# Bohr model (bohr-modeli): verification report

File: `simulasyonlar/bohr-modeli.html` (built from `sablon/sablon.html`; CSS and the Q4 core are byte-identical to the current template).
Tests in the Doğrulama tab: **24/24 pass**.

## 1. Audit of the original (`kaynak-eski/bohr-modeli.html`, Gemini 2025)

| # | Problem | Evidence | Correct physics / fix |
|---|---|---|---|
| 1 | UV and IR photons are given rainbow colours. Lyman α (121.6 nm) and all Lyman lines are shown violet. Paschen lines (e.g. 1875 nm) are shown red. | `getColorForWavelength`, l. 77–85: `return "hsl(270…)"` for everything < 450 nm, `"hsl(0…)"` for everything > 635 nm | Non-visible light has no colour. The new version uses `Q4.wlCss` (null outside 380–750 nm) and draws UV/IR as grey dashed lines with "UV"/"IR" labels. A test checks that `wlCss(121.567)` and `wlCss(1875)` are null. |
| 2 | Hβ (486 nm) is coloured blue (hsl 240), not cyan/blue-green; the colour buckets are coarse. | l. 82–83 | The CIE-based `Q4.wl2rgb` is used. |
| 3 | No resonance condition. The flux slider always moves the electron to n+1, whatever the photon energy. Photons have no energy before absorption; the "photon" is created from the transition, not the other way round. | l. 201–203: `const next = Math.min(5, this.n + 1); this.setLevel(next)` | Absorption only if \|hν − ΔE\| ≤ Γ for some n → n′ (n′ ≤ 7). Otherwise the photon passes. Above the ionisation threshold the atom is ionised, with K = hν − Eᵢ. A "photon gun" sets hν explicitly. |
| 4 | Flux from n = 4 jumps to "level 5", which the code treats as ionisation, with a photon of exactly 13.6/16 eV (zero kinetic energy). Excitation to n ≥ 5 is impossible. | l. 202, 230–233 | Levels n = 1…7 plus a continuum. Ionisation can happen from any level, and any surplus becomes kinetic energy. |
| 5 | Arbitrary decay law: rate 0.4/n² s⁻¹. This makes higher levels decay *faster* in relative terms per level, and the rate has no physical basis. | l. 205: `const decayChance = (0.4 / (this.n * this.n)) * (dt / 1000)` | Real lifetimes grow roughly as n^4.5 (2.13, 10.0, 33.1, 86.5, 193, 382 ns for n = 2…7, from NIST A values). The new version uses τₙ = 1/ΣA, with one global, user-adjustable slowdown (default 10⁸) so the ratios stay real. |
| 6 | Spontaneous emission chooses the lower level uniformly at random. | l. 207: `Math.floor(Math.random() * (this.n - 1)) + 1` | Branching ratios are A(n→n′)/ΣA from NIST n→n′ averaged A coefficients. For example, n = 3 gives 55.8 % to n = 1 and 44.2 % to n = 2; n = 7 gives 28.9 % to n = 6. |
| 7 | hc = 1240 eV·nm and E = −13.6/n². Wavelengths are printed as `toFixed(0)` (Hα → "656", but Hβ → "486" and Lyα → "122"). | l. 72, 96, 223, 364 | hc = 1239.84198 eV·nm (CODATA 2018). With the reduced mass, R_H = R∞/(1 + mₑ/mₚ) and Eᵢ = 13.59829 eV. Without it, the Balmer lines are off by 5.4 × 10⁻⁴ (Hα by 0.36 nm), about 25 times the residual fine-structure error. |
| 8 | Ionisation is a dead end. `update()` returns early once ionised, so only Reset recovers the atom. | l. 176 | Recombination: you can capture into a chosen level (manual), and in lamp mode capture happens automatically, with levels weighted ∝ 1/n (Kramers). The photon energy is K + Eᵢ, which gives a continuum below each series limit. |
| 9 | The dropdown is labelled "uyar" (excite), but choosing a lower level also triggers it, and the "photon" just appears. Absorption and emission photons are drawn identically (glowing dots with `shadowBlur`, i.e. neon). | l. 32–39, 318–338 | Kept as "Elektronu bir yörüngeye uyar". Going up now fires the resonant photon from the gun, and going down emits one. Photons are drawn as small wave packets with no glow. |
| 10 | No energy-level diagram, no spectrum, no wavelengths in air vs. vacuum, no lifetimes, no explanation of why Bohr needed postulates (no classical contrast). | whole file | All added. See §2. |
| 11 | Hard-coded dark Tailwind theme loaded from a CDN; `requestAnimationFrame` runs forever even when not visible; orbital angular speed is ∝ 1/n (Bohr: ω ∝ 1/n³). | l. 7, 151, 171, 214 | Template theming; `Q4.loop` stops when the tab is hidden; the visual rotation uses n^−1.5 and is labelled as visual (real period ∝ n³, shown at Lisans). |
| 12 | Text claims "Uyarılmış atom kararsızdır ve … daha kararlı bir hale döner" with no time scale. | l. 63 | Lifetimes and the slowdown factor are shown; the 2s metastability is discussed at İleri. |

