- Email: [email protected]
Electron thermalization and electron–ion recombination in liquid argon
Chemical Physics Letters 379 (2003) 20–27 www.elsevier.com/locate/cplett
Electron thermalization and electron–ion recombination in liquid argon Mariusz Wojcik a
a,*
, M. Tachiya
b
Institute of Applied Radiation Chemistry, Technical University of Lodz, Wroblewskiego 15, 93-590 Lodz, Poland National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8565, Japan
b
Received 13 June 2003; in final form 16 July 2003 Published online: 3 September 2003
Abstract The processes of electron thermalization and geminate electron–ion recombination in liquid argon at 87 K are studied by computer simulations based on the Cohen–Lekner theory of electron transport in liquids. The mean thermalization time of 1.8 ns is obtained, which is in agreement with experiment, and the mean thermalization distance is calculated as 2600 nm with the standard deviation of 1100 nm. The probability of geminate recombination is found to be very low (103 ). Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Electronic processes in ionized liquid argon (LAr) have been a subject of significant fundamental and applied research. Theoretical studies [1–9] were aimed at interpreting the measured high values of the electron mobility [10–12] and free-ion yield [13–15] in this system. On the other hand, the high free-ion yield and other favorable physical properties of LAr made this liquid an important material for constructing high-resolution ionization detectors [15,16]. As in other condensed systems, post-ionization electronic processes in LAr are conventionally
*
Corresponding author. Fax: +48-42-6840043. E-mail address: [email protected] (M. Wojcik).
divided into several phases. First, the electrons ejected from atoms undergo thermalization, in which they dissipate their initially high-kinetic energy to the surrounding medium and become near-thermal. Then, the electrons perform a diffusive motion under the influence of the Coulomb field of their parent cations, and may geminately recombine, or escape and become free charges. In the next stage, spatially uncorrelated electrons and cations undergo bulk recombination. The charges may also be collected by an external electric field, if it is applied to the system. Theoretical description of the electron thermalization requires a model of electron scattering processes in the considered system. As a liquefied rare gas, LAr is very specific. It lacks vibrational degrees of freedom that normally play an important role in electron collisions. One of the applicable
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.08.006
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
models of the electron scattering in LAr is that of Cohen and Lekner [1], in which spatial correlations in the liquid are represented by the structure factor, and different rates of electron energy and momentum transfers are postulated. Approximate analytical calculations of the electron thermalization in LAr based on this model were done by Mozumder [17], who estimated the thermalization time in LAr as 1.4 ns, and the mean thermalization distance as 1400 nm. Due to the high electron mobility of LAr, the mechanism of electron transport in this system cannot be described as an ideal diffusion [18]. Therefore, both the geminate and bulk electron– ion recombination in LAr cannot be treated by the standard theory [19] of diffusion-controlled reactions between ions. Specific models of electron transport and reactions are required to describe the electron–ion recombination in high-mobility systems. In our recent work [7], we have proposed a model of the electron scattering in LAr, which is based on the Cohen–Lekner theory and especially suitable for simulations. We have successfully applied this model to calculations of the fielddependent electron mobility, as well as the bulk electron–ion recombination rate constant. Considering very good agreement between our calculated results of electron mobility and experiment in a wide range of the electric field strength, we believe that the simulation model of [7] properly describes the electron scattering processes in LAr in a range of energies up to at least several electronvolts. Therefore, this model should also be applicable to calculations of the electron thermalization in LAr. Moreover, such calculations could naturally be extended to modeling of the geminate phase of electron–ion recombination. We have pursued this idea and carried out a simulation study of both the electron thermalization and geminate electron–ion recombination in LAr.
2. Model of electron scattering in liquid argon The Cohen–Lekner theory [1] of hot electrons in liquids predicts that the energy and momentum of an electron gained from an electric field are transferred to the medium at different rates, each
21
characterized by its own ‘‘mean free path’’. This theory was further developed by Atrazhev and Iakubov [3], who constructed energy-dependent cross-sections for both the energy and momentum transfers in liquid Ar, Kr, and Xe. In [7] we have proposed a mechanistic model of electron motion in LAr, in which the processes of energy and momentum transfer are separated from each other, in accordance with the Cohen–Lekner theory, and the electron scattering is described in terms of individual electron collisions. This makes the model applicable to simulation calculations. Following Atrazhev and Iakubov [3], we assume that the transfer of electron energy can be described in terms of an effective cross-section rE . We represent this process by a special type of electron collisions, with the mean free path kE ¼ 1=nrE , where n is the density. In these collisions the change of the electron kinetic energy is calculated from the classical expression for isotropic elastic scattering by gas atoms. Unlike normal elastic scattering, however, we assume that these collisions change only the absolute value of the electron velocity, and not its direction. Therefore, the change of the electron momentum in these events is negligible compared to that in normal elastic scattering. Physically, these ‘‘energy transfer’’ collisions may be to some extent identified with the electron scattering by interatomic vibrations (phonons). The transfer of electron momentum is also described in terms of a cross-section, which we denote here as rp . We may expect that the momentum is most efficiently exchanged in large angle scattering by rare gas atoms, and in the simulation we represent this process by isotropic elastic electron–atom collisions, with the mean free path kp ¼ 1=nrp . For the cross-sections rE and rp we assume the forms proposed in [3]. rE is assumed to be constant at energies lower than several electronvolts. At higher energies, it is assumed to be equal to the momentum transfer cross-section in the low-pressure Ar gas, rAr . We plot rE and rAr in Fig. 1 as the solid and dotted lines, respectively. The crosssection rp is calculated from the equation rp ðEÞ ¼ rE ðEÞS 0 ðEÞ;
ð1Þ
22
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
merge into one type of collisions, which is the elastic electron scattering by rare gas atoms.
