Nuclear

name: nuclear

description: > Expert nuclear physics assistant for physicists and students. Use this skill whenever the user needs: help with nuclear structure, radioactive decay, nuclear reactions, fusion and fission energy, or radiation detection. Includes both theoretical foundations and applications. trigger: Any nuclear physics question - from structure to reactions to medical and energy applications. license: MIT compatibility: opencode metadata: audience: physicists category: physics

Nuclear Physics — Structure, Decay, and Reactions

Covers: Nuclear Structure · Radioactive Decay · Nuclear Reactions · Fusion · Fission · Radiation Detection

Nuclear Structure

Nuclear Composition

Nucleus = protons + neutrons (nucleons):

• Atomic number Z: number of protons
• Neutron number N: number of neutrons
• Mass number A = Z + N

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

Nuclear Sizes

Charge radius:

R = r₀ A^{1/3}, r₀ ≈ 1.2 fm

Matter radius slightly larger: r₀ ≈ 1.4 fm.

Nuclear Density

Roughly constant inside nucleus:

ρ ≈ 0.16 nucleons/fm³

Uniform density → Fermi gas model.

Nuclear Spin

Total angular momentum I = combination of:

• Orbital angular momentum l of each nucleon
• Spin s of each nucleon (½)

Nuclear Moments

Magnetic dipole moment:

μ = g_I μ_N I

μ_N = eℏ/2m_p (nuclear magneton)

Electric quadrupole moment: Q measures deviation from spherical.

Nuclear Shell Model

Nucleons in shells (like electrons):

• Magic numbers: 2, 8, 20, 28, 50, 82, 126
• Extra stability at magic numbers

Spin-orbit interaction causes shell splitting.

Magic Numbers and Shell Closures

Collective Model

Deformed nuclei:

• Vibrational states
• Rotational bands
• Quadrupole moments

Liquid Drop Model

Semi-empirical mass formula:

B(A,Z) = a_V A - a_S A^{2/3} - a_C Z²/A^{1/3} - a_A (A-2Z)²/A + δ(A,Z)

Mass Excess and Binding Energy

Mass excess: ΔM = [M - A u] c² Binding energy: B = Z m_p + N m_n - M

Nuclear Forces

Short-range (~1 fm):

• Strong interaction
• Spin-dependent
• Charge independent
• Spin-orbit term

NN potential: Reid, Argonne, Nijmegen.

Deuteron Properties

• Bound state: B = 2.22 MeV
• Quadrupole moment
• Spin triplet (S=1)

Nucleon-Nucleon Scattering

Phase shift analysis:

• Singlet, triplet channels
• Partial wave expansion

Radioactive Decay

Decay Modes

Alpha Decay

Tunneling through Coulomb barrier:

T ∝ exp(-2G), G = (2m|Q|R/ℏ²)^{1/2}

Geiger-Nuttall law:

log T = aZ/√A + b

Beta Decay

Fermi's Golden Rule:

λ = (2π/ℏ) |M|² ρ(E)

Allowed transitions:

• Fermi: Δπ = no, ΔS = 0
• Gamow-Teller: Δπ = no, ΔS = 1

Forbidden transitions: ΔL > 0, suppressed.

Neglect of Parity

Weak interaction violates parity:

• Only left-handed neutrinos
• Maximum parity violation

Electron Capture

p + e⁻ → n + ν_e:

• Competes with β⁺
• Q must be > 0.511 MeV

Gamma Decay

Electromagnetic transitions:

• E1: Δπ = yes, ΔL = 1
• E2: Δπ = no, ΔL = 2
• M1: Δπ = no, ΔL = 0,1

Weisskopf estimates for transition rates.

Internal Conversion

Energy → e⁻ instead of γ:

• Competes with γ
• Higher for low-energy transitions

Decay Chains

Long chains of decays:

• ²³⁸U → ... → ²⁰⁶Pb
• ²³⁵U → ... → ²⁰⁷Pb
• ²³²Th → ... → ²⁰⁸Pb
• ⁴⁰K → ⁴⁰Ar, ⁴⁰Ca

Decay Constants

Activity:

A = λN = A₀ e^{-λt}

Half-life: t_{1/2} = ln 2 / λ

Mean lifetime: τ = 1/λ

Branching Ratios

Multiple decay modes possible:

• Branching ratio = fraction in each mode
• Sum = 100%

Secular Equilibrium

For daughter with shorter half-life: N_2(t) = λ_1 N_1 / (λ_2 - λ_1) (e^{-λ_1 t} - e^{-λ_2 t})

Transient Equilibrium

When parent half-life comparable.

Nuclear Reactions

Q-Value

Energy released:

Q = (m_initial - m_final) c²

Positive: exothermic Negative: endothermic

Threshold Energy

For endothermic reactions:

E_th = -Q (m_final/m_projectile)

For charged particles, must overcome Coulomb barrier.

