Nuclear Structure: The Heart of Matter

The Oppenheimer Trinity Test Connection 🎬

“Now I am become Death, the destroyer of worlds.” - Oppenheimer, quoting Bhagavad Gita after Trinity test

Before Oppenheimer could split the atom, scientists had to understand what’s inside! The nucleus, discovered by Rutherford, contains 99.9% of atomic mass in a space 100,000 times smaller than the atom itself.

Discovery of Nuclear Components

Timeline

1911: Rutherford discovers nucleus (positive, massive, tiny) 1919: Rutherford discovers proton

• Bombarded nitrogen with α-particles
• Observed hydrogen nuclei (protons) ejected
• ¹⁴N + α → ¹⁷O + p (first transmutation!)

1932: Chadwick discovers neutron

• Bombarded beryllium with α-particles
• Observed uncharged radiation (not γ-rays!)
• ⁹Be + α → ¹²C + n
• Nobel Prize 1935

Nuclear Notation

Standard Form

where:

• X = chemical symbol
• A = mass number (nucleons = protons + neutrons)
• Z = atomic number (protons)
• N = neutron number (N = A - Z)

Examples:

• $^{12}_6C_6$ → Carbon-12: 6 protons, 6 neutrons
• $^{14}_6C_8$ → Carbon-14: 6 protons, 8 neutrons
• $^{235}_{92}U_{143}$ → Uranium-235: 92 protons, 143 neutrons

Simplified Forms

Often written as:

• $^AX$ (when Z is obvious from element)
• C-12, U-235, etc.

Nuclear Building Blocks

Proton (p)

Neutron (n)

Electron (e⁻)

Mass Relationships

Atomic Mass Unit (u)

:::box Definition

1 u = 1/12 × mass of $^{12}C$ atom

Energy Equivalent:

This comes from Einstein’s $E = mc^2$ :::

Why 1 u ≈ 931.5 MeV/c²?

Isotopes, Isobars, Isotones

Isotopes

Same Z, different A (same element, different mass)

¹²C₆ and ¹⁴C₆  → Carbon isotopes
   ↓        ↓
  6p+6n   6p+8n

Same chemistry, different nucleus!

Examples:

• Hydrogen: ¹H (protium), ²H (deuterium), ³H (tritium)
• Carbon: ¹²C (stable), ¹⁴C (radioactive, used in dating)
• Uranium: ²³⁵U (fissile), ²³⁸U (fertile)

Isobars

Same A, different Z (different elements, same mass number)

¹⁴C₆ and ¹⁴N₇  → Mass 14, but different elements
   ↓        ↓
 6p+8n    7p+7n

Different chemistry, same total nucleons!

Examples:

• $^{14}_6C$ and $^{14}_7N$
• $^{40}_{18}Ar$, $^{40}_{19}K$, $^{40}_{20}Ca$

Isotones

Same N, different Z and A (same neutron number)

¹⁴C₆ and ¹⁵N₇  → Both have 8 neutrons
   ↓        ↓
 6p+8n    7p+8n

Examples:

• $^{14}_6C$ and $^{15}_7N$ (both N=8)
• $^{3}_1H$ and $^{4}_2He$ (both N=2)

Isodiaphers

Same (A-2Z) (neutron excess)

Less commonly used in JEE.

Comparison Table

Nuclear Size

Empirical Formula

:::box Nuclear Radius

where:

• R₀ = 1.2 × 10⁻¹⁵ m = 1.2 fm (fermi or femtometer)
• A = mass number :::

Implications:

• Volume ∝ A (each nucleon occupies same space)
• Density is constant for all nuclei!
• Nuclear matter is incompressible

Nuclear Density

This is constant for all nuclei! 🤯

Amazing fact: 1 cm³ of nuclear matter weighs 230 million tons!

Interactive Demo: Visualize Nuclear Decay Processes

Explore how unstable nuclei undergo radioactive decay through alpha, beta, and gamma emissions.

Mass Defect (Δm)

When nucleons combine to form a nucleus, the mass of the nucleus is LESS than the sum of its constituents!

:::box Mass Defect Formula

or

where:

• Z = number of protons
• N = number of neutrons (A - Z)
• M = actual mass :::

The “missing” mass has been converted to binding energy!

