How to compute Wien's displacement law: fundamental principle of thermal radiation physics linking peak wavelength of an ideal blackbody emission to its absolute temperature. Formula: lambda_max = b / T, where b = 2.8977719×10⁻³ m·K (Wien constant CODATA 2019), T absolute temperature (Kelvin), lambda_max wavelength of maximum spectral emission. Example Sun T=5778 K: lambda_max=502 nm (green, visible peak). Example human body T=310 K: lambda_max=9.4 um (mid-infrared, FLIR thermal cameras). Essential tool for astrophysics (star classification, CMB Big Bang Penzias-Wilson Nobel 1978/Mather-Smoot Nobel 2006), lighting design (color temperature Philips Cree GE), thermography (FLIR Bosch security), James Webb Space Telescope JWST IR, industrial applications (furnaces, foundries), LED development Cree Samsung Nichia, digital cinema (HMI Arri).

Wien's law and blackbody radiation formulas

Wien displacement law (1893): lambda_max * T = b, b = 2.8977719×10⁻³ m·K. Example Sun T=5778 K: lambda_max = 0.002898/5778 = 5.02×10⁻⁷ m = 502 nm (green). Reciprocal proportionality: temperature increases -> emission peak shifts toward shorter wavelengths (blue->violet->UV for hot stars).

Wien for frequency: nu_max = b' * T, b' = 5.8789×10¹⁰ Hz/K. Example Sun T=5778: nu_max = 3.4×10¹⁴ Hz. Difference from lambda_max: lambda_max * nu_max ≠ c due to Planck nonlinearity (Wien displacement not equivalent in nu vs lambda).

Stefan-Boltzmann law (1879): M = sigma * T^4, sigma = 5.67×10⁻⁸ W/m²K⁴ Stefan-Boltzmann constant. Total power emitted per unit area. Example Sun 5778 K: M = 6.32×10⁷ W/m² surface. Combined with Wien's law: complete Planck spectrum integrated over all lambda.

Planck's law (1900): B(lambda,T) = (2hc²/lambda^5) * 1/(exp(hc/(lambda*k*T))-1). Complete blackbody spectrum. Wien is peak approximation. Wien limit (high frequency): B ≈ (2hc²/lambda^5) * exp(-hc/(lambda*k*T)). Rayleigh-Jeans limit (low frequency): B ≈ 2ck*T/lambda^4 (classical UV catastrophe resolved by Planck quantization).

Color temperature application: CIE 1931 chromaticity standard. Tungsten incandescent bulb 2700 K: lambda_max=1073 nm (NIR, perceived warm). Daylight LED bulb 6500 K: lambda_max=446 nm (blue, perceived cool). Standard CIE D65: 6500 K. CIE D50 5000 K photographic printing. CIE D55 5500 K mid-day sun. CIE A 2856 K tungsten reference.

History and context

Wilhelm Wien (Germany 1864-1928) Nobel Physics Prize 1911 for discoveries on thermal radiation, formulated the displacement law in 1893 at Berlin Physikalisch-Technische Reichsanstalt PTR (today PTB Braunschweig). Wien empirically analyzed blackbody emission at various temperatures, establishing hyperbolic relation between lambda_max and T, anticipating quantum theory. Max Planck (Germany 1858-1947) Nobel 1918 resolved classical "ultraviolet catastrophe" in 1900 introducing quantized energy h*nu, founding quantum mechanics. Friedrich Paschen, Otto Lummer, Ferdinand Kurlbaum worked at PTR measuring precise spectra. USA: NIST National Institute Standards Technology Gaithersburg MD reference standards, NASA James Webb Space Telescope JWST 2021 IR observatory L2, Hubble Space Telescope HST 1990 NASA-ESA, Spitzer IR Caltech NASA. Astrophysics: Hertzsprung-Russell diagram star classification (Ejnar Hertzsprung Denmark 1911, Henry Norris Russell USA Princeton 1913), Annie Jump Cannon (USA Harvard 1901) classified 350,000 stars, Cecilia Payne-Gaposchkin Harvard 1925. CMB Cosmic Microwave Background discovered 1964 Penzias-Wilson Bell Labs Holmdel NJ Nobel 1978, COBE 1989 Mather-Smoot Nobel 2006 perfect blackbody T=2.725 K Big Bang residue. Lighting industry global 100+ billion USD/year: Philips Lighting Eindhoven (Signify), GE Lighting USA, Cree Durham NC LED, Eaton Lighting USA.

