Imaging the atomic structure of activated carbon

The precise atomic structure of activated carbon is unknown, despite its huge commercial importance in the purification of air and water. Diffraction methods have been extensively applied to the study of microporous carbons, but cannot provide an unequivocal identification of their structure. Here we show that the structure of a commercial activated carbon can be imaged directly using aberration-corrected transmission electron microscopy. Images are presented both of the as-produced carbon and of the carbon following heat treatment at 2000 degrees C. In the 2000 degrees C carbon clear evidence is found for the presence of pentagonal rings, suggesting that the carbons have a fullerene-related structure. Such a structure would help to explain the properties of activated carbon, and would also have important implications for the modelling of adsorption on microporous carbons

Equilibrium problems for Raney densities

The Raney numbers are a class of combinatorial numbers generalising the Fuss--Catalan numbers. They are indexed by a pair of positive real numbers $(p,r)$ with $p>1$ and $0 < r \le p$, and form the moments of a probability density function. For certain $(p,r)$ the latter has the interpretation as the density of squared singular values for certain random matrix ensembles, and in this context equilibrium problems characterising the Raney densities for $(p,r) = (\theta +1,1)$ and $(\theta/2+1,1/2)$ have recently been proposed. Using two different techniques --- one based on the Wiener--Hopf method for the solution of integral equations and the other on an analysis of the algebraic equation satisfied by the Green's function --- we establish the validity of the equilibrium problems for general $\theta > 0$ and similarly use both methods to identify the equilibrium problem for $(p,r) = (\theta/q+1,1/q)$, $\theta > 0$ and $q \in \mathbb Z^+$. The Wiener--Hopf method is used to extend the latter to parameters $(p,r) = (\theta/q + 1, m+ 1/q)$ for $m$ a non-negative integer, and also to identify the equilibrium problem for a family of densities with moments given by certain binomial coefficients.Comment: 13 page