In this final lab of Physics 1140, you will use your gamma-ray spectroscopy skills to measure the amount, in grams, of (natural) potassium in readily-available organic fertilizer. The strategy is to measure gamma rays emitted in the decay of the rare isotope \(^{40}\rm{K}\).

Note that the number of radioactive nuclei decays exponentially, $$N(t) = N_{0}e^{-\lambda t}. $$ The activity, or decay rate (number of decays per second), of a radioactive isotope is given by $$R(t) = -\frac{dN(t)}{dt} = \lambda N(t), $$ where \(N(t)\) is the number of undecayed radioactive nuclei at time \(t\) and the decay constant \(\lambda\) (not a wavelength!) is related to the half-life \(T_{1/2}\) by $$\lambda = \frac{\ln 2}{T_{1/2}}. $$ The SI unit of activity is the becquerel, \(1\:\rm{Bq} = 1\:\rm{decay/s} \). Activity is also given in curies, where \(1\:\rm{Ci} = 3.7\times 10^{10}\:\rm{Bq}\).

Your fertilizer sample has a low activity, so you’ll need to collect data over a longer period of time than you did last week. You should collect the background spectrum over an equal amount of time. To make good use of the data collection time, figure out the steps for computing the mass of potassium in the fertilizer sample based on the results of your measurement. In addition to the information above, you’ll need information in the data sheets at your bench and on the ch ar ts around the lab.

Equipment Data Collection and Analysis Summary Questions