The threshold belongs to the atom

A shell’s s and $p_0$ levels, pushed together by an electric field, have a minimum in their separation exactly when the d level above them sits close enough to p. The condition has a closed form: the d level must lie within a threshold fraction of the s–p gap above p, and that fraction is $\tfrac12|\langle p|z|d\rangle|^2 / |\langle s|z|p\rangle|^2$ — a ratio of two dipole matrix elements.

When the comparison was made with measured quantum defects, the offsets came from the atoms and the threshold came from hydrogen. Lithium passed, and sodium, potassium, rubidium and caesium failed at every shell, by factors of two to seven. That essay named the gap in its own reasoning: the threshold was computed from hydrogenic dipole elements, and for the heavier alkalis the offset is so far above the threshold that no plausible change in the matrix elements brings it back.

The matrix elements can be computed from the same defects. The threshold then turns out to be a property of the atom, and one of the four verdicts changes.

Integrals from nothing but a defect #

An alkali’s valence electron, once it is outside the core, sees a charge of one. There its radial function is a Coulomb function — hydrogen’s equation — at the energy the quantum defect sets, $-1/2n^{*2}$ with $n^* = n - \delta_l$. For non-integer $n^*$ that function cannot be continued to the nucleus; it is only an alkali’s function outside the core, which is the only place the dipole integral has much weight anyway, since the integrand carries a factor of $r$.

That is the Coulomb approximation of Bates and Damgaard. The function is integrated inward from far outside the atom, where it decays, until it stops looking like a bound state: to the origin for an s function, which stays finite there, and for p and d functions to the point inside their centrifugal barrier where the inward solution starts to grow. The radial integral $\int u_a\, r\, u_b\, dr$ is then a sum over that range, and each dipole element is the radial integral times an angular factor — $1/\sqrt3$ between s and $p_0$, $2/\sqrt{15}$ between $p_0$ and $d_0$.

Two things say whether numbers made this way can be trusted, and they are independent of each other.

At integer $n^*$ the inward Coulomb function is the hydrogen function, so the integrals must reproduce hydrogen’s closed form $\tfrac32 n\sqrt{n^2 - (l+1)^2}$. They do, to six figures, at the third, fourth and sixth shells, and the threshold they give is $2(n^2-4)/5(n^2-1)$ to the fifth decimal.

And the same integrals must give real lines. An absorption oscillator strength is $\tfrac23\,\Delta E\,|R|^2$ for an s-to-p line, so the resonance line of each alkali is a direct test of one of the elements this argument needs. From measured term values the integrals give 0.741 for lithium’s 2s–2p line, 0.962 for sodium’s 3s–3p and 1.049 for rubidium’s 5s–5p, against measured strengths of 0.747, 0.960 and 1.038 — summed over each line’s two fine-structure components. That is one per cent, on the lowest members of each series, which is where a core-blind approximation ought to be at its worst.

Lithium’s margin was hydrogen’s #

For lithium the offset is 0.103 at the third shell and the hydrogenic threshold a quarter, a margin of 59 per cent. Lithium’s own threshold is 0.378 — half as large again — and the margin against it is 73 per cent.

That is the answer to the question the previous comparison left: whether lithium’s room was room against the atom or against hydrogen. It was against hydrogen, and the atom gives more. The reason is in the s–p element. Lithium’s 3s function, with a defect of 0.40, is pulled inward relative to its 3p, and the overlap that makes the s–p integral large in hydrogen is reduced — so the denominator of the threshold shrinks while its p–d numerator, between two functions with nearly hydrogenic defects, stays hydrogen’s to within one per cent.

The difference grows with the shell. Hydrogen’s threshold rises towards two fifths; lithium’s own passes one half by the fifth shell and settles near 0.54. The margin against the atom reaches 77 per cent while the margin against hydrogen stops near 68. The single atom that kept its minimum keeps it more comfortably than the comparison said, and for a second reason the comparison did not name.

Sodium’s third shell crosses back #

Sodium’s valence shell was the case the earlier comparison was most confident about. Its offset is 0.705 — the d level sits seven tenths of an s–p gap above p — against a hydrogenic threshold of a quarter, nearly three times over.

Its own threshold is 0.773. The offset sits below it, by nine per cent of the threshold, and the shell has a coupled minimum.

The threshold is three times hydrogen’s for the opposite of lithium’s reason. Sodium’s 3s defect is 1.35 and its 3p defect 0.86, so the two functions have effective quantum numbers of 1.65 and 2.14 — both far from hydrogen’s three, and far enough from each other that the 3s–3p integral is 4.44 bohr against hydrogen’s 12.7. The p–d integral is reduced too, to 6.2 against 10.1, but less. The ratio of squares moves by a factor of three.

It is a statement about one shell. From the fourth shell up sodium’s own threshold falls below hydrogen’s — 0.44, 0.32, 0.26, down to 0.13 at the thirtieth — while its offset climbs from 0.93 towards 1.6. So the verdict that changes is exactly one entry in the earlier table, and it is the entry for the shell that is sodium’s ground state.

