Half Life Calculated

Given $T$,

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 $v_{boom}$.