Nuclei

1. The Nucleus

• Discovered by Rutherford (α-scattering experiment).
• Constituents : neutrons (n) and protons (p) – together called nucleons.
• Neutron : neutral particle, discovered by J. Chadwick.
• Proton : charge = +e, discovered by Goldstein.
• Masses are nearly the same (m_n slightly > m_p) :

2. Nuclide Notation

e.g. 126C → 6 p, 6 n ;  42He → 2 p, 2 n ;  23892U → 92 p, 146 n

3. Size of the Nucleus

R₀ ≈ 1.2 fm = 1.2 × 10⁻¹⁵ m (some books use 1.25 fm) ;  1 fm = 10⁻¹⁵ m

Volume :

V = 43πR³ = 43πR₀³ A

Density :

ρ = massvolume = A m_p43πR₀³ A

• Independent of A ⇒ all nuclei have nearly the same density.
• With R₀ = 1.2 fm, m_p = 1.67 × 10⁻²⁷ kg :

(≈ 10¹⁴ times the density of water)

4. Nuclear Force

• Attractive ; holds nucleons together in spite of the repulsion between protons.
• Acts between n–n, n–p and p–p (i.e. between nucleons).
• Strongest force within nuclear dimensions (F_n ≈ 100 F_e).
• Short range – acts only inside the nucleus (range ≈ 2–3 fm).
• Charge independent – same for n–n, n–p, p–p (does not depend on the nature of nucleons).

5. Mass–Energy Equivalence

• A rest mass m is equivalent to energy E (Einstein) :

c = 3 × 10⁸ m/s (speed of light)

Energy of 1 u :

E = (1.66 × 10⁻²⁷)(3 × 10⁸)² J = 1.49 × 10⁻¹⁰ J

   = 1.49 × 10⁻¹⁰1.6 × 10⁻¹⁹ eV ≈ 931 × 10⁶ eV

i.e. (1 u) × c² = 931.5 MeV

6. Mass Defect (Δm)

• Rest mass of a nucleus is smaller than the sum of rest masses of its nucleons. The difference is the mass defect.
• Δm = (expected mass) − (actual mass)

For AZX : nuclear mass m_X = M_X − Z m_e  (M_X = atomic mass)

Δm = [Z m_p + (A − Z) m_n] − [M_X − Z m_e]

     = Z(m_p + m_e) + (A − Z) m_n − M_X

M_H = m_p + m_e = mass of H-atom  (lowercase m → nuclear mass, capital M → atomic mass)

7. Binding Energy (B.E.)

• Energy required to break the nucleus into its constituent nucleons.
• = Energy released when the nucleus is formed from its nucleons.

8. B.E. per Nucleon & Stability

• Higher B.E./A ⇒ more tightly bound ⇒ more stable nucleus.

• B.E./A rises sharply for light nuclei, has a maximum ≈ 8.8 MeV near A = 56 (⁵⁶Fe), then slowly falls (≈ 7.6 MeV for ²³⁸U).
• For a wide range (A ≈ 30–170) B.E./A ≈ 8 MeV is nearly constant (nuclear force is short range – saturation).
• ⁴He (≈ 7.1 MeV) lies above its neighbours (⁶Li ≈ 5.3 MeV) ⇒ very stable.
• Heavy nuclei → move towards A ≈ 56 by fission ; light nuclei → by fusion. Both release energy.

9. Fission & Fusion

Fission :

• A heavy nucleus breaks into 2 (or more) medium nuclei to become more stable.

10n + 23592U → 23692U* → 14456Ba + 8936Kr + 3 10n + Q

(check : A : 236 = 144 + 89 + 3 ; Z : 92 = 56 + 36) ; Q ≈ 200 MeV per fission.

Fusion :

• Light nuclei fuse to form a heavier (more stable) nucleus.

21H + 31H → 42He + 10n + 17.6 MeV

(deuterium + tritium → helium + neutron)

10. Radioactivity

• Spontaneous disintegration of unstable nuclei to form more (energetically) stable nuclei.
• Types : (i) α decay (ii) β⁻ and β⁺ decay (iii) γ decay ; along with emission of neutrino / antineutrino in β decays.

1 u = 1.66 × 10⁻²⁷ kg ; e = 1.6 × 10⁻¹⁹ C

11. Neutrino & Antineutrino

• Fundamental particles, no charge and extremely tiny mass.
• Called "ghost particles" – they interact so weakly that trillions pass through the Earth without interacting.

12. Antiparticles

• An antiparticle has the same mass but opposite charge (and opposite other quantum numbers) as the particle.
• A particle and its antiparticle can annihilate – their mass converts completely into energy.
• Electron (β⁻) & positron (β⁺) are antiparticles ; neutrino (ν) & antineutrino (ν̄) are antiparticles.

13. General Nuclear Reaction

AZX → A₁Z₁Y + n₁α + n₂β⁻ + n₃β⁺ + n₄ν + n₅ν̄ + n₆γ + Q

• Given X and the emitted particles, find Y using conservation of Z and A :

14. Conserved in a Nuclear Reaction

• Atomic number (total charge number Z)
• Mass number A (total no. of nucleons)
• Charge
• Linear momentum
• Total (mass-energy + energy) – rest mass alone is not conserved
• Angular momentum (spin)

15. Energy Released (Q)

A + B → C + D + Q

• Q > 0 → exothermic, energetically favourable (can occur spontaneously).
• Q < 0 → endothermic, energetically not favourable (needs energy input).

