Isotope Production and Burnup Calculator

This formula gives the radioactivity of a product isotope at time t during neutron irradiation. N is the number of target atoms, σ is the activation cross-section in cm² (1 barn = 10^-24 cm²), φ is the neutron flux in n/cm²/s, and λ = ln(2)/T1/2 is the decay constant of the product. The term (1 - exp(-λt)) is the saturation fraction, ranging from 0 at t=0 to 1 as t grows much larger than T1/2. At t = T1/2, the saturation fraction is exactly 0.5.

Saturation activity A_sat = N·σ·φ is the maximum possible activity from a given target at a given flux. It equals the rate of production of the radioactive product at time zero, before any decay has occurred. At saturation, the production rate exactly equals the decay rate and the activity stops increasing. Saturation activity sets the ceiling for isotope production: to increase it you must increase the flux, the target mass, or use a target material with a larger cross-section.

Setting the saturation fraction to 0.90 gives 1 - exp(-λt) = 0.90, so λt = ln(10) = 2.303, and t = 2.303/λ = 2.303 × T1/2 / ln(2) = 3.32 × T1/2. To reach 99% of saturation requires 6.64 half-lives. For a short-lived isotope like Mo-99 (T1/2 = 65.94 hr), 90% saturation takes about 9.2 days of continuous irradiation.

Neutron burnup tracks the depletion of a target nuclide by neutron absorption without tracking radioactive ingrowth of a product. The burnup fraction B = 1 - exp(-σ·φ·t) measures what fraction of the original target atoms have been consumed. Burnup mode is appropriate when the product is stable or when you are only interested in fuel or poison depletion. Production mode is used when you care about the radioactivity that builds up in the product isotope.

Rearranging A_sat = N·σ·φ: the required flux is φ = A_sat / (N·σ). For Au-198 production with 1 g of Au-197 (N = 3.07×10^21 atoms), σ = 98.7 barns = 9.87×10^-23 cm², and a target of 10 GBq saturation activity: φ = 10^10 / (3.07×10^21 × 9.87×10^-23) = 10^10 / 0.303 = 3.3×10^10 n/cm²/s. This is achievable in a low-flux research reactor.

Co-59 (natural cobalt) is loaded into a reactor and irradiated for 18 months to 3 years at a thermal neutron flux of about 3×10^13 n/cm²/s. The activation cross-section of Co-59 is 37.2 barns. Co-60 has a half-life of 5.27 years. Typical irradiation produces 10-20% of saturation activity, yielding 50-100 TBq per kilogram of cobalt target. The product is used in food irradiation, industrial gamma radiography, and Gamma Knife radiosurgery.

The activation cross-section (also called the radiative capture cross-section, σ_γ) specifically quantifies the probability of neutron capture that produces a radioactive product via the (n,γ) reaction. The absorption cross-section σ_abs includes all reactions that remove a neutron from the beam: radiative capture plus fission (for fissile nuclei) plus other inelastic reactions. For non-fissile materials, σ_activation and σ_abs are nearly equal. For U-235, σ_abs = 683 b while σ_fission = 582 b and σ_capture = 101 b.

B-10 (natural boron is 20% B-10) has a thermal neutron absorption cross-section of 3840 barns, one of the highest of any stable nuclide. Control rods and neutron poisons use B-10 to absorb neutrons and suppress reactivity. During reactor operation, B-10 is consumed at a rate proportional to σ·φ·N. After 1000 hours at a flux of 10^14 n/cm²/s, the burnup fraction is about 26%. Tracking B-10 depletion is essential for predicting control rod worth over a fuel cycle.

Use N = (m × N_A) / A, where m is mass in grams, N_A = 6.022×10^23 atoms/mol (Avogadro constant), and A is the nuclide atomic mass in g/mol (approximately the mass number for most nuclides). For 1 mg of pure Mo-98 target (A = 97.91 g/mol): N = (0.001 × 6.022×10^23) / 97.91 = 6.15×10^18 atoms. Enter this as mantissa 6.15, exponent 18 in the calculator.

Research reactors (e.g., NIST, ILL): thermal flux 10^14 to 2×10^15 n/cm²/s. Power reactor fuel center: 3-5×10^13 n/cm²/s. Medical cyclotron neutron beam: 10^8 to 10^12 n/cm²/s. Am-Be or Cf-252 neutron sources: 10^5 to 10^8 n/cm²/s. High-flux material test reactors (e.g., HFIR): up to 2×10^15 n/cm²/s in the reflector. The flux directly scales the production rate and burnup rate for a given target.

Production mode can approximate the activity of a fission product if you treat fission as the source of the target atoms. However, for direct fission product inventory calculations, the proper approach is the Bateman equations including branching ratios and chain transitions. This calculator is most accurate for single-step activation reactions of the form target + n to product + gamma, where the product is the isotope of interest and no significant chain feeding occurs.

The saturation fraction S(t) = 1 - exp(-λt) depends only on the ratio t / T1/2 of the irradiation time to the product half-life. At t = T1/2, S = 0.500. At t = 2×T1/2, S = 0.750. At t = 3×T1/2, S = 0.875. Each additional half-life adds half of the remaining gap to saturation. This diminishing return explains why irradiating beyond 3-4 half-lives is rarely economical, since you must double the irradiation time to close half the remaining gap.

In burnup the target atoms are removed by neutron absorption, not by spontaneous radioactive decay. The removal rate is proportional to σ·φ·N(t), giving dN/dt = -σ·φ·N, whose solution is N(t) = N_0 × exp(-σ·φ·t). The product σ·φ plays the role of an effective decay constant. The key difference from radioactive decay is that σ·φ depends on the reactor operating condition and can be changed by adjusting the flux, whereas the radioactive decay constant λ is fixed by nuclear physics.