Structure of Atoms and Nuclei
Easy Overview
What is matter made of? At the centre of every atom is a tiny, dense nucleus containing protons and neutrons. Around it, electrons exist in quantised orbits — not like planets around the Sun, but as probability clouds described by quantum mechanics. The journey to understanding atomic and nuclear structure is one of the most fascinating stories in physics. The atom was once thought to be indivisible. Dalton proposed it as a solid sphere. J.J. Thomson discovered the electron and proposed the 'plum pudding' model: electrons embedded in a positive sphere. But Rutherford's gold foil experiment (1911) shattered that model. He fired alpha particles at a thin gold foil. Most passed through, but some bounced back — impossible if positive charge was spread uniformly. Rutherford concluded that the positive charge and most of the mass is concentrated in a tiny nucleus. The atom is mostly empty space. Bohr combined Rutherford's nuclear model with Planck's quantum theory. He proposed that electrons revolve in specific orbits without radiating energy. Energy is absorbed/emitted only when an electron jumps between orbits: ΔE = hf = E_final − E_initial. The Bohr model explains the hydrogen spectrum perfectly — Balmer series, Lyman series, Paschen series. But it failed for multi-electron atoms, leading to quantum mechanics. Nuclear physics goes deeper — inside the nucleus. The nucleus contains Z protons and (A − Z) neutrons. The strong nuclear force holds them together against electrostatic repulsion. Einstein's E = mc² explains nuclear binding energy: the mass of a nucleus is less than the sum of its constituent masses. This mass defect Δm corresponds to binding energy. Nuclear fission (splitting heavy nuclei) and fusion (combining light nuclei) release enormous energy. Radioactive decay — alpha, beta, and gamma — transforms one element into another, with applications in medicine, archaeology (carbon dating), and power generation.
Rutherford's Alpha-Particle Scattering Experiment
Alpha particles (He²⁺) from a radioactive source bombarded thin gold foil. Most passed through undeflected. A few scattered at large angles — some even bounced back. This meant the positive charge was concentrated in a very small volume (nucleus). Rutherford derived: N(θ) ∝ 1/sin⁴(θ/2). Radius of nucleus ≈ 10⁻¹⁴ m. Atom radius ≈ 10⁻¹⁰ m — mostly empty.
Bohr's Atomic Model — Postulates
Postulate 1: Electrons revolve in stable orbits called stationary states without radiating. Postulate 2: Angular momentum is quantised: mvr = nħ (n = 1,2,3...). Postulate 3: Electrons jump between orbits: hf = E_i − E_f. Explains hydrogen spectrum: Rydberg formula 1/λ = R(1/n_f² − 1/n_i²). R = 1.097 × 10⁷ m⁻¹.
Hydrogen Spectrum — Spectral Series
Lyman series (n_f = 1): ultraviolet. Balmer (n_f = 2): visible (Hα 656 nm, Hβ 486 nm, Hγ 434 nm, Hδ 410 nm). Paschen (n_f = 3): infrared. Brackett (n_f = 4), Pfund (n_f = 5): far IR. The Balmer series is visible in the solar spectrum and was discovered first.
Energy Levels of Hydrogen Atom
E_n = −13.6/n² eV. Ground state (n=1): −13.6 eV. First excited (n=2): −3.4 eV. Ionisation energy: 13.6 eV. Radius r_n = n²r₀, r₀ = 0.529 Å (Bohr radius). Velocity v_n = v₀/n, v₀ = c/137 ≈ 2.18 × 10⁶ m/s. These match experimental hydrogen spectrum.
Atomic Excitation and Ionisation
An atom can be excited by (1) absorbing a photon of exactly the right energy, or (2) collision with an energetic electron (Franck-Hertz experiment). Spontaneous emission: excited atom returns to ground emitting random-phase photon. Stimulated emission: photon triggers emission of identical photon (laser). Ionisation: electron removed completely.
Lasers — Principle and Applications
Population inversion: more atoms in excited state than ground state. Metastable states allow population build-up. Stimulated emission produces coherent, monochromatic, directional light. Three-level and four-level laser systems. Ruby laser (694 nm), He-Ne laser (632.8 nm). Applications: barcode scanners, surgery, optical communication, lidar.
Nuclear Structure — Protons and Neutrons
Nucleus: Z protons, N = A − Z neutrons. Notation: ᴬZX. Isotopes: same Z, different A (¹H, ²H, ³H). Isotones: same N. Isobars: same A (⁴⁰Ar, ⁴⁰Ca). Nuclear size: R = R₀A^(1/3), R₀ ≈ 1.2 × 10⁻¹⁵ m. Nuclear density ≈ 2.3 × 10¹⁷ kg/m³ — incredibly dense.
