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A perovskite is any crystal with the general formula $\ce{ABX3}$ where if the $\ce{B}$ atoms form a cube, an $\ce{A}$ atom sits in its center and the $\ce{X}$ atoms sit between neighboring $\ce{B}$ atoms.

Is the perovskite structure symmetric under exchange of $\ce{A}$ and $\ce{B}$ atoms? That is, if I replace every $\ce{A}$ atom with a $\ce{B}$ atom and vice versa, will I end up with the same exact structure as before?

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No. In the perovskite structure, the "A" atoms are larger than the "B" atoms. The B atoms are 6-fold coordinated, while the "A" atoms are in 12-fold cuboctohedral coordination.

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    $\begingroup$ I'd say the size is not the reason - the coordination/symmetry difference does the job to explain why the sites are not equivalent. The larger volume available in that center certainly helps explain why a larger atom likes it, but if it can't hack the coordination requirements it won't want to be a perovskite. $\endgroup$ – Jon Custer Jul 29 '16 at 0:40

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