The shell was the wrong unit

An alkali’s s level and the p level with no angular momentum about a field are coupled by that field, and whether they first approach or first separate as it is turned on decides whether their separation has a minimum. The threshold built for that question used three levels of one shell — ns, np and nd — and asked whether the d level sat close enough above p to push it down faster than the s–p pair pushes itself apart. Compared first with hydrogen’s matrix elements and then with each atom’s own, it gave lithium and sodium’s third shell the minimum and denied it to potassium, rubidium and caesium at every shell, for two reasons at once: their d levels sit far above p, and their p–d integrals nearly cancel.

That essay ended on the obvious hole in a three-level argument. Sodium’s 4s level lies between 3p and 3d and pushes 3p exactly as 3d does, and nothing in a threshold built from one shell can hold it. The criterion that holds every level has been available all along, and it is not a threshold at all. It is the sign of a difference of two polarisabilities — a number that is measured.

Computed, it confirms the one verdict that looked most fragile, by a margin thirty times larger, and reverses the three that looked safest.

A threshold is a truncated difference of polarisabilities #

In a weak field $F$ every nondegenerate level moves by $-\tfrac12\alpha F^2$ — quadratically, because only a degeneracy like hydrogen’s makes the shift linear — where $\alpha$ is that level’s static polarisability — a sum over every level it couples to, each term $2|\langle a|z|b\rangle|^2/(E_b - E_a)$. A level pushed by partners above it moves down and has a positive term; a partner below pushes it up and contributes a negative one. The separation of the s and $p_0$ levels therefore changes by

$$\Delta(F) - \Delta(0) = -\tfrac12\bigl(\alpha(p_0) - \alpha(s)\bigr)F^2,$$

and it falls at first exactly when the $p_0$ level is the more polarisable.

Keep only three levels and the sign reduces to the threshold. The s level’s polarisability is then its one term with p. The $p_0$ level’s has that same term with the opposite sign — s sits below it and pushes it up — plus the push from d. The difference is minus twice the pair’s term plus the d term, and asking for it to be positive is asking for the d level to sit within $\tfrac12|\langle p_0|z|d_0\rangle|^2/|\langle s|z|p_0\rangle|^2$ of an s–p gap above p. That is the threshold, level for level, and the two verdicts agree at every shell of every atom where the shell has a d level of its own — the three-level arithmetic is the same arithmetic, written as a ratio.

What the threshold was, then, is a sum over states that stopped after two terms. Whether stopping there was safe is a question about the terms it dropped, and those can be added one at a time.

Summed, and checked against the atoms #

Each term needs a radial integral and an energy. The integrals come from the same Coulomb-approximation functions the previous comparison used: outside the core an alkali’s valence electron sees a charge of one, its radial function is a Coulomb function at the energy its quantum defect sets, and the dipole integral is taken over the region where that holds. The energies of the lowest members of each series — where a defect depends most on $n$ — are measured term values; everything above them uses the asymptotic defects. The angular factors, which the one theorem behind every selection rule restricts to neighbouring values of $l$, are $\tfrac13$ between s and $p_0$, $\tfrac{4}{15}$ between $p_0$ and $d_0$, and $\tfrac15$ between $p_{\pm1}$ and $d_{\pm1}$. Every s, p and d level of each series is included from its lowest member to twenty-five shells above the one asked about; the tail beyond is below a part in ten thousand.

The check that matters is the one a three-level threshold could never be given. Ground-state polarisabilities of all five alkalis are measured, by atom interferometry and beam deflection, to a per cent or better. Summed over every p level and with the positive ion’s own small core polarisability added — 0.19 atomic units for lithium’s, 15.8 for caesium’s — the sums give 161, 163, 295, 323 and 407 against measured values of 164.2, 162.7, 290.6, 318.8 and 401.0. Sodium’s is within a tenth of a per cent. The worst, lithium’s, is 1.7 per cent low, which is where a compact 2s function inside a small core is least like a Coulomb function.

