J. Phys. Chem. 1995, 99, 10540-10548
10540
Electron Energy Loss Distributions in Solid and Gaseous Hydrocarbons Jay A. Laverne* and Simon M. Pimblott" Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 Received: March 23, 1995; In Final Form: April 28, 1995@
The dipole oscillator strength distributions for solid and for gas phase cyclohexane, cyclohexene, 1,3-cyclohexadiene, 1,4-cyclohexadiene, and benzene were constructed from experimentally derived optical constants and from atomic X-ray absorption cross sections. Monte Carlo simulations of the energy loss by electrons of initial energy from 10 keV to 1 MeV in these media were performed using cumulative inelastic cross sections obtained from a formulation incorporating the constructed dipole oscillator strength distributions. In the solid phase, the energy loss distributions, the most probable energy losses, and the mean energy losses for electrons show little effect due to the conjugation of n bonds. However, there are large differences between the gases, and there is a considerable effect due to condensation. The most probable and the mean energy losses for 1 MeV incident electrons in solid cyclohexane, cyclohexene, 1,3-~yclohexadiene, 1,4-cyclohexadiene, and benzene are in the ranges 22-24 and 47-48 eV, respectively. Comparison with data for water suggests that the values for the solid phases of hydrocarbons are acceptable approximations for the liquid phase. Density normalized stopping powers, inelastic mean free paths, and ranges for electrons in the various hydrocarbons are also presented.
1. Introduction A large part of our knowledge of the radiation chemistry of hydrocarbons comes from the many studies examining the end products formed.',* Much of the recent experimental work has focused on the short time chemistry of the transient ions, excited states, and radicals that lead to the observed final product^.^ However, virtually nothing is known about the physical and physicochemicals aspects of the radiolysis of hydrocarbons. A number of deterministic diffusion-kinetic ~ t u d i e s ~and - ~crude stochastic Monte Carlo of the radiolysis of hydrocarbons have been made; however, these calculations were hindered by the lack of information about the physical dimensions of the clusters of radiation-induced reactants produced by fast electrons and the columnar tracks produced by densely ionizing particles. Electron tracks in hydrocarbons are frequently modeled using data for water. The assumptions made about the initial number and spatial distributions of reactive species limit the kinetic studies to the determination of relative effects. The calculations are not suitable for predicting absolute radiation chemical yields. Energy loss by all ionizing radiation eventually leads to the production of low-energy electrons. Consequently, fundamental information on the transport and on energy loss of electrons in hydrocarbons is of great importance in the understanding of the radiation chemical processes occurring in these systems. Most radiation chemical studies are performed on liquids; however, with the possible exception of benzene, knowledge of the energy loss properties (inelastic cross sections, etc.) of electrons in hydrocarbons is limited. Recently, with the availability of cyclotron-based radiation sources, there has been an increasing amount of cross section data published on the gas and the solid phases of hydrocarbons.'0 The optical constants obtained from light absorption or reflection experiments provide much of the information required to estimate the dielectric response function of a medium, and so it is possible to obtain the energy loss function, which is related to the dipole oscillator strength distribution. I I The relationship between the cx Abstract published in Advance ACS Ahstrucrs. June 1. 1995.
0022-365419512099- 10540$09.0010
dipole oscillator strength distribution of a medium and the stopping power, the range, and the distribution of energy deposition events of energetic electrons has been thoroughly detailed in earlier publications.' Using this formulation, it is straightforward to compare the energy loss properties of electrons for a number of different media and phases since all the information required about a particular medium is contained within its dipole oscillator strength distribution. An earlier examination of the different phases of water showed that condensation from the gas to the liquid phase had a large effect while electron energy loss processes in the liquid and solid were virtually identical. Similar studies on electron energy loss processes in the different phases of hydrocarbons are centrally important for use in the development of models designed to understand the large amount of experimental data available in the literature on the radiolysis of liquid hydrocarbons.
In the next section, the dipole oscillator strength distributions for gas and solid phases of cyclohexane, cyclohexene, 1,3cyclohexadiene, 1,4-cyclohexadiene, and benzene are constructed from experimentally derived optical constants and from atomic X-ray absorption cross sections. These cyclic hydrocarbons are the subjects of a large number of the radiation chemical studies. In addition, these compounds will allow the study of any effects arising from the conjugation of the n-bonding systems. A brief summary of the formulation for the cumulative inelastic energy loss cross section and other related energy loss properties is then presented as well as the techniques used to simulate energy loss distributions for energetic electrons. The following section presents the results obtained and contains a discussion of their significance. The Results and Discussion section is broken into two parts. The first part includes the predictions for a variety of the energy loss properties of electrons (such as stopping power, inelastic mean free path, and range) which can be formulated in terms of the dipole oscillator strength distribution. The second part addresses the distribution of energy loss events for energetic electrons in the different phases of the hydrocarbons studied. The results are discussed, and they are compared to those
0 1995 American Chemical Society
Electron Energy Loss Distributions in Hydrocarbons
J. Phys. Chem., Vol. 99, No. 26, 1995 10541
obtained previously for gaseous n-hexane and water. The final section of the paper contains a summary of the significant conclusions.
2. Dipole Oscillator Strength Distribution Construction The imaginary parts,
€2,
of the dielectric response function,
~ ( w )for , the solid phases of cyclohexane, cyclohexene, 1,3-
cyclohexadiene, 1,4-cyclohexadiene, and benzene have been obtained by KillatI4 up to 40 eV. In these studies, a KramersKronig analysis of energy loss measurements of electrons was performed to obtain the energy loss function, Im(-l/c(w)); however, only the values derived for €2 are reported. The energy loss function for a medium is related to the real, E I , and imaginary parts of the dielectric response function by
The energy losses of electrons in gaseous cyclohexane, cyclohexene, 1,3-~yclohexadiene,1,4-cyclohexadiene, and benzene were also measured by Killat for energies up to 40 eV.I4 In the gas phase, the experimental measurements give the energy loss function directly, and the energy dependence of E ( @ ) up to 40 eV was reported. With this data it is straightforward to calculate the dipole oscillator strength distribution up to 40 eV. Appropriate summation of the atomic X-ray absorption cross sections of Veigele17 using eqs 3 and 4 provides the dipole oscillator strength distribution from 100 eV to 1 MeV. The gap from 40 to 100 eV is easily bridged by interpolation using the same power law as in the solid phase. The dipole oscillator strength distribution for gaseous n-hexane has been published by Jhanwhar et a l . I 8 No suitable data for solid n-hexane has been found in the literature.
