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💡 Quick rules: k (rad/m) = 2π / λ(m). Spectroscopic ˜ν (cm⁻¹) = 1 / λ(cm) = 10000 / λ(µm).
Angular Wavenumber
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Sources & Methodology
Wavenumber definitions from IUPAC Green Book (Quantities, Units and Symbols in Physical Chemistry). Speed of light from NIST CODATA.
IUPAC Green Book — Quantities, Units and Symbols in Physical Chemistry
Official IUPAC definitions for angular wavenumber k = 2π/λ (rad/m) and spectroscopic wavenumber ν̃ = 1/λ (cm⁻¹) used as the standard in all spectroscopy and optics applications.
NIST CODATA — Speed of Light c = 299,792,458 m/s
Exact value of c used for the frequency-to-wavenumber conversion: k = 2πf/c, and spectroscopic wavenumber from frequency: ν̃ = f/c.
Methodology: Angular wavenumber k = 2π/λ (rad/m). Spectroscopic wavenumber ν̃ = 1/λ(cm) = 1/(λ(m)×100) (cm⁻¹). From frequency (EM): λ = c/f, then apply above formulas. Reduced wavenumber κ = k/(2π) = 1/λ (m⁻¹). Relation to angular frequency: ω = v×k. Photon momentum p = ℏ×k = h/λ. Conversion: 10000 cm⁻¹ = 1 µm⁻¹.
⏱ Last reviewed: April 2026
How to Calculate Wavenumber — Angular and Spectroscopic
Wavenumber is a measure of spatial frequency — how many oscillations occur per unit distance. Unlike temporal frequency (oscillations per second), wavenumber counts oscillations per metre or per centimetre. There are two conventions in common use: the angular wavenumber k = 2π/λ used in physics, and the spectroscopic wavenumber ν̃ = 1/λ used in chemistry and spectroscopy.
The Two Wavenumber Formulas
k = 2π / λ (rad/m) — Angular wavenumber
ν̃ = 1 / λ(cm) (cm⁻¹) — Spectroscopic wavenumber
Green light 550 nm: k = 2π/(550×10⁻⁹) = 1.142×10⁷ rad/m • ν̃ = 18182 cm⁻¹
Mid-IR CO₂ 15 µm: k = 2π/(15×10⁻⁶) = 4.19×10⁵ rad/m • ν̃ = 667 cm⁻¹
FM radio 3 m: k = 2π/3 = 2.094 rad/m • ν̃ = 0.0333 cm⁻¹
Mid-IR CO₂ 15 µm: k = 2π/(15×10⁻⁶) = 4.19×10⁵ rad/m • ν̃ = 667 cm⁻¹
FM radio 3 m: k = 2π/3 = 2.094 rad/m • ν̃ = 0.0333 cm⁻¹
Spectroscopic Wavenumber Reference Table
| Region | Wavelength | ν̃ (cm⁻¹) | Application |
|---|---|---|---|
| Far-IR / THz | 25–1000 µm | 10–400 | Molecular rotations, lattice vibrations |
| Mid-IR fingerprint | 7–25 µm | 400–1400 | C-C, C-O, C-N bond stretches |
| Mid-IR functional | 2.5–7 µm | 1400–4000 | C=O, N-H, O-H stretches |
| Near-IR | 0.78–2.5 µm | 4000–12800 | Overtone bands, food analysis |
| Red visible | 700 nm | 14286 | Visible red light |
| Green visible | 550 nm | 18182 | Green light, peak eye sensitivity |
| Violet visible | 380 nm | 26316 | Violet light |
| UV-C | 200–280 nm | 35714–50000 | Germicidal sterilisation |
Key Molecular Absorption Wavenumbers (FTIR)
| Bond / Vibration | ν̃ (cm⁻¹) | Molecule Example |
|---|---|---|
| O-H stretch | 2500–3700 | Water, alcohols, carboxylic acids |
| C-H stretch (alkyl) | 2850–3100 | Hydrocarbons |
| CO₂ asymmetric stretch | 2349 | Carbon dioxide (greenhouse gas) |
| C=O stretch | 1680–1750 | Ketones, esters, aldehydes |
| CO₂ bending | 667 | Carbon dioxide |
| C-H bending | 600–1400 | Fingerprint region |
💡 Spectroscopy shortcut: ν̃(cm⁻¹) = 10000 / λ(µm). For CO₂ bending at 15 µm: ν̃ = 10000/15 = 667 cm⁻¹. For CO₂ stretch at 4.26 µm: 10000/4.26 = 2347 cm⁻¹. This is the most common conversion used in infrared spectroscopy. Note: spectroscopic wavenumber ν̃ = 1/λ (NOT 2π/λ). Angular wavenumber k = 2πν̃ in m⁻¹.
