Coulomb interaction recombination

An early theory of ionic recombination in liquids was developed by Jaffe (1913) for application at a relatively high LET. However, in Jaffe s theory, coulombic interactions are ignored and the positive and negative ions are assigned the same mobilities and distribution functions. Therefore, its use in a... 

The course of the recombination processes in a particular system depends on several factors. One of the most important ones is the polarity of the system. Both geminate and bulk recombination processes are strongly influenced by the Coulomb attraction between electrons and cations, and the range of this interaction in condensed matter is determined by the dielectric constant e. The range of the Coulomb interaction in a particular system is usually represented by the Onsager radius, r, which is defined as the distance at which the electrostatic energy of a pair of elementary charges falls down to the thermal level kj,T. 

In Chap. 2, the analysis of diffusion-limited reaction rates of Smolu-chowski, Collins and Kimball, and that of Noyes is followed. The considerable literature on reaction rates between solute species is also presented. Additional and important other factors which influence the rate of reaction are a coulomb interaction between reactants, long-range energy or electron transfer and an angular dependence of the rate of reaction. These topics are considered in the Chaps. 3—5. The experimental and theoretical work are compared and contrasted. When the reactants are formed in pairs (by bond fission of a precursor), the rate or probability of recombination can be measured and is of considerable interest. Chapters 6 and 7 discuss the theoretical aspects of the recombination of neutral and ionic radical pairs and also appeal to the extensive literature on the experimentally measured rate of recombination. The weaknesses of this theoretical... 

The potential of mean force due to the solvent structure around the reactants and equilibrium electrolyte screening can also be included (Chap. 2). Chapter 9, Sect. 4 details the theory of (dynamic) hydro-dynamic repulsion and its application to dilute electrolyte solutions. Not only can coulomb interactions be considered, but also the multipolar interactions, charge-dipole and charge-induced dipole, but these are reserved until Chap. 6—8, and in Chaps. 6 and 7 the problems of germinate radical or ion pair recombination (of species formed by photolysis or high-energy radiolysis) are considered. 

To show this connection, consider an ion-pair as above (Sect. 2.1). Not only may the ion-pair diffuse and drift in the presence of an electric field arising from the mutual coulomb interaction, but also charge-dipole, charge-induced dipole, potential of mean force and an external electric field may all be included in the potential energy term, U. Both the diffusion coefficient and drift mobility may be position-dependent and a long-range transfer process, Z(r), may lead to recombination of the ion-pair. Equation (141) for the ion-pair density distribution becomes... 

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. 

Lexp[-L/ro]. For the weak Coulomb interaction, as L —> 0, we naturally obtain from equations (3.2.56) and (3.2.57) that ro the reaction rate again is controlled by the recombination at the black sphere radius. At last, both effective radii - for repulsion and attraction - are trivially related [50, 71] ... 

In many cases of interest tunnelling recombination of defects is accompanied by their elastic or Coulomb interaction, which is actual, e.g., for F, H and Vk, A0 pairs of the Frenkel defects in alkali halides, respectively. In these cases the equation defining the steady-state recombination profile is... 

As it was said above (Section 3.2), for the elastic interaction this coefficient coincides with the effective radius of recombination, Reff = b, whereas for the Coulomb interaction Re ff is defined in equation (3.2.51). Therefore the problem of obtaining the steady-state reaction rate is reduced to the finding the asymptotic coefficient b of the solution of equation (4.2.25). Formally it coincides with the quantum-mechanical scattering length on the potential... 

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]. 

In principle, such approximations may serve as a basis of the description of partially ionized plasmas, if we have to take into account ionization and recombination. However, because of the long range of Coulomb interaction, the Landau collision integral (3.110) and such integrals of type (3.119) are divergent. Such divergencies may be avoided by an appropriate screening. The simplest way to do this is to replace the... 

A long time ago, Hong, Noolandi, and Street [16] investigated geminate electron-hole recombination in amorphous semiconductors. In their model they included the effects of tunneling, Coulomb interaction, and diffusion. Combination of tunneling and diffusion leads to an S(t) oc t 1/2 behavior. However, when the Coulomb interactions are included in the theory, deviations from the universal t /2 law are observed—for example, in the analysis of photoluminescence decay in amorphous Si H, as a function of temperature. 

Due to their close proximity, the electron-hole pair is bound by their mutual Coulomb interaction as shown in Fig. 8.7. When the potential is strong enough, the particles diffuse together, giving geminate recombination. Otherwise the electron and hole diffuse apart and any subsequent recombination is non-geminate. The Onsager (1938) model... 

Fig. 8.7. Onsager model of geminate recombination showing the Coulomb interaction of the photoexcited electron and hole.

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