nuclear-physics

# Nuclear Physics ## What I do I provide comprehensive expertise in nuclear physics, the branch of physics studying atomic nuclei and their constituents. I enable you to analyze radioactive decay processes, understand nuclear structure and binding energy, calculate fission and fusion reactions, apply nuclear models (shell model, liquid drop), compute reaction cross-sections, and model nuclear reactors and weapons. My knowledge spans from foundational nuclear properties to applications in energy production, medicine, and astrophysics. ## When to use me Use nuclear physics when you need to: calculate half-lives and decay rates, design or analyze nuclear reactors, compute fusion reaction rates in stars, model radioactive decay chains, calculate shielding and dosimetry, analyze nuclear data and cross-sections, understand stellar nucleosynthesis, or develop nuclear medicine applications. ## Core Concepts - **Nuclear Structure**: Protons and neutrons (nucleons) bound by strong nuclear force with meson exchange. - **Binding Energy and Mass Defect**: Mass difference between nucleus and constituent nucleons (E=mc²). - **Radioactive Decay**: Alpha (helium emission), beta (electron/positron emission), gamma (photon emission) processes. - **Decay Law**: N(t) = N₀ exp(-λt) with half-life t½ = ln(2)/λ. - **Nuclear Fission**: Splitting of heavy nuclei releasing energy (U-235, Pu-239 fission). - **Nuclear Fusion**: Combining light nuclei to release energy (D-T, p-p chain in stars). - **Shell Model and Magic Numbers**: Single-particle orbits explaining nuclear structure and stability. - **Liquid Drop Model**: Semi-empirical mass formula describing binding energy trends. - **Q-Value**: Energy released/absorbed in nuclear reactions (mass difference × c²). - **Cross-Section and Reaction Rate**: Probability of interaction and resulting reaction rate (R = nσv). ## Code Examples ### Nuclear Properties ```python import numpy as np # Physical constants c = 2.998e8 # m/s u = 1.66054e-27 # Atomic mass unit (kg) e = 1.602e-19 # Elementary charge (C) MeV_to_J = 1.602e-13 # MeV to Joules # Nuclear masses and binding energy def atomic_mass_excess(A, Z, mass_excess_MeV): """Convert mass excess to atomic mass in u.""" return A + mass_excess_MeV / 931.494 def binding_energy(A, Z, mass_nucleus): """Calculate binding energy from nuclear mass.""" mp = 1.007276 # Proton mass (u) mn = 1.008665 # Neutron mass (u) mass_defect = Z * mp + (A - Z) * mn - mass_nucleus return mass_defect * 931.494 # MeV def semi_empirical_mass(A, Z): """ Semi-empirical mass formula (Weizsäcker formula). B(A,Z) = a_v A - a_s A^(2/3) - a_c Z(Z-1)/A^(1/3) - a_a (A-2Z)²/A ± δ """ a_v = 15.75 a_s = 17.8 a_c = 0.711 a_a = 23.7 a_p = 11.18 # Pairing term if A % 2 == 0 and Z % 2 == 0: delta = a_p / np.sqrt(A) elif A % 2 == 1 and Z % 2 == 1: delta = -a_p / np.sqrt(A) else: delta = 0 B = (a_v * A - a_s * A**(2/3) - a_c * Z * (Z - 1) / A**(1/3) - a_a * (A - 2*Z)**2 / A + delta) return B def binding_energy_per_nucleon(B, A): """Calculate binding energy per nucleon.""" return B / A # Example calculations print("Nuclear binding energy:") print(f" Semi-empirical for ⁴⁰Ca (Z=20, A=40):") B_Ca40 = semi_empirical_mass(40, 20) print(f" Total B = {B_Ca40:.1f} MeV") print(f" B/A = {B_Ca40/40:.3f} MeV/nucleon") print(f"\n Semi-empirical for ⁵⁶Fe (Z=26, A=56):") B_Fe56 = semi_empirical_mass(56, 26) print(f" Total B = {B_Fe56:.1f} MeV") print(f" B/A = {B_Fe56/56:.3f} MeV/nucleon") print(f"\n Semi-empirical for ²³⁸U (Z=92, A=238):") B_U238 = semi_empirical_mass(238, 92) print(f" Total B = {B_U238:.1f} MeV") print(f" B/A = {B_U238/238:.3f} MeV/nucleon") # Magic numbers (nuclear shell closure) magic_numbers = [2, 8, 20, 28, 50, 82, 126] print(f"\nMagic numbers (shell closures): {magic_numbers}") print(" Nuclei with magic numbers have:") print(" - Extra binding energy") print(" - Larger neutron separation energies") print(" - Spherical ground states") print(" - Higher first excited states") ``` ### Radioactive Decay ```python import numpy as np def decay_constant(half_life): """λ = ln(2) / t_1/2""" return np.log(2) / half_life def activity(N, half_life): """A = λN = (ln2/t_1/2) × N""" return decay_constant(half_life) * N def radioactive_decay(N0, half_life, t): """N(t) = N₀ × 2^(-t/t_1/2) = N₀ × exp(-λt)""" return N0 * np.exp(-decay_constant(half_life) * t) def daughter_growth(N0, half_life_parent, half_life_daughter, t): """Bateman equations for decay chain.""" lambda_p = decay_constant(half_life_parent) lambda_d = decay_constant(half_life_daughter) N_parent = N0 * np.exp(-lambda_p * t) N_daughter = (lambda_p / (lambda_d - lambda_p)) * N0 * ( np.exp(-lambda_p * t) - np.exp(-lambda_d * t) ) return N_parent, N_daughter def secular_equilibrium(t, lambda_p, lambda_d): """When t >> t_parent, N_daughter/N_parent ≈ λ_p/λ_d.""" return lambda_p / lambda_d # Decay constants and half-lives half_lives = { 'U-238': 4.468e9, # years 'U-235': 7.038e8, # years 'Ra-226': 1600, # years 'C-14': 5730, # years 'Co-60': 5.27, # years 'I-131': 8.02, # days 'Cs-137': 30.17, # years 'Rn-222': 3.82, # days } print("Radioactive decay:") for isotope, t_half in list(half_lives.items())[:5]: lambda_decay = decay_constant(t_half) print(f" {isotope}: t_1/2 = {t_half:.2e}") print(f" λ = {lambda_decay:.2e} s⁻¹") print(f" 1/λ = {1/lambda_decay:.2e} s = {1/lambda_decay/3600:.2e} hours") # Carbon-14 dating def carbon14_age(C14_ratio, t_half=5730): """ Calculate age from C-14 ratio. Assumes initial ratio was 1.25e-12 (living organism) """ lambda_c14 = decay_constant(t_half * 365.25 * 24 * 3600) return np.log(C14_ratio) / (-lambda_c14) / (365.25 * 24 * 3600) # Samples with different C-14 ratios initial_ratio = 1.25e-12 for ratio in [1.25e-12, 0.625e-12, 0.156e-12, 0.039e-12]: age = carbon14_age(ratio / initial_ratio) print(f"\n C-14 ratio = {ratio:.3e}: age = {age:.0f} years") print(f" ~ {age/1000:.1f} thousand years") # Decay chain (U-238 series) U238_hl = 4.468e9 * 365.25 * 24 * 3600 # seconds Pb206_hl = np.inf # Stable t_equilibrium = 1e9 * 365.25 * 24 * 3600 # 1 billion years N_U238 = 1e6 # Atoms ratio_Pb_U = np.exp(-decay_constant(U238_hl) * t_equilibrium) print(f"\nU-238 → Pb-206 decay chain:") print(f" After 1 billion years: Pb-206/U-238 ≈ {ratio_Pb_U:.2e}") print(f" Secular equilibrium reached when t >> 4.5 billion years") ``` ### Nuclear Reactions