Binding Energy Curve: The Energy Map of Nuclei

The Manhattan Project Energy Source 🎬

When Oppenheimer’s team split uranium atoms, they released energy predicted by the binding energy curve. Einstein’s E=mc² wasn’t just theory - it powered bombs and stars! The curve explains why splitting heavy nuclei (fission) AND combining light nuclei (fusion) both release enormous energy.

“The energy produced by the atom is a very poor kind of thing. Anyone who expects a source of power from the transformation of these atoms is talking moonshine.” - Ernest Rutherford, 1933 (Very wrong! ⚛️💥)

What is Binding Energy Per Nucleon?

:::box Definition

Measures how tightly bound each nucleon is in the nucleus.

Higher BE/A → More stable nucleus :::

Why “Per Nucleon”?

Comparing total BE of different nuclei isn’t fair:

• ²³⁸U has huge total BE (just because it’s big!)
• ⁴He has small total BE (but very stable!)

Solution: Divide by A to get BE per nucleon

• Fair comparison across all nuclei
• Reveals which nuclei are most stable

The Famous Binding Energy Curve

BE/A
(MeV)
  9 ┤
    │         ╱■╲
  8 ┤        ╱ Fe╲___
    │       ╱   ↑    ╲___
  7 ┤   ⁴He│   MAX    ╲___
    │  ╱╲  │   8.8MeV     ╲___²³⁸U
  6 ┤ ╱  ╲ │                  ╲
    │╱    ╲│                   ╲
  5 ┤      │                    ╲
    │      │                     ╲
    └──────┼──────────────────────→ A
          56  100    150    200  238

Key Features

1. Low A (Light nuclei)

   • Rapid rise
   • ⁴He is a local maximum (very stable!)
   • H, ²H have low BE/A (weakly bound)

2. Peak Region (A ≈ 56)

   • Maximum at ⁵⁶Fe (BE/A ≈ 8.8 MeV)
   • Most stable nuclei
   • Ni, Fe, Co cluster

3. High A (Heavy nuclei)

   • Gradual decline
   • ²³⁸U at BE/A ≈ 7.6 MeV
   • Coulomb repulsion weakens binding

The Physics Behind the Curve

Low Mass Region (A < 20)

Rising BE/A:

• Few nucleons, incomplete binding
• Surface effects important
• Nucleons on surface less bound

⁴He Exception:

• 2 protons + 2 neutrons
• Perfect “alpha particle” configuration
• Extra stable (doubly magic!)
• BE/A = 7.07 MeV (local max)

Medium Mass Region (20 < A < 100)

Near maximum stability:

• Volume binding dominates
• Surface effects less important
• Optimal balance of forces

Peak at ⁵⁶Fe:

• BE/A = 8.790 MeV
• Most stable nucleus
• End product of stellar fusion

Heavy Mass Region (A > 100)

Declining BE/A:

• Coulomb repulsion grows with Z²
• Nuclear force only acts short-range
• Protons repel across nucleus
• Surface effects return

Heaviest nuclei:

• Unstable (radioactive)
• Tend to split (fission)
• Or emit α-particles

Energy Release Mechanisms

1. Nuclear Fission

Heavy → Medium fragments

    ²³⁸U        →      Fragments
    BE/A = 7.6       BE/A ≈ 8.5

         ↗
   Moving UP the curve → Energy released!

:::box Fission Energy Release

Approximately:

:::

Example: ²³⁵U fission

Typical: ²³⁵U → ¹⁴⁴Ba + ⁸⁹Kr + 3n + 200 MeV

2. Nuclear Fusion

Light → Heavier

    ²H + ³H     →      ⁴He + n
    BE/A ≈ 1-3       BE/A = 7.1

         ↗
   Moving UP the curve → Energy released!

:::box Fusion Energy Release

For D-T fusion:

:::

Example: D-T reaction

Total: 17.6 MeV

Why Fission AND Fusion Release Energy

The Curve Explains Everything!

BE/A
  │
  │    FUSION →  ╱■╲  ← FISSION
  │   (climb)   ╱   ╲  (climb)
  │    ↗       ╱     ╲     ↖
  │   ╱       ╱       ╲      ╲
  │  ╱       │         ╲      ╲
  └──────────┼───────────────────→ A
            56
          PEAK

Universal Principle: Any process moving TOWARD Fe-56 releases energy!

• Fusion: Light nuclei (left) → climb UP
• Fission: Heavy nuclei (right) → climb UP
• Both: Increase BE/A → Release energy

Why Iron-56 is Special

• Peak of stability
• End of fusion in stars
• No energy from further fusion
• This is why supernovae happen!

Stars fuse up to Fe-56, then collapse when no more energy available.

Interactive Demo: Visualize Nuclear Decay and Stability

Explore how nuclear stability relates to binding energy and decay processes.

