Radioisotope Activity Calculator

Activity (A) is the number of radioactive decays per second in a given amount of material, measured in becquerels (Bq). It is a property of the source itself. Dose rate (D˙) is the amount of energy deposited per unit mass per unit time (e.g., mSv/h) at a specific location in the body. Activity does not directly tell you the biological effect; it depends on the type of radiation, energy, distance, and shielding. For example, 1 Bq of alpha-emitting plutonium and 1 Bq of beta-emitting tritium have very different dose rates. The activity is the starting point for calculating dose using conversion factors specific to the radionuclide, chemical form, and exposure pathway.

Since the decay constant λ = ln(2) / t₁/₂, the activity is inversely proportional to the half-life for a fixed number of atoms N. A = (ln(2) / t₁/₂) × N. This means that for the same N, a short-lived isotope (small t₁/₂) has a much higher activity than a long-lived one. For example, 1 gram of ⁶⁰Co (t₁/₂ = 5.27 years) has an activity of ~4.2×10¹³ Bq, while 1 gram of ²³⁸U (t₁/₂ = 4.47×10⁹ years) has an activity of only ~1.24×10⁴ Bq. This relationship is fundamental for sourcing and managing radioactive materials.

The becquerel is the SI unit for activity, defined as 1 decay per second (1 Bq = 1 s⁻¹). The curie is the older unit, defined as the activity of 1 gram of ²²⁶Ra, which is 3.7×10¹⁰ Bq. The becquerel is used because it is a direct SI unit and avoids the cumbersome large numbers associated with curies for most practical activities. To convert: 1 Ci = 3.7×10¹⁰ Bq; 1 Bq ≈ 2.7×10⁻¹¹ Ci. For example, a 100 MBq source (10⁸ Bq) is approximately 2.7 mCi. The formula A = λN works seamlessly with Bq; if you have values in Ci, convert to Bq before using the formula.

Specific activity is the activity per unit mass (Bq/g) or per unit molar quantity (Bq/mol). It can be derived from A = λN by dividing by the mass. For a pure radionuclide, specific activity = λ × (Avogadro's number / molar mass). This value decreases with longer half-life. For example, ¹⁴C has a specific activity of ~1.65×10¹¹ Bq/g (or 165 GBq/g), while ²³⁸U has only ~1.24×10⁴ Bq/g. Specific activity is critical for preparing radioactive standards, for radioactive dating, and for determining the purity of radioactive compounds. The formula essentially tells you how many decays you get per gram of material.

First, convert the mass to number of atoms: N = (mass in grams / molar mass) × N_A (Avogadro's number, 6.022×10²³). Then calculate λ = ln(2) / t₁/₂ (ensuring t₁/₂ is in seconds for Bq). Finally, A = λN. For example, to find the activity of 0.1 g of ⁶⁰Co: molar mass = 59.93 g/mol, t₁/₂ = 5.27 yr = 1.66×10⁸ s. N = (0.1/59.93)×6.022×10²³ = 1.005×10²¹ atoms. λ = 0.693/1.66×10⁸ = 4.17×10⁻⁹ s⁻¹. A = 4.17×10⁻⁹ × 1.005×10²¹ = 4.19×10¹² Bq = 4.19 TBq. This is the standard calculation used in radiation source production.

Yes, rearranging the formula: N = A / λ = A × t₁/₂ / ln(2). This is used to determine the amount of a radioisotope in a sample (e.g., in bioassay or environmental sampling). For example, if a sample shows an activity of 100 Bq of ¹⁴C (t₁/₂ = 5730 years = 1.81×10¹¹ s), then N = 100 / (0.693/1.81×10¹¹) = 100 × (1.81×10¹¹/0.693) = 2.61×10¹³ atoms. This atom count can be converted to mass (N × molar mass / N_A) to find the amount of ¹⁴C present. This method is used in liquid scintillation counting and for material balances in nuclear facilities.

The activity A is the absolute decay rate of the source. A detector measures a count rate (cps or cpm) that is always less than A due to: (1) geometric efficiency (not all emitted particles hit the detector), (2) intrinsic efficiency (detector doesn't detect every particle that hits it), (3) absorption by the sample matrix and air, and (4) dead time losses. To convert count rate to activity, you need a calibration factor that accounts for all these effects: A = (count rate) / (efficiency). This calibration is performed using a standard source of known activity (traceable to a primary standard). Without calibration, the count rate is a relative measure, not an absolute activity.

Yes, but you must use the total decay constant λ = λ₁ + λ₂ + ... for all branches. The total activity is A = λN, where λ is the sum of partial decay constants. The formula A = λN gives the total disintegration rate. If you are interested in the activity of a specific decay mode (e.g., the beta branch), you would multiply by the branching ratio: A_branch = λ_branch × N = f_branch × A_total. For example, ⁴⁰K decays to ⁴⁰Ar (electron capture, 10.7%) and ⁴⁰Ca (beta minus, 89.3%). The total λ is the sum of both branches; the total activity is A = λN; the beta activity is 0.893×A.

