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How to use this blackbody radiation calculatorBlackbody radiationPlanck's law for blackbody radiationCalculating blackbody spectrum radiation with wavelength or wavenumberHow to calculate blackbody radiation in photon spaceCalculating peak wavelength of blackbody radiationEmissivity of a real body

Our blackbody radiation calculator will help you calculate the radiation spectrum of a blackbody or a body that can be closely approximated as one. All you need to provide are the body's temperature and emissivity to determine its radiance and peak spectral radiance. You can even specify a particular wave's wavelength, frequency, or wavenumber to find its spectral radiance!

You can also calculate blackbody radiation in wave space (i.e., watts) or photon space (i.e., photons per second)!

You would be forgiven if any of the words above sounded alien to you. But blackbody radiation is the fundamental concept that led us towards quantum physics! The following article will help you familiarize yourself with these terms as we learn Planck's law and how to calculate blackbody radiation from its temperature.

Before we dive in, we recommend you give our wavelength calculator a quick read to understand the terms wavelength, frequency, and wavenumber.

How to use this blackbody radiation calculator

Our blackbody radiation spectrum calculator is a versatile tool:

1. Select which spectral parameter to use for calculation. You can choose between wavelength, frequency, and wavenumber. For example, select wavelength if you want to use wavelength to calculate blackbody radiation.

2. Select whether you wish to calculate in wave space (i.e., in watts) or photon space (i.e., in photons/second).

3. Enter the characteristics of the blackbody — its temperature and emissivity.

4. The blackbody radiation calculator will instantly determine the total radiance and radiance emittance in wave space. If you select the photo space option, it will also determine the total photon radiance and photon radiance emittance in photon space.

5. It will also automatically calculate the peak spectral parameter (peak wavelength, frequency, or wavenumber) of the radiation curve and the corresponding peak spectral radiance. Similarly, by selecting the photon space option, the calculator determines the peak spectral parameter and the peak spectral photon radiance in photon space.

6. To calculate the spectral radiance (or spectral photon radiance), provide the corresponding spectral parameter.

A blackbody is an ideal body that absorbs all electromagnetic radiation incident on it, reflecting nothing back. A perfect blackbody doesn't exist in nature, but some objects can approximate a blackbody.

Blackbody radiation is the electromagnetic radiation emitted spontaneously by a blackbody in thermal equilibrium with its surroundings. The radiation spectrum is continuous and depends only on the body's temperature.

When the body is at room temperature, most of the spectrum lies in the infrared region, invisible to the naked eye. If the body were hotter than 500 °C (930 °F), some of the emission would fall in the visible light region, and the body would glow dull red.

Although they may not be perfect black bodies, the thermal radiation emitted by many objects can be approximated as blackbody radiation, especially as an initial approximation.

The classical theory of radiation could not explain blackbody radiation successfully because it assumed that energy is continuous. According to classical theory, the energy radiated would tend to infinity with higher frequency (i.e., shorter wavelength), a problem known as the ultraviolet catastrophe.

In 1901, Max Planck proposed that energy is not continuous but quantized — it can only increase or decrease in small increments. He successfully explained blackbody radiation and calculated its radiation curve. This paved the way for what would become quantum physics.

According to Planck's law, the spectral radiance of a body for a given radiation frequency is given by:

$B_\nu (\nu, T) = \frac{2h \nu^3}{c^2} \frac{1}{e^{h \nu/k_\text{B} T} - 1}$

where:

• $B_\nu$Spectral radiance of frequency $\nu$;

• $\nu$ — Frequency of the radiation;

• $T$ — Temperature of the blackbody;

• $h$ — Planck's constant, $6.62607015 \times10^{-34} \text{ J/Hz}$;

• $c$ — Speed of light in vacuum, $299792458 \text{ m/s}$;

• $e$ — Euler's number, $\approx 2.71828$; and

• $k_\text{B}$ — Boltzmann constant, $1.380649 \times10^{-23} \text{ J/K}$.

Spectral radiance refers to the power (or energy per second) the blackbody radiates in the frequency spectrum $\nu$. Its units are watts per solid angle in steradians (sr) per square meter of the body surface area per frequency $(\rm W/sr/m^2/Hz)$.

