Positronium Formation Dynamics in Radiolytic Tracks: A Computer

1124

J. Phys. Chem. B 2002, 106, 1124-1130

Positronium Formation Dynamics in Radiolytic Tracks: A Computer Simulation Study Alfonso Alba Garcı´a,† Simon M. Pimblott,‡ Henk Schut,† Anton van Veen,† and Laurens D. A. Siebbeles*,† IRI, Delft UniVersity of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands, and Radiation Laboratory, UniVersity of Notre Dame, Notre Dame, Indiana 6556-0579 ReceiVed: July 11, 2001; In Final Form: October 17, 2001

Positron track structures were simulated by stochastic modeling of the collision-by-collision slowing down of positrons in n-hexane. The details of the inelastic scattering of positrons and of electrons produced in ionizing collisions were taken into account until the particle energies had degraded to less than 25 eV. Further slowing down to thermal energy was assumed to result in a spherically symmetric thermalization distribution around the position where the particle energy becomes less than 25 eV. The dynamics of positronium formation in competition with electron-cation recombination and positron annihilation was followed by a simulation of the diffusive motion of the charged species in each other’s Coulomb field. Positronium formation was found to involve reaction of the thermalized positron with electrons produced along the last few hundred nm of its track-end, corresponding to several keV of energy attenuation. Positronium formation occurs mainly during the first few tens of picoseconds after positron implantation and extends to hundreds of picoseconds. Simulation reproduces the experimental positron lifetime spectra reported in the literature. The experimentally reported electric-field dependence of the ortho-positronium yield in liquid n-hexane could be reproduced with an exponentially decaying positron thermalization distribution with a mean distance of 50 nm, while a Gaussian positron thermalization distribution was found to be inappropriate.

I. Introduction (e+,

antiparticle of the electron) When an energetic positron is injected into a material it slows down by scattering on the molecules in the medium, and it eventually decays by annihilation with an electron to produce γ quanta. The positron can decay by direct annihilation with a bound electron from the molecules of the material. In a variety of insulating materials some of the injected positrons form positronium (Ps) prior to annihilation. Ps is a bound positron-electron pair, which can be considered as a hydrogen atom in which the proton has been replaced by a positron. Two different Ps states can be distinguished: para-positronium (p-Ps), in which the positron and electron spin have opposite directions, and ortho-positronium (o-Ps) in which the spins are parallel. Direct annihilation of a thermalized positron with a bound electron in the material occurs on a time scale of 100-500 ps.1 If the positron and the electron were to be at rest, the annihilation would result in the production of two 0.511 MeV γ-quanta, which would move in opposite directions. However, due to the finite momentum distribution of the positron and the electrons in a material, the annihilation quanta exhibit a Doppler shift, which can be as large as 1 keV, and a deviation from exact collinearity of the order of milliradians. Measurements of Doppler broadening (DB) and angular correlation of annihilation radiation (ACAR) can thus be used to investigate the momentum distribution of electrons in a material.1-4 Since p-Ps has zero spin angular momentum it can decay into two γ-quanta by intrinsic annihilation with a lifetime of * Corresponding author. E-mail: [email protected]. Fax: ++ 31152787421. † Delft University of Technology. ‡ University of Notre Dame.

