Section 31–4 Absorption

 (Absorption index / Absorption of light / Absorption spectrum)

 

In this section, the three interesting concepts are absorption index, absorption of light, and absorption spectrum.

 

1. Absorption index:

“We can see what such a complex index means by going back to Eq. (31.6), which is the equation of the wave after it goes through a plate of material with an index n (Feynman et al., 1963, p. 31–8).”

 

Feynman suggests that we can write n = n′−in′′ in which n′ and n″ are real numbers in order that n″ will turn out to be a positive number. In Volume II, he modifies the symbol slightly: “Let’s say that we write n as the sum of a real and an imaginary part: n = n_R−in_I, (32.35) where n_R and n_I are real functions of ω (Feynman et al., 1964).” However, one may clarify that this is a matter of convention, e.g., an alternative is using n = n′ + in′′ (instead of n = n′ - in′′), but n′′ will be a negative number that corresponds to a loss of energy. In addition, the amplitude of light wave in the material is gradually decreased and thus n′′ may be known as the absorption index. In essence, the use of complex index corresponds to the light wave that is represented by a complex function.

 

“We see that the imaginary part n″ of a complex index of refraction represents an absorption (or “attenuation”) of the wave. In fact, n″ is sometimes referred to as the ‘absorption index’ (Feynman et al., 1963, p. 31–8).”

 

It is good that Feynman relates the absorption index to the absorption of the light wave. Perhaps some may elaborate that there is a profound connection between the real index of refraction and the absorption index. Specifically, the real index (n′) and absorption index (n″) are related through the Kramers–Kronig relations. This was deduced independently by Ralph Kronig in 1926 and by Hans Kramers in 1927. In a sense, the relation between n′ and n″ should be expected because the Lorentz Oscillator model assumes the existence of “friction force.” Simply put, the friction force affects the absorption of light in the medium (including the absorption index) and the dispersion of light with respect to different frequency of light (as shown in the refractive index curve).

 

2. Absorption of light:

“As the wave goes through the material, it is weakened. The material is ‘absorbing’ part of the wave. The wave comes out the other side with less energy. We should not be surprised at this, because the damping we put in for the oscillators is indeed a friction force and must be expected to cause a loss of energy (Feynman et al., 1963, p. 31–8).”

 

Feynman says that the damping we put in for the oscillators is a friction force, but it causes a loss of light energy. However, the term friction force may be considered as a metaphor or an assumption of the Lorentz oscillator model, and it is not needed to explain the absorption of light. From the perspective of quantum physics, the energy of a light wave is absorbed by an electron provided the frequency of the light wave through a medium is equal to a resonant frequency of an electron in an atom. After the absorption of light, the electron interacts quantum mechanically with nearby atoms and converts its vibrational energy into thermal energy. In short, a photon is absorbed when an electron in an atom transits from one energy level to another.

 

“We may also point out that an imaginary part to the index n corresponds to bending the arrow E_a in Fig. 31–3 toward the origin. It is clear why the transmitted field is then decreased (Feynman et al., 1963, p. 31–8).”

 

It may not be clear to some why Feynman points out that an imaginary part to the index n corresponds to bending the arrow E_a toward the origin (See Fig. 31–3). One may explain that the arrow should be almost perpendicular in order that the resultant arrow E_a is decreased slightly. Perhaps some may prefer Feynman’s (1985) explanation in his public lecture on QED: “For substances that absorb light, the minor arrows are at less than right angles to the main arrow (Fig. 69b). This causes the final arrow to be shorter than the main arrow, indicating that the probability of a photon going through partially opaque glass is smaller than through transparent glass… (p. 109).” In other words, the almost perpendicular arrow E_a corresponds to the absorption of light in a medium such as transparent glass.

 

3. Absorption spectrum:

“It is just this effect that gives the dark lines in the spectrum of light which we receive from the sun. The light from the solar surface has passed through the sun’s atmosphere (as well as the earth’s), and the light has been strongly absorbed at the resonant frequencies of the atoms in the solar atmosphere (Feynman et al., 1963, p. 31–9).”

 

Feynman explains that the light from the solar surface has been strongly absorbed by the atoms in the solar atmosphere depending on their resonant frequencies. However, one may clarify that the absorption of light occurs in the cooler atmosphere that is further away from the solar surface (Dwivedi & Phillips, 2001). To be specific, the sun’s corona (the outer solar atmosphere) is hundreds of times hotter than the solar surface. (It is analogous to feeling warmer when you are walking farther away from a fireplace.) Furthermore, the atoms in the cooler (and farthest) atmosphere will re-emit absorbed photons in random directions subsequently. Thus, the dark lines of the absorption spectrum are not really black lines, but dimmer lines because much lesser photons are re-emitted in the same direction as the original photons.

 

“The observation of such spectral lines in the sunlight allows us to tell the resonant frequencies of the atoms and hence the chemical composition of the sun’s atmosphere. The same kind of observations tell us about the materials in the stars. From such measurements we know that the chemical elements in the sun and in the stars are the same as those we find on the earth (Feynman et al., 1963, p. 31–9).”

 

Feynman claims that the chemical elements in the sun and in the stars are the same as those we find on the earth. However, some may ask whether all atoms are exactly the same because these atoms could be compressed to a smaller size due to the very high pressure in the sun and their life span may appear to be different because of stronger gravitational forces. It is worth mentioning that some physicists study spectral lines from distant quasars to investigate whether physical laws remain constant. For example, the study of the fine structure constant in distant quasars suggests a modification of electromagnetic force. Interestingly, the study of the quasar spectrum also indicates that the absorption of light depends on the density of hydrogen in the Universe, as predicted by Gunn and Peterson (1965).

 

Review Questions:

1. Would you represent the complex index using n = n′ + in′′ or n = n′ - in′′?

2. Would you explain the absorption of light in a medium using the concept of frictional force?

3. Do you agree with Feynman that the chemical elements in the sun and in the stars are exactly the same as those we find on the earth?

 

The moral of the lesson: An absorption index is a complex number because it corresponds to the absorption of light that is represented by a complex function, whereas the absorption spectrum is due to the absorption of light that occurs in the cooler atmosphere further away from the solar surface.

 

References:

1. Dwivedi B. N., & Phillips K. J. (2001). The paradox of the sun's hot corona. Scientific American, 284(6), 40-7.

2. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

5. Gunn, J. E. & Peterson, B. A. (1965). On the density of neutral hydrogen in intergalactic space. Astrophysical Journal, 142, 1633‒1636.