 In this lecture we will learn the following things. We'll learn about the structure of the atom, as it was known in the late 1800s and very early 1900s. We'll learn about how matter itself can have wave aspects to its behavior. We'll learn about Louis de Broglie's experimentally verified conjectures about the wave properties of matter. And we'll learn about how to conduct experiments that reveal the wave aspects of matter's behavior. To review, let's take a look at the things that describe the wave aspects and the particle aspects of electromagnetic radiation. Recall that the wave description of light was really the first and most formally developed part of the description of its behavior set. And these are included in Maxwell's equations. Now, they describe a spatially and temporally distributed phenomenon. You can boil the wave equations in Maxwell's equations down to this set of equations describing space and time variations in electric and magnetic fields. And these variations propagate at the speed of light in the material under configuration for simplicity we can assume empty space or simply the vacuum. And in that case, the solutions in empty space are the famous electromagnetic wave solutions and oscillating electric and magnetic disturbance that travels at the speed of light perpendicular to the variations in electric and magnetic components. And then, of course, there's an energy per unit area of an electromagnetic wave in empty space. There's no one place where the energy is concentrated. There's more in some place and less in others. And one can think about the energy density or the energy per unit area of a traveling electromagnetic wave phenomenon. Now, the particle description of light, which emerged from evidence based on the black body radiation spectrum and the photoelectric effect, these are descriptions of something that has definite energy and definite momentum at a definite location in space and time. That's what a particle is. It's a localized phenomenon at a very specific place in space time, whereas a wave is a spread out and distributed phenomenon that isn't only in just one place in space time. Now, from Albert Einstein's work on the photoelectric effect which built upon Max Planck's work with the black body spectrum, we have sort of a combined description of the particle-like aspects of light's behavior set. So, for instance, in special relativity, we have massless phenomena whose energy and momentum are related by the speed of light, E equals P times C. But unfortunately, in special relativity, we couldn't glimpse where the energy or the momentum of light came from. We got nonsense answers from pure special relativity. However, Max Planck's work with the black body spectrum revealed another relationship that the energy of a quantum of light, a photon, is related to the frequency of the light, E equals Hf, and the constant of proportionality is Planck's famous constant. We can plug in the relationship between wavelength and frequency for a light wave, C equals F lambda, and we can get a relationship, for instance, between energy and frequency, energy and wavelength, and more interestingly, between momentum and wavelength, which we get by combining the energy and momentum relationships between special relativity and the black body spectrum. So, we find that for a quantum of light, a localized packet-like unit of light, energy is related to the wave properties of that same phenomenon by E equals Hf, and momentum is related to the wave properties by P equals H over lambda. Let's model a long wavelength interaction and a short wavelength interaction using a beaker sitting in a tank of water. This identical beaker is going to be smashed into by a wave under two different conditions. On the left, a long wavelength disturbance that's much larger than the size of the beaker, and on the right, a short wavelength disturbance whose size is comparable to that of the beaker. This will illustrate the difference between wave-like phenomena and particle-like phenomena. The motion on the left, as the wave develops in the tank, will be more gradual, with the beaker gently falling and then rising, whereas on the right, when the short wave crashes into the beaker, it's almost as if it's been struck by something small and fast moving. Look how violent the collision on the right is compared to the one on the left. We might argue the beaker on the right has been struck by something more particle-like, whereas the beaker on the left, which is bobbing around, has been struck by something more wave-like. This helps to illustrate why a wavelength phenomena short compared to the target will exhibit particle-like behavior, whereas a wavelength long compared to the target will exhibit wave-like behavior. But what if? What's so special about electromagnetic radiation? Why does electromagnetic radiation get to have all the fun of having particle-like and wave-like aspects to its behavior set? What if matter, electrons, protons, neutrons, whole atoms also could exhibit wave-like behaviors? They had been experienced primarily as particulate objects, definite things with locations in space and time. But maybe it was just because nobody had observed the wave aspects of the behavior up to a certain point, such as in the early 1900s. There were hints, of course, that something funny was going on with matter at the scale of the atomic size. So, for instance, it had been long known, since