Atomic and Nuclear Structure: Comprehensive Study Notes

Evolution timeline (chronological order)
- Dalton’s "Billiard-Ball" Model (1803)
- J. J. Thomson’s "Plum-Pudding" Model (1897)
- Rutherford’s Nuclear Model (1911)
- Bohr’s Planetary Model (1913)
- Modern Quantum/Cloud Model (1920s →)
Significance
- Each model arose to explain new experimental evidence (mass measurements, discharge-tube work, scattering data, line spectra)
- Demonstrates the scientific method: hypothesis → experiment → refinement

Motivated by two empirical laws
- Law of Conservation of Matter (Lavoisier, 1780s)
- Law of Definite Proportions (Proust, ~1800)
Core Postulates
- Matter is composed of tiny, indivisible, indestructible particles called atoms
- Atoms of the same element are identical in mass, size, and other properties
- Atoms of different elements differ in these properties
- Atoms cannot transmute into atoms of another element
- Chemical change = breaking/formation of bonds and rearrangement of atoms
Limitations / Later Modifications
- Could not explain variations in valency & mass between elements
- Discovery of isotopes (same Z, different $A$ ) & isobars (different Z, same $A$ )
- Sub-atomic particles (e⁻, p⁺, n⁰) proved atoms are divisible

Experimental trigger: cathode-ray deflection → discovery of the electron (first sub-atomic particle)
Postulates
- Atom overall electrically neutral
- Positive charge spread uniformly throughout a sphere
- Electrons ("negatively electrified corpuscles") embedded like plums in a pudding
- Electrons are mobile inside the positive matrix → heating yields radiation
Limitations
- Mass treated as uniformly distributed; ignores concentrated nucleus
- Static: no provision for electron motion stability
- Cannot explain large-angle scattering of $\alpha$ -particles (Rutherford)
- Fails to describe atomic spectra or bonding mechanism

Method
- Narrow beam of high-speed $\alpha\,(\text{He}^{2+})$ particles fired at thin gold foil
- Scattered particles detected via a movable fluorescent screen/microscope
Key Observations
- ≈ 99 % passed straight through (little/no deflection)
- Small fraction deflected at small angles
- Very few deflected at large angles; some rebounded (
  $\theta \approx 180^{\circ}$ )
Conclusions
- Atom mostly empty space
- Positive charge & almost all mass concentrated in a tiny, dense core → nucleus
- Electrons orbit the nucleus to preserve electrical neutrality
- Greater atomic number ⇒ more protons ⇒ stronger repulsion ⇒ larger deflection
Rutherford/Nuclear Model
- Miniature solar system: electrons orbit like planets; nucleus at center
Limitation
- Classical electrodynamics: orbiting (accelerating) charges should radiate energy continuously → electrons should spiral into nucleus, yet atoms are stable

Introduced to resolve Rutherford’s stability & spectral issues
Arbitrary Postulates
1. Electrons occupy "stationary" orbits in which they do NOT radiate
2. Allowed orbits possess discrete energies; energy is quantized
3. Radiation emitted/absorbed only when an electron transitions between orbits: $E = E{\text{high}} - E{\text{low}} = h f$
4. Angular momentum quantized: $L = n\hbar$ where $\hbar = \dfrac{h}{2\pi}$ and $n = 1,2,3,\dots$
Successes
- Correctly predicted discrete hydrogen spectrum lines
- Explained Rydberg formula
Shortcomings
- Works exactly only for one-electron systems (H, He⁺, Li²⁺)
- Lacked mechanism for quantization → remedied by full quantum mechanics

Electrons described by wavefunctions (orbitals) rather than fixed orbits
Each orbital specified by three quantum numbers (n, l, $m_l$ ) that set size, shape, orientation
1. Principal quantum number $n$ (1, 2, 3…)
- Determines energy level & average radial distance
1. Angular momentum (azimuthal) quantum number $l$ (0 → $n-1$ )
- Determines orbital shape (s, p, d, f …)
1. Magnetic quantum number $m_l$ (– $l$ … 0 … + $l$ )
- Determines orientation; number of allowed $m_l$ values = $2l+1$
Worked Example (n = 3)
- $l = 0,1,2$
- $m_l$ possibilities: 0 | –1,0,+1 | –2,–1,0,+1,+2 → 9 orbitals total
Energy shells vs. subshells
- Shell = fixed n; Subshell = fixed n & l (e.g., 3p has n = 3, l = 1)

Three fundamental particles
- Electron (e⁻): charge = –e, mass $9.11\times10^{-31}\,\text{kg}$ , outside nucleus
- Proton (p⁺): charge = +e, mass $1.67\times10^{-27}\,\text{kg}$ , inside nucleus
- Neutron (n⁰): charge 0, mass $1.67\times10^{-27}\,\text{kg}$ , inside nucleus
Mass hierarchy: $mp \approx mn \approx 2000\,m_e$ ⇒ nucleus accounts for > 99.9 % of atomic mass

Radius ≈ $10^{-15}\,\text{m}$ (femtometre); entire atom ≈ $10^{-10}\,\text{m}$ → nucleus is ~ $10^{5}$ times smaller in diameter yet holds almost all mass
Composition: Z protons + N neutrons
Charge neutrality ⇒ # electrons = Z under normal conditions
Ion: atom missing/extra electron(s) → net charge

Electromagnetic force: like charges repel, unlike attract; binds e⁻ to nucleus
Nuclear (strong) force
- Attractive between nucleons (p & n) at ranges ≲ 2–3 fm
- ≈ 100× stronger than EM at that scale; acts as "glue" preventing Coulomb explosion of multi-proton nuclei
Practical impact: stability of heavy nuclei, radioactivity, nuclear energy & medicine

Atomic number: $Z$ = # protons ⇒ element identity
Neutron number: $N$ = # neutrons
Mass number: $A = Z + N$
Nuclide symbol: ${}^{A}{Z}\text{X}$ where X = chemical symbol (e.g., $^{238}{92}\text{U}$ )
Element: collection of atoms with same Z
Isotopes: same Z, different A (e.g., $^{12}\text{C}, ^{13}\text{C}, ^{14}\text{C}$ )
- Identical chemical behaviour (depends on electrons)
- Varied nuclear behaviour (stability, radioactivity)
- Medical imaging uses: PET (\beta⁺ emitters), cancer radiotherapy (radio-isotopes)
Isobars: different Z, same A (e.g., $^{40}{18}\text{Ar}$ & $^{40}{20}\text{Ca}$ )

Emission or absorption occurs at discrete frequencies $f$ such that
$E = h f$
Atomic line spectra (e.g., hydrogen Balmer series) validate quantization; contradict classical expectation of continuous spectra

Scientific progress: iterative refinement; disproving "indivisible" atom led to modern electronics, nuclear medicine
Responsible use of nuclear knowledge: radiation therapy vs. nuclear weapons
Imaging technology (context of course): PET, SPECT, MRI depend on nuclear & electronic properties explored by these models

Dalton → conservation laws → foundation of stoichiometry
Thomson → discovery of e⁻ → beginning of particle physics
Rutherford → nuclear model → pathway to nuclear energy & medicine
Bohr & quantum numbers → underpin modern quantum chemistry, periodic table structure, semiconductor physics
Strong vs. electromagnetic forces → basis for nuclear stability, reactor design, radiological safety