Writing a Decay Equation

A decay transforms a "parent" nucleus into a "daughter" nucleus, emitting a particle. We write it as an equation, where a nucleus is denoted with its mass number A (at the top) and its atomic number Z (at the bottom): ᴬ_Z X.

Two rules, Soddy's laws, govern everything: the mass number A and the charge Z are conserved (the sum is the same on both sides).

   α radioactivity :   A_Z X  ->  (A-4)_(Z-2) Y  +  4_2 He
   β- radioactivity :  A_Z X  ->  A_(Z+1) Y      +  0_(-1) e
   γ radioactivity :   A_Z X* ->  A_Z X          +  γ        (A and Z unchanged)

In alpha decay, the nucleus loses 2 protons and 2 neutrons: A decreases by 4, Z by 2. In beta-minus decay, a neutron turns into a proton (emitting an electron): A does not change, but Z increases by 1. In gamma emission, the nucleus changes neither A nor Z: it only loses energy.

A concrete example, uranium-238 decaying into thorium:

   238_92 U  ->  234_90 Th  +  4_2 He        (check: 238 = 234+4, 92 = 90+2)

The check is mechanical: add up the A values on each side, then the Z values. If they match, the equation is correct, and the daughter nucleus is identified by its Z (which gives the element). These conservation rules are the basic tool of all nuclear physics: they tell us what can happen, before even talking about speed or probability.