﻿ The Photoelectric Effect and Einstein’s Explanation - Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer ﻿

Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)

Chapter 4. The Photoelectric Effect and Einstein’s Explanation

AT THE END OF THE NINETEENTH CENTURY, classical electromagnetic theory was one of the great triumphs of classical mechanics. It was capable of explaining a wide variety of experimental observations. But early in the twentieth century, new experiments were causing problems for the classical wave picture of light. One experiment in particular, along with its explanation, showed a fundamental problem with the seemingly indestructible wave theory of light.

THE PHOTOELECTRIC EFFECT

The experiment is the observation of the photoelectric effect. In the photoelectric effect, light shines on a metal surface and, under the right conditions, electrons fly out of the metal. For our purposes here, electrons are electrically charged particles. The electron charge is negative. (Later we will see that electrons are not strictly particles for the same reason that light is not a wave.) Because electrons are charged particles, they are easy to detect. They can produce electrical signals in detection equipment. Figure 4.1 shows a schematic of the photoelectric effect with the incoming light viewed as a wave.

It is possible to measure the number of electrons that come out of the metal and their speed. For a particular metal and a given color of light, say blue, it is found that the electrons come out with a well-defined speed, and that the number of electrons that come out depends on the intensity of the light. If the intensity of light is increased, more electrons come out, but each electron has the same speed, independent of the intensity of the light. If the color of light is changed to red, the electron speed is slower, and if the color is made redder and redder, the electrons’ speed is slower and slower. For red enough light, electrons cease to come out of the metal.

THE WAVE PICTURE DOESN’T WORK

The problem for classical theory with these observations is that they are totally inconsistent with a wave picture of light. First, consider the intensity dependence. In the wave picture, a higher light intensity means that the amplitude of the wave is larger. Anyone who has been in ocean waves knows that a small wave hits you gently and a big wave hits you hard. As illustrated in Figure 4.2, low-intensity light is an electromagnetic wave with small amplitude. Such a wave should “hit” the electrons rather gently. The electrons should emerge from the metal with a relatively slow speed. In contrast, high intensity light has associated with it a large amplitude wave. This large amplitude wave should “hit” the electrons hard, and electrons should fly away from the metal with a high speed. FIGURE 4.1. The photoelectric effect. Light impinges on a metal, and electrons (negatively charged particles) are ejected. In the classical picture, light is a wave, and the interaction of the wave with the electrons in the metal causes them to fly out.

To put this more clearly, the light wave has associated with it an oscillating electric field. The electric field swings from positive to negative to positive to negative at the frequency of the light. An electron in the metal will be pulled in one direction when the field is positive and pushed in the other direction when the electronic field is negative. Thus, the oscillating electric field throws the electron back and forth. According to classical theory, if the wave has large enough amplitude, it will throw the electron right out of the metal. If the amplitude of the wave is bigger (higher intensity), it will throw the electron out harder, and the electron should come out of the metal having a greater speed. However, this is not what is observed. When the intensity of light is increased, electrons come out of the metal with the same speed, but more electrons come out. FIGURE 4.2. A wave picture of the intensity dependence of the photoelectric effect. Low-intensity light has a small wave amplitude. Therefore, the wave should “hit” the electrons gently, and they will come out of the metal with a low speed. High-intensity light has a large wave amplitude. The large wave should hit the electrons hard, and the electrons will come out of the metal with a high speed.

Furthermore, when the light color is shifted to the red (longer wavelength), the electrons come out of the metal with a lower speed no matter how high the intensity is. Even in the wave picture, longer wavelength light is less energetic, but it should be possible to turn up the intensity, making a bigger amplitude wave, and therefore increase the speed of the electrons that fly out of the metal. But, as with a bluer wavelength, turning up the intensity causes more electrons to emerge from the metal, but for a given color, they all come out moving with the same speed.

An additional problem is that when the color is shifted far enough to the red, the electrons stop coming out. The electrons have some binding energy to the metal, that is, the negatively charged electrons are attracted to the positively charged metal atom nuclei. (Atoms will be discussed in detail beginning in Chapter 9 and metals in Chapter 19.) This binding energy is what keeps the electrons from flying out of the metal in the absence of light. In the wave picture, it should always be possible to turn up the intensity high enough, and therefore make the amplitude of the oscillating electric field large enough, to overcome the binding energy. If you are standing in the ocean, a small wave may not knock you off of your feet, but if the waves get bigger and bigger, eventually they will be big enough to break the binding of your feet to the ocean floor and send you flying. But with light, for a red enough color, no matter how big the wave is, the binding of the electrons to the metal is not overcome.

