Summerfield (Arnold Sommerfeld): Primer

Primer author (STEP 4), 2026-10-09. Built from arbiter.md (the final verdicts), the three reviews (physicist.md, mathematician.md, contrarian.md), claims.md, the deep reads in deep/, sites.md and the research/ notes. No new verdicts are given here; every verdict below is the arbiter's. Revised 2026-10-09 after QE (qe.md, verdict MINOR) and the arbiter's "QE revision" (arbiter.md section 11), using the new deep read of Paschen 1916 (deep/Paschen-1916-BohrsHeliumlinien-AnnPhys50-iss16.md). The main changes: Paschen's 1916 measurement of the circular-orbit term (the quantity that separates relativity from Weber's law) is now stated, where this file had dropped it; Paschen's own 1916 remark that his hydrogen doublet was probably measured too small is quoted; and several citations and verdict labels are corrected (listed in "Corrections in this revision" before the Answers). The folder name spells the name "Summerfield"; the physicist is Arnold Sommerfeld (1868-1951), professor of theoretical physics in Munich [memory, high], and this file uses the correct spelling.

Source tags: [read] from the project's document or a file the Investigator saved (page numbers are PDF pages of the folder scan unless "book p." is said); [web] with URL; [memory] recalled, with a confidence level; [computed] arithmetic done in this review and reproduced by example.py; [Arbiter] / [Physicist] / [Mathematician] / [Contrarian] for a reviewer's finding.

Source note (carried from claims.md and the arbiter). The project holds one document: the 1923 English translation (by H. L. Brose, Methuen) of the third German edition of Atombau und Spektrallinien (Atomic Structure and Spectral Lines, preface January 1922) [read]. The PDF has no text layer; all 324 pages were OCR'd and most equations came out garbled, so every equation used here comes from page images read by the deep readers, who recomputed the numbers. What has been read in depth and what has not: 1. Deep read from page images: Ch. IV (phase integrals, elliptic orbits, spatial quantisation, magnetons), Ch. V part 1 (radiation, selection rule, Stark effect), Ch. VIII (relativistic fine structure, X-ray doublets, the constants e and h) and the Mathematical Notes 1-17. Ch. V sections 6-7 (the normal Zeeman effect by Larmor precession and the adiabatic hypothesis, book pp.296-311) were read by the arbiter in a cleaner OCR of the same edition that the Investigator saved [Arbiter, section 0]. Not a manifest document, but now deep read from page images: Paschen's 1916 paper "Bohrs Heliumlinien" (Ann. Phys. 50, 901-940), which the Investigator saved [deep/Paschen-1916-...md; Arbiter, section 11]. 2. Deep read later (QE revision 2, 2026-10-09): the parts of Ch. I-III and Ch. VI that the review relies on: the Sommerfeld-Debye accumulation theory and the nebulium passage (Ch. I-II), nuclear physics (Ch. II section 6), Moseley's law and the X-ray doublet law (Ch. III sections 4-7), all of Ch. VI (series spectra, electron-impact tests, inner quantum numbers, the anomalous Zeeman effect, the "doublet-atom" model) and the Ch. VIII footnote table of direct hydrogen measurements (deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch1-2.md, .ch2-s6.md, .ch3.md, .ch6-part1.md, .ch6-part2.md, .ch8-notes.md; page images for tables and formulas) [Arbiter, section 12]. Still digest only: the rest of Ch. I-III (electrons, isotopes, crystal structure, early X-ray methods) and Ch. VII (band spectra); nothing below depends on them. 3. Not obtained: Kragh's 1985 study of the fine-structure dispute (paywalled), Robotti 1986, Lau's 1924 report, the 1930s papers of Houston, Hsieh, Williams and Pasternack, and the Nobel committee's evaluation reports [Arbiter, sections 5-7].

Framing. Sommerfeld is not an outsider. This book was the standard text of the "old quantum theory" (the Bohr-Sommerfeld theory of 1913-1925); Eckert calls it the "bible" of atomic physics, with over 4,000 copies of the third edition sold within a year [web: Eckert, via the Investigator, research/2026-10-09-mprl-eckert.md]. There is no suppression story of the usual kind here. What makes it worth studying is unusual: its central formula, for the fine structure of hydrogen, turned out to be exactly right (Dirac obtained the identical energy levels in 1928 from a completely different theory), while the picture that produced it (electrons on precessing elliptical orbits, with no spin) is wrong. Along the way there was one real experimental dispute (direct measurements of the hydrogen doublet came out about 25 per cent too small), one alternative theory that gives the same splittings (Weber's 1846 electrodynamics), and one long silence (84 or so Nobel nominations and no prize). Claims are judged twice: against what was known in 1921-23, and against today's physics. The difference is the lesson.

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1. The idea in one paragraph

Niels Bohr's 1913 atom put the electron of hydrogen on circular orbits of fixed size, and explained the colours (spectral lines) of hydrogen light as jumps between them. Sommerfeld generalised Bohr's rule so that it could quantise any motion that repeats, including elliptical orbits and orbits tilted in space. He then asked what happens when the electron's mass grows with speed, as Einstein's special relativity says it must. An electron on an ellipse moves faster near the nucleus than far from it, so its mass changes around the orbit, and the ellipse no longer closes: it slowly turns (precesses), tracing a rosette. Orbits with the same Bohr energy but different shapes then have very slightly different energies, so each hydrogen line should split into a close group of lines, the "fine structure". Sommerfeld calculated the size of this splitting from the electron's charge, Planck's constant and the speed of light alone (a few hundred-thousandths of the line's own frequency: 0.365 wave-numbers for the basic hydrogen doublet, against about 15,000 for the red hydrogen line), predicted the pattern for ionised helium, and used Friedrich Paschen's photographs to argue that relativity was confirmed inside the atom. His book applies the same machinery to atoms in electric fields (Stark effect), in magnetic fields (Zeeman effect), to X-ray spectra of heavy elements, and to a purely spectroscopic measurement of the electron's charge and Planck's constant, and it states plainly where the theory fails (helium, molecules, heavier atoms).

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2. What you need to know first

The Bohr atom and spectral lines. An atom radiates light of frequency f when it drops from a state of energy W_a (the initial state) to a lower state W_e (the final state): h f = W_a - W_e, where h = 6.626 x 10^-27 erg s is Planck's constant. For hydrogen the allowed energies are W_N = -R h c / N^2, where N = 1, 2, 3, ... is the principal quantum number, c = 2.998 x 10^10 cm/s is the speed of light and R = 109,678 cm^-1 is the Rydberg constant for hydrogen. The Balmer series is the set of jumps that end on N = 2; its first line, H-alpha (N = 3 to 2), is red, at 6563 Angstrom. Read Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, the chapter on Bohr's model, which also covers the Wilson-Sommerfeld rule [memory, medium for that coverage].

Spectroscopists' units. Spectroscopists measure lines by wave-number, written nu-tilde here as "nu" for short: the number of wavelengths per centimetre, nu = 1/lambda, in cm^-1. Energy and wave-number are linked by W = h c nu. Useful conversions [memory, high]: 1 cm^-1 = 29.979 GHz; a small wave-number gap d(nu) at wavelength lambda is a wavelength gap d(lambda) = lambda^2 d(nu). At H-alpha (lambda = 6.563 x 10^-5 cm), 0.365 cm^-1 is 0.157 Angstrom [computed]. One Angstrom (A) is 10^-8 cm. Theorists also use "atomic units", in which the energy unit is the hartree: 1 hartree = 2 R h c, twice the binding energy of hydrogen's ground state (27.2 eV, or 219,475 cm^-1 as a wave-number) [memory, high].

