THEORY OF LIGHT SCATTERING AND ITS APPLICATION TO PROBLEMS OF ATMOSPHERIC AND FOG TRANSPARENCY

I. A. Khvostikov, Moscow

Part I

MOLECULAR SCATTERING OF LIGHT

1. Rayleigh’s theory. The starting point of modern ideas about the mechanism of light scattering in the atmosphere is Rayleigh’s theory, developed by him in five memoirs123, published in 1871–1899. The main conclusions of this theory are widely known; nevertheless, we shall dwell on its principal points, since this is necessary for a clear understanding of the difficulties encountered in applying the theory to the real atmosphere, and of the significant improvements that the theory subsequently underwent.

The origin of Rayleigh’s theory is connected with the question of the color of the sky. Why is the sky blue? Attempts to answer this question have their own interesting history.

The hypothesis proposed by Leonardo da Vinci is characteristic of that time. From Aristotle to Newton, color in general was regarded as the result of a “mixture of light and darkness” in one proportion or another. And Leonardo supposed that the blue color of the sky is a mixture of white light, reflected by the atmosphere, with the blackness of world space.

Newton, by analogy with the appearance of colors in the interference of rays reflected from the front and rear surfaces of transparent objects (for example, the coloration of soap bubbles), concluded that the blue color of the sky arises owing to the presence in the atmosphere of the finest drops of water; this color is the closest to the central dark spot in “Newton’s rings” blue color, which can arise through interference4. Newton’s explanation, based on analogy, is erroneous, but, like much else that remained from Newton, it was accepted by everyone for a very long time—more than a century and a half.

Facts directly contradicting Newton’s view had been known for a long time; they concern the polarization properties of the light of the sky. As early as 1811 Arago\(^5\) discovered that the light of the celestial vault is partially polarized and that this polarization is greatest at an angle of \(90^\circ\) from the Sun. On the other hand, polarization upon reflection had already been studied by that time (Brewster’s law was discovered in 1815), and it could easily have been shown that the polarization properties of the celestial vault cannot be explained if one proceeds from Newton’s view of reflections from droplets of water.

Nevertheless, as late as 1847 Clausius shared Newton’s view, even attempting to improve the theory somewhat. By calculation\(^6\) he showed that a cloud consisting of droplets of small size should greatly increase the visible sizes of the heavenly bodies. He further showed that these difficulties could be avoided if, instead of the small droplets assumed by Newton, one proceeded from the notion of thin-walled bubbles of large radius.

The essence of the phenomenon was understood only as a result of the experimental study of the scattering of light by the smallest particles. The first investigations belong to Brücke\(^7\), who in 1853 established experimentally that the blue color of the sky can arise owing to the scattering of light by particles too small to reflect specularly. He found that a transparent medium with very small particles suspended in it appears blue when illuminated by white light. Soon Tyndall\(^8\), who carefully studied this phenomenon, found that the blue light scattered by very small particles is polarized, and that the greatest polarization was observed at a right angle to the incident rays.

Two years after the publication of Tyndall’s results, Rayleigh printed his first work on the theory of scattering\(^1\).

In his early investigations Rayleigh proceeded from the idea that the centers of scattering are small foreign particles suspended in the medium. Rayleigh treated the phenomenon by considering light waves as waves in an elastic ether. Later he gave a calculation of this phenomenon on the basis of the electromagnetic theory of light.

A decisive step was taken by Rayleigh in a work of 1899: he advanced the bold hypothesis that the centers of scattering are not foreign suspended particles, but the molecules of the medium themselves. Since air (the sky) appears the bluer the purer it is, the blue scattered light should be attributed to the air itself\(^3\).

The conception of molecules as centers of light scattering had a definite influence in that period on the formation of the molecular theory of matter.

Rayleigh greatly simplified the problem of calculating the scattering of light by assuming that the scattering is caused by spherical particles whose dimensions are extremely small in comparison with the wavelength of light.

If we have a homogeneous and unbounded electric field \(E\), then, introducing into it a sphere of radius \(a\) with dielectric po-

constant \(\varepsilon\), we shall also obtain inside the sphere a homogeneous field with intensity

\[ E_1 = E - E \frac{\varepsilon - 1}{\varepsilon + 2}. \]

As for the space external to the sphere, there the field will be

\[ E_2 = E + E', \]

where \(E'\) is the field produced by the sphere in the external space. The latter may be represented approximately by the field of a dipole with moment

\[ p = E a^3 \frac{\varepsilon - 1}{\varepsilon + 2}, \tag{1} \]

located at the center of the sphere.

If the particle is very small in comparison with the wavelength \(\lambda\), then the field may be regarded as the same at all its points. If the particle is in the field of an electromagnetic wave, then the field \(E\) (homogeneous) will be periodically variable. The dipole moment \(p\) in formula (1) will also be periodically variable, and thus the problem is reduced to the radiation of a dipole.

Fig. 1

Let us suppose that a monochromatic wave of frequency

\[ \omega = \frac{2\pi c}{\lambda} \]

is incident on the particle. Writing the equation of oscillation with frequency \(\omega\) in the form

\[ x = \xi e^{i\omega t}, \tag{a} \]

we have for the acceleration

\[ \ddot{x} = -\omega^2 \xi e^{i\omega t} = -\omega^2 x. \tag{b} \]

Let \(P\) in Fig. 1 be the scattering particle, and \(Q\) the point of observation, separated from \(P\) by the distance \(PQ = r\). A plane wave of frequency \(\omega\) propagates in the direction \(OP\), so that the angle \(OPQ = \varphi\) is the scattering angle. Assuming the particle \(P\) to possess a charge \(e\), executing forced oscillations described by formula (a), we can characterize the radiation of the particle \(P\) at the point \(Q\) by the Poynting vector, directed along \(PQ\) and equal to

\[ S = \frac{e^2 \ddot{x}^2}{4\pi c^3 r^2} \sin^2 \vartheta, \]

or, substituting \(\ddot{x}\) from formula (b) and taking the time average, we obtain:

\[ \overline{S} = \frac{\omega^4}{4\pi c^3 r^2} \frac{\sin^2 \vartheta}{2} (e\xi)^2, \]

where \(\vartheta\) is the angle formed by the direction of the radius vector \(r\) with the direction of oscillation of the charge \(e\) (the dipole moment). Let us decompose the electric vector of the incident wave into two components, of which

one perpendicular to the scattering plane \(OPQ\) (direction I in Fig. 1), and the other lies in this plane (direction II). For I: \(\vartheta=\dfrac{\pi}{2}\) and \(\sin\vartheta=1\), for II: \(\vartheta=\dfrac{\pi}{2}+\varphi\) and \(\sin\vartheta=-\cos\varphi\). The field (intensity) of the scattering particle \(P\) at the point \(Q\) as the sum of both components can be represented in the form

\[ \overline{S}=\frac{\omega^{4}}{4\pi c^{3}r^{2}}\frac{1+\cos^{2}\varphi}{2}(e\overline{\xi})^{2} \]

or

\[ \overline{S}=\frac{4\pi^{3}c}{\lambda^{4}r^{2}}\frac{1+\cos^{2}\varphi}{2}(e\overline{\xi}^{2})^{2}. \]

For the radiation of a unit volume of the substance we obtain

\[ \overline{S}_{1}=\frac{4\pi^{3}c}{\lambda^{4}r^{2}}\frac{1+\cos^{2}\varphi}{2}\sum(e\overline{\xi})^{2}, \]

where the sign \(\sum\) denotes ordinary summation, since the different particles radiate incoherently. But \(\sum(e\overline{\xi})\) is the electric moment of a unit volume, which for a sufficiently rarefied gas in a field \(E\) will be equal to

\[ \sum(e\overline{\xi})=\frac{\mu^{2}-1}{4\pi}E, \tag{2} \]

where \(\mu\) is the refractive index of the medium. For a gas of high density or for a liquid, in formula (2) the expression \(\mu^{2}-1\) would have to be replaced by the expression \(\dfrac{3(\mu^{2}-1)}{\mu^{2}+2}\).

From expression (2) we obtain \(e\overline{\xi}=\dfrac{\mu^{2}-1}{4\pi N}E\), where \(N\) is the number of particles in a unit volume, and consequently,

\[ \sum(e\overline{\xi})^{2}=N(e\overline{\xi})^{2}=\frac{(\mu^{2}-1)^{2}}{16\pi^{2}N}E^{2} \]

and

\[ \overline{S}_{1}=\frac{\pi c(1+\cos^{2}\varphi)}{8r^{2}N\lambda^{4}}(\mu^{2}-1)^{2}E^{2}. \]

But the time-averaged value of the Poynting vector for the incident wave will be \(\overline{S}_{0}=\dfrac{c}{4\pi}E^{2}\). Consequently, for the ratio \(\dfrac{\overline{S}_{1}}{\overline{S}_{0}}\), characterizing the scattering capacity of the medium, we have

\[ \frac{\overline{S}_{1}}{\overline{S}_{0}}=\frac{\pi^{2}(\mu^{2}-1)^{2}}{2Nr^{2}\lambda^{4}}(1+\cos^{2}\varphi). \tag{3} \]

This is Rayleigh’s formula, showing that the coefficient of light scattering decreases in proportion to the fourth power of the wavelength and depends on the scattering angle \(\varphi\). Thus there is an extremely sharp dependence of the scattering coefficient on the wavelength. If for red light (\(\lambda=7000\ \text{Å}\)) the scattering coefficient is conventionally taken to be equal to unity, then for \(\lambda=6200\) (orange-

... it will be equal to 1.6; for 5700 (yellow), 2.2; for 5200 (green), 3.3; for 4700 (blue), 4.9; for 4400 (violet), 6.4; and for ultraviolet rays \((\lambda = 3000\,\text{Å})\), 30.0.

Rayleigh’s theory explains the blue color of the sky. The distribution of energy in the spectrum of the scattered light of the sky must be determined by a curve whose ordinates are equal to the product of the ordinates of the curves for the functions \(f_1(\lambda)\) and \(f_2(\lambda)\), where \(f_1\) is the distribution of energy in the spectrum of the solar rays illuminating the atmosphere and scattered by it, and \(f_2 = \dfrac{1}{\lambda^4}\). Of course, in doing this one must take into account changes in the spectrum that are called absorption, and the absorption will also be caused by the scattering under consideration.

2. Spectral transparency of the atmosphere. Only in the middle of the second decade of the present century was it possible to prove experimentally the presence of molecular scattering of light in gases. Because of the low intensity of the scattered light, observation of this effect is rather difficult. Whereas Tyndall’s observations, reported in 1869, that carefully purified air “has no effect whatever on light and in this respect is like a vacuum,” were made even before the appearance of Rayleigh’s theory, the first attempts to detect light scattering by pure gases, undertaken by so modern and so good an experimentalist as Wood, also proved unsuccessful. The main difficulty consists in excluding any extraneous light from the field of view. The first successful experiments belong to Cabannes,^9 who began these investigations in 1913 under the direction of Fabry. Independently (although somewhat later), similar experiments were successfully carried out by Smoluchowski^10 and Strutt^11 (Rayleigh’s son). Soon Wood also demonstrated the presence of molecular scattering of light in gases.^12

It is characteristic that the results of all these works were presented by the authors from the point of view of the “laboratory reproduction of the blue sky.”

It should be noted that verification of the consequence following from Rayleigh’s law concerning the inverse proportionality of the coefficient of light scattering to the fourth power of the light wavelength is especially difficult: the spectral decomposition required for this causes an additional weakening of the light. In this respect, data on the spectral transparency of the atmosphere are of fundamental importance, since the atmosphere may be regarded as a laboratory of enormous dimensions, in which light rays pass through layers of gas tens of kilometers thick, making it possible to detect even insignificant effects.

The rays of the Sun, passing through the atmosphere, are weakened as a result of scattering. If \(I_{0\lambda}\) is the intensity of solar rays of wavelength \(\lambda\) upon entering the Earth’s atmosphere, and \(I_\lambda\) is the intensity after passing through the atmosphere, then their ratio \(p_\lambda = \dfrac{I_\lambda}{I_{0\lambda}}\) determines the transparency of the atmosphere. According to Rayleigh’s law, the weakening of light caused by scattering increases as \(\lambda\) decreases, and one should expec-

date that \(p_\lambda\) will decrease noticeably with decreasing \(\lambda\), and moreover according to a definite law.

But how is \(p_\lambda\) to be determined? If \(I_\lambda\) is accessible to direct measurement, then \(I_{0\lambda}\) cannot be measured directly. Sunlight reaches us on Earth only after it has already passed through the atmosphere.

This difficulty can be overcome by measuring the intensity of the solar rays at different zenith distances of the Sun. Taking the mass of air pierced by a ray coming from the zenith as unity, for an oblique passage of the rays through the atmosphere we obtain larger masses, namely: at a solar zenith distance \(Z_\odot = 20^\circ\), the mass is \(m = 1.06\); at \(Z_\odot = 70^\circ\), \(m = 2.99\); at \(Z_\odot = 80^\circ\), \(m = 5.72\), and so on. The slope of the curve giving the magnitude of \(I_\lambda\) for different \(m\) determines the absorption in the atmosphere for \(m = 1\).

Fig. 2

Fig. 3

A large many-year series of such measurements, the results of which have now become classical, was carried out by the Astrophysical Observatory on Mount Wilson. Regular determinations of transparency began in 1905 and are due chiefly to Fowle. In all cases without exception there is a significant decrease of transparency toward the short wavelengths. In Fig. 2, using Fowle’s average data\(^{13}\), the atmospheric-transparency curves \(p_\lambda\) for the visible spectrum are shown for various zenith distances of the sun \(Z_\odot\) (reduced to sea level and to completely dry air\(^{14}\)).

Figure 3 presents data\(^{15}\) that make it possible to compare the changes in atmospheric transparency over the spectrum with the law \(\lambda^{-4}\). Along the abscissa are plotted \(\lambda^{-4}\) (\(\lambda\) is measured in Å), and along the ordinate the optical densities of the atmosphere,

\[ \Delta_\lambda = \lg \frac{I_{0\lambda}}{I_\lambda} = \lg \frac{1}{p_\lambda}. \]

The dashed line shows the theoretical curve corresponding to the scattering law \(\lambda^{-4}\), i.e. in the present case simply a straight line drawn from the origin. The solid line gives the experimental curve (according to Fowle’s sixteen-year averages\(^{16}\) for the vertical ray at the height of Mount Wilson) for the short-wavelength part of the visible spectrum (outside the region of selective absorp-

absorption by water vapor and ozone). For orientation in wavelengths, we give a small table of values of \(\lambda^{-4}\) for different \(\lambda\):

\(\lambda\)	3 838	3 974	4 000	4 127	5 000
\(10^{-17}\lambda^{-4}\)	4.609	4.009	3.906	3.447	1.600

The data of Fig. 3 undoubtedly show that the spectral absorption of light in the earth’s atmosphere is in agreement with Rayleigh’s scattering law \(\lambda^{-4}\), if the observations are made under conditions of clean air.

How significantly atmospheric absorption changes the intensity and spectral composition of rays passing through the atmosphere can be seen from Fig. 4, where curves are given for the distribution of energy in the spectrum of the sun’s radiation at the earth’s surface for different zenith distances of the sun. In this graph the intensities are given in conventional units\(^ {14}\).

Fig. 4.

Fig. 5.

Having data on the absorption of light in the atmosphere, one can determine the distribution of energy in the solar spectrum before the solar rays enter the earth’s atmosphere. In Fig. 5 the curve (solid) is given, calculated by Abbot, Fowle, and Aldrich\(^ {17}\). The dotted curve shows the distribution of energy in the spectrum of a black body at \(6\,000^\circ\) K, calculated by Planck’s formula.

To estimate the attenuation of a ray traveling horizontally near the earth’s surface, let us recall that the path of a vertical ray through the entire atmosphere is equivalent to the path through a layer of air at normal pressure \(8\) km thick (the height of the homogeneous atmosphere). Thus, the ratio of the curves for \(Z_{\odot}=20^\circ\) (\(m=1.06\)) and \(Z_{\odot}=80^\circ\) (\(m=5.72\)) in Fig. 4 corresponds to the absorption ratio for two horizontal rays that have traversed paths of about 8 and 45 km. These are only approximate ratios for a horizontal ray, since the layers of air near the earth’s surface are less transparent because of dustiness, high humidity, etc.

Molecular scattering of light, causing attenuation of the direct solar rays, at the same time accounts for the great brightness of the entire firmament. As a result, the illumination of the earth’s sur-

surface consists of two parts: direct sunlight and scattered light from the sky. It is clear that the greater the attenuation of the solar rays due to scattering, the greater will be the intensity of the scattered light. If, at a zenith distance of the Sun \(Z_\odot = 20^\circ\), the ratio of the luminous fluxes from the Sun \(\Phi_\odot\) and from the sky \(\Phi_{\text{sky}}\) is (according to Kimball’s data) 5.7, then as \(Z_\odot\) increases the share of sky light increases noticeably\(^{14}\): at \(Z_\odot = 40^\circ\), \(\dfrac{\Phi_\odot}{\Phi_{\text{sky}}} = 4.0\); at \(60^\circ\) it is 3.0; at \(70^\circ\) it is 1.6; and at \(80^\circ\) it is 1.2.

Such characteristics are very important for questions of natural illumination, in view of the difference in the spectral composition of sunlight and of scattered sky light. However, if absorption causes a considerable decrease in the relative share of sunlight as \(Z_\odot\) increases, then the spectral characteristics of this same absorption, on the contrary, determine a comparative stability of the spectral composition of daylight as \(Z_\odot\) changes. Fig. 6 presents data giving the relative distribution of energy in the spectrum of the combined radiation of the Sun and the sky for different \(Z_\odot\). To facilitate comparison of the curves with one another, their maximum ordinates have been set equal to one and the same value. The curves were constructed by Rautian\(^{14}\) on the basis of Kimball’s\(^{18}\) measurements of total daylight illumination on a horizontal surface and illumination from the sky.

Fig. 6

The small change in the spectral composition of the combined radiation is connected with the fact that, as \(Z_\odot\) increases, the fraction of sky radiation grows, which causes the light to “turn blue,” but owing to the increase in absorption of the solar rays the light “turns yellow.”

