Nuclear Energy

[Decay] Radioactive decay law

N = N₀·e^(−λt). Same form for activity A = A₀·e^(−λt) and mass m = m₀·e^(−λt). λ = decay constant (s⁻¹): constant probability of decay per nucleus per second, independent of chemistry/temperature

[Decay] Half-life vs decay constant

T½ = ln2/λ = 0.693/λ. Mean lifetime τ = 1/λ = T½/0.693

[Decay] Activity

A = λN (decays per second). Unit: Becquerel (Bq) = 1 decay/s. 1 Ci = 3.7×10¹⁰ Bq. Use λ in s⁻¹ to get Bq

[Decay] Number of atoms from a mass

N = m·N_A/M (M in g/mol ≈ mass number A; N_A = 6.022×10²³ per mol)

[Decay] 'Activity must drop by a factor 256' - how long?

256 = 2⁸ → 8 half-lives. t = 8·T½ (I-131: 8 × 8.02 d ≈ 64 d). No logarithm needed when the factor is a power of 2

[Decay] Time for N (or A) to fall from N₀ to N

t = ln(N₀/N)/λ = T½·log₂(N₀/N). Keep λ and t in the same time unit

[Decay] I-131 facts

Half-life 8.02 days. Major fission product released in reactor accidents (health hazard). 1 µg = 4.6×10¹⁵ atoms = 4.6×10⁹ Bq = 0.12 Ci

[Decay] Activity of a fission product created by F fissions

N = (fission yield) × F, then A = λN. E.g. 10¹⁹ fissions, yield 0.03, T½ = 8 d → N = 3×10¹⁷, A ≈ 3×10¹¹ Bq

[Nuclear] Isotopes / isobars / isotones

Isotopes: same Z. Isobars: same A. Isotones: same N

[Nuclear] Nucleus notation ᴬ_ZX

A = Z + N = mass number (nucleons), Z = atomic number (protons), N = neutrons

[Nuclear] Mass unit conversion

1 u = 931.494 MeV/c². 1 eV = 1.6×10⁻¹⁹ J. 1 MeV = 1.602×10⁻¹³ J

[Nuclear] Nuclear radius and density

R = r₀·A^(1/3) with r₀ ≈ 1.25 fm. Density is roughly constant for all nuclei (nucleons only feel nearest neighbours)

[Binding] Mass defect and binding energy

Δm = Z·M_H + N·M_n − M(Z,A). B = Δm·c². Use ATOMIC masses with the hydrogen atom mass M_H (not m_p) so electron masses cancel

[Binding] What does binding energy mean?

Energy needed to split a nucleus into its separate nucleons; tells you about nuclear forces, stability, and energy released in decays/reactions

[Binding] B/A curve - what does it imply?

B/A rises steeply for light nuclei, peaks near Fe/Ni (A≈56-60), falls slowly for heavy nuclei. So fusing light nuclei and splitting heavy nuclei both release energy

[Binding] Binding energy per nucleon of He-4

B = 28.3 MeV, so B/A ≈ 7.07 MeV per nucleon

[Binding] Separation energy of the last neutron

S_n = B(A,Z) − B(A−1,Z) = [M(A−1) + M_n − M(A)]·c². E.g. last neutron in C-13: (12 + 1.008665 − 13.003355) u × 931.494 = 4.95 MeV

[Binding] Terms of the semi-empirical (liquid drop) mass formula

Volume (+), surface (−), Coulomb (−), asymmetry (−), pairing (+ for even-even nuclei, − for odd-odd)

[Binding] Magic numbers

2, 8, 20, 28, 50, 82, 126: nuclei with these Z or N are extra stable (shell structure)

[Decay] Alpha decay

X(A,Z) → Y(A−4, Z−2) + He-4. Heavy nuclei (A ≳ 180). Alpha energy is far below the Coulomb barrier, so it escapes by quantum tunnelling. Alpha carries ≈98% of Q

[Decay] Beta-minus decay

n → p + e⁻ + ν̄. A unchanged, Z → Z+1. For neutron-rich nuclei. E.g. C-14 → N-14 + e⁻ + ν̄

[Decay] Beta-plus decay

p → n + e⁺ + ν. A unchanged, Z → Z−1. For proton-rich nuclei. Q = [M(Z) − M(Z−1) − 2mₑ]c² (note the 2mₑ)

[Decay] Electron capture

p + e⁻ → n + ν. Z → Z−1, A unchanged. Proton-rich nuclei; competes with β⁺. Q = [M(Z) − M(Z−1)]c². E.g. C-11 + e⁻ → B-11 + ν

