Nuclear and Quantum Physics PDF

# E Nuclear and quantum physics The Standard Model of elementary particles consists of quarks (which make up baryons, such as protons and neutrons, and mesons), leptons (like electrons and neutrinos) and bosons (which transmit forces). The Higgs boson was predicted to exist in the 1960s precisely b...

# E Nuclear and quantum physics The Standard Model of elementary particles consists of quarks (which make up baryons, such as protons and neutrons, and mesons), leptons (like electrons and neutrinos) and bosons (which transmit forces). The Higgs boson was predicted to exist in the 1960s precisely because of the Standard Model and eventually discovered through ATLAS and CMS collaboration in 2012 at CERN. Chemical reactions involve electrons. Nuclear reactions involve nucleons. All are found in atoms, the building blocks of matter. The structure of atoms has been investigated for centuries, and the contemporary nuclear model is based on experiments in which only a small proportion of charged particles were found to deflect when fired toward a thin sheet of gold. Experiments also provide the evidence for the energy levels in which electrons reside; the light that is absorbed and released by atoms due to electron energy level transfers has specific wavelengths unique to the element. Quantum physics is the study of how the smallest particles in the Universe interact. You will find out that all is not as it seems with light; just as there is ample evidence for the wave model of light, there is also experimental evidence that light behaves as discrete particles (or 'quanta'). Unsurprisingly, electrons (and other well-established particles) are just the same! They exhibit wave-like behaviors when manipulated in certain scenarios. Overall, we call these properties wave-particle duality. Looking at samples of thousands of atoms, the number of physical phenomena only continues to grow - because individual nuclei can undergo decay, fission or fusion in order to become more stable. Radioactive decay is a random and spontaneous process with the format (α, β, β+ or y) depending on the ratio of neutrons to protons present and in which a constant half-life can be determined for each given isotope. Fission is the route to stability for the heaviest nuclei and fusion the route for the lightest nuclei; binding energy per nucleon increases to a maximum value for iron-56. Chain fission reactions are harnessed using fuel rods, moderators and heat exchangers in nuclear power stations (with control rods and shielding for safety). Fusion is the source of energy in stars, where the density and temperature are high enough for positively-charged nuclei to come together under the strong nuclear force. Stars can be classified according to their temperature, luminosity, radius and lifecycle stage with the Hertzsprung-Russell diagram being the ultimate for displaying them. You will come to be amazed by the stellar detective work that can be carried out based on just one or two pieces of information. ## E.1 Structure of the atom <start_of_image> Schematic of the periodic table with highlighted areas showing the chemical elements: Cobalt, Nickel, Copper, Ruthenium, Rhodium, Palladium, Silver, Osmium, Iridium, Platinum, and Gold. The periodic table is an arrangement of types of atom - the chemical elements. Chemists make use of the reactivity patterns across the periods and down the groups, while physicists pay more attention to the composition of atomic nuclei and the energies of the electrons that surround them. ### Guiding Questions * What is the current understanding of the nature of an atom? * What is the role of evidence in the development of the model of the atom? * In what ways are previous models of the atom still valid despite recent advances in understanding? The development of atomic theory is often used as an example of how the scientific method works, but the process as presented in this chapter is a very selective view of events. This was not the only research being carried out in atomic physics. Trying to model the atom is like trying to work out what is in a present without opening the packaging. Imagine you received the object shown in Figure 1. If it was placed in a large box and you rolled it around inside the box, you might think it was a solid cube. However, if you fired pellets at the box, they would pass straight through, except for the occasional pellet that would hit the larger central ball. This would imply that the object was mostly space with a heavy center. The next step might be to blow up the box and attempt to catch the pieces. This would reveal finer details, but only if you were standing in the path of a piece as it moved outward. Students should understand: * the Geiger-Marsden-Rutherford experiment and the discovery of the nucleus * nuclear notation $^A_ZX$ where A is the nucleon number, Z is the proton number and X is the chemical symbol * emission and absorption spectra provide evidence for discrete atomic energy levels * photons are emitted and absorbed during atomic transitions * the frequency of the photon released during an atomic translation depends on the difference in energy level as given by $E = hf$ * emission and absorption spectra provide information on the chemical composition * the relationship between the radius and the nucleon number for a nucleus as given by $R = R_0A^{1/3}$ and implications for nuclear densities * deviations from Rutherford scattering at high energies * The distance of closest approach in head-on scattering experiments. * the discrete energy levels in the Bohr model for hydrogen as given by $E = \frac{-13.6}{n^2}$ eV * the existence of quantized energy and orbits arise from the quantization of angular momentum in the Bohr model for hydrogen as given by $mvr = \frac{nh}{2\pi}$. ### The arrangement of charge in the atom * A schematic of a Thomson's plum pudding model of an atom with a positively charged nucleus with negatively charged electrons. * A schematic diagram showing that the electron is a fundamental particle with charge -1.60 x 10^-19 C and a mass of 9.110 x 10^-31 kg. We already know that matter is made up of particles (atoms) and we used this model to explain the thermal properties of matter. We also used the idea that matter contains charges to explain electrical properties. Since matter contains charge and is made of atoms, it seems logical that atoms must contain charge. But how is this charge arranged? There are many possible ways that charges could be arranged in the atom, but since atoms are not themselves charged, they must contain equal amounts of positive and negative charge. Maybe half the atom is positive and half is negative, or perhaps the atom is made of two smaller particles of opposite charge? The discovery of the electron by J. J. Thomson in 1897 added a clue that helps to solve the puzzle. The electron is a small negative particle that is responsible for carrying charge when current flows in a conductor. By measuring the charge-to-mass ratio of the electron, Thomson realized that electrons were very small compared to the whole atom. He therefore proposed a possible arrangement for the charges as shown in Figure 2. This model was called the 'plum pudding' model. This model was accepted for some time until, under the direction of Ernest Rutherford, Geiger and Marsden performed an experiment that proved it could not be correct. ### Scattering experiments * A schematic diagram showing 4 boxes, each containing a different object: "glass ball", "steel ball", "air" and "sand". A pellet is used as a projectile to determine the contents of each box: a glass ball would shatter, a steel ball would bounce back, air would let the pellet pass through, and sand would stop the pellet. The problem with trying to find out what is inside an atom is that the atom is far too small to see. Imagine you have four identical boxes and each contains one of the following: a large steel ball, a glass ball, air or sand. You have to find out what is inside the boxes without opening them. One way of doing this is to fire a pellet at each. Here are the results: 1. Shattering sound → glass ball 2. Bounces back → steel ball 3. Passes straight through → air 4. Does not pass through → sand Different situations need different projectiles. If, for example, one box contained a large cube, then a projectile smaller than the cube would be fine. If the big cube was made out of smaller cubes, you would need a projectile so small that it could pass between the cubes or one with so much energy that it would knock some of the small cubes out of the box. ### The Rutherford model * A schematic diagram showcasing Rutherford's experiment: alpha particles are fired at a thin sheet of gold foil, and the alpha particles' deflection patterns are observed. Rutherford's idea was to shoot alpha particles at a very thin sheet of gold to see what would happen. In 1909, very little was known about alpha particles - only that they were fast and positive. In accordance with normal scientific practice, Rutherford applied the model of the day to predict the result of the experiment. The current model was that the atom was like a small plum pudding, so a sheet of gold foil would be like a wall of plum puddings, a few puddings thick. Shooting alpha particles at the gold foil would be like firing pellets at a wall of plum puddings. If we think what would happen to the pellets and puddings, it will help us to predict what will happen to the alpha particles. If you shoot a pellet at a plum pudding, it will pass through and out the other side. If you were to shoot a positive alpha particle at 'plum pudding' atoms, within which negative charges are evenly distributed, the alpha particles, overall, should be undeflected. What actually happened was, most alpha particles passed through without changing direction, but a significant number were deflected and a few even came straight back, as shown in Figure 4. This was so unexpected that Rutherford said: 'It was quite the most incredible event that ever happened to me in my life. It was almost as incredible as if you had fired a 15-inch shell at a piece of tissue paper and it came back and hit you.' We know from our study of collisions that you can only get a ball to bounce off another ball if the second ball is much heavier than the first. This means that there must be something heavy in the atom. The fact that most alpha particles pass through means that there must be a lot of space. If we put these two findings together, we conclude that the atom must have something heavy and small within it. This small, heavy thing is not an electron since they are too light. It must therefore be the positive part of the atom. We call this dense, positive center of the atom the nucleus. This would explain why the alpha particles come back, since they are also positive and are repelled from it. ### Charge and mass * A table showing the mass and charge of a proton and a neutron, highlighting the difference in their mass, but noting that both are approximately equal. Nuclear research was undoubtedly hurried along by the strategic race to be the first to build an atom bomb. Many other advancements in science have also been pushed forward due to military, political or financial concerns. In