3. Polarization of the sky vault. The need to improve the theory of scattering. According to Rayleigh’s theory, scattered light observed from the side must be polarized. For the intensities \(i_I\) and \(i_{II}\) of two mutually perpendicular polarized components (see Fig. 1) we may write:

\[ i_I = I \frac{\pi^2(\mu^2 - 1)^2}{2Nr^2\lambda^4}, \]

\[ i_{II} = I \frac{\pi^2(\mu^2 - 1)^2}{2Nr^2\lambda^4}\cos^2\varphi, \]

where \(I\) is the intensity of the incident rays. At a scattering angle \(\varphi = 90^\circ\), we obtain \(i_{II} = 0\), i.e., the scattered light will be completely polar-

polarized in the plane of incidence (the electric vector is perpendicular to the plane of scattering). Forming the expression for the degree of polarization \(p\) of the scattered light, we obtain

\[ p=\frac{i_I-i_{II}}{i_I+i_{II}}=\frac{1-\cos^2\varphi}{1+\cos^2\varphi}. \tag{4} \]

Observations of the scattered light of the sky show that the condition for maximum polarization at \(\varphi=90^\circ\) is always fulfilled. However, complete polarization is never observed.

Polarization at different \(\varphi\) was measured under conditions of especially clean air on Elbrus, at an altitude of 3000 m above sea level\({}^{19}\). The general course of the curve corresponds to the theoretical one (as regards the dependence on the scattering angle); the maximum polarization occurs at \(\varphi=90^\circ\); however, the absolute values of the polarization are smaller than the theoretical ones.

The greatest values of polarization are usually observed when the zenith distance of the sun is \(Z_\odot=90^\circ\) (when the sun is high, light reflected from the earth’s surface is scattered in the atmosphere and reduces the polarization\({}^{20}\)). The greatest polarization was measured by Tikhanovskii\({}^{21}\) on Mount Ai-Petri in the Crimea: \(p=84.7\%\). Among the extreme values of polarization observed in other places, one may mention \(p=83.4\%\) in Sarapul, in the Ural region (Tikhanovskii), \(p=82.0\%\) in Slutsk (Kalltin). According to many years of observations by Dorno in Davos, the greatest value of \(p\) amounts to only \(80.8\%\).

A number of factors can be indicated that certainly lower the polarization of the scattered light of the sky, but they cannot account for the whole observed deficiency of polarization. Secondary scattering, and in general scattering of higher orders, gives unpolarized light: each given volume of air is illuminated by the scattered light of the atmosphere from all sides, and there is no predominant direction of illumination; however, the polarization of the scattered light depends on the scattering angle.

Algrim\({}^{22}\) constructed a theory of the scattering of light by the atmosphere with allowance for secondary scattering (1914). The formulas he obtained make it possible to estimate the depolarizing action of secondary scattering. Subsequently his theory was considerably improved, as will be discussed below; here we shall merely point out that the depolarizing action of secondary scattering is insufficient to explain the observed magnitudes of polarization.

Another essential depolarizing factor is the action of particles suspended in the air (dust, water droplets, etc.). Tikhanovskii carried out a series of measurements of polarization at different humidities and different dust contents (the amount of dust was determined directly with Aitken’s dust counter). By the formula

\[ p=a+\frac{\Delta p}{\Delta n}n \]

(\(p\) is the degree of polarization, \(n\) is the number of dust particles in \(1\ \mathrm{cm}^3\)) from all series of observations, by the method of least squares, were determined ...

of the quantities \(a\) and \(\dfrac{\Delta p}{\Delta n}\), and the value \(p\) corresponding to an atmosphere deprived of dust was found. Further, by the formula

\[ p' = p_0 + \frac{\Delta p}{\Delta e} e \]

(\(e\) is absolute humidity) the value of the polarization for an “absolutely dry” atmosphere was found analogously\(^{23}\). Tikhanovsky’s calculation is based on the assumption that if at the place of observation \(n = 0\) and \(e = 0\), then throughout that part of the atmosphere which determines the magnitude of the observed polarization, the dust content and humidity are also equal to zero. Tikhanovsky found \(p_0 = 85.6\%\), which is also too small for agreement with Rayleigh’s theory.

The polarization of light scattered by pure gases has been subjected to careful study under laboratory conditions. The investigations of Strutt (Rayleigh), Cabannes, Gans, Raman and Rao, Narayan, Martin, and others led to the establishment of the fundamental fact that, in the scattering of light by gases, the polarization, when observed in a direction perpendicular to the incident light, contrary to Rayleigh’s theory, is never complete.

Strutt\(^{24,25}\) was the first to discover this fact, in 1918. At the present time 89 gases and vapors\(^{26–32,15}\) have been investigated, and in all cases it has been found that \(p < 100\%\). Table 1 gives data for some vapors and gases\(^{15}\), and, in addition to the limiting polarization

\[ p = \frac{i_I - i_{II}}{i_I + i_{II}} \]

the corresponding values of the depolarization \(\rho = \dfrac{i_{II}}{i_I}\) are also indicated

\[ \left(\text{obviously, } p = \frac{1-\rho}{1+\rho} \text{ and } \rho = \frac{1-p}{1+p}\right). \]

The data given in Table 1 testify to facts of primary importance. For air, even under especially favorable (laboratory) conditions, the degree of polarization at a scattering angle \(\varphi = 90^\circ\) is equal to \(92\%\) instead of \(100\%\). This is connected with the discrepancies in the theory of the polarization of skylight mentioned above. If, moreover, the depolarization of light scattered by gases is always different from zero, although according to Rayleigh’s theory at a scattering angle \(\varphi = 90^\circ\) the depolarization \(\rho\) should be equal to zero, then it is clear that Rayleigh’s theory, while correctly describing a number of the basic properties of scattered light, is not entirely exact.

The data of Table 1 indicate a connection between the magnitude of the depolarization and the structure of the molecule. Indeed, monatomic molecules give a very small depolarization, less than 0.001. Conversely, triatomic homeopolar molecules give, as a rule, a very large depolarization, about 0.1 and more. Of the homeopolar molecules, diatomic ones have a smaller depolarization than triatomic ones. However, the matter here is evidently not in the complexity of the molecule, since, for example, complex alcohol molecules give a smaller depolarization than simple diatomic molecules of oxygen, nitrogen, hydrogen, and others.

Table 1

Gas name	Gas designation	\(p\), in %	\(100\rho\)	Gas name	Gas designation	\(p\), in %	\(100\rho\)
Monatomic molecules				Triatomic homopolar molecules
Helium	He	\(>88\)	\(<6.5\)	Carbon dioxide	CO\(_2\)	82.2	9.8
Neon	Ne	\(>98\)	\(<1\)	Nitrous oxide	N\(_2\)O	77.8	12.5
Argon	A	99	0.5	Carbon disulfide	CS\(_2\)	79.4	11.5
Krypton	Kr	99	0.5
Xenon	Xe	99	0.5
Heteropolar molecules				Vapors of hydrocarbon compounds
Water vapor	H\(_2\)O	96.1	2.0	Benzene \(^{1)}\)	C\(_6\)H\(_6\)	91.96	4.45
Ammonia	NH\(_3\)	97.5	1.3	Toluene \(^{2)}\)	C\(_7\)H\(_8\)	91.7	4.3
Nitric oxide	NO	95.1	2.6	Methane \(^{3)}\)	CH\(_4\)	97.1	1.5
Carbon monoxide	CO	96.1	2.1	Cyclohexane	C\(_6\)H\(_{12}\)	98.0	1.0
Diatomic homopolar molecules
Hydrogen	H\(_2\)	94.9	2.7	Methyl alcohol \(^{4)}\)	CH\(_3\)OH	96.8	1.7
Chlorine	Cl\(_2\)	91.9	4.3	Ethyl alcohol \(^{5)}\)	C\(_2\)H\(_5\)OH	98.2	0.9
Oxygen	O\(_2\)	87.9	6.4
Nitrogen	N\(_2\)	93.2	3.6
Air	—	92.03	4.15

\(^{1)}\) At 80–100°.
\(^{2)}\) At 110–120°.
\(^{3)}\) At 20°.
\(^{4)}\) At 65°.
\(^{5)}\) At 80°.

The further development of the theory of light scattering showed that the Rayleigh theory is only approximate. The more general scattering theories subsequently created include Rayleigh’s theory as a special case. The results of some of them have very great practical significance. For example, for the study of the transparency of fogs and clouds (in particular, for the section on the passage of infrared rays through fogs), Rayleigh’s theory is completely insufficient. But even for a perfectly pure atmosphere the correct values of transparency can be calculated only on the basis of a theory more general than Rayleigh’s.

A substantially new view of the nature of light scattering was provided by the investigations of Smoluchowski and Einstein, who developed the fluctuation theory of scattering\(^{33,34}\). These investigations were carried out in order to explain the phenomenon of critical opalescence.

As the temperature of a liquid approaches the critical temperature, the brightness of the scattered light increases extraordinarily. For example, for ether near the critical temperature the intensity of scattering is 750 times greater than for liquid ether at a temperature of 35°, and 20,000 times greater than for ether vapor at a temperature of 35°. Under intense illumination of ether located in the critical

states, the brightness of the light beam observed from the side can reach one candle per \(1\ \text{cm}^2\). Smoluchowski in 1908 showed\(^{33}\) that the cause of the phenomenon lies in the rapid increase in the compressibility of a substance as its temperature approaches the critical one. With very great compressibility, two neighboring regions of a liquid may, with respect to one another, be in states differing little from equilibrium even with a large difference in their densities. Owing to the disordered thermal motion of particles, groups of molecules continuously arise here and there, forming regions with a density very different from the mean. As a result of the dependence of the refractive index on density, the fluctuations of the latter make the medium optically nonuniform. A substance, even one very transparent under ordinary conditions, near the critical point becomes a turbid medium that strongly scatters light.

In 1910 Einstein\(^{34}\) developed the fluctuation theory of critical opalescence. For the theory of light scattering it is essential that density fluctuations, though of smaller magnitude, also occur at temperatures below the critical one. The theory of fluctuations, developed by Einstein for the case of an ideal gas, gives Rayleigh’s formula. This circumstance, in addition to deriving the scattering formula by another, independent route on the basis of different physical ideas, is of great importance also in the following respect. Rayleigh, in calculating the intensity of light scattered by a collection of molecules, simply added the intensities from individual molecules. This procedure, within the framework of Rayleigh’s theory, remains somewhat arbitrary. Einstein’s theory in fact confirms the legitimacy of this procedure, since it already solves a rigorous volume problem (the scattering medium is continuous), and the same result is obtained as from the simple summation of the action of separate discrete Rayleigh scattering centers.

However, the presence of depolarization of the scattered light cannot be explained by the theories considered. The features in molecular structure that affect light scattering (as is evident from Table 1) remained outside these theories. Taking into account the structure of the molecules of the scattering medium was the starting point for the further development of the theory of scattering. It was based on Born’s ideas\(^{35}\) on the anisotropy of molecules (1918). Allowance for molecular anisotropy made it possible to develop a more rigorous theory of light scattering, allowing, in particular, the optical properties of the pure atmosphere to be calculated in better agreement with observations. The development of this theory belongs chiefly to Cabannes\(^{36,15}\), and its application to the atmosphere—to Tikhonovsky.

For the questions considered in the present article, this theory is very important, in particular because it allows one to determine correctly the absolute value of the transparency coefficient of the pure atmosphere.

4. Theory of light scattering taking account of the optical anisotropy of molecules. Cabannes, for the derivation of the scattering formula—

of light with allowance for the anisotropy of molecules[^15],[^36], one uses the static model of the molecule, corresponding to Langevin’s model in the theory of magnetism. In this connection it may be pointed out that the use of the static model in the theory of light scattering is quite expedient.

Phenomena connected with the emission of light by matter and with the accompanying changes in the states of the atom (a decrease of internal energy), and phenomena connected with the selective absorption of light (an increase of the internal energy of atoms), require the introduction of a dynamic model and the notion of the quantum character of the change of energy. Another group of phenomena, concerning the action of molecules on the propagation of light, if frequencies close to the absorption region are not considered, can be explained with the aid of the electromagnetic (and even elastic) theory of light, without using quantum theory, by likening molecules to a resonator executing oscillations under the action of the incident light. In this way the accuracy of the calculations is preserved and the intuitive clarity of such a model is fully utilized. The absence, in these phenomena, of essential intra-atomic changes makes it possible not to enter into the essence of the internal dynamics of the atom.

We shall imagine a molecule as an aggregate of particles (positive nuclei and electrons), bound to one another by forces tending to return each particle to its equilibrium position if it has been disturbed. The force arising when a point is displaced by \(\Delta \mathbf{r}\) from its equilibrium position shall be taken to be equal to \(k\cdot \Delta \mathbf{r}\). We assume the molecule to be uncharged, i.e. the sum of the positive charges \(\sum e^{+}\) is equal to the sum of the negative charges \(\sum e^{-}\).

The center of the positive charges \(P\) and the center of the negative charges \(M\) may coincide with one another (a homeopolar molecule). If they do not coincide (a heteropolar molecule), then the molecule has a permanent electric moment. Under the action of an external electric force \(\mathbf{E}_0\) the charges are displaced. If, as a result, the center of gravity of the positive charges has moved from \(P\) to \(P'\), and that of the negative charges—from \(M\) to \(M'\), then the molecule acquires an induced additional electric moment equal to the vector sum of the moments

\[ \mathbf{PP}' \cdot \sum e^{+} \quad \text{and} \quad \mathbf{M}'\mathbf{M}\cdot \sum e^{-}. \]

Let us imagine a system of rectangular coordinates \(OXYZ\), connected in an unchanging manner with the molecule, so that the position of the point \(O\) and the direction of the axes determine the position and orientation of the molecule. The projections of the external force \(\mathbf{E}_0\) on the axes will be denoted by \(E_x, E_y, E_z\), and the projections of the induced moment by \(P, Q, R\). We shall calculate \(P, Q, R\) as functions of \(E_x, E_y, E_z\).

A particle of the molecule having coordinates \((x_i, y_i, z_i)\) and charge \(e_i\), after moving to the point \((x_i+\Delta x_i,\ y_i+\Delta y_i,\ z_i+\Delta z_i)\), imparts to the molecule an electric moment \((e_i\Delta x_i,\ e_i\Delta y_i,\ e_i\Delta z_i)\). At the point \(x_i y_i z_i\) there arises, as it were, a dipole. At the point \(A\) with coordinates

\((x_a, y_a, z_a)\), located at a distance \(r_{ai}\) from the dipole, the field has the component:

\[ (E_x)_{ai}= e_i \Delta x_i \left[\frac{3(x_i-x_a)^2}{r_{ai}^5}-\frac{1}{r_{ai}^3}\right]+ \]

\[ + e_i \Delta y_i \frac{3(x_i-x_a)(y_i-y_a)}{r_{ai}^5} + e_i \Delta z_i \frac{3(x_i-x_a)(z_i-z_a)}{r_{ai}^5}. \tag{5} \]

The expressions for \((E_y)_{ai}\) and \((E_z)_{ai}\) are obtained from formula (5) by a circular permutation of the coordinates.

If at the point \(A\) there is a particle whose charge is equal to \(e_a\), then it is displaced from its equilibrium position under the action of the external force \(\mathbf E_0\) \((E_x, E_y, E_z)\) and of all the forces \([(E_x)_{ai}, (E_y)_{ai}, (E_z)_{ai}]\). If the force tending to return the particle to its initial equilibrium position does not depend on the direction of the displacement of the particle, then we may write:

\[ \begin{aligned} k_a \Delta x_a &= e_a\left[E_x+\sum (E_x)_{ai}\right],\\ k_a \Delta y_a &= e_a\left[E_y+\sum (E_y)_{ai}\right],\\ k_a \Delta z_a &= e_a\left[E_z+\sum (E_z)_{ai}\right], \end{aligned} \tag{6} \]

where the summation sign \(\sum\) extends over all \(i\), except \(i=a\). Substituting for \([(E_x)_{ai}, (E_y)_{ai}, (E_z)_{ai}]\) their expressions in terms of \((e_i\Delta x_i, e_i\Delta y_i, e_i\Delta z_i)\) from (5) and writing analogous equations for all \(n\) particles forming the molecule, we obtain \(3n\) linear equations with \(3n\) unknowns \((e_a\Delta x_a, e_a\Delta y_a, e_a\Delta z_a)\). Solving the equations, we shall find the components of the induced moment of the molecule that interest us as the sums

\[ \begin{aligned} P &= \sum e_a \Delta x_a,\\ Q &= \sum e_a \Delta y_a,\\ R &= \sum e_a \Delta z_a. \end{aligned} \]

Let us consider the case when these three components can be represented in the form of linear functions of the field components \((E_x, E_y, E_z)\):

\[ \begin{aligned} P &= A_xE_x+B_xE_y+C_xE_z,\\ Q &= A_yE_x+B_yE_y+C_yE_z,\\ R &= A_zE_x+B_zE_y+C_zE_z. \end{aligned} \tag{7} \]

Equations (7) apply to weak fields, when the displacements of the particles are small in comparison with the distances between the particles (the restoring forces are proportional to the displacements). The quantities \(A_x, B_x, \ldots, C_z\) are the components of the electrostatic deformation tensor. The coefficients \(A_x, B_y, C_z\) are longitudinal coefficients, and the remaining six are transverse coefficients.