Gaussian (cgs) units. The book uses centimetre-gram-second units with charge in electrostatic units (esu). The electron charge is e = 4.803 x 10^-10 esu, and Coulomb's law is F = e^2/r^2 with no 4 pi epsilon_0. The fine-structure constant is alpha = 2 pi e^2/(h c) = 1/137.036 = 7.2974 x 10^-3 [memory, high]; in Bohr's model it is the speed of the electron in the smallest orbit divided by c, which is how Sommerfeld introduces it [read: Ch. VIII deep read, 2.2]. alpha^2 = 5.325 x 10^-5. A useful identity: R h c = (1/2) alpha^2 m c^2, where m = 9.109 x 10^-28 g is the electron mass. Griffiths, Introduction to Electrodynamics, Appendix C, gives the conversions to SI.

Action and phase integrals. For a coordinate q with momentum p that repeats a cycle, the "phase integral" (also called the action) is J = closed integral of p dq over one cycle: the area enclosed by the motion in a plot of p against q. For a harmonic oscillator of frequency f and energy E, J = E/f. The old quantum rule is J = n h with n a whole number; for the oscillator that gives Planck's E = n h f. Read Goldstein, Classical Mechanics, the chapter on Hamilton-Jacobi theory and action-angle variables, or Landau and Lifshitz, Mechanics, the sections on action variables and adiabatic invariants [memory, high for the topics].

Adiabatic invariants. If a parameter of a periodic system (the length of a pendulum, a field strength) changes very slowly compared with one period, the action J stays almost constant even though the energy changes. For a pendulum whose string is slowly shortened, E/f stays constant. This is why "quantum numbers" could be argued to survive slow changes. Same references.

Special relativity: mass and energy. A particle of rest mass m moving at speed v has momentum p = m v / sqrt(1 - v^2/c^2) and kinetic energy T = m c^2 (1/sqrt(1 - v^2/c^2) - 1). At small speeds, T = (1/2) m v^2 (1 + (3/4) v^2/c^2 + ...). Read Taylor and Wheeler, Spacetime Physics, or the special relativity chapter of Griffiths' Introduction to Electrodynamics.

Angular momentum, spin and selection rules (modern view). In quantum mechanics an electron in hydrogen has orbital angular momentum quantum number l = 0, 1, ..., N - 1 (letters s, p, d, f for l = 0, 1, 2, 3), and spin 1/2. The two combine into a total angular momentum j = l +/- 1/2. A photon emitted in an ordinary (electric dipole) transition carries one unit of angular momentum h/(2 pi), so l changes by exactly one (Delta l = +/-1) and j by 0 or +/-1. A level is written N l_j, for example 2p_3/2 (N = 2, l = 1, j = 3/2). Read Griffiths, Introduction to Quantum Mechanics, the chapters on the hydrogen atom, angular momentum and spin, and time-independent perturbation theory (which covers the fine structure, the Zeeman effect and the Stark effect).

Doppler broadening. Atoms in a hot gas move toward or away from the observer, so each line is smeared into a bell-shaped (Gaussian) profile. Its full width at half maximum is FWHM = 7.16 x 10^-7 x nu x sqrt(T/M), where T is the temperature in kelvin and M the atomic mass in atomic mass units [memory, high]. Light atoms at room temperature have wide lines; hydrogen is the worst case. Any optics or spectroscopy text (e.g. Hecht, Optics, or Demtroder, Laser Spectroscopy) covers it.

Error bars and systematic errors. A quoted uncertainty (e.g. +/-0.009) usually covers only random scatter. A systematic error (a wrong calibration, a wrong constant used in the analysis) shifts every measurement the same way and is not reduced by repeating. A result "9 sigma" away from the truth means it missed by 9 of its own quoted error bars: the bar was too small, not the arithmetic wrong. Taylor, An Introduction to Error Analysis, is the standard first book.

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3. The textbook view

3.1 The quantisation rule (claim 1)

Sommerfeld's rule (1915-16). For each coordinate q_k of a motion that separates into independent cycles, closed integral of p_k dq_k = n_k h, with n_k a whole number. The same rule was found independently by William Wilson (completed November 1915) and by Jun Ishiwara (1915) [read: Ch. IV deep read, A.5; web: arXiv 1708.04676, research/2026-10-09-arxiv.md]. For hydrogen in polar coordinates (r, phi) there are two numbers: the azimuthal number n (angular momentum p_phi = n h/(2 pi)) and the radial number n'. The book's result is

W = -2 pi^2 m e^2 E^2 / (h^2 (n + n')^2)

where E is the nuclear charge (E = e for hydrogen, 2e for He+), m the electron mass and W the energy [read: Ch. IV deep read, E.1]. Only the sum N = n + n' matters, so this is Bohr's formula again. The orbit for each pair is a Kepler ellipse with eccentricity epsilon given by 1 - epsilon^2 = n^2/(n + n')^2; n = N is the circle, smaller n are more elongated ellipses. n = 0 (a straight line through the nucleus) is excluded "by permanence of atoms" [read].

The modern view. The Schroedinger equation replaced the rule in 1925-26. The rule survives as the first approximation of the semiclassical (WKB) method. (WKB, after Wentzel, Kramers and Brillouin, is the standard way of solving the Schroedinger equation approximately when the wavelength of the particle changes slowly from place to place; to lowest order it gives back the old phase-integral rule.) Its corrected form, Einstein-Brillouin-Keller (EBK) quantisation, adds half-units (the "Maslov index") to the integers [Physicist, claim 1; memory, high]. For hydrogen the half-units cancel in the sum, so the old rule gives the exact energies. What it gets wrong is everything about the states: the true ground state has zero orbital angular momentum (l = 0), whereas Sommerfeld's has one unit; the true number of orbital states with principal number N is N^2, against the old theory's N(N+1)/2; there are no definite orbits [Physicist; Mathematician].

3.2 The fine structure (claim 2)

The modern formula. The Dirac equation (1928) gives hydrogen energy levels that depend on N and j only. To order alpha^2,

W(N, j) = -(R h c/N^2) [1 + (alpha^2/N^2) (N/(j + 1/2) - 3/4)]

In the modern explanation this is the sum of three effects of the same size: the relativistic increase of kinetic energy, the spin-orbit interaction (the electron's spin magnet in the magnetic field it feels while moving past the nucleus, with Thomas's factor of 1/2) and the Darwin term (a contact effect for s states) [memory, high]. For N = 2 the levels j = 1/2 (2s_1/2 and 2p_1/2, which have the same Dirac energy) and j = 3/2 (2p_3/2) differ by

Delta nu_H = R alpha^2/16 = 0.365 cm^-1 = 10.94 GHz [computed].

In 1947 Lamb and Retherford found that 2s_1/2 lies about 1057.8 MHz (0.035 cm^-1) above 2p_1/2, a quantum-electrodynamic effect (the Lamb shift) that the Dirac formula does not contain [web: https://en.wikipedia.org/wiki/Lamb_shift; Arbiter, section 9]. The measured 2p_3/2 - 2p_1/2 interval is 10969.13 MHz, 0.24 per cent above the bare formula; the reviewers attribute the excess to the electron's anomalous magnetic moment [memory, medium; not web-confirmed: Arbiter, correction 4]. Note that this 0.24 per cent is not the Lamb shift; an early deep read lumped the two together and the physicist corrected it.

Sommerfeld's formula (1916). For a nucleus of charge E, with n the azimuthal and n' the radial number,

1 + W/(m c^2) = {1 + (alpha E/e)^2 / [n' + sqrt(n^2 - (alpha E/e)^2)]^2}^(-1/2)

[read: Ch. VIII eqn (23), page image p.246]. Expanded to order alpha^2, for hydrogen (E = e) and N = n + n',

W/(m c^2) = -alpha^2/(2 N^2) - alpha^4/(2 N^3 n) + 3 alpha^4/(8 N^4)

[computed: example.py Part A; same expansion derived independently by the mathematician and the physicist]. The middle term depends on the shape of the orbit (n), so ellipses and circles of the same N separate. The splitting pattern follows from the 1/n dependence: N = 2 gives a doublet; N = 3 a triplet with intervals in the ratio 1:3; N = 4 a quartet 1:2:6; N = 5, 3:5:10:30 [read; computed]. With Sommerfeld's n read as j + 1/2, his formula and Dirac's are the same expression [computed: example.py Part C shows the symbolic difference is zero].

