Diffusion-annihilation equation

The evolution of the positron spatial distribution /(r t) is conventionally described by the diffusion-annihilation equation [72]... 

Fick s second law states the conservation of the diffusing species i no i is produced (or annihilated) in the diffusion zone by chemical reaction. If, however, production (annihilation) occurs, we have to add a (local) reaction term r, to the generalized version of Fick s second law c, = —Vjj + fj. In Section 1.3.1, we introduced the kinetics of point defect production if regular SE s are thermally activated to become irregular SE s (i.e., point defects). These concepts and rate equations can immediately be used to formulate electron-hole formation and annihilation... 

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 ... 

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.

Traditionally, experimental values of Zeff have been derived from measurements of the lifetime spectra of positrons that are diffusing, and eventually annihilating, in a gas. The lifetime of each positron is measured separately, and these individual pieces of data are accumulated to form the lifetime spectrum. (The positron-trap technique, to be described in subsection 6.2.2, uses a different approach.) An alternative but equivalent procedure, which is adopted in electron diffusion studies and also in the theoretical treatment of positron diffusion, is to consider the injection of a swarm of positrons into the gas at a given time and then to investigate the time dependence of the speed distribution, as the positrons thermalize and annihilate, by solving the appropriate diffusion equation. The experimentally measured Zeg, termed Z ), is the average over the speed distribution of the positrons, y(v,t), where y(v,t) dv is the number density of positrons with speeds in the interval v to v + dv at time t after the swarm is injected into the gas. The time-dependent speed-averaged Zef[ is therefore... 

Assuming that all positrons in the swarm have energies below the positronium formation threshold and that only elastic collisions and annihilation are possible, the speed distribution may be derived theoretically as the solution of the following diffusion equation (Orth and Jones, 1969) ... 

Several studies have been made of the behaviour of low energy positrons in gases under the influence of a static electric field e. The broad aim of this work has been to study the diffusion and drift of positrons in order to understand better the behaviour of the momentum transfer and annihilation cross sections at very low energies. The theoretical background has been given in section 6.1, and the diffusion equation with an... 

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. 

Let us examine now the set of equations controlling the creation and annihilation of neutral A and B particles in the Euclidean space which have equal diffusion coefficients D = Dq = D [93]. It has the form of equations (2.2.20) to (2.2.21). Here K is a reaction rate of bimolecular recombination in particular, it can be equal to K = SirDro. Also, and... 

The constants of the equation were obtained from simulations and the process applies only to generation of singlet excited states at diffusion-limited annihilation rates. Nonetheless, the expression provided an experimental approach for determining efficiencies of the production of emitting excited states in annihilation reactions. Simulations for systems that react via triplet formation and subsequent triplet-triplet annihilation were also developed [40b] and they illustrated that the two mechanisms can be distinguished by analysis of intensity-time profiles. 

Chemical systems are traditionally modeled by reaction-diffusion systems on suitable domains. As was explained above, our main modeling assumption is that the domain is actually unbounded, that is, we consider governing partial differential equations on the entire plane. This assumption may seem unrealistic neither experiments nor numerical simulations can be performed on unbounded domains. In our particular context of spiral waves in the BZ reaction, however, experiments indicate that spiral waves behave much as if there were no boundaries. Therefore until boundary annihilation sets in - typically within only one to two wavelengths from the boundary itself - we consider reaction-diffusion systems ... 

Several points in this general treatment require further comment. In the first place we have neglected interaction between dislocations, except for the multiplication equation (8.45). One might have expected A in (8.43) to depend on the dislocation density n as in metals, where such interaction impedes dislocation motion and leads to work-hardening. This does not occur in ice. Secondly, if we consider a normal creep experiment with exponential increase of strain with time. This does not occur and Cp tends to a constant. The probable explanation is that, when the dislocation density becomes high, dislocations can climb by a diffusion mechanism (Weertman, 1957) to annihilate each other after a limited amount of motion, thus maintaining n constant. 

The reactions of deposition or crystal growth are surface reactions. The reactants are adsorbed, more or less mobile molecules, e.g., A and in the fictitious reaction A -h B 0. These adsorbates form the substrate surface and growth is the annihilation reaction between the adsorbed reactants. The reaction rate r is expressed as usual (Chapter 6) in the reactant concentrations as r = k[ A][ B]. This can be done if the surface (the reaction space) can be considered to be a well-stirred reactor. In other words, the mobilities of A and B are high compared to the rate of the growth reaction. If that is no longer true and there is diffusion limitation the reaction can still be fitted to the above rate equation except that the reaction rate coefficient k is replaced by kit in which h — i —jS (with S being the spectral dimension). A characteristic value for his for the case of a reaction on a percolation cluster with a spectral dimension of... 

The thermalised positron is further scattered by phonons and diffuses until it annihilates with an electron (or is eventually trapped). During its lifetime (r > 10 s), it can diffuse over a volume of about 1000 A [83]. The diffusion process of a positron e" " can by characterised by a diftiisivity D+ and a mobility /x+, which are related by the Einstein equation... 

Consider Equation 21.6 in which the diffusion of holes from the p- to the n-type material is balanced by the gradient of the contact potential. If the p-side is connected to the positive terminal of a battery (holes flow from + to —, electrons from — to +), some of the electrons that had diffused into the p-side are taken up by the battery. This applied voltage V reduces the contact potential as seen in Figure 21.4 as well as the electric field across the depletion zones as seen in Figure 21.5 and more electrons can diffuse across the jimction. The positive terminal of the battery continues to supply holes to the p-material by taking up excess electrons while the electrons flowing from the negative terminal of the battery continue to annihilate the excess holes that were injected into the n-type material. 

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