Nuclear and particle physics

In short

Nuclear physics studies the structure, binding and transformations of the atomic nucleus — the ~10⁻¹⁴ m core that holds 99.9% of an atom's mass — while particle physics studies the quarks, leptons and gauge bosons out of which nuclei and all matter are built. The two ideas that run through the whole topic are the mass-energy equivalence E = mc² (mass defect becomes binding energy) and the conservation laws (energy, charge, baryon and lepton number) that decide which reactions are allowed. It is a heavy-weight section in B.Sc Modern Physics papers, in GATE-PH, IIT-JAM, CSIR/UGC-NET and JEST, and in the written tests for BARC OCES/DGFS, NPCIL, IGCAR and ECIL. Applied knowledge of decay law, shielding, detectors and dose units is directly used by radiation safety officers, nuclear medicine and radiotherapy physicists, NDT technicians and reactor operators in India.

Explained

Nuclear structure, size and nuclear force

• A nucleus of symbol ᴬ𝓏X has Z protons (atomic number), N = A − Z neutrons; A is the mass number. Protons and neutrons are jointly called nucleons.
• Isotopes same Z, different A (¹H, ²H, ³H); isobars same A (⁴⁰Ar, ⁴⁰K, ⁴⁰Ca); isotones same N; isomers same Z and A but different energy state (⁹⁹ᵐTc).
• Nuclear radius R = r₀A^1/3 with r₀ ≈ 1.2 fm (1 fm = 10⁻¹⁵ m), found from electron scattering, mirror nuclei and alpha scattering. Volume ∝ A, so nuclear density is nearly constant (~2.3 × 10¹⁷ kg/m³) for all nuclei.
• Nuclear force properties: strongly attractive, very short range (~1-2 fm, negligible beyond), charge independent (n-n ≈ p-p ≈ n-p), spin dependent, saturated (a nucleon interacts only with near neighbours), and repulsive below ~0.5 fm (hard core).
• Yukawa (1935): the force is mediated by exchange of mesons; range ≈ ħ/(mc), which for the 140 MeV pion gives ≈ 1.4 fm. In the modern view it is the residual effect of the colour (strong) force between quarks.
• Stability: light stable nuclei have N ≈ Z; heavy ones need N > Z (N/Z ≈ 1.5 at lead) because Coulomb repulsion grows as Z². No stable nuclide exists beyond Z = 83 (bismuth-209 is effectively stable but very weakly alpha active).

Binding energy and the nuclear models

• Mass defect Δm = [Zm_H + (A−Z)m_n] − M_atom; binding energy BE = Δm·c², with 1 u = 931.494 MeV/c². BE is the energy needed to split a nucleus into free nucleons.
• BE/A curve: rises steeply to ~8 MeV by A ≈ 20, peaks at ≈ 8.8 MeV/nucleon near ⁵⁶Fe-⁶²Ni, falls to ≈ 7.6 MeV/nucleon at uranium. Hence fusion of light nuclei and fission of heavy nuclei both release energy. ⁴He, ¹²C, ¹⁶O stand above the trend (alpha-particle nuclei).
• Liquid drop model (Weizsäcker SEMF): treats the nucleus as an incompressible charged drop; BE = volume − surface − Coulomb − asymmetry ± pairing terms. Explains the BE/A curve, fission and the mass parabola of isobars; cannot explain magic numbers or spins.
• Shell (independent particle) model: nucleons fill quantised levels in a mean potential with strong spin-orbit coupling. Closed shells at the magic numbers 2, 8, 20, 28, 50, 82, 126 give extra stability, zero quadrupole moment, high separation energy. Explains ground-state spin and parity (for example ¹⁷O has 5/2⁺ from the d5/2 neutron) and the abundance of doubly magic ⁴He, ¹⁶O, ⁴⁰Ca, ²⁰⁸Pb.
• Fermi gas model: nucleons as two degenerate Fermi gases, gives the asymmetry term. Collective/unified (Nilsson) model: mid-shell nuclei are deformed and show rotational and vibrational bands.
• Nuclear spin I (integer for even A, half-integer for odd A), parity, magnetic dipole moment in nuclear magnetons (μ_N = 5.051 × 10⁻²⁷ J/T) and electric quadrupole moment (zero only for I = 0 or ½) are the measurable structure data.

