Key Points

A nucleus contains protons and neutrons, collectively called nucleons. The atomic number $Z$ is the number of protons, and the mass number $A$ is the total number of protons and neutrons ($A = Z + N$).

A nuclide is represented as ${}_{Z}^{A}\text{X}$. Isotopes are nuclides with the same atomic number ($Z$) but different neutron numbers ($N$). Isobars have the same mass number ($A$), and isotones have the same neutron number ($N$).

The atomic mass unit (u) is defined as $\frac{1}{12}$th of the mass of a neutral carbon-12 atom. Its value is approximately $1 \text{ u} = 1.660539 \times 10^{-27} \text{ kg}$.

The radius of a nucleus is proportional to the cube root of its mass number, given by the formula $R = R_0 A^{1/3}$, where the constant $R_0 \approx 1.2 \times 10^{-15} \text{ m}$ (or 1.2 fm).

The density of nuclear matter is nearly constant for all nuclei, independent of mass number $A$, and is extremely high, approximately $2.3 \times 10^{17} \text{ kg/m}^3$.

Einstein's famous relation $E = mc^2$ states that mass is a form of energy. In nuclear reactions, mass can be converted into energy and vice-versa.

The mass of a nucleus is always less than the sum of the masses of its constituent protons and neutrons. This difference in mass is called the mass defect, $\Delta M = [Z m_p + (A-Z) m_n] - M$.

The binding energy ($E_b$) is the energy equivalent of the mass defect, given by $E_b = \Delta M c^2$. It represents the energy required to break a nucleus into its individual nucleons.

A plot of binding energy per nucleon ($E_{bn}$) versus mass number ($A$) shows that nuclei with intermediate mass numbers (around $A=56$ for iron) are the most stable. The peak value is about $8.75 \text{ MeV/nucleon}$.

The nuclear force is the strong, short-range force that holds nucleons together. It is much stronger than the electrostatic force and is charge-independent, acting equally between p-p, n-n, and p-n pairs.

Radioactivity is the spontaneous decay of an unstable nucleus. The main types of decay are alpha ($\alpha$) decay (emission of a helium nucleus), beta ($\beta$) decay (emission of an electron or positron), and gamma ($\gamma$) decay (emission of high-energy photons).

Fission is a nuclear reaction in which a heavy nucleus, like ${}_{92}^{235}\text{U}$, splits into two or more lighter nuclei, releasing a large amount of energy (about 200 MeV per fission) and several neutrons.

Fusion is a nuclear reaction in which two or more light nuclei combine to form a heavier, more stable nucleus, releasing enormous amounts of energy. This process powers the sun and other stars.

Nuclear energy is released when reactions move towards greater binding energy per nucleon. Fission of heavy nuclei and fusion of light nuclei both result in products that are more tightly bound.

In nuclear physics calculations, a useful conversion is the energy equivalent of one atomic mass unit. $1 \text{ u} = 931.5 \text{ MeV}/c^2$.