This is the "Sommerfeld puzzle". Heisenberg called it a "miracle" and Schroedinger "a fortuitous computational accident" [web: arXiv 2401.07485 abstract; https://pmc.ncbi.nlm.nih.gov/articles/PMC7418904/]. Kragh: "Sommerfeld's celebrated fine-structure formula was correct for the wrong reasons" [web: https://www.ilnuovosaggiatore.sif.it/article/402]. Keppeler's semiclassical analysis finds that the Maslov half-units and the electron spin cancel each other, and Biedenharn (1983) traced the coincidence to a symmetry of the Coulomb problem [web, second-hand: cited in arXiv 1902.09612 and Vickers 2018; not read by the reviewers in the original].

3.3 Fields: Stark and Zeeman effects; the adiabatic principle (claim 4)

Stark effect. In an electric field F, a hydrogen level of principal number N splits linearly in F. In parabolic coordinates, with parabolic quantum numbers n1 and n2 (and magnetic quantum number m), the first-order shift is Delta W = (3/2) e a0 F N (n1 - n2), where a0 = 0.529 x 10^-8 cm is the Bohr radius [memory, high]. The book's formula, Delta W = -(3 h^2 F/(8 pi^2 m E)) (n2 - n1)(n1 + n2 + n3), is the same thing in different notation (n3 is the equatorial number) [read: Ch. V deep read 6.4; checked with SymPy by the deep reader]. Schwarzschild and Epstein derived it in 1916; Schroedinger got the same result from wave mechanics in 1926. At Stark's field of 104,000 V/cm the unit of splitting is 6.66 cm^-1, about 2.9 Angstrom at H-alpha [Physicist; computed].

Normal Zeeman effect. In a magnetic field B (gauss), the orbit precesses about the field at the Larmor rate omega_L = e B/(2 m c). Each line splits into three, shifted in frequency by 0 and +/- e B/(4 pi m c) (Gaussian units, e in esu). In wave-numbers that is +/- (e/m) B/(4 pi c) with e/m in electromagnetic units (1.7588 x 10^7 emu/g): 4.67 x 10^-5 cm^-1 per gauss [computed: example.py Part F; Contrarian 4.70 x 10^-5 with the book's e/m]. Planck's constant cancels: Sommerfeld's phrase is that "the quantum theory has in a certain sense become latent" [read: IA OCR, book pp.296-311, via Arbiter, claim 4]. The real "anomalous" Zeeman effect of most atoms needs electron spin (1925) and is outside what the orbit theory could explain [memory, high; Ch. VI is digest only].

Adiabatic principle (Ehrenfest 1916). Quantum numbers do not change when an external parameter is changed infinitely slowly. The book states three conditions (infinitely slow; acting through a parameter, not on the coordinates; unsystematic with respect to the phase), works the pendulum example (E/f constant), and states that "transitions through degenerate systems are excluded, as Bohr has shown" [read: via Arbiter]. In quantum mechanics this becomes the adiabatic theorem (Born and Fock, 1928), with the same restriction that levels must not cross [memory, high; not web-checked].

3.4 Light carries angular momentum: the selection rule (claim 3)

A circularly polarised wave carries angular momentum equal to its energy divided by its angular frequency, N/W = 1/omega [memory, high; Note 9 derives it, read]. If one act of emission carries energy h f, it carries at most h f/(2 pi f) = h/(2 pi), one unit. Rubinowicz (1918) used this to argue that the atom's azimuthal number can change by at most one, with circular light for a change of +/-1 and linear light for 0 [read: Ch. V deep read 4]. The modern rule is Delta l = +/-1 (Laporte's parity rule) with Delta m = 0 (linear) or +/-1 (circular) [memory, high]. (Parity says whether a state's wave function keeps or flips its sign when every position vector r is replaced by -r; a hydrogen state has parity (-1)^l. The photon of an ordinary dipole transition flips parity, so l must change by an odd number, and with one unit of angular momentum that means by exactly one.) Same answer, different reason.

3.5 X-rays (claim 5)

Moseley (1913) found that the square root of an X-ray line's frequency rises in a straight line with the atomic number Z; for the K-alpha line the book prints nu/R = (3/4)(Z - 1)^2, the slope sqrt(3/4) = 0.866 and a screening number "almost equal to 1" [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch3.md, book p.157, page image]. The book itself says the law is not exact: the line curves upward at large Z, which it puts down to relativity; a 2018 measurement of K-alpha for Z = 29 to 92 confirms that the curvature is relativistic [web: https://arxiv.org/abs/1809.10480]. Worked check from the book's own Table 11: for copper (Z = 29) 0.866 x 28 = 24.25 against the tabulated 24.35; for tungsten (Z = 74) 0.866 x 73 = 63.2 against 66.1, 4.5 per cent low [computed by the deep reader]. In heavy atoms the inner electrons feel almost the full nuclear charge, so the fine-structure formula with E/e replaced by (Z - s), s a "screening number", predicts doublets that grow as (Z - s)^4. The L-shell doublets do: Delta nu_L = Delta nu_H (Z - s)^4 with one s = 3.49 for Z = 41-92 [read: Ch. VIII deep read 1.4]. For uranium the factor is (92 - 3.5)^4 = 6.1 x 10^7, Sommerfeld's "gigantic magnification" of the hydrogen doublet [computed by the physicist; the observed ratio is 8.4 x 10^7 because the higher-order relativistic terms matter at alpha Z = 0.67]. Today these are spin-orbit doublets (same l, different j), and the "irregular" doublets that obey Hertz's law are screening doublets (same j, different l) [memory, high].

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4. Where Sommerfeld departs

Sommerfeld did not depart from the textbook of his day: he wrote it. The departure is between his 1916-22 theory and what physics settled on after 1925. Side by side:

Two points of departure deserve more words, because a learner can check them by hand.

The rigid electron (Abraham). For the rigid-sphere electron of Max Abraham, the kinetic energy at small speed is T = (1/2) m v^2 (1 + (3/5) v^2/c^2 + ...) instead of relativity's (1 + (3/4) v^2/c^2) [read: book, as given by the mathematician]. Writing p^2 = 2 m T (1 + gamma T/(m c^2)), relativity has gamma = 1/2 and the rigid sphere gamma = 2/5. The fine-structure terms are proportional to 2 gamma alpha^2, so with the same alpha every interval would be 4/5 as large: 0.292 cm^-1 instead of 0.365 [Mathematician, re-derived with SymPy; computed: example.py Part D]. The idea of this test was W. Lenz's, and Glitscher's 1917 Munich dissertation carried it out against Paschen's helium data, comparing the alpha^2 needed by each model (relativity 5.32 +/- 0.22 x 10^-5, Abraham 6.65 +/- 0.28 x 10^-5) with alpha^2 from independently measured e and h, and concluded that Abraham's value could "auf keine Weise" (in no way) be reconciled. QE checked the figures in the dissertation text: Glitscher widened Paschen's range to 0.350-0.380 cm^-1 "without doubting these observations", and compared the two spectroscopic alpha^2 values with the one from e and h, alpha = 7.29 x 10^-3, alpha^2 = 5.30 x 10^-5. Even with deliberately extreme constants (e = 4.8 x 10^-10, "surely too large", and h = 6.3 x 10^-27, "surely too small") he got alpha^2 = 5.86 x 10^-5, still below Abraham's minimum of 6.37 x 10^-5 [read: Glitscher 1917, research/files, via qe.md section 7 correction 2; computed: 2 pi (4.8 x 10^-10)^2/(6.3 x 10^-27 x 3 x 10^10) = 7.66 x 10^-3]. So Abraham fails even on Glitscher's most favourable assumptions for it. (An earlier version of this primer said the independent value "needs checking" and the contrarian gave "about 5.8 x 10^-5"; that is the extreme bound, not the best value, 5.30 x 10^-5 [Arbiter, 11.5].)

