Potassium's turn was a fine-structure question

Whether an alkali’s s level and the p level with no angular momentum about a field first approach or first separate as the field is turned on — whether their separation has a minimum at all — is a question about two polarisabilities. In a field F each level moves by $-\tfrac12\alpha F^2$, so the s–p separation shrinks at first exactly when the p level is the more polarisable. Summed over every s, p and d level, with radial integrals that reproduce five measured ground-state polarisabilities to two per cent, the p level wins at every valence shell from sodium down — and potassium was the one atom whose verdict changed along its own Rydberg series. Its ratio $\alpha(np_0)/\alpha(ns)$ is 2.75 at the fourth shell and falls steadily, and on the tabulated asymptotic defects it dropped below one at the thirty-seventh.

That number came with a warning attached. The ratio near the turn changes by about seven thousandths a shell, so a defect moved by a thousandth moved the turn by several shells — to anywhere between the thirty-first and the forty-ninth — and a thousandth was said to be the size of the Ritz corrections the asymptotic defects leave out. The turn was located, honestly, as somewhere between the thirtieth and fiftieth shell — a number of the kind that looked like a limit elsewhere in this argument and turned out not to be one.

The Ritz corrections for potassium are not estimates. They were measured.

A defect that depends on the shell #

A quantum defect is the amount by which an alkali level sits below the hydrogen level of the same principal quantum number, written as a shortfall in that number: the level is at $-R/(n - \delta)^2$. Far up a series the defect settles to a constant $\delta_0$, set by how deeply the series’ orbits penetrate the ionic core. Lower down it drifts, because a lower level has less kinetic energy outside the core and the core’s influence on it is not quite the asymptotic one. The drift is written as a series in the inverse square of the effective quantum number,

$$\delta(n) = \delta_0 + \frac{\delta_2}{(n - \delta_0)^2} + \frac{\delta_4}{(n - \delta_0)^4} + \frac{\delta_6}{(n - \delta_0)^6},$$

the expansion Ritz introduced and spectroscopists have fitted to every alkali’s series since.

For potassium-39 the coefficients come from a single campaign: Peper, Merkt and co-workers measured np Rydberg transitions from ultracold atoms against a frequency comb, and the s, d, f and g series by millimetre-wave spectroscopy between Rydberg states, published in 2019. Their $\delta_0$ values carry eight decimals and their $\delta_2$ values five. Averaged over the fine-structure components with the degeneracy weights the asymptotic defects used — 1 : 2 for $p_{1/2}$ and $p_{3/2}$, 2 : 3 for $d_{3/2}$ and $d_{5/2}$ — they give 2.180208 for s, 1.711894 for p and 0.277088 for d, with $\delta_2$ of 0.1345, 0.2319 and −1.0257.

The first thing they settle is which input was uncertain. The four-decimal asymptotic defects agree with the measured $\delta_0$ to within $1.2 \times 10^{-5}$, and replacing them moves the turn from 36.93 to 36.99 — six hundredths of a shell. The constant part of the defects was never the problem.

The drift is. At the shells where the turn falls, the s correction is $1.1 \times 10^{-4}$, the p correction $1.9 \times 10^{-4}$, and the d correction $-7.6 \times 10^{-4}$, the largest by a factor of four because potassium’s d series has a $\delta_2$ near −1. None reaches the thousandth used as a nudge. All of them are of a size that a ratio moving by seven thousandths a shell, through a balance of polarisabilities that cancel to a few parts in a thousand, can feel.

The thirty-fourth shell #

With every level above the measured low terms placed by its Ritz-corrected defect, the ratio passes one at n = 33.72: between the thirty-third shell, where the p level is still the more polarisable by about half a per cent, and the thirty-fourth, where it has stopped being. The turn has moved more than three shells down from where the asymptotic defects put it, and every input that moved it is measured.

How firmly it is now placed is a separate question with three parts. Truncating the Ritz series after $\delta_2$ moves the turn by three thousandths of a shell, so the higher coefficients, whose stated uncertainties are the largest, do not matter. The stated uncertainties of $\delta_0$ and $\delta_2$ are parts in a million and a few in a hundred thousand, two orders of magnitude below anything that moves the turn. And the radial integrals, computed in the Coulomb approximation from the same defects, have one adjustable choice: where the dipole integral starts, since the approximation does not hold inside the core. Moving that lower limit from the origin to one bohr and to two changes the ratio at the thirty-fourth shell in the seventh decimal. At these shells the orbits spend almost no time near the core, and the integrals are as good as the energies they are built from.

So on its own terms, the spinless sum now has an answer. Potassium’s p level stops being the more polarisable at the thirty-fourth shell.