Why Mass Defect Occurs

When nucleons bind:

• Strong nuclear force pulls them together
• Energy is released
• Mass decreases by Δm = E/c²

Analogy: Like a ball falling into a valley:

• Loses potential energy
• Gains binding (negative potential energy)
• Total mass-energy decreases

Binding Energy (BE)

Energy required to completely disassemble a nucleus into separate protons and neutrons.

:::box Binding Energy

In convenient units:

:::

Binding Energy Per Nucleon

Most important quantity in nuclear physics!

• Measures nuclear stability
• Higher BE/A → more stable
• Maximum at Fe-56 (~8.8 MeV)

Packing Fraction (f)

where:

• M = actual atomic mass (in u)
• A = mass number

Physical meaning:

• Measures deviation from integer mass
• Related to BE/A
• Negative f → more tightly bound

Less commonly used than BE/A in modern physics.

Interactive Demo: Nuclear Calculator

const NuclearCalculator = () => {
  const [Z, setZ] = useState(6);
  const [A, setA] = useState(12);

  const mp = 1.007276; // u (proton mass)
  const mn = 1.008665; // u (neutron mass)
  const me = 0.000549; // u (electron mass)

  // Example atomic masses (would need database for real app)
  const atomicMasses = {
    '1,1': 1.007825, '1,2': 2.014102, '1,3': 3.016049,
    '2,3': 3.016029, '2,4': 4.002603,
    '6,12': 12.000000, '6,13': 13.003355, '6,14': 14.003241,
    '7,14': 14.003074, '7,15': 15.000109,
    '8,16': 15.994915, '8,17': 16.999132, '8,18': 17.999160,
    '92,235': 235.043923, '92,238': 238.050788
  };

  const N = A - Z;
  const key = `${Z},${A}`;
  const Matom = atomicMasses[key] || (Z * mp + N * mn + Z * me);
  const Mnucleus = Matom - Z * me;

  // Mass defect
  const massDefect = (Z * mp + N * mn) - Mnucleus;

  // Binding energy
  const BE = massDefect * 931.5; // MeV
  const BEperA = BE / A;

  // Packing fraction
  const packingFraction = (Matom - A) / A;

  // Nuclear radius
  const R = 1.2 * Math.pow(A, 1/3); // fm

  return (
    <div>
      <h3>Nuclear Structure Calculator</h3>

      <label>Atomic Number (Z): {Z}</label>
      <input
        type="range"
        min="1"
        max="92"
        value={Z}
        onChange={(e) => setZ(Number(e.target.value))}
      />

      <label>Mass Number (A): {A}</label>
      <input
        type="range"
        min={Z}
        max={Z + 150}
        value={A}
        onChange={(e) => setA(Number(e.target.value))}
      />

      <div className="results">
        <h4>Nuclear Composition:</h4>
        <p>Protons: {Z}</p>
        <p>Neutrons: {N}</p>
        <p>Total Nucleons: {A}</p>

        <h4>Masses:</h4>
        <p>Expected mass (separate): {(Z*mp + N*mn).toFixed(6)} u</p>
        <p>Actual nuclear mass: {Mnucleus.toFixed(6)} u</p>
        <p>Mass defect: {massDefect.toFixed(6)} u</p>

        <h4>Binding Energy:</h4>
        <p>Total BE: {BE.toFixed(2)} MeV</p>
        <p>BE per nucleon: {BEperA.toFixed(3)} MeV</p>

        <h4>Size:</h4>
        <p>Nuclear radius: {R.toFixed(2)} fm</p>
        <p>Atomic radius: ~100,000 fm (0.1 nm)</p>

        <h4>Stability:</h4>
        {BEperA > 8 ? (
          <p style={{color: 'green'}}>✓ Highly stable nucleus</p>
        ) : BEperA > 7 ? (
          <p style={{color: 'orange'}}>Moderately stable</p>
        ) : (
          <p style={{color: 'red'}}>⚠ Less stable nucleus</p>
        )}
      </div>
    </div>
  );
};

Memory Tricks 🧠

“ZIPPER” for Nuclear Notation

Z = atomic number (protons) Isotopes have same Z Protons and neutrons in nucleus Packing fraction = (M-A)/A Einstein: E = mc² Radius = R₀A^(1/3)