Weber's electrodynamics. Wilhelm Weber's 1846 force between charges depends on their relative speed and acceleration; it was superseded by Maxwell's field theory. Frauenfelder and Weber (2019; the second author is not the 19th-century Weber) applied Sommerfeld's own quantisation to it and found, in atomic units (energy unit 1 hartree = 2 R h c) [read: arXiv 1902.09612, Theorem A and eq. 1.2]:

• Weber: -1/(2N^2) - alpha^2/(2 N^3 l) + alpha^2/(2 N^4)
• Sommerfeld (and Dirac, with l -> j + 1/2): -1/(2N^2) - alpha^2/(2 N^3 l) + 3 alpha^2/(8 N^4)

The term that depends on the orbit's shape is identical; only a term that depends on N alone differs, by alpha^2/(8 N^4) hartree, 0.091 cm^-1 at N = 2 [computed by the arbiter, physicist and mathematician; the mathematician also checked it by numerical quantisation]. Every interval within one N is therefore the same in both theories. So the fine structure measured in 1916-24 could rule out Abraham's electron but could not, by itself, single out special relativity among all velocity-dependent dynamics [Arbiter, claim 6]. Weber's theory fails other tests (no radiation, no spin) [memory, high; physicist's statement, not from the paper]. Frauenfelder and Weber are writing in favour of Weber's electrodynamics (their stake; the arbiter checked their mathematics, not their physical interpretation).

The measurement that does separate them: the circular-orbit term. Put n = l = N (a circular orbit) in the two lines above. Sommerfeld gives -alpha^2/(2N^4) + 3 alpha^2/(8N^4) = -alpha^2/(8N^4) hartree; Weber gives -alpha^2/(2N^4) + alpha^2/(2N^4) = zero [computed: example.py Part G]. For N = 2 the relativistic circle term is R alpha^2/64 = 0.091 cm^-1. It moves a whole line rather than splitting it, so it has to be found by comparing the absolute positions of different lines. The book says such a shift had been seen: "Very slight deviations from the formula ... have already been occasionally observed earlier, and ... these deviations, according to a paper by Paschen ... are of the order of magnitude of our relativity correction R alpha^2/64 required by theory and bear the correct sign" [read: book p.488, IA OCR lines 42967-42984, via qe.md 3.1]. Paschen's 1916 paper gives the number [read: Paschen 1916 pp.931-933, page images, deep read section 3]: - Method: each He+ line, divided by its Bohr factor 4(1/i^2 - 1/k^2), gives an "effective Rydberg constant". Without the circular term every line would give the same value; with it, the value from 4686 (4 -> 3) should be larger than the value from 4339 (10 -> 4). - Result: observed difference 0.569 +/- 0.25 cm^-1; calculated from "Bohr's relativity term" 0.590; Weber predicts 0. Paschen's own summary: "findet die richtige Groessenordnung" (finds the right order of magnitude). Leaving the term out would put 4339 0.025 A too low and leave "no place for the observed weak component" [read: pp.931-933]. - At face value the result is 2.3 of Paschen's "possible errors" from Weber's zero (about 2.9 if his two errors are added in quadrature) and 0.08 from relativity [computed: example.py Part G].

Caveats that belong beside the figure [Arbiter, 11.2; deep read sections 3-5]: - It is one measurement, never repeated, by the theory's collaborator (Paschen reduced his plates "with the friendly help of [the theory's] creator"). - It rests on one weak component at 4339 (4338.536 A) and its assignment, and on the position of the circular component inside a 12-component scheme for 4686 that was itself fitted with the theory. The +/-0.25 covers only the two wavelength settings (0.07 + 0.18, added linearly), not the assignment or the fit. With Paschen's "less certain" alternative wavelength the difference becomes about 0.516. - It cannot separate relativity from Abraham, even in principle: Paschen took alpha^2 from the measured splitting, and in either electron model the circle term and the splitting scale together, so Abraham predicts the same 0.590. Abraham is excluded only by the absolute scale (alpha against independently measured e and h; see above), not by this term. - Counter-evidence: Sommerfeld's draft letter of 29 Dec 1915 found the absolute values of Balmer-line differences off by more than a factor of two (0.102 against 0.24). The mathematician reads this draft as concerning the circular-orbit shift; the arbiter has not confirmed which quantity it concerns. It was a working draft, before the final scheme [Mathematician; Arbiter, 11.2].

The arbiter's weighing: as an independent, neutral test between relativity and Weber, Never properly tested; beside that verdict stands one low-significance result, from an interested party, that favours relativity. An earlier version of this primer left this measurement out, and the physicist and arbiter had said the absolute shift was "not measured at the needed accuracy then" / "not measurable then". That was wrong: the book and Paschen report it [qe.md 3.1; Arbiter, 11.1].

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5. Worked example: the n = 2 hydrogen doublet, and why the 1921 measurements came out small

Every number below is reproduced by example.py (run command in its header). The aim is to see (a) where 0.365 cm^-1 comes from, (b) that it is the right number today, (c) why the electron model matters, and (d) why a careful experimenter in 1921 could measure 0.272 and see no error.

Step 1. The doublet from Sommerfeld's formula

From the expansion in 3.2, take N = 2. The circle has n = 2 and the ellipse n = 1. Only the middle term differs:

W(n = 2) - W(n = 1) = m c^2 x alpha^4/(2 x 8) x (1 - 1/2) = m c^2 alpha^4/32.

Use R h c = (1/2) alpha^2 m c^2, so m c^2 alpha^4/32 = R h c x alpha^2/16. In wave-numbers:

Delta nu_H = R alpha^2/16.

Step 2. Put in the numbers

• With the book's constants, alpha^2 = 5.32 x 10^-5 and R = 1.097 x 10^5 cm^-1: Delta nu_H = 0.36475 cm^-1, which the book rounds to 0.365 [read: Ch. VIII deep read 4.1; computed].
• Small slip in the book: the printed e = 4.77 x 10^-10 and h = 6.55 x 10^-27 give alpha^2 = 5.29 x 10^-5, not 5.32; the effect on the doublet is 0.002 cm^-1, inside Paschen's error [computed; deep read].
• With modern constants, alpha = 7.29735 x 10^-3 and R_H = 109,677.58 cm^-1: Delta nu_H = 0.36503 cm^-1 = 10.943 GHz [computed].
• As a wavelength gap at H-alpha: (6.5646 x 10^-5 cm)^2 x 0.36503 cm^-1 = 1.57 x 10^-9 cm = 0.157 Angstrom [computed]. That is the whole doublet: about one sixth of a tenth of an Angstrom, inside a red line.
• Paschen's 1916 value from the helium spectrum, 0.3645 +/- 0.0045 cm^-1, is 0.12 of its error bar from the modern 0.3650 [computed].
• Today's measured 2p_3/2 - 2p_1/2 interval, 10969.13 MHz, is 0.24 per cent above the formula [web, via Arbiter section 9; computed].

Step 3. It is Dirac's number

Replace n by j + 1/2 and n' by N - (j + 1/2) in Sommerfeld's exact formula and subtract Dirac's: SymPy simplifies the difference to 0. For N = 2: W(j = 1/2) = -6.6565301 x 10^-6 m c^2 and W(j = 3/2) = -6.6564415 x 10^-6 m c^2 [computed: example.py Part C]. Their difference, times m c^2 = 2 R h c/alpha^2, is again 0.365 cm^-1.

