Four alkalis the model cannot hold

In an electric field the s and $p_0$ levels of one shell couple, and whether the gap between them has a minimum as the field grows depends on a third level. The condition was found in closed form: the d level must sit above p by less than a threshold fraction of the s–p gap, $2(n^2 - 4)/5(n^2 - 1)$ — zero at the second shell, a quarter at the third, rising to two fifths. On the other side of the comparison was the shell’s own offset, from a screening model in which a core is represented by an extra $-\lambda/r^2$ potential. In that model’s small-defect limit every shell’s offset is exactly one fifth, so every shell above the second passes.

That conclusion came with its own limitation stated: a real alkali’s quantum defects are not given by any one-parameter law, and a real atom’s offset would have to be computed from its own measured defects. It also came with a prediction about what would happen when it was: that a large defect pushes the offset below a fifth, so real atoms have more room than the model does.

The defects are measured, for every alkali and every series. So the comparison can be made with them, and the prediction tested.

Lithium passes and nobody else does #

A level of principal quantum number $n$ and quantum defect $\delta_l$ lies at $-1/2(n - \delta_l)^2$, so a shell’s offset is $(E_d - E_p)/(E_p - E_s)$ with each energy from its own defect. The defects used are the asymptotic values of each alkali’s s, p and d Rydberg series, the p and d averaged over their fine-structure components.

Lithium’s offset is 0.103 at the third shell, against a threshold of a quarter, and it stays below its threshold at every shell to the thirtieth. It is the one alkali whose shells behave as the model said. Its second shell has no d level and so no minimum, as for every atom; from the third shell up it has one.

Sodium’s offset is 0.705 at its valence shell, the third — nearly three times the threshold. Potassium’s is 1.071 at the fourth, rubidium’s 0.997 at the fifth, caesium’s 0.953 at the sixth. And none of them comes back below the threshold at any larger shell. The offsets rise with $n$, the threshold rises too, and the offsets rise faster: by the thirtieth shell the four heavier alkalis sit between 4 and 7 times above it.

So four of the five alkali atoms have no coupled minimum at any shell. The statement the closed form licensed — that every shell above the second has one — is true of hydrogen, true of the screening model, and true of lithium. It is false of sodium, potassium, rubidium and caesium, which is most of the column.

The ratio that decides it #

For a large shell every level gap is proportional to a difference of defects divided by $n^3$, so the offset tends to a single number for each atom:

$$\text{offset} \to \frac{\delta_p - \delta_d}{\delta_s - \delta_p}.$$

In the screening model the defects go as $1/(l + \tfrac12)$ in the small-defect limit, in the ratio 2 : ⅔ : ⅖, and that ratio makes the limit exactly one fifth — which is why the model’s offset had no $n$ in it. For lithium the measured ratio is 0.128: its p defect is small and its d defect smaller, so the p–d difference is a small fraction of the s–p difference, and the offset sits below even the model’s fifth.

For every heavier alkali the ratio is above one and a half: 1.70 for sodium, 3.06 for potassium, 2.68 for rubidium, 2.29 for caesium. The p defect is not two thirds of a third of the s defect, as the law has it; it is most of the s defect. Sodium’s p defect is 0.855 against an s defect of 1.348; caesium’s is 3.570 against 4.049. The s and p levels are pulled down nearly together and the d level is left behind, so the d level is far above p compared with how far p is above s.

The division between lithium and the rest is the division in their cores. Lithium’s core is a single 1s pair, compact and without angular structure, so a valence p electron — held away from the nucleus by its centrifugal barrier — barely reaches it. Sodium’s core has a whole second shell, including a filled 2p subshell, and extends far enough out that the valence p electron penetrates it substantially. From potassium on the core is larger again and the d electron penetrates too, which is why the d defect climbs from 0.016 in sodium to 2.47 in caesium. Penetration is what the defects measure, and it depends on the core’s size and shape in a way no single-parameter potential tracks.