## 2. What the new version models

**Bohr atom (hydrogen-like, Z = 1, 2, 3; Z selector at İleri).**
- Eₙ = −Z² (μ/mₑ) R_y / n², with R_y = 13.605693 eV and μ/mₑ = 1/(1 + mₑ/M). The nuclear masses are mₚ, m_α and ⁷Li (atomic mass minus 3mₑ).
- rₙ = n² a₀ mₑ/(μZ), vₙ = Zαc/n, L = nħ, Tₙ = 2πrₙ/vₙ.
- λ_vac = hc/ΔE. For display, wavelengths above 200 nm are converted to air with Edlén's formula as standardised by the IAU (Morton 2000, ApJS 130, 403, eq. 8).
- Levels n = 1…7 plus the continuum.

**Spontaneous emission.**
- Uses hydrogen n → n′ averaged A coefficients (n ≤ 7) from Wiese & Fuhr, JPCRD 38, 565 (2009), Table 4, which are identical to NIST ASD. I also reproduced them independently from the analytic radial integrals with the reduced-mass factor (agreement to 4–5 digits).
- For hydrogen-like ions, A ∝ Z⁴ μ.
- Lifetime τ = 1/ΣA. Branching ratio = A/ΣA. Waiting times are drawn from an exponential distribution.
- Display time = real time × 10^s, with s from 7 to 9.3 (default 8, so 10 ns → 1 s).

**Absorption (photon gun).**
- Deterministic: absorbed if \|hν − ΔE\| ≤ Γ (default 0.01 eV, adjustable 0.001–0.1 eV at İleri; the real natural width of Lyα is 4 × 10⁻⁷ eV and the Doppler width at 300 K is about 10⁻⁴ eV).
- Ionised if hν ≥ Eᵢ(n).
- Stimulated emission if hν matches a downward transition: an identical twin photon leaves in the same direction.
- Optional ±0.6 % "snap" to lines, so the coarse log slider can hit them. A zoomable energy ruler gives fine tuning.

**Lamp (broadband, the "Rastgele foton bombardımanı" slider).**
- Flat photon spectrum per unit energy over 380–750 nm, 10–380 nm or 10 nm–5 µm.
- Event rates ∝ flux × f_lu for absorption, ∝ (g_l/g_u) f_lu for stimulated emission, and ∝ the continuum oscillator strength in the band for photo-ionisation. The continuum shape is Kramers (σ ∝ E⁻³).
- f is computed from A with f_lu = (g_u/g_l) A ε₀mₑc³/(2πe²ν²), using g = 2n².
- The continuum oscillator strengths come from the TRK sum rule (Σf = 1), using the analytic discrete f(n→n′) for n′ ≤ 45 plus an n′⁻³ tail: 0.4350, 0.2308, 0.1568, 0.1187, 0.0955, 0.0798 and 0.0686 for n = 1…7. The n = 1 value matches Bethe & Salpeter's 0.4350.
- Visual photons that do not interact are drawn as well. Physically motivated result: ground-state H is transparent to visible light.

