Physicist review (Step 2): Summerfield (Arnold Sommerfeld), Atomic Structure and Spectral Lines

Reviewer: PhD physicist role. Date: 2026-10-09. Source folder: source folder: Summerfield (read only).

Tags: [read] = taken from the document or a saved file; [web] = from a web page (URL given); [memory] = recalled, with confidence; [computed] = I calculated it (Python, numpy).

0. Read this first

Naming. The folder is spelled "Summerfield"; the author is Arnold Sommerfeld. The document is the 1923 English translation (H. L. Brose, Methuen) of the third German edition (preface January 1922) [read]. This is the standard textbook of the old quantum theory (the Bohr-Sommerfeld theory, 1913-1925), not the work of an outsider. I judge it twice: against what was known in 1921-23, and against today's physics. The two judgements often differ, and the difference is the lesson.

Source gap (repeated from claims.md). The PDF has no text layer; all pages were OCR'd and most equations are garbled. What I rely on: - Page-image deep reads (by earlier agents, who recomputed numbers): Ch. IV, Ch. V part 1 (Stark effect), Ch. VIII (fine structure, X-ray doublets, constants), Mathematical Notes 1-17. - Digest only, no deep read: Ch. I-III (electrons, nuclei, Moseley's law, crystal structure) and PDF pp.157-235 (Zeeman effect, adiabatic hypothesis, Ch. VI series spectra, Ch. VII bands). My statements about the Zeeman effect, the adiabatic hypothesis, series spectra and the doublet-atom model rest on digests of garbled OCR plus [memory]; I say so each time. The Investigator saved a single-page scan of the same translation (research/files/Sommerfeld-1923-AtomicStructure-IA-atomicstructures0000somm.pdf), which would make those deep reads easier. - Where I say "the book says", I mean the deep readers' page-image quotation, which I took as read (I checked their arithmetic where noted, not every page image myself). I did read the saved primary sources myself where marked.

What I checked myself. The Gehrcke-Lau 1921 passage (p.575 of Ann. d. Phys. 65), Paschen's 1916 wording, Sommerfeld's letter to Paschen of 23 May 1924, the Frauenfelder-Weber arXiv paper's formulas, plus a set of numerical checks (section 9).

1. Short verdict table

2. Corrections to Step 1

Step 1 (claims.md, facts.json, the deep reads) is careful and mostly right. I found no error that changes a verdict. The points below sharpen or correct it. "Correction" means the earlier text should be changed; "addition" means pertinent material to carry forward.

1. Correction (small), claims.md claim 6 / deep read Ch. VIII section 4.7. The deep reader writes that the 4/5 test "needs an independent alpha". True, but the book itself supplies one on the same pages: Millikan's e = 4.774e-10 esu plus Planck's h = 6.55e-27 erg s give alpha^2 about 5.32e-5, hence a relativistic doublet of 0.365 cm^-1 and an Abraham doublet of about 0.292 [computed: 0.8 x 0.365 = 0.292]. So the test is genuine at roughly the 1-2 per cent level of those inputs. What the claim leaves out is that it then rests on Paschen's He+ value, which he obtained by fitting the theory's own interval scheme (section 4 below).
2. Addition, Ch. VIII section 4.2 deep read (direct doublet data). The direct hydrogen H-alpha values are not all "unexplained shortfall": they lie between the relativity-plus-line-blending prediction and the Abraham prediction. Relativity with Sommerfeld's selection lag gives 0.329 cm^-1; the same lag applied to the Abraham doublet would give 0.292 x (1 - 8/81) = 0.263 [computed]. Observed H-alpha: Michelson 0.32, Fabry-Buisson 0.306, Meissner-Paschen 0.288, Gehrcke-Lau 0.272 [read: footnote table in the deep read]; mean 0.297 [computed], close to the midpoint 0.296 of the two predictions. The 1921 direct data therefore did not discriminate. That is why the He+ route carried the argument, and why Gehrcke (an anti-relativist) could read his own data as support.
3. Addition/clarification, deep read Ch. V section 5.2 vs Ch. VIII section 4.4. The symbol n (azimuthal quantum number) plays two roles with two different modern mappings: in the selection rule Delta n = +/-1, n behaves like l + 1; in the energy formula n corresponds to j + 1/2 [memory, high]. The deep readers state both mappings but not that they differ. This is why the book's count of 5 allowed He+ 4686 components disagrees with the modern 8 (the deep reader's inference, which I confirm is consistent with the Dirac rules Delta l = +/-1, Delta j = 0, +/-1 [memory, medium]).
4. Refinement, deep read Ch. VIII section 4.1. It attributes the small excess of the measured 2P3/2 - 2P1/2 interval (about 0.3659 cm^-1) over 0.3650 to "QED (Lamb shift) effects". The Lamb shift proper is the 2S1/2 - 2P1/2 splitting (about 1058 MHz = 0.035 cm^-1), which is not in the Dirac formula. The 2P3/2 - 2P1/2 excess (about 0.0007 cm^-1, 0.2 per cent) comes mainly from the electron's anomalous magnetic moment [memory, medium-high; not web-checked]. My value of the formula: R alpha^2/16 = 0.36503 cm^-1 with R_H, 0.36524 with R_inf [computed]. No verdict changes.
5. Addition, claims.md claim 4 / deep read Ch. V section 6.6. Sommerfeld says the field cannot be measured "hardly to within 1 per cent", yet the field inferred from the theory (Epstein) differs from Stark's measured one by 3 per cent (107,000 against 104,000 V/cm: (107-104)/104 = 2.9 per cent [computed]). The deep reader notes the circularity; the 3 per cent is also larger than the author's own stated 1 per cent uncertainty. See claim 4.
6. Addition, claims.md claim 5. "Magnification of up to 7e7" is 92^4 = 7.16e7 [computed]. With the book's own screening (Z - 3.5)^4 = 6.13e7, and the actual L-doublet of uranium (3.06e7 cm^-1) is 8.4e7 times the hydrogen doublet 0.365 [computed from Table 52]. The extra factor 1.37 is the higher-order relativistic terms, which the book includes. This is not an error, only a difference between three numbers that a reader will meet.
7. Addition, from the Investigator, claims.md claim 2: the saved 1924 letter and the Gehrcke-Lau data show Sommerfeld himself separating "structure secure" from "size 0.365 not equally secure" two years after the book (section 4).
8. Confirmed, not corrected: the deep readers' misprint list (e.g. "alpha^4/4" for alpha^2/4, 191/32 for 2807/1024, the "four octaves" slip) and their corrections of the Step 1a digests. I checked the numerical items I reuse: alpha, Z^4 values, the 0.365 and 0.329 figures, the Stark scale C, the error propagations in section 4.4 of the Ch. VIII deep read (1/81 -> 1/54, 5/162; 5/3 x relative error in e) [computed].
9. No error found in the deep readers' Note 9 check (N/W = h/2 pi for circular polarisation) or Note 16 (Mercury 7.16" and 42.98" per century [computed in the deep read; formula 4 pi^3 a^2/(tau^2 (1-e^2) c^2) agrees with general relativity's with factor 24, the ratio 6 is exact]).

