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The-Photo-electric-effect
The curious case of the nature of light Summary. Light exhibits both diffraction and interference and therefore exhibits wave-like properties. In this lecture topic, we will see that light can also exhibit particle properties. These “ light ” particles” are called photons. Light can behave like a wave or like a particle. Background reading: Tipler sections 34.1, 34.2 Is light a wave (for a definition of a wave, see http://www.kettering.edu/~drussell/Demos/waves- intro/waves-intro.html ) or a particle? These are very different entities. Particles are discrete, with their energy concentrated into a finite space. They exist at a specific location. Interactions between particles are governed by laws, such as the conservation of energy and momentum. The energy of waves cannot be considered to exist in a single place. A wave can propagate until it exists in all locations. The energy carried by a wave depends on its intensity. One of the best-known interactions between waves is the phenomenon of interference. By the end of the 19th century, it had been firmly established that light was a wave. Then came the photoelectric effect. When light falls on a metal surface, electrons are emitted (photoelectrons). monochromatic light evacuated tube A anode cathode variable voltage V The experiment is shown above. Light falls on the electrode called the cathode and electric current flows in the external circuit. It is found that Current is directly proportional to light intensity Current appears within nanoseconds of light hitting the cathode Photoelectrons are emitted only if the frequency of the light is greater than some threshold frequency f0 If V > 0 (anode positive with respect to cathode), current does not change as V increases. If V < 0, current decreases until V = -Vstop, the stopping potential Vstop is independent of light intensity http://phet.colorado.edu/simulations/sims.php?sim=Phot oelectric_Effect Under classical wave theory, an atom will absorb energy from an incident wave and will continue to absorb energy until it has enough energy to be emitted from the solid. Classical wave theory expected that the energy of the light source should be determined by its intensity. Hence, the energy required to eject a photoelectron should be supplied by light of high intensity, no matter how low the frequency of the radiation. Thus, there should be no threshold frequency, below which no electrons are emitted. Moreover, the kinetic energy of the electrons should increase with intensity, not with light frequency. These predictions are not observed, so the results are counter to physical intuition. Einstein saw the incident light as a stream of discrete bundles of energy or photons. It is these photons which are absorbed or emitted when a body absorbs from or emits energy to the incident light. A photon is one emitted or absorbed quantum of electromagnetic energy. The energy of a photon is related to the frequency of the light by E = hf h is known as Planck’s constant and has a value of 6.63 x 10-34 Js. These photons also carry momentum, which, from relativity, can be shown to be E = cp Using this concept, we can explain the features of the photoelectric effect thus. We assume that a metal contains large numbers of free electrons. To remove an electron from the surface of a metal takes a minimum energy, known as the work function Electrons that are in the bulk of the metal take more energy. Material (eV) Na 2.28 Al 4.08 Co 3.90 Cu 4.70 Zn 4.31 Ag 4.73 Pt 6.35 Pb 4.14 An electron will be emitted whenever a photon is absorbed of energy E = hf ≥ . If hf > , an electron will be emitted and the difference in energy between the photon energy and the energy needed to free the electron from the metal will appear as kinetic energy. The maximum kinetic energy of the electrons is Kmax = hf – This kinetic energy can be determined from the stopping potential since Kmax = eVstop. The cut-off frequency occurs when the photon energy just equals the work function. hf0 = Worked example The work function of tungsten metal is 4.52 eV. What is the cut-off wavelength for tungsten? (b)What is the maximum kinetic energy of photoelectrons when light of wavelength 200 nm falls on a tungsten surface? (c) What is the stopping potential in this case? Solution (a) hf0 = hc/= giving 0 = 274 nm (b) Kmax = hf - 6.2 eV - 4.52 eV = 2.68 eV (c) The stopping potential is Vstop = Kmax/e = 2.68 volts The Compton Effect The