The formula A = λN applies to a single nuclide. In a decay chain, the activity of a daughter nuclide builds up over time. The total activity of a mixture is the sum of the individual activities of each radionuclide present. For secular equilibrium (when parent half-life ≫ daughter half-life), the activity of the daughter equals the activity of the parent, i.e., λ_parent N_parent = λ_daughter N_daughter. The formula cannot be applied to the chain as a whole without accounting for the ingrowth equations (Bateman equations). The simple A = λN is only valid for a single isotope at a given instant, not for a mixture or transient equilibrium.

Cyclotron production of ¹⁸F (t₁/₂ = 109.8 min) requires precise calculation of activity. The target is irradiated for a certain time, and the produced activity is calculated from the production rate. The formula A = λN is used to convert the number of ¹⁸F atoms (from the irradiation yield) into the activity at the end of bombardment (EOB). For example, if the irradiation produces 1×10¹⁴ atoms of ¹⁸F, the activity at EOB is A = (ln(2)/109.8 min) × 1×10¹⁴ = 0.00631 min⁻¹ × 1×10¹⁴ = 6.31×10¹¹ Bq = 6.31 GBq. This activity then decays during transport and synthesis; the formula A(t) = A₀e^{-λt} is used to predict the activity at the time of patient injection.

In waste management, the activity of a radioactive waste package at any future time is calculated using A(t) = A₀e^{-λt}, which is derived from A = λN. By knowing the initial activity (A₀) or the number of atoms of each radionuclide (N) in the waste, the decay constant λ gives the current activity and the future decay. This is used to assess the heat generation (decay heat), the radiotoxicity over time, and to design disposal strategies. For a mixture of radionuclides, the total activity is the sum of individual activities. The formula is fundamental to the safety case for geological disposal, where the decay of the inventory must be calculated over thousands of years.

From A = λN = (ln(2)/t₁/₂)N, if A is the same for both, then N must be proportional to t₁/₂. A longer-lived isotope has more atoms for the same activity. Since mass = N × (molar mass / N_A), the mass also scales with half-life (assuming similar molar mass). For example, 1 GBq of ¹⁸F (t₁/₂ = 1.83 h) requires ~1.7×10¹² atoms (~0.05 µg), while 1 GBq of ⁶⁰Co (t₁/₂ = 5.27 yr) requires ~1.4×10¹⁵ atoms (~0.14 mg). This illustrates why long-lived isotopes have very low specific activity and are present in relatively large masses for a given activity. It's a crucial concept for source manufacturing and dosimetry.

The formula A = λN gives the activity at the instant the atoms N are present. If a sample is counted over a finite time Δt, the activity is not constant during the count because decays reduce N. For precise work, you calculate the activity at the midpoint of the counting interval (or decay-correct back to the start of the count) using A(t_mid) = A(t_start) × e^{-λ(t_mid - t_start)}. The number of counts collected over Δt is C = ∫₀^{Δt} A(t) dt = (A(t_start)/λ)(1 - e^{-λΔt}). This is rearranged to find A(t_start). For short-lived isotopes or long counting times (relative to half-life), this correction is essential. For long-lived isotopes (λΔt << 1), the activity is nearly constant and A ≈ C/Δt.

Decay heat is the thermal power generated by the decay of fission products and actinides in a nuclear reactor after shutdown. Each decay of a radioactive nucleus releases energy (beta, gamma, alpha). The total decay heat is H = Σ (A_i × E_i), where A_i is the activity of each nuclide and E_i is the average energy released per decay (including neutrinos, which carry away some energy). Since A_i = λ_i N_i, the formula is the starting point for predicting the activity of the fission product inventory and thus the decay heat. This is critical for designing cooling systems and for safety analysis of spent fuel pools and transport casks. The activity and its decay over time are calculated using the Bateman equations, which extend A = λN to include production and depletion in the reactor core.

The alternative form uses the half-life directly: A = (ln 2 / t₁/₂) × N. This is convenient because half-lives are more commonly tabulated and intuitive than decay constants. For quick calculations, you can use: A (Bq) = 0.693 × N / t₁/₂ (with t₁/₂ in seconds). For example, if you have N = 10²⁰ atoms of a nuclide with t₁/₂ = 10 years (3.15×10⁸ s), A = 0.693 × 10²⁰ / 3.15×10⁸ = 2.2×10¹¹ Bq. This avoids the intermediate step of computing λ. It's particularly useful for radiation protection and source handling when you have the atom count or mass.