The residue that is two numbers

A size a confound cannot supply priced a closed-shell repulsion as an energy and asked whether it displaces two ions by the tenths of an ångström the additive radii are wrong by. It does not: the balance of a Madelung attraction against six computed repulsions predicts six separations to 0.242 ångström where adding two radii predicts them to 0.183, and the displacement it produces ranks at 0.14 against the shortfall it was proposed to explain.

It closed by turning the question round. If the shortfall is what a fit could not absorb rather than a physical effect, that is a statement about the fitting — and it can be tested by refitting a set of radii to these distances alone and asking what residue is left.

The residue is not a fitted quantity. It is two numbers, obtained by subtraction, and they are three hundredths of an ångström and six.

Eight separations are not eight constraints #

The eight measured separations join four cations to four anions, and which pairs are present matters more than how many there are.

Sodium and potassium are paired with fluoride and chloride; magnesium and calcium with oxide and sulfide. Nothing joins the two groups, so nothing measured in one constrains anything in the other — and within each group there is a further redundancy that is easy to miss: adding a constant to both cations and subtracting it from both anions leaves every sum unchanged.

So each block has four measurements, four radii, one arbitrary constant, and therefore exactly one degree of freedom an additive model cannot fit.

That constant is why a radius is not an observable and why two sets of radii cannot be compared by comparing radii. Only their residuals against measured distances are comparable, and that is the comparison everything below is.

The residue has a closed form #

The degree of freedom an additive model cannot fit is written down rather than found. For a complete two-by-two block,

$$\Delta = d(AX) - d(AY) - d(BX) + d(BY)$$

is exactly zero for every assignment of radii, so whatever the measurements make it is what no radii can reproduce.

For the alkali halides it is −0.03100 ångström; for the alkaline-earth chalcogenides, −0.05600. A least-squares fit spreads each over its four separations with alternating sign, so the best any set of radii can do is 0.00775 and 0.01400 ångström respectively.

There is no fitting in either number. Both are differences of four measured distances, and the least-squares result is checked against them rather than reported on its own — a general solver and a closed form disagreeing would be the one way this could be wrong without looking wrong.

So the shortfall is the model’s #

The floor is 0.0098 ångström. The tabulated radii miss by 0.0237 — a factor of 2.4 above it. The radii read off the ions’ own densities miss by 0.1828, nineteen times the floor, and the balance of a Madelung attraction against a computed repulsion by 0.2421, twenty-five times it.

That answers the question and answers it cleanly. These separations are additive to about a hundredth of an ångström, so the shortfall being chased is not a property of the pairs. It is a property of the two computed ways of assigning radii, both of which are nearly twenty times worse than an assignment fitted to nothing but the distances themselves.

What that leaves of the earlier arguments #

It is worth saying plainly what survives, because two arguments were spent on a quantity that has just been shown to be an artefact of the models rather than of the compounds.

The argument that ranked a computed repulsion against the shortfall is untouched: it found a rank correlation of 0.857 that a control beat, and concluded that the correlation was not evidence of a mechanism. That conclusion is right and this strengthens it — the thing being correlated against is now known to be the enclosure radii’s own error rather than a physical shortfall, and a quantity computed from the ions’ densities correlating with another quantity computed from the ions’ densities is not surprising.

The control that outranked the mechanism is the sharper of the two and it is also untouched. Its point was that a correlation with a confound is not a mechanism, and the confound has now been identified: both the ranked quantity and the ranking one are functions of ionic size as the enclosure computes it.

What does not survive is the framing. Both arguments described the shortfall as something to be explained, and named candidate mechanisms — polarisation, a closed-shell repulsion, a covalent contribution. There is nothing of that size to explain. The separations are additive to a hundredth of an ångström, and everything larger than that belongs to whoever assigned the radii.

And a universal table gives up almost nothing #

The other half of the question was whether the shortfall is the compromise a universal table makes. It is not that either.

