Core Concepts - Nuclei
Build from basics to advanced. Every concept exam-focused.
1. Nuclear Structure & Composition
Atomic Nucleus Basics
The nucleus is the tiny, dense core of an atom containing protons and neutrons (collectively called nucleons).
- Protons (p): Positively charged, mass ≈ 1.6726 × 10⁻²⁷ kg ≈ 1.007276 u
- Neutrons (n): Electrically neutral, mass ≈ 1.6749 × 10⁻²⁷ kg ≈ 1.008665 u
- Atomic Number (Z): Number of protons = Number of electrons in neutral atom
- Mass Number (A): Total number of nucleons = Z + N (where N = number of neutrons)
NEET/JEE frequently test isotope/isobar identification. Master the notation: AZX where X is element symbol.
Nuclear Size & Density
Nuclear Radius
R = R₀ A^(1/3)
where R₀ = 1.2 × 10⁻¹⁵ m (fermi constant), A = mass number
Key Points:
- Nuclear radius ∝ A^(1/3), so volume ∝ A
- Nuclear density is constant ≈ 2.3 × 10¹⁷ kg/m³ for all nuclei
- Nuclear size is ~10⁻¹⁵ m, atomic size is ~10⁻¹⁰ m (ratio: 1:100,000)
Students often confuse R ∝ A (wrong!) with R ∝ A^(1/3) (correct!). This error appears in 20% of JEE Main problems.
Isotopes, Isobars, Isotones
Isotopes
Same Z, different A
Example: ¹H, ²H, ³H (Hydrogen isotopes)
Isobars
Same A, different Z
Example: ¹⁴C, ¹⁴N (both have A=14)
Isotones
Same N, different A & Z
Example: ³H, ⁴He (both have N=2)
Nuclear Force
The strong nuclear force binds nucleons together, overcoming electromagnetic repulsion between protons.
Characteristics of Nuclear Force (Must Remember for CBSE Theory):
- Strongest force in nature (~100 times electromagnetic force)
- Short-range force (effective up to ~10⁻¹⁵ m)
- Charge-independent (same between p-p, n-n, p-n)
- Saturated force (acts only between nearest neighbors)
- Non-central force (depends on spin orientation)
2. Mass-Energy Equivalence & Binding Energy
Einstein's Mass-Energy Relation
Mass-Energy Equivalence
E = mc²
where c = 3 × 10⁸ m/s (speed of light)
Important Unit Conversion:
1 u = 931.5 MeV/c² = 1.66 × 10⁻²⁷ kg
This is the MOST USED conversion in Nuclei chapter.
JEE Advanced often tests unit conversion fluency. Practice converting between u, kg, and MeV instantly.
Mass Defect (Δm)
The difference between the sum of masses of constituent nucleons and the actual mass of the nucleus.
Mass Defect
Δm = [Z·mₚ + N·mₙ] - M_nucleus
mₚ = proton mass, mₙ = neutron mass, Z = atomic number, N = neutron number
Why does mass defect occur? When nucleons bind together, energy is released. This energy comes from converting some mass (E = mc²). The "missing mass" is the mass defect.
Binding Energy (BE)
Energy required to completely separate all nucleons in a nucleus. It equals the energy released when the nucleus was formed.
Total Binding Energy
BE = Δm × c² = Δm × 931.5 MeV (if Δm in u)
Binding Energy Per Nucleon
BE/A = (Total BE) / A
This is the KEY parameter for nuclear stability.
Most students use atomic mass instead of nuclear mass. Remember: Nuclear mass = Atomic mass - Z × (electron mass). However, for calculations, atomic mass is often provided and sufficient due to electron binding energy being negligible.
Nuclear Stability Curve
A plot of Binding Energy per nucleon (BE/A) vs Mass Number (A).
Key Features of BE/A Curve:
- Peak at Fe-56 (most stable nucleus, BE/A ≈ 8.8 MeV)
- Light nuclei: BE/A increases with A (fusion possible)
- Heavy nuclei: BE/A decreases with A (fission possible)
- Helium-4 shows a local peak (extra stable)
Graph-Based Questions (NEET/JEE):
You'll be asked to identify which nuclei can undergo fusion/fission by looking at their position on the BE/A curve. Master this graph interpretation.
Why is Fe-56 Most Stable?
Fe-56 has the highest BE/A, meaning nucleons are most tightly bound. Moving to heavier or lighter nuclei releases energy.
- Lighter than Fe: Fusion releases energy (Sun's energy source)
- Heavier than Fe: Fission releases energy (Nuclear reactors)
- At Fe-56: Maximum stability, no energy gain from fusion or fission
3. Radioactivity & Nuclear Decay
Radioactivity is the spontaneous disintegration of unstable nuclei with emission of radiation.
Why do nuclei decay? Unstable nuclei have excess energy or unfavorable neutron-to-proton ratio. They decay to reach a more stable state (closer to the stability line).