What the defects are a record of #

A quantum defect is the shortest summary there is of how an electron in one angular momentum feels a core. It is zero for a hydrogenic electron, which feels a bare point charge at every radius, and it grows as the electron spends more of its time inside the core, where the screening that makes the far field look like a charge of one has not yet happened. That is the same fact an electron’s effective charge records in Slater’s rules, and the part a single screening constant cannot capture is precisely the dependence on angular momentum.

The radial distributions say where that dependence comes from. Across the table an s function of a high shell has inner lobes all the way in to the nucleus, a p function has one fewer and is held off the centre by its barrier, and a d function has fewer still. Whether a p function’s inner lobe falls inside the core depends on how large the core is. Lithium’s core is a 1s pair at about half a bohr; the valence p electron’s innermost density is outside it. Sodium’s core reaches the 2p shell at around a bohr, and the 3p’s inner lobe sits squarely in it.

The same defects predict an ordering chemistry already knows. Potassium’s 4s level, with a defect of 2.18, lies at −0.151 hartree; its 3d level, with a defect of 0.28, lies at −0.067. So in potassium the 4s is filled before the 3d, which is an ordering that belongs to the atom and not to the shell: a d electron in potassium penetrates little and an s electron a great deal. The quantity that decides a coupled minimum in a field and the quantity that decides where the fourth period starts are the same three numbers read two ways.

So the comparison of offsets is a comparison of penetrations, and it has the answer penetration gives. An atom whose p electrons penetrate almost as deeply as its s electrons — every alkali with a p shell in its core — has its s and p levels close together and its d level far away, and in a field its s and p couple like two isolated levels with nothing above to make a minimum. An atom whose core is too small for p penetration keeps the hydrogenic structure the closed form describes.

The prediction went the wrong way #

The earlier argument about large defects went like this: a defect enters the energy as $-1/2(n-\delta)^2$, a level’s shift grows faster than linearly in $\delta$, it grows fastest for the s level, so a large defect stretches the s–p gap more than the p–d gap and the offset falls. Every step is right. What it assumed without saying so was that the three defects keep their proportions as they grow.

They do not. Scaled to each atom’s own s defect, the law would give sodium a p defect of 0.45 and a d defect of 0.27; the measured ones are 0.86 and 0.016. Caesium’s law values would be 1.35 and 0.81; the measured ones are 3.57 and 2.47. The p defect is far too large a share and, until potassium, the d defect far too small — both in the direction that moves the offset up, not down.

So the margin argument was a correct statement about the model and an incorrect one about atoms, and nothing inside the model could have shown the difference, because the model has exactly one parameter and the thing that went wrong is a ratio between three defects. The same kind of failure was found when the defect was swept across two decades inside the model and every dimensionless quantity settled: a sweep of one parameter explores one line through a three-dimensional space of defects, and the atoms are not on it.

The model has no member like sodium #

It is worse than a wrong ratio. The screening model is not even able to produce the heavier alkalis’ s defects.

Adding $-\lambda/r^2$ to the potential replaces $l(l+1)$ in the centrifugal term by $l(l+1) - 2\lambda$, which turns the exactly soluble hydrogen problem into an approximately soluble alkali one with an effective angular momentum $l'$ and a defect $l - l'$. For an s electron that defect is $\tfrac12 - \sqrt{\tfrac14 - 2\lambda}$, and when $\lambda$ reaches an eighth the square root vanishes and the barrier with it. The largest s defect the model can produce is one half. Lithium’s, 0.40, lies below it. Sodium’s is 1.35 and caesium’s 4.05, and there is no strength of the model that gives either.

So the model is not a small-defect approximation to sodium that becomes inaccurate at sodium’s defect; it is a family of atoms that stops before sodium begins. Even for lithium, where a strength exists, fitting it to the s defect puts the p defect at 0.082 against a measured 0.047 and the d defect at 0.048 against a measured 0.002 — more than twenty times too large. The model’s lithium has the right s level and a d level in the wrong place, which is the level the comparison is about.

Lithium, alone #

For the one alkali that keeps the minimum, the margin behaves as the closed form suggested it should — only for a different reason. Lithium’s offset starts at 0.103 at the third shell and creeps up to 0.126 at the thirtieth, approaching its limit of 0.128 from below; the threshold rises from a quarter to nearly two fifths. The margin is 59 per cent of the threshold at the third shell, opens to 69 per cent by the tenth and holds there.

