## Thursday, December 13, 2007

### The Photoelectric Effect

A couple of years ago I tried to get a group of undergrads excited about the photoelectric effect. Some of them got so excited they fell asleep. Others built impressive constructs with roller pens. A few typed away on their cell phones.

When I was through with my exciting lecture one of them looked up from the display, and asked me what it's good for. Well, to show that the energy of the light is proportion to the frequency, I explained pa-ti-ent-ly. Ah, he said, but isn't the frequency of light the same as the energy?

See, that's what happens if ħ is equal to one in textbooks from the school level on. But more seriously, I figured the students had just learned from the very beginning on that frequency is essentially the same as energy. So then what's the big deal with the photoelectric effect? And why on earth did somebody get a Nobel Prize for it?

Well, until the last century students didn't have cellphones, h didn't have a bar, and light was a wave. A wave has an amplitude and a frequency. If you turn up the volume of your stereo its the amplitude of the sound waves that you change, not the frequency. If you turn the dimmer of your living room light, it's the light's amplitude that you change, not the frequency*.

In 1899 Thomson established that ultraviolet light caused electrons to be emitted from a metal surface. This was believed to be due to the atoms being shaken around by the infalling light waves, such that an electron could escape. In this case however, a higher light intensity should result in more emitted electrons and with more intensity the electrons should have a larger (average) kinetic energy.

So it came as a surprise what von Lenard found in 1902 when he studied how the energy of the emitted electrons varied with the intensity of the light. For this, he placed a negatively charged plate, the collector, opposite to the plate on which the light fell. The electrons that were emitted were repelled by the plate, and could only reach it if they had sufficiently high kinetic energy. If they reached the plate, they would cause a current that was measured.

Lenard found that there was a minimum voltage Vstop at the collector at which no electrons would reach it. The expectation was that increasing the light's intensity would then equip the electrons with more kinetic energy, and thus raise the repelling voltage necessary to stop them from reaching the collector. But it turned out Vstop did not depend on the intensity of the light. Instead it varied with the light's frequency.

In 1905 Einstein explained these findings by suggesting that the light should be thought of as quanta of frequency hf, with f the frequency that kick out the electrons from the plate. The electron would then carry the light quanta's energy, minus some constant energy that needs to be provided to get the electron off the metal surface. If the voltage is adjusted such that it stops the electrons from reaching the collector, e Vstop should be linear in the light's frequency with the constant of proportionality being Planck's constant. The plot below shows this dependence. On the y-axis you see the stopping voltage; the x-axis shows the frequency of the infalling light. The box in the corner is the computation of the curve's slope which gives Planck's constant.

Source: Robert A. Millikan's Nobel Lecture The Electron and the Light-Quant from the Experimental Point of View, The Nobel Foundation, 1923.

A. Einstein received the Nobel Prize in 1921 for "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect", and R.A. Millikan received the Nobel Prize two years later "for his work on the elementary charge of electricity and on the photoelectric effect".

And now you can set ħ = 1 again.

* Roughly speaking. I guess the spectrum of the emitted frequencies depends somewhat on the voltage.

This post is part of our 2007 advent calendar A Plottl A Day.

1. The Plottl posts are a highlight of my day at the moment. I think you have better things to do than write them up at greater length, but they might work as a book. Thanks for these tidbits and your nice explanations!

2. In the quantum optics texts and articles, the "old" fashioned classical quantum way to imagine the photoelectric effect takes a beating. What does anyone think of the idea to be found in such treatments, that we can at least model light as a classical wave, but imagine the absorption of photons as being discrete? Is that just one of those other-way-to-do-it things, or is there something more there?

3. Neil, in what Optics book do they say you can "model light as a classical wave, but imagine the absorption to be discrete"? I guess you can, but QED doesn't work like that...

4. Hi Peter:

Thanks for the kind words :-) I am about to put the posts together in a pdf file, so who is interested can just download and print them all together.

Hi Neil:

I actually don't know what you mean with 'model light as a classical wave' but 'imagine the absorption as being discrete'. What happens to the 'classical' part that is not absorbed? Or are you essentially talking about a dynamical collapse model? Best,

B.