Step 4. The rigid electron would give 0.292

With gamma = 2/5 instead of 1/2, every interval is multiplied by 2 gamma = 4/5: 0.8 x 0.36503 = 0.292 cm^-1. Paschen's 0.3645 +/- 0.0045 then excludes Abraham by (0.3645 - 0.292)/0.0045 = 16 standard errors, if Paschen's error bar is honest [computed: example.py Part D; Arbiter, claim 6]. Two caveats from the record: Paschen reduced his data "mit freundlicher Hilfe ihres Schoepfers" (with the friendly help of the theory's creator), so the scale 0.3645 is a one-parameter fit to a pattern that the theory fixed in advance [read: Paschen 1916, by all three reviewers]; and the test needs alpha from somewhere else, which Millikan's e with Planck's h, Glitscher's dissertation and the X-ray doublets supply [Arbiter, claim 6].

Step 5. Why the direct hydrogen measurements came out small

The book prints, in a footnote table, the direct measurements of the H-alpha doublet [read: Ch. VIII deep read 4.2]: Michelson 0.32, Fabry and Buisson 0.306, Meissner and Paschen 0.288, Gehrcke and Lau 0.272 cm^-1 (and 0.283 for H-beta, 0.271 for H-gamma). Every one is below 0.365. Gehrcke and Lau wrote: "Bemerkenswert ist, dass der Betrag fuer Delta nu etwa um 25 Proz. kleiner gemessen wird, als er aus der theoretischen Feinstrukturkonstanten sich nach Sommerfeld berechnet ... in Anbetracht der Groesse der gefundenen Abweichung [ist uns] ein Irrtum bei der Messung nicht ersichtlich" (remarkably, Delta nu is measured about 25 per cent smaller than Sommerfeld's theory gives; given the size of the deviation, no measurement error is apparent to us) [read: Ann. Phys. 65, p.575, research/files; quoted by all three reviewers]. Check: 0.272/0.365 = 0.745 [computed].

The shortfall was not new in 1921, and nor was its likely explanation. In 1916 Paschen himself resolved H-alpha into 6562.853 and 6562.732 A (0.281 cm^-1), and Meissner, with an interferometer in Paschen's laboratory, got 6562.849 and 6562.725 A (0.288 cm^-1), both about 21-23 per cent below the theory's 0.1571 A. Paschen wrote: "Die Messung war aber schwierig, und es ist wegen der uebergreifenden Raender der unscharfen Komponenten wahrscheinlich, dass der Abstand der eingestellten schwaerzesten Stellen zu klein ist" (the measurement was difficult, and because the edges of the blurred components overlap, it is probable that the distance between the darkest points set on is too small) [read: Paschen 1916 p.933, page image, deep read section 6; Arbiter, 11.3]. The book names Doppler and Stark broadening as the causes of the blurring [read: IA OCR lines 42402-42424, via qe.md 3.2]. So the experimenter who made the measurement gave the blending explanation five years before the dispute. That supports the explanation below; it does not prove it, because a shared real shortfall would also make two observers agree. Paschen, remember, was the theory's collaborator.

(a) The book's own explanation: the "selection lag". H-alpha joins N = 3 to N = 2. With Sommerfeld's selection rule only three of its six old-theory components survive, and the observers measure from the strongest of group I to the strongest of group II, which is Delta nu_H (1 - 8/81) = 0.329 cm^-1, not 0.365 [computed from the book's scheme by the deep reader]. That brings Michelson (0.32) and Fabry-Buisson (0.306) close but leaves Gehrcke-Lau 17-19 per cent low [Mathematician]. The same lag applied to Abraham's doublet gives 0.8 x 0.329 = 0.263. The mean of the four H-alpha values is 0.296, exactly the midpoint of 0.329 and 0.263 [computed: example.py Part D]. In 1921 the direct data could not decide between the two electron models; that is why the argument rested on helium and X-rays [Arbiter, claim 2].

(b) The Doppler width. At 300 K the Doppler FWHM of H-alpha is 7.16 x 10^-7 x 15,233 x sqrt(300/1.008) = 0.188 cm^-1; at liquid-air temperature (77 K) it is 0.095 cm^-1 [computed]. The doublet is only about twice the width of each line, so the two halves overlap heavily.

(c) The modern line list. In today's theory H-alpha has seven components (all Delta l = +/-1, Delta j = 0, +/-1 transitions from the five N = 3 levels to the three N = 2 levels). example.py builds them from the Dirac levels plus the Lamb shifts (the same input assumptions as the arbiter: s levels +1057.8 MHz x 8/N^3, 2p_1/2 and 3p_1/2 -12.8 MHz x 8/N^3 [web for 2S; memory, medium for the scaling]), with relative strengths from the standard angular-momentum (6j) formula and exact hydrogen radial integrals [computed]. (A component's "line strength" is the square of the transition's dipole matrix element, summed over the sub-levels; it sets how bright the component is when the upper levels are equally populated. It factors into a radial integral, which depends on the wave functions' sizes, and an angular factor for how l, the spin and j recombine; that angular factor is a tabulated coefficient called a Wigner 6j symbol, which example.py takes from SymPy.)

Look at the structure. The red group is dominated by 3d_5/2 -> 2p_3/2 at 0.134; the violet group is a blend of 3d_3/2 -> 2p_1/2 at 0.463 and 3p_3/2 -> 2s_1/2 at 0.428 (which the Lamb shift has pulled toward the middle), with a weaker 3p_1/2 -> 2s_1/2 at 0.319 filling the gap. Even with perfectly sharp lines the two strongest components are 0.329 apart, not 0.365; once each is widened by Doppler motion, the violet group's centre is dragged toward the red by its weaker members.

(d) Add Doppler widths and find the two maxima. Summing seven Gaussians of the FWHM above and measuring the distance between the two highest peaks [computed: example.py Part E; reproduces the arbiter's table]:

The apparent doublet comes out 0.26-0.32 cm^-1 under plausible conditions. That covers Michelson, Fabry-Buisson and Meissner-Paschen, and Gehrcke-Lau only if the 3d levels were strongly under-populated in their tube (Gehrcke and Lau worked near 320 K with a Lummer-Gehrcke interferometer and fitted Gaussian profiles [read by the contrarian: pp.572-573]). Nobody measured the level populations in the 1920s tubes, and the model leaves out Stark and pressure broadening in the discharge, the instrument function and the observers' fitting method. So the arbiter's reading is: the shortfall is "largely but not fully explained" by blending; the residual for Gehrcke-Lau is "explainable but not shown" [Arbiter, claim 2].

Step 6. What the example teaches

1. The formula's number (0.365 cm^-1 = 10.94 GHz) is right, and is Dirac's number.
2. A measured distance between two peaks of a blended line is not the distance between two energy levels. Gehrcke and Lau's data were real and honestly reported; their numbers were the separation of two blended maxima. Their conclusion that the theory's value was 25 per cent too high did not follow; but neither did the data confirm 0.365 [Contrarian; Arbiter].
3. The electron model enters as a single factor (2 gamma), so a crucial test needs alpha fixed independently; and even then it only decides between the models you compared (Abraham versus relativity). Weber's law needs a different measurement, the circular-orbit term (section 4).

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6. Evidence table

Verdicts are the arbiter's final verdicts (arbiter.md section 1, as revised in section 11 after QE), on the STEP 2 scale. The rows that earlier said "pending deep read" now cite the deep reads done in QE revision 2 [Arbiter, section 12]; none of their verdicts changed.

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7. What a learner should take away

What is solid. - Sommerfeld's fine-structure formula. It gives the hydrogen and He+ levels exactly as Dirac's theory does, and the n = 2 doublet of 0.365 cm^-1 (10.94 GHz) agrees with today's measurement to 0.24 per cent. Its pattern rules (1:3, 1:2:6, 3:5:10:30) are correct. - The first-order Stark effect of hydrogen, the normal Zeeman triplet, the X-ray doublet laws, and the dipole selection rule (as Delta l = +/-1). These survive with new labels. - The book's honesty. It concedes Bohr's objection ("incontestable"), calls the orbit models of X-ray screening "hopeless", withdraws several earlier views (at least five: the L1/L2 levels in different atoms; Kossel's defects; the "A-doublets", owed to "a false interpretation by the author" and disproved by G. Hertz's absorption limits, together with the extra L-limits they implied, which are "refuted by Hoyt's measurements"; the identification of theta with x'; and his first triplet Runge-denominator scheme, called "erroneous" [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch3.md, book pp.159, 171-172, 181; .ch6-part2.md, book p.392; digests .23 p.262, p.272]), says one step is "not inevitable" and another "certainly rather risky", prints the dissenting hydrogen measurements in a footnote, and says the crude intensity rule is "contradicted by experience" [read]. That is the standard a reader should hold both the author and his critics to.