If in equations (6) we replace \((E_x)_{ai}, (E_y)_{ai}, (E_z)_{ai}\) by their expressions from (5), then the result obtained may be written in the form:

\[ \begin{aligned} k_a \Delta x_a &-\sum \alpha_{ai}^{x}\Delta x_i -\sum \alpha_{ai}^{y}\Delta y_i -\sum \alpha_{ai}^{z}\Delta z_i = e_aE_x,\\ k_a \Delta y_a &-\sum \beta_{ai}^{x}\Delta x_i -\sum \beta_{ai}^{y}\Delta y_i -\sum \beta_{ai}^{z}\Delta z_i = e_aE_y,\\ k_a \Delta z_a &-\sum \gamma_{ai}^{x}\Delta x_i -\sum \gamma_{ai}^{y}\Delta y_i -\sum \gamma_{ai}^{z}\Delta z_i = e_aE_z, \end{aligned} \tag{6'} \]

where

$$ \left. \begin{aligned} \alpha_{ai}^{x} &= e_a e_i\left[\frac{3(x_i-x_a)^2}{r_{ai}^{5}}-\frac{1}{r_{ai}^{3}}\right], \qquad \alpha_{ai}^{y}= e_a e_i\,\frac{3(x_i-x_a)(y_i-y_a)}{r_{ai}^{5}},\\ \alpha_{ai}^{z} &= e_a e_i\,\frac{3(x_i-x_a)(z_i-z_a)}{r_{ai}^{5}},\\[4pt] \beta_{ai}^{x} &= e_a e_i\,\frac{3(y_i-y_a)(x_i-x_a)}{r_{ai}^{5}}, \qquad \beta_{ai}^{y}= e_a e_i\left[\frac{3(y_i-y_a)^2}{r_{ai}^{5}}-\frac{1}{r_{ai}^{3}}\right],\\ \beta_{ai}^{z} &= e_a e_i\,\frac{3(y_i-y_a)(z_i-z_a)}{r_{ai}^{5}},\\[4pt] \gamma_{ai}^{x} &= e_a e_i\,\frac{3(z_i-z_a)(x_i-x_a)}{r_{ai}^{5}}, \qquad \gamma_{ai}^{y}= e_a e_i\,\frac{3(z_i-z_a)(y_i-y_a)}{r_{ai}^{5}},\\ \gamma_{ai}^{z} &= e_a e_i\left[\frac{3(z_i-z_a)^2}{r_{ai}^{5}}-\frac{1}{r_{ai}^{3}}\right]. \end{aligned} \right\} \tag{8} $$

From these equalities it is immediately evident that $\alpha^y=\beta^x$, $\alpha^z=\gamma^x$, $\beta^z=\gamma^y$, and consequently $B_x=A_y$, $C_x=A_z$, $C_y=B_z$, i.e. the tensor with components $A_x$, $B_x,\ldots,C_z$ is symmetric. Let us denote the longitudinal coefficients $A_x$, $B_y$, $C_z$ simply by $A$, $B$, $C$, and the transverse ones by $B_x=A_y=C'$, $C_x=A_z=B'$ and $C_y=B_z=A'$. In the new notation relations (7) take the form:

$$ \left. \begin{aligned} P &= AE_x + C'E_y + B'E_z,\\ Q &= C'E_x + BE_y + A'E_z,\\ R &= B'E_x + A'E_y + CE_z. \end{aligned} \right\} \tag{7'} $$

Consider the ellipsoid whose equation is

$$ AE_x^2+BE_y^2+CE_z^2+2A'E_yE_z+2B'E_zE_x+2C'E_xE_y=a, \tag{9} $$

where $a$ is some number.

If the principal axes of the ellipsoid are chosen as the coordinate axes, then in the new variables the coefficients $A'$, $B'$, $C'$ in the equation of the ellipsoid become zero. According to (7′) we obtain

$$ P=AE_x,\quad Q=BE_y,\quad R=CE_z, \tag{10} $$

if by $E_x$, $E_y$, $E_z$ we understand the projections of the external field $\mathbf{E}_0$ on the new coordinate axes.

From the point of view of optics this means that the model of the molecule considered by us (the linear dependence of the components of the induced moment on the components of the external field) possesses the same symmetry elements as the ellipsoid (9). Three directions are associated with the molecule (the principal directions of the molecule) which have the property that the induced electric moment of the molecule coincides with the direction of the external field if it is directed along one of these directions. The coefficients $A$, $B$, and $C$ are equal to the magnitude of the induced moment when the external field is equal to unity and is directed parallel to one of the principal directions.

If $A=B=C$, then the molecule is isotropic. The inequality of these coefficients indicates the optical anisotropy of the molecule.

Let us now suppose that the external field \(E_0\) is not constant, but varies sinusoidally with time. If the mass of the particle is \(m\), then the inertial force which must be added to the first terms of equations (6) will be \(m\dfrac{d^2}{dt^2}\Delta x_a\), and for steady-state oscillations we may write

\[ \Delta x_a=\frac{e_a}{k_a-\dfrac{4\pi^2 m}{T^2}} \left[E_x+\sum (E_x)_{ai}\right] \]

and analogously for \(\Delta y_a\) and \(\Delta z_a\) (\(T\)—period). Denoting by

\[ \omega=\frac{2\pi}{T} \]

the frequency of the incident light, and by

\[ \omega_a=\sqrt{\frac{k_a}{m}} \]

the frequency of the particle’s natural oscillations, we obtain

\[ \Delta x_a=\frac{e_a}{m(\omega_a^2-\omega^2)} \left[E_x+\sum (E_x)_{ai}\right] \]

and analogously for \(\Delta y_a\) and \(\Delta z_a\).

Let us compare the magnitude of the displacement \(\Delta x'\) of the nucleus with the magnitude of the displacement \(\Delta x''\) of the electron. If \(e\) is the charge of the electron, \(+n'e\) the charge of the nucleus, and \(\omega'\) and \(\omega''\) the natural frequencies of oscillation of the nucleus and the electron, then

\[ \frac{\Delta x'}{\Delta x''} = \frac{n'e}{-e}\cdot\frac{m''}{m'}\cdot \frac{(\omega''^{2}-\omega^{2})}{(\omega'^{2}-\omega^{2})}. \]

But \(\omega'<\omega\), if we assume that the molecule is illuminated by visible radiation, since \(\omega'\) always lies in the infrared region of the spectrum; conversely, \(\omega''>\omega\), since \(\omega''\) corresponds to the ultraviolet region of the spectrum. Neglecting \(\omega'^2\) in comparison with \(\omega^2\) and \(\omega^2\) in comparison with \(\omega''^2\), we obtain

\[ \frac{\Delta x'}{\Delta x''} = \frac{n'm''}{m'}\cdot\frac{\omega''^{2}}{\omega^{2}} = \frac{n'm''}{m'}\left(\frac{\lambda}{\lambda''}\right)^2. \]

The mass of the nucleus of some atom is approximately equal to the mass of the hydrogen atom multiplied by twice the atomic number \(n'\), while the mass of the hydrogen atom is 1846 times greater than the mass of the electron. Therefore

\[ \frac{n'm''}{m'}\simeq \frac{1}{2\cdot 1846}=\frac{1}{3692}. \]

Taking \(\dfrac{\lambda}{\lambda''}\) to be approximately equal to 3, we find that the displacement of the nucleus is always at least 400 times smaller than the displacement of the electron, and it may be practically neglected. Thus, under the action of a light wave in a molecule there occurs a displacement of the center of gravity of the negative charges; the center of gravity of the nuclei remains practically immobile.

For one electron the coefficient \(\dfrac{e}{m\omega^2}\) will be of order \(10^{-21}\) CGSE, as is easy to verify if \(\dfrac{\omega}{2\pi}\) is replaced by \(\dfrac{c}{\lambda}\) (where \(c\) is the speed of light), the value of \(\dfrac{e}{m}\) is taken in CGSE units, and \(\lambda=0.15\mu\) is put. Suppose that the molecule is illuminated by solar

rays. We have agreed to consider phenomena only for rays whose frequency differs significantly from the resonant frequency. Therefore let us assume that the solar rays have been passed through a suitable light filter. If we suppose that the light filter reduces the energy of the solar rays by a factor of ten, then the energy carried by the rays will be about \(0.2\ \mathrm{cal}/\mathrm{min}\,\mathrm{cm}^2\). Calculating, for the light field corresponding to this, the value of the Poynting vector, we find for the electric-field strength \(E\) approximately \(10^{-2}\) CGSE, or \(3\ \mathrm{V}/\mathrm{cm}\). Under the action of this force the displacement of the electron will be \(\Delta x = 10^{-21}\cdot E = 10^{-23}\ \mathrm{cm}\). This displacement is very small in comparison with the dimensions of the atom (\(10^{-8}\ \mathrm{cm}\)), of the electron and the nucleus (\(10^{-13}\ \mathrm{cm}\)), which justifies the use of the model of the molecule adopted by us, satisfying relations (7).

This model makes it possible at once to calculate the scattering of light by an individual molecule. Since the frequency of the incident light is small in relation to the electron’s natural frequency, the latter, under the action of the light wave, will execute forced oscillations of the same period and in the same phase as the incident wave. The center of gravity of the negative charges of the molecule will oscillate about the equilibrium position, and the molecule will behave like a Hertz oscillator.

Rayleigh’s theory, which has in view an entirely different model of the scattering particle (an isotropic sphere whose size \(a \ll \lambda\)), also ultimately leads to a Hertz oscillator, and we can immediately obtain the Rayleigh dependence of the scattering coefficient on \(\lambda\): proportionality to \(\lambda^{-4}\).

If the molecule is immobile, then the induced electric moment of the molecule \(\mathbf{M}\), whose components along the coordinate axes fixed in the molecule are \((P, Q, R)\), will preserve its amplitude and direction unchanged.

The amplitude of the linear light oscillations of the scattered wave at a point \(B\) at a distance \(r\) (assuming \(r \gg \lambda\)) at the time \(t + \dfrac{r}{c}\) will be obtained by projecting onto the plane of the wave the vector

\[ -\frac{1}{rc^2}\frac{d^2}{dt^2}\mathbf{M} = \frac{\omega^2}{rc^2}\mathbf{M} = \frac{4\pi^2}{r\lambda^2}\mathbf{M}. \tag{11} \]

The intensity of the light is proportional to the square of the amplitude, i.e. \(\sim \lambda^{-4}\). But in reality, owing to thermal motion, the molecule is continually changing its direction, and moreover in a disorderly manner. Because of this the vector \(\mathbf{M}\) continually changes its magnitude and direction.

We have assumed that the induced moment of the molecule is proportional to the field \(\mathbf{E}\). We shall write the projections of the moment on the axes \(OXYZ\) in the form \((HE, JE, KE)\), where the coefficients \(P, J, K\) depend only on the structure of the molecule and its orientation relative to the external field. We shall suppose that the incident ray is parallel to the axis \(OX\), and that the incident wave is linearly polarized, with \(\mathbf{E}\) parallel to \(OZ\). If we consider the scattering of light in the direction \(OY\), then the amplitude of the light oscillations at the point \(B\) \((\overline{OB}=r)\) at the mo-

ment \(t-\dfrac{r}{c}\) will have components \(\dfrac{4\pi^2}{r\lambda^2}HE\) (parallel to the axis \(OX\)) and \(\dfrac{4\pi^2}{r\lambda^2}KE\) (parallel to the axis \(OZ\)). Denoting by \(I\) the illumination produced on the plane \(YZ\) by the beam of incident rays, and by \(i_I\) and \(i_{II}\) the illuminations produced by the scattered light, the amplitude of whose oscillations is equal to \(\dfrac{4\pi^2}{r\lambda^2}KE\) and, respectively, \(\dfrac{4\pi^2}{r\lambda^2}HE\), we may write, to within one and the same factor:

\[ I=\overline{E^2}, \qquad i_I=\frac{16\pi^4}{r^2\lambda^4}\,\overline{K^2E^2} \quad \text{and} \quad i_{II}=\frac{16\pi^4}{r^2\lambda^4}\,\overline{H^2E^2}, \]

where the bar over the algebraic expressions means that the mean of these expressions is taken. Since there is no correlation between the light oscillations and the thermal motion of the molecule, the mean of the product of the squares is equal to the product of the means of the squares:

\[ \overline{H^2E^2}=\overline{H^2}\cdot \overline{E^2}, \qquad \overline{K^2E^2}=\overline{K^2}\cdot \overline{E^2}, \]

and therefore

\[ i_I=\frac{16\pi^4}{r^2\lambda^4}\,\overline{K^2}J \quad \text{and} \quad i_{II}=\frac{16\pi^4}{r^2\lambda^4}\,\overline{H^2}J. \]

The obtained scattering formulas contain the coefficients \(H\) and \(K\), which depend on the structure of the molecule. The sum \(i_I+i_{II}\) is a measure of the light energy scattered in the direction \(OY\) by the molecule, which in the present case plays the role of a point source of light.

Fig. 7

This light will be partially polarized, with depolarization

\[ \rho=\frac{i_{II}}{i_I}=\frac{\overline{H^2}}{\overline{K^2}}. \]

The sum

\[ \overline{H^2}+\overline{K^2} = \frac{r^2\lambda^4}{16\pi^4}\,\frac{i_I+i_{II}}{I} \]

Kabanov\(^{15}\) denotes by \(f\left(\dfrac{\pi}{2}\right)\): it characterizes the intensity of the lateral scattering of light, i.e. scattering in a direction perpendicular to the direction of the incident ray. Let us now compute the scattering of light in an arbitrary direction \(OM\), which we shall define by the angles (Fig. 7) \(\alpha=\angle NOZ\), \(\beta=\angle MOX\), and \(\gamma=\angle MOZ\).

Denote by \(P\) and \(R\) the projections of the induced electric moment \(\mathbf{M}\) onto two mutually perpendicular directions \(MP\) (the normal to the plane \(MOZ\)) and \(MR\) (the normal to \(OM\) in the plane \(MOX\)).

The depolarization of the light scattered in the direction \(OM\) is equal to \(\dfrac{\overline{P^2}}{\overline{R^2}}\), and its intensity is proportional to the sum \(\overline{R^2}+\overline{P^2}\). But in view of

by symmetry the intensity and depolarization will be the same for all directions lying on the surface of the cone described by the direction \(OM\) about the vector of the light oscillations \(\mathbf E\) of the incident wave (\(\mathbf E\) parallel to \(OZ\)) and having vertex angle \(2\gamma\). Therefore, for the direction \(OM'\) lying in the plane \(YOZ\) and making an angle \(\gamma\) with \(OZ\), we have

\[ \overline{P'^2}=\overline{P^2} \quad\text{and}\quad \overline{R'^2}=\overline{R^2}. \]

But

\[ P'=H,\qquad R'=K\sin\gamma+J\cos\gamma. \]

Therefore

\[ \overline{P^2}=\overline{P'^2}=\overline{H^2} \]

and

\[ \overline{R^2}=\overline{R'^2}=\overline{K^2}\sin^2\gamma+\overline{J^2}\cos^2\gamma =\overline{K^2}\sin^2\gamma+\overline{H^2}\cos^2\gamma . \]

Thus, the intensity of the light scattered in the direction \(OM\) will be proportional to

\[ f(\gamma)=(\overline{K^2}+\overline{H^2})-(\overline{K^2}-\overline{H^2})\cos^2\gamma, \]

and the depolarization will be equal to\({}^{15}\)

\[ \rho(\gamma)=\frac{\rho}{1-(1-\rho)\cos^2\gamma}, \]

where \(\rho\) corresponds to scattering along \(OY\).

Until now we have assumed that the molecule is illuminated by linearly polarized light. If a beam of natural light propagates along the axis \(OX\) (Fig. 7), then it may be considered that the vector \(\mathbf E\) has all possible directions (all the time changing its direction at random) in the plane \(YOZ\).

Since \(\cos^2\gamma=\cos^2\alpha\cdot\sin^2\beta\), as a measure of the intensity of the scattered light one may take

\[ \overline{R^2}+\overline{P^2} =(\overline{K^2}+\overline{H^2})-(\overline{K^2}-\overline{H^2})\cos^2\alpha\sin^2\beta. \]

But the mean value of \(\cos^2\alpha\) is \(1/2\), and for the intensity of light scattered at an angle \(\beta\) to the direction of the incident ray we obtain the measure

\[ f(\beta)=\frac{\overline{K^2}+3\overline{H^2}}{2} +\frac{\overline{K^2}-\overline{H^2}}{2}\cos^2\beta . \]

For lateral scattering in the direction \(\beta=\dfrac{\pi}{2}\)

\[ f\left(\frac{\pi}{2}\right)=\frac{\overline{K^2}+3\overline{H^2}}{2}. \tag{12} \]

This relation becomes obvious if one regards the incident unpolarized light as consisting of two mutually perpendicular polarized incoherent rays of equal intensity: in one of them the vector \(\mathbf E\) is parallel to \(OZ\) and gives in the scattered light \(\dfrac12\overline{K^2}\) for the component along \(OZ\) and \(\dfrac12\overline{H^2}\)—along \(OX\); in the other, \(\mathbf E\) is parallel...

\(OY\) and gives in the scattered light \(\frac{1}{2}\overline{H^2}\) both along the \(OZ\) axis and along the \(OX\) axis, owing to symmetry. For depolarization in scattering along \(OY\) we have

\[ \rho=\frac{2\overline{H^2}}{\overline{K^2}+\overline{H^2}}. \tag{13} \]

We can now calculate the scattering into all the surrounding space by summing the scattering over all directions. If \(d\Omega\) is an elementary solid angle about \(OM\), then the luminous flux scattered within this cone is proportional to \(f(\beta)\,d\Omega\). For the total flux we have:

\[ \iint f(\beta)\,d\Omega = 2\int_{0}^{\frac{\pi}{2}} f(\beta)\cdot 2\pi\sin\beta\,d\beta = \frac{8\pi}{3}\left(\overline{K^2}+2\overline{H^2}\right) = \frac{8\pi}{3}f\left(\frac{\pi}{2}\right)\frac{2+\rho}{1+\rho}. \]

The last expression is obtained by substituting the values of \(\overline{K^2}\) and \(\overline{H^2}\) taken from (12) and (13).

Let us summarize the results obtained. If a molecule is illuminated by unpolarized light of intensity \(I\), then:

1. The depolarization of light scattered at an angle of \(90^\circ\) is

\[ \rho=\frac{i_{II}}{i_I}=\frac{2\overline{H^2}}{\overline{K^2}+\overline{H^2}} \tag{14} \]

and its intensity is

\[ i=i_I+i_{II}=\frac{8\pi^4 I}{\lambda^4}\left(\overline{K^2}+\overline{H^2}\right). \tag{15} \]

1. For scattering at an angle \(\beta\) to the incident ray, the intensity is

\[ i(\beta)=I\left(1+\frac{1-\rho}{1+\rho}\cos^2\beta\right) \tag{16} \]

and the depolarization is

\[ \rho(\beta)=\rho\sin^2\beta+\cos^2\beta. \tag{17} \]

1. For the luminous flux scattered into all the surrounding space, we have

\[ L=\frac{8\pi}{3}\,i\,\frac{2+\rho}{1+\rho}. \tag{18} \]

Such are the properties of the light scattered by an individual isolated molecule.

