11.3 Modern Physics

11.3 Modern Physics on NMAT Physics

CEM NMAT Physics includes modern physics at introductory college depth: light as photons, discrete atomic energies, nuclear structure and decay, and the idea that mass and energy are linked. Medical school relevance is intentional — imaging and radiation therapy rest on these concepts — but items still reward clear physics reasoning, not clinical protocols.

Quick frame: Classical waves explain interference; photons explain particle-like energy packets. Nuclei explain isotopes and radioactivity. Always track what is conserved (charge, nucleon number in many decays, energy including rest energy when needed).

Photons and the photoelectric effect

Electromagnetic radiation can be modeled as photons, each with energy:

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

where h is Planck’s constant (≈ 6.63 × 10⁻³⁴ J·s), f is frequency, and λ is wavelength. Higher frequency (shorter wavelength) means more energy per photon.

Photoelectric effect (conceptual essentials):

• Electrons are ejected from a metal only if photon energy exceeds the work function φ (binding energy of the least-bound electron)
• Maximum kinetic energy of photoelectrons: K_max = hf − φ
• Below threshold frequency f₀ = φ/h, no electrons leave, no matter how intense the light
• Higher intensity (above threshold) increases number of photoelectrons per second, not K_max for a fixed frequency

This pattern supports the photon model: energy arrives in packets of size hf, not as a continuous drip that eventually accumulates to free an electron at arbitrarily low f.

Worked example A
A metal has φ = 2.0 eV. Light of photon energy 3.0 eV strikes it. K_max = 3.0 − 2.0 = 1.0 eV. If intensity doubles at the same frequency, K_max stays 1.0 eV but more electrons may be emitted per second. If photon energy falls to 1.5 eV, emission stops (below threshold).

Worked conceptual scenario B
Red light fails to eject electrons; weak ultraviolet succeeds. Explanation: UV photons each have enough energy; many red photons still each fall short of φ. Intensity is not a substitute for frequency in the threshold condition.

Atomic models and spectra (Bohr intro)

Classical Rutherford nuclear atom could not stably explain discrete line spectra. The Bohr model (hydrogen-like intro) postulates stationary orbits with quantized angular momentum and energy levels. For hydrogen, energies are often written:

$E_n = -\frac{13.6\ \text{eV}}{n^2} \quad n = 1, 2, 3, \ldots$

• Ground state n = 1: E = −13.6 eV
• Excited states n ≥ 2: higher (less negative) energy
• Ionization from ground state requires 13.6 eV to reach E = 0 continuum

Photon emission: electron drops from n_i to n_f (n_i > n_f); photon energy:

$hf = E_{n_i} - E_{n_f}$

Absorption is the reverse: atom absorbs a photon whose energy matches a level spacing.

Line spectra are fingerprints of elements because level structures differ. Continuous spectra arise from thermal radiation of hot solids or free-free processes; discrete lines from bound transitions.

Worked example C — energy levels
Hydrogen transition n = 3 → n = 2:

$E_3 = -13.6/9 ≈ -1.51\ \text{eV}, \quad E_2 = -13.6/4 = -3.40\ \text{eV}$

$hf = E_3 - E_2 ≈ 1.89\ \text{eV}$

This is the Balmer series H-α line (visible red) — classic exam connection: visible hydrogen lines involve transitions ending at n = 2.

Worked conceptual scenario D
Why does a gas discharge tube show bright lines rather than a rainbow? Electrons excite atoms to discrete levels; only specific ΔE values are emitted. A hot solid’s continuum is different physics (many interacting oscillators / thermal distribution).

Nuclear structure: isotopes

The nucleus contains protons (charge +e, number = atomic number Z) and neutrons (charge 0). Mass number A = Z + N (N = neutron number).

Isotopes of an element: same Z, different N (hence different A). Chemical behavior is nearly identical (same electron count in neutral atoms); nuclear stability and mass differ.

Worked conceptual scenario E
(^{235}\mathrm{U}) and (^{238}\mathrm{U}) are uranium isotopes (both Z = 92). They differ in neutron count and fission/decay properties, not in the identity of the element.

Radioactive decay types

Unstable nuclei transform by emitting particles or radiation:

Exam habits: Balance nuclear equations by conserving charge (Z) and nucleon number (A) for α and β processes as taught at intro level. γ often follows α or β when the daughter is left excited.

Worked example F — alpha decay equation
(^{238}{92}\mathrm{U} \rightarrow\ ^{234}{90}\mathrm{Th} +\ ^{4}_{2}\mathrm{He}). Check: 92 = 90+2; 238 = 234+4.

Worked conceptual scenario G
Which radiation type is most penetrating among α, β, γ at comparable activity framing? Gamma (electromagnetic), requiring lead/concrete-type shielding discussions — while α is dangerous if internal (ingested/inhalated) despite low external penetration.

Half-life problems

Half-life T½ is the time for half of a radioactive sample’s nuclei (or activity, under standard assumptions) to decay. After n half-lives, remaining fraction:

$\left(\frac{1}{2}\right)^n$

Activity (decays per time) also halves each half-life for a pure exponential sample.

Worked example H
A sample has activity 800 Bq and T½ = 3.0 h. After 9.0 h (three half-lives):

$A = 800 \times \left(\frac{1}{2}\right)^3 = 800 \times \frac{1}{8} = 100\ \text{Bq}$

Worked example I
A radioisotope falls from 64 mg to 4 mg. Remaining fraction = 4/64 = 1/16 = (1/2)⁴ → 4 half-lives. If T½ = 2 days, elapsed time = 8 days.

Worked conceptual scenario J
Does a single nucleus decay at exactly T½? No — half-life is statistical for a large ensemble. One nucleus has a constant decay probability per time; half-life characterizes the population.

Mass–energy and medical relevance (conceptual)

Einstein’s mass–energy relation:

$E = mc^2$

Rest energy associated with mass is enormous (c² is huge). Nuclear binding energies and mass defects in fission/fusion convert small mass differences into large energies. Particle–antiparticle annihilation converts mass fully to photon energy (e.g., e⁺e⁻ → γ rays).

Medical awareness (not clinical dosing):

• X-rays / CT: high-energy photons for imaging density differences; ionization hazard managed by protocols
• Nuclear medicine: radioactive tracers (e.g., technetium isotopes, PET with β⁺ emitters) map physiology; half-life chosen for procedure timing and waste
• Radiation therapy: beams or implanted sources damage tumor DNA more than surrounding tissue when planned carefully; γ, x-rays, particles used depending on modality
• PET conceptual link: positron emission → annihilation photons detected in coincidence — modern physics meets imaging

NMAT-level goal: connect photon energy, nuclear decay, and half-life to why radiation can image or treat, without claiming physician-level expertise.

Worked conceptual scenario K
Why might a PET tracer need a short half-life? Sufficient activity during the scan with rapid decrease afterward reduces long-term dose and residual radioactivity — a half-life design tradeoff.

Error traps checklist

Study protocol

1. Memorize E = hf and K_max = hf − φ structure; run two threshold thought experiments.
2. Compute three hydrogen ΔE transitions including ionization from n = 1.
3. Balance five nuclear equations (α, β⁻, γ).
4. Drill half-life charts: start amount → remaining after 1–5 half-lives in under a minute.

Section checkpoint

You are ready when you can: (1) explain photoelectric threshold with photons, (2) compute simple Bohr transition energies for hydrogen, (3) define isotopes and contrast α/β/γ, (4) solve multi-step half-life quantity problems, and (5) state E = mc²’s conceptual role in nuclear energy and medical radiation contexts.