Exercise:
In a threstage decay chain isotope decays o isotope with a decay constant c_ isotope o isotope with a decay constant c_ and isotope o isotope with a decay constant c_. abcliste abc Derive the system of differential s describing the decay chain. abc Determine the eigenvalues and eigenvectors for the system. Write down the fundamental solutions. abc In the ning there are only nuclei of isotope . Derive the corresponding solution and graph the time evolution of all three isotopes for decay constants caO cbO and ccO. abcliste

Solution:
abcliste abc Isotope decays with a decay constant c_: dot N_ -c_ N_ For isotope the number of nuclei increases due to the decay of isotope but it decays itself with a decay constant c_: dot N_ +c_ N_ - c_ N_ For isotope the number of nuclei increases due to the decay of isotope but it decays itself with a decay constant c_: dot N_ +c_ N_ - c_ N_ The system of differential s can be written as fracddtleftmatrixN_ N_ N_ matrixright leftmatrix-c_ & & c_ & -c_ & & c_ & -c_ matrixright leftmatrixN_ N_ N_ matrixright abc The eigenvalues and eigenvectors can be calculated by hand with a CAS e.g. Mathematica or with Python using the package SymPy. The eigenvalues are lambda_c_ lambda_c_ and lambda_c_ with the corresponding eigenvectors bf v_ leftmatrixfracc_-c_c_-c_c_ c_ fracc_-c_c_ matrixright bf v_ leftmatrix fracc_-c_c_ matrixright bf v_ leftmatrix matrixright The fundamental solutions are therefore y_t bf v_ e^-c_ t y_t bf v_ e^-c_ t y_t bf v_ e^-c_ t abc We have to find the coefficients a_ a_ a_ such that leftmatrixN_ matrixright a_ leftmatrixfracc_-c_c_-c_c_ c_ fracc_-c_c_ matrixright + a_ leftmatrix fracc_-c_c_ matrixright + a_ leftmatrix matrixright It follows that a_ N_ fracc_ c_c_-c_c_-c_ a_ N_ fracc_ c_c_-c_c_-c_ a_ N_ fracc_ c_c_-c_c_-c_ The decay for the three isotopes is therefore given by N_t NaF N_t NbF N_t NcF abc The solution can be confirmed with the graph. center includegraphicswidthtextwidth#image_path:threstagdecay-chain# center abcliste