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Summerfield (Arnold Sommerfeld): central claims

Naming. The project folder is spelled "Summerfield". The one document is Arnold Sommerfeld's Atomic Structure and Spectral Lines (English translation by H. L. Brose of the third German edition, Methuen, 1923). This is a mainstream textbook of the old (Bohr-Sommerfeld) quantum theory, not a work of an outsider. Later steps should say so, and should judge it against what was known in 1921-23 as well as against today's physics.

Source gap (partial; updated after QE 2026-10-09). The PDF has no text layer. All 324 pages were OCR'd. Prose is mostly readable, but almost every equation, table and subscript is garbled. Each PDF page is a two-page spread of the book. Therefore: - Cleaner text exists: the Internet Archive OCR of the same 1923 edition (research/files/Sommerfeld-1923-AtomicStructure-IA-...djvu.txt) is far more legible than the folder OCR. The arbiter read Ch. V sections 6-7 (PDF pp.157-166, Zeeman effect and adiabatic hypothesis) in it, so those are no longer digest-only. After QE, Paschen's 1916 paper "Bohrs Heliumlinien" (not a manifest document) was deep-read; its numbers appear in claims 2 and 6. - Every numerical claim below comes from page images read by the deep readers, who recomputed the numbers (SymPy/NumPy). They cover PDF pp.107-156 (Ch. IV; Ch. V part 1), pp.236-275 (Ch. VIII), and pp.276-318 (Mathematical Notes 1-17). - QE revision 2 (orchestrator fix, 2026-10-09): the QE-ordered deep reads were done (Sonnet, IA OCR plus page images): PDF pp.29-32 and 41-45 (Ch. I sections 6-7, Ch. II section 3), pp.54-58 (Ch. II section 6), pp.76-106 (Ch. III sections 4-7, Moseley's law and X-ray doublets), pp.167-216 (all of Ch. VI and its Addendum) and pp.250-257 (Ch. VIII section 4 footnotes). Files: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch1-2.md, .ch2-s6.md, .ch3.md, .ch6-part1.md, .ch6-part2.md, .ch8-notes.md. Claim 1's electron-impact checks, claim 5's Moseley's law and the peripheral claims below now rest on these reads. - Digest only, no deep read: PDF pp.1-28, 33-40, 46-53, 59-75 (Ch. I sections 1-5, Ch. II sections 1-2 and 4-5, Ch. III sections 1-3) and pp.216-235 (Ch. VII band and continuous spectra). No central claim depends on them. - Recommended further deep reads (Sonnet, IA OCR with page images for tables): (a) done by the arbiter for the text of PDF pp.157-166 (eqns (14)-(16) not re-derived); (b) done in QE revision 2: PDF pp.167-216 (Ch. VI), inner quantum numbers, Landé formula and the doublet-atom model; (c) PDF pp.80-106 (Ch. III), Moseley's law and the absorption-edge tables; (d) optionally PDF pp.216-235 (Ch. VII). - Reviews should state plainly when a conclusion rests on a digest or on [memory] alone. The book's printed misprints and the digest corrections are listed at the end.

The central claims

  1. Quantisation of orbits. Allowed atomic states are fixed by one quantum condition on each coordinate of the motion (the integral of p dq over a cycle equals n h). For a hydrogen-like atom quantised in polar coordinates, the energy depends only on the sum n + n' of the azimuthal and radial numbers, W = -2 pi^2 m e^2 E^2 / (h^2 (n + n')^2); orbits with n = 0 are excluded, orbits are Kepler ellipses, and their orientation in space is also quantised (cos alpha = n1/(n1+n2)). Sommerfeld himself states the gaps: the choice of coordinates is not justified, the Kepler problem is degenerate (parabolic coordinates give different orbits; Bohr's objection is conceded), and the exclusion of n = 0 and n1 = 0 rests on physical arguments, not derivation. Independent electron-impact checks of the Z^2 energy formula for He+ (Ch. VI section 3): predicted second-order ionisation of He 79.3 V, observed 79.5 +/- 0.3 V (Franck and Knipping) and 80 V (Horton); predicted excitation of the 4686 line printed 75.4 V (the book's own arithmetic slip; (15/16) x 54 = 50.6 gives 75.9 V [computed]), observed absent at 75 V and at full intensity at 80 V (Rau). [read, Ch. IV deep read; electron-impact figures: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch6-part1.md, page image PDF p.182, book pp.344-345]