[Decay] Gamma decay

Excited nucleus → lower state + photon. Z and A unchanged (no transmutation). E.g. Co-60* → Co-60 + γ

[Decay] Spontaneous fission

Very heavy nucleus splits into 2-3 fragments plus neutrons/photons without being hit (e.g. Cf-252, U-238). Source of neutrons in unirradiated fuel

[Decay] Checking a nuclear equation

Sum of A equal on both sides and sum of Z equal on both sides. e⁻ has Z = −1, e⁺ has Z = +1, neutrinos have A = Z = 0

[Decay] Natural decay chains

U-238 → Pb-206; U-235 → Pb-207; Th-232 → Pb-208

[Q-value] Definition

Q = (ΣM_initial − ΣM_final)c² = ΣT_final − ΣT_initial (atomic masses). Q > 0: energy released (exothermic). Q < 0: energy required (endothermic)

[Q-value] Example

Be-9 + He-4 → C-12 + n has Q = +5.70 MeV (used in Be-alpha neutron sources)

[Reactions] Elastic scattering

(n,n): neutron bounces off, nucleus unchanged and unexcited. This is how a moderator slows neutrons

[Reactions] Inelastic scattering

Target absorbs the neutron then re-emits a lower-energy neutron, leaving the nucleus excited (then it emits a γ)

[Reactions] Radiative capture

(n,γ): neutron absorbed, compound nucleus de-excites by emitting γ. E.g. n + U-238 → U-239 + γ

[Cross section] Types and units

σtot = σ_scatter + σ_absorption; σ_a = σγ (capture) + σ_f (fission). Unit barn: 1 b = 10⁻²⁴ cm²

[Cross section] Macroscopic cross section

Σ = N·σ (cm⁻¹) = probability of interaction per cm of neutron travel. N = number density (atoms/cm³)

[Cross section] Number density

N = ρ·N_A/M (ρ in g/cm³, M in g/mol)

[Cross section] Mixtures / alloys

N_i = (weight fraction w_i)·ρ·N_A/M_i for each element, then Σ = Σ N_i·σ_i (do it separately for absorption and scattering)

[Cross section] Mean free path

λ = 1/Σ (cm). Not the decay constant

[Cross section] Getting Σ_f and σ_f from a measured fission rate

Σ_f = R_f/φ, then σ_f = Σ_f/N. E.g. φ = 3×10¹³, R_f = 1.29×10¹², N = 10²⁰ → Σ_f = 0.043 cm⁻¹, σ_f = 4.3×10⁻²² cm²

[Rates] Neutron flux

φ = n·v (neutron density × speed), unit n/(cm²·s)

[Rates] Reaction rate density

R = Σ·φ (reactions per cm³ per s). Fission: R_f = Σf·φ. Absorption: Σ_a·φ = Σγ·φ + Σ_f·φ. Multiply by V for the whole reactor

[Rates] Reactor power

P = E_f·R_f·V = E_f·Σ_f·φ_av·V, with E_f = 200 MeV = 3.204×10⁻¹¹ J (1 MeV = 1.602×10⁻¹³ J, 1 W = 1 J/s)

[Rates] Fission rate of the entire reactor

F = P/E_f (fissions per second). Each fission consumes one fissile nucleus, so this is the U-235 consumption rate in nuclides/s

[Rates] Fission rate at the centre (per cm³)

Σ_f·φ_max (fissions per cm³ per s)

[Fission] Fissile vs fertile

Fissile (U-235, U-233, Pu-239): fission with neutrons of any energy. Fertile (U-238, Th-232): capture a neutron and decay into fissile Pu-239 / U-233; fission only with MeV neutrons

[Fission] Why thermal neutrons fission U-235

Binding energy of the extra neutron (6.4 MeV) exceeds the critical energy (5.3 MeV), so even zero-energy neutrons cause fission. σ_f is largest at low energy (1/v law)

[Fission] 1/v law

At low energy σ ∝ 1/v: slower neutrons spend longer near the nucleus (and have a larger de Broglie wavelength), so interaction is more likely

[Fission] Energy released per fission

≈200 MeV: fragments' kinetic energy 165, prompt neutrons 5, prompt γ 7, β electrons 7, antineutrinos 10, decay γ 6. Antineutrino energy escapes

[Fission] Fission products

Two humps of fragment mass (A ≈ 95 and ≈ 140), then long beta-decay chains. ν ≈ 2.4-2.5 neutrons per fission of U-235

[Fission] Prompt neutrons

99% of fission neutrons; emitted within ~10⁻¹⁴ s; energies 0.1-10 MeV (≈2 MeV average)

[Fission] Delayed neutrons

[Fission] Why does a chain reaction work?

