Titanium vacancy defects in sol–gel prepared anatase

Rietveld refinement of powder X-ray diffraction data for nanocrystalline anatase samples prepared by different sol–gel methods shows that the samples contain high concentrations (⩽ 20%) of titanium vacancies, the levels of which decrease with increasing crystallite size. Debye function modelling of anatase clusters with different well-defined sizes, shapes and stoichiometries confirmed that the titanium vacancy concentrations obtained from the Rietveld refinements are correct. However, the Debye modelling showed that for nanocrystals smaller than ∼4 nm, the Rietveld modelling gives artificially high cell parameters. Density function theory calculations show that the titanium vacancies are stable defects when the vacancy sites are charge-balanced by incorporation of protons.