Spent Fuel Activity

Spent fuel contains hundreds of different fission products and actinides, each with its own half-life ranging from seconds to millions of years. If you used a single exponential, you would severely overestimate or underestimate the activity at different cooling times. The sum over all isotopes, A_sf(t) = Σ A_i e^{-λ_i t}, is essential because the short-lived isotopes dominate the activity immediately after discharge, while long-lived isotopes dominate after decades. For example, at 1 day, ¹³¹I (8-day half-life) contributes significantly; at 10 years, ¹³⁷Cs and ⁹⁰Sr dominate; at 1000 years, actinides like ²⁴¹Am become important. The summation captures this multi-component decay behavior accurately.

Spent fuel activity (Bq) is the total number of decays per second, a fundamental source term. The dose rate (Sv/h) at a distance depends on the activity, the gamma energy spectrum, and the shielding geometry. The formula A_sf = Σ A_i e^{-λ_i t} gives the activity, but to design shielding, you need to know the gamma and neutron emission rates, which require the isotopic inventory and their specific emission probabilities. Dose rate calculations use A_sf as an input, but the formula alone is insufficient; you also need the photon and neutron source terms (e.g., using ORIGEN or SCALE codes) that provide the energy spectrum. Shielding design uses the activity to determine the source strength, then applies attenuation and buildup factors.

Immediately after reactor shutdown, the fission product inventory is rich in isotopes with half-lives of seconds to days (e.g., ⁹⁷Zr (17h), ¹³¹I (8d), ¹⁴⁰Ba (12.8d)). These have high specific activities because λ is large. Their activity decays exponentially; after about 10 half-lives, their contribution drops below 0.1% of their initial activity. For example, after 1 year, most short-lived isotopes have decayed away, and the activity is dominated by medium-lived isotopes like ¹³⁷Cs (30 yr) and ⁹⁰Sr (28.8 yr). The formula automatically handles this because the sum of exponentials naturally shifts dominance to the longest-lived isotopes as time passes.

No, because the decay is not a single exponential. The ratio of activity at 10 years to activity at 1 year depends on the isotopic composition. For typical LWR spent fuel, the activity at 10 years is roughly 1/3 to 1/4 of the activity at 1 year, but this ratio varies with burnup and initial enrichment. The decay is dominated by ¹³⁷Cs and ⁹⁰Sr, which have similar half-lives (~30 years), so after the short-lived isotopes have decayed (beyond 1 year), the decline is roughly exponential with an effective half-life of about 30 years. However, the precise ratio requires the full summation. A common approximation: A(10 yr) ≈ A(1 yr) × 2^{-9/30} ≈ A(1 yr) × 0.81, but this is only valid if the short-lived isotopes are already gone. The correct approach is to use the sum over all isotopes with their specific λ.

The initial activity A_i at discharge is the product of the fission yield for each isotope and the reactor's neutron flux and irradiation time. In a depletion calculation (e.g., with ORIGEN or SCALE), the concentration of each isotope is tracked over the irradiation period, accounting for production by fission, neutron capture, and decay. The A_i at the end of irradiation (EOB) is λ_i × N_i(discharge). The value depends on: (1) fission yields from U-235 and Pu-239 fission, (2) the neutron spectrum (which affects capture rates), (3) the burnup (GWd/tU), (4) the enrichment, and (5) the power history (steady-state vs. load-following). A higher burnup generally produces more actinides and higher A_i for certain isotopes, changing the sum's composition.

At discharge (zero cooling time), the activity of 1 tonne of spent fuel from a PWR (burnup ~50 GWd/tU) is about 10¹⁶ to 10¹⁷ Bq. After 1 year of cooling, it drops to about 10¹⁵ Bq; after 10 years, about 10¹⁴ Bq; after 100 years, about 10¹² Bq; and after 1000 years, about 10¹⁰ Bq. These are rough estimates; the exact values depend on the initial enrichment and burnup. The activity decreases by about 2-3 orders of magnitude in the first 10 years due to the decay of short-lived fission products, then declines more slowly as ¹³⁷Cs and ⁹⁰Sr dominate. The formula A_sf = Σ A_i e^{-λ_i t} is used to generate these decay curves for specific fuel inventories.

Fission products with half-lives up to 30 years (¹³⁷Cs, ⁹⁰Sr) have mostly decayed after 300-500 years. At that point, the remaining activity is dominated by actinides with half-lives of hundreds to thousands of years, such as ²⁴¹Am (432 yr), ²⁴³Am (7370 yr), and ²³⁹Pu (24,110 yr). These have relatively low specific activities, but because they are long-lived, their contribution to the sum becomes significant after the short-lived isotopes have faded. The formula, being a sum of exponentials, naturally shows that as t increases, the terms with the smallest λ (longest half-life) dominate the sum, because e^{-λ_i t} decays more slowly. So, at 1000 years, the actinide terms are the ones that remain.

