. (6) Freeman, 0 . R. In Kinetics of Nonhomogeneous Processes. A Practical Introductionfor Chemists, Biologists. Physicists and Maierial Scientists; Freeman, G. R., Ed.; Wiley-lnterscience: New York, 1987. (7) Laverne, J. A.; Mozumder, A. J . Phys. Chem. 1986, 90, 3242. (8) Green, N . J. E.; Laverne, J. A.; Mozumder, A. Radiat. Phys. Chem. 1988, 32, 99. (9) Pimblott, S. M. D.Phil Thesis, Oxford University, 1988. (IO) Pimblott, S. M.; Laverne, J. A.; Mozumder, A.; Green, N . J. B. J . Phys. Chem. 1990, 94,488. ( I 1.) Lind, S.C. The Chemical Effects of Alpha Particles and Electrons; American Chemical Society Monograph Series; Chemical Catalogue Company Inc.: New York. 1928.
0022-3654/9 1/2095-3907$02.50/0
addressed. The work has included discussions about the spatial distribution of the secondary electrons,l* about the mobility of the electrons and their partner ~ t t i o ~ .and , ' ~about the very nature of the reactive ions in some systems.I4 One other region of considerable interest is the period of fast kinetics that immediately follows r a d i o l y s i ~ . ~This ~ J ~ short-time chemistry provides the most direct information about the energy loss by the radiation and it involves a significant fraction of the total amount of reaction. The fast kinetics depend strongly upon the spatial distribution of the reactive species and so are heavily influenced by the structure of the radiation track. This sensitivity to the track structure means that an accurate picture of the energy loss by ionizing radiation is of primary importance. The only estimate of the distribution of energy loss events for electrons in hydrocarbons is due to Mozumder and Magee,17 who used an unsophisticated Monte Carlo technique based upon simple cross sections and an empirically constructed dipole oscillator strength distribution (DOSD). The absence of an acceptable description of the energy loss in hydrocarbons is emphasized by the fact that kinetic calculations describing the chemistry following hydrocarbon radiolysis are frequently performed'* using spur size distributions obtained for moist air or for ~ a t e r . l ~The - ~ energy ~ loss properties of these media may be very different from the alkanes. In this paper we examine the structure of high-energy electron tracks in the series of gaseous straight-chain alkanes C,,H,+,,(n = 2, 4, 6, 8). The following section describes the relationships between the DOSD and the inelastic mean free path, the linear energy transfer, and the range of energetic electrons. Energy loss (12) See: Freeman, G. R. Annu. Rev. Phys. Chem. 1983, 34, 463 and references therein. (13) Holroyd, R. A.; Schmidt, W. F. Annu. Reo. Phys. Chem. 1989,40, 439. (14) Trifunac, A. D.; Sauer, M. C.; Jonah, C. D. Chem. Phys. Lett. 1985, 113, 316. (15) Sumiyoshi, T.; Tsugaru, K.; Katayama, M. Chem. Lett. 1982, 1983. (16) LeMotais, E. C.;Jonah, C. D. Radiat. Phys. Chem. 1989, 33, 505. (17) Mozumder, A.; Magee, J . L. J . Chem. Phys. 1967, 47,939. (18) Baker, G. J.; Brocklehurst, B.; Hayes, M.; Hopkirk, A,; Holland, D. M. P.; Munro, 1. H.; Shaw, D. A. Chem. Phys. Lett. 1989, 161, 327. (19) Wilson, C. T. R. Proc. R . Soc. London, A 1923, 104, 192. (20) Beckman, W. J. Physica 1949, 15, 327. (21) Ore, A.; Larsen, A. Radio?. Res. 1964, 21, 331. (22) Mozumder, A.; Magee, J. L. J . Chem. Phys. 1966,45, 3332.
