Photoelectric Effect

Discussion

dilemma

Under the right circumstances light can be used to push electrons, freeing them from the surface of a solid. This process is called the photoelectric effect (or photoelectric emission or photoemission), a material that can exhibit this phenomena is said to be photoemissive, and the ejected electrons are called photoelectrons; but there is nothing that would distinguish them from other electrons. All electrons are identical to one another in mass, charge, spin, and magnetic moment.

The photoelectric effect was first observed in 1887 by Heinrich Hertz (1857–1894) during experiments with a spark-gap generator (the earliest form of radio receiver). In these experiments, a spark is generated between two small metal spheres in the transmitter to induce a similar spark to jump between two different metal spheres in the receiver. Compared to later radio devices, the spark-gap generator was notoriously difficult to work with. The air gap would often have to be smaller than a millimeter for a the receiver to consistently reproduce the spark of the transmitter. Hertz found that he could increase the sensitivity of his spark-gap device by illuminating it with visible or ultraviolet light. Later studies by J.J. Thomson (1856–1940) showed that this increased sensitivity was the result of light pushing on electrons (which he discovered in 1897).

While this is interesting, it is hardly amazing. All forms of electromagnetic radiation transport energy and it is quite easy to imagine this energy being used to push tiny particles of negative charge free from the surface of a metal where they are not all that strongly confined in the first place. The era of modern physics is one of completely unexpected and inexplicable discoveries, however. Subsequent investigations into the photoelectric effect yielded results that did not fit with the classical theory of electromagnetic radiation. When it interacted with electrons, light just didn't behave like it was supposed to. Repairing this tear in theory required more than just a patch. It meant rebuilding a large portion of physics from the ground up.

It was Philipp Lenard (1862–1947), an assistant of Hertz, who performed the earliest, definitive studies of the photoelectric effect. Lenard used metal surfaces that were first cleaned and then held under a vacuum so that the effect might be studied on the metal alone and not be affected by any surface contaminants or oxidation. The metal sample was housed in an evacuated glass tube with a second metal plate mounted at the opposite end. The tube was then positioned or constrained in some manner so that light would only shine on the first metal plate — the one made out of photoemissive material under investigation. Such a tube is called a photocell (formally) or an electric eye (informally). Lenard connected his photocell to a circuit with a variable power supply, voltmeter, and microammeter as shown in the schematic diagram above. He then illuminated the photoemissive surface with light of differing frequencies and intensities.

Knocking electrons free from the photoemissive plate would give it a slight positive charge. Since the second plate was connected to the first by the wiring of the circuit, it too would become positive, which would then attract the photoelectrons floating freely through the vacuum where they would land and return back to the plate from which they started. Keep in mind that this experiment doesn't create electrons out of light, it just uses the energy of the light to push electrons that are already there around the circuit. The photoelectric current generated by this means was quite small, but could be measured with the microammeter (a sensitive galvanometer with a maximum deflection of only a few microamps). It also serves as a measure of the rate at which photoelectrons are leaving the surface of the photoemissive material.

Note how the power supply is wired into the circuit — with its negative end connected to the plate that isn't illuminated. This sets up a potential difference that tries to push the photoelectrons back into the photoemissive surface. When the power supply is set to a low voltage it traps the least energetic electrons, reducing the current through the microammeter. Increasing the voltage drives increasingly more energetic electrons back until finally none of them are able to leave the metal surface and the microammeter reads zero. The potential at which this occurs is called the stopping potential. It is a measure of the maximum kinetic energy of the electrons emitted as a result of the photoelectric effect.

What Lenard found was that the intensity of the incident light had no effect on the maximum kinetic energy of the photoelectrons. Those ejected from exposure to a very bright light had the same energy as those ejected from exposure to a very dim light of the same frequency. In keeping with the law of conservation of energy, however, more electrons were ejected by a bright source than a dim source.

Later experiments by others, most notably the American physicist Robert Millikan (1865–1953), found that light with frequencies below a certain cutoff value, called the threshold frequency, would not eject photoelectrons from the metal surface no matter how bright the source. These result were completely unexpected. Given that it is possible to move electrons with light and given that the energy in a beam of light is related to its intensity, classical physics would predict that a more intense beam of light would eject electrons with greater energy than a less intense beam no matter what the frequency. This was not the case, however.

Results of a Typical Photoelectric Experiment

Red light does not eject photoelectrons
(even if it is very bright).

Green light does eject photoelectrons
(even if it is very dim).

Blue light ejects photoelectrons with more energy than green light
(even if it is very dim).

Actually, maybe these results aren't all that typical. Most elements have threshold frequencies that are ultraviolet and only a few dip down low enough to be green or yellow like the example shown above. The materials with the lowest threshold frequencies are all semiconductors. Some have threshold frequencies in the infrared region of the spectrum.

The classical model of light describes it as a transverse, electromagnetic wave. Of this there was very little doubt at the end of the Nineteenth Century. The wave nature of light was confirmed when it was applied successfully to explain such optical phenomena as diffraction, interference, polarization, reflection and refraction. If we can imagine light as waves in an electromagnetic ocean and be quite successful at it, then it wouldn't be much of a stretch for us to image electrons in a metal surface as something like tethered buoys floating in an electromagnetic harbor. Along come the waves (light) which pull and tug at the buoys (electrons). Weak waves have no effect, but strong ones just might yank a buoy from their mooring and set it adrift. A wave model of light would predict an energy–amplitude relationship and not the energy–frequency relationship described above. Photoelectric experiments describe an electromagnetic ocean where monstrous swells wouldn't tip over a canoe, but tiny ripples would fling you into the air.