3. Claim 1: Quantisation of orbits

What the claim says. For each coordinate of a periodic motion, the "phase integral" (the integral of momentum p over its coordinate q around one cycle, written as the contour integral of p dq) equals a whole number times Planck's constant h. For hydrogen in polar coordinates this gives energy W = -2 pi^2 m e^2 E^2 / (h^2 (n + n')^2), where n is the azimuthal number (angular momentum, in units of h/2 pi) and n' the radial number; orbits with n = 0 are excluded; orientations are quantised by cos alpha = n1/(n1+n2).

The mainstream view, stated fairly. Quantum mechanics replaced this rule, but did not simply discard it. The rule is the leading approximation of the WKB (semiclassical) method, in a refined form called EBK quantisation that includes a "Maslov index" (a half-unit shift). Hydrogen's spectrum is exact in the old theory because two errors cancel: the correct angular momentum is sqrt(l(l+1)) h/2 pi, about (l + 1/2) h/2 pi in the semiclassical limit (not l + 1), and the correct radial number is shifted by 1/2; their sum n_r + l + 1 gives the exact principal quantum number [memory, high]. So the formula is exactly right and the picture behind it is not. Keppeler's semiclassical analysis finds that the Maslov index and the electron spin cancel each other in the fine-structure formula too; this is cited in Frauenfelder and Weber 2019 [web: research/files/arXiv-1902.09612, introduction; secondary description of Keppeler].

a. Is the mathematics internally consistent? Yes within its stated assumptions, with gaps the author states. The deep readers checked every step of Ch. IV and Notes 6-8 (the radial integral by contour integration, 1 - e^2 = n^2/(n+n')^2, energy independent of the angle, all checked with SymPy), and found only misprints (q_max with a stray 2; "b = a sqrt 2" for a/2). The conceded gaps: - The choice of coordinates in "the integral of p dq" is not justified. Hydrogen's non-relativistic Kepler problem is degenerate (the energy depends only on n + n'), so polar and parabolic coordinates give different quantised orbits with the same energy. Bohr's objection is conceded as "incontestable" [read: Ch. IV deep read, section E.3]. Sommerfeld's defence is that relativity lifts the degeneracy and picks polar coordinates (Note 8). That defence is an appeal to a later chapter, not a proof. - Exclusion of n = 0 ("permanence of atoms", the electron would fall into the nucleus) and of n1 = 0 (for the equatorial orbit) are physical arguments, not derivations; n1 = 0 is "taken for granted" until Ch. V. - The a priori probabilities (all orientations equally likely) are an assumption. - A hidden restriction: p = dE_kin/dq-dot fails when forces depend on velocity (magnetic fields); the general definition needs the Lagrangian. The author concedes it only for forces without a potential.

I add one observation. The limit F -> 0 of the Stark (parabolic) orbits does not equal the polar-coordinate orbits (Ch. V: "only one of our two methods of quantising can be physically true"). In quantum mechanics there is no conflict: the degenerate level is a multiplet, and parabolic and spherical states are two different bases of the same multiplet. A field defines the right basis. So Sommerfeld's two statements ("polar for weak fields, parabolic for strong fields, with the split set by comparing the Stark splitting to the relativistic doublet") are essentially correct: they describe which basis diagonalises the perturbation [memory, high].

b. Consistent with established experiments? The hydrogen spectrum (Balmer, Lyman, Paschen series; R = 109,677.7 cm^-1) and He+ (nuclear charge 2) follow; the book's Table 24 matches Balmer lines to about 1e-5 relative in air wavelengths [read: Ch. IV deep read, B]. The counting of states disagrees with experiment: the old theory gives s(s+1)/2 orbital states for principal number s (3 for s = 2), reality has s^2 orbital states (4), twice that with spin [memory, high]; and it assigns hydrogen's ground state a non-zero angular momentum (one unit), whereas the true ground state has l = 0 [memory, high]. The old theory fails for any atom beyond hydrogen (claim 7).

c. Has it been tested? By whom? Replicated? The rule was proposed independently by Sommerfeld, W. Wilson (Phil. Mag. 1915) and J. Ishiwara (1915); the book credits Wilson (completed November 1915) and Bohr's account credits Ishiwara [read: Ch. IV deep read, A.5; web: arXiv 1708.04676]. Tests were spectroscopic (Paschen, Fowler, Pickering series reassigned to He+; Stark and Zeeman effects) and mostly successful for one-electron atoms. Replicated by many laboratories. Failed for helium, molecules, and anomalous Zeeman effects; superseded by matrix and wave mechanics in 1925-26 [memory, high].

d. If it were true, what in mainstream physics would have to change? If the orbit picture were literally true: classical orbits with sharp radii and angular momentum would exist (in conflict with the uncertainty principle), the ground state would carry angular momentum h/2 pi, and electrons in n = 0 states could not exist. The measured s-wave states (l = 0) would be absent. For hydrogen's energy numbers nothing needs to change: they are right.

Verdict. Split. Hydrogen-like energy formula: Established. "Integral of p dq = nh" as a fundamental law, planar elliptic orbits, spatial quantisation counting and the n = 0 and n1 = 0 exclusions: Historically superseded but instructive (reasonable in 1916, replaced since; survives as EBK semiclassics).

