Chapter: Nuclei

• Atomic Nucleus

  • Composition
    • Protons and Neutrons (collectively called nucleons)
  • Basic Terms
    • Atomic Number ($Z$) = Number of protons
    • Mass Number ($A$) = Number of protons + neutrons
    • Neutron Number ($N$) = $A - Z$
  • Nuclide Representation
    • Notation: $_{Z}^{A}\text{X}$

• Nuclear Terms & Isotopes

  • Isotopes
    • Same $Z$, different $A$ (e.g., ${1}^{1}\text{H}$, ${1}^{2}\text{H}$)
  • Isobars
    • Same $A$, different $Z$ (e.g., ${1}^{3}\text{H}$, ${2}^{3}\text{He}$)
  • Isotones
    • Same $N$, different $Z$ or $A$

• Size and Density of Nucleus

  • Nuclear Radius
    • Formula: $R = R_0 A^{1/3}$
    • $R_0 \approx 1.2 \text{ fm}$
  • Nuclear Density
    • Independent of mass number $A$
    • Extremely high order of $10^{17} \text{ kg/m}^3$

• Mass-Energy & Mass Defect

  • Einstein's Mass-Energy Equivalence
    • Formula: $E = \Delta m c^2$
  • Atomic Mass Unit ($\text{amu / u}$)
    • $1\text{ u} = \frac{1}{12} \times \text{mass of C-12 atom} \approx 931.5 \text{ MeV}$
  • Mass Defect ($\Delta m$)
    • Difference between rest mass of nucleus and sum of constituent nucleons
    • $\Delta m = [Zm_p + (A-Z)m_n] - M$
  • Binding Energy ($\text{BE}$)
    • Energy required to separate nucleons to infinity
    • $\text{BE} = \Delta m \times c^2$

• Binding Energy per Nucleon ($\text{BE}/A$)

  • Significance
    • Measure of nuclear stability
  • BE Curve Features
    • Rapid rise for light nuclei
    • Flat broad maximum ($8.8 \text{ MeV}$) for intermediate nuclei ($A \in [30, 170]$)
    • Gradual decrease for heavy nuclei ($A > 170$)

• Nuclear Force

  • Characteristics
    • Strongest fundamental force in nature
    • Short-range (effective up to few femtometers)
    • Charge-independent (acts equally between p-p, p-n, n-n)
    • Attractive nature (responsible for stability)
    • Non-central and spin-dependent

• Radioactivity

  • Definition
    • Spontaneous emission of radiation (${\alpha}, {\beta}, {\gamma}$) by unstable nuclei
  • Radioactive Decay Law
    • Formula: $N(t) = N_0 e^{-\lambda t}$
    • $\lambda$ = Decay constant
  • Half-life ($T_{1/2}$)
    • Time for half the nuclei to decay
    • Formula: $T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$
  • Mean Life ($\tau$)
    • Formula: $\tau = \frac{1}{\lambda}$
  • Types of Decay
    • Alpha ($\alpha$) decay (loss of He nucleus)
    • Beta ($\beta$) decay ($\beta^-$ and $\beta^+$ emission, neutrino/antineutrino)
    • Gamma ($\gamma$) decay (emission of high-energy photons from excited states)

• Nuclear Energy

  • Nuclear Fission
    • Splitting of a heavy nucleus into two lighter nuclei
    • Example: Uranium-235 bombarded with slow neutrons
    • Releases enormous energy and secondary neutrons
    • Principle behind Nuclear Reactors and atomic bombs
  • Nuclear Fusion
    • Combining two light nuclei to form a heavier nucleus
    • Example: Fusion of hydrogen isotopes in the Sun
    • Requires extreme high temperature and pressure
    • Principle behind Hydrogen bomb and stellar energy source