If that wasn't enough, the photoelectrons seem to pop out of the surface too quickly. When light intensities are very low, the rate at which energy is delivered to to the surface is downright sluggish. It should take awhile for any one particular electron to capture enough of this diffuse energy to free itself. It should, but it doesn't. The instant that light with an appropriate frequency of any intensity strikes a photoemissive surface, at least one electron will always pop out immediately (t < 10−9 s). Continuing with the ocean analogy, imagine a harbor full of small boats (electrons). The sea is calm except for tiny ripples on the surface (low intensity, short wavelength light). Most of the boats in the harbor are unaffected by these waves, but one is ripped from the harbor and sent flying upward like a jet aircraft. Something just ain't right here. No mechanical waves behave like this, but light does.

new idea

The two factors affecting maximum kinetic energy of photoelectrons are the frequency of the incident radiation and the material on the surface. As shown in the graph below, electron energy increases with frequency in a simple linear manner above the threshold. All three curves have the same slope (equal to Planck's constant) which shows that the energy–frequency relation is constant for all materials. Below the threshold frequency photoemission does not occur. Each curve has a different intercept on the energy axis, which shows that threshold frequency is a function of the material.

The genius that figured out what was going on here was none other than the world's most famous physicist Albert Einstein. In 1905, Einstein realized that light was behaving as if it was composed of tiny particles (initially called quanta and later called photons) and that the energy of each particle was proportional to the frequency of the electromagnetic radiation that it was a part of. Recall from the previous section of this book that Max Planck invented the notion of quantized electromagnetic radiation as a way to solve a technical problem with idealized sources of electromagnetic radiation called blackbodies. Recall also that Planck did not believe that radiation was actually broken up into little bits as his mathematical analysis showed. He thought the whole thing was just a contrivance that gave him the right answers. The genius of Einstein was in recognizing that Planck's contrivance was in fact a resonable description of reality. What we perceive as a continuous wave of electromagnetic radiation is in reality a stream of discrete particles.

Es scheint mir nun in der Tat, daß die Beobachtungen über die „schwarze Strahlung‟, Photolumineszenz, die Erzeugung von Kathodenstrahlen durch ultraviolettes Licht und andere die Erz­eu­gung bez. Verwandlung des Lichtes betreffende Erschein­ungs­gruppen besser verstandlich erscheinen unter der An­nahme, daß die Energie des Lichtes dis­kontinu­ier­lich im Raume verteilt sei. Nach der hier ins Auge zu fassenden Annahme ist bei Aus­brei­tung eines von einem Punkte aus­gehen­den Licht­strahles die Energie nicht kon­tinu­ier­lich auf größer und größer werden­cle Räume ver­teilt, sondvern es bes­teht dies­elbe aus einer end­li­chen Zahl von in Raum­punk­ten lokal­isier­ten Ener­gie­quan­ten, welche sich b­ewegen, ohne sich zu teil­en und nur als Ganze ab­sorb­iert und er­zeugt wer­den kön­nen.   In fact, it seems to me that the observations on "black-body radiation", photoluminescence, the production of cathode rays by ultraviolet light and other phenomena involving the emission or conversion of light can be better understood on the assumption that the energy of light is distributed discontinuously in space. According to the assumption considered here, when a light ray starting from a point is propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localised in space, which move without being divided and which can be absorbed or emitted only as a whole.
Albert Einstein, 1905   Albert Einstein, 1905

equations

Einstein described the photoelectric effect using a formula that relates the maximum kinetic energy (Kmax) of the photoelectrons to the frequency of the absorbed photons (ƒ) and the threshold frequency (ƒ0) of the photoemissive surface.

Kmax = h(ƒ − ƒ0)

or if you prefer, to the energy of the absorbed photons (E) and the work function (ϕ) of the surface

Kmax = E − ϕ

where the first term is the energy of the absorbed photons (E) with frequency (ƒ) or wavelength (λ)

E = hf =  hc
λ

and the second term is the work function (ϕ) of the surface with threshold frequency (ƒ0) or threshold wavelength (λ0)

ϕ = hf0 =  hc
λ0

The maximum kinetic energy (Kmax) of the photoelectrons (with charge e) can be determined from the stopping potential (V0).

V0 =  W  =  Kmax
q e

Thus…

Kmax = eV0  
 

When charge (e) is given in coulombs, the energy will be calculated in joules. When charge (e) is given in elementary charges, the energy will be calculated in electron volts. This results in a lot of constants. Use the one that's most appropriate for your problem.

  SI units acceptable
non SI units
h 6.63 × 10−34 J s 4.14 × 10−15 eV s
hc 1.99 × 10−25 J m 1240 eV nm
Planck's constant with variations

Lastly, the rate (n/t) at which photoelectrons (with charge e) are emitted from a photoemissive surface can be determined from the photoelectric current (I).

I =  q  =  ne
t t

Thus…

n  =  I
t e

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