about 125 ps. In some materials Ps may become localized in small cavities. Due to this confinement, the p-Ps has a finite momentum distribution, which also results in a Doppler shift and angular deviation of the annihilation quanta. However, these effects are smaller than for the case of direct positron annihilation. The Doppler broadening and angular correlation of the radiation produced by annihilation of p-Ps can provide information on the size and shape of voids in materials.5,6 The total spin angular momentum of o-Ps is equal to one and in vacuum conservation of spin angular momentum makes annihilation of o-Ps into three γ-quanta the only possible decay channel. In a vacuum, this process occurs with a lifetime of 142 ns. However, in condensed matter the positron in o-Ps usually decays on a much shorter time scale of 1-5 ns by annihilation with an electron from the medium that has a spin opposite to that of the positron. This process is referred to as “pick-off” annihilation. The precise lifetime for pick-off annihilation of o-Ps depends on the size and the shape of the cavity in which o-Ps resides. Hence, positron annihilation lifetime (PAL) spectroscopy can be used as a tool to study the occurrence and nature of free volume and voids in condensed matter.5,7 Taking into consideration the annihilation processes described above, a PAL spectrum thus consists of contributions due to annihilation of p-Ps with a lifetime of ∼125 ps, direct annihilation of positrons on a time scale of 100-500 ps, and pick-off annihilation of o-Ps with a lifetime of 1-5 ns. For a more complete understanding of DB, ACAR, and PAL experiments, in which Ps is used as a probe of the microscopic structure of materials, insight into the factors that govern Ps formation is essential. According to the Ore model “hot” Ps is formed during the slowing down of the positron at energies in the “Ore gap”, which typically ranges from several to a few tens of eV.1 However, the inhibition of Ps formation in organic

10.1021/jp012611z CCC: $22.00 © 2002 American Chemical Society Published on Web 01/09/2002

Positronium Formation Dynamics in Radiolytic Tracks liquids by electron scavengers1,8-13 and the observed effects of an external electric field on the yield of Ps in nonpolar liquids and polymers14-16 cannot be explained in terms of the Ore model. These effects are explained in terms of Ps formation by reaction of a thermalized positron with an electron produced along its radiolysis track in competition with cation-electron recombination. According to the “spur” model,16-18 the positron interacts only with the last few electrons produced in the final spur of the positron, when it slows down from typically 100 eV to thermal energy over a distance of several nm. However, the positron and electrons could diffuse over longer distances so that the positron can react with electrons that are produced along the final track segment corresponding to more than 100 eV of energy attenuation.19,20 Note that the formation of Ps requires the presence or formation of a cavity, which is sufficiently large to accommodate Ps. The aim of the present work is to provide theoretical insight into the formation of Ps by reaction of a thermalized positron with the electrons produced along its track. Ps formation via track processes is of particular importance for the interpretation of positron annihilation studies on materials with a low dielectric constant, such as nonpolar hydrocarbons and polymers, since in these materials the Coulomb interactions between the charged species extend over relatively long distances. In the present work liquid n-hexane is considered as an example, since for this material the data necessary to perform a computer simulation of the positron track structure and Ps formation kinetics are known. The track structure, i.e., the initial spatial distribution of the cations and electrons produced by ionizing events of the incident positron and/or secondary electrons with sufficient energy, is obtained from stochastic computer simulations as described in section IIA. The procedure to simulate the motion of the charged species in each other’s Coulomb field leading to charge recombination and Ps formation is described in section IIB. The results of the computer simulations concerning the effect of the positron track length, the kinetics of Ps formation, and consequences for the interpretation of PAL spectra, as well as the effects of an external electric field on Ps formation, are discussed in section III. The conclusions are presented in section IV. II. Theoretical Framework A. Initial Track Structure. The structures of tracks produced by the slowing down of a positron in the target medium were obtained by stochastic simulation of collision-by-collision slowing down of the positron and the secondary, tertiary, etc. electrons produced in ionizing collisions, analogous to the work in refs 21 and 22. The medium was considered as a continuum and the distance between successive collisions was sampled from a Poisson distribution with an energy dependent mean distance. This mean distance is the inverse of the product of the total cross section for scattering on a molecule and the density of the molecules. The total cross section is the sum of the elastic scattering cross section and the inelastic cross sections for excitation and ionization of the molecules. The probabilities for an elastic collision, excitation, or ionization are equal to the relative value of the corresponding cross sections. In case of an elastic event, the direction of motion of the incident particle after the collision was sampled from the angular dependence of the cross section. For inelastic collisions, the energy loss was sampled from the differential energy loss distribution and the energy of the incident particle is reduced by the energy loss. The angular deviation of the incident particle and the direction of motion of the electron that is produced in the case of an