3. Simulation method
Fig. 1. Cross-sections rE (solid line) and rp (dashed line) that are used in the simulation model of electron transport in liquid argon at 87 K. The dotted line shows the momentum transfer cross-section rAr for argon gas [28]. The inset shows the function S 0 ðEÞ for liquid argon [3].
where S 0 ðEÞ is obtained from the structure factor SðkÞ determined for the momentum transfer hk on the assumption of isotropic electron scattering [3]. At small k the structure factor is known experimentally (Sð0Þ ¼ 0:048 for LAr), while at large momentum transfers SðkÞ ! 1. The assumed form of S 0 ðEÞ [3] is shown in the inset to Fig. 1, and the cross-section rp ðEÞ calculated from Eq. (1) is plotted in Fig. 1 as the dashed line. A more detailed description of the cross-sections used in our model was given in [7]. The model of electron scattering in LAr described above can be readily applied in a simulation in the following way. Both types of the electron collisions in LAr are related to elastic electron scattering by rare gas atoms. Since rp 6 rE (except in a narrow region of energies around 5 eV), we can simply generate elastic electron collisions, with the mean free path kE , and assume that a fraction (rp =rE ) of these collisions change the direction of the electron motion, while the others change only the magnitude of the electron velocity. With increasing electron energy the crosssections rE and rp approach each other and, finally, become equal to the gas phase cross-section. Therefore, at high energies the two types of electron collisions introduced in our simulation model
Modeling of the electron thermalization and geminate electron–ion recombination processes requires a simulation method to calculate electron trajectories in the Coulomb field of a cation and to simulate electron collisions according to the assumed scattering model. In this work, we apply the simulation method that was originally designed for modeling the bulk electron–ion recombination in dense rare gases [20] and adapted to the case of LAr in [7]. In this method, we calculate classical electron orbits in the Coulomb field of a cation and solve numerically the Kepler equation [21] to find the position of an electron on its orbit at a time corresponding to a collision. The details of the method can be found in [20]. Time intervals between the electron collisions depend on the scattering cross-section rE ðEÞ. We use the null-collision method [22–24], which is an efficient method of modeling collisions in a system where the collision cross-section depends on energy. In this method, in addition to real electron collisions with the cross-section rE , we introduce fictitious null collisions that do not change the electron velocity v. The cross-section rn ðvÞ for the null collisions is chosen in such a way that the total collision rate Kmax is independent of velocity vrE ðvÞ þ vrn ðvÞ ¼ Kmax ¼ const:
ð2Þ
The times between the collisions are generated from the exponential distribution with the mean value s¼
1 ; nKmax
ð3Þ
and the decision as to whether a particular collision is real or null is made at random, with the probability of real collisions given by pr ¼
vrE ðvÞ : Kmax
ð4Þ
The real electron collisions are modeled in the following way. First, we determine the velocity,
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
vM , of an atom of mass M that takes part in the collision by generating a random vector according to the Maxwell–Boltzmann distribution. We also calculate the relative velocity u ¼ v vM . Then, we calculate the new velocity from the classical expression for an isotropic elastic collision v1 ¼
Mu mv þ MvM nþ ; mþM mþM
ð5Þ
where n is a random vector of unit length. Since our model of electron scattering in LAr involves two types of real electron collisions, we have to decide whether a given collision is an ‘‘energy transfer’’ collision or a ‘‘momentum transfer’’ collision. This decision is made at random, with the acceptance probability for ‘‘momentum transfer’’ collisions given by pp ¼
rp ðvÞ : rE ðvÞ
ð6Þ
If a ‘‘momentum transfer’’ collision occurs, we set the electron velocity to v0 ¼ v1 :
23
4. Results and discussion The calculations described in this work were carried out for LAr at T ¼ 87 K and n ¼ 2:11 1022 cm3 . The dielectric constant e ¼ 1:49 calculated from the Clausius–Mosotti equation [11] was used. In the simulations of the electron thermalization process electrons were started with the initial energy E0 in the subexcitation range (<12 eV). The dissipation of electron energy above this range is known to be very fast [17], and the electrons ejected from atoms in ionization events are expected to reduce their energy to the subexcitation range at distances from the cations not larger than several angstroms. In the simulations we assumed the initial electron–cation distance to and used two values of the initial enbe r0 ¼ 5 A ergy E0 ¼ 1 and 5 eV. These values correspond to the initial electron kinetic energies of 2.93 and 6.93 eV, respectively. The simulation results for the mean electron kinetic energy hEk i as a function of time are shown in Fig. 2. We see that hEk i
ð7Þ
In the other case, when an ‘‘energy transfer’’ collision takes place, the electron velocity is set to v0 ¼
v1 v: v
ð8Þ
In the calculations of the electron thermalization process we start an electron with a total energy E0 at a distance r0 from the cation and set its initial velocity in a random direction. Then, we calculate the electron trajectories and collisions using the simulation method described above. In the course of the simulation we record the electron kinetic energy and its distance from the cation at given time steps. The simulation is repeated for a large number of independently generated electrons, which allows us to calculate the mean electron kinetic energy and mean electron–cation separation as functions of time. During the simulation the electrons may undergo geminate recombination with the cation. We use the energy criterion of recombination [18,20] and assume that the reaction takes place when the total relative energy of the electron–cation pair falls below Ecrit ¼ 10kB T , where kB T is the thermal energy.
Fig. 2. Time dependence of the mean kinetic energy of electrons undergoing thermalization in LAr. The results represented by the solid and dashed lines were obtained for electrons started with the energy E0 ¼ 1 and 5 eV, respectively. The inset shows the transient conductivity of LAr resulting from pulsed irradiation measured by Sowada et al. [27]. The horizontal dotted lines show the mean thermal energy Eth ¼ ð3=2ÞkB T , and the energy of 1:1 Eth that is used as the operational thermalization criterion.
