Atomic Models and Spectra
Atomic spectra were measured for a century before anybody could say what an atom was. Fraunhofer mapped and lettered the dark lines of the solar spectrum in 1817 [Fraunhofer:1817]; Kirchhoff and Bunsen showed in 1860 that each element carries its own fixed set of lines and thereby turned spectroscopy into chemical analysis [Kirchhoff:1860b]. By the end of the century the wavelengths were known to six figures and were the most precise numbers in physics — and they were pure numerology. Balmer found that the visible hydrogen lines fit a two-integer formula [Balmer:1885], Rydberg extended it to whole series in many elements with one constant [Rydberg:1890], and Ritz observed that every observed frequency is a difference of two terms drawn from a single list [Ritz:1908]. Nothing in classical physics explains why a bound system should radiate at a discrete set of frequencies at all, still less why those frequencies should be differences.
This chapter follows the two lines of evidence — spectroscopic and mechanical — to the point where they meet. The mechanical line runs from Thomson's electron [Thomson:1897] through the large-angle \(\alpha\) scattering of Geiger and Marsden [Geiger:1909] [Geiger:1913] to Rutherford's nuclear atom [Rutherford:1911], which is dynamically impossible in classical electrodynamics: an orbiting charge radiates and the atom collapses in about \(10^{-11}\,\mathrm{s}\). Bohr's 1913 trilogy [Bohr:1913a] [Bohr:1913b] [Bohr:1913c] joins the two lines by postulating stationary states, and buys with that postulate the Rydberg constant computed from \(e\), \(m_{\text{e}}\) and \(h\). The old quantum theory that grew from it [Sommerfeld:1916] [Bohr:1920] explained the X-ray regularity of Moseley [Moseley:1913] [Moseley:1914] and the fine structure, and failed on helium, on the anomalous Zeeman effect [Paschen:1912] and on intensities. It is a scaffold, not a theory — but everything it got right is a constraint that Part IX — Quantum Mechanics must reproduce, and the direct demonstration that its energy levels are real is Experiment: Franck–Hertz.
Atomic Models and Spectra: all derivations of this chapter are pending.
Spectra before any model
The Fraunhofer lines
[Reserved: Wollaston's 1802 sighting of dark gaps in the solar spectrum, and Fraunhofer's systematic mapping of some 570 of them with a theodolite telescope and a diffraction grating of his own making, lettered A through K in order of decreasing wavelength [Fraunhofer:1817]. The measurement was metrological in intent — Fraunhofer needed fixed wavelengths to characterize optical glass — and the lines became the first absolute wavelength standards. The coincidence of the D line with the yellow of a sodium flame, noted but not explained.]
Spectral analysis
[Reserved: Kirchhoff and Bunsen's demonstration that every element emits a characteristic line spectrum, that the same element absorbs at the same wavelengths, and that a spectrum therefore identifies a substance [Kirchhoff:1860b]; the immediate discovery of caesium and rubidium by this means, and the identification of the Fraunhofer D lines with sodium in the solar atmosphere. Absorption versus emission as an instance of Kirchhoff's radiation theorem [Kirchhoff:1860a], so that Black-Body Radiation and Planck's Hypothesis and this chapter share a founding result. Astrophysical consequence: stellar composition becomes measurable (Stellar Structure and Nucleosynthesis).]
A chemical element heated to incandescence in the vapour phase emits light only at a fixed, discrete set of wavelengths: the same set every time, independent of the temperature, of the chemical compound the element was liberated from and of everything else about its preparation, and different for every element. The same element, interposed cold in front of a continuous source, absorbs at exactly those wavelengths and at no others. A spectrum therefore identifies a substance without touching it, which is how caesium and rubidium were discovered [Kirchhoff:1860b]. The dark lines Fraunhofer had mapped and lettered in the solar spectrum are of this kind [Fraunhofer:1817]: the D pair coincides with the emission of a sodium flame, so there is sodium in the solar atmosphere.