certainly the work of Anders Angstrom in the early part of the 1800s, that the emission spectra of elements like hydrogen gas or helium gas, that they emitted only certain colors. White light comes in a full rainbow. And if you stare at white light closely enough, for instance, from the sun, you'll see that there are missing lines of color in its spectrum, but they're hard to detect. If you take a pure gas and you excite it so that it emits light, it's much easier to see that it's only allowed to emit certain wavelengths, certain colors of light. Here I show you the hydrogen emission spectrum revealed using a optical disc, a DVD or a CD-ROM. The many scattering surfaces for light on the surface of the disc will spread any light that strikes it out like a prism into a rainbow. But we see an incomplete rainbow from hydrogen. There's a bright red line, there's a bright blue-green or cyan line, there's a blue line, and then there's a fainter purple line, which you can actually see part of down here reflected in the disc. Now, this is a more classical way of laying out the emission spectrum of a gas like hydrogen, flattening it all out on a plane. Here's that red line, the blue-green line, it's very hard to see the faint dark blue line, and then there's a violet or purple line down here. These are the long wavelength emissions, these are the short wavelength emissions, but there are big gaps in between these things. And no two elements have the same spectral fingerprints, excite helium, excite neon, excite argon, and you'll get a very different pattern of colored lines out of each of those. Why? Why are atoms only allowed to emit certain kinds of light? That was a source of curiosity in the 1800s that could not be resolved. Whole mathematical patterns were observed out of the relationships in wavelength or frequency between these colored lines, but nobody could make sense of why these relationships existed and where they came from. Now, to understand what's going on with matter at size scales like that of the atom, it's very valuable to dig back a little bit into the history of the discovery and description of the atom as a real phenomenon in nature. Now, of course, thousands of years ago, philosophers and mathematicians and perhaps what would now be considered proto-scientists and engineers thought deeply about matter and they argued endlessly about whether it was continuously distributed, made of only a finite number of substances or atomic in nature that is coming in small units that could be built up into the structures we experience in nature. But that was a lot of argument without a lot of evidence and our modern understanding of how to understand the natural world and the scientific method reflects the reality that speculation is fine, but the final arbiter is observation of nature and the testing of your claims. The discovery of atoms as a real feature of nature or at least a potentially real feature of nature goes back to the early 1800s when chemist John Dalton discerned that not only elements have weight and that the weights are specific to each element in proportion to, for instance, the weight of hydrogen, but also that when you react one element with another element, you'll only get products from the reaction that completely use up the reactants if you have the right proportions of reactants. For instance, you might try reacting two things one to one, but have an incomplete reaction. React them in a ratio of two to one and you completely eliminate all of the original reactants that went into the process. That was something that Dalton characterized and it was a strong hint that the elements come in units and that those units have rules of combination that only allow certain proportions of them to completely react and disappear into other final products. Now it wouldn't be until 1897, although speculation had preceded in the decades before this work, that Joseph John or J.J. Thompson would reveal the first component of what would come to be known as atoms. Atoms themselves were not completely firmly established as the correct description of nature in 1897, but Thompson found out by experimenting on a kind of radiation known as cathode rays in his day that they actually possess of mass, but they possess of a unit of mass that is about a thousand times less than that of hydrogen. Now this would imply that either there's a lighter element than hydrogen or perhaps one had ripped something out of hydrogen and isolated it in the first place to be studied. He observes that these cathode rays with their very tiny masses also possess of electric charge and can be made to, for instance, accelerate in electric fields or bend in magnetic fields. In 1905, based on the idea that this electron, which composes the cathode rays, which is the identity of the cathode rays, is a piece of what we call atoms, he proposed a model of the atom. Imagine a central large positive charge with negative charges embedded in it, and this was known as the plum pudding model because it looked very much like a British dessert known as a plum pudding where you have a whole bunch of raisins or other fruits embedded in sort of a uniform distribution of dough which is cooked up into a dessert. So imagine that the raisins are the negative charges and the positive charge is the dough and the negative charges are spread throughout the dough. This was Thomson's model of the atom. Now that may sound ludicrous and