The grating that splits sodium

Add more slits to Young's two — why do soft fringes sharpen into spectral lines, and what sets whether a grating can split two almost-identical colours? Is Rayleigh's resolving power R = λ/Δλ = m·N really the law that decides, order by order, whether the sodium doublet is one line or two?

Units: metres (fed doublet splitting 0.60 nm; NIST Na D₂ 588.99509, D₁ 589.59244, Δλ = 0.59735 nm — the module rounds each line to 0.1 nm)

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The finding

The grating that splits sodium: from a raw 600-slit phasor sum with no closed form coded, the doublet splitting comes back Δλ̂ = 0.6001 ± 0.0001 nm (fed 0.60, 1.24 SE; noiseless 2.1e-7) with each line to 2e-7 — and Rayleigh's R = mN is the whole story: R=600 (1st order) leaves one merged line, R=1200 (2nd) splits it, the measured saddle at N=982 lands on 8/π² = 0.8106 to 3e-4, line width shrinks as 1/N (slope −0.9999), the census steps 11/5/3 with m=3 evanescent, and both rivals die (corpuscular light has no orders at contrast 5e-11; Young's N=2 never resolves at any order)

Method

Generate the screen intensity from FIRST PRINCIPLES: a raw phasor sum A(u) = Σ_j e^{ikr_j}/√r_j over the N = 600 individual slits (d = 1/600 mm) with EXACT path lengths r_j = √(R² + x_j² − 2R·x_j·u) to a screen arc of radius R = 50 m — no grating equation, no [sin(Nγ)/(N·sinγ)]² closed form, no sinc anywhere in the generator (the module's analytic factor is verified only as the far-field limit of the sum, RMS 3.7e-6). The two sodium lines (fed 589.0 / 589.6 nm) are mutually incoherent: intensities add across wavelengths, amplitudes across slits. Recovery reads peak angles off the pattern and inverts d·sinθ = mλ; the doublet splitting is de-biased against peak interaction by least-squares fitting TWO copies of the self-calibrated single-line profile (measured from a one-wavelength run of the same generator). 24 seeded runs add ±2% reading noise. The known 0.60 nm splitting is loaded only to score. ?world=grating.

The law it recovers

principal maxima at d·sinθ = mλ, line half-width λ/(Nd) shrinking as 1/N, resolving power R = λ/Δλ = m·N — the slit COUNT, jointly with the order, decides whether two colours are one line or two

Measurements, controls & cross-checks

Recovered uncertainty

1.0000e-13

Recovered se distance

1.24

Noiseless rel offset

2.0600e-7

Per line recovery

LamA rel

6.7500e-8

LamB rel

6.7700e-8

Single lambda fit rel

2.8800e-11

Note

each wavelength individually to ~7e-8 from its 2nd-order angle; the 4-order through-origin grating-equation fit hands back the fed 589.0 nm to 3e-11

Resolving power

R needed

982

Order1 R600

1 peak, all 24 seeds (merged)

Order2 R1200

2 peaks, all 24 seeds (resolved)

Rayleigh crossing

N at rayleigh

982

Single line first zero vs doublet sep

0.000177

Measured saddle

0.8109

Expected saddle

0.8106

Note

at N = 982 in 1st order the single-line first zero (measured from the phasor sum) equals the doublet separation to 1.8e-4 and the saddle-to-peak ratio lands on 2·sinc²(π/2) = 8/π²; N=600 merges (1 peak), N=1300 splits with dip 0.34

Width scaling

Loglog slope vs N

-0.9999

Fwhm Nd over lambda

0.886

Note

FWHM ∝ 1/N across N ∈ {150,300,600,1200}, with the sinc² constant 0.886 emerging unbidden

Perturbation

U1 loglog slope vs d

-1

Census by ruling

• 11
• 5
• 3

Note

1st-order angle tracks 1/d across 300/600/1200 lines/mm and the line census steps discretely as 2·floor(d/λ)+1 — m=3 at 600 lines/mm needs sinθ = 1.06 > 1 and is simply absent from the raw sum

Rival falsified

Corpuscular contrast

4.9900e-11

Coherent peak to mean

678

Young N2 peaks m1 m2

• 1
• 1

Note

intensities-adding light (Newton) is structureless — no orders at all; and Young's own N=2 pair at the same d (R = 2m ≤ 4 ≪ 982) never splits the doublet at any existing order — amplitude superposition is load-bearing for the lines, the slit count for the resolution

Module cross check

Mirror pins

Peaks order1

1

Peaks order2

2

LamA display

589.0

LamB display

589.6

DLam display

0.62

Module split bias

0.042

Scrape verified

5/5 (?world=grating HUD matches the mirror pins exactly; the HUD itself discloses 'true 0.60')

Note

exact headless replica of the on-screen measurement (analytic doublet + per-order LCG + 11-point smoothing + dominance peak-finder): the displayed Δλ = 0.62 nm reads +4.2% HIGH because at R/R_needed = 1.22 each line's smoothed-argmax maximum rides the other line's first SIDELOBE, whose gradient pushes the peaks apart; the oracle's two-profile fit is free of this pull (0.6000). Disclosed, not hidden — the per-line displays are unaffected at 0.1 nm precision. The '+4.2%' claim is no longer an estimate: gates G12–G17 execute the shipped module and decompose it (see honesty_certificate)

