A is called the mass number gives, roughly, the mass of the nucleus or atom in atomic mass units = amu = u

5/5 A is called the mass number gives, roughly, the mass of the nucleus or atom in atomic mass units = amu = u The number of neutrons in the nucleus is given by the symbol N. Clearly, N = A Z. Isotope: Two nuclei are isotopes if they contain the same number of protons but different numbers of neutrons. Same Z but different N or A. Here are some isotopes: Hydrogen: Heavy Hydrogen or Deuterium: Tritium: Uranium-235: Uranium-238: Isotone: Two nuclei are isotones if they contain the same number of neutrons and different number of protons same N, different Z or A. Example: Isobar: Same A but different Z and N. Example: Isomer: Same Z and same N but in different electrical energy state - proton excited like an electron in the atom. Note that we believe that the neutron and proton are two different charge states of a particle called a nucleon that is, a particle found in the nucleus of the atom. What holds the nucleus together? Like charges repel, so the protons in a nucleus should fly apart. There must be a force stronger than electromagnetism holding the nucleus together strong nuclear interaction. If this force is so much stronger 137 times stronger than the electrostatic interaction, why don t we experience it in our everyday lives? The strong nuclear interaction is a ranged force it is very strong but, beyond a certain range roughly the size of a nucleus, it drops to zero. There is another force called the weak nuclear interaction, about 100,000 times weaker than electromagnetism. This interaction is responsible for certain types of radioactive decay. Since the protons in the nucleus are constantly pushing against each other, many nuclei are

unstable. They will decay. Properties of the Strong Nuclear Interaction: 1. As noted, it is a ranged force. 2. Depends on the direction of the spins of the particles if parallel, stronger than if antiparallel. We know this because the deuterium nucleus is a combination of a neutron and a proton is stable. Since the particles are different, the can be in a state with spins aligned. The diproton and dineutron are both unstable because they must have spins in opposite directions. 3. It has an angular dependence. 4. The force saturates only nearest neighbor nucleons apply forces to each other. 5. It is a many-body force The electrostatic force is a single-body force. Suppose two charged particles 1 and 2 are located in a region of space. Particle 2 applies a given force on particle 1. Add a third particle, particle 3, to the system. It will add to the total force on particle 1, but it won t affect the force of 2 on 1. For nucleons, this is not the case. If I bring an additional nucleon into a system, it will affect the force between the first two. 6. Can t write down a formula for it the way we can for gravity. Radioactivity: Three Types of Radioactivity: Alpha, Beta, and Gamma It was discovered that the alpha and beta were deflected in opposite directions in a magnetic field and the gamma, not at all. Alpha - positive, Beta- negative, Gamma - neutral Beta deflected more than the alpha, suggesting alphas much slower and more massive. It turns out the alphas are helium nuclei, betas are electrons, and gammas are high-energy photons. Typical emissions of these particles: Alpha U-238 Note that the A s and Z s must individually add up on both sides: Note that the total number of nucleons after the decay is the same as that before. The same goes for the total number of protons charge must be conserved.

Beta H-3 The tritium nucleus decays into a He-3 nucleus with the emission of an electron and an electron antineutrino. Note the small change in notation. The electron does not contain 1 protons; we now interpret the subscript as giving the charge of the particle in units of the fundamental charge. The antineutrino is necessary to conserve energy and to conserve angular momentum. A nucleus with three nucleons has half-odd-integer spin as does the electron. Without the antineutrino, the right-hand side of the decay would have integer spin. Gamma Ba-137* Often a nucleus that results from a radioactive decay is electromagnetically excited and emits a photon just as an excited atom emits a photon. The asterisk on the left-hand side of the decay indicates that the nucleus is an isomer of the nucleus in its ground state. That is, it is an excited state of the nucleus. The symbol indicates the gamma ray or photon. One more term: the beginning nucleus is called the parent nucleus and the final nucleus is called the daughter nucleus. Nuclear Stability Based on E = mc 2. For a decay to occur, the parent nucleus must have more energy than the daughter nucleus plus the emitted particles since reactions go from higher energy to lower energy. This means that the mass of the parent must be greater than the total mass of the products. You can t use the atomic mass number (since these masses are averages over all isotopes) need a table of atomic masses which gives the mass of each isotope. Radioactivity is a statistical process: In a given time interval there is a given probability that a nucleus will decay. If we choose a time interval so that the probability is one-half, the time is called the half-life of the nucleus. The half-life is the time for half of the nuclei in a sample to decay. Note: after one half-life, half left; after two half-lives, one quarter left, after three half-lives, one eighth left, and so on. We can see how alpha decay is a statistical process as follows. Recall that an alpha particle is a helium nucleus.