What is wrong, and why the mistake is instructive. - The orbit picture. A wrong model gave the right energies twice over: once because the half-units of semiclassical quantisation cancel for the 1/r potential, and again because (per Keppeler) the half-units and the electron's spin cancel in the fine structure. Agreement of a formula with experiment does not prove the mechanism behind it. Kragh's phrase is "correct for the wrong reasons" [web]. - "Experimentum crucis for relativity." The test decided between the two electron models Sommerfeld compared, not between relativity and every alternative; Weber's velocity-dependent force gives the same splittings. Only the circular-orbit term separates the two, and it was measured once, weakly, by Paschen (section 4). A crucial experiment is only as crucial as the list of alternatives considered. - The "truest value" of h. The arithmetic was right; the narrow error bar left out a systematic error (Millikan's air viscosity). The wide, honest bars of the purely spectroscopic route contained the truth. Since h goes as e^(5/3) at fixed R and e/m, a 0.6 per cent error in e becomes 1.0 per cent in h [Arbiter, computed]. Sommerfeld saw the weak point in the spectroscopic route: "If, then, Paschen has not under-estimated the accuracy of his measurement ..." [read: IA OCR line 47880, via qe.md 3.11]. The author flagged the error bar of his key datum himself. - Circular support. Paschen's scale was fitted with the theory's interval scheme ("with the friendly help of its creator"), Epstein corrected Stark's measured field from the theory, and the "forbidden" components were blamed on unmeasured stray fields. Each is reasonable, and each means the agreement was partly built in. The book half-admits the first ("served Paschen as a means of calculating the ideal hydrogen doublet") [read]. Bohr is quoted as calling it "a piece of good fortune" that the selection rule failed in Paschen's photographs, since otherwise the theory "would probably have been delayed for years" [read: p.257]. - Peak separation is not level spacing (Step 5). Both sides of the 1921 dispute could have been helped by modelling the full line shape. Paschen had said as much of his own hydrogen measurement in 1916 ("probably too small"). - The "doublet riddle" of 1924. Optical doublets had been put down to magnetic coupling and X-ray doublets to relativity, until Millikan's (and Bowen's) 1924 ultraviolet spectra suggested one relativistic doublet law for both [web: Eckert, via sites.md and the contrarian, claim 5]; Lande meanwhile argued against the relativistic reading of the X-ray doublets. Sommerfeld wrote to Paschen (23 May 1924) that Lande's polemic seemed to end in "eine ungeheure Erweiterung des Gueltigkeitsbereiches" (an enormous extension of the range of validity) of the relativistic formula, "jedenfalls ebenso unverstaendlich" (in any case just as incomprehensible) in the visible as in the X-ray region [read: research/pages/trans02183.html; web: https://sommerfeld.userweb.mwn.de/TransDat/02183.html]. The formula kept working where its picture could not apply: the same "right formula, wrong picture" lesson, a year before spin explained it. - Reported effects that theory and replication set aside. Barkla's J-radiation (1917), reported by a leading experimenter, was dropped because "repeated tests by other observers have ... not been able to confirm" it and "the theory [has no] place for such radiation" [read: IA OCR lines 10915-10919, via qe.md 3.5]: both reasons are stated, and only the first is evidence about the effect. The book also used the nuclear model to exclude "nebulium", "coronium" and "protofluor", new elements of atomic weight 2 or 3 proposed from unidentified nebular and coronal lines: He would then need a nuclear charge of 4 or 5, "irreconcilable with our general experiences of a-rays" [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch1-2.md, book pp.70-71]. The book does not say what the lines are. Bowen showed in 1927 that the nebulium lines are forbidden lines of doubly ionised oxygen in very thin gas [web: https://en.wikipedia.org/wiki/Nebulium], and Grotrian and Edlén traced the coronal lines to iron atoms that have lost about half their electrons [web: https://en.wikipedia.org/wiki/Walter_Grotrian]; protofluor was not checked. Here theory was right that there were no new elements, but the lines were real: they are "forbidden" lines that appear only where collisions are rare. The book's footnote also left room for H or He of mass 2 or 3, and both exist (deuterium, helium-3) [memory, high]. A selection rule can fail under conditions a theory has not considered, which is worth remembering beside the stray-field explanation in row 2e. - The author's own failed alternatives and rejected right answers. The Sommerfeld-Debye accumulation theory of 1913 (the atom stores radiation until it holds h nu) gave Einstein's law only "under certain conditions (chosen in rather an artificial manner)" and predicted "accumulation times ... of the order of magnitude of years" for X-rays, where emission starts at once [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch1-2.md, book pp.44-45]. In Ch. VI the author rejected Fues's half-integer numbering of the s-term constants, because it would spoil the regular ratios of his Table 43 [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch6-part2.md, book p.377]; yet the Addendum he wrote with Heisenberg uses half-integer phase integrals and half-integer magnetic numbers [read: same, book pp.406-407]. The half-integers that won are Landé's angular-momentum numbers j and m, which Sommerfeld and Bohr at first did not accept and which electron spin explained in 1925 [web: https://arxiv.org/abs/2203.07833]; whether Fues's s-term numbering corresponds to anything physical is not settled by our sources [memory, medium]. The doublet-atom model (Li splittings 0.32 cm^-1 for p, said to "agree excellently" with the measured 0.31-0.34, and 0.035 cm^-1 for d, which the book has no measurement for) gave right numbers from a core with half a Bohr magneton, a forerunner of spin [read: same, book p.409; memory, high].

