Black-Body Radiation

Suppose that we were to make a small hole in the wall of the enclosure described in the previous section, and were then to 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?

The power density inside the enclosure can be written

$$u(\omega)\,d\omega = \hbar\,\omega\,\, n(\omega)\, d\omega,$$ (5.472)

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 subtend an angle in the range $\theta$ to $\theta+d\theta$ with the normal to the hole, is

$$n(\omega, \theta)\,d\omega\,d\theta = n(\omega)\,d\omega\,g(\theta)\,d\theta,$$ (5.473)

where $g(\theta)\,d\theta = (1/2)\,\sin\theta\,d\theta$ is the fractional range of solid angle in the specified range of directions. (See Section 5.3.2.) The previous two equations give

$$\hbar\,\omega\,n(\omega,\theta)= \frac{1}{2}\,u(\omega)\,\sin\theta.$$ (5.474)

Photons travel at the speed of light, so the power per unit area escaping from the hole in the frequency range $\omega$ to $\omega+d\omega$ is

$$P(\omega)\, d\omega =\int_0^{\pi/2} c\,\cos\theta\,\,\hbar\,\omega\,\,n(\omega, \theta) \,d\omega\,d\theta,$$ (5.475)

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

$$P(\omega)\, d\omega = c \,u(\omega)\, d\omega\,\frac{1}{2}\!\int_0^{\pi/2} \cos\theta\,\sin\theta\,d\theta = \frac{c}{4} \,u(\omega)\,d\omega,$$ (5.476)

so

$$P(\omega)\,d\omega = \frac{\hbar}{4\pi^{2}\, c^{2}} \frac{\omega^{3}\,d\omega} {\exp(\hbar\,\omega/k_B\,T)-1}$$ (5.477)

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 $\hbar\,\omega\,\simeq 3 \,k_B\,T$, implying 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, after Wilhelm Wein who derived it in 1893, and allows us to estimate the surface temperatures of stars from their colors (surprisingly enough, stars are fairly good black-bodies). Table 5.2 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 5.2: Physical properties of some well-known stars.

Name	Constellation	Surface Temp. (K)	Color
Antares	Scorpio	3300	Very Red
Aldebaran	Taurus	3800	Reddish Yellow
Sun		5770	Yellow
Procyon	Canis Minor	6570	Yellowish White
Sirius	Canis Major	9250	White
Rigel	Orion	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, but its spectrum remained invariant). Nowadays, this primordial radiation is detectable as a faint microwave background that pervades the whole universe. The cosmic microwave background was discovered accidentally by Arno Penzias and Robert Wilson in 1964. For many years, it was difficult to measure the full spectrum of the microwave background with any degree of precision, because of strong absorption and scattering of microwaves 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 5.8. The data shown in the figure 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: 5.8 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.