# Cation-anion radius ratio of manganese difluoride

How to determine the cation-anion radius ratio for $$\ce{MnF2}?$$

I tried to calculate the $$r_\mathrm{c}/r_\mathrm{a}$$ ratio using the relationships: $$2r_\mathrm{c} = c$$

and

$$(2r_\mathrm{c} + 4r_\mathrm{a})^2 = 2a^2,$$

but that gives $$r_\mathrm{c}/r_\mathrm{a} > 1$$ with the given $$a$$ and $$c$$ parameters, which is impossible. $$\ce{MnF2}$$ has a rutile structure, so the predicted ratio should be between $$0.414$$ and $$0.732.$$

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You assumed that the cations touch along the $$c$$ direction. They don't. In most ionic compounds you have only anion to cation contact and, therefore, any attempt to determine ionic radii from lattice parameters gives only the sum of the two radii, not the individual values. There are exceptions such as lithium iodide, in which the lattice parameters enable determination of an ionic radius for iodide, but then they carry no information about the lithium ion.