The quarter, generalised

One minimum was found in a Stark shell, between levels the field couples and found what decides whether it exists: the d level must sit within a quarter of the s–p gap above the p level, and the shell’s own d level sits at a fifth. It got the quarter two ways — from a second-order expression whose bracket changes sign at 27 over 108, and from a numerical search bisected to 0.24999886 — and both were about one shell.

Its closing paragraph said the condition is general and that both sides of it move with the principal quantum number. Only one side does.

Half a ratio of two squared elements #

The threshold is ½|⟨p|z|d⟩|²/|⟨s|z|p⟩|², and both elements were computed there by quadrature because that is what this collection’s orbitals support. Both have closed forms. For two m = 0 functions of one shell differing by one in angular momentum,

$$\langle n,l+1 \,|\, z \,|\, n,l \rangle = -\tfrac32 \, n \sqrt{n^2 - (l+1)^2} \; \frac{l+1}{\sqrt{(2l+1)(2l+3)}}$$

The closed form is used rather than derived here, and what makes that safe is that it can be checked. At n = 2 it gives exactly −3 for the s–p element and the quadrature gives −3.000000. At n = 3 it gives −7.348469 and −5.196152 and the quadrature gives −7.348470 and −5.196153. Three elements, two shells, two routes with nothing in common — one a three-dimensional integral on a mapped grid, the other a line of algebra in the quantum numbers.

Substituting and cancelling leaves a ratio with no quadrature in it:

$$\text{threshold} = \frac{2(n^2 - 4)}{5(n^2 - 1)}$$

At n = 3 that is 2 × 5 over 5 × 8, which is a quarter exactly — the number that bisection stopped at, and the number the second-order bracket changes sign at. At n = 4 it is 8/25, at n = 5 seven twentieths, and as the shell grows it rises to two fifths and stops.

The other side does not move #

That calculation expected both sides to change with the shell, and said so: the offset through the screening formula, the matrix elements through the radial functions. The matrix elements do. The offset does not.

Its own derivation is the reason. As the quantum defect goes to zero it tends to λ/(l + ½), so the three levels’ shifts are in the ratio 2 : two thirds : two fifths — and that ratio has no n in it. The s–p gap is 4λ/3 and the p–d gap is 4λ/15, so the offset is their quotient: one fifth, for every shell there is.

So the comparison is not two moving quantities but a constant against a curve, and a constant against a monotone curve changes answer exactly once. It changes at n between two and three.

Setting the threshold equal to a fifth gives $n^2 = 7$ exactly, so the crossing is at n = 2.6458 — between the only two integers that matter and close to neither. A comparison whose crossing landed near an integer would be a comparison whose answer depended on details; this one does not come near one.

Zero, and not small #

At n = 2 the threshold is exactly zero, and the reason is worth separating from the arithmetic that produces it.

The numerator of the ratio is the squared element between p and d. The second shell has no d function. Its m = 0 levels are 2s and 2p and stop there — and the degeneracy that makes a hydrogen shell what it is is a degeneracy of two levels there — so the numerator is not a small number, it is an absent one, and the formula’s factor of $(n^2 - 4)$ is the algebra noticing.

A minimum between two coupled levels exists only because a third level pushes. The two-level half of that was established there: two coupled levels only ever separate, so no minimum between them can be theirs. What makes one is a third level driving one of the pair down faster than the coupling drives them apart, and a shell with only two such levels has no third.

So no shell of principal quantum number two has a coupled minimum at any quantum defect, in any atom, at any screening. That is a structural statement and it takes no calculation past counting the functions.

Why the ratio has a limit at all #

The threshold rises and stops, and the stopping is worth a paragraph because it is not obvious that a ratio of two growing quantities should settle.

Both elements grow with the shell — the s–p one from 3 at n = 2 to 86 at n = 10, the p–d one from nothing to 76. Each carries a factor of n from the overall scale and a factor of $\sqrt{n^2 - (l+1)^2}$ from the radial overlap, so each grows roughly as $n^2$. Their ratio therefore tends to the part that does not scale: the angular factor $(l+1)/\sqrt{(2l+1)(2l+3)}$, which is $1/\sqrt{3}$ for s–p and $2/\sqrt{15}$ for p–d.