cartoonish, but the beauty of science is that that's a conjecture that can be tested. For instance, you might imagine trying to do experiments that verify whether or not the negative charge and the positive charge are uniformly spread out in something of the volume of an atom. That's an experiment that couldn't necessarily be conducted at that moment in 1905, but it was certainly possible shortly thereafter. Now cherry picking my way through the story, I want to focus for a moment on Marie and Pierre Curie. Now they, among many other people, came to understand that unstable elements or radioactive elements that emit radiation when they decay away results in a new kind of radiation that had not yet up till that point been understood. Now they experimented on these radiations and it was finally Ernest Rutherford who classified them into the modern way that we usually talk about emission of radiation from unstable atomic nuclei. And those three classes of radiation are alpha, beta, and gamma for the first three letters of the Greek alphabet. Alpha radiation would eventually be revealed to be a whole hydrogen nucleus entirely ejected from a very heavy nucleus of a very heavy atom. So this would be two protons and two neutrons bound together in a very stable little unit and it can be spat out of an unstable nucleus that spontaneously radioactively decays. Now alpha radiation is highly electrically charged. It has plus two units of the elementary charge because of its two protons. And that means that it can't penetrate very far into material, but it can get into material and it can dump a lot of energy along the way. As I mentioned, Ernest Rutherford came up with this classification scheme. He was another physicist who is considered to be one of the greatest experimentalists, if not of his day, perhaps even of all time. Working in conjunction with the physicist Hans Geiger and Ernest Marsden, he scattered alpha radiation off metallic targets and he found out by looking at the scattering process that the plum pudding model of J.J. Thompson did not describe what happened when you scattered alpha particles off of atomic nuclei. The Thompson model would have postulated that because all the charges very spatially spread out, the probability of striking any of the positive charge or any of the negative charge is extremely small. And so for the most part you'd expect to find your alpha radiation traveling through the atom lightly scattered but mostly coming out on the other side of the target. But when Rutherford asked Geiger and Marsden to look at what's called back-scattered alpha particles, that is, look for alpha particles that strike the metallic target and then reflect almost exactly back at the original emitter of the alpha radiation, they were surprised to find out that there are a significant number of alpha particles that bounce back off of the metal target as if they're striking a huge target of positive charge concentrated somewhere in the center of every atom. And this in fact was a picture that Rutherford used to build his own model of the atom modifying J.J. Thompson's model and concentrating all the positive charge in each atom at the center of the atom. This forms the first sort of planetary model of the atom as electrons orbiting a central tightly packed nucleus with a huge positive charge of course depending on the element in question. But it was this picture that adequately described the back-scattering process with its higher rate than expected from the Thompson model observed by Rutherford, Geiger and Marsden. This is now known as the Rutherford model of the atom and it would be further modified as more experiments were conducted on this system. Now how do we know the sizes of atoms? Well skipping ahead a little bit in the story of the atom you can look at the scattering of x-rays. For instance we looked at Compton scattering in a previous lecture but imagine scattering x-rays with slightly longer wavelengths than we would have been talking about when talking about Compton scattering. Here the x-ray is it turns out comparable in size to the atoms off of which it's scattering you know with wavelengths of about 0.1 nanometer or so. Smashing these x-rays into crystalline solids like table salt sodium chloride it was observed that specific patterns will appear in the scattered x-rays. So for instance this image on the right is the very first x-ray diffractogram made by Max von Lau Paul Nipping and Walter Friedrich in 1912. Now not long after Rutherford's experiments revealed that the atom was composed of electrons with a tightly packed positively charged nucleus. Now von Lau, Nipping and Friedrich noticed that there were bright spots where the x-rays tended to accumulate and dark regions where no scattered x-rays tended to be observed and this interestingly enough looked like a interference pattern that you would expect from light interfering and scattering in different ways off of a target. So using these interference patterns and especially through the work of William Henry Bragg and William Lawrence Bragg, the only father and son team to ever won the Nobel Prize in Physics, they were able to explain the scattering of the the x-rays as being off of small objects albeit with comparable size to the x-rays in question and separations in space that were similarly comparably sized. So William Henry Bragg and Lawrence Bragg did their own scattering experiments and Lawrence