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When people talk about oxides like $\ce{SiO2}$ or $\ce{Al2O3}$, they use expressions like stoichiometric or under-stoichiometric. I understand that this refers to the relative composition of materials, but I don't really know what it exactly means.

Say for $\ce{SiO2}$, does under-stoichiometric mean that it isn't exactly $\ce{SiO2}$, but more like $\ce{SiO_{1.8}}$? Does the term usually refer to the oxygen composition?

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Yes, the convention is to refer to the oxygen atoms when talking about over- and under-stoichiometric in metaloxides. Normally they also indicate this explicitly by writing $\ce {SiO}_x$ (like in the article here for example) or $\ce {Al_2O}_x$. Being explicit about it is a good habit because it avoids ambiguity.

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