EINSTEIN GIVES THE EXPLANATION

The upshot of these experimental observations is that the wave picture of light that describes the interference pattern of Figure 3.4 so well does not properly describe the photoelectric effect. The explanation for the photoelectric effect was given by Einstein in 1905 (Albert Einstein, 1879-1955). Einstein won the Nobel Prize in Physics in 1921 “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.” It may seem surprising that Einstein, known for his Theory of Relativity, won the Nobel Prize for explaining the photoelectric effect, which was an important step in the transition from classical to quantum theory. Einstein’s prize demonstrates the importance of the explanation of the photoelectric effect in modern physics.

Einstein said that light is not composed of waves, but rather of photons or quanta of light. In the photoelectric effect, a photon acts like a particle rather than a wave. So Einstein said that a beam of light is composed of many photons, each of which is a discrete particle. (As discussed in detail later, these are not particles in the classical sense of a particle.) As shown in Figure 4.3, one photon “hits” one electron and ejects it from the metal. The process is in some sense like the cue ball in a game of pool hitting a stationary ball and sending it across the table. The cue ball hitting the stationary ball transfers energy to it in the form of kinetic energy, that is, energy of motion. The collision causes the cue ball to give up energy and the target ball to gain energy. A light beam is composed of many photons, but one photon ejects one electron from the metal. FIGURE 4.3. Einstein described light as composed of discreet quanta of light “particles” called photons. In the photoelectric effect, one photon hits one electron and knocks it out of the metal.

When the intensity of light is increased, the light beam is composed of more photons. As illustrated in Figure 4.4, more photons impinging on the metal can hit and eject more electrons from the metal. Because one photon hits one electron, increasing the intensity of the light beam does not change the speed of the electron that is ejected. In pool, the speed of a target ball is determined by how fast the cue ball was moving. Imagine two cue balls were simultaneously shot at the same speed at two different target balls. After being hit, the two target balls would move with the same speed. When more photons of a particular color hit the metal, more electrons come out, but all with the same speed. In contrast to the wave picture, increased intensity does not produce a harder hit on an electron; increased intensity only produces more photons hitting more electrons. Each photon hits an electron with the same impact whether there are many or few. Therefore, electrons come out with the same speed independent of the intensity. FIGURE 4.4. An increase in the intensity of a light beam corresponds to the beam being composed of more photons. More photons can hit and eject more electrons, so an increase in intensity results in more electrons flying out of the metal.

RED LIGHT EJECTS SLOWER ELECTRONS THAN BLUE LIGHT

To explain why changing the color of the light to red (longer wavelength, lower energy) caused electrons to be ejected with a lower speed, Einstein used a formula first presented by Planck (Max Karl Ernst Ludwig Planck, 1858-1947). Planck first introduced the idea that energy comes in discreet units, called quanta, while he was explaining another phenomenon involving light, called black body radiation. When a piece of metal or other material is heated to a high temperature it will glow; it is emitting light. If it is quite hot, it will glow red. An example is the heating element of an electric stove or space heater when turned to high. As its temperature is increased, the color shifts toward blue. This is not only true of a piece of metal but also of stars. Red stars are relatively cool. A yellow star, such as our own sun, is hotter. A blue star is very hot. In 1900, classical physics could not explain the amount of light that came out at each color from a hot object. Planck found the explanation that still stands today by introducing a new concept, that the electrons in a piece of metal could only “oscillate” at certain discreet frequencies. The energy steps between these frequencies are called quanta. Planck won the Nobel Prize in Physics in 1918 “in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta.” Planck’s discovery of energy quanta led to the name Quantum Mechanics.

In his work, Planck introduced the formula that related the frequency of the electrons to their energy, E = hν, where ν is the frequency as discussed in Chapter 3, and h is called Planck’s constant. In the equation, h = 6.6 × 10-34 J-s, J is the unit of energy Joule, and s is seconds. In the formula, the units of ν are Hz or 1/ s; so h times ν gives the units of energy, J. In his description of black body radiation, Planck postulated that the energy E could only change in discreet steps. It could be hν, of 2hν, or 3hν, and so on, but the energy could not have values between these integer step changes. The recognition that energy changes in discreet quanta at the atomic level marked the beginning of quantum mechanics.