4. Claim 2: Relativistic fine structure

What the claim says. The electron's mass grows with speed, so a quantised Kepler ellipse does not close but precesses (a "rosette"). The energy depends on both quantum numbers: 1 + W/(m0 c^2) = {1 + (alpha E/e)^2 / [n' + sqrt(n^2 - (alpha E/e)^2)]^2}^(-1/2), with alpha = 2 pi e^2/(h c) ("fine-structure constant", about 1/137; alpha^2 = 5.3e-5). For n + n' = 2 in hydrogen this gives a doublet of R alpha^2/16 = 0.365 cm^-1 (about 11 GHz); n = 3 a triplet (intervals 1:3), n = 4 a quartet (1:2:6), n = 5 (3:5:10:30). Evidence offered: Paschen's photographs of He+ 4686 (12 components expected), Paschen's indirect value 0.3645 +/- 0.0045 cm^-1, and X-ray L-doublets (claim 5).

The mainstream view. The formula is correct and is exactly the Dirac (1928) result: with Sommerfeld's n read as j + 1/2 (j = total angular momentum), the energies are identical [computed: I evaluated both for n = 2 and alpha = 7.2974e-3 and they agree to the printed 7 digits, -6.6564e-6 and -6.6565e-6 in units of m c^2 for the two levels respectively]. The underlying picture is wrong: electrons have spin, the orbits are not classical rosettes, and in Dirac's theory two levels (2S1/2 and 2P1/2) share an energy that Sommerfeld's scheme splits differently. This coincidence is called the "Sommerfeld puzzle": Heisenberg called it a "miracle", Schroedinger "a fortuitous computational accident" [web: arXiv 2401.07485 abstract]; Biedenharn (1983) traced it to an underlying symmetry [web: https://pmc.ncbi.nlm.nih.gov/articles/PMC7418904/]. Kragh: "Sommerfeld's celebrated fine-structure formula was correct for the wrong reasons. ... whereas the theory was wrong the energy formula deduced from it was not" [web: https://www.ilnuovosaggiatore.sif.it/article/402]. Heisenberg and Jordan (1926) found the new mechanics needed both relativity and spin to reproduce it [web: same page]. Experiment later found the Lamb shift (2S1/2 about 1058 MHz above 2P1/2, 1947), which Dirac's formula lacks [web: https://en.wikipedia.org/wiki/Lamb_shift].

a. Is the mathematics internally consistent? Yes. The deep reader reproduced the expansion coefficients (1/4, 1/8, 5/64; 5/4, 21/8, 429/64), the component ratios (1:3, 1:2:6, 3:5:10:30) and the exact formula [read + computed]. My own check: R alpha^2/16 with the book's alpha^2 and R gives 0.36475 cm^-1; with modern constants 0.36503 (R_H) [computed]. A one-line numeric error: alpha^2 = 5.32e-5 does not quite follow from the printed e and h (they give 5.29e-5), a 0.6 per cent rounding that changes the doublet by 0.002 cm^-1, inside the quoted +/-0.0045 [read, computed in the deep read]. One internal inconsistency: Sommerfeld's special-relativistic treatment of Mercury gives 7" per century against 43" observed, and he says so; only general relativity fits [read: Note 16; computed 7.16" and 42.98"].

b. Consistent with established experiments? The pattern (number and position of components) and the formula: yes, for He+ and, indirectly, hydrogen. The absolute size of the hydrogen doublet is where the 1920s record is not clean: - The book itself prints direct measurements: Michelson H-alpha 0.32, H-beta 0.33; Fabry-Buisson H-alpha 0.306; Meissner-Paschen 0.288; Gehrcke-Lau 0.272 (H-alpha), 0.283 (H-beta), 0.271 (H-gamma) cm^-1, every one below 0.365. Sommerfeld says the difference is "partly explained" by the lag from which components are strong (his selection principle leaves three of the six H-alpha lines; the measured interval is then IIb - Ia = 0.329 [computed]). The remaining shortfall (Gehrcke-Lau, Meissner-Paschen) is 0.04-0.09 cm^-1 below that, and the book leaves it [read: Ch. VIII deep read section 4.2]. - I read the Gehrcke-Lau passage myself: "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. ... uns in Anbetracht der Groesse der gefundenen Abweichung ein Irrtum bei der Messung nicht ersichtlich ist; uebrigens fanden auch frueher Beobachter Werte fuer Delta nu, die sich den von uns gefundenen gut annaehern." [read: research/files/Gehrcke-Lau-1921..._djvu.txt, p.575, lines 6116-6123]. 0.272/0.365 = 0.745, so "about 25 per cent" is right [computed]. Gehrcke was a prominent anti-relativist (Investigator: de.wikipedia); that is his stance, recorded beside his data. The data stand or fall on their own. - What explains the shortfall? Doppler broadening is large: at 300 K the H-alpha Doppler width is about 0.19 cm^-1 (0.08 Angstrom), at 77 K about 0.095 [computed]; the doublet 0.33 cm^-1 is then barely resolved and its apparent peaks depend on source conditions and the unresolved sub-components (the modern H-alpha has seven fine-structure components, not three). A simple two-line model with those widths keeps the peaks at the true separation [computed], so unresolved blending of just two lines does not by itself account for 25 per cent; the shortfall needs the other components and the discharge fields (Stark effect in the tube, which the book mentions). I could not derive the 1920s numbers from first principles here, and no source I have does. After 1925, Houston and Hsieh (1933) found H-alpha-region splittings "far too small" by about 3 per cent [web: Lamb shift article, which names Houston and Hsieh 1933, Gibbs and Williams, and Pasternack 1938]. So later, cleaner measurements shrank the discrepancy from about 25 per cent to a few per cent, and the few per cent was real physics (the Lamb shift). That is evidence that the early direct data were dominated by experimental difficulty, but I did not find a source that explains the 1921 numbers specifically (open item). - The He+ evidence: Paschen's spark photograph of 4686 shows the predicted pattern (fused pairs, a strong II-b component), and the book calls it "a striking confirmation". The book also records: the crude intensity rule is "contradicted by experience" (II-b is the strongest component); the weak "forbidden" components are attributed to unintended fields (Kramers estimates 100 V/cm in the direct-current tube and 300 V/cm in the spark); and Paschen's 0.3645 +/- 0.0045 was obtained by using the theory's interval scheme to read the plate [read: Ch. VIII deep read, 4.4]. Bohr is quoted as saying it was "a piece of good fortune" that the selection principle failed in Paschen's photographs (otherwise the components used would not have appeared) [read].