  2. Relativistic fine structure. Because electron mass varies with velocity, quantised Kepler ellipses precess. The resulting energy formula (with the fine-structure constant alpha = 2 pi e^2/(hc), alpha^2 = 5.3e-5) splits n = 2 hydrogen into a doublet of R alpha^2/16 = 0.365 cm^-1, n = 3 into a triplet (1:3), n = 4 into a quartet (1:2:6), n = 5 into 3:5:10:30. Evidence offered: Paschen's photographs of the 12-component He+ 4686 pattern ("a striking confirmation"), Paschen's indirect 0.3645 +/- 0.0045 cm^-1, and X-ray L-doublets. Counter-evidence printed in the book itself: direct hydrogen doublet measurements give 0.27-0.33 cm^-1, all below 0.365 (the book says "partly explained"; the arbiter's seven-component blend recomputation indicates the shortfall is largely explainable as an unresolved blend, apparent splitting 0.26-0.32 cm^-1, with the Gehrcke-Lau residual open; Paschen himself wrote in 1916 that his measured H-alpha spacing, 0.121 A against 0.157 A from theory, is "wahrscheinlich ... zu klein" because the edges of the blurred components overlap, and the book names Doppler and Stark broadening as the cause of the blurring); the crude intensity rule is "contradicted by experience"; the weak "forbidden" components are attributed to stray electric fields; Paschen's value was obtained by fitting the theory's own interval scheme. [read, Ch. VIII deep read]

  3. Wave and quantum pictures joined by conservation laws. The atom emits light only in a single act with h nu = W_a - W_e (Bohr). Energy, momentum and angular momentum are all conserved between the atom and the "ether" (the optical field, which Sommerfeld defines as having "no properties other than such as light itself possesses"). A spherical wave of energy h nu can carry at most the angular momentum h/2 pi, so the azimuthal quantum number changes by at most one unit (selection principle) and the emitted light is circularly polarised for Delta n = +/-1 and linear for Delta n = 0 (Rubinowicz 1918). The wave theory is used for propagation as a stated assumption ("repress all doubts"), the atom is assumed not to recoil, and Sommerfeld says the wave and light-quantum views each give "only one-half of the truth". The weak points he states: the Delta n = 0 case is "not inevitable", one step is "certainly rather risky", 2 of 7 determining elements of a spherical wave (including coherence length) remain unfixed, atom and ether cannot be wholly decoupled in stationary orbits (he concedes this in a footnote), and the rule is "rescinded" in dense discharges and weakened in fields. [read, Ch. V part 1 and Note 9]

  4. Electric and magnetic field effects. The first-order Stark splitting of hydrogen, Delta W = -(3 h^2 F/(8 pi^2 m E)) (n2 - n1)(n1+n2+n3) in parabolic coordinates, gives the component positions and polarisations observed by Stark and Lo Surdo ("the whole of the experimental facts" for hydrogen; perfect agreement for H-gamma and H-delta). Admitted mismatches: for H-alpha the predicted components at Delta = 5, 6, 8 were not observed (Sommerfeld says they are faint, citing Kramers' correspondence-principle intensities), and for H-beta four observed components are not predicted; the field scale was corrected by Epstein from the theory itself (107,000 against Stark's measured 104,000 V/cm). The normal Zeeman triplet follows from Larmor precession of the quantised orbit (h cancels), and the adiabatic hypothesis (quantum numbers invariant under infinitely slow changes) links the Stark and Zeeman orbits. [read, Ch. V deep read for Stark; Zeeman and adiabatic hypothesis read by the arbiter in the IA OCR (Ch. V sections 6-7, arbiter section 0); adiabatic proof is in Note 12 with the degeneracy gap stated by Sommerfeld]

  5. X-ray spectra mirror the hydrogen fine structure. Nuclear charge, not atomic weight, orders the elements (Moseley). The regular X-ray doublets follow Delta nu = Delta nu_H (Z - s)^4 with one fitted screening number s = 3.49 over Z = 41-92, "a gigantic magnification" of the hydrogen doublet by up to 7e7. Irregular doublets obey Hertz's law (sqrt(nu/R) differences independent of Z). The full K/L/M/N/O level scheme of 21 levels is built with Wentzel's complete selection principle, and Kossel's combination defects are explained as irregular doublets. Sommerfeld states the limits: the screening numbers cannot be explained by orbital models ("hopeless"; 105 atomic species would be needed), he retracts several positions of earlier German editions (at least five in the book: L1/L2 levels in different atoms; Kossel's defects; the "A-doublets", owed to "a false interpretation by the author", with the extra L-limits they implied, "refuted by Hoyt's measurements"; theta = x'; and his first triplet Runge-denominator scheme, "erroneous"), and the L-doublet formula is "an empirical equation ... brilliantly confirmed", not derived. Moseley's law as printed is nu/R = (3/4)(Z - 1)^2 for K-alpha (slope 0.866, screening number 'almost equal to 1'), which the author says is not exact at high Z (curvature assigned to relativity); a modern measurement over Z = 29-92 confirms the relativistic curvature [read, Ch. VIII part 2 deep read; Moseley: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch3.md, book pp.156-158; web: arXiv 1809.10480]