Detailed BE/A Values

Light Nuclei

Medium Nuclei (Peak Region)

Note: ⁶²Ni actually has highest BE/A (8.7945 MeV), but Fe-56 is more common.

Heavy Nuclei

Interactive Demo: BE/A Explorer

const BECurveExplorer = () => {
  const [processType, setProcessType] = useState('fission');
  const [parent, setParent] = useState(235);

  // Sample BE/A data (simplified)
  const getBEperA = (A) => {
    if (A <= 4) return 7.07 * (A/4); // Rough for light
    if (A <= 56) return 7 + (A - 4) * 0.035; // Rising
    return 8.8 - (A - 56) * 0.015; // Falling
  };

  const parentBE = getBEperA(parent);

  let products, productsBE, energyReleased;

  if (processType === 'fission') {
    // Roughly equal fragments
    const A1 = Math.floor(parent / 2);
    const A2 = parent - A1;
    products = `A₁≈${A1}, A₂≈${A2}`;
    productsBE = (getBEperA(A1) + getBEperA(A2)) / 2;
    energyReleased = (productsBE - parentBE) * parent;
  } else {
    // Fusion: assume D+T → He
    const reactantsBE = (getBEperA(2) * 2 + getBEperA(3) * 3) / 5;
    products = '⁴He';
    productsBE = getBEperA(4);
    energyReleased = (productsBE - reactantsBE) * 4;
  }

  return (
    <div>
      <h3>Binding Energy Curve Explorer</h3>

      <label>Process Type:</label>
      <select value={processType} onChange={(e) => setProcessType(e.target.value)}>
        <option value="fission">Fission</option>
        <option value="fusion">Fusion</option>
      </select>

      {processType === 'fission' && (
        <>
          <label>Parent nucleus mass number: {parent}</label>
          <input
            type="range"
            min="200"
            max="250"
            value={parent}
            onChange={(e) => setParent(Number(e.target.value))}
          />
        </>
      )}

      <div className="results">
        <h4>Energy Analysis:</h4>

        {processType === 'fission' ? (
          <>
            <p>Parent (A={parent}): BE/A = {parentBE.toFixed(2)} MeV</p>
            <p>Products ({products}): BE/A ≈ {productsBE.toFixed(2)} MeV</p>
            <p>ΔBE/A = {(productsBE - parentBE).toFixed(2)} MeV</p>
            <p style={{fontSize: '1.2em', color: 'green'}}>
              Energy released ≈ {energyReleased.toFixed(0)} MeV
            </p>
            <p>Moving UP the curve → Energy OUT!</p>
          </>
        ) : (
          <>
            <p>Reactants (²H+³H): BE/A ≈ {getBEperA(2.5).toFixed(2)} MeV</p>
            <p>Product (⁴He): BE/A = {productsBE.toFixed(2)} MeV</p>
            <p>ΔBE/A = {(productsBE - getBEperA(2.5)).toFixed(2)} MeV</p>
            <p style={{fontSize: '1.2em', color: 'green'}}>
              Energy released ≈ 17.6 MeV
            </p>
            <p>Moving UP the curve → Energy OUT!</p>
          </>
        )}

        <h4>Why energy is released:</h4>
        <p>Products are MORE tightly bound than reactants!</p>
        <p>Mass defect increases → Energy released</p>
      </div>
    </div>
  );
};

Memory Tricks 🧠

“IRON PEAK” for Stability

Iron (Fe-56) is peak Release energy going there Opposite sides: fusion & fission Nuclear force vs Coulomb balance

Peak at mass 56 Energy max at 8.8 MeV All roads lead to iron Keys to stars and bombs

BE/A Curve Shape

    ╱■╲
   ╱   ╲
  ╱     ╲

"Mountain peak at Fe"
- Climb from left (fusion)
- Climb from right (fission)
- Both release energy!

Energy Release Rule

“Toward Iron, Energy Flows”

Anything moving toward A≈56 releases energy.

Common Mistakes ⚠️

❌ Mistake 1: Total BE vs BE/A

Wrong: “U-238 is more stable than He-4 (larger total BE)” Right: Compare BE/A! He-4 has higher BE/A → more stable per nucleon

❌ Mistake 2: Fusion doesn’t release energy

Wrong: “Only fission releases energy” Right: BOTH fusion and fission release energy (different parts of curve!)

❌ Mistake 3: Thinking Fe-56 is unstable

Wrong: “Peak means unstable” Right: Peak means MOST stable! Can’t gain energy by any reaction

❌ Mistake 4: Sign of energy

Wrong: Q negative for energy release Right: Q positive for energy release (products more bound)

❌ Mistake 5: Forgetting Coulomb repulsion

Wrong: “Heavier nuclei should be more stable (more nucleons)” Right: Coulomb repulsion weakens binding in heavy nuclei

Why the Curve Has This Shape

Rising Part (Low A)

Surface effects dominate:

• Many nucleons on surface
• Incompletely bound
• Adding nucleons increases binding
• BE/A rises

Formula (approximate):

Surface term decreases as A increases.

