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Summerfield (Arnold Sommerfeld): Science Contrarian review (Step 2)

Reviewer role: Science Contrarian, same PhD standard as the physicist and mathematician. Claims are judged by experimental evidence, not by consensus or reputation. Mallove is a model for the standards (follow the evidence, check how a claim was handled, hold the inventor to the same rigour as the critics), not a persona.

Tags: [read] taken from the project documents or the Investigator's saved files; [web] from an Internet source (URL given); [memory] recalled, with confidence; [computed] my own calculation. Where a conclusion rests on [memory] alone, I say so.

0. Naming, status of the source, and source gap

  • The folder is spelled "Summerfield"; the author is Arnold Sommerfeld. The one document is the 1923 English translation (H. L. Brose, Methuen) of the third German edition (preface January 1922) of Atomic Structure and Spectral Lines [read: claims.md; deep read p.1-3]. This is a mainstream textbook by one of the creators of the Bohr-Sommerfeld theory. There is no outsider and no suppression story in the usual sense. The contrarian task here is therefore narrower: (i) separate what was tested from what was asserted; (ii) examine the one real dispute on record (the hydrogen doublet size, 1920-26); (iii) say what survived and why; (iv) hold the book to the same standard as its critics. Judge it against 1921-23 knowledge first, then against today.
  • Source gap (repeated from claims.md). The PDF has no text layer; all 324 pages were OCR'd and nearly every equation is garbled. Numbers below come from deep readers' page images. Digest only, no deep read: PDF pp.1-106 (Ch. I-III, including Moseley's law itself) and pp.157-235 (Zeeman effect, adiabatic hypothesis, Ch. VI series spectra and anomalous Zeeman effect, Ch. VII bands). The Zeeman and Ch. VI parts of claims 4 and 7 therefore rest on digests; I checked only the one number I could (the Zeeman constant, see claim 4). Equations in those parts are unread.
  • What I read myself: the full deep read of the book (all sections); Gehrcke and Lau 1921 pp.572-575 (Gaussian-profile fits, the "25 Proz." passage) [read: research/files/Gehrcke-Lau-1921...djvu.txt lines ~5960-6130]; Paschen 1916 (the "Unstimmigkeiten" and "freundlicher Hilfe" passage) [read]; Glitscher 1917 (the Abraham versus Lorentz test) [read, OCR]; Sommerfeld to Paschen 23 May 1924 (full transcription, fetched and read in German) [web: https://sommerfeld.userweb.mwn.de/TransDat/02183.html]; Frauenfelder and Weber 2019 pp.1-3 [read: research/files/arXiv-1902.09612...pdf]. I did not obtain Kragh 1985, Robotti 1986, Lau's 1924 report, Houston/Williams/Pasternack papers, or the German 1921/1922 editions.

1. Corrections to Step 1 and to the Investigator

These follow "CORRECTIONS ARE WELCOME". I name the document and passage, give the evidence, and do not soften.

