The photoelectric effect and duality
Einstein won the Nobel Prize for this, not for relativity. A one-line equation explained three facts that wave-like light could not, and in passing gave a name to the particle that a photomultiplier now counts one at a time.
01The experiment that waves cannot explain
Light shining on a metal plate inside an evacuated tube knocks out electrons, which a collector picks up as current. None of that surprises wave theory: a wave carries energy, and that energy can free charges. What wave theory does not predict is how that current depends on the light that produces it.
| What is observed | What the wave predicts | What is measured |
|---|---|---|
| Cutoff frequency | It should not exist: just wait long enough | Below \( f_0 \) no electron comes out, whatever the intensity |
| Energy of the electrons | It should grow with intensity | It depends only on the frequency; intensity changes the number, not the energy |
| Emission time | With weak light there should be a delay | It is immediate, even with the weakest light that can be measured |
The three facts point in the same direction: there is something in light that is not distributed continuously. That was Einstein's way in, in 1905, the same year he published special relativity, and he used the same idea of Planck's that until then had been taken as a mathematical device for the blackbody.
02Einstein's equation
Einstein took Planck's quanta seriously: if light of frequency \( f \) comes in packets of energy \( hf \), an electron at the surface absorbs one whole photon or none. Part of that energy is spent escaping from the metal —the work function \( \phi \), different for each material— and the rest remains as kinetic energy.
This single equation explains the three facts in the table above. Below \( f_0 = \phi/h \) a photon cannot pay the work function and nothing comes out, no matter how many photons arrive. Above it, each absorbed photon gives exactly \( hf - \phi \) of kinetic energy, so more intensity means more photons and more current, but the same energy per electron. And there is no delay because absorption happens one photon at a time, not as a gradual accumulation.
You choose the metal and the light, and the graph plots the stopping potential against frequency. The five lines have the same slope —it is \( h/e \), and it does not depend on the metal— and cross the axis at a different point depending on the work function. It is the experiment with which Millikan, between 1914 and 1916, measured \( h \) by a completely different route from Planck's and obtained the same number.
03Wave-particle duality and the Compton effect
Interference and diffraction —the whole previous unit— can only be explained if light is a wave. The photoelectric effect can only be explained if light arrives in packets. Light is not one thing instead of the other: it is both, and which one is convenient to use depends on the experiment. This is called wave-particle duality, and it is not a flaw of the theory but the first symptom that a new physics is needed: quantum mechanics.
The final confirmation that the photon is a particle with momentum of its own came in 1923, when Arthur Compton made X-rays collide with free electrons and measured that the scattered light came out with a lower frequency than the incident light, lower the larger the deflection angle. That is exactly what happens in a billiard-ball collision between two particles.
The photoelectric effect measures \( h \) from energies (electronvolts and optical frequencies). The Compton effect measures it from momentum (X-ray wavelengths). Blackbody radiation, in the previous topic, measures it from a thermal spectrum. Three experiments that look nothing alike give the same \( h = 6.626\times 10^{-34} \) J·s: that coincidence is what convinced physics that the quantum was real and not a calculation trick.
04Where it appears in electronics
| Device | What it does | The effect at work |
|---|---|---|
| Photomultiplier tube | Detects photons one at a time, with an electron cascade | External photoelectric effect: the photon knocks the electron out of the photocathode |
| Camera tubes (vidicon, iconoscope) | First TV cameras, before the CCD | Photoelectric effect on a mosaic of photocathodes |
| Photodiode and CCD/CMOS | Convert light into current inside a semiconductor | Internal photoelectric effect: the electron does not leave the material, it moves up a band |
| Solar cell | Generates a voltage from light, without an external battery | Photovoltaic effect: a relative of the internal photoelectric effect, in a p-n junction |
| Photoresistor (LDR) | Changes its resistance with light | Photoconductive effect: the electron does not escape either, it just changes band |
The photomultiplier is the direct heir of Millikan's experiment: it multiplies the signal of a single photoelectron by a million or more, with a chain of electrodes at increasing voltage (dynodes), and today it remains the reference detector when photons have to be counted one at a time, as in a positron emission tomography scanner or a gamma-ray telescope. The photodiode and the CCD in an ordinary camera solve the same problem —converting light into an electrical signal— with the internal photoelectric effect: the electron does not need to escape from the material, it is enough for it to jump from the valence band to the conduction band. That conduction band is, precisely, the topic this subject ends with.