The excited level can be checked too, in one atom. Sodium’s 3p polarisability has a measured scalar part of about 360 and a tensor part of −88.4, which for a p level without spin put the $m = 0$ state at 537 and the $m = \pm1$ states at 272. The sums give 527 and 271. The 3p fine structure is seventeen wavenumbers, small beside every gap in the sum, so the states without spin are the right objects here; the measured numbers are about as direct a test of the whole calculation as exists.

Two things are missing from any bound-state sum and both have been estimated. The continuum adds a term for every energy above the ionisation limit; if the oscillator-strength density does not rise above the limit, the continuum can add at most that density divided by the level’s binding energy, which is under 2.1 atomic units for every ground state and under 74 for every $p_0$ level. The core’s polarisability is the same in both levels and cancels from the difference: a closed shell is spherical, and a field distorts it by the same amount whichever valence level sits outside it.

Sodium: the same verdict, thirty times the margin #

Sodium’s third shell was the verdict that had changed once already. Against hydrogen’s matrix elements it had no minimum; against its own it had one by nine per cent of the threshold, and by four per cent when the offset came from measured term values.

The ledger explains why the margin was thin. The 3s–3p pair, counted in both levels, takes 322 atomic units from the difference. The shell’s own 3d level gives back 334. Three levels leave twelve. A number that small is the difference of two numbers of three hundred, and it is exactly the four-per-cent margin in another currency.

The 4s level adds 315. Its 3p–4s integral, 4.35 bohr, is nearly the 3s–3p integral, and it sits 8,773 wavenumbers above 3p — closer than 3d’s 12,206. Its push on the $p_0$ level is only six per cent smaller than 3d’s, the push the threshold was built from, and the threshold had no place to put it. Two levels cannot make a minimum, and the argument since has been about which third level makes it; for sodium’s valence shell the answer is two third levels of almost equal weight, one of them from the next shell. Every other level adds another 38, and the total is 365 — thirty times what the three levels gave, and fourteen times the most the continuum could add.

The verdict does not change; its footing does. A comparison that sat four per cent from its own boundary now sits a factor of three from it. And the difference is measured: 537 against 162.7 is 374, within three per cent of the sum.

Three verdicts reverse #

For potassium, rubidium and caesium the three-level differences at the valence shell are −575, −613 and −644 atomic units: firmly negative, which is the “no minimum at any shell” the threshold reported. Summed over every level they are +506, +779 and +1,484. The $p_0$ polarisability of each heavy alkali’s resonance level is between two and five times its ground state’s, and the separation falls at first in all three.

The reversal holds at every shell the table drew, from each valence shell to the thirtieth. Rubidium’s $p_0$ polarisability runs from 3.2 to 6.2 times its s level’s; caesium’s from 4.8 to 21. Potassium’s falls towards one and is still 1.06 at the thirtieth shell, which makes it the subject of a later section.

Lithium goes the other way at its own valence shell, which the three-level comparison never drew because the second shell has no d level. Its 2p level is pushed up by 2s and down by 3s and 3d, and the sum leaves it less polarisable than 2s — 112 against 161. The separation of lithium’s resonance pair widens in a field. The measured Stark shift of lithium’s $\mathrm{D_1}$ line agrees in sign: the 2p level is about 37 atomic units less polarisable than 2s, against 43 from the sum’s scalar parts.

The d level beside p belongs to the shell below #

The heavy alkalis do not reverse because of the next s level. It helps — 5s adds 391 to potassium’s 4p, 6s adds 486 to rubidium’s 5p, 7s adds 552 to caesium’s 6p — but alone it would not quite outweigh the pair. What reverses them is a d level that was in the atom all along and was never in the threshold.

A shell, in hydrogen, is a set of levels with one principal quantum number and one energy — a degeneracy no rotation explains, and one that holds only for a potential going exactly as $1/r$. In an alkali the defects pull the levels of one $n$ apart by different amounts, and by potassium they have been pulled past each other. Potassium’s 4p has an effective quantum number of 2.23 and its 4d one of 3.80; they are, in energy, levels of different shells. The d level that actually sits just above 4p is 3d, at an effective quantum number of 2.85 — above 4p by 0.62, where the shell’s own 4d is 1.56 above it. The same holds down the column: rubidium’s 5p has 4d just above it, caesium’s 6p has 5d at only 0.20 above.

