Cross Relaxation and Spatial Distribution of Radicals

The cross relaxation is the energy transfer between A and B spins through dipolar interactions. Let us consider a pair of electron spins (an A spin and a B spin), the rate of the cross relaxation between them is given by 17, 18] 

When the A spin is surrounded by many B spins with different r and 0, the cross relaxation rate is given by the superposition of the effect from all the B spins with different r and 0, as 

The location of the B spins with respect to the A spin is actually not the same for all the A spins. Moreover, the precession frequency of the B spins is not all the same. The observed relaxation kinetics are therefore given by the superposition of Eq. (3) with a different rate constant. 

When the electron spins are coupled with nuclear spins, the cross relaxation accompanying the change of the nuclear spin state can occur. In this case the apparent spectral overlap of the A and the B spins is not necessary. The spectral averaging of Eq. (3) is therefore a difficult task. Instead, we assume that the spectral overlap function in Eq. (2) is given by a constant F. Then, the spatial averaging of Eq. (3) is necessary for correlating the observed relaxation kinetics with the theory. The result of the spatial averaging will be shown for the two extreme cases of the spatial distribution of radicals in solids. 

One of the extreme cases is a random distribution in which an A spin is surrounded by statistically-distributed B spins. This is the distribution for homogeneously-dispersed radicals. Summation of Eq. (3) under the random distribution gives 

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