0 199 1 American Chemical Society
3908 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 by ionizing radiation is in its very nature a stochastic process, and so it is amenable to simulation using Monte Carlo methods. A simulation techniqueqJOhas been developed incorporating a differential energy loss cross section based upon the quadratic extension of the dielectric response function into the energy momentum The cross section used includes the effects of exchange and binding energy but not relativistic effects, which are not significant at the primary energies of interest?JO The only experimental parameter in the cross section is the DOSD. Section 2 also describes the calculation of the distribution of energy deposition events along an electron track. In section 3 the experimentally based DOSDsZ5st6are used to calculate the inelastic mean free paths, the linear energy transfer, and the ranges of energetic electrons in the homologous series of straight-chain alkanes CnHZnC2 ( n = 2,4,6,8) and so to examine the character of electron tracks in gaseous hydrocarbons. Also included are the results for water vapor and for liquid water, which are compared with the calculations for the hydrocarbons. The DOSDs of the liquid alkanes are not presently available, and the comparison of the results for the gaseous alkanes with those for gaseous and liquid H 2 0 gives an impression of the phase effects that may be observed in the liquid alkanes. Section 3 includes calculations for the distribution of energy loss events along electron tracks. We consider the change in the distribution with the electron's incident kinetic energy ( E ) and the value of n. For convenience the results are divided into four types of energy loss event: spurs (0-100 eV), blobs (100-500 eV), short tracks (500-5000 eV) and branched tracks (>SO00eV).I9 This division is arbitrary and was originally introduced to reflect some of the spatial aspects of the reactant distribution in the liquid p h a ~ e . ~ J ~ The calculations are limited to a spur size distribution in energy and not in species. While it is the latter that is of importance in kinetic calculations, the conversion between energy loss and number of reactants produced is complex. The number of each primary species is not merely proportional to the energy loss but is determined by the thresholds and the efficiencies of the relevant processes. Extension to a spur size distribution in primary species will be considered in a future publication.
Pimblott and LaVerne wherefle) is the dipole oscillator strength a t energy loss e, eo is the lowest excitation energy level of the gas, and ,e, is the maximum energy transfer (- E/2 for nonrelativistic electrons). For large energy losses Icrrand Zcrrapproach the quantities I and Z of the standard Bethe theory. Two of the most important factors in determining the structure of a particle track are the rate of energy loss and the range distribution of the secondary electrons. Both these quantities can be expressed in terms of the dipole oscillator strength distribution.29.30 Our approach relies upon Ashley's extension of the generalized dielectric response function into the energy-momentum This approximation is used to relate the value of the function for nonzero momentum transfer to the value in the zero momentum transfer limit. The rate of energy loss and the range distribution of an electron are both related to the differential cross section for an energy loss ho and a momentum transfer hq. This cross section is proportional to the imaginary part of the response function, Im (-l/e(q,o)) and according to Ashley Im (-l/e(q,o)) can be approximated by Im (-l/e(0,w-hq2/2m)). As Im (-l/ e(O,wh$/2m)) is proportional to the DOSD, the differential cross section, the rate of energy loss and the range of an electron are all directly related to the DOSD. Ashley's zero momentum transfer approximation has been tested for both A1 and Cu, and it has been shown to be extremely accurate." Furthermore, it should be remembered that in all the calculations presented in this paper, the DOSD is used under integration, and this will serve to lessen any minor errors introduced by the Ashley approximation. The approximation of Ashley has provided the basis of a number of different formulations of the inelastic mean free path, the stopping power, and the other energy loss properties of an electron The various developments in terms of the DOSD.7*8J0*23.24.29*30 of the expressions have included exchange, binding energy, and relativistic effects. We have chosen to overlook relativistic effects in the following calculations as our previous results for waterlo show that even for a 1-MeV electron these effects are very small, and the calculations are simplified considerably by this mild approximation. The nonrelativistic expression describing the stopping power, S, of an electron with kinetic energy E is23924
2. Methodology
S = -dE/dx = 2 x N
We recently presented a technique for determining the energy loss along electron tracks in terms of the DOSD of the medium, and we used this formalism to examine electron tracks in water."I0 In this and the following sections we extend this study to gaseous hydrocarbons. The central feature of all the following calculations is the role of the DOSD. Our previous studies on water showed that an inappropriate DOSD can lead to appreciable errors in the calculated parameter^.^.'^ The mean excitation potential, I , , the mean energy loss during a collision, 7, and the effective number of electrons that can receive energy during a transfer, Zcff,are all determined by the DOSD. These quantities and the "mean excitation energy for straggling", are defined by the following integrak2*