**Classical atom (Larmor).**
- dr/dt = −(4/3)Z rₑ²c/r², so r³ = r₀³ − 4Z rₑ²c t and τ = r₀³/(4Z rₑ²c). For r₀ = a₀, τ = 1.556 × 10⁻¹¹ s.
- The emitted frequency is the orbital frequency, ∝ r^−3/2.
- The spectrum is accumulated exactly: the energy E(r_a) − E(r_b) radiated in each frame is distributed over log-λ bins analytically, using λ(r).
- Starting radius: a₀, 4a₀, 9a₀ or 16a₀. Time is slowed so that the full collapse lasts about 9 s.
- The E(t) plot replaces the level diagram in this mode. Energy that falls outside the spectroscope (< 85 nm) is reported as a percentage.
- The visual spiral is stopped at 0.02 a₀.

**Views.**
- Orbit view, true scale r ∝ n², with automatic zoom.
- Energy-level diagram (Grotrian style): columns per series, recent transitions as arrows, ionisation and recombination column.
- Spectroscope: three panels (UV log-λ, visible linear, IR log-λ, adapting to Z). Line brightness scales with count. Optional NIST reference strip.
- "Anlık bilgiler" keeps every readout of the original: location, energy, incoming/outgoing energy, wavelength, colour (or "görünmez · morötesi") and state. Lisans adds rₙ, vₙ, L, Tₙ and τₙ, and İleri adds R and μ/mₑ.
- Event log. The "Simülasyon hakkında" text is kept in a fold.

## 3. Verification table (all pass, run in the browser on load)

| Test | Expected | Model | Tolerance | Source |
|---|---|---|---|---|
| Hα 3→2, air | 656.279 nm | 656.288 | 5e-5 rel | NIST ASD (observed, n-averaged) |
| Hβ 4→2, air | 486.135 nm | 486.138 | 5e-5 | NIST ASD |
| Hδ 6→2, air | 410.1734 nm | 410.178 | 5e-5 | NIST ASD |
| Lyα 2→1, vacuum | 121.56701 nm | 121.5684 | 5e-5 | NIST ASD; Wiese & Fuhr 2009 |
| Lyman series limit | 1e7/109678.77174 cm⁻¹ = 91.1753 nm | 91.1763 | 5e-5 | NIST ionisation energy of H |
| Balmer limit (air) | 364.56 nm | 364.60 | 3e-4 | Balmer 1885, "Grundzahl" 3645.6 Å |
| Ionisation energy | 13.598434600 eV | 13.598287 | 2e-5 | NIST ASD |
| Reduced mass needed | infinite-mass Hα (655.93 nm) is > 20× further from 656.279 than the μ-corrected value | – | check | NIST |
| a₀ = 4πε₀ħ²/(mₑe²) | CODATA a₀ | computed | 1e-8 | CODATA 2018 |
| v₁ = αc | 2.18769126364e6 m/s | | 1e-9 | CODATA atomic unit of velocity |
| mₑv₃r₃ = 3ħ | 3ħ | | 1e-8 | Bohr's quantum condition |
| Classical collapse time | numerical RK4 integral of dE/dt = −P_Larmor (in ln E) | a₀³/(4rₑ²c) = 1.5562e-11 s | 1e-4 | analytic vs numerical; textbook ≈ 1.6e-11 s |
| Classical λ at r = a₀ | hc/(2R_y) = 45.563 nm | c/f_orb | 1e-6 | analytic |
| He⁺ 4→3, air | 468.570 nm | 468.585 | 5e-5 | NIST He II (blend) |
| He⁺ 7→4 (Pickering), air | 541.152 nm | 541.161 | 5e-5 | NIST Handbook He II |
| R(He⁺)/R(H) | 4.00163 | 4.001629 | 2e-6 | Bohr, Nature 92, 231 (1913) |
| A(2→1) = ¾ A(2p→1s) | 4.6987e8 s⁻¹ | 4.6986e8 | 2e-4 | Wiese & Fuhr 2009 (A(2p) = 6.2649e8) |
| f(Lyα) from A | 0.41641 | 0.41642 | 1e-3 | Wiese & Fuhr 2009, Table 4 |
| f(Hα) from A | 0.64108 | 0.64109 | 1e-3 | Wiese & Fuhr 2009, Table 4 |
| Continuum f, 1s | 0.4350 | 0.4350 | 1e-3 | Bethe & Salpeter (sum rule) |
| Branching ratios sum to 1, n = 2…7 | – | – | check | conservation |
| Correspondence, n = 1000: f_orb/f_photon | 1 | 0.9985 | 2e-3 | Bohr's correspondence principle (deviation ≈ 3/(2n)) |
| Colour: Lyα and 1875 nm uncoloured, Hα red | – | – | check | fix of original bug 1 |
| Energy conservation: (3→2) + (2→1) = (3→1) | – | – | check | conservation |

