There's two different kinds of contrast, amplitude contrast and phase contrast. When we thought about amplitude contrast, we envisioned the incoming electrons as particles that were scattered by dense objects in the sample. Now because some of these electrons were scattered to high angle and some of them lost energy within the sample. Some of them would, were removed either by the aperture of the lens or by the energy filter and this is a simplified view of amplitude contrast by removing the total number of electrons arriving at the image plane. There is contrast built up between dense objects and less dense objects. Phase contrast is very different. To understand phase contrast, it's imperative to think about the incoming electrons as waves that are simultaneously scattered into all of the scattering directions and then these scattered rays undergo phase shifts relative to each other. And then they are recombined at the image plane and the interference of all these scattered rays together produces phase contrast, because that interference alters the probability of each particular electron being detected at any particular spot on the image plane. So what do I mean by thinking about the electron as a wave? Well, instead of thinking of the electron as a single particle with a condensed charge, instead we're going to think of the electron as an array. We'll think of it as a plain wave. Mathematically, we could keep track of it as an array of numbers. And each pixel within this plane wave, each pixel would have an amplitude and it will have a phase. Now in a plane wave, all the amplitudes across the array are equal. So let's just say that all the amplitudes are one. And let's say, at a particular position in space in this plane wave, all of the pixels have the same phase and let's say that phase is zero degrees. Then as the wave moves downwards through the electron microscope, we might draw it later. Okay.So this is the same plane wave. But after it's moved downward in the electron microscope some time later. Now, all of the pixels still have the same amplitude, but now their phases have advanced maybe we'll say to 90 degrees. So just as a wave passes through a position with time, the phase would change. So to this wave as it passes down the microscope, the phase of each pixel moves around and around a phase circle. So for instance, then the time later as this plane wave passes lower into the microscope. [SOUND] Now again, all the amplitudes of all the pixels might be one. But now, the phase might be a 180 degrees. And sometime later, the phases would advance to 270 degrees. Sometimes later, they would come back to 360 degrees and the cycle would start again. And so as this plain wave moves down through space, propagates through space, the amplitudes of all the pixels stay the same. But their phases move forward around and around, the face circle from 0 to 360 degrees and back. Okay. Now let's step back and think about instead of the plain wave traveling through a vacuum, what if the plane wave hit a sample and interacted with it? So I'm going to draw a little sample here and let's suppose that there was an object here in the sample that started to scatter the electron wave. Now, in this way, scattering can be thought of an analogy to this scattering would be like having a flat pond and taking a pebble and throwing it into the middle of the pond and as it hit the pond, you would see a little circular ripple expand away from the place that the pebble had hit. In somewhat similar manner, when a plane electron wave interacts with a scattering center like an atom. The atom causes the phase of that part of the wave to be slowed down. And as a result, there's a little phase ripple that emal, that emanates from that scattering center, so that later on in the plane wave, let's say, here, you would see a ripple in the phases. If you were to plot a surface of constant phase, it would have little ripples in it and also perturbs the amplitudes. At, and as this wave propagated further down the microscope, these ripples would expand and start to perturb the amplitude and phase throughout of greater portion of that plane length. Now, if there was another scattering center in that object say, like another atom, it would also induce phase ripples. And these ripples here in that plane wave right there, might be expanding from that position and they would start to interfere with the ripples from the first. Just like if you threw multiple pebbles in the lake at the same time, each would have these circular ripples that eventually interfere with each other. So later on down the column, you would have these phase ripples interfering with each other and so on if you had yet another scattering center here for instance, it would have it's own phase ripples and all of these would interfere together later on down the column. Now, one of the important principles that this analogy conveys is that just like a single flat lake, each electron feels each atom in the sample simultaneously. And the scattering that each produces interferes with the scattering from all the other perturbations of that wave. And so, each electron is by itself in the microscope. It's not that you have one electron in the microscope that's scattered and another scattered from a different part of the sample and these two electrons interfere. No, if you look at how many electrons are used to form a typical image and how fast their moving down the column, and how long the column is. It turns out, that on average, each electron is produced by the gun passes through