PHYS 202 Notes, Week 13/14 Greg Christian April 19, 21 & 26, 2016 Last updated: 04/19/2016 at 14:25:34 This week we learn about atomic structure, nuclei, radioactivity, and nuclear reactions. Atomic Structure Wave Functions As we ve learned, atoms consist of electrons orbiting in specific sets of stable orbits around a nucleus. It turns out that the existence of these stable orbits is no accident: it comes from fundamental properties of the microscopic world. As we discussed last time, wave-particle duality states that all object have both wave and particle properties, with the wave properties summarized by the de Broglie Wavelength, λ = h/p. More fundamentally, every particle can be described by something called a Wave Function, Ψ. This is a mathematical equation which governs the wave nature of subatimic particles. It s analogous to the wave function you used in PHYS-201 to describe waves on a string. Important points Wave functions fundamentally describe the dynamics of microscopic particles; solutions to the complete wave function equation involve three quantum numbers (plus a fourth, spin, which is intrinsic to the particle). Important equations Angular momentum L = l (l + 1)h/2π L z = m l h/2π Figure 1: Examples of wavefunctions. For electrons orbiting a nucleus, Ψ is a function of the three position coordinates, (x, y, z) and time. The total form is often written as Ψ(x, y, z, t). What Ψ actually describes is a probability: it tells the probability of finding an electron in a particular point in space and time, (x, y, z, t). Figure?? shows what some wavefunctions can look like for spherically symmetric systems. When Ψ is large, the probability of
phys 202 notes, week 13/14 2 finding the electron there is large; when it s small, the probability of finding the electron there is small. We can only ever talk about the position of an electron when we actually do an experiment to look and see where it is otherwise a probability is the best description we ve got. The wave function is determined by an equation called Schrödinger s Equation, which is a complicated differential equation whose solution is outside the scope of this course. However, we can still discuss some important properties of the solution. The first is that a solution is only possible when some physical quantity, such as the energy, has one of a specific set of values. This, then, is the fundamental reason for the quantization of things like energy levels of electrons we introduced in the last chapter: it arises naturally from the mathematical solutions of Schrödinger s Equation. As it turns out, these allowed energies agree exactly with those predicted by the Bohr model, which themselves agree with experiment. Thus the Schrödinger equation is able to predict complicated experimental quantities in a natural, fundamental way. Aside: Schrödinger s Cat Figure 2: Schrödinger s cat thought experiment. The idea of subatomic particles being fundamentally described in terms of probabilities only is a difficult one to accept. In particular, the fact that something like an electron does not even have a position until someone decides to look at it can be hard to swallow. To elucidate the seeming ridiculousness of this, Schrödinger himself came up with a thought experiment, which he described as follows, One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter, there is a tiny bit of radioactive substance, so small, that perhaps in the course of the
phys 202 notes, week 13/14 3 hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer that shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts. In other words, quantum mechanics and the Schrödinger equation state that the cat is neither dead nor alive until someone bothers to look at. This is, of course, a ridiculous proposition, and it serves to show just how strange the microscopic world is. Quantum Numbers One consequence of the Schrödinger equation solutions is that angular momentum is quantized. The allowed range depends on the principal quantum number n that defines the energy (recall: E n = 13.6 ev/n 2 ). In particular, only values of L that satisfy the following equation are allowed: L = l(l + 1) h 2π, (1) where l = 0, 1, 2,..., n 1. Note that the maximum value of l is limited to being n 1. Furthermore, the component of l in a specific direction (defined to be the z direction) is quantized: L z = m l h 2π, (2) where