That is more room than the model gave — the model’s fifth had a twenty per cent margin at the third shell — and lithium has it because its d defect is tiny, not because its s defect is large. The one atom for which the earlier prediction came true came true for a reason the prediction did not name.

How the offsets were computed #

Each atom’s s, p and d quantum defects are the asymptotic values of its Rydberg series as tabulated from spectroscopy, quoted to four decimals; the p and d values are averages over the two fine-structure components with the degeneracy weights 1 : 2 and 2 : 3. Each level is $-1/2(n-\delta_l)^2$ in hartrees, and each shell’s offset is the ratio of its p–d gap to its s–p gap. The threshold is the closed form from hydrogenic dipole elements. Nothing is fitted.

The checks, run wherever these figures are drawn: the screening model’s own small-defect offset is one fifth, which is the number the earlier comparison used; lithium’s offset is below the threshold at every shell from the third to the thirtieth; each heavier alkali’s is above it at every shell, by more than a factor of two; each heavier alkali’s ratio of defect differences exceeds one and a half, and its offset rises towards that ratio and is within an eighth of it by the thirtieth shell; the screening model has no strength that gives any heavier alkali’s s defect; and fitted to lithium’s it overstates the d defect more than tenfold. The refusal is a set of defects in exactly the model’s proportions, which must give one fifth at a large shell whatever their scale — so the difference from one fifth for real atoms is the proportions and not the arithmetic.

Three assumptions still in it #

The threshold uses hydrogenic dipole elements. A real alkali’s s–p and p–d radial matrix elements differ from hydrogen’s, most for the low shells, and the threshold is their ratio. For the heavier alkalis the offset is so far above the threshold that no plausible change in the matrix elements brings it back; for lithium the margin is sixty per cent, which is large but is a margin against a hydrogenic number.

The defects are asymptotic. At the bottom of a series a defect depends weakly on $n$, through Ritz terms, and the valence shells are exactly where that matters most. For sodium’s third shell the difference moves the offset by a few hundredths, against a gap to the threshold of nearly half.

Hydrogen is outside the comparison altogether. With every defect zero its three levels are degenerate and the offset is a ratio of two zeros; the whole argument needs a core to split the shell before a field can do anything with the splitting, and the model’s fifth was always a statement about a small core rather than about no core.

And a level structure is not a field response. Whether a coupled minimum appears depends on the three levels and their couplings; a real atom in a field has every other level of every other shell coupled in as well, and at fields strong enough for shells to mix the three-level argument is not the calculation.

A one-parameter model explores a line #

The general lesson is about what a model’s parameter sweep can establish. The screening model was swept across two decades of its strength and its conclusions held throughout, and that robustness was reported and was real. It was robustness along the one direction the model can move in. Real atoms differ from one another in a space of several defects, and the ratio between them — the thing a d-level comparison depends on — is exactly what a single parameter fixes.

That is also why the measured table was the right test and a larger sweep was not. A sweep of the model can only ever find the model; the question of whether atoms are like it has to be asked of atoms. Lithium is, and its compact core is the reason. The others are not, and what the model left out — how deeply each angular momentum penetrates a structured core — is what decides the answer.

Still open: the real matrix elements, and the field #

The obvious open question is the threshold’s side. It was computed from hydrogenic dipole elements, and for the alkalis the real s–p and p–d radial elements are known from the same quantum-defect theory that gives the energies. Recomputing the threshold with them would test whether lithium’s sixty per cent margin is a margin against the atom or against hydrogen, and it would say whether any heavy alkali’s threshold is itself large enough to change a verdict — which the size of the gaps above suggests it is not, but a suggestion is not a computation.

The nearer question is what sodium does instead. With its d level far above its p level, a field couples sodium’s 3s and 3p like a two-level system and the gap only widens; the coupled minimum is gone, and what replaces it is the ordinary quadratic Stark shift of an isolated pair. How far into the field that two-level description holds before the d level and the next shell arrive is a calculation with numbers already in hand, and it would say at what field sodium begins to look like the shell the closed form describes.