Summary

Nuclear composition: A nucleus is made of protons and neutrons (nucleons). Atomic number $Z$ = number of protons; mass number $A$ = total nucleons. Isotopes have same $Z$, different $A$; isobars have same $A$, different $Z$.

Nuclear size and density: Nuclear radius $R = R_0 A^{1/3}$, where $R_0 \approx 1.2 \times 10^{-15} \,\text{m}$. Nuclear density is roughly constant ($\sim 2.3 \times 10^{17} \,\text{kg/m}^3$), independent of $A$.

Mass defect and binding energy: Mass defect $\Delta m = [Z m_p + (A-Z) m_n] - m_{\text{nucleus}}$. Binding energy $BE = \Delta m \, c^2$. Higher binding energy per nucleon ($BE/A$) means greater stability; iron ($^{56}\text{Fe}$) has the maximum $BE/A$.

Radioactivity: Spontaneous decay of unstable nuclei. Three common types:

• $\alpha$-decay: emits $^4_2\text{He}$ nucleus; $A$ decreases by 4, $Z$ by 2.
• $\beta^-$-decay: $n \rightarrow p + e^- + \bar{\nu}_e$; $Z$ increases by 1, $A$ unchanged.
• $\gamma$-decay: emits high-energy photon; $Z$ and $A$ unchanged.

Decay law: $N = N_0 e^{-\lambda t}$, where $\lambda$ is decay constant. Half-life $T_{1/2} = \frac{\ln 2}{\lambda}$; mean life $\tau = 1/\lambda$.

Nuclear fission: Heavy nucleus (e.g., $^{235}\text{U}$) splits into two lighter nuclei, releasing energy ($\sim 200 \,\text{MeV}$ per fission) and neutrons. Chain reaction possible if at least one neutron per fission causes another fission.

Nuclear fusion: Light nuclei combine to form a heavier nucleus, releasing energy (e.g., $^2_1\text{H} + ^3_1\text{H} \rightarrow ^4_2\text{He} + n$). Requires extremely high temperature ($\sim 10^7 \,\text{K}$) to overcome Coulomb repulsion.

Energy equivalence: $1 \,\text{u} = 931.5 \,\text{MeV}/c^2$. This conversion is essential for calculating binding energies and reaction $Q$-values.

Nucleus vs. atom: the nuclear radius is smaller than the atomic radius by a factor of about $10^4$, yet the nucleus holds more than $99.9\%$ of the atom's total mass — essentially all of an atom's mass and positive charge is concentrated in this tiny central volume.

Proton and neutron masses: $1\,\text{u} = 1.660539 \times 10^{-27}\,\text{kg}$; a neutron is electrically neutral, and its mass is almost the same as a proton's ($m_n$ is only about $0.1\%$ larger than $m_p$). …