  1. claims.md claim 2 and facts.json: the hydrogen-doublet shortfall is "only partly explained"; the record now supports more than that. The book says the gap between direct measurements (0.27 to 0.33 cm^-1) and 0.365 is "partly explained by the influence of the variable term"; the deep reader's own estimate of that effect is 0.329 (Hα) and 0.350 (Hβ), leaving Gehrcke-Lau's 0.27-0.28 unexplained [read: deep read Ch. VIII §4.2]. Using the modern line list I computed what a Doppler-broadened Balmer line should show (method and caveats in §3.2). With the Dirac fine-structure levels, Lamb shifts, dipole line strengths and equally populated sublevels, the two blended maxima of Hα, Hβ and Hγ at about 320 K come out 0.31-0.32 cm^-1 apart, and 0.28-0.30 cm^-1 when the d-state population is lowered [computed]. That covers Michelson, Fabry-Buisson, Meissner-Paschen and, at the low end, Gehrcke-Lau (0.271-0.283). It also reproduces Gehrcke and Lau's observation that Δν is nearly the same for Hα, Hβ and Hγ, which the book's own 1921 scheme does not (it predicts 0.329 rising to 0.350). So: the direct measurements were largely blend artefacts of a doublet whose components overlap at room temperature (Doppler FWHM 0.20 cm^-1 at Hα against a 0.37 cm^-1 true spacing). Not an error in Step 1, but Step 1's "partly explained, residual shortfall" should be updated to "largely explained by line blending, residual up to ~0.04 cm^-1 depending on assumptions". The computation is mine and crude (no Stark or collision broadening; assumed populations); treat it as a plausibility check, not a fit.
  2. Investigator's lead on Sommerfeld's 1924 letter, framing. sites.md says the dissent "was answered privately with doubt about 0.365 rather than in the book." The letter is dated 23 May 1924 and states that the new edition "an der fleissig gedruckt wird" (is being printed) [web: TransDat/02183]. The book in the folder is the 1922 text, translated in 1923, and cannot contain a 1924 doubt. The fair statement: in the book Sommerfeld printed the dissenting measurements in a footnote table and said "partly explained" [read: deep read §4.2]; two years later he privately separated the secure structure (number and position of components) from the less secure absolute size; whether the 4th edition said so I did not check. The letter is real and pertinent (kept below); the implication of concealment is not supported.
  3. Deep read, Ch. VIII §4.7 and claims.md claim 6: "the 4/5 test needs an independent α, which Sommerfeld does not raise." True of the book's text, but the source the book cites for the test, Glitscher's 1917 dissertation (Ann. Phys. 52, 608), does raise it. Glitscher takes α from the helium pattern for each electron model (relativistic α^2 = 5.32 +/- 0.22 e-5; Abraham 6.65 +/- 0.28 e-5, a 20% difference) and compares both with α^2 computed from independently measured e and h, concluding that Abraham's value "auf keine Weise" can be reconciled even with extreme e and h [read: Glitscher, OCR lines ~4340-4400; OCR is poor, so the exact figures for the e,h-derived value (given as about 5.8 e-5) need checking against the page]. Modern α^2 = 5.3251e-5 [memory, high] agrees with the relativistic value and excludes the Abraham one. The test is therefore not circular as Glitscher states it; but 1917 e and h were uncertain by several per cent, so the discrimination then was weaker than the "experimentum crucis" language of the book.
  4. facts.json mainstream_conflict, "Millikan's e carried an unrecognised systematic error [memory, high]". Retag [web]: the standard account is that Millikan used a wrong value for the viscosity of air; his 4.774(5)e-10 esu is about 0.6% below the accepted value; Bäcklin's 1928 X-ray value 4.793(15)e-10 is within uncertainty of the exact value; 4.800 by 1937 [web: https://en.wikipedia.org/wiki/Oil_drop_experiment]. Consistent with Step 1.
  5. facts.json: the deep reader's medium-confidence inference that the weak He+ 4686 "forbidden" component III d is allowed in Dirac theory. I checked the count: He II n=4 to n=3 has four upper and three lower j levels; the Dirac electric-dipole rules (Δl = +/-1, Δj = 0, +/-1) allow 8 of the 12 (j,j') pairs, all 8 realisable [computed]. The deep reader's mapping of Sommerfeld's labels (I-III, a-d) to specific j values I did not verify. Count confirmed; mapping remains unverified. Retag medium.
  6. Step 1 omission (not an error): extra model-failure evidence in the Investigator's letter. The same 1924 letter calls the finding that the ground term of carbon is a p-term "ein harter Schlag für Bohrs sonst so überzeugende Modellfolge" and says the arithmetic regularities "reach much further than the model pictures lead one to expect" (Sommerfeld to Landé, 20 April 1924) [web: TransDat/02183; sites.md]. Add to claim 7.
  7. claims.md claim 2: "Paschen's value was obtained by fitting the theory's own interval scheme." Confirmed by Paschen's own words: the earlier mean e/mu was 1.74 against the known 1.76 +/- 0.01; "Eine Klärung der Unstimmigkeiten brachte die neue Theorie von A. Sommerfeld"; the material was brought "mit freundlicher Hilfe ihres Schöpfers" into "genügende Übereinstimmung" [read: Paschen 1916, text lines ~4236-4262]. I add what Step 1 left unsaid: the fit has one free parameter (the scale Δν_H) but the pattern it must match is fixed by the theory (ratios 1:3, 1:2:6 and the placement of 12 components), so the pattern is a real test and the absolute scale is a one-parameter fit. The scale turned out right (see claim 2).

No other errors found in Step 1. The misprint list in claims.md is consistent with what I read.

2. Standard used for verdicts

For each claim I separate: the formula or number (what is measured), the mechanism the author attaches to it, and the assumptions used to derive it. Several claims split on that line. Where a verdict is "Established" it means the formula or relation survives in today's theory and measurement, often with different quantum-number labels and a different physical mechanism, and I say so.

3. Claims

Claim 1. Quantisation of orbits (phase integrals, elliptic orbits, n + n', n = 0 excluded, spatial quantisation)

a. Mathematics. Internally consistent where checked. The deep reader reproduced W = -2π²me²E²/(h²(n+n')²), the eccentricity relation 1 - ε² = n²/(n+n')², the Kepler conic integral (to 1e-12) and the spatial-quantisation relation cos α = n1/(n1+n2) with SymPy [read: deep read Ch. IV §E]. Admitted gaps, in the author's words: the choice of coordinates is not justified; the Kepler problem is degenerate so parabolic coordinates give "quantised ellipses selected from the group of Kepler orbits different from" the polar ones; Bohr's objection that both conditions are not needed is "incontestable"; the exclusion of n = 0 ("permanence of atoms") and of n1 = 0 is by physical argument and "taken for granted"; the rule's positivity is proved for the oscillator only; "how to proceed in cases which do not fall under this heading is not yet known" [read: deep read Ch. IV §A.5, §E.2-E.5]. Hidden restrictions found by the deep reader: p = ∂E_kin/∂q̇ fails in a magnetic field; the zero of quantisation (area 0 versus h/2) is a convention; negative n1 is not counted. Misprints (q_max factor 2, "p = φ", b = a√2 for a/2, "four octaves" for two) do not change conclusions.

b. Consistency with experiment. Balmer, Ritz, Paschen, Lyman and the He+ Pickering series are reproduced exactly (Table 24 agrees to 0.03 Å) [read]. The energy depends only on n + n', which is the right degeneracy for hydrogen [memory, high]. The reduced-mass shift between H and He+ (1.000412 predicted, 1.000412 measured) is a clean quantitative success [read: deep read §D.2]. The book states its own failures: He ionisation 28.75 V computed against 25.3 V observed; H2 and H2+ models untenable [read: Note 14]. Not consistent with modern knowledge: the hydrogen ground state has zero orbital angular momentum, not h/2π; the s(s+1)/2 state count (modern s²); the counting of n1 ≥ 1 orientations only [read: deep read §E.5]. Stern and Gerlach's 1922 experiment showed discrete deflection (two lines for silver) and was taken as support for spatial quantisation [web: https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment]; that the two lines come from electron spin, not orbital orientation, is [memory, high] and illustrates "right result, wrong mechanism".

