Nuclei

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Nuclear Physics — Key Facts

Atomic nucleus contains protons (Z) and neutrons (N). Mass number A = Z + N. Isotopes have same Z, different N. Isobars have same A, different Z. Isotones have same N, different Z.

Nuclear Mass and Binding Energy:

Mass of nucleus is less than sum of masses of constituent nucleons — this mass defect (Δm) is converted to binding energy (E = Δmc²).

$$\Delta m = [Zm_p + Nm_n - M_{nucleus}]c^2$$

Binding energy per nucleon peaks at iron-56 (~8.8 MeV/nucleon), meaning Fe is the most stable nucleus.

Radioactive Decay:

• Alpha decay: $_Z^AX \rightarrow _{Z-2}^{A-4}Y + _2^4He$ (α-particle = helium nucleus)
• Beta minus: $_Z^AX \rightarrow _{Z+1}^AY + e^- + \bar{\nu}_e$ (neutron → proton)
• Beta plus: $_Z^AX \rightarrow _{Z-1}^AY + e^+ + \nu_e$ (proton → neutron)
• Gamma: $_Z^AX^* \rightarrow _Z^AX + \gamma$ (excited nucleus loses energy)

Decay Law: $$N = N_0 e^{-\lambda t}$$

Half-life: $T_{1/2} = \frac{\ln 2}{\lambda} = 0.693\tau$ Activity: $A = \lambda N = A_0 e^{-\lambda t}$ (in Becquerel, Bq)

⚡ JEE Exam Tip: In JEE Advanced 2023, a question on half-life calculations with sequential decays appeared. Remember: for sequential decay A → B → C, at equilibrium, activity of intermediate equals activity of parent if half-life of parent is much larger than that of B.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Nuclear Forces:

Nucleons are held together by strong nuclear force, which is:

• Attractive between nucleons at distances ~1 fm (10⁻¹⁵ m)
• Repulsive at very short distances (< 0.5 fm)
• Charge-independent (proton-proton, neutron-neutron, proton-neutron all similar)
• Short-range (saturates — each nucleon interacts only with neighbours)

Radioactive Series:

Uranium-238 series (4n+2): $^{238}U \rightarrow ^{206}Pb$ (half-life 4.5 × 10⁹ years) Thorium-232 series (4n): $^{232}Th \rightarrow ^{208}Pb$ (half-life 1.4 × 10¹⁰ years) Uranium-235 series (4n+3): $^{235}U \rightarrow ^{207}Pb$ (half-life 7 × 10⁸ years)

Carbon Dating:

$^{14}C$ (half-life 5730 years) is produced in atmosphere, taken up by living organisms. After death, $^{14}C$ decays. Ratio of $^{14}C$ to $^{12}C$ gives age of sample.

$$t = \frac{\ln(R_f/R)}{\lambda}$$

Nuclear Fission and Fusion:

Fission: Heavy nucleus splits (e.g., $^{235}U + n \rightarrow ^{141}Ba + ^{92}Kr + 3n + 200$ MeV) Fusion: Light nuclei combine (e.g., in sun: 4p → He + 2e⁺ + 2ν + 26.7 MeV)

Q-Value of Nuclear Reaction: $$Q = (M_{initial} - M_{final})c^2$$

Positive Q = exothermic (releases energy); negative Q = endothermic.

⚡ JEE Exam Tip: For Q-value problems, use atomic masses (including electrons) when calculating initial and final masses. The electron masses often cancel out but you must be careful.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Detailed Decay Kinetics:

For parent-daughter decay A → B → C (with decay constants λ₁ and λ₂):

$$N_B(t) = \frac{\lambda_1}{\lambda_2 - \lambda_1} N_{1,0}(e^{-\lambda_1 t} - e^{-\lambda_2 t})$$

Special Cases:

• If λ₁ << λ₂ (parent long-lived): $N_B \approx \frac{\lambda_1}{\lambda_2} N_{1,0}(1 - e^{-\lambda_2 t})$ → Transient equilibrium
• If λ₁ << λ₂ and t >> 1/λ₂: $N_B \approx \frac{\lambda_1}{\lambda_2 - \lambda_1} N_{1,0}e^{-\lambda_1 t} = \frac{\lambda_1}{\lambda_2 - \lambda_1} N_{1}(t)$ → Secular equilibrium (activities equal: A₁ = A₂)
• If λ₁ > λ₂: No equilibrium; B grows continuously

Nuclear Shell Model:

Magic numbers (2, 8, 20, 28, 50, 82, 126) correspond to filled shells and extra stability:

• ²He, ⁸O, ²⁰Ca, ⁴⁰Ca, ⁴⁸Ca, ⁸⁸Sr, ¹³²Sn, ²⁰⁸Pb are “doubly magic”
• Nuclei with magic Z or N have higher binding energy per nucleon

Mass Defect and Binding Energy Calculation:

For $^{12}C$: Z=6, N=6 Mass of 6 protons: 6 × 1.007276 u = 6.043656 u Mass of 6 neutrons: 6 × 1.008665 u = 6.051990 u Total mass of nucleons: 12.095646 u Actual atomic mass of ¹²C: 12.000000 u Mass defect: Δm = 0.095646 u Binding energy: E = 0.095646 × 931.5 MeV = 89.1 MeV Binding energy per nucleon: 89.1/12 = 7.4 MeV

Fermi Model of Beta Decay:

In beta minus decay, a neutron transforms into a proton, electron, and antineutrino: $$n \rightarrow p + e^- + \bar{\nu}_e$$

The neutrino (ν) was proposed by Pauli in 1930 to conserve energy and momentum — it has essentially zero rest mass and negligible interaction with matter.

Nuclear Reactors:

Components:

1. Fuel: Enriched uranium-235 (3-5%)
2. Moderator: Slows neutrons (H₂O, D₂O, or graphite)
3. Control rods: Absorb excess neutrons (cadmium or boron)
4. Coolant: Removes heat (water or liquid sodium)
5. Reflector: Bounces neutrons back into core

Critical mass: minimum mass for sustained chain reaction Neutron reproduction factor k = neutrons in one generation / neutrons in previous generation

• k = 1: critical (controlled reaction)
• k > 1: supercritical (explosion)
• k < 1: subcritical (reaction dies out)

⚡ JEE Advanced 2022 Analysis: Questions on sequential decay attaining equilibrium (secular and transient) appeared in 2022. The Q-value calculation using mass defect and binding energy per nucleon curve are also frequently tested. Also prepare for fusion reactions in stars and the proton-proton chain.

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