Section 37–4 An experiment with electrons

(Idealized conditions / Observing electrons arrival / Interference pattern)

 

In this section, Feynman discusses the idealized experimental conditions, observations of electrons arrival, and interference pattern of double slit experiment involving electrons (or single-electrons).

 

1. Idealized conditions:

“We make an electron gun which consists of a tungsten wire heated by an electric current and surrounded by a metal box with a hole in it. If the wire is at a negative voltage with respect to the box, electrons emitted by the wire will be accelerated toward the walls and some will pass through the hole. All the electrons which come out of the gun will have (nearly) the same energy. In front of the gun is again a wall (just a thin metal plate) with two holes in it. Beyond the wall is another plate which will serve as a “backstop.” In front of the backstop we place a movable detector. The detector might be a geiger counter or, perhaps better, an electron multiplier, which is connected to a loudspeaker (Feynman et al., 1963, p. 37–4).”

 

Feynman’s description of the double slit experiment involving electrons could be summarized as three idealized conditions: (1) Ideal electron gun: Electrons emitted from a heated tungsten wire are accelerated through a hole, ensuring they have nearly the same energy (or wavelength). (2) Ideal double slit: A thin metal plate with two holes is placed in front of the electron gun that allows an electron (or electrons) to have two alternative paths. (3) Ideal detector: The detector, such as a Geiger counter, registers the arrival of electrons as audible clicks. It seems that Feynman considered the three idealized conditions were difficult to be achieved with the technology available in his time. Remarkably, the double slit experiment involving electrons was performed in 1961 by Claus Jönsson, but he was not able to reduce the electron firing rate to one at a time.  

 

“We should say right away that you should not try to set up this experiment (as you could have done with the two we have already described). This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossibly small scale to show the effects we are interested in (Feynman et al., 1963, p. 37–5).”

 

Although Feynman suggested the concept of nanotechnology in 1959, he advised not to set up the experiment possibly because he was unaware of the latest technologies and Jönsson’s (1961) paper was published in German. However, to observe electrons’ interference, the slit width should be comparable to the de Broglie wavelength of the electrons, which is on the scale of nanometers to micrometers, depending on their energy. Jönsson created slits about 10 micrometers wide, small enough to cause interference when electrons passed through. In addition, electrons are sensitive to irregularities on the surface of the substrate or the edges of the slits. If the slits had any roughness or contamination, electrons could scatter unpredictably, blurring the interference pattern. Jönsson’s experience in clean materials ensured that the substrate and slits were free from microscopic debris or irregularities that could “distort” the path of the electrons.

Photograph of diffraction pattern of a single slit and interference pattern of two slits (Jönsson, 1974)

 

2. Observing electron arrival:

“As we move the detector around, the rate at which the clicks appear is faster or slower, but the size (loudness) of each click is always the same. If we lower the temperature of the wire in the gun the rate of clicking slows down, but still each click sounds the same. We would notice also that if we put two separate detectors at the backstop, one or the other would click, but never both at once (Feynman et al., 1963, p. 37–5).”

 

While hearing clicks (as suggested by Feynman) tells us that electrons are being detected, seeing the interference pattern provides spatial information that shows the probabilistic nature of quantum mechanics. Historically, Merli and his collaborators’ (1974) experiment used an electron biprism and TV image intensifiers to directly observe single-electron interference patterns. Firstly, by passing single-electrons through the electron biprism, which is essentially a very thin wire placed in the path of the electron beam. Applying a voltage to the biprism, it behaves like two virtual slits (instead of physical slits). Secondly, after passing through the biprism, the electrons were detected using image intensifiers that are highly sensitive and capable of detecting the impact of single-electron. These devices amplify the weak signal generated by each electron, which help to observe the gradual build-up of the interference patterns.

We would also notice that the “clicks” come very erratically. Something like: click ….. click-click … click …….. click …. click-click …… click …, etc., just as you have, no doubt, heard a geiger counter operating. If we count the clicks which arrive in a sufficiently long time—say for many minutes—and then count again for another equal period, we find that the two numbers are very nearly the same (Feynman et al., 1963, p. 37–5).”

