 Hello, and welcome to our How Small Is It segment on atoms. The ancient Greeks had two schools of thought on atoms. One was that you could take a substance like water and divide it and divide it and divide it infinitely. The other school said you can divide it and divide it and divide it to a point at which point you have the smallest piece of water and if you separate that it's not water anymore. In those days they didn't have any testing capabilities to verify one theory or another. But today we do. This segment is all about the atom and the pieces that go together to form an atom. In our first segment we saw carbon atoms through the eye of an electron microscope. Their diameter was 0.14 nanometers. That's very small. There are more atoms in the breath of air that I just took than there are stars in the entire visible universe. But what do we really know about atoms and the parts that go together to make an atom? That's what this segment is all about. We'll start with J.J. Thompson's guess as to the structure of these things, atoms. It's a fascinating story and it'll put us on a path to understanding elementary particles and the Higgs boson. In the 19th century it was well understood that the chemistry of substances consisted of atoms. But we knew very little about atoms themselves. It was the discovery of the electron by J.J. Thompson that first introduced the idea that an atom had parts. In 1898 with the electron being so light compared to the atom Thompson suggested what is called the plum pudding model of the atom with a uniform mass of positively charged matter containing spots of electrons embedded in it like plums in a pudding. A way to find out if this model is correct or not is to probe the pudding. But you need to probe with something smaller than the object being probed. For example you can't probe a grain of sand with your finger. In 1898 there simply wasn't anything smaller than atoms that could be used to probe an atom. But around that time radioactivity was discovered by the French scientist Henri Bacarelle. Using uranium salts he was able to blacken a photographic plate. Here's a photograph of the plate. Further research by Bacarelle, Rutherford, Curie and others discovered three types of radiation. Here's how they did it. A radiation source shines on a lead plate with a small hole in it to create a beam. The beam is directed at a fluorescent screen. The screen flashes when it is struck. Without any electric field present the beam illuminates a single point on the screen. When an electric field is applied the beam is separated into three components. One is deflected upward by the electric field indicating that it is negatively charged. These were named beta rays. One is deflected downward but not as far as the beta rays were deflected upward indicating that it consists of positively charged particles that are more massive than the beta rays. These were named alpha rays. The radiation that continued to hit the center was not affected by the electric field and therefore as no charge. These emissions were named gamma rays. It turned out the beta rays are high speed electrons. The alpha particles were later found to be helium atoms without their electrons. And the gamma rays turned out to be high energy photons more energetic than x-rays. With alpha particles Rutherford had something to fire at atoms to see if they were indeed like a positive pudding with embedded electrons. Here's a graphic of the apparatus used to run the experiment. An alpha particle emitting substance is placed behind a lead screen with a single small hole in it to enable a narrow beam of particles to flow through. This beam is directed at a very thin gold foil. A movable zinc sulfide screen is placed on the other side of the foil. Zinc sulfide flashes when hit by an alpha particle. A microscope swivels to view all scattering angles. If the Thompson model was correct, the positively charged alpha particles would pass through the distributed and therefore diluted positive charge in the gold atoms with little or no deflections. But after days of observation, here's what they found. While most of the alpha particles do go right through with only minor deflections, some were scattered through very large angles. A few were even scattered in the backward direction. Rutherford described this as almost as incredible as if you fired a 15 inch shell at a piece of tissue paper and it came back at you. To explain these results, Rutherford was forced to picture an atom as being composed of a tiny nucleus with a positive charge and nearly all of its mass are concentrated with electrons some distance away. Note that the closer the alpha particle is to the nucleus, the greater the angle of the deflection. We can use this angle to measure the maximum possible size of the nucleus. Here we have an alpha particle trajectory with an impact parameter b that scatters at the angle theta and reaches a closest distance labeled d. The number of protons in the alpha particle is 2 and the number of protons in gold is 79. The energy of the naturally occurring alpha