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 that 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

$$\overline{u}(\omega) d\omega = \hbar \omega n(\omega) d\omega,$$ (8.119)

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

$$n(\omega, \theta) d\omega d\theta = \frac{1}{2} n(\omega) d\omega \sin\theta d\theta,$$ (8.120)

where $\sin\theta$ is proportional to the solid angle in the specified range of directions, and

$$\int_0^\pi n(\omega, \theta) d\omega d\theta = n(\omega) d\omega.$$ (8.121)

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

$$P(\omega) d\omega =\int_0^{\pi/2} c \cos\theta \hbar \omega n(\omega, \theta) d\omega d\theta,$$ (8.122)

where $c \cos\theta$ is the component of the photon velocity in the direction of the hole. This gives

$$P(\omega) d\omega = c \overline{u}(\omega) d\omega \frac... ... \cos\theta \sin\theta d\theta = \frac{c}{4} \overline{u}(\omega) d\omega,$$ (8.123)

so

$$P(\omega) d\omega = \frac{\hbar}{4\pi^{ 2} c^{ 2}} \frac{\omega^{ 3} d\omega} {\exp(\beta \hbar \omega)-1}$$ (8.124)

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 $\hbar \omega \simeq 3 k T$ . (See Exercise 15.) 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 colors (surprisingly enough, stars are fairly good black-bodies). Table 8.4 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 colors (which correspond quite well to the colors of the peak radiation) scan the whole visible spectrum, from red to blue, as the stellar surface temperatures gradually rise.

Table 8.4: Physical properties of some well-known stars.

Name	Constellation	Spectral Type	Surface Temp. (K)	Color
Antares	Scorpio	M	3300	Very Red
Aldebaran	Taurus	K	3800	Reddish Yellow
Sun		G	5770	Yellow
Procyon	Canis Minor	F	6570	Yellowish White
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 fairly 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--see Exercise 13), but its spectrum remained invariant. Nowadays, this primordial radiation is detectable as a faint microwave background that 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 Figure 8.3. This data can be fitted to a black-body curve of characteristic temperature $2.735$ K. In a very real sense, this can be regarded as the ``temperature of the universe.''

Figure: Cosmic background radiation spectrum measured by the Far Infrared Absolute Spectrometer (FIRAS) aboard the Cosmic Background Explorer satellite (COBE). The fit is to a black-body spectrum of characteristic temperature $2.735\pm 0.06$ K. [Data from J.C. Mather, et al., Astrophysical Journal Letters 354, L37 (1990).]