Types of Radioactive Decay
Alpha (α) Decay
Process: Emission of Helium nucleus (²He⁴ or α-particle)
ᴬ_Z X → ᴬ⁻⁴_{Z-2} Y + ⁴_2 He
Mass number decreases by 4, Atomic number decreases by 2
Properties:
- Charge: +2e
- Mass: 4 u
- Speed: ~5% speed of light
- Penetration: Stopped by paper (least penetrating)
- Ionization: Highest ionizing power
CBSE often asks: "Why do heavy nuclei undergo α-decay?" Answer: Heavy nuclei have too many nucleons. α-decay reduces both A and Z, moving toward stability.
Beta (β⁻) Decay
Process: Neutron converts to proton, emitting electron and antineutrino
ᴬ_Z X → ᴬ_{Z+1} Y + ⁰_{-1} e + ν̄
Mass number unchanged, Atomic number increases by 1
Inside Nucleus
n → p + e⁻ + ν̄
Properties:
- Charge: -1e
- Mass: ~1/2000 u (electron mass)
- Speed: ~90% speed of light
- Penetration: Stopped by aluminum foil
- Ionization: Moderate ionizing power
Don't confuse β⁻ (electron) with β⁺ (positron). JEE tests this. β⁻ decay increases Z, β⁺ decay decreases Z.
Beta Plus (β⁺) Decay / Positron Emission
Process: Proton converts to neutron, emitting positron and neutrino
ᴬ_Z X → ᴬ_{Z-1} Y + ⁰_{+1} e + ν
Mass number unchanged, Atomic number decreases by 1
Inside Nucleus
p → n + e⁺ + ν
Occurs in proton-rich nuclei. Positron quickly annihilates with an electron producing two gamma photons.
Gamma (γ) Decay
Process: Excited nucleus releases energy as high-energy photons
ᴬ_Z X* → ᴬ_Z X + γ
No change in A or Z (only energy state changes)
Properties:
- Charge: 0 (electromagnetic radiation)
- Mass: 0 (photon)
- Speed: Speed of light
- Penetration: Highest (requires thick lead to stop)
- Ionization: Lowest ionizing power
γ-decay often accompanies α or β decay when daughter nucleus is in excited state. Pure γ-decay is rare in exam problems.
Comparison Table (Memorize This)
| Property | α | β⁻ | γ |
|---|---|---|---|
| Nature | He nucleus | Electron | EM wave |
| Charge | +2e | -1e | 0 |
| Penetration | Lowest | Medium | Highest |
| Ionization | Highest | Medium | Lowest |
| Effect on A | A - 4 | No change | No change |
| Effect on Z | Z - 2 | Z + 1 | No change |
4. Radioactive Decay Law & Kinetics
This is the MOST IMPORTANT section for numerical problems. 60% of Nuclei numericals come from decay law. Master this.
Decay Law
Radioactive decay is a random process but statistically predictable for large numbers of nuclei.
Rate of Decay (Activity)
dN/dt = -λN
where λ = decay constant, N = number of nuclei at time t
The negative sign indicates N decreases with time. The rate is proportional to current number of nuclei—this is first-order kinetics (similar to chemical reactions).
Exponential Decay Law
Number of Nuclei
N(t) = N₀ e^(-λt)
N₀ = initial number of nuclei, N(t) = nuclei remaining at time t
Number Decayed
N_decayed = N₀ - N = N₀(1 - e^(-λt))
Common Error: Using N instead of N₀ in decay formulas. Always identify initial condition correctly. If problem says "initially 1000 nuclei," that's N₀, not N.
Half-Life (T½)
Time taken for half the nuclei to decay. N(T½) = N₀/2
Half-Life
T½ = (ln 2)/λ = 0.693/λ
After n Half-Lives
N = N₀/(2^n) where n = t/T½
This is faster for mental calculations!
Quick Calculation Trick:
- After 1 half-life: N = N₀/2 (50% remaining)
- After 2 half-lives: N = N₀/4 (25% remaining)
- After 3 half-lives: N = N₀/8 (12.5% remaining)
- After 10 half-lives: N ≈ N₀/1000 (practically zero)
Mean Life (τ)
Average lifetime of a radioactive nucleus.
Mean Life
τ = 1/λ = T½/ln2 = 1.44 T½
NEET loves asking relation between T½ and τ. Remember: τ = 1.44 T½ (mean life is always longer than half-life).
Activity (A)
Number of decays per unit time. Unit: Becquerel (Bq) = 1 decay/second
Activity
A = λN = λN₀ e^(-λt) = A₀ e^(-λt)
Important Units:
- Becquerel (Bq): 1 decay/second (SI unit)
- Curie (Ci): 3.7 × 10¹⁰ Bq (older unit)
- Rutherford (Rd): 10⁶ Bq (rarely used)
Relation with Half-Life
A = (0.693 N)/T½
Summary of Key Relations
Decay Constant Relations
λ = (ln 2)/T½ = 1/τ
Time Relations
τ = 1.44 T½
Exponential Form
N = N₀ e^(-λt)
A = A₀ e^(-λt)
A = A₀ e^(-λt)
Half-Life Form
N = N₀/(2^n)
where n = t/T½
where n = t/T½
5. Nuclear Reactions & Energy Calculations
Nuclear reactions involve changes in nuclear composition, releasing or absorbing large amounts of energy.
