Question

(3) Prove the Compton scattering formula: Δλ--( 1-cos θ) for the wavelength shift Δλ of a photon of wavelength λ scattering from an electron of mass me and is scattered through angle θ, c is the velocity of light. What value of θ is the maximum shift experienced?

Answer 1

To find the result just apply conservation of linear momentum and then apply conservation of energy in relatisrelat terms.

Let λ1 and λ2 be the wavelengths of the incident and scattered x rays, respectively, as shown in Figure 3-18. The corresponding momenta are Pi usingfA- c. Since Compton used the K line of molybdenum A-0.0711 nm; see Figure 3-15b), the energy of the incident x ray (17.4 keV) is much greater than the binding energy of the valence electrons in the carbon-scattering block (about 11 eV); therefore, the carbon electrons can be considered to be free. Conservation of momentum gives or 3-26 where Pe is the momentum of the electron after the collision and θ is the scattering angle of the photon, measured as shown in Figure 3-18. The energy of the electron before the collision is simply its rest energy Eomc (see Chapter 2). After the co sion, the energy of the electron is (E% + p.c2)12. FIGURE 3-18 The scattering of x rays can pe L NE2E2 be treated as a collision of a photon of initial momentum hand a free electron. Using conservation of momentum and energy, the momentum of the scattered photon h/l2 can be related to the initial momentum, the electron mass, and the scattering angle. The resulting Compton equation for the change in the wavelength of the x ray is Equation 3-25.

Conservation of energy gives Transposing the term p2c and squaring, we obtain or 3-27 Eliminating p2 between Equations 3-26 and 3-27, we obtain Eo(P1-P2) = PiPa (1-cos θ) Multiplying each term by hc/PP-Eo and using λ = h/p, we obtain Compton's equation: hc 0 mc or cos θ 3-25 mc
also the shift is maximum when
cosa)
is maximum and is at