the column, scatters from the sample and contributes to the image all by itself and then there's a long period of nothing before the next electron comes. And individually, fields the entire object is scattered by many scattering centers. That scattering is collected by the lenses and contributes to the image at the bottom. In fact, this calls to mind the famous double slit experiment that first proved that electrons have a wavelike nature. In the experiment,. Electrons were fired at a small slit and produce a particular pattern behind that slit. And then when two slits were introduced, the pattern was different. And the pattern evidenced that each electron as it came thru had to be feeling both slits simultaneously. It had to go thru both slits simultaneously and interfere with itself to produce the pattern that it did. Like manner an electron microscope is like a billion slit experiment. Each electron individually fills all the atoms with the sample and collects all of their scattering to produce an image. It's an, it's an amazing thing. Now, I told you that what the scattering centers did is produce a phase ripple. There's one context in which this is easier to understand, which is the scattering of an X-ray by the electrons in an atom. So, I'm going to draw an atom, it has a nucleus and it has an electron cloud surrounding it and will have a single electron in the cloud. And now, I'll draw an incoming X-ray. Now the X-ray can be understood as an oscillating electric field. So this is an electric field that is pointing in one direction and then the other direction and the other direction. And that electric field is oscillating strong to weak, to strong to weak and changing direction. And as it passes by one of the electrons in a sample, because electrons experience a force when they're in an electric field. One might imagine that this electron starts balancing up and down and up and down. A moving electron as it experiences this, this oscillating electric field going by it. As the photon fly's by it, you might imagine the electron is oscillating up and down. Now we know from physics that any charge that's oscillating, moving up and down and up and down is going to create an oscillating electric field around it that then propagates outwards. So you have an oscillating electric field in response to its own movement. And this is a way to think about scattering. The energy coming in as the alos, oscillating electric field of the photon is converted by the movement of the electron up and down and spread into all these directions, scattered. It's scattered and spread into these directions by the movement of the electron. Now this is an analogy, what happens when the incoming particle is actually an electron like in an electron microscope? We don't actually know what it is about the electron that is oscillating, if anything is oscillating. And so we don't know exactly what it means when that electron is scattered by some other electron in an atom of the sample. But we believe that it's in some similar fashion that the energy of this incident electron is scattered into all directions as it interacts with an electron of the sample. Now this idea leads us to the second way to understand how the scattering from all the different scattering centers interferes. Imagine that the incoming electron wave, now I'm going to draw it as a series of straight lines, moving in this direction. You can think of these lines ass wavefronts where the wave all has a uniform phase like where the phase is all zero, say. And as it moves forward again, here again the phases are all zero and here again the phases are all zero, phases are all zero. So, as this wave propagates forward, the phases are advancing around the phase circle. And then if it impinges upon the sample, it hits a large number of gathering centers. So these dots are supposed to represent individual scattering centers within the sample. And because the wavefront hits them all at the same time, they're all, they're scattering is induced in phase. And so then, they start producing phase ripples away from their centers. That one does and this one does. This one does as well, I was just skipping it to, for speed. Each induces phase ripples emanating from its center. Now, if you follow these ripples as they expand outwards and the ripples from each of these scattering centers. What you find is that there's some directions where they all constructively interfere. So for instance all of these phase ripples you'll see start to be in phase in a forward direction. And the further out you go, the flatter that line of ripples that are all in phase will appear. And so you can see that the incoming wave is essentially transmitted through it into another wave that's headed straight on forward. But in addition to that, there's other directions where the scattering occurs, you can see that this, these phase ripples are in this direction. And the phase ripple from this scattering center has a front moving in that direction as does this one moving in this direction. And so you can see that there's going to be wavefronts moving in this direction as well, where there's significant constructive interference. So we can think of this as a scattered ray moving at a scattering angle and we can call that theta. Some scattering angle and it moves forward and it turns out the angle there depends critically on the distance between these individual scattering centers as we'll see later. But of, of course, there's other directions as well. In this case, you know, the ripples from this scattering center