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Quantum Field Theory

Compton Scattering

The quantum theory of light postulates that photons behave like particles except for the absence of any rest mass. If this is true, then it should be possible for us to treat collisions between photons and, say, electrons in the same manner as billiard-ball collisions are treated in classical mechanics. Figure 02g shows how such a collision might be represented, with an X-ray photon striking an electron at rest and being scattered away from its original path while the electron receives an impulse and begins to move. Using just kinematics the change in the wavelength of the X-ray can be expressed in the form:

Figure 02g Compton Scattering [view large image]

' - = o (1 - cos) ---------- (38f)

where o = (h/mc) = 2.43x10-10 cm is the Compton wavelength, which is actually the de Broglie wavelength for the electron moving at a speed of c/.
The Compton scattering belongs to the second order graphs in the Feynman diagrams. Since there is no loop diagram involved, it is still susceptible to classical treatment. The cross section has been computed by J. J. Thomson in the 1900's:

d()/d = (ro2/2) (1 + cos2) ---------- (38g)

where ro = e2/mc2 = 2.82x10-13 cm stands for the classical radius of the electron.

Figure 02h Scattering Cross Section[view large image]

The experimental scattering cross section at different incident energy are shown in Figure 02h. At the low energy range of about l Kev, the cross section in Eq.(38g) shows good agreement with minima at = 90o and 270o, and maxima at = 0o and 180o. Asymmetric patterns appear at higher energy with the maximum shifted to the forward direction.
The relativistic correction has been derived within the framework of QED. It is known as the Klein-Nishina formula:

d()/d = (ro2/2) ('/)2 (/' + '/ + sin2) ---------- (38h)

where , and ' is the frequency before and after the scattering as related in Eq.(38f). Already in the late 1920's, the angular distribution of -ray scattering had been measured to show that the observed deviation from the Thomson formula is precisely what one expects from Eq.(38h). The agreement achieved then is one of the earliest quantitative triumphs of the Dirac theory, which is an attempt to unify the theories of quantum mechanics and special relativity. It is the linearized version of the Schrodinger equation in consistence with special relativity, i.e., it is Lorentz invariant as shown in Eq.(13). The Dirac equation describes particles with spin-1/2. Note also that the deep inelastic scattering is just a special kind of Compton scattering with virtual photon hitting a necluon (such as the proton) instead of real photon colliding an electron as shown in Figure 02e.

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