How nearly a broken symmetry survives

The symmetry that is not a rotation built the operator hydrogen has and nothing else does — the one that connects states of different angular momentum inside a shell, and whose conservation is why a hydrogen shell is degenerate beyond what rotations require. That extra degeneracy is the one well-known case where a degeneracy is not a group theorem about the rotations, and it is the reason the shell has anything to lose. Its most visible consequence is that hydrogen’s Stark effect is linear in the field where every other atom’s is quadratic.

It closed by naming the question it could not reach. “What happens to this symmetry when it is broken slightly rather than badly.” A screened potential splits a shell by an amount that can be computed, and the field at which the linear effect returns follows from the splitting. That gives a curve of crossover field against screening, which says how nearly the symmetry survives — and it is one figure.

Here it is, and one thing about it is a surprise: the curve is a straight line on logarithmic axes, exactly, because the thing the screening moves and the thing that overcomes it are decoupled.

The screening, which is exactly soluble #

A real atom’s electron sees a nucleus screened by the others, and the standard way to turn that into an exactly soluble problem is to add $-\lambda/r^2$ to the potential. It has one property that makes it worth using: it goes into the centrifugal term rather than beside it, so $l(l+1)$ becomes $l(l+1) - 2\lambda$ and the whole hydrogen solution survives with $l$ replaced by an effective $l'$.

The energy is then $-Z^2/2(n-\delta)^2$ with $\delta = l - l'$, which is exactly the quantum defect a spectroscopist quotes for an alkali. So the abstract parameter $\lambda$ has a name in the laboratory, and the essay’s horizontal axis can carry both. It is the same screening Slater’s rules approximate with an effective charge, written as a radial term rather than as a reduced nuclear charge — and the difference matters here, because a reduced charge scales a shell without splitting it and this term splits it.

The splitting between the shell’s s and p levels is that expression evaluated twice and subtracted, and it is exact rather than perturbative. Its first-order limit is $\lambda\langle 1/r^2\rangle$, which is the form usually quoted — and having both means the approximation can be checked rather than trusted.

Where the approximation goes #

The first-order form is good to a per cent up to a defect of about four thousandths and is wrong by half at a defect of a fifth. That the exact answer is available at all is a property of this particular screening, and what a screening model cannot see is where the price of using one is counted. A fifth is small by the standards of real atoms — sodium’s 3s defect is 1.37 — so the linear approximation to the splitting has already failed well before any alkali is reached.

That is worth separating from the essay’s main result because it is a caution about a different thing. The natural proposal is to compute the splitting from $\langle 1/r^2\rangle$, which is a first-order recipe, and the exact answer is available at no extra cost.

The curve, and why it is a straight line #

The crossover is where a two-state problem stops being dominated by its gap and starts being dominated by its coupling: $F_c = \Delta E / 2d$, with $d$ the dipole matrix element between the two states.

The dipole is 3.000000 atomic units for the n = 2 shell, and it is 3.000000 at every screening tested. The screening changes the energies and leaves the coupling exactly alone, because $-\lambda/r^2$ is a radial operator and the matrix element is set by the angular parts and the radial functions, which the model does not move.

So the crossover field is exactly proportional to the splitting — checked here to a part in a billion at every point — and the curve of one against the screening is the curve of the other rescaled by a single number.

That is a stronger statement than it looks. It says the two halves of the question are not coupled: how badly the symmetry is broken and how hard it is to overcome are the same information. A quantum defect measured spectroscopically therefore gives the crossover field with no further calculation, once the dipole is known.

What the numbers say about a laboratory #

The fields are large and the range is enormous.

At a quantum defect of 0.0004 — which is a very nearly hydrogenic state, of the kind a highly excited Rydberg level of a light atom can have — the crossover is 4.3 × 10⁴ V/cm, which is a field a laboratory produces.

At a defect of 0.21 it is 2.9 × 10⁷ V/cm, which is not: it is within a factor of ten of the field that ionises the atom outright, and an atom cannot show a Stark effect in a field that takes its electron away.