 

Perhaps Feynman could have included Taylor’s (1909) double-slit experiment in his lecture because it serves as a historically significant and pedagogically valuable demonstration of “single photon interference.” (Taylor had to run the exposure for 3 months, i.e., counting the “clicks” for many minutes as suggested by Feynman may not seem to be a good time-scale.) Essentially, smoked glass screens were used as a filter to reduce the intensity of light to the point where it was dim enough that only one photon was emitted at a time. Although it significantly decreased the number of photons reaching the slits, the cumulative result of many photon impacts led to the formation of an interference pattern, similar to what would happen with continuous light. The feeblest light source could form the interference pattern led to Dirac’s dictum that “Each photon interferes only with itself. Interference between different photons never occurs.”

 

3. Interference pattern:

“Just as for our experiment with bullets, we can now proceed to find experimentally the answer to the question: “What is the relative probability that an electron ‘lump’ will arrive at the backstop at various distances x from the center?” As before, we obtain the relative probability by observing the rate of clicks, holding the operation of the gun constant. The probability that lumps will arrive at a particular x is proportional to the average rate of clicks at that x. The result of our experiment is the interesting curve marked P₁₂ in part (c) of Fig. 37–3 (Feynman et al., 1963, p. 37–5).”

Just like the interference pattern for water waves, when the two slits are closer to each other, the overlapping water waves form a higher central peak as shown in the middle of the graph (See Fig 37-3 above). In his Cornell lecture, the two slits are drawn further apart, thus there is no central peak due to the reduced overlapping in the middle, but there are two separate peaks corresponding to each of the two slits (See Fig 30 above). In addition, Feynman’s (1965) diagram suggests that the diffraction pattern for a slit due to electron waves has no secondary maxima. It implies that the intensity distribution of the diffraction pattern would be different for electron waves and light waves. However, the diffraction pattern of electron waves for single slit has secondary maximas on either side of the central maximum. The intensity of the first maxima is not high as shown in the diagram below. 

(Young & Freedman, 1996, p. 1274)

In a Berkeley Symposium, Feynman (1951) writes: “The probability distribution must now be (d) instead of the interference pattern (a)… The uncertainty principle assures us that the vertical uncertainty Dq in the screen position must exceed h/Dp and hence exceed h/2dp so that the maxima and minima of the diffraction pattern (a) are completely smeared out and the resulting distribution is that of (d) (p. 538).” This shows that Feynman was inconsistent in describing the same probability distribution using the terms diffraction pattern and interference pattern. It is unsurprising because he explains that no one has ever been able to define the difference between interference and diffraction satisfactorily in chapter 30. Currently, physicists prefer the term interference pattern for the phenomenon involving two sources, but if there is a large number of sources (or only one slit), the term diffraction pattern is used instead. However, the probability distribution of arrival of electrons as drawn in the Berkeley Symposium (Fig 2 below) is slightly better than Fig 37-3 and Fig 30.

 

Review Questions:

1. Why did Feynman advise against setting up the double slit experiment involving electrons?

2. How would you observe the arrival of electrons?

3. How would you explain the interference pattern of the double slit experiment involving electrons?

 

The moral of the lesson: We can set up the double slit experiment involving single-electrons using two real slits or virtual slits (electron biprism or ultrathin wire) and TV image intensifiers.

Youtube Video: "Electron Interference" (english version) - YouTube

 

References:

1. Feynman, R. P. (1951). The concept of probability in quantum mechanics. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (Vol. 2, pp. 533-542). University of California Press.

2. Feynman, R. P. (1965). The character of physical law. Cambridge: MIT Press.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Jönsson, C. (1961). Elektroneninterferenzen an mehreren künstlich hergestellten Feinspalten. Zeitschrift für Physik, 161(4), 454-474.

5. Jönsson, C. (1974). Electron diffraction at multiple slits. American Journal of Physics, 42(1), 4-11.

6. Merli, P. G., Missiroli, G. F., & Pozzi, G. (1974). Electron interferometry with the Elmiskop 101 electron microscope. Journal of Physics E: Scientific Instruments, 7(9), 729–732.

7. Taylor, G. I. (1909). Interference fringes with feeble light. Proceedings of the Cambridge Philosophical Society, 15(1), 114-115.

8. Young, H. D., & Freedman, R. A. (1996). University physics. Reading, MA: Addison-Wesley.