Radioactivity and modes of decay

• Decay law: the number of disintegrations per second is proportional to the number of nuclei present — dN/dt = −λN, so N = N₀e^−λt. Half-life t½ = ln2/λ; mean life τ = 1/λ = 1.44 t½. Activity A = λN, unit becquerel (1 Bq = 1 disintegration/s); 1 curie = 3.7 × 10¹⁰ Bq.
• Alpha decay: emission of a ⁴He nucleus, A → A−4, Z → Z−2; occurs in heavy nuclei because the alpha is tightly bound. Energies 4-9 MeV, discrete spectrum, explained by quantum tunnelling through the Coulomb barrier (Gamow); the Geiger-Nuttall law links log λ linearly to Q (or 1/√E), so a 1 MeV change in Q alters the half-life by many orders of magnitude.
• Beta decay: β⁻ (n → p + e⁻ + ν̄ₑ) when N/Z is too high; β⁺ (p → n + e⁺ + νₑ) and competing electron capture when N/Z is too low. Spectrum is continuous up to an end-point energy — the missing energy and spin led Pauli to postulate the neutrino, mediated by the weak interaction (Fermi theory, W boson). Free neutron half-life ≈ 10.2 min.
• Gamma decay: a nucleus in an excited state drops to a lower state emitting a photon (keV-MeV), with internal conversion as the competing process (electron ejected instead). Discrete energies — the basis of gamma spectroscopy for identifying isotopes.
• Other modes: spontaneous fission (²³⁸U, ²⁵²Cf), proton emission, double beta decay; neutrinoless double beta decay is searched for but not observed.
• Decay chains and secular equilibrium: in a chain with t½(parent) ≫ t½(daughter), the daughter's activity equals the parent's (²²⁶Ra/²²²Rn, the ²³⁸U series); when the parent is only moderately longer-lived the result is *transient* equilibrium, where the daughter's activity slightly exceeds the parent's (⁹⁹Mo/⁹⁹ᵐTc generators). Radioactive dating: ¹⁴C (t½ 5,730 y) for organic material up to ~50,000 y; ⁴⁰K-⁴⁰Ar and ²³⁸U-²⁰⁶Pb (4.47 × 10⁹ y) for rocks.
• Displacement law (Soddy-Fajans): alpha shifts two groups left, beta⁻ one group right in the periodic table.