Bragg in particular developed a model of the scattering process of scattering x-rays off of regular layers of atoms in a crystalline solid that beautifully explained these patterns of light and dark that were observed at first by von Lau, Nipping and Friedrich in 1912. And this actually led to the ability to determine the approximate size of atoms using these x-ray diffraction patterns. Let's take a look at the model that Lawrence Bragg developed, because it will help us to understand how we can detect wave properties in general going forward. Let's begin by modeling a crystal as a series of regularly arranged atoms layered in planes. We'll come back to the separation between the planes later, but they could be represented by some distance d, which will appear later in this example. Let's then imagine that we draw an incoming x-ray that scatters off of one particular atom in a plane at the top of the system. Now, from the place where this ray has been emitted, the x-ray will strike an atom and scatter off of it. This will have a certain path length associated with it, the default length that this x-ray had to travel during the scattering process. We can imagine then that this x-ray came from the plane of emission shown here, which makes a 90-degree angle to the original x-ray. A second ray emitted from very close by from the plane of emission, which also makes an angle of 90 degrees with respect to that surface, strikes another atom nearby, missing the first one, but hitting one in the layer below it. That ray also scatters and is detected at another point where the first x-ray is also detected, photographic film or a camera or some system like that. Now, because the second ray did not strike the same atom as the first x-ray, there's going to be an extra bit of distance that the second x-ray has to travel before coming back to the plane where the first x-ray is also detected. So, we can imagine considering what that extra length is by drawing another line parallel to the line of emission, the plane of emission, that represents the extra distance that the x-ray would have to go. That's highlighted here in red. This is the extra length that the x-ray, the second one scattering off the second atom, has to travel before it returns to the same location where the first x-ray also strikes a detection system. On each side of the scatter off the second atom, we have an extra length, capital L, that the x-ray had to travel. And we can start doing some geometry to figure out how one relates that extra length L to the displacement D between atoms and the planes of the crystal. Notice that the angle between the black lines, which are parallel to the plane of emission, and the red lines here must also be 90 degrees. This is some geometry that you yourself could work through to verify. But that ray will always remain perpendicular to the plane of emission. Now, the scattered x-rays will make an angle, theta, with respect to the surface of the crystal. And if one works through the trigonometry and the geometry of the problem, you'll find that there is one interior angle inside the little triangle whose hypotenuses D and who each have a side of length L. And the similar angle is indicated here. Now we can relate the length L, this is half the extra length the ray has to travel, to the distance D and the angle theta of scattering by simply noting that in this triangle the sign of theta is equal to L, the opposite side, divided by D, the hypotenuse of the right triangle. Now let's think about what's going to happen if these two waves, one scattered off of one atom on the surface of the crystal and one scattered off of an atom in the next layer of the crystal, meet at the same place on the detection screen at the same time. One of these x-rays, the first one for instance, is a wave and it's going to have crests and troughs just like any other electromagnetic wave. Now it's partner x-ray that arrives at the same time will interfere constructively or destructively depending on the alignment of the second ray with the first one. Let's imagine we want to figure out what the condition is for completely constructive interference, that is where the peaks of x-ray one line up with the peaks of x-ray two and the condition for that is that they be shifted relative to each other by exactly an integer number of wavelengths. This is the condition for constructive interference. The waves can be shifted in distance by some distance 2L with respect to each other, but the condition is that that distance 2L has to be an integer multiple of the x-rays wavelengths after scattering. So n times lambda, so that is an integer number n times the wavelength of the x-ray lambda, meets the condition for constructive interference when n is an exact integer multiple of lambda. As I said, the condition for constructive interference is that n times lambda is some distance d, and that's the extra distance that the second x-ray has to travel, and from our picture that's twice L. Now we can relate this extra length L to the angle of scatter of the x-rays theta using the trigonometric relationship derived earlier, and that relationship was just that sine theta, the sine of the scattering angle, equals L, the side opposite that angle, divided by d, the hypotenuse of the triangle. This allows us to solve for L in terms of d and sine theta. L is equal to d times sine theta. Now plugging that into our constructive interference condition, we