particles used by Rutherford were 7.7 million electron volts. An electron volt is the energy it takes to move one electron across one volt. If we consider the direct hit trajectory, the initial kinetic energy of the alpha particle will drop to zero when it reaches its closest possible distance to the nucleus. All its kinetic energy would have been converted to electric field potential energy. Conservation of energy tells us that these two numbers must be equal. Rutherford's calculations show that the radius of a gold atom nucleus cannot be any larger than 0.00003 nanometers. That was 10,000 times smaller than the size of a gold atom. Here's a picture of the test apparatus Hans Geiger of Geiger Counter fame and Ernst Marsden built to look for scattered alpha particles from every angle. This was the first experiment that fired a beam of particles at a target to detect the scattering effects and deduce what is going on. That was around 100 years ago. This is exactly what we are doing today at the European Center for Nuclear Research, CERN, analyzing the Higgs Boson. We'll return for a deeper look at Rutherford's scattering when we get to particle accelerators. The Rutherford model of the atom left one outstanding problem. In the Thomson model, the electrons were stationary in the positively charged pudding. But what keeps a negatively charged electron from falling into the positively charged nucleus? Given that the opposite charges attract each other, the first proposed solution was to assume that the electron is in orbit around the nucleus, like the earth around the sun. Just as we can use gravitational and centripetal forces to calculate the radius and velocity of a planet around the sun, we can use electric and centripetal forces to calculate the radius, circumferences, velocity, and revolutions per second of an electron around the nucleus. For hydrogen, we get a very small circumference of around a third of a nanometer, and a very large velocity, almost 1% of the speed of light. That combination gives us a fantastically large 660 trillion revolutions per second. Now think about that for a second. But classical electromagnetic theory points out that an accelerating charge radiates energy. Theoretically, the electron should collapse into the nucleus in less than a trillionth of a second. And yet, we see that it does not collapse. You'll recall from our How Far Away Is It segment, Untistant Stars, that the light spectrum from stars was covered by thousands of dark lines called Fraunhofer lines, or spectral lines. Although these lines had been studied for over a hundred years, no one understood what they were. In 1885, Johann Ballmer broke out a subset of these lines for hydrogen, and developed some mathematical interrelationships between them. Then almost 30 years later, Nielus Bohr developed a quantized momentum theory for the atom. It partially explained these lines. His models still had the electrons orbiting the nucleus, but they could only orbit at certain specific distances from the nucleus, called shells. Each shell had its own unique energy level, N, where N was a positive integer, equal to one, two, three, etc. These were called the atom's quantum numbers. Electrons radiate or absorb energy when they change energy levels. The emitted or absorbed light has the energy difference between the two levels. This energy is equal to Planck's constant times the frequency of the light. Here's how it works. When a photon with an energy E hits an electron in a shell around a nucleus that has a higher shell it can reach with the same exact energy, the photon's entire energy is transferred to the electron instantaneously. This jumps the electron to a higher energy level with a larger quantum number. The photon is eliminated. This creates absorption lines in a star's spectrum as light from the star travels through the star's atmosphere. When the electron drops from this excited state back to a lower energy level, a photon with the exact difference between energy levels is emitted. This creates emission lines that we can see in the lab. Bohr's model explained the Ballmer series for hydrogen spectra. In addition, it provided the physical mechanism for Planck's quantized emission black body radiation and Einstein's quantized absorption photoelectric effect. It was also momentous for astronomy. By observing the absorption lines in a star's spectrum, we can tell what the star is made of. And not only that, by analyzing how these lines shift, we can calculate star radial velocities via the Doppler effect and even use them to measure their distances and the expansion of the universe. Indeed, Bohr's model explained a great deal, but there was no explanation for why the shell distances from the nucleus were as described. And there was no explanation for why the orbiting electrons didn't radiate away their energy and collapse into the nucleus. In 1925, Lewis De Broglie came up with the model that explained how electrons avoid falling into the nucleus. Earlier, we calculated the circumference and velocity of the electron, so like we did for