J. Phys. Chem. B, Vol. 106, No. 5, 2002 1125 ionizing collision were calculated from classical kinematics. The trajectories of the incident positron and the electrons produced by ionizing collisions were followed until their energy had degraded to less than 25 eV, since no appropriate cross sectional data is available for lower energies. It was assumed that the further slowing down of the particles to thermal energy occurs without inducing ionizations. The position where the particles reach thermal energy was sampled from a spherically symmetric thermalization distribution centered at the position where the energy of the particle became less than 25 eV. In the case that an incident positron or electron induces an ionization of a target molecule involving an energy transfer such that the kinetic energy of the ejected daughter electron is less than 25 eV, the center of the thermalization distribution of the daughter electron corresponds to the position of the cation produced. From a comparison of simulated and experimental ion escape yields from high-energy electron tracks in n-hexane, it was found that the radial electron thermalization distribution can be described by an exponentially decaying function

fexp(r)r2 dr )

1 exp(-r/b)r2 dr br2

(1a)

with the average electron thermalization distance b ) 8 nm.22 The positron thermalization distribution is unknown. The exponentially decaying function of eq 1a was used for the positron with the average thermalization distance, b, as an adjustable parameter. However, it cannot a priori be excluded that the shape of the positron thermalization distribution differs from that in eq 1a. It has been demonstrated theoretically23 that the distribution in eq 1a is obtained for a random walk involving a single step, while an infinite number of steps gives a Gaussian distribution of the form

fGauss(r)r2 dr )

( )

32 4r2 2 exp r dr π2b3 πb2

(1b)

The distribution functions in eqs 1a and 1b can thus be considered as extreme cases of a thermalization process involving a single collision or a large number of collisions. It is one of the aims of the present study to determine the positron thermalization distribution by comparing the theoretical data with those from experiments. Therefore, both the exponentially decaying function in eq 1a and the Gaussian distribution in eq 1b were considered for the positron. The cross sections for elastic scattering of positrons or electrons on the molecules were calculated from the atomic cross sections using the independent atom model.24 According to this model, the elastic cross section for scattering on a molecule is expressed in terms of the atomic scattering amplitudes fi according to

dσ dΘ

N

) 2π

∑ fi*fj i,j)1

sin(κrij) κrij

sin Θ

(2)

where rij is the distance between atoms i and j in the molecule, κ ) 2k sin(Θ/2) is the momentum transfer of the incident positron or electron, k is the incident particle momentum, and θ is the scattering angle. The atomic scattering amplitudes, fi, for carbon and hydrogen were calculated using the computer program PWASCH,25 which employs the static field approximation with nonrelativistic partial wave analysis to describe elastic scattering of electrons and positrons by neutral atoms. The positron and electron scattering cross sections differ due to