Two things must be derived and only the second is classical. That a bound system radiates at a discrete set of frequencies at all follows only from the quantisation of its internal energies, and is taken up in the quantum-mechanics part; that emission and absorption occur at the same wavelengths is the detailed-balance content of Kirchhoff's radiation theorem, and follows from the equilibrium argument of the black-body chapter applied frequency by frequency.
Balmer's formula
[Reserved: Balmer's fit to the four visible hydrogen lines, \(\lambda=h_{\text{B}}m^{2}/(m^{2}-4)\) with \(m=3,4,5,6\) and \(h_{\text{B}}=364.56\,\mathrm{nm}\), matching the measured wavelengths to five significant figures [Balmer:1885], together with its successful prediction of the fifth line before Balmer was told it had already been seen in stellar spectra. The point to make: this is an exact empirical regularity with integers in it, found sixty-five years before it was understood, and it is the reason hydrogen became the test case for every atomic model.]
The Rydberg formula and the combination principle
[Reserved: Rydberg's generalization from wavelengths to wavenumbers, \(\tilde{\nu}=R(1/n_{1}^{2}-1/n_{2}^{2})\) for hydrogen and the term formula with quantum defects for the alkalis, all series in all elements sharing one constant \(R\) [Rydberg:1890]. Ritz's combination principle [Ritz:1908]: the wavenumber of every observed line is the difference of two terms, and sums and differences of observed lines are themselves observed lines. The principle is stronger than any particular series formula, it is exact, and it says — before any model — that a spectrum is a set of levels and not a set of frequencies. The other hydrogen series: Lyman in the ultraviolet [Lyman:1906], Paschen in the infrared [Paschen:1908].]
The wavenumbers of the hydrogen lines are given, to the full precision of the measurements, by two integers and one constant,
Balmer fitted the four visible lines with \(n_{1}=2\) to five significant figures and predicted the fifth before learning it had already been seen [Balmer:1885]; the series with \(n_{1}=1\) lies in the ultraviolet [Lyman:1906] and that with \(n_{1}=3\) in the infrared [Paschen:1908]. More generally, and in every element examined, the wavenumber of each observed line is the difference of two members of a single list of terms belonging to that element, and sums and differences of observed wavenumbers are themselves observed lines [Rydberg:1890] [Ritz:1908]. The combination principle is exact, it is stronger than any particular series formula, and it says — before any model of the atom exists — that a spectrum is a set of levels and not a set of frequencies.
The term structure: that there exists a set of stationary energies of the atom whose differences, divided by Planck's constant, are exactly the observed frequencies. The stationary-state postulate assumed in the Bohr section below turns the combination principle into a law rather than deriving it; the derivation proper is the eigenvalue problem for the atomic Hamiltonian, solved for hydrogen in the quantum-mechanics part.
The constituents of the atom
The electron
[Reserved: Thomson's cathode-ray experiments [Thomson:1897] — deflection by crossed electric and magnetic fields, the charge-to-mass ratio some \(1800\) times that of hydrogen, and its independence of the cathode material and the residual gas, which is what licenses the claim that the corpuscle is a universal constituent of matter. The subsequent measurement of \(e\) itself by the oil-drop method, and the current values [Mohr:2025]. This is the first subatomic particle, and its charge quantization is an observed fact that Part XI — Quantum Field Theory and the Standard Model still does not explain.]
Thomson's model and its failure
[Reserved: the “plum pudding” atom of electrons in stable rings inside a uniform sphere of positive charge [Thomson:1904], which was a serious dynamical model and not a straw man: it predicted the periodicity of chemical properties from ring closure, and it predicted small-angle multiple scattering of \(\alpha\) particles with a calculable and very small tail. That prediction is quantitative and falsifiable, which is exactly why the experiment of the next section settled the question. What it could not do is produce sharp spectral lines.]