Passing to an aggregate of molecules—to a gas—we assume only that the gas is at not too high a pressure (obeys Boyle–Mariotte’s law). In this case the positions and orientations of the individual molecules are completely random: if we consider two neighboring molecules, then the position and orientation of one of them do not depend on the presence nearby of the other. It is clear that in this case there is no correlation whatever between the amplitudes and phases of the induced moments \(M\) of the individual molecules.

Consequently, the molecules of the illuminated gas behave as incoherent sources of light. Therefore we can simply add the intensities of the light scattered by the individual molecules

THEORY OF LIGHT SCATTERING AND ITS APPLICATION

If the superscript “0” is used to denote quantities relating to the scattering of light by each of the \(Nv\) molecules contained in the volume \(v\), then we can write

\[ i_I = Nv i_{0I}, \qquad i_{II} = Nv i_{0II}, \qquad \rho = \frac{i_{II}}{i_I} = \frac{i_{0II}}{i_{0I}} = \rho_0 . \]

The scattering coefficient (per unit volume) is

\[ R = \frac{i_I+i_{II}}{vI} = N\frac{i_{0I}+i_{0II}}{I} = NR_0 . \]

Let \(I\) be the illumination of a surface normal to \(OX\), produced by a beam of light propagating parallel to \(OX\). The decrease of illumination over the distance \(dx\) will be \(-dI=kI\,dx\). The ratio \(-\dfrac{dI}{dx}\) gives the luminous flux scattered in all directions by \(1\ \mathrm{cm}^3\), i.e. by \(N\) molecules. \(k\) is the absorption coefficient (apparent absorption, caused by the scattering of light in all directions, as distinct from true absorption). According to (18),

\[ k = \frac{8\pi}{3} R \frac{2+\rho}{1+\rho}. \tag{19} \]

The scattering constants must be expressed through the macroscopic optical characteristics of the medium, primarily through the refractive index.

For calculations it is convenient to pass from the parameters \(\overline{H^2}\) and \(\overline{K^2}\) to other parameters \(\varepsilon\) and \(\eta\), defined as follows:

\[ \varepsilon=\frac{\overline{H^2}}{(\overline{K})^2}, \qquad \eta=\frac{\overline{K^2}}{(\overline{K})^2}-1 . \tag{20} \]

Between \(\varepsilon\) and \(\eta\) there is a simple relation; in deriving it, in addition to the coordinate axes \(OXYZ\), we shall also consider axes \(OUVW\), coinciding with the principal directions of the molecule.

As before, let \(P, Q, R\) denote the projections of the induced moment on the axes \(OUVW\), and let the field \(E\) be directed along \(OZ\). We have: \(P=AE\cos uz\), \(Q=BE\cos vz\), \(R=CE\cos wz\), and for the parameters \((H, I, K)\):

\[ H = A\cos ux\cos uz + B\cos vx\cos vz + C\cos wx\cos wz, \]

\[ I = A\cos uy\cos uz + B\cos vy\cos vz + C\cos wy\cos wz, \]

\[ K = A\cos^2 uz + B\cos^2 vz + C\cos^2 wz. \]

Passing to mean values, we shall use the fact that:

\[ \left. \begin{aligned} \overline{\cos^2 uz} &= \overline{\cos^2 vz} = \overline{\cos^2 wz} = \frac{1}{3},\\ \overline{\cos ux\cos uz} &= \overline{\cos vx\cos vz} = \ldots = 0,\\ \overline{\cos^4 uz} &= \overline{\cos^4 vz} = \overline{\cos^4 wz} = \frac{1}{5},\\ \overline{\cos^2 ux\cos^2 uz} &= \overline{\cos^2 vx\cos^2 vz} = \ldots = \frac{1}{15},\\ \overline{\cos ux\cos uz\cos vx\cos vz} &= \ldots = -\frac{1}{30}. \end{aligned} \right\} \tag{21} \]

(We omit the proof of these relations; it can be found in Kabanov’s book[^15].)

Hence we have:

\[ \overline{H}=\overline{I}=0;\qquad \overline{K}=\frac{A+B+C}{3}, \]

\[ \overline{H^2}=\overline{I^2}=\frac{1}{15}(A^2+B^2+C^2-BC-CA-AB), \]

\[ \overline{K^2}=\frac{1}{15}(3A^2+3B^2+3C^2+2BC+2CA+2AB). \tag{22} \]

It is not difficult to see that

\[ \overline{K^2}-(\overline{K})^2=\frac{4}{45}(A^2+B^2+C^2-BC+CA+AB)=\frac{4}{3}\overline{H^2}, \]

and consequently,

\[ 3\{\overline{K^2}-(\overline{K})^2\}=4\overline{H^2}, \]

or, on the basis of (20),

\[ 3\eta=4\varepsilon . \tag{23} \]

Replacing \(H\) and \(K\) by \(\varepsilon\) and \(\eta\), we may rewrite formulas (14) and (15) in the form:

\[ \rho=\frac{2\varepsilon}{1+\eta+\varepsilon}, \tag{14'} \]

\[ i=\frac{8\pi^4 l}{\lambda^4}(\overline{K})^2(1+\eta+3\varepsilon). \tag{15'} \]

Let us consider a volume of gas \(\Delta v\), sufficiently large that the number of molecules in it \(N\Delta v\) is large, but small in comparison with \(\lambda\). The gas, under the action of the electric field of the wave, becomes polarized (dielectric polarization). The projections on the axes \(OXYZ\) of the induced moment of the volume \(\Delta v\) will be:

\[ (H_1+H_2+\ldots)E=N\Delta v\,\overline{H}E=0, \]

\[ (I_1+I_2+\ldots)E=N\Delta v\,\overline{I}E=0, \]

\[ (K_1+K_2+\ldots)E=N\Delta v\,\overline{K}E=F\Delta v, \]

where \(F=N\overline{K}E\) is the induced electric moment per unit volume of the gas. But the dielectric constant1 \(\nu\) of the medium is related to \(F\) by the relation \(\nu=1+4\pi F'\), where \(F'=\frac{F}{E}\) is the electric moment per unit volume of a medium situated in an external field \(E=1\). Consequently, \(\nu=1+4\pi N\overline{K}\). But \(\nu=\mu^2\), where \(\mu\) is the refractive index of the medium. Thus, we obtain

\[ \nu-1=\mu^2-1=4\pi N\overline{K}=\frac{4}{3}\pi N(A+B+C). \]

We have related the refractive index of the scattering gas to the parameters \((A,B,C)\), which determine the anisotropy of the gas molecules.

For the intensity of lateral scattering \(\left(\beta=\frac{\pi}{2}\right)\) we had [formula (15)]

\[ i=\frac{8\pi^{4}I}{\lambda^{4}}\,(K)^2(1+\eta+3\varepsilon) \]

for one molecule, and \(N\) times greater for \(1\ \mathrm{cm}^3\); consequently, the coefficient of lateral scattering (per \(1\ \mathrm{cm}^3\)) will be

\[ R=\frac{i}{I}=\frac{8\pi^{4}N}{\lambda^{4}}(\overline{K})^2(1+\eta+3\varepsilon). \]

Substituting instead of \(\overline{K}=\frac{\mu^2-1}{4\pi N}\), we have

\[ R=\frac{\pi^2(\mu^2-1)^2}{2N\lambda^4}(1+\eta+3\varepsilon). \]

Adding to this relation expression (14) for \(\rho\), we can express \(\varepsilon\) and \(\eta\) through quantities obtained directly from measurements:

\[ \varepsilon=\frac{2N\lambda^4 R}{\pi^2(\mu^2-1)^2}\cdot\frac{\rho}{2(1+\rho)},\qquad \eta=\frac{2N\lambda^4 R}{\pi^2(\mu^2-1)^2}\cdot\frac{2-\rho}{2(1+\rho)}. \]

Let us now use the relation (23) derived earlier:

\[ 3\eta=4\varepsilon. \]

It makes it possible to find the basic expression, for the scattering theory under consideration, for the coefficient of lateral scattering:

\[ R=\frac{\pi^2(\mu^2-1)^2}{2N\lambda^4}\cdot\frac{6(1+\rho)}{6-7\rho}. \tag{24} \]

Thus, the presence of anisotropy in the scattering molecules not only leads to a decrease in the polarization of the scattered light, but also changes (increases) the scattering coefficient. The multiplier

\[ \frac{6(1+\rho)}{6-7\rho} \]

is the anisotropy factor. It is equal to unity if \(\rho=0\); in this case we obtain Rayleigh’s classical formula—formula (3) of § 1. There is every reason to call formula (24) the Rayleigh–Cabannes formula.

If the scattering angle is equal to \(\beta\), then the scattering coefficient, on the basis of formula (16), is

\[ R(\beta)=\left(1+\frac{1-\rho}{1+\rho}\cos^2\beta\right)R \tag{25} \]

and the depolarization is

\[ \rho(\beta)=\rho\sin^2\beta+\cos^2\beta. \tag{17} \]

For the absorption coefficient \(k\) we obtain, substituting \(R\) from (24) into (19):

\[ k=\frac{8\pi^3(\mu^2-1)^2}{3N\lambda^4}\cdot\frac{6+3\rho}{6-7\rho}. \tag{26} \]

It is very convenient that these formulas, obtained on the basis of taking molecular anisotropy into account, do not contain the parameters \(A, B, C\), which determine the anisotropy. They contain only the quantities \(\mu\) and \(\rho\), determined directly from experiment.

5. Absolute values of atmospheric transparency

The theory of molecular scattering of light set forth in the preceding section agrees with the results of measurements much better than the original Rayleigh theory.

If, under laboratory conditions, one measures the lateral scattering of light by various gases, then according to Rayleigh’s formula (3) the ratio of the intensities \(\dfrac{i_1}{i_2}\) for two gases must be equal to the ratio

\[ \frac{(\mu_1^2 - 1)^2}{(\mu_2^2 - 1)^2}. \]

Strutt was the first to measure the intensities \(i\) of a number of gases relative to air, and although in general he obtained satisfactory results, the discrepancies with the calculated values were nevertheless quite considerable. Table 2 gives Strutt’s results\(^{11}\) (relative to air).

Table 2

Name of gas	Hydrogen \(H_2\)	Air	Carbon dioxide \(CO_2\)	Nitrous oxide \(N_2O\)	Ethyl chloride	Ether vapor
\(\dfrac{(\mu_1^2 - 1)^2}{(\mu_2^2 - 1)^2}\)	0.23	1	2.36	3.12	16.1	27.1
\(\dfrac{i_1}{i_2}\)	0.25	1	2.5	3.75	16.0	26.0

However, more accurate subsequent measurements by Cabannes\(^{36}\) showed that the agreement with Rayleigh’s formula is less satisfactory than was obtained by Strutt. Table 3 gives his results. In the second row \(U^R\) denotes the quantity \((\mu^2 - 1)^2\) for the given gas, and in the third row \(\Delta\) denotes the relative deviation

\[ \frac{\dfrac{i_1}{i_2} - \dfrac{U_1^R}{U_2^R}}{\dfrac{U_1^R}{U_2^R}}. \]

The deviations lie within the range from 4 to 23%, i.e., they are very large.

Cabannes, simultaneously with the intensity, measured the depolarization \(\rho\) of the scattered light. According to the Rayleigh–Cabannes formula (24), the ratio \(\dfrac{i_1}{i_2}\) must be equal to the ratio

\[ (\mu_1^2 - 1)^2 \frac{6(1+\rho_1)}{6 - 7\rho_1} : (\mu_2^2 - 1)^2 \frac{6(1+\rho_2)}{6 - 7\rho_2}. \]

Table 3

$\dfrac{i_A}{i_{N_2}} = 0.829$	$\dfrac{i_{\mathrm{CO}_2}}{i_A} = 3.31$	$\dfrac{i_{\mathrm{CO}_2}}{i_{\text{air}}} = 2.62$	$\dfrac{i_{\mathrm{CO}_2}}{i_{O_2}} = 2.93$	$\dfrac{i_{H_2}}{i_{O_2}} = 0.255$
$\dfrac{U^R_A}{U^R_{N_2}} = 0.90$	$\dfrac{U^R_{\mathrm{CO}_2}}{U^R_A} = 2.53$	$\dfrac{U^R_{\mathrm{CO}_2}}{i_{\text{air}}} = 2.35$	$\dfrac{U^R_{\mathrm{CO}_2}}{U^R_{O_2}} = 2.80$	$\dfrac{U^R_{H_2}}{U^R_{O_2}} = 0.276$
$\Delta = -0.079$	$\Delta = +0.23$	$\Delta = +0.10$	$\Delta = +0.044$	$\Delta = -0.082$

Let us denote this ratio by $U^C$. Table 4 gives the results of Kaban’s calculations$^{36}$.

Table 4

$\dfrac{i_A}{i_{N_2}} = 0.829$	$\dfrac{i_{\mathrm{CO}_2}}{i_A} = 3.31$	$\dfrac{i_{\mathrm{CO}_2}}{i_{\text{air}}} = 2.65$	$\dfrac{i_{\mathrm{CO}_2}}{i_{O_2}} = 2.93$	$\dfrac{i_{H_2}}{i_{O_2}} = 2.55$
$\dfrac{U^C_A}{U^C_{N_2}} = 0.823$	$\dfrac{U^C_{\mathrm{CO}_2}}{U^C_A} = 3.12$	$\dfrac{U^C_{\mathrm{CO}_2}}{U^C_{\text{air}}} = 2.65$	$\dfrac{U^C_{\mathrm{CO}_2}}{U^C_{O_2}} = 3.07$	$\dfrac{U^C_{H_2}}{U^C_{O_2}} = 0.255$
$\Delta = +0.007$	$\Delta = +0.057$	$\Delta = -0.011$	$\Delta = -0.047$	$\Delta = 0$

From this table one sees a very good agreement of the measurement results with theory; the deviations do not exceed a few percent.

Formula (26) of the preceding paragraph determines the absorption coefficient $k$, caused by the scattering of light. The formula includes the depolarization quantity $\rho$, which for air, according to numerous measurements, is $0.042$. Using this formula, one can determine the attenuation of light rays that have passed through the entire thickness of the atmosphere and thereby determine the magnitude of the absolute transparency of the atmosphere.

Let us consider a mixture of gases $1, 2, 3, \ldots$, whose refractive indices are $\mu_1, \mu_2, \mu_3, \ldots$ under normal conditions. If the rela-

relative volume content of each gas is \(v_1, v_2, v_3,\ldots\), with \(v_1+v_2+v_3+\cdots=1\), then the absorption coefficient \(k\) of this gas mixture under normal conditions will be

\[ k_0=v_1k_1+v_2k_2+v_3k_3+\cdots, \]

where \(k_1, k_2, k_3,\ldots\) are the absorption coefficients of the individual gases under normal conditions (normal pressure \(P_0\) and temperature \(0^\circ\mathrm{C}\)). If the mixture is at temperature \(t\) and pressure \(P\), then the absorption coefficient is

\[ k=\frac{P}{P_0(1+\alpha t)}k_0, \]

where \(\alpha=\dfrac{1}{273}\).

If the point at which transparency is measured is at height \(H\) above sea level and \(I\) is the measured illumination on a horizontal surface from a beam of monochromatic rays coming from the zenith, then the vertical transparency of the atmosphere at this point is the ratio \(\dfrac{I}{I_0}\) (\(I_0\) is the illumination beyond the limits of the atmosphere), determined by the formula

\[ \Delta=-\lg\frac{I_0}{I}=M\int_H^\infty k\,dh. \tag{27} \]

The logarithm is common, where \(M=\lg e=0.4343\), \(k\) is the absorption coefficient of a layer of air at height \(h\), measured in centimeters, and \(\Delta\) is the optical density of the atmosphere in the vertical direction.

Let us recall that the coefficient of transparency of a medium is the quantity \(p\), defined by the formula \(e^{-k}=p\), whence \(I=I_0p^x\), since \(I=I_0e^{-kx}\). The coefficient of transparency is a proper fraction showing what part of the energy passes through a layer of the given medium of unit thickness.

In the case of a medium whose density changes with distance \(x\), in the formula \(I=I_0p^x\), instead of \(x\) one takes the mass \(m\) of the layer of the substance, i.e. the calculations are carried out by Bouguer’s formula \(I_m=I_0p^m\). If the mass of the atmosphere in the vertical direction is taken as unity, then the value \(p\) calculated by Bouguer’s formula is called the coefficient of transparency of the atmosphere.

It is clear that the quantities \(k\), \(\Delta\), and \(p\) are uniquely related to one another. Knowing one of them, the other two can be computed. In what follows we shall most often use the quantity optical density.

Since the atmosphere is a mixture of gases, whose temperature and pressure may vary, formula (27) must be rewritten in the form

\[ \Delta = M\int_H^\infty \frac{p\,(v_1k_1+v_2k_2+v_3k_3+\cdots)\,dh} {P_0(1+\alpha t)} = M\sum \left\{ k_i\int_H^\infty \frac{pv_i\,dh}{P_0(1+\alpha t)} \right\}. \tag{28} \]

For calculations of the optical density of the atmosphere by this formula it is necessary to know the numerical values of \(k\) for the principal constituents

parts of the atmosphere. These data are given in Table 5, which also indicates the depolarizations \(\rho_i\) and the percentage content \(v_i\) by volume for the dry atmosphere of air (for the troposphere at middle latitudes). The absorption coefficients are given for three wavelengths, 3 974, 4 127, and 5 893 Å. Table 5 gives, for these wavelengths, the values \(\mu_i-1\), which may be regarded as known to an accuracy of \(1/1000\).

In calculating the coefficients \(k\) for the number of molecules in \(1\ \mathrm{cm}^3\) of gas under normal conditions, the value\({}^{15}\) \(N=2.90\cdot 10^{19}\) was adopted.

Table 5

Name of gas	\(100v_i\)	\(\rho_i\)	\((\mu_i-1)\cdot 10^5\), 3 974 Å	\(k_i\cdot 10^8\), 3 974 Å	\((\mu_i-1)\cdot 10^5\), 4 127 Å	\(k_i\cdot 10^8\), 4 127 Å	\((\mu_i-1)\cdot 10^5\), 5 893 Å	\(k_i\cdot 10^8\), 5 893 Å
Nitrogen	78.03	0.037	30.55	45.44	30.45	38.81	29.78	8.91
Oxygen	20.99	0.065	27.48	38.59	27.42	33.03	27.02	7.71
Argon	0.94	0.005	28.91	38.55	28.84	32.98	28.37	8.15
Carbon dioxide	0.03	0.098	46.00	114.67	45.87	98.03	45.02	22.60
Hydrogen	0.01	0.022	14.26	9.65	14.21	8.24	13.90	1.88
Neon	0.0012	0.005	—	—	—	—	6.71	0.41
Helium	0.0004	?	—	—	—	—	3.50	0.12(?)