Just as the atom helium is extremely stable, so it its nucleus. It is the only doubly extremely stable atom/nucleus known. Consider the nucleus uranium-238 with 92 protons and 146 neutrons. Within the nucleus, the protons and neutrons are constantly moving around. Periodically, helium nuclei will form because the structure is so stable but they are broken apart after a while. We need to look at the energetics of the nucleus we will look at a model of the potential energy seen by a helium nucleus in the uranium nucleus. In the picture at right, the strong potential energy of attraction due to the strong nuclear interaction is represented by the deep square well. The symbol r 0 represents the range of the interaction. The decreasing tail is the electrostatic repulsion seen by the alpha particle once it leaves the nucleus. Inside the nucleus there is a certain probability that two protons and two neutrons will come together to form an alpha particle for a short period of time. The alpha particle a helium nucleus, remember is represented by the combination of two blue protons and two yellow neutrons. While the alpha particle is together, there is a small probability that it will leak out of the nucleus or tunnel out of the nucleus. This is shown by the wave function for the alpha particle tunneling out of the potential barrier. Why don t individual protons and neutrons leak out? To be energetically favorable, the sum of the masses of the individual decay products must be less than the mass of parent nucleus. This follows from Einstein s famous equation E = mc 2, which means in this case that the total energy of parent nucleus must be greater than the sum of the energies of the decay products. The difference in energy is the kinetic energy of the decay products. Proton emission from a nucleus is not energetically favorable The mass of a uranium-238 nucleus is less than the mass of a uranium-238 nucleus minus a proton protactinium-237 plus the mass of a free proton. 238.0507882 u < 237.05115 + 1.00782503207 = 238.05898 u However, the energy of the uranium-238 nucleus is greater than the energy of a uranium-238 nucleus minus an alpha particle thorium-234 and a free alpha particle. 238.0507882 u > 234.043601 u + 4.00260325415 u = 238.045204 u The mass used here are actually the masses of the atoms the masses of the electrons subtract out. You can find these masses at the NIST (National Institute of Standards and Technology) Web site.

Radioactive Dating Consider Carbon Dating There is carbon-14 created by cosmic ray collisions with carbon atoms in carbon dioxide in the upper atmosphere. This carbon-14 is constantly being replenished in our environment. Living organisms are constantly taking in and expelling carbon-14 the carbon-14 content of a living organism is the same as that in its environment. When a living organism dies, it stops taking in carbon-14. The ratio of carbon-14 to carbon-12 within the body begins to drop. The half-life of carbon-14 is roughly 6000 years, so after 6000 years, the ratio of carbon-14 to carbon-12 will have dropped to half of its original value. By measuring the ratio of carbon-14 to carbon-12 in a once living organism, we can find the age of that organism. Doesn t work well if the organism died within the past, say, thousand years or so. It also doesn t work well for organisms that died more than six or seven half-lives ago. Carbon-14 dating works for roughly a thousand years to sixty thousand years. To measure the ages of ancient rocks, we use uranium-238, which has a half-life of 4.5 billion years. The universe is, roughly, 13.7 billion years old, and we can measure rocks that old. Oldest measured rocks on Earth 3 billion years. Moon rocks 4 billion Meteorites 4.5 billion years old. Chapter 34 - Nuclear Fission and Fusion We can get energy by taking a large nucleus and splitting it into two pieces - fission. We can get energy by taking two small nuclei and putting them together to make a larger nucleus - fusion If we plot the stability of nuclei binding energy per nucleon against the atomic number, we find that it

rises to a maximum at iron and then drops for large nuclei. Note: the binding energy of a nucleus is the total energy needed to separate the nucleus into its constituent nucleons. The Bomb Some nuclear species will spontaneously fission: uranium-235 and plutonium-239 In the fission process, the large nucleus is divided into two smaller nuclei with the emission of a few neutrons. If these neutrons hit other large nuclei, they can induce them to fission as well. If they do, we get a chain reaction. To generate a chain reaction, you need a certain amount of material called the critical mass. The critical mass is determined by the following condition: If we have a small amount of uranium-235, the neutrons that are produced in a spontaneous fission will escape from the material before they cause another fission. No chain reaction. If, on the average, more than one neutron per spontaneous fission causes a second fission, a chain reaction will occur. The critical mass is the mass at which on the average one neutron per spontaneous fission will cause an additional fission. How do we make a bomb? Put 60% of the critical mass of uranium at one end of the bomb and another at the other end of the bomb. At the front end of the bomb, set a charge TNT. Set off the TNT, and it pushes the two 60% critical masses together and BOOM! Note that the products from a nuclear fission reaction are extremely radioactive. Small nuclei tend to more stable if they have equal numbers of protons and neutrons. In larger nuclei, though, the electrostatic repulsion between protons tends to make such nuclei less stable, so more neutrons are required. The products of a fission event will have way to many neutrons to be stable and will be highly radioactive. The will be beta emitters. Nuclear Fission Reactor for Producing Electric Power Uranium-238 is used in reactors. Note that uranium-238 does contain some uranium-235. At the core of the reactor, we have some moderating material either water or heavy water. The purpose of the moderator is to slow the neutrons emitted when a uranium-235 nucleus