Nuclear reactions, fission and fusion

• Written a(X,Y)b or X + a → Y + b; Q value = (sum of initial masses − sum of final masses)c². Q > 0 exothermic; Q < 0 endothermic and needs a threshold energy E_th ≈ |Q|(1 + m_a/M_X) in the lab frame.
• Conserved in all nuclear reactions: charge, mass number (nucleon number), energy-momentum, angular momentum, parity (except weak interactions).
• Cross-section σ measures reaction probability; unit barn = 10⁻²⁸ m². Reaction rate = n·σ·φ (φ = flux, n/m²·s). Thermal fission cross-section of ²³⁵U ≈ 583 b against ≈ 2.7 b capture in ²³⁸U — the reason enrichment or heavy-water moderation is needed.
• Fission: a thermal neutron absorbed by ²³⁵U, ²³³U or ²³⁹Pu (fissile) splits it asymmetrically, releasing ≈ 200 MeV, two fragments (ternary fission is rare, <1%) and on average 2.4 fast neutrons, plus ~0.65% delayed neutrons whose seconds-long timescale makes a reactor controllable. ²³⁸U and ²³²Th are fertile — they breed ²³⁹Pu and ²³³U by neutron capture.
• Chain reaction and criticality: multiplication factor k_eff = (neutrons in one generation)/(previous generation). k < 1 subcritical, k = 1 critical (steady), k > 1 supercritical. Four-factor formula k∞ = η·f·p·ε (reproduction factor, thermal utilisation, resonance escape probability, fast fission factor); k_eff = k∞ × (non-leakage probabilities). Critical mass depends on enrichment, geometry and reflector.
• Reactor components: fuel, moderator (D₂O, H₂O, graphite — slows neutrons by elastic collision; D₂O has the best moderating ratio, low absorption), coolant, control rods (boron, cadmium, hafnium — high absorption), reflector, shielding, containment.
• Indian programme: a three-stage plan — Stage 1 natural-uranium PHWRs (220, 540 and 700 MWe; Kakrapar-3 was the first 700 MWe unit) with D₂O moderator, coolant and on-power refuelling; Stage 2 fast breeder reactors burning the plutonium (FBTR 40 MWt at IGCAR Kalpakkam; the 500 MWe PFBR at Kalpakkam reached first criticality on 6 April 2026, taking India into Stage 2, and is being raised towards grid power); Stage 3 thorium-U233 systems (AHWR design) to use India's large monazite thorium reserves. Kudankulam uses imported VVER-1000 light-water reactors. Installed nuclear capacity is of the order of 8 GWe from about two dozen units and changes as projects commission.
• Fusion: D + T → ⁴He + n releases 17.6 MeV and is the easiest reaction (largest cross-section at ~10 keV); D-D and p-p chain (stellar) are harder. Needs confinement against Coulomb repulsion — Lawson criterion/triple product nTτ. Magnetic confinement (tokamak: Aditya-U and SST-1 at the Institute for Plasma Research, Gandhinagar; India is one of the seven ITER partners) and inertial confinement (laser). Advantages: abundant fuel, no long-lived actinides, no runaway; limitation: not yet net-energy-positive at power-plant scale.

Detectors, accelerators and radiation protection

• Gas detectors: ionisation chamber (proportional to energy, no gas multiplication), proportional counter, Geiger-Müller counter (avalanche, large pulse, no energy information, has a dead time of ~100-300 µs).
• Scintillators: NaI(Tl) with photomultiplier for gamma counting; organic plastics for fast timing. Semiconductor detectors: Si surface-barrier for charged particles, HPGe for high-resolution gamma spectroscopy. Track/visualising: cloud chamber, bubble chamber, nuclear emulsion, spark and drift chambers; neutron detectors use BF₃ or ³He.
• Accelerators: Van de Graaff and Cockcroft-Walton (DC), cyclotron (f = qB/2πm, limited by relativistic mass increase), synchrocyclotron, betatron, linac, synchrotron, and colliders (LHC). Indian facilities: Pelletron-LINAC at TIFR/BARC Mumbai, Variable Energy Cyclotron Centre Kolkata, IUAC Delhi; India has been an Associate Member of CERN since 2017 and contributes to CMS and ALICE.
• Dose quantities: absorbed dose in gray (1 Gy = 1 J/kg; old unit rad = 0.01 Gy); equivalent dose in sievert = Gy × radiation weighting factor w_R (1 for X, γ, β; ~20 for alpha; 1 rem = 0.01 Sv); effective dose weights organs.
• Protection rests on time, distance (inverse-square), shielding (I = I₀e^−µx) and containment. Regulation in India is by the Atomic Energy Regulatory Board (AERB), with licensing through the eLORA portal. The SHANTI Act, 2025 (Sustainable Harnessing and Advancement of Nuclear Energy for Transforming India, assented 20 December 2025) replaces both the Atomic Energy Act, 1962 and the Civil Liability for Nuclear Damage Act, 2010, opens civil nuclear power to private operators, puts AERB on a statutory footing and sets graded operator liability; the Atomic Energy (Radiation Protection) Rules, 2004 apply until rules under the new Act replace them. AERB occupational limit: 20 mSv/year averaged over five consecutive years, not exceeding 30 mSv in any single year; public limit 1 mSv/year — check AERB's current regulations, which are being re-issued under the SHANTI Act.