5. OK, try O. Scully, "The Photoelectric Effect without Photons" - I haven't got the ref data yet. It's supposed to be a minor classic of cutting-edge quantum optics called the semi-classical approximation. More later, and look at

www.phys.uconn.edu/~chandra/05-Fringe Energy-SPIE-V.5866.pdf

6. "In 1899 Thomson established that ultraviolet light caused electrons to be emitted from a metal surface."
What ever Thomson did in 1899, the fact of
negative charges emitted from a metal surface caused by light of sufficient
wavelength was shown much earlier by
Hallwachs:
http://www.acolytescience.co.uk/origins/herz.html
Georg

7. Thanks for this one too! I really like this series.. it is the best thing I've ever seen in a physics blog.

Your readers might be interested to know that the photoelectric emission is an important charging process for interstellar and interplanetary dust. That's the 'Q' in the Lorentz force, which, if the particle is small enough, dominates the other forces to affect the dust particles' dynamics.

Some examples of research of this process, as applied to dust is: here and here and here.

8. Hi Georg:

Yes, thanks for your comment. That's why I wrote 'established' and not 'discovered'. I didn't intend to cover the full history, I just wanted to explain the plot, but it's good you clarified this.

Hi Amara:

That's quite interesting, I didn't know that. What is the typical frequency for which this works most efficiently? Best,

B.

9. The photoelectron energy is a few eV, so say 3 eV which would equal a photon wavelength ~4 microns and a frequency of about 10^{14} Hz.

10. When one thinks of all the thought and effort and sleepless nights and devising of apparatus and conceptual breakthroughs that underlie each of the simple graphs in your Plottls, it is both humbling and extremely inspiring. You are privileged indeed to carry forward this quest for another generation.

11. Hi George,

thanks for the pointer at Hallwachs and Hertz. To my knowledge, Hertz stumbled over the effect when experimenting with electromagnetic waves, and left the systematic study to his students, with Lenard as one of them, but I've never followed the details...

Best, Stefan

12. Hi Neil', markk,

that we can at least model light as a classical wave, but imagine the absorption of photons as being discrete?

I remember that my professor in undergraduate atomic physics has mentioned this, that it was quite confusing ;-), and I have never looked at any details... I've just spent a little time on it right now... The full reference to the Scully and Lamb paper you mention is the following contribution to a conference proceedings:

Lamb, Willis E., Jr., and Scully, Marlan O. "The Photoelectric Effect without Photons," pages 363-369 in “Polarisation, Matière et Rayonnement”, Jubilee Volume in Honor of Alfred Kastler (Presses Universitaires de France, Paris 1969)

That's pretty inaccessible ;-), I guess... But it is discussed in some detail in this preprint (PDF) by Geoffrey Hunter and Camil Alexandrescu: Photons in the Photo-Electric Effect. Hunter acknowledges "the gracious hospitality of Marlan Scully" for "this
work to be done", so it may be considered a serious write-up.

After my so far superficial reading it seems that what Scully and Lamb had done in 1969 was to apply time-dependent perturbation theory for a quantum system by a classical perturbation - that's what results in Fermi's Golden Rule. The classical perturbation in this case is the plane wave of the electromagnetic field describing the incident light, and the electrons in the material hit by the light are the quantum system. This can account for the frequency-dependence of the effect and the linear relationship between photocurrent and light intensity.

The crucial point seems to be the time delay between the incidence of light and the emission of the electron - there is no such delay observed in experiment. Lamb and Scully argue in their 1969 paper that their calculation shows no time delay either, thus, that a classical electromagnetic wave can account fully for the photoelectric effect. However, this has been contested. Hunter and Alexandrescu write:

The conclusion of that 1969 study [...] was implicitly refuted by an elaborate theoretical and experimental study by Clauser published in 1974. Furthermore, the analysis presented in Lamb and Scully's paper is essentially the same as that in a 1964 paper by Mandel, Sudarshan and Wolf, and they explicitly state that their theoretical analysis is only valid after a time which is much longer than the period of oscillation of the light, thus this analysis cannot claim to yield the conclusion of `no time delay'. ...

The Clauser paper referred to is John F. Clauser: Experimental distinction between the quantum and classical field-theoretic predictions for the photoelectric effect, Phys. Rev. D 9 (1974) 853-860. The abstract says that the experiment has measured various coincidence rates between photo-electrons emitted by light created with a beam splitter. This experimental configuration is said to be sensitive to differences between the classical and quantum field-theoretic predictions for the photoelectric effect. The results contradict the predictions by any classical or semiclassical theory in which the probability of photoemission is proportional to the classical intensity.

Maybe the Hunter/Alexandrescu paper and the references therein help a bit to clarify the situation, which seems to have more subtle twists than are mentioned in standard textbooks.

