Select Elements

Display Options

Line σ (nm) 1.5

Brightness 100

CMF overlay Labels Rainbow BG Normalise

Rydberg / Balmer Calculator

n₁ n₂

Keyboard

R Reset C Copy JSON H Toggle H ←→ Adjust σ

Emission Spectrograph

Perceived Colour (CIE 1931)

No element selected

Emission Line Data

Element	Line	λ (nm)	Rel. Intensity	Colour	Region	eV

Hydrogen Energy Levels & Balmer Transitions

CIE 1931 Chromaticity Diagram

SPD × CMF Integration Curve

NIST Atomic Spectra Database

The NIST Atomic Spectra Database (ASD) is the authoritative source for atomic energy levels, wavelengths, and transition probabilities. Version 5.11 provides data for over 10 000 species. Wavelengths are vacuum wavelengths (below 200 nm) or standard air wavelengths (above 200 nm) per IAU convention.

Key parameters: wavelength λ (nm), relative intensity, transition probability A_ki (s^-1), energy levels E_i/E_k (cm^-1), statistical weights g_i/g_k, and quantum numbers (configuration, term, J).

CIE 15:2004 — Colour Matching Functions

The CIE 1931 2° Standard Observer colour matching functions &xbar;(λ), &ybar;(λ), &zbar;(λ) define the spectral sensitivity of a “standard” human observer. They are the basis for all colorimetric calculations in this tool. The piecewise Gaussian approximation (Wyman et al., 2013) provides continuous analytic CMFs accurate to <0.5% of the tabulated data.

Standard illuminant: Equal-energy (E), normalised so ∫&ybar;(λ) dλ = 1. Tristimulus values X, Y, Z are computed by spectral integration: X = ∫ S(λ)·&xbar;(λ) dλ.

Bohr Model & Quantum Numbers

Niels Bohr (1913) proposed that electrons orbit the nucleus at discrete energy levels E_n = −13.6 / n² eV for hydrogen. Transitions between levels emit photons with energies ΔE = 13.6 × (1/n₁² − 1/n₂²) eV. The four quantum numbers (n, l, m_l, m_s) fully specify each electron state.

Selection rules for electric-dipole transitions: Δl = ±1, Δm_l = 0, ±1. Hydrogen transitions are the only ones exactly predicted by the Bohr/Rydberg model; multi-electron atoms require quantum-mechanical corrections.

Spectral Series Classification

Hydrogen spectral series are named for their lower level: Lyman (n₁=1, UV), Balmer (n₁=2, visible/UV), Paschen (n₁=3, IR), Brackett (n₁=4, IR), Pfund (n₁=5, far IR). Each series converges to a series limit λ_lim = n₁²/R_∞.

For multi-electron atoms, series are classified by term symbols (e.g., ²S, ²P, ²D) describing orbital angular momentum and multiplicity.

IEC 61966-2-1 sRGB Colour Space

sRGB (IEC 61966-2-1:1999) defines the standard colour space for web and consumer displays. The XYZ→sRGB conversion uses a 3×3 matrix (D65 reference white) followed by gamma encoding: C′ = 12.92·C (C ≤ 0.0031308) or 1.055·C^1/2.4−0.055.

Primaries: R(0.64, 0.33), G(0.30, 0.60), B(0.15, 0.06). White point: D65 (0.3127, 0.3290). Out-of-gamut emission colours are clipped to the nearest sRGB boundary for display.

ISO 11664 — Colorimetry

ISO 11664-1:2007 defines CIE standard colorimetric observers. ISO 11664-2:2007 specifies CIE standard illuminants. ISO 11664-4:2008 defines the CIE 1976 L*a*b* and CIE 1976 L*u*v* colour spaces. ISO 11664-6:2014 defines CIEDE2000 colour difference.

These standards underpin the tristimulus and chromaticity calculations used when rendering perceived colours from emission line spectra.

SI Units & Physical Constants

Wavelength λ in nanometres (nm, 10⁻⁹ m). Frequency ν in terahertz (THz, 10¹² Hz). Photon energy E = hc/λ in electron-volts (eV). Rydberg constant R_∞ = 1.0973731568×10⁷ m⁻¹. Planck constant h = 6.626×10⁻³⁴ J·s. Speed of light c = 2.998×10⁸ m/s.

Hydrogen ionisation energy: 13.6 eV (1 Rydberg). Bohr radius a₀ = 0.0529 nm.

Rydberg Formula

The universal formula for hydrogen emission wavelengths:

1/λ = R_∞ × (1/n₁² − 1/n₂²)
λ = 10&sup9; / (R_∞ × (1/n₁² − 1/n₂²))  [nm]

R_∞ = 1.0973731568 × 10&sup7; m^-1
n₁ < n₂   (both positive integers)

Valid only for hydrogen-like (single-electron) atoms. For heavier elements, quantum-defect corrections apply: E_nl = −13.6 × Z_eff² / (n − δ_l)² eV.

Bohr Energy Levels

E_n = −13.6 / n²  [eV]      (hydrogen)
ΔE = 13.6 × (1/n₁² − 1/n₂²)  [eV]

Photon wavelength: λ = hc / ΔE = 1240 / ΔE  [nm]
Photon frequency:  ν = c / λ  [Hz]

Ground state (n=1): −13.6 eV. First excited (n=2): −3.4 eV. Ionisation: 0 eV (n→∞).