Since the body releases energy in all frequencies, the total radiance $B$ (called radiance) emitted by a blackbody is an integration over all frequencies:

$B = \int_0^\infin B_\nu d\nu = \frac{2\pi^4 k_\text{B}^4}{15h^3c^2}T^4$

The radiance is measured in $\rm W/sr/m^2$.

The radiance emittance or the radiance exitance $M$ is the total power radiated per unit surface area. For an ideal blackbody, the total radiance emittance is given by:

$M = \pi \cdot B = \frac{2\pi^5 k_\text{B}^4}{15h^3c^2}T^4$

The radiance emittance is measured in $\rm W/m^2$.

🔎 Notice that the radiance and radiance emittance depend only on the body temperature. So, we need only the body temperature to calculate the blackbody radiation's total radiance or radiance emittance. Meanwhile, the spectral radiance depends on the temperature and the spectral parameter (frequency, wavelength, or wavenumber).

Calculating blackbody spectrum radiation with wavelength or wavenumber

We have seen Planck's law used to calculate blackbody spectrum radiation using its frequency. But the spectral radiance can also be calculated using the spectrum wavelength or wavenumber.

An alternative form of Planck's law that uses wavelength to calculate blackbody radiation is:

$B_\lambda (\lambda, T) = \frac{2h c^2}{\lambda^5} \frac{1}{e^{h c/\lambda k_\text{B} T} - 1}$

where:

• $B_\lambda$Spectral radiance based on the radiation's wavelength; and
• $\lambda$ — Wavelength of the radiation spectrum.

The spectral radiance based on wavelength is measured in $\rm W/sr/m^2/\text{μ} m$ if the wavelength is in $\rm \text{μ} m$.

We can also calculate blackbody radiation using its wavenumber:

$B_{\overline{\nu}} = 2h c^2\overline{\nu}^3\frac{1}{e^{h c\overline{\nu}/ k_\text{B} T} - 1}$

where:

• $B_{\overline{\nu}}$Spectral radiance based on the radiation's wavenumber; and
• $\overline{\nu}$ — Wavenumber of the radiation spectrum.

The spectral radiance based on wavenumber is measured in $\rm W/sr/m^2/cm^{-1}$ when the wavenumber is in $\rm cm^{-1}$.

How to calculate blackbody radiation in photon space

So far, we have looked at radiation in "wave space" — we calculated the power in a blackbody radiation wave. We can also calculate the photons emitted per second. To do this, we divide the equations in the previous sections with the energy of a photon.

🔎 If you're curious about the wave-particle duality, explore our De Broglie wavelength calculator.

Dividing the spectral radiance based on frequency with the energy of a photon, $h\nu$, we get:

$B_\nu^P (\nu, T) = \frac{2 \nu^2}{c^2} \frac{1}{e^{h \nu/k_\text{B} T} - 1}$

where $B_\nu^P$ is the spectral photon radiance in the given frequency, measured in $\rm photons/s/sr/m^2/Hz$.

Similarly, dividing the spectral radiance based on wavelength with the energy of a photon, $hc/\lambda$:

$B_\lambda^P (\lambda, T) = \frac{2 c}{\lambda^4} \frac{1}{e^{h c/\lambda k_\text{B} T} - 1}$

where $B_\lambda^P$ is the spectral photon radiance in the given wavelength, measured in $\rm photons/s/sr/m^2/\text{μ} m$.

Further, dividing the spectral radiance based on wavenumber by a photon's energy $hc\overline{\nu}$ gives us:

$B_{\overline{\nu}}^P = 2 c\overline{\nu}^2\frac{1}{e^{h c\overline{\nu}/ k_\text{B} T} - 1}$

where $B_{\overline{\nu}}^P$ is the spectral photon radiance in the given wavenumber, measured in $\rm photons/s/sr/m^2/cm^{-1}$.

Similar to how we obtain the radiance in wave space, the photon radiance is given by:

$\begin{split} B^P &= \int_0^\infin\!\! B_\nu^P (\nu, T)\\[1.2em] &= \frac{4 \zeta(3)k^3}{h^3c^2}T^3 \end{split}$

where:

• $B^P$ — Total photon radiance over all frequencies, in $\rm photons/s/sr/m^2$; and
• $\zeta$ — Reimann zeta function, $\zeta(3) \approx 1.202056903159594$.

