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Lesson Video: Electron Diffraction and Microscopy Physics

In this video, we will learn how to describe electron beam diffraction, how it is used in electron microscopy, and how other forms of electron microscopy compare to it.


Video Transcript

In this lesson, we’re going to learn about some of the physics underlying transmission electron microscopes, as well as a little bit about their construction and relationship to other kinds of electron microscopes.

Let’s start by talking about one of the most important physical phenomena at play in a transmission electron microscope, which is electron diffraction. Wavelength is one of the quantities that determines how something will diffract. Electrons are particles, so thinking about them as having wavelength is very counterintuitive. Before we resolve this seeming contradiction, let’s recall that diffraction is observed when a wave passes through two or more slits or openings between barriers, which causes the wave to start interfering with itself both constructively and destructively.

If we place a screen reasonably far away from the barrier with the slits and then observe the intensity of the wave at each point on the screen, we will see a characteristic pattern known as a diffraction pattern. If we plot this diffraction pattern on a graph, whose horizontal axis is positioned along the screen and whose vertical axis is intensity of the wave, we get a symmetric pattern with a series of maxima and minima. Let’s make a note of a few generally true facts about diffraction patterns.

First, as we can see from this picture, the middle of the screen corresponds to the middle of the pattern, where the most intense maximum is located. This is generally true for diffraction patterns. For the second fact, let’s recall that the distance between successive crests of a wave is called the wavelength and is given the symbol 𝜆. Let’s also call the width of the slits in the barrier 𝑑. It is then the case returning to our pattern that the distance between successive maxima or successive minima is a function of the ratio of the wavelength 𝜆 to the slit width 𝑑.

Lastly, the strongest diffraction, the clearest patterns, will be observed when the wavelength is approximately the same size as the barrier width. All of this is well and good. But diffraction is a wave phenomenon, and we’re interested in electrons, which are particles. This is where one of the fundamental observations that led to the development of quantum mechanics comes in. The theory of quantum mechanics grew out of the observation that waves can exhibit particle properties and particles can exhibit wave properties. One of the earliest quantitative relationships between these quantities was given by the French scientist Louis de Broglie in the early 20th century.

At the time, it was known that one could connect the wavelength of a wave to its momentum by the relationship 𝜆, the wavelength, is equal to ℎ, the Planck’s constant, divided by 𝑝, the momentum of the wave. De Broglie suggested that the same formula could be used in the other direction. Anything which had a momentum, including particles, would have an associated wavelength given by this relationship. It is for this reason that we often call the wavelength associated with a particle the de Broglie wavelength. Thanks to the de Broglie relationship, our discussion about diffraction is also relevant when it comes to particles like electrons. This will be true as long as the wavelength that we use in describing our diffraction phenomenon is the de Broglie wavelength of the particle.

Okay, so, electrons can have wave-like properties and they can even undergo diffraction. Let’s see if we can use this diffraction to figure out other things that we might like to know. Remember that the distance between the maxima in a diffraction pattern depends on the ratio of the wavelength of the wave to the width of the slits in the barrier. It’s pretty easy to measure a diffraction pattern. So, if we know the distance between the maxima of the pattern and the wavelength of the wave producing the pattern, we could work backwards to figure out 𝑑, the width of the slits.

Of course, to make the best possible measurement, we would want the strongest, clearest pattern, which will occur when the wavelength is comparable to the slit width. So, the distances that we would best be able to measure are those that are similar to the de Broglie wavelengths for electrons. The momentum of a particle is given by its mass times its velocity. So, the de Broglie wavelength of the electron is inversely proportional to its speed. And so, the limit to how small we can make this wavelength is really just how large we can make the speed. The speed of an electron is related to its kinetic energy. And it turns out that if the kinetic energy of an electron is around one electron volt, the de Broglie wavelength of that electron will be about 1.2 nanometers.

And how hard is it to give an electron a kinetic energy of one electron volt? Well, one electron volt is the kinetic energy that an electron would have if it was accelerated from rest through a potential difference of one volt. One volt is less than the voltage of a small battery. So, it’s quite easy to give electrons a kinetic energy of one electron volt or more and, thus, create electrons with a wavelength of 1.2 nanometers or less. So, electron diffraction would be best for determining the width of slits, whose size is some fraction of a nanometer.

Length scales that are comparable to fractions of a nanometer are typical for the sizes of atoms and the distances separating atoms in crystals. In our diffraction drawing, if we replace our series of slits with a crystal, we can see that the regular spacing between the atoms forms a series of slits just like we had before. We would, therefore, expect to see the same kind of diffraction pattern on the screen, where the intensity at a point on the screen is the number of electrons that hit that point per unit time. We could then go ahead and use the spacing between the maxima of this pattern, along with our known de Broglie wavelength for the electrons, to determine the spacing between the atoms in the crystal lattice. So, we see that we can use electron diffraction to help determine crystal structure.