Three corrections pulling two ways #

The three series do not all push the same way, and taking them apart shows why the answer is not simply “the largest correction wins”. Applied alone, the d correction moves the turn from 36.99 to 34.80; the p correction to 35.35; the s correction moves it the other way, up to 37.63. Together they land at 33.72, and the pairwise combinations sit where adding the single moves would put them, to within a few tenths of a shell.

The directions follow from what each correction does to the energy gaps that set the polarisabilities. Potassium’s p level is pushed by levels of the shell below and the next shell up, and the most important of them is the d level of the shell below, which sits just above the p level because the d series’ defect is small and the p series’ large. The d correction is negative, so it lowers the d defect a little, raises the d level’s effective quantum number and moves that d level slightly further above p. A larger gap is a weaker push, the p level’s polarisability falls, and the turn comes sooner. The s correction is positive and moves the s level down relative to its neighbours, and the s level’s own polarisability is dominated by the p levels just above it — the gap to them opens, its polarisability falls, the ratio rises and the turn goes later.

The bookkeeping is the point. A ratio that is a near-cancellation responds to every level’s position in the same currency, and the response to each is the same order: a few hundred-thousandths of a defect, a fraction of a shell. The threshold built from three levels of one shell had one level to blame for each verdict. The sum over every level has to be read the way a balance sheet is.

The interval the average hides #

That would close the question if potassium’s p series were one series. It is two. The $np_{1/2}$ and $np_{3/2}$ levels have defects of 1.713926 and 1.710879, apart by 0.003047, and the defect the sum uses is their weighted average — the same convention the asymptotic defects followed, and the natural one for a spinless calculation.

Three thousandths is three times the nudge that moved the turn by shells.

Placing every p level at $p_{1/2}$'s defect puts the turn at n = 22.8. Placing it at $p_{3/2}$‘s puts it at 45.2. The average sits between them at 33.7, and nothing about it is privileged except that it is an average. The d series’ fine structure is two orders of magnitude smaller — its components differ by 0.00017 — and moves the turn by only a quarter of a shell either way.

What this calculation does not do is the full fine-structure problem. It keeps the spinless angular factors and changes only where the p levels sit, so its two outer curves are not the polarisabilities of the $p_{1/2}$ and $p_{3/2}$ states; those have their own angular coupling, a $p_{1/2}$ level has no tensor polarisability at all, and a $p_{3/2}$ level’s depends on its projection. What it does show is a bound on the claim the spinless sum makes. The turn it locates at the thirty-fourth shell is sensitive to the p energies at the level of the fine-structure interval, and that interval is part of the atom.

The scales make the situation plain. At the thirty-fourth shell the gap from 34s to 34p is 3.12 cm⁻¹, and the 34p fine-structure interval is 0.0199 cm⁻¹ — 596 MHz, a hundred and fiftieth of the gap. In almost any other question about these levels an interval that small would be a detail. Here the quantity being decided is a difference of two polarisabilities, each of order $2.5 \times 10^{10}$ atomic units, that cancel to a few parts in a thousand, and a detail at the half-per-cent level is the whole answer.

A turn that is harder to place the higher it falls #

The two outer turns are not symmetric about the average, and the asymmetry is a property of the series rather than of the fine structure. Placing the p levels at $p_{1/2}$'s defect raises the p defect by 0.0020 and brings the turn down 10.9 shells; placing them at $p_{3/2}$'s lowers it by only 0.0010 and sends the turn up 11.5. Half the change in the input, slightly more change in the answer.

The reason is the slope the turn is read off. Near the twenty-third shell the ratio falls by 0.0147 a shell; near the thirty-fourth by 0.0074; near the forty-fifth by 0.0045. It halves roughly every time the shell number grows by half, because both polarisabilities grow as the seventh power of the effective quantum number and their ratio settles towards a limit, approaching it more and more slowly. A fixed error in the ratio therefore moves the turn by more shells the further up the series the turn lies — twice as many at the thirty-fourth shell as at the twenty-third, and more again beyond.

That is an uncomfortable property for a quantity whose interest is its location. Every input to this sum is known better at high $n$ than at low: the levels lie closer to the ionisation limit, the Ritz corrections shrink, the Coulomb functions spend less time near the core. And the turn is located worse there, because the ratio it is read from has nearly stopped changing. The same thing made the shell’s own degeneracy a poor guide in an alkali: whatever is left of a hydrogen-like structure at high $n$ is left in the form of near-equalities, and a sign decided by a near-equality is decided by the smallest structure the levels carry. In potassium, at these shells, that structure is the spin–orbit splitting of p.

When the spinless question is the right one #

There is a regime in which the spinless sum is the physically correct description, and it is set by the field. The s level’s quadratic Stark shift equals the 34p fine-structure interval at a field of about 14 V/cm. Well above that, the field mixes the two fine-structure components more strongly than the spin–orbit interaction holds them apart, the spin decouples, and the p level with no orbital angular momentum about the field is a real state — the spinless picture’s $p_0$. Well below it, the fine-structure levels are the states, and the question has to be asked of $p_{1/2}$ and $p_{3/2}$ separately.

