With "b mode" activation, the directly activated parent with a short half-life decays into an active daughter product with a longer half-life.
During exposure the build up of the daughter from the decay of the parent is properly accumulated, but after exposure the decay of the parent is is not added to the daughter activity level. This is currently flagged in the result table with a warning that "b mode" activity is underestimated.
The Bateman equation (wikipedia) can be used to properly account for daughter buildup after exposure ends: $A_d(t) = \frac{λ_d}{λ_d - λ_p} A_p(0) (e^{-λ_p t} - e^{-λ_d t})$, where $λ$ is the decay rate (not half-life) and $A(t)$ is the activity level at time $t$. We need to add a correction for the decay of the initial daughter activity: $A_c(t) = A_d(0) e^{-λ_d t}$ to get the full activity level.
We don't have to worry about cancellation errors since $|λ_p - λ_d| \gg 0$ for all "b mode" pairs.
The "time to decay below" calculation will need to be updated.
Using the example of 1 g Bi with exposed for 10 h with flux 1e8, the parent product Bi-210t has a half-life of 5 days, and the daughter Po-210 has a half-life of 138 days. The activity of Po-210 after 15 days is underestimated by 30%, asymptotically approaching $A_d(t)/A_c(t) = 134$ % at longer times.
With "b mode" activation, the directly activated parent with a short half-life decays into an active daughter product with a longer half-life.
During exposure the build up of the daughter from the decay of the parent is properly accumulated, but after exposure the decay of the parent is is not added to the daughter activity level. This is currently flagged in the result table with a warning that "b mode" activity is underestimated.
The Bateman equation (wikipedia) can be used to properly account for daughter buildup after exposure ends:$A_d(t) = \frac{λ_d}{λ_d - λ_p} A_p(0) (e^{-λ_p t} - e^{-λ_d t})$ , where $λ$ is the decay rate (not half-life) and $A(t)$ is the activity level at time $t$ . We need to add a correction for the decay of the initial daughter activity: $A_c(t) = A_d(0) e^{-λ_d t}$ to get the full activity level.
We don't have to worry about cancellation errors since$|λ_p - λ_d| \gg 0$ for all "b mode" pairs.
The "time to decay below" calculation will need to be updated.
Using the example of 1 g Bi with exposed for 10 h with flux 1e8, the parent product Bi-210t has a half-life of 5 days, and the daughter Po-210 has a half-life of 138 days. The activity of Po-210 after 15 days is underestimated by 30%, asymptotically approaching$A_d(t)/A_c(t) = 134$ % at longer times.