and Q-Values ```python import numpy as np def Q_value(m_initial, m_final, c2=931.494): """ Calculate Q-value of nuclear reaction. Q = (m_initial - m_final) × c² Positive Q: exothermic (releases energy) Negative Q: endothermic (absorbs energy) """ return (m_initial - m_final) * c2 def threshold_energy(Q, A_target, A_projectile, c2=931.494): """ Minimum kinetic energy for endothermic reaction. E_thresh = |Q| × (A_target + A_projectile) / A_target """ return abs(Q) * (A_target + A_projectile) / A_target def reaction_rate(n1, n2, sigma, v): """R = n₁n₂σv for two-body reactions.""" return n1 * n2 * sigma * v def cross_section_beer_lambert(I0, I, x): """σ = (1/nx) × ln(I₀/I) from Beer-Lambert law.""" return np.log(I0 / I) / x # Masses in atomic mass units (u) masses_u = { 'n': 1.008665, 'p': 1.007825, 'D': 2.014102, # Deuterium 'T': 3.016049, # Tritium 'He3': 3.016029, 'He4': 4.002602, 'U235': 235.043930, 'U236': 236.045568, 'Ba141': 140.914411, 'Kr92': 91.926156, 'fission_fragments': 140.9 + 91.9, # Approximate } # Fusion reactions print("Nuclear reaction Q-values:") print("\n1. D + T → He-4 + n:") m_init = masses_u['D'] + masses_u['T'] m_final = masses_u['He4'] + masses_u['n'] Q_DT = Q_value(m_init - m_final) print(f" Q = {Q_DT:.1f} MeV (releases energy!)") print("\n2. D + D → He-3 + n:") m_init = 2 * masses_u['D'] m_final = masses_u['He3'] + masses_u['n'] Q_DDn = Q_value(m_init - m_final) print(f" Q = {Q_DDn:.1f} MeV") print("\n3. D + D → T + p:") m_init = 2 * masses_u['D'] m_final = masses_u['T'] + masses_u['p'] Q_DDp = Q_value(m_init - m_final) print(f" Q = {Q_DDp:.1f} MeV") print("\n4. U-235 + n → U-236*:") m_init = masses_u['U235'] + masses_u['n'] m_final = masses_u['U236'] Q_fission = Q_value(m_init - m_final) print(f" Q = {Q_fission:.1f} MeV") print("\n5. U-236 fission products (typical):") Q_fission_products = Q_value(masses_u['U236'] - (masses_u['Ba141'] + masses_u['Kr92'] + 3*masses_u['n'])) print(f" Ba-141 + Kr-92 + 3n: Q = {Q_fission_products:.1f} MeV") print(f" Total fission Q ≈ 200 MeV") # Threshold energy A_target, A_proj = 12, 1 # C-12 + n E_thresh = threshold_energy(-5 MeV), A_target, A_proj) print(f"\nThreshold energy (¹²C + n → ¹³C):") print(f" Q = -1.95 MeV (endothermic)") print(f" E_thresh = {E_thresh:.2f} MeV") ``` ### Nuclear Reactors ```python import numpy as np # Neutron transport def neutron_diffusion(D, Σa, phi): """Diffusion equation: D∇²φ - Σaφ = -S""" # Simplified 1D slab solution L = np.sqrt(D / Σa) # Diffusion length return phi def four_factor_formula(k_inf, epsilon, p, f): """ Four-factor formula for infinite multiplication. k∞ = η × f × ε × p η = neutrons per absorption in fuel f = thermal utilization factor ε = fast fission factor p = resonance escape probability """ return eta * f * epsilon * p def effective_multiplication(k_inf, k_eff): """k_eff = k∞ × P_non_leakage""" return k_inf * (1 - k_non_leakage) def reactor_period(T, rho): """Reactor period from reactivity.""" # inhour equation (simplified) return T / (1 - 1/k_eff) # Criticality calculations k_inf = 1.32 # For U-235 thermal reactor L = 0.10 # Non-leakage probability for large reactor k_eff = k_inf * L print("Nuclear reactor criticality:")