Conservation Laws
In ALL nuclear reactions, these are ALWAYS conserved:
- Mass-Energy (total mass-energy remains constant)
- Charge (total atomic number Z)
- Nucleon number (total mass number A)
- Momentum (linear and angular)
Q-Value of Reaction
Energy released (or absorbed) in a nuclear reaction.
Q-Value (Energy Released)
Q = (Total mass of reactants - Total mass of products) × c²
Q = [(Mᵢₙᵢₜᵢₐₗ - Mfᵢₙₐₗ)] × 931.5 MeV (if masses in u)
Sign Convention:
- Q > 0: Exothermic (energy released), reaction spontaneous
- Q < 0: Endothermic (energy required), minimum KE needed
60% of JEE Main Nuclei numericals test Q-value calculation. Master mass-to-energy conversion (1 u = 931.5 MeV).
Nuclear Fission
A heavy nucleus splits into two lighter nuclei, releasing energy.
Example: Uranium Fission
²³⁵U + ¹n → ⁹⁰Sr + ¹⁴³Xe + 3¹n + Energy (~200 MeV)
Key Points:
- Used in nuclear reactors and atomic bombs
- Requires slow neutrons (thermal neutrons) to initiate
- Produces more neutrons → chain reaction possible
- Energy per fission ≈ 200 MeV (very large)
- Critical mass needed for sustained chain reaction
Students confuse critical mass with threshold energy. Critical mass is the minimum fuel needed for chain reaction, not energy.
Nuclear Fusion
Two light nuclei combine to form a heavier nucleus, releasing energy.
Example: Deuterium-Tritium Fusion
²H + ³H → ⁴He + ¹n + 17.6 MeV
Key Points:
- Energy source of stars (including our Sun)
- Requires extremely high temperature (~10⁷ K) to overcome electrostatic repulsion
- Cleaner than fission (less radioactive waste)
- Still not commercially viable on Earth
- Hydrogen bomb uses fusion
Fusion vs Fission (CBSE Theory Question):
- Fission: Heavy → Light nuclei | Used in reactors | Produces radioactive waste
- Fusion: Light → Heavy nuclei | Powers stars | Requires extreme temperature
- Both: Convert mass to energy | Harness BE/A curve
Threshold Energy
Minimum kinetic energy of projectile required to initiate an endothermic reaction.
Threshold Energy (for stationary target)
KEₜₕ = -Q[1 + mₚᵣₒⱼ/Mₜₐᵣ + |Q|/(2Mₜₐᵣc²)]
This is a JEE Advanced level formula. For JEE Main/NEET, simplified version is sufficient.
For NEET/JEE Main, remember: If Q < 0, minimum energy ≥ |Q| is needed. Detailed threshold calculation is JEE Advanced territory.
6. Applications of Nuclear Physics
Radiocarbon Dating
Used to determine age of organic materials (up to ~50,000 years old).
Principle: Living organisms maintain constant C-14/C-12 ratio by exchanging carbon with atmosphere. After death, C-14 decays (T½ = 5730 years) while C-12 remains constant. Measuring remaining C-14 gives age.
Age Calculation
t = (2.303/λ) log(N₀/N) = (T½/0.693) ln(N₀/N)
where N₀ = initial C-14, N = current C-14
NEET frequently asks: "C-14 method cannot date rocks." Why? Rocks are inorganic—never had C-14 to begin with!
Nuclear Reactor
Device to harness energy from controlled nuclear fission.
Components (CBSE Theory):
- Fuel: Enriched Uranium (U-235) or Plutonium (Pu-239)
- Moderator: Slows down fast neutrons (heavy water D₂O, graphite)
- Control Rods: Absorb excess neutrons (cadmium, boron)
- Coolant: Carries away heat (water, liquid sodium)
- Shielding: Protects from radiation (concrete, lead)
Reactor vs Bomb: Reactor has controlled chain reaction (k ≈ 1), Bomb has uncontrolled chain reaction (k >> 1). Where k = neutrons produced/neutrons absorbed.
Medical Applications
Diagnosis
- PET Scan: Uses positron emitters (F-18)
- Gamma Camera: Tc-99m for organ imaging
- Thyroid Test: I-131 uptake measurement
Treatment
- Radiotherapy: Co-60 gamma rays kill cancer cells
- Brachytherapy: Radioactive seeds near tumor
- Thyroid Treatment: I-131 for hyperthyroidism
Other Applications
- Agriculture: Mutation breeding, food preservation (gamma irradiation)
- Industry: Thickness gauges, crack detection, sterilization
- Research: Tracers in chemistry/biology, neutron activation analysis
- Power: Nuclear submarines, spacecraft (RTGs)
CBSE loves asking "Write any 4 applications of radioactivity." Memorize at least 6 applications with examples for 5-mark questions.
Safety & Hazards
Radiation Effects:
- Ionizing radiation damages DNA and cells
- Acute exposure: radiation sickness, burns
- Long-term exposure: cancer, genetic mutations
- Half-life determines hazard duration
Next Steps: You've completed Core Concepts! Now move to the Formula Bank to consolidate all equations, then practice with Problem Types. Remember: Understanding > Memorization.