The remaining 1–1.4 × 10⁻⁵ differences on the measured lines are physical. They come from fine structure, the Lamb shift and relativistic corrections, none of which are in the Bohr model. The tab text explains this.

## 4. Historical claims and sources

Primary texts read where possible: Balmer 1885 (Zenodo scan), Bohr Part I (Project Gutenberg), Rydberg 1890 (Lund) and the nobelprize.org pages.

- **Balmer (1885).** *Ann. Phys. Chem.* 25, 80–87 (a reprint of *Verh. Naturforsch. Ges. Basel* 7, 548).
  - Born 1 May 1825; he taught at the Basel girls' school.
  - He worked with the "Aufmunterung" (encouragement) of E. Hagenbach.
  - Ångström's lines: 6562.10, 4860.74, 4340.1 and 4101.2 Å. His formula gave 6562.08 for the first; h = 3645.6 Å.
  - He predicted the fifth line at 3969.65 Å. Hagenbach then pointed him to Vogel's laboratory lines and Huggins' white-star lines.
- **Rydberg.** He found the formula in late 1888. The memoir was presented on 13 Nov 1889 and published in 1890 (*KVA Handl.* 23 No. 11; *Phil. Mag.* 29, 331).
- **Pickering.** *ApJ* 4, 369 (1896) and 5, 92 (1897), on ζ Puppis.
  - The lines were found by Williamina Fleming (arXiv:2402.14734).
  - Pickering attributed them to "some element not yet found in other stars or on earth".
- **Fowler (1912).** *MNRAS* 73, 62: H + He tubes under a strong discharge; 4686 Å.
- **Rutherford and Bohr's travels.**
  - Rutherford's nuclear-atom paper: *Phil. Mag.* 21, 669 (1911).
  - Bohr was in Cambridge from autumn 1911, then in Manchester from mid-March to late July 1912 (nobelprize.org biography; Heilbron 2013).
- **Hansen and the Balmer quote.**
  - Heilbron & Kuhn (1969, pp. 264–266) date Hansen's conversation with Bohr to early February 1913.
  - The quote "As soon as I saw Balmer's formula…" comes from Rosenfeld's introduction (1963), p. xxxix. It is marked in the story as "Bohr'un sık sık söylediğine göre / anlattığına göre", since it is not a verified primary utterance.
- **Rutherford's letter.** 20 March 1913, *Bohr Collected Works* II, p. 583. It contains the "grave difficulty … knows beforehand where it is going to stop" passage and the advice to shorten the paper. The draft had been sent on 6 March. The Turkish rendering is a translation, not a quote of a Turkish source.
- **The trilogy.** *Phil. Mag.* 26, 1–25, 476–502, 857–875 (1913).
  - Bohr computed the Rydberg constant as 3.1 × 10¹⁵ s⁻¹ against an observed 3.290 × 10¹⁵, agreeing "inside the uncertainty due to experimental errors in the constants".
  - Bohr gives **no** collapse time. The 10⁻¹¹ s figure is presented as a modern textbook estimate.
- **He⁺.**
  - Evans, *Nature* 92, 5 (1913): 4686 Å in helium tubes with hydrogen removed.
  - Fowler's objection: *Nature* 92, 95.
  - Bohr's reply: *Nature* 92, 231 (ratio 4.00163). Fowler conceded on pp. 232–233.