If it is assumed that the composition of the air is the same at all altitudes (a mixed atmosphere), and if for the height of the homogeneous atmosphere one takes the value \(7.991\cdot 10^5\ \mathrm{cm}\) for sea level, then the calculations give\({}^{15}\):

\(\lambda\), Å	\(k_0\)	\(\Delta\)	\(\dfrac{I}{I_0}\)
3 974	\(43.95\cdot 10^{-8}\)	0.1525	0.7039
4 127	37.56	0.1303	0.7408
5 893	8.65	0.03003	0.9332

The relative error for \(k_0\) and \(\Delta\) is of the order of \(1/500\), and for \(\dfrac{I}{I_0}\) of the order of \(1/2000\), if \(\Delta=0.1\).

The hypothesis of an atmosphere mixed at all altitudes (in any case up to 120–150 km) is finding more and more confirmations. However, as is known, from the point of view of the theory of static equilibrium of the atmosphere one should expect changes in the composition of the air with altitude in the direction of an increase in the relative content of light gases at great heights. For comparison one may calculate \(\Delta\) and \(\dfrac{I}{I_0}\) for this case as well.

If it is assumed that the air temperature is the same at all altitudes and that the density of each gas decreases with altitude according to the barometric formula (as a function of the molecular weight of the gas), then the calculations give:

\(\lambda,\ \text{Å}\)	\(\Delta\)	\(\dfrac{I}{I_0}\)
3 974	0.1538	0.7018
4 127	0.1314	0.7339
5 893	0.03025	0.9327

Finally, if one simply assumes that air is a gas (and not a mixture of gases) whose molecules give rise to depolarization \(\rho=0.042\), then formula (26) may be used. For sea level under normal conditions the calculations give:

\(\lambda,\ \text{Å}\)	\(\mu-1\)	\(k\)	\(\Delta\)	\(\dfrac{I}{I_0}\)
3 974	\(2.9824\cdot 10^{-4}\)	\(43.68\cdot 10^{-8}\)	0.1516	0.7053
4 127	2.9748	37.36	0.1297	0.7418
5 893	2.9260	8.69	0.03017	0.9329

Thus, the transparency in general changes very little depending on what composition of air is assumed at different altitudes, and the calculations by formula (26) give quantities that differ little from those obtained in detailed calculations (for \(\Delta\)—by \(1/200\)).

Formula (26) is convenient to use for rapid calculations; at the same time the accuracy is comparatively high.

Such are the results of the calculations. For comparison with the results of measurements let us turn again to the classical data of the Mount Wilson observatory. According to the measurements of 1910–1911, the vertical optical thickness of the atmosphere above Mount Wilson is:

\[ \Delta_{3838\ \text{Å}}=0.1469,\qquad \Delta_{3974\ \text{Å}}=0.1238,\qquad \Delta_{4127\ \text{Å}}=0.1062. \]

After reducing the calculated values to the altitude of Mount Wilson \((P=623.5\ \text{mm Hg})\), one obtains:

\[ \Delta_{3838\ \text{Å}}=0.1451,\qquad \Delta_{3974\ \text{Å}}=0.1254,\qquad \Delta_{4127\ \text{Å}}=0.1071. \]

The results of measurement agree with the calculated values quite satisfactorily; the deviations amount to approximately \(1\%\).

In the formula for the absorption coefficient there enters the number of molecules in \(1\ \text{cm}^3\), \(N\). If the numerical values of \(N\) are calculated from the values found for Mount Wilson, one obtains\({}^{15}\):

\(\lambda,\ \text{Å}\)	\(N\)
3 838	\(2.86\cdot 10^{19}\)
3 974	2.94
4 127	2.92

and on the average \(N=(2.91\pm0.08)\cdot 10^{19}\), which gives for the Avogadro–Millikan number \((6.52\pm0.18)\cdot 10^{23}\). This somewhat exceeds the value \((6.062\pm0.006)\cdot 10^{23}\), determined accurately by other methods, but at the same time all these data, like the data on the spectral transparency of the atmosphere presented in § 2, undoubtedly prove that under the conditions of the pure atmosphere of Mount Wilson and for wavelengths far from the regions of selective absorption of light by the atmosphere

(absorption by ozone, water vapor, etc.), the attenuation of rays as they pass through the atmosphere is undoubtedly caused mainly by molecular scattering of light and can be calculated with sufficient accuracy by the formulas of the Rayleigh–Cabannes theory of molecular scattering.

This theory also makes it possible to explain the magnitude of the polarization of sky light \(^{20,23}\), scattered at an angle of \(90^\circ\), which was discussed in § 3.

To what extent can the results obtained be used for practical purposes—for calculating the absorption of light rays passing horizontally through the lower layers of the atmosphere? Owing to the turbidity of the lower layers of air by dust, water droplets, etc., somewhat different regularities occur here, even in perfectly clear weather. We now turn to the consideration of this question; moreover, passing over the numerous measurements carried out in various places and at various times, we shall turn to the works of the most recent years, first of all to the works of Tien Kiu and A. and E. Vassy. We shall consider the results of a number of other investigations only as needed.

1. Transparency of the lower layers of the atmosphere. A. Vassy and E. Vassy \(^{37-39}\) carried out in 1937 a series of careful measurements of transparency (in Morocco). The results of their measurements are of great value because they applied the most modern methods of photometry, ensuring high reliability and accuracy of the results (the work was advised by Ch. Fabry), and also because, at one and the same location, they carried out absolute measurements of transparency over the entire visible and part of the ultraviolet and infrared spectra, both vertically through the whole thickness of the atmosphere and horizontally for a ray traveling along its entire path close to the earth’s surface. This makes it possible to draw a number of essential conclusions about the specific character of light absorption in the lower layers of the atmosphere.

All measurements were made under night-time conditions \(^{37}\) (measurements of horizontal transparency, which require the use of artificial light sources at a great distance, are difficult to carry out by day because of interference produced by bright sunlight). To measure the transparency of the whole thickness of the atmosphere, spectrophotometric observations were made of the light of stars (most often the star Vega in the constellation Lyra). To determine the absorption per unit mass (vertical ray), the measurements, as indicated in § 2, are made at different zenith distances of the star, i.e. at different masses \(m\), and from the derivative of the optical density \(\dfrac{d\Delta}{dm}\) the optical density for \(m=1\) is found.

The horizontal measurements were made with the aid of a powerful artificial light source placed at a great distance (up to \(24\) km). To determine the absolute values of the transparency, after measurements at a large distance the apparatus was quickly transported by automobile to another place, and the same light source was spectrophotometered from a smaller distance.

For convenience in comparing the vertical and horizontal absorptions, the results are given in the form of optical densities \(d\),

Fig. 8

calculated for a layer of air \(1\ \text{km}\) thick, reduced to normal conditions of temperature and pressure (\(0^\circ\) and \(760\ \text{mm Hg}\)).

Fig. 9

In Fig. 8 are shown the results of measurements for the entire thickness of the atmosphere (by star) for the night of October 4, 1937, which was distinguished by the greatest transparency of all the nights when the measurements were carried out\(^{38}\). In Fig. 9 are given the results for the night of August 12,

which seemed quite clear, but in fact yielded the largest value of absorption.

In these curves one can see a very rapid decrease in transparency toward short wavelengths, as is to be expected from the laws of molecular scattering, but the main course of the curves is disturbed by individual absorption bands.

In Fig. 10 data are given for absorption in the horizontal direction for the night of September 11, when the atmosphere in the lower layers was especially transparent; in Fig. 11—for the night of September 4, when the transparency was especially low. For this latter case, measurements in the short-wavelength region of the spectrum either did not succeed at all (in the ultraviolet part), or are of little reliability, since, owing to the large absorption, the spectrograms proved underexposed.

Fig. 10

Fig. 11

These curves show that the transparency of the lower layers is considerably less than for the whole atmosphere as a whole (even on the clearest, completely cloudless nights). If, for example, for the region around \(5000\) Å the atmosphere as a whole has an optical density from \(0.010\) to \(0.020\) per \(1\) km, then for the lower layers \(d\) amounts to from \(0.020\) to \(0.110\). This circumstance has great practical

is important for questions of the visibility of distant objects, the transmission of light signals, etc., since here we usually deal with the passage of light rays through the lower layers of the atmosphere. In aerial photography, too, one has to deal with the oblique passage of rays through the lower layers.

In this respect it is very important to clarify the role and share of absorption due to the scattering of light, since scattering, in addition to absorption, creates a luminous veil that interferes with observations. Suppose that we are considering an ideally black target (for example, a vertical screen) situated against the background of the sky, while gradually moving away from it. At some distance \(a\) the target will cease to be visible. \(a\) is the visibility range of the given target. The target has ceased to be visible because the difference between the visible brightnesses of the target and of the surrounding background has reached the threshold of perception of brightness difference by the given human eye. Denoting this threshold by \(\varepsilon\), we can define it as follows:

\[ \frac{H_h-H_z}{H_h}=\varepsilon, \tag{29} \]

where \(H_z\) is the brightness of the target, \(H_h\) the brightness of the background.

The thickness of air between the observer and the target not only weakens the light coming from the target, thereby reducing its visible brightness, but itself, being illuminated by the light of the sun and the sky, creates an additional veil through scattering and thereby reduces contrast. How great the role of scattering is in this is shown by successful photographs of distant objects in infrared rays: the amount of scattered light rapidly decreases as the wavelength of light increases. However, as we shall see below, not all scattering can be reduced to molecular scattering.

Among the nine curves of the spectral transparency of the lower layers of the atmosphere (for 9 nights), measured by Vassy\(^38\), five curves are distinguished by the following essential feature: despite a considerable difference in transparency, the curves are similar to one another in the sense that one curve is obtained from another by adding to the optical density \(d\) a definite quantity, the same throughout the entire spectrum. Thus, if we start from the curve for September 11 (Fig. 10), it is necessary to add:

Date	Addition
For July 17	0.002
“ 30 “	0.011
“ August 30	0.022
“ 3 “	0.026

These data show that, besides the weakening of light caused by molecular scattering and selective absorption, in the lower layers of the atmosphere there often occurs neutral absorption. This neutral absorption can reach a considerable magnitude, and in practice it must be taken into account.

In order to clarify the role of molecular scattering in the weakening of rays passing through the lower layers of the atmosphere, let us try to separate the various factors that determine absorption. First of all we shall analyze the factor of selective absorption.

7. Selective Absorption by the Atmosphere

For the visible part of the spectrum, among the gases composing air, oxygen, ozone, and water vapor exhibit selective absorption.

For oxygen, since the time of Janssen,^40 two systems of absorption bands have been known, differing from one another in that in one the absorption increases in proportion to the density, while in the other it increases in proportion to the square of the density. As a result of numerous investigations^41–44, both under laboratory conditions and in the atmosphere, it has been established that the first system belongs to the neutral oxygen molecule

\[ \mathrm{O}_2\left(A^1\Sigma \to X^3\Sigma\right). \]

This system consists of the following bands:

Band	From, Å	To, Å	Maximum near, Å
\(A\)	7,594	7,703	7,596
\(A'\)	7,602	7,650	—
\(B\)	6,946	6,868	6,869
\(\alpha\)	6,276	6,319	6,278
\(\alpha'\)	5,788	5,835	5,790
\(\alpha''\)	about 5,380	5,380	—
and a weak band		7,710

According to the measurements of Babcock and Dieke,^41 the total intensities of the individual bands are as follows (if the intensity of the \(A\) band is conventionally taken as unity): \(B\)—approximately 0.5; \(\alpha\)—0.05, while the intensity of the remaining bands is vanishingly small.

The intensities of the bands of the second system were studied most completely in the work of Salow and Steiner,^45 who made measurements with a tube 2.4 m long at oxygen pressures from 40 to 160 atm.

The measurement data are collected in Table 6; in the second row are given optical densities reduced to a layer of 1 km at a pressure of 1 atm.^38 In the third row are given Herman’s data,^46 likewise recalculated^38 to 1 km at a pressure of 1 atm.

Table 6

\(\lambda\), Å	6,299	5,773	5,223	4,770	4,464	3,803	3,607	3,436	3,282
\(d\), according to Salow and Steiner	0.0215	0.0315	0.0019	0.0215	0.0015	0.0086	0.0147	0.0041	0.0006
\(d\), according to Herman	0.0170	0.0350	—	0.0210	—	—	—	—	—

When using these data to calculate the absorption of rays as they pass through the entire thickness of the atmosphere, one must take into account the decrease of the absorption coefficient with height, owing to the quadratic dependence of the amount of absorption on pressure. Assuming that the oxygen pressure \(p\) decreases with height \(h\) according to the law \(p=p_0 e^{-bh}\), and considering the absorption \(dA\) in an elementary layer \(dh\) at height \(h\) as \(dA=kp^2\,dh\), we obtain the total absorption in the atmosphere in the form:

\[ A=\int kp_0^2 e^{-2bh}\,dh, \]

the integral being taken from the height of the observation site to infinity.

For ozone, in addition to the Hartley and Huggins bands, which lie in the ultraviolet region, the Chappuis band is known in the green-red region of the spectrum. However, the absorption in this ozone band is small (for the maximum at 6020 Å the absorption coefficient is \(k = 0.068\)), and, because of the small concentrations of ozone in the lower layers of the atmosphere, its absorption for horizontal rays may be neglected. If the total amount of ozone in the atmosphere corresponds, at normal pressure and temperature, to a layer \(0.15\)—\(0.40\) cm thick, then near the earth there is only \(0.0003\) cm or less of it per 1 km.

The absorption of ozone in the visible part of the spectrum need be taken into account only when rays pass through the entire thickness of the atmosphere.

Absorption by water vapor was first studied by Janssen\(^{47}\), and more recently by Mecke, Baumann, and Freudenberg\(^{48,49}\). In the visible and near infrared regions of the spectrum there are the following absorption bands:

Band \(Z\)	8399—7843 Å
“ \(a\)	7304—6850
“	6660—6280
Rain band	6060—5860
Band \(\delta\)	5780—5670
“	5478—5420
“	5111—4981

The circumstance that selective absorption is distributed over the spectrum in separate, sometimes well-pronounced bands facilitates the identification of the bands and makes it possible to separate selective absorption from absorption that extends continuously and uniformly over the spectrum and is due to molecular scattering. According to formula (26), the absorption coefficient \(k\), due to molecular scattering, varies over the spectrum proportionally to the multiplier

\[ y = \frac{(\mu_0^2 - 1)^2}{\lambda^4}. \]

Formula (26) may be written in the form

\[ k = C \frac{y}{N}, \tag{30} \]

where

\[ C = \frac{8\pi^3}{3}\frac{6+3\rho}{6-7\rho}, \]

and \(N\) is the number of molecules in \(1\ \mathrm{cm}^3\). For air \(\rho = 0.042\), and \(A\) is simply a certain number. Bassi used the most recent measurements and calculated \(\mu\) as a function of \(\lambda\). Their results are presented in Fig. 12. If the absorption of light in the atmosphere were caused only by molecular scattering, then, representing graphically the optical density \(d\) as a function of \(y\), we should have to obtain, according to formula (30), a straight line passing through the origin. The presence of absorption of another nature causes the real graph to deviate from a straight line, as is seen from Figs. 13 and 14, constructed from data taken from Figs. 9 and 11; Fig. 14 corresponds to absorption through the entire thickness of the atmosphere on the most transparent night of October 4,

and Fig. 15—to absorption in the lower layers on the transparent night of September 11 (to clarify the role of molecular scattering it is advisable to use data referring to the most transparent nights).

In Fig. 13 the straight line \(d=A+By\), constructed by Vassy\({}^{38}\) from calculations taking into account the influence of selective absorption, is shown; here \(A\) was found to be \(0.001\), and \(B=1.41\cdot 10^{-13}\). Such a small value of \(A\) indicates that, for rays passing through the entire atmosphere, on a clear night neutral absorption is very insignificant.

For absorption in the lower layers it is more difficult to separate out the influence of molecular scattering. In Fig. 14 a straight line is drawn with the same slope as in Fig. 13, but here the neutral absorption is much greater: \(A=0.007\).

From the value found, \(B=1.41\cdot 10^{-13}\), it can be determined that the Avogadro–Millikan number is \(6.11\cdot 10^{23}\), which is very close to the generally accepted value \(6.06\cdot 10^{23}\).

Fig. 12

Extensive research on the transparency of the atmosphere in the direction of interest to us was also carried out by Tcheng-Kiou\({}^{49a}\). He made a careful analysis of measurements of the Smithsonian Institution, carried out over 10 years (from 1920 to 1930) on Mount Montezuma in Chile (at an altitude of \(2711\) m). These data refer to a much smaller number of wavelengths, but on the other hand they cover a much longer period of time. Tcheng-Kiou found that in most cases the optical density of the atmosphere at this great altitude can be represented by the formula

Fig. 13

Fig. 14

\[ d=A+Bf(\lambda), \]

where \(f(\lambda)=(\mu^{2}-1)^{2}\lambda^{-4}\).

For a dry atmosphere the neutral absorption is very small: \(A=0.0034\). For the Avogadro number Tcheng-Kiou found (from the value for \(B\))

\[ N=(6.136\pm0.085)\cdot 10^{23}. \]

It is very interesting to estimate the relative share of molecular scattering in the total absorption of the lower layers of the atmosphere for the case

of clearest weather. This can be done using Vassy’s data (Fig. 14). For this night (September 11) \(A=0.007\), as was indicated above, and for \(y<100\) (\(\lambda>4400\) Å, see Fig. 12) molecular scattering produces absorption, and only for \(y>100—130\) (\(\lambda<4400—4000\) Å) does absorption due to molecular scattering prove predominant, even in relation to the sum of neutral + selective absorption. The absorption of the atmosphere as a whole, i.e. absorption for a vertical ray (Fig. 13), gives a substantially different picture: absorption due to molecular scattering proves predominant already for \(y>25—30\), i.e. for \(\lambda<6000\) Å (throughout the visible spectrum, with the exception of its red part).

