The spectrum is a subtraction, not a list of values

Pass the light from a hydrogen discharge through a prism and what comes out is not a smear. It is a small number of sharp lines at fixed wavelengths, identical in every hydrogen lamp anywhere, and stable to a precision that made them the definition of the metre for eighty years.

The crowding is the clue. A set of arbitrary sharp frequencies would be a list; a set that piles up at a limit is a sequence, and a sequence like that comes from subtracting terms of a convergent series from each other.

The formula before the physics #

Balmer found the visible four in 1885 by fitting, with no theory whatever. He was a schoolteacher, the wavelengths were 656.3, 486.1, 434.0 and 410.2 nanometres, and he noticed they were fitted by

$$\lambda = B\,\frac{n^2}{n^2-4}, \qquad n = 3,4,5,6.$$

Rydberg generalised it three years later into a form that covered other elements’ series too, and it is the generalised version that gives the game away:

$$\frac{1}{\lambda} = R_\infty\left(\frac{1}{n_2^2} - \frac{1}{n_1^2}\right).$$

A difference of two terms, each of the same form, each indexed by a whole number. Nobody knew what the terms were. But the structure says that whatever an atom’s states are, a spectral line is not one of them — it is a gap between two.

Turning the formula into a picture #

Combine that with light arriving in quanta of hf and the reading is forced. If an atom has a set of allowed energies and can drop from one to another, emitting the difference as a single quantum, then

$$\frac{hc}{\lambda} = E_{n_1} - E_{n_2},$$

and matching to Rydberg’s expression gives E_n = −13.606 eV/n².

The minus sign and the crowding both come from the potential. A Coulomb well is not a box with hard walls; it is a funnel that flattens out with distance, so a highly excited electron is barely bound and the levels bunch up as they approach freedom. The ladder has a top, at zero, and the energy needed to climb from the bottom to it — 13.6 eV — is hydrogen’s ionisation energy, which is measured independently and agrees.

Why the families exist, and why only one is visible #

Each family is the set of transitions ending on the same level. Ending on n = 1 gives the Lyman series, on n = 2 the Balmer, on n = 3 the Paschen.

Because the level being landed on fixes the smallest possible energy drop, it fixes the family’s longest wavelength; and because the levels above crowd onto zero, the family’s shortest wavelength is the drop from the very top, which is the series limit. Lyman runs from 121.5 down to 91.1 nanometres, Balmer from 656.1 to 364.5, Paschen from 1875 to 820.

Now compare those ranges with the band an eye responds to, 380 to 750 nanometres. Lyman is entirely ultraviolet. Paschen is entirely infrared. Balmer alone straddles the visible, and only its first four lines fall inside — which is why the visible spectrum of hydrogen has exactly four lines in it, why they were the ones found first, and why Balmer was fitting four numbers rather than fifty.

That is a piece of luck with consequences. The energy scale of atomic transitions is set by the Rydberg energy, the visible band is set by the peak of the Sun’s blackbody curve and by what evolution built a detector for, and there is no deep reason for the two to overlap. They overlap by about a factor of three, and the entire nineteenth century of spectroscopy — and with it the discovery of helium in the Sun before it was found on Earth — happened inside that overlap.

The same levels, without any light #

The strongest evidence that the levels are real, rather than a bookkeeping device for the lines, comes from an experiment that never looks at a spectrum.

Franck and Hertz in 1914 accelerated electrons through mercury vapour and measured the current arriving at a collector. As the accelerating voltage rose the current rose with it, then fell abruptly at 4.9 volts, then rose again, then fell again at 9.8, and again at 14.7. Evenly spaced drops, every 4.9 volts.

The explanation is that an electron below 4.9 eV cannot give any energy to a mercury atom at all — there is no level within reach, and the collision is elastic. At 4.9 eV it can excite the atom, loses all its kinetic energy doing so, and no longer reaches the collector. At 9.8 it can do so twice. The spacing of the current drops measures a level difference directly, in volts, with the electron as the probe.

The experiment matters because it removes the photon from the argument. The lines could conceivably have been a fact about how light interacts with matter; the current drops are a fact about the atom’s internal energies, established with no light in the apparatus. Two independent routes to the same ladder is what makes it a ladder rather than a fit.

Why the lines are the same everywhere #

A hydrogen lamp in a laboratory, a hydrogen cloud ten billion light years away and the hydrogen in a star’s atmosphere emit at the same wavelengths, and the constancy is not a small claim.

It says that the constants entering R∞ — the electron’s mass and charge, Planck’s constant, the permittivity of free space — are the same there as here. Astronomical spectra are therefore a test of the constancy of physical law across distance and time, and one that has been pushed hard: comparisons of transitions with different sensitivities to the fine-structure constant limit any drift in α to a few parts in 10¹⁷ per year.

The precision also had a metrological use. From 1960 to 1983 the metre was defined as 1,650,763.73 wavelengths of a particular orange line of krypton-86, chosen because a spectral line is a length that any competent laboratory can reproduce without possessing a prototype bar. That definition was replaced only when the speed of light became a defined constant and a stabilised laser became the better ruler — but the principle stands, and it began with the observation that atoms of one element agree with each other to more decimal places than anything manufactured does.