3. Energy Loss Parameters where w is the photon energy. In order to reconstruct the energy loss function, E(o), the Kramers-Kronig relation~hip'~
is used to obtain € 1 from €2. ( P denotes the principal part of the integral.) To do this, the value of €2 must be known at all energies. At high photon energies, or for the gas phase, €2 *: E I x 1. Therefore, at high energies €2 approaches the value of the energy loss function. The relationships between the energy loss function, the dipole oscillator strength distribution, Aw), at an energy w , and the photoabsorption cross section, a(o), are
f ( o ) / w = 2m/(h2e2N)Im(-l/c(w)) a(w) = ne2h/(mc)f(o)
(3)
(4)
where the energy loss is w , m and e are the electron mass and charge, respectively, N is the number density of molecules, c is the speed of light, and h is Planck's constant. These relationships are well-known and are summarized in a number of standard reviews such as that of Fano and Cooper.I6 For all of the calculations reported, a density of 1 g/cm3 was used to aid comparison between the different hydrocarbons. The gas phase X-ray absorption cross sections for the elements have been compiled by VeigeleI7 for energies from 100 eV to 1 MeV. At these relatively high energies, the photoabsorption cross section for a molecule is equivalent to the sum for the individual elements. For each of the hydrocarbons, the atomic cross sections above 100 eV were summed appropriately and converted to values of €2. The gap in the available data for energies from 40 to 100 eV was bridged by interpolating the value of €2 using a simple power law of the form ez(w) = ~2(40)(4O/w)p. The value of p was determined so the estimated data agreed with the experimental data at 100 eV. The constructed optical constant and the Kramers-Kronig relationship were then used to obtain the value of € 1 at all energies. The appropriate summation of the two optical constants, €1 and €2, gives the energy loss function distribution, which in tum provides the dipole oscillator strength distribution via eq 3. The accuracies of the constructed dipole oscillator strength distributions were tested by checking that they integrate to give the appropriate total number of electrons. Where discrepancies were found, the data of Killat were scaled accordingly, and the entire procedure was repeated until better agreement was obtained. The scaling was always of the order of a few percent, which is within the uncertainty of the experimental data.
A number of different electron energy loss properties can be calculated from the dipole oscillator strength distribution. The effective number of electrons, &ff, which may receive an energy transfer from an incident electron is given by the sum rule
while the effective mean excitation energy, Zeff, can be evaluated from the relationship
Here wg and wmaxare the minimum and the maximum energy transfer. As the value of wmaxincreases, the values of Zeffand Zeff approach the standard quantities Z and Z, respectively. Nonrelativistic formulations for the stopping power, S, and the reciprocal inelastic mean free path, A-I, of an electron with kinetic energy E have been described earlier'g,20and are
and
x
where = 2ne4/(mv2)and v is the velocity of the incident electron. These relationships rely upon an approximation for the differential energy loss cross section that is based on the quadratic extension of the dielectric response function into the energy-momentum plane.21-22It is this assumption of the quadratic extrapolation in the energy-momentum plane, due to and verified by Ashley,21.22which makes it possible to determine the energy loss properties of electrons from the dipole oscillator strength distribution. The functions G(w/E)and L(w/ E ) are described in refs 19 and 20. When the function G is expanded in powers of w/E, the limiting terms give the standard nonrelativistic Bethe result for a fast electron. The inelastic mean free path, A, is a measure of the average distance an electron of a given energy travels between inelastic collisions, while the range determines the distance the electron travels until it reaches some well-defined lower energy. The most frequently quoted estimate of the range is the continuous slowing down approximation (CSDA) range, which is equal to the total path length of the electron. The CSDA range is obtained by integrating the inverse of the stopping power from the incident electron energy to the defined final energy. In the calculations presented here the final energy was taken to be 25 eV. Furthermore, it is also straightforward to calculate the mean
Laverne and Pimblott
10542 J. Phys. Chem., Vol. 99, No. 26, 1995
range of an electron from the mean free path and the stopping power. First, the mean free path of the electron is calculated, and then the stopping power is integrated to estimate the energy loss after traveling this distance. This energy is subtracted from the electron energy and a new mean free path is calculated. The process is repeated until the electron energy reaches the predetermined terminal energy (25 eV in the calculations reported here). The passage of an electron through a medium leaves a trail of energy deposition events in its wake. For an electron with an energy, E, the probability of an energy loss less than a given value, w',is determined by the ratio of the cumulative inelastic cross section up to w',a(w',E), to the total inelastic cross section, u(wmax,E),1.e. (9) where
+
o(o',E)= [K'&/E)x"'"'f(u) M(u/E,u'/E) du]/N (10) The function M(w/E,w'/E)and the formulation of the cumulative and total cross sections in terms of the dipole oscillator distribution employed are the same as used previously. I ' . 1 9 m The inelastic cross sections incorporate exchange and binding effects, but not relativistic effects which have been shown to be negligible for energy attenuation in water when E < 1 MeV." A stochastic Monte Carlo simulation technique has been developed for modeling the energy loss of energetic electrons. This technique has been described in detail' 1.19.20and involves simulating the energy degradation of a primary electron and all of its secondaries of energy greater than 5 keV down to this short-track limit. The simulation is performed by sampling from a tabulation of the Y distribution expressed as a function of energy loss and electron energy. Branch tracks, that is, secondary electrons of energy greater than 5 keV, associated with each primary track are attenuated in the same manner as the primary track, and the energy loss events occurring along each secondary track are added to the distribution obtained for the main track. Each calculation typically considers lo5 different primary tracks. As only the energy degradation of electrons of energy greater than 5 keV is considered, these simulations require no assumptions concerning the ionization cross sections of the compound.
4. Results and Discussion 4.1. Energy Loss by Electrons. The derived dipole oscillator strength distributions for each of the gaseous and solid hydrocarbons are shown in Figure la-e. Each distribution has been normalized to the total number of electrons to facilitate comparison. The data of Killati4 were used to construct the dipole oscillator strength distributions of the hydrocarbons as these experiments covered the widest range of energies for the compounds of interest. Furthermore, by selecting a single source, any systematic errors in the measurements have the least effect when comparing the derived energy loss properties of electrons in the different hydrocarbons. Experimental studies of the absorption spectra for gaseous cyclic hydrocarbons in addition to those of Killat have been reported. Pinkett and c o - ~ o r k e r smeasured ~~ the absorption spectra of gaseous cyclohexane, cyclohexene, and benzene from about 5.5 to 8 eV. In addition, Kumar and MeathZ4 have compiled a complete dipole oscillator strength distribution for gaseous benzene from a number of experimental absorption studies. The data from both these sources agree with the gas phase dipole oscillator strength distributions presented here.
