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alby
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(Apologies for cross-posting this in the Nuclear and Differential Equations forums. I'm new to posting on Physics Forums and not entirely sure where it belongs. Mods, please feel free to delete/move as appropriate.)
I am trying to create a decay chain simulator in Excel that my pupils can use to create graphs similar to those created by the Nucleonica Decay Engine: the idea being that they can find a decay chain and enter the isotopes and their half-lives and Excel will create the graph.
The problem I'm having is with the equations required to calculate the population of each isotope at each stage in time. My initial approach was far too simplistic and whilst it created correct *looking* graphs, the numbers weren't correct.
As an example, I'm using the last four stages of the actinium series uranium decay chain:
{}^{211}\mathrm{Bi}\rightarrow {}^{211}\mathrm{Po}\rightarrow {}^{207}\mathrm{Tl}\rightarrow {}^{207}\mathrm{Pb}
I can calculate the population of the Bi-211 at time t with:
N_{Bi}=N_0 e^{-\lambda t}
and the population of Po-211 using:
N_{Po} = \frac{\lambda_{Po}}{\lambda_{Bi} - \lambda_{Po}} N_0(e^{-\lambda_{Bi}t}-e^{\lambda_{Po}t})
The problem is with the equations for the population of Tl-207 and Pb-207. I know I should be looking at Bateman's work on the subject, but as a lowly teacher I can't access the literature.
I am trying to create a decay chain simulator in Excel that my pupils can use to create graphs similar to those created by the Nucleonica Decay Engine: the idea being that they can find a decay chain and enter the isotopes and their half-lives and Excel will create the graph.
The problem I'm having is with the equations required to calculate the population of each isotope at each stage in time. My initial approach was far too simplistic and whilst it created correct *looking* graphs, the numbers weren't correct.
As an example, I'm using the last four stages of the actinium series uranium decay chain:
{}^{211}\mathrm{Bi}\rightarrow {}^{211}\mathrm{Po}\rightarrow {}^{207}\mathrm{Tl}\rightarrow {}^{207}\mathrm{Pb}
I can calculate the population of the Bi-211 at time t with:
N_{Bi}=N_0 e^{-\lambda t}
and the population of Po-211 using:
N_{Po} = \frac{\lambda_{Po}}{\lambda_{Bi} - \lambda_{Po}} N_0(e^{-\lambda_{Bi}t}-e^{\lambda_{Po}t})
The problem is with the equations for the population of Tl-207 and Pb-207. I know I should be looking at Bateman's work on the subject, but as a lowly teacher I can't access the literature.