1126 J. Phys. Chem. B, Vol. 106, No. 5, 2002 exchange effects that are included for electrons. The distances between the atoms in an n-hexane molecule needed to evaluate eq 2 were obtained assuming C-C and C-H bond lengths equal to RC-C ) 1.54 Å and RC-H ) 1.09 Å, respectively, and an angle of 110° for both the C-C-C and H-C-H bonds.26 The cross sections for inelastic scattering of positrons and electrons in liquid n-hexane were calculated using the methodology described in ref 27, which incorporates effects of exchange for electron scattering. According to this formalism the generalized oscillator strength distribution is obtained by quadratically extending the optical dipole oscillator strength distribution to nonzero momentum transfer. Integration of the generalized oscillator strength distribution over all possible momentum transfers for a given energy loss gives the inelastic scattering cross section. In the calculations the experimental optical dipole oscillator strength distribution for solid cyclohexane28 was used, due to the lack of data for n-hexane. To determine whether an energy transfer from a positron or an electron to a molecule gives rise to an ionization, the ionization efficiency must be known. In liquid n-hexane the ionization threshold is close to 9 eV.29 The dependence of the ionization efficiency of liquid n-hexane on the energy transfer was estimated from experimental photoionization data for cyclohexane in the gas phase.30 According to these experimental data the ionization efficiency increases approximately linearly from the ionization threshold to become equal to one for energy transfers, leading to excitation of 2 eV or more above the ionization onset. For each inelastic collision a random number uniformly distributed in the interval [0,1> was sampled. If this random number was less than the ionization probability, the collision was considered to result in ejection of an electron from the target molecule or otherwise to give rise to an excitation. The kinetic energy of the electron produced by an ionizing collision is equal to the energy transfer reduced by the electron binding energy in the target molecule. In the track structure simulations ionization was assumed to occur from outer and inner valence molecular orbitals and from the 1s atomic core shell on the carbon atoms. The binding energies of the outer valence orbitals were assumed to be equal to 9 eV; i.e., the onset for ionization. According to X-ray photoelectron spectroscopy data for gaseous n-hexane, the inner valence electrons have binding energies in the range 18-30 eV.31 To bring into account the fact that ionization energies in the condensed phase are usually ∼1.5 eV lower than in the gas phase,29 the binding energies of the inner valence electrons were taken equal to 16.5 eV. The binding energies of the core electrons of carbon atoms were taken equal to 284 eV. The relative probabilities for ionization from the outer and inner valence orbitals were estimated from experimental data on photodissociation of n-butane.32 In the work of ref 32 the branching ratios for fragmentation into different products are given as a function of photon energy. Fragments produced at photon energies less than 20 eV are considered to be due to ejection of electrons from the outer valence shell, while fragments that are produced only at higher energies are considered to result from ejection of electrons from the inner valence shell. The relative abundance of the different fragments for different photon energies was taken as the probability for ejection of an electron from the outer and inner valence orbitals. Energy losses exceeding 284 eV may result in ejection of a valence electron but may alternatively cause ionization from a 1s core orbital on a carbon atom. For energy losses larger than 284 eV, the probability for ionization from the valence orbitals was estimated to be 0.44 by extrapolation of the inelastic

Alba Garcı´a et al. scattering cross section for energy losses below 284 eV to higher energy losses. In case of an ionizing collision, the orbital from which the electron is ejected was determined by comparison of a uniformly distributed random number in the interval [0,1> with the probabilities for ionization from the different orbitals. Ionization from the carbon core shell was assumed to be followed by Auger decay, resulting in the production of a double charged cation. The energy of the Auger electron that is ejected was taken equal to the difference of the binding energies of the core and the outer valence electron. B. Computer Simulation of Charge Recombination and Positronium Formation Kinetics. After the initial track structure has been determined, computer simulation of the motion of the charges is performed in order to obtain the kinetics of positronium formation in competition with cation-electron recombination and direct positron annihilation. The computer simulation method has been described previously.33 Briefly, the displacement δr of a charged particle during a small time step δt is given by

δr ) µ E δt + x6 D δt R

(3)

The first term represents the drift of the particle in the electric field, E, due to its Coulomb interactions with all the other particles and its interaction with any externally applied electric field. The mobility of the particle, denoted by µ, is related to the diffusion coefficient, D, by µ ) eD/kT, where e is the charge of an electron, k the Boltzmann constant, and T the temperature, taken as 293 K. The second term in eq 3 describes the motion of the particle due to random Brownian diffusion. The vector R is spherically symmetrically distributed and has a uniformly distributed length, chosen such that 〈R2〉 ) 1. The time step δt was taken such that the diffusive displacement of the charges never exceeded 10% of the smallest distance between two charges in the system. Further reduction of the time step gave similar results. After each time step the Coulomb interactions between the charges at their new positions were recalculated. The electrons can recombine with a cation or can react with a positron to form positronium. These reactions were assumed to occur when the distance between the oppositely charged species became smaller than 1 nm. Since the electron and positron spins are random, the probabilities for formation of p-Ps and o-Ps were taken equal to the statistical probabilities of 1/4 and 3/4, respectively. These probabilities correspond to the spin degeneracy of 1 for p-Ps (singlet state) and 3 for o-Ps (triplet state). In addition, the decay of the positron by direct annihilation was taken into account using a typical lifetime of τe+ ) 450 ps. The lifetime for annihilation of p-Ps was taken equal to τp-Ps ) 125 ps, while a pick-off lifetime equal to τo-Ps ) 4 ns was assumed for o-Ps. The time dependent fractions of cations, electrons, positrons, and Ps were obtained with a statistical error less than ∼5% by averaging over typically 1000 simulations of the initial track structure and dynamics of the thermalized species. The mobilities of electrons and the cations in n-hexane were taken to be 0.09 cm2 V-1 s-1,34,35 and 0.68 × 10-3 cm2 V-1 s-1,36 respectively. The positron mobility was taken equal to the experimental value of 53 cm2 V-1 s-1 reported recently.37 The Coulomb interactions between the charged species were calculated using 1.89 for the dielectric constant of n-hexane.26 III. Results and Discussion A. Initial Track Structure. The distance traveled by the positron can be characterized by the vector distance |Xf - Xi| between the initial position of the positron (Xi) and the position

