CalculatorPod’s nuclear section covers one of the deepest free collections of nuclear physics tools available online - from basic decay kinetics and binding energy to reactor criticality analysis, radiopharmaceutical dosimetry (MIRD), uranium enrichment SWU, and cyclotron vault shielding per NCRP 151. Every calculator shows the governing equation with all variables defined in SI units, cites the relevant standard or publication, and walks through worked examples with real isotopes and realistic numbers.

What These Calculators Cover

Decay kinetics and nuclear reactions. The Radioactive Decay Calculator applies N(t) = N₀ · e^(−λt) to find remaining activity after any elapsed time. The Half-Life Calculator converts between t½, λ, and mean lifetime τ. The Nuclear Binding Energy Calculator computes mass defect and binding energy per nucleon for any nuclide - iron-56 tops at ~8.79 MeV/nucleon. The Q-Value Calculator handles both reaction and decay Q-values, computing threshold energy for endothermic reactions and flagging whether a process is energetically forbidden.

Multi-step decay chains. The Bateman Equations Solver goes beyond single-nuclide decay to solve two- and three-member chains (A → B and A → B → C) analytically, returning daughter activity at any time, peak daughter activity, equilibrium ratios, and total atom inventories. The Secular/Transient Equilibrium Calculator classifies the equilibrium type from the parent-to-daughter half-life ratio, computes the A_B/A_A activity ratio at equilibrium, the time to peak daughter activity, and the time to reach 99% of equilibrium - quantities that appear in Mo-99/Tc-99m generator design and Ra-226 source characterisation.

Reactor physics. The Four-Factor Formula Calculator computes the infinite multiplication factor k∞ = η · ε · p · f for a thermal reactor lattice, explaining the physical meaning of each factor. The k-eff Calculator extends this with geometric buckling B² and migration area M² to give the effective multiplication factor for finite reactors. The Moderator-to-Fuel Ratio Calculator sweeps the N_mod/N_fuel ratio R to find R_opt where the product p(R) × f(R) is maximised - the lattice design optimisation that every reactor physics textbook derives but rarely provides a calculator for.

Fuel cycle and isotope production. The Nuclear Fuel Burnup Calculator computes discharge burnup in MWd/tU from power, irradiation time, and capacity factor, alongside U-235 depletion and Pu-239 buildup estimates. The Uranium Enrichment (SWU) Calculator applies the value function V(x) = (1 − 2x) · ln((1−x)/x) to compute separative work units, feed mass, and tails mass for any product enrichment level - a calculation used in fuel cycle cost analysis and non-proliferation studies. The Isotope Production and Burnup Calculator and Neutron Activation Analysis Calculator both use A(t) = Nσφ(1 − e^(−λt)) but serve different purposes: the former models in-pile production toward saturation activity, the latter estimates trace-element concentration from measured induced activity in NAA experiments.

Radiation protection and shielding. The Radiation Dose Calculator converts absorbed dose (Gy) to equivalent dose (Sv) via radiation weighting factors wR per ICRP 103, and to effective dose using tissue weighting factors. The Gamma Ray Attenuation and Shielding Calculator applies the narrow-beam exponential I(x) = I₀ e^(−μx) for six materials (lead, concrete, iron, water, polyethylene, aluminium). The HVL/TVL Calculator works the inverse problem: given a transmission target, find the required shield thickness. The Neutron Shielding for Medical Cyclotrons Calculator implements the NCRP Report 151 two-component model H(d, x) = H₁ · d^(−2) · 10^(−x/TVL₁) to design vault walls for PET cyclotron facilities - a niche but high-stakes calculation that most commercial packages lock behind expensive licenses.

Medical nuclear physics. The PET/SPECT Isotope Activity Planner solves the reverse decay problem: given a required activity at injection time and the scan delay, what calibration activity must be prepared? It covers 11 clinical isotopes (F-18, Tc-99m, Ga-68, I-123, I-131, Lu-177, Y-90, and more) with correct physical half-lives. The Radiopharmaceutical Dosimetry (MIRD) Calculator implements the MIRD formalism D = 576.7 · A_tilde · E · φ / m to compute organ absorbed dose and ICRP 128 effective dose for Tc-99m, F-18, I-131, and Lu-177 - the calculation at the core of therapeutic radiopharmacy and nuclear medicine regulatory submissions.

Who Uses These Calculators

Undergraduate and postgraduate students in nuclear engineering, medical physics, and radiochemistry use these tools for problem-set verification and exam preparation. NEET and JEE aspirants studying modern physics rely on the radioactive decay, half-life, and nuclear binding energy calculators - all are high-weightage standard-exam topics. Medical physicists, nuclear medicine technologists, and radiopharmacists use the MIRD dosimetry, PET/SPECT activity planner, and radiation dose calculators to cross-check clinical calculations. Reactor physics students and nuclear fuel cycle analysts use the four-factor formula, k-eff, moderator-to-fuel ratio, fuel burnup, and SWU calculators. Health physicists and cyclotron facility designers use the shielding and dose-rate tools for facility planning and licence applications.