Isotope vs Isobar

Iso-TOPE → same TOP number (Z, at top) Iso-BAR → same MASS (A, the bar/weight)

Mass Defect Memory

Nucleons together → Lighter
Binding makes → Tighter
Missing mass → Energy
E = mc² → The key

Atomic Mass Unit

“931.5 is the magic number” 1 u = 931.5 MeV/c²

“Carbon-12 defines the scale” 1 u = (1/12) × mass of ¹²C

Common Mistakes ⚠️

❌ Mistake 1: Confusing mass defect sign

Wrong: Δm = M_nucleus - [Zm_p + Nm_n] (positive) Right: Δm = [Zm_p + Nm_n] - M_nucleus (positive)

❌ Mistake 2: Using wrong particle mass

Wrong: Using atomic mass instead of nuclear mass Right: Subtract electron mass: M_nucleus = M_atom - Zm_e

❌ Mistake 3: Forgetting c² in BE

Wrong: BE = Δm (in Joules) Right: BE = Δm × c² OR Δm(u) × 931.5 MeV

❌ Mistake 4: Nuclear radius formula

Wrong: R = R₀A or R = R₀A² Right: R = R₀A^(1/3)

❌ Mistake 5: Neutron number

Wrong: N = Z Right: N = A - Z (not always equal to Z!)

Important Numerical Values

Particle Masses

Conversion Factors

Nuclear Sizes

Derivations You Must Know

1. Nuclear Density

Independent of A!

2. Binding Energy Calculation

For $^{4}_2He$:

3. Ratio of Nuclear Radii

Example: $^{27}Al$ vs $^{64}Cu$

Problem-Solving Strategy

Step 1: Identify Given Information

• Element (Z is fixed)
• Mass number (A)
• Or: specific isotope notation

Step 2: Calculate Basic Quantities

• N = A - Z
• Expected mass = Zm_p + Nm_n (+ Zm_e for atom)
• Actual mass (given or looked up)

Step 3: Find Mass Defect

• Δm = Expected - Actual
• Must be positive!

Step 4: Calculate BE

• BE = Δm × 931.5 MeV
• BE/A = BE / A

Step 5: Check Units

• Mass in u, BE in MeV
• OR all in SI (kg, Joules)

Practice Problems

Level 1: JEE Main Basics

Q1. Find the number of protons, neutrons, and electrons in $^{23}_{11}Na$.

Solution:

Protons (Z) = 11
Electrons = 11 (neutral atom)
Neutrons (N) = A - Z = 23 - 11 = 12

Q2. Calculate the nuclear radius of $^{27}Al$.

Solution:

R = R₀A^(1/3)
R = 1.2 × (27)^(1/3)
R = 1.2 × 3 = 3.6 fm

Q3. The binding energy of $^{4}He$ is 28.3 MeV. Find the binding energy per nucleon.

Solution:

BE/A = 28.3 / 4 = 7.075 MeV

Level 2: JEE Main/Advanced

Q4. Calculate the mass defect and binding energy of deuterium ($^2_1H$). Given: m_p = 1.007276 u, m_n = 1.008665 u, m_d = 2.014102 u

Solution:

Deuterium: 1 proton + 1 neutron

Expected mass = 1.007276 + 1.008665 = 2.015941 u
Actual atomic mass = 2.014102 u
Actual nuclear mass = 2.014102 - 0.000549 = 2.013553 u

Δm = 2.015941 - 2.013553 = 0.002388 u

BE = 0.002388 × 931.5 = 2.224 MeV
BE/A = 2.224 / 2 = 1.112 MeV per nucleon

(This is quite low - deuterium is weakly bound!)

Q5. The radii of two nuclei are in the ratio 1:2. Find the ratio of their mass numbers.

Solution:

R₁/R₂ = 1/2
R₀A₁^(1/3) / R₀A₂^(1/3) = 1/2
(A₁/A₂)^(1/3) = 1/2
A₁/A₂ = (1/2)³ = 1/8

Mass numbers are in ratio 1:8

Q6. Find the density of nuclear matter.

