Wien's Displacement Law

The law in one line

Every opaque object whose temperature is above absolute zero emits a continuous spectrum of electromagnetic radiation. The shape of that spectrum, derived from Planck's radiation law, is a smooth curve that rises from zero at short wavelengths, climbs to a maximum, then falls off slowly toward longer wavelengths. Wien's displacement law locates the maximum:

λ_max × T = b = 2.8977719 × 10⁻³ m·K

Here T is the absolute temperature in kelvins and b is Wien's displacement constant. The relation is sometimes written λ_max = b/T to emphasize the inverse proportionality. Doubling the temperature halves the wavelength at which the spectrum peaks. Quadrupling the temperature shifts the peak to a quarter of its original wavelength. The phrase "displacement" refers to this lateral sliding of the peak along the wavelength axis.

The result was published by Wilhelm Wien in 1893, seven years before Planck wrote down the full spectrum. Wien obtained it by combining a thermodynamic argument from Wilhelm Schoenberg with empirical curve-fitting; Planck's later derivation made the underlying constant b a derived combination of fundamental physical constants rather than an empirical fit.

Deriving the displacement constant from Planck's law

Planck's law for the spectral radiance of a black body, expressed as energy per area per solid angle per wavelength interval, is:

B(λ, T) = (2hc²/λ⁵) × 1 / (exp(hc/λkT) − 1)

To find the wavelength at which B is maximum, we set the partial derivative of B with respect to λ equal to zero. Carrying out the differentiation and dividing by 2hc² yields, after algebra:

5 (e^x − 1) − x e^x = 0,    where x = hc/(λkT)

This is equivalent to x = 5(1 − e^−x). The transcendental equation has one non-trivial positive root, found numerically:

x_max = 4.9651142317...

Rearranging x = hc/(λkT) gives λT = hc/(x_max × k). Plugging in the modern recommended values h = 6.62607015 × 10⁻³⁴ J·s, c = 2.99792458 × 10⁸ m/s, k = 1.380649 × 10⁻²³ J/K:

b = hc/(x_max k) = 2.8977719 × 10⁻³ m·K

The constant therefore packages three fundamental physical constants and one transcendental number. There is no separate experimental measurement of b; it is computed from h, c and k, all of which are now defined to be exact in the 2019 SI revision.

Worked example: from Sun to CMB

Wien's law is most useful for back-of-envelope estimates of object temperature given a peak wavelength, or vice versa. Working through five very different objects spanning fourteen orders of magnitude in wavelength:

Sun's photosphere   T = 5778 K   →  λ_max = 2.898e-3 / 5778       = 5.016e-7 m  =  502 nm  (green-yellow visible)
Tungsten lamp       T = 2900 K   →  λ_max = 2.898e-3 / 2900        = 9.99e-7  m  ~ 1.0 μm  (near-infrared)
Forge steel         T = 1300 K   →  λ_max = 2.898e-3 / 1300        = 2.23e-6  m  =  2.23 μm (mid-infrared, glows red-orange)
Human body          T = 310 K    →  λ_max = 2.898e-3 / 310         = 9.35e-6  m  =  9.35 μm (long-wave infrared)
Room air            T = 295 K    →  λ_max = 2.898e-3 / 295         = 9.82e-6  m  =  9.82 μm (LWIR)
Liquid nitrogen     T = 77 K     →  λ_max = 2.898e-3 / 77          = 3.76e-5  m  =  37.6 μm (far infrared)
Cosmic microwave    T = 2.725 K  →  λ_max = 2.898e-3 / 2.725       = 1.064e-3 m  = 1.06 mm  (microwave)

The Sun's peak at 502 nm is famously close to the centre of the visible band (380–750 nm) and to the wavelength at which human photopic vision is most sensitive (~555 nm). This has been read by some as evolutionary tuning of the eye to the available light. The numerical match is partly coincidence — the eye is also adapted for daylight scattered through atmosphere — but it is at least a suggestive match.

The cosmic microwave background, residual radiation from when the universe was 380000 years old, has cooled to T = 2.725 K through 13.8 billion years of cosmic expansion. Wien's law puts its peak at 1.06 mm, exactly where the COBE FIRAS instrument measured it in 1990 — the most precise blackbody spectrum ever recorded.

Peak wavelengths across the temperature ladder

Object	Temperature	λ_max	Spectrum band
Cosmic microwave background	2.725 K	1.06 mm	Microwave
Pluto's surface	40 K	72 μm	Far infrared
Liquid nitrogen	77 K	37.6 μm	Far infrared
Earth's atmosphere	255 K	11.4 μm	Long-wave infrared
Human skin	308 K	9.41 μm	Long-wave infrared
Boiling water	373 K	7.77 μm	Mid-wave infrared
Domestic oven	500 K	5.80 μm	Mid-wave infrared
Glowing iron (dull red)	900 K	3.22 μm	Short-wave infrared
Tungsten lamp filament	2900 K	1.00 μm	Near infrared
Sun's photosphere	5778 K	502 nm	Visible (green-yellow)
Sirius A	9940 K	292 nm	Ultraviolet
Hottest O-class star	50000 K	58 nm	Extreme ultraviolet

The table traces a clean inverse line on a log-log plot: nine orders of magnitude in temperature span nine orders of magnitude in wavelength, exactly as λT = constant requires.

