How Big is Big and How Small is Small

by Smith, Timothy Paul

  As you add electrons we expect the atom to get bigger. But hydrogen, with one electron, has a radius of 0.53 Å, whereas helium, with two electrons, has a radius of 0.31 Å. As we have seen, you can put two electrons into a single wavefunction, which might lead you to think hydrogen and helium should have the same size. But helium also has a second proton, which pulls the electrons into a tighter orbit. Lithium is bigger than hydrogen, but as you march across the periodic table—beryllium, boron, carbon, nitrogen, oxygen, fluorine and finally neon—the atoms get smaller. Jump to the next row and we find that sodium is larger than lithium. The trend continues through the whole periodic table. Across a row the atoms get smaller, down a column they get bigger (see Figure 10.7).

  Helium is the smallest atom with a radius of 0.31 Å and cesium (used in atomic clocks), with 55 electrons, is the biggest atom with a radius of nearly 2.98 Å; that is, about six times the radius of hydrogen.

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  One last question. The heaviest element in nature is uranium and its most common isotope is 238U. It is build out of 92 electrons, 92 protons and 146 neutrons. Why do we not find more massive atoms in nature? (We have synthesized a few elements that are heavier, but they tend to be unstable and short lived.) The reason atoms are not heavier is not a limit set by the electrons. It is a limit on the number of protons you can shove into a nucleus. But that is a problem of nature that takes place at a scale 100,000 times smaller than an atom, the scale of the next chapter.

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  All the things that we have looked at in this chapter—crystals, molecules and atoms—have a special type of beauty. They all have a complexity, a richness and a wide range of variations that arise from a very few axioms. Firstly, the important force holding all these things together is the electromagnetic force: opposite charges attract, like charges repel, and the force becomes weaker at a distance. Secondly, electrons are best described by wavefunctions, with all the properties of waves such as interference, resonances, harmonics and nodes. Given these two principles and a pile of electrons and protons (and neutrons) we can build all the elements of the periodic table. Given these atoms we can build all of chemistry and stitch together things like DNA or even macroscopic crystals.

  The starting point is simple; that is why I say that the resulting complexity is beautiful. It is not messy and obscure. The chemistry of life, the radiant beauty of a diamond, all arise from a few simple principles, none of which surprise us.

  11

  How Small Is Small

  We ended the last chapter by looking at the way the electron’s wave-function traces out the size and shape of the atom. Common atoms have a diameter of about an angstrom (10−10 m) and are never larger than three times that. We now continue to look at the world at smaller and smaller scales. The break between these two chapters also reflects an important boundary in nature.

  What objects in nature are a tenth of the size of the atom (10−11 m)? Nothing. This is curious, because so far we have never discussed a scale size for which there are no naturally occurring objects or systems. So let us look a bit smaller. What is there that is one hundred times smaller than an atom? Nature entertains us with … nothing again. This void continues. There are no objects in nature with a size of 10−13 m either. Finally, the largest nucleus, the core of the heaviest natural element (uranium-238) has a diameter of about 1.5 × 10−14 m, about 10,000 times smaller than an atom.

  At the very beginning of this book there was plot of a number of objects in nature laid out on a logarithmic scale. Between 10−10 and 10−14 there is nothing. Other parts of the plot are crowded and things had to be left out. But in this range nothing was omitted; rather, there really was nothing to plot. The gap must be telling us something; it jumps out at us like a missing tooth in a smile. It is significant because it marks the boundary between two great organizing principles of nature. For the moment we only comment upon its presence but will leave an explanation to the end of the book when we have all the threads in hand.

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  The next object of study is the nucleus of an atom. The nucleus is made up primarily of protons and neutrons. As we have said before, since an atom is usually electrically neutral the number of protons and electrons will be the same. So, even if it is the electrons that interact, it is the number of protons that defines an element and therefore its chemistry.

