Ion survival probability

The Incident Ion beam Intensity can be measured, and there are several tabulations of cross-section calculations. ( ) Also, the analyzer parameters, T, D, and d6 can be determined. The three aspects of this equation, which are not well understood nor easily determined. Include the number of atoms of a particular kind, the Ion survival probability, and the shadowing or geometric term. The first quantity Is quite often that which you would like to determine. The second two are often difficult to separate. Shadowing can be particularly Important when trying to observe second layer effect or when trying to determine the location of adsorbates.( ) However, shadowing for polycrystalline samples, though Important, Is very difficult to deal with quantitatively. 

More qualitatively, we can plot the reciprocal of the perpendicular velocity versus the log of the Ion survival probability. Ions arising from different depths should have dif-... 

Figure 3.57. The ion survival probability as a function of time at To = 0.5 ns with a great excess of acceptors. In line with UT and IET (above) and Markovian theory (below) (dashed curve), the contact approximation (dashed-dotted line in the middle) and exponential model with fcjep = A et = 1.0 ns-1 (dotted line) are also shown. The horizontal thick lines indicate the free-ion quantum yield ((). The concentrations and ionization parameters are the same as in Figure 3.56, while wy = 3.4ns-1, D = D = 1.2 X 10-6 cm2/s, k1 — 7S4 A3/ns, and kr — 4S6 A3/ns. (From Ref. 195.)...

Many empirical models have been proposed to treat both resonant and Auger electron transfer processes at surfaces. Hagstrum assumed that the rate of neutralization has an exponential dependence on ion/surface distance.He then showed that the ion survival probability, P+, can be approximated by ... 

Writing the ion yield scattered from a species i and the experimental quantity of transmission T of the detecting system, including the detector sensitivity and the solid angle of detection A 2, ion survival probability for the scattering... 

The ion survival probability factor is the most important difference between the RBS and MEIS regime and low energy spectroscopy (LEIS). It depends on the individual projectile-target combination used. The probabilities of ion survival are... 

Taking into account that neutralization means tunneling of a target conduction-band electron to the ion, the time integral can easily be replaced by integration over the distance from the surface, s, by use of the identity dt = ds/v , where Vj is the component of the ion velocity perpendicular to the surface. Prom this, the velocity-dependence of the survival probability, P , is obtained ... 

In the preceding part of this section, we have concentrated on the electron escape probability, which is an important quantity in the geminate phase of recombination, and can be experimentally observed. However, modern experimental techniques also give us a possibility to observe the time-resolved kinetics of geminate recombination in some systems. Theoretically, the decay of the geminate ion pairs can be described by the pair survival probability, W t), defined by Eq. (4). One method of calculating W t) is to solve the Smoluchowski equation [Eq. (2)] for w r,t) and, then, to integrate the solution over the space variable. Another method [4] is to directly solve Eq. (7) under relevant conditions. 

The analytical solution of the Smoluchowski equation for a Coulomb potential has been found by Hong and Noolandi [13]. Their results of the pair survival probability, obtained for the boundary condition (11a) with R = 0, are presented in Fig. 2. The solid lines show W t) calculated for two different values of Yq. The horizontal axis has a unit of r /D, which characterizes the timescale of the kinetics of geminate recombination in a particular system For example, in nonpolar liquids at room temperature r /Z) 10 sec. Unfortunately, the analytical treatment presented by Hong and Noolandi [13] is rather complicated and inconvenient for practical use. Tabulated values of W t) can be found in Ref. 14. The pair survival probability of geminate ion pairs can also be calculated numerically [15]. In some cases, numerical methods may be a more convenient approach to calculate W f), especially when the reaction cannot be assumed as totally diffusion-controlled. 

Figure 2 Survival probability of geminate ion pairs as a function of time. The two solid lines correspond to two different values of the initial electron-cation distance. The broken lines show the asymptotic kinetics calculated from Eq. (25). The value of the escape probability for Tq = O.Sr is indicated by

The probability that the ions have a separation r, given that one ion is at r0, at time t is the density p(r, t r0, t0), which is also related to the Green s function of the diffusion equation (see Sect. 2.3 and the discussion in Appendix A). As before, the survival probability is the integral of the density over all space [eqn. (123)] and this may be related to the flux, J, crossing the encounter surface [eqn. (124)], which is (following Chap. 3, Sect. 1.1)... 

Survival probability of an ion-pair in an applied electric field... 