~~
G ( c / E ) de
(2a)
60
where G(a) = In
(1 - a)(l + a ) (1-a-s)(l+o+s)
I+ (k), E]
ra In [ ((1I - +- a)(l a x 1+ -a + + s) ] - darcsin ~
+
c
1 s
Q
rfmfle) J
= ( 1 - 2a)1/2
J (2c)
and F(0,k) is an elliptic integral of the first kind in Legendre normal form.32 Here and throughout the rest of the paper x = 2 ~ t ? / m d e, and m are the charge and the rest mass of an electron, u is the velocity of an electron with kinetic energy E , and N is the number density of molecules. If the function G is expanded in powers of € / E ,the leading terms yield the nonrelativistic Bethe stopping power formula for an electron. It is easily shown that the corresponding equation for the inelastic mean free path is
to
~~
(23) Ashley, J. C.; Williams, M. W. Report RADC-TR-83-87; Rome Air Development Center: Griffiss Air Force Base, NY, 1983; p 5 . (24) Ashley, J. C. J . Electron Specrrosc. Relar. Phenom. 1988, 46, 199. (25) Jhanwar, B. L.; Meath, W. J.; MacDonald, J. C. F. Can. J . Phys. 1981. 59. 185. (26) Jhanwar, B. L.; Meath, W. J.; MacDonald, J . C. F. Radiar. Res. 1986, 106, 288. (27) Fano, U. Annu. Rev. Nucl. Sci. 1963, 13, I .
(28) Laverne, J. A.; Mozumder, A. J . Phys. Chem. 1985,89, 4216. (29) Mozumder, A.; Laverne, J. A. J . Phys. Chem. 1984,88. 3926. (30) Mozumder, A.; Laverne, J. A. J . Phys. Chem. 1985,89,930. (31) Ashley, J. C., private communication. Also sec discussion in ref 33. (32) Gradshteyn, 1. S.;Ryzhik, 1. M. Tables of Integrals, Series and Products; Academic Press: New York, 1980.
The Journal of Physical Chemistry, Vol. 95. No. 10, 1991 3909
Energy Loss by Electrons in Gaseous Hydrocarbons where
+
+ a - s)(l - a s) (1 - a - s ) ( l + a + s )
(1
1-
LF[arcsin l+a
1"
I-a' 1+a
(3b)
The inelastic mean free path is related to the distance traveled by an electron between collisions, and it is important in determining the gross structure of an electron track. However, it is the range of low-energy electrons that is the more important parameter when considering the local structure as it describes the length of an electron's path as it is slowed down to thermal energy. The range, like the inelastic mean free path, is related to the inelastic collision cross section and so can be expressed in terms of the DOSD. The most frequently used estimate of the range is that calculated by using the continuous slowing down approximation (csda). Under this approximation the csda range of an electron is merely the integral of the inverse of the stopping power from the initial, incident electron energy to some acceptably defined final energy. In previous calculations we have taken the final energy to be 25 eV (about twice the ionization potential), and we shall continue with this prescription. The energy transfer from an electron to the medium through which it is travelling is a stochastic phenomenon. Consequently, there is a statistical variation in the length of the path, which is commonly known as range straggling. For low-energy electrons that suffer only a moderate number of energy loss interactions, the distribution of the path length will not be Gaussian as is the case for high-energy electrons. Recently, Mozumder and LaVerne29*30 developed a collision-by-collision convolution method for computing the range distribution. This treatment involves the calculation of the inelastic mean free path and the stopping power over a series of energies. The technique has been described in detail in refs 29, 30, and 33 and so will not be discussed further here. The passage of an electron through a gas or a liquid leaves a trail of energy deposition events. For an electron with a particular energy the magnitude of an energy loss is described by a probability distribution determined by the inelastic cross section. The probability distribution function is generally denoted Y, and it is the ratio of the cumulative inelastic cross section up to an energy loss e, a(E,e),to the total inelastic cross section, a(E),9*10*22 that is (4)
In a recent series of papers we described the calculation of the differential track structure of ~ater.~-IOThese calculations rely upon an accurate yet simple inelastic cross section in which the only parameter is the DOSD of the medium. The development of this formulation for the inelastic cross section has been described in detail.8J0*23*24-34 In the following calculations, for the homologous series of straight-chain alkanes CnH2,,+2,we use a cross section that includes the influence of exchange and binding energy but not relativistic effects. In our previous studies, we showed that relativistic effects did not play a significant role at energies of less than 1 MeV in water, and a similar absence is expected for hydrocarbons. The expression for the nonrelativistic inelastic cross section for an event with energy loss less than c for an electron of energy E is
a
(b - a)(l + a - b) b(l - 6 )
I+
(33) Laverne, J. A.; Pimblott, S.M.; Mozumder, A. Radiat. Phys. Chem., in press. (34) Green, N . J. B., manuscript in preparation.