Positronium Formation Dynamics in Radiolytic Tracks

Figure 1. Energy dependence of the vector distance and axial distance of positrons in liquid n-hexane. The inset schematically shows the definition of these distances.

where the positron has reached an energy below 25 eV (Xf). The drawn curve in Figure 1 shows the mean vector distance as a function of the initial positron energy, obtained by averaging over 104 track structure simulations. The dashed curve in Figure 1 shows the axial distance, which is the component of the vector Xf - XI parallel to the initial direction of the positron trajectory. The axial distance is somewhat smaller than the vector distance due to angular deflection (straggling) of the positron during slowing down. The vector distance increases from 40 nm for a positron energy of 100 eV to 40 µm for an initial energy of 50 keV. For energies above 1 keV the distance increases with energy, E, as En with n ) 1.76. Such energy dependence has been found earlier for simulated positron tracks in carbon38 and for electrons in polyethylene.39 The latter reflects the fact that for higher energies the energy loss properties for positrons and electrons in saturated hydrocarbons are similar.27 In the simulated tracks the average yield of ionizations was found to be 4.6 per 100 eV of energy deposition. This implies that, for instance, a 6 keV positron produces around 275 cation-electron pairs. B. Ps Formation Dynamics. Figure 2 shows the simulated fraction of the incident positrons that react to form o-Ps for different energies of the incident positron. The results shown were obtained using the exponentially decaying thermalization distribution in eq 1a for the positron with mean distances, b+, as indicated. Use of the Gaussian distribution in eq 1b with the same mean thermalization distances gave similar results. The o-Ps fraction first increases with energy and reaches a constant value for energies above about 2 keV. According to the results in Figure 1, this implies that Ps formation involves the reaction of a positron with electrons that were produced along the last 200 nm of the positron track. For a positron mobility of 53 cm2 V-1 s-1 the time needed to diffuse over this distance is t ) r2/6D ∼ 40 ps. This time is significantly shorter than the direct annihilation lifetime of 450 ps. The amount of Ps formed is not limited by the positron lifetime but by the time scale for recombination of electrons and cations, as will be discussed below. The results in Figure 2 demonstrate that a positron interacts with a significant part of its track end and that it is inappropriate to consider only the final positron spur or blob as was done in the work of refs 16-20. The simulated Ps fraction becomes smaller as the mean positron thermalization distance is increased, see Figure 2. This result is due to the fact that for a larger thermalization range

J. Phys. Chem. B, Vol. 106, No. 5, 2002 1127

Figure 2. Fraction of incident positrons that form o-Ps as a function of the initial kinetic energy of the positron. The error bars correspond to the statistical uncertainty in the results and the lines are drawn as a guide to the eye.