and this ripple from that scattering center and this one from that scattering center. These all build up to create another set of another series of wavefronts that's moving in this direction. So this is yet, another scattered wave. So, in this way, we can think of scattering from a large number of centers as resulting in separate scattered beams each going in a particular direction. Now letâs take it one step more complicated and imagine that instead of our sample being a series of equidistant scattering centers, instead letâs imagine that our sample is a set of proteins. A very small crystal of proteins. Letâs imagine that these proteins have some alpha helices. Okay. And then they're connected by a loop and then another alpha helix and, and that's it. And in our sample, there's one of the proteins is here. Another of the proteins is there. Another of the proteins are there. And they're spaced at a particular distance. Now, as the incoming electron wave, impinges on this sample. There are a number of scattering centers around here in this protein and a number of scattering centers here. And a number of scattering centers here. And each one of these will cause phase ripples to emerge away from itself. And as these fav, phase ripples move forward, it turns out that there is a particular direction where the scattering will be constructive. So if we mark the forward direction like that, the forward direction, it turns out that there's going to be a particular angle, we'll call it theta here. And if we look at the path length of the scattering from here forward, and from here forward, we'll notice that the extra distance that this scattering travels compared to this one, the extra distance is this distance right here. Now, if that distance is exactly one wavelength, then the ripples emanating from this point will be in phase with the ripples emanating from this point as they move forward in this direction, characterized by the angle alpha. So the equation that describes that situation, if this is the scattering angle alpha, this is also the scattering angle alpha. And we see that the sine of alpha is equal to, if we characterize this distance between the scattering centers, this distance here as d, the sine of that angle is equal to one wavelength over d. And so, scattering will constructively interfere at the special angle whose sine is equal to a wavelength over the distance between the scattering centers. But in a protein, it's more complicated than just each protein being a single scattering center. In this case, we've imagined that the protein has an alpha helix here and an alpha helix there. And so you might think of each alpha helix as a scattering density. So one is there and one is here, and the scattering from these two densities, we'll notice that the distance between them now is smaller. And so if we again draw the horizontal and call this the angle theta for that scattering, here again we notice that this distance, in order for there to be constructive interference in this direction. This distance, which is the extra length that this ray travels beyond this ray, that extra length has to be an integral number of wavelengths. Or if it's one wavelength, then again the sine of alpha is going to be equal to lambda over this new distance. Let me call this an intermediate distance, okay, to distinguish it from the distance we drew in red here and this, let's call this alpha intermediate, alpha medium. So the point is that when these densities are closer together, the angle at which their scattering will constructively interfere is now a higher angle. It's this angle compared to when the dis, the distances we were considering were large distances like between the proteins. Then the scattering angle was small, that was alpha small, this is now alpha medium. And we can continue this line of thought into the next situation which is that in a protein, in addition to just alpha helices, we also have, say, particular atoms on side chains of amino acids that might be very close together. In this case, if we were to draw the horizontal. Now the, the angle at which the scattering will constructively interfere from this atom and this atom and this atom is now a very high angle, let's label that angle high, alpha h, for alpha high. And that's because the distance between this atom and this atom and this atom now, that distance, I'll label this d high, is very small. And so what we see, is that when scattering centers are far apart, like this protein compared to that protein compared to that protein, their scattering interster, interferes constructively at low angles. Again, I'll draw the horizontal, at low scattering angles. But when the densities are closer together, like this alpha helix compared to that alpha helix, their scattering constructively interferes at higher angles, namely that angle. And as distances are even smaller, like between this atom, that atom, and that atom, now they're scattering their ripples, the phase ripples they induce constructively interfere at even higher angle. Now if we were to think about this protein crystal and the density of that crystal and all the locations, we're going to try to decompose it into a series of sine waves that, when added up, would reconstruct the same density pattern. In other words, if we were to try to calculate a Fourier transform of this density pattern here, it would include a number of sine waves. And so what I'm going to draw now, is that series, is several example sine waves that would be required to represent this density. So on this axis I'm going