Question Video: Calculating Layer Separation Distance from Electron Diffraction Pattern | Nagwa Question Video: Calculating Layer Separation Distance from Electron Diffraction Pattern | Nagwa

# Question Video: Calculating Layer Separation Distance from Electron Diffraction Pattern Physics • Third Year of Secondary School

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A beam of electrons that have a velocity of 2.85 × 10⁶ m/s 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 × 10⁻³¹ kg for the mass of the electrons and a value of 6.63 × 10⁻³⁴ J ⋅ s for the Planck constant.

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### 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.

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