24
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
decreases with time and asymptotically approaches the thermal level Eth ¼ ð3=2ÞkB T . When we use the operational thermalization criterion, which assumes that the electrons are thermalized when their kinetic energy reaches the level higher than Eth by no more than 10%, we obtain the mean electron thermalization time in LAr as tth ¼ 1:8 ns. We see from Fig. 2 that the value of tth practically does not depend on the initial energy E0 . This means that the simulation results should not be very sensitive to the choice of r0 either. In Fig. 3, we show the time dependence of the mean electron–cation distance hri obtained from the same simulation runs as those described in Fig. 2. Again in this case the results show little dependence on the value of E0 . When we define the mean thermalization distance, rth , as the value of hri at t ¼ tth , we obtain rth ¼ 2600 nm. Fig. 4 shows the contour maps of the probability distribution function f ðr; EÞ that the electrons undergoing thermalization in LAr will have energy E and the electron–cation distance r calculated for t ¼ 100 ps, 500 ps, 1 ns, and 1.8 ns. The calculations of f ðr; EÞ were carried out for 105 electrons generated with the initial energy E0 ¼ 5 eV. In each plot f ðr; EÞ is normalized to its maximum at a given time. From the data obtained at
Fig. 3. Time dependence of the mean electron–cation distance. The results represented by the solid and dashed lines were obtained for electrons started with the energy E0 ¼ 1 and 5 eV, respectively.
t ¼ 1:8 ns the standard deviation of the thermalization distance is calculated as about 1100 nm. By extending the simulations to later times we have also determined the recombination probability, Prec , and the escape probability, Pesc ¼ 1 Prec , which is related to the experimentally observed free-ion yield. In this phase of the calculations we used the same simulation method as that applied in modeling of the electron thermalization, except that the simulation runs were not terminated at a certain time but were carried on until the electron either reacted with the cation or escaped to a large distance rmax . This allowed us to obtain the apparent recombination probability Pr , which is somewhat different from the true recombination probability Prec because of the finite value of rmax . The results of the calculations carried out for electrons started with the initial energy E0 ¼ 5 eV are presented in Fig. 5. For each value of rmax at least 105 independent electron trajectories were simulated. The standard error of the calculated recombination probability is about 104 . The true recombination probability Prec may be found as the asymptotic value of Pr ðrmax Þ as rmax ! 1. To determine Prec we plotted lnð1 Pr Þ against rc =rmax , as it is shown in Fig. 5. Here, rc denotes the Onsager radius defined as e2 =ekB T (rc ¼ 128:9 nm in LAr at 87 K). We have found that for small values of rc =rmax the dependence of lnð1 Pr Þ on rc =rmax is very well approximated by a straight line. By extrapolating the straight line to rc =rmax ¼ 0 we obtained the true recombination probability in LAr as Prec ¼ 0:0016. The results presented above were obtained using the simulation method based on the electron scattering model that is derived from the Cohen–Lekner theory and verified in our earlier calculations of the electron mobility and bulk recombination rate constant in LAr [7]. The good agreement between those calculations and experiment, especially in the case of the mobility, allows us to expect that the present results are also of quantitative value. We have calculated the thermalization time in LAr as tth ¼ 1:8 ns, which is not very different from the value of 1.4 ns obtained by Mozumder [17] from approximate analytical calculations also based on the Cohen–Lekner theory. However, the mean thermalization distance obtained in our study,
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
25
Fig. 4. Contour maps of the probability distribution function f ðr; EÞ for electrons undergoing thermalization in LAr calculated at t ¼ 100 ps, 500 ps, 1 ns, and 1.8 ns. Each distribution is normalized to its maximum, and the contours are shown for the heights of 0:1; 0:2; . . .. The solid line shows the Coulomb potential energy e2 =er in LAr (crossing of this line by one of the contours at t ¼ 1:8 ns is a graphical artifact).
Fig. 5. Results of the simulation of geminate electron–ion recombination in LAr at 87 K. Plots of Pr vs rmax =rc (circles), and lnð1 Pr Þ vs rc =rmax (squares) are shown. The error bars correspond to the standard error of the simulation results. The solid line shows a linear regression fit of lnð1 Pr Þ ¼ cðrc =rmax Þ calculated for rmax =rc P 30. rc ¼ e2 =ekB T ¼ 128:9 nm.
