# 1.1: Blackbody Radiation Cannot Be Explained Classically

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All normal matter at temperatures above absolute zero emits electromagnetic radiation, which represents a conversion of a body's internal thermal energy into electromagnetic energy, and is therefore called *thermal radiation*. Conversely, all normal matter *absorbs *electromagnetic radiation to some degree. An object that absorbs ALL radiation falling on it, at all wavelengths, is called a blackbody. When a blackbody is at a uniform temperature, its emission has a characteristic frequency distribution that depends on the temperature. This emission is called *blackbody radiation*.

A room temperature blackbody appears black, as most of the energy it radiates is infra-red and cannot be perceived by the human eye. Because the human eye cannot perceive light waves at lower frequencies, a black body, viewed in the dark at the lowest just faintly visible temperature, subjectively appears grey, even though its objective physical spectrum peaks in the infrared range. When it becomes a little hotter, it appears dull red. As its temperature increases further it becomes yellow, white, and ultimately blue-white.

Blackbody radiation has a characteristic, continuous frequency spectrum that experimentally depends only on the body's temperature. In fact, we can be much more precise:

A body emits radiation at a given temperature and frequency

exactlyas well as it absorbs the same radiation.

This statement was proved by Gustav Kirchhoff: the essential point is that if we *instead* suppose a particular body can absorb better than it emits, then in a room full of objects all at the same temperature, it will absorb radiation from the other bodies better than it radiates energy back to them. This means it will get hotter, and the rest of the room will grow colder, contradicting the second law of thermodynamics. *Thus, a body must emit radiation exactly as well as it absorbs the same radiation at a given temperature and frequency in order to not violate the second law of thermodynamics.*

Any body at any temperature above absolute zero **will **radiate to some extent, the intensity and frequency distribution of the radiation depending on the detailed structure of the body. To begin analyzing heat radiation, we need to be specific about the body doing the radiating: *the simplest possible case is an idealized body which is a perfect absorber, and therefore also (from the above argument) a perfect emitter. *So how do we construct a perfect absorber in the laboratory? In 1859 Kirchhoff had a good idea: a small hole in the side of a large box is an excellent absorber, since any radiation that goes through the hole bounces around inside, a lot getting absorbed on each bounce, and has little chance of ever getting out again. So, we can do this *in reverse*: have an oven with a tiny hole in the side, and presumably the radiation coming out the hole is as good a representation of a perfect emitter as we’re going to find (Figure \(\PageIndex{2}\)).

By the 1890’s, experimental techniques had improved sufficiently that it was possible to make fairly precise measurements of the energy distribution of blackbody radiation. In 1895, at the University of Berlin, Wien and Lummer punched a small hole in the side of an otherwise completely closed oven, and began to measure the radiation coming out. The beam coming out of the hole was passed through a diffraction grating, which sent the different wavelengths/frequencies in different directions, all towards a screen. A detector was moved up and down along the screen to find how much radiant energy was being emitted in each frequency range. They found a radiation intensity/frequency curve close to the distributions in Figure \(\PageIndex{3}\).

By measuring the blackbody emission curves at different temperatures (Figure \(\PageIndex{3}\)), they were also able to construct two important phenomenological Laws (i.e., formulated from experimental observations, not from basic principles of nature): **Stefan-Boltmann’s Law** and **Wien’s Displacement Law**.

## The Stefan-Boltzmann Law

The first quantitative conjecture based on experimental observations was the **Stefan-Boltzmann Law*** *(1879) which states the total power (i.e., integrated over all emitting frequencies in Figure \(\PageIndex{3}\)) radiated from one square meter of black surface goes as the *fourth power* of the absolute temperature (Figure \(\PageIndex{4}\)):

\[P = \sigma T^4 \label{Eq1}\]

where

- \(P\) is the total amount of radiation emitted by an object per square meter (\(Watts\; m^{-2}\))
- \(\sigma\) is a constant called the Stefan-Boltzman constant (\(5.67 \times 10^{-8}\, Watts\; m^{-2} K^{-4}\))
- \(T\) is the absolute temperature of the object (in K)

The Stefan-Boltzmann Law is easily observed by comparing the integrated value (i.e., under the curves) of the experimental black-body radiation distribution in Figure \(\PageIndex{3}\) at different temperatures. In 1884, Boltzmann derived this \(T^4\) behavior from theory by applying classical thermodynamic reasoning to a box filled with electromagnetic radiation, using Maxwell’s equations to relate pressure to energy density. That is, the tiny amount of energy coming out of the hole (Figure \(\PageIndex{2}\)) would of course have the same temperature dependence as the radiation intensity inside.