  But what about the neutrons? They appear at first to have a very passive role; they do not entrap electrons. Their role also seems a bit ambiguous in so far as an element such as carbon can have a variable number of neutrons. As was mentioned when we talked about carbon-14 dating, carbon can have six, seven, or even eight neutrons. Neutrons seem to not be very rigid about their number of partners. And then there is the case of hydrogen. Most, but not all, hydrogen atoms have no neutrons. How important can the neutron be if the most common element in the universe does not need one?

  The fact that hydrogen, the element with a single proton is also the only type of atom that can exist without a neutron is a clue to the neutron’s role. Helium, the next simplest atom after hydrogen, has two electrons, two protons, and one to eight neutrons, but never zero. Helium usually has two neutrons, one out of a million will have one neutron, and the versions with three or more neutrons are much rarer, being artificially created and having very short lives. Elements with different number of neutrons are called isotopes. So hydrogen has three isotopes and helium has eight isotopes.

  The point is that if an atom has two or more protons in its nucleus, there needs to be a neutron. If a helium nucleus had two unaccompanied protons it would be very unstable, and unstable things fall apart very quickly.

  In the world of the nucleus the neutron and proton are in many ways very similar. They are about the same size and have the same mass. Neutrons and protons attract each other by the nuclear force. Also the nuclear force between two neutrons is the same as between two protons or even between a neutron and a proton. Therefore neutrons and protons are often collectively called nucleons. The nuclear force collects nucleons (neutrons and protons) and forms them into the nucleus. But neutrons and protons are not exactly the same. Protons still have that positive charge. Two protons will attract each other via the nuclear force and also repel each other via their electromagnetic force. And so, as pointed out above, a nucleus of two protons will quickly fall apart.

  Neutrons provide the extra nuclear force needed to hold things together. We can see this especially in heavier elements. With more and more protons, nature also adds more and more neutrons. For the light elements, such as oxygen or carbon, the ratio of protons to neutrons is about 1 to 1. For gold the ratio is 79:118 (about 1:1.5) and for the ultraheavy elements such as uranium it is 92:146 (about 1:1.6). This increasing ratio in the heavier elements is because the neutrons have an increasingly hard time keeping the unruly protons from pushing on each other, potentially breaking the nucleus apart.

  Figure 11.1 Detail from a small part of the table of nuclides. The number of neutrons increases going to the right. The number of protons increases going up. The dark cells are the stable isotopes.

  In nuclear physics there is a chart called the table of nuclides (see Figures 11.1 and 11.2). It is the equivalent of the periodic table of the elements for chemistry. On it are plotted all the different isotopes. Vertically, the number of protons (or electrons) increases, so each row of the table is a new element. Starting from the bottom the rows are hydrogen (1 proton), then helium, lithium and so forth. Going across the table the number of neutrons increases. Usually each square on the table is marked with the name of the isotope. Sometimes lifetime or decay modes are included in the fine print. In isotope notation a name like 14C (read as “carbon-14”) tells us that there are 14 nucleons, or protons plus neutrons, and since we know that carbon has 6 protons and electrons (since it is carbon), then it must have 8 neutrons. Alternatively, take a more extreme case such as 238U, uranium-238. From the table of elements, uranium has 92 electrons and protons, and so it has 23
8 − 92 = 146 neutrons.

  Figure 11.2 The table of nuclides. All isotopes are plotted. The dark ones are the stable isotopes. In the bottom left part of the table, the light elements tend to have about the same number of neutrons and protons. The heavier elements have 50–60% more neutrons than protons.

  However, I really only want to talk about the big trend in this table. In the light elements there is a one-to-one neutron-to-proton ratio that nicely offsets the proton–proton repulsion. But that does not work for bigger, heavier nuclei. This is because the electromagnetic force is long range whereas the nuclear force has a short reach. In fact we can estimate the range of the nuclear force just by looking at the chart. The one-to-one ratio goes up to less than twenty protons and twenty neutrons. After that the trend becomes more horizontal; there are more neutrons then protons. A nucleus with twenty protons has a radius of about 3–4 × 10−15 m, which tells us that the nuclear binding force has cut off. In reality the nuclear force is a bit shorter then this, extending out only about 2 × 10−15 m, effectively only binding the neighbor or near neighbors. This is reminiscent of chemistry. I have visions of a nucleus as a crowd, with protons as feisty characters pushing and shoving and neutrons mixed in, trying to calm things down and hold everything together but only talking in whispers to their nearest neighbors.

