Video Transcript
A beam of electrons that have a
velocity 𝑣 passes through a crystal in which the atoms have an average separation
of 𝑑 equals 1.2 times 10 to the negative 10th meters, as shown in the diagram. A diffraction pattern of concentric
rings is formed on the screen, recording the positions of electrons that arrive at
it, behind the crystal. Maximum diffraction occurs when the
beam is incident normal to the crystal and then a single spot is observed. For maximum diffraction, wavelength
equals two times 𝑑. Calculate 𝑣 in the case of maximum
diffraction. Use a value of 9.11 times 10 to the
negative 31st kilograms for the mass of the electrons and use a value of 6.63 times
10 to the negative 34th joule-seconds for the Planck constant.
In this scenario, we have a beam of
electrons approaching atoms in a crystal lattice. The atoms in this lattice are
represented by these black dots, and we see that they’re separated by a distance
𝑑. We’re told that the electron beam
is incident normally at an angle of 90 degrees on the crystal. The electrons pass through the
crystal and are diffracted or redirected. When the electrons reach a screen
that we could put right behind the crystal, they form what’s called a diffraction
pattern. We’re told that when the electron
beam is incident on the crystal normally, like it is here, maximum diffraction
occurs and a single spot appears on the screen.
Knowing all this, we want to solve
for the velocity 𝑣 of the incoming electrons. To start doing that, let’s recall
from our problem statement that the separation distance between atoms in this
crystal is 1.2 times 10 to the negative 10th meters. We’re also told that the wavelength
that these electrons have, and we’ll call this wavelength 𝜆, is equal to two times
𝑑 in a case of maximum diffraction like we have here. Now, since the electrons are
particles, it may seem strange to say that they have a wavelength. This fact is confirmed, though, by
what’s called the de Broglie relationship. This relationship says that any
object that has momentum has a wavelength 𝜆 that’s equal to the Planck constant ℎ
divided by the object’s momentum.
Since our incoming electrons do
have momentum, that is, they have mass and velocity, then the de Broglie
relationship tells us it also makes sense to talk about their wavelength. Classically speaking, the momentum
of an object is equal to its mass times its velocity. This equation is accurate so long
as an object’s velocity is not close to the speed of light in vacuum 𝑐. Here, we want to solve for the
electron velocity, so we won’t know until the end whether 𝑣 is nearly equal to the
speed of light. Until we find that out, though,
let’s proceed on the assumption that it’s not. Let’s say that the electron
velocity is significantly less than the speed of light 𝑐 and so that 𝑃 really does
equal 𝑚 times 𝑣.
Working under that assumption, we
can say that the wavelength of one of our electrons equals Planck’s constant divided
by the electron mass times its velocity. Multiplying both sides of this
equation by 𝑣 over 𝜆, on the left-hand side 𝜆 cancels out, and on the right the
velocity 𝑣 does. We have then that electron velocity
𝑣 equals Planck’s constant over 𝑚 times 𝜆. Let’s recall at this point that 𝜆
can be replaced by two times 𝑑, the separation distance between atoms in our
crystal. With this equation for velocity 𝑣,
we recognize that we know Planck’s constant ℎ, electron mass 𝑚, and separation
distance 𝑑. We substitute in 6.63 times 10 to
the negative 34th joule-seconds for the Planck constant, 9.11 times 10 to the
negative 31st kilograms for the mass of an electron, and 1.2 times 10 to the
negative 10th meters for 𝑑.
When we calculate this fraction, to
two significant figures, we get a result of 3.0 times 10 to the sixth meters per
second. Note that we’ve only kept two
significant figures because that’s how many our separation distance 𝑑 has.
Before we box this as our final
answer, let’s confirm that our assumption that the velocity 𝑣 is significantly less
than the speed of light 𝑐 is correct. The speed of light in vacuum is
approximately three times 10 to the eighth meters per second. Using the velocity 𝑣 we just
solved for, 𝑣 divided into 𝑐 is about 100. This means that 𝑐 is roughly 100
times greater than the electron velocity 𝑣 we’ve solved for. This, we can say, is a safely
nonrelativistic velocity. And therefore, our assumption that
𝑃 is equal to 𝑚 times 𝑣 is correct. The velocity of the electrons in
the beam is 3.0 times 10 to the sixth meters per second.
