Atoms and Nuclei

Thomson's Plum Pudding Model (1898): Atom = uniform positive sphere with electrons embedded like raisins in pudding. Failed to explain Rutherford's scattering and atomic spectra.

Rutherford's Gold Foil Experiment (1911): Alpha particles scattered by thin gold foil; most pass through, few deflected by large angles (some bounce back). Conclusions:

• Atom mostly empty space
• Tiny dense positive nucleus at center (radius $\sim 10^{-15}$ m vs atomic radius $10^{-10}$ m)
• Electrons orbit nucleus (planetary model)

Distance of Closest Approach: $r_0 = \dfrac{1}{4\pi\epsilon_0}\cdot\dfrac{2Ze^2}{K}$ for $\alpha$-particle of KE $K$ hitting nucleus head-on.

Impact Parameter: $b = \dfrac{Ze^2\cot(\theta/2)}{4\pi\epsilon_0 K}$ for scattering angle $\theta$.

Failures of Rutherford Model:

• Classical EM theory: accelerating electron should radiate, spiral into nucleus → atom unstable
• Cannot explain discrete atomic spectra

Bohr's Postulates (1913):

1. Electron revolves only in certain stationary orbits where it does not radiate energy
2. Angular momentum quantized: $L = mvr = nh/(2\pi) = n\hbar$, $n = 1, 2, 3, \ldots$
3. Energy emitted/absorbed when electron jumps between orbits: $h\nu = E_i - E_f$

For Hydrogen-like Atom (charge $Ze$, one electron):

For hydrogen ($Z = 1$): $r_1 = 0.529$ Å (Bohr radius), $E_1 = -13.6$ eV, $v_1 = c/137$.

When electron jumps from higher orbit $n_i$ to lower $n_f$, photon emitted with: $$h\nu = E_{n_i} - E_{n_f}, \quad \frac{1}{\lambda} = R_H Z^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$

$R_H = 1.097 \times 10^7$ m⁻¹ is Rydberg constant.

Spectral Series of Hydrogen:

Series limit = wavelength when $n_i \to \infty$: $\lambda_{\text{limit}} = n_f^2/(R_H Z^2)$.

Number of Spectral Lines when electron transitions from $n$th level to ground: $$N = \frac{n(n-1)}{2}$$

Ionization Energy and Potential:

• Ionization energy: $|E_1| = 13.6 Z^2$ eV
• Ionization potential: $V_i = 13.6 Z^2$ V
• Excitation energy from $n_1$ to $n_2$: $E_{n_2} - E_{n_1}$
• Excitation potential: corresponding $V$

Limitations of Bohr Model:

• Cannot explain fine structure (splitting due to spin-orbit coupling)
• Cannot explain Zeeman/Stark effects (magnetic/electric field splitting)
• Cannot account for relative intensities of spectral lines
• Cannot extend to multi-electron atoms
• Inconsistent with quantum mechanics' uncertainty principle

These led to the modern quantum mechanical model (Schrödinger, Heisenberg).

Nucleus consists of protons ($p$, charge $+e$) and neutrons ($n$, neutral); collectively called nucleons.

Notation: $^A_Z X$, where $A$ = mass number (total nucleons), $Z$ = atomic number (protons), $N = A - Z$ = neutrons.

Isotopes: Same $Z$, different $A$ (e.g., $^1$H, $^2$H, $^3$H). Isobars: Same $A$, different $Z$ (e.g., $^{14}$C, $^{14}$N). Isotones: Same $N$, different $Z$ (e.g., $^{13}$C, $^{14}$N).

Nuclear Size: $R = R_0 A^{1/3}$, where $R_0 \approx 1.2 \times 10^{-15}$ m = $1.2$ fm. Density: $\rho \approx 2.3 \times 10^{17}$ kg/m³ (constant for all nuclei).

Mass-Energy Equivalence: $E = mc^2$. Atomic mass unit: $1$ u = $1.66 \times 10^{-27}$ kg = $931.5$ MeV/$c^2$.

Mass Defect: $\Delta m = Zm_p + (A-Z)m_n - M_{\text{nucleus}}$ (always positive).

Binding Energy (BE): Energy needed to dissociate nucleus into individual nucleons. $$\text{BE} = \Delta m \cdot c^2 = \Delta m \cdot 931.5 \text{ MeV (if } \Delta m \text{ in u)}$$

Binding Energy per Nucleon (BE/A): Measure of nuclear stability.

BE/A curve features:

• Rises rapidly for light nuclei
• Peaks at $\sim 8.8$ MeV/nucleon near $^{56}$Fe
• Slowly decreases for heavy nuclei
• Explains why fission of heavy / fusion of light → energy release

Nuclear Forces:

• Strong, short-range (~$1$ fm), attractive (charge independent)
• Much stronger than EM force (factor $\sim 100$)
• Saturated (each nucleon interacts with few neighbors only)

Radioactivity: Spontaneous emission of particles/radiation from unstable nuclei. Discovered by Becquerel (1896); systematically studied by Marie and Pierre Curie.

Types of Radiation:

In $\beta^-$ decay: $n \to p + e^- + \bar\nu$; in $\beta^+$: $p \to n + e^+ + \nu$.

Radioactive Decay Law: $$\frac{dN}{dt} = -\lambda N, \quad N(t) = N_0 e^{-\lambda t}$$ $\lambda$ = decay constant. Activity: $A = \lambda N$ (decays per second; unit: becquerel, Bq, or curie: $1$ Ci $= 3.7 \times 10^{10}$ Bq).

Half-life: $T_{1/2} = \ln 2/\lambda = 0.693/\lambda$.

Mean life: $\tau = 1/\lambda = T_{1/2}/\ln 2 = T_{1/2}/0.693$.

After $n$ half-lives: $N = N_0/2^n$.

Nuclear Fission: Heavy nucleus splits into lighter ones, releasing energy and neutrons.

Example: $^{235}\text{U} + n \to ^{141}\text{Ba} + ^{92}\text{Kr} + 3n + \sim 200$ MeV.

Chain reaction: neutrons released cause further fission. Critical mass = minimum mass for sustained chain reaction. Used in nuclear reactors (controlled) and atomic bombs (uncontrolled).

Reactor components: fuel ($^{235}$U, Pu), moderator (slows neutrons: water, graphite), coolant (water, sodium), control rods (Cd, B; absorb neutrons), shielding (concrete).

Nuclear Fusion: Light nuclei combine to form heavier (e.g., H → He in stars). Requires very high temperature ($\sim 10^7$ K) to overcome Coulomb repulsion.

Example: $^2\text{H} + ^3\text{H} \to ^4\text{He} + n + 17.6$ MeV.

Powers the Sun and stars. Hydrogen bomb uses fusion (uncontrolled). Controlled fusion is still under research (tokamaks, ITER project).