So “does potassium’s separation first rise or fall at the thirty-fourth shell” has two answers depending on the field range, and the computation here gives the one for the decoupled regime. In the weak-field limit proper — the limit the original question named — the relevant comparison is between $\alpha(ns_{1/2})$ and the $j$-resolved polarisabilities of each p component, and the spread from 22.8 to 45.2 says those will not agree with the spinless answer or with each other.

That is the same lesson the earlier alkali essays kept relearning from the other end. A threshold built on hydrogen’s degeneracy failed because an alkali’s shells are not degenerate; a sum built on a shell failed because the shell was not the unit that mattered; and a sum over spinless levels finds a turn whose position is decided by the splitting it averaged away. Each refinement located the answer more precisely and showed which structure, smaller than the last, it depended on.

How the turn was located #

Every polarisability is a sum over levels of $2|\langle a|z|b\rangle|^2/(E_b - E_a)$: for the ns level over every p level, for the $np_0$ level over every s and d level, each series from its lowest member to well past the shell in question, with angular factors 1/3 for s–p and 4/15 for p–d. The lowest levels of each series are placed at measured term values; everything above at the defect under test. The radial integrals are Coulomb-approximation functions of each level’s effective quantum number, integrated by Numerov’s method on a grid in $\sqrt r$, the same functions that reproduced the five measured ground-state polarisabilities. The turn is the first shell from the twentieth at which the ratio is below one, refined by linear interpolation between that shell and the one before.

The checks, made wherever these figures are drawn. On the asymptotic defects this sum is the earlier one, to the last digit at three shells, and its turn is at the thirty-seventh. The measured $\delta_0$ alone moves the turn by under two tenths of a shell; the Ritz series puts it at the thirty-fourth; the terms past $\delta_2$ move it by under a twentieth; the d series’ correction, alone, moves it furthest of the three. The integral’s lower limit, from 0 to 2 bohr, moves the ratio at the turn by under a millionth. Either p component in place of the average moves the turn by more than ten shells, and the interval between them is under one per cent of the s–p gap. The refusal is the Ritz series cut to one term, which must reproduce the measured $\delta_0$ exactly for every series, or the expansion is not the one the coefficients were fitted with.

Where the sum stops #

Spinless angular factors. The fine-structure curves move only the p energies. A proper $j$-resolved calculation needs the coupled angular factors and the tensor polarisability of $p_{3/2}$, and it would replace the spread from 22.8 to 45.2 with two definite answers, one per component.

The Coulomb approximation. Its integrals are insensitive to the core here, but they are not exact; they reproduce measured ground-state polarisabilities to two per cent, and at high shells there is no measured polarisability of the same pair to check a ratio against. A ratio is kinder than either polarisability — errors common to both cancel — but the turn depends on what remains.

Core polarisation. The sum treats the valence electron alone. The core’s own polarisability adds to both levels almost equally and the core’s polarisation by the valence electron modifies the dipole operator slightly; both enter at a level the turn could feel and neither is included.

One isotope. The series are ³⁹K’s. Potassium-41’s differ by isotope shifts far below anything here, and nothing in the comparison depends on them.

Who measured the series #

The Ritz expansion is Walther Ritz’s of 1908. The coefficients used here are from Peper, Helmrich, Merkt and co-workers, who in 2019 measured potassium-39’s np series against a frequency comb and its other series by millimetre-wave spectroscopy, and fixed its ionisation energy at 35 009.813 971 cm⁻¹; the low-lying term values are from the NIST Atomic Spectra Database. The Coulomb approximation is that of Bates and Damgaard. The sum with the measured Ritz series, the decomposition of the turn by series and the fine-structure bound are computed here.

Still open: the components themselves, and where the minimum sits #

The obvious open question is the one this essay bounded and did not answer: potassium’s $ns_{1/2}$ against $np_{1/2}$ and against each projection of $np_{3/2}$, with the angular factors that belong to them. The $p_{1/2}$ level is the cleaner case — no tensor part, one projection — and its polarisability is a sum over $s_{1/2}$ and $d_{3/2}$ levels only, so its turn, which the spinless proxy puts near the twenty-third shell, needs one change of angular algebra rather than a new calculation.

The nearer question is the one the shell-by-shell argument has carried since sodium: where the minimum of the separation sits once the separation does fall. With every level included the first approach is certain wherever the ratio is above one, and the field at which the separation stops shrinking needs the next order — the hyperpolarisability — or the few levels that matter diagonalised in the field directly. At the thirty-fourth shell the answer to that would also say whether the minimum arrives below 14 V/cm, where the fine structure still holds, or above it.