Three events, and a ratio of two dipoles

The field at which a broken symmetry returns — where a shell’s quadratic Stark shift becomes linear — moves when the shell has three levels instead of two. Three levels give two coupled pairs, so there are two such fields rather than one, and they came out 3.66 apart with both of them below the two-level shell’s single one.

Those two fields belong to the m = 0 levels of that shell: 3s, 3pz, 3dz2. The rest of the shell is the m = ±1 half, which is a p and a d — a two-level set with its own gap and its own dipole — and where its crossover falls decides how many distinct events a whole shell has: either four or three.

It is three, and the reason it is three is worth more than the number.

Two halves that do not talk #

A field along z changes the angular momentum by one and leaves m alone. So the n = 3 shell splits, for this purpose, into two pieces that have no matrix element between them at all: the m = 0 states — 3s, 3pz, 3dz2 — and the m = ±1 states, which are a 3p and a 3d.

Within the m = 0 half two of the three possible pairs are coupled — s with p and p with d — and the third, s with d, is not: two steps in the angular momentum, and the integral vanishes by parity. Within the m = ±1 half there are only two states and one pair.

So the shell has three coupled pairs, not four. The m = ±1 subshell occurs twice over — once at m = +1 and once at m = −1 — but the two are related by a rotation and give the same gap, the same dipole and therefore the same field, so they are one event and not two.

The third event falls between the other two #

The three fields are $9.5965 \times 10^{-6}$, $1.1081 \times 10^{-5}$ and $3.5087 \times 10^{-5}$ atomic units — 4.93 × 10⁴, 5.70 × 10⁴ and 1.80 × 10⁵ volts a centimetre. The middle one is the m = ±1 subshell’s.

That settles the choice between four and three, and the answer is neither of the shapes the question implied. The events do not separate into an m = 0 group and an m = ±1 group; they interleave, with the m = ±1 one nine per cent above the lowest m = 0 one and a factor of three below the highest.

All three are still below the n = 2 shell’s single crossover of $2.9205 \times 10^{-4}$, which extends the two-level finding — a higher shell’s events are at lower fields, because its gaps are smaller and its dipoles larger, and both push the same way — from half a shell to the whole of one.

Both trends have simple origins. The dipole between neighbouring l in a shell grows roughly as n², because the orbitals themselves spread out as n²; and a gap made by quantum defects shrinks roughly as 1/n³, because level spacings near a series limit do. A crossover field that goes as gap over dipole therefore falls roughly as n⁻⁵, which is why each shell up gives back its degeneracy at a much weaker field.

Two of them share a gap exactly #

The interesting part is the pair of fields that are close together, and why.

The m = 0 subshell’s lower crossover joins 3pz to 3dz2. The m = ±1 subshell’s joins 3px to 3dxz. Those are the same two levels — a p and a d of the same shell, at the same quantum defects — so the gap between them is the same number, to twelve figures, in both.

A crossover is a gap divided by twice a dipole. With the gaps identical, the ratio between the two fields is the ratio between the two dipoles and nothing else — and that ratio is

$$\frac{\langle 3p_z | z | 3d_{z^2}\rangle}{\langle 3p_x | z | 3d_{xz}\rangle} = \frac{5.196152}{4.500000} = 1.1547007 = \frac{2}{\sqrt3}.$$

Nothing in the calculation knows that number. Both dipoles are integrals over hydrogenic functions computed by quadrature on a mapped grid, and the ratio comes out at $2/\sqrt3$ to seven figures because the radial parts are identical and the angular parts differ by a Clebsch–Gordan factor.

So two of a shell’s three events are separated by a pure angular number, fixed before any screening, any nuclear charge or any field is chosen. The third is separated from them by a quantum defect, which is none of those things.

Why the m = ±1 dipole is the smaller one #

The two dipoles differ because the angular factors differ, and it is worth saying which way and why, because the direction is not obvious.

