Electron subexcitation

The molecular time scale may be taken to start at 10 14 s following energy absorption (see Sect. 2.2.3). At this time, H atoms begin to vibrate and most OH in water radiolysis is formed through the ion-molecule reaction H20+ + H20 H30+ + OH. Dissociation of excited and superexcited states, including delayed ionization, also should occur in this time scale. The subexcitation electron has not yet thermalized, but it should have established a quasi-stationary spectrum its mean energy is expected to be around a few tenths of an eV. 

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). 

Frohlich and Platzman (1953) developed a detailed electromagnetic theory for the rate of energy loss of a subexcitation electron in a polar medium due to dielectric loss. Their final result may be expressed as... 

At the end of the physical stage, which is within about 10 sec of the passage of the ionizing particle through the liquid, the track made by the particle contains H20", subexcitation electrons e , and electronically excited water molecules H2O in small clusters called spurs. From about 10 to 10 sec, the following processes are thought to occur and comprise the physicochemical stage [9,10] ... 

C. The Theory of Retardation of Subexcitation Electrons in Condensed Media... 

The subexcitation electrons lose their energy in small portions, which are spent on excitation of rovibrational states and in elastic collisions. In polar media there is an additional channel of energy losses, namely, the dipole relaxation of the medium. The rate with which the energy is lost in all these processes is several orders of magnitude smaller than the rate of ionizaton losses (see the estimates presented in Section II), so the thermalization of subexcitation electrons is a relatively slow process and lasts up to 10 13 s or more. By that time the fast chemical reactions, which may involve the slow electrons themselves (for example, the reactions with acceptors), are already in progress in the medium. For this reason, together with ions and excited molecules, the subexcitation electrons are active particles of the primary stage of radiolysis. 

As a rule, the lowest excitation level in molecules is a triplet level. Since such states are efficiently excited by slow electrons (see Section IV.B.3), it is the energy of the lowest triplet level that one should take as a boundary energy for subexcitation electrons rather than the energy of the first singlet excitation level, as was done in Ref. 23. 

The role of subexcitation electrons is most important when the irradiated medium contains small amounts of impurity molecules the excitation energy ha) 0j (or the ionization potential I ) of which is below h(o0l. Such additive molecules can be excited or ionized by the subexcitation electrons the energy of which is between h(o 0j and fuom, and, consequently, the relative fraction of energy absorbed by an additive will be different from what it should be if the distribution of absorbed energy were solely determined by the relative fraction of valence electrons of each component of the mixture.213 214 According to estimates of Ref. 215, this effect is observed when the molar concentration of the additive is of the order of 0.1%. This selective absorption with ionization of additives has been first pointed out by Platzman as an explanation for the increase in the total ionization produced by alpha particles in helium after small amounts of Ar, C02, Kr, or Xe were added (the so-called Jesse effect).216... 

The subexcitation electrons are characterized by a certain distribution function 17(E) defined in the range 00l. The first speculations concerning the form of t](E) where made by Magee and Burton,217 who analyzed the energy distribution of electrons ejected during photoionization of atoms. Later, Platzman has proposed the following general form... 

The energy distribution of subexcitation electrons in liquids has not been studied thoroughly, and there is only a small number of papers on the matter.143,220 In Ref. 220 the authors have found 77(E) considering the liquid as a dense gas and using the formulas for the cross sections obtained within the theory of binary collisions.146 In Ref. 143 the spectrum of subexcitation electrons was calculated using the Monte Carlo method. Apparently, it was in this study the influence the state of aggregation of water has on the energy distribution of subexcitation electrons was considered for the first time. 

It is convenient to present the energy spectrum of subexcitation electrons in dimensionless units as a function hwm rj(E) of the variable Elhwater vapor each dot represents the average value for a small energy interval. One can note that the spectra... 

Fig. 13. Spectrum of subexcitation electrons in helium2, > (solid line), in water vapor (O), and in liquid water (A).143 Al, is the number of subexcitation electrons.

Regarding the formation of subexcitation electrons, many authors consider only two possibilities they assume that each subexcitation electron either has been ejected during ionization (and happened to have the energy below fcco01), or it is one of the fast electrons that has slowed down to energies below hw(n. However, there may be other possible ways for subexcitation electrons to be formed. One of them is via a decay of a superexcitation state according to the ionization channel. Such a process may occur both in gaseous and in condensed media, and the majority of electrons produced in this case are the subexcitation ones. 

In condensed media the slow electrons may be additionally formed via the decay of collective excited states of the plasmon type. However, initially, the energy of such electrons is still high enough to excite or even ionize the molecules. For instance, the energy of an electron produced with decay of the ho)p = 21.4 eV plasmon-type state in water is E = hwp - /c = 12.64 eV. An electron with such an energy is still capable of exciting or ionizing one molecule, and only after that it becomes a subexcitation electron. 

Thus, it is not easy to calculate the energy spectrum of subexcitation electrons taking into account all these processes, and, apparently, it can only be done using computer simulation. 

In the problem of retardation of subexcitation electrons, the two important characteristics are the thermalization time and the thermalization path length. In condensed media the key role is played by thermalization path length, which determines how far can an electron travel away from its parent ion when it is thermalized. The thermalization path length determines the probability of formation of a free ion. 

The thermalization path length of subexcitation electrons has been the object of many discussions from the time the first track models appeared up to this day. The reason is that for quite a long time there were no direct methods of measuring the path lengths of slow electrons, while the corresponding theoretical analysis is very difficult owing to the need to take into account all the processes relevant to retardation of subexcitation electrons. 

Samuel and Magee250 were apparently the first to estimate the path length /th and time rth of thermalization of slow electrons. For this purpose they used the classical model of random walks of an electron in a Coulomb field of the parent ion. They assumed that the electron travels the same distance / between each two subsequent collisions and that in each of them it loses the same portion of energy A E. Under such assumptions, for electrons with energy 15 eV and for AE between 0.025 and 0.05 eV, they have obtained Tth 2.83 x 10 14 s and /th = 1.2-1.8 nm. At such short /th a subexcitation electron cannot escape the attraction of the parent ion and in about 10 13 s must be captured by the ion, which results in formation of a neutral molecule in a highly excited state, which later may experience dissociation. However, the experimental data on the yield of free ions indicated that a certain part of electrons nevertheless gets away from the ion far enough to escape recombination. 

The subsequent theoretical calculations of the rate of energy loss and of the path lengths of subexcitation electrons were based on formula (5.22). Since the velocity of subexcitation electrons is much smaller than the speed of light, from (5.22) we get the following expression... 

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