The field an atom calls strong

The line that is really two found that sodium’s yellow line is split before any magnet is brought near it. The electron’s magnetic moment sits in the magnetic field it experiences because it is moving through the nucleus’s electric field, and the two ways the moment can point make two levels, 515 GHz apart. It ended by stating what an applied field would add: a further splitting, in one pattern if the applied field is weaker than the internal one and in another if it is stronger.

Stated like that, the two regimes sound like two approximations. They are two ends of one calculation, and doing the calculation — six states, one matrix, every field strength — shows why the patterns are what they are, what happens in between, and why a field that is strong for one atom is weak for another.

An electron with two moments #

An applied field $B$ acts on both of the electron’s magnetic moments. Its orbital motion gives one moment, of one Bohr magneton per unit of orbital angular momentum. Its spin gives another, of very nearly two Bohr magnetons per unit of spin. That factor of two belongs to an angular momentum that is not the angular momentum of anything going round; it is the moment that splits a beam of atoms in two in a Stern–Gerlach magnet, and taken at face value in the electron’s own frame it once made the fine structure come out twice too large. For a p electron, with orbital angular momentum $l = 1$ and spin $s = \tfrac12$, the energy is

$$H = \zeta\,\mathbf{L}\cdot\mathbf{S} + \mu_B B\,(L_z + g_s S_z),$$

where the first term is the fine structure, with $\zeta$ fixed by the measured 515.5 GHz between sodium’s two p levels, and $g_s = 2.0023$.

The six states $|m_l, m_s\rangle$ are the natural basis for the second term and the wrong one for the first. $\mathbf{L}\cdot\mathbf{S}$ mixes states with the same total $m_j = m_l + m_s$, and the field term is diagonal in $m_l$ and $m_s$ separately. No basis makes both simple, so the only honest procedure is to write the six-by-six matrix at each field and diagonalise it. Every figure here does exactly that, with nothing about the weak or strong patterns put in, and then checks the answer against both.

Ten lines where there were two #

In a field of one tesla the magnetic energy, 14 GHz, is a thirtieth of the fine structure. The spin and the orbit stay locked to each other by their coupling, and the field sees only their combination: a total angular momentum $j$ with $2j + 1$ orientations. Each orientation shifts by $g_j\,m_j\,\mu_B B$, where the Landé factor $g_j$ is the moment of the locked pair along $j$, per unit of $j$.

The factor is a projection, and it is not one. The orbital moment points along $\mathbf{L}$ with a factor of one and the spin moment along $\mathbf{S}$ with a factor of two, so the total moment does not point along $\mathbf{J}$; as the pair precesses about $\mathbf{J}$, only the component of the moment along $\mathbf{J}$ survives the average. For $j = \tfrac32$, where spin and orbit are aligned, that component is $4/3$ per unit; for $j = \tfrac12$, where they are opposed, it is $2/3$; for the s ground state, with no orbit, it is the spin’s own 2.

A line joins an upper sublevel to a lower one — a spectrum is a subtraction — so its shift is the difference $(g_j m_j - 2m_s)\,\mu_B B$, and a photon can change $m$ by one or leave it alone. For D1, upper $j = \tfrac12$ and lower $s = \tfrac12$, the allowed differences are $\pm 2/3$ and $\pm 4/3$: four components. For D2 they are $\pm 1/3$, $\pm 1$ and $\pm 5/3$: six. The diagonalisation finds all ten at those positions without being told, and it finds their strengths as well, as the squared overlaps of each upper eigenstate with the one orbital a photon of each polarisation can reach from the s state. The D2 components together carry exactly twice the strength of D1’s, as four upper states against two require.

This is the pattern Pieter Zeeman’s successors saw when they resolved the sodium lines in a magnet, and it was called anomalous because it was not the one theory predicted. Lorentz’s classical electron, orbiting without spin, gives every line three components at $-1$, $0$ and $+1$ times $\mu_B B$ — the normal Zeeman effect — and a spectrum of ten components at thirds of that spacing had no explanation until Landé fitted the factors empirically in 1921 and spin, four years later, supplied their origin.

A moment that changes its mind #

As the field grows toward the fine structure, the locking weakens and the pattern stops being Landé’s. The cleanest way to see how is to ask each state how large a magnetic moment it has.

A state’s moment along the field is the slope of its energy against the field, and the figure computes it as the numerical derivative of each eigenvalue. At a thousandth of the crossover field, every slope is a Landé value, to two parts in a thousand. At a thousand times the crossover, every slope is $m_l + g_s m_s$: the orbit and spin have unlocked, and each responds to the field on its own. Two states keep the same moment throughout, $\pm 2$, because they are the stretched states with orbit and spin both fully aligned with the field, which are simultaneously eigenstates of everything.