Fig. 15

Fig. 16

Vassy’s calculations\(^ {39}\) showed that even the sum of absorptions—selective + neutral + molecular scattering—often does not cover all the observed absorption. In Fig. 15 a curve is given for August 12, showing the course of this “residual” absorption over the spectrum for the complete atmosphere on August 12—a night with poor transparency, see above. In Fig. 16 the same is given for the lower layers (night of September 11). In both cases a maximum is observed near \(5000\) Å, for which the optical density due to residual absorption is approximately \(0.01\).

What is the nature of this residual absorption, connected neither with selective absorption by the atmospheric constituents known to us, nor with molecular scattering? In seeking the carrier of this absorption, one should turn to the laws of scattering of light by large particles.

Attention should also be concentrated in this same direction in connection with neutral absorption. Indeed, neutral absorption is characteristic of the lower layers of the atmosphere. If for the complete atmosphere \(A=0.001\), then for the lower layers, according to Vassy’s data\(^ {38}\), \(A\) is much larger:

Date	\(A\)
September 11	0.007
July 17	0.009
? 30	0.018
August 30	0.029
September 3	0.033

It should be expected that this absorption is caused by some large particles which, in accordance with the barometric formula, should be located precisely in the lower layers of the atmosphere. Such large suspended particles in the air may be dust particles, droplets of water, etc.

Scattering of light by large particles must take place in an entirely different way than scattering by molecules. This is evident already from the fact that the calculations connected with molecular scattering were carried out by us under the assumption that the external electric field is homogeneous, which corresponds to the supposition that the dimensions \(a\) of the scattering particles are very small in comparison with the wavelength \(\lambda\). If, however, \(a\) is not small in comparison with \(\lambda\), and still more if \(a > \lambda\), then the field strength of the light wave is substantially different at different points of the scattering particle, and the scattering process can no longer be likened to the radiation of a dipole executing forced oscillations.

Fig. 17

1 and 2 — sky light, 3 — scattering by an infinitely small particle according to Rayleigh.

The theory of light scattering by large particles shows, in particular, that scattering at different angles \(\varphi\) ceases to satisfy the simple symmetric relation of the Rayleigh theory \((1+\cos^2\varphi)\), and that the scattered light is directed predominantly forward. In this respect, the presence in the scattered light of the sky of the so-called Mie effect is indicative; this effect consists in the brightness of the sky increasing for points close to the sun to a far greater degree than is expected from Rayleigh’s formula. In Fig. 17, Pokrovsky’s\(^ {50}\) data are given for the brightness of the sky at different angles relative to the sun (the lengths of the segments are a measure of the brightness for the indicated angles). For comparison, below is given a diagram corresponding to Rayleigh’s law \((1+\cos^2\varphi)\). The considerable deviations from Rayleigh’s law (the Mie effect) testify to the appreciable influence on the total scattering exerted by light scattered by large particles suspended in the air. As is seen from Fig. 17, the Mie effect is more strongly expressed for long-wave radiation, which, as we shall see below, also agrees with the conclusions of the theory of light scattering by large particles.

In proceeding to the exposition of this theory, it should immediately be pointed out that its results are extremely important for questions of light absorption, especially for the practical questions belonging here. Thus, for example, the entire theory of the passage of light through clouds, fogs, and smoke is based on the consideration of light scattering by large particles.

Part II

SCATTERING OF LIGHT BY LARGE PARTICLES

8. Mie theory. This theory arose, as is known, in connection with a problem not directly related to the range of atmospheric questions. This is evident from the very title of Mie’s work[^51], “On questions of the optics of turbid media, especially colloidal solutions of metals.” It was necessary to explain the varied coloration acquired by the aforementioned solutions, especially solutions of gold, under different conditions. Later, however, Mie’s theory acquired fundamental importance for atmospheric optics and itself underwent substantial development.

Fig. 18

Following Mie, for simplicity we shall denote the coordinate axes \(X, Y, Z\) by the numerals \(1, 2, 3\). We choose a right-handed coordinate system; \(r\) is the radius vector of the point \(x, y, z\); the angle subtended by \(r\) with axis \(1\) is \(\vartheta\); the angle of the projection of \(r\) onto the plane \((2,3)\) with axis \(2\) is \(\varphi\) (Fig. 18). Thus, \(r, \vartheta, \varphi\) are the polar coordinates of the point. The components of the electric and magnetic vectors in these polar coordinates will be denoted by \(E_r, E_\vartheta, E_\varphi\) and \(H_r, H_\vartheta, H_\varphi\). Maxwell’s equations in polar coordinates have the form:

\[ \left. \begin{aligned} r^2 \sin\vartheta\left(k \frac{\partial E_r}{\partial t}+\Lambda E_r\right) &= \frac{\partial(r\sin\vartheta H_\varphi)}{\partial\vartheta} - \frac{\partial(rH_\vartheta)}{\partial\varphi}, \\ r\sin\vartheta\left(k \frac{\partial E_\vartheta}{\partial t}+\Lambda E_\vartheta\right) &= \frac{\partial H_r}{\partial\varphi} - \frac{\partial(r\sin\vartheta\cdot H_\varphi)}{\partial r}, \\ r\left(k \frac{\partial E_\varphi}{\partial t}+\Lambda E_\varphi\right) &= \frac{\partial(rH_\vartheta)}{\partial r} - \frac{\partial H_r}{\partial\vartheta}, \\ -r^2\sin\vartheta\cdot\mu\frac{\partial H_r}{\partial t} &= \frac{\partial(r\sin\vartheta\cdot E_\varphi)}{\partial\vartheta} - \frac{\partial(rE_\vartheta)}{\partial\varphi}, \\ -r\sin\vartheta\cdot\mu\frac{\partial H_\vartheta}{\partial t} &= \frac{\partial E_r}{\partial\varphi} - \frac{\partial(r\sin\vartheta\cdot E_\varphi)}{\partial r}, \\ -r\mu\frac{\partial H_\varphi}{\partial t} &= \frac{\partial(rE_\vartheta)}{\partial r} - \frac{\partial E_r}{\partial\vartheta}. \end{aligned} \right\} \tag{31} \]

Here \(k, \Lambda, \mu\) denote the dielectric constant, conductivity, and magnetic permeability (all in one and the same system of units). In a nonconducting medium \(k\mu = \dfrac{1}{v^2}\), where \(v\) is the velocity of propagation of electromagnetic waves in the medium.

We further suppose that

\[ E_r = E_{r0} e^{2\pi int},\quad H_r = H_{r0} e^{2\pi int}\quad \text{and so on,} \tag{32} \]

where \(E_{r0}\) and \(H_{r0}\) already depend only on the coordinates, but not on time, and \(n\) is the number of oscillations per second.

Next,

\[ 4\pi^2 n^2 \mu k - 2\pi i n p\lambda = \frac{4\pi^2 m^2}{\lambda^2}, \tag{33} \]

where \(m\) is the complex refractive index of the medium for light of wavelength \(\lambda\). Finally, let us introduce the following further notation:

\[ -\frac{i n \mu}{m} H_{r0}=M_r, \quad \text{etc.}, \tag{34} \]

\[ \frac{2\pi m r}{\lambda}=x, \tag{35} \]

which will allow Maxwell’s equations to be written in a form in which the quantities \(E\) and \(M\) enter in exactly the same way:

\[ \left. \begin{aligned} x^2\sin\vartheta\, E_r &= \frac{\partial\left(x\sin\vartheta\, M_\varphi\right)}{\partial\vartheta} - \frac{\partial\left(x M_\vartheta\right)}{\partial\varphi}, \\ x\sin\vartheta\, E_\vartheta &= \frac{\partial M_r}{\partial\varphi} - \frac{\partial\left(x\sin\vartheta\, M_\varphi\right)}{\partial x}, \\ x E_\varphi &= \frac{\partial\left(x M_\vartheta\right)}{\partial x} - \frac{\partial M_r}{\partial\vartheta}, \\ x^2\sin\vartheta\, M_r &= \frac{\partial\left(x\sin\vartheta\, E_{\varphi0}\right)}{\partial\vartheta} - \frac{\partial\left(x E_{\vartheta0}\right)}{\partial\varphi}, \\ x\sin\vartheta\, M_\vartheta &= \frac{\partial E_{r0}}{\partial\varphi} - \frac{\partial\left(x\sin\vartheta\, E_{\varphi0}\right)}{\partial x}, \\ x M_\varphi &= \frac{\partial\left(x E_{\vartheta0}\right)}{\partial x} - \frac{\partial E_{r0}}{\partial\vartheta}. \end{aligned} \right\} \tag{36} \]

We shall regard the scattering particle as a small sphere of radius \(b\), whose center coincides with the origin of coordinates. Quantities referring to the field inside the sphere will be supplied with the subscript \(i\), and those referring to the medium surrounding the sphere with the subscript \(a\); the corresponding refractive indices will be denoted by \(m\) and \(m_0\). With respect to the permeability we shall assume that \(\mu_i=\mu_a\). The variable \(x\) undergoes a jump at the surface of the sphere, since

\[ x_i=\frac{2\pi m}{\lambda}r,\quad x_a=\frac{2\pi m_0}{\lambda}r=\frac{2\pi}{\lambda'}r, \]

if \(\lambda'\) denotes the wavelength in the medium surrounding the sphere. On the surface of the sphere the following boundary conditions must be satisfied:

\[ \left. \begin{aligned} E_{\vartheta a}&=E_{\vartheta i},& E_{\varphi a}&=E_{\varphi i},\\ (xM_\vartheta)_a&=(xM_\vartheta)_i,& (xM_\varphi)_a&=(xM_\varphi)_i. \end{aligned} \right\} \tag{37} \]

To solve equations (36) under the boundary conditions (37), Mie uses the method that had previously been developed by Rayleigh in his well-known work “Theory of Sound.”

By means of suitable substitutions, (36) can be transformed into second-order equations:

\[ \left. \begin{aligned} \frac{\partial^2\left(x^2 E_r\right)}{\partial x^2} &+ \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left( \sin\vartheta\,\frac{\partial E_{r0}}{\partial\vartheta} \right) + \frac{1}{\sin^2\vartheta}\frac{\partial^2 E_{r0}}{\partial\varphi^2} + x^2 E_{r0} =0, \\ \frac{\partial^2\left(x^2 M_r\right)}{\partial x^2} &+ \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left( \sin\vartheta\,\frac{\partial M_r}{\partial\vartheta} \right) + \frac{1}{\sin^2\vartheta}\frac{\partial^2 M_r}{\partial\varphi^2} + x^2 M_r =0. \end{aligned} \right\} \tag{38} \]

We shall consider two different groups of particular solutions of the system of equations (36). The first group corresponds to the assumption that the radial component of the magnetic field is absent:

\[ M_r=0,\quad E_{r0}\ne 0, \]

and the second group—to the assumption that the radial component of the electric field is absent:

\[ E_{r0}=0,\quad M_r\ne 0. \]

Thus, the first group of solutions is formed by waves arising owing to the “electric oscillations” of the sphere, and the second owing to its “magnetic oscillations.”

Suppose that we have in some way found an expression for \(E_{r0}\); the remaining components in this case are easily determined as follows: in the second and third equations of the system (36) we put \(M_r=0\) and substitute into them the values for \(M_\varphi\) and \(M_\vartheta\) taken from the fifth and sixth equations. Then we obtain equations for calculating \(E_{\vartheta0}\) and \(E_{\varphi0}\) from the known \(E_{r0}\). Knowing \(E_{r0}\), \(E_{\vartheta0}\), \(E_{\varphi0}\), we find \(M_\varphi\), \(M_\vartheta\) from the last two equations of the system (36). The calculation for the second group of solutions is carried out in an analogous manner.

As for \(E_{r0}\), it is obtained in the form of a sum of terms, each of which satisfies equation (38) and is the product of a function of \(x\) by a function of the angles \(\vartheta,\varphi\); the \(\nu\)-th term has the form

\[ E_{r0}^{(\nu)}=\frac{K_\nu(x)}{x^2}P_\nu(\vartheta,\varphi). \]

The functions \(K_\nu\) and \(P_\nu\) must satisfy the equations:

\[ \frac{d^2K_\nu}{dx^2}+\left(1-\frac{c_\nu}{x^2}\right)K_\nu=0, \tag{39} \]

\[ \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left[\sin\vartheta\,\frac{\partial P_\nu}{\partial\vartheta}\right] +\frac{1}{\sin^2\vartheta}\frac{\partial^2P_\nu}{\partial\varphi^2} +c_\nu P_\nu=0, \tag{40} \]

where

\[ c_\nu=\nu(\nu+1). \tag{41} \]

The function \(P_\nu\) is a spherical function of the variables \((\vartheta,\varphi)\) with integer index \(\nu\); the function \(K_\nu\) is related to a cylindrical function with fractional index (denominator 2). The solution of Maxwell’s equations is obtained in the form:

\[ \left. \begin{aligned} E_{r0}^{(\nu)}&=\frac{K_\nu(x)}{x^2}P_\nu(\vartheta,\varphi),\\ E_{\vartheta0}^{(\nu)}&=\frac{1}{\nu(\nu+1)}\,\frac{K'_\nu(x)}{x}\, \frac{\partial P_\nu}{\partial\vartheta},\\ E_{\varphi0}^{(\nu)}&=\frac{1}{\nu(\nu+1)}\,\frac{K'_\nu(x)}{x\sin\vartheta}\, \frac{\partial P_\nu}{\partial\varphi},\\ M_r^{(\nu)}&=0,\\ M_\vartheta^{(\nu)}&=\frac{1}{\nu(\nu+1)}\,\frac{K_\nu(x)}{x\sin\vartheta}\, \frac{\partial P_\nu}{\partial\varphi},\\ M_\varphi^{(\nu)}&=-\frac{1}{\nu(\nu+1)}\,\frac{K_\nu(x)}{x}\, \frac{\partial P_\nu}{\partial\vartheta}. \end{aligned} \right\} \tag{42} \]

Similarly one finds the second group of solutions for the case \(E_{r0}^{(\nu)}=0\). Representing the functions \(P_\nu\) and \(K_\nu\) in the form of series, Mie solves the problem for a plane wave incident on a scattering particle—a sphere. Having found the expression for the incident plane wave in the form (42), Mie then determines the components \(E_{ri}, E_{\vartheta i}, \ldots, M_{\varphi i}\) of the refracted wave and the components \(E_{ra}, E_{\vartheta a}, \ldots, M_{\varphi a}\) of the reflected wave. Each component is represented by an infinite series whose terms are products of certain coefficients \(a_\nu\) and \(P_\nu\) by the functions \(P_\nu\) and \(K_\nu\) and their derivatives. Restricting the problem to the case of sufficiently small dimensions of the scattering particles \((b<\lambda)\), Mie showed that light scattered by small spheres is practically formed by the superposition of a finite number of partial waves, and the number of partial waves increases as the dimensions of the scattering particles increase. In doing so, one should sum the waves corresponding both to the “electric oscillations” and to the “magnetic oscillations” of the particle.

Mie further showed that the magnetic partial oscillations obey the same laws as the electric ones, and the \(\nu\)-th magnetic oscillation approximately corresponds to the \((\nu+1)\)-st electric one.

Restricting the calculations to the case \(b \leq 180\ \mathrm{m\mu}\), Mie finds that in practice it is quite possible to limit oneself to the first and second electric \((\nu=1\ \text{and}\ 2)\) and the first magnetic \((\nu=1)\) oscillations. Oscillations of higher orders are vanishingly small in comparison with the first ones indicated.

It is noteworthy here that, for infinitely small spheres \((b\to 0)\), only the first partial electric wave remains, which coincides exactly with the Rayleigh wave of scattered light. Thus Mie’s theory is a more general theory and includes Rayleigh’s theory as a special case.

In Figs. 19–26 are given diagrams of the electric lines of the field on the surface of the scattering sphere for the first four electric and the first four magnetic oscillations. The plane of the drawing is taken to be the plane of oscillation in the incident beam. It is a plane of symmetry of the process.

For a more visual representation of Mie’s results, let us consider a large particle of radius \(b\) as a system of bound charges. Let us calculate the potential energy of this system when it is placed in an external field (for charges that are not at rest, one may use the time-averaged value). Suppose that this system contains a certain number of charges \(e_i\) with coordinates \(\xi_i,\eta_i,\zeta_i\) in the Cartesian system \(x,y,z\); the origin of this coordinate system is assumed to lie inside the volume occupied by the charges. If we denote by \(\varphi\) the potential of the external field at the origin, then the potential \(\varphi_i\) at the point \((\xi_i,\eta_i,\zeta_i)\) can be expressed as:

\[ \varphi_i=\varphi+\left(\xi_i\frac{\partial\varphi}{\partial x}+\eta_i\frac{\partial\varphi}{\partial y}+\zeta_i\frac{\partial\varphi}{\partial z}\right) +\frac{1}{2}\left(\xi_i^2\frac{\partial^2\varphi}{\partial x^2}+\eta_i^2\frac{\partial^2\varphi}{\partial y^2}+\zeta_i^2\frac{\partial^2\varphi}{\partial z^2}+\right. \]

\[ \left. +2\xi_i\eta_i\frac{\partial^2\varphi}{\partial x\partial y} +2\eta_i\zeta_i\frac{\partial^2\varphi}{\partial y\partial z} +2\zeta_i\xi_i\frac{\partial^2\varphi}{\partial z\partial x} \right)+\cdots \]

Fig. 19
First electric
oscillation.

Fig. 20
First magnetic
oscillation.

Fig. 21
Second electric
oscillation.

Fig. 22
Second magnetic
oscillation.

Fig. 23
Third electric
oscillation.

Fig. 24
Third magnetic
oscillation.

Fig. 25
Fourth electric
oscillation.

Fig. 26
Fourth magnetic
oscillation.