  - Einstein's "one of the greatest discoveries" is in Hevesy's letter to Rutherford of 14 Oct 1913 (Eve, *Rutherford*, 1939, p. 226).
- **Franck–Hertz.** *Verh. DPG* 16, 457 (1914). The result was first interpreted as ionisation. The Nobel Prize for 1925 was awarded in 1926.
- **Sommerfeld.** *Ann. Phys.* 51, 1 and 125 (1916): elliptical orbits, a relativistic correction and fine structure.
- **Bohr's Nobel Prize (1922).** Citation: "for his services in the investigation of the structure of atoms and of the radiation emanating from them". Nobel lecture "The Structure of the Atom", 11 Dec 1922.
- **Data.**
  - NIST ASD H I (n-averaged rows, ref. T8637 = Wiese & Fuhr 2009), He II and Li III.
  - Ionisation energies: H 13.598434599702 eV; He II 54.4177655 eV.
  - 2s two-photon rate: 8.229 s⁻¹.
  - Lifetimes from the fine-structure A values: 3s 158 ns, 3p 5.27 ns, 3d 15.5 ns.
- **Statements in the text that rest on general knowledge** (standard textbook facts):
  - Bohr was 27 in 1913.
  - Lyman/Paschen/Brackett/Pfund series were found in 1906–1924.
  - Balmer lines are strongest in A-type stars (~10⁴ K).
  - The Lamb shift is 1.06 GHz.

## 5. Known limitations

- **Levels.** n ≤ 7. Absorption to n > 7 is ignored, so photons near the series limit pass through.
- **ℓ substructure.** ℓ sublevels are statistically averaged, so the metastable 2s is not modelled (it is discussed at İleri). The real branching depends on how the ℓ sublevels are populated.
- **Gun.** The absorption probability is 1 inside the resonance window, which is enlarged to about 10⁴× the Doppler width for visibility. There is no recoil and no Doppler shift.
- **Lamp.** The source spectrum is flat in energy. Photo-ionisation follows the Kramers E⁻³ shape (no Gaunt factor). Recombination uses Kramers 1/n level weights with a slow-electron energy of about Exp(0.4 eV).
- **Classical mode.** Non-relativistic Larmor radiation with an infinite nuclear mass. The spiral stops at 0.02 a₀. The number of visual turns is compressed (the real figure is about 2 × 10⁵ turns from a₀). Only the fundamental frequency is radiated, which is exact for a circular orbit.
- **Animation.** Orbital rotation speed is visual (n^−1.5). Real periods are given in the readouts.
- **Air wavelengths.** These are used above 200 nm, including in the IR panel, following the usual spectroscopic convention. The reference strip uses NIST observed air values.

## 6. Core-template changes

None. The CSS and Q4 core are identical to the current `sablon/sablon.html`, including the lead's two later fixes: `cv.redraw` clears the canvas, and `details > *{box-sizing:border-box}`. A few redundant `ctx.clearRect` calls remain at the top of the simulation's own draw functions; they are harmless.