m l = 0, 1, 2,..., l. Note that the quantity h/2π is used so often that it s given its own symbol ħ ( h-bar ), ħ = h 2π = 1.045 10 34 J s. (3) Fundamentally, the solution to the Schrödinger Equation is defined by three quantum numbers, as opposed to just the one in the Bohr model. They are called: The principal quantum number n; The angular momentum quantum number l; and The magnetic quantum number m l. There s also a fourth quantum number called the spin, which is an angular momentum that s intrinsic to the electron itself. As the name suggests, it s analogous to something like a planet spinning around its axis (while the orbital angular momentum would be the planet orbiting around the sun). However, this analogy only goes so far; in reality the electron isn t a perfect sphere; it s more like something smeared out at different locations outside the nucleus. Fundamentally,
phys 202 notes, week 13/14 4 the spin is just a quantity that goes along with every electron, sort of like a color on some ordinary object. The spin of an electron can be defined in terms of its z component, which is constrained to be S z = ± 1 ħ. (4) 2 Note that when speaking of spin, the ħ is often dropped. Thus a spin of +ħ/2 is often called plus one-half (sometimes written +1/2) and ħ/2 is often called minus one-half ( 1/2). Figure 3: Electron probability distribution ( cloud ) shapes for different quantum numbers. Remember that we said that electrons aren t actually perfect spheres circling around the nucleus. Rather, they re more like clouds or smeared out distributions in space. When fully described in terms of their four quantum numbers, the electron clouds can take on a variety of shapes, as shown in Figure??. Pauli Exclusion Principle & Atomic Structure One principle which fundamentally defined how microscopic systems (like atoms) are built is the Pauli Exclusion Principle. Basically, what this says is that no two electrons in an atom 1 can have exactly the same quantum numbers. This enforces a hierarchy from atomic structure naturally arises. When filling up the available slots in an atom, electrons prefer to go into the slots with the lowest quantum numbers first, as these have the lowest energy, and nature tends to prefer the lowest energy configuration whenever possible. According to the Pauli principle, for each set of n, l, l quantum numbers, two electrons are allowed, since one can have spin +1/2 and the other can have spin 1/2. These arrangements of quantum numbers are called shells, with each value of the principal quantum number n defining a new shell. A listing of the quantum numbers for the first four shells is shown in Figure??. 1 Although defined for an atom, it applies equally well to other systems, such as nuclei which we will learn about next.
phys 202 notes, week 13/14 5 As the atomic number increases, so does the number of electrons orbiting the nucleus. These electrons fill up the shells as outlined in Figure??. The chemical properties of each atom are defined by the behavior of their outer electrons, i.e. those in the final shell. Inner electrons are effectively shielded from the outside world. This is what leads the the periodic table: similar elements (in the same column) have effectively the same outer-shell configuration, leading to similar chemical behavior. Nuclei We ve talked so far about atomic structure, treating the nucleus, the thing at the center of the atom, mostly as a black box. But the nucleus has structure and properties, as well, which can be understood. Nuclei are made up of protons and neutrons, with the following symbols used to describe proton and neutron numner: Proton number, Z Neutron number, N Mass number, A = Z + N Although they can be uniquely classified by using 2/3 of the above numbers, it s most typical to classify nuclei by Z and A. And since proton number Z also defines the element, we often represent nuclei in terms of their elemental symbol, with extra numbers added to denote the proton an neutron number. For example, for the nucleus with Figure 4: Available quantum orbitals in atoms. Important points Nuclei have less mass than the sum of their parts; this supplies the energy that binds them together. Some (most) nuclei have binding energies which allow for radioactive decay. Important equations Energy-mass equivanence Radius E = mc 2 R = R 0 A 1/3, R 0 = 1.2 10 15 m Mass defect M = Zm p + Nm n M.