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  I would like to “see” what a neutron or nucleus or quark looks like, but the whole concept of seeing is going to be very different as we move to smaller and smaller scales. But it is worth pausing a moment to dissect what we normally mean by seeing, so we can compare it to the methods needed to bring the subnuclear world into focus.

  When you look at a maple tree in autumn and see the glowing red leaves, what is really happening is that light of all color from the sun has shined on the leaves. Only the light with wavelength of about 6500 Å(6.5 × 10−7 m) has been reflected back. These photons go in all directions, but a tiny fraction enter your eyes and stimulate the retina. In turn, your brain is told that a photon of about 6500 Å has arrived and then the brain figures out that it came from an Acer rubrum, a red maple. In this whole process we have a light source (sun), and object of study (tree), detector (eye) and interpreter (brain). Laboratories where people study particle and nuclear physics mimic these same steps.

  We could not see protons or even a nucleus with classical microscopes, even if we had amazing lenses. The problem is that to see a nucleus we need the right sort of light. Common light is the wrong size. The light our eyes are sensitive to has wavelengths between 3800 Å(violet) and 7400 Å(red) in length. The things we are trying to see are 10−15 m across, 100 million times smaller than the light waves.

  If we tried to use regular light to see a nucleon it would be like trying to see the effect of an ocean wave with a 10-m wavelength scattering off of a virus. The wave would roll past the virus and would not be scattered or deflected. It we had a stick a centimeter thick, a thousandth of the size of the wave, the effect is barely perceptible. A post half a meter thick starts to scatter waves, whereas a boulder a few meters across finally breaks up the waves and creates interesting scattering patterns. To see something with scattering techniques you need to match the size of the wave with the size of the object of study. To see a nucleus we need light that is 10−15 m across or shorter. This is what accelerators can provide.

  Back in Chapter 10, when talking about the atom, we said that in a quantum system there is a wave associated with a particle such as an electron, and that if the wavelength was right the wave could wrap itself around an atom in a snug-fitting orbit. We also said that energy was related to orbit and wavelength. This is generally true for all waves, whether they are sound waves, light waves, or waves associated with an electron. In light, the longest waves are radio waves, which are not very energetic. Among the shorter waves are infrared, visible and ultraviolet, which have more energy. The shortest are X-rays and gamma rays, which have the highest energy per photon and the greatest penetration. The shorter the wavelength, the more energetic the photon. The same is true for waves associated with electrons, or any other particle. So if we want to create a wave about the size of a nucleon we can do it with an electron with about 3 × 10−11 J of energy. That does not sound like much, but it is a lot of energy for something as small as an electron. The machines that can produce these energies are called accelerators because of their ability to push particles up to extreme speeds, and so to short wavelengths.

  The original accelerators worked by having one end electrically grounded and the other end at a very high voltage. A electron at one end would be repelled by the voltage at that end and attracted towards the other end. A television video tube (things that are vanishing and being replaced by flat screens) works this way, with about 20,000 V between the electron source and the screen. So when the electron hits the front of the screen it has 20,000 electron volts, abbreviated eV. The electron volt is the normal unit of energy used in atomic, nuclear and particle physics. An electron with a wavelength about the size of a nucleon has an energy of 200 million eV. This is usually written as 200 MeV or 0.2 GeV (G = giga = billion). This electron is now traveling at 99.9997% of the speed of light.

  Accelerators are complex machines. The ones that use high voltages are called Van de Graff accelerators and can obtain energies of about 30 MeV, which is about 70 times too low to see a nucleon. They are, however, powerful enough to excite a nucleus into new configuration. However, higher voltages are very hard to control and even Van de Graff energies are approaching the level of lightning bolts. Higher-energy Van de Graff accelerators would spark, arc and randomly discharge.

