nuclear-physics

name	nuclear-physics
description	Nuclear physics fundamentals including radioactive decay, nuclear structure, fission, fusion, particle interactions, and nuclear models for energy and research applications.
category	physics
tags	["physics","nuclear-physics","radioactive-decay","fission","fusion","nuclear-structure","particle-physics"]
difficulty	advanced
author	neuralblitz

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

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 %  ==   Z %  == :
        delta = -a_p / np.sqrt(A)
    :
        delta = 
    
    B = (a_v * A 
         - a_s * A**(/) 
         - a_c * Z * (Z - ) / A**(/) 
         - a_a * (A - *Z)** / A 
         + delta)
    
     B

 ():
    
     B / A


()
()
B_Ca40 = semi_empirical_mass(, )
()
()

()
B_Fe56 = semi_empirical_mass(, )
()
()

()
B_U238 = semi_empirical_mass(, )
()
()


magic_numbers = [, , , , , , ]
()
()
()
()
()
()

Radioactive Decay

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': ,         
    : ,         
    : ,       
    : ,         
}

()
 isotope, t_half  (half_lives.items())[:]:
    lambda_decay = decay_constant(t_half)
    ()
    ()
    ()


 ():
    
    lambda_c14 = decay_constant(t_half *  *  * )
     np.log(C14_ratio) / (-lambda_c14) / ( *  * )


initial_ratio = 
 ratio  [, , , ]:
    age = carbon14_age(ratio / initial_ratio)
    ()
    ()


U238_hl =  *  *  *   
Pb206_hl = np.inf  

t_equilibrium =  *  *  *   
N_U238 =   
ratio_Pb_U = np.exp(-decay_constant(U238_hl) * t_equilibrium)
()
()
()

Nuclear Reactions and Q-Values

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': ,
    : ,
    :  + ,  
}


()
()
m_init = masses_u[] + masses_u[]
m_final = masses_u[] + masses_u[]
Q_DT = Q_value(m_init - m_final)
()

()
m_init =  * masses_u[]
m_final = masses_u[] + masses_u[]
Q_DDn = Q_value(m_init - m_final)
()

()
m_init =  * masses_u[]
m_final = masses_u[] + masses_u[]
Q_DDp = Q_value(m_init - m_final)
()

()
m_init = masses_u[] + masses_u[]
m_final = masses_u[]
Q_fission = Q_value(m_init - m_final)
()

()
Q_fission_products = Q_value(masses_u[] - (masses_u[] + masses_u[] + *masses_u[]))
()
()


A_target, A_proj = ,   
E_thresh = threshold_energy(- MeV), A_target, A_proj)
()
()
()

Nuclear Reactors

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:")
print(f"  k∞ = {k_inf:.2f}")
print(f"  k_eff = {k_eff:.2f}")
print()

 k_eff > :
    ()
 k_eff < :
    ()
:
    ()


neutron_generations = []
N_thermal =   

 gen  ():
     gen == :
        N_thermal = 
    :
        N_thermal *= k_eff
    neutron_generations.append(N_thermal)

()
 i, N  (neutron_generations[:]):
    ()


 ():
    
     P_thermal / (E_fission * )

 ():
    
     P / volume


P_gw =   
fissions_per_sec = thermal_power(P_gw)
()
()
()


 ():
    
     N_235 *  / (N_235 *  + N_238 * ) * 

 N_235_perc  [, , , ]:
    N_238_perc =  - N_235_perc
    wt_235 = enrichment(N_235_perc, N_238_perc)
    ()

Stellar Nucleosynthesis

import numpy as np

def pp_chain_rate(T):
    """
    Proton-proton chain reaction rate.
    Approximate: ε ∝ T⁴ for pp-I
    """
    T9 = T / 1e9  # Temperature in billions of K
    if T9 < 0.01:
        return 1e-26 * T9**4
    return 1e-6 * np.exp(-33.8 / T9**0.5)

def CNO_cycle_rate(T):
    """CNO cycle rate dominates at T > 1.5e7 K."""
    T9 = T / 1e9
    return 1e-25 * np.exp(-152 / T9**0.5)

def triple_alpha_rate(T):
    """
    Triple-alpha reaction (3⁴He → ¹²C).
    Rate ∝ T⁻³ for helium burning.
    """
    T9 = T / 1e9
    return 1e-8 * np.exp(-4.4 / T9**3)

# Stellar temperatures
temps = [1.5e7, 2e7, 3e7]  # K (Sun core, hotter stars)

print("Stellar nucleosynthesis rates:")
for T in temps:
    T9 = T / 1e9
    pp_rate = pp_chain_rate(T)
    cno_rate = CNO_cycle_rate(T)
    triple_alpha = triple_alpha_rate(T)
    
    print(f"\n  T = {T:e} K (×10⁹ K):")
    ()
    ()
    ()


 ():
    
    T8 = T / 
      * T8**

 ():
    
    T8 = T / 
      * T8**

()
 T  [, ]:
    eps_pp = energy_generation_pp(T)
    eps_cno = energy_generation_CNO(T)
    
    ()
    ()
    ()


 ():
    
      * M_solar**(-) / L_solar

 M  [, , , , ]:
    L = M**  
    t = main_sequence_lifetime(M, L)
    ()


 ():
    
    


 ():
    
    k = 
    G = 
    m_H = 
    
    T_K = T
    rho_kg = rho
    
     ( * k * T_K / (G * m_H))**(/) * ( / ( * np.pi * rho_kg))**(/)

()
T_Jeans =   
rho_Jeans =   
M_J = jeans_mass(T_Jeans, rho_Jeans)
()
()

Best Practices

• Use consistent mass units (u or MeV/c²) when calculating Q-values and binding energies.
• For half-life calculations, be careful with time unit conversions (seconds, years, etc.).
• In reactor physics, distinguish between infinite multiplication factor (k∞) and effective multiplication factor (k_eff).
• Account for both prompt and delayed neutrons in reactor kinetics; delayed neutrons make control possible.
• For shielding calculations, consider gamma ray buildup factors and neutron moderation.
• In decay chain calculations, use the Bateman equations for secular and transient equilibrium.
• For stellar nucleosynthesis, remember that reaction rates depend strongly on temperature (Arrhenius behavior).
• When calculating cross-sections, distinguish between microscopic (σ) and macroscopic (Σ) cross-sections.
• Use Monte Carlo methods (MCNP, Geant4) for complex radiation transport problems.
• Always consider radiation safety (ALARA principle) when working with radioactive materials.