Reaction Types

Cross Section

σ = (rate)/(flux × target nuclei)

Units: barns (10⁻²⁴ cm²)

Compound Nucleus

High-energy intermediate:

• Equilibrium reached
• Statistical decay

Direct Reactions

Peripheral collisions:

• Stripping, pickup
• Few nucleons transferred

Optical Model

Complex potential:

U(r) = V(r) + iW(r)

Describes elastic scattering.

Compound Elastic

Through compound nucleus, interferes with direct.

Resonance Scattering

Breit-Wigner formula:

σ(E) = σ₀ (E_R/E) [(E_R/2)² + Γ²/4] / [(E-E_R)² + Γ²/4]

Fusion Reactions

D + T → α + n (14.1 MeV) D + D → T + p (4.0 MeV) D + D → ³He + n (3.3 MeV) p + ¹¹B → 3α (8.7 MeV)

Fission

Nuclei split:

• Heavy (A > 230) typically
• Energy ~200 MeV

Neutrons released → chain reaction.

Fission Products

Most at probable A ~ 95 and 140. Neutron-rich → β decay chains.

Fission Barriers

Double-humped for deformed:

• Outer barrier
• Inner barrier
• Isomeric states

Nuclear Models

Shell Model

Independent particles in potential:

• Harmonic oscillator
• Woods-Saxon potential
• Spin-orbit term

Configuration Mixing

Mix of Slater determinants. Shell model wavefunctions.

Cluster Model

Preformed clusters:

• α-clustering in ⁸Be, ¹²C
• α + ¹²C in ¹⁶O

Mean Field

Hartree-Fock approach:

• Self-consistent potential
• Single-particle states

RPA (Random Phase Approximation)

Excited states:

• Correlated particle-hole
• Collective modes

Interacting Boson Model

IBM:

• s and d bosons
• SU(3), O(6) limits

Ab Initio Methods

Start from NN interaction:

• No-core shell model
• Green's function Monte Carlo
• Coupled cluster

Density Functional Theory

EDF approaches:

• Skyrme, Gogny forces
• Global fits to data

Radiation Detection

Detector Types

Energy Resolution

σ/E ∝ 1/√E

Better with semiconductor.

Pulse Height

Proportional to energy deposited. Calibration needed.

Photomultiplier Tubes

Light → photoelectron → cascade. Gain ~10⁶-10⁸.

Spectroscopy

• HPGe: high resolution
• NaI(Tl): high efficiency
• Si(Li): X-rays

Particle Identification

Neutron Detection

Via reactions:

• ³He(n,p)³H
• B(n,α)
• Fission

Background

Cosmic rays:

• Muons, neutrons
• Shielding needed

Internal contamination:

• Natural radionuclides
• Cosmogenic activation

Dead Time

Paralyzable vs. non-paralyzable. Correct for high rates.

Coincidence Detection

Reduce background:

• Time coincidence
• Compton suppression

Pulse Shape Discrimination

Separate particles:

• n/γ in liquid scint
• α/β in some scintillators

Applications

Nuclear Energy

• Fission reactors
• Fusion (future)
• Radioisotope generators

Medical

• Radiotherapy (γ, e⁻)
• Imaging (PET, SPECT)
• Radioisotope production

Dating

• ¹⁴C dating (t₁/2 = 5730 y)
• K-Ar (t₁/2 = 1.25 Gy)
• U-Pb

Tracers

• Metabolic studies
• Industrial tracers
• Environmental

Astrophysics

• Nucleosynthesis (r, s, p processes)
• Solar neutrinos
• Cosmic rays

Materials

• Ion implantation
• Neutron activation analysis
• Radiation damage studies

Security

• Portal monitors
• SNM detection
• Arms control verification

Reactor Physics

Neutron Economy

• Production: fission
• Loss: absorption + leakage
• Breeding possible with fast reactors

Criticality

k_eff = 1:

• Prompt critical: exponential growth
• Delayed neutrons needed for control

Moderation

Slow neutrons:

• Hydrogen, deuterium
• Graphite
• Slowing down power

Cross Sections

• Capture: σ_γ
• Fission: σ_f
• Scattering: σ_s

Resonances important.

Reactor Types

Fuel Cycle

• Mining, enrichment
• Burnup in reactor
• Spent fuel
• Reprocessing
• Waste disposal

Criticality Safety

• Keep systems subcritical
• Double contingency
• Geometry control

Common Errors to Avoid

• Confusing mass number with atomic mass
• Using wrong units (keV vs MeV)
• Forgetting binding energy in Q-value
• Confusing decay constant with half-life
• Misapplying decay chain equations
• Confusing cross section with probability
• Forgetting Coulomb barrier in fusion
• Using classical instead of quantum tunneling
• Confusing neutrons and protons in strong force
• Ignoring parity violation in weak decays

Key References

• Introductory Nuclear Physics by Krane — Standard text
• Nuclear Physics by Wong — Theory
• Nuclear Physics: Principles and Applications — Modern
• Radiation Detection and Measurement — Knoll — Detection