c. Tested? Yes, extensively, for one-electron atoms: Balmer-Rydberg spectra, He+ (Fowler, Paschen), Franck-Hertz electron-impact excitation of stationary states [read: claims.md peripheral list; digest only], X-ray series. Independent discovery of the rule by W. Wilson (1915), Ishiwara (1915) and Sommerfeld [read: deep read §A.5; web: arXiv 1708.04676 per Investigator], which is itself weak corroboration of the rule's naturalness. Not replicated for many-electron atoms: the book says so.

d. If true, what changes? The literal claim (sharp classical trajectories, exactly quantised, with an unexplained non-radiating state) would contradict Maxwell inside the atom and, later, the uncertainty principle. This is exactly what mainstream physics did change in 1925-26; the reviewer's point is that the old theory's success was partly inherited: the semiclassical quantisation of the Coulomb problem is exact for the energies [memory, high; Suslov and Barley et al. 2024-25 re-derive this, web per Investigator: arXiv 2401.07485, 2506.00408; I did not read them].

Verdict (split). - Energy rule for one-electron (hydrogen-like) atoms: Established (formula survives; labels differ). - Orbit picture, exclusion of n = 0, n1 = 0, state counting, spatial quantisation as orientations of orbits: Historically superseded but instructive. - Extension to He, H2, H2+ and heavier atoms by orbit models: Tested, not supported (by the book's own tests). - The coordinate-choice question: not a testable claim as posed; settled in the relativistic limit (claim 2), as the author argues.

Claim 2. Relativistic fine structure (Δν_H = Rα²/16 = 0.365 cm^-1; component patterns; He+ 4686)

a. Mathematics. Consistent. The deep reader reproduced the exact formula (28), all coefficients of the expansions (1)-(2) of §5, the component ratios 1:3, 1:2:6, 3:5:10:30, and 0.36476 cm^-1 from the book's constants (0.36504 with modern α² and R_H) [read; computed]. Slips: "α⁴/4 = 1.3e-5" should be α²/4; α² = 5.32e-5 is 0.6% above what the printed e and h give; Darwin's nuclear-motion term is below the data's precision [read]. The "non-degenerate relativistic problem" removes the coordinate gap of claim 1, a real point in the author's favour.

b. Consistency with experiments. - For: (i) Paschen's He+ 4686 photographs, 12 components expected, pattern in the right places; intervals ratio 1:3 observed also in the higher He+ analogues (Sommerfeld to Paschen 1924) [read; web]. (ii) X-ray L-doublets scale as (Z - s)^4 over Z = 41-92 with the Δν_H read off the table extrapolating to 0.365 (Fig. 118) [read: deep read Ch. VIII §1.4]. (iii) The formula equals Dirac's 1928 energy with n = j + 1/2, so its numbers are in today's theory: Heisenberg called the coincidence a "miracle", Schrödinger "a fortuitous computational accident" [web: Vickers 2018, PMC7418904, and Suslov arXiv 2401.07485, per Investigator]. I computed the n = 2 doublet from the Dirac formula: 0.3650 cm^-1 [computed]. With Lamb shifts [memory, high for the 1057.8 MHz 2S1/2 value; Wikipedia Lamb shift page says "about 1057 MHz": web https://en.wikipedia.org/wiki/Lamb_shift] the 2P3/2-2P1/2 interval is unchanged at the 2P level, to the accuracy I need. - Against (as printed in 1923 and in the record): (i) direct hydrogen measurements 0.27-0.33 cm^-1 against 0.365; Gehrcke and Lau: "Bemerkenswert ist, daß der Betrag für Δν etwa um 25 Proz. kleiner gemessen wird, als er aus der theoretischen Feinstrukturkonstanten sich nach Sommerfeld berechnet"; they could see no measurement error, and earlier observers found values close to theirs [read: Gehrcke-Lau 1921 p.575]. Their method: Lummer-Gehrcke interferometer, each component fitted by a Gaussian (Rayleigh's Doppler profile, temperature near 320 K), carriers shown to be H atoms not H2 [read: pp.572-573]. (ii) The crude intensity rule "is contradicted by experience" (component IIb of 4686 is strongest, not Ia) [read]. (iii) The "forbidden" components in Paschen's photographs are explained after the fact by stray fields (Kramers: about 100 V/cm direct current, 300 V/cm spark); no field was measured independently [read: deep read §4.4]. Bohr's remark in the book that it was "a piece of good fortune" that the selection principle failed in Paschen's photographs, because otherwise the theory "would probably have been delayed for years" [read: p.257] is an honest admission that the favourable photographs were not ideal conditions. (iv) Lyman α is a doublet in reality, not "strictly simple" as the book says [memory, high]. (v) Later: Houston and Hsieh (1933) found Balmer-line separations "far too small to agree with the ordinary theory", about 3% low; Gibbs and Williams (Cornell) the same shortly after; Pasternack 1938 proposed a shift of 2S1/2; the work "received little attention for more than a decade" until Lamb and Retherford (1947) [web: https://en.wikipedia.org/wiki/Lamb_shift]. The direction matches the 1920s dissent; the size (3%, 0.03 cm^-1 scale) is an order of magnitude smaller than 25%. - My reconstruction of the 1920s hydrogen data with the correct modern line list (§3.2 below): the apparent splitting of Hα, Hβ, Hγ at 320 K lies in 0.28-0.32 cm^-1 depending on level populations, nearly independent of the line, as Gehrcke and Lau found. So the measured shortfall is mostly the blend of seven fine-structure components, not a failure of 0.365. I therefore do not rest anything on the dissent being a refutation; I rest the point that it was a real and correctly reported measurement, whose explanation was only partly available in the book.

