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be no area directly associated with it, physically. It is merely a way of expressing
the answer to a certain kind of problem; it tells us what area the incident beam
--- Trang 566 ---
would have to hit in order to account for that much energy coming off. Thus, for
OUT Ca§@,
8mrổ 3
Ø = + (2—oŸ)? (32.19)
(the subscript s is for “scattering”).
Let us look at some examples. First, if we go to a very low natural frequency œ0,
or to completely unbound electrons, for which œọ = 0, then the frequency œ
cancels out and the cross section is a constant. 'This low-frequency limit, or the
free electron cross section, is known as the Thomson scatfering cross seclion. IW
is an area whose dimensions are approximately 10~!5 meter, more or less, on a
side, i.e., 10—9 square meter, which is rather smalll
Ôn the other hand, ïf we take the case of light in the air, we remember that for
aïr the natural frequencies of the oscillators are higher than the frequency of the
light that we use. This means that, to a frst approximation, we can disregard ¿2
in the denominator, and we fñnd that the scattering is proportional to the ƒourth
pouer oÊ the frequency. hat is to say, light which is of higher frequency by, say,
a factor of two, is siz‡een tứmes more intensely scattered, which is a quite sizable
diference. 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. Thus when we look at the sky it looks that glorious blue that we see all
the timel
There are several points to be made about the above results. One interesting
question is, why do we ever see the clowds? Where do the clouds come from?
tverybody knows it is the condensation of water vapor. But, of course, the
water vapor Is already in the atmosphere 0eƒfore it condenses, so why don” we
see it then? After it condenses it is perfectly obvious. It wasnt there, now it 2s
there. 5o the mystery of where the clouds come from is not really such a childish
mystery as “Where does the water come from, Daddy?,” but has to be explained.
W© 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 fremendouslu greater amownt of light?
Consider what would happen If, instead of a single atom, we had an agglom-
erate of atoms, say Ewo, very close together compared with the wavelength of the
light. Remember, atoms are only an angstrom or so across, while the wavelength
of light is some 5000 angstroms, so when they form a clump, a few atoms together,
they can be very close together compared with the wavelength of light. Then
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when the electric fñeld acts, bo#h, oƒ the atoms tuiÏH tnoue together. he electrie
fñeld that is scattered will then be the sum of the two electric fields in phase, ï.e.,
double the amplitude that there was with a single atom, and the enerøgu which
is scattered is therefore ƒour: tưnes what it is with a single atom, not twicel So
lumps of atoms radiate or scatter more energy than they do as single atoms. Ôur
argument that the phases are independent is based on the assumption that there
is a real and large difference in phase bebween any ÿwo atoms, which is true only
1f they are several wavelengths apart and randomly spaced, or moving. 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.
Tf we have atoms in a lump, which is a tiny droplet of water, then each one
will be driven by the electric field in about the same way as before (the efect of
one atom on the other is not important; it is Just to get the idea anyway) and
the amplitude of scattering from each one is the same, so the total field which is
scatered is /-fold increased. The 7m#ensitu of the light which is scattered is then
the square, or WZ-fold, increased. We would have expected, if the atoms were
spread out in space, only Ñ times as much as 1, whereas we get W2 times as
much as 1l "That is to say, the scattering of water in lumps of ) molecules each
is / times more intense than the scattering of the single atoms. So as the water
agglomerates the scattering increases. Does it increase øở ?nfimitum2? Nol When
does this analysis begin to fail? How many atoms can we put together before
we cannot drive this argument any further? Ansuer: IÝ the water drop gets so
big that om one end to the other is a wavelength or so, then the atoms are
no longer all in phase because they are too far apart. 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. Eurthermore, the
blue disappears, because for long wavelengths the drops can be bigger, before
this limit is reached, than they can be for short wavelengths. Although the short
waves scatter more per atom than the long waves, there is a bigger enhancement
for the red end of the spectrum than for the blue end when all the drops are
bigger than the wavelength, so the color is shifted from the blue toward the red.
Now we can make an experiment that demonstrates this. We can make
particles that are very small at frst, and then gradually grow in size. We use a
solution of sodium thiosulfate (hypo) with sulfuric acid, which precipitates very
fine grains of sulfur. As the sulfur precipitates, the grains frst start very small,
and the scattering is a little bluish. Äs it precipitates more it gets more intense,
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and then it will get whitish as the particles get bigger. In addition, the light
which goes straight through will have the blue taken out. hat is why the sunset
1s red, of course, because the light that comes through a lot of air, to the eye has
had a lot of blue light scattered out, so i% is yellow-red.
Finally, there is one other important feature which really belongs in the next
chapter, on polarization, but it is so Interesting that we point it out now. “This
1s that the electric fñeld of the scattered light tends to vibrate in a particular
direction. The electric feld in the incoming light is oscillating in some way, and
the driven oscillator goes in this same direction, and if we are situated about at
right angles to the beam, we will see polarzcởd light, that is to say, light in which
the electric feld is going only one way. In general, the atoms can vibrate in any
direction at right angles to the beam, but if they are driven directly toward or
away from us, we do not see it. 5o if the incoming light has an electric ñeld which
changes and oscillates in any direction, which we call unpolarized light, then the
light which is coming out at 909 to the beam vibrates in only one direction! (See
Eig. 32-3.)
—X Electron
moVe€S In
4“ plane L k
Incident beam +
(unpolarized)
-L k ¡is plane polarized