Of course, we might also wonder why bother using electrons? Why not just use a light with a short enough wavelength? As we saw before, it’s pretty easy to create electrons with a wavelength that’s already quite close to what we need. However, light with wavelengths less than around 200 nanometers get increasingly difficult to produce, detect, and manipulate. And 200 nanometers is still more than 100 times longer than the wavelength of electrons at even relatively small kinetic energies. So, the reason to prefer electrons to light for determining crystal structure is that it is far easier to produce, manipulate, and detect electrons of appropriate wavelengths than it is to do the same for light of similar wavelengths.

Before we discuss what tools we might use in performing an electron diffraction experiment, let’s describe how what we would observe on our screen relates to the particle nature of the electrons. As the electrons travel towards the crystal, diffract through the crystal, and then move towards the screen, they’re exhibiting wave-like properties, which is why they can diffract. However, at the screen itself, when we observe an electron, we only ever observe a single particle at a single point in space. So when the electrons hit the screen, they’re behaving as particles.

The connection between this particle nature of observing individual electrons at the screen and the wave nature of the electrons diffracting through the crystal will come if we count the total electrons at each point on the screen over a given interval of time. Although the location of each individual electron will appear random, the electrons will cluster at certain areas of the screen, while leaving other areas of the screen relatively empty.

If we make a graph showing the number of electrons that hit each position along the screen, we would get exactly the graph that we got for diffraction of a wave through a series of slits. Except this time on the vertical axis, the intensity means the number of electrons that hit at a particular point. So even though our measurements will show each individual electron as a single particle, on the whole, combining many individual measurements, we’ll also be able to see the wave properties of these electrons.

Let’s now talk about the transmission electron microscope, which actually uses electron diffraction to determine crystal structure. We’ll begin by reviewing transmission light microscopy to get an idea of what kind of components we would expect to find in a transmission electron microscope. In a transmission light microscope, we would have a source of light, a sample that we were looking at, and an image collector where we would see the image of our sample. The image collector could be a screen or a computer or even our eyes. This setup is called a transmission optical microscope. Because a beam of light leaves the source, travels to the sample. It then passes or transmits through the sample before continuing on to the image collector for viewing.

There is one other key component of all microscopes that actually allows them to perform their function of magnifying the sample. Lenses help shape and focus the beam of light, either between the source and the sample or between the sample and the image collector. The specific way the lenses shape the beam will depend on a particular application, but the general idea is the same. Lenses between the source and the sample help shape the beam from the source so that it hits the sample in the desired way. For example, the source may emit light traveling in several different directions, and the lens can redirect this light so that it all hits the sample traveling in the same direction.

Similarly, lenses between the sample and the image collector redirect the light so that it hits the image collector in the desired way. For example, part of the sample may scatter light in many different directions, and the lens can redirect this light so that it all falls onto the same point in the image collector. By analogy to the optical setup, we can now describe the setup of a TEM, a transmission electron microscope. We’ll have a source that will emit and accelerate a beam of electrons until they have enough kinetic energy to have the desired de Broglie wavelength. Such a source is often called an electron gun.

We’ll have a sample which in order to transmit electrons must be quite thin, usually around 100 nanometers thick or less, and we’ll have a collector. However, unlike the image collector in the optical setup that collects an image of the sample, the collector in the TEM will collect a diffraction pattern from the electrons diffracting through the sample. Like the optical setup, a beam of electrons will leave the source, travel to the sample, transmit through the sample, and then travel to the collector where it will be observed. To reiterate, what we will observe at the collector is the diffraction pattern of the beam passing through the sample. Finally, we’ll need lenses to help shape and focus the electron beam. However, the lenses that work for light won’t work for electrons, so we need some other way to shape and focus this beam.

An electron beam is made up of many electrons, each traveling with its own trajectory. The trajectories determine the shape of the beam. For example, a cylindrical beam consists of electrons that are all traveling more or less in the same direction. On the other hand, if the electrons all had angled trajectories, they would create a beam that converges to a single point. So, in order to change the shape of the beam, we need to be able to change the trajectories of electrons. In order to change the trajectory of a particle, we need to apply a force to that particle.

Since electrons are charged particles, we can apply a force by either applying an electric field or a magnetic field to the region of space where the electrons are located. If an electric field is applied, the electrons will feel an electric force given by 𝑞, the charge on the electron, times the electric field. And since the charge on the electron is negative, this force and, hence, the direction of deflection will be opposite the direction of the electric field. If the electrons are moving and a magnetic field is applied, the electrons will feel a magnetic force. The magnetic force is given by 𝑞 times the electron’s velocity vector cross the magnetic field vector.

Note that the electric force, magnetic force, and electric field are also vector quantities. The reason we’ve specifically put half-arrows on the velocity vector and magnetic field vector is just to highlight the vector nature of the cross product. In this case, the electrons will be deflected in a direction that is perpendicular to both their velocity and the magnetic field. Specifically, in the direction opposite to the vector 𝐕 cross 𝐁, the directions will be opposite the electric field and opposite 𝐕 cross 𝐁 because the charge on the electron is negative.