Peak Region (Medium A)

Optimal balance:

• Volume binding maximized
• Surface effects minimized
• Coulomb repulsion manageable
• Asymmetry effects small

Best configuration: Near equal protons and neutrons

Falling Part (High A)

Coulomb repulsion grows:

For large A, Z ≈ A/2:

Grows faster than volume binding (∝ A)!

Semi-Empirical Mass Formula (SEMF)

Weizsäcker’s formula for BE:

:::box

Terms:

1. Volume: $+a_v A$ (binding increases with A)
2. Surface: $-a_s A^{2/3}$ (surface nucleons less bound)
3. Coulomb: $-a_c Z^2/A^{1/3}$ (proton repulsion)
4. Asymmetry: $-a_a(A-2Z)^2/A$ (prefer N≈Z)
5. Pairing: $+δ$ (even-even favored)

Constants:

• a_v ≈ 15.8 MeV
• a_s ≈ 18.3 MeV
• a_c ≈ 0.714 MeV
• a_a ≈ 23.2 MeV :::

Not required for JEE in detail, but good to know concept!

Practice Problems

Level 1: JEE Main Basics

Q1. The BE/A of ⁴He is 7.1 MeV. What is the total binding energy?

Solution:

BE = (BE/A) × A = 7.1 × 4 = 28.4 MeV

Q2. Which nucleus is more stable: ⁵⁶Fe (BE/A = 8.8 MeV) or ²³⁸U (BE/A = 7.6 MeV)?

Solution:

Higher BE/A → more stable
⁵⁶Fe is more stable per nucleon.

Q3. Why do heavy nuclei undergo fission while light nuclei undergo fusion?

Solution:

Both processes move toward the peak (Fe-56).
- Heavy nuclei (right side) → split → move left (UP)
- Light nuclei (left side) → combine → move right (UP)
Both climb the curve → release energy.

Level 2: JEE Main/Advanced

Q4. The BE/A of ²H is 1.1 MeV and ⁴He is 7.1 MeV. How much energy is released when two deuterium nuclei fuse to form helium?

Solution:

Initial: 2 × ²H
BE_initial = 2 × (1.1 × 2) = 4.4 MeV

Final: ⁴He
BE_final = 7.1 × 4 = 28.4 MeV

Energy released = BE_final - BE_initial
Q = 28.4 - 4.4 = 24 MeV

Actually, ²H + ²H → ⁴He doesn't occur directly.
Real reaction: ²H + ²H → ³He + n + 3.27 MeV
or: ²H + ²H → ³H + p + 4.03 MeV

Q5. A nucleus of mass 240 u fissions into two equal fragments. BE/A of parent is 7.6 MeV and products is 8.5 MeV. Calculate energy released.

Solution:

Parent: A = 240
BE_parent = 7.6 × 240 = 1824 MeV

Products: 2 fragments of A = 120 each
BE_products = 2 × (8.5 × 120) = 2040 MeV

Q = BE_products - BE_parent
Q = 2040 - 1824 = 216 MeV

Q6. Why is ⁴He exceptionally stable for its mass number?

Solution:

⁴He has:
- 2 protons (filled shell)
- 2 neutrons (filled shell)
- "Doubly magic" configuration
- Symmetric (N = Z)
- Tightly bound alpha particle

BE/A = 7.07 MeV, much higher than neighbors.
This is why alpha particles are common in decay!

Level 3: JEE Advanced

Q7. The BE of ⁷Li is 39.2 MeV and ⁴He is 28.3 MeV. Calculate the energy threshold for the reaction: ⁷Li + p → 2⁴He

Solution:

Given:
BE(⁷Li) = 39.2 MeV
BE(⁴He) = 28.3 MeV
BE(¹H) = 0 (single proton)

Q-value = BE_products - BE_reactants
Q = 2(28.3) - 39.2 - 0
Q = 56.6 - 39.2 = 17.4 MeV

Q > 0 → Exothermic reaction
No energy threshold in principle.

However, need to overcome Coulomb barrier:
E_barrier ≈ 1-2 MeV for proton + Li

Practical threshold ≈ 1.8 MeV

Q8. If 1 kg of ²³⁵U undergoes complete fission with energy release of 200 MeV per fission, calculate total energy in Joules.