3.2 How the computation was done (so a reader can check it): hydrogen levels from the Dirac formula, reduced mass included; Lamb shifts for s levels (+1057.8 MHz x 8/n³) and p1/2 (-12.8 MHz x 8/n³) [memory, high for 2S; medium for the n-scaling]; seven allowed Hα components (3s1/2, 3p1/2, 3p3/2, 3d3/2, 3d5/2 to 2s1/2, 2p1/2, 2p3/2); line strengths from 6j symbols and numerically integrated radial matrix elements, sublevels equally populated; each component a Gaussian with Doppler FWHM = 7.16e-7 ν √(T/M) (0.195 cm^-1 at Hα, 320 K); separation of the two maxima (and a two-Gaussian fit) reported. Results [computed]: Hα 0.314 cm^-1 (Dirac only 0.324), Hβ 0.319, Hγ 0.315 with equal populations; 0.29-0.30 when the d states are down-weighted by 0.3 or s,p up-weighted by 3. Not included: Stark or pressure broadening, unequal populations measured in the tube, instrument function, the actual fit procedure of Gehrcke and Lau. Scratch scripts are in the session scratchpad and are not deliverables.

c. Tested? By whom? Replicated? Tested by Paschen (1916; He+, photographic and photometric), Michelson, Fabry and Buisson, Meissner and Paschen, Gehrcke and Lau (1920-21), McLennan (reported confirming; Sommerfeld says it is unreliable), Lau (1924 report, not obtained), Hansen and Houston (1925; Houston criticised by Hansen per letter summaries), Houston and Hsieh and Williams (1933-38), Lamb and Retherford (1947) [read; web]. The He+ pattern and the X-ray doublets were repeated by others; the direct hydrogen values were repeated by several observers and agreed with each other (all low), which is a replication of a blended-line artefact rather than of a failure of the theory.

How the dispute was handled. The dissenters were Gehrcke, Lau and Lenard, prominent anti-relativists (Gehrcke worked at the Physikalisch-Technische Reichsanstalt 1901-46 and is described as one of the best-known anti-relativists [web: de.wikipedia Ernst Gehrcke, per Investigator]). That is a stake, recorded beside their data. Sommerfeld, who as DPG chairman defended Einstein against them in 1920 [web: Sommerfeld-Projekt, per Investigator], privately called their motives "fanatischen Geist" but also told Paschen that Lau's result "hat mich sehr nachdenklich gemacht" [web: TransDat/02183]. Paschen worked "with the friendly help of" the theory's author; so both sides of the key measurements had interests. What I find sound: the book printed the dissenting measurements in a footnote and said "partly explained"; the 1924 letter shows the author asking for better measurements ("Die Theorie dazu werden wir schon nachliefern"). What I find weak: the explanation offered in the book (only three of six Hα lines survive; shortfall 0.329) was not tested against Hβ and Hγ, where it predicts rising values the data did not show; and Kragh's summary is that the agreement "was far from perfect and questioned by a minority", but "did not affect the consensus view" [web: https://www.ilnuovosaggiatore.sif.it/article/402, per Investigator]. No evidence of omitted data or blocked publication was found; Gehrcke and Lau published in the Annalen at W. Wien's suggestion [read: Investigator, archive.org note].

d. If true. Nothing in mainstream physics has to change; it is already inside it, relabelled. The mechanism is what changed: Kragh calls the theory "correct for the wrong reasons"; the l-dependence comes from spin-orbit and the Darwin term together with relativistic kinetic energy, not from precessing ellipses; Keppeler's semiclassical analysis (cited by Frauenfelder and Weber) says the Maslov index and spin "mutually cancelled out" [read: arXiv 1902.09612 p.3; web: Kragh 2025]. Biedenharn traced the coincidence to an underlying symmetry [web: Vickers 2018].

Verdict (split). - Energy formula and the size of the n = 2 doublet (0.365 cm^-1): Established (survives as Dirac; 1923 evidence = He+ pattern, X-ray doublets; the direct hydrogen shortfall is largely explained by blending; the Lamb shift, 1947, is a further small departure of about 0.03 cm^-1 in the 2S1/2 level). - Mechanism (precessing quantised relativistic ellipses without spin), the selection principle stated for n, the crude intensity rule, the attribution of the "forbidden" components to stray fields: Historically superseded but instructive (and the intensity rule: Tested, not supported, by the book's own He+ data). - The 1921-23 direct measurements as a challenge to 0.365: they were tested and were an honest anomaly at the time; now largely resolved (not a refutation).

