Wavenumber — Definition, Units, Symbol & Formula | Wavenumber vs Wavelength
If you have studied spectroscopy — or even glanced at an FTIR spectrum — you have encountered two different ways to describe electromagnetic radiation: wavelength and wavenumber. UV-Vis spectra are plotted in nanometers. FTIR spectra are plotted in cm⁻¹. Raman spectra use wavenumber shifts. The same photon can be described using either scale, yet different fields of spectroscopy have strong preferences for one or the other.
Understanding the relationship between wavelength and wavenumber — why they are inversely related, why different disciplines prefer different scales, and how to convert between them — is a fundamental skill for anyone working with spectroscopic data. This guide covers the definitions, the conversion formulas, the physical reasons behind the conventions, and provides a quick-reference table for common values.
What Is Wavelength?
Wavelength (symbol: λ, the Greek letter lambda) is the distance between two consecutive peaks — or any two corresponding points — of an electromagnetic wave. It is a spatial measurement: how long, in physical units, one complete cycle of the wave is.
The units depend on the spectral region. In UV-Vis spectroscopy, wavelength is expressed in nanometers (nm, 10⁻⁹ meters), typically spanning the range from about 200 nm (deep ultraviolet) to 800 nm (near-infrared). In infrared spectroscopy, wavelength is expressed in micrometers (μm, 10⁻⁶ meters), covering roughly 2.5 to 25 μm for the mid-IR range. In telecommunications and fiber optics, wavelength is also given in nm, but in the 1300–1600 nm near-IR window.
Wavelength is intuitively appealing because it directly corresponds to the physical “size” of the wave. Visible light wavelengths correspond to colors we can see: 400 nm is violet, 550 nm is green, 700 nm is red. This concreteness makes wavelength the natural choice for optics, color science, and UV-Vis spectroscopy.
What Is Wavenumber?
Wavenumber (symbol: ν̃, pronounced “nu-tilde”) is the number of complete wave cycles that fit into one centimeter of space. Its unit is cm⁻¹ (reciprocal centimeters, also read as “inverse centimeters” or informally as “wavenumbers”).
The defining relationship is simply:
ν̃ = 1 / λ
where λ is the wavelength in centimeters
A wavenumber of 1000 cm⁻¹ means that 1000 complete wave cycles fit within one centimeter. A wavenumber of 4000 cm⁻¹ means 4000 cycles per centimeter — a shorter wavelength and higher energy.
The crucial physical property of wavenumber is that it is directly proportional to both the frequency and the energy of the photon: E = hcν̃, where h is Planck's constant and c is the speed of light. This direct proportionality is the primary reason that infrared and Raman spectroscopists prefer wavenumber over wavelength.
Wavenumber Symbol
The standard symbol for wavenumber is ν̃(the Greek letter nu with a tilde), pronounced “nu-tilde.” The tilde distinguishes wavenumber (ν̃, in cm⁻¹) from frequency (ν, in Hz). In published spectra, the x-axis is typically labeled “ν̃ / cm⁻¹” or simply “Wavenumber / cm⁻¹.”
Some older texts and certain physics subdisciplines use σ (sigma) or k as the wavenumber symbol. In physics, k = 2π/λ is the angular wavenumber (in radians per unit length), which differs from the spectroscopic wavenumber ν̃ = 1/λ by a factor of 2π. In spectroscopy contexts, ν̃ is virtually universal.
Wavenumber Units
The unit of wavenumber is cm⁻¹(reciprocal centimeters), also read as “inverse centimeters” or colloquially as “wavenumbers.” A value of 1000 cm⁻¹ means 1000 complete wave cycles fit within one centimeter.
In the SI system, the proper unit would be m⁻¹ (reciprocal meters), but cm⁻¹ is the universal convention in infrared and Raman spectroscopy because it produces conveniently sized numbers for the mid-IR range (400–4000 cm⁻¹). To convert: 1 cm⁻¹ = 100 m⁻¹.
Important spectral ranges in wavenumber units: far-IR is below 400 cm⁻¹, mid-IR spans 400–4000 cm⁻¹, near-IR spans 4000–12,500 cm⁻¹, and visible light spans approximately 13,000–26,000 cm⁻¹.