CIE 1931 CMF (Piecewise Gaussian)

g(wl, μ, σ₁, σ₂) = exp(−0.5 × d² / σ(d)²)
  where d = wl − μ, σ(d) = σ₁ if d<0 else σ₂

&xbar;(λ) = 1.056 × g(wl, 599.8, 37.9, 31.0)
             + 0.362 × g(wl, 442.0, 16.0, 26.7)
             − 0.065 × g(wl, 501.1, 20.4, 26.2)

&ybar;(λ) = 0.821 × g(wl, 568.8, 46.9, 40.5)
             + 0.286 × g(wl, 530.9, 16.3, 31.1)

&zbar;(λ) = 1.217 × g(wl, 437.0, 11.8, 36.0)
             + 0.681 × g(wl, 459.0, 26.0, 13.8)

Source: Wyman, Sloan, Shirley (2013). Accurate to <0.5% vs. CIE tabulated data.

Spectrum → XYZ Integration

X = (1/k) × ∑ S(λ_i) × &xbar;(λ_i) × Δλ
Y = (1/k) × ∑ S(λ_i) × &ybar;(λ_i) × Δλ
Z = (1/k) × ∑ S(λ_i) × &zbar;(λ_i) × Δλ

k = ∑ &ybar;(λ_i) × Δλ   (equal-energy normalisation)

Integration range: 380–780 nm at 1 nm intervals (401 points). Normalisation factor k ensures Y = 1 for a flat (equal-energy) spectral power distribution.

XYZ → sRGB Conversion

[R_lin]   [ 3.2405  -1.5372  -0.4986] [X]
[G_lin] = [-0.9693   1.8760   0.0416] [Y]
[B_lin]   [ 0.0556  -0.2040   1.0572] [Z]

Gamma encode:
  C' = 12.92 × C          if C ≤ 0.0031308
  C' = 1.055 × C^(1/2.4) - 0.055   otherwise

sRGB = round(clamp(C', 0, 1) × 255)

Negative linear values are clipped to 0 before gamma encoding. Out-of-gamut spectral colours (e.g., monochromatic 520 nm green) are clipped after matrix multiplication.

Monochromatic Colour Rendering

// Build narrow Gaussian SPD at target wavelength
SPD[i] = exp(-0.5 × ((WL[i] - λ) / σ)²)   σ = 1.5 nm

// Integrate against CIE CMF
{X, Y, Z} = spectrumToXYZ(SPD)

// Normalise to peak channel (preserve hue, maximise brightness)
max = max(R_lin, G_lin, B_lin)
R_lin /= max; G_lin /= max; B_lin /= max

// Gamma encode → sRGB byte

This gives the perceptually brightest sRGB representation of any monochromatic wavelength. Deep violet (<400 nm) and deep red (>700 nm) approach black.

Gaussian Line Profile

I(λ) = I₀ × exp(-0.5 × ((λ - λ₀) / σ)²)

FWHM = 2√(2 ln 2) × σ ≈ 2.3548 × σ

The Gaussian profile models Doppler broadening of spectral lines. The σ parameter (adjustable 0.3–8 nm) controls visual line width. Real spectral lines also exhibit Lorentzian (natural) and Voigt (combined) profiles.

Limitations & Approximations

Rydberg formula exact only for hydrogen; multi-electron atoms require quantum-defect or Hartree–Fock corrections.
CIE 1931 piecewise Gaussian is an analytic approximation (<0.5% error vs. tabulated CMFs).
Emission line wavelengths are standard air values from NIST (may differ slightly from vacuum values).
Relative intensities are approximate and depend on discharge conditions (pressure, current, temperature).
sRGB gamut cannot reproduce many monochromatic spectral colours; out-of-gamut values are clipped.
The tool assumes incoherent addition of emission lines (no interference effects).

Academic & Standards References

NIST Atomic Spectra Database (ASD), Version 5.11. A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team (2023). https://physics.nist.gov/asd
Rydberg, J.R. (1888). “Recherches sur la constitution des spectres d'émission des éléments chimiques.” Kungliga Svenska Vetenskapsakademiens Handlingar, 23(11).
Bohr, N. (1913). “On the Constitution of Atoms and Molecules.” Philosophical Magazine, 26(151), 1–25.
Balmer, J.J. (1885). “Notiz über die Spectrallinien des Wasserstoffs.” Annalen der Physik, 261(5), 80–87.
CIE 15:2004. Colorimetry, 3rd edition. Commission Internationale de l'Éclairage.
Wyman, C., Sloan, P.-P., & Shirley, P. (2013). “Simple Analytic Approximations to the CIE XYZ Color Matching Functions.” Journal of Computer Graphics Techniques, 2(2), 1–11.
IEC 61966-2-1:1999. Multimedia systems and equipment — Colour measurement and management — Part 2-1: sRGB colour space.
ISO 11664-1:2007. Colorimetry — Part 1: CIE standard colorimetric observers.
Planck, M. (1901). “Über das Gesetz der Energieverteilung im Normalspektrum.” Annalen der Physik, 309(3), 553–563.
Herzberg, G. (1944). Atomic Spectra and Atomic Structure. Dover Publications.
Cowan, R.D. (1981). The Theory of Atomic Structure and Spectra. UC Press.
Lyman, T. (1906). “The Spectrum of Hydrogen in the Region of Extremely Short Wave-Length.” The Astrophysical Journal, 23, 181.

Batch Element Survey

Compute perceived colour, spectral span, dominant wavelength, and total intensity for all 11 elements simultaneously. Results are computed on-device.

Elements: — Total lines: — Visible lines: — Avg span: —

Sym	Name	Z	Lines	Visible	Span (nm)	Dominant	Total I	Perceived

Hydrogen Series Convergence

Each hydrogen spectral series converges to a limit wavelength as n₂→∞. Below: first member, series limit, energy, and spectral region for all five named series.

Series	Lower n	First λ	Limit λ	ΔE (eV)	Region	Colour

Atomic Emission Lines