Shannon’s radii were fitted to hundreds of compounds and are used unchanged on these eight — and what they enclose of an ion’s own density varies by a factor of several between them, which is a standing finding about them and is untouched by their accuracy here.

Used unchanged on these eight, they reproduce them to 0.0237 ångström against a floor of 0.0098. A factor of 2.4 for a table that was never asked about these particular compounds is a small price, and on two of the six the table is within a hundredth of the floor.

So there are two negative answers and they are different. The shortfall is not the pairs’ non-additivity, which is a hundredth of an ångström. It is not the universal table’s compromise, which is another hundredth. What is left is the two computed assignments, and what is wrong with those is that they are computed from the wrong thing — which is what the radius that was tabulated establishes directly and this establishes by elimination.

One number is worth taking away from the comparison as a working figure. The best any additive assignment of ionic radii can do on rock-salt separations is about a hundredth of an ångström, and Shannon’s table achieves two and a half hundredths. Anybody quoting a sum of two tabulated radii for a rock-salt separation should expect that error and no more, and anybody explaining a discrepancy of a tenth of an ångström is explaining their own radii.

Why the two blocks differ by a factor of two #

The two interaction terms are −0.031 and −0.056, and the second is nearly twice the first. That is one comparison and it should not be over-read, but its direction is worth recording because it is the only chemical statement the residue makes.

The second block is the doubly charged ions — magnesium and calcium against oxide and sulfide — and the first is the singly charged ones. A larger charge means a stronger field at the neighbour, and a stronger field means more polarisation, which is the one effect that would make a large ion with a large partner sit closer than additivity predicts. Both terms are negative and the larger charges give the larger one.

That is consistent and it is two data points. What would settle it is a third block at a third charge, or the same two blocks with the sign checked against a set that does not share the polarisability trend — and neither is available from eight measurements. The honest statement is that the residue is a hundredth of an ångström, is negative twice, and is larger where the charges are larger.

What completeness is doing, and what a hole would cost #

The closed form is not a property of additive models. It is a property of additive models on a complete block, and the difference is worth stating because it is the one assumption the subtraction rests on.

Four measured separations, two cations against two anions, carry four numbers. An additive model on them has four parameters — two cation radii and two anion radii — and one exact redundancy, because adding a constant to both cations and subtracting it from both anions changes nothing. So three of the four parameters are free, four measurements less three free parameters leaves one degree of freedom, and that one number is the alternating sum. It is available by subtraction because there is exactly one way to combine four measurements so that every row effect and every column effect cancels, and the alternating sum is it.

Remove one of the four and the arithmetic collapses in a specific way: three measurements against three free parameters is an exact fit, the residual is zero, and the model looks perfect while having been told nothing. Remove one from a larger block and the count still works but the combination that cancels the row and column effects is no longer unique, so the floor becomes a least-squares quantity again — found by fitting, which is what the second refutation above says is unnecessary. It is unnecessary here. On an incomplete design it is the only way.

That is the practical warning here. The cheap answer is bought with completeness, and a table of separations with holes in it does not supply the discount.

What was computed, and how #

The separations are quoted crystal separations for rock-salt compounds, the same eight used throughout. The tabulated radii are Shannon’s and Bondi’s, quoted. The enclosure radii and the computed equilibria are recomputed here rather than carried, so the four columns are four predictions of the same six numbers.

The additive fit is a least-squares solution constructed from row and column means, which is the exact solution for a complete block, with the gauge fixed by splitting the grand mean equally between the two sides. That split is a choice and is stated: any other gives the same sums and the same residuals, which is the property the gauge figure demonstrates rather than claims.

Every claim about the floor is checked twice — once from the fit and once from the closed-form alternating sum — and the two must agree to $10^{-12}$.

The refusal is the arbitrary constant. Shifting every radius in a block by a tenth of an ångström must change no predicted separation and no residual, and a fit that reported a different quality afterwards would be fitting the gauge rather than the distances.