The m = 0 pair is 3pz with 3dz2. Both functions are concentrated along z, which is the direction the field acts in, so the operator connects them strongly. The m = ±1 pair is 3px with 3dxz. The p function lies in the xy plane and the d function has lobes tilted out of it, so the z operator connects them through a smaller angular overlap.

The ratio comes out at exactly $2/\sqrt3$. That number is what the angular integrals give and it is the same number for every shell: the ratio of the m = 0 to the m = ±1 dipole between a p and a d does not depend on n at all, because the radial parts cancel. So the pattern found here — two p–d events separated by 15.47 per cent — should be a feature of every shell with a p and a d in it, which is a prediction with no free parameter.

The two remaining differences between the shells are the quantum defects, which move the third event, and the radial extent, which moves all three together. The defect is what screening produces and it is the only chemistry in the whole arithmetic.

What that means for counting events #

The first finding was that a rule counting levels gets the number of crossovers wrong: the n = 2 shell has four states and one crossover, because only one pair is coupled. The full shell adds a second way the count goes wrong, and it goes the other way.

Counting coupled pairs in the whole shell gives three, and counting distinct fields gives three as well — but only because the m = ±1 subshell’s double occurrence collapses. If the field were not along a symmetry axis, m would not be a good label, every state would couple to every other, and the count would be different again.

The honest statement is that neither levels nor pairs is the thing to count. What produces an event is a gap with a dipole across it, and how many of those a shell has depends on the direction of the field as much as on the shell.

There is a practical form of that. Anybody looking at a Stark map for an avoided crossing is looking for one field and expecting one number; what the shell actually supplies is a small set of fields, two of which are within a sixth of each other and belong to different m states. Whether an experiment resolves them depends on whether it resolves m — which a field along a well-defined axis does, since m is conserved — so the two nearby events are separately observable in principle and would look like one broad feature to anything that averaged over m.

The degeneracy no group predicts is the reason any of this is interesting: in a bare Coulomb potential all three levels are degenerate, all three gaps are zero, and all three crossovers are at zero field. Every number above is a measurement of how far screening has taken the atom from that, and the three fields are three independent readings of it.

One further consequence follows and it is about what a Stark map is evidence for. The three fields here are set by two things multiplied together — a gap and a dipole — and the essay has been able to separate them only because both are computed. An experimenter has the fields and not the factors, so a measured crossover field is a single number constraining a product, and two different screenings with two different dipoles can give the same field. The ratio is what breaks that: the m = 0 and m = ±1 events share a gap exactly, so their ratio of fields is a ratio of two dipoles with the gap divided out, and it is a measurement of the dipoles alone.

That is the argument for quoting a pair of crossovers rather than one. A single field mixes the atom’s level structure with its transition strengths and cannot be inverted; two fields across the same gap invert to a pure dipole ratio, and a dipole ratio is a much sharper test of a wavefunction than an energy is, because energies are what every model is fitted to and dipoles are what none of them are. Two events sharing a denominator are worth more than three events that do not.

What was computed, and how #

Each subshell’s zero-field levels come from a quantum defect: the screened potential shifts an orbital’s effective principal number down by an amount that depends on how far into the core it penetrates, so an s is shifted most and a d least. The defects at a screening of 0.01 are 0.020417, 0.006682 and 0.004003, and the p and d defects are the same numbers in both halves of the shell — which they must be, since a defect depends on l and not on m.

Every dipole is a matrix element of z between two hydrogenic functions, by the same mapped quadrature the rest of this collection uses, and an element that symmetry forbids comes back at $10^{-18}$ rather than at something small — the same distinction a symmetry-forbidden overlap makes, where the integrand is odd about a plane and the contributions cancel in pairs. That distinction is the site’s standing example of an exact zero and it is what lets the s–d pair be excluded rather than merely deprioritised.

The refusal is the pairs the field does not couple. The s and d of one subshell must have exactly no dipole, and the two subshells must have exactly no matrix element between them — a version of this that reported a small number for either would produce two extra crossovers out of quadrature noise.