Reflection and absorption spectra of solid benzene from about 5.5 to 7 eV have been measured by Brith et al.25 The dipole oscillator strength distribution derived from their published data is almost twice as large at 7 eV as that presented here. In order to convert their measurements into optical constants, Brith et al. extrapolate to infinite energy and used a Kramers-Kronig relationship. Such an extrapolation must be able to predict the large peak in the dipole oscillator strength distribution at 2025 eV. As no suitable extrapolation technique exists, the discrepancy in the data is not surprising. A number of absorption and reflection spectra of liquid benzene have also been r e p ~ r t e d . ~ ~As - ? ~with the solid phase studies, these experiments only extend to about 10 eV, and extrapolation to higher energies requires more information than is experimentally available. Given these reservations, the optical constants derived in all of the solid and liquid phase studies of benzene agree reasonably well. The agreement between liquid and solid supports the argument that the dipole oscillator strength distributions for solids and liquids are sufficiently close that they can be interchanged in energy loss calculations. Previous work on the three phases of water, where more complete distributions are available for all phases, arrived at the same conclusion.".'? It can be seen in Figure la-e that the dipole oscillator strength distributions for the gaseous hydrocarbons exhibit wide variations. The thresholds of the dipole oscillator strength distribution are considerably different for each gas. The distribution for 1,3-~yclohexadienehas the lowest threshold at -4.2 eV while the thresholds for the saturated compounds cyclohexane (-7.1 eV) and n-hexane (-7.8 eV) occur at the highest energy. Several sharp peaks are observed in the distributions at low energies: at 4.9 and 8.0 eV for 1,3cyclohexadiene, at 7.9 eV for 1,4-cyclohexadiene, and at 6.9 eV for benzene. All of the hydrocarbons have a broad peak centered in the range 15-18 eV, and some of the compounds show a small shoulder peak at 10- 11 eV. By 40 eV all of the dipole oscillator strengths have decreased to less than 20% of their peak value and are dropping rapidly. Conjugation of the n bonds of the hydrocarbon appears to have no predictable effect on the dipole oscillator strength distribution. Figure la-e show a remarkable similarity in the dipole oscillator strength distributions for the solid hydrocarbons. Except for the small peak at about 7 eV in benzene, the dipole oscillator strength distributions are dominated by a single broad band with a maximum at about 23-24 eV. The thresholds of the dipole oscillator strength distributions vary from 3.6 eV for 1,3-cyclohexadiene to 6.0 eV for cyclohexane. As observed with the gases, by 40 eV the distributions have dropped to less than 20% of their peak values. The similarity of the dipole oscillator distributions of the solid hydrocarbons implies that many of the electron energy loss parameters will be the same in the condensed phase. While the dipole oscillator strength distributions of the solid hydrocarbons are very similar, they differ considerably from the gas phase distributions. For each hydrocarbon, the distribution is shifted to higher energies in the solid phase. Daniels30 suggested that a similar shift observed in water was due to changes in the energy levels of the states on condensation. Comparison of the dipole oscillator strength distributions for the solid hydrocarbons with those published previously for liquid and solid water and for solid DNA3' reveals very significant differences. While all the distributions are characterized by a single broad band, the band for the hydrocarbons is higher and narrower than those for liquid and solid water and for solid DNA. For instance, the peak for cyclohexane is about 40% taller than that for liquid water, and at 40 eV the dipole oscillator strength distribution of liquid water has dropped to only 40% of its peak maximum compared to a
Electron Energy Loss Distributions in Hydrocarbons
J. Phys. Chem., Vol. 99, No. 26, 1995 10543
-
cyclohexane gas
. . 0.04
0'05
n-hexane - gas
,.' '.. :' '.,/
0.03
kN,
-
1,4-cycIohexadiene gas
cyclohexane - solid '
v
.¶
1
I
I
0.02
h
0 00
I
0
.
,
.
20
10
,
30
.
50
40
60
Energy, w (eV)
Energy, o (eV)
cyclohexene - gas
0.04
k
--
0.04
-
benzene - gas
cyclohexene solid
0.03
2
0.02
3 .c
0.01
".""
1 0
10
20
30
40
50
60
Energy, o (eV)
Energy, o (eV)
1,3-~yclohexadiene- gas
0.04 0.05[
..,
Energy, o (eV) Figure 1. Dipole oscillator strength distributions normalized to the total number of electrons for gaseous (dotted lines) and for solid (solid lines) hydrocarbons: (a) cyclohexane and gaseous n-hexane (dashed line), (b) cyclohexene,(c) 1,3-~yclohexadiene, (d) 1,4-~yclohexadiene, and (e) benzene.
