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Deriving the Bateman equation of Nuclear Decay Chains

by Elariel
Tags: bateman, chains, decay, deriving, equation, nuclear
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Mar13-07, 08:31 PM
P: 1
1. The problem statement, all variables and given/known data

Derive Bateman equation for a decay chain
a->b->c->d where each decays with a given mean life let decay constant be L, where L=1/mean life
Na(0)=No, Nb(0)=Nc(0)=Nd(0)=0

2. Relevant equations

Want to derive Nb(t)={(No)(La)/(Lb-La)}*{exp[-La*t]-exp[-Lb*t]}
extend for Nc(t)

3. The attempt at a solution



this is a none homogenous differential equation. I can't find a way to solve it.


I'm really not sure where to go from here.

If anyone could lend a hand it would be greatly appreciated.
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Mar14-07, 11:09 AM
Sci Advisor
HW Helper
P: 25,228
Ok, so what you are really asking is how to solve y'(t)+L*y(t)=f(t). You know the homogenous solution is exp(-L*t). Guess the solution will be of the form y(t)=g(t)*exp(-L*t). Put this guess into your original equation and get:


So g'(t)=exp(L*t)*f(t) and you can just integrate to get g(t).

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