phys 202 notes, week 13/14 6 A = 9 and Z = 4, we are dealing with beryllium, chemical symbol Be. Hence we write this as 9 4 Be. This general format, Z A El is followed for any nucleus we want to represent. One important property of the nucleus is its radius, which is approximately given by the formula 2 R = R 0 A 1/3, (5) where R 0 = 1.2 10 15 m = 1.2 fm. This number is significantly smaller than the atomic radius, by about five orders of magnitude. Hence most of the atom, and by extension, all matter, is by far empty space. Mass is also an important property of the nucleus. Owing to its small size, the nuclear mass is often discussed in terms of either the unified mass unit, 1 u = 1.6605 10 27 kg. (6) Nuclear masses are also often discussed in terms of energy-mass equivalent. What does this mean? It basically comes from Einstein s famous equation E = mc 2. This says that energy and mass are essentially the same thing, just related by a constant c 2. Hence we can discuss the mass of a nucleus in terms of the mega-electron volt, 1 MeV = 10 6 ev. To relate MeV to the unified mass unit, use the following: 1 u = 931.494 MeV. Why is the mass of the nucleus such an interesting thing? You might just think it should be sum of it s constituent parts, 2 Though there are some massive deviations from this in certain cases; for example 11 3 Li has roughly the same radius as 208 82 Pb! You can ignore these for this course, though. M = Zm p + Nm n. (7) However, this is not correct: the total mass of the nucleus is always less than that given by Eq. (??). We can introduce a concept called the mass defect, M to represent this, M = Zm p + Nm n M. (8) A related concept is the binding energy, E b which is the energy equivalent of the mass defect, E b = ( M)c 2. (9) This is the same thing as the amount of energy required to break apart the nucleus into its constituent particles. For example, the deuteron, 2 1H, has a mass of 2.014101 u; hence its mass defect is M = 1.007825 u + 1.008665 u 2.014101 u (10) = 0.00239 u, (11)
phys 202 notes, week 13/14 7 and the binding energy is (0.00239 u) (931.5 MeV/u) = 2.23 MeV. (12) So if you wanted to break a deuteron apart into a proton and a neutron, you d have to supply it with at least 2.23 MeV of energy. Nuclear Forces and Binding Energy. As mentioned, nuclei consist of protons and neutrons packed together at very close radius. But recall that protons are positively charged and repel each other, especially when very close together. So why doesn t the nucleus fly apart? It turns out there is another force in play, the strong nuclear force, which only acts at very close distances and it always attractive between protons and neutrons. 3 This force is able to overcome the electrical repulsion of the protons and hold the nucleus together. The strong force has a number of unique properties, summarized on page 965 of your textbook. 3 Note that it s attractive for all possible pairs, i.e. p-p, n-n, and p-n. Figure 5: Nuclear binding energies. Depending on the total number of protons and neutrons and the way in which the strong and electrical forces work to bind the nucleus, the total binding energy will differ for different nuclei. Figure?? shows a plot of binding energy across the range of nuclear masses. Another effect of the nuclear force is that not all nuclei are created equal. Some have higher binding energies than others, which causes them to be particularly stable, i.e. to be the nuclei that you see around you in the world. The ratio of neutron:proton number for stable nuclei changes as the mass A increases. This is illustrated on something
phys 202 notes, week 13/14 8 called a Segré chart, shown in Figure??. As you can see, stable nuclei tend to have relatively higher neutron numbers, i.e. N/Z increases as Z increases. For the lightest nuclei N = Z (or N/Z = 1) forms the stable configurations, but by the heaviest the ratio is more like N/Z = 1.5. Figure 6: A segre chart showing changing N/Z ratio for stable nuclei. Radioactivity The differing binding energies mentioned in the last chapter leads to the phenomena of radioactivity, wherein nuclei can release energetic particles. The reason they do this is to seek higher binding energy: effectively nuclei can spontaneously turn themselves into an-
phys 202 notes, week 13/14 9 other species in order to increase their binding energy. The two most common decay modes (types of ground-state radioactivity) are alpha (α) and beta (beta) decay. Alpha decay Alpha decay occurs when the nucleus spontaneously emits an α particle, which is just another name for a 4 2He nucleus, consisting of two protons and two neutrons. The net result is that the nucleus loses two protons and two neutrons. For example, the nucleus 226 88 Ra decays by alpha emission, becoming 222 86 Rn via the following process (also shown in Figure??), 226 88 Ra 222 86 Rn +4 2 He. Alpha decay always occurs such that the final system gains energy; this energy gain is a result of the final nucleus having larger binding energy than the initial one. This extra energy is carried off in the form of kinetic energy of the α particle. Since its energy is set by the binding energy differences, for any given alpha-emitting nucleus, the α particles will always exit with the same kinetic energy (or speed). Figure 7: Alpha decay of 226 88 Ra. Beta decay Beta decay occurs when the nucleus emits a high-energy electron, which can also be called a beta-minus particle (β ). Effectively what this does is turn a neutron into a proton. The total mass remains the same, but we swap a neutron for a proton in the nucleus. For example, 19 8 O emits a β particle to become 19 9 F. The process for this is as follows: 19 8 O 19 9 F + β + ν e. You probably noticed that there s an extra, unexpected, particle in the equation above. This particle is called the anti-neutrino, and it s always emitted along with the β particle during beta-minus decay. This was somewhat of a surprise in the early days of studying radioactivity, 4 but has now been well established theoretically. Similar to β-minus decay is another decay process called beta-plus (β + ) decay. In this process, the nucleus releases something called a beta-plus particle, or an anti-electron. This particle is exactly the same as an electron in every way, except is has charge +e rather than e. In the original nucleus, this effectively turns a proton into a neutron. Additionally, a neutrino, ν e is also released. This fills the same role as the anti-neutrino in beta plus decay. 5 An example of β + decay would be 15 8 O decaying into 15 7 N: 15 8 O 15 7 N + β+ + ν e. 4 One famous physicist, Wolfgang Pauli, is rumored to have said, Who ordered that? when he learned about the antineutrino. 5 Some theories claim that the antineutrino and neutrino are exactly the same particle. This is a subject of massive experimental research efforts, which usually involve looking for very rare decay events in old mine shafts located miles underground.