Claim 3. Wave and quantum pictures joined by conservation laws (Rubinowicz selection principle; polarisation rule; "ether")

a. Mathematics. Internally consistent where derived: N = (W/2πν) 2ab sin γ/(a²+b²) was checked from the Hertz-vector solution in Note 9 and reduces to ħ per photon for circular light; the steps |Δn| ≤ 1 follow from (a-b)² ≥ 0 [read: deep read §3.4-3.5, §4.1; memory, high for the classical result]. The author himself marks the weak points: Δn = 0 gives "nothing at all happens" by a conclusion that is "not inevitable"; the n_a = n_e case needs an assumption he calls "certainly rather risky"; the field-case Δn = 0 argument is "not, however, inevitable"; two of seven wave elements (including coherence length) are undetermined ("5/7ths ... solved"); he concedes in a footnote that decoupling atom and ether in stationary orbits "is justified" as an objection [read]. Note 10(f) misprints Δn = 0 for Δn = ±1. The no-recoil assumption is explicit and conflicts with the directed-quantum picture which he footnotes.

b. Consistency with experiment. The ratio angular momentum/energy = 1/ω for circular light and the photon spin ħ are standard [memory, high], and the later direct measurement of the angular momentum of circularly polarised light (Beth 1936) [memory, high; not web-checked]. The dipole selection rule Δl = ±1 (Laporte) is exact for single photons in the modern account, so Sommerfeld's Δn = ±1 with Δn = 0 excluded corresponds to it; but the deep reader notes his exclusion rests on the "not inevitable" step, not on parity [read: deep read §5.2]. The statement that the rule is "rescinded" in dense discharges, via microscopic fields, agrees with modern Stark-mixing accounts [memory, high]. The no-recoil assumption fails: Compton scattering (1923) and later work show photon momentum and atomic recoil [memory, high]. The "mildest modification" the author mentions (conserve energy only on average, p.136-137) was proposed in the Bohr-Kramers-Slater theory (1924) and refuted by the Bothe-Geiger and Compton-Simon coincidence experiments (1925) [memory, high; not web-checked]; the author rejected that option in 1922, correctly as it turned out.

c. Tested? The polarisation rule was said to elude test in field-free atoms; in fields it was tested in the Stark and Zeeman effects (claim 4) [read]. The selection principle was tested by Paschen's discharge photographs, where it holds at low currents and fails at high [read: p.255]. Replication for the wave-quantum bookkeeping as a whole: none needed in 1923; later tests (photon spin, Compton) bear on it.

d. If true. If the atom and the optical field were coupled only at emission and radiated with no recoil, momentum conservation for single events would fail; mainstream physics (Compton, QED) says the atom recoils and the field carries momentum from the start. The author's "ether" is defined to have "no properties other than such as light itself possesses", so it adds nothing testable. Anyone reading "ether" here as a medium should note he explicitly denies that [read: p.139].

Verdict (split). - Angular momentum carried by a circular photon is at most h/2π; |Δ(angular momentum)| ≤ ħ per emission; circular polarisation for Δm = ±1: Established. - The derivation that Δn = 0 gives no emission and linear polarisation along the field: Plausible, untested as a derivation (the result agrees with parity; the argument is flagged by its author as not inevitable). - No recoil of the atom: Tested, not supported (Compton and later work; [memory, high]). - The "ether" statements: Not testable as stated.

Claim 4. Electric and magnetic field effects (Stark, Zeeman, adiabatic hypothesis)

a. Mathematics. Stark: the first-order formula ΔW = -(3h²F/(8π²mE))(n2 - n1)(n1+n2+n3) was checked and equals (3/2) e a0 F n (n1 - n2), the wave-mechanical first-order result; every entry of Tables 27-34 recomputed (two clerical slips, same component set); eq. (28) of Note 11 has the wrong sign, the energy is unaffected [read: deep read §6.4-6.5, Note 11]. Zeeman: h cancels in Larmor precession (digest only); I checked the one constant: e/m = 1.77e7 emu/g gives 1.77e7/(4π × 2.998e10) = 4.70e-5 cm^-1 per gauss, the printed value [computed]. Adiabatic hypothesis: the proof (Note 12) bounds an integral of a Fourier series and needs divisors to stay away from zero; the author concedes the failure at degeneracy, and the near-resonant case is not treated [read: deep read Note 12].

b. Consistency with experiments. Stark's and Lo Surdo's eight experimental regularities are reproduced; the hydrogen splitting at Hγ and Hδ is "absolutely perfect", Hα agrees up to Δ = 4 [read]. Mismatches the author states: theory predicts Hα components at Δ = 5, 6, 8 not seen (faint by Kramers' intensities); four Hβ observed components not predicted (he suspects they are unreal or secondary). The absolute scale is a ~3% consistency between Stark's measured field (104,000 V/cm) and the field Epstein inferred from the theory (107,000), not an independent test [read: deep read §6.6]. The normal Zeeman triplet is observed for singlet lines; the anomalous Zeeman effect (Preston, Runge, Landé) is the case where orbital Larmor precession fails and which spin explains [memory, high; Ch. VI is digest only here]. The modern result: first-order Stark spectrum of hydrogen identical [memory, high]. In 1923 the Hα and Hβ anomalies were open items; I could not determine from the files how they were later settled.

c. Tested? By Stark (1913-14), Lo Surdo (1914), with theory by Schwarzschild and Epstein (1916), all independent of each other in the sense that the experiments preceded the theory; the field in Stark's apparatus is hard to measure to better than 1%, as the book says. Hansen and Jacobsen (1921) found He+ 4686 in a magnetic field "essentially looks normal" after reducing stray electric fields [read: deep read §4.6]. Replicated in later spectroscopy [memory, high].

d. If true. Mainstream unchanged: wave mechanics reproduces the first-order Stark effect with the same numbers.

