Thermalization distances

In liquefied rare gases (LRG) the ejected electron has a long thermalization distance, because the subexcitation electrons can only be thermalized by elastic collisions, a very inefficient process predicated by the small mass ratio of the electron to that of the rare gas atom. Thus, even at a minimum of LET (for a -1-MeV electron), the thermalization distance exceeds the interionization distance on the track, determined by the LET and the W value, by an order of magnitude or more (Mozumder, 1995). Therefore, isolated spurs are never seen in LRG, and even at the minimum LET the track model is better described with a cylindrical symmetry. This matter is of great consequence to the theoretical understanding of free-ion yields in LRG (see Sect. 9.6). 

On the other hand, electron thermalization, although fast on the scale of thermal reactions, can still be discerned experimentally. In the gas phase, it exhibits itself through the evolution of electron energy via time-dependent reaction rates. In the liquid phase, the thermalization distance in the field of the positive ion is the all-important quantity that determines the probability of free-ion generation (see Chapter 9). In this chapter, we will deal exclusively with electron thermalization. 

In a nonattaching gas electron, thermalization occurs via vibrational, rotational, and elastic collisions. In attaching media, competitive scavenging occurs, sometimes accompanied by attachment-detachment equilibrium. In the gas phase, thermalization time is more significant than thermalization distance because of relatively large travel distances, thermalized electrons can be assumed to be homogeneously distributed. The experiments we review can be classified into four categories (1) microwave methods, (2) use of probes, (3) transient conductivity, and (4) recombination luminescence. Further microwave methods can be subdivided into four types (1) cross modulation, (2) resonance frequency shift, (3) absorption, and (4) cavity technique for collision frequency. 

In hydrocarbon liquids other than n-hexane, the procedure for obtaining the thermalization distance distribution could conceivably be the same. However, in practice, a detailed theoretical analysis is rarely done. Instead, the free-ion yield extrapolated to zero external field (see Chapter 9) is fitted to a one-parameter distribution function weighted with the Onsager escape probability, and the mean thermalization length (r ) is extracted therefrom (see Mozumder, 1974 ... 

FIGURE 8.4 Electron thermalization distance distribution in n-hexane at 290K starting from an initial separation 23A. See text for details. Reproduced from Rassolov (1991). 

Over the temperature interval 165 K to 300 K, the calculations of Silinsh and Jurgis (1985) indicate that the thermalization rate in pentacene decreases from 3 X 1012 to 0.8 X 1012 s 1. The trend is opposite to what would be expected in liquid hydrocarbons and may be attributed to the rapid increase of mcff with temperature. The calculated mean thermalization distance increases with incident photon energy fairly rapidly, from 3 nm at 2.3 eV to 10 nm at 2.9 eV, both at 204 K. With increasing temperature, (r) decreases somewhat. These thermalization distances have been found to be consistent with the experimental photogeneration quantum efficiency when Onsager s formula for the escape probability is used. 

Lekner, 1967 Lekner and Cohen, 1967). From the experimental viewpoint, LRGs are excellent materials for the operation of ionization chambers, scintillation counters, and proportional counters on account of their high density, high electron mobility, and large free-ion yield (Kubota et al., 1978 Doke, 1981). Since the probability of free-ion formation is intimately related to the thermalization distance in any model (see Chapter 9), at least a qualitative understanding of electron thermalization process is necessary in the LRG. 

In all liquids, the free-ion yield increases with the external electric field E. An important feature of the Onsager (1938) theory is that the slope-to-intercept ratio (S/I) of the linear increase of free-ion yield with the field at small values of E is given by e3/2efeB2T2, where is the dielectric constant of the medium, T is its absolute temperature, and e is the magnitude of electronic charge. Remarkably S/I is independent of the electron thermalization distance distribution or other features of electron dynamics in fact, it is free of adjustable parameters. The theoretical value of S/I can be calculated accurately with a known value of the dielectric constant it has been well verified experimentally in a number of liquids, some at different temperatures (Hummel and Allen, 1967 Dodelet et al, 1972 Terlecki and Fiutak, 1972). 

With an increase of E beyond a certain value specific to the liquid, the free-ion yield increases sublinearly with the field, eventually showing a saturation trend at very high fields (see Mathieu et al.,1967). Freeman and Dodelet (1973) have shown that a fixed electron-ion initial separation underestimates the free-ion yield at high fields, and that a distribution of thermalization distance must be used to explain the entire dependence of Pesc on E. Therefore, the theoretical problem of the variation of free-ion yield with external field is inextricably mixed with that of the initial distribution of electron-cation separation. 

Here e is the electron charge, is the Boltzmann constant, and T is temperature. The value of rc in nonpolar liquids at room temperature may be as high as —30 nm, while in water it is only 0.7 nm. Because the electron thermalization distances are usually on the order of a few nanometers, the effect of the Coulomb attraction between the ionization products in water will be much weaker than that in, e.g., liquid hydrocarbons. This will result in much lower probabilities of geminate recombination in polar liquids compared to those in nonpolar ones. 

We have derived the escape probability for a pair of charges initially separated by a given distance for various cases. However, in real systems, the electron thermalization distance is distributed. If we denote the distribution of thermalization distances by /(r), the total averaged escape probability, (ptot, can be calculated from... 

This equation shows that at low electric fields, the escape probability is a linear function of F, and the slope-to-intercept ratio of this dependence is given by erJlk T. It is worth noting that this ratio is independent of /-q. Therefore plots of (p F)l(p 0) vs. 7 may be used to test the applicability of the presented theory to describe real systems, even if the distribution of electron thermalization distances is unknown. 

As would be expected, these results indicate that the thermalization distances and spatial distribution of the hydrated electron are key parameters in modelling the radiation chemistry of water. Although the stochastic approach is the more logical one to adopt, its present status does not appear to outweigh the advantages of using the simpler deterministic model to represent the essential features of water radiolysis over a wide range of conditions. 

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