## Photonelectron interactions

The interactions wherein an electron disrupts the serene straight-line travel of a photon are usually distinguished by the different energies of the incident photon. Usually, one may think of the electrons as being nearly at rest, but in certain cases, the electron may be very energetic and even relativistic such as in a hot plasma. The electrons may be free or part of an atom. But, in each of the three cases discussed here, Rayleigh, Thomson, and Compton scattering, the interaction is primarily between the photon and electron.

### Rayleigh scattering

A relatively low-energy photon may be absorbed and immediately re-emitted by an atom without causing the atom or molecule to change its energy state. This is known as Rayleigh scattering which may be understood classically as the interaction of the photon and one of the electrons. Consider the electron to be bound by a mechanical spring to the atom or molecule and approach the problem as a classical oscillator. The electric field of the incoming electromagnetic wave (photon) causes the bound electron to oscillate. The electron reradiates energy in the form of electromagnetic waves, as any accelerating charged particle must.

The radiated power of an accelerated charge is, you may recall, P = e2a2/(6tc£oc3) a a2. For frequencies well below resonance, the acceleration of the oscillator is a a m2, from the second derivative of x = x0 cos Mt. Thus the radiated power varies as P a m4. Since this radiated power at frequency m must equal that absorbed from the incoming beam, the scattering is highly frequency dependent, with high frequencies being scattered the most. The scattering of sunlight from molecules in our atmosphere is an example of this. The blue light is scattered more than the red, which gives rise to a blue sky.

### Thomson scattering

If the photon energy is sufficiently high, substantially higher than the ionization energy, then the target electrons will appear to be "free", and Thomson scattering applies. Like Rayleigh scattering, this too can be described with classical physics. In fact, Thomson scattering is the high-frequency limit of Rayleigh scattering. It describes the scattering of photons by free electrons, namely those that are not bound to an atom. However, as noted, it also applies to those that appear to be free because their binding energy is much smaller than the photon energy.

In this (classical) process, the photon is absorbed and immediately reradiated by the electron into a different direction, but it retains essentially all of its initial energy. This is because the photon is assumed to have a much smaller equivalent mass mp = hv/c2 (from mpc2 = hv), than the mass me of the electron. Similarly, a bouncing basketball retains most of its energy but changes its direction each time it collides with the much more massive earth.

The above requirement on the equivalent mass of the photon is equivalent to saying that its energy hv is much less than the electron rest energy mec2, namely that hv ^ mec2 where h is the Planck constant, v the frequency of the photon, me the electron mass, and c is the speed of light. The rest energy of an electron is mec2 = 0.51 MeV. The photon energies must be less than this but more than the few electron volts that bind the outer electrons to an atom or molecule because we require the photon energy in Thomson scattering to be substantially greater than the binding energy.

In the case of Thomson scattering, the probability for scatter turns out to be independent of frequency. The probability of an interaction is often described as a "cross section for interaction" a which is an equivalent area or bull's-eye of the target object. (The cross section is further described in Section 4 of this chapter.) The constant cross section for Thomson scattering turns out to be

aT = — re = 6.65 x 10 9m (Thomson cross section) (10.1)

where e

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