The other four change. The state that starts with a moment of $2/3$ ends with 1, the one that starts at $1/3$ ends at 0, and so on, and the passage is not abrupt. It occupies about two decades of field centred on the crossover, the field $B_c$ at which $\mu_B B_c$ equals the fine-structure splitting. In between, a state has a moment that is neither Landé’s nor Paschen and Back’s, because it is a mixture of the two $|m_l, m_s\rangle$ states that share its $m_j$, in proportions the field is still deciding.

Levels that refuse to cross #

The energies make the mixing visible as geometry. The two straight lines are the stretched states. Every other level curves, and the curves obey a rule: two levels with the same $m_j$ never cross. The pair with $m_j = -\tfrac12$ heads toward each other, comes within 486.0 GHz near 12.3 T, and turns away. That closest approach is $\sqrt2\,\zeta$ exactly, because the spin–orbit term connecting the two states has that size and sets the minimum separation of the two eigenvalues of a two-by-two matrix. Meanwhile the $m_j = -\tfrac32$ state falls through the lower $m_j = +\tfrac12$ state at 16.6 T as if it were not there.

The difference is symmetry. The field has rotational symmetry about its own direction, so $m_j$ is the quantity that symmetry conserves at every strength, and nothing in the Hamiltonian connects states with different $m_j$; their energies can pass through one another freely. States with the same $m_j$ are connected, and connected states cannot become degenerate by varying a single parameter — the rule von Neumann and Wigner stated in 1929. What looks like two levels swapping at an avoided crossing is the two states exchanging character: the upper one arrives as mostly one $|m_l, m_s\rangle$ combination and leaves as mostly the other, which is exactly the moment exchange of the previous figure seen from the side.

From ten lines to three #

The spectrum follows from the levels, and it ends where Lorentz started. Well above the crossover, the orbit and spin respond to the field separately, and a photon, which couples to the electron’s position rather than its spin, changes $m_l$ by at most one and leaves $m_s$ alone. A transition that does not flip the spin does not feel the spin’s moment, so every line is shifted only by the change in the orbital moment: $-1$, $0$ or $+1$ times $\mu_B B$, the normal triplet. The residual spin–orbit energy splits each band into close pairs of width about $\zeta$, and at thirty crossover fields every component with any strength lies inside those bands.

Paschen and Back saw the reorganisation in 1912, in atoms whose fine structure was small enough for their magnets to cross. The anomalous pattern of the weak field and the normal one of the strong field turned out not to be two effects, a normal one and an anomalous exception, but the two limits of one — and the pattern a theory without spin predicts is the one it gets right only when the field is strong enough to make spin irrelevant to the light.

When a field counts as strong #

The crossover field is the fine structure divided by the Bohr magneton, and the fine structure varies enormously from atom to atom. Hydrogen’s n = 2 splitting is Dirac’s $\alpha^2\,\mathrm{Ry}/16$, 10.95 GHz, and its crossover is 0.78 T — a field an ordinary laboratory electromagnet exceeds. Caesium’s outer p electron dips so far into a weakly screened core that its splitting is 16.6 THz and its crossover is 1187 T, beyond any field made on Earth except fleetingly in explosively compressed magnets. The fourth power of the effective charge that made sodium’s doublet visible in a school spectrometer is the same fourth power that makes its Paschen–Back regime almost unreachable.

Strong and weak are therefore properties of the atom, not the magnet. The strongest steady field a laboratory can maintain, about 45 T, is a strong field for hydrogen and lithium, puts sodium squarely in the crossover, and is still a weak field for potassium, rubidium and caesium. An experiment choosing an atom to probe a field is choosing which of these regimes it will read.

That choice is made most often by people measuring the Sun. In 1908 George Ellery Hale found the lines in sunspot spectra split, and measured fields of a few tenths of a tesla — the umbral field whose magnetic pressure holds a sunspot’s gas at bay. Those fields are weak for every atom in the photosphere, so the splitting is Landé’s, and solar magnetographs now choose lines with unusually large effective Landé factors, such as an iron line with a factor of 2.5, because a larger factor turns the same field into a larger, more measurable shift. The laboratory use runs the other way: the Zeeman slower that feeds atom traps uses a field shaped so the Zeeman shift cancels a changing Doppler shift, which works cleanly only because it drives a transition between stretched states, whose moments are fixed at every field and whose shift is therefore a straight line.

