Three-dimensional diffusion in a spherically symmetric system

THREE-DIMENSIONAL DIFFUSION IN A SPHERICALLY SYMMETRIC SYSTEM 

For diffusion in a sphere the propagator should be independent of the azimuth angle (p, then we can write for R  

For = n(n + 1) Eq. (44) is the differential equation of Bessel and Eq. (45) the Legendre differential equation with solutions  

The general solution of the diffusion equation for a system with spherical symmetry can therefore be written as  

For a complete definition of Eq. (53) we need to determine the constants Cnk from the conditions (17)-(19) and then calculate the Fourier integral Eq. (1) for the echo signal. To avoid the tedious algebra we compare the three published solutions numerically, but first reproduce these solutions here using our notation. Two of these solutions resulted from a calculation that included the effect of surface relaxation. To make a correct comparison we eliminate from the equations the terms due to relaxation. Then we have the following formulae for the echo intensity for diffusion in a sphere with radius a and reflecting walls  

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