Industrial thermography: FLIR Systems Boston USA leader (acquired Teledyne 2021), Fluke USA Test Equipment, Land Instruments UK Raytek. CIE Commission Internationale Eclairage Vienna 1913 founded, color standardization. Universities: MIT Boston, Caltech Pasadena, UT Austin, Harvard Cambridge MA, Princeton NJ. Famous applications: HMI Arri 6000K cinema lighting since 1969 Munich Germany, used Hollywood productions standard.

Temperatures and peak wavelengths typical table

• Cosmic Microwave Background (CMB): 2.725 K, lambda_max=1.06 mm (microwaves)
• Cold interstellar space: 10-50 K, lambda_max=58-290 um (FIR)
• Pluto surface: 40 K, lambda_max=72 um (FIR)
• Mars surface average: 210 K, lambda_max=13.8 um (MIR)
• Earth surface (geo-IR): 288 K, lambda_max=10.06 um (MIR atmospheric window)
• Human body: 310 K (37°C), lambda_max=9.35 um (MIR, FLIR thermal cam)
• Home oven: 473 K (200°C), lambda_max=6.13 um
• Molten steel furnace: 1873 K (1600°C), lambda_max=1.55 um (NIR)
• Halogen incandescent bulb: 3000 K, lambda_max=966 nm (NIR)
• High-efficiency halogen: 3200 K, lambda_max=905 nm
• Sun surface (photosphere): 5778 K, lambda_max=502 nm (green peak visible)
• Star Sirius A (Canis Major): 9940 K, lambda_max=292 nm (UV)
• Star Vega (Lyra): 9602 K, lambda_max=302 nm (UV)
• Star Rigel (Orion B): 12100 K, lambda_max=240 nm (UV)
• O class massive stars: 30000-50000 K, lambda_max=58-97 nm (VUV)
• Thermonuclear fusion plasma: 100×10⁶ K, lambda_max=29 pm (hard X-rays)
• Wien constant CODATA 2019: b = 2.8977719×10⁻³ m·K

Worked numerical examples

Example 1 - Sun-Earth spectrum. T=5778 K (Sun photosphere). lambda_max = 2.898×10⁻³/5778 = 5.02×10⁻⁷ m = 502 nm. Green visible peak emission solar light. Atmosphere optical window 380-700 nm transmits practically all green. Photosynthesis chlorophyll a/b absorbs blue (450) and red (660), reflects green (green leaves). Planck integrated curve: 50% solar energy visible 0.4-0.7 um, 47% NIR 0.7-2.5 um, 3% UV <0.4 um. NASA Solar Dynamics Observatory SDO 2010 satellite heliophysics monitoring.

Example 2 - human body FLIR thermography. T=310 K (37°C; fever 38°C=311 K). lambda_max = 2.898×10⁻³/310 = 9.35×10⁻⁶ m = 9.35 um. MIR mid-infrared, atmospheric window 8-13 um transparent. FLIR (Forward-Looking Infrared) thermal cameras detect this band. USA hospitals COVID-19 screening 2020: Honeywell FLIR thermal scanners airport entrances. Medical research: thermography breast suspected cancer early diagnosis. CDC Centers Disease Control USA airport screening protocol. ANSI/IEC 80601-2-59 medical device safety standard.

Example 3 - vintage incandescent bulb. T=2700 K (Edison 60W tungsten bulb). lambda_max=2.898×10⁻³/2700=1073 nm=1.07 um. NIR near-infrared (beyond visible), continuous spectrum emits also visible light: ~5% luminous efficacy. 95% IR heat lost. Compare modern LED: 8-10x more efficient. Energy Independence Security Act 2007 USA phased out incandescent bulbs 2012-2014. USA: GE Lighting Cleveland OH, Philips USA, Sylvania (Osram) Hillsboro NH. Cinema HMI Arri 6000K daylight for movie set since 1969.

Example 4 - Cosmic Microwave Background CMB. T=2.725 K (Big Bang residue). lambda_max=2.898×10⁻³/2.725=1.06×10⁻³ m=1.06 mm=160 GHz peak frequency. Discovered Penzias-Wilson 1964 Bell Labs Holmdel NJ Nobel 1978, precisely measured COBE FIRAS 1989 Mather Nobel 2006 perfect blackbody T=2.72548±0.00057 K. Fluctuations 1 part 100,000 (galaxy ripples) WMAP NASA 2001-2010, Planck satellite ESA 2009-2013. NASA Goddard Space Flight Center Greenbelt MD COBE design.