Best, Stefan

13. Thanks again, Stefan, about the subtleties of the photoelectric effect - I think you folks here appreciate the "twists" of things better than do most bloggers and commenters. So many others present a sort of standard picture in what I consider to be over-confident terms.

One point I read about photon emission from atoms: the atom oscillates between the excited and de-excited states at the frequency of the light for a certain length of time, and that in effect emits the photon. That time is not "instantaneous" or even merely one full period of the light, it is typically millions of vibrations. Indeed, that is what provides the coherence time and length of that photon - how many "humps" are in the wave function, correlated by Fourier transform to the energy uncertainty spread of the wave. That spread can actually me measured statistically, with a splitter/recombiner interferometer in which one leg is delayed relative to the other. The chance of getting a hit in the ideally dark channel B depends on whether the delay can push one split wave train past being able to superpose with the other one.

As tacky of a crude model as that sounds, it actually works - for example, when light is emitted by atoms having a natural interval of vibrating of about 10^-8 s (millions of vibrations of green light), that is the actual delay needed in the interferometer to get substantial hits in Ch. B.

But what I wonder is, how does that affect the extent to which we can or "should" model the related photoelectric effect in any semi-classical manner?

14. “In 1905 Einstein explained these findings by suggesting that the light should be thought of as quanta of frequency hf, with f the frequency that kick out the electrons from the plate.”

I also find it interesting that in the same year, he would publish another paper that would have this aspect of frequency or energy per light quanta become relative between the quanta and the plate being struck. That is to say that the quanta’s energy or frequency is not strictly a distinct or innate quality but relative to the motion to what it is reverenced to. As a consequence one of the greatest dangers in having humans travel at speeds approaching that of light is that the relative intensity (frequency) of even the background radiation would become that of gamma rays. Another reason not to worry about the implications of the so called twin paradox

15. Hi,

I think it is worth mentioning that there is a not very well-known experiment in connection with the photoelectric effect which can hardly be reconciled with a particle model: the point is that the photoelectrons are actually primarily emitted in the direction of the electric field vector but not in the direction of propagation of the 'photons' (as one should expect it from a particle model) (see this link).

A further problem I see with assuming photons as localized particles is that in order to interact with an atomic electron, it must actually be within the atom (from far away, the atom appears as overall neutral and the photon couldn't interact with the electron). Now considering that a photon must move with the speed of light, the interaction time would thus be a mere t=10^-18 sec (the time for a photon to traverse an atomic diameter). During this time the electron would have to be accelerated to a velocity of the order v=10^6 m/sec (corresponding to an energy of the order of 1 eV). Now from v=a*t one finds that this requires an acceleration of a=v/t=10^24 m/sec^2 , i.e. a force of F=m*a=9*10^-31 *10^24 = 10^-6 N and thus an electric field of E=F/e = 10^-6/1.6*10^-19 = 6*10^12 V/m. This is more than the inner-atomic electric field, and even f one could make plausible where such a strong photon field should originate from, it should effectively blow the whole matter apart.

Also, the photoelectron should actually nowhere near be able to acquire a sufficient energy for photoionization in a momentum conserving photon-electron collision. I have addressed this point on my page Wave and Particle Theory of Light applied to the Photoelectric Effect, which also shows that a proper wave-atom interaction model for the photoionization process could account for the almost instantaneous release times observed (by coincidence, the times I obtained with a semi-classical model seem to be similar to the ones obtained in the Lamb and Scully paper mentioned above, which have been refuted in the Hunter and Alexandrescu paper, but the point is that for realistic values one has to consider the photoionization process actually in more detail; I have done this on my page Photoionization Theory for Coherent and Incoherent Light, which gives an argument why the release times may in practice appear as much shorter than predicted with the simple theory (on the basis of this I have also suggested a Classical Interpretation of
EPR- Bell Test Photon Correlation Experiments
, which might be of interest with regard to the Clauser paper mentioned above).

I hope this sheds some new light on the issue.

Thomas

16. Thomas,

First, you must excuse me for I should truly be considered a novice in all this. However, after reading what you have said and looking at your web page; it seems to me that you have left out one important aspect of quantum mechanics as a consideration of your analysis and that being the “uncertainty principle”. it seems you take position and momentum as absolutes. Also, correct me if I’m wrong, but you appear to consider the interaction of a photon with electron analytic with the collision of billiard balls or something. However, the photon has no rest mass and in such circumstances surrenders all of its energy/momentum to the electron, not a part of it. In essence the photon is no more. As I see it , when this is considered there should be sufficient energy transferred to free the electron. I am therefore curious to hear the more qualified responses in regards to your contention's merits.