rth ¼ 2600 nm, is significantly larger than the standard deviation of the thermalization length b ¼ 1400 nm reported in [17]. The distribution of thermalization distances in LAr was derived by Freeman et al. [25,26] from an analysis of the electric field dependence of free-ion yields in LAr based on the Onsager theory of geminate electron–ion recombination. Their values of the mean thermalization distance are an order of magnitude smaller than that obtained in our study. This suggests that the method they used to extract the thermalization lengths from experimental data might not be appropriate for LAr, as it has already been pointed out elsewhere [17]. The electron thermalization in LAr was studied experimentally by Sowada et al. [27], who measured the microwave conductivity in this system resulting from pulsed irradiation. The electron mobility in LAr is known to increase with decreasing electron energy [10], so the conductivity is
26
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
expected to increase during the thermalization. A plot of the transient conductivity in LAr obtained in [27] is presented in the inset to Fig. 2. We see that the conductivity initially increases, reaching a maximum at about 1.8 ns. This value is identical with tth obtained in our calculations. At longer times the conductivity decays due to the electron– ion recombination. The measured thermalization time in LAr is listed in Table 1 of [27] as 0.9 0.2 ns. It was obtained as the time required for 10–90% development of the conductivity corrected for the finite time response of the microwave detector and for the recombination effect. The difference between the result of [27] and our tth is partly accounted for by different definitions of the thermalization time used in the two studies. We have found that the probability of geminate electron–ion recombination in LAr is very low (Prec ¼ 0:0016). The Onsager theory of diffusioncontrolled geminate ion recombination predicts the 0 recombination probability Prec ¼ 1 expðrc =r0 Þ. Taking the mean thermalization distance rth as r0 0 we have Prec ¼ 0:048. The geminate recombination probability in LAr obtained in this study is much 0 lower than Prec , as it can be expected in the system of high-electron mobility, where the electron transport cannot be described as ideal diffusion [18]. Considering the facts that the rate constant of bulk electron–ion recombination in LAr calculated by us in [7] was lower than the available experimental results, and that the methodology of the present study is similar to that of [7], we cannot exclude the possibility that Prec obtained in this work may underestimate the recombination probability to some extent. But even if we take this into account, the geminate recombination probability in LAr should still be expected well below 0.01. Discussion of the geminate electron–ion recombination in LAr is complicated by the fact that the distances between successive ionizations in radiation tracks in this system are normally much shorter than the mean electron thermalization distance, so the geminate electron–cation pairs constituting the track are not completely isolated from one another. Theoretical models were proposed to take into account this effect [8,9], in which the cylindrical symmetry of the charge distribution was assumed. In those models, however, the electron motion in
LAr was approximated by diffusion. It would be interesting to apply the present model of the electron motion in LAr to calculations of the track recombination processes in this system. The calculations presented in this work are directly applicable to modeling of the recombination of geminate electron–ion pairs created by photoionization of photosensitive dopants in LAr by scintillation light [29,30]. Photosensitive molecules are added to LAr in order to increase the amount of collected charge in ionization chambers and thus improve their energy resolution. The results of our calculations indicate that practically all electrons in such systems escape geminate recombination. The use of the Onsager theory in interpretation of the experiments with photosensitive dopants in LAr [30] is therefore not justified.
Acknowledgements This work was supported by Grant No. 3 T09A 049 19 from the Polish State Committee for Scientific Research and by the COE development program of MEXT of Japan.
References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
M.H. Cohen, J. Lekner, Phys. Rev. 158 (1967) 305. S. Basak, M.H. Cohen, Phys. Rev. B 20 (1979) 3404. V.M. Atrazhev, I.T. Iakubov, J. Phys. C 14 (1981) 5139. K. Kaneko, Y. Usami, K. Kitahara, J. Chem. Phys. 89 (1988) 6420; J. Chem. Phys. 90 (1989) 6810. R.A. Holroyd, W.F. Schmidt, Annu. Rev. Phys. Chem. 40 (1989) 439, and references therein. A.F. Borghesani, D. Iannuzzi, G. Carugno, J. Phys.: Condens. Matter 9 (1997) 5057. M. Wojcik, M. Tachiya, Chem. Phys. Lett. 363 (2002) 381. W.G. Burns, A. Mozumder, Chem. Phys. Lett. 142 (1987) 381. A. Mozumder, Chem. Phys. Lett. 238 (1995) 143. S.S.-S. Huang, G.R. Freeman, Phys. Rev. A 24 (1981) 714. K. Shinsaka, M. Codama, T. Srithanratana, M. Yamamoto, Y. Hatano, J. Chem. Phys. 88 (1988) 7529. P. Lamp, G. Buschhorn, Phys. Rev. B 50 (1994) 16824. R.T. Scalettar, P.J. Doe, H.-J. Mahler, H.H. Chen, Phys. Rev. A 25 (1982) 2419. T. Doke, A. Hitachi, J. Kikuchi, K. Masuda, S. Tamada, A. Mozumder, E. Shibamura, T. Takahashi, Chem. Phys. Lett. 115 (1985) 164.
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27 [15] J. Thomas, D.A. Imel, Phys. Rev. A 36 (1987) 614. [16] E. Aprile, W.H.-M. Ku, J. Park, H. Schwartz, Nucl. Instrum. Methods A 261 (1987) 519. [17] A. Mozumder, Fundamentals of Radiation Chemistry, Academic Press, New York, 1999 (Chapter 8). [18] M. Tachiya, J. Chem. Phys. 87 (1987) 4108; J. Chem. Phys. 89 (1988) 6929. [19] S.A. Rice, in: C.H. Bamford, C.F.H. Tipper, R.G. Compton (Eds.), Diffusion Limited Reactions, Comprehensive Chemical Kinetics, vol. 25, Elsevier, Amsterdam, 1985, p. 47. [20] M. Wojcik, M. Tachiya, J. Chem. Phys. 110 (1999) 10016. [21] H. Goldstein, Classical Mechanics, Addison-Wesley, London, 1980, p. 99.
[22] [23] [24] [25] [26] [27] [28] [29] [30]
27
H.R. Skullerud, J. Phys. D 1 (1968) 1567. H.D. Rees, J. Phys. Chem. Solids 30 (1969) 643. S.L. Lin, J.N. Bardsley, J. Chem. Phys. 66 (1977) 435. G.R. Freeman, J. Chem. Phys. 80 (1984) 3506. S.S.-S. Huang, N. Gee, G.R. Freeman, Radiat. Phys. Chem. 37 (1991) 417. U. Sowada, J.M. Warman, M.P. de Haas, Phys. Rev. B 25 (1982) 3434. K. Kowari, K. Leung, B.D. Shizgal, J. Chem. Phys. 108 (1998) 1587. D.F. Anderson, Nucl. Instrum. Methods A 242 (1986) 254. K. Masuda, T. Doke, A. Hitachi, H. Ichinose, J. Kikuchi, E. Shibamura, T. Takahashi, Nucl. Instrum. Methods A 279 (1989) 560.
