[MUSIC] During this second module, we deal with nuclear physics and its applications. In this second video, we will summarize what is known about the size and the spin of nuclei. The goals for you are to know how nuclear size is measured and what the results are. To know general facts about the spin of nuclei. And to be able to describe the valley of stability of nuclei as a function of the number of protons and neutrons. The size of a subatomic object must be carefully defined. In a quantum system, it is given by the root mean square of the eigenvalue of the coordinate operator in its ground state. For an atom, this is the root mean square of the radial position of the electron, which is farthest away from the nucleus. This distance can be calculated because we know perfectly the binding electromagnetic force and because it is defined with respect to a static reference point, the position of the nucleus. For the nucleus, we do not have a simple description of these actions. So we must interpret the results of experiments, which probe the distribution of nucleons inside the nucleus. It will be unwise to use hadronic probes to do so because they are sensitive to the nuclear force. High energy electrons, on the contrary, can penetrate inside the nucleus and their scattering maps out the charge distribution of the target. The cross section for a point-like target without spin is given by the Mott formula displayed here. If the charge instead is distributed according to a volume density rho(x), leaving the total charge intact, the cross section will be reduced by form factor F(q), which is a function of the momentum transfer q. For a static target, F(q) is the Fourier transform of the spatial charge distribution. For small momentum transfers, we can develop the form factor in a Taylor series. If the distribution is spherically symmetric, the terms with an odd power drop out. And the dominant second term is proportional to the mean square radius, <r^2> of the charge distribution. And this can serve as a size estimator for the nucleus. For an exponential charge distribution, the form factor takes what is called a dipolar form. The dependence of the electron nucleus cross section on the momentum transfer for small angle scattering at high electron energies is thus used to measure the size of the nucleus. Scattering experiments establish a simple relation between the radius of a nucleus R and the number of nucleons A. The radius is proportional to the cube root of the number of nucleons, with a universal proportionality constant of 1.2 femtometers. Nuclei are thus indeed small compared to the atomic size. The nuclear volume is proportional to A. This corresponds to densely packed and incompressible nucleons, which do not fuse. The mass density of nuclear matter is of the order of 10^14 grams per cm^3. Nuclear spin is the sum of the spin S of all individual nucleons and the relative angular momentum, L. Protons and neutrons are fermions with spin one-half. Like in atoms, the nuclear angular momentum L follows an integer quantum number. The total should therefore be a half integer number if A is odd, an integer if A is even. Indeed, all nuclei with both N and Z even have nuclear spin 0. Heavy nuclei have rather small nuclear spin in their ground state. We conclude that neutrons and protons tend to arrange in pairs of opposite spin direction. Every charged particle has a magnetic dipole moment associated with its spin, including the nuclei. The order of magnitude for an electron is given by the Bohr magneton, µ_B. The nuclear magneton is three orders of magnitude smaller due to the larger proton mass. The gyromagnetic factor g measures the ratio between the angular momentum and the magnetic moment. For a point charge, g is about 2 with small deviation of order 10^-3 for electrons and muons as we will see in module 4. The magnetic moments of proton and neutron are considerably different, +2.79 and -1.91 nuclear magnetons, respectively. This is the first indication of a charged substructure of the nucleons. In fact, since the neutron has zero net charge, it must contain charged particles. All nuclei have measured magnetic moments between minus three and ten nuclear magnetons, thus, relatively small ones. This is a consequence of the spin pairings of nucleons, leading to a limited total nuclear angular momentum. Most nuclei and their isotopes are unstable. Stable nuclei are found in a narrow band, in the N-Z diagram which is called the valley of stability. The valley has the following shape. For lighter nuclei with atomic mass less than 40, the number of neutrons equals the number of protons. For heavy nuclei with atomic mass bigger than 40, the number of neutrons is about 1.7 times the number of protons. This indicates that for heavy nuclei, the charge density, and thus the Coulomb repulsion must be diluted by additional neutrons. The decay of unstable nuclei is the source of nuclear radioactivity. Alpha radioactivity is the emission of He-4 nuclei or so-called alpha particles. Beta radioactivity is the emission of electrons or positrons together with neutrinos. And gamma radioactivity is the emission of photons. Nuclei with a surplus of neutrons can be stabilized by converting a neutron into a proton. And those with the surplus of protons can convert a proton into a neutron. These are isobar decays of type beta plus or minus. Heavy nuclei often decay into a pair of lighter nuclei. This corresponds to spontaneous fission, often by emitting a He-4 nucleus. This is the alpha decay. This decay is normally related to an excited state of the daughter nucleus. And they're often followed by a gamma decay towards the ground states. We will enter into more detail on these processes in the fourth and fifth video of this module. So let us summarize what we have learned about the nuclear force. It has a very short range limited to the nuclear size. The binding energy per nucleon it leads to is independent of the size of the nucleus. The nucleon thus interacts only with its nearest neighbors. The nuclear force is attractive and much stronger than the Coulomb repulsion between protons. But it must also have a repulsive component at distances compatible to the size of the nucleon, which is about 1 femtometer. This is due to the existence of quarks inside the nucleon. The repulsive component is necessary to prevent the fusion among nucleons. QCD is a quantum field theory which describes the interaction among colored particles, so quarks inside hadrons, via the exchange of equally colored gluons. But the strong force acts only inside hadrons, which thus have a zero net color. Nucleons cannot exchange gluons. It is by the exchange of colorless objects like mesons that they bind together. This makes the nucleus a complex multibody object, which is difficult to understand. The appropriate theoretical methods are effective field theories like Chiral Perturbation Theory or a numerical calculation like Lattice QCD. All of this goes beyond the scope of this course. In the next video, we will rather concentrate on much simpler models, which describe the gross features of nuclei. [MUSIC]