The probability of any particle attaining \(v_{boom}\) is \(p_v\). If radioactive decay is the result of particles colliding at \(v_{boom}\), then the portion of particles decaying away at any one time is,
\(\cfrac{dN}{dt}=-p_v.N\) --- (*)
where \(N\) is the number of particles.
\(N(t)=N_oe^{-p_vt}\)
where \(N_o\) is the number of particles at time \(t=0\). Consider,
\(\cfrac{1}{2}N_o=N_oe^{-p_vt_{1/2}}\)
\(t_{1/2}=\cfrac{ln(2)}{p_v}\)
Half life can be calculated from a graph of Probability, \(p\) vs Velocity, \(v\). Unfortunately such plots are often illustrative. The velocities profile of free particles in a solid has never been done.
Note: Expression (*) requires that \(p_v=\cfrac{N_v}{N}\) where \(N_v\) is the number of particles with velocity equals to \(v_{boom}\) and \(N\) is the total number of particles; that the probability of a particle having \(v_{boom}\) is the fraction of particles having \(v_{boom}\).