Consequently, the potential energy of the system of charges \(e_i\), determined by the action of the external field, can be represented as follows:

\[ \sum e_i'\varphi_i'=\sum e_i\varphi+\sum\left(e_i\xi_i\frac{\partial\varphi}{\partial x} +e_i\eta_i\frac{\partial\varphi}{\partial y} +e_i\zeta_i\frac{\partial\varphi}{\partial z}\right) +\frac{1}{2}\sum\left(e_i\xi_i^2\frac{\partial^2\varphi}{\partial x^2}+\right. \]
\[ \left. +e_i\eta_i^2\frac{\partial^2\varphi}{\partial y^2} +e_i\zeta_i^2\frac{\partial^2\varphi}{\partial z^2} +2e_i\xi_i\eta_i\frac{\partial^2\varphi}{\partial x\,\partial y} +2e_i\eta_i\zeta_i\frac{\partial^2\varphi}{\partial y\,\partial z} \right. \]
\[ \left. +2e_i\zeta_i\xi_i\frac{\partial^2\varphi}{\partial z\,\partial x}\right)+\cdots \tag{43} \]

Here the quantities \(\varphi,\dfrac{\partial\varphi}{\partial x},\dfrac{\partial^2\varphi}{\partial x^2},\ldots\) should be regarded as constants characterizing the external electric field; the electrical properties of the system of charges are determined by sums composed of products of the charges by their coordinates. The first of these sums, \(\sum e_i\), denotes the total charge of the system. The second term in (43) determines the energy by means of the sums \(\sum e_i\xi_i,\ \sum e_i\eta_i,\ \sum e_i\zeta_i\). These three sums are the components of the electric dipole moment along the coordinate axes \(x,\ y,\ z\). The third term in (43) contains six sums:

\[ \begin{gathered} \theta_{11}=\sum e_i\xi_i^2;\quad \theta_{22}=\sum e_i\eta_i^2;\quad \theta_{33}=\sum e_i\zeta_i^2,\\ \theta_{12}=\sum e_i\xi_i\eta_i;\quad \theta_{23}=\sum e_i\eta_i\zeta_i;\quad \theta_{31}=\sum e_i\zeta_i\xi_i. \end{gathered} \tag{44} \]

These six sums are the electric moments of inertia, or quadrupole moments. If in our system the sums of the positive and negative charges are equal to one another (the system is not charged, \(\sum e_i=0\)), then in a homogeneous field the potential energy is determined by the dipole moment. The potential energy due to moments of higher orders (quadrupole, octupole, etc.) can arise only in an inhomogeneous field. It is clear that the more inhomogeneous the field is, the greater the significance of the terms in (43) due to moments of higher orders.

If the external electric field under consideration is the field of a light wave, then the potentials in (43) may be regarded as depending on the coordinates and time according to the sine or cosine law. Obviously, in this case we obtain the same law also for the first, second, and all subsequent derivatives \(\dfrac{\partial\varphi}{\partial x},\ \dfrac{\partial^2\varphi}{\partial x^2},\ldots\). If the dimensions \(b\) of the volume containing the system of charges under consideration are very small in comparison with the wavelength \(\lambda\) \((b\ll\lambda)\), then correspondingly very small will be the products \(\xi_i\dfrac{\partial\varphi}{\partial x},\ \eta_i\dfrac{\partial\varphi}{\partial y},\ldots;\) the products \(\xi_i^2\dfrac{\partial^2\varphi}{\partial x^2},\ \eta_i^2\dfrac{\partial^2\varphi}{\partial y^2},\ldots\) will evidently be still smaller; they will correspond to an even higher order of smallness. One may therefore restrict oneself to the first term in (43), i.e., consider the field of the light wave homogeneous (within the limits of the small volume under consideration). But it is clear that with increasing

for dimensions of the volume \(b\) comparable with \(\lambda\), the field can no longer be regarded as completely homogeneous, and the subsequent terms in series (43) will have to be taken into account.

These illustrative relations help one to understand the results of Mie’s theory. Let us recall that the derivation of Rayleigh’s formula was based on the assumption of the homogeneity of the electric field in which the scattering particle is situated, and that, on the other hand, according to Mie’s theory one must take the more partial waves, the larger the dimensions of the scattering particles. We can clearly imagine why the first partial electric wave in Mie’s theory coincides exactly with the Rayleigh wave: it corresponds to the case in which only dipole radiation need be taken into account, while quadrupole, octupole, etc., radiations may be neglected. This is the case of a homogeneous field. But if the dimensions of the scattering particles become very small in comparison with the length of the light wave (and in precisely this lies the difference between the conditions of the problem solved by Mie and the conditions of the problem in Rayleigh’s theory), then one cannot restrict oneself to the Rayleigh wave, and it becomes necessary to introduce the more subsequent terms, the larger the particle size. From Mie’s diagrams in Figs. 19–26 one directly sees the increase in the order of the moments that influence the process of light scattering by large particles.

Fig. 27              Fig. 28

Mie’s partial waves of higher order destroy the symmetry of Rayleigh scattering, and the properties of light scattered by large particles differ substantially from the properties of light that has undergone molecular scattering. Mie carried out a number of numerical calculations, and although they refer to the special case of light scattering by colloidal solutions of gold, nevertheless the results are in many respects characteristic in general of scattering by large particles.

First, the polarization properties of the scattered light change substantially. In Fig. 27 two curves are given: for the degree of polarization

\[ P=\frac{I_{\perp}-I_{\parallel}}{I_{\perp}+I_{\parallel}} \]

and depolarization

\[ \rho=\frac{I_{\parallel}}{I_{\perp}} \]

as functions of the diameter \(d\) of the particles, for scattering at an angle of \(90^\circ\), for green light \((\lambda=5500\,\text{\AA})\).

Beginning with \(d=90\,\mathrm{m}\mu\), the polarization ceases to be complete, decreasing rapidly for larger \(d\). In this, the spatial—

linear symmetry of polarization; the maximum of polarization is observed already at angles different from 90°. In Figs. 29 and 30 the curves give the magnitude of the polarization at different scattering angles for particles of diameter 160 and 180 mμ. In Fig. 28, for comparison, a symmetric curve is shown corresponding to an infinitely small particle size (Rayleigh scattering). It is noteworthy that negative polarizations of the scattered light can exist, as is seen from Fig. 30.

Fig. 29

Fig. 30

Further, the spatial distribution of the scattered light in terms of its intensities changes completely. In Fig. 17 we saw the symmetric curve corresponding to Rayleigh’s law. In a more complete form this curve is given in Fig. 31—this diagram determines the value of the scattering coefficient as a function of the scattering angle separately for the two components \(I_{\perp}\) and \(I_{\parallel}\). The extent to which the spatial symmetry of scattering is violated in scattering by large particles is seen from the curves in Figs. 32 and 33, constructed by Mie for

Fig. 31

Fig. 32

\(d = 160\) and \(180\) mμ. The scattering of light occurs predominantly forward, in the direction of the incident ray.

In § 7 we encountered phenomena proving the existence of a similar Mie effect in the scattering of light in the atmosphere.

These are the principal results of Mie’s theory, with applications of which to questions of atmospheric optics we shall have occasion to meet later.

It should also be noted that Mie’s theory was somewhat supplemented and extended by Jobst, who used in the mathematical part

of the theory, Debye’s previous results. Mie’s calculations, as we have seen, were carried out up to a particle size of \(180\,m\mu\). For still larger particles the calculations became difficult to perform, since it was necessary to take too large a number of terms in the series giving the solution of the problem.

Jobst\(^{52}\) was able to advance further by making use of the asymptotic expansions of cylindrical functions, which had been given by Debye\(^{53}\) in his work devoted to the calculation of the light pressure on an individual particle.

Using analytic expressions for sums of series representing cylindrical functions, Jobst continued Mie’s calculations up to a particle size of \(600\,m\mu\), specifically (as did Mie) for the case of colloidal gold solutions: he determined the absorption and scattering spectra of such solutions.

Fig. 33

The results obtained by Debye and Jobst are of fundamental importance for the theory of the spectral transparency of clouds and fogs: using them, Stratton and Houghton were able to construct their theory, which will be discussed below.

A number of calculations were carried out by Shuleikin\(^{54}\) in connection with questions of the optics of seawater. Having simplified Mie’s formulas for the case of nonconducting particles, he calculated several cases, namely when the particle diameter is \(0.32\lambda\), \(0.96\lambda\), and \(2.87\lambda\). Similar numerical computations in especially large numbers were performed by Blumer, who solved problems for spheres with diameters up to \(3.8\lambda\), and moreover with various refractive indices \(\mu\). In Blumer’s work\(^{55}\) one can find numerical results for the following cases

\[ \left(a=\frac{2\pi b}{\lambda},\ \text{where } b \text{ is the particle radius}\right): \]

\[ \begin{aligned} \mu &\simeq 1 \qquad && a=5;\ 10;\ 12.5;\ 13.9;\ 17.3.\\ \mu &=1.2 \qquad && a=1;\ 1.5.\\ \mu &=1.25\ \text{(ice particles at low temperature)} \qquad && a=0.01;\\ &&& 0.1;\ 0.2;\ 0.25;\ 0.3;\ 0.4;\ 0.5;\ 0.6;\ 0.8;\ 1.2;\ 1.6;\ 2;\ 2.5;\ 3;\ 4;\\ &&& 5;\ 6;\ 8.\\ \mu &=1.3\ \text{(water droplets in air)} \qquad && a=0.1;\ 1.5;\ 3;\ 12.\\ \mu &=1.4661 \qquad && a=5.\\ \mu &=1.5 \qquad && a=0.01;\ 0.1;\ 0.25;\ 1;\ 1.5;\ 1.75;\ 2;\ 2.25;\ 4.\\ \mu &=\infty\ \text{(spheres of an opaque absolutely nonconducting substance)} \qquad && a=0.01;\ 0.1;\ 0.5;\ 1;\ 3;\ 5;\ 10. \end{aligned} \]

Figure 34 gives, from Blumer’s data, a polar diagram for scattering in various directions by ice particles of radius \(0.2\lambda\). The intensities are indicated separately for both polarized components \(i_I\) and \(i_{II}\). How much the scattering pattern becomes complicated as the particle size increases is clear from Fig. 35, where a diagram is given for scattering by a water droplet whose radius is about \(2\lambda\).

Fig. 34

Fig. 35

9. Theories constructed taking account of reflection, refraction, and diffraction.

Before turning to the concluding part of the present review—the theory of the transparency of fogs—it is necessary to clarify the question of whether, in calculating the passage of light through an aggregate of the smallest particles, one may not get by simply by taking account of reflection and refraction of rays, or, in any case, by taking account of reflection, refraction, and diffraction. Mie’s theory is a rigorous theory; it considers the problem of light scattering in all its breadth, analyzing the electromagnetic fields excited by forced oscillations on the spherical surface of separation that arise under the action of the incident electromagnetic wave. As a result, the difficulties that arise in carrying out numerical calculations are so great that it proves possible to perform calculations only for a very limited number of cases. In application to the problem of the transparency of fogs, Mie’s theory proves so complex that one is, willy-nilly, still forced to confine oneself to the calculation of only the most elementary cases. If it were possible to replace this complex theory by taking account of reflection, refraction, and diffraction, then the solution of the problem of the transparency of fogs would be substantially facilitated.

Fig. 36

Let us therefore consider the influence exerted by the three factors named.

A fairly large number of works have been devoted to taking account of reflection and refraction. Clausius\(^{56}\) already calculated double refraction on spherical drops of water. Roth\(^{57}\) took into account single reflection, double refraction, and single total internal reflection. Richard\(^{58}\) and Mirdel\(^{59}\) took into account only reflection. Mecke\(^{60}\), taking into account single reflection and double refraction, determined, in addition to the intensity, also the phase difference. Shuleikin\(^{54}\), using Fresnel’s formulas, likewise took into account single reflection and double refraction. Brücke\(^{7}\), in his work of 1853, gave essentially the same calculation as Shuleikin.

The most complete investigation belongs to Wiener\(^{61}\), who for droplets of water calculated the influence of fivefold reflection and refraction; the droplets were here regarded as perfectly transparent. The results of his calculations are represented by the polar diagram in Fig. 36. The reflected light is partially polarized, and at an angle of \(100^\circ\) the polarization is complete, as is seen from Table 7, where Wiener’s data are given separately for the intensities of the two components \(i_\perp\) and \(i_\parallel\) at various scattering angles \(\varphi\) (in Fig. 36 the data indicated are for the total intensity of the scattered light \(i = i_\perp + i_\parallel\)).

As regards taking diffraction into account, this problem was solved by Fraunhofer\(^{62}\), Schwerdt\(^{63}\), Airy\(^{64}\), and Mecke\(^{60}\). Obviously, in this case also the result must depend essentially on the size of the sphere

compared with the wavelength of light. In Fig. 37 polar diagrams are given for three cases: \(a = 1\) (curve 1), \(a = 3\) (curve 2) and \(a = 5\) (curve 3). For the different curves the intensity scale is not the same, as is shown in the figure.

Table 7

\(\varphi\)	\(i_I\)	\(i_{II}\)	\(\varphi\)	\(i_I\)	\(i_{II}\)	\(\varphi\)	\(i_I\)	\(i_{II}\)
\(0^\circ\)	0,051	0,051	\(70^\circ\)	0,009	0,002	\(135^\circ\)	0,184	0,224
\(10^\circ\)	0,053	0,048	\(80^\circ\)	0,005	0,001	\(150^\circ\)	0,556	0,609
\(20^\circ\)	0,053	0,041	\(90^\circ\)	0,006	0,001	\(160^\circ\)	1,031	1,076
\(30^\circ\)	0,051	0,033	\(100^\circ\)	0,008	0,000	\(170^\circ\)	1,682	1,690
\(45^\circ\)	0,050	0,009	\(110^\circ\)	0,016	0,015	\(180^\circ\)	2,043	2,043
\(60^\circ\)	0,017	0,003	\(120^\circ\)	0,048	0,064

Let us now compare the numerical data obtained by applying the rigorous Mie theory and by taking account of reflection, refraction, and diffraction. It is obvious that such a comparison can be made for particles that are not too large, for which it is still possible to carry out a calculation by Mie theory.

Fig. 37

In Figs. 38—40 the corresponding data according to Blumer\(^ {65}\) are presented for particles of different size: small particles of radius 150 mµ—Fig. 38, medium ones (300 mµ)—Fig. 39, and somewhat larger ones (1200 mµ)—Fig. 40. In these graphs, along the axis of abscissas are plotted

Fig. 38. Comparison of numerical data obtained by applying the rigorous Mie theory and by taking reflection, refraction, and diffraction into account for small particles

Fig. 39. Comparison of numerical data obtained by applying the rigorous Mie theory and by taking reflection, refraction, and diffraction into account for medium-sized particles

scattering angles in degrees, and along the ordinate axis the scattering coefficients in arbitrary units. On each graph four curves are drawn: 1—\(\Gamma_M\), calculated according to Mie; 2—\(\Gamma_W\), reflection and refraction, calculated according to Wiener; 3—\(\Gamma_{Bg}\), diffraction; 4—\(\Gamma_W+\Gamma_{Bg}\), the combined action of reflection, refraction, and diffraction.

In addition, on each graph an arrow indicates the value

\[ z=\Gamma_M-(\Gamma_W+\Gamma_{Bg}), \]

representing the difference of the scattering coefficients determined: 1) according to Mie and 2) with allowance for reflection, refraction, and diffraction.

Fig. 40

These data show quite definitely that reflection and refraction, taken both separately and together with diffraction, cannot give results consistent with rigorous theory. Therefore, in analyzing all phenomena of any significance that pertain here, it is necessary either to resort to rigorous theory, or to risk making a considerable error.

After these remarks we shall turn to the question of the transparency of mists and clouds, indicating only in conclusion that, besides the authors mentioned earlier, the theory of light scattering by large particles was also studied by Kleiner^66, Gans^67, Stratton^68 (Rayleigh), Ray^69, Raman^70, Schaefer^71, Meredith and Willmott^72, Senftleben and Benedict^73, Shirmann^74, Möbius^75, Grüner^76, Bromwich^77, Pokrovskii^78.

10. Theory of the transmission of light through mist (Stratton and Houghton theory). As early as 1909, Wood showed the advantages of photographing distant objects in infrared rays^79. The Rayleigh scattering coefficient is proportional to \(\lambda^{-4}\),

therefore the scattering of long-wave radiation is small, as a result of which the brightness of the atmospheric haze that interferes with the observation of distant objects is low for infrared rays. In 1923 the American high-altitude flier Stevens demonstrated the enormous practical importance of infrared photography, obtaining, with the aid of a suitable light filter, a distinct image of a mountain range from a distance of more than 500 km. If in clear weather visibility is already great, then the improvement of visibility in foggy weather is especially important. Attempts to use in this case as well photographic plates sensitive to infrared rays, and the corresponding light filters, according to reports by a number of authors, supposedly gave positive results. However, it soon became clear that these measures often do not lead to any improvements, and that in general the whole question of the advantages of using infrared rays in photography in fog is far more confused than had previously seemed. And in fact, the transfer of the laws of Rayleigh scattering of light from the conditions of a pure atmosphere to fog would be justified only if the size of the fog particles were sufficiently small in comparison with the wavelength of light. But measurements have shown that this is not so; the radii of droplets of fog and clouds lie within the range from several microns to several tens of microns, i.e. they are much greater than the wavelength of those light rays that could be used for photographic purposes (up to 1.3 μ).

Fig. 41

In the two preceding paragraphs we had sufficient material for judging how substantially the properties of scattered light change with an increase in the size of the scattering particles. In connection with the problem of using infrared rays in fog, the question arises of the spectral characteristics of the light scattered by fog droplets. Stratton and Houghton attempted to solve this problem.

The immediate impetus for carrying out their extensive theoretical investigation was the unexpected experimental result of Houghton^80, who discovered a maximum of transparency of artificial fogs for blue rays. His results are presented in Fig. 41, where the wavelengths in microns are plotted on the abscissa axis, and transparency in arbitrary units on the ordinate axis. A maximum of transparency is seen at 0.49 μ instead of a monotonic increase of transparency toward larger λ. But a number of other investigators, who also studied before Houghton the transparency of artificial and natural fogs for visible light, did not detect such a maximum. Granath and Hulburt^81 found for natural fog a quite definite monotonic increase of transparency toward larger λ. Anderson^82 obtained the same for artificial fog. Having no grounds to cast doubt upon

correctness of one result or another, Stratton and Houghton sought the cause of the discrepancies in differences in the experimental conditions, and the only factor that changed appreciably in all these measurements could, in their view, be the size of the fog droplets. But if a certain maximum of transparency is observed in the visible region of the spectrum for one droplet size, while for another size it is absent in the visible region, then one may expect that it has shifted into another region of the spectrum. This would mean that, in general, transparency has an extremal course along the spectrum, and that the spectral characteristics shift when the radius of the droplets changes. The prospect of revealing such a substantial new spectral effect prompted Stratton and Houghton to undertake their study. However, the difficulties connected with this are so great that the authors were forced to introduce considerable simplifications into the conditions of the problem: they regarded the fog droplets as nonconducting spheres with refractive index 1.33 for all wavelengths. The method of solution used by them\(^83\) was taken from Mie theory with the improvements introduced by Debye and Jobst, of which we spoke above\(^{51—53}\).