The nuclear atom
Large-angle alpha scattering
[Reserved: Geiger and Marsden's observation that a small fraction of \(\alpha\) particles incident on a gold foil are deflected through more than \(90^\circ\), some returning towards the source [Geiger:1909] — in Rutherford's later phrase, as incredible as an artillery shell rebounding from a sheet of tissue paper — against the Thomson model's prediction of essentially none, since a diffuse positive charge exerts no large force anywhere. The quantitative campaign of 1913 [Geiger:1913]: counts against scattering angle, foil thickness, foil material and \(\alpha\) velocity, verifying every dependence of the formula below. Apparatus and uncertainties belong here; the modern descendant of the technique is deep inelastic scattering (Experiment: Deep Inelastic Scattering), where the same logic finds structure inside the proton.]
A beam of \(\alpha\) particles traversing a thin metal foil is overwhelmingly deflected through small angles, but a small, definite and measurable fraction is deflected through more than \(90^\circ\), some of it returning towards the source [Geiger:1909]. The number scattered into unit solid angle about the angle \(\theta\) follows
with \(E\) the kinetic energy of the incident particle and \(Z_{1}\), \(Z_{2}\) the charges of projectile and target in units of \(e\). Geiger and Marsden verified every dependence separately — on the scattering angle, on the foil thickness, on the material of the foil and on the velocity of the \(\alpha\) particles [Geiger:1913]. A positive charge spread uniformly through the volume of the atom, as in [Thomson:1904], exerts nowhere a force large enough to produce such deflections, so the positive charge and essentially all the mass of the atom occupy a region smaller than \(10^{-14}\,\mathrm{m}\) [Rutherford:1911].
The Rutherford cross-section: the relation between impact parameter and scattering angle for a repulsive inverse-square force, read off the Kepler hyperbola of the classical two-body problem, and the Jacobian that converts it into a differential cross-section; the summation over foil thickness that gives the counting rate actually measured; and the remark, which must be made honestly, that the same expression is also the exact Born approximation for a Coulomb potential, so that a mechanics now known to be wrong gave the right answer here for a reason no one could have known in 1911.
Rutherford's scattering formula
[Reserved: Rutherford's derivation of the differential cross section \(\dd\sigma/\dd\Omega\propto Z^{2}/(E^{2}\sin^{4}(\theta/2))\) for a point charge [Rutherford:1911], from the Kepler hyperbola of Central Forces and Statics, and the inference that the positive charge and essentially all the mass occupy a region smaller than \(10^{-14}\,\mathrm{m}\). The remarkable accident that the classical result coincides with the Born approximation of Scattering Theory, so that a wrong mechanics gave the right answer; and the deviation at high energy that measures the nuclear radius (Nuclear Forces and Nuclear Structure).]
Why the classical nuclear atom cannot exist
[Reserved: an accelerated charge radiates, by the Larmor formula of Radiation and Scattering of Electromagnetic Waves. An electron in a classical orbit of radius \(5.29\times 10^{-11}\,\mathrm{m}\) therefore spirals into the nucleus in about \(1.6\times 10^{-11}\,\mathrm{s}\), radiating a continuous spectrum of rising frequency as it goes. Both consequences contradict observation outright: atoms are stable for the age of the universe and their spectra are discrete. The estimate should be done explicitly, because it is the cleanest statement in the whole treatise that classical physics is not merely inaccurate but excluded.]
Every atom of a given element in its ground state is indistinguishable from every other, has the same size to within the precision of any measurement, and persists indefinitely: matter does not run down, and no atom has ever been observed to radiate its electrons into its nucleus. An excited atom does not radiate a continuum of rising frequency as it decays either; it radiates the sharp lines of Phenomenon 70.1. Classical electrodynamics requires the opposite on all three counts, because an electron on any bound orbit about the nucleus of Phenomenon 70.3 is accelerated and therefore radiates by the Larmor formula of Radiation and Scattering of Electromagnetic Waves, losing energy continuously and shrinking its orbit without limit.
Stability of the atomic ground state: the existence of a lowest energy level, which no classical bound system in an inverse-square potential possesses, and the bound relating the localisation of the electron to its kinetic energy which fixes the size and energy of that level. The derivation belongs to the quantum-mechanics part, where the hydrogen atom is solved; what can be established here is only the negative statement of the remark below.
