J. Phys. Chem. B 2000, 104, 9607-9614
9607
Energy Loss by Nonrelativistic Electrons and Positrons in Polymers and Simple Solid Hydrocarbons Simon M. Pimblott,*,† Jay A. LaVerne,† A. AlbaGarcia,‡ and Laurens D. A. Siebbeles‡ Notre Dame Radiation Laboratory, Notre Dame, Indiana 46556-0579, Interfaculty Reactor Institute, Mekelweg 15, 2629 JB Delft, The Netherlands ReceiVed: March 28, 2000; In Final Form: July 20, 2000
The energy loss and transport of electrons and positrons in gaseous cyclohexane, and solid cyclohexane, benzene, polyethylene, and polystyrene are determined and discussed. A formalism for the energy loss properties of nonrelativistic positrons, analogous to that previously developed for electrons, is presented. This approach makes use of the dipole oscillator strength distributions of the medium, which are constructed here for polyethylene and polystyrene from their optical properties and from atomic X-ray photoabsorption coefficients. Inelastic collision cross-sections, mean free paths, stopping powers, energy loss distributions, mean energy losses, and csda ranges are evaluated for electrons in gaseous and solid cyclohexane, solid benzene, polyethylene, and polystyrene. Those properties depending on the zeroth energy moment of the inelastic collision cross-section are similar for the solid media, while those depending on the first energy moment show significant differences. The data for electrons are discussed and compared and contrasted with the results of similar calculations for positrons. For energies greater than ∼1 keV, the energy loss properties of electrons and positrons are similar, but for lower energies there are very apparent differences.
Introduction Energetic positrons play an important role in a number of techniques employed for material characterization, for instance, lifetime spectroscopy, positron reemission microscopy, and measurement of angular correlation and Doppler broadening of the annihilation rate.1-5 Analysis of positron experiments is usually performed in two stages: (i) positron implantation, and then (ii) diffusion, reaction, and annihilation of the thermalized positron.6 To understand the effects of all types of ionizing radiation, it is important to develop knowledge of the energyloss processes of the particles as they pass through the medium of concern. There have been a large number of Monte Carlo studies of the track structure of nonrelativistic electrons in water;7-10 however, developments in other media have been limited.11 A number of standard packages for calculating the physical effects of high-energy electrons and positrons do exist, such as GEANT (available from the CERN Program Library), ITS (NERSC High Performance Computing Facility) and EGS.12 However, these simulation tools are not appropriate for investigating the collision-by-collision track structure of low and intermediate energy ( γ′. In addition, in the case of an incident electron, if it is assumed that the more energetic of the two electrons after the collision is the primary, then V12/2 > V22/2 > 0 (or γ′ > 2γ - V2/2). Conservation of momentum gives V - V1 < q < V + V1 or equivalently -(V2 - 2γ) + V(V2 - 2γ)1/2 > γ′ > 0 > -(V2 2γ) - V(V2 - 2γ)1/2 since γ ) γ′+q2/2 and V1 ) (V2 - 2γ)1/2. These inequalities reduce to
0 < γ′ < - (V2 - 2γ) + V1/2(V2 - 2γ)1/2 ) γ′max or alternatively,
γ- ) 1/4V2[1 + 2γ′/V2 - (1 - 4γ′/V2)1/2] < γ < 1/4V2 [1 + 2γ′/V2 + (1 - 4γ′/V2)1/2] ) γ+ Combination of the energy and momentum conservation criteria gives the constraints shown in Figure 1 for an electron and a positron. The allowed regions of the γ-γ′ energy plane are shaded as denoted in the key. Notice the significant reduction of the allowed region for an energetic electron compared to an energetic positron because of the possibility of electron exchange. A formulation for evaluating the nonrelativistic energy loss parameters for an energetic electron has been discussed
Figure 1. Allowed region of the γ-γ′ plane for an electron and for a positron. Electron ) left area under the curve; positron ) total area under the curve.
