# Negative numbers for ionic radii?

I was looking for a good table of ionic radii for an experiment on the van't-hoff factor when I stumbled upon R. D. Shannon, Acta Cryst. 1976, A32, 751.

In table 1, it lists 1 of the ions of carbon with as so:

C +4 1S 2 III .06 -.08

The bold numbers are of concern here. The first number, 0.06 Å is what the paper refers to as the crystal radius, while the second number, -0.08 Å, is referred to as the effective ionic radius. I was also unable to find any good distinction between the two in the paper, except that the radius of $\ce{O^2+}$ for the crystal radius is 1.40 Å, while the effective ionic radius for $\ce{O^2+}$ is 1.26 Å.

I also doubt that this is a typo since it agrees with this statement made later in the paper:

crystal radii differ from traditional radii only by a constant factor of 0.14 [ Å].

• Interestingly $\ce{H+}$ is listed as having both a negative crystal radius as well as a negative effective ionic radius. Only light, highly positively charged ions appear to give negative values ($\ce{H+}$, $\ce{D+}$, $\ce{C^{4+}}$, $\ce{N^{5+}}$). It is likely that these values are calculated in a way which cannot handle the extremely high demand for at least some covalency from these strongly polarising cations, so the model breaks down. – Nicolau Saker Neto Feb 10 '16 at 10:46
• Though I'm not 100% sure about this, the paper seemed to not be a model, but instead was directly measured. I couldn't find a real model anywhere so I can only assume that it must have been directly measured. These values were also not discussed anywhere in the paper as far as I can tell. – purepani Feb 10 '16 at 13:36
• Also, that about negatively charged ions: why are they not affected? – purepani Feb 10 '16 at 14:15
• Is it possible that the radii were measured in referential unit like amu? – Agyey Arya Feb 10 '16 at 16:42
• I'm going to take a much closer look at the paper and then come back to this question. – purepani Feb 11 '16 at 0:31

## 1 Answer

Ionic radii are not measured on individual ions by some sort of mini-micrometer, but bond distances can be measured and then radii are assigned to the individual atoms/ions. The fuzziness of the assignments can be seen by the variation in ionic radii as the coordination number changes: if you give the ion more room (greater coordination number), it will fill the space - if you assume that the ligands have the same size as usual (whatever that is).

(There are materials (ferroelectrics) in which the coordinated ion does not completely fill the space allotted. In $$\ce{BaTiO3}$$, the $$\ce{Ti^{+4}}$$ ion is so small that it can shift its position in the octahedral cage of oxygen anions in response to an external electric field.)

The data in the Shannon paper are extensive, certainly a credit to the Central Research Department of DuPont in 1976, but the trends reported seem to be much more valuable than any specific datum.

Nicolau mentioned covalency as one factor in determining radii. Purepani brought up negative ions not having an effect of negative radii. Assigning a radius to a proton other than zero suggests that you are allowing it to have some electron density surrounding it, able to repel an anionic ligand. Electrostatically, a positive ion will attract some electron body (anion or atom) and come closer; depending on your initial assignment of radii, you can adjust to keep your idea of proton radius constant, or adjust, even into the negative region (because we assume you have a good imagination). Negative ions have a larger size than protons, and are less hard, so they can dent a little. One reason to use negative radii for the cations is that the anion may be dented only at one site, but has a "normal" radius everywhere else around it.

My approach to handling a negative radius is to conclude that the proton (or other mini- cation) is so electrostatically demanding that it can begin to bury itself in the outer shell of the anionic ligand, if the coordination number allows such a deviation in distance to be measured. And since we don't want to redefine the anionic radius, a negative radius is assigned to the proton, and the mathematics works out OK.

I guess you could say atoms and ions are somewhat mushy, and mathematics can deal with it.