What is genuinely open (from the arbiter's sections 5-7). - How much of the Gehrcke-Lau shortfall is blending. A re-fit of their published intensity curves (Ann. Phys. 65, p.564 ff.) with the seven-component model, tube temperature free and level populations constrained by the Balmer intensity ratios, would settle it without a new experiment. - Sommerfeld's own later doubt. On 23 May 1924, two years after the book's text, he wrote to Paschen: "Ich bin ueberzeugt, dass die relativistische Theorie richtig ist; sie wird durch die Anzahl und Lage der Komponenten sichergestellt. Aber die Groesse von [Delta nu] = 0,365 moechte ich nicht mit gleicher Sicherheit behaupten" (I am convinced the relativistic theory is right; it is secured by the number and position of the components. But I would not assert the size 0.365 with equal certainty) [read: research/pages/trans02183.html, by the physicist; web: https://sommerfeld.userweb.mwn.de/TransDat/02183.html]. He called the Abraham alternative "natuerlich indiscutabel" and its proponents (Lenard, Gehrcke, Lau) "fanatisch" (his private characterisation, not a neutral one), yet said Lau's finding that the higher He+ analogues of 4686 show noticeably smaller Delta nu while their ratio becomes "very exactly" 1:3 had made him "sehr nachdenklich" (very thoughtful), suggested the mass formula might need an electron-nucleus interaction term, and urged Paschen to measure the constant again. This rests on one documented source (the letter); Lau's 1924 report and the plates have not been found. Competing explanations: unresolved components (the hydrogen blending mechanism), Stark or pressure effects, or a real departure [Arbiter, section 7b]. The letter post-dates the book, so an earlier reading that the doubt was kept "privately ... rather than in the book" implied a concealment the record does not show [Arbiter, correction 1]. - Earlier in the collaboration, Sommerfeld's draft of 29 Dec 1915 found the ratios of Balmer-line differences agreed "vortrefflich" (excellently) but "die Absolutwerte nicht" (not the absolute values: 0.102 predicted against 0.24 observed). The mathematician reads this draft as concerning the circular-orbit shift, a different quantity from the doublet (the arbiter has not confirmed which quantity it concerns; see section 4 for Paschen's later measurement of that shift); its arithmetic has two small slips (27/36 divided by 7/16 is 1.714, not 1.73; and 0.159 x 16/25 should be 0.169 x 16/25 = 0.108), which are Sommerfeld's own [web: https://sommerfeld.userweb.mwn.de/TransDat/00585.html; Mathematician]. - Whether Paschen's 0.3645 would survive a neutral re-reduction of his plates (their location is unknown). The same plates hold the circular-orbit figure (0.569 +/- 0.25): a re-reduction of He+ 4339 and 4686 with the weak 4338.536 component's assignment tested, the fitted scheme varied and errors propagated properly would say whether it is a real detection; so would a modern He+ measurement of the l-independent term [Arbiter, 11.11]. - Birge's 1921 paper (Phys. Rev. 17, 589) has not been obtained. The book cites it only for testing "again very carefully in detail" that the weak satellite does not shift the centre of the long-wave H-alpha group, and gives no Birge number [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch8-notes.md, book p.487]; whether Birge's own doublet value sided with Gehrcke and Lau or with Michelson is unknown [Arbiter, 12.1]. - The "forbidden" He+ 4686 components. Sommerfeld's rule allows 5 of the 12 positions; the Dirac rules allow 8 of the 12 (j, j') pairs [computed: example.py Part F; also the contrarian]. Which of Sommerfeld's labels (I-III, a-d) corresponds to which j has not been verified, so whether, say, III d is an allowed line rather than a stray-field effect is open [Arbiter, correction 8]. - How the 1920s H-beta extra Stark components were later settled. - The Nobel silence. Sommerfeld was nominated many times and never won: 84 nominations 1917-1951 by one count, 81 or 74 by others [web: https://www.bbvaopenmind.com/en/sommerfeld-the-eternal-nobel-candidate/ (search summary); other counts from the same search]. A 1926 nomination jointly with Paschen is known only from search snippets. This is a record that goes quiet where it would be expected to continue (his formula was vindicated in 1928). Competing explanations the arbiter sets out, none yet tested against records: (i) the committee saw the old quantum theory as provisional, overtaken by quantum mechanics; (ii) the prizes went to experimenters (Stark 1919, Siegbahn 1924, Franck and Hertz 1925, Compton 1927) and later theorists; (iii) politics or personalities (Sommerfeld, as chairman of the German Physical Society, defended Einstein in 1920 against anti-relativists who included the laureates Stark and Lenard). The deciding record is the committee's evaluation reports for 1919-1951 in the Nobel archive (now open, not obtained) [Arbiter, section 7a]. - The 1930s gap. Houston and Hsieh (1933) found Balmer fine-structure separations "far too small" by about 3 per cent, Gibbs and Williams the same, and Pasternack (1938) proposed a shift of 2s_1/2; this "received little attention for more than a decade" until Lamb and Retherford (1947) [web: https://en.wikipedia.org/wiki/Lamb_shift]. That 3 per cent was real new physics, but it is a different size from the 25 per cent of 1921; one does not explain the other [Arbiter, section 7c].

Interested parties. Sommerfeld (author of the theory); Paschen (his measurements, reduced with the theory's author's help); Gehrcke, Lau and Lenard (opponents of relativity; Gehrcke worked at the Physikalisch-Technische Reichsanstalt and is described as a leading anti-relativist [web: de.wikipedia, via the Investigator]); Millikan (his own e); Frauenfelder and Weber (advocates of Weber electrodynamics). Each side's data are weighed on their merits, apart from their stakes [Arbiter, section 8].

Other pertinent material kept in the record. The magneton numbers: Sommerfeld's spatial-quantisation correction gave 8.7 and 13.7 Weiss magnetons against 9 and 14 observed (for NO and O2), but in quantum mechanics the averaging "correction" vanishes and O2's real moment comes from spin (2.83 Bohr magnetons = 14.1 Weiss magnetons), so the agreement was partly coincidental; the author hedged [Mathematician; memory, medium]. Mercury: special relativity alone gives a perihelion advance of only 7" per century against 43" observed, and the book says so; general relativity gives six times more [read: Note 16; computed 7.16" and 42.98"]. Sommerfeld, writing to Paschen on 23 May 1924, called carbon's p ground term "ein harter Schlag fuer Bohrs sonst so ueberzeugende Modellfolge" (a hard blow to Bohr's otherwise so convincing sequence of models) [web: TransDat/02183, via the contrarian]. A separate letter, to Lande on 20 April 1924, says the arithmetical regularities reach "viel weiter ... als nach den Modellvorstellungen zu erwarten waere" (much further than the model ideas would lead one to expect) [web: SBPK Nachlass Lande, via research/2026-10-09-sommerfeld-projekt-lmu.md item 7]. The contrarian and the arbiter had merged the two letters [Arbiter, 11.6]. The book has a long list of misprints and translation slips (alpha^4/4 for alpha^2/4; "the higher the temperature the lower the velocities due to heat motion", where "higher" is meant [read: IA OCR lines 42415-42416]; 191/32 for 2807/1024; sign slips in the Notes; Note 13 and Note 17 numbers wrong as printed); none changes a physical conclusion, and whether they are in the German original is unchecked (the German 1921 and 1922 editions are online) [claims.md; Arbiter].

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8. Check yourself

1. Paschen used ionised helium (He+, nuclear charge 2, mass 4) instead of hydrogen. By what factor is the He+ N = 2 doublet larger than hydrogen's, by what factor is its Doppler width smaller at the same temperature, and so by what factor does the ratio "splitting / line width" improve?
2. The normal Zeeman shift is (e/m) B/(4 pi c) in wave-numbers. Find it for B = 10,000 gauss, and explain in one sentence why Planck's constant does not appear.
3. Weber's force law and Sommerfeld's relativistic theory give level energies that differ by alpha^2/(8 N^4) hartree. Why can no measurement of the splitting of a hydrogen line between levels of the same N distinguish them? What would you have to measure instead?
4. Millikan's e was 0.6 per cent low. Sommerfeld's "truest" h was computed as h proportional to e^(5/3) (R and e/m held fixed). How far off should h be, and why did the wide error bars of the "purely spectroscopic" route contain the true value while the narrow ones did not?
5. Gehrcke and Lau measured 0.272 cm^-1 for the H-alpha doublet. Give two reasons, from Step 5, why a measured distance between the two maxima of H-alpha is smaller than the 2p_3/2 - 2p_1/2 level spacing.

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9. Further reading

Mainstream textbooks (for the physics). - D. J. Griffiths, Introduction to Quantum Mechanics: the hydrogen atom, spin, and the perturbation-theory chapter (fine structure, Zeeman and Stark effects). - R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles: Bohr's model, the Wilson-Sommerfeld rule and its relativistic fine structure, then spin. - H. Goldstein, Classical Mechanics, or L. D. Landau and E. M. Lifshitz, Mechanics: action-angle variables and adiabatic invariants (the classical machinery of the old quantum theory). - G. Herzberg, Atomic Spectra and Atomic Structure (Dover): fine structure, Zeeman effect and X-ray doublets in the spectroscopist's language. - H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms: the reference for the Dirac levels, Lamb shift and line strengths used in Step 5. - J. R. Taylor, An Introduction to Error Analysis: random versus systematic errors.