electron microscopes in the previous segment, he calculated its wavelength. He found that it was exactly the length of the electron orbit's circumference, as enumerated by Bohr. In other words, the wavelength of the electron is exactly the length of one revolution. This would create a standing wave. Here are a couple of standing waves on a string. Here's what a standing wave looks like in water. A standing wave is a wave constrained to vibrate in a distance that's exactly one multiple of its wavelength. Anything more or less would create destructive interference and the wave would collapse. So the first energy shell would have to have the radius that creates the circumference that exactly fits one wave. The second shell would have to have the radius that creates the circumference that exactly fits two wavelengths. The third shell would have to have the radius that creates the circumference that exactly fits three wavelengths, and so on. So De Broglie answered the question, how can the electron sit way outside the nucleus without orbiting away its energy? The answer is that the electrons exist as standing waves that envelop the nucleus. No orbital motion is required, and therefore no radiation is emitted. So it's important to remember, electrons in an atom do not orbit the nucleus. They are not like planets around the sun. They exist as standing waves. De Broglie's simple geometry elegantly explained the reason for each energy shell's distance from the center and its corresponding energy. But it didn't scale to explain the spectra of more complex atoms that have more electrons. Given that particles travel as waves and are confined in atoms as standing waves, it followed that a generalized wave equation was needed to describe them. Depending on the works of Planck, Einstein, Rutherford, Bohr, De Broglie, and others, Erwin Schrodinger, an Austrian physicist, developed just such an equation, now bearing his name. Here's a simple sine wave in water. It's described by a wave function. The function tells us the displacement of every water molecule in the wave at any time t. If we take the change in a particle's displacement with respect to time, we get its velocity. And if we take the change in a particle's velocity with respect to time, we get its acceleration. For example, if we take a look at the particle 7 meters down the line and take a snapshot at the 11 second mark, we see that it is just above the line, heading up rapidly and slowing down slightly. A generalization called the wave equation describes how a wave function evolves over time. Schrodinger used the fundamental relationships between energy and wave frequency and quantized momentum to develop a quantum-mechanical equivalent of the wave equation. Importantly, it had one critical difference with the classical model that did not produce a location for a particle. In fact, it did not represent an observable physical quantity at all. Instead, it produced a probability curve for particle location. For free particles, the square of the wave function gives us the probability of experimentally finding the particle at a particular location at a particular time. For example, suppose we had a particle moving from left to right at a specific speed. From Newton's equations, the distance is equal to the speed times the time. After 24 seconds, we would say that the particle is here, but because the particle moves as a matter wave, we need to use Schrodinger's equations. So when you touch the wave at a particular time, it collapses into a particle. Where classical physics says it is here, quantum mechanics says here is the most likely place. But there is a smaller probability that it might be here, or an even smaller probability that it's here. In fact, there is a chance that it may be anywhere along the probability curve, with the probabilities dropping rapidly as we move away from the most probable point. This brings us to the Heisenberg Uncertainty Principle. In our quest to understand how small things can get, we need to know if there is a measure of size below which we can't go. We see from our little thought experiment that as a wave, a particle's location is not fixed. The wave is spread out. Here we see three different wave packets, or an electron. The wave packet at the top is narrow and therefore easier to locate, but it is less than one wavelength so its momentum is impossible to figure out. The bottom wave packet contains plenty of wavelength information, but it is quite spread out and its location is more uncertain. The wave packet in the middle has enough wavelength information to make its momentum less uncertain and it is less spread out than the one on the bottom, making its location less uncertain. But due to the spread out nature of matter waves, we still can't know both the location and momentum at the same time. Mathematically stated, the uncertainty in position times the uncertainty in momentum is always greater than or equal to Planck's constant divided by 4 pi. This is the Heisenberg Uncertainty Principle. It has nothing to do with the accuracy of our instruments