Investigator's findings bearing on claim 2 (kept whatever my verdict). - Paschen 1916, in his own words: the hydrogen lines taken at face value give e/mu = 1.74 against the known 1.76 +/- 0.01; "Eine Klaerung der Unstimmigkeiten brachte die neue Theorie von A. Sommerfeld"; "mit freundlicher Hilfe ihres Schoepfers" the helium and hydrogen material was brought "in genuegende Uebereinstimmung" [read: Paschen 1916 djvu text, lines 4232-4262 and 4251-4258; the 0.3645 +/- 0.0045 value and its "Gleichstrombild/Funkenbild" weighting at lines 4534, 4835-4848]. Two readings: (i) a clean experimenter-theorist collaboration that resolved a real discrepancy; (ii) the data were interpreted through the theory they were then said to confirm. Both are consistent with the text. What decides: the plate and an independent reduction of it. - Paschen to Sommerfeld, May 1916: "agree everywhere most beautifully" (Kragh/Eckert, [web]); Sommerfeld to Paschen, 29 Dec 1915 (draft): ratios of Balmer-line differences agree "vortrefflich", but "die Absolutwerte nicht" (0.102 predicted against 0.24 observed) [web: https://sommerfeld.userweb.mwn.de/TransDat/00585.html; Investigator's reading]. - Sommerfeld to Paschen, 23 May 1924, which I read in full [read: research/pages/trans02183.html]: "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." He suggests the mass formula, derived for the isolated electron, might need an interaction term with the nucleus that would shrink the doublet. He calls the Abraham alternative "natuerlich indiscutabel" and its proponents (Lenard, Gehrcke, Lau) "fanatisch", but adds that Lau's demonstration that for the higher analogues of 4686 the Delta nu values stay noticeably smaller while their ratio becomes very exactly 1:3 made him "sehr nachdenklich", dismisses McLennan's "brilliant confirmation", and urges Paschen to obtain new, more exact values of the fine-structure constant. The letter is private and two years after the book. It supports a careful reading of the public text: "partly explained" was meant. - Kragh (2025): "the agreement was far from perfect and questioned by a minority of physicists who disliked either Bohr's quantum atom or Einstein's relativity theory, or both ... it did not affect the consensus view" [web: Il Nuovo Saggiatore]. The standard study of the dispute is Kragh 1985 (HSPS 15:67-125, paywalled, not obtained) and Robotti 1986 (not obtained). - Frauenfelder and Weber (2019), read by me: applying Sommerfeld's method to Weber's electrodynamics (a 19th-century action-at-a-distance force law with velocity dependence, superseded by Maxwell) gives levels -1/(2n^2) - alpha^2/(2 n^3 l) + (1/2) alpha^2/n^4, against Sommerfeld's -1/(2n^2) - alpha^2/(2 n^3 l) + (3/8) alpha^2/n^4 (atomic units, to second order in alpha). The part that depends on l, which sets the splitting between levels of the same n, is identical; the two differ only in a term depending on n alone, alpha^2/(8 n^4) hartree [read: arXiv 1902.09612, introduction, formulas 1.2 and Theorem A]. In wavenumbers that is R alpha^2/(4 n^4), which for n = 2 is 0.091 cm^-1 [computed]. This matches the order of the "circular-orbit relativity correction" R alpha^2/64 that Sommerfeld's book says is "much less accessible experimentally" and that it claims Paschen found with the right sign and size [read: Ch. VIII deep read 4.3]. The authors themselves say the difference "seems difficult to actually measure". Consequence for the reader: the fine-structure splitting of 1916-1924 cannot tell special relativity from Weber's law; only the overall level shift (absolute Balmer wavenumbers) could, and that was not measured at the needed accuracy then. Modern evidence excludes Weber's law for other reasons (it does not reproduce radiation, particle accelerators or the Dirac spectrum with spin) [memory, high; my statement, not from the paper].

c. Tested? By whom? Replicated? Tested by Paschen (He+, 1916), Michelson, Fabry-Buisson, Meissner-Paschen, Gehrcke-Lau (hydrogen, 1910s-20s), McLennan (1924, per Sommerfeld's letter, reliability questioned by him), Hansen and Jacobsen (Zeeman of He+, 1921), Houston-Hsieh and Williams (1933-38), Lamb and Retherford (1947) [web for the last three: Wikipedia Lamb shift; memory for Hansen and Jacobsen from the book]. Replication of the pattern: yes. Replication of the absolute doublet size: not in the 1920s; achieved to about 3 per cent by 1933, to parts per million by radiofrequency work after 1947 [memory, high: 2P3/2-2P1/2 about 10,969 MHz].

d. If true, what must change? Nothing: it is true, as a formula. The conceptual change is that "relativistic mass increase of a classical ellipse" is not the correct reason; spin-orbit coupling and the Thomas factor in the Dirac theory replace it. The agreement with the Dirac formula is the surprise (see Keppeler above).

Verdict. Split: level formula and pattern, Established. 1920s direct hydrogen doublet (0.27-0.33 against 0.365): Tested, not supported at the time; later measurements agree with the formula to a few per cent and then better; the 25 per cent 1921 shortfall is not explained by any source I have (open). Crude intensity rule: Tested, not supported. Mechanism (spinless mass variation): Historically superseded but instructive.

5. Claim 3: Wave and quantum pictures joined by conservation laws

What the claim says. Light is emitted in one act with h nu = W_a - W_e (Bohr). Energy, momentum and angular momentum are all conserved between atom and "ether", where "ether" means only the optical field. A classical spherical wave carrying energy h nu has an angular momentum N = (W/omega) x 2ab sin(gamma)/(a^2 + b^2) (Note 9), at most h/2 pi when the wave is circularly polarised (a = b, gamma = +/- pi/2). Equating this with the change in the atom's angular momentum (Delta p = (h/2 pi) Delta n) gives |Delta n| <= 1 (the selection principle) and circular light for Delta n = +/-1, linear for Delta n = 0 (Rubinowicz 1918).

