From lifetimes to pore sizes

In order to convert the measured lifetimes into dimensions (and shapes) of the pores, additional information is required. A basic shape of pores has to be assumed. The single parameter (lifetime) is not sufficient to determine a three dimensional object. Here simple spherical shapes for isolated pores and channels (tubes) with circular cross sections are assumed. The diameter of the pores can be related to the classical mean free path ( ) of a particle in such objects of volume V and surface area f [46], 

This relation applies generally and specifically to spheres and cylinders. Empirical relations have been developed to convert the measured lifetime (x) to this mean free path ( ). In case of small pores ( 2 nm) an earlier version is adequate [43, 44], The model is based on the assumptions of spherical potential wells with infinite depth and radius r that traps the positronium. 

The porosity (P) is P = nV for each pore size. With pores of the size i=l,2,3 (1 and 2 are spheres and 3 are channels) that yields 

A shape dependent factor (f=l/6 for spheres, f=l/4 for channels when D L) is introduced [9]. This model is rather simple and excludes effects like detrapping or quantum mechanical tunneling among neighboring pores [60], 

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