Video Transcript
A beam of electrons that have a
velocity of 2.85 times 10 to the sixth meters per second passes through a
crystalline material. The diffraction of the electrons
produces a pattern containing a single spot. A single spot diffraction pattern
occurs when the electrons are normally incident on the plane of the crystal lattice
and the separation 𝑑 of the planes of the crystal lattice is half the wavelength of
the electrons. Find 𝑑. Use a value of 9.11 times 10 to the
negative 31st kilograms for the mass of the electrons and a value of 6.63 times 10
to the negative 34th joule-seconds for the Planck constant.
Okay, so in this example, we’re
working with a crystalline material. And let’s say that this is the view
of that material from the side. So this is one layer of the
crystal. This is another layer, another
layer, another layer, and so on. The distance between adjacent
layers of this crystal is called 𝑑, and that’s what we want to solve for. On this crystalline material, a
beam of electrons is incident. In fact, we’re told the beam is
normally incident, meaning there’s a 90-degree angle between the beam and the
layers. Our problem statement tells us that
the electrons in the beam have a wavelength. Indeed, they do have a wavelength
according to what is known as the de Broglie relationship. This relationship says that the
inverse of the wavelength of some object equals that object’s linear momentum
divided by Planck’s constant ℎ.
We can use this relationship as
well as the information given to solve for 𝑑, the distance between the layers of
our crystal. First, let’s record some given
information. First, the velocity of the incoming
electrons, we’ll call that velocity 𝑣, is 2.85 times 10 to the sixth meters per
second. Next, the mass of each individual
electron, we’ll call that mass 𝑚, is 9.11 times 10 to the negative 31st
kilograms. And lastly, we’re told that
Planck’s constant, which we’ll call ℎ, is 6.63 times 10 to the negative 34th
joule-seconds.
Now, even though we don’t yet know
specifically what 𝑑 is, we do have some information about it. We’re told that the diffraction
pattern of these electrons incident on the crystalline material is a single
spot. This happens, we’re told, when the
distance 𝑑 is one-half the wavelength of the electrons. So whatever value 𝑑 has, we know
that it is equal to the wavelength of the electrons, we’ll call that 𝜆, divided by
two.
Let’s now clear some space on the
screen and think about how we can use the de Broglie relationship to solve for
𝑑. Recall that we’re applying this
relationship to electrons. So 𝜆 is the wavelength of the
electrons and 𝑝 is their momentum. Classically speaking, the momentum
of an object is equal to its mass times its velocity. This is a nonrelativistic way of
writing momentum. The equation holds true whenever
the velocities of the objects involved are much less than the speed of light. That speed, we recall, is
approximately three times 10 to the eighth meters per second. What we want to do then is evaluate
the speed of our electrons 𝑣 and compare it to the speed of light 𝑐.
Note that the given speed of our
electrons is about three times 10 to the sixth meters per second. In other words, this velocity is
about one one hundredth of the speed of light. This is a safely nonrelativistic
speed, which means we can use the classical formulation of momentum in our de
Broglie relationship equation. We write then that one over the
wavelength of the electrons equals the electron mass times the electron velocity
divided by Planck’s constant. We recall now that it’s 𝑑 we want
to solve for. And note that two times 𝑑 is equal
to the electron wavelength 𝜆. Therefore, we can replace 𝜆 in our
equation with two times 𝑑. We have then this equation, and
we’ll rearrange it so that 𝑑 is the subject.
Let’s multiply both sides of the
equation by two times 𝑑 times ℎ divided by 𝑚 times 𝑣. When we do this, lots of
cancellation happens. First, on the left, two times 𝑑
cancels from numerator and denominator. And on the right, 𝑚 and 𝑣 and ℎ
all cancel. What we get is ℎ over 𝑚 times 𝑣
equals two 𝑑 or 𝑑 equals ℎ over two 𝑚𝑣. Note that we have a value for
Planck’s constant ℎ, the mass of the electron 𝑚, and the electron velocity 𝑣. When we substitute all these values
in, there is no need to convert any of our units. We can simply calculate this entire
fraction to solve for our distance 𝑑 in units of meters. Keeping three significant figures,
𝑑 is 1.28 times 10 to the negative 10th meters. That’s the separation distance
between adjacent layers in our crystalline lattice.