Solution:

ρ = 3m_u / (4πR₀³)

ρ = 3 × 1.67 × 10⁻²⁷ / [4π × (1.2 × 10⁻¹⁵)³]

ρ = 5.01 × 10⁻²⁷ / (2.17 × 10⁻⁴⁴)

ρ = 2.3 × 10¹⁷ kg/m³

This is the same for ALL nuclei!

Level 3: JEE Advanced

Q7. Calculate the energy required to remove a neutron from $^{13}_6C$ nucleus. Given: m($^{12}C$) = 12.000000 u, m($^{13}C$) = 13.003355 u, m_n = 1.008665 u

Solution:

Separation energy = BE of removed neutron

¹³C → ¹²C + n

Energy required = [m(¹²C) + m_n - m(¹³C)] × c²

Δm = 12.000000 + 1.008665 - 13.003355
Δm = 0.005310 u

E = 0.005310 × 931.5 = 4.95 MeV

(This is the neutron separation energy)

Q8. The binding energy per nucleon of $^{16}_8O$ is 7.97 MeV. Find: (a) Total binding energy (b) Mass defect in u (c) Atomic mass of $^{16}O$

Solution:

(a) Total BE = 16 × 7.97 = 127.52 MeV

(b) Δm = BE / 931.5 = 127.52 / 931.5 = 0.1369 u

(c) Expected mass = 8m_p + 8m_n + 8m_e
    = 8(1.007276) + 8(1.008665) + 8(0.000549)
    = 16.131920 u

    Atomic mass = Expected - Δm
    = 16.131920 - 0.1369
    = 15.995 u

    (Actual value: 15.994915 u - good match!)

Q9. A nucleus has radius R. If its mass number becomes 8 times, what happens to: (a) Nuclear radius (b) Nuclear density (c) Total mass

Solution:

(a) R' = R₀(8A)^(1/3) = R₀A^(1/3) × 2 = 2R
    Radius doubles

(b) ρ' = ρ (density remains same!)

(c) M' ≈ 8M (mass scales linearly with A)
    (approximately, ignoring mass defect changes)

Q10. Show that the radius of nucleus is proportional to the cube root of its volume, assuming constant nuclear density.

Solution:

Nuclear density: ρ = M/V

For constant ρ:
M ∝ V

Also: M = Am_u (approximately)
∴ V ∝ A

Volume: V = (4/3)πR³
∴ R³ ∝ A
∴ R ∝ A^(1/3)

This is the empirical formula R = R₀A^(1/3)!

Physical meaning: Nuclear matter is incompressible,
so volume simply scales with number of nucleons.

Cross-Links to Related Topics

• Binding Energy - Detailed BE curve and stability
• Radioactivity - Unstable nuclei and decay
• Nuclear Reactions - Fission, fusion, transmutations
• Rutherford Model - Discovery of nucleus

Quick Revision Checklist ✓

• Proton: +e, 1.007276 u, 938.27 MeV/c²
• Neutron: 0, 1.008665 u, 939.57 MeV/c²
• 1 u = 931.5 MeV/c² = 1.66 × 10⁻²⁷ kg
• Notation: $^A_ZX_N$ where N = A - Z
• Isotopes: same Z, different A
• Isobars: same A, different Z
• Nuclear radius: R = 1.2A^(1/3) fm
• Nuclear density: 2.3 × 10¹⁷ kg/m³ (constant!)
• Mass defect: Δm = [Zm_p + Nm_n] - M_nucleus
• Binding energy: BE = Δm × 931.5 MeV
• BE/A measures stability

Final Tips for JEE

1. Master conversions: u ↔ MeV/c² ↔ kg
2. Mass defect is positive: Expected > Actual
3. Nuclear vs atomic mass: Remember to subtract electrons
4. BE formula: Δm(u) × 931.5 = BE(MeV)
5. Radius formula: R ∝ A^(1/3), not A or A²
6. Density is constant: Same for all nuclei!
7. Isotope chemistry: Same Z → same chemistry
8. Know famous isotopes: ²H, ³H, ¹⁴C, ²³⁵U, ²³⁸U
9. Units matter: MeV for energy, fm for size
10. Physical meaning: Mass “disappears” into binding

Last updated: May 2, 2025 Previous: Hydrogen Spectrum Next: Binding Energy Curve