Electron thermalization and electron–ion recombination in liquid argon Mariusz Wojcik a
a,*
, M. Tachiya
b
Institute of Applied Radiation Chemistry, Technical University of Lodz, Wroblewskiego 15, 93-590 Lodz, Poland National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8565, Japan
b
Received 13 June 2003; in final form 16 July 2003 Published online: 3 September 2003
Abstract The processes of electron thermalization and geminate electron–ion recombination in liquid argon at 87 K are studied by computer simulations based on the Cohen–Lekner theory of electron transport in liquids. The mean thermalization time of 1.8 ns is obtained, which is in agreement with experiment, and the mean thermalization distance is calculated as 2600 nm with the standard deviation of 1100 nm. The probability of geminate recombination is found to be very low (103 ). Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction Electronic processes in ionized liquid argon (LAr) have been a subject of significant fundamental and applied research. Theoretical studies [1–9] were aimed at interpreting the measured high values of the electron mobility [10–12] and free-ion yield [13–15] in this system. On the other hand, the high free-ion yield and other favorable physical properties of LAr made this liquid an important material for constructing high-resolution ionization detectors [15,16]. As in other condensed systems, post-ionization electronic processes in LAr are conventionally
*
Corresponding author. Fax: +48-42-6840043. E-mail address: [email protected] (M. Wojcik).
divided into several phases. First, the electrons ejected from atoms undergo thermalization, in which they dissipate their initially high-kinetic energy to the surrounding medium and become near-thermal. Then, the electrons perform a diffusive motion under the influence of the Coulomb field of their parent cations, and may geminately recombine, or escape and become free charges. In the next stage, spatially uncorrelated electrons and cations undergo bulk recombination. The charges may also be collected by an external electric field, if it is applied to the system. Theoretical description of the electron thermalization requires a model of electron scattering processes in the considered system. As a liquefied rare gas, LAr is very specific. It lacks vibrational degrees of freedom that normally play an important role in electron collisions. One of the applicable
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.08.006
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
models of the electron scattering in LAr is that of Cohen and Lekner [1], in which spatial correlations in the liquid are represented by the structure factor, and different rates of electron energy and momentum transfers are postulated. Approximate analytical calculations of the electron thermalization in LAr based on this model were done by Mozumder [17], who estimated the thermalization time in LAr as 1.4 ns, and the mean thermalization distance as 1400 nm. Due to the high electron mobility of LAr, the mechanism of electron transport in this system cannot be described as an ideal diffusion [18]. Therefore, both the geminate and bulk electron– ion recombination in LAr cannot be treated by the standard theory [19] of diffusion-controlled reactions between ions. Specific models of electron transport and reactions are required to describe the electron–ion recombination in high-mobility systems. In our recent work [7], we have proposed a model of the electron scattering in LAr, which is based on the Cohen–Lekner theory and especially suitable for simulations. We have successfully applied this model to calculations of the fielddependent electron mobility, as well as the bulk electron–ion recombination rate constant. Considering very good agreement between our calculated results of electron mobility and experiment in a wide range of the electric field strength, we believe that the simulation model of [7] properly describes the electron scattering processes in LAr in a range of energies up to at least several electronvolts. Therefore, this model should also be applicable to calculations of the electron thermalization in LAr. Moreover, such calculations could naturally be extended to modeling of the geminate phase of electron–ion recombination. We have pursued this idea and carried out a simulation study of both the electron thermalization and geminate electron–ion recombination in LAr.
2. Model of electron scattering in liquid argon The Cohen–Lekner theory [1] of hot electrons in liquids predicts that the energy and momentum of an electron gained from an electric field are transferred to the medium at different rates, each
21
characterized by its own ‘‘mean free path’’. This theory was further developed by Atrazhev and Iakubov [3], who constructed energy-dependent cross-sections for both the energy and momentum transfers in liquid Ar, Kr, and Xe. In [7] we have proposed a mechanistic model of electron motion in LAr, in which the processes of energy and momentum transfer are separated from each other, in accordance with the Cohen–Lekner theory, and the electron scattering is described in terms of individual electron collisions. This makes the model applicable to simulation calculations. Following Atrazhev and Iakubov [3], we assume that the transfer of electron energy can be described in terms of an effective cross-section rE . We represent this process by a special type of electron collisions, with the mean free path kE ¼ 1=nrE , where n is the density. In these collisions the change of the electron kinetic energy is calculated from the classical expression for isotropic elastic scattering by gas atoms. Unlike normal elastic scattering, however, we assume that these collisions change only the absolute value of the electron velocity, and not its direction. Therefore, the change of the electron momentum in these events is negligible compared to that in normal elastic scattering. Physically, these ‘‘energy transfer’’ collisions may be to some extent identified with the electron scattering by interatomic vibrations (phonons). The transfer of electron momentum is also described in terms of a cross-section, which we denote here as rp . We may expect that the momentum is most efficiently exchanged in large angle scattering by rare gas atoms, and in the simulation we represent this process by isotropic elastic electron–atom collisions, with the mean free path kp ¼ 1=nrp . For the cross-sections rE and rp we assume the forms proposed in [3]. rE is assumed to be constant at energies lower than several electronvolts. At higher energies, it is assumed to be equal to the momentum transfer cross-section in the low-pressure Ar gas, rAr . We plot rE and rAr in Fig. 1 as the solid and dotted lines, respectively. The crosssection rp is calculated from the equation rp ðEÞ ¼ rE ðEÞS 0 ðEÞ;
ð1Þ
22
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
merge into one type of collisions, which is the elastic electron scattering by rare gas atoms.