  6. Relativity versus the "absolute" electron, and the universal constants. The fine structure is offered as a spectroscopic "experimentum crucis" for relativity: the rigid-sphere (Abraham) electron would reduce every fine-structure interval by 4/5 (hydrogen doublet 0.29 cm^-1), which "is in no wise compatible with Paschen's measurements". From the hydrogen doublet, R_H and R_He he derives e = 4.766 (1 +/- 1/54) e-10 esu, h = 6.526 (1 +/- 5/162) e-27 erg s (bars "unsatisfactorily wide"); with Millikan's e = 4.774 +/- 0.004 he gives h = 6.545 +/- 0.009 e-27 as "the truest value". The deep reader notes the 4/5 test needs an independent alpha, and that modern values lie inside the wide bars but outside the narrow ones. The part of the fine structure that separates relativity from Weber's law is the circular-orbit (l-independent) term, which is zero in Weber's law and -alpha^2/(8 N^4) hartree in relativity [computed, QE]. The book (p.~488) says Paschen found deviations from the Balmer formula "of the order of magnitude of our relativity correction R alpha^2/64 ... and bear the correct sign"; Paschen 1916 gives N_He(4686) - N_He(4339) = 0.569 +/- 0.25 cm^-1 against 0.590 calculated (Weber 0, Abraham 4/5 of 0.590 = 0.472). Caveats: a single source; Paschen worked with the theory's creator; the error bar is a linear-sum possible error that omits the assignment of one weak 4339 component; and because Paschen took alpha from the same He+ splitting, this test cannot separate relativity from Abraham (Abraham is excluded only through the absolute scale, alpha with e and h; Glitscher 1917: alpha^2 5.32 +/- 0.22 e-5 for relativity, 6.65 +/- 0.28 e-5 for Abraham, against 5.30 e-5 from e and h). Sommerfeld himself wrote "If, then, Paschen has not under-estimated the accuracy of his measurement". Sommerfeld's draft of 29 Dec 1915 found absolute Balmer differences off by more than a factor of two, counter-evidence kept in the record. [read, Ch. VIII deep read; deep/Paschen-1916 read; IA OCR lines 42967-42984, 47878-47882]

  7. Honest limits and unexplained facts as the book records them. Bohr's models of He (28.75 V against 25.3 V observed; printed 28.8/25.4 in Ch. II and 28.7/25.3 in Ch. VI), H2 and H2+ are untenable; models of heavier atoms are "provisional"; the series and Zeeman regularities (Ch. VI) are "mainly empirical" and the model interpretation is doubtful; Bohr's 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) for reasons Sommerfeld ascribes to "the size of the energy quantum"; the expansions for non-hydrogenic term constants are not always convergent (Weinacht, unpublished). These are claims about the state of knowledge in 1921-22 and are kept here because they are pertinent evidence of how the theory was held. [read, Ch. V part 1 and Notes 2, 10, 12-14]

Peripheral claims (not central; deep-read in QE revision 2 except where marked digest)