phys 202 notes, week 13/14 10 Like alpha decay, the total energy released in beta decay is determined by the binding energy differences between the initial and final nuclei. However, unlike alpha decay, this energy is shared between the beta particle and (anti-)neutrino: both exit the nucleus with some kinetic energy (speed). As a result, the kinetic energy of the beta particle can take on a range of values, and is not the same every time a decay occurs. Gamma decay There is a third type of radioactive decay called gamma decay (γ-decay). Unlike the other two, this occurs when the nucleus is in an excited state, which means that is has somehow been given extra energy. This extra energy is eventually released via emission of a γ-ray, which is a very high energy (short wavelength) photon. This process is almost exactly the same thing as light emission from atoms, except instead of the electron being promoted to an energy level, the protons/neutrons making up the nucleus are re-arranged. In both cases, the system de-excited by emitting a photon; in the case of atoms, this photon is in or near the visible spectrum, while in nuclei the energies involved are much higher (kev MeV), and the released photon is in the γ-ray regime. Decay rates Radioactive decay in nuclei is a spontaneous, statistical process. What this means is that if we have some sample of a radioactive nucleus, we can talk about the number of nucleons N in the sample that decay in some period of time t. This is given by the equation, N t = λn, (13) where λ is the decay constant for the nucleus. This can also be expressed in terms of the half life, T 1/2, or the amount of time it takes for half of a given sample to decay, T 1/2 = ln 2 λ = 0.693 λ. (14) Equation (??) can be rearranged to solve for the number of nuclei N remaining in the sample after time t, N = N 0 e λt, (15) where N 0 is the initial number of nuclei in the sample. A graph of N vs. time is shown in Figure??. Figure 8: Example plot showning number of nuclei in a sample N vs. time.