spontaneously fissions fast neutrons will not cause uranium-238 to fission, but slow neutrons will. We insert fuel rods of uranium-238 into the moderating liquid. A nucleus of uranium-235 within the uranium-238 will fission and produce fast neutrons that will escape from the fuel rod into the surrounding liquid. There they are slowed before entering another fuel rod to cause a uranium- 238 nucleus to fission. A second set of rods made of, say carbon, called control rods, can be inserted in the reactor core to absorb neutrons. These rods control the rate of the reaction. The moderating water flows through the core to carry the heat produced outside the reactor core. This water runs through a heat exchanger and heats other water that is used to run a turbine to turn a generator that produces electricity. A reactor will never explode, but if the reaction gets away, the reactor will become so hot that it will melt down the hot radioactive material will get into the environment and produce a major catastrophe. Nuclear Fusion How the Sun generates its energy Four protons two of which are converted to neutrons along the way combine together to form a helium nucleus. The mass of the helium nucleus is less than the mass of the four protons the mass difference is converted to energy. 4 million metric tons is converted to energy each second to power the Sun. Fusion Bomb hydrogen bomb. In the bomb, hydrogen is fused into helium with a huge release of energy. Fusion bombs can be made more explosive than fission bombs because there is no critical mass and they produce much less radiation. Most of the radiation produced when a fusion bomb explodes is due to the fact that a fission bomb is needed to provide the required high temperatures: the fission bomb is the detonator for the fusion bomb. What about fusion reactors? We have been working on developing controlled fusion reactions here on Earth since the 1950's not much success so far.

The temperatures required for fusion are so high that any substance on Earth would vaporize. To contain such a hot plasma, we must use a magnetic bottle we set up a magnetic field that will keep the hot plasma from contacting the walls of the container. We need a coil 2 km long not practical. Bend the coil around into a donut or torus. This localizes the fusion reaction but it introduces plasma instabilities that keep the plasma from lasting long enough. Called Tokamak machines. A different idea at Lawrence Livermore National Laboratory: laser fusion. Laser beam hits a pellet of deuterium and heats it to a high enough temperature to fuse. Advantages of Fusion Reactors over Fission Reactors: 1. No possibility of a major nuclear accident; if something goes wrong, it will stop. No melt down as would occur with fission reactor. 2. Fusion reactors have no radiation problems. Spent fuel from a fission reactor will remain highly radioactive for tens of thousands of years. 3. Fuel much more deuterium on Earth than there is Uranium on Earth. 4. Fusion produces much more energy per kg of deuterium than fission does per kg of uranium. The only problem they both share is thermal pollution. Chapter 35 - The Special Theory of Relativity Problem in the latter part of the 19 th century Newton s mechanics and Maxwell s electromagnetism were inconsistent. Maxwell s equations predict electromagnetic waves traveling at the speed of light. Consider the picture at right. An observer on the railroad flatcar is moving toward three people: a baseball pitcher, a trumpet player, and a kid with a flashlight. A baseball pitcher can throw a baseball at, say, 100 mph, which is about 50 m/s. Suppose the speed of the flatcar is also 50 m/s. The observer will measure the speed of the baseball to be 100 m/s. The speed of sound in air at room temperature is about 340 m/s. The observer will measure the speed of sound from the trumpet to be 390 m/s. The speed of light from the flashlight is 299,792,458 m/s. The observer on the flatcar should

measure the speed of light to be 299,792,508 m/s. The observer, though, measures the speed of light to be 299,792,458 m/s, the same as would be measured by the kid with the flashlight. Newtonian physics is the same in all frames of reference that are not accelerated. Maxwell s equations seem to suggest that there is a preferred reference frame, which is the one in which light has speed c. We can understand why the observer measures the speed of sound to be 390 m/s. The speed of sound in air the medium of the sound is 340 m/s. Assuming there is no wind, that is what the trumpet player will measure. But the observer is at 50 m/s relative to the air. Thus the speed of sound measured by the observer is the sum of the speed of sound relative to the air and the speed of the observer relative to the air. Physicists thought that light should have a medium as well: the luminiferous ether. The preferred frame of reference is the one in which the ether is at rest, which would be the frame in which light will have the speed predicted my Maxwell s equations.