Those are the levels that polarise the p state. Potassium’s 3d contributes 84 per cent of the whole $\alpha(4p_0)$, rubidium’s 4d 80 per cent of $\alpha(5p_0)$, caesium’s 5d 81 per cent of $\alpha(6p_0)$. The 3d–4p radial integral in potassium is 6.99 bohr — larger than the s–p integral of the pair. The shell’s own d level contributes almost nothing: under a thousandth of the total in potassium’s valence shell and under one per cent in rubidium’s; in caesium’s, where 6d is a little nearer, seven per cent at the valence shell and under one per cent by the tenth. Under one per cent at every potassium and rubidium shell drawn.

So the p–d cancellation found in the heavier alkalis is real, and irrelevant. It is the cancellation of an integral between two functions of different effective shells, and it made the threshold small. But the level whose integral cancelled is not the one doing the pushing. The threshold did not rule the minimum out twice over; it ruled it out once, with the wrong level, and the mismatch it found was the sign that the level was wrong.

Sodium above its third shell: a d level underneath #

The same ordering, one step less advanced, is what decides sodium’s higher shells. Sodium’s d defect is almost zero and its p defect 0.86, so the $(n-1)$d level has an effective quantum number of $n - 1.02$ and $np$ one of $n - 0.86$. From the fourth shell up, the shell below’s d level sits just under np — 0.16 of a unit below it — where it cannot push p down, only up.

It pushes hard. At the fourth shell the 3d term alone is minus three times $\alpha(4s)$; by the thirtieth it is minus eighteen and a half. The next shell’s s level still adds a positive term of between 1.7 and 4.5 times $\alpha(ns)$, and the shell’s own d level between a half and one times it, but neither is close. $\alpha(np_0)$ is negative at every sodium shell from the fourth, and the separation widens in a field.

That is the verdict the threshold gave — “sodium sits above its threshold at every higher shell” — and on the arithmetic of three levels it was right, since the three-level differences are negative there too. But the reason the atom has no minimum there is a level the threshold did not contain, below p rather than above it. A three-level model that had included the right d level would have given the same verdict for a reason that could be checked; the one that was drawn gave it for a reason that happened to agree.

Potassium’s sign turns where its defects cannot say #

One atom’s ratio does not settle. Potassium’s $\alpha(np_0)/\alpha(ns)$ is 2.75 at its valence shell, near two through the next few, 1.35 at the fifteenth and 1.06 at the thirtieth, and it drops below one at the thirty-seventh. On the tabulated defects, then, potassium’s high Rydberg shells have a rising separation and its low ones a falling one.

That crossing is not a result in the same sense as the others. Near the thirty-seventh shell the two polarisabilities differ by a few parts in a thousand, and the ratio moves by about seven thousandths a shell. Change any one of the three asymptotic defects by a thousandth and the crossing moves: to the forty-fifth or the thirty-second for the s defect, the thirty-first or the forty-ninth for the p, the fortieth or the thirty-fifth for the d. Thousandths are the size of the Ritz corrections the asymptotic defects leave out at these shells — potassium’s d series has one of the largest of any alkali. So the survey can say that potassium falls at every shell to about the thirtieth, firmly, and that somewhere between the thirtieth and fiftieth its sign turns. It cannot say where, and says so. A number that moves this much under the last decimal of its inputs is the kind that looked like a limit elsewhere in this argument and turned out not to be one.

The contrast with the other atoms is worth stating because it is the same arithmetic. Rubidium’s and caesium’s ratios rise along the series, towards six and twenty-one at the thirtieth shell, because their $(n-1)$d levels sit close above p — 0.30 and 0.10 of a unit — and those terms grow faster than the pair’s. Potassium’s sits 0.43 above, and its next-shell s level 0.53 above; the pushes that keep the ratio above one are the weakest of the three atoms’, and at high $n$ the pair very nearly balances them.