Einstein proposed that Planck’s formula also applied to photons, so that the energy of a photon was determined by its frequency ν as E = hν. Using this formula, Einstein explained the reason that red light generates slower electrons than blue light. Red light is lower frequency than blue light. Therefore a red photon is lower energy than a blue photon. In the pool ball analogy, a blue photon hits the electron harder than a red photon, and therefore, the blue photon produces an electron that has a higher speed than a red photon. With this picture, it is clear why using redder and redder light produces slower and slower electrons emerging from the metal.

VERY RED LIGHT DOES NOT EJECT ELECTRONS

The one observation left to explain is why do the electrons stop coming out of the metal when the light is tuned far enough to the red? Einstein resolved this as well. When an electron is ejected from a metal by a photon, it has a certain kinetic energy. Kinetic energy means the energy associated with its motion. The higher the energy, the faster the electron moves. The kinetic energy is Ek where the subscript k stands for kinetic. The formula for kinetic energy is given by where m is the mass and V is the velocity. Then the velocity of an electron that emerges from a metal is related to its energy, which in turn is related to the energy of the photon that knocked it out of the metal. A higher energy photon will give the electron more kinetic energy, and the electron will move faster (have a larger V). As mentioned, electrons are held in a metal by a binding energy, call it Eb, where the subscript b stands for binding. Therefore, some of the energy that is carried by the photon has to go into overcoming the binding energy. The kinetic energy of the electron that comes out of the metal is just the photon energy, E = hν, minus the binding energy, Eb. Thus, the electron’s kinetic energy is Ek = hν - Eb. For an electron to be ejected from the metal, the photon energy hν must be larger than the binding energy Eb. As the light is tuned further and further to the red (longer wavelength, λ), ν becomes smaller and smaller because ν = c/λ, where c is the speed of light. At some red enough color, hν is less than Eb, and electrons are no longer ejected from the metal. Turning up the intensity causes more photons to impinge on the metal, but none of these photons has enough energy to eject an electron.

The fact that electrons stop coming out of the metal when the photon is tuned far enough to red (has low enough energy) can be understood by thinking of the child’s game, Red Rover. In Red Rover, a line of kids stands across a field holding hands. A kid on the other team runs at the line. If he runs very fast (high energy), he breaks through the line and keeps going, although he is slowed down. If he runs somewhat slower, he will still break through the line. However, if he runs slow enough, he will not break through the line because his energy is insufficient to overcome the binding energy of the hands holding the line together.

HOW FAST IS AN EJECTED ELECTRON

It is interesting to get a feel for how fast an electron moves when it is ejected from a piece of metal. Different metals have different binding energies called work functions. A binding energy for a metal can be determined by tuning the color redder and redder and seeing the wavelength of light at which photons cannot eject electrons. For a metal with a small binding energy, a typical cutoff wavelength for electron ejection is 800 nm. For λ = 800 nm, ν = 3.75 × 1014 Hz, and Eb = hν = 2.48 × 10-19 J. If we shine green light on the metal with a wavelength of 525 nm, the energy of the photon is 3.77 × 10 - 19 J. The kinetic energy of the electron that will be ejected from the metal is Ek = hν-Eb= 1.30 × 10 - 19 J. We can find out how fast the electron is moving using  , where me is the electron mass, me = 9.11 × 10-31 kg (kg is kilograms, that is, 1000 grams). Multiplying the equation for Ek by 2 and dividing by me gives V2 = 2(1.30 × 10-19 J)/me = (2.60 × 10-19 J)/(9.11 × 10-31 kg) = 2.85 × 1011 m2/s2. This value is the square of the velocity. Taking the square root, V = 5.34 × 105 m/ s, which is about one million miles per hour. In this example of the photoelectric effect, the ejected electrons are really moving.

Classical electromagnetic theory describing light as waves seems to work perfectly in describing a vast array of phenomena including interference, but it can’t come close to explaining the photoelectric effect. Einstein explains the photoelectric effect, but now light can’t be waves, so what happens to the classical description of interference? Reconciling the photoelectric effect and interference brings us to the cusp of quantum theory and back to Schrödinger’s Cats.

﻿

﻿