Derivation of Decay Chain Formulae

[charliec2uk]

I'm afraid I'm suffering from a bit of brain block in try to get from the simple statement of change in the number of daughter nuclei arising from the decay of parent nuclei. The basic statement is straight forward...

$$\frac {dN_d}{dt} = \lambda_pN_p - \lambda_dN_d$$

Subscripts d and p denote parent and daughter nuclei and lambda is the activity.

However I'm struggling to derive the following relationship from the above identity - I can't seem to find any pointers in any textbooks... At face value I think this should be a pretty easy first order ODE to solve, but I think I'm probably missing something blindingly obvious, but any tips would be gratefully recieved.

$$N_d = \frac{\lambda_pN_{p0}}{\lambda_d - \lambda_p} (e^{-\lambda_pt}-e^{\lambda_dt}) + N_{d0}e^{\lambda_dt}$$

 

[Petr Mugver]

The ODE has two variables,Np and Nd, so you need another equation otherwise your system is undetermined.

 

[Petr Mugver]

Oh yeah I get it! The equation you are missing is simply

$$\frac{dN_p}{dt}=-\lambda_pN_p$$

This is easy to solve:

$$N_p=N_{p0}e^{-\lambda_pt}$$

Substitute this into the equation tou wrote, and you'll find exactly the solution you claimed.
(This is simply a system in which the species p decays into the species d, which in turn also decays through other channels)