  • Nuclear charge = atomic number (Chadwick 1920: 77.4, 46.3, 29.3 for Pt, Ag, Cu); radioactive displacement laws; isotopes (Aston) [digest, Ch. II]. He nucleus as 4 H nuclei plus 2 electrons with mass defect 0.03 per gram-atom; the book prints the heat of formation as 6.3 x 10^9 major calories, ten times the arithmetic value of 6.4 x 10^8 [computed]; Lenz's "inverted oxygen molecule" model, reported "without committing ourselves"; Rutherford's disintegrations: N, B, F, Na, Al, P shattered, Li, Be, Mg, Si, Cl, K uncertain, none from C, O, S, Ca, Ti, Mn, Cu, Sn [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch2-s6.md, page images PDF pp.56-58]. Lenard's dynamides [digest].
  • Series spectra, inner quantum numbers (n_i -> n_i +/- 1 or n_i; 0 -> 0 excluded by a convention from Landé), spectroscopic displacement law (spark spectrum like the arc spectrum of the preceding element, R replaced by 4R), Wechselsatz, anomalous Zeeman effect (Preston, Runge, Landé; splittings 2, 4/3, 2/3, 6/5, 4/5, 8/7, 6/7 of the normal), Paschen-Back effect, doublet-atom model (valency electron plus a trunk with half a Bohr magneton), Franck-Hertz excitation voltages [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch6-part1.md, .ch6-part2.md]. Selection-rule breaks blamed on fields with no field strength quoted: anomalous He series under deliberately applied fields (Stark, Liebert, Hardtke); b-p combinations ("quite possible"); He 1S-2S at 585 A ("presumably"); Na/K 1s-3d at high current density; Zeeman components at 2 and 3 times normal (Ch. V, Bohr's "unintended" electric field); Paschen's forbidden lines "called forth by a magnetic field". The Hg 1S-2p1 and 1S-2p3 collision steps are excused because the rule "restricts only ... monochromatic emission" [read: same; IA OCR lines 28030-28100]. Fues's half-integer s-term constants are rejected by the author (book p.377), although his Addendum with Heisenberg uses half-integer phase integrals and magnetic numbers m*; the half-integer angular momenta later vindicated by spin are Landé's j and m [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch6-part2.md; web: arXiv 2203.07833]. Doublet-atom numbers Li 0.32 cm^-1 (p, compared with Kent's 0.306 and 0.326 and 0.34) and 0.035 cm^-1 (d, no measurement in the book: a prediction) [read: same, page image PDF p.214].
  • Reported effects and alternatives set aside, as the book records them: Barkla's 1917 J-radiation ("Repeated tests by other observers have ... not been able to confirm"; "Nor has the theory a place for such radiation"; footnote Richtmyer, Phys. Rev. 17, 433, 1921) [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch3.md, book p.145]; Nebulium, Coronium and Protofluor, lines that "seem to hint at elements having an atomic weight of 2 or 3", excluded by the nuclear model; the book does not identify the lines (Bowen 1927: doubly ionised oxygen; Grotrian and Edlén: highly ionised iron) [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch1-2.md, book pp.70-71; web: https://en.wikipedia.org/wiki/Nebulium, https://en.wikipedia.org/wiki/Walter_Grotrian]; the Sommerfeld-Debye accumulation theory (1913), "impossible 'accumulation times' ... of the order of magnitude of years" for X-rays against immediate emission [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch1-2.md, book pp.44-45].
  • Records named by the book but not examined: Birge, Phys. Rev. 17, 589 (1921), cited only for testing "again very carefully in detail" that the weak satellite does not shift the centre of gravity of the long-wave H-alpha group (no Birge number given) [read: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch8-notes.md, book p.487]; Birge's alpha^2 = 5.308e-5; Grossmann, Z. Phys. 5, 280 (1921) on Newcomb's Mercury numbers; Franck and Knipping, Phys. Zeitschr. 20, 481 (1919); Horton, Proc. Roy. Soc. 95, 408 (1920); Hoyt, Proc. Nat. Acad. 6, 639 (1920).
  • Band spectra: rotation-vibration of HCl, missing zero line, cyanogen bands, hydrogen many-lines spectrum, continuous spectra and electron capture. [digest, Ch. VII]
  • Planck's h from the photoelectric effect (6.57-6.58e-27) and the X-ray short-wave limit (6.53-6.55e-27); Weiss and Bohr magnetons with Pauli's correction (NO, O2 at 8.7 and 13.7 against 9 and 14 observed). [digest and Ch. IV deep read]

Corrections to earlier documents

The deep readers recorded these corrections to the Step 1a digests; facts.json uses the corrected readings. No other earlier review file exists yet for this project.