The spent fuel activity determines both the gamma and neutron source terms. The gamma activity from fission products and activation products dictates the gamma shielding (steel, concrete, lead) needed to reduce the external dose to acceptable levels. The neutron emission comes from spontaneous fission of actinides (e.g., ²⁴⁴Cm, ²⁴²Cm) and from (α,n) reactions in the fuel and cladding. The total activity (Bq) is used to calculate the source strength; then, the shielding thickness is designed to attenuate the radiation to meet dose limits at the cask surface. The formula A_sf(t) provides the source term as a function of cooling time; longer cooling reduces the activity and thus allows thinner or less expensive shielding.

The activities of different isotopes are additive because radioactive decay from different species is independent. There is no interference or interaction between the decays of different isotopes; each nucleus decays independently. The total activity is the arithmetic sum of the activities of all isotopes present. The formula separates the contributions, so you can calculate the fission product activity (sum over fission products) and the actinide activity (sum over actinides) separately. However, actinides are produced from neutron capture by U-238 and other heavy isotopes during irradiation, so their initial activities A_i are determined by the reactor physics, but once discharged, they decay independently. The additivity holds because the decay processes are linear.

If isotopes are removed (e.g., via reprocessing to separate uranium, plutonium, and minor actinides), the formula becomes A_sf_remaining(t) = Σ A_i (1 - f_i) e^{-λ_i t}, where f_i is the fraction removed of isotope i. For example, if 99% of the plutonium is removed, the remaining plutonium activity is 1% of the original. The sum then only includes the remaining fractions. This is used to estimate the activity and decay heat of reprocessed waste streams. The formula remains a sum of exponentials, but with reduced initial activities for the removed isotopes. The same approach applies if specific fission products are separated for transmutation or disposal.

The cooling time is the time elapsed since the fuel was discharged from the reactor (EOC). The activity at discharge is the result of the irradiation period, and it's from that point that the decay starts. The fuel's activity during irradiation is not described by this formula (it involves production and decay simultaneously); the formula applies only after the reactor is shut down and no new fission products are being produced. The cooling time is the time available for decay before handling, transport, storage, or disposal. In safety analyses, the cooling time is a critical parameter because it determines the source term for shielding and heat generation. For example, fuel that has cooled for 10 years has much lower activity than fuel cooled for 1 year.

The activity (Bq) is the decay rate, but radiotoxicity (the potential harm from ingestion or inhalation) depends on the specific radionuclides and their dose coefficients (Sv/Bq). The formula gives the activity of each isotope, but to assess radiotoxicity, you multiply the activity of each isotope by its committed effective dose coefficient (from ICRP) and sum: Radiotoxicity(t) = Σ A_i(t) × e_i, where e_i is the dose coefficient. This is often reported as the radiotoxicity in Sv or as the equivalent mass of the respective isotope. The activity formula is the first step; toxicity calculations are a weighted sum of the activities. For example, ¹³⁷Cs and ⁹⁰Sr have high dose coefficients per Bq, so their radiotoxicity contribution is significant despite their relatively short half-lives.

Yes, all isotopes present in the spent fuel are included in the sum, including neutron capture products like U-236 and Pu-240. These are not fission products but actinides formed by neutron capture. They may have very low specific activities (e.g., U-236 has a half-life of 2.3×10⁷ years) but contribute to the long-term activity. Their initial activities A_i are determined by the depletion calculation. The sum includes every nuclide that has a non-negligible decay rate, from short-lived fission products to long-lived actinides. For example, Pu-240 (T₁/₂=6560 yr) contributes to the activity at long cooling times, and its daughter U-236 (also radioactive) must be included if significant ingrowth occurs.

The rule states that after 10 half-lives, the activity of a specific isotope decays to ~0.1% of its initial value. The formula for a single isotope shows that after 10 half-lives, A_i(10T) = A_i(0) × (1/2)^10 ≈ 0.001 A_i(0). However, for a mixture like spent fuel, you cannot apply this rule to the total activity because different isotopes have different half-lives. After 10 half-lives of the shortest-lived isotope, longer-lived isotopes still contribute significantly. For example, ¹³⁷Cs has a half-life of 30 years; after 300 years (10 half-lives), its activity is 0.1% of its initial, but other isotopes like ²⁴¹Am (432 yr) and Pu-239 (24,000 yr) remain, so the total activity doesn't drop to 0.1% of its initial value. The formula shows that you must evaluate each isotope individually; the total activity never reaches zero in finite time.

The initial activities A_i are not calculated directly from the formula; they are the output of a depletion code (e.g., ORIGEN, SCALE, CASMO). These codes solve the Bateman equations for neutron irradiation, using the reactor's flux history, fission yields, and cross-section libraries. The output is the atom density N_i (or activity A_i) of each isotope at the end of the burnup. The formula A_sf = Σ A_i e^{-λ_i t} is then used to decay-correct these activities to any cooling time. So, the formula is the post-processing step; the initial activities come from a detailed neutronic and burnup calculation that accounts for the reactor's operating conditions, including power level, control rod movements, and spectral variations.