Verdict (split). - First-order Stark pattern of hydrogen, polarisations and component positions: Established. - The few unobserved or unpredicted Hα and Hβ components: Never properly tested in the 1923 record (no intensity measurement was reported against the correspondence-principle intensities; resolution later is not documented in the files). - Normal Zeeman triplet from Larmor precession: Established (singlet lines; digest-only reading, equations unread). - Anomalous Zeeman effect: outside this claim; the book calls the model interpretation doubtful. - Adiabatic hypothesis: Plausible, untested as proved in the book (incomplete proof, stated limit); the quantum adiabatic theorem later made it a theorem for non-degenerate states [memory, high; not web-checked].

Claim 5. X-ray spectra mirror the hydrogen fine structure (regular and irregular doublets; level scheme)

a. Mathematics. Internally consistent with identified slips. Δν_L = Δν_H (Z - s)^4 is correct from the expansion; all coefficients (5/4, 21/8, 429/64) reproduced; Table 52 reproduced to 0.2%; Table 53's mean s = 3.492 with standard deviation 0.043 over 38 elements, two rows off (Os, Yb); Hertz's law Δ√(ν/R) = (σ1 - σ2)/n and Δν linear in Z follow [read: deep read Ch. VIII §1-2]. Inconsistencies in the book: Table 54 first-column entries (Pt, Au, Pb, Bi, U) do not match Table 55; the definition of m says "irregular" where the table shows "regular"; "Δλ increases with the third power" should read "inversely"; coefficient 191/32 where 2807/1024 is derived. The author labels the L-doublet formula "an empirical equation ... brilliantly confirmed", not derived, because s is fitted.

b. Consistency with experiments. Moseley, Siegbahn, Coster, Hertz, Wentzel, Kossel, Duane and Stenström data [read]. Spot check of Table 55 against modern edges, 0.2% [memory, medium]. In modern terms the regular doublets are relativistic and spin-orbit splittings (same l, different j), and the irregular doublets are screening doublets (same j, different l); Hertz's law is the textbook screening-doublet law [memory, high; deep-read §5]. The author's own conclusion that no orbital model of screening can work ("hopeless", 105 species of atom needed) is a strong, honest negative result, and he retracts two positions of earlier German editions [read: deep read §1.6, §3.5].

c. Tested? By multiple independent laboratories over 1913-21, with a large dataset (Z = 12-92) compared with a one-parameter law. The "doublet riddle" (Forman; Eckert) shows the interpretation was disputed: optical doublets were assigned to magnetic coupling and X-ray doublets to relativity, until Millikan's 1924 ultraviolet spectra suggested a single relativistic law [web: Eckert chapter in MPRL Studies 2, per Investigator; I did not read it]. Sommerfeld to Landé (1924): the relativity formula "extends its validity into the optical realm", which he describes as "ebenso unverständlich im sichtbaren wie im Röntgengebiet" in the letter to Paschen [web: TransDat/02183].

d. If true. Nothing; mainstream atomic theory contains it (spin-orbit and screening).

Verdict (split). - The regular- and irregular-doublet laws and the 21-level scheme with selection rules: Established (with the reversed subscript convention and the mapping n = j + 1/2, m = l + 1 in the deep read [memory, high]). - Orbital-model explanations of the screening numbers: Tested, not supported (by the author's own calculation of K-excitation mismatch, 6.4 / 8 / 45 ν/R units against at most 0.2 observed). - Moseley's law itself: Established (digest only here; rests on [memory, high]).

Claim 6. Relativity versus the "absolute" electron; universal constants e, h, α

a. Mathematics. The 4/5 factor (2γ with γ = 2/5 against 1/2) is correct: hydrogen doublet 0.292 against 0.365, deep reader 0.29; the series expansions agree with the exact Abraham formula to the order kept; the author notes that the coincidence of the relativistic case with the exact formula beyond first order "must be regarded as an accident" [read: deep read §4.7]. Error propagation is correct (1/81 to 1/54 and 5/162; 5/3 × Millikan's relative error) but carries one error source per route and sets the errors of R_inf and e/m0 to zero by assertion [read: deep read §4.4-4.6]. The text arithmetic reproduces e = 4.766e-10, h = 6.526e-27, 6.545e-27.

b. Consistency with experiments. (i) Abraham rigid-sphere electron versus Lorentz-Einstein: Glitscher found the Abraham value of α² (6.65e-5) well above the relativistic one (5.32e-5) and above values from independent e and h; modern α² = 5.3251e-5 selects the relativistic number [read: Glitscher; memory, high for α]. Rigid electron models are excluded by modern work anyway [memory, high]. (ii) Does the fine structure single out special relativity? Not on this evidence alone. Frauenfelder and Weber (2019) apply Sommerfeld's own quantisation to Weber's 1846 velocity-dependent electrodynamics and get the same l-dependent term, -α²/(2n³ℓ), as Sommerfeld's relativistic formula; only the l-independent α²/n⁴ term differs (1/2 against 3/8), which they say is "difficult to actually measure" [read: arXiv 1902.09612 pp.1-3]. They also note that Weber's rosette orbits have the same shape as the relativistic ones (Bush 1926) and that spin is not included. So the doublet size and the component pattern do not discriminate relativity from this alternative at the order considered, and the "experimentum crucis" (a term used by the book) is an experimentum crucis only between the two electron models the author compared (Abraham and Lorentz-Einstein), not against all velocity-dependent dynamics. I have no source saying Weber's theory survives other tests, and mainstream physics regards it as superseded by Maxwell's theory (the paper says as much); I do not claim otherwise. (iii) The constants: modern e = 4.803e-10 esu, h = 6.626e-27, e/m0 = 1.7588e7 emu/g [memory, high]. The "purely spectroscopic" values with their wide bars (e = 4.766 +/- 0.088, h = 6.526 +/- 0.200) contain the modern values; the narrow-bar "truest value" h = 6.545 +/- 0.009 does not (about 9 sigma low) [computed from deep read table]; the reason is Millikan's air-viscosity error [web: Oil drop experiment page]. This is a clean lesson about error budgets: the honest wide bars were right; the narrow bars omitted a systematic error. α itself came out within 0.03% because errors in e and e/m0 nearly cancel in α [computed].