to label this rho, for density, and this will be a spatial coordinate representing position along this, along this line. And first of all, we would need a slowly varying wave to represent the fact that we basically have one protein here, another protein here, and another protein here. So this density wave would rise to a maximum here, fall to a minimum, another maximum, a minimum, another maximum and so on, to represent that we have protein here and a gap between proteins here. Now, but in addition to that, we would need another sine wave to represent the fact that here we have an alpha helix, and an alpha helix, and they're close together, but. So we're going to have to have a more rapidly oscillating sine wave that would look something like this. As it moved across that variable, representing that we had an alpha helix in this position and here's a gap, and another alpha helix here. And more gaps, and then another alpha helix here, et cetera. Now, in addition to these waves, we would need, need yet another wave here, to represent the fact that we have one atom here, another atom here and another atom here. And this will need to be a verily rapidly oscillating wave, oscillating that rapidly as it goes through the protein, to represent the exact position of an atom here, here and here, as we move across. And so this, these are the the sine components of this density function. Now importantly, what you can see from this diagram is that these scattered waves. For instance the, the wave's scattered in this low angle direction, the red waves, these waves carry with them the information about how much of this large slowly oscillating red wave is present in the density function. In other words, the more spacings in the sample, the more scattering centers that are this far apart in the sample, the more scattering we will observe in this particular direction at this scattering angle. And it represents how much of this sine wave would be needed to represent the density function. Likewise, the amount of scattering in this direction reveals how many scattering centers in the object are at this distance apart. So because we have a bunch of them in these two helices, and a bunch of them in these two helices, and a bunch of them in these two helices, we'll have strong scattering in this direction. And so the amount of scattering tells us how many of those distances are present within the sample. And finally the strength of the scattering, the amount of the scattering in this direction reveals how many scattering centers were just this far apart within the sample. So wherever they occur in the sample, if they're just that distance apart, they will lead to constructive interference in this direction. And so as we measure the amount of scattering this direction, we measure the number of scattering centers that were that far apart, in the original sample. Therefore the physical process of scattering reveals the amount of each Fourier component that was originally present in the sample, and scattering physically is like a Fourier transform in that it physically reveals the amount of each Fourier component that was present in the sample. And it spreads each component out into a different scattering angle. At low scattering angle, we see the information about low special frequencies within the sample. At intermediate scattering angles, is the information for intermediate frequencies. And finally, high scattering angle reveals the amount of high frequency detail present in that sample. Now, in x-ray crystallography, we put a detector here, very far away from the protein crystal and because the distance between the crystal and the detector is very long compared to all the distances that I'm showing here in the picture. The scattering in each particular ray, angle appears as one spot here on the detector. And so this scattering appears as one spot. And this scattering appears as one spot over here. And this scattering would appear as another spot way off the top of the screen. So this for example, is an actual diffraction pattern produced by x-rays hitting a protein. We call the pattern of scattering that is produced a diffraction pattern. And so the x-ray beam went through the protein crystal and it was recorded on film. And you can see a number of discrete spots here on the film. Now this should remind you of something that we've seen before, namely we looked at what would the Fourier transform of a single sine wave, 2D sine wave look like. And when we plotted this Fourier transform, it was along spatial frequencies x and y, spatial frequency in x and spatial frequency in y. And we label those with the Miller indices h and k. And so an image that had a single Fourier component, it's power spectrum was just two spots on, on the h axis, representing the amplitude of that sine wave in the image. In a similar fashion each of the spots in an x ray crystal diffraction pattern represents one sine wave that was present in the original crystal. And from a diffraction pattern like this we measure the amplitude of the Fourier component that it represents. Each spot represents a single Fourier component. In the case of a protein crystal it's a single 3D sinewave. Each is identified by a particular set of Miller indices h, k and l. Each has an amplitude and a phase. The amplitude can be measured directly here from the intensity, and finally you would have to know both the amplitude and the phase of each of these Fourier components to be able to do an inverse Fourier transform and recalculate the density of the protein crystal itself.