3. Simulation method
Fig. 1. Cross-sections rE (solid line) and rp (dashed line) that are used in the simulation model of electron transport in liquid argon at 87 K. The dotted line shows the momentum transfer cross-section rAr for argon gas [28]. The inset shows the function S 0 ðEÞ for liquid argon [3].
where S 0 ðEÞ is obtained from the structure factor SðkÞ determined for the momentum transfer hk on the assumption of isotropic electron scattering [3]. At small k the structure factor is known experimentally (Sð0Þ ¼ 0:048 for LAr), while at large momentum transfers SðkÞ ! 1. The assumed form of S 0 ðEÞ [3] is shown in the inset to Fig. 1, and the cross-section rp ðEÞ calculated from Eq. (1) is plotted in Fig. 1 as the dashed line. A more detailed description of the cross-sections used in our model was given in [7]. The model of electron scattering in LAr described above can be readily applied in a simulation in the following way. Both types of the electron collisions in LAr are related to elastic electron scattering by rare gas atoms. Since rp 6 rE (except in a narrow region of energies around 5 eV), we can simply generate elastic electron collisions, with the mean free path kE , and assume that a fraction (rp =rE ) of these collisions change the direction of the electron motion, while the others change only the magnitude of the electron velocity. With increasing electron energy the crosssections rE and rp approach each other and, finally, become equal to the gas phase cross-section. Therefore, at high energies the two types of electron collisions introduced in our simulation model
Modeling of the electron thermalization and geminate electron–ion recombination processes requires a simulation method to calculate electron trajectories in the Coulomb field of a cation and to simulate electron collisions according to the assumed scattering model. In this work, we apply the simulation method that was originally designed for modeling the bulk electron–ion recombination in dense rare gases [20] and adapted to the case of LAr in [7]. In this method, we calculate classical electron orbits in the Coulomb field of a cation and solve numerically the Kepler equation [21] to find the position of an electron on its orbit at a time corresponding to a collision. The details of the method can be found in [20]. Time intervals between the electron collisions depend on the scattering cross-section rE ðEÞ. We use the null-collision method [22–24], which is an efficient method of modeling collisions in a system where the collision cross-section depends on energy. In this method, in addition to real electron collisions with the cross-section rE , we introduce fictitious null collisions that do not change the electron velocity v. The cross-section rn ðvÞ for the null collisions is chosen in such a way that the total collision rate Kmax is independent of velocity vrE ðvÞ þ vrn ðvÞ ¼ Kmax ¼ const:
ð2Þ
The times between the collisions are generated from the exponential distribution with the mean value s¼
1 ; nKmax
ð3Þ
and the decision as to whether a particular collision is real or null is made at random, with the probability of real collisions given by pr ¼
vrE ðvÞ : Kmax
ð4Þ
The real electron collisions are modeled in the following way. First, we determine the velocity,
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
vM , of an atom of mass M that takes part in the collision by generating a random vector according to the Maxwell–Boltzmann distribution. We also calculate the relative velocity u ¼ v vM . Then, we calculate the new velocity from the classical expression for an isotropic elastic collision v1 ¼
Mu mv þ MvM nþ ; mþM mþM
ð5Þ
where n is a random vector of unit length. Since our model of electron scattering in LAr involves two types of real electron collisions, we have to decide whether a given collision is an ‘‘energy transfer’’ collision or a ‘‘momentum transfer’’ collision. This decision is made at random, with the acceptance probability for ‘‘momentum transfer’’ collisions given by pp ¼
rp ðvÞ : rE ðvÞ
ð6Þ
If a ‘‘momentum transfer’’ collision occurs, we set the electron velocity to v0 ¼ v1 :
23
4. Results and discussion The calculations described in this work were carried out for LAr at T ¼ 87 K and n ¼ 2:11 1022 cm3 . The dielectric constant e ¼ 1:49 calculated from the Clausius–Mosotti equation [11] was used. In the simulations of the electron thermalization process electrons were started with the initial energy E0 in the subexcitation range (<12 eV). The dissipation of electron energy above this range is known to be very fast [17], and the electrons ejected from atoms in ionization events are expected to reduce their energy to the subexcitation range at distances from the cations not larger than several angstroms. In the simulations we assumed the initial electron–cation distance to and used two values of the initial enbe r0 ¼ 5 A ergy E0 ¼ 1 and 5 eV. These values correspond to the initial electron kinetic energies of 2.93 and 6.93 eV, respectively. The simulation results for the mean electron kinetic energy hEk i as a function of time are shown in Fig. 2. We see that hEk i
ð7Þ
In the other case, when an ‘‘energy transfer’’ collision takes place, the electron velocity is set to v0 ¼
v1 v: v
ð8Þ
In the calculations of the electron thermalization process we start an electron with a total energy E0 at a distance r0 from the cation and set its initial velocity in a random direction. Then, we calculate the electron trajectories and collisions using the simulation method described above. In the course of the simulation we record the electron kinetic energy and its distance from the cation at given time steps. The simulation is repeated for a large number of independently generated electrons, which allows us to calculate the mean electron kinetic energy and mean electron–cation separation as functions of time. During the simulation the electrons may undergo geminate recombination with the cation. We use the energy criterion of recombination [18,20] and assume that the reaction takes place when the total relative energy of the electron–cation pair falls below Ecrit ¼ 10kB T , where kB T is the thermal energy.
Fig. 2. Time dependence of the mean kinetic energy of electrons undergoing thermalization in LAr. The results represented by the solid and dashed lines were obtained for electrons started with the energy E0 ¼ 1 and 5 eV, respectively. The inset shows the transient conductivity of LAr resulting from pulsed irradiation measured by Sowada et al. [27]. The horizontal dotted lines show the mean thermal energy Eth ¼ ð3=2ÞkB T , and the energy of 1:1 Eth that is used as the operational thermalization criterion.