Frequently Asked Questions

What is the difference between secular equilibrium and transient equilibrium?

Both describe a steady state in a parent-daughter decay chain, but they arise under different half-life conditions. Secular equilibrium occurs when the parent half-life is so much longer than the daughter's that the parent activity is effectively constant over many daughter half-lives (t½_parent >> t½_daughter, typically by a factor of 100 or more). At equilibrium, daughter activity equals parent activity. Transient equilibrium occurs when the parent is only moderately longer-lived than the daughter; the daughter activity eventually exceeds the parent activity by the factor t½_parent / (t½_parent − t½_daughter) before both decay at the parent rate. The Mo-99/Tc-99m generator (t½ 66 h / 6 h) is the classic transient equilibrium example. Use the Secular/Transient Equilibrium Calculator to compute the activity ratio, time to peak, and time to 99% equilibrium for any parent-daughter pair.

How do I calculate separative work units (SWU) for uranium enrichment?

SWU quantifies the thermodynamic work required to enrich uranium from natural feed (0.711% U-235) to a product assay, with depleted tails as the byproduct. The formula is SWU = P · V(x_P) + W · V(x_W) − F · V(x_F), where V(x) = (1 − 2x) · ln((1−x)/x) is the value function and P, W, F are the product, waste, and feed masses. For 1 kg of 4.5% LEU from natural feed at 0.3% tails, you need roughly 7.0 SWU and about 8.8 kg of natural uranium. Use the Uranium Enrichment (SWU) Calculator to solve for any enrichment level and tails assay.

How do I use the MIRD dosimetry calculator for Tc-99m?

The MIRD formalism computes organ absorbed dose as D = 576.7 · A_tilde · E · φ / m, where A_tilde is the cumulated activity in the source organ (MBq·h), E is the mean energy emitted per disintegration (MeV), φ is the absorbed fraction for the target organ, and m is the target organ mass (g). For a standard 740 MBq Tc-99m HMPAO brain scan, the critical organ (brain) receives roughly 6–7 mGy, and the effective dose is approximately 6–7 mSv. Enter the administered activity, organ uptake fraction, and residence time into the Radiopharmaceutical Dosimetry (MIRD) Calculator to get organ doses and ICRP 128 effective dose.

What is k-eff and what does a value above or below 1.0 mean?

k-eff (effective neutron multiplication factor) is the ratio of neutrons produced in one generation to neutrons lost (absorbed + leaked) in the same generation. k-eff = 1.0 means criticality - a self-sustaining chain reaction. k-eff < 1.0 means subcritical - the chain reaction dies out. k-eff > 1.0 means supercritical - the reaction grows exponentially. The k-eff Calculator computes this from the four-factor formula result k∞ corrected for geometric leakage via k_eff = k∞ / (1 + M²B²), where M² is the migration area and B² is the material buckling.

What is the difference between gray (Gy) and sievert (Sv)?

Gray measures absorbed dose - the energy deposited per unit mass of tissue (1 Gy = 1 J/kg), regardless of radiation type. Sievert measures equivalent dose, which weights the absorbed dose by a radiation weighting factor (wR) that accounts for the biological effectiveness of the radiation type: wR = 1 for gamma/X-ray and beta, wR = 2 for protons, wR = 20 for alpha particles, and wR = 2.5–20 for neutrons depending on energy. Effective dose further weights equivalent doses by tissue weighting factors to give a single whole-body risk metric. See the Radiation Dose Calculator for full wR and wT tables per ICRP 103.

How is nuclear binding energy related to nuclear stability?

Binding energy per nucleon is the most direct measure of nuclear stability - the higher it is, the more energy is required to disassemble the nucleus. Iron-56 sits at the peak of the binding energy curve (~8.79 MeV/nucleon), making it the most stable nucleus. Nuclides lighter than iron can release energy by fusion; nuclides heavier than iron can release energy by fission. The Nuclear Binding Energy Calculator computes mass defect and binding energy per nucleon for any nuclide given Z, N, and atomic mass.

How much energy is released in uranium-235 fission?

A single U-235 fission event releases approximately 200 MeV (3.2 × 10⁻¹¹ J), distributed across fission fragments (~168 MeV), prompt neutrons (~5 MeV), prompt gamma rays (~7 MeV), and delayed beta/gamma from fission products (~20 MeV). Per gram of U-235, this is roughly 8.2 × 10¹⁰ J - equivalent to about 20 tonnes of TNT. Use the Nuclear Fission Energy Calculator to compute Q-values from reactant and product masses for any specific fission reaction.

What calibration activity do I need to prepare for a PET scan if the scan is delayed?

Because F-18 has a 110-minute half-life, activity decays significantly between synthesis and injection. If you need 370 MBq at the time of injection (t_inj) and the calibration time is 45 minutes earlier, you must prepare A_cal = 370 × e^(λ × 0.75 h) ≈ 370 × e^(0.693/1.833 × 0.75) ≈ 420 MBq. The PET/SPECT Isotope Activity Planner handles this for 11 clinical isotopes and also solves for scan delay, injected activity, or calibration activity depending on which quantity you need.

Nuclear Science Calculators