Coupled nuclear decay rate equations

[jmz34]

Homework Statement

If we have the following partial decay chain:

N1 -> N2 -> N3 where N1 is the number of nuclei of species 1, etc.

and N1 -> N2, not via a decay but by the reaction such as N1 + neutron -> N2 + photon
and we know this rate of formation of N2, say 'a'.

I then get the following rate equation:

dN2(t)/dt=at-R2N2(t)=at-R2N2(0)exp(-R2t) where R2 is the decay rate from N2->N3

This would be simple to solve if the RHS wasn't coupled. By this I mean in a certain time, dt, there will be an increase in N2, dN2, which will couple into the N2(0) term and subsequently decay.

How would I go about solving this? Thanks.

 

[gneill]

If you're assuming that the rate of production of N2 is constant (i.e., the supply of N1 is constant) then you're really only dealing with one substance, N2, and its rate of production versus its rate of decay. So, dropping the "2" and just calling it "N",

dN/dt = a - rN

where r is the decay rate of substance N. This is a first order linear differential equation, namely

N' + rN - a = 0

 

[epenguin]

Also I think your second equation seems to be wrong, I think I can see where you mistakenly got it from.

Just write a completely separate equation for dN₃/dt