Elementary particles and the Standard Model

• Classification by statistics: fermions (half-integer spin, obey Pauli exclusion, Fermi-Dirac) and bosons (integer spin, Bose-Einstein). By interaction: hadrons feel the strong force, leptons do not.
• Hadrons = baryons (three quarks, B = 1: p = uud, n = udd, Λ⁰ = uds) + mesons (quark-antiquark, B = 0: π⁺ = ud̄, K⁻ = sū).
• Matter fermions: six quarks (u, d, s, c, b, t with charges +2/3 and −1/3 and three colours) and six leptons (e, µ, τ and their neutrinos), in three generations.
• Gauge bosons: photon (electromagnetic), eight gluons (strong/colour), W⁺, W⁻, Z⁰ (weak), and the Higgs boson (spin 0, discovered at CERN in 2012) which gives mass via the Higgs mechanism. The graviton is hypothetical; gravity is not part of the Standard Model.
• Conservation laws: energy, momentum, charge, baryon number B, and separate lepton numbers Lₑ, L_µ, L_τ in every interaction. Strangeness, charm and parity are conserved by strong and electromagnetic interactions, isospin only by the strong interaction, and all of them are violated by the weak interaction (ΔS = 0, ±1); CP violation is seen in the K and B systems.
• Gell-Mann-Nishijima: Q/e = I₃ + Y/2 with hypercharge Y = B + S. Quark confinement: free quarks are never observed; separating them creates new hadrons (asymptotic freedom at high energy).
• Antiparticles have opposite charge and quantum numbers, same mass (positron, antiproton); annihilation gives photons. Neutrino oscillation (Super-Kamiokande and SNO, Nobel 2015) proves neutrinos have small non-zero mass; the India-based Neutrino Observatory at Theni is a proposed underground detector awaiting clearances.
• Classic processes to recognise: π⁰ → γγ (EM), π⁺ → µ⁺ + ν_µ (weak), Δ⁺⁺ → p + π⁺ (strong), n → p + e⁻ + ν̄ₑ (weak).

Key points

• Mass number A - total nucleons; nuclear volume ∝ A, so nuclear density is the same for all nuclei.
• R = r₀A^1/3 - nuclear radius with r₀ ≈ 1.2 fm; 1 fm = 10⁻¹⁵ m.
• 1 u = 931.494 MeV/c² - the conversion that turns mass defect into binding energy.
• BE/A peak ≈ 8.8 MeV at ⁵⁶Fe - why fusion below it and fission above it both release energy.
• Magic numbers 2, 8, 20, 28, 50, 82, 126 - closed shells, extra stability, shell-model evidence.
• Liquid drop model - volume, surface, Coulomb, asymmetry, pairing terms; explains fission, not spins.
• Decay law N = N₀e^−λt - t½ = 0.693/λ, mean life τ = 1/λ = 1.44 t½, activity A = λN.
• Alpha decay is barrier tunnelling - Geiger-Nuttall law makes half-life hugely sensitive to Q.
• Beta spectrum is continuous - the evidence for the neutrino; weak interaction changes quark flavour.
• Gamma decay - de-excitation only, no change in Z or A; competes with internal conversion.
• Fissile vs fertile - ²³⁵U, ²³³U, ²³⁹Pu fission with thermal neutrons; ²³⁸U and ²³²Th breed them.
• ≈200 MeV and ~2.4 neutrons per fission - and ~0.65% delayed neutrons make control possible.
• k_eff = 1 - critical reactor; four-factor formula k∞ = ηfpε.
• Moderator vs absorber - D₂O/graphite slow neutrons; boron/cadmium rods absorb them.
• India's three-stage programme - PHWR → fast breeder → thorium-U233; regulator is AERB.
• Barn = 10⁻²⁸ m² - unit of cross-section; reaction rate = nσφ.
• Gray vs sievert - Gy is energy deposited (J/kg), Sv is Gy × w_R for biological effect.
• Standard Model - 6 quarks, 6 leptons, 12 gauge bosons plus the Higgs; gravity is outside it.
• Conserved always - charge, energy-momentum, baryon number, each lepton number; strangeness and parity only in strong and EM processes.

