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Disperse fine equiaxed α-Al2O3 nanoparticles with narrow size distribution are important materials in nanotechnology and nanomaterials, but syntheses of disperse fine equiaxed α-Al2O3 nanoparticles usually result in fine γ-Al2O3 nanoparticles or large α-Al2O3 nanoparticles larger than 15 nm. α-Al2O3 has a higher surface energy than γ-Al2O3 and becomes thermodynamically not stable with respect to γ-Al2O3 at specific surface areas larger than 100 m(2)/g (at sizes smaller than 15 nm for spherical particles) at room temperature. So disperse fine equiaxed α-Al2O3 nanoparticles smaller than 15 nm with narrow size distribution are extremely difficult to synthesise. Here we show the successful synthesis of disperse fine equiaxed α-Al2O3 nanoparticles with average sizes below 10 nm and narrow size distribution by selective corrosion and refined fractionated coagulation separation. An almost fully dense nanocrystalline α-Al2O3 ceramic with a relative density of 99.5% and an average grain size of 60 nm can be sintered from disperse fine equiaxed α-Al2O3 nanoparticles with narrow size distribution.

The problem of enumerating spanning trees in lattices with Klein bottle boundary condition is considered here. The exact closed-form expressions of the numbers of spanning trees for 4.8.8 lattice, hexagonal lattice, and 3(3) · 4(2) lattice on the Klein bottle are presented.