c. Tested? Mass-velocity law by Kaufmann, Bucherer, Neumann, Guye and Lavanchy (cited; the book says the 1914 and 1916 refinements succeeded where Kaufmann's early work did not settle it) [read]. Fine structure used as a test by Lenz's suggestion and Glitscher's dissertation [read; web]. Replicated: relativistic mass increase is among the best-tested effects [memory, high].

d. If true. Nothing new for mainstream physics. If one wanted the fine structure to be a unique test of special relativity, the alternative dynamics would have to be excluded by further data (the l-independent term, or non-hydrogenic systems); that is not in the record.

Verdict (split). - Abraham's rigid electron versus Lorentz-Einstein, decided by the fine structure: Tested, supported relativity over Abraham (Established as a conclusion); the argument as a unique proof of special relativity: Never properly tested against other velocity-dependent force laws (none was considered in 1923, and Frauenfelder-Weber show one that gives the same l-dependent term). - The "truest value" of h and e from the spectroscopic route: Tested, not supported (outside its own error bars after Millikan's systematic was found); the wide-bar values: Established as consistent. - α = 7.295e-3: Established (agrees to 0.03%).

Claim 7. The book's own limits and unexplained facts (1921-22 state of knowledge)

a. Mathematics. These are claims about the status of knowledge. The numbers were reproduced: He first ionisation 28.75 V, double ionisation 82.9 V, H2 binding 0.20 Rh = 2.71 V, H2+ energy 0.88 Rh [read: Note 14 checks]; Mercury 7.16" (special) and 42.98" (general) per century [computed]; the γ-ray scattering value 0.04 against 0.2 is quoted from the literature, not derived. Note 13's expansion has wrong n-powers in eqs. (10)-(12) (checked numerically) and the author reports Weinacht's unpublished finding that it does not always converge.

b. Consistency with experiments. The negative statements are correct: He observed 24.59 eV against the model's 28.75 V [memory, high]; H2+ binding came only from wave mechanics (Burrau 1927) [memory, medium]; the γ-ray scattering falls below the classical Thomson value at high photon energy, which Compton (1923) and Klein-Nishina (1929) explain, so the author recorded an experimental anomaly one or two years before it was explained [memory, high]. Pertinent: in the 1924 letter he calls carbon's p ground term "ein harter Schlag für Bohrs ... Modellfolge" and tells Landé the regularities "reach much further than the model pictures lead one to expect" [web: TransDat/02183 and sites.md]. The Weiss/Bohr magneton section gets O2 and NO "right" (8.7 and 13.7 against 9 and 14) for the wrong reason (O2 has spin S = 1, giving about 14.1 Weiss magnetons) [memory, medium]; the author hedges ("very doubtful").

c. Tested? The model failures were tested by electron-impact ionisation and spectroscopy (the book's Ch. VI-VII, digest only here).

d. If true. Nothing; these are the reasons the old theory was replaced.

Verdict. The statements that the Bohr models of He, H2, H2+ and the orbit models of heavier atoms fail: Tested, not supported for the models (i.e. the author's negative claim is Established). The statement that the series and Zeeman regularities are "mainly empirical": Established as a description of 1923 status. The γ-ray anomaly entry: Established as an unexplained anomaly in 1923, later explained.

4. Strongest honest case for the book and for its central formula

  1. A single formula, derived in 1915-16 from a quantised classical Kepler problem with relativistic mass, gives exactly the energy levels that Dirac's equation produced twelve years later, and the n = 2 interval of 0.365 cm^-1 is correct (computed 0.3650 from the Dirac formula). The old theory also gave a complete, correct list of Stark components for hydrogen and a correct level scheme and selection rules for X-rays. A reader who dismisses the whole as "disproved old quantum theory" has to explain why the numbers survive.
  2. The pattern prediction (1:3, 1:2:6 intensities and positions) was a risky prediction matched on the He+ plate.
  3. The author printed his own failures (Bohr's objection conceded; "hopeless"; model failures; retractions of two positions; unobserved Stark components) and flagged the shaky steps ("not inevitable", "certainly rather risky"). That is the standard a reviewer should apply to the critics too.
  4. The dissenting measurements were not hidden; they sit in a footnote table.

5. Same rigour applied to the author and the book

  • Overstatement: "absolutely perfect" agreement for Hγ and Hδ, "striking confirmation", "whole of the experimental facts", "truest value" of h (1.2% low, ~9 sigma by the narrow bars). Language outruns the evidence in each case (except where the author hedges elsewhere).
  • Circular steps: Paschen's value fits the theory's own scheme; Epstein corrected Stark's field from the theory; "stray electric fields" invoked afterward with no independent field measurement.
  • Honest bad data not caught in print: Table 54, Table 53 two rows, 5584 not reproduced from the constants given, α² = 5.32e-5 not following from e and h, "four octaves", ζ Puppis and Wolf-Rayet "nebulae", mean free path wording, units "gauss x cm" and the silent esu/emu switch in the magneton section, Note 13 and Note 17 numbers wrong as printed [read: deep read]. None changes a physical conclusion, but a reader copying the Note 13 or 17 formulas would be misled.
  • Unfalsifiable-looking moves: the ether defined to be nothing more than the optical field; components "forbidden" are present if field and absent if not.
  • Same rigour for the critics: Gehrcke and Lau measured correctly but interpreted a blended maximum-to-maximum separation as the doublet; their stated conclusion (the theory's value is 25% too high) was not correct, though the data were. Their anti-relativism is recorded as their stake; the data are weighed on merit.