find the following. That if the second x-ray is shifted by an integer number of wavelengths with respect to the first, n times lambda, then this will simply equal to 2 d sine theta constrained by the scattering requirements in the system for constructive interference. And this condition, this mathematical condition in order to obtain constructive interference, is known as the Bragg condition, as derived by Lawrence Bragg originally in thinking about this x-ray scattering process. So all one has to do is look at angles where you see bright spots in the interference pattern, and this will tell you, given the wavelength of the x-rays, what is the space separation of the planes of atoms in the crystal. Now in the specific case of the sodium chloride x-ray scattering that I hinted at earlier, if you take regular crystals of sodium chloride and expose them to a beam of x-rays, you can look to see where in scattering angle, relative to the incident beam, the bright spots and dark spots appear. So for instance we have here an x-ray spectrometer. The vertical axis is the number of x-rays per second that are detected, and the horizontal axis is the angle with respect to the incident beam of x-rays. Now theta here is the scattering angle with respect to say the surface of the material, but this can be related via 2 theta back to the original angle to the beam. You'll notice that there are in fact places where there are build-ups of intensity of scattered x-rays, so for instance just before 30 degrees, around 28 degrees or so, and just around 32 degrees, and then there's another clump of peaks over here. There's a clump just around 60 or so degrees, and so forth, and then there's another clump over here, there's a very low bump, and then a larger bump, and you'll notice that these bumps come with different intensities. Well what's going on here is that a copper emitter is being used to generate the x-rays, and because of the properties of copper, it generates two kinds of x-rays in the beam, the so-called copper k-alpha line and the copper k-beta line. The k-alpha line has a wavelength of about 0.15 nanometer, and the k-beta line has a wavelength of about 0.14 nanometer. So they're not exactly the same wavelength, and that explains why the first bright fringe in the x-ray has two peaks, one from each of the k-alpha and k-beta lines, the second bright spot in the x-ray scatter has two peaks, again one from the alpha and one from the beta line, and so forth. Now if you take the Lawrence Bragg scattering approach and you relate the locations and angle space of bright spots, constructive interference locations, back to the size of the scattering distance between scatterers in the crystal lattice, you can actually estimate the separation of the atoms or molecules that make up the crystal lattice, and you find out that this comes in at about 0.28 nanometers regardless of which of these x-ray lines you consider. So we find out that the spacing of the scatterers inside a sodium chloride crystal is about the same scale as the x-ray wavelengths. It's only about a factor of 2 or so larger than the x-ray wavelengths. That's easy then for us to see the wave nature of the scattered x-rays emerge, because they are a little bit bigger then, but comparable in size to the things off of which they're scattering. It's no wonder we don't see strong Compton scattering here. The particle nature of the x-rays is not in effect the wave nature of the x-rays because they're large compared to the size of the things they're scattering off of is in effect. But this is nice because it tells us roughly the scale of the size of the scattering objects, and that comes in at about a fraction of a nanometer. So this roughly tells us that the size of atoms or atomic distance scales is at that level of about a fraction of a nanometer. Now this tells us something about the sizes of atoms. Atoms come in at sizes around 10 to the minus 10 meters or so. This unit is not in the system international, but it's known as the Angstrom in honor of Anders Angstrom. The Angstrom is about 10 to the minus 10 meters, and that roughly corresponds to the size of, say, a hydrogen atom, or an atom that's slightly larger than that. Now going back to atomic emission spectra, that is, you know, heating or ionizing a gas, an elemental gas like hydrogen or helium or neon or something like that, we get these patterns of light that come out. It's as if only certain energies are permitted for the electrons in an atom. Why would that be? Well, in your mind, you might start modeling the electron in orbit around the central nucleus of its parent atom as a string on a guitar. A string on a guitar is confined at two ends. It's bolted down at two ends and tensioned. And once you set the tension of a guitar string, all the primary and secondary frequencies of its vibration are fixed, and that's how you can tune the tension of a guitar string and get a specific note. A note consists of a specific fundamental frequency and then a whole bunch of other frequencies layered on top of it with regular intervals. And what determines the frequency is the length of the string and the tension of the string, and that that basically says how many of each kind of standing wave with a certain wavelength can actually be found on a guitar string. So perhaps, like guitar strings, confined at two ends, electrons are wave-like and find themselves confined in a specific volume