Question Video: Comparing the Paths of Electrons in a Diffracted Electron Beam Physics • Third Year of Secondary School

Video Transcript

The diagram shows some parts of an electron beam that is passing through a crystal lattice. The lattice has parallel planes separated by a perpendicular distance 𝑑. Some of the electrons in the beam are scattered by the atoms of the lattice. The electrons all have a wavelength 𝜆. Each of the blue dotted lines in the diagram corresponds to a separate wave. The waves at points A and B are in phase with each other, and the waves at points B and C are in phase with each other. Lines 𝐿 one and 𝐿 two are parallel. Which of the following correctly describes the length of the path traveled by electrons between point A and point C? (A) The length is equal to 𝑑. (B) The length is equal to 𝑛 times 𝜆 times 𝑑, where 𝑛 is an integer. (C) The length is equal to 𝑛 times 𝜆, where 𝑛 is an integer. (D) The length is equal to 𝑛 times 𝜆 divided by two, where 𝑛 is an integer. (E) The length is equal to 𝑛 times 𝜆 divided by 𝑑, where 𝑛 is an integer.

In our diagram, we locate point A right here and point C right here. We want to find out which of these five descriptions correctly describes the path difference of electrons that travel from point A to point C. And specifically, we’re talking about the path followed by electrons as they follow this dotted blue line. This means we want to find what is the length of this line segment added to the length of this line segment. Our diagram shows us that there are other paths through this crystal lattice that electrons follow. For example, there’s also this path here shown again by dotted blue lines.

On the second electron path that we’ve picked out, there’s a point B right here. As electrons follow these paths, they have a wavelength depending on their speed. This is in agreement with the de Broglie relationship, which says that the wavelength of a particle is equal to the Planck constant divided by that particle’s mass times its velocity. In our problem statement, we’re told that all the electrons incident on this crystal have the same wavelength.

Say that we have two such waves where not only do the waves have the same wavelength, but the peaks of one wave line up with the peaks of the other and the troughs of one wave line up with the troughs of the other. When this happens, these two waves are said to be in phase.

Looking again in our diagram, we’re told that the wave at point A is in phase with the wave at point B. And not only that, but the wave at point B is in phase with the wave at point C. Let’s call the wave along this path to the far right, the one that passes through point B, wave one. We’ll say that this corresponds to the orange wave that we’ve drawn here. And then let’s call the electron path that passes through points A and C wave two. That’s represented here by the blue line.

What we’re going to do is pick a point on our orange wave and call that point point B. Let’s say that it’s this peak right here. So that’s point B. And we’re told that point A on our blue wave is in phase with this point. Really, we can pick any peak on our blue wave to represent point A. Let’s say we choose this one. We know that wave two continues on for some distance and that farther along on that wave, there’s a point C that is also in phase with point B. Let’s say that this peak here is that point.

Our question comes down to “what is the path length difference between these two points A and C on this blue wave?” Because both of these points are peaks on the wave, we know that they must be separated by some number of wavelengths. Knowing that fact helps us to eliminate some of our answer options.

For example, answer option (A) says that the length between A and C is equal to 𝑑. 𝑑, we know, is the separation distance between adjacent layers of the crystal. In general, 𝑑 could have any value. It doesn’t necessarily need to equal some multiple of our wavelength. We can’t assume that 𝑑 relates to the wavelength in any particular way. So, we’ll cross out answer option (A).

Similarly, option (B) says that that length is equal to 𝑛 times 𝜆 times 𝑑, where 𝑛 is an integer. If we imagine that 𝑛 in this expression is, say, one, then 𝑛 times 𝜆 times 𝑑 simplifies to 𝜆 times 𝑑. But what if 𝑑, say, is one-half or three-quarters? In that case, we’d be saying that the points A and C are not separated by some number of full wavelength. Once again because the value 𝑑 could take on any value, we can’t choose answer option (B).

For this same reason, we can eliminate answer option (E). 𝑛 times 𝜆 divided by 𝑑, where 𝑛 is an integer suffers from the same shortcoming of answer option (B). This leaves us with options (C) and (D) remaining.

Consider answer choice (D), which says that the length between A and C is equal to 𝑛 times 𝜆 divided by two, where 𝑛 is an integer. Since this equation says that 𝑛 could be any integer, let’s once again imagine that it’s equal to one. In that case, the path length difference between points A and C would be one-half of one wavelength. But that would mean that if A is a peak, then C is a trough on the wave, rather than them both being peaks. Answer option (D) then can’t satisfy the condition that both A and C are in phase with point B.

Lastly, option (C) says that the length between A and C is equal to 𝑛 times 𝜆, where 𝑛 is an integer. This agrees with our sketch, where we know that the distance between A and C is some number of wavelengths, but we don’t know how many. Therefore, the most general way we can describe this path length difference is as some integer multiple of 𝜆. This agrees with answer option (C). The length of the path traveled by electrons between point A and point C is equal to 𝑛 times 𝜆, where 𝑛 is an integer.

Let’s look now at part two of this question.

Which of the following correctly describes the relationship between angles 𝜃 one and 𝜃 two? (A) 𝜃 one equals 𝜃 two. (B) 𝜃 one is greater than 𝜃 two. (C) 𝜃 one is less than 𝜃 two.

In our diagram, we see the angle 𝜃 one identified here and the angle 𝜃 two here. We want to figure out how these angles are related. And to help us do that, we’re told that this line 𝐿 two and this line 𝐿 one are parallel. We can notice that 𝐿 one aligns with an undeflected electron path. In other words, electrons from the original beam pass through the crystal aren’t deflected. And this path aligns with 𝐿 one. That means that this original undeflected path of the beam that we’ve called wave two also aligns with 𝐿 one. If we trace this line out, we can say that it too is parallel with 𝐿 one and 𝐿 two.

That’s helpful because if we look at this path here, which follows the deflected path of an electron, we see that it helps us define an angle, this angle right here, which is equal to 𝜃 two. Knowing that that angle is 𝜃 two, let’s consider this four-sided shape, where one of the interior angles we see is 𝜃 one. Because this shape has four sides, we know that the sum of all the interior angles on the shape is 360 degrees. We know that one of these angles is 𝜃 one, two of these angles are 90 degrees, and one is an unknown angle that we’ll call 𝛼 for now.

All of this lets us write that 360 degrees equal 𝜃 one plus 90 degrees plus 90 degrees plus 𝛼, or equally 360 degrees equal 𝜃 one plus 180 degrees plus 𝛼. Let’s now notice something about the angle that includes 𝛼 and 𝜃 two. If we follow this entire arc, then that arc defines an angle of 180 degrees. This is so because the arc starts and ends on two different points of the same line. The total angle measure of the green arc must be 180 degrees, so 180 degrees equals 𝛼 plus 𝜃 two. Or subtracting 𝜃 two from both sides of this equation, we find that 𝛼 equals 180 degrees minus 𝜃 two.

We can then substitute this result in for 𝛼 in our earlier expression. And that gives us this result. And if we add together these two 180-degree measures, we find that 360 degrees equal 𝜃 one plus 360 degrees minus 𝜃 two. If we subtract 360 degrees from both sides of this equation, then this measure cancels out entirely from our expression. We find that zero equals 𝜃 one minus 𝜃 two. Or adding 𝜃 two to both sides of the equation so that it cancels out on the right, we discover that 𝜃 one equals 𝜃 two. This agrees with our answer option (A). The two angles 𝜃 one and 𝜃 two, shown in our diagram, are equal to one another.