Why the peak shifts: a microscopic picture

The Planck distribution counts the number of photons in each wavelength bin. At low temperature most thermal energy quanta are small — kT is the order of magnitude — so photons of high energy are exponentially suppressed by the Boltzmann factor exp(−hν/kT) = exp(−hc/λkT). As T rises, the Boltzmann factor opens the high-energy doorway: more short-wavelength photons fit into the population. Simultaneously, the long-wavelength tail (which is approximately Rayleigh-Jeans, B ∝ T/λ⁴) does not grow as fast as the short-wavelength surge. The net effect is that the peak — the wavelength at which the spectrum is highest — slides toward shorter wavelengths.

The condition x_max = hc/(λ_max kT) ≈ 4.965 says that the typical photon at the peak has energy ~5 kT. This is significantly more than the thermal mean kT and reflects the fact that the spectrum is asymmetric in λ: there are more low-energy photons in the long-wavelength tail than there are high-energy photons in the short-wavelength tail, but the peak in λ-space sits at a higher energy than the equipartition value.

Frequency form: a different peak

If you write Planck's law per unit frequency rather than per unit wavelength, the peak does not occur at c/λ_max. The spectral density transforms by B_ν |dν| = B_λ |dλ| with |dν/dλ| = c/λ², which deforms the curve. Maximizing the frequency form yields a different transcendental equation x = 3(1 − e^−x) with root x ≈ 2.821, giving:

ν_max / T = 5.879 × 10¹⁰ Hz/K

For the Sun this gives ν_max ≈ 3.4 × 10¹⁴ Hz, corresponding to a wavelength of about 882 nm — well into the near-infrared, not the green peak the wavelength form gives. Both numbers are correct: they answer different questions ("what wavelength bin contains the most energy?" vs. "what frequency bin contains the most energy?"). The lesson is that "the peak of the Planck distribution" is not a wavelength-versus-frequency-invariant statement.

Where Wien's law shows up

• Infrared forehead thermometers. A non-contact medical thermometer reads the long-wave infrared emission from skin, peaking near 9.5 μm at 310 K. The detector is a thermopile or microbolometer with a Fresnel lens; calibration uses Stefan-Boltzmann and Wien together to invert the measured intensity to a temperature with about 0.1 K precision.
• Stellar spectroscopy. Star colors map directly onto temperature via Wien. The OBAFGKM classification, with O around 30000 K and M around 3000 K, is essentially a temperature ladder. Apparent stellar colour-index B−V is calibrated against effective temperature using Planck/Wien fits.
• Pyrometry in steel mills. Optical pyrometers read incandescent steel at 1300–1800 K through narrowband filters near 650 nm. The ratio of intensities through two different colour filters gives a "two-colour" temperature that does not depend on emissivity — a Wien-trick that turns a tricky measurement into a colour-balance reading.
• Climate forcing budgets. Earth radiates to space mostly at 8–14 μm where the atmospheric window is most transparent. Greenhouse gases that absorb in this window (CO₂, methane, water vapour) directly reduce outgoing flux. The relevant peak wavelengths are exactly what Wien predicts for 250–290 K emission.
• CMB cosmology. The cosmic microwave background's 1.06 mm peak fixes the present-day photon temperature at 2.725 K. Tiny fluctuations around this temperature, mapped by COBE, WMAP and Planck, encode the seeds of cosmic structure. Wien's law is the lever that converts a dish-antenna intensity reading into a meaningful temperature map.

Variants and extensions

• Planck's radiation law. The full spectral radiance B(λ, T). Wien's displacement law is the location of its peak; Stefan-Boltzmann's law is the area underneath. All three are different observables of the same underlying distribution.
• Stefan-Boltzmann law. Total power per area emitted by a black body: σT⁴, with σ = 5.670 × 10⁻⁸ W/m²K⁴. Doubling the temperature increases peak wavelength by a factor of 2 (Wien) and total emitted power by a factor of 16 (Stefan-Boltzmann).
• Wien's approximation. The high-frequency limit of Planck's law: B(λ, T) ≈ (2hc²/λ⁵) exp(−hc/λkT). Valid for hν ≫ kT — the short-wavelength tail. It was Wien's original spectrum from 1896 and works well for visible Sun-light, but underestimates the infrared by orders of magnitude.
• Rayleigh-Jeans law. The low-frequency limit: B(λ, T) ≈ 2ckT/λ⁴. Valid for hν ≪ kT. Predicts the "ultraviolet catastrophe" if extrapolated; Planck's interpolation between Wien and Rayleigh-Jeans gave birth to quantum mechanics.
• Wien's law in frequency form. ν_max/T ≈ 5.879 × 10¹⁰ Hz/K. Same physics, different abscissa, different numerical constant.

Common pitfalls

• Forgetting that absolute temperature is required. Use kelvins, not Celsius. A "warm" 30 °C object is at T = 303 K, giving λ_max = 9.56 μm — not a wavelength you'd compute from T = 30.
• Mixing the wavelength and frequency forms. The constants and the peak positions differ. ν_max × λ_max ≠ c. Pick which form your detector samples in (a grating spectrograph is wavelength; a heterodyne radio receiver is frequency) and use the matching formula.
• Assuming a strong selective emitter is a black body. A sodium-vapour lamp peaks at 589 nm because of an atomic emission line, not because Wien's law forces it there. Wien's law applies only to thermal continuum emission from optically thick, near-black materials.
• Using Wien's approximation outside its range. Wien's pre-Planck formula B ≈ (2hc²/λ⁵) exp(−hc/λkT) underestimates the long-wavelength radiance dramatically. Use full Planck for any quantitative work that touches the Rayleigh-Jeans tail.
• Confusing peak wavelength with average wavelength. The Planck distribution is asymmetric in λ. The mean wavelength of emitted photons is longer than λ_max — the long tail outweighs the short side when you compute number-weighted averages. For energy-weighted averages the relationship inverts again. State which average you mean.