with only specific frequencies allowed. That would certainly help explain why these patterns of light are so specific to each atom. So we might draw in our mind a model of the atom as an electron confined to a volume, like a spherical volume, with a radius that's about the size of an atom, 0.1 nanometer or so. Maybe it's there that these wave-like properties of electrons, which you couldn't really notice at larger scales, clearly emerge. And maybe that's why atomic spectra have the properties that they have with these regularly spaced and in fact mathematically related colored lines. This certainly would be consistent with observations of other phenomena, like the black-body cavity emitter, where only certain vibrational frequencies of the walls of the cavity appeared to be allowed, and that constrained the radiation that the cavity could emit. So this isn't totally alien, the black-body spectrum, an atomic emission spectra, maybe two aspects of the same behavior trying to tell us something about matter. So if matter can be wave-like, as well as particle-like, what is it that determines the wave properties of matter? Remember, for light, we had Maxwell's equations. They were built up from the careful study of the electric and magnetic forces and fields, and emerged as wave equations that when solved an empty space told us that light was an electromagnetic wave, an oscillatory phenomena with wave-like characteristics. We have no wave equation for matter. There is no first principles thing that we've experienced up through the end of the 1800s that tells us, oh well of course, there's a wave equation for matter, too. So we don't have a starting point for the wave properties of matter, assuming they're even real at all. So in his 1924 PhD thesis, French physicist Louis de Broglie postulated, postulated in the same way that Einstein postulated that the speed of light was the same for all observers, that matter also has wave properties, and not only that, drawing from Planck's relationship between energy and frequency for light, and the relationship between momentum and wavelength that results from special relativity, de Broglie asserted the hypothesis that the very same facts would be true for matter if it had wave-like properties. So the energy of a piece of matter would be related to the frequency of the matter wave by e equals hf, that's a conjecture, that the momentum of a piece of matter would be related to the wavelength of the corresponding matter wave by h divided by the wavelength, that's a conjecture. So how would one prove this? Recall, Einstein made the conjecture based on the Michelson-Morley experiment that the speed of light was the same for all observers, regardless of the state of motion of the source of the light, or the observer of the light, relative motion did not change the speed of light. That could be tested by conducting experiments looking at the constancy of the speed of light with respect to motion. Now that conjecture, along with the other postulate of relativity, had other predictive consequences for this description of space and time, and those consequences were verified, think about time dilation and the lifetime of the muon. So how would one prove de Broglie's conjecture? Well, Bragg's scattering offers the possibility to test this hypothesis. We could, for instance, compute the matter wave properties of electrons. And then we might try to find a system off of which we might scatter them and see if we can see the wave properties of electrons revealed by the scattering process. All we have to do is find a scattering system whose size scale is slightly smaller than or roughly comparable to whatever the corresponding matter wavelengths of an electron would be. So just as x-rays scattered from crystals allows the wave nature of x-rays to reveal to us the structure of the crystals, once we know the structure of crystals themselves, regular arrangements of atoms, we can then look at electrons scattering and see if it reveals any wave properties of electrons. For instance, interference. Well, this is precisely what was done. So consider the electron with its mass of 9.11 times 10 to the minus 31 kilograms. Now imagine accelerating it up to some momentum. Now we're going to be fully relativistic here. We're going to use the correct definition of momentum because we might have to accelerate electrons to extremely high speeds to achieve the kinds of properties, the wave properties we would need in order to see if those wave properties exist. So we're going to use the fully relativistic momentum equation, the gamma factor of the electron times its mass times its velocity. Which we can set by accelerating the electron. Now by de Broglie's postulates, the momentum of an electron accelerated up to some speed u is going to be related to its matter wavelength by h over lambda e. So what momentum would we need to accelerate an electron to? To probe the scale of a crystal whose spacing is going to be somewhere around the level of 0.1 nanometers or so. Well, we would ideally want to achieve an acceleration that gets our wavelength down to something comparable to that, about 0.1 nanometer. Now notice that momentum, according to de Broglie's postulates, is inversely proportional to wavelength. So if we want to get the wavelength down to something the size of 0.1 nanometer, we've got to get the momentum up high to some target value. Now if you crunch the numbers on this, this