Where the model stops #

One further limit is about what the alternating sum can and cannot be blamed on. It is a combination of four measured separations, so every uncertainty in those measurements is in it — and at three hundredths of an ångström it is not enormously larger than the spread between crystallographic determinations of the same compound. Nothing here propagates that, because the separations are quoted without error bars, and a residue of a hundredth of an ångström computed from numbers good to a few thousandths is a residue worth reporting rather than a residue worth interpreting.

Six separations, not eight. Two of the eight are absent from the comparison because the quadrature that computes their repulsion refuses them, and the rule is that a pair whose integrals cannot be done is excluded rather than approximated. The closed-form interaction terms use all eight, because they need no integrals at all — so the two blocks’ Δ values are the whole story and the residual comparison is over six.

Whether a repulsion of the computed size can move an ion at all was the earlier question and its answer stands: it can, by tenths of an ångström. What has changed is what those tenths have to explain, which is now nothing.

Every separation is a crystal separation, so the additive model being tested is the one crystallography uses: a radius that is the same in every compound. A model in which a radius depended on coordination number, or on the partner, would fit better and would not be an additive model.

And a hundredth of an ångström is not zero. What has been shown is that these eight compounds are additive to that, not that ionic radii are additive in general — the radius that was tabulated is about what a radius means, and nothing here bears on it.

The generalisation #

Before asking how well a model fits, find out how well any model of that form could fit.

For an additive model on a complete block that question has an answer in closed form, and the answer costs one subtraction. Once it is known, every comparison changes character: a model missing by 0.18 where the floor is 0.01 is not slightly worse than one missing by 0.02, it is failing at something the form of the model was never responsible for.

The same counting settles a redundant coordinate set, where the number of internal coordinates less the number of redundancies is what a force field can be determined along. The general procedure is to count. Measurements, parameters, and the redundancies among the parameters; what is left over is what the model form cannot absorb, and it can usually be written down. Here it was one number per block, and writing it down turned a long chase after a shortfall into a subtraction.

The same move appears in a parameter that never finds a value, where a parameter with no interior optimum was diagnosed by counting rather than by fitting. The difference is that this one is exact: an alternating sum of four measured numbers has no uncertainty of method in it at all.

Who found it, and when #

The gauge freedom is Pauling’s problem and it is why his radii and Goldschmidt’s differ by a constant rather than by a pattern: both fit the same separations, and a fit to sums cannot see a constant added to one side and taken from the other. That is stated in every account of the tables and is worth restating here, because it is the exact reason a comparison of two sets of radii has to be a comparison of residuals.

Additive ionic radii are Landé’s and Goldschmidt’s, from the nineteen-twenties, and Shannon’s revision of 1976 is the table in use. That the additivity is good to a few hundredths of an ångström is the reason the tables exist and is stated in every account of them; what is less often stated is that the residual non-additivity is a countable quantity rather than a scatter.

The alternating combination is the interaction term of a two-way analysis of variance, from Fisher, and its use here is elementary. That a set of measured separations decomposes into row effects, column effects and one interaction is a fact about the design of the measurement rather than about chemistry — which is exactly why it bounds every chemical model of the same form.

Still open: a larger block, and the sign #

The obvious open question is a block that is not two by two. Three cations against three anions gives nine measurements, five free parameters and four degrees of freedom left over, so the residue stops being one number and becomes a matrix — and the pattern in that matrix is a chemical statement rather than a single one. The alkali halides supply a five-by-four block from the standard tables, and the interaction terms in it would say whether the non-additivity is random or systematic in the periodic table.

The nearer question is the sign. Both interaction terms here are negative, and with two of them that is a coin toss. A larger block would settle it, and a systematic sign would mean something: it would say that a large cation with a large anion is closer than additivity predicts, which is a statement about polarisability rather than about size and is exactly the kind of thing the radius arguments were looking for.