decrease to 15% for cyclohexane. These differences will have important consequences on the energy loss properties of energetic electrons. The calculated total number of electrons, 2, capable of receiving energy in an energy loss event are given in Table 1 for the gaseous and solid hydrocarbons. Agreement with the expected limiting values is good, as the distributions were appropriately normalized in their construction. Table 1 also includes the values for the effective number of electrons accessible for energy losses of up to 9 and 40 eV, &. Although several of the dipole oscillator strength distributions for the gaseous hydrocarbons show peaks at low energies, these peaks are responsible for at most a few electrons. Even the low-energy exciton peak for solid benzene accounts for less than one electron. On the other hand, more than 20 electrons are represented by the broad peaks centered from 15 to 18 eV in
the gases and at 23 eV in the solids. By about 40 eV, the effective number of electrons is nearly independent of phase for all the hydrocarbons considered. Calculated mean excitation energies, I , are also listed in Table 1. They show relatively large variations among the gases ranging from 46.9 eV for cyclohexene to 60.5 eV for benzene, but the values are very similar for the solids varying from 58.5 eV for cyclohexane to 64 eV for benzene. With the exception of benzene, the mean excitation energy increases -20% on condensation. Agreement of the calculated mean excitation energies with the experimental data available is good.'s,24,32 Furthermore, notice that the values determined here for solid cyclohexane and solid benzene match the literature values for the corresponding liquids. All the properties of a medium, including any effects of phase, necessary for consideration of the energy loss by electrons are
Laverne and Pimblott
10544 J. Phys. Chem., Vol. 99, No. 26, 1995 TABLE 1: Effective Number of Electrons, &e, at 9 and 40 eV, Total Number of Electrons, 2, and Mean Excitation Energy, I , for n-Hexane, Cyclohexane, Cyclohexene, l,J-Cyclohexadiene, 1,4-Cyclohexadiene, and Benzene Z
Zff
this work
9 e V 40eV
I (ev) literature
~
hexane 0.47 gas cyclohexane 0.69 gas 0.09 solid c yclohexene 1.36 gas 0.15 solid 1,3-cyclohexadiene 2.19 gas 0.26 solid 1,4-~yclohexadiene 2.32 gas 0.29 solid benzene 1.23 gas 0.36 solid
31.2
50.01
49.06
49.1"
31.0 29.7
48.02 48.07
47.84 58.52
56.4b (liquid)
30.3 28.1
46.01 46.03
46.85 60.41
27.8 27.3
44.00 43.98
49.18 61.42
27.8 26.8
44.00 44.01
48.78 61.39
23.5 24.4
42.10 41.98
60.53 64.04
60.3' 63.4b (liquid)
Reference 18. Reference 32. Reference 24
contained within the dipole oscillator strength distribution. I ' * I 2 Therefore, phase effects are automatically incorporated in the energy loss properties reported here. The density normalized stopping powers of gaseous and solid benzene, 1,3-cyclohexadiene, 1,4-~yclohexadiene,cyclohexene, cyclohexane, and gaseous n-hexane for electrons of energy from 25 eV to 1 MeV are given in Table 2. As expected from the wide variation in the dipole oscillator strength distributions, the stopping powers
of the gases for low-energy electrons are quite different. The peak maxima vary from about 60 eV for cyclohexene to about 80 eV for benzene, and the magnitude for cyclohexene at its peak is about 40% greater than that of benzene at its peak. In contrast to the gases, the stopping powers for the solids are very similar; they exhibit a variation of only about 10% over the range of electron energies from 30 eV to 1 MeV. For a 1 MeV electron, the difference in the density normalized stopping powers of all of the hydrocarbons, both solid and gaseous, is only about 10%. The effect of phase on the density normalized stopping power of a particular hydrocarbon is very significant for low-energy electrons. The maximum stopping power in the solid phase occurs at about 90 eV for all the hydrocarbons, which is a considerably higher electron energy than in the gas phase. The maximum value is about the same for gas and solid benzene, but it is about 20% higher for cyclohexane, cyclohexene, 1,3-~yclohexadiene,and 1,4-cyclohexadiene. Experimental stopping powers of the hydrocarbons examined here are not available; however, the ICRU33 values for solid paraffin wax (mean excitation energy 55.9 eV) agree very well with the results calculated for cyclohexane which has a similar mean excitation energy (58.5 eV). As expected from the dipole oscillator strength distributions, there is no definitive effect of increasing n bond conjugation on the stopping power of the hydrocarbons. However, at high electron energies the completely conjugated benzene has a slightly lower stopping power in both phases than that of the other hydrocarbons. The density normalized inelastic mean free paths for electrons in the different hydrocarbons are listed in Table 3. Comparison of the data shows that density normalized inelastic mean free