previously.18-20 The energy loss parameters for a nonrelativistic positron are somewhat more straightforward to describe than those of an electron. For a positron, it is not necessary to introduce exchange, and at nonrelativistic energies (less than ∼0.1 MeV), virtual annihilation and subsequent recreation of an electron-positron pair can be ignored. Making use of the optical approximation of Ashley21 and the appropriate energy and momentum transfer constraints gives the following expression for the “differential cross-section”,
τ(E,γ) )
1 2πE
∫γ′γ′
max
Im[-1/(0,γ′)]G(γ,γ′)γ′ dγ′
low
with γ′max as defined above. For a positron
Gpos(γ,γ′) )
1 γ(γ - γ′)
and γ′low ) 0 for all values of γ. For an electron16
Gelec(γ,γ′) )
1 1 + γ(γ - γ′) (V2/2 - γ)(V2/2 - γ + γ′) 1
xγ(γ - γ′)(V2/2 - γ)(V2/2 - γ + γ′) (because of exchange), and γ′low ) 0, when 0 < γ < V2/4 or γ′low ) 2γ - V2/2 when V2/4 < γ < 3V2/8. The expression for τ can now be inserted into the equations above to evaluate the inverse mean free path and the stopping power. Since there are no singularities in the region of the γ-γ′ plane over which the integration is performed, the order of integration can be reversed. Ultimately, the mathematical analysis gives
1 Λpos-1(E) ) χ 2
∫0E/2Im[-1/(0,γ′)]ln
[
]
(1 - a + s)(1 + a - s) dγ′ (1 - a - s)(1 + a + s)
[
]
(1 - a + s)
∫0E/2Im[-1/(0,γ′)]ln (1 - a - s) γ′ dγ′
1 Spos(E) ) χ 2
for a positron,23,24 and
Energy Loss by Nonrelativistic Electrons and Positrons
∫
[
J. Phys. Chem. B, Vol. 104, No. 41, 2000 9609
1 1 E/2 Λelec-1(E) ) χ 0 Im[-1/(0,γ′)] ln 2 a (1 - a + s)(1 + a - s) 2 F arcsin(s/(1 1+a (1 - a - s)(1 + a + s) 1-a γ a)), dγ′ 1+a E
[
]
∫0
1 Selec(E) ) χ 2
[
[
]
E/2
Im[-1/(0,γ′)][Σ]γ′ dγ′
] [ ] [
]
(1 + a) 1-a 1 + ln + ln [Σ] ) ln 1-a-s a (1 + a + s) (1 - a + s)(1 + a) 1-a - F arcsin(s/(1 - a)), + 1+a (1 - a)(1 + a + s)
[
]
[
1 1 ln 1 + a/2 - xa(1 + a/4) 2 a
∫0γ′
dγ′γ′ Im[-1/(0,γ′)]
∫0γ′
dγ′ Im[-1/(0,γ′)] ln
1χ 2N
)
1χ 2N
max
]
∫γE′dγ Gpos(γ,γ′) -
[
]
for 0 < E′ < 3E/4, while for 3E/4 < E′ < E,
∫0E/2dγ′γ′ Im[-1/(0,γ′)]∫γγ dγ Gpos(γ,γ′) ∫0γ′ dγ′γ′ Im[-1/(0,γ′)]∫E′γ dγ Gpos(γ,γ′)
1χ 2N 1χ 2N
+
-
+
max
) Λpos-1(E)/N -
1χ 2N
∫γ′γ
+
max
dγ′ Im[-1/(0,γ′)] ln
[
∫0γ′
max
dγ′ Im[-1/(0,γ′)][Σ2]γ′/E
with
[Σ2] ) 1/a ln (b - a)(1 - b + a)(1 + a - s)(1 - a + s) 2 + F b(1 - b)(1 - a - s)(1 + a + s) (1 + a) 2 1 + a - 2b 1 - a arcsin , F arcsin(s/(1 1-a 1+a 1+a
[
[
(
)
]
]
[
1-a a)), 1+a
]
σelec(E,E′) ) Λelec-1(E)/N -
∫γ′2E′ - Edγ′ Im[-1/(0,γ′)]
1χ 2N
max
[Σ3]γ′/E with
[
]
[
(b - a)(1 - b + a) 1 2 + [Σ3] ) ln F arcsin a 1+a b(1 - b)
(1 +1 a--a 2b), 11 +- aa]
These expressions are considerably more complicated than those for the positron, due to the possibility of electron exchange and its effect on the bounds. In the formalism described above, there is no division between inelastic events leading to ionization and to excitation. While, in the gas phase it is possible to measure photoionization crosssections, this is presently not feasible in the condensed phase. To partition the dipole oscillator strength distribution into states leading to ionization and excitation in all the hydrocarbons and polymers considered here would be a very complex task. In the absence of suitable experimental data, it would require a large number of unverifiable approximations. Results and Discussion
max
(b - a)(1 + a - s) b(1 - a - s)
σpos(E,E′) )
1χ 2N
while for E/2 < E′ < 3E/4,
for an electron.18 The variables a and s are given by a ) γ′/E, and s ) (1 - 2a)1/2. The function F(x,y) is an incomplete elliptic integral of the first kind, and the parameter χ is equal to 1/(πE) in the reduced units employed here. Another energy loss parameter of interest in many problems, in particular in track structure simulation studies, is the ratio of the cumulative inelastic cross-section, σ(E,E′), to the total inelastic cross-section, σ(E). This ratio [commonly known as the Y function25 and denoted here, Y(E,E′)] describes the probability of an energy loss smaller than a value, E′. Since calculation of the cumulative inelastic cross-section involves an energy loss E′ less than E′max careful consideration of the bounds is necessary when performing the integration. For a positron the bounds are given by 0 < γ′ < -(V2- 2γ) + V(V2 - 2γ)1/2 with the maximum possible value of γ′ ()E/2) occurring when γ ) 3E/4. This value of γ′ does not correspond to the maximum energy loss possible, which is E. Consequently,
