Exponential decay

A quantity $E(t)$ undergoes exponential decay with rate $\gamma >0$ when

For a constant-rate radiation measurement through $n$ identical absorbing sheets, a simple Poisson distribution model has expected counts ${\mu }_n=t[N_0{e}^{-bn}+B]$. Here $N_0$ is the unattenuated signal rate, $b$ is attenuation per sheet, $B$ is an additive background rate and $t$ is exposure. At low attenuation the signal is approximately exponential; at strong attenuation the observed rate approaches $B$. Consequently a logarithmic plot of raw counts bends towards a plateau rather than remaining a straight line. Measure background independently with the source absent, or fit it with its uncertainty; subtracting it and then taking a logarithm is unreliable when the net count is small or negative.

If a measured unattenuated reference rate includes the same additive background as subsequent measurements, then $N_{\mathrm{ref}}=S_0+B$, where $S_0$ is signal alone. Substituting $S_0=N_{\mathrm{ref}}-B$ into an attenuation count model with an additive background gives $N(n)=(N_{\mathrm{ref}}-B)e^{-bn}+B$. This automatically returns the measured reference at $n=0$ and tends to $B$ at strong attenuation. Adding $B$ directly to $N_{\mathrm{ref}}{e}^{-bn}$ would count background twice at zero attenuation. A joint count likelihood can fit the reference and attenuated data with their separate exposure times, avoiding the need to regard the reference estimate as exact.

The time constant $\tau >0$ in exponential decay is the time over which the decaying quantity is multiplied by $e^{-1}$. A response approaching equilibrium as $q(t)=q_{\infty }+(q(0)-q_{\infty })e^{-t/\tau }$ has the same time constant. For a resistor-inductor circuit it is $L/R$.

A stretched exponential has the form $e^{-c{x}^{\alpha }}$ with $c>0$ and $0<\alpha <1$. A boundary cost proportional to the square root of an enclosed area naturally produces exponent $\alpha =1/2$.

The half-life is the time at which an exponentially decaying quantity reaches half its initial value. For decay rate $\gamma$,