So the practical statement is that a linear Stark effect is observable in an atom whose defect is very small and not otherwise, and the boundary between the two is a factor of a few in the defect rather than a matter of degree. Which is why hydrogen’s linear Stark effect is a textbook curiosity rather than a general technique.

There is a second reading of the same numbers that is more useful. Turn it round, and a measured crossover field is a measurement of the splitting, which is a measurement of the defect — and the defect is what a level position gives anyway. So the crossover is not a new instrument; it is a second route to a number spectroscopy already has, and its interest is that the two routes go through different physics. Symmetry decides what a spectrum can show and this decides how strong a field it takes to change what symmetry there is.

What survives, and what a symmetry is worth #

The extra operator is exact for hydrogen and exact for nothing else, which makes it look like a curiosity. The measurement here says what it is worth in the neighbourhood.

The degeneracy is fragile and the coupling is not. A screening of a thousandth splits the shell by a part in ten thousand of its own binding; the dipole between the states is unmoved. So what a small perturbation destroys is the equality of two energies, and everything else about the structure survives.

Which means the symmetry can be restored by force. Apply a field larger than the splitting and the shell behaves hydrogenically again — the equal spacing returns, the shifts become linear, the whole-number dipoles come back. A broken symmetry is not gone; it is outvoted, and a large enough field overturns the vote.

And the price of restoring it goes as the breaking. Exactly, with no coefficient of its own. That is the cleanest possible answer to how nearly does it survive: it survives above a field proportional to how badly it was broken, and the constant of proportionality is one matrix element.

What is quoted, and what is computed #

Nothing is quoted. There is no atom named in this essay and no measured defect used. The screening strengths are the model’s parameter; every effective angular momentum, defect, energy, splitting, matrix element and field is computed.

Two quantities are computed twice by different means. The ⟨1/r²⟩ of each orbital comes from this collection’s own numerical integration and from the closed form $Z^2/n^3(l+\tfrac12)$, and they agree to a part in a hundred thousand. And the splitting comes from the exact energy expression and from the first-order product, which agree as the screening vanishes and part company as it grows — the second being the result rather than a check.

The dipole is a third number of the same kind. It comes out at 3.000000 atomic units from a numerical integration over the orbitals, and the closed form for a hydrogenic 2s–2pz element is exactly 3 — so a quantity the whole essay is divided by is checked against an integer, which is the best sort of check, because a tolerance cannot absorb a disagreement with one.

What this cannot say #

One electron. The screening here is a term in a potential, not a set of other electrons, so there is no exchange, no correlation and no self-consistency. Orbitals are not where the electron is applies with full force: this is a one-electron picture of a many-electron atom, and the defect is standing in for everything it leaves out. A real quantum defect comes out of a many-electron calculation and is not a parameter anybody chooses.

A $1/r^2$ screening specifically. It is chosen because it is exactly soluble and because it produces a quantum defect, which is the quantity spectroscopy reports. A different screening with the same defect would give the same energies and could give a different dipole — and the essay’s neat proportionality is a property of this one.

And the two-state arithmetic is two states. The n = 2 shell has four, and the crossover computed from the s–p pair is the field at which that pair mixes strongly. The full manifold involves all four and the crossover is a description of where the behaviour changes rather than a sharp boundary — the same caution why a character table stops raises about reading a two-level formula into a degenerate set.

And the defect is taken as a property of the potential rather than of the state. A real atom’s quantum defect depends on n as well as on l, weakly, and the model here gives one defect per angular momentum for every shell. Over a Rydberg series that is nearly right and it is not exact.

No spin, and no relativity. A real n = 2 shell is split by spin–orbit coupling and by the Lamb shift before any screening is applied, and for hydrogen itself those are the splittings that matter. This is a model of the effect screening has, not a model of hydrogen’s actual level structure — and a spin–orbit coupling is what returns the orbital contribution in the magnetic case, where the same competition between a splitting and a field is played out.

The atoms for which the field is small #

The curve runs from forty thousand volts a centimetre to thirty million, which is a range from achievable in a teaching laboratory to achievable with difficulty. There is a family of atoms for which the field required falls off the bottom of that range entirely, and they are used as field sensors for exactly that reason.