Squaring and halving those gives (4/15)/(2/3), which is two fifths. So the limit is an angular number and the approach to it is a radial one, and the difference between the third shell’s quarter and the limit is entirely the difference between $\sqrt{n^2-4}$ and $\sqrt{n^2-1}$ — the d function’s radial overlap suffering more than the p function’s from the shell being small.

That is why the second shell is the extreme case rather than an exception to a pattern: at n = 2 the radial factor for the p–d element is $\sqrt{4-4}$, which is zero, and the angular factor has nothing to multiply.

What a shell with a large defect would do #

The comparison above is between a threshold that depends on the shell and an offset that does not, and the offset’s independence rests on a limit — the small-defect limit, where the quantum defect tends to λ over l plus a half.

Away from that limit the offset moves, and which way is worth knowing. The defect enters the energy as $-Z^2/2(n-\delta)^2$, so a level’s shift from the hydrogenic value grows faster than linearly in δ — and it grows fastest for the level with the largest defect, which is always the s. So a large defect stretches the s–p gap more than it stretches the p–d one, and the offset falls below a fifth.

Falling below a fifth moves a shell further inside its threshold, not closer to it. So the twenty per cent margin the third shell has at small defect is the smallest margin it has at any defect, and every real atom’s third shell has more room than the model’s does.

The defect was swept across two decades and found every dimensionless quantity of the shell settling rather than moving, which is the same statement seen from the other end: the features are robust to the defect, and the direction of what movement there is points away from the edge.

So the one place this argument is fragile is not the defect. It is the shells above the third, where the formula has been checked at no point at all.

The first shell rather than a special one #

The m = 0 levels of a shell number n — one for each angular momentum from zero to n − 1 — so three is the smallest shell with the three a minimum needs.

That makes the third shell’s minimum the first and not a curiosity. Every shell above it has one too, and with more room: the margin below the threshold is twenty per cent at n = 3, thirty-seven at n = 4, forty-three at n = 5 and approaches a half.

Two facts that the bars and the level diagram show separately are one fact when the shell is the thing being moved. A shell of principal quantum number n has n levels with m = 0, so the third level a minimum needs arrives exactly when the d function does; and the threshold that third level has to sit below is a function of n alone. Moving the shell up adds a level and raises the threshold in the same step, which is why there is no shell that has the levels and fails the condition.

The margin grows for a reason that is now visible: the threshold rises and the offset does not. Whatever makes the offset a fifth — the reciprocal-of-l-plus-a-half law, which is a property of a screened potential and not of a shell — is indifferent to how large the shell is, while the matrix elements care about nothing else.

So that twenty per cent is the narrowest margin any shell with a minimum has, and it belongs to the shell that was available rather than to a shell chosen for being tight.

There is a reading of that worth resisting. A margin that is narrowest where the calculation was done looks like the calculation having been lucky, and it is the opposite: the third shell is the only one whose functions are available here, so the margin found there is forced rather than chosen, and it happens to be the least comfortable one in the family. A finding at the tightest available margin is a stronger finding than the same one at a comfortable margin, because the comfortable case can be produced by a sloppy calculation and the tight one cannot. The minimum’s depth was measured at four thousandths of a per cent — a feature that shallow surviving a calculation is evidence about the calculation as much as about the shell.

And it sets what a check would look like. The fourth shell’s minimum, if its functions were available, ought to be deeper and to sit at a different field, and the formula says by how much: its threshold is 8/25 against a fifth, so the bracket whose sign decides the minimum is further from zero by a factor of (0.32 − 0.2)/(0.25 − 0.2), which is two and two fifths.

What was computed, and how #

Nothing here is a diagonalisation. The two dipole elements come from a closed form, the threshold is their ratio halved, the offset comes from the screening model’s own small-defect limit, and the comparison is a subtraction.