0
20
40
60
80
IO0
Energy ( e V )
Figure 1. Dipole oscillator strength distributions for the homologous series o f gaseous straight-chain hydrocarbons CRHw2, n = 2, 4, 6 , 8. Data from ref 25.
The second term in eq Sa is negative, which is immediately obvious when one remembers that A-I(E) = N a ( E ) and Y 5 1 by definition. As the energy loss by an electron is probabilistic, it is amenable to Monte Carlo simulation, and a number of calculations of this type have been performed for ~ a t e r . ~ We ~ ~have ~ * described, ~ ~ * ~ ~ in detail, the development and the use of Monte Carlo simulation techniques to obtain the energy loss distribution describing an electron track?JO The technique involves the simulation and the accumulation of all the energy loss events along the tracks of a large number of electrons. The number of tracks must be sufficient that statistical fluctuations in the energy loss distribution become insignificant. We have performed Monte Carlo simulations using Y functions calculated by using eq 4 and the experimentally based DOSD and following the methodology detailed in ref 9. The branched tracks associated with each primary track were folded into the parent track and both the main and branched tracks were degraded in energy until the short track limit ( 5 keV) was attained. Each simulation involved at least 1O5 different primary tracks.
3. Results and Discussion There have been numerous experimental studies of the energy dependence of the DOSDs for a variety of different molecules.37 M a t h and ~ ~ - ~ ~ r k ehave r staken ~ ~ these * ~ experimental ~ J ~ ~ ~data and constructed complete DOSDs for numerous systems, in particular they have presented the DOSDs for the homologous series of straight-chain alkanes, CnHw2.25While there is often minor discrepancies between the M a t h compilations and recently measured partial D O S D S ? the ~ ~differences are generally within experimental accuracy.33 Figure 1 shows the compiled DOSDs for the saturated hydrocarbons CnHZ,,+2,n = 2, 4, 6, 8. The distributions for ethane and butane are based upon experimental data up to 70 eV, and for higher energies they were constructed by using a mixture rule and the DOSDs for atomic carbon, molecular hydrogen, and methane. The same techniques were used for hexane, but in this case experimental data are available only (35) Paretzke, H. G.;Turner, J. E.; Hamm, R. N.; Wright, H. A.; Ritchie, R. H. J . Chem. Phys. 1986.84, 31 82. (36) Zaider, M.; Brenner, D. J.; Wilson, W. E. Radiur. Res. 1983,95,231. (37) See references within refs 25, 26, and 38. (38) Thomas, G. F.; Meath, W. J. Mol. Phys. 1977, 34, 113. (39) Zeiss, G. D.; Meath, W. J.; MacDonald, J. C. F.; Dawson, D. J. Can. J . Phys. 1977, 55, 2080. (40) Jhanwar, B. L.; Meath, W. J.; MacDonald, J. C. F. Can. J . Phys. 1983, 61, 1027. (41) Kumar, A.; Meath, W. J. Con.J . Phys. 1982,67, 185. (42) Koizumi, H.; Yoshimi, T.; Shinsaka, K.; Ukai, M.; Morita, M.; Hatano, y.; Yagishita, A,; Ito, K. J . Chem. Phys. 1985, 82, 4856. (43) Koizumi, H.; Hironaka, K.; Shinsaka, K.; Arai, S.; Nakazawa, H.; Kimura, A.; Hatano, Y.; Ito, Y.; Zhang, Y.; Yagishita, A.; Ito, K.; Tanaka, K. J . Chem. Phys. 1986.85.4276. (44) Koizumi, H.; Shinsaka, K.; Hatano, Y. Radiar. Phys. Chem. 1989, 34, 87.