The quantity that has to be overcome is the splitting between levels of the same shell and different angular momentum — the splitting screening produces, measured by the quantum defect. The quantity doing the overcoming is the dipole coupling, which grows with how large the atom is.

Both scale steeply with the principal quantum number, and they scale in opposite directions.

The splitting falls as the shell rises, because the outer electron spends less of its time near the core where the screening differs between one angular momentum and another. It falls roughly as the inverse cube of the shell number.

The dipole grows, because the atom is larger and a larger charge separation couples to a field more strongly — as the square of the shell number.

So the critical field, which is the ratio of the first to the second, falls as the fifth power of the shell number. Going from a valence shell to the thirtieth shell is a factor of thirty to the fifth — about twenty-four million — and a field that would have needed a megavolt per centimetre needs a fraction of a volt.

That is why highly excited atoms are used to measure electric fields. Their levels are so nearly degenerate, and their dipoles so large, that a field of a few volts per centimetre is enough to restore the linear behaviour this essay is about — and a linear Stark shift is an easy thing to measure and a direct report of the field.

It also says what such an atom is, in the language of broken symmetry. A Rydberg atom is an atom in which the broken symmetry has very nearly returned: the screening that split the shell has become negligible, the levels have collapsed back together, and the extra symmetry of the pure $1/r$ problem is restored to whatever accuracy the shell number provides.

The same scaling sets a limit that has a name and is worth knowing about, because it is where the restoration stops being useful. As the shell rises the levels of adjacent shells also crowd together, and at high enough field the Stark manifolds of neighbouring shells overlap. Beyond that point the field is no longer perturbing a shell; it is mixing several, the linear behaviour ends, and the spectrum becomes a dense mesh rather than a fan of lines.

So the field window in which a highly excited atom shows a clean linear Stark effect is bounded at both ends — below by the splitting computed here, and above by the spacing between shells — and the window narrows as the shell rises, because the lower bound falls as the fifth power and the upper as the fifth power of something else. Which of the two wins is a calculation the same method could do, and it is not done here.

What was checked #

⟨1/r²⟩ agrees with its closed form for both orbitals, to a part in a hundred thousand — the tie between the abstract screening parameter and the orbitals the rest of this collection draws.

At a vanishing screening the exact splitting is the first-order one, to a part in a thousand, which is what says the two expressions are of the same thing.

And at a real one it is not, by more than five per cent — so the first-order form is an approximation rather than the answer, and the essay’s table is a measurement rather than a restatement.

The shell has a dipole matrix element between its own states, which is what the degeneracy is for and what the field exploits.

A more badly broken symmetry needs a larger field to restore it, at every step of the scan — the shape of the answer rather than only its existence, and the check that the two ends of the table are not an accident of the two ends.

And the shell has a Stark manifold to return to, with whole-number dipoles and shifts that cancel — the result for the unscreened shell, which is the behaviour a large enough field restores and is checked there rather than here.

And the crossover is exactly proportional to the splitting, to a part in a billion, because the dipole does not move — which is the essay’s structural result and is checked rather than observed.

So a factor of three hundred in the screening is a factor of hundreds in the field.

And the refusal is a screening that removes the centrifugal barrier. Past λ = 1/8 an s state’s effective angular momentum is complex, the potential has no bound spectrum of this form, and the calculation stops rather than returning a meaningless number — which is also why the scan stops at 0.1, since past there “slightly broken” has stopped describing anything.

Still open: the level pattern past the crossover, and higher shells #

The obvious open question is the field beyond the crossover. This essay computes where the behaviour changes and stops; what happens above it is a shell that is neither cleanly quadratic nor cleanly linear, with four levels whose positions are the eigenvalues of a matrix containing both the splitting and the field. Diagonalising that matrix across the crossover would give the actual level pattern a screened atom shows — the thing a spectrum would look like — rather than the two limits it interpolates between.

The nearer question is which shell. Everything here is n = 2, where the shell has one s and one p and the two-state arithmetic is nearly exact. The n = 3 shell has a d as well, three different defects, and dipoles that run over four whole numbers rather than two — so the crossover is not one field but several, and whether they are close together or spread out decides whether “the linear effect returns” is a single event at all.