TABLE 2: Effect of Electron Energy on the Density Normalized Stopping Power ( S , MeV cm*/g) for n-Hexane, Cyclohexane, Cyclohexene, 1,3-Cyclohexadiene, 1,4-Cyclohexadiene, and Benzene energy (eV) 25 50 100 200 500 103 2x 5x 104 2x 5x 105 2x 5x 106
103 lo1 104 104 105 105
hexane gas
87.0 44 1 499 387 224 143 88.5 44.4 25.8 14.9 7.44 4.62 3.13 2.25 2.03
cyclohexane gas solid 142 523 530 393 224 143 87.8 43.9 25.5 14.8 7.34 4.56 3.09 2.22 2.00
22.7 254 424 345 206 134 83.3 42.2 24.6 14.3 7.14 4.45 3.02 2.17 1.96
cyclohexene pas solid 179 563 542 393 222 141 86.6 43.3 25.1 14.5 7.23 4.49 3.04 2.18 1.97
23.1 24 1 410 333 199 130
81.0 41.1 24.0 13.9 6.98 4.35 2.95 2.13 1.92
1,3-cyclohexadiene gas solid 215 533 515 376 213 136 83.8 42.0 24.4 14.1 7.04 4.37 2.96 2.13 1.92
29.9 243 406 325 194 126 78.9 40. I 23.4 13.6 6.8 1 4.25 2.88 2.08 1.88
1,4-~yclohexadiene gas solid 212 547 519 378 2 14 136 83.9 42.1 24.4 14.2 7.04 4.38 2.96 2.13 1.92
30.4 249 405 325 194 126 79.0 40.1 23.4 13.6 6.82 4.25 2.88 2.08 1.88
gas
benzene solid
110 373 419 323 189 123 77.5 39.4 23.0 13.4 6.70 4.18 2.83 2.04 1.85
38.8 248 384 31 1 186 122 76.3 38.9 22.7 13.2 6.63 4.14 2.8 1 2.02 1.83
TABLE 3: Effect of Electron Energy on the Density Normalized Inelastic Mean Free Path (A, pg/cm*) for n-Hexane, Cyclohexane, Cyclohexene, 1,3-Cyclohexadiene, 1,4-Cyclohexadiene, and Benzene energy (eV)
hexane gas
25 50 100 200 500 103 2 x IO' 5 x IO3 10' 2 x 10' 5 x 104 105 2 x 105 5 x 105 106
0.28 0.06 0.06 0.08 0. I4 0.23 0.40 0.86 1.53 2.72 5.65 9.27 13.9 19.8 22.2
cyclohexane gas solid 0.17 0.05 0.05 0.07 0.12 0.21 0.36 0.78 1.40 2.49 5.16 8.48 12.8 18.1 20.3
1.09 0.14 0.08 0.10 0.18 0.30 0.52 1.11 1.97 3.50 7.23 11.8 17.8 25.2 28.2
cyclohexene gas solid 0.11 0.04 0.04 0.06 0.12 0.20 0.35 0.74 1.33 2.37 4.93 8.09 12.2 17.3 19.4
0.93 0.15 0.09 0.11 0.19 0.32 0.54 1.15 2.04 3.62 7.49 12.3 18.4 26.0 29.2
1,3-cyclohexadiene gas solid
1,4-cyclohexadiene
gas
solid
0.08 0.04 0.04 0.06 0.12 0.20 0.35 0.75 1.35 2.40 4.99 8.20 12.4 17.5 19.7
0.09 0.04 0.04 0.06 0.12 0.20 0.35 0.75 1.35 2.40 5.00 8.21 12.4 17.5 19.7
0.63 0.14 0.09 0.11 0.19 0.32 0.54 1.15 2.05 3.64 7.52 12.3 18.5 26.2 29.4
0.60 0.14 0.09 0.1 1 0.19 0.32 0.55 1.16 2.06 3.65 7.55 12.4 18.6 26.3 29.4
benzene gas solid 0.15 0.07 0.06 0.09 0.16 0.26 0.46 0.98 1.75 3.12 6.48 10.6 16.0 22.7 25.4
0.46 0.13 0.09 0.1 1 0.20 0.33 0.56 1.19 2.11 3.75 7.75 12.7 19.1 27.0 30.3
Electron Energy Loss Distributions in Hydrocarbons
J. Phys. Chem., Vol. 99, No. 26, I995 10545
TABLE 4: Density Normalized CSDA Ranges of Electrons in n-Hexane, Cyclohexane, Cyclohexene, 1,3-Cyclohexadiene, l,rl-Cyclohexadiene,and Benzene in Units of glcm* hexane -
energy (eV) 50 100 200 500 103 2 103 5 x 103 104 2 x 104 5 x 104 105 2 105 5 x 105
gas 1.05E-7" 2.06E -7 4.34E- 7 1.49E-6 4.38E-6 1.36E-5 6.48E-5 2.19E-4 7.52E-4 3.82E-3 1.27E-2 4.00E-2 1S9E- 1 3.96E-1
106
cyclohexane gas solid 7.69E-8 3.22E-7 1.68E-7 4.54E-7 3.89E-7 7.14E-7 1.88E-6 1.44E-6 4.99E-6 4.34E-6 1.48E-5 1.36E-5 6.88E-5 6.53E-5 2.31E-4 2.21E-4 7.88E-4 7.6OE-4 3.99E- 3 3.87E-3 1.32E-2 1.29E-2 4.16E-2 4.05E-2 1.64E- 1 1.61E-1 4.02E- 1 4.10E-1
cyclohexene gas solid 6.72E-8 3.44E-7 1.54E-7 4.80E-7 3.73E-7 7.49E-7 1.44E-6 1.95E-6 4.37E-6 5.17E-6 1.37E-5 1.52E-5 6.62E-5 7.08E-5 2.24E-4 2.37E-4 7.71E-4 8.08E-4 3.93E-3 4.09E-3 1.31E-2 1.36E-2 4.12E-2 4.25E-2 1.63E- 1 1.68E- 1 4.08E- 1 4.19E-1
1.3-cyclohexadiene gas solid 6.56E-8 3.04E-7 1.57E-7 4.41E-7 3.87E-7 7.15E-7 1.50E-6 1.95E-6 4.54E-6 5.26E-6 1.42E-5 1.56E-5 6.83E-5 7.25E-5 2.31E-4 2.43E-4 7.94E-4 8.27E-4 4.04E-3 4.18E-3 1.34E-2 1.39E-2 4.23E-2 4.35E-2 1.68E- 1 1.72E- 1 4.19E-1 4.29E-1
1,4-~yclohexadiene gas solid 6.44E-8 2.99E-7 1.55E-7 4.34E-7 3.83E-7 7.09E-7 1.49E-6 1.95E-6 4.52E-6 5.25E-6 1.42E-5 1.56E-5 6.82E-5 7.25E-5 2.31E-4 2.43E-4 7.92E-4 8.27E-4 4.03E-3 4.18E-3 1.34E-2 1.39E-2 4.22E-2 4.35E-2 1.67E- 1 1.72E- 1 4.18E-1 4.29E- 1
benzene gas solid 1.15E-7 2.64E-7 2.34E-7 4.05E-7 5.06E-7 6.93E-7 1.77E-6 1.99E-6 5.15E-6 5.42E-6 1.57E-5 1.61E-5 7.37E-5 7.49E-5 2.47E-4 2.50E-4 8.42E-4 8.52E-4 4.26E-3 4.30E-3 1.41E-2 1.43E-2 4.43E-2 4.47E-2 1.75E- 1 1.77E- 1 4.36E- 1 4.40E-1
The notation E-a stands for x lo-". 1.6 a C
d 2 -0
1.4
1
+ hexane \
gases
1
a C
d
solids
ki
0
.-c
1.2
a P)
C
. (d
r
1,4 cyclohexadiene
1.0
a
1,3 cyclohexadiene 1,4cyclohexadtene
P,
c
(d
a
0.8 1 0'
102
1o3
1o4
Electron Energy (eV) Figure 2. Density normalized CSDA ranges of electrons in gaseous
Electron Energy (eV)
cyclohexene, 1,3-cyclohexadiene, 1,4-cyclohexadiene, benzene, and n-hexane relative to that of cyclohexane as a function of electron energy.
Figure 3. Density normalized CSDA ranges of electrons in solid cyclohexene, 1,3-~yclohexadiene,1,4-~yclohexadiene,and benzene relative to that of cyclohexane as a function of electron energy.
