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What is the physical meaning when we have an empirical formula with fractional subscripts such as $\ce{LiMn_{1-x}Fe_{x}PO4}$ $(0 \le x \le 1)$?

This an example of the paper the notation above is used:

Bai, J.; Hong, J.; Chen, H.; Graetz, J.; Wang, F. J. Phys. Chem. C 2015, 119 (5), 2266–2276. DOI 10.1021/jp508600u.

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These indicate "doping" or non-stoichiometric compounds.

In this case, there's some fraction of Fe, and some fraction of Mn in a lithium iron phosphate battery. If I remember correctly, you need to dope a different ion to increase the conductivity of $\ce{LiFePO4}$.

See also here: How are non-Stoichiometric compounds determined?

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    $\begingroup$ I think it is a bit more than that: the nomenclature here is used when the ($Mn$,$Fe$) site in the lattice is fully occupied, and $x$ is telling you how many have $Fe$ vs $Mn$ in the sample. This is indicating that there is a single phase across $LiMnPO_4$ to $LiFePO_4$, trading out $Mn$ for $Fe$. Non-stoichiometry ($Mn$+$Fe$ isn't 1) would be indicated by a $\delta$. If $Mn$ did not substitute perfectly for $Fe$, it might be shown as $LiMn_{1-x-\delta}Fe_{x}PO_4$. $\endgroup$
    – Jon Custer
    Oct 6, 2014 at 22:51
  • $\begingroup$ Yes, that's a useful clarification. I agree that in this particular case, there's a single phase and x vs. 1-x is giving you a ratio of $\ce{Fe}$ to $\ce{Mn}$ in this compound. However, I think most people would consider this case as a part of the broader subset of non-stoichometry, since there's no simple integer to define the formula. The OP's title question referred to the general question of fractional subscripts, which does go to non-stoichiometry. $\endgroup$
    – Geoff Hutchison
    Oct 7, 2014 at 1:14

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