  Above these energies, accelerators generally push the electrons (or protons, ions, or even other charged particles) with pulses of microwaves. I used to work at an accelerator at the Massachusetts Institute of Technology. It was about 190 m long with an energy of nearly 1 GeV (one billion electron volts), which means the electrons this accelerator produced had wavelengths of about 0.2 × 10−15 m, or a quarter of the radius of a nucleon. In other words, this is about as small as an accelerator can be and just start to probe inside a nucleon, which is what we did.

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  The radius of a neutron or proton is about 0.8 × 10−15 m. In fact the distance 10−15 m shows up so often in these types of studies that it has been given a special name, much like the angstrom for atomic distances. It is called the fermi, named after the Italian–American physicist Enrico Fermi (1901–1954). If instead we had applied normal metric prefixes this distance would be called the femtometer, and thus by either name the abbreviation is fm. It was Robert Hofstadter (1915–1990), a Stanford physicist, who saw the common abbreviation and coined the term fermi in 1956. He also made one of the first measurements of the size of the proton by scattering electrons off of it. He reported its radius as 0.8 fm.

  Scattering can tell us more than just the size of a nucleon. Sometimes the accelerator’s beam actually knocks out a fragment of the target. We might knock a neutron or proton out of a nucleus, and then from its energy and angle deduce how deep it was packed and how tightly it was bound into that nucleus. Sometimes we knock out a new type of particle. The most common is the pion. Pions are particles in a nucleus with a unique role of holding the nucleus together. In fact there are three different types of pions, all with almost the same mass, but with different electrical charges: π +, π 0, π −. Pions are particles in a nucleus with a unique role. Pions are what hold the nucleus together.

  So is it pions or the nuclear force that hold a bunch of protons and neutron together into a nucleus? The answer is either or both. By now we have gotten use to the idea that particles have waves; the reverse is also true. A force, usually described as a field can also be described as particles. Actually we have already encountered this idea before. The particle associated with the electromagnetic force and light is the photon. To understand pions, and later in this chapter gluons, a quick note on the relationship between forces, particles and fields may be u
seful.

  The simplest force to understand is the electrostatic force. Since magnetism happens when charges move, this is really the electromagnetic force, just without motion. It is the model upon which we build our vocabulary for forces. The electrostatic forces act between objects that have electric charge. The force radiates outward in all directions going on forever. It gets weaker as it radiates out as the inverse of the square of the distance: Alternatively we could say photons radiate outward in all directions going on forever. The photons become less dense as they radiate out as the inverse of the square of the distance. In other words, describing the force as a field or as particles are two sides of the same coin.

  So the general features of a force are that it acts between things that have some sort of “charge.” Also there are intermediate particles that travel between the principal particles. These intermediate particles are the carriers of the force and technically called “bosons.” They are fleeting, without permanence. They are created by one particle, and travel to another, where they are absorbed.

  Gravity is similar. Its charge is called mass; the greater the masses involved the greater the forces. The carrier of the force is a particle called the graviton. It is true that we have not isolated a lone graviton, but that is not surprising since the effects of a single one are truly minuscule. They are also expected to travel forever, like a photon.

  Now we can talk about the nucleus and the nuclear force. The nuclear force may not be a fundamental force, but it does create a unique structure in nature—the nucleus—and so is most certainly worthy of our attention. All nucleons have the same nuclear charge, which was part of the reason the term nucleon was introduced. The force is carried by a pion. Now the pion is very different than either the photon or the graviton in one very important way: it has mass. To make a pion takes a lot of energy, which is borrowed from the neighboring nucleus and must be returned soon. Actually the relationship between this borrowed mass–energy and lifetime is dictated by the Heisenberg uncertainty relationship. In fact in 1935 Hideki Yukawa (1907–1981) used the range of the nuclear force and the uncertainty relationship to predict the mass of the pion. It has a mass that is about 15% of the mass of the nucleons and the nuclear force has a range of only a few fermi (see Figure 11.3).