Section 32–5 Scattering of light

(Rayleigh scattering / Mie scattering / Tyndall scattering)

 

In this section, Feynman discusses the mechanism of Rayleigh scattering, Mie scattering, and Tyndall scattering. Thus, the section could be titled “Rayleigh scattering, Mie scattering, and Tyndall scattering.”

 

1. Rayleigh scattering:

“The electric field of the incoming beam drives the electrons up and down, and they radiate because of their acceleration. This scattered radiation combines to give a beam in the same direction as the incoming beam, but of somewhat different phase, and this is the origin of the index of refraction (Feynman et al., 1963, p. 32–6).”

 

Elastic scattering: “Why is the sky blue?” is commonly explained using the term Rayleigh scattering by many, but it does not mean they really understand the phenomenon. On the contrary, Feynman did not mention Rayleigh, but we should acknowledge his works related to the scattering of light by particles (primarily nitrogen and oxygen) that are much smaller than the wavelength of the incoming light. Alternatively, we may use the term elastic scattering because the wavelength of scattered light is predominantly the same as the incoming light. Importantly, the word scattering means the absorption of light by a particle and the re-emission of light in almost all directions due to the oscillation of the particle. In other words, the electric field of the incoming light oscillates the electrons, and then the oscillating electrons emit scattered light.

 

“That is to say, light which is of higher frequency by, say, a factor of two, is sixteen times more intensely scattered, which is a quite sizable difference. This means that blue light, which has about twice the frequency of the reddish end of the spectrum, is scattered to a far greater extent than red light (Feynman et al., 1963, p. 32–8).”

 

Wavelength-dependent scattering: The scattering of light is frequency-dependent or wavelength-dependent, i.e., the shorter wavelengths of light (blue and violet) are strongly deflected, whereas the longer wavelengths (red and orange) are slightly deflected. By using Larmor’s formula, the power or intensity (I) of the scattered light is directly proportional to the square of the acceleration (a²) of oscillating electrons in the field of the incoming light (I µ a²). Furthermore, the acceleration of electrons is directly proportional to the square of the frequency of the incoming light (a µ -w²x). In short, one may write I µ a² µ ω⁴. Thus, the intensity of scattered blue light is 1.49⁴ (= 4.9) times more than red light if we assume the wavelength of red light and blue light are about 700 nm and 470 nm respectively (use l = c/f).

 

But if the objects are randomly located, then the total intensity in any direction is the sum of the intensities that are scattered by each atom, as we have just discussed (Feynman et al., 1963, p. 32–6).”

 

Random scattering: Strictly speaking, the objects (or scatterers) are not only randomly located and the total intensity is not simply the sum of the intensities that are scattered by each atom. Some may use the term random scattering because the air molecules are randomly oriented, in random molecular motion, and there are random microscopic fluctuations that scatter more light in one direction than another. Historically, Einstein (1910) deduced that the random thermal motion of the air results in rapid density fluctuations and causes a similar effect to Rayleigh scattering. It can be described as density fluctuation scattering because the fluctuations in the density of air would result in fluctuations in the refractive index of the medium. Based on this model, the refractive index fluctuations behave like molecular scatterers.

2. Mie scattering:

“We have just explained that every atom scatters light, and of course the water vapor will scatter light, too. The mystery is why, when the water is condensed into clouds, does it scatter such a tremendously greater amount of light? …… But if they are right next to each other, they necessarily scatter in phase, and they have a coherent interference which produces an increase in the scattering (Feynman et al., 1963, p. 32–8).”

 

Coherent scattering: Feynman discusses another mystery pertaining to the scattering of a greater amount of light by clouds. It is known as Mie scattering or coherent scattering because the light waves are coherent in the forward direction due to a lump of particles. Specifically, Mie scattering is due to aerosol particles, such as water droplets and ice crystals, but it may include dust, pollen, and smoke that are present in the atmosphere. One may define Mie scattering as the scattering of light whereby the size of a lump of “almost in-phase particles” is the same or more than the wavelength of the incoming light. Although Rayleigh scattering may be considered as the scattering of light in which the size of the particles is less than 1/10 of the light’s wavelength, it is a limiting case of Mie scattering.

 

“So as we keep increasing the size of the droplets we get more and more scattering, until such a time that a drop gets about the size of a wavelength, and then the scattering does not increase anywhere nearly as rapidly as the drop gets bigger (Feynman et al., 1963, p. 32–8).”