The result obtained is very remarkable: the scattering coefficient, considered as a function of the wavelength of the light wave, varies nonmonotonically, but has two maxima, and the position of these maxima in the spectrum depends on the radius of the scattering particles \(a\). When \(a\) changes, the entire spectral curve shifts as a whole; when \(a\) increases, the maxima shift toward longer wavelengths.

If \(I_0\) denotes the intensity of the light beam entering the fog, and \(I\) its intensity after passing through a layer of fog of thickness \(z\), then we may write

\[ \frac{I}{I_0}=e^{-kz}. \tag{45} \]

Stratton and Houghton write their formula in the form

\[ \frac{I}{I_0}=e^{-2\pi a^2 zK}. \tag{46} \]

Here, \(n\) is the number of particles in \(1\ \mathrm{cm}^3\), and the function \(K\) is equal to

\[ K=\frac{\lambda^2}{4\pi^2 a^2 N_a^2}\operatorname{Re}\sum_{u=1}^{\infty}(-1)^u(2u+1)(C_u^1+C_u^2), \tag{47} \]

where

\[ C_u^1=(-1)^u \frac{ N_a\psi_u(x)\psi'_u(y)-N_i\psi'_u(x)\psi_u(y) }{ N_a\Phi_u(x)\psi'_u(y)-N_i\Phi'_u(x)\psi_u(y) }, \]

\[ C_u^2=(-1)^u \frac{ N_i\psi_u(x)\psi'_u(y)-N_a\psi'_u(x)\psi_u(y) }{ N_i\Phi_u(x)\psi'_u(y)-N_a\Phi'_u(x)\psi_u(y) }. \]

Here \(\lambda\) is the wavelength of the incident light;

\[ x=2\pi a\frac{N_a}{\lambda};\qquad y=2\pi a\frac{N_i}{\lambda}; \]

$N_a$ is the complex refractive index of the external medium (relative to the sphere) (taken equal to unity); $N_i$ is the same for the substance of the sphere (taken equal to 1.33)

\[ \psi_u(x)=\left(\frac{\pi x}{2}\right)^{\frac12}\cdot I_{u+\frac12}(x); \qquad \Phi_u(x)=\left(\frac{\pi x}{2}\right)^{\frac12}\cdot H_{u+\frac12}(x) \]

and $\operatorname{Re}$ is the real part. Primes denote the first derivative with respect to $x$ or, respectively, with respect to $y$. The functions $I_{u+\frac12}$ and $H_{u+\frac12}$ are Bessel functions and, respectively, Hankel functions with half-integer index.

The function $K(x)$ is shown in Fig. 42.

If, as an example, we consider fog droplets of radius $3\mu$, then for such droplets

\[ x=\frac{2\pi\cdot 3}{\lambda}. \]

Hence we can determine the wavelength $\lambda_{\max}$ for which the transparency is greatest ($K$ is smallest). This, according to Fig. 42, corresponds to $x=11.2$, and consequently

\[ \lambda_{\max}=\frac{6\pi}{11.2}\simeq 1.8\,\mu. \]

Fig. 42

The smallest transparency will occur at $x=6$, i.e.

\[ \lambda_{\min}\simeq 3.1\mu. \]

The second minimum of transparency will occur at $x=15$, i.e.

\[ \lambda'_{\min}\simeq 1.2\,\mu, \]

and for still smaller $\lambda$ (larger $x$) the transparency, according to the curve in Fig. 42, will slowly increase. Only for very large wavelengths from $4\mu$ to $\infty$ (the range of $x$’s from 5 to 0) will a rapid increase of transparency be observed as $\lambda$ increases, similar to what is observed in Rayleigh scattering.

If we consider a smaller particle, $a=1\,\mu$, then all the spectral points considered will shift toward shorter wavelengths:

\[ \lambda_{\max}\simeq 0.6\,\mu,\qquad \lambda_{\min}\simeq 1.0\,\mu \]

and

\[ \lambda'_{\min}\simeq 0.4\,\mu. \]

Consequently, in the wavelength interval $0.4$—$0.6\,\mu$ and for $\lambda\gg 1.3$ the dependence of transparency on $\lambda$ will be “similar” to the Rayleigh one. Transparency rapidly increases toward larger $\lambda$, but for $\lambda$ from $1\,\mu$ to $0.6\,\mu$ and for $\lambda<0.4\,\mu$ the behavior of the transparency will be the reverse.

Finally, for very large particles all the extremal points shift into the far infrared region, and within the range of wavelengths that can be used in visual, photographic-

…in photographic and photoelectric observations, we shall be on the almost flat part of the curve corresponding to large \(x\) (\(x>15\)). Such is the theoretical picture, obtained by Stratton and Houghton, of the transmission of light of different wavelengths through fog.

11. Experimental data on the transmission of light through fog. The results of Houghton’s measurements mentioned at the beginning of the preceding paragraph\(^{80}\) (Fig. 41) can be directly compared with theory, since the particle size of the artificial fog with which the measurements were made was known.

Since in Fig. 41 the ordinates represent the magnitudes of transparency, the maximum on the curve in Fig. 41 must correspond to the minimum on the curve in Fig. 42. Stratton and Houghton believe that the maximum of transparency at \(0.49\,\mu\) found by the latter corresponds to the minimum of the curve in Fig. 42 at \(x=11.2\). Substitution into the formula

\[ x=\frac{2\pi a}{\lambda} \]

of the values \(x=1\) and \(\lambda=0.49\) gives

\[ 2a=\frac{11.2\cdot 0.49}{\pi}=1.75\,\mu . \]

The diameter of the fog droplets, found by measurement (from observation of the diffraction pattern in the fog), according to Houghton’s estimate, is from 2 to 3 \(\mu\). Thus the agreement with theory in this respect may be regarded as sufficiently satisfactory.

Fig. 43

Figure 43 shows both curves—the theoretical one (points) and the experimental one (circles)—which have been reduced to the same scale at the maximum. The agreement of the curves must be regarded as rather unsatisfactory. This is due to the simplifications that Stratton and Houghton were forced to make in solving this problem (§ 10).

Unfortunately, Houghton’s curve (Fig. 41) is almost the only case in which a maximum of transparency was observed that could be identified with the minimum on the curve in Fig. 42. Only in the work of Japanese investigators were similar results obtained\(^{87}\). In all other numerous measurements\(^{84}\) one usually observes a monotonic increase of transparency as one moves into the infrared region; this increase is sometimes quite appreciable, while sometimes it is negligibly small. The difference in results may depend to a considerable degree on the difference in the sizes of fog particles: they range from 5–10 \(\mu\) to several tens of microns. Unfortunately, measurements of the transparency of natural fogs are by no means always accompanied by measurements of droplet sizes.

A very interesting question is the transparency of fogs in the far infrared region of the spectrum. Granath and Hulburt\(^{81,85}\) carried out meas…

measurements at a distance of 400 m of the transparency of natural fogs for wavelengths from 0.4 μ to 7 μ. In measurements in the far infrared region they very cleverly simplified the spectral problem by using simply thermal radiators with very different temperatures. Thus, a nichrome coil, heated by a current in air, had an emission maximum near 3 μ (curve 2 in Fig. 44), while another radiator, consisting of an iron vessel blackened on the outside and filled with water under high pressure at a temperature of 161°C, had an emission curve with a maximum at 7 μ (curve 1, Fig. 44). Below are the data of Granath and Hülburt for a fog in which the ordinary visual visibility of a black object was 600 m. The figures in the second column show the distance at which the intensity of the beam is attenuated by the fog one hundredfold:

Fig. 44

Fig. 45

Wavelength in μ	Distance in m
0.4	710
0.5	843
0.6	910
1.0	970
2.0	980
3.0	980
7.0	1,140

The general course of the transparency \(t = I/I_0\) over the spectrum, according to Hülburt’s data[^85], is shown in Fig. 44 (curve 4), where wavelengths in microns are plotted along the abscissa. Fig. 45 gives data on the relation of the transparency \(t\) in fog for \(\lambda = 7\ \mu\) to the ordinary visibility (in kilometers).

Leaving aside the numerous and sometimes mutually contradictory, in their results, works on determining the transparency of fogs (we refer those interested to special surveys[^84,^86]), let us turn to the most recent work, carried out by the Institute of Theoretical Geophysics of the Academy of Sciences of the USSR in the summer of 1939. These measurements were made on Elbrus at an altitude of 3,000 m, where fogs were especially frequent. This study compares favorably with the overwhelming majority of previous ones in that in it a serious

attention was paid, simultaneously with the study of transparency, to the careful determination of the structure of the fog (the size and number of droplets). This latter part of the work was carried out by the Leningrad Institute of Experimental Meteorology.^88

The work studied the transparency of fogs for ultraviolet, visible, and infrared rays from 250 to 1200 mμ, special attention being given to testing the Stratton–Houghton formula (46) with respect to the dependence of transparency on the number of droplets \(n\), which appears in the exponent. According to formula (46), the quantity \(\lg \dfrac{I}{I_0}\) should vary proportionally to \(n\), which is well confirmed for all wavelengths, as can be seen from Fig. 46.

Fig. 46

Fig. 47

These data refer to a fog with an average droplet size \(2a = 16\ \mu\). The slope of the curves depends rather strongly on \(\lambda\). An interesting behavior as a function of \(a\) was obtained for the “relative coefficient” of fog absorption, for which the value^88

\[ R=\frac{\lg\left(\dfrac{I}{I_0}\right)_{1200}}{\lg\left(\dfrac{I}{I_0}\right)_{400}} . \tag{48} \]

was adopted.

This dependence is presented in Fig. 47. It is clear that the “relative coefficient” of absorption characterizes the degree of advantage in using infrared rays as compared with visible ones.

When considering the dependence of one or another optical characteristic of fog on droplet size, it must be borne in mind that natural fogs are usually inhomogeneous, and the radii of their droplets are not identical.

The Stratton–Houghton theory, although only a rough approximation, must serve as the starting point for the development of a more complete theory. The need for such a theory, in view of the enormous practical importance of the questions belonging here, is very great.

12. Rocard’s theory. Attempts to interpret “residual absorption.” King’s formula. In 1930 Rocard published a theory of the scattering of light by relatively large particles, which differs substantially from the theories considered above both in its initial assumptions and in its results—

there.^89 Pocart, solving the problem on the basis of the classical electromagnetic theory of light, simplified the conditions of the problem by assuming that all molecules of the scattering particle behave in the electric field of the incident light wave as though neighboring molecules did not exist at all.

In other words, he completely neglects the field arising owing to polarization in the dielectric, and regards the field inside the dielectric (inside the scattering particle) as equal to the external field (the field of the light wave). It is clear that this conception differs substantially from the rigorous theory of Mie. Pocart considers his assumption correct in the first approximation and bases this conclusion on the results of comparatively recent work by Mallemann,^90 who studied the question of the polarization of a dielectric.

Fig. 48

The simplification of the problem carried out by Pocart enabled him to reduce the problem of the scattering of light to the comparatively simple case of interference and, thanks to this, to find a general expression for the scattering coefficient that makes it possible to carry out all calculations quite simply. It was precisely this last circumstance that could serve as justification for carrying out simplified calculations after the rigorous Mie—Debye—Iobst theory had already been created, for, as we have seen, the formulae of this theory are very complicated and in any case compel one to confine oneself to approximate calculation.

It may be said that in Pocart’s theory the simplifications are introduced directly into the initial physical conceptions; whereas in the Mie—Debye—Iobst theory the initial physical conceptions are completely exact, but mathematical difficulties connected with the complexity of the final formulae force one to resort to approximate calculations.

Let \(P\) be the scattering particle, \(OX\) the direction of the incident wave (Fig. 48). The scattered light is observed in a direction forming an angle \(\varphi\) with the direction of the incident ray. As the axis \(OZ\) we choose the direction of the external bisector of the angle \(\varphi\), as shown in Fig. 48.

Let us now imagine the scattering particle to be cut into plane-parallel thin layers \(dz\) perpendicular to \(OZ\). The position of each such layer will be determined by the coordinate \(z\).

We shall take the phase of the forced oscillations performed by the molecule at the point \(O\) in the field of the incident light wave to be equal to zero. Then the phase of the oscillations of a molecule in the layer with coordinate \(z\) (the phase lag relative to the molecule at \(O\)) will be

\[ \frac{2\pi\left(2z\sin\frac{\varphi}{2}\right)}{\lambda}. \]

A wave scattered by a layer \(z\) (of thickness \(dz\)) in the direction \(\varphi\) will be described by the equation

\[ y=f(z)\sin\left(\omega t-\frac{4\pi\sin \frac{\varphi}{2}\,z}{\lambda}\right)dz, \]

where \(f(z)\) is a quantity proportional to the number of molecules in a layer \(dz\) with coordinate \(z\) and to the amplitude of the electric field of the wave scattered by one molecule.

The intensity of the light scattered by the entire particle \(P\) in the direction \(\varphi\) will be determined by the square of the integral

\[ \int f(z)\sin\left(\omega t-\frac{4\pi\sin \frac{\varphi}{2}\,z}{\lambda}\right)dz, \]

extended from \(0\) to the surface bounding the particle.

Rocard solves the problem completely for a spherical particle. In this case the expression for the intensity of the scattered light is obtained in the form

\[ i=A^2 16\pi^2\nu^2\left(\frac{\lambda}{a}\right)^6 \left[ \sin\left(\frac{aR}{\lambda}\right) -\left(\frac{aR}{\lambda}\right)\cos\left(\frac{aR}{\lambda}\right) \right]^2, \]

where \(A\) is the scattering power of one molecule, \(\nu\) is the number of molecules in \(1\ \mathrm{cm}^3\) of the scattering particle, \(R\) is the radius of the particle,

\[ a=4\pi\sin\frac{\varphi}{2}. \]

These formulas make it possible to determine the dependence of the scattering coefficient on the angle \(\varphi\). The results of such calculations are presented in the polar diagram of Fig. 49. A petal-shaped curve is obtained, showing an extremely sharp dependence of the scattering coefficient on the scattering angle \(\varphi\), which is in contradiction with experiment.

Fig. 49

Rocard finds an ingenious way out of the situation, pointing out that for a spherical particle of another radius the scattering diagram will be deformed in a definite manner, and that one must take the sum of such diagrams for particles of different size, taking into account the law of distribution of the particles by size.

After considering the question of possible distributions of particles by radii, Rocard adopts the distribution law shown in Fig. 50, where the radii of the particles \(R\) are plotted along the abscissa axis, and along the ordinate axis—the relative number \(n\) of particles with a given radius. Using this form of distribution, Rocard finds the final dependence of the scattering coefficient of such a cloud of spherical particles on the angle \(\varphi\), which is shown in Fig. 51. As for the dependence on the wavelength of the scattered light, it turns out that it can approximately be expressed by means of the factor \(\lambda^{-2.5}\) (instead of the Rayleigh \(\lambda^{-4}\)).

Having obtained the scattering law, Rocard calculates the coefficient of absorption of light in the atmosphere.

It must be said that Rocard’s theory has so far found no substantial applications in atmospheric optics (although, for the time being, there are no grounds for regarding its results as incorrect); therefore we shall not present detailed numerical calculations concerning the coefficient of absorption of light in the atmosphere, but shall refer those interested to the works of Rocard89 and Vassy39, which discuss the applicability of Rocard’s formula for explaining the observed absorption of light in the atmosphere.

Fig. 50

Fig. 51

In the first part of the present review, at the end of § 7, we mentioned the “residual absorption” found by Vassy on spectral absorption curves (Figs. 15 and 16). Vassy39 attempted to interpret this absorption both from the point of view of the Stratton–Houghton theory and from that of Rocard’s theory.

On the Stratton–Houghton curve for spectral transparency (Fig. 42) there are two absorption maxima. By bringing the maximum of the residual absorption (Figs. 15 and 16) into coincidence with the left-hand maximum (for \(x = 6\)) on the Stratton–Houghton curve, Vassy obtains for the radius of the particles \(a = 0.5\); coincidence with the right-hand maximum gives \(a = 1.3\). The radii thus obtained themselves have a fairly plausible (though in general too small) value, but if, from the Stratton–Houghton curve, one determines the course of the absorption over the whole spectrum for these two radii, the curves shown in Fig. 52 are obtained. Both of these curves are inconsistent with the observed ones, as is immediately evident if they are compared with the curves in Figs. 15 and 16.

Fig. 52

With the aid of Rocard’s theory, too, it is not possible to explain the facts established by Vassy. As a result, in his paper Vassy39 (November 1939) was compelled to state that none of the theories can as yet give a complete interpretation of all the phenomena connected with the absorption of light in the atmosphere.

In our review we have left without mention the formula proposed at one time by King for describing the absorption of light in a turbid atmosphere. King91 proceeded from Rayleigh’s formula, supplementing it

as its term accounting for neutral absorption

\[ I = I_0 e^{-(\alpha+\beta\lambda^{-4})x}, \]

where \(x\) is the thickness of the layer. This formula is empirical and corresponds to the simplest assumption that absorption is composed of Rayleigh and neutral absorption. In this form, this formula is now only of historical interest.

Although we have not considered everything related to the topic formulated in the title of the article, in any case many fundamental questions have been touched upon. The applications of scattering theory to absorption phenomena in the atmosphere undoubtedly have great practical significance, which makes it necessary to continue work on the further development of the theory. If attempts of this kind prove successful, this will undoubtedly bear fruit both with regard to the use of optical methods for the weather service, and in questions connected with work in fog, and in the range of important problems connected with aerial photography at long distances, and also, possibly, in other problems.

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1. We have retained the notation of the dielectric constant by \(\nu\), adopted by Kabanov, despite the fact that the generally accepted notation is \(\varepsilon\). — I. Kh. ↩↩