Wavenumber Formula
The wavenumber formula is the defining equation that relates wavenumber to wavelength:
ν̃ = 1 / λ
where λ is wavelength in cm; for λ in nm, use ν̃ = 10⁷ / λ
From this, all other wavenumber relationships follow. Wavenumber connects to frequency (f) and energy (E) through:
f = c · ν̃ (frequency in Hz)
E = h · c · ν̃ (energy in joules)
E(eV) = 1.2398 × 10⁻⁴ × ν̃(cm⁻¹)
These formulas show why wavenumber is so useful: it is directly proportional to both frequency and energy, unlike wavelength which has an inverse relationship with both.
Why Do Spectroscopists Prefer Wavenumber?
The preference for wavenumber in FTIR and Raman spectroscopy is not arbitrary tradition — there are three concrete physical and practical reasons.
1. Linear with Energy
Wavenumber is directly proportional to photon energy: E = hcν̃. If one absorption band appears at 1000 cm⁻¹ and another at 2000 cm⁻¹, the second photon has exactly twice the energy of the first. On a wavelength scale, this relationship is inverse (E = hc/λ), so equal energy spacings produce unequal wavelength spacings. A wavenumber axis makes it immediately obvious which transitions require more energy, and energy differences between peaks can be read directly from the axis.
2. Additive for Combination Bands
In vibrational spectroscopy, combination bands arise when two fundamental vibrations are excited simultaneously. The wavenumber of the combination band is approximately the sum of the two fundamental wavenumbers. For example, if two fundamentals appear at 1000 cm⁻¹ and 1500 cm⁻¹, their combination band appears near 2500 cm⁻¹. This simple addition does not work with wavelengths — you cannot meaningfully add 10 μm and 6.67 μm to predict a combination band position.
3. Natural Output of FTIR Instruments
FTIR instruments produce interferograms — signals recorded as a function of optical path difference in the interferometer. The Fourier transform of this interferogram naturally yields a spectrum in wavenumber space, not wavelength space. Plotting in cm⁻¹ is the native representation of the data, requiring no additional transformation. Converting to wavelength would distort the uniform point spacing that the Fourier transform produces.
Wavenumber to Wavelength Conversion
The conversion formulas are reciprocal relationships. The exact form depends on the wavelength unit you are working with.
Wavelength in Nanometers (nm)
ν̃ (cm⁻¹) = 10⁷ / λ (nm)
λ (nm) = 10⁷ / ν̃ (cm⁻¹)
The factor of 10⁷ comes from converting centimeters to nanometers (1 cm = 10⁷ nm).
Wavelength in Micrometers (μm)
ν̃ (cm⁻¹) = 10⁴ / λ (μm)
λ (μm) = 10⁴ / ν̃ (cm⁻¹)
The factor of 10⁴ converts centimeters to micrometers (1 cm = 10⁴ μm).
Worked Example
A carbonyl (C=O) stretch (e.g., ketone) typically absorbs at 1715 cm⁻¹. What is the corresponding wavelength?
λ (nm) = 10⁷ / 1715 = 5,831 nm
λ (μm) = 10⁴ / 1715 = 5.83 μm
Notice the inverse relationship: halving the wavenumber (from 1715 cm⁻¹ to about 858 cm⁻¹) doubles the wavelength (from 5.83 μm to about 11.66 μm). This nonlinear scaling is why the same spectrum looks very different when plotted on a wavelength axis versus a wavenumber axis — the high-wavenumber region gets compressed and the low-wavenumber region gets stretched.
Quick Reference Table
Common wavenumber-wavelength pairs for important spectral positions:
| Wavenumber (cm⁻¹) | Wavelength (nm) | Wavelength (μm) | Region |
|---|---|---|---|
| 4000 | 2,500 | 2.5 | Near-IR / Mid-IR boundary |
| 3000 | 3,333 | 3.3 | C-H stretch region |
| 1715 | 5,831 | 5.8 | C=O stretch |
| 1000 | 10,000 | 10.0 | Mid-IR |
| 400 | 25,000 | 25.0 | Far-IR boundary |
Related Units
Wavenumber and wavelength are not the only ways to describe electromagnetic radiation. Several other units appear in spectroscopy, and all are interconvertible:
- Frequency (Hz, THz): The number of wave cycles per second. Related to wavenumber by f = cν̃, where c is the speed of light. Terahertz (THz = 10¹² Hz) is common in far-IR and THz spectroscopy. 1 THz ≈ 33.36 cm⁻¹.