The mainstream view. The content is right and is standard: a photon carries spin angular momentum h-bar (= h/2 pi) and an electric-dipole transition changes the atom's orbital angular momentum number l by exactly +/-1 (Delta l = +/-1, Laporte parity rule) with magnetic quantum number change 0 or +/-1 giving linear or circular light [memory, high]. The derivation differs: the modern rule follows from parity and the dipole operator, and conservation of total (atom + photon) angular momentum, with the photon treated quantum mechanically. There is no recoil-free atom; photon recoil of an atom exists and is measured (Frisch 1933) [memory, high], though it is tiny (about 1e-9 of the photon energy for optical lines). The "ether" in the book is explicitly not a material medium: "has no properties other than such as light itself possesses" [read].

a. Mathematics consistent? Note 9 is correct step by step (deep reader: Hertz-vector solution, far fields, angular-momentum flux, N/W = 1/omega for circular polarisation) [read, checked]. The classical result N/W = 1/omega is standard for a circularly polarised wave [memory, high]. Weak points the author states himself: the exclusion of Delta n = 0 is "not inevitable"; the move for n_a = n_e uses an assumption he calls "certainly rather risky" (the initial orbital plane fixes a preferred axis); the proof is for a single Kepler electron and is "extended" to the general case without proof; two of the seven determining elements of a wave (including the coherence length) are unfixed ("5/7ths ... solved"); decoupling of atom and ether between transitions is conceded to fail in a footnote. In a field, only the field-direction component is conserved, so the rule is weakened and, in dense discharges, "rescinded" (his word). That qualification is correct physics: strong local fields (the Stark effect in a plasma) mix states of different l.

b. Consistent with experiments? Selection rules in series spectra and in Stark/Zeeman polarisation: yes (the deep reader's modern mapping works). The book's claim that a circular orbit "can never transform into a circular orbit" for H-beta onward is a consequence of the rule and is, in modern terms, an artefact of the orbit picture. The statement "Delta n = 0 gives no emission" is right for a single electron in a central field (parity) but his reason (no preferred direction) is not why. The prediction that unpolarised free atoms give unpolarised light "eludes experimental proof" in field-free cases, which is also what the modern theory says.

c. Tested? The rule was tested in series spectra (inner quantum number rules, Ch. VI; digest only, not deep-read), Stark and Zeeman polarisation (deep-read), and Bohr's correspondence principle independently (Bohr 1918, Kramers 1919: intensities "agree remarkably well", per the book). Sommerfeld himself judges the correspondence principle "completely superior" for polarisation and intensity. Replicated: yes, widely [memory, high].

d. What would have to change? Nothing. A recoil-free atom and a classical coherent spherical wave are approximations; the book itself says the wave and quantum views each give "only one-half of the truth". Einstein's directed-quantum picture (needing recoil) was not rejected by the author, only set aside as an assumption.

Verdict. Selection rule and polarisation rule: Established (as Delta l = +/-1). The derivation from a classical spherical wave: Historically superseded but instructive.

6. Claim 4: Stark and Zeeman effects, adiabatic hypothesis

What the claim says. In an electric field F the hydrogen energy shifts to first order by Delta W = -(3 h^2 F / (8 pi^2 m E)) (n2 - n1)(n1 + n2 + n3) in parabolic coordinates (n1, n2 parabolic numbers, n3 equatorial number). Component positions and polarisations then follow from the selection rules and match Stark and Lo Surdo's observations ("the whole of the experimental facts" for hydrogen). The normal Zeeman triplet follows from Larmor precession of the orbit. The adiabatic hypothesis (an infinitely slow change of an external field keeps the quantum numbers) links Stark and Zeeman orbits to the field-free ones.

The mainstream view. First-order Stark splitting of hydrogen: correct and exactly the quantum-mechanical result, with the energy shift (3/2) e a0 F n (n1 - n2) (a0 = Bohr radius) found by Schroedinger and Epstein in 1926 [read: deep-reader SymPy check of the coefficient; memory, high for the equality]. Larmor's theorem: correct classical result for the normal Zeeman effect (shift of +/- e B/(4 pi m c) in Gaussian units; it contains no h) [memory, high; digest only for the book's treatment]. Adiabatic invariance of action integrals: a theorem of classical mechanics (Ehrenfest 1916; Burgers 1917), valid except when the system passes through degenerate states or resonances [memory, high].

a. Mathematics consistent? Yes: separation in parabolic coordinates, Hamiltonian, first-order energy (checked symbolically by the deep reader), and the Tables 27-34 of components (every printed Delta reproduced by script; two clerical slips: Table 27 row 111 -> 101 should be 111 -> 011, Table 29 rows' N signs swapped) [read]. Misprints: the sign of beta in Note 11 eqn (28), Note 4 (22c). For the adiabatic argument (Note 12): the author's proof sets the rate of change "constant" as a choice, and the author states that it "falls to the ground" for degenerate systems; near-resonances (small divisors) are not treated [read; the gap is real, and was closed only by later KAM-era mathematics [memory, medium]].

b. Consistent with experiments? The Stark pattern: perfect for H-gamma and H-delta, for H-alpha "as far as Delta = 4" agree; predicted components at 5, 6 and 8 were not observed (the book says the intensities are faint, citing Kramers's dissertation: "completely confirmed"). For H-beta four observed components (p at Delta = 4 and 12, s at 0 and 8) are not in the theory; the author suggests they may be "unreal or produced by a secondary effect". Quantum mechanics predicts the same position set as the old theory to first order; the observed extra components are therefore a question of intensities and second-order or field-inhomogeneity effects, not of the formula [memory, medium]. The scale: at the measured field 104,000 V/cm the unit interval is C = 3 e a0 F / (2 h c) = 6.66 cm^-1, i.e. 2.87 Angstrom per unit at H-alpha and the outermost observed H-alpha component (Delta = 4) lies 11.5 Angstrom from the line [computed; matches the deep reader's value]. Circularity: the absolute field cannot be measured to better than "hardly 1 per cent"; Epstein therefore corrected it from the theory ("trusting, justifiably, in the truth of the theory"), 107,000 against 104,000 V/cm. The 3 per cent difference is bigger than the author's stated field uncertainty (section 2, item 5). So the position of the lines relative to each other is a test; the scale relative to the field is a consistency check at about 3 per cent (a measured field and a derived field of the same order), not independent confirmation of the coefficient. Independent field calibrations in modern Stark spectroscopy confirm the coefficient [memory, medium].

