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When an electromagnetic wave is incident on a charged particle, the
electric and magnetic components of the wave exert a Lorentz force
on the particle, setting it into motion. Since the wave is periodic
in time, so is the motion of the particle. Thus, the particle is accelerated
and, consequently, emits radiation. More exactly, energy is
absorbed from the incident wave by the particle
and reemitted as electromagnetic radiation. Such
a process is clearly equivalent to the scattering of the electromagnetic
wave by the particle.
Consider a linearly polarized, monochromatic, plane wave incident on a
particle carrying a charge . The electric component of the wave can be
written

(1013) 
where is the peak amplitude of the electric field, is
the polarization vector, and is the wave vector
(of course,
).
The particle is assumed to undergo small
amplitude oscillations about an equilibrium position which coincides with
the origin of the coordinate system. Furthermore,
the particle's velocity is assumed
to remain subrelativistic, which enables us to neglect the magnetic
component of the Lorentz force. The equation of motion of
the charged particle is approximately

(1014) 
where is the mass of the particle,
is its displacement from the origin, and denotes
.
According to Eq. (2.321), the timeaveraged power radiated per unit solid
angle by an accelerating, nonrelativistic, charged particle is given by

(1015) 
where
denotes a time average. However,

(1016) 
Hence, the scattered power per unit solid angle is given by

(1017) 
The timeaveraged Poynting flux of the incident wave is

(1018) 
It is convenient to define the scattering cross section as the equivalent area of the incident wavefront which delivers the same power as that
reradiated
by the particle:

(1019) 
By analogy, the differential scattering cross section is
defined

(1020) 
It follows from Eqs. (5.63), (5.64), and (5.66) that

(1021) 
The total scattering cross section is then

(1022) 
The quantity appearing in Eq. (5.67) is the angle
subtended between the direction
of acceleration of the particle and the direction of the outgoing radiation
(which is parallel to the unit vector ). In the present case,
the acceleration is due to the electric field, so it is parallel to the
polarization vector . Thus,
.
Up to now, we have only considered the scattering of linearly polarized
radiation by a charged particle. Let us now calculate the
angular distribution of scattered radiation for the commonly
occurring case of randomly polarized incident radiation. It is helpful
to set up a righthanded coordinate system based on the three
mutually orthogonal unit vectors ,
,
and . In terms of these unit vectors, we can write

(1023) 
where is the angle subtended between the direction of
the incident radiation and that of
the scattered radiation, and is an angle which
specifies the orientation of the polarization vector in the plane
perpendicular to (assuming that is known).
It is easily seen that

(1024) 
so

(1025) 
Averaging this result over all possible polarizations of the
incident wave (i.e., over all possible values of
the polarization angle ), we obtain

(1026) 
Thus, the differential scattering cross section for unpolarized incident
radiation (obtained by substituting
for
in Eq. (5.67)) is given by

(1027) 
It is clear that the differential scattering cross section is
independent of the frequency of the incident wave, and is also symmetric
with respect to forward and backward scattering. The frequency of
the scattered radiation is the same as that of the incident radiation.
The total scattering cross section is obtained by integrating over the entire
solid angle of the polar angle and the azimuthal
angle . Not surprisingly, the result
is exactly the same as Eq. (5.68).
The classical scattering cross section (5.73) is modified by quantum
effects when the energy of the incident photons, , becomes
comparable with the rest mass of the scattering particle, . The
scattering of a photon by a charged particle is called Compton
scattering, and the quantum mechanical version of the Compton scattering
cross section is known as the KleinNishina formula. As the photon
energy increases, and eventually becomes comparable with the rest mass energy
of the particle, the KleinNishina formula predicts that forward scattering
of photons becomes increasingly favored with respect to backward scattering.
The KleinNishina cross section does, in general, depend on the
frequency of the incident photons.
Furthermore, energy and momentum conservation demand a shift in the
frequency of scattered photons with respect to that of the incident photons.
If the charged particle in question is an electron then Eq. (5.68)
reduces to the well known Thomson scattering cross section

(1028) 
The quantity
m is
called the classical electron radius (it is the radius of spherical
shell of total charge whose electrostatic energy equals the rest mass
energy of the electron). Thus, as a scatterer the electron acts rather like
a solid sphere whose radius is of order the classical electron
radius. Since this radius is extremely small, it is clear that scattering
of radiation by a single electron (or any other charged particle) is
a very weak process.
Next: Rayleigh scattering
Up: Radiation and scattering
Previous: Antenna arrays
Richard Fitzpatrick
20020518