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Black-body radiation

Suppose that we were to make a small hole in the wall of our enclosure, and observe the emitted radiation. A small hole is the best approximation in Physics to a black-body, which is defined as an object which absorbs, and, therefore, emits, radiation perfectly at all wavelengths. What is the power radiated by the hole? Well, the power density inside the enclosure can be written
\overline{u}(\omega)\,d\omega = \hbar\,\omega\,\, n(\omega)\, d\omega,
\end{displaymath} (647)

where $n(\omega)$ is the mean number of photons per unit volume whose frequencies lie in the range $\omega$ to $\omega+d \omega$. The radiation field inside the enclosure is isotropic (we are assuming that the hole is sufficiently small that it does not distort the field). It follows that the mean number of photons per unit volume whose frequencies lie in the specified range, and whose directions of propagation make an angle in the range $\theta$ to $\theta + d\theta$ with the normal to the hole, is
n(\omega, \theta)\,d\omega\,d\theta =
\end{displaymath} (648)

where $\sin\theta$ is proportional to the solid angle in the specified range of directions, and
\int_0^\pi n(\omega, \theta)\,d\omega\,d\theta = n(\omega)
\end{displaymath} (649)

Photons travel at the velocity of light, so the power per unit area escaping from the hole in the frequency range $\omega$ to $\omega+d \omega$ is
P(\omega)\, d\omega =\int_0^{\pi/2} c\,\cos\theta\,\,\hbar\,\omega\,\,n(\omega, \theta)
\end{displaymath} (650)

where $c\,\cos\theta$ is the component of the photon velocity in the direction of the hole. This gives
P(\omega)\, d\omega = c \,\,\overline{u}(\omega)\, d\omega\,...
...\theta\,d\theta = \frac{c}{4} \,\overline{u}(\omega)\,d\omega,
\end{displaymath} (651)

P(\omega)\,d\omega = \frac{\hbar}{4\pi^2\, c^2} \frac{\omega^3\,d\omega}
\end{displaymath} (652)

is the power per unit area radiated by a black-body in the frequency range $\omega$ to $\omega+d \omega$.

A black-body is very much an idealization. The power spectra of real radiating bodies can deviate quite substantially from black-body spectra. Nevertheless, we can make some useful predictions using this model. The black-body power spectrum peaks when $\hbar\,\omega\,\simeq 3 \,k\,T$. This means that the peak radiation frequency scales linearly with the temperature of the body. In other words, hot bodies tend to radiate at higher frequencies than cold bodies. This result (in particular, the linear scaling) is known as Wien's displacement law. It allows us to estimate the surface temperatures of stars from their colours (surprisingly enough, stars are fairly good black-bodies). Table 9 shows some stellar temperatures determined by this method (in fact, the whole emission spectrum is fitted to a black-body spectrum). It can be seen that the apparent colours (which correspond quite well to the colours of the peak radiation) scan the whole visible spectrum, from red to blue, as the stellar surface temperatures gradually rise.

Table 9: Physical properties of some well-known stars
Name Constellation Spectral Type Surf. Temp. ($^\circ $ K) Colour
Antares Scorpio M 3300 Very Red
Aldebaran Taurus K 3800 Reddish
Sun   G 5770 Yellow
Procyon Canis Minor F 6570 Yellowish
Sirius Canis Major A 9250 White
Rigel Orion B 11,200 Bluish White

Probably the most famous black-body spectrum is cosmological in origin. Just after the ``big bang'' the Universe was essentially a ``fireball,'' with the energy associated with radiation completely dominating that associated with matter. The early Universe was also pretty well described by equilibrium statistical thermodynamics, which means that the radiation had a black-body spectrum. As the Universe expanded, the radiation was gradually Doppler shifted to ever larger wavelengths (in other words, the radiation did work against the expansion of the Universe, and, thereby, lost energy), but its spectrum remained invariant. Nowadays, this primordial radiation is detectable as a faint microwave background which pervades the whole universe. The microwave background was discovered accidentally by Penzias and Wilson in 1961. Until recently, it was difficult to measure the full spectrum with any degree of precision, because of strong microwave absorption and scattering by the Earth's atmosphere. However, all of this changed when the COBE satellite was launched in 1989. It took precisely nine minutes to measure the perfect black-body spectrum reproduced in Fig. 9. This data can be fitted to a black-body curve of characteristic temperature $2.735~^\circ$ K. In a very real sense, this can be regarded as the ``temperature of the Universe.''

Figure 9: Cosmic background radiation spectrum measured by the Far Infrared Absolute Spectrometer (FIRAS) aboard the Cosmic Background Explorer satellite (COBE).
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next up previous
Next: The Stefan-Boltzmann law Up: Quantum statistics Previous: The Planck radiation law
Richard Fitzpatrick 2006-02-02