There is a caveat worth stating precisely, because it is the exception that makes the rule useful. Lines do shift: a source moving away stretches them, a strong magnetic field splits them, pressure moves them. Every one of those shifts is a measurement of something about the source, and each is separable from the others because it has its own signature. What does not change is the rest-frame wavelength, and the whole of astronomical spectroscopy is the arithmetic of comparing the two.

What a spectrum measures #

Once the lines are understood as differences, a spectrograph becomes an instrument for reading energy levels, and level structure is a fingerprint.

Composition. Every element has its own level scheme and therefore its own line list. Kirchhoff and Bunsen established in 1860 that the pattern identifies the element regardless of what it is mixed with, and the technique found caesium and rubidium within two years.

Absorption as well as emission. A cool gas in front of a continuous source removes exactly the wavelengths it would emit, because the same energy difference works in both directions. Fraunhofer had catalogued 574 dark lines in the solar spectrum by 1814 without knowing what they were; they are the Sun’s outer atmosphere subtracting its own emission lines from the light coming through it.

One correction is worth separating from the rest because it affects only half the measurement. The atmosphere scatters blue far more strongly than red, so the blue end of a spectrum reaches a ground-based observer much attenuated — a correction every measurement of a line’s strength has to make and no measurement of its position does. The lines sit where they sit; what changes with the airmass is how bright they look, and the two halves of a spectroscopic measurement have quite different error budgets.

Conditions. Line widths and relative intensities report temperature, pressure and velocity. A line’s Doppler width gives the emitting gas’s temperature; its pressure broadening gives the density; a bulk shift gives the source’s radial velocity. A spectrum measured well is a diagnostic of an object nobody can visit, and it is the only such diagnostic there is.

The lamp that is not a blackbody #

A discharge lamp and a hot filament are both sources of light and their spectra have nothing in common, which is worth setting side by side because the contrast is the cleanest statement of what a line spectrum is.

A continuous spectrum is the contrast that makes the point. A hot solid emits at every wavelength, with a smooth hump whose position reports its temperature and whose shape reports nothing else — the curve of the previous rung. A hydrogen discharge emits at four visible wavelengths and nothing whatever between them. The difference is not one of degree, and it is the observation that made a ladder of levels necessary rather than merely convenient.

The reason for the difference is the density. In a solid the atoms are packed close enough that their levels are perturbed by their neighbours, broadened into bands, and smeared into a continuum; the emission is thermal and its spectrum is the universal blackbody curve. In a thin gas the atoms are far enough apart to be undisturbed between collisions, their levels stay sharp, and what comes out is a list.

Everything between the two extremes exists. A high-pressure sodium lamp shows the sodium doublet broadened into a wide trough with a self-absorbed dip in the middle; a star’s photosphere shows a continuum with absorption lines cut into it; a fluorescent tube shows mercury lines plus a broad phosphor emission. Reading which parts of a spectrum are lines and which are continuum is reading how dense the emitting material is, and it is often the first thing an astronomer wants to know.

Where the sharpness comes from, and its limits #

An energy level is not perfectly sharp, and the reason is a general one worth having.

A state that decays after a typical time τ does not have a precisely defined energy: the trade between duration and frequency gives it a spread of about ħ/τ. An excited hydrogen level lives about a nanosecond, which corresponds to a natural linewidth of a few times 10⁻⁷ eV — a relative width of 10⁻⁸, which is what makes the lines look infinitely sharp on any ordinary instrument.

The ground state is different. It does not decay, so τ is infinite and its width is zero — which is why atomic clocks use ground-state hyperfine transitions rather than optical ones, and why “the second” is currently defined by a caesium line whose fractional width is 10⁻¹⁶.

Three other mechanisms broaden a line in practice, and all three are larger than the natural width in an ordinary lamp: Doppler broadening from the emitters’ thermal motion, collisional broadening from interruptions to the emission, and instrumental broadening from the spectrograph itself. Distinguishing them is most of the craft of quantitative spectroscopy.

The rule that said so before anyone knew why #

The claim in this essay’s title was not a consequence of quantum mechanics. It was an empirical regularity, established from tables of wavelengths, five years before Bohr wrote down a model that explained it.

Rydberg had noticed by 1888 that the lines of an element could be organised by writing each wavenumber as a difference of two terms drawn from a list. Ritz sharpened it in 1908 into a rule with predictive content: if a spectrum contains a line at $T_a - T_b$ and another at $T_b - T_c$, then it contains a line at $T_a - T_c$, and the wavenumbers of the first two add to give the third.

That is testable without any theory whatever. Take a table of measured wavelengths, convert to wavenumbers, and look for triples that add. They are there, in every element, far more often than chance allows, and the rule was used to predict lines that were then found.

The Rydberg–Ritz combination principle is therefore the discovery that a spectrum is a subtraction, made from the data alone. What it did not supply was any idea of what the terms were. They were positive numbers with units of inverse length, arranged in series, and nothing in the spectroscopy said they corresponded to anything an atom possessed.

