Figure 3.6: Simple model of an atom as a stationary positive nucleus (black sphere) surrounded by an equal but oppositely charged continuous (gray) electronic cloud. When the cloud is displaced a distance x, it experiences an electrostatic restoring force.

source of light, what is observed is described to good approximation by the laws of specular reflection and refraction. Although textbooks rarely say so, these are laws of scattering by coherent arrays. But there is always a residual incoherent component to the total scattering. When water is illuminated, most (but not all) of what is observed is accounted for by the laws of specular reflection and refraction. In particular, the reflected and refracted rays all lie in the plane determined by the normal to the surface of the water and the direction of the incident wave. Yet if we look in directions outside this plane we can observe scattered light, weak but measurable, and not necessarily the result of junk in the water. Water is intrinsically junky in that it is made up of discrete pieces (molecules), highly correlated pieces to be sure but not perfectly correlated.

Light from the sky is a consequence of scattering by molecules essentially uncorrelated in position. The spectrum of this light is given approximately by Eq. (3.9), Rayleigh's scattering law. Air molecules are certainly small enough compared with the wavelengths of visible light that this condition for the validity of the law is satisfied. But there is another condition, evident from Eq. (3.8): the frequencies for which Eq. (3.9) is a good approximation must be much less than a resonant frequency of the scatterer. What is this frequency for an air molecule?

To answer this question we resorted to a crude model of an atom: a continuous (negative) charge distribution (electron cloud) surrounding a positively charged point nucleus (see Fig. 3.6). When the center of the electron cloud coincides with the nucleus, there is no net force, but when the cloud is displaced it experiences a restoring force. Let q be the charge on the nucleus. The charge density of the electron cloud is therefore —3q/4nR3, where R is the atomic radius. When the cloud is displaced a distance x, it experiences an attractive force that pulls it back toward the center. This force is that between two point charges, the nucleus and one at the center with charge equal to the total (negative) charge within the sphere of radius x, which is -qx3/R3. From Coulomb's law we therefore have for the restoring force where e0 (the permittivity of free space) is a constant (8.85 x 10~12 in SI units). This equation should look familiar: it is the restoring force for a linear harmonic oscillator (Sec. 2.6). Thus the resonant frequency follows immediately from Eq. (3.76):

where we take m to be the mass of the electron cloud. If we take R = 0.15 nm for the atomic radius (strictly, this is the radius of molecular nitrogen), we obtain a resonant frequency that corresponds to a wavelength of about 0.1 pm, well into the ultraviolet.

Eq. (3.101) is an example of a plasma frequency. Plasma frequencies pop up all over the place, including in the ionosphere, for which plasma frequencies correspond to wavelengths of tens of meters. The plasma frequency is a cutoff frequency: below it a medium is reflecting, above it transmitting. This is why reflection of radio waves (for frequencies below the plasma frequency) by the ionosphere can be put to use for communication on Earth. To communicate beyond Earth would require waves above the plasma frequency. The late cosmologist, Fred Hoyle, wrote The Black Cloud, considered to be a classic of science fiction. Earth is menaced by an intelligent interstellar cloud that communicates with scientists on Earth by adjusting its plasma frequency.

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