## Wien’s Displacement Law

The second phenomenological observation from experiment was **Wien’s Displacement Law**. Wien's law identifies the dominant (peak) wavelength, or color, of light coming from a body at a given temperature. As the oven temperature varies, so does the frequency at which the emitted radiation is most intense (Figure \(\PageIndex{3}\)). In fact, that frequency is directly proportional to the absolute temperature:

\[\nu_{max} \propto T \label{Eq2}\]

where the proportionality constant is \(5.879 \times 10^{10} Hz/K\).

Wien himself deduced this law theoretically in 1893, following Boltzmann’s thermodynamic reasoning. It had previously been observed, at least semi-quantitatively, by an American astronomer, Langley. This upward shift in \(\nu_{max}\) with \(T\) is familiar to everyone—when an iron is heated in a fire (Figure \(\PageIndex{1}\)), the first visible radiation (at around 900 K) is deep red, the lowest frequency visible light. Further increase in \(T\) causes the color to change to orange then yellow, and finally blue at very high temperatures (10,000 K or more) for which the peak in radiation intensity has moved beyond the visible into the ultraviolet.

Another representation of Wien's Law (Equation \(\ref{Eq2}\)) in terms of the peak wavelength of light is

\[\lambda_{max} = \dfrac{b}{T} \label{Eq2a}\]

where \(T\) is the absolute temperature in kelvin and \(b\) is a constant of proportionality called Wien's displacement constant, equal to \(2.89 \times 10^{−3} m\, K\), or more conveniently to obtain wavelength in micrometers, \(b≈2900\; μm \cdot K\). This is an inverse relationship between wavelength and temperature. So the higher the temperature, the shorter or smaller the wavelength of the thermal radiation. The lower the temperature, the longer or larger the wavelength of the thermal radiation. For visible radiation, hot objects emit bluer light than cool objects.

Remember that thermal radiation always spans a wide range of wavelengths (Figure \(\PageIndex{2}\)) and Equation \ref{Eq2a} only specifies the single wavelength that is the **peak **of the spectrum. So although the Sun appears yellowish-white, when you disperse sunlight with a prism you see radiation with all the colors of the rainbow. Yellow just represents a characteristic wavelength of the emission.

## The Rayleigh-Jeans Law

Lord Rayleigh and J. H. Jeans developed an equation which explained blackbody radiation at low frequencies. The equation which seemed to express blackbody radiation was built upon all the known assumptions of physics at the time. *The big assumption which Rayleigh and Jean implied was that infinitesimal amounts of energy were continuously added to the system when the frequency was increased*. Classical physics assumed that energy emitted by atomic oscillations could have any continuous value. This was true for anything that had been studied up until that point, including things like acceleration, position, or energy. Their resulting **Rayleigh-Jeans Law **was

\[ \begin{align} d\rho \left( \nu ,T \right) &= \rho_{\nu} \left( T \right) d\nu \\[4pt] &= \dfrac{8 \pi k_B T}{c^3} \nu^2 d\nu \label{Eq3} \end{align}\]

Experimental data performed on the black box showed slightly different results than what was expected by the Rayleigh-Jeans law (Figure \(\PageIndex{5}\)). The law had been studied and widely accepted by many physicists of the day, but the experimental results did not lie, something was different between what was theorized and what actually happens. The experimental results showed a bell type of curve, but according to the Rayleigh-Jeans law the frequency diverged as it neared the ultraviolet region (Equation \(\ref{Eq3}\)). Ehrenfest later dubbed this the "ultraviolet catastrophe".

## Contributors

Michael Fowler (Beams Professor, Department of Physics, University of Virginia)

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")

Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/85abf193-2bd...a7ac8df6@9.110).

- ACuriousMind (StackExchange)