Modern Physics - Lecture 12 - X-Rays and the Compton Effect

In this lecture we will learn about the following things. We will learn about the nature of a kind of radiation called x-rays. We'll learn a little bit about the production of x-rays and finally we'll look at the scattering of x-rays by matter and the implications for the nature of electromagnetic radiation. Now x-rays were discovered serendipitously in 1895 while Wilhelm Röntgen was experimenting with what are then known as cathode rays and which we would now simply know as electrons. He was using a device that would boil electrons off a metal using a very strong electric field and he observed some distance away from the apparatus that a special phosphorescent screen was glowing even though there should be no radiation from the experiment actually reaching the screen. And so he became obsessed with trying to understand this phenomenon and after careful experimentation he decided that he had isolated a new kind of radiation that was heretofore unknown and using the variable for an unknown quantity in math which is usually x he coined the term x-rays to describe these. Now one of the things that he observed during his experiments was that if he allowed the x-rays to pass through his hand it would cast a shadow on a screen behind the hand that showed only the bones of his hand and in fact this led to him attempting to make the first what we would now call medical x-ray in 1895. He used the hand of his then spouse Anna Ludwig and her hand famously is the first medical x-ray ever known to have been recorded in the history of science. You can see here the dark areas that look very much like the bones of the hand. The knuckles are up here, she's clearly wearing a ring or something around her finger here, and the tips of the fingers are up here, the thumb is off to the side. For public presentation Röntgen made a much nicer version of this picture using a different hand and a different experimental apparatus but essentially this is the birth of medical imaging as we think of it now, non-invasive imaging using radiation or something else to see inside the body. Now we now know that x-rays are a kind of electromagnetic radiation. They're a very short wavelength of light. You can't see them with your eyes but if you have the right instrumentation which Röntgen did when he serendipitously discovered them, you can induce a signal in something that can be seen with the eyes. They have wavelengths that range at their smallest between 0.01 nanometer all the way up to 10 nanometers. Now as Röntgen discovered, they easily penetrate common low-density materials. Think cardboard, skin, muscle. Most x-rays will pass through those undeflected, unstopped. Now if you use more dense material between you and the source of x-rays, then of course what he observed was that more of the x-rays are stopped. So the light regions here are places where x-rays easily made it through. The dark regions are places where many fewer x-rays penetrated through the hand in order to get to the imaging device on the other side, but in this case photographic film. So lead, bone, this is more dense than skin, muscle, paper, cardboard, and so it's more likely to stop or scatter x-rays. Now you can imagine that these are insanely useful not just for practical applications but for all kinds of interesting studies of the natural world, and they would themselves become a key object of study and ultimately would lead the way toward understanding more about the particle-like aspects of light's behaviors. Now let's talk about Arthur Holly Compton and x-ray scattering experiments. As I mentioned in the previous lecture, in the late 1800s and early 1900s there were not many notable physicists from the United States. Now that's as compared to the then European powerhouse of both education and research that was long and well established across the Atlantic Ocean. Now one of the physicists who became very well known in the early 1900s was Ohio-born Arthur Holly Compton. Interestingly, his Ph.D. thesis was in part on the reflection of x-rays. After his Ph.D. he received National Research Council support and was then free to travel and do research abroad, and he selected to conduct work at the then famous Cavendish Laboratory in Cambridge, England, and he did this in 1919. Now there he would experiment with very short wavelength light including x-rays and gamma rays, laying the groundwork for his eventual discovery of what is now known as the Compton Effect. He returned to the United States in the early 1920s and became faculty at Washington University in St. Louis, and it was there that he observed definitively and methodically now what we refer to as the Compton Effect, that x-ray quanta scattered by free electrons experience a lengthening of their wavelength after the scattery, and that this lengthening is a strong function of the angle at which the light is scattered. So how to explain this? This is a very particle-like picture of an x-ray striking a free electron, causing the electron to scatter, and itself being scattered. Now in classical electromagnetism, a wave would come in, it would start an electron oscillating. The electron would oscillate sympathetically, but this shouldn't result in a change in the wavelength of the radiation. Like waves on the surface of a pond, it'll make things on the surface start to bounce up and down, but the wave itself doesn't change wavelength when it scatters through these things. On the other hand, Compton could only explain this phenomenon by analyzing this scattering process from a more particle-like viewpoint, where the x-ray quanta have energy and momentum before and after the collision with the free electron, and that because the energy and momentum of the quantum is changed, the wavelength is changed. And he came up with a precise mathematical formula to relate