will require an electron momentum of about 7 times 10 to the negative 24 kilogram meter per second. That doesn't really tell us much. So for instance, if we used an accelerating electric potential difference, a voltage, to get our electrons up to this momentum, what voltage would be needed to achieve that for an electron? Now I'm going to leave the math to you if you would like to play around with this, but you need to make sure that you're careful and use special relativity to answer these questions. Remember the relationship between energy and the gamma factor, total energy and the gamma factor for an electron. That's written down here. And remember also from special relativity that that can be related to the momentum and the rest mass of the electron through this equation. And keep in mind also the special relativistic definition of kinetic energy. You're going to need to combine all of these things to get the answer to that question. What voltage would be needed to achieve this for an electron? But it turns out that this corresponds, this momentum corresponds to a gamma factor that's actually quite modest. It's only 1.0003. That's only a small fraction of the speed of light. And that shouldn't be hard to achieve for something as low mass as the electron. That corresponds to a kinetic energy of about 2 times 10 to the negative 17 joules. And if you remember your conversion of electron volts, an electron volt is roughly 10 to the negative 19 joules or so, this isn't many electron volts worth of kinetic energy. And so if you crunch the numbers and you relate the kinetic energy to the accelerating potential that would be required to achieve that for an electron with its one unit of elementary charge, you very quickly find out that this only requires about 150 volts. That is no problem at all. Certainly in the days when this experiment was done, and this experiment was done in 1927, achieving 150 volt electric potential difference for electrons was quite a trivial activity in that day. So that scattering experiment was famously done by two physicists, Lester Germer, shown on the right-hand side of the photo, and Clinton Davison, shown on the left. And this is in fact a piece of the equipment of their scattering experiment with the electron emitter and the nickel crystal that they used to target in 1927 to do the scattering. And then they looked at the pattern of scattered electrons to see if any wave-nature effects emerged. What's the most obvious wave-nature effects? Well, if you see an interference pattern in the scattered locations of the electrons, that is if you see places where there are intense locations where electrons scatter to and other dark regions where they don't scatter to, there's some evidence for the wave-nature of electrons. Matter wave properties could in fact be real. So just as an x-ray scattering, if you scan over the scattering angle of the electrons from the crystal, and if wave properties manifest, then constructive and destructive wave interference should occur at different angles for a fixed wavelength, and thus a fixed momentum. So this is an analogy to the x-ray scattering process, of course, that we looked at earlier with Bragg scattering. So you could what you could do, of course, is you could set your voltage to accelerate the electrons to something specific to achieve a specific momentum for the incoming beam, and then you could look at different angles of scattering relative to the beam to see if you see intense regions and less intense regions of scattering. In that case, the Bragg scattering formula just applies. If you want to see the nth bright fringe of constructive interference, the first, the second, the third, and so forth, then all you have to do knowing the wavelength of the thing you're scattering is look at a specific angle, knowing the size of the crystal, the space in between the scatterers and the crystal D. And then the wavelength would simply be determined using de Broglie's hypothesis using the momentum of the electron. But actually instead of scanning over scattering angle, in fact when you can control very easily the momentum of the electrons, then it's actually easier to simply vary the momentum of the electron beam and observe at a fixed angle theta. So don't move around where you're looking. Just observe at a fixed angle theta and scan through voltage, which changes the momentum of the beam and thus changes the degree of the wave properties of the beam as a function of voltage. And as you scan over the voltage, sometimes you'll make the electrons have just the right wavelength to interfere totally constructively when they scatter. And sometimes as you keep tuning the voltage around, you'll make them interfere totally destructively with each other and you'll see no scattered electrons at that same angle theta. This is what Davison and Germer did and here's what they saw. So this is the intensity of scattered electrons versus the square root of the voltage of their instrument. And what you notice is that there is a place of course where there's a bright intensity peak and then it falls off to a minimum and then there's another bright intensity peak at a different voltage and then it falls off to a minimum and so forth. You see that there are these increases in electron intensity at a certain voltage and then you crank the voltage up a little bit more and the intensity decreases down to a minimum. You keep cranking it and