17. Hi Phil: Thanks.
Hi Thomas: The photon is neither a wave nor a particle.

18. Hi Phil,

The zero rest mass of a photon (assuming for a moment such an entity exists) is irrelevant as it is never at rest. What is relevant here is the relativistic mass, which for a photon of energy E is m=E/c^2. The associated momentum of the photon would thus be p=m*c=E/c. As the photon is destroyed in the photoionization process, momentum conservation would require that this is equal to the momentum M*V imparted to the photoelectron, so E/c=M*V or V=E/M/c. On the other hand, energy conservation requires (neglecting the work function here) E=M/2*V^2 = E^2/M/c^2/2 (having inserted the previous expression for V) and thus E=2M*c^2 . So momentum and energy conservation require that the initial photon energy is twice the rest energy of the electron, i.e. about 10^6 eV. But the photoeffect is in fact observed for light corresponding to much smaller energies (visible light corresponds to about 1 eV).

Thomas

19. Hi Thomas,

"energy conservation requires (neglecting the work function here) "

Hi Thomas,

Yet this is precisely the point. The only relevant additional energy required to free the electron is simply the work function (φ = hfo), and the energy you indicate in visible light (per quanta) corresponds to this. This of course also relates to the material involved. As such I don’t see how this indicates that the photoelectric effect is wrong.

20. Hi Phil,

The work function is not material for my argument. I mentioned it only in order to make clear that the velocity V of the photoelectron should not be understood as the one observed after the electron has completely escaped from the atom, but as the initial velocity imparted by the photon (this will be reduced somewhat as the electron does work against the field of the atom, but the resulting velocity is not the one we are interested in here; alternatively, you can just assume the work function as very small, in which case the velocity would not be reduced at all (and thus it would be the finally observed one)).

Thomas

21. Hi Phil,

Hi Thomas,

I guess one has to keep in mind that the electron kicked out of a material in the photoelectric effect has both initial energy and momentum due to quantum mechanical effects. Imagining photon and electron as Newtonian point particles falls short of properly taking into account quantum mechanics - it may work in some circumstances, but certainly not in all.

BTW, there is a whole bunch of experimental techniques used in condensed matter physics to study electronic structure, energy bands and densities of states using the photoelectric effect, going under the name tag of photoemission spectroscopy. Here is a nice introductory review article: Photoemission spectroscopy - from early days to recent applications, by Friedrich Reinert and Stefan Hüfner, New J. Phys. 7 (2005) 97. As a side note, Hüfner is a well-known expert in photoemission spectroscopy, and was the PhD advisor of this year's Nobel Prize winner Peter Grünberg.

Best, Stefan

22. Hi Stefan,

If you look at the Reinert and Hüfner article, you can see that in fact the radiation field goes simply as a classical electromagnetic field into the perturbation approach. So the quantization effects in the photoemission are solely due to the quantization of the atom, not the radiation field (which is assumed as continuous). Such a perturbation approach can also show (it does not come out that well in the article) that the photoelectron emission is primarily observed parallel to the electric field vector (see this link but not in the photon propagation direction (as a particle model would suggest).

Thomas

23. Well, it looks like I opened up a little can of worms by asking about how can we think of the photoelectric effect *without* photons instead of the usual presumption about what Einstein showed. I am thinking, if we imagine the field impinging on a detector as if classical, how does that effect the way we imagine the quantum mechanics of the impinging light? In particular, what about decoherence? Decoherence seems to me to be a circular argument, since you develop the point using concepts of probability, that take in effect the collapse for granted to begin with.

24. To all,

Wave or particle, particle or wave; I think the most sensible thing to do is go ask some broccoli. Now I don’t mean Louis de Broglie , yet rather a head of broccoli. If you haven’t already heard, a team lead by biophysicist Gregory Engel discovered that photosynthesis in plants owes its almost 100% efficiency to utilizing processes that relate very closely to those studied by those researching quantum computing. Seeing that the plants have had a 2 ½ billion year head start on us, I find it only logical that we take advantage of this experience :-)

25. In case someone end up in this discussion wondering "where photons are actually needed", there is an interesting posting at Uncertain Principles on this topic. Photon correlations are a key, and especially "anti-bunching".

26. pretty inaccessible but not quite... try http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680009569_1968009569.pdf