all of these changes. So we can hypothesize, as Compton did based on Einstein's 1905 photoelectric effect work, which was itself based on Max Planck's blackbody spectrum work, that the x-ray incident on the electron, which I've labeled here I for the purposes of the coming notation, before scattering carries a total momentum that's given by E equals Pc for the incident momentum and energy. This can be related through Planck's relationship to the frequency of the radiation, so h Planck's constant times fi, the initial frequency of the radiation. And if we want to get wavelength into this to consider shifts in the wavelength of the scattered light, then we can convert this into hc over lambda I, where lambda I is the initial wavelength. Now this allows us to write the momentum of the incident x-ray as the initial momentum is Planck's constant divided by the wavelength. So E is most neatly equal to h times f, but P, the momentum of the quantum, is most neatly related to the wavelength by h divided by the wavelength lambda. The scattering process then occurs, and the final scattered light quantum carries a different momentum, P final equals h over lambda final. Now the initial electron state we could take as being at rest, and so it has no velocity in the initial state. But the final state involves an electron that's now been scattered at some angle phi that we'll write down later. Now it has a total speed u final, and thus it has a total momentum. Now I'm going to be careful here. I'm not going to assume that this is necessarily a slow-moving electron, and in fact in reality in the Compton scattering experiments, these electrons come out with a whopping great amount of momentum, putting them very close to the speed of light, so close that it is obviously safe to use the relativistic definition of momentum, that is the gamma factor that's a function of the speed of the ejected electron times the mass of the electron times its speed. Now to analyze this as a scattering process involving the collision of a x-ray with a stationary electron leading to a moving electron and a scattered light quantum, we need to only conserve total energy and then momentum in the x and y directions. So x is here clearly labeled as the horizontal direction, positive to the right, y is the vertical direction, and it would be positive upward vertically. So let's go ahead and do this. Let's start from conservation of momentum in the horizontal or x direction. This is a closed and isolated system, so the initial momentum of the system, that includes the x-ray and the stationary electron, must be equal to the final momentum of the system, that now involves the scattered light quantum and the moving electron. So if we substitute in with the equations for the initial and final momentum of the light quantum, and we put in the component of the velocity of the electron along the horizontal axis, we wind up with this equation, which, removing the zero because the electron in the initial state is not moving, simplifies to this equation here. Now let me comment on a few things. First of all, the initial momentum of the x-ray is entirely along the x-axis, but only a part of its final momentum lies along the x-axis, and so that's given by hc over lambda f, its total final momentum, times the cosine of the scattering angle theta. Now there's another angle in the problem, it's the angle between the horizontal and the electron that gets scattered, and that's denoted phi. And because of the picture up here, we're only considering the horizontal component of the electron's momentum, and that's given by gamma mu cosine phi. So, so far nothing exciting going on, it's just breaking down the kinematics of the x-ray and the final state light quantum and the scattered electron all along the x-axis, and that's about as far as we can go right now without knowing things like phi, the scattering angle of the electron. We need more equations, and so we're going to turn to conserving momentum in y. So let's go ahead and write down the vertical conservation of momentum for the same problem. I'll proceed through this relatively quickly. Again, the initial total momentum in the y direction must be equal to the final total momentum in the y direction. There is no initial momentum in the y direction, the x-ray is moving entirely along the x-axis, the unscattered electron has no velocity, so the initial state is all zeros, and the final state has two pieces, the positive vertical component of the scattered light quantum and the negative vertical component of the scattered electron. And so we can consolidate the zeros on the left-hand side, and we wind up with this equation here. Now we have sines instead of cosines for the two scattering angles in the problem. Now, we could use this to solve for phi, or at least sine of phi. Where it's already looking a bit nasty, we can already see that this is going to be a bit of a lift in algebra. Let's see if the conservation of total energy in the system holds any comfort for us in attempting to get at a singular equation that relates the initial wavelength, the final wavelength, and the scattering angle of the light quantum. Well, we're going to conserve total energy. Total initial energy must be equal to total final energy. We can plug in the total energy of the initial x-ray, hc over lambda I. Now, remember, the total energy of the unmoving electron is not zero. This is special relativity. Mass energy is internal energy, and is therefore just energy. So we have to put in the rest mass energy of the electron. The final light quantum has an energy hc over lambda f, and the final scattered electron has a total energy given by gamma times the mass of the electron times c squared. This involves both kinetic and internal mass energy. Now, we can then just rewrite this equation without the conservation of energy stuff on the left, and