and everything to do with the wave nature of matter. A good way to illustrate this is to look at an electron in an energy well too deep for it to get out. But remembering that the electron has a wave function that gives the probability of finding it at any given point, and some of those points, admittedly with very low probability, can be found outside the walls of the well, as if it had tunneled through the wall when in fact it did not. In our How Small Is It chapter on the microscopic, we covered scanning electron microscopes that mapped the surface of an object by using the wave nature of electrons and analyzing their scattering properties. Here we will cover scanning tunneling microscopes, or STM for short, that used the quantum mechanical tunneling property. Here is an STM at the Max Planck Institute. It has a small pin head that is actually one single atom at its tip. The tip is brought close enough to the object or electrons to tunnel across the space, exactly in accordance with Schrodinger's equation. This creates an electric current. As the tip scans across the object, the current will go up or down depending on whether an atom is under the tip or not. This is repeated over and over till the entire surface is mapped. What we are doing is actually feeling the surface of the object, to see and measure the atoms. With a little stronger pull, we can even dislodge and move atoms. Here we see that the scientists at the Max Planck Institute move the atoms one by one to spell their institute's initials, MPI. The tag is just six nanometers wide. Before we get back to the atom, it is helpful to examine macroscopic orbital systems to see what varies with respect to energy, angular momentum and orientation. In Newtonian mechanics, we can calculate the angular momentum L for an elliptical orbit and its energy E. We see that there are no limits to the number of angular momentum values L that can be associated with any particular system energy E. We also note that there are no restrictions on the orientation or azimuth angle for any angular momentum L. As we are dealing with a spherical system with the bulk of the mass at the center, it is common practice to use spherical coordinates. R is the vector specifying the position of the electron relative to the proton. Its length is the distance between the two and the direction is the orientation of the vector pointing from the proton to the electron. R is the polar angle most closely related to angular momentum and phi is the azimuth angle associated with orientation. But when we move from matter systems to matter wave systems, we move from Newtonian equations to Schrodinger's equations. The relationships between energy, angular momentum and orientation are quite different. With Schrodinger's equation for the hydrogen atom in spherical coordinates, we can separate the variables R, theta and phi. Solving Schrodinger's equation yields multiple wave functions and solutions. They define an electron's probability density cloud. Energy is quantized into electron shells designated by the letter n. It determines the distance the electron is from the nucleus. These energy levels match the ones proposed by Bohr. For each energy level n, the associated angular momentum is also quantized into electron sub-shells designated by the letter l. It determines the shape of the orbital. And surprisingly, for each quantized angular momentum sub-shell, even the allowed orientations are quantized into orbitals and designated by the letters m sub-l. It determines the orientation of the orbital. In chemistry, an atomic orbital is defined as the region within an atom that encloses where the electron is likely to be 90% of the time. It is these radii with their binding energies and interesting geometries that give atoms their chemical properties. Although Schrodinger's equation went a lot further than Bohr and De Broglie, there were still a couple of things about the atom that were not completely explained. One was, when examined very closely, many spectral lines showed up as pairs instead of single lines as called out by Schrodinger's equation. And two, the splitting of spectral lines by magnetic fields was not accounted for. This is known as the Zeeman effect. This effect is now used to measure the strength of magnetic fields around distant stars. And three, it was not understood why all the electrons did not move to the innermost lowest energy orbital. In order to deal with these issues, Wolfgang Pauli proposed a fourth quantum number and his exclusion principle. In classical physics, the exclusion principle states that no two objects can occupy the same space at the same time. Pauli's exclusion principle stated that no two particles could occupy the same quantum state at the same time. But Pauli could find no explanation for the fourth quantum number. The physical explanation turned out to be electron spin. Electrons have an intrinsic property that is best observed with the modern version of the Stern-Gerlach experiment that used silver atoms. Here we use magnets and electrons directly. The device