Recitation 9 Chapter 28 Problem 28.14. Calculate the energy and momentum of a photon of wavelength 700 nm. You should be familiar with these equations by now (after our time with the Bohr atom and relativity). The energy is given by (Eq’s 28.3 and 11.15) 1240 eV·nm hc = = 1.77 eV . (1) E= λ 700 nm For momentum you can either use the relativistic energy-momentum equation (Eq. 9.22) E 2 = p2 c2 + m20 c4 Ephoton = pc E p= = 1.77 eV/c = 9.46 · 10−28 kg·m/s c (2) (3) (4) or the de Broglie formula (Eq. 28.10) p= 6.63 · 10−34 J·s h = = 9.46 · 10−28 kg·m/s . λ 700 nm (5) Problem 28.15. X-rays having an energy of 300 keV undergo Compton scattering from a target. The scattered rays are detected at 37.0◦ relative to the incident rays. Find (a) the Compton shift at this angle, (b) the energy of the scattered x-ray, and (c) the energy of the recoiling electron. (a) From the Compton shift equation (Eq. 28.8) h (1 − cos θ) me c ∆λ = 2.43 pm(1 − cos 37.0◦ ) = 489 fm . λ0 − λ0 = (6) (7) (b) The wavelength of the incoming photon was λ0 = hc 1240 eV·nm = 4.13 pm . = E0 300 keV (8) The scattered wavelength is thus λ0 = λ0 + ∆λ = (4.13 + 0.489) pm = 4.62 pm , (9) and the energy of the scattered photon is E0 = hc 1240 eV·nm = = 268 keV . 0 λ 4.62 pm (10) (c) All the energy lost by the photon must go into the recoiling electron so Ee = E0 − E 0 = (300 − 268) keV = 31.7 keV . (11) Problem 28.16. A 0.110 nm photon collides with a stationary electron. After the collision, the electron moves forward and the photon recoils backward. Find the momentum and the kinetic energy of the electron. The photon scatters by 180◦ , so from the Compton shift equation λ0 = λ0 + h 2h (1 − cos 180◦ ) = λ0 + = (110 + 4.85) pm = 115 pm . me c me c (12) The kinetic energy of the electron is given by the change in photon energy (just like problem 28.15). K = E0 − E 0 = hc hc − 0 = 11.3 keV − 10.8 keV = 476 eV . λ0 λ (13) We conserve momentum to find the electron’s momentum, using pphoton = E/c. E0 E0 = pf = pe − c c E0 + E 0 pe = = (11.3 + 10.8) keV/c = 22.1 keV/c = 1.18 · 10−23 kg·m/s . c pi = (14) (15)