Chapter 38 Photons Light Waves Behaving as Particles 38.1 The Photoelectric Effect The photoelectric effect was first discovered by Hertz in 1887, and was explained by Einstein in 1905. The photoelectric effect is the ejection of electrons from a metallic surface by the action of light.! e 1
"V o # The Stopping Potential! the electric potential (voltage) at which the photoelectric current drops to zero. The maximum kinetic energy KEmax of the ejected electrons is thus obtained using conservation of energy: KE max = e "V o There are 3 major experimental features of the photoelectric effect: 1. The maximum kinetic energy KEmax of the ejected electrons is independent of the light intensity. Higher light intensity means that more electrons are ejected, not necessarily with each electron having more kinetic energy. 2. For each surface, there exists a characteristic threshold frequency f c. For values of f < f c, the photoelectric effect does not occur. For values of f! f c, the photoelectric effect occurs. 2
3. There is no time lag as the electrons absorb energy from the incident light. Albert Einstein proposed that the radiant energy is quantized into concentrated bundles called photons. He assumed that the energy content E of a photon is related to the frequency of the wave you associate with the radiation by E = h f Where h is Planck s constant. 3
Einstein also assumed that in the photoelectric effect one photon is completely absorbed by one bound electron in the photocathode (metal surface). In the case of loosest binding the photoelectron emerges with maximum kinetic energy such that h f = KE max + " where " # the work-function of the metal = minimum energy needed to remove an electron from the metal. Note from the above equation that KE max = h f " # 4
Einstein s equation explained the photoelectric effect beautifully. For this work he was awarded the Nobel Prize in physics in 1921. Energy, Mass, and Momentum of a Photon A photon has energy hc E = h f = (photon)! The linear momentum of a photon is given by E p = (photon) c Combining the last two equations yields the result that the wavelength of a photon of momentum p is h! = (photon) p 5
38.2 Light Emitted as Photons: X-Ray Production The inverse of the photoelectric effect is x-ray production. This is so because instead of releasing electrons from a metal by shining electromagnetic radiation on it, a surface is caused to emit radiation by bombarding it with electrons. The German physicist Wilhelm Roentgen discovered x- rays in 1895. The wavelength of x-rays is less than 1 nm. X-rays are produced when a beam of free energetic electrons accelerated through a potential difference of thousands of Volts is stopped upon striking a solid target. In a typical x-ray tube, free electrons are emitted by a hot cathode and accelerate through vacuum toward a positively charged metal anode. The voltage between the cathode and anode is about ΔV = 35,000 Volts. The anode metal is a tungsten or molybdenum disk, which spins rapidly to keep it from melting. When the free electrons collide with the metal target, they emit both x-rays that consist of two components: bremsstrahlung and characteristic x-rays. Bremsstrahlung: This process usually refers to the case when a charged particle decelerates extremely rapidly resulting in the emission of a high-energy photon. In an x- ray tube, a fast moving free electron arcs around a 6
massive nucleus and decelerates so abruptly that it emits an x-ray photon. The x-ray photon carries off some of the electron s kinetic energy. Applying conservation of energy to the fundamental process responsible for the continuous x-ray spectrum depicted in the figure above yields: where KE = K E " + h f KE = kinetic energy of incident electron K E " = kinetic energy of scattered electron h f = energy of emitted x-ray photon 7
Thus the energy of the emitted x-ray photon is hf = KE " K E # The spectrum of the emitted x-ray photons is broad and continuous because the free electron is brought to rest after many encounters with the nuclei of the target. The shortest wavelength λ min x-ray photon is emitted when the electron loses all of its kinetic energy in one single encounter, for which K " E = 0. Thus hf max = KE hc " min = KE but KE = e "V so that " min = hc e #V 8
38.3 Light Scattered as Photons A. The Compton Effect (1923) A photon of wavelength "and frequency f is incident on a free electron at rest. Upon collision, the photon is scattered at angle! with increased wavelength!", while the electron moves off at an angle!. According to the principle of conservation of energy, the energy of the incident photon must equal the sum of the energies of the scattered photon and the recoil electron. That is, h f = h f " + KE electron 9
Conservation of total relativistic momentum requires that the momentum of the incident photon must equal the sum of the momenta of the scattered photon and the recoil electron. Compton predicted that the wavelength of the scattered photon should depend on the scattering angle! as or #" = # + h m e c "# $ #%& # = h m e c ( 1$ cos% ) ( 1& cos' ) The Compton wavelength for the electron is defined as h " C! = 0. 00243 m c e nm Compton s effect works beautifully for x-rays scattered from various targets like graphite. Since the cos (! ) varies between -1 and +1, the shift $! =!# "! in the 2 h wavelength can vary between zero and, depending me c on the value of!. 10
This was further justification that light is made up of photons. For this work, Arthur Compton, who performed this experiment at the University of Chicago, was awarded the Nobel Prize in physics in 1927. 11
B. Pair Production We discussed pair production in the chapter on the special theory of relativity (chapter 37). 38.4 Wave-Particle Duality, The Uncertainty Principle Light has a dual nature. It exhibits both wave and particle characteristics. Interference and diffraction effects demonstrate conclusively the wave nature of light. The photoelectric effect and Compton scattering demonstrate conclusively the particle nature of light. We cannot treat though a photon as a point object because there are fundamental limitations on the precision with which we can simultaneously determine the position and momentum of a photon. Many aspects of a photon s behavior can be stated only in terms of probabilities (quantum mechanics). 12
A. Heisenberg Uncertainty Principle for Position and Momentum It is impossible to specify simultaneously and with infinite precision a photon s linear momentum and the corresponding position. If the position x has an uncertainty Δx and if the corresponding linear momentum component p x has an uncertainty Δp x then "x "p x # h 2 where h " h 2# =1.055x10$34 J % s The uncertainties in the two quantities play complementary roles. The more precise a photon s position is, the more uncertain its momentum is, and viseversa. 13
B. Heisenberg Uncertainty Principle for Energy and Time The uncertainty ΔE in the energy of a state that is occupied for a time Δt obeys the following uncertainty relation: "E "t # h 2 That is, the energy of an object (photon) can be given with infinite precision (ΔE = 0) only if the object exists for an infinite time ("t = #). 14