has a north and south pole shaped to create a magnetic field that is stronger near the tip. This varies the force on charged particles passing through. A magnet is sent through with the north pole up and the south pole down. The magnetic field creates a force that deflects the magnet upward as it passes through the field. As we change the orientation of the magnets being sent through, we see the change in the amount and direction of the deflections. The deflections depend on the orientation. This is as expected. When electrons are sent through the field, they too are deflected. But they always arrive at the screen deflected either up or down, never in between like the magnets. Each electron behaves as a magnet, but with only one of two possible orientations, up or down. This intrinsic property of an electron is called spin. It is interesting to note that whenever an electron in an atom changes state, the atom's angular momentum changes. For example, here an electron moves from a higher energy orbital with angular momentum to a lower orbital with no angular momentum. We see that the emitted photon carries away both the energy and the angular momentum, giving it a spin equal to one. With the poly exclusion principle and spin as the fourth quantum number, the full set of spectral lines, orbitals and geometries and interactions with each other fell into place. In fact, when we add this fourth quantum number to Schrodinger's equations, we can generate the entire periodic table of the elements. Now that we have a handle on the electrons around the atom, let's take a quick look at the nucleus. For atoms to be neutral, the number of protons with a positive charge must equal the number of electrons with their negative charge, but mass spectrometers showed that atoms have more mass than the number of protons alone could account for. For example, carbon has six protons and six electrons, but its mass is just a tad more than the mass of twelve protons. In the 1920s, it was assumed that electron-proton pairs existed in the nucleus to account for the increase in mass without an increase in charge. But with the advances in quantum mechanics, it became clear that an electron couldn't exist in a volume as small as the nucleus. Ernst Rutherford and James Chadwick proposed that a new particle, the neutron, must exist in the nucleus to account for the data. Between 1932, Chadwick and others performed a series of experiments verifying his suggestion. They began by beaming alpha particles into beryllium. This produced a radiation that was not affected by applied electric fields. In other words, it was electrically neutral. At first this radiation was thought to be gamma rays, but when this new radiation was used to bombard a hydrogen-rich substance like paraffin, a proton radiation was produced. The energy acquired by those protons was measured and found to be more than a gamma ray could possibly impart to a proton. In fact, the protons ejected from the paraffin on the right was equal to the energy of the radiation coming out of the beryllium on the left. The conclusion was that the particles hitting the paraffin were of the same mass and energy as the protons, but without any charge. At this point it was generally accepted that indeed the neutron had been discovered. In this segment we developed the basic quantum mechanics for electrons around the atom and measured the size of atomic components. At the end of our previous segment we used a scanning electron microscope to see carbon atoms 14 hundredths of a nanometer in diameter. Using Rutherford scattering techniques covered in this segment, we measured the size of a proton at 1.76 millionths of a nanometer, that's 20,000 times smaller than the atom. At this scale we find that the neutron is about the same size and mass as the proton. Also in this segment we added spin as an intrinsic property of particles to go along with mass and electric charge. Protons and neutrons both display the same spin properties as electrons when they traverse the Stern-Gerlach apparatus, so their spin is one half as well. The notable difference between these particles is that the proton has a positive charge with the same magnitude as the electron's negative charge, but the neutron is neutral with no charge at all. For electrons it's hard to talk about their size because their wave packets are different for varying circumstances, from standing waves in thin shells to scattered waves in electron microscopes. What we did in this segment was to calculate its length around the hydrogen nucleus at 0.033 nanometers. For photons we see that they have no mass at all, no charge, and a spin equal to one. For the gamma rays coming out of uranium the wavelength is one one hundredth of a nanometer. That's 51,000 times smaller than the wavelength of green light. Looking at the atoms nucleus we see one main question. How do positively charged protons pack together in the nucleus when their repulsive positive charges would have them flying apart? We'll make some headway on this question in our next segment on elementary particles.