Annihilation radius

Now the effective recombination (annihilation) radius could be defined similarly to that in the continuous approximation... 

As it follows from equation (4.2.29), at low temperatures decreases with temperature as T-1/3 [61-63] and approaches an annihilation radius at high temperatures. 

Making use of the elastic constant entering equation (3.1.4) for F, H centres inKBr a = 3eVA3 [69], one can estimate easily that the effective radius of annihilation stimulated by elastic interaction, (equation (4.2.29)) varies from 11 A down to 7 A as the temperature increases from 40 K to 200 K (and then is independent of the annihilation radius R 4 A). On the other hand, the effective radius of tunnelling recombination, equation (4.2.17), decreases from 10 A (at 40 K) down to 5 A (60 K). It coincides with the elastic radius, 7 eh at37 K, where diffusion is very slow and the binary approximation, equation (4.2.19), does not hold any longer. 

The sizes and concentration of the free-volume cells in a polyimide film can be measured by PALS. The positrons injected into polymeric material combine with electrons to form positroniums. The lifetime (nanoseconds) of the trapped positronium in the film is related to the free-volume radius (few angstroms) and the free-volume fraction in the polyimide can be calculated.136 This technique allows a calculation of the dielectric constant in good agreement with the experimental value.137 An interesting correlation was found between the lifetime of the positronium and the diffusion coefficient of gas in polyimide.138,139 High permeabilities are associated with high intensities and long lifetime for positron annihilation. 

Because the electron should also lose energy as a result of this radiation, the radius of its orbit should continuously decrease in size until it spirals into the nucleus. This predicted annihilation of the atom would occur in a fraction of a second. However, atoms were not seen to destabilize as predicted by this model. 

Fig. 3.2. Two principal mechanisms of defect recombination in solids, (a) Complementary defect annihilation, r is the clear-cut (black sphere) radius, (b) distant tunnelling recombination due to overlap of wave functions of defects. Two principal kinds of hole centres - H and Vk...

Fig. 3.2(a). Its typical value varies between 3-5 ao for metals [3, 4] and nearest neighbours for alkali halides. In some semiconductors, e.g., In2Te3, the radius of the instability zone could be very large (e.g., [19-22]). The relevant physical mechanism is annihilation of interstitial atoms with their own vacancies, which occurs in the time interval of several lattice vibrations, 10-13 s, and results in the restored perfect crystalline lattice. This mechanism takes place for all kinds of solids. Thus we can write down phenomenologicaly for the recombination probability (per unit time)... 

Tunnelling recombination of primary F, H pairs can result either in closely spaced v+,i pairs (the so-called a, I centres) which annihilate immediately due to Coulomb interaction and a consequently large instability radius. However some i ions occur in crowdion configurations, and leave vacancy moving away up to 4-5 ao even at 4 K [31]. The distinctive feature of tunnelling recombination is its temperature independence, which makes it one of the major low-temperature secondary processes in insulating solids with defects. 

To describe quantitatively the diffusion-controlled tunnelling process, let us start from equation (4.1.23). Restricting ourselves to the tunnelling mechanism of defect recombination only (without annihilation), the boundary condition should be imposed on Y(r,t) in equation (4.1.23) at r = 0 meaning no particle flux through the coordinate origin. Another kind of boundary conditions widely used in radiation physics is the so-called radiation boundary condition (which however is not well justified theoretically) [33, 38]. The idea is to solve equation (4.1.23) in the interval r > R with the partial reflection of the particle flux from the sphere of radius R ... 

On the other hand, the effective radius of the annihilation stimulated by the Coulomb interaction is well known to be [8, 10, 22, 64]... 

Let us consider now several variational estimates of the effective radius taking into account annihilation, tunnelling and an elastic interaction. If tunnelling term in equation (4.2.25) is large in comparison with others in brackets, we can use for the upper estimate equation (4.2.15) as a trial function y(r) which leads to... 

Arrhenius law, K oc exp(—Ees/(k T)), does not hold here. (The same is true for the Coulomb interaction [39, 46, 68].) Variational estimates of the effective radius at high temperatures, when the recombination is controlled predominantly by annihilation, are discussed in [60]. Variational estimates of the effective radius taking into account annihilation, tunnelling and an elastic interaction were discussed in detail in [33]. 

Making use of the elastic constant entering equation (3.1.4) for F, H centres in KBr a = 3 eVA3 [69], one can estimate easily that the effective radius of annihilation stimulated by elastic interaction, (equation (4.2.29)) varies... 

Fig. 4.7. Temperature dependence of the effective radius of H, A0 recombination in KBr controlled by an elastic interaction, diffusion and tunnelling. Curve 1 - exact result, 2 - effect of tunnelling and annihilation, 3 - isotropic attraction and annihilation, 4 - pure annihilation. Variational estimates upper bound when (i) tunnelling dominates (equation (4.2.32) - curve 5) or an elastic interaction dominates (equation (4.2.34) - curve 6). Curve 7 - lower bound estimate, equation (4.2.36), when an elastic interaction is a predominant factor.

For a critical concentration of excitons 17 = 7/. the critical radius (, below which bimolecular annihilation process predominates over singlet exciton recombination can be expressed as [5],... 

Hence the Markovian distribution is common for all non-Markovian ones but only in the limit X[> —> oo. This distribution completely ignores the nonstationary annihilation and therefore does not depend on the exciton concentration and lifetime. The difference between /o(r) and fm(r) becomes more pronounced when xD is reduced (Fig. 3.99). Under diffusion control both of them have a well-pronounced maximum near the effective ionization radius. However, the nearcontact contribution of the nonstationary annihilation increases with shortening xD, on account of the main maximum. Finally (as xD > 0), the UT distribution tends to become exponential, as W/(r), while the Markovian one remains unchanged. 

Figure 16.25. Density profiles of spikes that form adiabatically around the black hole at the center of our Galaxy. The position of the Sun is indicated by a cross. Four models for the halo profile are shown two with cores ( PS by Persic, Salucci Stel (1996) and can by Bahcall Soneira (1980)) and two with cusps ( NFW by Navarro, Frenk White (1996) and M by Moore et al.(1998)). The spikes form within the radius of influence of the black hole, rinn 1 pc. In the annihilation plateau neutralino annihilations have been so rapid as to deplete the number of neutralinos. (Figure from Buckley et al.(2001).)...

Abstract. Free-volume structure in the lanthanum salt of laurinic acid in crystalline and liquid-crystalline states and an effect of dissolved Cgo molecules on the mean nanovoid radius and concentration were studied by means of the positron annihilation technique. La(Ci2H25COO)3 clathrate compound with dissolved C6o molecules was obtained, which is thermodynamically more stable than a simple mixture of components. Increased mean nanovoid radius (from 0.28 to 0.39 nm) after the inclusion of C6o molecules and concomitant decrease of the positronium annihilation rate by a factor of 2.7 indicate the decrease of the smallest nanovoid concentration. 

At the same time the mean nanovoid radius, sampled by Ps, increases to 0.39 nm (Fig. 2). This is an evidence that annihilation in small nanovoids is strongly suppressed. Observed changes are accompanied by the increase in the positron annihilation probability on oxygen anions from 77.3 to 87.7 % and the decrease in their radius from 0.160 ( 0,001) to 0.156 nm. 

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