phys 202 notes, week 13/14 11 The term activity refers to the number of decays per second that a sample undergoes, i.e. N/ t. The SI unit of activity is the becquerel, or Bq, 1 Bq = 1 decay/second. Another unit is commonly employed, the Curie (Ci), which is equal to 3.7 10 10 Bq. 6 Nuclear Reactions 6 This is approximately the activity of one gram of Radium, which in the early days of studying radioactivity made more sense than the becquerel. So far all the processes we ve discussed pertaining to nuclei, such as radioactivity, have been natural and spontaneous. In other words, they occur with out any human intervention; all we can do is sit back and observe. However, there s also the possibility to induce nuclear reactions by smashing two nuclei together with some amount of kinetic energy. In the early days, nuclear reactions typically involved bombarding some sample with alpha particles from a radioactive sample. The first nuclear reaction study was undertaken by Rutherford, who bombarded 14 7 N nuclei with alpha particles (4 2H nuclei). In doing this, ha observed the following process, 4 2He +14 7 N 17 8 O +1 1 H. What s going on is that the 14 7 N and 4 2 He merge together, briefly form- H. Nowadays, ing a compound system that then decays into 17 8 O and 1 1 nuclear reactions are still heavily studied, but most research replaces the alpha decaying source with a nucleus accelerated by some type of particle accelerator, such as a cyclotron. Nuclear reactions always obey conservation principles for charge, momentum, angular momentum, and energy. For example, in the reaction above, the total number of protons (9) and neutrons (9) in the system doesn t change. Reaction Energy Because they re changing nuclear species, the total mass (or energy equivalent) on the left and right sides of the reaction will in general be different. This difference (expressed as an energy) is referred to as the Q-value of the reaction. For the reaction A + B C + D, the Q-value is Q = (M A + M B M C M D ) c 2. (16) Note that in this equation, the masses M A, M B, M C, M D are the neutral atom masses, i.e. the mass calculated with the contribution of atomic electrons included. Q-values can either be positive or negative. When Q is positive, the reaction is exoergic (or more commonly, exothermic), while if Q
phys 202 notes, week 13/14 12 is negative the reaction is endoergic (endothermic). In the case of endothermic reactions, the reaction cannot occur at all unless the initial kinetic energy (caused by accelerating one or both of the involved nuclei) is greater than or equal to Q. For this reason, Q is sometimes called the threshold energy of the reaction. Fission and Fusion Two categories of nuclear reaction are very important for practical (energy) applications. The first in nuclear fission. This occurs when some heavy nucleus, such as 235 92 U, breaks apart into two roughly equal-mass nuclei. This process can release very large amounts of energy, as the Q-value makes it a highly exoergic reaction. Typically, fission is induced by a neutron, and there are some leftover neutrons at the end of the reaction. For example, the total process for 235 92 U breaking up into, say, 141 56 Ba plus 92Kr is 235 36 92 U +1 0 n 141 56 Ba +92 36 Kr + 3 10 n (where the notation 3 10 n means three separate neutrons). Fission has been employed practically by creating what is called a chain reaction. The basic idea is to make use of the extra neutrons emitted in a reaction like the one shown above. These neutrons can go on to interact with other 235 92 U,7 which themselves fission, releasing energy along with more neutrons which fission other nuclei and so on. Left uncontrolled, this process results in an enormous release of energy in a very small amount of time, i.e. a nuclear bomb. 8 Fortunately, fission chain reactions can also be controlled by moderating the neutrons with special materials. This essentially assures that the chain reaction can t get out of control, leading to a stable slow release of energy. This is the basis of a conventional (fission) nuclear power plant, which are used around the world to generate electricity. Another important reaction is fusion, where light nuclei combine to create a heavier one. For example, tritium ( 3 1H) and deuterium ( 2 1 H) can combine (fuse) to form an alpha particle (4 2He) plus a neutron, 7 Or whatever nucleus is being used as the fuel for the chain reaction; for example, 239 94 Pu is another nucleus that can sustain a chain reaction. 8 Which is unfortunately often called an atomic bomb a largely incorrect term, especially considering that conventional explosives, such as dynamite are far more atomic in nature than are nuclear weapons. 3 1 H +2 1 H 4 2 He +1 0 n (17) Again, this process is highly exoergic, with a large positive Q-value. For example, in the reaction above, the Q-value is 17.6 MeV. Fusion reactions have enormous potential for peaceful energy generation. They require only hydrogen (extractable from seawater) for fuel and produce very little of the dangerous radioactive contamination that results from fission power plants. However, despite 60+ years of effort, no one has yet been able to harness them for energy. The difficulty lies in the fact that despite the reaction being exoergic,
the coulomb repulsion between the initial positively-charged hydrogen nuclei prevents them from ever coming close enough for the reaction to take place. This can be overcome by accelerating one (or both) of the nuclei to a kinetic energy that s greater than the coulomb repulsion. However, the energy required to do this is more than that released by the reaction. Current efforts at using fusion for power generation rely on confining the nuclei either with very powerful lasers or with a magnetic field. This results in conditions that allow the nuclei to become close enough for the fusion to occur. A natural example of a fusion reactor is stars, such as our sun. In this case, the massive gravitational force due to the star itself packs the nuclei close enough together that they can overcome the Coulomb repulsion. phys 202 notes, week 13/14 13