- Energy in electron volts (eV): E = 1.2398 × 10⁻⁴ × ν̃(cm⁻¹). Electron volts are used in X-ray spectroscopy, photoelectron spectroscopy, and semiconductor physics. The mid-IR range (400–4000 cm⁻¹) corresponds to roughly 0.05–0.50 eV.
- Energy in kJ/mol: E = 0.01196 × ν̃(cm⁻¹). This is the “molar” energy — the energy per Avogadro's number of photons. It connects spectroscopic transition energies to thermodynamic quantities like bond enthalpies. A typical C-H stretch at 3000 cm⁻¹ corresponds to about 35.9 kJ/mol.
SpectralBench's unit converter handles all 8 spectroscopic units simultaneously. Enter a value in any unit and get instant conversions to all others.
Summary
Wavenumber and wavelength describe the same physical reality — the spatial periodicity of electromagnetic radiation — on reciprocal scales. Wavelength (nm, μm) is intuitive and dominant in UV-Vis spectroscopy and optics. Wavenumber (cm⁻¹) is linear with energy and dominant in FTIR and Raman spectroscopy.
The conversion is straightforward: λ(nm) = 10⁷ / ν̃(cm⁻¹). Remember that it is a reciprocal relationship — doubling the wavenumber halves the wavelength.
- Convert online — instant wavenumber, wavelength, frequency, and energy conversions
- FTIR Interpretation Guide — put your unit knowledge into practice with spectrum analysis
- Beer-Lambert Law — the fundamental equation for quantitative spectroscopy
- NIR Spectroscopy Explained — near-infrared spectroscopy, where wavenumber units are essential
Frequently Asked Questions
What is wavenumber?
Wavenumber (symbol: ν̃, pronounced 'nu-tilde') is the number of complete wave cycles per centimeter, measured in cm⁻¹ (reciprocal centimeters). It is defined as ν̃ = 1/λ, where λ is the wavelength in centimeters. Wavenumber is directly proportional to the energy and frequency of electromagnetic radiation (E = hcν̃), which makes it the preferred unit in infrared and Raman spectroscopy.
What are the units of wavenumber?
The unit of wavenumber is cm⁻¹ (reciprocal centimeters, also called inverse centimeters or simply 'wavenumbers'). A value of 1000 cm⁻¹ means 1000 complete wave cycles fit within one centimeter. In the SI system, wavenumber can also be expressed in m⁻¹, but cm⁻¹ is the universal convention in spectroscopy. The mid-infrared range spans approximately 400–4000 cm⁻¹.
What is the symbol for wavenumber?
The standard symbol for wavenumber is ν̃ (the Greek letter nu with a tilde), pronounced 'nu-tilde.' Some texts use σ (sigma) or simply k. The tilde distinguishes wavenumber (ν̃, in cm⁻¹) from frequency (ν, in Hz). In FTIR and Raman spectroscopy, the x-axis is labeled ν̃ (cm⁻¹) or simply 'Wavenumber / cm⁻¹.'
How do you convert wavenumber (cm⁻¹) to wavelength (nm)?
To convert wavenumber to wavelength in nanometers: λ (nm) = 10⁷ / ν̃ (cm⁻¹). For wavelength in micrometers: λ (μm) = 10⁴ / ν̃ (cm⁻¹). The relationship is reciprocal — doubling the wavenumber halves the wavelength. For example, a C=O stretch at 1715 cm⁻¹ corresponds to 5,831 nm or 5.83 μm.
Why do spectroscopists prefer wavenumber over wavelength?
Spectroscopists prefer wavenumber for three reasons: (1) wavenumber is linearly proportional to energy (E = hcν̃), so equal cm⁻¹ spacings represent equal energy differences; (2) wavenumbers are additive for combination bands in vibrational spectroscopy; and (3) wavenumber is the natural output of FTIR instruments, whose Fourier transform directly yields spectra in cm⁻¹ without additional conversion.
What is the difference between wavenumber and wavelength?
Wavelength (λ) measures the physical distance between consecutive wave peaks (in nm or μm), while wavenumber (ν̃) counts the number of wave cycles per centimeter (cm⁻¹). They have a reciprocal relationship: ν̃ = 10⁷/λ(nm). Wavelength is used in UV-Vis spectroscopy and optics, while wavenumber dominates in FTIR and Raman spectroscopy because of its linear relationship with energy.