Atomic, Nuclear, and Modern Physics

Atomic Structure and Energy Levels

Electrons occupy orbitals defined by quantum numbers: principal n (shell), angular momentum l (subshell: s,p,d,f for l = 0,1,2,3), magnetic m_l (orientation), and spin m_s (±½). Filling follows Aufbau (lowest first), Pauli exclusion (unique set), and Hund's rule (degenerate orbitals fill singly). Z_eff = Z − S controls trends: ionization energy rises across a period, falls down a group. Bohr (E_n = −13.6·Z²/n²) applies only to H-like one-electron systems.

Photoelectric Effect

Light on metal ejects electrons: (1) emission requires hf > work function φ — threshold f₀ = φ/h; below this, zero electrons regardless of intensity; (2) above threshold, intensity raises photocurrent but not KE; (3) frequency raises maximum KE: K_max = hf − φ. Photon energy E = hf = hc/λ. Stopping potential eV_s = K_max. Einstein's photon model explained this where classical wave theory failed.

Photon Energy and the Bohr Model

Bohr yields E_n = −13.6·Z²/n². Transitions: ΔE = −13.6(1/n_f² − 1/n_i²). Series: Lyman (n_f=1, UV), Balmer (n_f=2, visible), Paschen (n_f=3, IR). λ = hc/|ΔE|; Rydberg: 1/λ = R(1/n_f² − 1/n_i²). Fails for multi-electron atoms, but quantization carries forward.

Nuclear Structure and Stability

The nucleus (Z p⁺, N n⁰, A = Z+N) is bound by the strong force (~10⁻¹⁵ m). Band of stability: light nuclei N/Z ≈ 1; heavy nuclei N/Z ≈ 1.5. Z > 83: all radioactive. Binding energy E_b = [Zm_p + Nm_n − m_nucleus]c²; per-nucleon peak at Fe-56 (~8.8 MeV). Fusion and fission both release energy toward the iron peak.

Radioactive Decay: Alpha, Beta, Gamma

Alpha: ⁴₂He (A↓4, Z↓2). Low penetration. β⁻: n→p⁺+e⁻+ν̄ (Z↑1, neutron-rich). β⁺: p⁺→n+e⁺+ν (Z↓1, proton-rich). EC: p⁺+e⁻→n+ν_e. Gamma: photon (no ΔA/ΔZ). β⁻ electron is created, not orbital.

Half-Life and Decay Kinetics

First-order: N = N₀(1/2)^(t/t₁/₂). t₁/₂ is constant — independent of T, P, or amount. λ = 0.693/t₁/₂; N = N₀e^(−λt); A = λN (Bq). After n half-lives: (1/2)ⁿ remains. ¹⁴C dating (5,730 yr). Parent-daughter equilibrium: t₁/₂(parent) >> t₁/₂(daughter).

Nuclear Reactions and Mass-Energy Equivalence

E=mc²: Q=Δm·c²; 1 u = 931.5 MeV/c². Fission: ²³⁵U (~200 MeV). Fusion: 4H→⁴He+26.7 MeV. Products near iron peak. ~10⁶× chemical bond energies.

Particle Physics Fundamentals

Fermions: 6 quarks (fractional charge), 6 leptons (e,μ,τ,νe,νμ,ν_τ). Bosons: photon (EM), W/Z (weak), 8 gluons (strong). p = uud (+1), n = udd (0). β decay: d→u via W. e⁺+e⁻→2γ (511 keV). Leptons: no strong force.

Medical Applications of Nuclear Physics

PET: β⁺ emitter (¹⁸F-FDG, 110 min) → annihilation → two 511 keV photons 180° apart → metabolic map. SPECT: γ emitter (⁹⁹ᵐTc, 6 hr). Radiation therapy: ionizing radiation → DNA damage. Tracers: ¹³¹I (thyroid), ²°¹Tl (cardiac). X-ray/CT: bremsstrahlung + characteristic X-rays from metal target — atomic, not nuclear. Isotope: half-life long enough for procedure, short to limit dose. Dose: gray (Gy=J/kg), sievert (Sv=bio-weighted).

