Nuclear Physics Notes

Nuclear Structure

• Nucleus: Dense region with nucleons (protons and neutrons).

• Nucleons: Protons (positive charge) and neutrons (no charge).

• Atomic Number (Z): Number of protons.

• Isotopes: Same number of protons, different number of neutrons.

• Nuclide Notation: $^{A}_{Z}X$ where X is the element, Z is the atomic number, and A is the mass number.

• Isotopes: Nuclides with same number of protons.

• Isobars: Nuclides with same mass number.

• Isotones: Nuclides with the same number of neutrons.

• Isomers: Compounds with the same molecular formula but different structure.

• Atomic Mass Unit (AMU): 1 amu = $1.66054 x 10^{-27}$ Kg = 931.49 MeV.

• Nuclear Magnetons (µN): $µN = eh/4πmp$ where mp is the proton mass.

Nuclear Size

• Nuclear Radius: $R = R<em>0 A^{1/3}$, where $R</em>0 = 1.25 x 10^{-15}$ m.

Nuclear Binding Energy

• Binding Energy Formula: $ΔE = Δmc^2 = (Zmp + Nmn – Mnuc)c^2$

Nuclear Models

• Liquid Drop Model: Nucleus as an incompressible liquid drop.

  • Accounts for volume, surface, Coulomb, asymmetry, and pairing energies.

  • Semi-empirical mass formula: $m = Zmp + (A-Z)mn - EB/c^2$

• Shell Model: Nucleons arranged in energy levels based on the Pauli principle, leading to greater stability with filled shells.

• Magic Numbers: 2, 8, 20, 28, 50, 82, 126 (more tightly bound).

• Energy levels obtained by solving the Schrodinger equation.

Radioactive Decay

• Decay Law: $N = N_0 e^{-λt}$, where λ is the decay constant.

• Half-life: $T_{1/2} = ln(2)/λ = 0.693/λ$

• Activity: $|dN/dt| = λN$

Types of Radioactive Decay

• Alpha Decay: Emission of an alpha particle ($^4_2He$).

• Beta Decay:

  • Beta-minus ($β^−$): Neutron converts to proton, emitting an electron and antineutrino ($n → p + e^- + \bar{ν}$).

  • Beta-plus ($β^+$): Proton converts to neutron, emitting a positron and neutrino ($p → n + e^+ + ν$).

• Electron Capture: Proton captures an electron, converting to a neutron ($p + e^- → n$).

• Gamma Decay: Emission of a gamma ray photon.

Radioactivity

• Natural Radioactivity: Spontaneous nuclear reactions.

• Artificial Radioactivity: Induced nuclear reactions by bombarding nuclei with particles.