Honesty certificate

Executed

src/modules/DiffractionGratingModule.ts, sha256 e9614130…, 27 strip pairs + bulk 15/9, zero module edits

Exec equals mirror

the executed init measurement (_peaks1/_peaks2/_lamA/_lamB/_dLam/_dipRatio) === the statics replica BIT-FOR-BIT: λ_A 588.9833 nm, λ_B 589.6083 nm, Δλ 0.6250 nm, saddle 0.2385; executed _grating/_doublet match the op-order replica at 201 sample angles and the self-seeded per-order LCG makes a fresh executed _findPeaks re-run bit-identical

Screen bias decomposed

the displayed Δλ = 0.625 nm (+4.17% vs the fed 0.60) is DETERMINISTIC estimator pull, measured by execution: exact continuous argmax of the module's OWN analytic doublet puts the raw peaks +3.84% apart (each maximum rides the other line's first sidelobe at 1.22× Rayleigh — the physics of overlapping profiles, not a bug), the symmetric 11-sample smoothing leaves that at +3.84% (it does NOT widen the split), and the grid scan lands within the derived quantization ceiling d·Δu_step = 0.0042 nm of the continuous value (no fitted tolerance); a 24-fresh-LCG-base ensemble gives 0.6229 ± 0.0157 nm per-seed SD with the R=mN verdict (1 peak in 1st order, 2 in 2nd) unanimous at every seed — the shipped seed sits +0.13 SD, a typical draw, and the HUD discloses 'true 0.60' on screen

Display liveness

600 engine calls at fl(1/120): _acc/_phase/_frame accumulator and all 5 sodium-spot emissive triples bit-equal the statics replica at every call (spot-0 shimmer changed 300× — the live element); 150 %4-cadence HUD writes each bit-equal to the replica (HUD content is statics-by-design: the measurement is fixed at init, young/newton-style); the %8 chart branch is DEAD (post-increment _frame always ≡ 1 mod 4 — same defect class as malus/young/newton) and proven HARMLESS: chart written exactly once at init, frozen SVG === replica === a fresh executed _buildChart() bit-for-bit, disclosed not patched

Answer free

the executed measurement block (_findPeaks + init recovery, comments stripped) contains no 588/589/590/982/0.59/0.60 literals and ZERO L1/L2 references; the single disclosed M.LM reference only centres the scan window (the recovery is λ = d·u/m from the apparatus geometry alone); a planted-violation self-test proves the scanner live; 3 tamper tests are surgical (known_value → only scoring gates with recovery unchanged; cert sha → only G12; exec-pin ulp → only G15)

Seeds

24

What it reduces to

Fraunhofer's ruled grating (1821) and Rayleigh's spectroscope resolution theory (Phil. Mag. 1879): N-slit coherent superposition concentrates light at d·sinθ = mλ with line width λ/(Nd) and resolving power R = mN — the law behind every spectrometer since. It VALIDATES, not derives: the lab assumes scalar wave superposition from N point slits (exact path lengths to a 50 m arc, far field a RESULT not an assumption) and shows (i) the orders, the hard m=3 cutoff, the 1/N sharpening with the sinc² constant 0.886, and the 8/π² Rayleigh saddle all EMERGE from the raw sum with no closed form coded; (ii) the pattern inverts back to what was fed — each sodium line to 2e-7 and the 0.60 nm splitting to 1.24 SE under reading noise — with the full ruler content (u₁ ∝ 1/d, width ∝ 1/N, census 2·floor(d/λ)+1) holding to slopes ±1e-4; and (iii) both rivals are falsified by the same machinery: corpuscular intensity-addition has no orders (contrast 5e-11), and N=2 at the same spacing never resolves the doublet — resolution is bought by slit count, exactly R = mN (600 fails at m=1, succeeds at m=2). NON-CIRCULARITY: the generator contains only e^{ikr} phasors and geometry; the grating equation appears solely in the inversion step of the recovery, and the module's analytic [sin(Nγ)/(N sinγ)]² only on the scoring side as the verified far-field limit (RMS 3.7e-6). It does NOT model blaze angles, finite slit width (no envelope — slits are points), overlapping-order white-light spectra, or the D₂:D₁ ≈ 2:1 intensity ratio (lines fed equal); the fed doublet is the module's 0.1 nm-rounded 589.0/589.6, disclosed against NIST 588.995/589.592.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- grating (scripts/grating-derisk.mjs)

Oracle

scripts/oracles/grating.reference.json

Sources

J. von Fraunhofer, Denkschriften der Königlichen Akademie der Wissenschaften zu München 8, 1–76 (1821/22) — the ruled grating and d·sinθ = mλ. Lord Rayleigh, 'Investigations in optics, with special reference to the spectroscope', Phil. Mag. (5) 8, 261–274 (1879) — the resolution criterion and R = mN. A. Kramida, Yu. Ralchenko, J. Reader, NIST ASD (ver. 5.11): Na D lines. E. Hecht, Optics (5th ed., Pearson 2017), §10.2.7.