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# Half-life and Math

Sun 22 Jan, 2017 04:16 pm
I'm looking for verification that I'm doing this correctly.

On the topic of radionuclides, I was hoping to map out, with a simple algebraic equation, what the expected quantities of a parent nuclide and its progeny as time progresses (assuming only one decay mode for each nuclide).

I think it should look something like this, but I'm hoping someone could point out any mistakes (or a more elegant way of writing this):

Let 'x' represent the number of half-lives experienced by the parent nuclide. Let 'G' represent the ratio of the rate of decay of the parent and the daughter (example: <1 represent slower half-lives & >1 are faster than the parent). Let 'H' represent the ratio for the granddaughter nuclide relative to the parent. Etc.

Parent nuclide: f(x) = 0.5^x

Daughter nuclide: g(x) = (1-0.5^x)(0.5^(G*x))
g(x) = (the total minus the parent) times (the rate of decay of the daughter)

Granddaughter nuclide: h(x) = (1- f(x) - g(x) ) (0.5^(H*x))
h(x)= (production rate) times (decay rate)
h(x)= (the total minus the parent quantity, minus the daughter) times (the rate of decay for the granddaughter)
h(x) = (1-0.5^x - (1-0.5^x)(0.5^(G*x)) (0.5^(H*x))

3rd progeny: p(x) = (1- f(x) - g(x) - h(x)) (0.5^(P*x))
(...and so on)

Does this seem correct so far?

Thank you
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