Separation distance exponential distribution

The theoretical treatment to correlate the separation distance and the local concentration presumes that the relaxation is expressed by the exponential function of t /2. The observed exponent of tj is 0.62 and deviates from the theoretically expected value of 0.5. This difference is attributed to the effect of the energy transfer within the radicals on resonance (between A spins), which has not been taken into account in the theory developed in Sect. 2.6. Nevertheless, the heterogeneous distribution has been unequivocally demonstrated by the ESE method. 

Hill et al. [117] extended the lower end of the temperature range studied (383—503 K) to investigate, in detail, the kinetic characteristics of the acceleratory period, which did not accurately obey eqn. (9). Behaviour varied with sample preparation. For recrystallized material, most of the acceleratory period showed an exponential increase of reaction rate with time (E = 155 kJ mole-1). Values of E for reaction at an interface and for nucleation within the crystal were 130 and 210 kJ mole-1, respectively. It was concluded that potential nuclei are not randomly distributed but are separated by a characteristic minimum distance, related to the Burgers vector of the dislocations present. Below 423 K, nucleation within crystals is very slow compared with decomposition at surfaces. Rate measurements are discussed with reference to absolute reaction rate theory. 

When integrating this equation (which is readily accomplished for small and large values of zFy IRT separately) we can find the distribution of potential /(x) relative to distance x. At low values of / this distribution is exponential. 

As a final remark it must be mentioned that theoretical and experimental works have been dedicated to investigating the effect of the finite size of the chains [65]. In fact, as grows exponentially, at low temperatures it can become comparable with the distance between two consecutive defects (e.g. impurities and vacancies) which are always present in real systems and hardly separated by more than 103 -104 elementary units. In case of Z < , the nucleation of the DW is energetically favoured if occurring at the boundaries, because the energy cost is halved. However the probability to have a boundary spin is inversely proportional to L thus the pre-exponential factor becomes linearly dependent on L, as experimentally found in doped SCMs. As doping occurs at random positions on the chain, a distribution of lengths is observed in a real system. However, as the relaxation time is only linearly dependent on L, a relatively narrow distribution is expected. 

The range parameter was used in an exponential function representation of the initial distribution of distance, w(r0) = exp — r0/ /47rr2b, together with the scavenging probability of an ion-pair of separation, rn,p(r0, es) in eqn. (173). 

Until recently very little quantitative experimental data concerning the distance dependence of electron-transfer rates were available. From experiments on electron transfer between statistically distributed donor and acceptor species in an inert glassy matrix it had been concluded (Miller et al., 1984) that the rate falls of sharply with increasing donor-acceptor separation and that at a given edge-to-edge separation Re (in A) the fastest rate (k in s ) achievable under optimally exothermic conditions would be given by the exponential expression eq. (1) ... 

A widely-used model in this class is the direct-interaction with product repulsion (DIPR) model [173—175], which assumes that a generalised force produces a known total impulse between B and C. The final translational energy of the products is determined by the initial orientation of BC, the repulsive energy released into BC and the form of the repulsive force as the products separate. This latter can be obtained from experiment or may be assumed to take some simple form such as an exponential decay with distance. Another method is to calculate this distribution from the quasi-diatomic reflection approximation often used for photodissociation [176]. This is called the DIPR—DIP model ( distributed as in photodissociation ) and has given good agreement for the product translational and rotational energy distributions from the reactions of alkali atoms with methyl iodide. 

Dispersive transport was analysed by Scher and Montroll (1975) for a random array of hopping sites in a regular crystal lattice. The transition rate between sites is an exponential function of both inter-site distance and activation energy. The distribution of transition rates may arise from a variation in separation with fixed activation energy, a variation in activation energy for... 

---