Energy and momenta balances¶

\[\Delta t=24\unit{hr}\]

\[N_{0}=\frac{1683\unit{counts}}{10\unit{min}}\]

\[N_{1}=\frac{1683\unit{counts}}{20\unit{min}}\]

\[\frac{N_{0}}{N_{1}}=\exp\left(-\lambda\Delta t\right)\]

\[\lambda=\frac{1}{\Delta t}\ln\left(\frac{N_{1}}{N_{0}}\right)\]

\[t_{\nicefrac{1}{2}}=\ln\left(2\right)\Delta t\frac{1}{\ln\left(N_{1}\right)-\ln\left(N_{1}\right)}\]

\[\sigma_{t_{\nicefrac{1}{2}}}=\sqrt{\cancelto{0}{\left(\frac{\partial t_{\nicefrac{1}{2}}}{\partial\Delta t}\right)^{2}\sigma_{\Delta t}}+\left(\frac{\partial t_{\nicefrac{1}{2}}}{\partial N_{1}}\right)^{2}\sigma_{N_{1}}+\left(\frac{\partial t_{\nicefrac{1}{2}}}{\partial N_{0}}\right)^{2}\sigma_{N_{0}}}\]