Question #42010

A digital ratemeter measures the count rate $R$ of a radioactive sample by counting the number of emissions $N$ over a fixed sampling window $\tau$, such that $R = N/\tau$. The ratemeter's display has a least count of $R_0$ (in counts per second). Due to the statistical nature of radioactive decay, the number of counts $N$ follows a Poisson distribution with standard deviation $\sqrt{N}$. A student determines the decay constant $\lambda$ by observing the expected drop in the count rate, $\Delta R$, over a time interval $\Delta t$ (where $\tau \ll \Delta t \ll 1/\lambda$). For the measurement to be valid, the expected drop $\Delta R$ must be at least equal to the sum of the ratemeter's least count and the Poisson standard deviation of the count rate. If the experiment is performed at an instant when the Poisson standard deviation of the rate exactly equals the least count $R_0$, what is the minimum time interval $\Delta t$ required for a valid measurement?