the 1860s, chemists calculated the relative mass of many elements by measuring how they combine to form compounds. If placed in order of atomic mass, the chemical properties of the elements seemed to periodically repeat themselves. This led to the periodic table that chemistry students will be familiar with. There were, however, some anomalies where the order in terms of chemical properties did not match the order of mass. In 1911, Rutherford's scattering experiments not only revealed the existence of the nucleus but made it possible to calculate the charge of the nucleus. This was found to be the same whole number of positive electron charges for all atoms of the same element. This number is not the same as the mass number, but when the elements were placed in order of this 'charge number', then the anomalies were sorted. To summarize: * The mass of all atoms is (approximately) a multiple of the mass of a hydrogen atom, the 'mass number'. * The charge of all nuclei is a multiple of the charge of a hydrogen nucleus, the 'charge number'. * The 'charge number' is not the same as the 'mass number'. Since all nuclei are multiples of hydrogen nuclei, one might imagine that all nuclei are made of hydrogen nuclei. If this was the case, then the charge number would be the same as the mass number, but it is not. For example, helium has a relative atomic mass of 4 but a charge of +2e. There appear to be two extra particles that have approximately the same mass as a hydrogen nucleus but no charge. This particle is the neutron, which was discovered by Chadwick in 1932. So the nucleus contains two types of particle, shown in Table 1. | Particle | Mass/kg | Mass/u | Charge/C | |---|---|---|---| | Proton | 1.673 x 10^-27 | 1.007276 | +1.60 x 10^-19 | | Neutron | 1.675 x 10^-27 | 1.008665 | 0 | Nucleons are the particles of the nucleus (protons and neutrons). A particular combination of nucleons is called a nuclide. Each nuclide is defined by three numbers: * nucleon number (A) = number of protons + neutrons (defines the mass of the nucleus) * proton number (Z) = number of protons (defines the charge of the nucleus) * neutron number (N) = number of neutrons (A - Z). Isotopes are nuclides with the same proton number but different nucleon numbers. ### Size of the nucleus * A schematic diagram of an alpha particle approaching a gold nucleus head-on, showing the path the alpha particle takes and the variables used in the calculations: velocity (ν), mass (m), charge (Q) and distance (r). The size of the nucleus can be determined by conducting a similar experiment to the alpha scattering experiment of Geiger and Marsden. The alpha particles that come straight back off the gold foil must have approached a nucleus head-on, following a path as shown in Figure 6. Applying the law of conservation of energy to this problem, we can deduce that at point P, where the alpha particle stops, the original kinetic energy has been transferred to electrical potential energy. $\frac{1}{2}mv^2 = k\frac{Qq}{r}$ Where: * Q = the charge of the nucleus (+Ze) * q = the charge of the alpha particle (+2e) The kinetic energy of the alpha particle can be calculated from the change in the mass of a nucleus when it is emitted (more about this later), so, knowing this, the distance *r* can be calculated. To determine the size of the nucleus, faster and faster alpha particles are sent toward the nucleus until they no longer come back. The fastest ones that return have got as close to the nucleus as possible. This is just an estimate of the nuclear radius, especially since (as is the case for all particles) the position of the particles that make up the nucleus is determined by a probability function. This will make the definition of the edge of the nucleus rather fuzzy. From the results of experiments like this, we know that the radius of a nucleus is approximately 10-15 m. By measuring the radii of different nuclei, it has been found that $R = R_0A^{1/3}$, in which $R_0$ is a constant known as the Fermi radius, 1.20 × 10-15 m. This experimental result reveals that the density of all nuclei is the same. Cubing both sides of the equation shows that $R^3$ is proportional to A, which means that volume is proportional to mass, while density is constant. This also tells us that the nuclear force is very short range. If it obeyed an inverse square law, like gravity, the nucleons on the surface would be attracted to all the nucleons inside, which would have the effect of squashing the inside, making heavier nuclei more dense (as happens in stars). ### Electrons So, the atom consists of a heavy but very small positive nucleus surrounded by negative electrons. But what stops the electrons falling into the nucleus? One idea could be that the atom is like a mini solar system with electrons orbiting the nucleus, similar to how the planets orbit the Sun. The circular motion of the electrons would make it possible for them to accelerate toward the center without getting any closer. One problem with this model is that if an electron were to move in this way, it would create a changing electric and magnetic field, resulting in emission of electromagnetic radiation. This would lead to a loss of energy and the electron would spiral into the nucleus. To gain more insight into the structure of the atom, we need to look in detail at the relationship between light and matter. ### The connection between atoms and light There is a very close connection between matter and light. For example, if we give thermal energy to a metal, it can give out light. Light is an electromagnetic wave so must come from a moving charge. Electrons have charge and are able