Question Video: Determining Electron Velocity from Electron Diffraction Pattern Physics

A beam of electrons that have a velocity, 𝑣, passes through a crystal in which the atoms have an average separation of 𝑑 = 1.2 × 10⁻¹⁰ m, as shown in the diagram. A diffraction pattern of concentric rings is formed on a 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 = 2𝑑. Calculate 𝑣 in the case of maximum diffraction. Use a value of 9.11 × 10⁻³¹ kg for the mass of the electrons and use 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 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.

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