c. Tested? Stark (1913-14), Lo Surdo (1914), then Epstein and Schwarzschild's theory (1916), Takamine and Kokubu (second-order effect, cited by the book via Sommerfeld's own 1921 paper). Replicated many times; the Stark effect of hydrogen is textbook-level [memory, high].

d. What would have to change? Nothing for the Stark effect. For the adiabatic hypothesis as an explanation of why the quantum numbers stay: modern quantum mechanics replaces it by the adiabatic theorem, which has the same restriction (no level crossings). The old theory could not handle the Zeeman anomalous case without spin; this lies in Ch. VI (digest only, not reviewed here beyond [memory]).

Verdict. Stark first order, hydrogen: Established. Normal Zeeman: Established. Adiabatic hypothesis: Established for non-degenerate systems (book states the exception); as the physical reason the quantum numbers are what they are: Historically superseded but instructive.

7. Claim 5: X-ray spectra mirror the hydrogen fine structure

What the claim says. Moseley's law: the square root of an X-ray line's frequency rises linearly with the atomic number Z. The doublet splitting of the L shell follows Delta nu = Delta nu_H (Z - s)^4, with one fitted screening number s = 3.49 over Z = 41-92 ("gigantic magnification" of the hydrogen doublet by up to 7e7). Irregular doublets obey Hertz's law (the difference of square roots of the energies is independent of Z). A 21-level K, L, M, N, O scheme is built with Wentzel's selection rule; Kossel's "combination defects" are level differences.

The mainstream view. All this is correct and permanent, with a changed reading. The regular doublets are today's spin-orbit (fine-structure) doublets (same l, different j), which scale as (Z - s)^4; the irregular doublets are screening doublets (same j, different l); Hertz's law is the standard statement for these [memory, high]. Sommerfeld's Table 57 maps onto modern notation as n = j + 1/2 and ground number m = l + 1 [read: deep reader; memory, high]. The ordering of subscripts is reversed from modern usage (his L1 is today's L3). Wentzel's selection rule (n changes by 0 or -1, m by -1 or +1) are the dipole rules Delta j = 0, +/-1 and Delta l = +/-1.

a. Mathematics consistent? Yes, with a handful of printing slips the deep reader found and recomputed: eqn (14) coefficient 191/32 should be 2807/1024; Table 54 L3 - L2 entries for Pt, Au, Pb, Bi, U inconsistent with Table 55 (the root differences, the quantities argued from, are consistent); a printed definition that says "irregular" where "regular" is meant; "increases with the third power" for Delta lambda should be "inversely" [read]. The Z^4 scaling and the 5/2, 53/8 coefficient expansion were verified symbolically; Table 52's 18 rows were recomputed to 0.2 per cent. The mean of the 38 screening numbers in Table 53 is 3.492 with spread 0.043 [computed by the deep reader; two rows disagree (Os)].

b. Consistent with experiments? Moseley (1913) and the Siegbahn school (K, L series), Coster (precision L-series), Hjalmar (M doublets), C. Hertz (L2L3 for Cs-Nd), Smekal, Duane and Stenstroem, de Broglie, Dauvillier are cited as the experimental base [read]. Modern X-ray absorption edges agree at the 0.2 per cent level with Table 55 (for U, L3 modern 1262 against 1264.3 in ν/R units; K 8497 against 8477) [deep reader, memory, medium]. One theoretical caution: for Z near 90, alpha Z = 0.67; the point-nucleus formula still works for the L shell, but the K shell of heavy elements is sensitive to nuclear size [memory, medium].

c. Tested? Replicated? Many independent laboratories, over several years, by crystal spectrometers. The relation "Delta nu_H = Delta nu_L / (Z - 3.5)^4 rises from 0.38 to 0.50 cm^-1" is explained by the higher-order relativistic terms: I checked that the first two correction terms give about 1.33 times the leading term at Z - s = 88.5 against the observed 1.37, the rest being still higher terms and the choice of s [computed, rough]. The extrapolation of Delta nu_H(Z) back to small Z approaches 0.365 (Fig. 118), the best argument that the same physics runs from hydrogen to uranium. Important: in 1921 the X-ray doublets were explained relativistically and the optical doublets magnetically (the "doublet riddle"); Millikan's 1924 ultraviolet data showed both relativistic, and Sommerfeld wrote to Landé that the relativity formula "far from being obsolete ... extends its validity into the optical realm" [web: Eckert chapter, research/2026-10-09-mprl-eckert.md; Investigator]. That riddle (relativistic doublets in alkali atoms) is the next chapter and is historically important for the discovery of spin.

Honesty in the book. Sommerfeld says the orbital models of screening are "hopeless" (grouped ellipses would need "105 different species" of atom, and a calculation shows K and L doublets would then differ by up to 45 units in ν/R at W against the observed 0.2); the L-doublet formula "cannot ... be regarded as an equation derived from theory ... it stands as an empirical equation that has been brilliantly confirmed"; and two positions of earlier German editions are retracted in footnotes (grouped ellipses; Kossel's reading) [read]. I note these here because retraction and explicit empirical status are exactly what a reader should expect of a reliable textbook.

d. What would have to change? Nothing. The reinterpretation does not alter a single number.

Verdict. Relations and level scheme: Established. Orbit models of the screening: Historically superseded but instructive (the author abandons them). The statement that the L-doublet "is a macroscopic copy of the microscopic hydrogen doublet" is a fair description of the Dirac-theory Z^4 law.

8. Claim 6: Relativity against the rigid electron; universal constants

What the claim says. If the electron is a rigid sphere (Abraham), its mass grows as m0 (1 + (2/5) beta^2) at small speeds beta = v/c, against the relativistic m0 (1 + (1/2) beta^2). Every fine-structure interval would shrink by 4/5 (the hydrogen doublet to about 0.29 cm^-1), "in no wise compatible with Paschen's measurements". From the doublet, R_H and R_He he derives e = 4.766 (1 +/- 1/54) e-10 esu and h = 6.526 (1 +/- 5/162) e-27 erg s; adding Millikan's e = 4.774 +/- 0.004 he gives h = 6.545 +/- 0.009 as "the truest value".