Anyway, as we can see, to make a lens for our electron microscope, we simply need to create an electric field or a magnetic field in a particular region of space that as the electron beam passes through will shape the beam as we desire. A lens that creates an electric field to shape the beam is called an electrostatic lens, and a lens that creates a magnetic field to shape the beam is called a magnetic lens. This latter type of lens is the most commonly used lens in electron microscopes.

Let’s now see an example of how a magnetic lens can change the shape of an electron beam.

The two rectangles represent the walls of our magnetic lens. The field inside is represented by a series of circles. Where the magnetic field points into the page, circles have x’s. And where the magnetic field points out of the page, the circles have dots. The size of the circles represents the magnitude of the magnetic field, where larger circles represent larger magnitudes and smaller circles represent smaller magnitudes. The dots in the middle represent that the magnetic field there has zero magnitude.

Let’s now imagine that a cylindrical electron beam enters this lens. We’ll represent the beam by the trajectories of several electrons, all pointing in the same direction. As the electrons enter the lens, they will begin to deflect. From the charge of the electrons and the right-hand rule for cross products, we’ll see that electrons on the right of the lens will deflect towards the left. And electrons on the left of the lens will deflect towards the right. Electrons passing through the middle portion of the lens won’t deflect at all because the magnetic field is zero in the middle of the lens. While the electrons passing through the outer portions of the lens will deflect the most because the magnetic field is largest at the outer portions of the lens.

Once the electrons pass through the lens, they will stop deflecting and travel straight in the direction they were traveling when they left the lens. As we can see, this will result in all of the electrons focusing to a single point before diverging again. In other words, this lens is a converging lens. If we swap the direction of the magnetic fields in this lens, the direction of the deflections would also swap, and this will become a diverging lens. In fact, the magnetic lens we just described is actually a cross section of a real magnetic lens that’s used in electron microscopes, called a quadruple. Alright, so, we can indeed make lenses for transmission electron microscopes.

Let’s now contrast TEM to two other kinds of electron microscopy. The first will be scanning electron microscopy, often abbreviated SEM. Like TEM, SEM uses accelerated electrons because their short de Broglie wavelength allows for the resolution of nanometer-sized features. Like TEM, SEM shoots a beam of electrons at a sample. Unlike TEM, the SEM collects the electron that’s scattered back rather than pass through the sample. As a consequence of this, the thickness limitations of TEM samples doesn’t apply to SEM samples. Similar to light microscopy, the SEM forms an image of the sample surface. This is distinct from the TEM, which forms a diffraction pattern based on the sample’s interior structure.

The second technique that we’ll contrast to TEM is scanning tunneling microscopy, also known as STM. Unlike SEM and TEM, STM does not beam electrons at a sample. Rather, the scanning tunneling microscope collects electrons that are escaping from a sample surface by the process known as quantum tunneling. Also, unlike SEM and TEM, the resolution of an STM does not depend on the electron’s wavelength. Like SEM, STM forms an image of the sample surface. However, unlike SEM and TEM, the only kinds of samples that can be effectively imaged are those that are conducting.

So, SEM and STM both form a direct image of a sample’s surface. TEM, on the other hand, forms a diffraction pattern that reveals information about the internal structure of the sample. And neither SEM nor STM give appreciable information about the internal structure of a sample.

Let’s review some of the key points that we’ve covered in this lesson. We saw that even though electrons are particles, the principles of quantum mechanics teach us that they still have wave-like properties, like wavelength, given by the de Broglie relationship wavelength is equal to Planck’s constant divided by momentum. Furthermore, we saw that this tells us that for electrons with a kinetic energy of a little bit more than one electron volt, their wavelength will be just about one nanometer.

Because wavelengths of a nanometer or less are similar in size to the spacing between the atoms in a crystal, electron diffraction is a very useful technique for helping to determine crystal structure. Electrons are preferable for this task because they’re much easier to work with than light of similar wavelengths.

We saw that we could actually build a piece of equipment to perform such diffraction experiments, called a transmission electron microscope. During a measurement, a beam of electrons is shot through a sample and onto a collector. The sample has to be less than about 100 nanometers thick to make sure the electrons can pass through it. The electrons are emitted from a source called an electron gun. There, they’re accelerated until they have the speed corresponding to the desired de Broglie wavelength. The beam is observed at the collector, where we’ll see a diffraction pattern caused by the electrons diffracting through the sample. It is this diffraction pattern that helps us determine the crystal structure.

Finally, the beam is shaped and focused so that the experiment can proceed with a series of lenses. Since electrons are charged particles, we can either use the electric force as provided by an electrostatic lens or the magnetic force as provided by a magnetic lens to shape the beam. By choosing the fields appropriately, we can make converging, diverging, and many other kinds of electron lenses.

Finally, we noted a key difference between transmission electron microscopy, often abbreviated TEM, and scanning electron microscopy and scanning tunneling microscopy, often abbreviated SEM and STM, respectively. TEM can provide information about the internal structure of a sample, while SEM and STM are limited to providing information about a sample’s surface.

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