Section 2.5 Photons Interacting with Matter
In the previous section we argued why the photon picture resolved the dilemma of the UV catastrophe, but many physicists at the time that Einstein proposed photons were skeptical of the idea. But contained in Einstein's brilliant suggestion was the potential for a quantitative test of the idea, through a phenomenon called the photoelectric effect. This is the work for which he was awarded his only Nobel Prize. 1
Subsection 2.5.1 Photoelectric Effect
Metals are materials in which electrons are free to move around — this is why metals are good conductors, why they are shiny, etc. But the electrons cannot just freely hop off the surface. They are bound to remain inside the material by a binding energy \(U_\text{ bind }\text{.}\) It was discovered at the close of the 19\(^\text{ th }\) century that some electrons can be made to fly off the surface of a metal by shining light on it, as sketched in Figure 2.3. This should seem reasonable: the EM wave gives some extra energy to the electrons, allowing them to break free of their binding to the metal. The electron “pays off” its binding energy debt and leaves with the remaining energy in the form of kinetic energy.
Let's focus on how this effect should depend on the intensity and the frequency of the light. With classical EM waves, increasing the amplitude of the wave should cause a stronger force on the electrons and give them more energy. This would suggest that increasing the intensity of light should increase the energy given to the electrons and they should come off with a higher final speed. Also, the classical EM wave picture would predict that it should be possible to eject electrons from the metal for any frequency of light, as long as the intensity is high enough.
By 1905 there were qualitative indications that this was not what was happening in experiments, and Einstein's photon hypothesis provided a quantitative theory that could be tested experimentally. In Einstein's theory, which was quantitatively verified by experiments in 1916, the intensity of the light has no effect on the kinetic energy of the emitted electrons — it is the frequency of the light that is related to the kinetic energy. Shining low frequency light on the material results in no electrons being emitted, regardless of the intensity, and when electrons are emitted, the kinetic energy increases linearly with the frequency of the light. The experimental results are illustrated graphically in the Figure 2.4.
Einstein's theory of the photoelectric effect is based on the following key assumption:
In the microscopic world of atoms and subatomic particles, light interacts with matter in the form of single photons.
This means light of frequency \(f\) can only give a single photon's energy to an electron. That photon, with energy \(E_\text{ ph } =hf\text{,}\) might or might not have enough energy to free the electron from the metal. If the photon energy is below the binding energy (\(E_\text{ ph } \lt U_\text{ bind }\)), then no electrons will escape. Since \(E_\text{ ph } = hf\text{,}\) this sets the cutoff frequency \(f_c\) when \(E_\text{ ph } = hf_c = U_\text{ bind }\text{.}\) If the photon energy is greater than the binding energy holding the electron to the metal (\(E_\text{ ph } > U_\text{ bind }\)), then the electron might escape (it can always squander the energy and head off in the wrong direction, or bump into an impurity in the metal). But of those electrons that do escape, there will be an upper limit to their kinetic energy given by
That upper limit is obtained when the electron uses the energy absorbed from the photon optimally. Note that the maximum electron kinetic energy depends only on the frequency of the light and not on its intensity. This was Einstein's bold prediction, and it was confirmed experimentally by Millikan, who measured \(K_\text{ max }\) for various frequencies and found data of the form sketched on the right in Figure 2.4.
Thus we see that the photon picture explains the experimental results for the photoelectric effect perfectly. It also provides a quantitative measurement of Planck's constant \(h\) from the slope of the \(K_\text{ max }\) versus \(f\) plot.
Example 2.5. Electrons emitted from copper.
Copper has a binding energy of \(4.7\Xunits{eV}\text{.}\) For light of (a) \(200\Xunits{nm}\) and (b) \(400\Xunits{nm}\) shining on a piece of copper, determine whether electrons are emitted and, in the case they are emitted, calculate their maximum kinetic energy.
First we calculate the photon energy. A handy trick for getting photon energies in electron volts is the following: 2
For (a), \(\lambda = 200\Xunits{nm}\text{,}\) so the photon energy is
When the photon is absorbed by the electron, the electron gains enough energy to pay off its binding energy debt and escape with
For (b) we have a photon energy of \(1240/400= 3.1\Xunits{eV}\text{.}\) This is less than the binding energy, so no electrons will be emitted.
Subsection 2.5.2 Ionization
Einstein's basic idea that photons are the energy chunks in the interaction of light with matter is able to explain more phenomena than just the photoelectric effect. For example, an electron in an atomic orbital is held in the atom by some binding energy. For the case of hydrogen — which is just a single electron orbiting a single proton — the electron is usually in the lowest energy state possible, called the ground state. In the hydrogen ground state the binding energy for the electron is \(13.6\Xunits{eV}\text{.}\) When lower frequency radiation (EM waves) shines on a gas of hydrogen, then none of the electrons are freed from their atomic orbitals. Radiation of higher frequency is able to liberate electrons from their host nucleus, creating ionized hydrogen. This higher frequency radiation is called, rather sensibly, ionizing radiation.
This idea extends beyond simple hydrogen. EM waves frequently induce chemical reactions, from a photographic plate to photosynthesis to mutation of DNA. That is, the energy of the absorbed photon is being used to break and possibly re-arrange some chemical bonds. In the case of DNA mutation, the relevant energy scale of the chemical bonds is in the 0.1 to \(10\Xunits{eV}\) range; EM waves with photon energies comparable to or greater than these binding energies can cause mutations. This is a known mechanism for explaining why exposure to EM radiation with certain frequencies can cause cancer in humans. The photon nature of light is also critical for understanding various photo-detectors, including the sensors in the digital camera in your cell phone. We will explore these ideas in the assigned problems and in subsequent chapters.
Finally, we conclude this section with an important point: light had previously been considered to be exclusively a wave phenomenon, but with photons it has been shown to have particle characteristics. But it's not an either/or situation. It is not the case that the wave description is wrong; after all, you have seen the interference phenomena yourself in lab. So we are led to acknowledge that light can be both wave-like and particle-like. What it does in a particular experiment depends on what the experiment is measuring. This is very strange!