to move, so it would be reasonable to think that the production of light is something to do with electrons. But what is the mechanism inside the atom that enables this to happen? ### Atomic spectra * A schematic of a discharge tube containing bromine, hydrogen, and helium gas. * A schematic of the line spectrum for hydrogen, showcasing the discreet energy levels of its electrons. Before we can answer that question, we need to consider the nature of light, in particular, light that comes from isolated atoms. We must look at isolated atoms because we need to be sure that the light is coming from single atoms and not the interaction between atoms. A single atom would not produce enough light for us to see, but low-pressure gases have enough atoms far enough apart not to interact. To analyze the light coming from an atom, we need to give the atom energy. This can be done by adding thermal energy or electrical energy. The most convenient method is to apply a high potential to a low-pressure gas contained in a glass tube (a discharge tube). This causes the gas to give out light, and already you will notice (see Figure 7) that different gases give different colors. To see exactly which wavelengths make up these colors, we can split up the light using a prism (or diffraction grating). To measure the wavelengths, we need to know the angle of refraction. This can be measured using a spectrometer. **The hydrogen spectrum** Hydrogen has only one electron, so it is the simplest atom and the one we will consider first. Figure 8 shows the spectrum obtained from a low-pressure discharge tube containing hydrogen. The first thing you notice is that, unlike a usual rainbow, which is continuous, the hydrogen spectrum is made up of thin lines. Light is a form of energy, so whatever the electrons do, they must lose energy when light is emitted. If the color of light is associated with different energies, then, since only certain energies of light are emitted, the electron must only be able to release certain amounts of energy. This would be the case if the electron could only have certain amounts of energy in the first place. We say the energy is **quantized**. To help us understand this, we can consider an analogous situation of buying sand. You can buy sand loose or in 50kg bags, and we say the 50 kg bags are quantized, since the sand comes in certain discrete quantities. So if you buy loose sand, you can get any amount you want, but if you buy quantized sand, you have to have multiples of 50 kg. If we make charts showing all the possible quantities of sand you could buy, then they would be as shown on Figure 9; one has discrete values and the other is continuous. If the electron in the hydrogen atom can only have discrete energies, then when it changes energy, these changes must also be in discrete amounts. We represent the possible energies on an energy level diagram (Figure 10), which looks rather like the sand diagram. For this model to fit together, each of the lines in the spectrum must correspond to a different energy change. Light therefore must be **quantized** and this does not tie in with our classical view of light being a continuous wave that spreads out like ripples in a pond. ### The quantum nature of light * A schematic diagram showing the energy levels of a hydrogen atom. Light definitely has wave-like properties; it reflects, refracts, diffracts and interferes. But sometimes light does things that we do not expect a wave to do, and one of these things is the photoelectric effect. The photoelectric effect is an interaction between particles of light (photons) and electrons, in which photons of sufficient energy can cause the emission of electrons from a metal surface after being absorbed by atoms. The energy of a photon is related to the frequency of the electromagnetic radiation. This is the mechanism found in solar panels. Higher Level students will explore the photoelectric effect in detail in E.2. ### Quantum explanation of atomic spectra We can now put our quantum models of the atom and light together to explain the formation of atomic spectra. To summarize what we know so far: * Atomic electrons can only exist in certain discrete energy levels. * Light is made up of photons. * When electrons lose energy, they give out light. * When light is absorbed by an atom, it gives energy to the electrons. We can therefore deduce that when an electron changes from a higher energy level to a lower energy level, a photon of light is emitted. Since the electron can only exist in discrete energy levels, there are a limited number of possible changes that can take place. This gives rise to the characteristic line spectra that we have observed. Each element has a different set of lines in its spectrum because each element has different electron energy levels. To make this clear, we can consider a simple atom with electrons in the four energy levels shown in Figure 11. ### Ionization Ionization occurs when the electrons are added to or removed from an atom, leaving a charged atom called an ion. Electrons can be removed if the atom absorbs a high-energy photon or the electron could be 'knocked off by a fast-moving particle like an alpha particle. These interactions are quite different. When a photon interacts with an atom, it is absorbed. However, alpha particles are positively charged, which means they can attract electrons. Removal of an electron from the ground state of a hydrogen atom requires 13.6 eV of energy (Figure 13). ### Absorption of light A photon of light can only be absorbed by an atom if it has exactly the right amount of energy to excite an electron from one energy level to