it goes up to a maximum again. We are seeing exactly what would have been predicted from Bragg scattering with the matter wave hypothesis. This did not have to be this way but it turns out that matter also has wave like properties that can be revealed under the right conditions. Just to really drive this home, in two dimensions now scanning over scattering angle rather than fixing the scattering angle and scanning over electron momentum this is what an electron diffractogram looks like. You see this pattern of bright and dark spots separated by gaps here. We can very clearly see that electrons will intensely build up in the scattering process in some places and not at all in other places with big gaps in between both vertically and horizontally. There are very clearly bright spots and dark spots just like a laser beam that interferes with itself through passing through two slits for instance. Only waves can interfere with each other in this manner. In this case it's because the crystal and solids like nickel for instance off of which the electrons are scattered have structures that can accommodate an easily tuned electron momentum that yields a wavelength comparable to the size of the scattering system or a little bit larger and that's easy to do with even modestly accelerated electrons on a metal target. So here's what scattering and interference tell us about the true nature of both matter and electromagnetic radiation. Electromagnetic radiation already has a wave equation that describes its wave nature. It comes from Maxwell's equations. So again we come back to this question well if matter can be revealed through experiment and observation to have wave properties under certain conditions then where's the wave equation? Where's the equivalent of the thing that comes from Maxwell's equations that describes the wave properties of electrons, protons, neutrons, whole atoms, etc. Where is it? What is it? Electromagnetic fields and light propagating through empty space. These are the solutions to Maxwell's equations. If we had an equivalent matter wave equation what will the solutions to the matter wave equation look like? And these are all excellent questions. And these are the questions that after these kinds of experiments had been done physicists really began to struggle with in the 1920s and into the 1930s. Now we're going to get to the answer to this question very soon but we have some hints for ourselves already. The solutions to the matter wave equation whatever they are, whatever specific form they take for a very specific system an electron scattering off of a nickel crystal, an electron confined in a hydrogen atom, whatever the solutions to the matter wave equation are going to be they're going to be probabilistic in nature. And we can already see this revealed in the scattering intensity patterns from experiments like Bragg's scattering the Davis and Germer experiment and so forth. The intensity of the scattering pattern seems to have everything to do with the probability of finding a particle at a certain location in space and time after the scattering process has occurred. And that probability is controlled in some way by the original wave nature of the thing that experienced in this case the scattering phenomenon. Probability whatever our wave equation describes it's going to be probabilistic in nature waves are a spread out spatial and temporal phenomenon there's no one place where a wave is and where it is not. There are many places where a wave can be and probability and the wave equation whatever it is are going to play a fundamental and deep role with one another in describing matter and radiation. So let's review. In this lecture we have learned the following things. We've learned about the structure of the atom as it was known in the late 1800s and very early 1900s cherry picking our way through just a few scenes and the great story of the atom. We've learned about how matter itself can have wave aspects to its behavior first hinted at although no one really understood this at the time by the nature of atomic spectra and the black body spectrum. Now it was Louis de Broglie who conjectured that the same wave and momentum and energy descriptions that could be discerned from the black body spectrum and special relativity equally applied to matter like electrons that was a conjecture and that was experimentally verified using scattering experiments of matter off of other matter these the target had size scales that were comparable to the matter wavelength we were trying to assess and in fact tuning the beam of electrons to the right momentum to get the desired wavelength we actually see that the wave properties manifest in the scattering experiment. If electrons did not have wave like aspects to their behavior we would not have seen the diffractograms that can be discerned from scattering electrons off of crystal and targets. So that was also taught as how to conduct experiments both with light and with matter to reveal the wave aspects of matter's behavior and Compton scattering offers us a glimpse of how to reveal the particle aspects of the behavior of radiation and matter all we have to do is get the wavelength of the phenomenon to be much smaller than the size scale of the thing we're shooting it at and the particle nature should manifest again. These ideas are going to play key roles going forward in everything we're going to do with matter and radiation.