The contradiction is quantitative, not merely qualitative, and the estimate is short enough to be given in full. An electron on a circular orbit of radius \(r\) about a proton has centripetal acceleration \(a=e^{2}/4\pi\varepsilon_{0}m_{\text{e}}r^{2}\) and total energy \(E=-e^{2}/8\pi\varepsilon_{0}r\), and radiates at the Larmor rate \(P=e^{2}a^{2}/6\pi\varepsilon_{0}c^{3}\). Setting \(\dd E/\dd t=-P\) and writing \(r_{\text{e}}:=e^{2}/4\pi\varepsilon_{0}m_{\text{e}}c^{2}\) for the classical electron radius, every constant collapses into
for the time to fall from \(r_{0}\) to the nucleus. Taking for \(r_{0}\) the observed atomic size \(5.29177\times 10^{-11}\,\mathrm{m}\), and for \(r_{\text{e}}\) its measured value \(2.81794\times 10^{-15}\,\mathrm{m}\) [Mohr:2025], this is about \(1.6\times 10^{-11}\,\mathrm{s}\). Classical physics does not merely misdescribe the atom; it forbids the atom to exist for longer than a hundredth of a nanosecond, while radiating a continuous spectrum that is never seen.
The Bohr model
The 1913 trilogy
[Reserved: Bohr's postulates [Bohr:1913a] — that an atom possesses stationary states in which it does not radiate, and that radiation accompanies a transition between two of them with \(h\nu=E_{i}-E_{f}\), which is Ritz's combination principle [Ritz:1908] promoted to a law. Selection of the stationary states by quantized angular momentum \(L=n\hbar\), equivalently by the action condition of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy. Parts II and III of the trilogy [Bohr:1913b] [Bohr:1913c] extend the scheme to multi-electron atoms and to molecules and are much less successful; the section should say which parts survived.]
What the model gets right
[Reserved: energies \(E_{n}=-13.606\,\mathrm{eV}/n^{2}\), radii \(r_{n}=n^{2}a_{0}\) with the Bohr radius \(a_{0}=5.29177\times 10^{-11}\,\mathrm{m}\), and the Rydberg constant emerging as \(R_{\infty}=m_{\text{e}}e^{4}/8\varepsilon_{0}^{2}h^{3}c= 1.0973731568\times 10^{7}\,/\mathrm{m}\) — computed from independently measured constants and agreeing with the spectroscopic value, which is the single result that forced the model on a sceptical community. Also the \(Z^{2}\) scaling verified on ionized helium, and the correct identification of the Pickering series in stellar spectra as He\(^{+}\) rather than hydrogen.]
The terms whose differences Equation (70.1) organizes are, for hydrogen, inversely proportional to the square of an integer, so that the energies of the stationary states are
The Rydberg constant is known spectroscopically as \(1.0973731568\times 10^{7}\,/\mathrm{m}\), so the ground state lies \(13.605693\,\mathrm{eV}\) below the ionization limit [Mohr:2025]. The same expression with \(R_{\infty}\) replaced by \(Z^{2}R_{\infty}\) reproduces the observed spectrum of a one-electron ion, which is what identified the Pickering series seen in stellar spectra as belonging to He\(^{+}\) and not to hydrogen [Bohr:1913a].
Derivation. Take Bohr's two postulates [Bohr:1913a]: that the atom possesses stationary states in which it does not radiate, and that radiation accompanies a transition between two of them, carrying away \(h\nu=E_{i}-E_{f}\) — which is Ritz's combination principle [Ritz:1908] promoted from a regularity to a law. Select the stationary states by requiring the orbital angular momentum to be an integer multiple of \(\hbar\). For a circular orbit of radius \(r\) about a fixed charge \(Ze\), Newton's second law with the Coulomb force and the selection rule read
Eliminating \(v\) between them gives the radii
the Bohr radius \(a_{0}\) being \(5.29177\times 10^{-11}\,\mathrm{m}\), which is the observed size of the hydrogen atom and was not put in. The first of the two displayed relations makes the kinetic energy half the magnitude of the potential energy, so the total energy is \(E=-Ze^{2}/8\pi\varepsilon_{0}r\), and substituting \(r_{n}\),
The wavenumber emitted in a transition is then
which is Equation (70.1) with \(R_{\infty}=m_{\text{e}}e^{4}/8\varepsilon_{0}^{2}h^{3}c\), together with the \(Z^{2}\) scaling. The decisive point is not that the formula has the right shape — Balmer's had that in 1885 — but that \(R_{\infty}\) is here computed from \(m_{\text{e}}\), \(e\) and \(h\), each of them measured in experiments having nothing to do with spectroscopy, and the number agrees with the spectroscopic value. That agreement is what forced a sceptical community to take the postulates seriously.