Formulas and examples

1. Binding energy of the alpha particle. For ⁴He (Z = 2): m_H = 1.007825 u, m_n = 1.008665 u, M(⁴He) = 4.002602 u.

• Δm = 2(1.007825) + 2(1.008665) − 4.002602 = 4.032980 − 4.002602 = 0.030378 u
• BE = 0.030378 × 931.494 = 28.30 MeV; BE/A = 28.30/4 = 7.07 MeV/nucleon — high for such a light nucleus, which is why alpha emission is favoured.

2. Alpha decay of ²³⁸U → ²³⁴Th + ⁴He. Masses 238.050788, 234.043601, 4.002602 u.

• Δm = 238.050788 − (234.043601 + 4.002602) = 0.004585 u
• Q = 0.004585 × 931.494 = 4.27 MeV (positive, so the decay is spontaneous)
• E_α = Q(A−4)/A = 4.27 × 234/238 = 4.20 MeV; recoil of the ²³⁴Th carries the remaining 0.07 MeV.

3. (a) Activity of 1 mg of ⁶⁰Co (t½ = 5.27 y, molar mass 59.93 g).

• N = (1 × 10⁻³/59.93) × 6.022 × 10²³ = 1.005 × 10¹⁹ nuclei
• t½ = 5.27 × 3.156 × 10⁷ = 1.663 × 10⁸ s → λ = 0.6931/1.663 × 10⁸ = 4.17 × 10⁻⁹ s⁻¹
• A = λN = 4.17 × 10⁻⁹ × 1.005 × 10¹⁹ = 4.19 × 10¹⁰ Bq ≈ 41.9 GBq ≈ 1.13 Ci (matches the known specific activity ~1,130 Ci/g).

(b) Carbon dating: a wooden sample shows ¹⁴C activity one-fourth that of living wood. One-fourth = (1/2)², so two half-lives have passed: t = 2 × 5,730 = 11,460 years.

4. Energy from 1 g of ²³⁵U at 200 MeV per fission.

• Number of nuclei = (1/235) × 6.022 × 10²³ = 2.563 × 10²¹
• Energy = 2.563 × 10²¹ × 200 × 1.602 × 10⁻¹³ J = 8.21 × 10¹⁰ J ≈ 82 GJ
• In reactor units: 8.21 × 10¹⁰/(8.64 × 10¹⁰) ≈ 0.95 MW·day — the standard "1 g of U-235 ≈ 1 MWd" rule. Burning 1 kg of coal gives ~30 MJ, so 1 g of ²³⁵U ≈ 2.7 tonnes of coal.

5. D-T fusion Q value and energy density. ²H (2.014102) + ³H (3.016049) → ⁴He (4.002602) + n (1.008665).

• Δm = 5.030151 − 5.011267 = 0.018884 u → Q = 0.018884 × 931.494 = 17.59 MeV
• Per nucleon: 17.59/5 = 3.5 MeV, against ~0.85 MeV/nucleon for fission.
• Per kg of D-T mix: 17.59 MeV ÷ (5.030 × 1.6605 × 10⁻²⁷ kg) = 2.818 × 10⁻¹² J / 8.35 × 10⁻²⁷ kg ≈ 3.4 × 10¹⁴ J/kg, about four times the 8.2 × 10¹³ J/kg of pure ²³⁵U fission.

Where it is used

• Nuclear power: NPCIL's PHWR fleet and the Kudankulam VVERs — reactor physics, fuel management, neutronics and health physics roles; BARC and IGCAR for fuel cycle, reprocessing and the breeder-thorium programme.
• Nuclear medicine and radiotherapy: ⁹⁹ᵐTc, ¹³¹I, ¹⁷⁷Lu diagnostics and therapy, ⁶⁰Co teletherapy and linacs, PET with ¹⁸F — medical physicist and RSO posts in hospitals, with AERB certification (BARC's radiological physics training route).
• Industry and agriculture: gamma radiography for weld NDT, nucleonic level and thickness gauges, ⁶⁰Co radiation processing of spices and onions (BRIT, Krushak facilities), sterilisation of medical products, mutation breeding.
• Research and detectors: VECC Kolkata, Pelletron-LINAC Mumbai, IUAC Delhi, the proposed INO; Indian groups in CMS and ALICE at CERN build and analyse detector data — strong overlap with instrumentation, FPGA and data-analysis jobs.
• Regulation and safety: AERB licensing and inspection, TLD/OSL personnel monitoring, environmental surveillance; dose records under the Atomic Energy (Radiation Protection) Rules, 2004.
• Allied skills: Monte Carlo transport codes (MCNP, GEANT4, FLUKA), gamma spectrometry with HPGe, and Python/ROOT analysis — the most commonly asked practical skills in M.Sc/PhD and PSU interviews.

