the decay idea and mathematic is simple. so, i just state it.

Number of particle (time) = Initial # of particle × Exp( – time / T )

or in formula

$N(t) = N(0) Exp \left( - \frac {t} {T} \right )$

where T is time constant, which has a meaning that how long we should wait before it decay. T also has another name, “mean-lifetime“, coz when you find out the mean of their life by usually statistical method, integrate the whole area of the graph of decay time and make it equal to initial # of particle × “mean lifetime”. that is what you got. ( $\int_0^\infty Exp( - \frac {t}{T}) dt= T$ )

some people like to write the equation is other way:

$N(t) = N(0) Exp \left( - R t \right )$

where R is the chance of decay in unit time. which is just the “invert” meaning of T.

we also have “Half-Life$t_\frac {1}{2}$, which is the time that only half of the particle left. by the equation, we have:

$t_\frac {1}{2} = ln(2) T$

thus, a longer T, the particle live longer, as what is the T mean!

But above mathematics only tell us the statistic result of the decay, not about the mechanism, or physics of what cause the decay happen. why there is decay? why particles come out from nucleus? how many kind of decay ?

the easiest question is, there are 3 decay happen in nature and a lot more different decay happened in lab. the reason for only 3 decay is that, only these 3 live long enough to let us know. the other, they decay fast and all of them are done.

and the reason for nucleus decay is same as the reason for atomic decay. excited nucleus is unstable (why?) they will emit energy to become stable again.

and the physics behind decay, we will come back to it later.