paths of low-energy electrons are dependent on the medium. The mean free paths of electrons in the gaseous hydrocarbons are somewhat different although the variation is smaller than that observed earlier for stopping power. In the solid phase, the density normalized mean free path is almost independent of the hydrocarbon for electron energies greater than 50 eV. Furthermore, density normalized mean free paths in a hydrocarbon vary by almost an order of magnitude between the solid and gas phases at low electron energies, and even at 1 MeV there is still a substantial effect of phase. The present results for solids are in excellent agreement with the previous calculations of Ashley on the inelastic mean free path in condensed organic matter.34 For example, he calculated inelastic mean free paths for 200 eV and 10 keV electrons in polyethylene to be 0.11 and 2.14 pg/cm2, respectively. This data further supports the assumption that electron energy loss processes are somewhat independent of the medium for all condensed organic materials. Table 4 contains the density normalized CSDA ranges of electrons in n-hexane, cyclohexane, cyclohexene, 1,3-cyclohexadiene, 1,4-cyclohexadiene,and benzene. As expected from the large variations observed in the stopping powers listed in Table 2, there are some significant differences in the density normalized CSDA ranges. At low electron energies large discrepancies are observed between the hydrocarbons, and the effects of phase are considerable; however, at high electron energies the ranges are very nearly independent of the hydrocarbon and its phase. Figure 2 illustrates the effect of electron energy on the density normalized CSDA ranges for the gaseous hydrocarbons. The data are presented relative to the range for
gaseous cyclohexane. In gaseous benzene and n-hexane, the density normalized ranges of low-energy electrons are much longer than in cyclohexane, while the ranges in the other hydrocarbons are comparable to cyclohexane. Similar calculations for the solid hydrocarbons are shown in Figure 3 with the ranges presented relative to those for solid cyclohexane. For the solid phase, the ranges in all the different hydrocarbons are similar (less than 10%difference) at electron energies above 1 keV, and except for benzene, the differences are also small for low-energy electrons. Comparison of the density normalized CSDA ranges for the solid hydrocarbons with those for water and for DNA shows significant discrepancies: even the ranges for high-energy electrons are smaller in the hydrocarbons than in water and DNA.3' The mean ranges of low-energy electrons are compared in Table 5 . The data extend over the range 100 eV to 2 keV. For greater electron energies the mean range and the CSDA range are essentially the same. The data show a considerable phase effect on the mean range of low-energy electrons; however, the variation between the different hydrocarbons is small. The density normalized mean range of a 100 eV electron in solid cyclohexane is 86% larger than in the gas. For a 2 keV electron this difference drops to only 8%. These values are to be compared with differences in the CSDA range of 170% and 9% at the respective energies. The range of the incident electron plays a large part in the spatial distribution of excited species produced by the energy deposition processes. The density normalized range data suggest that these spatial distributions are likely to be very similar for all solid (and liquid) hydro-
LaVeme and Pimblott
10546 J. Phys. Chem., Vol. 99, No. 26, 1995
TABLE 5: Density Normalized Mean Ranges of Electrons in n-Hexane, Cyclohexane, Cyclohexene, 1,3-Cyclohexadiene, 1.4-Cvclohexadiene. and Benzene in Units of dcmZ hexane
energy (eV)
___ gas
100
1.60E-7" 4.07E-7 1.49E-6 4.41E-6 1.37E-5
200 500 103 2 x lo3
cyclohexane gas solid 1.40E-7 3.7 1E-7 1.45E-6 4.39E-6 1.37E-5
cyclohexene gas solid 2.6OE-7 4.90E-7 1.73E-6 5.03E-6 1.52E-5
1.40E-7 3.68E-7 1.46E-6 4.42E-6 1.39E-5
2.6OE-7 4.73E-7 1.67E-6 4.86E-6
1.48E-5
1,3-cyclohexadiene gas solid 2.738-7 4.9OE-7 1.85E-6 S.20E-6 1.56E-5
1.47E-73.82E-7 1.52E-6 4.60E-6 1.44E-5
1,4-~yclohexadiene gas solid 1.4SE-7 3.79E-7 1.5 1E-6 4.57E-6 1.43E-5
2.72E-7 4.86E-7 1.84E-6 S.19E-6 1.56E-5
gas
benzene solid
1.86E-7 4.76E-7 1.76E-6 S.19E-6 1.58E-5
2.81E-7 S.74E-7 1.88E-6 S.3SE-6 1.61E-5
The notation E-a stands for x IO-".
t
cyclohexane - gas 006
-
1,4-cyclohexadiene - gas
2
x c .(I)
C Q1
n
Energy loss (eV)
F
0'08
7
>, - 0.06 ^
^
I
Energy loss (eV)
cyclohexene . gas
t
0.06
-
-
benzene - gas
h
fi
2
cyclohexene - solid
v
x
f-' .-
(I)
C
I
0.00
/
f
0
10
.
1
.
20
I
.
30
40
50
60
Energy loss (eV)
1,3-cyclohexadtene - gas
0.06 h 7
5
0.04
Energy loss (eV)
A
c
\
10
20
1,3-cyclohexadiene - solid
30
40
50
60
Energy loss (eV) Figure 4. Probability of a given energy loss event in the track of a 1 MeV electron in gaseous (dotted lines) and in solid (solid lines) hydrocarbons: (a) cyclohexane and gaseous n-hexane (dashed line), (b) cyclohexene, (c) 1,3-~yclohexadiene,(d) 1,4-cyclohexadiene, and (e) benzene.
carbons. Of course, the chemical nature of radiation-induced reactants formed from the parent hydrocarbons may be different. 4.2. Energy Deposition Distributions. Energy loss by electrons with initial energy from 10 keV to 1 MeV has been simulated in both gas and solid phases of benzene, 1,3-
-cyclohexadiene, 1,4-~yclohexadiene,cyclohexene, and cyclohexane using the Monte Carlo method described. The predicted distributions of energy loss events in the hydrocarbons for a 1 MeV electron are shown in Figure 4a-e. It is readily apparent that the differences in the dipole oscillator strength distributions
Electron Energy Loss Distributions in Hydrocarbons
J. Phys. Chem., Vol. 99, No. 26, 1995 10547
TABLE 6: Effect of Incident Electron Enerm on the Enerw Loss by Electrons in Gaseous Hydrocarbon@~ ~
energy (eV) 5 x 103 1 x 104 2 104
5 x lo4 105 2 x los 5 x los 1 x lo6 1
~
spur 0 30 44
52 54 56 57 58
cyclohexane blob (w)(eV) 0 5000 69 io 14 47 40 16 17 38 36 17 17 35 35 17
spur 0 30 44
52 54 56 57 57
cyclohexene blob ( w ) (eV) 0 5 o O 0 10 64 14 44 16 37 17 35 17 34 17 33 17 33
~
1,3-~yclohexadiene spur blob ( w ) (eV) 0 0 5 o O 0 30 10 63 44 14 43 51 17 37 54 17 35 55 18 34 56 18 33 57 18 32
~
~~
1,4-~yclohexadiene spur blob (w)(eV) 0 0 5000 30 10 63 44 14 43 51 17 37 54 17 35 55 18 34 56 18 33 57 18 32
spur 0 28 41
48 50 52 53 53
_
_
_
benzene blob (w)(eV) 0 11 16 18
19 19 19 20
5000 77 53 45 43 42 41 40
a Spur = percentage of energy deposited in events smaller than 100 eV, blob = percentage of energy deposited in events larger than 100 eV and smaller than 500 eV, and short track = 100 - spur - blob = percentage of energy deposited in events larger than 500 eV and smaller than 5 keV. Mean energy in deposition event.
