What is the Size of a Cation Vacancy in Tetragonal Tungsten Bronzes?

The tolerance factor for perovskites is a well-established and recognised tool to predict stable perovskite phases and inspired the formulation of a similar tool for the tetragonal tungsten bronzes (A2₄A1₂B₁₀X₃₀, TTBs). Despite the convenience of the tolerance factor for filled TTBs, difficulties arise when faced with unfilled TTBs, i.e., compositions that include A-cation vacancies. There are two “simple” ways to describe the size of a cation vacancy; the first one is to assume the vacancy to be zero dimensional, while the other one is to assume that the vacancy has approximately the same size as the average size of the present A-cations. Both assumptions lead to different tolerance factors and could tip the scale whether the studied TTB composition is predicted to be stable or not, especially since the stability window of TTBs is narrower than that of perovskites. Moreover, literature studies revealed that cation vacancies are larger than both these estimates. To be able to predict the stability of TTBs with cation vacancies, a mathematical model to determine the size of the cation vacancies has been developed and the results are compared to literature data for lead-free unfilled TTBs. Structural trends caused by the presence of cation vacancies in TTBs were identified and the impact of cation vacancy concentration on the TTB framework will be reported.