more positrons become thermalized away from the track end, and consequently encounter between a positron and an electron is less likely. For a mean positron thermalization distance equal to b+ ) 50 nm, the simulated o-Ps fraction is 42%. This value agrees with the experimental results reported in refs 11, 12, and 40, which were obtained by fitting a sum of three exponentials to PAL spectra measured for n-hexane,11,12,40 see the values of I3 in Table 1. Note that the mean positron thermalization distance needed to reproduce the experimental o-Ps fractions is about six times larger than that of electrons in n-hexane.22 This suggests that for low energies the scattering cross section of positrons in n-hexane is smaller than that of electrons. It is not unlikely that the positron scattering cross section for impact energies of 10-30 eV is smaller than the electron scattering cross section, since such an effect has been observed for scattering on (halogenated) methane and propane molecules.41-43 Figure 3a shows the simulated time dependence of the positron, electron, and Ps fractions on a linear time scale of 200 ps, while data on a longer logarithmic time scale are presented in Figure 3b. The data were obtained from simulations with an exponential positron thermalization distribution (eq 1a) with b+ ) 50 nm. Use of a Gaussian distribution (eq 1b) with the same mean distance gave similar results. The initial positron energy was taken to be 6 keV, so that the Ps yield is not limited by the track length, see Figure 2. The data in Figure 3a show that the majority of the electrons decay on a time scale of ∼20 ps, due to recombination with cations. The positron fraction shows a rapid decay on the same time scale, due to reaction of a positron with an electron to form Ps, which is most likely when most of the electrons have not yet recombined with a cation. Indeed, the initial positron decay correlates with the formation of Ps, which also takes place mainly during the first 20 ps after positron implantation. At longer times the positron mainly decays by direct annihilation with a lifetime of 450 ps, see Figure 3b. The p-Ps fraction starts to decay after about 20 ps, which is due to the fact that at longer times the annihilation of p-Ps with a lifetime of 125 ps dominates over the production of Ps. The decay of p-Ps is clearly visible in Figure 3b. The data in Figure 3a show that the o-Ps fraction continues to increase up to times exceeding a few hundred ps, due to reaction of positrons with excess electrons that have not recombined with molecular cations. At longer times the o-Ps fraction also decays

1128 J. Phys. Chem. B, Vol. 106, No. 5, 2002

Alba Garcı´a et al.

TABLE 1: Literature Data of Fractions and Lifetimes Attributed to p-Ps (I1, τ1), Positrons (I2, τ2) and O-Ps (I3, τ3), Obtained from a Fit of Eq 5 to Experimental Lifetime Spectraa expt.40 expt.11,12 calculated fractions (eq 4) fit of eq 3 calculated fractions (field of 20 kV/cm) fit of eq 3 (field of 20 kV/cm)

I1 (%)

I2 (%)

I3 (%)

t1 (ps)

τ2 (ps)

τ3 (ps)

20.6 ( 0.7 27.4 Fp-Ps ) 13.8 17.6 Fp-Ps ) 9.9 10.2

37.1 ( 0.7 30.8 Fe+ ) 44.0 40.6 Fe+ ) 60.9 59.63

42.4 ( 0.1 41.8 Fo-Ps ) 42.2 41.8 Fo-Ps ) 29.2 29.5

155 ( 7 189 τp-Ps ) 125* 143 τp-Ps ) 125* 126

475 ( 7 531 τe+) 450* 448 τe+ ) 450* 451

4010+10 3964 τo-Ps ) 4000* 4022 τo-Ps ) 4000* 4005

a The third row shows the calculated fractions obtained from the simulations using eqs 4b-4c. The lifetimes used as input in the simulations are marked with an asterisk (*). The fourth row shows the intensities and lifetimes obtained from a fit of eq 5 to the simulated lifetime spectrum. The fifth and sixth rows show the results obtained for the case of an external electric field of 20 KV/cm.