TABLE 7: Effect of Incident Electron Energy on the Energy Loss by Electrons in Solid Hydrocarbon& cyclohexane cyclohexene 1,3-~yclohexadiene 1,4-~yclohexadiene energy (eV) spur blob ( w ) (eV) spur blob (w)(eV) spur blob ( w ) (eV) spur blob (w)(eV) spur 5 x 103 0 0 5000 0 0 5000 0 0 5000 0 0 5000 0 1 x 104 29 10 93 29 10 94 29 10 92 29 10 92 28 42 15 42 15 2 x 104 43 15 64 63 63 41 42 15 64 49 17 54 48 5 104 50 17 54 49 18 54 49 18 53 52 18 51 1 x 105 52 18 52 52 18 51 51 53 18 51 2 x 105 54 18 49 53 18 50 52 53 19 49 53 19 49 54 19 48 55 18 49 53 5 x 105 55 18 48 19 48 54 55 19 47 1 x 106 55 18 48 54 55 19 47 56 18 47
benzene blob ( w ) (eV) 0 5000 11 92 15 63 18 54 19 51 19 49 19 48 19 47
Spur = percentage of energy deposited in events smaller than 100 eV, blob = percentage of energy deposited in events larger than 100 eV and smaller than 500 eV, and short track = 100 - spur - blob = percentage of energy deposited in events larger than 500 eV and smaller than 5 keV. Mean energy in deposition event.
for the gases are reflected in the energy loss distributions. Gaseous benzene has a sharp peak in its distribution at 7-8 eV followed by three overlapping peaks at 13, 18, and 22 eV. The first peak is due to the (exciton) peak at -7 eV in the dipole oscillator strength distribution. Energy losses of less than 100 eV account for 86% of the events in the electron track. The simulation for gaseous 1,3-~yclohexadienegives a distribution qualitatively similar to that for benzene. Its distribution has four energy loss peaks at 6, 9, 12, and 16 eV. However, the four peaks all overlap, and their spread is much narrower than found for the benzene distribution. About 91% of the energy loss events are smaller than 100 eV. The energy loss distribution for gaseous 1,.l-cyclohexadiene has two low-energy peaks, a sharp peak at 8-9 eV, and a broad peak at 13 eV. In contrast, cyclohexene has a broad, slightly structured, energy loss distribution with a maximum at 15 eV. Cyclohexane has two peaks (1 1 and 15 eV); however, these peaks overlap. The spread of the peaks in 1,Ccyclohexadiene, cyclohexene, and cyclohexane is the nearly the same as observed for gaseous 1,3cyclohexadiene and different from that for benzene. Energy loss events smaller than 100 eV account for 91%, 91%, and 90% of the events in the tracks of a 1 MeV electron in 1,4cyclohexadiene, cyclohexene, and cyclohexane, respectively. In contrast to the gases, the energy loss distributions of the solid hydrocarbons are very similar. The distributions are broad and essentially structureless consisting of a single broad peak with a maximum in the range 19-24 eV. The distribution for benzene does have a low-energy peak at 8 eV; however, it is much smaller than the main peak (and the corresponding lowenergy peak in the distribution for the gas phase). At least 96% of the energy loss events are smaller than 100 eV for a 1 MeV electron in solid benzene, 1,3-~yclohexadiene,1,4-cyclohexadiene, cyclohexene, and cyclohexane. The figures show a considerable effect of phase on the energy loss distributions for all the hydrocarbons studied. The most obvious effects are a loss of structure in condensation from gas to solid and a shift of the energy loss distribution to larger energy losses for the
solid phase. The magnitude of the shift depends slightly on the hydrocarbon. Both of the predicted phase effects on the energy loss are direct consequences of the influence of phase on the dipole oscillator strength distributions. In radiation chemistry, the structure of energetic electron tracks is frequently described in terms of three different types of energy loss event^.^^,^^ These events are known as spurs (w 100 eV), blobs (100 < w 500 eV), and short tracks (500 eV w 5 keV). Originally, the designation was made to aid in modeling the radiation chemical kinetics and was supposed to reflect of the effects of the relative spatial distributions of the reactants on the chemistry. While this view of the chemistry is simplistic, categorization of the energy loss events in this manner is still quite useful in the visualization of these processes. The effects of incident electron energy on the fraction of energy resulting in spurs, blobs, and short tracks and on the mean energy of an event are shown for the gas phase hydrocarbons in Table 6 and for the solids in Table 7. By definition, every electron track terminates in a short track. Consequently, the fractions of energy deposited in spurs and in blobs increase from zero at the 5 keV short track limit to asymptotic values for energetic electrons that depend on the medium. In the gas phase, the spur, blob, and short track fractions for a 1 MeV electron are 53%, 20%, and 27% for benzene. These values are somewhat different from those obtained for cyclohexane, which has a higher spur fraction (58%) and smaller blob and short track fractions (17% and 25%, respectively). The results for a 1 MeV electron in gaseous 1,3cyclohexadiene, 1,4-cyclohexadiene, and cyclohexene are essentially the same as for cyclohexane. In the solid phase, the fractions of energy deposited in the three different track entities are almost the same in all the hydrocarbons. For a 1 MeV electron, the spur fraction is 55 f 1% and the blob fraction is 18-19% with the remainder of the energy in short tracks. The fraction of energy deposited in spurs is larger in solid than in gaseous benzene; however, effect of condensation is to decrease
10548 J. Phys. Chem., Vol. 99, No. 26, 1995
the spur fraction for 1,3-~yclohexadiene,1,4-~yclohexadiene, cyclohexene, and cyclohexane. The magnitude of the track averaged mean energy loss by an electron decreases with increasing incident energy. It reaches a limiting value for high-energy electrons because of the final short track; i.e., for the same reason the spur and blob fractions increase with increasing incident electron energy. The mean energy loss of a 1 MeV electron in gaseous benzene is 40 eV and is considerably larger than in the other hydrocarbon gases considered. The mean energy losses of a 1 MeV electron in 1,3-~yclohexadiene,1,4-~yclohexadiene,and cyclohexene are 32 eV, while that for cyclohexane is 35 eV. In the solid phase, the mean energy loss of a 1 MeV electron is in the range 4748 eV in all five of the hydrocarbons. Condensation from gas to solid results in a considerable increase ('34%) in the mean energy loss. Comparison of the results obtained here for the solid hydrocarbons shows considerable differences from earlier predictions for liquid and solid water and solid DNA.3' The energy loss distributions for the hydrocarbons have a larger and narrower main peak. About 82% of the energy loss events in the solid hydrocarbons are smaller than 40 eV as compared to only 71% for liquid and solid water and solid DNA. Furthermore, the mean energy loss for an event in a 1 MeV electron track in the solid hydrocarbons is 47-48 eV, which is considerably smaller than the values of 60 eV in solid water and 57 eV in liquid water and solid DNA. The spur and blob fractions for a 1 MeV electron are 49% and 20% in solid water, 52% and 19% in liquid water, and 52% and 20% for solid DNA. These values are slightly different from those for the solid hydrocarbons. Notice that there is little difference between the energy loss properties of energetic electrons in the condensed phases of water. This similarity implies that the data obtained for the solid hydrocarbons should be an acceptable approximations for liquid hydrocarbons.