 

Multiple scattering: Feynman explains that there is more scattering of light if the size of the water droplets is increased till the wavelength of the incoming light. Instead of saying more and more scattering, we may use the term multiple scattering which depends on the density of particles, size of scatterers (air molecules or aerosol particles), and path of light. For example, the paths of sunlight near the horizon (sunrise or sunset) are longer than the path through the zenith (noon), i.e., we expect more scattering through longer paths. The color of the sky is not simply blue, but it continues to vary depending on the Sun’s position, atmospheric conditions, and locations of the observer (direction of viewing). Multiple scattering of light by water droplets may result in the appearance of a white cloud or dark cloud depending on the density and height of the clouds.

 

3. Tyndall scattering:

“We use a solution of sodium thiosulfate (hypo) with sulfuric acid, which precipitates very fine grains of sulfur. As the sulfur precipitates, the grains first start very small, and the scattering is a little bluish (Feynman et al., 1963, p. 32–9).”

 

Feynman ends the lecture using a demonstration to show the scattering of light by colloidal particles of sulfur. This may be described as Tyndall scattering experiment (See Fig. 1) that uses a glass tube to simulate the sky and a light source to represent the Sun. However, Tyndall scattering experiment does not completely explain how the blue sky varies due to different meteorological or humidity conditions. In his paper titled On the blue colour of the sky, and on the polarization of light, Tyndall (1869) writes: “[f]rom the illuminated bluish cloud, therefore, polarized light was discharged, the direction of maximum polarization being at right angles to the illuminating beam… (p. 224).” To acknowledge his findings, Tyndall scattering may be defined as the lateral scattering of unpolarized light by colloidal particles whereby maximum polarized light is observable at 90^o to the incoming light.

 

Fig. 1

So if the incoming light has an electric field which changes and oscillates in any direction, which we call unpolarized light, then the light which is coming out at 90^o to the beam vibrates in only one direction! (Feynman et al., 1963, p. 32–9).”

 

Lateral scattering: We may adopt the term lateral scattering because the scattered light is perpendicular to the incoming light, however, it is misleading to say that the light vibrates in only one direction. Perhaps Feynman could have emphasized that the unpolarized light vibrates in all planes that are perpendicular to the direction of light propagation and thus, the scattered light does not vibrate in the same direction as the incoming light. If the scattered light is moving upward in the vertical direction, it should be horizontally polarized (See Fig. 2). Additionally, we observe vertically polarized light if the scattered light emerges in the horizontal direction. You should try to connect the direction of lateral scattering of blue light to Brewster’s angle or Heaviside-Feynman’s formula for the electric field of an accelerated charge.

 

Fig. 2

According to Rayleigh scattering, the color of the sky should be violet instead of blue, but it could be clarified from the perspective of a light source and observer. Firstly, the sunlight is not an equal mix of all colors, but there is more blue than violet light. This is related to Planck’s radiation law that is formulated to explain the spectral-energy distribution of radiation emitted by a blackbody. Secondly, our eyes are more sensitive to blue than violet light. Based on Brown and Wald's (1964) experiments, there are three types of color cones in the retina of the human eye that are relatively more sensitive to red, green, and blue light. To summarize, the sky is blue partly because of more blue than violet light emitted by the Sun and our eyes are more sensitive to blue light than violet light.

 

Review Questions:

1. How would you explain the amount of scattering of light is inversely proportional to the fourth power of its wavelength?

2. How would you explain the scattering of light by clouds results in a greater amount of light?

3. How would you explain the scattered light is perpendicular to the incoming light?

 

The moral of the lesson: The phenomenon of blue sky is related to Rayleigh scattering that involves (1) elastic scattering, (2) wavelength-dependent scattering, (3) random scattering, (4) multiple scattering, (5) lateral scattering, (6) Sun emits more blue than violet light, and (7) eye-sensitivity to blue.

 

References:

1. Brown, P. K., & Wald, G. (1964). Visual pigments in single rods and cones of the human retina. Science, 144(3614), 45-52.

2. Einstein, A. (1910). The Theory of the Opalescence of Homogeneous Fluids and Liquid Mixtures near the Critical State. Annalen der Physik, 33, 1275-1298.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Tyndall, J. (1869). On the blue colour of the sky, and on the polarization of light. Phil. M., (4), 37, 384-394.