Where the model stops #

The quantum defect is a one-parameter caricature of screening and the three defects here come out of it rather than out of a measurement. A real alkali atom’s defects are fitted to its own spectrum and are not related by any single screening strength, so the three gaps would be three independent numbers rather than three values of one function. What survives that is the ratio, because the ratio has the gap divided out of it and is a property of the hydrogenic angular functions alone.

This is one electron in a screened Coulomb potential, so the quantum defects are a model of penetration rather than measurements of anything. A real alkali’s defects are larger and are fitted to spectra; the pattern — s shifted most, d least — is right, and the numbers are the model’s.

The field is along z throughout, which is what makes m a good label and the two halves independent. A field in any other direction mixes them, and the clean count above would become a single five-by-five problem with more coupled pairs and no subshell structure at all.

The two subshells are treated as two independent problems, which they are only because the perturbation is a single vector along one axis. What a symmetry settles about a splitting is settled here by which representations the operator connects, and that is a statement about the operator as much as about the shell.

And there is no spin and no fine structure. In a real n = 3 shell the spin–orbit splitting is comparable with the quantum defects for a light atom and much larger for a heavy one, so the level pattern this arithmetic is built on is not the pattern a real spectrum has. What the field competes with here is a screening gap, and in a real atom it would compete with that and with the spin–orbit coupling at once.

The generalisation #

A count of events is a count of coupled pairs, and which pairs are coupled is a property of the perturbation rather than of the system.

Half of that point is that counting levels overcounts, because the field does not couple every pair. The other half is that counting pairs can overcount too, because a symmetry can make several pairs give the same number. Between them they say that the only reliable procedure is to write the perturbation down, find which matrix elements it has, and count the distinct values of gap-over-twice-dipole.

The second finding — two events separated by an exact angular ratio — is a different kind of thing and is the more useful of the two. When two quantities share a factor exactly, their ratio is whatever is left, and here what is left is a number from the angular integrals with no chemistry in it at all. The same shape appears wherever an exactly shared denominator turns a comparison into an angular fact, and it is why a ratio is so often cleaner than either of the things it is a ratio of.

Who found it, and when #

The quadratic-to-linear Stark crossover is elementary two-state perturbation theory and is in every textbook. That an alkali’s shell has several such crossovers rather than one is implicit in the quantum-defect picture and is the reason a Stark map of a Rydberg series is drawn as a set of avoided crossings rather than as a fan.

The $2/\sqrt3$ between the two p–d dipoles is a Clebsch–Gordan coefficient and is in the tables; what is unusual here is arriving at it by quadrature on two three-dimensional integrals and reading it out of a ratio of two fields, neither of which was set up to produce it.

One more thing this does not settle and should not be read as settling. The three fields are the fields at which each pair’s behaviour turns over, computed from a two-state formula applied to a pair inside a larger problem. The shell’s actual exponent, from diagonalising the whole thing, passes through 1.5 between the two crossovers rather than at either — so the crossovers are landmarks in a continuous change rather than events with sharp edges. Adding a third landmark between them does not make the change any sharper, and the honest reading of three events is three fields worth naming, not three things that happen.

Still open: a tilted field, and the n = 4 shell #

The obvious open question is the field’s direction. Everything above depends on the field being along z, which keeps m a good label and the two halves apart. Tilting it mixes them, and the interesting question is what happens to the count: the two coincident m = ±1 events must separate, so the number of distinct fields should rise from three towards the number of coupled pairs in a five-by-five problem. Where it stops, and whether the separation is linear in the tilt, is one sweep of the same diagonalisation with a different perturbation matrix.

The nearer question is the n = 4 shell. It has four l values, so its m = 0 set has four levels and three coupled pairs, its m = ±1 set has three and two, and its m = ±2 set has two and one — six pairs and some number of distinct fields under six. Whether the pattern found here repeats, with each |m| contributing one fewer event than the last and the p–d pairs separated by angular ratios, is a prediction that can be made now and checked later. It needs the 4f functions, which are not in the table used here — and the same gap is what stops the shell whose orbitals were counted in momentum space from going one shell further.