Fe+ )

∫0∞

fe+(t) τe+

(4b)

Fp-Ps )

∫0∞

fp-Ps(t) dt τp-Ps

(4c)

Fo-Ps )

∫0∞

fo-Ps(t) dt τo-Ps

(4d)

and

respectively. The drawn line in Figure 4 is the PAL spectrum obtained from eq 4a with the time-dependent fractions shown in Figure 3. The total fractions calculated from eqs 4b-4d are presented in the third row of Table 1. In the case that Ps formation would occur promptly, i.e., directly after positron implantation on a time scale much smaller than τp-Ps, the annihilation rate could be described by the first-order decay of positrons and Ps according to

R(t) )

Figure 3. Time-dependence of the remaining fractions of positrons, electrons, and Ps on a linear time scale of 200 ps (a) and a longer logarithmic time scale (b).

due to pick-off annihilation with a lifetime of 4000 ps, see Figure 3b. The p-Ps fraction is initially about a factor of 3 smaller than the o-Ps fraction, due to the statistical probabilities of 1/4 and 3/4 for p-Ps and o-Ps formation, respectively (see section II.B). The ratio of the p-Ps and o-Ps fractions decreases with time due to the shorter annihilation lifetime of p-Ps. C. Analysis of a Simulated Lifetime Spectrum. The timedependent fractions of positrons, p-Ps and o-Ps, denoted by fe+(t), fp-Ps(t), and fo-Ps(t), respectively, which are shown in Figure 3, can be used to simulate a PAL spectrum. The positron annihilation rate is given by

R(t) )

fp-Ps(t) fe+(t) fo-Ps(t) + + τp-Ps τe+ τo-Ps

(4a)

Note that the total fraction of the incident positrons decaying by direct annihilation, by formation of p-Ps and o-Ps prior to annihilation, can be calculated according to

I1 exp(-t/τ1) I2 exp(-t/τ2) I3 exp(-t/τ3) + + (5) τ1 τ2 τ3

with I1, τ1, I2, τ2, and I3, τ3 the fractions and lifetimes of positrons, p-Ps and o-Ps, respectively. The annihilation rate obtained from fitting eq 5 with the intensities and lifetimes as fitting parameters to the simulated PAL spectrum is shown by the open circles in Figure 4. The result of the fit agrees very well with the simulated PAL spectrum. The intensities and lifetimes obtained from the fit are shown in Table 1. The intensities of the two long-lived components (I2 and I3) are close to the calculated fractions of positrons (Fe+) and o-Ps (Fo-Ps), while the lifetimes from the fit correspond to those used in the simulations. However, the fitted intensity (I1 ) 17.6%) and lifetime (τ1 ) 143 ps) of the first component differ from the calculated p-Ps fraction (Fp-Ps ) 13.8%) and lifetime (τp-Ps ) 125 ps). The reason is that p-Ps is formed on a time scale of the order of 20 ps, which is not negligible compared to the p-Ps lifetime of 125 ps. The ratio I1/I3, obtained from the fit is equal to 17.6/41.8 ) 0.42. This value is higher than 1/3, which is the statistical formation ratio of p-Ps and o-Ps that was used in the simulations, see section II.B. Such a deviation has been obtained from PAL experiments on a variety of organic liquids 1,11,12,40 and polymers.44-47 The data in Table 1 show that the intensities obtained from a fit of eq 5 to the simulated PAL spectrum are close to the experimental values for n-hexane from refs 11, 12, and 40. This demonstrates that the PAL spectrum obtained from the computer simulations agrees with the experimental results. D. Electric Field Effect on Ps Formation. The formation of Ps by reaction of a positron with an electron produced along its track should be affected by an external electric field.

Positronium Formation Dynamics in Radiolytic Tracks

Figure 4. The simulated PAL spectrum (solid line) together with a fit of eq 5 to the spectrum (open circles).