Example 5 - American steel mill furnace. T=1600°C=1873 K (molten steel). lambda_max=2.898×10⁻³/1873=1550 nm=1.55 um. NIR near-infrared, silica fiber telecom optical window (1310-1550 nm). Optical pyrometer non-contact T measurement: 1500 nm filter, emissivity ~0.9 steel. USA: Nucor Steel Charlotte NC, US Steel Pittsburgh PA, Cleveland-Cliffs Cleveland OH. Safety: pyrometers Land Instruments UK, Raytek Fluke Everett WA, Optris Berlin. Standards: ASTM E1256 pyrometer calibration, OSHA workers safety molten metal.

Limits and considerations

1. Wien's law valid for ideal blackbodies: emissivity epsilon=1. Real bodies (gray bodies) emissivity 0.1-0.95.
2. Continuous blackbody spectrum: atomic/molecular emissions (NaD, H-alpha) require atomic spectroscopy.
3. Low-T gas/plasma: rare collisions, not thermalized, no Maxwell-Boltzmann equilibrium.
4. Rayleigh-Jeans limit (low frequency, high T*lambda): B = 2ck*T/lambda^4 fails at UV (classical UV catastrophe resolved by Planck 1900).
5. Wien limit (high frequency): B ≈ exp(-hc/(lambda*k*T)) approximation visible for hot stars.
6. Wien displacement nu_max (frequency) and lambda_max (wavelength) NOT linked by c=lambda*nu due to Planck nonlinearity.
7. Atmospheric absorption: Earth captures H2O CO2 5-8 um, 13-19 um. Window 8-13 um for thermography.
8. Relativistic effects: fast stars, redshift z>1 shifts spectrum: lambda_observed = lambda_emitted * (1+z).
9. Color temperature CCT correlated: perceptual measure, uses CIE 1931 chromaticity diagram, not real Planck.
10. Stellar effective T vs photospheric T: difference for opacity (limb darkening, atmospheres).

Common errors

1. Confusing lambda_max with nu_max: lambda_max=b/T (b=2.898e-3 m*K), nu_max=b'*T (b'=5.879e10 Hz/K).
2. Neglecting emissivity epsilon < 1 for real bodies: M_real = epsilon * sigma * T^4.
3. Calculating lambda_max in mm/cm instead of nm/um: factor 10^6 error.
4. Using T in Celsius instead of Kelvin: huge error T_K = T_C + 273.15.
5. Confusing color temperature CCT (correlated, perceptual) with effective T blackbody (radiometric).
6. Neglecting differences CIE D65 (6500K, daylight) vs D50 (5000K, printing) vs A (2856K, incand bulb).
7. Stellar classification: O>30000K, B 10000-30000, A 7500-10000, F 6000-7500, G Sun 5200-6000, K 3700-5200, M <3700.
8. Demanding Wien valid for single atoms: needs emission line spectroscopy (Balmer, Lyman, etc.).
9. Cosmology: lambda_max CMB observed includes redshift z=1100 from matter-radiation decoupling.
10. Not considering stellar pulsation effects, magnetic activity (sunspots) modify effective spectrum.

Numbered applications

1. Astrophysics and stellar classification: NASA, ESA, Hertzsprung-Russell diagram, JWST IR.
2. Space telescopes: James Webb JWST IR 0.6-28 um, Hubble UV-Vis, Spitzer IR, Planck CMB.
3. Lighting and LED: Cree Durham NC, GE Lighting Cleveland, Philips USA, Nichia Tokushima.
4. Industrial thermography: FLIR Boston, Fluke Everett WA, Land Instruments UK pyrometers.
5. Medical diagnostics: thermography breast cancer, COVID-19 screening 2020 airports USA.
6. Steel mills and foundries: Nucor Charlotte NC, US Steel Pittsburgh, NIR melting pyrometers.
7. Cinema and photography: HMI Arri daylight 6000K, RED Camera, color temp settings.
8. Satellite meteorology: GOES NOAA, MODIS Terra/Aqua, IR-MIR atmospheric sensors.
9. Cosmology and Big Bang: COBE NASA Goddard, WMAP, Planck satellites CMB studies.
10. Educational: astrophysics MIT Caltech UT Austin Princeton Stanford courses.

For support

For technical questions or reports: info@calcoi.com. Educational tool for preliminary thermal radiation physics. For professional applications consult CODATA 2019 fundamental constants, NIST Standard Reference Database physical constants, Planck 1900 "Über das Gesetz der Energieverteilung im Normalspektrum", Wien 1893 "Über die Energieverteilung im Emissionsspektrum eines schwarzen Körpers", IUPAC Gold Book physical chemistry, NASA Goddard Space Flight Center, ESA European Space Agency. CIE Commission Internationale Eclairage Vienna lighting standards. NIST Gaithersburg MD reference standards. Caltech IPAC Spitzer IR archive.

Updated July 2026