∎Reduced mass and the isotope shift
[Reserved: replacing the electron mass by the reduced mass of the two-body problem (Central Forces and Statics) shifts every line by a part in \(2000\) between hydrogen and infinite nuclear mass, and by a computable amount between hydrogen and its isotopes. Urey's discovery of deuterium was exactly this: the predicted shifted Balmer lines were sought in evaporated liquid hydrogen and found [Urey:1932]. A quantitative prediction of a new isotope from a spectral formula is a strong argument, and it is worth setting out in full. Present-day accuracy of \(R_{\infty}\) [Mohr:2025] and the spectroscopy that delivers it (Experiment: Precision Spectroscopy and Atomic Clocks).]
Every hydrogen line is accompanied by a faint companion displaced slightly towards shorter wavelength, in the fixed proportion that follows from replacing the electron mass in the Rydberg constant by the reduced mass of the electron and the nucleus. Urey, Brickwedde and Murphy concentrated the heavy component by evaporating liquid hydrogen near its triple point, looked for the predicted companions of the Balmer lines, and found them at the predicted separations — about \(0.179\,\mathrm{nm}\) for H\(\alpha\), whose principal component lies at \(656.28\,\mathrm{nm}\) — thereby discovering deuterium [Urey:1932].
Derivation. The nucleus is not infinitely heavy, so the two-body reduction of Central Forces and Statics replaces \(m_{\text{e}}\) throughout the derivation of Equation (70.4) by the reduced mass \(\mu=m_{\text{e}}M/\left(m_{\text{e}}+M\right)\) of the electron and a nucleus of mass \(M\). Every term, and therefore every wavenumber, is multiplied by
For two isotopes of nuclear masses \(M\) and \(M'>M\) the fractional difference in wavenumber is, to first order in the small ratio \(m_{\text{e}}/M\),
With \(M\) the proton mass and \(M'\) the deuteron mass, very nearly twice it, this is about \(m_{\text{e}}/2M_{\text{p}}\), that is \(2.72\times 10^{-4}\). Applied to H\(\alpha\) it predicts a separation of \(0.179\,\mathrm{nm}\), which a grating spectrograph of 1932 could resolve comfortably — the whole difficulty of the experiment was to enrich the sample enough for the companion to be visible at all. The displacement is towards shorter wavelength because the heavier nucleus makes \(\mu\) larger and hence every term value larger. Note that the argument makes a quantitative prediction about a substance nobody had yet observed, and that this is a far stronger test of Equation (70.4) than a fit to lines already measured.
∎Where the model fails
[Reserved: no line intensities, no transition rates, no selection rules except by fiat; no account of the helium atom, whose ground-state energy the old quantum theory got badly wrong despite a decade of effort; no molecules; no explanation of why the postulates hold. The orbits themselves are wrong — the ground state has zero orbital angular momentum, not \(\hbar\) — and the reason the energies come out right anyway is explained only by The Hydrogen Atom. Bohr himself catalogued the failures [Bohr:1920]; the model is retained in this treatise as evidence about levels, not as a picture of an atom.]
The old quantum theory
Sommerfeld's elliptic orbits and the fine structure
[Reserved: Sommerfeld's extension [Sommerfeld:1916] of the quantization condition to each separable degree of freedom of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy, giving elliptic orbits with a radial and an azimuthal quantum number, degenerate in the nonrelativistic problem; the relativistic correction to the electron mass lifts the degeneracy and yields a fine-structure splitting of order \(\alpha^{2}\), with \(\alpha=e^{2}/4\pi\varepsilon_{0}\hbar c\approx 1/137\) introduced here for the first time. The formula agrees with the observed hydrogen fine structure, and it agrees for the wrong reason — the true splitting involves the electron spin of Angular Momentum and Spin, absent from Sommerfeld's theory. The coincidence is worth a remark: it is the standard warning against counting agreement alone as confirmation.]