Compton effect is a dramatic illustration of the particle-like nature of light. A beam of X-rays or -rays falls on a target (e.g. graphite) and the intensity of the radiation scattered at various angles is measured as a function of wavelength. (http://physics.berea.edu/~king/Teaching/ModPhys/QM/ Compton/compton.html) The results of such an experiment is shown below These data show that, although the incident beam consists essentially of a single wavelength, the scattered beam has intensity peaks at two wavelengths. There is a shifted component at longer wavelength. The Compton shift '– varies with the angle at which the scattered beam is observed. The shifted component at ’ cannot be explained classically. It can be interpreted by considering the incident beam as a collection of photons colliding with free electrons in the target as in a collision between billiard balls. In such a collision, the photon looses some energy to the (recoiling) electron and so shifts to longer wavelength. Note that the photon is scattered and not absorbed. (The electrons are not really free but are the loosely bound outer or valence electrons in atoms. Since the energy of the X-rays is very much greater than the electron binding energy, the electrons can be considered as free. The assumption is further justified by noting that the wavelength of the scattered beam ’ is independent of the target material.) Consider the collision between a photon of initial momentum p0, energy E0 with a stationary electron of rest mass m0. The scattered photon has a momentum p1, energy E1 whilst the electron recoils with momentum p and kinetic energy K. Conserving momentum in the x- and y-directions gives p0 = p1cos + pcos p1sin = psin Squaring and adding gives p02 + p12 –2p0p1cos = p2 Conservation of energy gives E0 – E 1 = K For a photon, E = cp. Therefore, K = c(p0 – p1) For the electron, the total energy E is related to momentum p by the relativistic formula E2 = c2p2 + m02c4 = (K + m0c2)2 which gives p2 = K2/c2 + 2Km0 = p02 + p12 –2p0p1cos from which we get 1 1 1 1 cos θ p1 p 0 m 0c Since E = hf = cp for a photon, then = '– = c(1 – cos) c is known as the Compton wavelength and equals 0.00243nm. The unshifted peak is the result of scattering from the whole atom. Since the mass of the atom is many thousands of times more massive than an electron, the resulting Compton shift from this scattering is negligible. Worked Example X-rays of wavelength 0.2400 nm are Compton scattered and the scattered beam is observed at an angle of 600 relative to the incident beam. Find (a) the wavelength of the scattered X-rays (b) the energy of the scattered X-ray photons (c) the kinetic energy of the scattered electrons (d) the direction of travel of the scattered electrons Solution (a) ’ – =c(1 – cos) Thus, ’ = + c(1 – cos) = 0.24 nm + 0.00243 nm( 1 – cos 60) = 0.2412 nm (b) The energy of the scattered photons is hc E1 = ' = 5141 eV λ (c) The kinetic energy of the electron is K = E0 – E1 = 5167 eV – 5141 eV = 26 eV (d) From conservation of momentum, p0 = p1cos + pcos p1sin = psin p1sinθ tan = p p1cosθ Now E = cp for the photons and the momentum of the recoiling electrons can b e calculated form their kinetic energy. Calculations give = 59.70. What is light? Light behaves with wave-like properties (diffraction, interference) and with particle-like properties (photoelectric effect). Light is not either particles or waves, but somehow both particles and waves. This is known as wave-particle duality. It shows only one or other aspect, depending on the kind of experiment we are doing. Neither the wave nor the particle picture is correct all the time; both are needed for a complete description of physical phenomena, and the two are complimentary with one another. Modern quantum theory emphasises the primacy of measurement and not attributing properties to objects beyond what can be measured. Hence the concept of wave-particle duality arose: it is not necessary, or useful, to say that light is a particle - or a wave - just that in certain circumstances it behaves like a wave, and in others like a particle. General Problems 1. When a metal is illuminated with light of wavelength < 388 nm, photoelectrons are observed. Determine the work function of the metal. (Ans. 3.2 eV) 2. A stopping potential of 2.0 V is measured when light of 290 nm illuminates a metal. (i) Determine the work function of the metal (ii) Determine the stopping potential if the intensity of the light is doubled (Ans. (i) 2.28 eV (ii) 2.0 V)