16. Q in terms of Mass

Q = [(m_A + m_B) − (m_C + m_D)] c²

17. Q in terms of B.E.

• Usually B.E./A is given ⇒ B.E. = B.E.A × A

18. K.E. of Products

X (at rest) → Y + Z + Q

• Q appears as K.E. of products : K_Y + K_Z = Q
• Momentum conservation : 0 = p_Z − p_Y ⇒ p_Y = p_Z = p
• K = p²2m ⇒ K ∝ 1m (same p)

The lighter product carries most of the energy.

19. α Decay

• Unstable nucleus emits an α-particle ⇒ mass number decreases and the nucleus moves towards stability.
• Shown mainly by heavy nuclei (A > 210) ; B.E./A increases, so Q is positive.

e.g. 23892U → 23490Th + 42He + Q

Sharing of Q (X at rest) :

Masses ∝ mass numbers : m_Y ∝ (A − 4), m_α ∝ 4

e.g. ²³⁸U : K_α = 234238 Q ≈ 0.983 Q

• α-particles from a given decay have a definite (discrete) energy (two-body decay).

20. β⁻ Decay

• In β decay the N/Z ratio changes (shown by unstable nuclei).
• In β⁻ decay a neutron is converted into a proton (nucleus with excess neutrons).

Basic process : 10n → 11p + 0−1e + ν̄ + Q

e.g. 146C → 147N + β⁻ + ν̄ + Q

• Q is shared randomly between e⁻ and ν̄ (three-body decay ; recoil of Y negligible) :

⇒ β-particles have a continuous energy spectrum.

Stability curve (N vs Z) :

Light stable nuclei have N ≈ Z (⁴He, ¹²C) ; heavy stable nuclei have N > Z (²³⁸U : 92 p, 146 n).

21. β⁺ Decay

• A proton is converted into a neutron (nucleus with excess protons).

Basic process (inside the nucleus) : 11p → 10n + 0+1e + ν

(a free proton cannot do this since m_p < m_n ; the energy comes from the nucleus)

e.g. 106C → 105B + β⁺ + ν + Q

= (M_X − M_Y)c² − 2m_ec² = (M_X − M_Y)c² − 2(0.511 MeV)

(using atomic masses ; 2m_ec² ≈ 1.02 MeV)

• Q is shared randomly between β⁺ and ν.

22. K-Capture (Electron Capture)

• Nucleus captures an electron from the nearest (K) shell ⇒ a proton becomes a neutron (N/Z increases).

Basic : 11p + 0−1e → 10n + ν

e.g. 4019K + 0−1e → 4018Ar + ν + Q

23. γ Decay

• After α or β decay, the daughter nucleus is often left in an excited state ; it comes to the ground state by emitting γ-photon(s).

e.g. AZX → A−4Z−2Y* + 42He, then Y* → Y + γ.  No change in A or Z.

24. Law of Radioactive Decay

• (Rutherford & Soddy) Rate of disintegration ∝ number of active (undecayed) nuclei present.

dNdt ∝ N ⇒

• λ = decay constant – depends only on the nature of the nucleus ; not on temperature, pressure, concentration etc.

∫_N₀^N dNN = −∫₀^t λ dt ⇒ ln N − ln N₀ = −λt

25. Activity (A or R)

• Number of nuclei decaying per second : A = |dN/dt|

• Same λ but more nuclei (bigger sample) ⇒ larger activity.

26. Half-life & Mean Life

• Half-life : time in which half of the active nuclei decay.

N₀2 = N₀e^−λT ⇒ e^−λT = ½ ⇒ λT = ln 2

After n half-lives (t = nT_½) :

N₀ → N₀2 → N₀4 → N₀8 → …

27. Writing Rate Equations

• Rule : dN/dt of a nuclide = (rate of formation) − (rate of decay).

Ex. 1 : Many inputs / outputs

Ex. 2 : Production at constant rate R

Factory → (R per second) → X —λ→ Y ;  take N_X = 0 at t = 0

Solving : N_X = Rλ(1 − e^−λt)  → R/λ as t → ∞ (steady state)

N_Y = ∫₀^t R(1 − e^−λt) dt = R[t + 1λ(e^−λt − 1)]

(total produced Rt = N_X + N_Y)

Ex. 3 : Chain X —λ₁→ Y —λ₂→ Z

28. Parallel Decay

29. Successive Decay X → Y → Z

X : dN_X/dt = −λ₁N_X ⇒

Y : dN_Ydt + λ₂N_Y = λ₁N₀e^−λ₁t  (1st-order linear)

Multiply by e^λ₂t : ddt(N_Ye^λ₂t) = λ₁N₀e^{(λ₂−λ₁)t}

Integrate (0 → t) : N_Ye^λ₂t = λ₁N₀λ₂ − λ₁(e^{(λ₂−λ₁)t} − 1)

• N_X continuously decreases ; N_Z continuously increases ; N_Y first increases, then decreases.

Maximum N_Y : dN_Y/dt = 0

λ₁N_X = λ₂N_Y ⇒ λ₁N₀e^−λ₁t = λ₂λ₁N₀λ₂ − λ₁(e^−λ₁t − e^−λ₂t)

⇒ λ₂e^−λ₂t = λ₁e^−λ₁t ⇒ e^{(λ₂−λ₁)t} = λ₂λ₁

Then N_Y = λ₁λ₂N_X = λ₁λ₂N₀e^−λ₁t_m , with e^−λ₁t_m = (λ₁/λ₂)^{λ₁/(λ₂−λ₁)}

Check : λ₁ = λ, λ₂ = 2λ ⇒ t_m = ln2/λ, (N_Y)_max = N₀(½)² = N₀/4 ✓