  • Digest .10, p.113: the second Paschen line is 12817.5 A (not 19817.5) and the Lyman base line 1215.7 A (not 1015.74). Digest .11, p.122: R_He = 109722.144 +/- 0.04 and R_H = 109677.691 +/- 0.06 (the digest labelled the first as R_inf); R_inf = 109737.11 is derived.
  • Digest .12-.13: the Stark intense p-components lie "on the outside" (not "red side"); the quantum sum is n1+n2+n3 (not "n1+n2+m+1"); N_z carries the factor h/2 pi; eqn (4) coefficients are legible (Delta nu = (3hF/8 pi^2 mE){...}); digest .13 azimuthal letters are psi, p_psi. The first Ch. V deep-read section wrote the field term with "me E"; the page images and Note 11 give "mE" (and 2 pi^2 for the Rydberg term, so the book has a misprint there).
  • Digest .21 (Ch. VIII): eqn mismatches are OCR; the triplet intensity rule reads 3:2:1 (not 4:1).
  • Digest .23, p.272-273: e = 4.766e-10 comes from eqn (10) with limits +/-1/54 (not "short-wavelength limit of X-rays, +/-5 alpha"); R_inf = 109737.11 (not 109787.11); e/m0 comes from R_H and R_He (not electrochemical data); alpha = 7.290e-3 is from the doublet and 7.295e-3 only from Millikan's e plus the spectroscopic h; the Kossel defects are M3 - M2 and N3 - N2; there is no sidebar box.
  • Digests .24-.26 (Notes): Note 1 not "Note 4" for the radiation note; the Mercury eqn is 4 pi^3 a^2/(tau^2 (1-e^2) c^2) (24 is the general-relativity factor); Note 12 credits Burgers, not Epstein; the double-ionisation 82.9 V is unnumbered; antimony angle is printed "563 deg 6 min".
  • Book misprints found by the deep readers, for the record: Ch. IV: q_max factor 2, "p = phi" for q = phi, b = a sqrt(2) for a/2, "four octaves", Haas "1900", unit "gauss x cm" for the magnetic moment; Ch. V: "i.e. pi = 0", "a = 0 and b = 0", Table 27 row 111-101, Table 29 N values, Epstein page 498; Ch. VIII: -3/8 beta^4 for +3/8, alpha^4/4 for alpha^2/4, 191/32 for 2807/1024 in eqn (14), "regular"/"irregular" in the definition of m, "increases with the third power" for Delta lambda, Table 54 L3 - L2 entries for Pt/Au/Pb/Bi/U; Notes: Note 4 eqn 22c sign, Note 6 eqns 27 and 29 sign, Note 2 low-frequency sign, Note 11 eqn 28 sign, Note 13 eqns 10-12 n-powers, Note 15 eqn 6 constant, Note 17 sign of +4c xi^3 and of the b^2 term, cerium c:a 1.62 for 1.633; the translation slip "the higher the temperature the lower the velocities due to heat motion" (book p.~485, Doppler broadening; should read "higher") [read: IA OCR lines 42415-42416]. None changes a physical conclusion; the Note 13 and Note 17 numbers are wrong as printed. Whether these appear in the German original was not checked.
  • QE (2026-10-09): digest .16 p.182 prints the second-order ionisation potential of He as 79.8 V; the book prints 79.3 V (25.3 + 54) [read: IA OCR line 31197].
  • QE (2026-10-09): the earlier statement that the circular-orbit shift "was not measurable then" (physicist, arbiter) is wrong as a statement of the record: the book (p.~488) and Paschen 1916 report it (0.569 +/- 0.25 against 0.590 cm^-1), with the caveats in claim 6. The Glitscher figures flagged "need checking" are confirmed (claim 6). Sommerfeld's "harter Schlag" sentence is in his letter to Paschen of 23 May 1924, not to Lande.
  • Paschen 1916 deep read: OCR misreadings corrected from page images (4686 scheme shift 0.003 A not 0.008; H-alpha components 6562.853 and 6562.732 A; Table I e/mu 1.760 and 1.753).
  • QE revision 2 (orchestrator fix, 2026-10-09), corrections to earlier review files: (1) the withdrawals list (claim 5) conflated the "A-doublets" with their "false interpretation" (one episode, disproved by G. Hertz's L-absorption limits) and attached "refuted by Hoyt" to the A-doublets, where the book attaches it to the extra L-limits Lambda1, Lambda2; the triplet Runge-denominator scheme ("erroneous", book p.392) is a further withdrawal, so "at least five" stands; (2) Birge 1921 was described as a retest of the H-alpha components, which the book does not say; (3) "Fues's half-integers ... later proved right" overstated the record (see Peripheral claims); (4) the Hg collision lines are 1S-2p1 and 1S-2p3, not 2P0/2P2 [read: QE revision 2 deep reads; arbiter section 12].
  • Book misprints found in QE revision 2: He+ 4686 excitation printed 25.3 + 50.1 = 75.4 V, arithmetic 50.6 and 75.9 V (book p.345); He heat of formation printed 6.3 x 10^9 major cals, arithmetic 6.4 x 10^8 (book p.95); "doubly ionised helium" for lithium in the Li++ paragraph (book p.71); "1S - 2s" for 1S - 2S at 21.25 V (book p.349); Ann. d. Phys. 41 dated 1918 in the folder OCR for 1913; Bohr's He model printed 28.8/25.4 (Ch. II), 28.7/25.3 (Ch. VI), 28.75/25.3 (Note 14); "odd" for "even" valency (book p.363) [read: QE revision 2 deep reads].
  • Folder name: "Summerfield" is a misspelling of Sommerfeld. The project name is kept for the folder; reader-facing documents should use "Sommerfeld".