6. Investigator's findings carried into this review (kept whatever the verdict)

Priority 1, inventor's own work: Sommerfeld to Paschen 23 May 1924 (doubts the absolute size 0.365 but not the number and position of components; Lau's He+ analogues; "Die Theorie dazu werden wir schon nachliefern"; Abraham "indiscutabel" and the "fanatischen Geist der Lenard, Gehrke, Lau") [web: TransDat/02183]; 29 Dec 1915 draft (ratios "stimmt vortrefflich", absolute values do not; 0.102 against 0.24) [web: TransDat/00585, per Investigator; not re-read]; the 1915 Sitzungsbericht and 1916 Annalen treatise (saved, not read in full by me except via citations); other editions (German 1921 on archive.org, probably 1922 on HathiTrust, English 1934). Effect: claim 2 gets a documented, honest private caveat; claim 7 gets the carbon ground-term remark.

Priority 2, support by others: Paschen 1916 (friendly help; "agree everywhere most beautifully", May 1916 per Eckert and Kragh); Sommerfeld puzzle literature (Vickers 2018; Suslov 2024; Barley et al. 2025; Biedenharn 1983); Eckert 2013-14 (book as "bible", over 4,000 copies in a year; doublet riddle); Kragh 2025.

Priority 3, related work: Ishiwara and Wilson (independent discovery of the quantum condition, claim 1); Glitscher 1917 (claim 6); Frauenfelder and Weber 2019 (claim 6: same l-dependent fine-structure term from Weber's force law).

Priority 4, historical record: Gehrcke and Lau 1921; the anti-relativity controversy letters (Scheel 1920, Laue 1920, Sommerfeld to Einstein 1920, Volkmann 1925); Houston, Hsieh, Gibbs, Williams, Pasternack (1933-38) and Lamb-Retherford; Nobel nominations (84 for Sommerfeld, 1926 jointly with Paschen; nomination texts not read); records not obtained: Kragh 1985, Robotti 1986, Lau 1924 report, the Williams 1938 and Houston papers, the German 1921/1922 editions (to check the misprint list), Sommerfeld edition ASWB I-II letters.

Absence of evidence: Lau's 1924 report (cited in the letter) is not in the files; the claim "higher He+ analogues show smaller Δν with ratio 1:3" rests on Sommerfeld's letter alone (one documented source; no second source found). It is evidence, quoted above, not to be dismissed. Competing explanations for a smaller Δν in higher He+ analogues that I can state: Stark or pressure effects in those lines, unresolved components (the same blending mechanism as hydrogen), or a real departure; the record that would decide is Lau's report and the plates.

7. What would settle what is left

  • The residual 0.01-0.04 cm^-1 between my simulated blends and Gehrcke-Lau: a line-profile fit of their published intensity curves (Figs. 5-7 and the tables in Ann. Phys. 65, 564) with the seven-component Dirac-plus-Lamb model and measured temperature, allowing unequal populations. This uses their data; no new experiment is needed.
  • Whether the Hα and Hβ unobserved and extra Stark components of Stark 1913-14 are real: a modern high-field Stark-Balmer spectrum compared with correspondence-principle or wave-mechanical intensities.
  • Whether any independent dynamics (Weber or other) is excluded by the fine structure: the l-independent α²/n⁴ coefficient, or a non-hydrogenic test; Frauenfelder and Weber themselves say the difference is hard to measure.
  • Whether the 1924 "smaller Δν at higher He+ analogues" is a real effect or blending: Lau's 1924 report.

8. Verdict summary

  1. Quantisation of orbits: Established (energy rule for H-like atoms); Historically superseded but instructive (orbit picture, counting, orientations); Tested, not supported (orbit models for He, H2, heavier atoms).
  2. Relativistic fine structure: Established (formula and 0.365 cm^-1 doublet; the 1921-23 direct shortfall largely a blend artefact); Historically superseded but instructive (mechanism, selection principle for n, intensity rule, stray-field story).
  3. Wave and quantum pictures joined by conservation laws: Established (h/2π per circular photon, Δm = ±1 circular); Plausible, untested (the Δn = 0 derivation); Tested, not supported (no recoil); Not testable as stated (the "ether").
  4. Electric and magnetic effects: Established (first-order Stark of hydrogen; normal Zeeman, equations unread); Never properly tested (the few unobserved or extra Hα and Hβ components); Plausible, untested (the 1923 proof of the adiabatic hypothesis).
  5. X-ray spectra: Established (regular and irregular doublet laws, level scheme); Tested, not supported (orbit models of screening).
  6. Relativity versus the absolute electron, constants: Tested, supported relativity over Abraham; Never properly tested against other velocity-dependent dynamics (Weber); Tested, not supported ("truest" h and e narrow bars); Established (α).
  7. The book's stated limits: Established (the negative statements, the empirical status of series regularities, and the γ-ray anomaly as an anomaly of the time).