Question

7-(15 points) A nucleus decays into three channels with the relative 0.60, respectively. The half-life of the nucleus is 180 into the three channels? probabilities 0.10,030, and seconds. What are the partial decay rates

Answer 1

The total decay constant is

The total decay constant is k = ln 2/t_1/2 = in 2/180 = 0.00385 s^-1 Let the decay rates of the 3 channels be k_1, k_2, k_3 then the total decay rate k_1 + k_2 + k_3 = k and given their relative probabilities, the partial decay rates are: k_1 = 0.1 k = 0.000385 s^-1 = 3.85 times 10^-4 s^-1 k_2 = 0.3 k .001155 s^-1 = 1.155 times 10^-3s^-1 k_3 = 0.6 k = 0.00231 s^-1 = 2.31 times 10^-3 s^-1

Let the decay rates of the 3 channels be $k_1,k_2,k_3$

then the total decay rate

$k_1+k_2+k_3=k$

and given their relative probabilities, the partial decay rates are:

$k_1 =0.1 k =0.000385\,\,s^{-1}=3.85\times 10^{-4}\,\,s^{-1}$

$k_2 =0.3 k =0.001155\,\,s^{-1}=1.155\times 10^{-3}\,\,s^{-1}$