### 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 π. Some of the electrons in the beam
are scattered by the atoms of the lattice. The electrons all have a wavelength
π. 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 πΏ one and πΏ 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 π. (B) The length is equal to π times
π times π, where π is an integer. (C) The length is equal to π times
π, where π is an integer. (D) The length is equal to π times
π divided by two, where π is an integer. (E) The length is equal to π times
π divided by π, where π 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, weβre 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, thereβs also this path
here shown again by dotted blue lines.

On the second electron path that
weβve picked out, thereβs 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 particleβs mass times its velocity. In our problem statement, weβre
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, weβre
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. Letβs call the wave along this path
to the far right, the one that passes through point B, wave one. Weβll say that this corresponds to
the orange wave that weβve drawn here. And then letβs call the electron
path that passes through points A and C wave two. Thatβs represented here by the blue
line.

What weβre going to do is pick a
point on our orange wave and call that point point B. Letβs say that itβs this peak right
here. So thatβs point B. And weβre 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. Letβs say we choose this one. We know that wave two continues on
for some distance and that farther along on that wave, thereβs a point C that is
also in phase with point B. Letβs say that this peak here is
that point.

Our question comes down to βwhat is
the path length difference between these two points A and C on this blue wave?β 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 π. π, we know, is the separation
distance between adjacent layers of the crystal. In general, π could have any
value. It doesnβt necessarily need to
equal some multiple of our wavelength. We canβt assume that π relates to
the wavelength in any particular way. So, weβll cross out answer option
(A).

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

For this same reason, we can
eliminate answer option (E). π times π divided by π, where π
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 π times π divided by two, where
π is an integer. Since this equation says that π
could be any integer, letβs once again imagine that itβs 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 canβt
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 π times π, where π 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
donβt know how many. Therefore, the most general way we
can describe this path length difference is as some integer multiple of π. This agrees with answer option
(C). The length of the path traveled by
electrons between point A and point C is equal to π times π, where π is an
integer.

Letβs look now at part two of this
question.

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

In our diagram, we see the angle π
one identified here and the angle π two here. We want to figure out how these
angles are related. And to help us do that, weβre told
that this line πΏ two and this line πΏ one are parallel. We can notice that πΏ one aligns
with an undeflected electron path. In other words, electrons from the
original beam pass through the crystal arenβt deflected. And this path aligns with πΏ
one. That means that this original
undeflected path of the beam that weβve called wave two also aligns with πΏ one. If we trace this line out, we can
say that it too is parallel with πΏ one and πΏ two.

Thatβs 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 π two. Knowing that that angle is π two,
letβs consider this four-sided shape, where one of the interior angles we see is π
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
π one, two of these angles are 90 degrees, and one is an unknown angle that weβll
call πΌ for now.

All of this lets us write that 360
degrees equal π one plus 90 degrees plus 90 degrees plus πΌ, or equally 360 degrees
equal π one plus 180 degrees plus πΌ. Letβs now notice something about
the angle that includes πΌ and π 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 πΌ plus π two. Or subtracting π two from both
sides of this equation, we find that πΌ equals 180 degrees minus π two.

We can then substitute this result
in for πΌ 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 π one plus 360 degrees minus π
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 π one
minus π two. Or adding π two to both sides of
the equation so that it cancels out on the right, we discover that π one equals π
two. This agrees with our answer option
(A). The two angles π one and π two,
shown in our diagram, are equal to one another.