Air molecules near sea level are separated on average by about 3 nm (~10 molecular diameters). That is, the average volume allocated to each of N molecules in a volume V is V/N, the cube root of which is defined as the average separation. The separation r j between the ith and jth molecules, however, is distributed statistically from some minimum (approximately a molecular diameter) to some maximum (approximately the cube root of V). Because the lateral coherence length of sunlight is around 50 pm, we have to add the waves, taking due account of phases, scattered by all the air molecules (about 109) in a volume with approximately this linear dimension, which is about 100 times the wavelengths of visible light.

Suppose that N time-harmonic waves, all with the same amplitude a but different phases, are superposed. For sake of argument we take them to be plane waves, and omit the common factor exp(ikx — iwt). The sum of these waves is

Because these are time-harmonic waves, the time-averaged power transmitted by their sum is proportional to

W* = |^|2 = a2^Yl exp{i(^j —^m)} = a2N+a2 ^ ^ exp{i(^ — pm)}. (3.98)

The sum in the rightmost side of this equation is the addition of phasors (vectors in the complex plane) of unit length but with different polar angles (phase differences A j = ^j — ). We take Eq. (3.98) to be the total light scattered by the N air molecules in a volume equal to the coherence length cubed. The phase differences are given by Eq. (3.94), and hence Eq. (3.98) becomes

|*|2 = a2N + a2 ^ exp{ik(ei — es) • rjm}. (3.99)

Air molecules move and their positions are almost completely uncorrelated. That is, a molecule can occupy almost any position in space regardless of the positions of other molecules because the volume accessible to a molecule is about 1000 times its volume. This in turn implies that all phase differences are equally probable, and hence the sum in Eq. (3.99) is approximately zero, which is readily evident if you draw a great many phasors with random directions emanating from a point, similar to the spokes in a bicycle wheel (laced carelessly). Because the positions of air molecules are uncorrelated, scattering by N (in a small volume) is N times scattering by one. As far as scattering by air molecules is concerned, it is as if phase does not exist. This is true for all scattering directions except the forward direction (es = e;), for which scattering is in-phase regardless of the molecular separation relative to the wavelength. In fact, this in-phase scattering is the source of refraction (see Sec. 8.3 on atmospheric refraction and mirages). Historically, refraction has been treated separately from scattering, as if the two had nothing to do with each other. This is not true. Refraction is scattering; it can be looked upon as the coherent part of scattering. Scattered light not associated with refracted light is the incoherent part.

In 1899 Lord Rayleigh attributed the blue of the sky (see Sec. 8.1) to scattering by air molecules. Underlying his theoretical expression for the amount of light scattered by N air molecules was the assumption that the phases of the separate scattered waves are "entirely at random", and hence scattering by N air molecules is N times scattering by one. But he also recognized that "When the volume occupied by the molecules is no longer very small compared with the whole volume, the fact that two molecules cannot occupy the same space detracts from the random character of the distribution."

Scattering by air molecules is almost, but not quite, the same in all directions (isotropic), the reason for which is best left for the chapter on polarization (see Sec. 7.3; also Prob. 7.18). Angular scattering by air is also almost, but not quite, the same as that by a small (compared with the wavelength) sphere (see Fig. 3.13), small differences arising because the dominant air molecules are not spherically symmetric.

The Blue of the Sky: Scattering by Fluctuations or by Molecules?

You sometimes encounter the assertion that the blue of the sky (scattering by air molecules) is "really" a consequence of "scattering by fluctuations." This is piffle, reflecting ignorance of physics and its history, tantamount to denying the existence of molecules. The origins of this piffle go back almost 100 years to the work of Smoluchowski (1908) and Einstein (1910) who developed theories of scattering by dense media (e.g., liquids) by considering matter to be continuous but with spatially varying properties to which they applied thermodynamic arguments. Thermodynamics does not explicitly invoke molecules and so, of course, neither do the theories of Smoluchowski and Einstein, which, not surprisingly, contain thermodynamic variables such as temperature and isothermal compressibility. But this does not mean that Smoluchowski and Einstein believed that molecules are not the agents responsible for scattering. Einstein, in particular, recognized that his theory circumvented the difficulties of a molecular theory of scattering in fluids. He noted after his labors that "It is remarkable that our theory does not make direct use of the assumption of a discrete distribution of matter." But the fluctuation theories of Smoluchowski and Einstein have been distorted over the years into the fatuous notion that fluctuations, not molecules, do the scattering.

For an ideal gas, the theories of Smoluchowski and Einstein yield Rayleigh's result (which Einstein acknowledged) that scattering by N molecules is N times scattering by one, but their theories go further. In particular, they account for critical opalescence. At the critical point, the distinction between gas and liquid disappears and scattering greatly increases (according to theory it becomes infinite) as a fluid teeters between the liquid and gas phases. We give Einstein's scattering formula in Problem 5.16 and include a few problems that require this formula.

Many years later in 1945 Bruno Zimm tackled the problem of scattering in dense media by explicitly considering scattering by molecules. Zimm's own words demonstrate that he understood this scattering whereas the it-is-really-scattering-by-fluctuations folks do not: "the difficulty [of calculating interference between waves scattered by different molecules] was elegantly circumvented by Smoluchowski and Einstein, who considered the liquid as a continuous medium troubled by small statistical fluctuations in density. The extent of these fluctuations could be calculated from the macroscopic compressibility of the medium, and the intensity of the scattered light was obtained without discussing the individual molecules at all."

What Zimm did, in essence, was evaluate the sum in Eq. (3.99), approximating it by an integral and accounting for the correlations between molecular separations r^ by using results from statistical mechanics. The first term in Eq. (3.99) is scattering by N isolated molecules, the second term a consequence of interference of waves from molecules correlated in position. Zimm's molecular theory of scattering reproduces the results of Einstein and Smoluchowski but goes a step further in that it does not yield infinite scattering at the critical point.

Now let us turn to an ordinary glass of water, absolutely free of contamination (setting aside the difficulty of preparing such water). Molecules in liquid water are separated by distances comparable with their diameter, and hence one molecule cannot move without pushing others aside. Because of this correlation in position, scattering by N water molecules is not simply N times scattering by one. Indeed, scattering per molecule in the liquid phase is appreciably less than in the gas phase. When clean water is illuminated by a monodirectional

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