3910 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 for energies less than 40 eV. The experimental information still covers a region containing the majority of the DOSD, and so any error in the constructed part of the distribution should not be significant. Experimental formulation of the DOSD for octane was possible only at energies smaller than 10.5 eV, and the remainder had to be built from the DOSDs of C, H2, and the smaller alkanes. The extended range of empirical construction means that there is an increased possibility of error. All four distributions are comparatively free of structure except at low energies. There are two discontinuities in the curve for octane (20, 24 eV) and one in the curve for butane (24 eV). These jumps appear in the compilation of Meath and co-workers. The former pair are a consequence of the method of construction and the changing between different mixture rules. The jumps are artifacts of the construction techniques and would not be expected to be found in an experimental DOSD. The discontinuity in the butane DOSD is well within the region defined by experiment, and its origin is unclear. In the following calculations we have taken the DOSDs as compiled by Meath and co-workers and have made no alterations to smooth the curves. Notice the real jump in all the DOSDs caused by the carbon K shell at 280 eV. This influence of the K shell will be mentioned again later as it has observable consequences. The variation in fen and Zcawith the maximum energy loss for the four alkanes C2H6, C4HI0,C6HI4,and CsH18are shown in Figure 2. The high-energy limits of Zenare 18.00, 34.01, 50.01, and 66.03, respectively, and those of fen are 45.55, 47.50, 49.06, and 49.56 eV, respectively. These values agree well with the results of Jhanwar et a1.25*26 with the exception of fer[ for C4HI0. Here there is a difference of about 2% between the two calculations. The difference is difficult to explain as the same DOSD was used in both sets of calculations. We have performed additional calculations of the DOSD sums and logarithmic sums for butane presented by Jhanwar et al.,25326 and our results show close agreement except for L(O), where we obtain 18.95 compared to the value 19.49 quoted by Jhanwar et al. It is this difference that leads to the discrepancy in the mean excitation energy, cf. eq 1. The examination of the two values for L(0) suggests a simple typographical mistake in ref 25. Comparison of the calculated mean excitation potentials with experimental estimates for C2H6, C4HI0, and C6H1445shows agreement to within about 5%. This agreement is surprisingly good considering that the experimental values are inferred from proton and a experiments. The variation, with electron energy, of the stopping power and the inelastic mean free path in the homologous series of alkanes CnH2n+2is described in Table I. The table also includes data for water vapor and liquid water, and it refers to media with the same (unit) density. In all five gases and liquid water the stopping power has a maximum and the inelastic mean free path a minimum a t about 100 eV. The values of S at 100 eV are between 49 and 54 eV/nm for the alkanes. This value is considerably larger than the maximum stopping power of water, which is 32 eV/nm in the gas phase and 30 eV/nm in the liquid. Over the majority of the range of energy considered, S decreases the higher the value of n. As mentioned previously, the function G of eq 2 can be expanded in powers of c / E . If we consider the leading term of the resulting expression, which is equivalent to the simple expression of Bethe for the stopping power
where e'is the base of natural logarithms, then it is apparent that this variation of S with n is in the expected direction as NZcrr decreases and In (fcrr) increases as n becomes larger. For an electron of kinetic energy 1 MeV and for gases of unit density the value of NZcrr is 0.600, 0.586, 0.581, and 0.579 L cm-3 in (45) Experimental values for mean excitation energies are obtained from ref 19 and from the review of: Huyton, D. W.; Woodward, T. W.Radiar.