The mainstream view. The rigid-electron theory is dead: the electron has no classical size, and the relativistic mass-velocity law is confirmed to high accuracy by accelerator and storage-ring work [memory, high]. The fine-structure comparison is a legitimate test, because the fine-structure splitting is proportional to the product (coefficient of beta^2) x alpha^2; the 4/5 is the ratio of the two coefficients [computed]. It is a weak test when alpha is taken from the doublet itself, and a strong one when alpha comes from elsewhere.

a. Mathematics consistent? The algebra is right: alpha-dependence e proportional to alpha^3, h to alpha^5 (propagation 1/81 -> 3/2 x 1/81 = 1/54 -> 5/2 x 1/81 = 5/162) [read, checked by the deep reader; I re-verified 0.0045/0.3645 = 1/81 = 0.0123 [computed]], and the semi-spectroscopic h = cube root of (2 pi^2 e^5 / (R_inf c (e/m0))) with error 5/3 x relative error of e = 0.0014, hence +/-0.009 [read, checked]. Two simplifications: only one error source per route (Paschen's doublet; Millikan's e) is propagated, R_inf and e/m0 treated as exact; ±0.005 for alpha in eqn (15) is not derivable from those inputs alone (deep reader: Millikan's error alone gives ±0.002e-3). The book states the pure route's bars as "unsatisfactorily wide" and says "If, then, Paschen has not under-estimated the accuracy of his measurement of Delta nu_H, the idea of purely spectroscopic units must for the present remain an intention."

b. Consistent with established experiments? Compare with modern values [memory, high; CODATA-type values], checked as follows [computed]:

Millikan's e was low by about 0.6 per cent, because of the value of air viscosity he used (Feynman's comment, quoted in the Wikipedia article: "a little bit off because he had the incorrect value for the viscosity of air") [web: https://en.wikipedia.org/wiki/Oil_drop_experiment]. Since h scales as e^(5/3), an error of -0.6 per cent in e gives -1.0 per cent in h [computed]; this is the reason the narrow bars fail and the wide bars hold. The error was never in the arithmetic; it was an unrecognised systematic. alpha came out well because e and e/m0 errors nearly cancel in alpha proportional to (e (e/m0))^(1/3) [computed in the deep read: (0.9939 x 1.0058)^(1/3) = 0.9999].

c. Tested? By whom? The relativity-versus-Abraham question was posed by W. Lenz; Glitscher's Munich thesis (1917) carried it out against Paschen's helium lines [read: Glitscher 1917 opening; web: research/files; Investigator]. Independently: Kaufmann (beta rays, early), Bucherer, Schaefer and Neumann (1914), Guye and Lavanchy (1916) are cited by the book for direct mass-velocity measurements [read]. In 1921-23, a second line of evidence (the direct hydrogen doublet) was in tension with the 0.365 value, near the midpoint between relativity and Abraham (section 2, item 2). Gehrcke (Abraham/anti-relativity side) and Lenard, Weyland are named in the Investigator's letters as opponents of relativity in a public dispute [web: Sommerfeld-Projekt summaries; Investigator], and Sommerfeld was in the other camp as DPG chairman. These are the parties' positions; the data are weighed separately. Strength of the book's argument for the crucial test: Paschen's 0.3645 +/- 0.0045 is a fit of the He+ pattern; the independence needed is from e and h, which carried about 1-2 per cent errors. The 20 per cent Abraham factor is well outside that. So the He+ test did exclude the rigid-sphere formula with its alpha; it did not exclude other velocity-dependent dynamics (Weber, section 4) or Sommerfeld's own 1924 suspicion that the mass law might be modified by interaction with the nucleus.

d. What would have to change? If Abraham were right: all the relativistic mass-velocity tests, special relativity, the Dirac equation and its precision agreement with hydrogen to parts per billion would have to be wrong. Not credible.

Verdict. The conclusion (relativity, not Abraham): Established today; the 1923 demonstration as stated: Historically superseded but instructive. e, h from spectroscopy: pure route consistent with modern values; "truest value of h": Tested, not supported (the narrow error bar excluded the truth; section 8b).

9. Claim 7: The book's own limits

What the claim says. (Records in the book.) Bohr's He model gives 28.75 V against 25.3 V observed; H2 and H2+ models are untenable; heavier-atom models are "provisional"; series regularities are "mainly empirical"; the correspondence principle is "completely superior" for polarisation and intensity but ambiguous about initial versus final orbit; gamma-ray scattering falls below the classical value (0.04 against 0.2), "probably due to the size of the energy quantum"; non-hydrogenic term expansions are not always convergent (Weinacht, unpublished).

The mainstream view. All of these are accurate reports. He first ionisation is 24.59 eV [memory, high: NIST], so the book's 25.3 V is close and the model's 28.75 V is wrong. The old theory has no way to treat two-electron atoms, as the three-body problem is not separable, and the ring models of heavier atoms are dynamically unstable (Note 14, Nicholson's stability argument) [read]. Gamma-ray scattering below the Thomson value is exactly what the Compton effect (1923) and later Klein-Nishina formula (1929) explain [memory, high]: at high photon energy the cross-section falls below the classical value. Sommerfeld's note, written one or two years before Compton, is an example of an honest record of an anomaly. The convergence statement (the s-term expansion "never" converges) means that the supplementary-field model reproduces the form of Ritz's formula, not its constants (the author's own wording, Note 13 caveats) [read]. For the Weiss/Bohr magneton: the correction of Weiss's formula via spatial quantisation produces 8.7 and 13.7 against measured 9 and 14 for NO and O2; the modern reason is electron spin (O2: S = 1, mu = 2.83 mu_B = 14.1 Weiss magnetons), so the 1921 agreement was partly coincidental [memory, medium; deep reader].

a-d. (a) Internally, the book flags its own incompleteness. The deep reader's list of errors (24 items in Ch. IV; 5 sign errors in the Notes; the Note 13 and Note 17 numbers wrong as printed) does not change any physical conclusion. (b) Experiments: yes, the unexplained facts are real; later physics explains them with spin, Pauli exclusion, wave mechanics and Compton scattering. (c) Tested: Franck-Hertz and electron-impact experiments (cited in Ch. VI), Lyman and Millikan UV work; replicated. (d) Nothing is required to change; the old theory was replaced.