we arrive at this equation here relating initial and final energies. Nothing's really simplified. So there's not a lot of comfort here. It's going to be an algebraic lift, but these are the pieces that Compton would have worked with, and in fact did work with, in order to try to understand his scattering experiments. From his experiment, he would have known three things. The incident x-ray wavelength, lambda I, the scattered light wavelength, lambda f, and the angle at which light is scattered, theta. So the question is, of course, can we use algebra, possibly pages of it, in order to relate these things using this hypothesis of a particle-like scattering process between light quanta and an electron? And can we then make a prediction for the relationship between these three things? Well, the answer is yes, and I'm going to leave the lengthy algebra to the viewer or reader of all of this stuff here. But basically, we're going to eventually find by working through all of this what Compton found, and that is that the predicted relationship between the final and initial wavelength and the scattering angle is given by this very nice looking equation here. In fact, what we find out is that from Compton's analysis of this process, it suggests that the difference in wavelengths after and before the scatter will depend only on the scattering angle of the light and some constants of nature, h, the mass of the electron, and the speed of light. Compton ultimately confirmed that this was a correct description of these experiments by doing his own experiments and testing this idea. Now, there are some implications from the Compton effect, which is described in this formula. An undeflected x-ray, that is an x-ray that goes straight through the system with an angle theta equals zero, will experience no shift in wavelength. The cosine of zero is one, one minus one is zero, there is no difference between the initial and final wavelength of the x-ray. That doesn't lead to a big surprise, but more interesting perhaps is that if you have a completely deflected x-ray, one whose scattering angle is 180 degrees, or pi radians, that is a so-called backscatter, comes straight back at the source of the x-rays, it will experience the maximal possible shift in wavelength, and that corresponds from an energy perspective to the largest achievable kinetic energy for the electron. That's the most kinetic energy a scattered electron is ever going to get, is when you have a perfect backscatter of the x-rays as a result of losing energy to the electron and coming out at this 180 degree scattering angle. Now, Compton, in the course of doing his experiments, did observe scattered light at angles other than those expected from simply scattering off the electrons, and from this he determined that some of the x-rays were scattering not off of just electrons in the atoms, but entire atoms themselves. That is, you could rework the algebra that would lead to the Compton scattering formula, not by putting in the mass of a scattered electron, but by putting in the mass of an entire scattered atomic nucleus or atom. And if you do that, you'll find that scattering at the same angle leads to a much smaller wavelength shift, because the mass of an atom is much bigger than the mass of an electron, and that causes the wavelength shift to get much much much smaller at the same angle. But nonetheless, you will see scattered light with an entirely different set of wavelengths, albeit at a lower rate, at that same angle, when sometimes the x-rays scatter off of whole atoms and not just electrons. What this also implied was that for light with wavelengths or frequencies at the level of x-rays, which does span a large space of wavelength ranges, scattering of the light behaves more like scattering particles off of other particles, like bouncing tennis balls off of bowling balls, or something like that, rather than waves off of particles, where waves would cause sympathetic oscillations in the particles but wouldn't change the wavelength of the original wave. So this flies in the face again of the purely wave hypothesis of light. And it seems that under these conditions, a much better and more accurate description of the way that light behaves is as if it behaves as a large collection of quanta, more than as a collection of waves. Now, this all has deeper implications that stem from the Compton Effect. Let's take everything from the last few lectures together into one coherent picture. All together, the Blackbody Problem, the Photoelectric Effect, and Compton Scattering point toward a complex set of aspects of light behavior. Light isn't just a wave, and it's not just a unit or discrete thing, like a particle. Under some conditions, light behaves exactly according to classical Maxwell Equation Theory. Waves scattering off of or otherwise interacting with and causing oscillations in matter. That behavior was well established by the late 1800s. Electromagnetic waves really can behave like waves. But under different conditions, suddenly one can observe that light behaves more accurately according to a particle description, a quantum description, that light is discretized in some way, not continuous like a wave. So, in that case, it's better described as a collection of quanta, the photon, so many photons all acting together, and that can be thought of as particles interacting with the particles that themselves compose matter, electrons, whole atoms, and so forth. So, what ultimately resulted from all of this was that there are particle-like aspects of light's behavior that tend to correspond more often to when the wavelength of the light was very short, that is, very high frequency, whereas the wave-like aspects of light's behavior seem to manifest or correspond more when the wavelength is very long, that is, the radiation has very low frequency. Somewhere in that