Frequently Asked Questions
Wavenumber measures spatial frequency — cycles or radians of phase per unit distance. Angular wavenumber k = 2π/λ (rad/m): number of radians per metre. Spectroscopic wavenumber ν̃ = 1/λ(cm) (cm⁻¹): number of wave cycles per centimetre. They differ by a factor of 2π: k = 2πν̃ (when ν̃ is in m⁻¹).
k = 2π/λ (rad/m). For 550 nm green light: k = 2π/(550×10⁻⁹) = 1.142×10⁷ rad/m. For FM radio at 3 m: k = 2π/3 = 2.094 rad/m. Angular wavenumber relates to angular frequency via the dispersion relation: k = ω/v. For EM waves: k = 2πf/c = ω/c.
ν̃ = 1/λ(cm) (cm⁻¹). For 550 nm = 0.0000550 cm: ν̃ = 1/0.0000550 = 18182 cm⁻¹. Quick formula: ν̃(cm⁻¹) = 10000/λ(µm). Spectroscopists use cm⁻¹ because it gives convenient numbers (100–10000 cm⁻¹) for molecular vibrations in the infrared, and energies in cm⁻¹ are proportional to frequency: E = hcν̃.
Red 700 nm: 14286 cm⁻¹, k = 8.98×10⁶ rad/m. Green 550 nm: 18182 cm⁻¹, k = 1.14×10⁷ rad/m. Violet 380 nm: 26316 cm⁻¹, k = 1.65×10⁷ rad/m. The visible range spans approximately 14000–26000 cm⁻¹. Electronic spectroscopy often uses cm⁻¹ or nm interchangeably to identify transitions.
Key mid-IR bands: C-H stretch 2850–3100 cm⁻¹, O-H stretch 2500–3700 cm⁻¹, C=O stretch 1680–1750 cm⁻¹, C=C stretch 1600–1680 cm⁻¹, C-C stretch 800–1300 cm⁻¹. CO₂: asymmetric stretch 2349 cm⁻¹, bending 667 cm⁻¹. Water vapour: O-H stretch ~3500 cm⁻¹. These define the fingerprint regions in FTIR spectroscopy.
λ(µm) = 10000/ν̃(cm⁻¹). For 1000 cm⁻¹: λ = 10 µm. For 18182 cm⁻¹: λ = 0.55 µm = 550 nm. For 667 cm⁻¹: λ = 14.99 µm. Alternatively: λ(m) = 1/(ν̃×100). This conversion is used daily in FTIR and Raman spectroscopy to switch between instrument output in cm⁻¹ and literature wavelength values.
CO₂ laser emits at 10.6 µm. Spectroscopic wavenumber = 10000/10.6 = 943 cm⁻¹. Angular wavenumber k = 2π/(10.6×10⁻⁶) = 5.93×10⁵ rad/m. This wavelength falls between CO₂’s bending mode (667 cm⁻¹) and asymmetric stretch (2349 cm⁻¹). The 10.6 µm output is strongly absorbed by organic materials, making CO₂ lasers ideal for cutting, engraving, and medical applications.
k = ω/v (dispersion relation for non-dispersive media). For EM waves in vacuum: k = ω/c = 2πf/c. For 545 THz light: k = 2π×545×10¹²/299792458 = 1.142×10⁷ rad/m. Angular wavenumber is the spatial equivalent of angular frequency: ω counts radians per second, k counts radians per metre. Together they define the wave’s phase: φ = kx − ωt.
Raman shift is the difference in spectroscopic wavenumber between the scattered photon and the excitation laser: Δν̃ = ν̃(laser) − ν̃(scattered) (cm⁻¹). For a 532 nm laser (18797 cm⁻¹) and a Raman peak at 536 nm (18657 cm⁻¹): Δν̃ = 140 cm⁻¹. Raman shifts up to 4000 cm⁻¹ reveal molecular vibrational modes. Common peaks: Si at 520 cm⁻¹, graphene G-band at 1580 cm⁻¹, diamond at 1332 cm⁻¹.
Photon momentum p = ℏk = h/λ. For 550 nm green light: p = 6.626×10⁻³⁴ / 550×10⁻⁹ = 1.205×10⁻²⁷ kg·m/s. In quantum mechanics, angular wavenumber k is the eigenvalue of the momentum operator divided by ℏ. This is the de Broglie relation generalised to wave mechanics: any particle with wavelength λ has momentum h/λ.
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