Video Transcript
Electrons in a beam all have the same wavelength, 𝜆. The beam is diffracted by a regular crystal lattice. After the electrons exit the crystal, what is the difference between the distances traveled by electrons that arrive at point I and electrons that arrive at point II? (A) 𝜆, (B) two times 𝜆, (C) 𝜆 divided by two.
Our diagram shows us a beam of electrons incident on a crystal. The crystal scatters the incoming electrons, causing them to diffract and then interfere with one another to form this intensity distribution on a screen, we can imagine, to the far right. There are two points labeled on the screen, point I and point II. Electrons that would travel to point I would follow this path, and electrons arriving at point II follow this one. We want to know what is the difference between the distances of these two paths. We can be helped in figuring this out by noting that the intensity distribution has a maximum value at point II, and then at point I it has a minimum value. This would be a dark spot on the screen.
The electrons in the electron beam and then also the electrons after they have passed by the crystal all have the same wavelength. We can think then of these electrons as waves. And since they are waves, that means they interfere with one another. The interference of waves that takes place to generate the dark spot at point I is different than the kind of interference that takes place to generate the bright spot at point II. A bright spot on the screen indicates that at that point, the waves coming from the crystal constructively interfere. On the other hand, a dark spot, like we have at point I, indicates that the waves at that point destructively interfere.
We know that in reality there are many, many waves coming from the crystal that interfere with one another to form our intensity distribution. As a simplifying step though, let’s imagine that it’s just two waves that are interfering with one another to generate our alternating series of bright and dark spots on the screen. In their current position, these waves are in phase with one another. That means that both the peaks of each wave line up as well as the troughs. We could say they have a constant phase relationship of zero. When waves align like this and they interfere, the result is a wave with amplitude greater than either of the waves that make it up. This we know as constructive interference. And when this happens, it creates a bright spot in our intensity distribution. As we’ve seen, this corresponds to the interference pattern at point II.
Creating a different type of interference pattern though is as simple as shifting one of our waves relative to the other. If we want our blue and our orange waves to interfere destructively, we could do that by shifting the orange wave so that now a trough on this wave lines up with a peak on the blue wave. These waves will combine to give a wave we could call it with amplitude zero. This is destructive interference, and it creates the dark spot that we see at point I on our screen. In order to transition from constructive to destructive interference, we shifted our waves a linear distance of 𝜆 divided by two relative to one another.
Note that if we double this shift distance to an entire wavelength 𝜆, then that would once again make the peaks of our blue and orange waves align. That would give us constructive interference, which we know is not represented by the dark spot at point I. To shift then from a bright spot to a dark spot, like we do in moving from point II to point I on our intensity distribution, our interfering waves must shift relative to one another a distance of 𝜆 over two. Note that if these waves instead shifted a distance of, say, three 𝜆 over two or five 𝜆 over two, then they would go from a bright spot to a dark spot. But those spots would not be adjacent to one another as points one and two are on our diagram.
We’re able to identify then our final answer. The difference between the distances traveled by electrons that arrive at point I and electrons that arrive at point II is one-half the wavelength of the electrons. This is answer option (C).