another. If light containing all wavelengths (white light) is passed through a gas, then the photons with the right energy to excite electrons will be absorbed. The spectrum of the light emitted will have lines missing. This is called an absorption spectrum and is further evidence for the existence of electron energy levels. The core of the Sun is very hot so the atoms move around with high speed. When the atoms collide with each other, they knock all of the electrons out of their energy levels. This material is called a plasma and because the atoms do not contain electrons, they cannot emit light of visible light frequencies. Instead, they emit gamma radiation, which is absorbed by the outer layer of the star, increasing the temperature. The outer layer atoms still have electrons in place so give out light of all wavelengths. This light passes through the outermost low-density gas, where certain wavelengths are absorbed by discrete energy changes in the atomic electrons. This leaves dark absorption lines in the spectrum of light from any given star, enabling us to identify the chemical composition (which elements are present). ### The Bohr model * A schematic of the Bohr model of a hydrogen atom: showing the nucleus with a positive charge, and the electrons with negative charges orbiting the nucleus at specific energy levels. We can see from the spectrum of hydrogen that the energy of atomic electrons can only have discrete values, but we do not have a model for how the electrons could be arranged around the atom. In 1913, Niels Bohr proposed that if the electrons were in certain specific orbits then they would not emit electromagnetic radiation. The radii of these orbits were defined in terms of the angular momentum of the electron: this is the angular equivalent of linear momentum. If a body, mass *m*, is traveling in a circle radius *r* with constant speed *v*, it will have angular momentum *mvr*. As with linear momentum, angular momentum is conserved, provided no tangential forces act on the body. According to the Bohr model, an electron will be in a stable orbit if $mvr = \frac{nh}{2\pi}$, where *n* is a whole number called the quantum number. We can show how this leads to a quantization of energy by considering the orbiting electron in Figure 14: $F_c = F_e$ Centripetal force = electrostatic attraction $\frac{mv^2}{r} = \frac{e^2}{4\pi\epsilon_0r^2}$ Rearranging and multiplying by *mr* gives: $m^2v^2r^2 = \frac{me^2r}{4\pi\epsilon_0}$ So, according to Bohr: $m^2v^2r^2 = \frac{n^2h^2}{4\pi^2}$ This gives that: $r = \frac{n^2h^2}{4\pi^2me^2\epsilon_0}$ This gives the radii of all allowed orbits. Now, the energy of the electron is: $E = E_k+E_p = \frac{1}{2}mv^2 - \frac{e^2}{4\pi\epsilon_0r}$ But from equation (1), $mv^2= \frac{e^2}{4\pi\epsilon_0r}$, so: $energy = \frac{e^2}{8\pi\epsilon_0r} - \frac{e^2}{4\pi\epsilon_0r}$ Substituting for *r* gives: $E= \frac{-me^4}{8n^2h^2\epsilon_0^2} = \frac{-13.6}{n^2}$ eV This predicts that the electron energies will have discrete values that get closer together as the energies increase. This closely matches the energy level diagram that was derived from spectral analysis. ### Exercise 1. Calculate the radius of the lowest orbit of an electron in a hydrogen atom based on the Bohr model. 2. Use the Bohr model to calculate the frequency of electromagnetic radiation emitted when an electron in a hydrogen atom moves from the second orbit down to the first. From close observation of spectral lines, we see that they are not all the same intensity. This implies that not all transitions are equally probable. Bohr's model cannot predict this detail and, although it works very nicely for hydrogen, it does not work for any other atom. To create a more accurate model, we would need to look at matter in a different way. ## Guiding Questions revisited * What is the current understanding of the nature of an atom? * What is the role of evidence in the development of the model of the atom? * In what ways are previous models of the atom still valid despite recent advances in understanding? In this chapter, we have explored the experimental evidence relating to the nature of the atom to give the current understanding that: * Positive protons are located in a dense central nucleus, based on the findings of the Geiger-Marsden-Rutherford experiment. * Neutral neutrons are also located in the nucleus because the masses and charges of consecutive nuclei do not increase in equal steps. * Nuclei have constant density irrespective of the number of nucleons, based on closest-approach measurements to estimate nuclear radii. * Electrons are located in quantized energy levels, and we can measure the differences between them using the emission spectra of excited atoms. * The Bohr model accurately predicts the energy levels for hydrogen. Although human understanding of atomic structure has improved in its precision over time because of scientific open-mindedness, some aspects of previous models (e.g. the existence of the electron and the balance of charges) remain. ### Practice questions 1. (a) The element helium was first identified from the absorption spectrum of the Sun. (i) Explain what is meant by the term *absorption spectrum*. (ii) Outline how this spectrum may be experimentally observed. (b) One of the wavelengths in the absorption spectrum of helium occurs at 588 nm. (i) Show that the energy of a photon of wavelength 588 nm is 3.38 x 10^-19 J. (ii) The diagram represents some the energy levels of the helium atom. Use the information in the diagram to explain how absorption at 588 nm arises. * A schematic diagram showing the energy levels of a helium atom.