13.1 Modern & Nuclear Physics

13.1 Modern & Nuclear Physics

Quick Answer: This introductory modern-physics review covers photon energy, the photoelectric equation, Bohr levels, radioactive decay, and energy from mass defect. Learn equations and units without treating the topic list as official PAF coverage.

Modern and nuclear physics provide useful formula practice: photoelectric stopping potential, Bohr transitions, half-life fractions, and radiation properties. Treat this section as a compact independent toolkit rather than a prediction of a PAF test.

Photons and the Photoelectric Effect

A photon has energy

$E = hf = \frac{hc}{\lambda}$

where h ≈ 6.63 × 10⁻³⁴ J·s (Planck’s constant), f is frequency, and λ is wavelength. In electron-volts, E (eV) ≈ 1240 / λ(nm) is a useful FSc approximation.

Photoelectric effect: light ejects electrons from a metal surface only if f exceeds a threshold frequency f₀. Einstein’s equation:

$hf = \phi + K_{\max}$

• Work function φ = hf₀ is the minimum energy to free an electron from the metal.
• Maximum kinetic energy of photoelectrons: K_max = hf − φ.
• Stopping potential V₀: eV₀ = K_max, so V₀ = (hf − φ)/e.

Classic trap: intensity does not raise K_max. Only frequency (or wavelength) does, once emission occurs. Intensity controls how many electrons leave per second.

Worked example — photoelectric. A metal has φ = 2.0 eV. Light of wavelength 400 nm strikes it. Photon energy ≈ 1240/400 = 3.1 eV. Then K_max = 3.1 − 2.0 = 1.1 eV, and V₀ = 1.1 V. If intensity doubles at the same λ, V₀ stays 1.1 V; current roughly doubles.

Atomic Models

Bohr hydrogen energies:

$E_n = -\frac{13.6\,\text{eV}}{n^2}$

Ground state (n = 1): −13.6 eV. First excited state (n = 2): −3.4 eV. Ionisation from ground state needs 13.6 eV.

A transition from nᵢ to n_f emits (or absorbs) a photon with

$|\Delta E| = hf = 13.6\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\text{ eV}$

(for emission, nᵢ > n_f). Lyman series ends at n = 1 (UV); Balmer at n = 2 (visible).

Worked example — Bohr. Photon for n = 3 → n = 2: |ΔE| = 13.6(1/4 − 1/9) = 13.6(5/36) ≈ 1.89 eV. Wavelength ≈ 1240/1.89 ≈ 656 nm (Balmer H-α)—a standard recognition item.

Radioactivity and Half-Life

Unstable nuclei decay spontaneously. Common emissions:

Decay law:

$N = N_0 e^{-\lambda t}, \quad A = \lambda N = A_0 e^{-\lambda t}$

Half-life T½ is the time for N (or A) to fall to half:

$T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$

After n half-lives, remaining fraction = (1/2)^n. After 3 half-lives, 1/8 remains; after 4, 1/16.

Worked example — half-life. A sample’s activity falls from 800 Bq to 100 Bq. Ratio = 1/8 = (1/2)³, so three half-lives elapsed. If T½ = 2 h, elapsed time = 6 h.

Mean life τ = 1/λ = T½ / ln 2 ≈ 1.44 T½—occasionally asked as a definition MCQ.

Mass Defect, Binding Energy, Fission and Fusion

Nuclear mass is less than the sum of free nucleon masses. Mass defect Δm relates to binding energy:

$BE = \Delta m\, c^2$

Higher binding energy per nucleon means a more stable nucleus. The curve peaks near iron/nickel; that is why:

• Fission: heavy nuclei (e.g. ²³⁵U) split into medium-mass fragments, releasing energy.
• Fusion: light nuclei (e.g. H isotopes) combine toward mid-mass, releasing even more energy per nucleon in stellar and experimental contexts.

Introductory exercises often use qualitative comparisons among fission, fusion, and α/β/γ radiation plus the scale sense of E = mc²—not reactor kinetics. Materials, propulsion, and sensors make radiation an interesting aeronautical context.

Practice Tips for This Topic

1. Convert eV ↔ joules carefully when asked (1 eV = 1.6 × 10⁻¹⁹ J).
2. Photoelectric: threshold + stopping potential problems dominate; intensity traps are common.
3. Half-life: prefer fraction (1/2)^n over messy continuous exponential unless λ is given.
4. Distinguish ionising power from penetrating power for α, β, γ.
5. Do not infer a PAF test blueprint or cut-off from this topic list; use the current official induction notice for any published assessment details.

Key Takeaways

• Photoelectric: frequency sets K_max; intensity sets emission rate.
• Bohr: E_n = −13.6 eV / n²; spectral series from ΔE = hf.
• Decay: exponential; T½ = 0.693/λ; remaining = (1/2)^n after n half-lives.
• α / β / γ differ in charge, mass, ionisation, and penetration.
• Fission and fusion release nuclear binding energy via mass defect.