J. Phys. Chem. B, Vol. 106, No. 5, 2002 1129 thermalization distribution is thus most accurately described by the exponentially decaying function given in eq 1a. The intensities and lifetimes obtained by fitting eq 5 to a PAL spectrum simulated in the presence of an external electric field of 20 kV/cm are presented in Table 1. The intensities are smaller than in the absence of an external electric field, as discussed above. It is interesting to note that the ratio I1/I3 is now equal to 1/3 and the intensities from the fit agree with the simulated fractions obtained using eqs 4b-4d. In addition, the lifetimes used in the simulations are recovered by the fit. This can be understood as follows. In the presence of an external electric field, the positron and the electrons drift away from each other and consequently Ps formation can only occur prior to separation, which reduces the effective time scale of Ps formation. For an external electric field strength of 20 kV/cm it was found that Ps is formed mainly during the first 10 ps after positron implantation, which is approximately a factor two smaller than in the absence of an electric field. This reduction of the Ps formation time makes it more appropriate to describe the PAL spectrum by a first order exponential decay of positrons and Ps, as is implicitly assumed in the use of eq 5 to describe the PAL spectrum. IV. Conclusion

Figure 5. Experimental field-dependence of the o-Ps yield in liquid n-hexane taken from ref 16, and simulated yields obtained using the exponentially decaying and the Gaussian positron thermalization distributions defined in eq 1 with b+ ) 50 nm.

Measurements of PAL spectra for different hydrocarbon liquids and polymers have indeed shown an electric-field dependence of o-Ps formation.16,19 In the experiments positrons are emitted from 22Na in a random direction with respect to the direction of the external electric field.16 To simulate the experimental conditions, the initial direction along which the positron moves was taken randomly from an isotropic distribution. The initial positron energy was taken to be 6 keV, which was found to be sufficiently high so that the o-Ps yield is not limited by the track length. The simulated electric-field dependence of the o-Ps yield is shown in Figure 5 together with the experimental results from ref 16. Both the simulations and the experimental results show a reduction of the o-Ps yield with electric field strength. The net effect of an external electric field is to cause a drift motion of the charges such that the positron is separated from the electrons produced along its track. The experimental results are reproduced by using the exponentially decaying positron thermalization distribution in eq 1a with a mean thermalization distance b+ ) 50 nm. Using the Gaussian positron thermalization distribution in eq 1b with the same average positron thermalization distance gives a comparable o-Ps fraction for zero external field, see Figure 5. However, in that case the simulated reduction of the o-Ps yield with electric field strength is significantly larger than in the experiment. The positron

Positron track structures were obtained from stochastic simulations of the slowing down of positrons in liquid n-hexane. The inelastic scattering cross sections of the incident positron and of the electrons produced in ionizing collisions were calculated using a formalism reported earlier. According to this formalism, the generalized oscillator strength for scattering of the charged particles is obtained by a quadratic extension of the optical dipole oscillator strength into the nonzero momentum plane. The effects of exchange on scattering of electrons were included. The elastic scattering cross sections were calculated using the independent atom model with atomic scattering amplitudes obtained from numerically solving the Schro¨dinger equation for positron and electron scattering on the atoms constituting the target molecule. Due to the lack of accurate cross sections for low energies, it was assumed that the positron and electrons become thermalized in a spherically symmetric distribution around the position where their energy has degraded to less than 25 eV. The dynamics of Ps formation and positron annihilation, in competition with electron-cation recombination, was obtained by simulation of the motion of the charged particles in their mutual Coulomb field. According to the simulations, Ps formation occurs by reaction of a thermalized positron with electrons produced along its radiolysis track. The positron was found to interact with a significant part of the track end, corresponding to several keV of energy attenuation, equivalent to a range of a few hundred nanometers. Ps formation occurs mainly on a time scale of several tens of picoseconds and extends to hundreds of picoseconds following positron implantation. Experimental positron lifetime spectra were reproduced by the computer simulations. It was shown that the delayed formation of Ps affects the lifetime spectrum in such a way that a fit of a sum of three exponentials to a lifetime spectrum does not give the correct p-Ps fraction and lifetime. The experimentally reported electric-field dependence of the o-Ps yield in liquid n-hexane could be reproduced with an exponentially decaying positron thermalization distribution with a mean distance equal to 50 nm. Simulations with a Gaussian positron thermalization distribution gave results that differ appreciably from experiment.

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