Two cautions go with it, and neither reverses it. The offset came from asymptotic defects, and the valence shell is where a defect depends most on $n$; recomputed from sodium’s measured 3s, 3p and 3d term values the offset is 0.719 and the threshold 0.746, still below, by a smaller margin of four per cent. And the integrals were started at the natural limit of the inward functions; the next section moves that start and asks what happens.

Where the core is taken to end #

The weakest part of the approximation is the inside. Within the sodium core, which ends at about a bohr, the valence functions are not Coulomb functions, and an integral that included that region with the wrong functions would carry the error into the threshold.

So the integrals can be started further out and the verdict watched. Starting them at one bohr moves sodium’s threshold from 0.773 to 0.774; at two bohr, to 0.784; at four bohr, well outside the core, to 1.16. Moving the start outward only raises the threshold. Both integrals lose weight as the inside is removed, but the 3s function is the most compact of the three, so the s–p integral loses more: at four bohr it has fallen by a fifth and the p–d integral by four per cent. The denominator falls faster than the numerator, and no choice of where the core ends takes sodium’s minimum away.

That direction was not obvious in advance and it is the result the verdict actually rests on. Had the threshold fallen as the start moved out, the verdict would have depended on a choice nobody can make precisely — how far the core extends — and the essay would have had to say undecided. It rises, so the verdict holds on the conservative side of every such choice.

The heavier alkalis keep their verdict for a different reason #

For potassium, rubidium and caesium the verdict is unchanged, and the earlier essay’s reason for it — offsets far above the threshold — is still true. But the thresholds themselves moved the other way. Potassium’s own threshold at its fourth shell is 0.010, against hydrogen’s 0.32. Rubidium’s is 0.065 at the fifth, caesium’s 0.23 at the sixth. From one and a half to thirty times smaller.

The cause is the numerator. Potassium’s p and d defects are 1.71 and 0.28, so its 4p and 4d functions have effective quantum numbers 2.29 and 3.72 — they are, in effect, functions of different shells, oscillating out of step through the region where both have weight, and their product integrates to almost nothing. Potassium’s p–d integral is four per cent of hydrogen’s at the fourth shell and changes sign between the fifth and sixth, passing through zero; rubidium’s changes sign between the twentieth and thirtieth. A zero in a radial integral is a familiar thing in photoionisation, where it is called a Cooper minimum, and it is the same interference between two functions whose nodes do not line up.

So for three of the five alkalis the coupled minimum is ruled out twice over: the d level is far from p, and d barely couples to p at all. The first reason is about where levels sit and the second is about what the field can do with them, and the earlier comparison, having only hydrogen’s elements, could see the first alone.

A race between two mismatches #

Every one of these thresholds is hydrogen’s multiplied by one factor, and the factor has two parts that pull in opposite directions. The threshold is the p–d element squared over the s–p element squared. Each element, compared with hydrogen’s at the same shell, is reduced by how badly its two functions’ effective quantum numbers match — and a reduced denominator raises the threshold while a reduced numerator lowers it.

Lithium loses a fifth of its s–p element — 10.40 bohr against hydrogen’s 12.73, a squared ratio of 0.667 — and nothing of its p–d element, whose defects of 0.047 and 0.002 are as close to hydrogen’s as an atom’s get. Only the denominator moves, and the threshold rises by a factor of 1.51.

Sodium loses 65 per cent of its s–p element and 39 per cent of its p–d element. Squared, that is 0.122 against 0.377. Both parts move and the denominator moves further, so the threshold rises by a factor of 3.09.

Potassium loses 77 per cent of its s–p element and 96 per cent of its p–d element: squared, 0.052 against 0.0017. Now the numerator has lost far more, and the threshold falls by a factor of thirty-one. Rubidium’s factor is 0.19 and caesium’s 0.64 — each, like potassium, a numerator that has lost more than its denominator, by less.

The pattern is in the defect differences, though not as a formula. Where an atom’s p and d defects differ by under one — lithium by 0.05, sodium by 0.84 — the p–d element keeps most of its size. Where they differ by more than one — caesium by 1.10, rubidium by 1.30, potassium by 1.43 — it keeps under a tenth. The s and p defects differ by about a half in every alkali past lithium, and that pair loses between two thirds and nine tenths of its element, more in the heavier atoms, whose shells are larger. Past a p–d difference of about one the two functions stop overlapping in step, the numerator collapses faster than the denominator, and the threshold falls instead of rising.

What the offsets never contained #

The whole of the earlier argument ran through one number per shell, the offset, and treated the threshold as the fixed thing it was compared with. The threshold is the larger variable. Across the five valence shells the offsets run from 0.10 to 1.07, a factor of ten. The atoms’ own thresholds run from 0.010 to 0.77, a factor of seventy-four.