ν ≈ 2.5 neutrons per fission: self-sustaining when enough neutrons survive captures and leakage to cause the next fission

[Moderation] Neutron energy stages

Born at ~1-10 MeV (fast) → elastic scattering in moderator → 1 eV-1 MeV: resonance region (U-238 captures) → thermal (<1 eV, ≈2200 m/s vs ≈9×10⁶ m/s for fast)

[Moderation] What makes a good moderator?

Light nuclei (large energy loss per collision), low neutron absorption, cheap. H₂O (good but absorbs), D₂O (best, expensive), graphite

[k] Definition of k

k = neutrons in generation n+1 / neutrons in generation n = production / (absorption + leakage)

[k] Criticality states

k < 1 subcritical (chain reaction dies out); k = 1 critical (self-sustaining, constant power); k > 1 supercritical (power grows)

[k] Six-factor formula

k = ε·P_FNL·p·P_TNL·f·η

[k] Four-factor formula

k∞ = ε·p·f·η (infinite reactor, no leakage). k = k∞·P_NL where P_NL = P_FNL·P_TNL = 1/(1+L²B²)

[k] ε - fast fission factor

(total fast neutrons)/(fast neutrons produced by thermal fission) ≥ 1. Extra fast fission in U-238 by MeV neutrons

[k] P_FNL - fast non-leakage probability

Fraction of fast neutrons that do not leak out of the core. ≈ 1/(1+τB²) with τ = neutron age

[k] p - resonance escape probability

Fraction of neutrons that slow down through the U-238 resonance region without being captured

[k] Doppler effect

Hotter fuel → resonance peaks broaden → more capture → p decreases → negative reactivity (inherently stabilising)

[k] P_TNL - thermal non-leakage probability

Fraction of thermal neutrons that do not leak out. = 1/(1+L²B²) with L² = D/Σ_a

[k] f - thermal utilisation factor

Thermal neutrons absorbed in the fuel / all thermal neutrons absorbed = Σ_a^fuel/(Σ_a^fuel + Σ_a^mod + Σ_a^rest). Lowered by rods, boron, poisons, structure

[k] η - reproduction factor

Neutrons produced per neutron absorbed in the fuel: η = νΣf^fuel/Σ_a^fuel = νΣ_f/(Σ_f + Σγ). Increases with enrichment

[k] Which factors depend on reactor size?

Only the two leakage factors P_FNL and P_TNL. Larger reactor → less leakage → higher k

[k] '1000 fast neutrons' recipe

Multiply step by step: ×ε → ×P_FNL → ×p → ×P_TNL → ×f → ×η. Each factor = (neutrons after)/(neutrons before). 'New neutrons' = increase; 'lost' = before − after

[k] Example 1000-neutron cycle

ε 1.04, P_FNL 0.865, p 0.80, P_TNL 0.861, f 0.799, η 2.02: 1000 → 1040 → 900 → 720 → 620 → 495 → 1000, so k = 1

[k] Exam-style: ε = 1.03, P_FNL = 0.96, 734 pass resonances

1000 → 1030 (30 NEW from fast fission) → 988.8 ≈ 989 do not leak → p = 734/989 = 0.742

[k] Neutron losses in a reactor

Fall back into fuel before thermalising / captured in resonances; captured by H in water; captured by structural metals and poisons; leak out of the core

[Diffusion] Fick's law

J = −D·∇φ: net neutron current flows from high to low flux. D in cm

[Diffusion] Diffusion coefficient

D = λ_tr/3, λ_tr = 1/Σ_tr = 1/[Σ_s(1−μ̄)], μ̄ = 2/(3A)

[Diffusion] Extrapolation distance

d = 0.71·λ_tr = 2.13·D. Bare-reactor flux extrapolates to zero at R + d (Fick's law fails at the boundary)

[Diffusion] Diffusion length

L = √(D/Σ_a) (cm), L² = D/Σ_a. Related to the distance neutrons travel before absorption

[Diffusion] One-group reactor equation

D∇²φ − Σ_aφ + (1/k)·νΣ_fφ = 0, i.e. ∇²φ + B²φ = 0: leakage + absorption balanced by fission source

[Buckling] Material buckling

B_m² = (νΣ_f − Σ_a)/D = (k∞ − 1)/L². Depends only on material properties

[Buckling] Geometric buckling

B² depends only on shape and size. General: B² = (1/D)(νΣ_f/k − Σ_a). For a critical reactor (steady power) use k = 1: B² = (νΣ_f − Σ_a)/D