How it works

Three energy tiers. Atomic (eV): E=hf=hc/λ, K_max=hf−φ, ΔE=−13.6(1/n_f²−1/n_i²). Nuclear (MeV): balance A,Z; check N/Z; half-life; 1u=931.5 MeV/c². Particle: quark composition (uud=+1), force carriers, conservation laws. Medical: isotope selection by half-life, decay mode, targeting; trace decay→image. Unifying: quantization from spectra to PET.

How it works

Three energy tiers. Atomic (eV): E=hf=hc/λ, K_max=hf−φ, ΔE=−13.6(1/n_f²−1/n_i²). Nuclear (MeV): balance A,Z; check N/Z; half-life; 1u=931.5 MeV/c². Particle: quark composition (uud=+1), force carriers, conservation laws. Medical: isotope selection by half-life, decay mode, targeting; trace decay→image. Unifying: quantization from spectra to PET.

Comparisons

• C/P (Gen Chem): Quantum numbers, electron configs, periodic trends, PES — all from atomic energy levels.
• C/P (Org Chem): HOMO-LUMO gap → λ_max — quantized levels applied to conjugated systems.
• B/B (Biochem): Radioisotope labeling (³²P, ³⁵S), X-ray crystallography, radiation DNA damage.
• B/B (Physiology): ¹³¹I thyroid, ¹⁸F-FDG PET brain, ⁹⁹ᵐTc cardiac, radiation therapy. Isotope: half-life, decay mode, tissue targeting.

Common confusions

• Using Bohr formula for multi-electron atoms — only valid for H-like one-electron systems.
• Confusing photoelectric intensity (current) with frequency (threshold and K_max). Below f₀: no emission at any intensity.
• Mixing β⁻ (n→p, Z↑, neutron-rich) and β⁺ (p→n, Z↓, proton-rich).
• Treating t₁/₂ as condition-dependent — it is constant, unlike chemical reaction rates.
• Writing 1 u = 931.5 MeV without c² — it is 1 u·c² = 931.5 MeV.
• Confusing X-ray (atomic) vs. gamma-ray (nuclear) origin — different physics, similar energies.

Quick review

• Four quantum numbers: n, l (s/p/d/f), m_l, m_s. Aufbau, Pauli, Hund's rule. Bohr (E_n = −13.6·Z²/n²) only for H-like atoms.
• Photoelectric: K_max = hf − φ, emission only if hf > φ. Intensity → current; frequency → threshold and K_max.
• Spectral series: Lyman (n_f=1, UV), Balmer (n_f=2, visible), Paschen (n_f=3, IR). Rydberg: 1/λ = R(1/n_f² − 1/n_i²).
• Band of stability: N/Z ≈ 1 (light), ~1.5 (heavy). Binding energy/nucleon peaks at Fe-56 (~8.8 MeV). Fusion and fission both release energy.
• α: ⁴₂He (A↓4, Z↓2). β⁻: n→p⁺+e⁻+ν̄ (Z↑1). β⁺: p⁺→n+e⁺+ν (Z↓1). γ: photon (no ΔA/ΔZ).
• Half-life: N = N₀(1/2)^(t/t₁/₂), constant, λ = 0.693/t₁/₂. After n cycles: fraction = (1/2)ⁿ.
• E = mc²: 1 u = 931.5 MeV/c². Nuclear reactions: MeV-scale (×10⁶ chemical). Q = Δm·c².
• Standard Model: proton = uud, neutron = udd. Forces: photon (EM), W/Z (weak), gluons (strong). Leptons feel no strong force.
• PET: β⁺ → annihilation → two 511 keV photons 180° apart → coincidence detection. ¹⁸F-FDG t₁/₂ 110 min.
• X-rays = atomic process; gamma rays = nuclear process. Different origin, potentially similar photon energy.