05In the lab
With the \( V_0 \) versus \( f \) data for a metal (or the data given by the lab on this page, read with precision), fit a straight line by least squares. The slope gives \( h/e \) and the y-intercept gives \( -\phi/e \). Compare the \( h \) obtained with the accepted value.
With the metal and the wavelength fixed, vary only the intensity and record how the saturation current and the stopping potential change. Repeat changing only the frequency, with the intensity fixed. Confirm which of the two variables moves each quantity.
Calculate the cutoff wavelength of the five metals in the lab and place them on the spectrum (visible, UV-A, UV-C). Explain why cesium and sodium are used in photocells for visible light and platinum only responds to vacuum UV.
06Common mistakes
- Believing that more intensity gives faster electrons. More intensity gives more electrons; the energy of each one depends only on the frequency.
- Thinking that any frequency will do if you wait long enough. Below \( f_0 \) nothing comes out, ever, whatever the intensity or the exposure time.
- Confusing the stopping potential with the supply voltage of the tube. \( V_0 \) is the one that makes the current drop exactly to zero, not just any voltage.
- Believing that the photoelectric effect shows that light “is not” a wave. It shows that it is also a particle. Interference is still true.
- Confusing the external photoelectric effect with the photoconductive or photovoltaic one. In the external one the electron escapes from the material; in the other two it changes band without leaving it.
- Treating the number \( h \) of this unit as different from the one in blackbody radiation. It is the same one, measured by another route.
07Self-assessment
Why does classical wave theory not explain the cutoff frequency?
Because in that theory the energy of light depends on its intensity, not on its frequency: with enough intensity, any frequency should eventually knock out electrons, and that does not happen below \( f_0 \).
What does the stopping potential \( V_0 \) measure?
The maximum kinetic energy of the photoelectrons, expressed in electronvolts: \( eV_0 = K_{max} \).
What changes when the light intensity is increased, with the frequency fixed?
The saturation current increases, that is, the number of photoelectrons per second. The energy of each one does not change.
What is the work function \( \phi \)?
The minimum energy needed to knock an electron out of the surface of a given metal.
What evidence does the Compton effect provide for the particle nature of light?
That the photon carries momentum and transfers it like a particle in a collision: the scattered light changes wavelength with the angle, just as a mechanical collision predicts.
Why did Einstein receive the Nobel Prize for the photoelectric effect and not for relativity?
Because by 1921 the photoelectric equation had already been confirmed with experimental precision by Millikan, whereas relativity still did not have such direct backing in the eyes of the committee.
08Further reading
- Robert Eisberg and Robert Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. 2nd ed., John Wiley & Sons, 1985 (Spanish edition, “Física cuántica: átomos, moléculas, sólidos, núcleos y partículas”, Limusa). Chapter 2 develops the photoelectric effect and the Compton effect with the same rigor with which it treated the blackbody.
- Robert A. Millikan. A Direct Photoelectric Determination of Planck's “h”. Physical Review, vol. 7, 1916. The paper that finished confirming Einstein's equation, after its author had spent years trying to refute it.
- Arthur H. Compton. A Quantum Theory of the Scattering of X-rays by Light Elements. Physical Review, vol. 21, 1923. The experiment that definitively gave light momentum of its own.
- Raymond Serway and John Jewett. Physics for Scientists and Engineers. 9th ed., Cengage, 2019 (Spanish edition, “Física para ciencias e ingeniería”). The table of work functions of the elements and the standard didactic treatment of this unit.