space of wavelengths and frequencies, between very long and very short, there's a transition between these sets of behavior, wave-like and particle-like. But what defines short and long? That's a very arbitrary distinction. Something that's hot to one person may actually be kind of chilly to another. Think about the way that offices are heated or air-conditioned. Some people find the temperature in a typical office setting perfectly fine and acceptable. Some people have to put a blanket over themselves to stay warm because they view it as chilly. Okay, but how do we define short and long to understand when the wave and when the particle-like phenomena are applicable? Well, it turns out that the answer has to do with the dimension, D, or size, or scale of the system with which the light interacts. So, the longness or shortness of wavelength, or the highness or lowness of frequency, is when compared to the size of the system with which the light interacts. If the wavelength of the light is much, much greater than the dimensions of the system, think long wavelengths that are far in excess of the size of atoms, for instance, then it turns out that wave-like behavior rules. The atom experiences light like a wave. If the wavelength is much, much smaller than the dimensions of the system, then the system experiences light more like being crashed into by a particle where all the momentum and energy is transferred at once. Particle-like behavior rules. Now, in the middle, as the wavelength becomes comparable to the size of the system, things get very complicated and you have to be extremely careful and have an accurate theory in order to actually predict what's going to happen in that case. So, there are some extreme cases. The wavelength is much smaller than the dimensions of the system. The wavelength is much longer than the dimensions of the system. Those are easy to handle when absolutely particle-like behavior or absolutely wave-like behavior manifests. In the middle, things get dicey, and in order to describe systems which have comparable sizes to the wavelength of light, for instance, you need the right theory. We don't quite have it yet. Now, let's talk about the sizes of things as a primer for what's to come in our thinking about the interactions of light and matter. To probe the scales of things with sizes larger than a virus, and we can see from this chart over here that a virus has a size scale that's roughly 100 nanometers. You can find that it's sufficient to use visible light. Bacteria have sizes of about 1 micron, 1,000 nanometers in size. Red blood cells, 10,000 nanometers or about 10 microns in size. Hares, about 100 microns or 100,000 nanometers in size. Ants are 10 to the 6th nanometers. Baseballs are about 10 to the 8th nanometers. We're in the realm of the macroscopic. Macroscopic here meaning larger than the wavelength of visible light. So, this helps us to understand a little bit about why it is that we didn't get ourselves into trouble with large-scale descriptions of motion and radiation, a la Newton's laws and Maxwell's equations, when we were dealing with things that had sizes that were much smaller than the wavelengths of light that we were using to interact with them. Looking at a bacterium or a red blood cell or a hair follicle with a microscope is straightforward because the wavelength of visible light is much smaller than all of those things, and so it simply scatters off of them and we can resolve the sizes of those structures quite easily. When you have a wavelength that's smaller than the structure you're looking at, you can resolve the features of that structure. But to probe viruses and DNA or hemoglobin or macromolecules, like glucose for instance, you need x-rays. You need to get down to sizes that are about the level of 1 to 5 to 10 nanometers or so. In those cases, you're going to need something like x-rays if you want to resolve the structure of glucose, hemoglobin, DNA, which are obviously essential to understanding modern biological functioning. So x-rays are your friend when you want to probe structures that are smaller than bacteria. X-rays will allow you to see, if you have the right instrumentation to reveal them to the eye, these sorts of distance scales. But if you want to probe atoms and molecules, you need to really push your x-rays. You need to go down to the shortest x-ray lengths, about 0.01 nanometer or so. The scale of atoms is at the level of 0.1 nanometers or 10 to the minus 10 meters. So x-rays can be comparable to or smaller than, but not by much, the size of an atom. And so the particle-like aspects of light begin to emerge naturally at this scale. It's no surprise that the behavior of x-rays, which is an electromagnetic radiation, became very particle-like when we started looking at them interacting with atomic systems like the atoms in metal and the electrons in those atoms. Those things turn out to have scales that are roughly comparable in size or a little bit bigger than the kinds of x-rays that were scattering off of them. And so that's when we got ourselves into trouble when it came to the theory of light and how it's supposed to behave when it interacts with matter. It's when the size of the light got to be smaller than the size of the thing that we were smashing the light into, and suddenly we needed a slightly different description of light in order to understand all that. Now just to tease things, if you wanted to probe the nucleus of an atom, there we're talking about sizes at the level of 10 to the negative 15 meter or one femtometer. And for that, x-rays are just too big. You're not going to resolve things of the size of a nucleus of an atom using x-rays. Instead, you need something with a wavelength that's really short, like gamma ray radiation, or even other things that, as we'll learn, turn out to have even smaller wavelengths than gamma