Bohr’s 1913 model is usually remembered for its orbits, which are wrong. Its enduring achievement is the identification: the terms are energies divided by $hc$, the atom has states with those energies, and a line is a transition between two of them. A rule that had been an unexplained pattern in a table became a statement about the object producing it.

There is a fossil of the pre-theory period in every chemistry textbook. The series of lines were named for how they looked — sharp, principal, diffuse and fundamental — and when the terms were later identified with states of definite angular momentum, the initials came along. The letters s, p, d and f labelling every atomic orbital are abbreviations of nineteenth-century descriptions of how blurry a spectral line appeared through a prism.

The element found in the Sun #

The strongest demonstration that a line identifies an atom is that an element was discovered in a place nobody could reach, and not found on Earth for another twenty-seven years.

During the total eclipse of August 1868, Janssen observed the spectrum of the Sun’s chromosphere from India and recorded a bright yellow emission line close to sodium’s familiar pair but not coincident with either. Lockyer, in London two months later, worked out how to observe the chromosphere without an eclipse at all — by putting a spectroscope’s slit on the limb of the Sun — and saw the same line. Measured carefully it sits at 587.49 nanometres, where sodium’s two lines are at 589.0 and 589.6.

One and a half nanometres is an enormous discrepancy by spectroscopic standards, and the line was labelled D₃ to keep it apart from sodium’s D₁ and D₂. Lockyer and Frankland concluded that it belonged to an element not known on Earth, and named it after the Sun.

The claim was regarded with considerable scepticism for a quarter of a century, and reasonably so: the entire evidence for the existence of a new element was one line in the spectrum of an object a hundred and fifty million kilometres away. Nobody had a sample, and nobody could get one.

Ramsay found it in 1895, by treating a uranium-bearing mineral with acid and collecting the gas that came off. Its spectrum showed D₃. Independent Swedish work found the same thing within weeks, and helium became an element with a bottle rather than a line.

Two things make the episode worth telling here. The first is what it establishes about the argument of this essay: a spectral line is a difference between two levels of one particular atom, and no other atom has the same pair, so the presence of a line is an identification and not a suggestion. The second is a comment on what was being trusted. To announce a new element on the strength of a wavelength requires believing that the wavelength belongs to the emitter rather than to the circumstances — that the same atom in the Sun’s chromosphere and in a laboratory would produce the same number. That belief was not yet supported by any theory, and it was correct.

Where the one-electron picture stops #

The −13.6/n² formula is exact for hydrogen and for nothing else, and the ways it fails are informative.

More than one electron. Helium’s levels cannot be written in closed form at all — two electrons repelling each other in a shared potential is a three-body problem, and the entire apparatus of atomic structure theory exists to handle it approximately. What survives is the shape of the argument: levels, differences, series. What does not survive is the formula.

Fine structure. Hydrogen’s lines are not single. At high resolution each splits into components separated by about 10⁻⁴ of the line’s energy, from the electron’s spin interacting with its own orbital motion and from relativistic corrections to its kinetic energy. The scale is set by α² ≈ 5 × 10⁻⁵, and the fine-structure constant is named for exactly this.

Selection rules. Not every pair of levels produces a line. A transition requires the two states to be connected by the interaction driving it, and for ordinary electric-dipole emission that forbids most pairs — which is why a level diagram with fifteen levels does not give 105 lines. The rules are a statement about symmetry, and the “forbidden” transitions that do occur weakly are how astronomers identify gas so thin that a forbidden decay has time to happen before a collision interrupts it.

Each level is not a single state. It is a family of shapes — $s$, $p$, $d$ — with the same energy in hydrogen and different energies in every other atom, and which pairs of shapes a photon can connect is what the selection rules are about. A level diagram flattens all of that into one horizontal line, which is why it predicts the positions of hydrogen’s lines exactly and says nothing about which of them are bright.

What the picture cannot show #

The series figure places every line at its wavelength and says nothing about how bright each one is. Intensities span orders of magnitude, depend on temperature and density, and are what most of the information in a real spectrum lives in. A figure of positions is a figure of what an atom can do, not of what it is doing.

The level diagram hides degeneracy. Hydrogen’s n = 2 level is four states — one s and three p — sharing an energy by an accident peculiar to the Coulomb potential, and drawing them as one line conceals both the accident and the fact that it stops being one in any other atom.

And neither figure shows time. An emission is a process with a duration, a direction and a polarisation, and a line drawn at a wavelength is the whole of what a spectrum retains of it.

Where the ladder goes next #

The rungs from here: the shapes behind the levels, which is what the n is counting; the selection rules, derived rather than quoted; fine and hyperfine structure, and the transition that defines the second; the Franck–Hertz experiment, which shows the same levels by firing electrons rather than by collecting light; the Pauli principle, which is why the periodic table has the shape it has once more than one electron is in play; and lasers, which run the same transitions backwards by arranging for more atoms to be up than down.

The claim to carry forward is the subtraction. A spectral line is never a property of an atom on its own — it is a difference between two of them, which is why the lines crowd where the levels crowd, why every family has a limit, and why a set of a hundred lines can be generated by a list of fifteen numbers.