5. Conclusions Dipole oscillator strength distributions have been constructed for solid and gaseous cyclohexane, cyclohexene, 1,3-cyclohexadiene, 1,4-~yclohexadiene,and benzene. These distributions have been used to predict various energy loss parameters for and the attenuation of energetic electrons. Monte Carlo simulations performed using cross sections obtained from the constructed dipole oscillator strength distributions predict that the energy losses by energetic electrons with incident energies from 10 keV to 1 MeV are similar for the solid hydrocarbons. The mean energy loss in an event produced by a 1 MeV electron is 47-48 eV with 55 i 1% of the energy deposited in events smaller than 100 eV. The simulations predict considerably different results for each of the gaseous hydrocarbons considered. The mean energy loss ranges from 32 eV for 1,3cyclohexadiene and 1,4-~yclohexadieneto 40 eV for benzene, while the fraction of energy loss in events smaller than 100 eV decreases from 58% for cyclohexane to 53% for benzene. Calculations of the density normalized stopping powers, inverse mean free paths, and CSDA ranges show large effects due to condensation, especially for low-energy electrons. In general, there is a large variation in the energy loss properties between the gases, but the majority of the parameters obtained for the solids are independent of the hydrocarbon.
LaVerne and Pimblott The results of the calculations presented suggest that, when appropriately density normalized, the physical dimensions of the clusters of highly reactive radicals and ions produced by fast electron radiolysis will be similar for all solid (and by inference liquid) hydrocarbons. Acknowledgment. The research described herein was supported by the Office of Basic Energy Sciences of the U.S. Department of Energy. This is contribution NDRL-3814 from the Notre Dame Radiation Laboratory. References and Notes (1) Foldiak, G., Ed. Radiation Chemist? of Hydrocarhons: Elsevier: Amsterdam, 1981. (2) Hummel, A. Radiation Chemistry of Alkanes and Cycloalkanes. In The Chemist? of Alkanes and Cycloalkanes: Patai, S., Rappopon, Z., Eds.: John Wiley and Sons: New York, 1992: p 743. (3) Tabata, Y . , Ed. Pulse Radiolysis; CRC Press: Boca Raton. FL, 1991. (4) Bums. W. G.; Jones, J. D. Trans. Faraday Soc. 1964, 60, 2022. (5) Bums. W. G.: Reed, C. R. V. Trans. Faradav Soc. 1970.66. 2159. (6) Bums, W G Hopper. M J , Reed, C R V Tranc Faradm Soc 1970, 66, 2159 (7) Brocklehurst, B. J . Chem. Soc., Faraday Trans. 1992. 88. 167. (8) Brocklehurst, B. J . Chem. Soc.. Faraday Trans. 1992, 88, 2823. (9) Bartczak. W. M.; Hummel, A. Radiat. Phys. Chem. 1994, 44, 335. (10) Hatano, Y . The Chemistry of Synchrotron Radiation. In Radiation Research: Fielden, E. M., Fowler, J . F., Hendry. J. H., Scott. D.. Eds.: Taylor & Francis: London, 1987; p 35. (1 1) Pimblott, S. M.; LaVeme, J. A,; Mozumder, A,; Green, N. J. B. J . Phys. Chem. 1990, 94, 488. (12) LaVeme, J. A , ; Mozumder, A. J , Phys. Chern. 1986, 90. 3242. (13) Mozumder, A.: LaVeme, J. A. J . Phys. Chem. 1984. 88, 3926. (14) Killat, U. J . Phys. C.: Solid State Phys. 1974. 7, 2396. (15) Frohlich. H. Theor?; ofDielectrics: Oxford: London, 1958: Chapter 1.
(16) Fano, U.; Cooper, J. W. Rev. Mod. Phys. 1968. 40. 441. (17) Veigele, Wm. J. At. Data Tables 1973, 5 , 51. (18) Jhanwar, B. L.; Meath. W. J.; MacDonald. J. C. F. Can. J . Phys. 1981, 59, 185. (19) Pimblott, S. M.; LaVeme. J . A. J . Phys. Chem. 1991, 95, 3907. (20) LaVeme. J. A,; Pimblott, S. M.: Mozumder, A. Radiar. Phys. Chern. 1991, 38, 75. (21) Ashley, J. C.; Williams, M. W. Studies ofthe lnteracfionsoflonizing Radiations with Communications Materials, Report RADC-TR-83-87: Rome Air Development Center: Griffiss Air Force Base, NY, 1983; p 5. (22) Ashley. J. C. J. Electron Spectrosc. Relat. Phenom. 1988, 46, 199. (23) Pinkett. L. W.: Muntz. M.: McPherson, E. M. J . Am. Chrm. Soc. 1951, 73, 4862. (24) Kumar, A , ; Meath, W. J. Mol. Phgs. 1992, 75. 3 11. (25) Birth. M.: Lubart. R.; Steinberger. I. T. J . Chem. Phys. 1971. 54. 5 104. (26) Williams, M. W.; MacRae. R. A,: Hamm. R. N.: Arakaua, E. T. Phys. Rev. Lett. 1969, 22, 1088. (27) Sowers. B. L.; Arakawa, E. T.: Birkhoff. R. D. J . Chem. Phys. 1971, 54, 2319. (28) MacRae. R. A,; Williams, M. W.: Arakawa. E. T. J . Chern. Phys. 1974, 61, 861. (29) Inagaki, T. J . Chem. Phys. 1973, 59, 5207. (30) Daniels, J. Opt. Commun. 1971, 3, 240. (31) LaVeme, J. A.: Pimblott. S. M. Radiat. Res. 1995, 141. 208. (32) Berger. M. J.; Seltzer, S. M. Stopping Powers and Ranges of Electrons and Positrons, 2nd ed.;Report NBSIR 82-2550-A; U S Department of Commerce, National Bureau of Standards: Washington. DC, 1983. (33) ICRU Report 37, Stopping Powers f o r Electrons and Positrons: International Commission on Radiation Units and Measurements: Bethesda. MD, 1984. (34) Ashley, J. C. J . Electron Spectrosc. Relar. Phenom. 1982, 28, 177. (35) Mozumder, A.: Magee, J. L. J . Chem. Phys. 1966, 45, 3332. (36) Mozumder, A,; Magee, J. L. Radiat. Res. 1966, 28. 203.
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