The photon emittance or photon exitance is:

$M^P = \pi \cdot B^P = \frac{4 \pi \zeta(3)k^3}{h^3c^2}T^3$

where $M^P$ is photon exitance in $\rm photons/s/m^2$.

Calculating peak wavelength of blackbody radiation

Understanding where the radiation curve peaks for a blackbody is helpful. We saw that the spectral radiance based on wavelength is given by:

$B_\lambda (\lambda, T) = \frac{2h c^2}{\lambda^5} \frac{1}{e^{h c/\lambda k_\text{B} T} - 1}$

To find the maximum point on the curve $B_\lambda$ (the peak), we differentiate $B_\lambda$ with respect to $\lambda$ and equate it to $0$:

$\begin{split} \frac{d B_\lambda }{d\lambda } &= \frac{d }{d\lambda }\left( \frac{2h c^2}{\lambda^5} \frac{1}{e^{h c/\lambda k_\text{B} T} - 1}\right)\\ &= 0 \end{split}$

Differentiating and solving for $\lambda$ gives us the peak wavelength — the wavelength at which the spectral radiance is maximum:

$\lambda_{\rm peak} = \frac{hc}{a_5 k_\text{B}T}$

where:

• $\lambda_{\rm peak}$ — Calculated peak wavelength of blackbody radiation; and
• $a_5$ — A constant approximately equal to $4.96511423174$.

To find the peak spectral radiance, we need only to use this peak wavelength in our spectral radiance by wavelength formula.

In a similar fashion, we can obtain the peak frequency and peak wavenumber for spectral radiance:

\begin{align*} \nu_{\rm peak} = \frac{a_3k_\text{B} }{h}T\\[1em] \overline{\nu}_{\rm peak} = \frac{a_3k_\text{B} }{hc}T \end{align*}

where:

• $\nu_{\rm peak}$ — Peak frequency of the blackbody radiation;

• $\overline{\nu}_{\rm peak}$ — Peak wavenumber of the blackbody radiation; and

• $a_3$ — A constant, $a_3 \approx 2.82143937212$.

Further, we can calculate the peak spectral parameters in the photon space in the same way, as summarized below:

\begin{align*} \lambda_{\rm peak}^P &= \frac{hc}{a_4k_\text{B} T}\\[1em] \nu_{\rm peak}^P &= \frac{a_2k_\text{B} }{h}T\\[1em] \overline{\nu}_{\rm peak}^P &= \frac{a_2k_\text{B} }{hc}T \end{align*}

where:

• $\lambda_{\rm peak}^P$ — Wavelength at which peak spectral photon radiance occurs;

• $a_4$ — A constant where $a_4 \approx 3.92069039487$;

• $\nu_{\rm peak}^P$ — Frequency at which peak spectral photon radiance occurs;

• $a_2$ — A constant where $a_2 \approx 1.59362426004$; and

• $\overline{\nu}_{\rm peak}^P$ — Wavenumber at which peak spectral photon radiance occurs;

Emissivity of a real body

A real body only emits a fraction of the power of an ideal blackbody. Emissivity $\varepsilon$ is the ratio of how much power a natural body emits to the power radiated by a perfect blackbody.

If you know the emissivity of a body, you can use it to adjust the equations we've discussed so far and calculate the radiation of the actual body. For example, the spectral radiance (by wavelength) of a natural body is:

\begin{align*} \varepsilon &= \frac{B_\lambda^*}{B_\lambda}\\[1em] \implies B_\lambda^* &= \varepsilon B_\lambda\\[1em] B_\lambda^* &= \frac{2\varepsilon h c^2}{\lambda^5} \frac{1}{e^{h c/\lambda k_\text{B} T} - 1} \end{align*}

where:

• $\varepsilon$ — Emissivity of the natural body; and
• $B_\lambda^*$ — Spectral radiance (by wavelength) of the natural body.

🙋 Emissivity is a ratio that lies between 0 and 1. The emissivity of an ideal blackbody is 1.

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