# Nuclear Physics decay

1. Oct 8, 2011

### SUDOnym

Problem is:

have a mother and daughter sample, $$A$$ and $$B$$ respectively. both are radioactive. The number of daughter nuclei at time t is given by (*):

$$n(t)=\frac{N_{0}\lambda_{A}}{\lambda_{B}-\lambda_{A}}[e^{-\lambda_{A}t}-e^{-\lambda_{B}t}]$$

where N_0 is number of mother nuclei at t=0 and n(t) is number of daughter nuclei at time t.

A has $$\tau_{\frac{1}{2}}=23minutes$$ and B has $$\tau_{\frac{1}{2}}=23days$$.

A is beta only emitter. B emits gamma and Beta. If A has been made and purified and 11.5minutes after this, the sample emits 1000 gammas/second and some time later the sample again emits 1000 gammas/second - how much time has elapsed?

My Thoughts:
I don't know how to handle this problem for the following reason: To find the rate of gammas being emitter, simply differentiate the equation I showed above (*) to get:

$$\frac{dn}{dt}=\frac{N_{0}\lambda_{A}}{\lambda_{B}-\lambda_{A}}[-\lambda_{A}e^{-\lambda_{A}t}+\lambda_{B}e^{-\lambda_{B}t}]$$

(note can find the lambda_A and lambda_B as we know the half-llife and can also solve (*) for N_0).

The time dependance of the above equation is a negative exponential... so to solve for t, do some rearranging, and take the natural log... but this will be a linear equation... ie. there will only be one value of t for any dn/dt so it is not clear to me how at 11.5mins can have 1000 gammas/second and then again some time later can also have 1000 gammas/second.

What to do?

Last edited: Oct 8, 2011