Physics – PSU‑GATE Study Notes
Topic: Atoms & Nuclei
Goal: Understand the structure, stability, and interactions of atoms and nuclei and be able to solve typical GATE‐style problems.
1. Simple Explanation
| Concept | What it means |
|---|---|
| Atom | The smallest unit of an element that retains its chemical properties. Consists of a positively charged nucleus (protons + neutrons) surrounded by negatively charged electrons. |
| Nucleus | Extremely dense region at the center of the atom. Protons have charge +e, neutrons are neutral. |
| Nuclear force | Short‑range, strongly attractive force that holds nucleons together, overcoming Coulomb repulsion between protons. |
| Binding energy | Energy required to disassemble a nucleus into its constituent nucleons. A larger binding energy per nucleon → more stable nucleus. |
| Decay | Spontaneous transformation of a nucleus to a more stable state, emitting particles or radiation. |
2. Key Points
2.1 Atomic Structure
| Item | Details | GATE relevance |
|---|---|---|
| Bohr model | Quantised orbits; energy levels (E_n = -\frac{Z^2 R_H}{n^2}). | Energy‑level calculations, selection rules. |
| Quantum mechanical (QM) model | Wavefunctions (\psi_{n\ell m}); quantum numbers (n,\ell,m,s). | Spectra, selection rules, Pauli exclusion. |
| Madelung rule | Order of orbital filling: (n + \ell) ascending. | Predicts ground‑state electron configuration. |
| Periodic trends | Ionization energy, electronegativity, atomic radius. | Understanding chemical behaviour. |
2.2 Nuclear Composition
| Symbol | Meaning | Typical values |
|---|---|---|
| (Z) | Atomic number (protons) | 1–118 |
| (N) | Neutrons | 0–200+ |
| (A = Z+N) | Mass number | integer |
| (m_n, m_p) | Mass of neutron/proton | 1.008665 u, 1.007276 u |
| (m_A) | Mass of nucleus | less than (A,u) (mass defect) |
2.3 Mass Defect & Binding Energy
[ \Delta m = Zm_p + Nm_n - m_A ] [ B = \Delta m,c^2 \quad (\text{in MeV}) ] [ \frac{B}{A} = \text{binding energy per nucleon} ]
- Peak at (^{56})Fe (~8.8 MeV/nucleon).
- Light nuclei: fusion releases energy.
- Heavy nuclei: fission releases energy.
2.4 Nuclear Models
| Model | Essence | Key Formula |
|---|---|---|
| Liquid‑drop model | Nucleus as incompressible fluid. | Semi‑empirical mass formula (SEMF). |
| Shell model | Nucleons in discrete shells; magic numbers (2,8,20,28,50,82,126). | Energy gaps at magic numbers. |
SEMF:
[
B(A,Z) = a_v A - a_s A^{2/3} - a_c \frac{Z^2}{A^{1/3}} - a_a \frac{(A-2Z)^2}{A} \pm a_p A^{-1/2}
]
- (a_v), (a_s), (a_c), (a_a), (a_p) are empirical constants.
2.5 Radioactive Decay Modes
| Mode | Typical products | Energy scale | Decay law |
|---|---|---|---|
| Alpha ((\alpha)) | He‑4 nucleus | ~5–10 MeV | (dN/dt = -\lambda N) |
| Beta‑ ((\beta^-)) | Electron + antineutrino | ~1 MeV | Same |
| Beta‑+ ((\beta^+)) | Positron + neutrino | ~1 MeV | Same |
| Gamma ((\gamma)) | Photon | 0.1–10 MeV | Same |
| Spontaneous fission | Two ~half‑mass fragments | >200 MeV | Same |
- Half‑life: (T_{1/2} = \frac{\ln 2}{\lambda}).
2.6 Nuclear Reactions & Q‑value
[ Q = \left(\sum m_{\text{initial}}\right) c^2 - \left(\sum m_{\text{final}}\right) c^2 ]
- (Q>0): exothermic (energy released).
- (Q<0): endothermic (energy absorbed).
2.7 Selection Rules (QM)
- Electric dipole (E1) transitions: (\Delta \ell = \pm 1), (\Delta s = 0).
- Forbidden if parity change not satisfied.
3. Worked Example
Problem
(GATE‑style)
Calculate the energy released in the alpha decay of (^{226})Ra:
[ ^{