The correspondence principle
[Reserved: Bohr's requirement [Bohr:1920] that in the limit of large quantum numbers the frequencies and intensities of quantum transitions approach those of the classical Fourier components of the orbital motion. It is the only tool the old quantum theory had for intensities and selection rules, it fixed \(\Delta l=\pm1\), and it was the heuristic from which Heisenberg's matrix mechanics was constructed. Its modern residue is the classical limit of Approximation Methods and the Ehrenfest theorem of The Postulates of Quantum Mechanics.]
Space quantization
[Reserved: the quantization of the orientation of the orbital plane in a magnetic field, introduced by Sommerfeld [Sommerfeld:1916] to account for the Zeeman pattern of Section 70.7.1, and taken by contemporaries as an absurdity too far. It is the prediction tested directly by the deflection of a silver beam in an inhomogeneous field [Stern:1921] [Gerlach:1922a] [Gerlach:1922b], whose two-component result confirmed the discreteness while contradicting the theory that predicted it — silver has no orbital angular momentum in its ground state, and the doubling is spin. The experiment and this honest history are Experiment: Stern–Gerlach [Friedrich:2003].]
X-ray spectra and the atomic number
X-rays and their wavelengths
[Reserved: Röntgen's discovery of a penetrating radiation from a discharge tube [Roentgen:1895]; the demonstration by Friedrich, Knipping and Laue that crystals diffract it, which established both that X-rays are short-wavelength electromagnetic radiation and that crystals are periodic lattices [Friedrich:1912]; the Braggs' reduction of the diffraction condition to \(n\lambda=2d\sin\theta\) and the crystal spectrometer built on it [Bragg:1913b], which made X-ray wavelengths measurable to a few parts in a thousand. Without this instrument neither Section 70.6.2 nor the Compton measurement of The Photon: Photoelectric and Compton Effects would have been possible.]
Moseley's law
[Reserved: Moseley's measurement of the characteristic K and L emission of the elements from aluminium to gold [Moseley:1913] [Moseley:1914] and the finding that the square root of the frequency is linear in an integer that advances by one from element to element, \(\sqrt{\nu}\propto(Z-\sigma)\) with \(\sigma\approx 1\) for the K\(\alpha\) line. The integer is the nuclear charge, not the atomic weight: the periodic table is thereby ordered by \(Z\), the inversions of the weight ordering (argon and potassium, cobalt and nickel) are resolved, and the number of missing elements is fixed at exactly the gaps found — three below gold, all later filled. Screening as the physical content of \(\sigma\), explained by Atoms and Molecules.]
The characteristic X-ray emission of the elements from aluminium to gold consists of a few sharp lines whose frequencies advance smoothly and regularly through the periodic table. The square root of the frequency of the K\(\alpha\) line is a linear function of an integer which increases by exactly one from each element to the next,
and the same holds, with a different slope and screening constant, for the L series [Moseley:1913] [Moseley:1914]. The integer is not the atomic weight but the charge on the nucleus: ordering the elements by \(Z\) resolves the inversions of the weight ordering, argon before potassium and cobalt before nickel, and the gaps in the sequence fix the number of elements then undiscovered — three below gold, all later found.
Derivation. Apply Equation (70.4) to the innermost electrons of a heavy atom. The K\(\alpha\) line is the transition from \(n=2\) to \(n=1\), and the electron making it moves in the field of the nucleus screened by the one remaining \(K\) electron, so that the effective charge is \(\left(Z-\sigma\right)e\) with \(\sigma\) of order unity. The \(Z^{2}\) scaling established with Equation (70.4) then gives
so that \(\sqrt{\nu}\) is linear in \(Z\) with slope \(A=\left(3cR_{\infty}/4\right)^{1/2}\), which is Equation (70.5). The inner electrons are the ones for which a hydrogen-like treatment is defensible in a many-electron atom, since the nuclear attraction dominates the mutual repulsion there; \(\sigma\) is what measures the residual screening. The straight line holding across sixty elements, with a slope containing no adjustable parameter, is what establishes that \(Z\) and not the atomic weight is the quantity that orders the table.