Nuclear Force
Strong nuclear force: attractive, short-range (~10⁻¹⁵ m). Charge-independent (proton-proton, proton-neutron, neutron-neutron same). Repulsive at very small distances (hard core). Mediated by pions (π mesons). Yukawa model: range ∝ 1/m_π. Much stronger than electromagnetic force at nuclear distances.
Mass Defect and Binding Energy
Mass of nucleus < sum of constituent masses. Δm = [Zm_p + (A−Z)m_n − M_nucleus]. Binding energy BE = Δmc². BE per nucleon ≈ 8 MeV (iron peak). Maximum BE per nucleon at A ≈ 56 (iron). Smaller and larger nuclei have lower BE/nucleon — explains energy release in fission and fusion.
Nuclear Fission
Heavy nucleus (²³⁵U, ²³⁹Pu) splits into two smaller nuclei + neutrons + ~200 MeV. Chain reaction: neutrons trigger more fissions. Critical mass: minimum mass for self-sustaining chain reaction. Applications: nuclear power plants (controlled chain reaction with control rods), atomic bombs (uncontrolled).
Nuclear Fusion
Light nuclei combine to form heavier nucleus + energy. Example: ²H + ³H → ⁴He + n + 17.6 MeV. Requires very high temperature (~10⁸ K) to overcome Coulomb barrier. Occurs in stars (proton-proton chain, CNO cycle). Controlled fusion: ITER (tokamak) aims for commercial fusion power.
Radioactivity — Alpha, Beta, Gamma Decay
Alpha decay: ᴬZX → ᴬ⁻⁴Z−₂Y + ⁴₂He + Q. Q-value = (M_parent − M_daughter − M_α)c². Beta decay: n → p + e⁻ + ν̄_e (β⁻). Or p → n + e⁺ + ν_e (β⁺). Electron capture. Gamma decay: excited nucleus emits γ-ray. Law: N = N₀e^(−λt). Half-life T_½ = ln2/λ = 0.693/λ. Mean life τ = 1/λ.
Carbon Dating
¹⁴C (T_½ = 5730 years) produced in atmosphere. Incorporated into living organisms. After death, ¹⁴C decays — ratio ¹⁴C/¹²C decreases. Measuring remaining ¹⁴C gives age up to ~50000 years. Calibration needed due to varying ¹⁴C production over time.
Nuclear Reactor
Components: fuel (²³⁵U), moderator (slows neutrons — heavy water, graphite), control rods (absorb neutrons — boron, cadmium), coolant (water, liquid sodium), shielding (concrete). Types: PWR (pressurised water), BWR, PHWR (CANDU). Breeder reactor: produces more fissile material than it consumes.
Key Points
- •Rutherford model: nucleus + electrons. Atom mostly empty.
- •Bohr model: quantised orbits. E_n = −13.6/n² eV.
- •Rydberg formula: 1/λ = R(1/n_f² − 1/n_i²).
- •Laser: population inversion, stimulated emission, coherent light.
- •Nuclear size: R = R₀A^(1/3), R₀ = 1.2 × 10⁻¹⁵ m.
- •Binding energy BE = Δmc². BE/nucleon ≈ 8 MeV (iron peak).
- •Fission: heavy splits → ~200 MeV. Fusion: light combine → energy.
- •Radioactive decay: N = N₀e^(−λt). T_½ = ln2/λ.
- •Carbon dating: ¹⁴C T_½ = 5730 years.
Practice Questions
- Describe Rutherford's alpha scattering experiment. State conclusions and shortcomings of this model.
- State Bohr's postulates. Derive radius and energy of nth orbit in hydrogen atom. Draw energy level diagram.
- Explain hydrogen spectrum. State spectral series. Calculate wavelength of Hα line (R = 1.097 × 10⁷ m⁻¹).
- What is binding energy? Plot BE per nucleon vs mass number. Explain how fission and fusion release energy.
- Explain nuclear fission. Draw schematic of a nuclear reactor. What is critical mass?
- State laws of radioactive decay. Derive N = N₀e^(−λt). Half-life of ²²⁶Ra is 1600 years. Initial activity 3.7 × 10¹⁰ Bq. Find activity after 3200 years.
- Distinguish between nuclear fission and fusion. Conditions required for fusion. Why is fusion difficult to achieve on Earth?
- Explain carbon dating. A wooden artefact has 25% ¹⁴C compared to living wood. Find age (T_½ = 5730 years).
- What are alpha, beta, gamma decays? Write balanced decay equations for (a) ²³⁸U → α, (b) ¹⁴C → β⁻.