σpos(E,E′) )
σelec(E,E′) )
b(1 - a + s) (b - a)(1 + a + s)
]
Here b ) E′/E, and a and s and the limits γ-, γ+, and γ′max are as defined above. The cumulative inelastic cross-section for an electron in a variety of different media has been discussed previously.16 The bounds of the domain of integration are given by the inequalities 0 < γ′ < - (V2 - 2γ) + V(V2- 2γ)1/2 and γ′ > 2γ - V2/2. The maximum permissible energy loss is 3E/4 when γ′ ) E/2. For an energy loss, E′, in the range 0 < E′ < E/2,
Oscillator Strength Distributions. The central feature of the methodology for evaluating the energy-loss properties of electrons and positrons in matter is the role of the energy loss function, Im[-1/(0,γ)], which is proportional to the dipole oscillator strength distribution, f(γ). Previous studies of water have shown that an inappropriate dipole oscillator strength distribution can result in appreciable errors in the calculated energy loss parameters.17,26 Construction of the dipole oscillator strength distribution from the optical properties of a medium has been described in detail previously.19 The dipole oscillator strength distributions of polyethylene and polystyrene were constructed following this methodology using the experimental data of ref 27 for polyethylene and of ref 28 for polystyrene, in addition to the compilation of photoabsorption cross-sections in ref 29 These dipole oscillator strength distributions are compared in Figure 2 with those of gaseous and solid cyclohexane and of solid benzene that were derived in ref 19. Cyclohexane is frequently used as a simple model for polyethylene, because it has the same base unit: cyclohexane has 3(CH2-CH2) units and polyethylene n. Benzene does not have the same base unit as polystyrene, C6H6 vs C6H5CHCH2, however, the two units are similar. The most obvious difference between the dipole oscillator strength distributions shown in Figure 2 is that due to condensation; there is a considerable
9610 J. Phys. Chem. B, Vol. 104, No. 41, 2000
Pimblott et al.
Figure 3. Effective number of electrons that can receive an energy transfer. Key as in Figure 2.
TABLE 1: Comparison of Calculated and Experimental Mean Excitation Energies Ieff (eV) (this work)
Figure 2. Dipole oscillator strength distributions of various hydrocarbon materials. (a) (Dash dot line) Gaseous cyclohexane; (solid line) solid cyclohexane; (long dash) polyethylene. (b) (Dotted line) Solid benzene; (short dash line) polystyrene.
shift (∼7-8 eV) to higher energy of the peak of the distribution for cyclohexane. The two solid-phase hydrocarbons and the two hydrocarbon polymers do not show significant differences from each other. They as well the other solid hydrocarbons considered in ref 19 exhibit a broad maximum peaked at ∼22-27 eV and a sharp rise corresponding to the carbon K-shell at ∼284 eV. There are a number of small differences between the dipole oscillator strength distributions of the solid materials. These differences are not immediately apparent in Figure 2 and are best demonstrated by considering the effective number of electrons that can receive an energy transfer, Zeff ) ∫f(γ)dγ. The quantity Zeff/Z, where Z is the total number of electrons per molecular unit, is plotted as a function of energy in Figure 3. The discontinuities in the all curves observed at ∼284 eV are real and arise from the electrons in the carbon K-shell. For energies smaller than 100 eV, the curve for gaseous cyclohexane is well separated from those for the solid-phase materials, because of the large shift in the dipole oscillator strength distributions. The curve for solid benzene has a somewhat different energy dependence from those for the other solid materials. Close inspection of the data reveals that at energies
