Key Takeaways — Chapter 1

Core Concepts

1. The Rutherford scattering cross section for Coulomb scattering of a charge $z_1 e$ from a point charge $z_2 e$ at kinetic energy $T$ is: $$\frac{d\sigma}{d\Omega} = \left(\frac{a}{2}\right)^2 \frac{1}{\sin^4(\theta/2)}, \quad a = \frac{k z_1 z_2 e^2}{2T}$$ The $\sin^{-4}(\theta/2)$ angular dependence was confirmed by Geiger and Marsden and revealed the nuclear atom.

2. The neutron was discovered by Chadwick in 1932 via the reaction ${}^{9}\text{Be}(\alpha,n){}^{12}\text{C}$, by analyzing recoil kinematics. The neutron mass $m_n = 939.565\,\text{MeV}/c^2$ is slightly greater than the proton mass $m_p = 938.272\,\text{MeV}/c^2$.

3. Nuclear notation: A nuclide ${}^{A}_{Z}\text{X}_N$ is specified by $Z$ (protons), $N$ (neutrons), and $A = Z + N$ (mass number). Isotopes share $Z$; isotones share $N$; isobars share $A$.

4. The chart of nuclides maps all ~3,300 known nuclides on an $N$–$Z$ grid. The valley of stability bends toward $N > Z$ for heavy nuclei. Only ~252 nuclides are truly stable.

5. Magic numbers: $2, 8, 20, 28, 50, 82, 126$. Nuclei with magic $Z$ or $N$ show enhanced stability (more stable isotopes/isotones, large separation energies). Doubly magic nuclei (${}^{4}\text{He}$, ${}^{16}\text{O}$, ${}^{40}\text{Ca}$, ${}^{48}\text{Ca}$, ${}^{208}\text{Pb}$) are exceptionally stable.

6. Nuclear radius and density saturation: $$R = r_0 A^{1/3}, \quad r_0 \approx 1.21\,\text{fm}$$ Because $V \propto A$, nuclear density is approximately constant: $\rho \approx 0.16\,\text{nucleons/fm}^3 \approx 2.3 \times 10^{17}\,\text{kg/m}^3$. This saturation implies the nuclear force is short-ranged.

7. Binding energy (the threshold concept of this chapter): $$B(A,Z) = \left[Z m_p + N m_n - M(A,Z)\right]c^2$$ Binding energy is the energy required to disassemble the nucleus — not energy stored inside it. More tightly bound nuclei have less mass per nucleon.

8. The binding energy per nucleon curve peaks near ${}^{62}\text{Ni}$ ($B/A = 8.795\,\text{MeV}$) and decreases for both lighter and heavier nuclei. This single curve explains why: - Fusion of light nuclei releases energy (moving up the curve from the left) - Fission of heavy nuclei releases energy (moving up the curve from the right)

9. Separation energies $S_n$ and $S_p$ measure the binding of the last neutron or proton. Sharp drops at magic numbers provide direct evidence for nuclear shell closures.

Essential Numbers to Remember