[Buckling] Criticality condition

B_m² = B² → critical. B_m² > B² (larger than critical size) → supercritical. B_m² < B² → subcritical. Larger B² = smaller reactor = more leakage

[Buckling] Non-leakage probability

P_NL = 1/(1+L²B²), k = k∞·P_NL. Leakage/absorption ratio = D·B²/Σ_a = L²B²

[Buckling] Sphere vs cube with same B²

Same material and same B² → same non-leakage probability (same leakage). For equal volume, the sphere has the smallest surface/volume so the least leakage

[Geometry] B² of a parallelepiped a×b×c

(π/a)² + (π/b)² + (π/c)². Flux: A·cos(πx/a)·cos(πy/b)·cos(πz/c), origin at the centre

[Geometry] B² of a sphere of radius R

(π/R)² → R = π/B. Flux φ(r) = A·sin(πr/R)/r

[Geometry] B² of a finite cylinder

(2.405/R)² + (π/H)². Flux: A·J₀(2.405r/R)·cos(πz/H)

[Geometry] B² of infinite slab and infinite cylinder

Slab thickness a: (π/a)². Infinite cylinder radius R: (2.405/R)²

[Flux] Average flux of a reactor

φ_av = P/(E_f·Σ_f·V) (P in W, E_f = 3.204×10⁻¹¹ J, V in cm³)

[Flux] Where is the flux maximum?

At the centre of a uniform bare reactor (all cosines = 1); it falls to ~0 at the extrapolated boundary

[Flux] Peak-to-average ratios

φ_max/φ_av: parallelepiped 3.87, finite cylinder 3.63, sphere 3.29 (π²/3), infinite slab 1.57 (π/2)

[Flux] Amplitude A for a parallelepiped

A = 3.87·P/(V·E_f·Σ_f), and φ_max = A (at the centre)

[Flux] Amplitude A for a sphere

A = P/(4R²·E_f·Σ_f). Centre flux from the limit sin(πr/R)/r → π/R: φ_max = A·π/R

[Flux] Amplitude A for a finite cylinder

A = 3.63·P/(V·E_f·Σ_f), φ_max = A at the centre

[Flux] Do the results make sense? (peak vs average)

Yes if φ_max is ≈3-4× φ_av: flux is cosine-shaped, maximum at the centre, ≈0 at the edge because neutrons leak out

[Flux] Full bare-reactor calculation order

1) cm units 2) B² from geometry 3) D or Σ_a from B² = (νΣ_f − Σ_a)/D 4) L 5) V 6) φ_av 7) φ_max 8) flux at r 9) fission rate P/E_f

[Flux] Worked check - sphere B² = 9.87×10⁻⁴ cm⁻², Σ_f = 0.0025, P = 50 MW

R = π/√B² = 100 cm; φ_av = 1.49×10¹⁴; φ_max = 4.9×10¹⁴; fission rate = 50 MW / 3.204×10⁻¹¹ J = 1.56×10¹⁸ per s

[Flux] Reflector

Layer around the core that scatters neutrons back: less leakage, better use of outer fuel, flatter flux (φ_av/φ_max increases). Needs high σ_s, low σ_a (graphite, heavy water)

[Flux] Ways to flatten the flux

Reflector; fuel enrichment zoning or poison loading (less fuel reactivity at the centre); control rod positioning. Aim: even temperatures and uniform burnup

[Diffusion] Group diffusion method

Neutron energies split into groups g = 1..N (1 = most energetic). Per group: leakage − absorption − scattering out + scattering in from higher groups + source = 0. No up-scattering assumed

[Kinetics] Reactivity

ρ = (k−1)/k = Δk/k: deviation from criticality. ρ > 0 power rises; ρ = 0 steady; ρ < 0 power falls. E.g. k = 0.98 → ρ = −0.0204

[Kinetics] Reactivity units

1 pcm = 10⁻⁵ Δk/k; 1 mk = 10⁻³; 1000 pcm = 1% = 10 mk = ρ of 0.01. Dollar = ρ/β_eff

[Kinetics] Prompt neutron lifetime

ℓ_p ≈ 10⁻⁴ s in thermal reactors: time from birth to death (leakage/absorption) = slowing-down time + diffusion time (diffusion time dominates)

[Kinetics] Reactor period without delayed neutrons

τ = ℓ_p/(k−1). E.g. k∞ = 1.001, ℓ_p = 1.08×10⁻⁴ s → τ = 0.1 s, power ×22 000 per second: uncontrollable

[Kinetics] Why delayed neutrons make control possible

They raise the mean generation time from ~10⁻⁴ s to ~0.1 s, so the same reactivity gives τ ≈ 85 s instead of 0.1 s: power changes slowly enough to control