Eleven things are checked. That the closed form reproduces the computed s–p element at n = 2 and the s–p and p–d elements at n = 3, each to the quadrature’s own precision — the only place the closed form can be tested and the reason it can be used elsewhere. That the threshold at n = 3 is exactly a quarter, which is the bisected 0.24999886 stated as a rational. That at n = 2 it is exactly zero and the shell has no d function, checked together so that the zero is reported with its cause. That no minimum exists there. That every shell above the second has one. That the threshold rises from each shell to the next. That it stays below two fifths at every shell. And that it approaches two fifths in the limit, which is what the closed form says and would catch a formula that merely happened to fit the first few.

The last two are the refusals. A ratio of squared matrix elements that grew without bound would mean the closed form is wrong, since both elements grow as the shell does and their ratio should not; and a formula that rose to some other limit would fit n = 2 and n = 3 and be wrong everywhere else. Checking the limit is the cheapest way to test a formula that has only two points to be checked against.

Where this stops #

The elements are hydrogenic and the offset is a screening model’s. The closed form is exact for a one-electron atom, and the offset comes from the $-\lambda/r^2$ model used throughout these essays, which is the standard way of turning an exactly soluble problem into an approximately soluble one. A real alkali atom’s quantum defects are not given by any one-parameter law, and its s-defect in particular is large — so the small-defect limit the offset is taken in is the limit this argument lives in, and a real atom’s offset would have to be computed from its own measured defects rather than from a formula.

Nothing above is a calculation for n greater than three. The functions are not in this collection’s library, and the shells above the third are described by a closed form checked at two shells rather than computed at any. That is the honest status: the pattern is a formula’s pattern, and its two anchor points are the two the formula was checked against.

And the threshold is a second-order statement. The bracket whose sign it comes from is the coefficient of the field squared in the levels’ separation, computed by perturbation theory and checked against the exact curvature. At a large field none of it applies, and the minimum’s location — as against its existence — is an algebraic number found by search rather than anything this essay reaches.

The generalisation #

The habit is to look for the closed form behind a number a search produced, and then to check it where the search can still run.

A bisection returning 0.24999886 is a bisection returning a quarter, and the difference between having the number and having the fraction is the difference between one case and every case. What made the fraction findable was that the number was a ratio of two quantities each of which is a standard integral — so the work was recognising what the search had been computing rather than computing anything new.

The corollary is about which side of a comparison to check. This one had two sides and both were expected to move; only one did, and the one that did not was the one that looked most like it should, since it came from a formula with the shell’s own quantum numbers in it. A quantity’s dependence on a parameter is a question to ask of its derivation and not of its appearance, and the cheapest version of the question is to write the quantity down and see whether the parameter survives the cancelling.

Who found it, and when #

The hydrogenic dipole matrix elements are Gordon, 1929. The quantum defect and its $-\lambda/r^2$ model are older than quantum mechanics in their empirical form and are Hartree’s in this one. The second-order bracket and the numerical bisection come from the essay before it. What is computed here is the threshold’s closed form, its check against the quadratures already computed at two shells, and the constancy of the offset it is compared against.

The number worth carrying is not two fifths. It is one fifth, at every shell — a quantity that looked like it belonged to a shell and belongs to a screening law.

Still open: a shell with two thresholds #

The obvious open question is what the fourth shell’s f level does. The condition here is about one push, from d onto p; a shell with an f level has a second, from f onto d, and nothing in this argument says whether it produces a second minimum or modifies the first. The offset of f above d is computable from the same law — the shifts go as the reciprocal of l plus a half, so f sits at two sevenths of λ and the d–f gap is 4λ/35 — and the ratio of that to the s–p gap is one thirty-fifth. Whether a push a fifth as far away as the one here produces anything is a second-order calculation of the same shape, and it needs 4f functions that are not available here.

The nearer question is what a real atom’s offsets are. The one fifth above is the small-defect limit of a one-parameter screening model, and every alkali atom’s s-defect is large — lithium’s is about 0.4 and caesium’s above four — so the limit is exactly where those atoms are not. Their measured defects are tabulated for every l and every n, and computing the offset from them directly would replace a model’s constant with seventeen numbers, one per shell per atom. Whether any real atom’s offset crosses its shell’s threshold is then a table lookup rather than a calculation, and it is the difference between a statement about a model and a statement about an element.