Choosing a line for its Landé factor is half of reading a field; choosing it for its wavelength is the other half. In frequency, a component’s shift is $g\,m\,\mu_B B/h$, 14 GHz per tesla per unit of $g m$, and it does not depend on which line it is. The width it has to be seen against does. A line in a hot gas is Doppler-broadened by a fixed fraction of its own frequency, so a line at twice the wavelength has half the frequency and half the Doppler width in gigahertz, while its Zeeman splitting is unchanged. An infrared line is a better magnetometer than a visible one of the same Landé factor, in proportion to its wavelength.

The numbers make the choice concrete. In a 0.3 T sunspot, the iron line at 617.3 nm, with an effective Landé factor of 2.5, splits its outer components by about 13 picometres, comparable with its thermal width, so the field is read from the line’s polarisation and shape rather than from separated peaks. The iron line at 1564.8 nm has a factor of 3; in the same field its components move about 100 picometres apart, several times its own width, and the splitting is resolved directly. That is why the most precise measurements of sunspot fields are made in the infrared, where the eye cannot follow.

There are places where fields are strong for everything. Magnetic white dwarfs carry fields from hundreds to a hundred thousand tesla, and their hydrogen lines are seen in the Paschen–Back regime and beyond it, where the field distorts the orbits themselves and the spectrum stops resembling an atom’s at all.

Where the pattern stops being this simple #

The nucleus has a moment too. Below about a tenth of a tesla, the hyperfine structure — the coupling of the electron to the nucleus’s own magnetic moment — is larger than the Zeeman energy for sodium’s ground state, and the pattern at such fields is a third one, in which the total angular momentum including the nucleus is the good quantum number. The same crossover happens one level down, at fields near half a tesla for the ground states of rubidium and caesium, and it is why atomic clocks built on those states are shielded to a small fraction of the Earth’s field and use the one transition whose frequency does not change to first order.

The quadratic term was dropped. A field also squeezes the electron’s orbit, adding an energy proportional to $B^2$ and to the orbit’s area. For sodium’s 3p level in laboratory fields it is negligible; for highly excited states, whose orbits are enormous, and for white dwarfs, it dominates.

The lines were given no width. Every excited state decays, and its width is its lifetime, but a sodium lamp’s lines are broadened far more by the Doppler effect, by about 1.5 GHz, so components 14 GHz apart at 1 T are resolved and components at a tenth of that field are not. When the splitting is smaller than the width, a field is detected instead through its effect on polarisation, which is how fields too weak to split a line are measured.

The light’s direction was ignored. Viewed along the field, the unshifted π components are absent and the σ components are circularly polarised in opposite senses; viewed across it, all are present and linearly polarised. The strengths in the figures are totals over directions, and a real spectrum is one direction’s share.

And one electron was treated as the whole atom. An alkali’s closed core contributes no angular momentum, which is what makes the six-state calculation exact up to the terms above. Atoms with several outer electrons have many more states to mix, and their Landé factors depend on how the electrons’ momenta couple among themselves.

What the figures leave out #

The weak-field figure draws two panels, each around its own line, and hides that the two lines are 515 GHz apart while the components spread over about 50 GHz. Drawn on one axis, the ten components would be two narrow clusters, and the pattern within each cluster would be invisible.

The crossover and spectrum figures run to 147 T, a field no laboratory holds steadily. The figures are about sodium because its crossover is near the edge of what can be reached; for hydrogen the same shapes appear below a tesla, and for caesium above a thousand, with nothing changed but the scale on the field axis.

Still open: the field the corona hides #

The Sun’s corona is threaded by magnetic fields that store the energy released in flares and eruptions, and those fields are, for the most part, not measured. At a ten-thousandth of a tesla to a hundredth, they split coronal emission lines by far less than the lines’ thermal widths, so the Zeeman splitting that measures sunspot fields is buried. What is measured instead is polarisation: the slight circular polarisation the unresolved splitting leaves, which requires large telescopes and long integrations, and the modification of scattering polarisation by fields too weak to split a line at all.

The direction of coronal fields has been mapped from such signals, and their strength at a few heights and places; a full three-dimensional measurement of the field in an active region before it erupts has not been made. How the field’s energy is stored and suddenly released, and whether an eruption can be predicted from the field beforehand, depends on that measurement, and large solar telescopes built in the last few years are the first with a chance of making it routinely.

The habit worth carrying away is to ask what a quantity is being compared with before calling it large. A field is strong or weak only relative to the energy an atom already has for the thing the field acts on, and the same magnet is a gentle perturbation for one atom, a complete reorganisation for another, and nothing in between for a third.