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Planck spectral radiance curves with Wien's peak, Stefan-Boltzmann total emittance, and emissivity correction
Planck's law describes the spectral radiance of electromagnetic radiation emitted by a blackbody in thermal equilibrium at temperature T:
B(λ, T) = (2hc² / λ⁵) × 1 / (e^(hc / λkT) − 1)
Here h is Planck's constant, c is the speed of light, k is Boltzmann's constant, λ is wavelength, and T is absolute temperature in Kelvin. The formula gives spectral radiance in W/sr/m²/m — the power emitted per unit solid angle, per unit area, per unit wavelength.
In spectroscopy, Planck's law matters because FTIR sources (globar, Nernst glower) and thermal emitters follow this distribution. The source spectrum sets the available signal intensity across the measurement range, directly affecting signal-to-noise ratios in thermal emission and absorption measurements.
Wien's displacement law gives the wavelength at which the Planck distribution peaks:
λ_max = b / T, where b = 2.8978 × 10⁻³ m·K
This inverse relationship means that hotter objects emit at shorter wavelengths. At room temperature (293 K), peak emission is at ~9.9 μm in the mid-IR — the basis for thermal imaging. The Sun (5778 K) peaks at ~0.50 μm in visible light. Astronomers use this law to estimate stellar temperatures from observed peak wavelengths.
The Stefan-Boltzmann law gives the total power radiated per unit area across all wavelengths:
P = εσT⁴, where σ = 5.6704 × 10⁻⁸ W/m²K⁴
Total emitted power scales as the fourth power of temperature — doubling the temperature increases radiated power by 16×. The emissivity factor ε accounts for real surfaces that radiate less efficiently than ideal blackbodies.
Emissivity (ε) is the ratio of radiation emitted by a surface to that emitted by a perfect blackbody at the same temperature. It ranges from 0 (perfect mirror) to 1 (perfect blackbody). Most non-metallic surfaces have high emissivity (>0.9), while polished metals can be as low as 0.03.
| Material | Emissivity (ε) |
|---|---|
| Polished aluminum | 0.04–0.06 |
| Polished copper | 0.03–0.05 |
| Oxidized steel | 0.60–0.85 |
| Concrete | 0.92–0.97 |
| Glass (smooth) | 0.92–0.94 |
| Human skin | 0.97–0.98 |
| Water | 0.95–0.98 |
| Vegetation | 0.92–0.96 |
| Matte black paint | 0.95–0.98 |
| Perfect blackbody | 1.00 |
FTIR sources: The globar (SiC, ~1300 K) and Nernst glower (ZrO₂/Y₂O₃, ~1900 K) are near-blackbody emitters. Their Planck curves determine the available signal across the mid-IR range. Globar emission drops sharply below 400 cm⁻¹, which is why far-IR FTIR often uses mercury arc lamps or synchrotron sources instead.
Detector response: MCT (HgCdTe) and DTGS detectors have wavelength-dependent sensitivity. Combining the source Planck curve with detector responsivity gives the effective signal-to-noise profile of an FTIR measurement. Regions where the source emission is weak (far-IR, near-UV) require more scans to achieve adequate SNR.
Thermal background in Raman: In Raman spectroscopy, the sample and optics emit thermal radiation following Planck's law. This thermal background becomes significant at low Raman shifts (<200 cm⁻¹) and high sample temperatures, sometimes obscuring weak Raman bands. Anti-Stokes/Stokes intensity ratios also follow a Boltzmann distribution governed by temperature.
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A blackbody is an idealized object that absorbs all incoming electromagnetic radiation and re-emits it as thermal radiation with a characteristic spectrum that depends only on its temperature. The spectral shape follows Planck's law — a single-peaked curve that shifts to shorter wavelengths as temperature increases. Real objects approximate blackbody behavior to varying degrees, described by their emissivity (0 to 1).
FTIR spectrometers use thermal infrared sources — typically a globar (silicon carbide rod heated to ~1300 K) or a Nernst glower (~1900 K). These sources emit near-blackbody radiation in the mid-IR range. Understanding their emission spectrum helps optimize detector sensitivity, plan measurement ranges, and interpret thermal background contributions in your spectra.
Emissivity (ε) is a dimensionless number between 0 and 1 that describes how efficiently a surface emits thermal radiation compared to a perfect blackbody. Polished metals have low emissivity (~0.05), oxidized metals are moderate (~0.6), and most organic materials and human skin are high (~0.95–0.98). A perfect blackbody has ε = 1.
Wien's displacement law states that the peak wavelength of blackbody emission is inversely proportional to temperature: λ_max = 2898 μm·K / T. As temperature increases, more energy is emitted at shorter wavelengths. At room temperature (~293 K), the peak is at ~9.9 μm (mid-IR). The Sun (~5778 K) peaks at ~0.50 μm (visible green-yellow). A blue star (~10,000 K) peaks in the UV.
Common FTIR infrared sources operate at 1000–2000 K. A globar (silicon carbide) typically runs at 1300–1500 K, peaking around 2.0–2.2 μm. A Nernst glower (zirconia/yttria) operates at 1800–2000 K, peaking around 1.5 μm. Both provide strong emission across the mid-IR fingerprint region (400–4000 cm⁻¹), which is why they are standard sources for FTIR measurements.
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