Verdict. The statements as a record: Established. Bohr's He, H2, H2+ models: Tested, not supported (experiments in the book and later). Models of heavier atoms and series: Historically superseded but instructive. The gamma-ray anomaly: an observation, later explained (Compton).

10. The Investigator's findings (kept in this record whatever the verdict)

• Priority 1, the inventor's own words. 1915 Sitzungsbericht and 1916 Annalen treatise saved; 1924 and 1915 letters to Paschen (above); other editions (German 1921 archive.org, German 1922 HathiTrust, English 1934 archive.org). Not yet used: comparing the German original for the misprints in claims.md (open). The 1916 treatise says Paschen's impulsive discharge "appears to lift the quantum inequalities" and uses it for the "ideal" hydrogen doublet [web via Investigator].
• Priority 2, support by others. Paschen 1916; "Sommerfeld puzzle" literature (Suslov 2024; Barley, Ruffing and Suslov 2025, arXiv 2506.00408; Vickers 2018; Biedenharn 1983); Eckert 2013, 2014; Kragh 2025; Lenz (field letter, 7 March 1916, derived the closed energy formula himself) [web, Investigator]. Weighing: the formula's success is beyond doubt; the explanation of why is debated, not the result.
• Priority 3, related work. Ishiwara/Wilson (independent rule); Glitscher; Frauenfelder-Weber (Weber's law gives the same l-dependence, section 4). The Investigator listed it as "not read"; I read it for the formulas and found the point in section 4.
• Priority 4, historical record. Gehrcke-Lau (dissent, measured, answered privately); Kragh 1985 and Robotti 1986 not obtained; Houston, Williams, Pasternack as the records that tie the shortfall to the Lamb shift (I used a secondary account only); the Nobel nomination record (84 nominations of Sommerfeld, never awarded; 1926 jointly with Paschen) [web: Google snippets via Investigator; I did not read the nominations]. This is the sort of silence the project rules call an anomaly to analyse, not a neutral gap. Competing explanations for "never awarded", none checked: (i) the committee judged the old theory provisional and superseded by quantum mechanics before it could be honoured; (ii) the prize went to the experimental discoveries or to the later theorists (Dirac, Heisenberg, Schroedinger); (iii) politics or personalities. The record that decides: the committee's evaluation reports in the Nobel archive for the years 1919-1951. I did not obtain them.
• Interested parties. Sommerfeld describes his own theory (stake: authorship); Paschen's measurements were interpreted with the theory's author's "friendly help" (stake: his own results); Gehrcke and Lau, Lenard, Weyland opposed relativity (stake: their stated position) and, in Sommerfeld's words in the 1924 letter, acted from "a fanatical spirit"; the letter is Sommerfeld's characterisation of them, written privately, not a neutral description. Millikan's e (stake: his own measurement) carried the systematic error in section 8.

11. Open items

1. The specific 1920s H-alpha shortfall (25 per cent, Gehrcke-Lau) has no explanation in the sources I have beyond the book's "partly explained" (0.329); Kragh 1985, Robotti 1986, Lau's 1924 Bericht, Houston 1933 and Williams 1938 would give the record.
2. The German original of the book (HathiTrust full view; archive.org 1921 2nd edition) should be compared with the English for the misprints listed in claims.md.
3. Chapters I-III (digest only) and VI-VII (digest only) are unreviewed in depth; my views on the Zeeman effect, the doublet-atom model and the series spectra are from digests and memory. A deep read of PDF pp.157-216 would settle them.
4. Whether Sommerfeld's n = l + 1 (selection rule) versus j + 1/2 (energy) is the whole story for the He+ 4686 component count (5 allowed in the book, 8 in the modern assignment) should be checked against a modern He+ treatment before the primer uses it (deep reader's own flag; I did not verify it independently).

12. Web verification log

Budget used: 3 web reads plus the saved Investigator files. - Lamb shift: Houston and Hsieh 1933 found Balmer fine-structure separations "far too small" by about 3 per cent; Gibbs and Williams reported similar results tied to the 2S1/2 level; Pasternack 1938; Lamb-Retherford found 2S1/2 about 1000 MHz above 2P1/2 (observed 1057 MHz) [web: https://en.wikipedia.org/wiki/Lamb_shift]. Retagged from [memory, medium]. - Millikan's oil-drop value 1.5924(17)e-19 C, about 0.6 per cent below the modern value; Feynman attributes it to the viscosity of air [web: https://en.wikipedia.org/wiki/Oil_drop_experiment]. Retagged from [memory, high]. - Not web-checked (memory): anomalous magnetic moment as the source of the 2P3/2-2P1/2 excess; Frisch 1933; Keppeler; the Stark second-order agreement; O2 spin explanation; the doublet-riddle chronology (taken from Eckert via the Investigator).

13. Numerical checks I ran

• R alpha^2/16 with modern alpha and R_H: 0.36503 cm^-1; with R_inf: 0.36524. Book values: 0.36475 (alpha^2 = 5.32e-5, R = 1.097e5).
• Sommerfeld's exact formula and the Dirac formula for n = 2 (j = 3/2 and 1/2) agree to the printed digits (alpha = 7.2974e-3).
• Doppler FWHM of H-alpha: 0.188 cm^-1 (300 K), 0.095 (77 K), 0.77 (5000 K).
• Stark unit C = 3 e a0 F/(2 h c) = 6.658 cm^-1 at 104,000 V/cm and 6.850 at 107,000.
• 92^4 = 7.16e7; (92 - 3.5)^4 = 6.13e7.
• Abraham doublet: 0.8 x 0.365 = 0.292; with the selection lag 0.263. Relativity with lag 0.329.
• Millikan e: 4.774/4.8032 - 1 = -0.61 per cent; implied h shift (5/3 power) -1.0 per cent.
• Frauenfelder-Weber difference: alpha^2/(8 n^4) hartree = R alpha^2/(4 n^4): 0.091 cm^-1 at n = 2.