24
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
decreases with time and asymptotically approaches the thermal level Eth ¼ ð3=2ÞkB T . When we use the operational thermalization criterion, which assumes that the electrons are thermalized when their kinetic energy reaches the level higher than Eth by no more than 10%, we obtain the mean electron thermalization time in LAr as tth ¼ 1:8 ns. We see from Fig. 2 that the value of tth practically does not depend on the initial energy E0 . This means that the simulation results should not be very sensitive to the choice of r0 either. In Fig. 3, we show the time dependence of the mean electron–cation distance hri obtained from the same simulation runs as those described in Fig. 2. Again in this case the results show little dependence on the value of E0 . When we define the mean thermalization distance, rth , as the value of hri at t ¼ tth , we obtain rth ¼ 2600 nm. Fig. 4 shows the contour maps of the probability distribution function f ðr; EÞ that the electrons undergoing thermalization in LAr will have energy E and the electron–cation distance r calculated for t ¼ 100 ps, 500 ps, 1 ns, and 1.8 ns. The calculations of f ðr; EÞ were carried out for 105 electrons generated with the initial energy E0 ¼ 5 eV. In each plot f ðr; EÞ is normalized to its maximum at a given time. From the data obtained at
Fig. 3. Time dependence of the mean electron–cation distance. The results represented by the solid and dashed lines were obtained for electrons started with the energy E0 ¼ 1 and 5 eV, respectively.
t ¼ 1:8 ns the standard deviation of the thermalization distance is calculated as about 1100 nm. By extending the simulations to later times we have also determined the recombination probability, Prec , and the escape probability, Pesc ¼ 1 Prec , which is related to the experimentally observed free-ion yield. In this phase of the calculations we used the same simulation method as that applied in modeling of the electron thermalization, except that the simulation runs were not terminated at a certain time but were carried on until the electron either reacted with the cation or escaped to a large distance rmax . This allowed us to obtain the apparent recombination probability Pr , which is somewhat different from the true recombination probability Prec because of the finite value of rmax . The results of the calculations carried out for electrons started with the initial energy E0 ¼ 5 eV are presented in Fig. 5. For each value of rmax at least 105 independent electron trajectories were simulated. The standard error of the calculated recombination probability is about 104 . The true recombination probability Prec may be found as the asymptotic value of Pr ðrmax Þ as rmax ! 1. To determine Prec we plotted lnð1 Pr Þ against rc =rmax , as it is shown in Fig. 5. Here, rc denotes the Onsager radius defined as e2 =ekB T (rc ¼ 128:9 nm in LAr at 87 K). We have found that for small values of rc =rmax the dependence of lnð1 Pr Þ on rc =rmax is very well approximated by a straight line. By extrapolating the straight line to rc =rmax ¼ 0 we obtained the true recombination probability in LAr as Prec ¼ 0:0016. The results presented above were obtained using the simulation method based on the electron scattering model that is derived from the Cohen–Lekner theory and verified in our earlier calculations of the electron mobility and bulk recombination rate constant in LAr [7]. The good agreement between those calculations and experiment, especially in the case of the mobility, allows us to expect that the present results are also of quantitative value. We have calculated the thermalization time in LAr as tth ¼ 1:8 ns, which is not very different from the value of 1.4 ns obtained by Mozumder [17] from approximate analytical calculations also based on the Cohen–Lekner theory. However, the mean thermalization distance obtained in our study,
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
25
Fig. 4. Contour maps of the probability distribution function f ðr; EÞ for electrons undergoing thermalization in LAr calculated at t ¼ 100 ps, 500 ps, 1 ns, and 1.8 ns. Each distribution is normalized to its maximum, and the contours are shown for the heights of 0:1; 0:2; . . .. The solid line shows the Coulomb potential energy e2 =er in LAr (crossing of this line by one of the contours at t ¼ 1:8 ns is a graphical artifact).
Fig. 5. Results of the simulation of geminate electron–ion recombination in LAr at 87 K. Plots of Pr vs rmax =rc (circles), and lnð1 Pr Þ vs rc =rmax (squares) are shown. The error bars correspond to the standard error of the simulation results. The solid line shows a linear regression fit of lnð1 Pr Þ ¼ cðrc =rmax Þ calculated for rmax =rc P 30. rc ¼ e2 =ekB T ¼ 128:9 nm.
