Wien’s displacement constant, b

The constant in Wien’s displacement law: divide it by a black body’s absolute temperature and you have the wavelength its spectrum peaks at. Hotter bodies peak at shorter wavelengths, which is why heated metal glows red before it glows white.

b = 2.897771955… × 10⁻³ m·K

Exact, derived. The decimal runs on past ten significant figures, so it is cut off there and marked …, the way NIST prints it, rather than rounded.

In other units

Wien’s displacement constant in SI units and in other units
Quantity	Value
b	2.897771955… × 10⁻³ m·K
b	2897.771955… µm·K
b	2,897,771.955… nm·K
Peak wavelength at 300 K	9.659239850… µm

b in µm·K

2.897771955… × 10⁻³ m·K × 10⁶ µm/m
= 2897.771955… µm·K

b in nm·K

2.897771955… × 10⁻³ m·K × 10⁹ nm/m
= 2,897,771.955… nm·K

Peak wavelength at 300 K in µm

2.897771955… × 10⁻³ m·K ÷ 300 K
= 9.659239850… × 10⁻⁶ m
= 9.659239850… µm

Where the value comes from

Planck’s law written per unit wavelength peaks where x = 5(1 − e⁻ˣ), with x = hc/(λkT). That equation has no solution in elementary functions; its root, x = 4.965114231…, has to be found numerically, and this site finds it by iteration rather than copying the digits in.

Because h, c and k are exact and x is a mathematical constant, b is exact too, though like π its decimal never ends. Wilhelm Wien found the displacement law in 1893, seven years before Planck’s law explained it.

b from h, c and k

b = hc ÷ (xk)
= 1.986445857… × 10⁻²⁵ ÷ (4.965114231… × 1.380649 × 10⁻²³)
= 2.897771955… × 10⁻³ m·K

Which form to use

Divide b by the absolute temperature. 2898 µm·K ÷ 300 K gives 9.66 µm for an object at room temperature, in the thermal infrared, and a 3000 K lamp filament peaks near 966 nm, in the near infrared, which is why a filament lamp gives out far more infrared than visible light.

Match the units of b to the wavelength you want out: m·K gives metres, µm·K micrometres and nm·K nanometres.

Common mistakes

Assuming the spectrum peaks at the same place on a frequency scale. Per unit frequency it peaks at ν = 5.878925757… × 10¹⁰ Hz/K × T, which at 300 K is 17.64 THz, a wavelength of about 17 µm rather than 9.66 µm. A band of fixed width in wavelength spans far more frequencies at short wavelengths than at long ones, so the two ways of plotting weight the spectrum differently and peak in different places.
Using a Celsius temperature. The law needs kelvin: 27 °C is 300.15 K.
Reading the peak as the only wavelength emitted. A black body radiates across the whole spectrum; the peak marks only where the most power per unit wavelength falls.

Tools that use it

Electromagnetic Spectrum Explorer places the peaks of body heat, sunlight and the cosmic microwave background with it.

Related constants

Stefan-Boltzmann constant, σ = 5.670374419… × 10⁻⁸ W/(m²·K⁴)
Planck constant, h = 6.62607015 × 10⁻³⁴ J·s
Boltzmann constant, k = 1.380649 × 10⁻²³ J/K
Speed of light in vacuum, c = 299,792,458 m/s