Interview questions

• Q: Why is binding energy per nucleon maximum around iron? A: Below it the surface term still costs a lot, above it Coulomb repulsion grows as Z²; the balance peaks at ≈ 8.8 MeV/nucleon near ⁵⁶Fe, making it the end point of both fusion and fission.
• Q: Why does nuclear density stay constant with A? A: Because the nuclear force saturates and nuclei are incompressible, R ∝ A^1/3, so volume ∝ A and mass/volume is independent of A.
• Q: Why is the beta spectrum continuous while the alpha spectrum is discrete? A: Beta decay is a three-body decay, so the available energy is shared randomly with the (anti)neutrino; alpha decay is two-body, so the energy split is fixed.
• Q: How can an alpha particle of 4 MeV escape a 25 MeV Coulomb barrier? A: By quantum tunnelling — the Gamow theory gives a small but finite transmission probability, which is why half-lives range from microseconds to 10⁹ years.
• Q: Why does a reactor need delayed neutrons? A: Prompt neutrons appear in ~10⁻⁴ s; the 0.65% of neutrons delayed by seconds stretch the response time enough for control rods to act.
• Q: What is the difference between a moderator and an absorber? A: A moderator (D₂O, graphite) slows neutrons to thermal energies with minimal capture; an absorber (boron, cadmium) removes them to reduce k_eff.
• Q: Why can India's PHWRs use natural uranium while light-water reactors cannot? A: Heavy water has a very low neutron absorption cross-section, so the neutron economy is good enough to sustain criticality at 0.7% ²³⁵U.
• Q: What are magic numbers and what evidence supports them? A: Nucleon numbers of closed shells (2, 8, 20, 28, 50, 82, 126); evidence includes high separation energies, natural abundance, zero quadrupole moment and sharp drops in neutron capture cross-section.
• Q: Is gravity in the Standard Model? A: No — the Standard Model covers the strong, electromagnetic and weak interactions; a quantum theory of gravity (graviton) is not part of it.
• Q: Which quantum numbers can the weak interaction violate? A: Strangeness (and other flavours), parity and CP; charge, energy-momentum, baryon number and lepton numbers are still conserved.
• Q: Why are free quarks never seen? A: Colour confinement — the potential grows with separation, so the energy supplied creates new quark-antiquark pairs (hadrons) instead of freeing a quark.
• Q: Difference between gray and sievert? A: The gray is physical energy absorbed per kg; the sievert weights it by radiation type (w_R ≈ 20 for alpha, 1 for gamma) to express biological harm.

Common mistakes

• Using the proton mass with the atomic mass in a binding-energy calculation — if M is an atomic mass, use m_H (hydrogen atom) so the Z electrons cancel.
• Taking Q as the alpha's kinetic energy and forgetting the recoil factor (A−4)/A.
• Mixing up mean life and half-life (τ = t½/0.693, not equal), or using t in different units from λ.
• Reporting activity in becquerel when the data were in curie, or forgetting that 1 Ci = 3.7 × 10¹⁰ Bq.
• Confusing fissile with fertile, or saying ²³⁸U and ²³²Th cannot fission at all (they do with fast neutrons, but cannot sustain a thermal chain).
• Applying strangeness or parity conservation to a weak decay, or forgetting lepton-number conservation when checking an allowed reaction.
• Treating a GM counter reading as an energy measurement, and ignoring dead-time and background correction in the lab record.

Related topics

Quantum mechanics basics · Atomic and molecular physics · Special relativity · Nuclear reactor physics and reactor engineering · Radiation detection, dosimetry and radiological safety · Astrophysics and nucleosynthesis