rth ¼ 2600 nm, is significantly larger than the standard deviation of the thermalization length b ¼ 1400 nm reported in [17]. The distribution of thermalization distances in LAr was derived by Freeman et al. [25,26] from an analysis of the electric field dependence of free-ion yields in LAr based on the Onsager theory of geminate electron–ion recombination. Their values of the mean thermalization distance are an order of magnitude smaller than that obtained in our study. This suggests that the method they used to extract the thermalization lengths from experimental data might not be appropriate for LAr, as it has already been pointed out elsewhere [17]. The electron thermalization in LAr was studied experimentally by Sowada et al. [27], who measured the microwave conductivity in this system resulting from pulsed irradiation. The electron mobility in LAr is known to increase with decreasing electron energy [10], so the conductivity is
26
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27
expected to increase during the thermalization. A plot of the transient conductivity in LAr obtained in [27] is presented in the inset to Fig. 2. We see that the conductivity initially increases, reaching a maximum at about 1.8 ns. This value is identical with tth obtained in our calculations. At longer times the conductivity decays due to the electron– ion recombination. The measured thermalization time in LAr is listed in Table 1 of [27] as 0.9 0.2 ns. It was obtained as the time required for 10–90% development of the conductivity corrected for the finite time response of the microwave detector and for the recombination effect. The difference between the result of [27] and our tth is partly accounted for by different definitions of the thermalization time used in the two studies. We have found that the probability of geminate electron–ion recombination in LAr is very low (Prec ¼ 0:0016). The Onsager theory of diffusioncontrolled geminate ion recombination predicts the 0 recombination probability Prec ¼ 1 expðrc =r0 Þ. Taking the mean thermalization distance rth as r0 0 we have Prec ¼ 0:048. The geminate recombination probability in LAr obtained in this study is much 0 lower than Prec , as it can be expected in the system of high-electron mobility, where the electron transport cannot be described as ideal diffusion [18]. Considering the facts that the rate constant of bulk electron–ion recombination in LAr calculated by us in [7] was lower than the available experimental results, and that the methodology of the present study is similar to that of [7], we cannot exclude the possibility that Prec obtained in this work may underestimate the recombination probability to some extent. But even if we take this into account, the geminate recombination probability in LAr should still be expected well below 0.01. Discussion of the geminate electron–ion recombination in LAr is complicated by the fact that the distances between successive ionizations in radiation tracks in this system are normally much shorter than the mean electron thermalization distance, so the geminate electron–cation pairs constituting the track are not completely isolated from one another. Theoretical models were proposed to take into account this effect [8,9], in which the cylindrical symmetry of the charge distribution was assumed. In those models, however, the electron motion in
LAr was approximated by diffusion. It would be interesting to apply the present model of the electron motion in LAr to calculations of the track recombination processes in this system. The calculations presented in this work are directly applicable to modeling of the recombination of geminate electron–ion pairs created by photoionization of photosensitive dopants in LAr by scintillation light [29,30]. Photosensitive molecules are added to LAr in order to increase the amount of collected charge in ionization chambers and thus improve their energy resolution. The results of our calculations indicate that practically all electrons in such systems escape geminate recombination. The use of the Onsager theory in interpretation of the experiments with photosensitive dopants in LAr [30] is therefore not justified.
Acknowledgements This work was supported by Grant No. 3 T09A 049 19 from the Polish State Committee for Scientific Research and by the COE development program of MEXT of Japan.
References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
M.H. Cohen, J. Lekner, Phys. Rev. 158 (1967) 305. S. Basak, M.H. Cohen, Phys. Rev. B 20 (1979) 3404. V.M. Atrazhev, I.T. Iakubov, J. Phys. C 14 (1981) 5139. K. Kaneko, Y. Usami, K. Kitahara, J. Chem. Phys. 89 (1988) 6420; J. Chem. Phys. 90 (1989) 6810. R.A. Holroyd, W.F. Schmidt, Annu. Rev. Phys. Chem. 40 (1989) 439, and references therein. A.F. Borghesani, D. Iannuzzi, G. Carugno, J. Phys.: Condens. Matter 9 (1997) 5057. M. Wojcik, M. Tachiya, Chem. Phys. Lett. 363 (2002) 381. W.G. Burns, A. Mozumder, Chem. Phys. Lett. 142 (1987) 381. A. Mozumder, Chem. Phys. Lett. 238 (1995) 143. S.S.-S. Huang, G.R. Freeman, Phys. Rev. A 24 (1981) 714. K. Shinsaka, M. Codama, T. Srithanratana, M. Yamamoto, Y. Hatano, J. Chem. Phys. 88 (1988) 7529. P. Lamp, G. Buschhorn, Phys. Rev. B 50 (1994) 16824. R.T. Scalettar, P.J. Doe, H.-J. Mahler, H.H. Chen, Phys. Rev. A 25 (1982) 2419. T. Doke, A. Hitachi, J. Kikuchi, K. Masuda, S. Tamada, A. Mozumder, E. Shibamura, T. Takahashi, Chem. Phys. Lett. 115 (1985) 164.
M. Wojcik, M. Tachiya / Chemical Physics Letters 379 (2003) 20–27 [15] J. Thomas, D.A. Imel, Phys. Rev. A 36 (1987) 614. [16] E. Aprile, W.H.-M. Ku, J. Park, H. Schwartz, Nucl. Instrum. Methods A 261 (1987) 519. [17] A. Mozumder, Fundamentals of Radiation Chemistry, Academic Press, New York, 1999 (Chapter 8). [18] M. Tachiya, J. Chem. Phys. 87 (1987) 4108; J. Chem. Phys. 89 (1988) 6929. [19] S.A. Rice, in: C.H. Bamford, C.F.H. Tipper, R.G. Compton (Eds.), Diffusion Limited Reactions, Comprehensive Chemical Kinetics, vol. 25, Elsevier, Amsterdam, 1985, p. 47. [20] M. Wojcik, M. Tachiya, J. Chem. Phys. 110 (1999) 10016. [21] H. Goldstein, Classical Mechanics, Addison-Wesley, London, 1980, p. 99.
[22] [23] [24] [25] [26] [27] [28] [29] [30]
27
H.R. Skullerud, J. Phys. D 1 (1968) 1567. H.D. Rees, J. Phys. Chem. Solids 30 (1969) 643. S.L. Lin, J.N. Bardsley, J. Chem. Phys. 66 (1977) 435. G.R. Freeman, J. Chem. Phys. 80 (1984) 3506. S.S.-S. Huang, N. Gee, G.R. Freeman, Radiat. Phys. Chem. 37 (1991) 417. U. Sowada, J.M. Warman, M.P. de Haas, Phys. Rev. B 25 (1982) 3434. K. Kowari, K. Leung, B.D. Shizgal, J. Chem. Phys. 108 (1998) 1587. D.F. Anderson, Nucl. Instrum. Methods A 242 (1986) 254. K. Masuda, T. Doke, A. Hitachi, H. Ichinose, J. Kikuchi, E. Shibamura, T. Takahashi, Nucl. Instrum. Methods A 279 (1989) 560.