History and the "Sommerfeld puzzle". - H. Kragh, article in Il Nuovo Saggiatore (2025), https://www.ilnuovosaggiatore.sif.it/article/402 ("correct for the wrong reasons") [web]; Kragh, Historical Studies in the Physical Sciences 15 (1985) 67-125, the standard study of the fine-structure dispute (not obtained). - M. Eckert, on how Sommerfeld extended Bohr's model, and the chapter on Atombau und Spektrallinien in MPRL Studies 2 (saved in research/files/). - P. Vickers (2018), on the Sommerfeld puzzle, https://pmc.ncbi.nlm.nih.gov/articles/PMC7418904/ [web]. - S. K. Suslov (2024), arXiv 2401.07485, and Barley, Ruffing and Suslov (2025), arXiv 2506.00408: modern re-derivations (abstracts read by the reviewers). - M. Jammer, The Conceptual Development of Quantum Mechanics: the old quantum theory in context [memory, high].

Related work. - J. Frauenfelder and G. Weber (2019), arXiv 1902.09612: Bohr-Sommerfeld fine structure from Weber's electrodynamics. - J. Ishiwara (1915), the independent quantum condition; see arXiv 1708.04676. - K. Glitscher, Ann. Phys. 52 (1917) 608: the spectroscopic comparison of the rigid and deformable electron.

The inventor's own key works and the primary record. - A. Sommerfeld, Atomic Structure and Spectral Lines (1923, English translation of the 3rd German edition): the project document; Ch. IV (phase integrals), Ch. V (selection rule, Stark, Zeeman, adiabatic principle), Ch. VIII (fine structure, X-rays, constants), Notes 6-16. - A. Sommerfeld, the 1915 fine-structure communication to the Bavarian Academy (Sitzungsberichte, saved as research/files/Sommerfeld-1915-Feinstruktur-SitzberBayAkad-badw003395523.pdf, CC BY) and "Zur Quantentheorie der Spektrallinien", Ann. Phys. 51 (1916): the first full statements. - F. Paschen, "Bohrs Heliumlinien", Ann. Phys. 50 (1916) 901-940: the He+ 4686 photographs, the circular-orbit term ("Das Relativitaetsglied", pp.931-933) and the H-alpha doublet "probably too small" (p.933); deep read in deep/Paschen-1916-BohrsHeliumlinien-AnnPhys50-iss16.md. - R. T. Birge, Phys. Rev. 17 (1921) 589: on the H-alpha components (cited by the book for the centre-of-gravity point; not obtained). - E. Gehrcke and E. Lau, Ann. Phys. 65 (1921) 564: the dissenting direct measurements. - Sommerfeld's letters (Sommerfeld-Projekt, LMU): https://sommerfeld.userweb.mwn.de/TransDat/02183.html (23 May 1924) and .../00585.html (29 Dec 1915).

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Corrections in this revision

Stated plainly, as the project's rule asks [qe.md section 7; Arbiter, section 11]: 1. Section 4 and answer 3 dropped a measurement. The earlier text said only an absolute shift could separate Weber from relativity and implied it had not been measured. The book (p.488) and Paschen 1916 report it (0.569 +/- 0.25 against 0.590; Weber 0). Now stated with its caveats. 2. Paschen's own 1916 "probably too small" remark on his H-alpha doublet was missing from Step 5. Added. 3. Evidence table row 2d carried the label "Tested", which is not on the verdict scale. Now Tested, not supported for the claim "0.365 is 25 per cent too large". 4. Row 6a cited Bucherer as a mass-velocity experiment named by the book. The book cites Schaefer and Neumann (1914) and Guye and Lavanchy (1916), not Bucherer. 5. Row 6b said "not in the record". One measurement is in the record; the verdict stays Never properly tested, with the result beside it. 6. "Two retractions" should be "several (at least five)". 7. Glitscher's figures were flagged "needs checking"; QE checked them and they hold (section 4). 8. The two 1924 letters (to Paschen, 23 May; to Lande, 20 April) had been merged in the reviews; each quotation is now cited to its own letter. 9. Terms used without definition (WKB, parity, 6j symbol, line strength, hartree) are now defined where first used.

QE revision 2 (orchestrator fix, 2026-10-09) [Arbiter, section 12]: 10. Withdrawals list. The "A-doublets" and "one doublet's false interpretation" were counted as two withdrawals; they are one episode. "Refuted by Hoyt" belongs to the extra L-limits, not to the A-doublets. The triplet Runge-denominator scheme is a further withdrawal, so "at least five" stands. 11. Half-integers. The earlier sentence "the half-integers later proved right" overstated the record (section 7 rewritten). 12. Birge 1921 was called a retest of the H-alpha components; the book does not say that. 13. Moseley's law, nebulium and coronium were stated from memory; they now cite the book and web sources. 14. He+ 4686 excitation. The book prints 75.4 V; its own arithmetic gives 75.9 V. Both lie in the observed 75-80 V window.

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Answers

1. The splitting scales as (nuclear charge)^4 = 2^4 = 16, so the He+ N = 2 doublet is 16 x 0.365 = 5.84 cm^-1. Doppler width scales as nu sqrt(T/M); for the same N-to-N transition nu is 4 times larger (Z^2) but M is 4 times larger, so the width scales as 4/sqrt(4) = 2: twice as wide in cm^-1. Splitting/width improves by 16/2 = 8. (The book's own reasons: four times heavier, so less Doppler-broadened, and a doubly charged nucleus, so less Stark-affected [read: Ch. VIII deep read 4.4].) For the line Paschen actually used, 4686 (N = 4 to 3), the whole 12-component pattern spans only 3.40 cm^-1 = 0.75 Angstrom [computed by the deep reader].
2. 4.67 x 10^-5 cm^-1 per gauss x 10,000 = 0.467 cm^-1 [computed]. The orbit's magnetic energy is its angular momentum times the Larmor frequency; angular momentum is quantised in units of h/(2 pi), the emitted frequency is energy divided by h, and the h cancels, so the shift is the classical Larmor frequency (Sommerfeld: the quantum theory "has in a certain sense become latent").
3. The difference depends only on N, so it shifts every level of one N by the same amount; a line's internal splitting is a difference between levels of the same N, and the shift cancels. You would need an absolute level energy: compare whole lines, not components within one line. The cleanest quantity is the circular-orbit shift, -alpha^2/(8 N^4) hartree under relativity and zero under Weber (0.091 cm^-1 at N = 2 in hydrogen). Frauenfelder and Weber say this "seems difficult to actually measure", but it had been measured once: Paschen (1916) compared effective Rydberg constants from the circular components of He+ 4686 and 4339 and found 0.569 +/- 0.25 cm^-1, against 0.590 for relativity and 0 for Weber, and the book (p.488) reports the shift as "of the order of magnitude of our relativity correction ... and [bearing] the correct sign". Caveats: one measurement, never repeated, by the theory's collaborator, resting on one weak component and a fitted scheme; at face value about 2.3 possible-errors from Weber's zero (example.py Part G). It also cannot distinguish Abraham's electron, because alpha was taken from the same splitting.
4. (5/3) x 0.6 = 1.0 per cent low (the actual miss was 1.2 per cent). The narrow bar (+/-0.009, 0.14 per cent) included only Millikan's random error, not the systematic one from air viscosity, so the truth lay about 9 bars away. The purely spectroscopic route used only Paschen's doublet, whose 1.2 per cent error grows to 3.1 per cent in h (h goes as alpha^5, and 5/2 x 1/81 = 5/162); that bar was wide enough to include the truth.
5. (i) The seven components are not two lines: the strong red component 3d_5/2 -> 2p_3/2 sits 0.13 cm^-1 above the lowest component, and the violet group is a blend of 3d_3/2 -> 2p_1/2, 3p_3/2 -> 2s_1/2 and a weaker 3p_1/2 -> 2s_1/2; even sharp, the two strongest are 0.33 apart. (ii) Doppler broadening (FWHM about 0.19 cm^-1 near room temperature) merges each group so that weaker inner components pull the two maxima toward each other; how far depends on temperature and on the populations of the 3s, 3p and 3d levels. (The Lamb shift also moves 3p_3/2 -> 2s_1/2 inward by 0.035 cm^-1.) Sommerfeld's own partial explanation, the "selection lag" to 0.329, is the old-theory version of point (i).