Quantum Physics

27.2 The Photoelectric Effect and the Particle Theory of Light 879A successful explanation of the photoelectric effect was given by Einstein in1905, the same year he published his special theory of relativity. As part of a generalpaper on electromagnetic radiation, for which he received the Nobel Prize in1921, Einstein extended Planck’s concept of quantization to electromagneticwaves. He suggested that a tiny packet of light energy or photon would be emittedwhen a quantized oscillator made a jump from an energy state E n nhf to thenext lower state E n1 (n 1)hf. Conservation of energy would require thedecrease in oscillator energy, hf, to be equal to the photon’s energy E, so thatE hf [27.5]where h is Planck’s constant and f is the frequency of the light, which is equal tothe frequency of Planck’s oscillator.The key point here is that the light energy lost by the emitter, hf, stays sharplylocalized in a tiny packet or particle called a photon. In Einstein’s model, a photonis so localized that it can give all its energy hf to a single electron in the metal.According to Einstein, the maximum kinetic energy for these liberated photoelectronsis Energy of a photonKE max hf [27.6] Photoelectric effect equationwhere is called the work function of the metal. The work function, which representsthe minimum energy with which an electron is bound in the metal, is on theorder of a few electron volts. Table 27.1 lists work functions for various metals.With the photon theory of light, we can explain the previously mentionedfeatures of the photoelectric effect that cannot be understood using concepts ofclassical physics:• Photoelectrons are created by absorption of a single photon, so the energy ofthat photon must be greater than or equal to the work function, else no photoelectronswill be produced. This explains the cutoff frequency.• From Equation 27.6, KE max depends only on the frequency of the light and thevalue of the work function. Light intensity is immaterial, because absorption ofa single photon is responsible for the electron’s change in kinetic energy.• Equation 27.6 is linear in the frequency, so KE max increases with increasingfrequency.• Electrons are emitted almost instantaneously, regardless of intensity, becausethe light energy is concentrated in packets rather than spread out in waves. Ifthe frequency is high enough, no time is needed for the electron to graduallyacquire sufficient energy to escape the metal.Experimentally, a linear relationship is observed between f and KE max , as sketchedin Figure 27.6. The intercept on the horizontal axis, corresponding to KE max 0,gives the cutoff frequency below which no photoelectrons are emitted, regardless oflight intensity. The cutoff wavelength c can be derived from Equation 27.6:KE max hf c 0 : hc hc[27.7]where c is the speed of light. Wavelengths greater than c incident on a materialwith work function don’t result in the emission of photoelectrons.cc 0TABLE 27.1Work Functions ofSelected MetalsMetal (eV)Na 2.46Al 4.08Cu 4.70Zn 4.31Ag 4.73Pt 6.35Pb 4.14Fe 4.50KE maxf c fFigure 27.6 A sketch of KE max versusthe frequency of incident light forphotoelectrons in a typical photoelectriceffect experiment. Photons withfrequency less than f c don’t havesufficient energy to eject an electronfrom the metal.INTERACTIVE EXAMPLE 27.3GoalPhotoelectrons from SodiumUnderstand the quantization of light and its role in the photoelectric effect.Problem A sodium surface is illuminated with light of wavelength 0.300 m. The work function for sodium is 2.46 eV.(a) Calculate the energy of each photon in electron volts, (b) the maximum kinetic energy of the ejected photoelectrons,and (c) the cutoff wavelength for sodium.