rays. Let's revisit the phenomenon of the interference of light. We've looked at this in class, in the context of the Michelson-Morley experiment. What we saw in an in-class demonstration was that light that is forced to pass through a very narrow opening will diffract. You'll get a pattern on a some distance away from the slit that the light passes through that shows light and dark spots. The bright spots are where the waves have constructively interfered and their amplitudes have added up. The dark spots are where the waves have been out of phase with each other and destructively interfere. Black areas are places where the waves completely cancel each other out. In this computer simulation, imagine sending in light waves toward a barrier that has two slits in it. The light can diffract through either slit. The resulting wave fronts that come out on the far side of the barrier then interfere with each other. If we put a screen up here on the right side, we could imagine imaging this and seeing bright spots and dark spots and bright spots and dark spots and then fainter bright spots and so forth. The pattern can be controlled by changing the geometry of this setup. For instance, if I increase the separation between the slits and wait a few moments for the light pattern to catch up, we'll notice that as we take more data with the screen on the right, the number of bright fringes has increased. We now see that what was once faint on the outside is much brighter, but nonetheless we have bright, dark, bright, dark, bright, dark, and so forth. This is a wave behavior. How can we reconcile the particle nature of light and the wave nature of light in a phenomenon like this, two-slit light diffraction? Instead of imagining waves of light coming into a system with two slits in the barrier, let's instead set up a situation where we can fire, say, photons that correspond to green light at a barrier with two slits in it, one at a time. If we do this, we imagine sending in one photon. That photon has to go through either the barrier on the left or the barrier on the right. We don't know which barrier it's going to go through, but we can look at the screen on the other side to see where it lands. And slowly, one photon at a time, as we look at the observing screen on the far side, we see green dots. The green dots indicate where the photon that we fired ultimately wound up on the screen. One photon at a time, we're building up an image on the far side of the screen. Now, this is rather tedious. We'd like to see if a pattern emerges in all of this. So, I'm going to speed this up, and then when sufficient information has been received by the viewing screen on the other side, I'll comment on the pattern. Sufficient time has passed that we can begin to comment on the pattern we observe on the detector screen. There are places where photons have clearly clumped after passing through the two-slit process. There are places where we find few or no photons on the detecting screen. For instance, those darker regions flank the bright region in the middle. This is akin to the interference pattern that we saw when we were thinking about light as a wave traveling through this system and interfering with itself. Now, single photon by single photon, we're building up a similar intensity pattern on the screen on the far side. There are bright bands, dark bands, bright bands, dark bands, and so forth, the same alternating pattern of high intensity and low intensity that we saw from the wave behavior. Indeed, it seems that the wave behavior is recovered in the limit of a large number of photons passing through the system. This reconciles the wave and particle behavior aspects of light in a single experiment, and in fact, this famous Young's two-slit experiment is one of the many ways that one can reconcile and understand these dual aspects of the existence of the phenomenon we call light. In fact, what seems to be true for the single photon experiment is that we're unable to predict with certainty where any single photon will wind up striking the screen on the other side, but the probability that a photon will strike in the middle is much higher than the probability that it will strike just to the right or just to the left of center. And from that, we can begin to build an understanding that the probability of where a photon goes on the screen seems to be somehow related to the intensity, the amplitude squared, of the light wave description of nature. Now in this lecture, let's review what we have learned. We've learned about the nature of x-rays and we've seen a little bit about how to produce them by experimenting with cathode rays, electrons, smashing into a target. We then looked at the scattering of x-rays by matter and following Arthur Compton's explanation of his scattering experiments, have come to understand something about the nature of radiation with very short wavelengths. From this, we've seen the implications not only for the nature of electromagnetic radiation as having both particle-like and wave-like aspects under different conditions, but something about the conditions themselves that trigger these different aspects of the behavior to be observed. When the wavelength of the radiation is much smaller than the scale of the thing that it's scattering off of, then we see the particle-like aspects of light's behavior emerge. When the wavelength is much greater than the size of the thing off of which the radiation is scattering, then we see the wave-like aspects of the radiation emerge. And in between, there's a transition, a place where we lack a theory so far to actually understand how to calculate. These are the foundations for what will happen next, as we depart the comfortable world of radiation with its wave-like behavior and now its newly understood particle-like behavior, and turn our eye from radiation to matter itself.