(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 = nR−inI, (32.35) where nR and nI 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 Ea 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 Ea toward the origin (See Fig. 31–3). One may explain that the arrow should be almost
perpendicular in order that the resultant arrow Ea 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 Ea 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.
