I am am performing a muon detection experiment at the University of Kansas. In the hyperphysics page entitled "Muon experiment" (http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/muon.html), they use an equation that relates survival rate ( I / I0 = 2^(-Tau) ). Where can I find information or a derivation of this equation?

Question

anonymous · Answer

The rate at which a radioactive material decays is characterized by what is known as a Half-Life.  It is the time that it takes a sample of radioactive material to decay to one half of its original amount.  It is specified by the equation 
$$N = N _{0}2^{-t/\tau }$$
Where No is the original number or amount
tau is the half life
t is the elapsed time

The equation is derived by noting that the rate of decay is proportion to the amount of material  that is in the notation of Calculus$$\frac{ dN }{ dt }=-kN$$

where dN/dt is the rate of change of the materials with time,  k is some constant depending on the material and the minus sign signifying that the sample is decreasing.

This simple differential equation can be integrated to get

$$\ln(N) =-kt +c$$

Using the initial condition that at time t=0, N= No  we get that C=ln(No)

so that we may write $$N=N _{0}e ^{-kt}$$
which can be transformed into $$N=N _{0}2^{-t/\tau} $$
by setting k= ln2/tau