The observed survival probability depends not only on the survival probability of an ion-pair formed with an initial separation, r0, but also on the distribution of initial separations, w(r0). As will be discussed in Sects. 3 and 4, the rate of loss of energy of ions after heterolytic bond fission (or, alternatively, the range of electrons formed by ionisation of a molecule) depends sensitively on the energetics of the solvent molecules. These separation distances can range up to 10 nm or more and are specifically discussed in Sect. 3. The survival probability of a collection of isolated ion-pairs, P(t), formed at a time t0 — 0, and with a distribution of initial distances w(r0) is... 

It has already been noted (Chap. 6, Sect. 2.2 and Sect. 2.1 of this chapter) that the survival probability of a pair of reactive molecules, radicals or ions is closely related to the density distribution of a vast excess of one of these reactants about the other (the homogeneous case which... 

Equation (164) describes the evolution with time t0 of the survival probability of an ion-pair formed at time t with a separation r. In the general, case this equation cannot be solved, but if no long-range transfer occurs and the transport coefficients are constant, this reduces to... 

Furthermore, the initial and outer boundary conditions are effectively identical [eqns. (3), (4) and (165)] as are also the partially reflecting boundary conditions [eqns. (46) and (165)]. This can be shown by substituting p by exp — p p in the boundary conditions (165). Consequently, the relationship between the survival probability of an ion-pair at a time t0 after they were formed at time t and separation r and the density distribution of an initial (time t0) homogeneous distribution of the majority ion species around the minority ionic species, p(r, f f0), is an identity. 

Hong and Noolandi [72], Berg [278], and Pedersen and Sibani [359] have also noted the connection between the survival probability and homogeneous density distribution. Finally, the escape probability of an ion pair formed with a separation, r0, with an arbitrary monotonically decreasing potential energy of interaction and with electric field-dependent mobility and diffusion coefficient ions was found by Baird et al. [350] to be (see also Tachiya [357])... 

Usually, experiments are performed with steady-state photolysis or radiolysis of the solution and the yield of scavenger products determined optically or by ESR methods. There is no direct interest in the actual time evolution of the density or recombination (survival) probability. Consequently, the creation of ion-pairs may be pictured as occurring at a constant rate, say 1 s 1, from time t0 = 0 to infinity. The steady-state ion-pair density distribution, which arises when dp/dt = 0, is the balance between continuous creation of ion pairs at a rate Is-1, recombination and scavenging. Removing the instantaneous creation of an ion-pair at time t = t0 (i.e. removing the 6(f — f0) in the source term), means that ion-pairs were continuously formed from time t = — 00 to t. At long times, f > — oo the density distribution is independent of t and, of course, t0. Let pss(r cs r0) = /i p(r, t cs t0, 0)d 0 be the steady-state ion-pair density distribution for ion pairs continuously formed at r0, and note d/dt J" f pd 0 = 0. The diffusion equation (169) becomes... 

Friauf et al. [326] have presented the general solution for the survival probability of an ion-pair in the presence of scavengers and an external electric field based on the analysis of the Debye—Smoluchowski equation by Hong et al. [72, 323—326] (Chap. 3, Sect. 1). Further comments have been made by Tachiya [357], Tachiya and Sano [358a], and Sano [358b]. The expression is only valid at low concentrations of scavenger, that is cs alkane solvents (see Sect. 2.5). 

The density and survival probability of radicals and ions in clusters... 

This Green s function displays the same reciprocity to interchange of the co-ordinate r, t with r0, f° as does the single ion-pair Green s function [72, 499] [eqn. (162)]. Using exactly the same techniques as used in Chap. 7, Sect. 2.3, the equation satisfied by the survival probability can be shown to be... 

An isolated ion-pair, of initial separation r0 at time t0 = 0, in a nonpolar solvent may recombine or separate and ultimately escape. At a time t, the probability that the ion-pair will have recombined is < (t r0, t0 = 0) and that it is still extant p(t r0) f0 = 0). A short while later, the probability that the ion-pair has not recombined is p t + df r0, t0 =0). The change in survival probability is the probability that the ion-pair recombined during the time interval t to t + df, that it had a lifetime between t and t -f df. Defining the lifetime distribution function as f(f), then... 

Against all odds, this concept constitutes the formal basis of the exponential model. The EM kinetic equations are written for the survival probabilities of ions in the reaction sphere f2jn and out of it f>oul ... 

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