Res. Reo. 1970, 2, 205.
Pimblott and LaVerne 251 20 -
e c
,M Ethane
15e e
N 10-
- 10 10
102
io4
103
M a x i m u m E n e r g y Transfer (eV) 50,
I
50
Butane 40
b
I
IO
102
io4
103
Maximum Energy Transfer (eV)
Ocione
d
ethane, butane, hexane, and octane, respectively, where L is Avogadro's constant. At this energy the increase in the logarithmic term of Bethe's equation is about 0.5% from ethane to octane. The stopping power of an electron in both gaseous and liquid water is always smaller than that in all the hydrocarbons at that energy. The value of NZcnis smaller and that of In (few) is larger for both phases of water than for the hydrocarbons. For the alkanes the change in In (Icfr)with n is small compared to In (E) except for low-energy electrons, and it is the parameter NZemthat primarily controls the variation of stopping power with n. However, the average excitation energy of water in both phases is very different from the alkanes, and so this also makes a significant contribution to the difference between the four alkanes and water. The minimal values of the inelastic mean free path, A, for the four hydrocarbons
The Journal of Physical Chemistry, Vol. 95, NO. 10, 1991 3911
Energy Loss by Electrons in Gaseous Hydrocarbons
TABLE I: Effect of Electron Energy on Stopping Power and Inelastic Mean
E,eV 25 50 100 200 500 1x 2x 5x I 04 2 x 5x
IO’ IO’ 103
lo4 104
IO’ 2 x 105 5 x IO’ I 06
ethane butane S,beV/nm A,Cnm S.eV/nm A,nm 2.98 1 0.559 0.513 0.693 I .267 2.137 3.716 7.922 14.16 25.20 52.27 85.78 129.0 182.9 205.2
9.38 48.76 54.05 41.42 23.90 15.17 9.31 4.65 2.70 I .56 0.776 0.482 0.326 0.234 0.21 1
9.18 45.41 50.83 39.56 22.93 14.62 9.00 4.50 2.61 1.51 0.753 0.468 0.317 0.228 0.206
2.778 0.596 0.547 0.736 1.341 2.260 3.926 8.365 14.95 26.59 55.15 90.50 136.1 192.9 216.4
Free Path for the Alkanes C,,Hk+2”
hexane S.eV/nm A.nm 8.70 44.06 49.92 38.67 22.44 14.34 8.85 4.44 2.58 1.49 0.744 0.462 0.313 0.225 0.203
Octane S,eV/nm A , n m
2.811 0.620 0.561 0.755 1.375 2.316 4.023 8.571 15.31 27.24 56.49 92.69 139.4 197.5 221.6
9.12 42.74 49.36 38.30 22.25 14.23 8.79 4.41 2.56 1.48 0.740 0.460 0.31 1 0.224 0.202
water vapor S,eV/nm A , n m
2.7 16 0.633 0.571 0.766 1.394 2.347 4.076 8.682 15.51 27.59 57.21 93.87 141.1 200.0 224.4
4.32 22.56 32.46 29.59 18.92 12.08 7.59 3.90 2.29 1.34 0.673 0.421 0.286 0.207 0.187
4.917 1.232 0.985 1.216 2.104 3.487 5.996 12.67 22.53 39.96 82.60 135.3 203.0 287.3 321.9
liquid water S,eV/nm ~ , n m 1.29 16.98 30.14 28.18 18.49 1 1.92 7.50 3.86 2.27 1.33 0.669 0.418 0.284 0.205 0.186
19.47 2.129 1.260 1.459 2.444 4.004 6.842 14.38 25.31 45.15 93.13 152.3 228.4 322.8 361.4
“For comparison purposes gases and liquid of unit density, 1 g ~ m -are ~ , considered. bStopping power. ‘Inelastic mean free path. TABLE 11: Range of Electrons in Caseous Alkanes“