∎Atoms in external fields
The Zeeman effect
[Reserved: Zeeman's observation that spectral lines broaden and then split in a magnetic field, with the components polarized as Lorentz's electron theory required [Zeeman:1897]; the normal triplet with splitting \(\Delta\nu=eB/4\pi m_{\text{e}}\), which yielded a charge-to-mass ratio agreeing with Thomson's [Thomson:1897] — an independent determination of the electron from optics. The Bohr magneton \(\mu_{\text{B}}=9.2740100657\times 10^{-24}\,\mathrm{J}/\mathrm{T}\) [Mohr:2025] as the natural unit; astrophysical use in measuring stellar and sunspot magnetic fields.]
A spectral line emitted by a source placed in a magnetic field \(B\) is resolved into several components. Their separations are proportional to \(B\), and their polarizations depend on whether the source is viewed along the field or across it [Zeeman:1897]. A minority of lines — the “normal” cases — split into exactly three components, the outer two displaced by
from which Zeeman obtained a charge-to-mass ratio for the radiating particle agreeing with the value Thomson measured on cathode rays [Thomson:1897] — an identification of the electron made optically and independently. Most lines, however, split into more components than three and in patterns Equation (70.6) does not describe. Paschen and Back mapped these anomalous patterns exhaustively, together with their collapse to the normal triplet in strong fields [Paschen:1912], and no theory available before 1925 fitted any of them.
Two derivations of different standing. The normal triplet follows classically, from the Larmor precession that a magnetic field imposes on a bound charge: the three normal modes of an isotropic oscillator in a field, their frequencies, and the circular and linear polarizations of the light each of them emits. The anomalous patterns do not follow at all without the electron spin and the coupling of spin to orbital angular momentum, and their treatment belongs to the angular-momentum chapter of the quantum-mechanics part; what must be recorded here is that the empirical splitting factor was fitted, not derived.
The anomalous Zeeman effect
[Reserved: most lines do not split into three. Paschen and Back's systematic study of the anomalous patterns and of their collapse to the normal triplet in strong fields [Paschen:1912] left a body of exact data that the old quantum theory could not fit at all, and that required the Landé \(g\) factor as a pure empirical parameter. This was the sharpest unsolved problem in atomic physics before 1925, and its solution is electron spin with \(g\approx2\) (Angular Momentum and Spin), whose own precise value is the most stringently tested prediction in physics (Experiment: The Electron Anomalous Magnetic Moment).]
The Stark effect
[Reserved: Stark's splitting of the hydrogen Balmer lines in an electric field of order \(10^{5}\,\mathrm{V}/\mathrm{cm}\), produced in the dark space just behind a perforated cathode [Stark:1914]; the splitting is linear in the field for hydrogen and quadratic for other atoms, a distinction with no classical explanation. Linearity requires the accidental degeneracy of the Coulomb problem, so the effect is direct evidence about the symmetry of the hydrogen Hamiltonian, and it is the standard worked example of degenerate perturbation theory in Approximation Methods. Stark broadening as a plasma diagnostic (Plasmas and Magnetohydrodynamics).]
Levels made visible
[Reserved: a short closing section stating what all the foregoing has and has not established. The spectroscopic evidence is about differences of energies, by Ritz's principle [Ritz:1908]; every model in this chapter fits differences, and no optical measurement can show that a level exists otherwise than as a term in a difference. The independent demonstration comes from electron impact, where energy is delivered to an atom mechanically and the threshold is measured directly: that is Experiment: Franck–Hertz, and it is placed immediately after this chapter for that reason. The theory that replaces the models assembled here begins at The Postulates of Quantum Mechanics, and the hydrogen atom is solved properly in The Hydrogen Atom.]