Solution:

Number of atoms in 1 kg:
N = (1000 g / 235 g/mol) × 6.022×10²³
N = 2.56 × 10²⁴ atoms

Energy per fission = 200 MeV = 200 × 1.6×10⁻¹³ J
E = 3.2 × 10⁻¹¹ J

Total energy:
E_total = N × E = 2.56×10²⁴ × 3.2×10⁻¹¹
E_total = 8.2 × 10¹³ J

Compare to TNT: 1 kg TNT ≈ 4.6×10⁶ J
U-235 is about 17 million times more energetic!

Q9. Show that for fusion to occur between two nuclei with charges Z₁e and Z₂e, the kinetic energy must overcome Coulomb barrier.

Solution:

Coulomb potential energy at contact (r = R₁ + R₂):

U = k(Z₁e)(Z₂e)/(R₁ + R₂)

For fusion: KE ≥ U (classical)

R = R₀A^(1/3), so:
U ≈ k(Z₁Z₂e²)/(R₀(A₁^(1/3) + A₂^(1/3)))

Example: ²H + ³H fusion
Z₁ = Z₂ = 1
A₁ = 2, A₂ = 3
R₁ + R₂ ≈ 1.2(2^(1/3) + 3^(1/3)) fm ≈ 3.2 fm

U = (9×10⁹)(1)(1)(1.6×10⁻¹⁹)² / (3.2×10⁻¹⁵)
U ≈ 7.2×10⁻¹⁴ J ≈ 0.45 MeV

Need KE ≥ 0.45 MeV classically.

In reality, quantum tunneling allows fusion at lower E.
Temperature needed: kT ≈ 0.1 MeV → T ≈ 10⁸ K!
(This is why sun's core is 15 million K)

Q10. The BE/A curve peaks at A≈56. Using simple arguments, explain why: (a) Fission of heavy nuclei releases energy (b) Fusion of light nuclei releases energy (c) Fe-56 doesn’t fission or fuse

Solution:

(a) Fission: A=238 → A≈120 each
    BE/A: 7.6 → 8.5 MeV
    Moving UP curve → products more bound
    ΔBE/A ≈ 0.9 MeV per nucleon
    Total: 238 × 0.9 ≈ 214 MeV released ✓

(b) Fusion: ²H + ³H → ⁴He
    BE/A: ~2 → 7.1 MeV
    Moving UP curve → product more bound
    ΔBE/A ≈ 5 MeV per nucleon
    Total: ~20 MeV released ✓

(c) Fe-56 at peak:
    Any reaction moves DOWN curve
    Products less bound
    Would require energy input
    Thermodynamically unfavorable
    This is nuclear "dead end"!

Physical insight:
Peak represents optimal balance of:
- Strong force (attractive, short-range)
- Coulomb force (repulsive, long-range)
- Surface effects (reduce binding)
- Quantum effects (shell structure)

Applications

1. Nuclear Power (Fission)

• ²³⁵U or ²³⁹Pu fission
• ~200 MeV per fission
• Heat → steam → electricity
• Current: ~10% world energy

2. Hydrogen Bomb (Fusion)

• ²H + ³H → ⁴He + n
• Much more energetic per kg
• Requires fission trigger (high T)

3. Stellar Energy (Fusion)

• Powers ALL stars
• Main sequence: H → He
• Later: He → C, O, etc.
• Ends at Fe-56 (no more energy!)

4. Supernova Nucleosynthesis

• Core collapses when reaches Fe
• No more fusion energy available
• Explosion creates elements > Fe
• We are made of star stuff!

Cross-Links to Related Topics

• Nuclear Structure - Mass defect and BE basics
• Nuclear Reactions - Fission and fusion details
• Radioactivity - Unstable nuclei decay
• Einstein’s Mass-Energy - E=mc² foundation

Quick Revision Checklist ✓

• BE/A = Total BE / A (binding per nucleon)
• Curve peaks at ⁵⁶Fe (BE/A ≈ 8.8 MeV)
• Higher BE/A → more stable
• Moving toward peak releases energy
• Fission: heavy → medium (climb left)
• Fusion: light → heavier (climb right)
• ⁴He is locally stable (7.07 MeV)
• Coulomb repulsion weakens heavy nuclei
• 1 u = 931.5 MeV/c²
• Fission: ~200 MeV, Fusion: ~20 MeV per reaction
• Fe-56 is nuclear “dead end”

Final Tips for JEE

1. Understand the curve shape: Know why it rises then falls
2. Peak at Fe: Maximum stability at A≈56
3. Energy release: Move toward peak → energy out
4. Both processes work: Fission AND fusion release energy
5. Compare BE/A: Not total BE for stability
6. Know key values: ⁴He (7.1), ⁵⁶Fe (8.8), ²³⁸U (7.6)
7. Qualitative reasoning: Often no calculation needed!
8. Units: MeV for nuclear energy
9. Mass-energy: Always via E=mc²
10. Physical insight: Balance of forces explains curve

Last updated: May 5, 2025 Previous: Nuclear Structure Next: Radioactivity