That inverts which part of the comparison deserved scrutiny. A defect enters an offset through energies, and energies are forgiving: a level’s position is a single number, and the defects that set it are measured to four decimals. A defect enters a threshold through wavefunctions, and wavefunctions are not forgiving: two functions whose effective quantum numbers differ by one and a half do not merely overlap less, they overlap with the wrong sign in places, and the integral can pass through zero.

It is the same lesson an extra degeneracy taught about hydrogen from the other side. Hydrogen’s levels of one shell share an energy and their functions are tuned to one another, which is what gives its dipole elements their closed form. Take the degeneracy away and the energies move by a little while the functions lose their tuning altogether — and the quantities built from pairs of functions, like this threshold, move by far more than the quantities built from energies.

How it was computed #

Each radial function is integrated by Numerov’s method on a uniform grid in $x = \sqrt r$, a grid on which a Coulomb function’s oscillations are nearly evenly spaced, from $r = 2n^*(n^* + 15)$ bohr inward. The effective quantum numbers come from the same asymptotic defects the earlier comparison used, averaged over fine structure with degeneracy weights. The offsets are unchanged from that comparison; only the threshold is new.

The checks, made wherever these figures are drawn: hydrogen’s closed-form elements and threshold at integer $n^*$; the three resonance oscillator strengths within two per cent of measurement; lithium below its own threshold at every shell and its own threshold above hydrogen’s; sodium’s third shell below its own threshold and above hydrogen’s, and every higher sodium shell above; potassium, rubidium and caesium above their own thresholds at every shell, each own threshold below hydrogen’s; the p–d sign changes where stated; and the threshold’s rise as the integrals’ start moves out. The refusal is an almost-hydrogenic atom — defects of a hundredth, in the screening model’s proportions — which must return the closed-form threshold to two per cent, so that everything the alkalis show is theirs and not the integration’s.

What a three-level threshold leaves out #

Other levels. The threshold is a statement about three levels, and sodium has a fourth where it matters. Its 4s level lies between 3p and 3d, and it couples to 3p exactly as 3d does. Computed with the same integrals, the push 4s gives the $3p_0$ level in a weak field is 190 in atomic units against 194 from 3d — as large as the push the threshold is built from. A three-level criterion cannot hold that, and for sodium’s third shell it is not a small correction; it is a second term of the same size.

Fine structure. Every defect here is averaged over its two fine-structure components, and at the field where sodium’s 3s and 3p would approach one another the 3p’s spin–orbit splitting is small by comparison, so the average is the right object. For caesium, whose 6p splitting is over five hundred wavenumbers, it would not be.

The field itself. A three-level argument is a weak-field argument. How far into the field sodium’s minimum would sit, and whether other levels of other shells have arrived by then, is a separate calculation with the same functions.

A threshold was a number about a shell #

The earlier essays found the threshold as a closed form in $n$, generalised it to every shell and quoted its limit of two fifths. Each of those was correct and each was about hydrogen, where every shell’s s, p and d functions share an effective quantum number and the dipole elements between them are fixed by that alone. The phrase the threshold at the third shell then means one number.

For an atom with a core it does not. The third shell of lithium, the third shell of sodium and hydrogen’s third shell have thresholds of 0.378, 0.773 and 0.25, and nothing about the shell’s label predicts which. The same habit this argument keeps finding — a model’s parameter sweep that explores one line, a limit that turned out not to be a limit — appears here as a number attached to the wrong object: a threshold that was attached to $n$ belongs to the atom.

Who computed what #

Bates and Damgaard gave the Coulomb approximation in 1949, with tables that were the standard source of alkali line strengths for a generation; the version here integrates the equation directly rather than using their tables, which is the practice since Numerov-based codes made it trivial. Cooper described the zero in the radial integral for photoionisation in 1962. The measured oscillator strengths are the resonance-line values from lifetime and absorption measurements, and the defects are the asymptotic Rydberg-series values used before.

What is computed here is each alkali’s own coupled-minimum threshold at every shell from its valence shell to the thirtieth, the verdict it gives against the offsets already computed, the sign changes in the heavier atoms’ p–d integrals, and the dependence of sodium’s verdict on where the integrals begin.

Still open: sodium’s fourth level, and a real field #

The obvious open question is sodium’s 4s. The threshold is built from one push on the p level, from d, and the same integrals say that 4s pushes as hard. The criterion that includes every level is the sign of a difference of polarisabilities: the s–p separation falls at first exactly when the $p_0$ level is more polarisable than the s level. Polarisabilities are sums over every level, they can be computed from the same integrals, and — unlike the three-level threshold — they are measured. Sodium’s 3s polarisability is known to a fraction of a per cent, which makes the all-level question the first one in this argument that has an experimental number on both sides of it.

The nearer question is where the minimum sits. With its own elements sodium’s third shell has the minimum, and the three-level model can say at what field and how deep. Whether that field is below the one at which the next shell’s levels arrive — sodium’s 4s sits half an s–p gap above its 3p — decides whether the minimum is a feature of sodium or of a three-level model of it.