electron energy, keV ethane csda range, nm mean range, nm most probable range, nm butane csda range, nm mean range, nm most probable range, nm hexane csda range, nm mean range, nm most probable range, nm octane csda range, nm mean range, nm most probable range, nm
0.1
0.2
0.5
1.0
2.0
5.0
10.0
20.0
50.0
100.0
200.0
500.0
1000.0
1.85 3.97 13.9 41.1 128 616 2.09 X IO3 7.19 X IO3 3.66 X IO4 1.45 3.74 13.9 41.5 128 616 2.09 X IO3 7.19 X IO3 3.66 X IO4 1.00 3.40 13.6 41.5 131 616 2.09 X IO3 7.19 X IO3 3.66 X IO4
1.22 X IO’ 3.84 X IO’ 1.53 X IO6 3.81 X IO6 1.22 x 10’ 3.84 x 105 1.53 X IO6 3.81 X IO6 1.22 X IO’ 3.84 X IO’ 1.53 X IO6 3.81 X IO6
1.99 4.22 14.6 42.8 133 637 2.16 X IO’ 7.41 X IO3 3.77 X IO‘ 1.55 3.97 14.6 43.1 134 637 2.16 X lo3 7.41 X IO3 3.77 X IO4 1.00 3.40 13.6 41.5 131 637 2.16 X IO3 7.41 X lo3 3.77 X IO4
1.26 X IO’ 3.95 X IO’ 1.57 X IO6 3.91 X lo6 1.26 X IO’ 3.95 X IO’ 1.57 X IO6 3.91 X IO6 1.26 X IO’ 3.95 X IO’ 1.57 X IO6 3.91 X IO6
2.06 4.34 14.9 43.8 136 648 2.19 X IO3 7.52 X IO3 3.82 X IO4 1.60 4.07 14.9 44.1 137 648 2.19 X IO3 7.52 X IO3 3.82 X IO4 1.10 3.40 13.8 42.5 134 648 2.19 X 10’ 7.52 X IO3 3.82 X IO4
1.27 X IO’ 4.00 X IO’ 1.59 X IO6 3.96 X IO6 1.27 x 10’ 4.00 x 105 1.59 X IO6 3.96 X IO6 1.27 X IO’ 4.00 X IOs 1.59 X IO6 3.96 X IO6
2.08 4.38 15.1 44.1 137 652 2.20 X IO3 7.56 X IO3 3.84 X IO4 1.63 4.12 15.0 44.5 138 652 2.20 X IO3 7.56 X IO3 3.84 X IO4 1.10 3.50 14.0 42.5 135 652 2.20 X IO3 7.56 X 10’ 3.84 X IO4
1.28 X IO’ 4.02 X IO’ 1.59 X IO6 3.98 X IO6 1.28 X IO’ 4.02 X IO’ 1.59 X IO6 3.98 X IO6 1.28 X IO’ 4.02 X IO’ 1.59 X IO6 3.98 X IO6
” For comparison purposes gases of unit density, 1 g ~ m -are ~ , considered. are in the range (5-6) X IO-* cm, whereas those for water are almost twice as large, being 9.85 X lo-* and 1.26 X lo-’ cm in the gas and liquid, respectively, at unit density. The inelastic free mean path of an electron in both phases of water is larger at all energies than in the hydrocarbons. Again this trend is in the expected direction. To gauge the accuracy of the results presented in Table I, we can make comparisons with the experimental data available for a number of other hydrocarbons. The stopping powers of electrons of energy IO keV to 1 MeV are available from the tabulations of Berger and Seltzer.4 The experimental measurements for both polyethylene and paraffin wax are the same to two significant figures. At IO keV, 100 keV, and 1 MeV the stopping powers are 25,4.4, and 1.9 MeV cm2 g-l, respectively. AI-Ahmad and Watt” have also measured the stopping power of electrons in polyethylene, considering energies between 1 and IO keV. At 1 and IO keV they get stopping powers of 161 and 25 MeV cm2 g-’. The discrepances between our calculations and these measurements (