Measured on both sides #

The earlier comparisons in this argument all set a model against a model: an offset from measured defects against a threshold from hydrogen’s elements, then against the atom’s own. Polarisabilities change the kind of statement available. For sodium’s third shell the difference $\alpha(3p_0) - \alpha(3s)$ has a measured value, about 374 atomic units, and the sum reproduces it to three per cent. For lithium’s valence shell the measured sign is negative and so is the sum’s. The verdict for those two shells no longer depends on a model of anything.

There is a connection here that the argument had not made. The difference of polarisabilities between an alkali’s resonance levels is the quantity that sets the DC Stark shift of its D lines, and the same differential polarisability at optical frequencies is what laser cooling and optical clocks engineer around — a trap wavelength where the two levels shift equally is chosen by making exactly this difference vanish. The question this argument arrived at from the geometry of three coupled levels is one experimental atomic physics has been measuring for decades for a different reason, and its sign was in the tables before the threshold was drawn.

What a sum over levels still leaves out #

The field beyond second order. A falling separation at weak field guarantees a minimum only if the separation rises again later, and in a real atom the pair is not alone at stronger fields: other levels arrive, and the next shell’s levels cross. For sodium the 4s level sits half an s–p gap above 3p, so the field at which 3s and 3p are closest has to be compared with the field at which 4s interferes. The sign computed here is a weak-field statement about which way the separation starts, which is what the threshold claimed to decide and all it claimed.

Fine structure. Every state here is a state without spin, which is right when the p level’s spin–orbit splitting is small beside the gaps in the sum. For sodium it is seventeen wavenumbers against gaps of thousands. For caesium it is 554 wavenumbers against a 6p–5d gap of 3,010, and there the sum over states without spin is an average over states that a real field sorts by $j$. The verdict is so large there — $\alpha(6p_0)$ nearly five times $\alpha(6s)$ — that averaging cannot plausibly reverse it, but the number itself is an average.

The Coulomb approximation’s inside. Every integral stops where the valence function stops being a Coulomb function, at the core. The measured ground-state polarisabilities are the evidence that this costs about two per cent; the core-penetrating 2s function of lithium is where it costs most.

The continuum, as an estimate. The continuum terms are bounded on the assumption that the oscillator-strength density does not rise above the ionisation limit. In the heavy alkalis it has a Cooper minimum above threshold, which only makes the bound more generous.

Who measured what #

Polarisabilities of the alkali ground states were measured by electric deflection of atomic beams from the 1970s and by atom interferometry since the 1990s: sodium’s 162.7 atomic units in 1995, lithium’s 164.2 in 2006, potassium’s and rubidium’s 290.6 and 318.8 in 2010, caesium’s 401.0 in a fountain experiment in 2003. Sodium’s excited-level values come from the Stark shift and splitting of its D lines, and lithium’s sign from the Stark shift of its $\mathrm{D_1}$ line. The core polarisabilities are those of the positive ions, from spectroscopy of high-angular-momentum Rydberg states. The term values are the standard tabulated ones, averaged over fine structure with degeneracy weights; each is checked against its series’ asymptotic defect before it is used.

What is computed here is the full sum over s, p and d levels for every alkali at its valence shell and up to the thirtieth, its agreement with every measured number above, the decomposition that shows which level pushes, and the dependence of potassium’s crossing on the last decimals of its defects.

Still open: where the minimum sits, and potassium’s turn #

The obvious open question is the field. With every level included sodium’s third shell is certain to start closing, and the depth and position of its minimum need the next order — the hyperpolarisability — or, more cleanly, the four levels that matter (3s, 3p, 4s, 3d) diagonalised in a field. Whether the minimum arrives before 4s and 3p begin to interact strongly decides whether it is a feature of sodium or only of its weak-field series.

The nearer question is potassium’s crossing. The asymptotic defects leave out Ritz corrections of exactly the size that moves it by ten shells, and those corrections are measured for every alkali’s s, p and d series. With them the crossing becomes a number, and it would say whether potassium’s Rydberg levels near the fortieth shell — where Rydberg experiments with potassium are routinely done — separate or approach in a weak field.