What makes an atom unstable for radioactivity?

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Nanaki - I've read some of the replys here and they confuse me too. First, you need to be familiar with what appears to be a basic principle of nature, and that is that nature likes to minimize potential energy (I realize guys this is oversimplified, but this is a basic explanation). Now, to radioactivity. As I am sure you're aware, the nucleus is made up of protons and neutrons. These neutrons and protons fill energy shells in the nucleus just like electrons fill energy shells around the nucleus. If some of these shells have too few or too many nucleons, the atom is not in it's most desirable energy state and acts to achieve that.

For too many neutrons, the atom converts a neutron into a proton and emits a negative particle known as a Beta-minus particle (it is essentially an electron)

For too few neutrons (in other words, too many protons), the atom converts a proton into a neutron and emits a positive particle known as a Beta-Plus particle (this is essentially an anti-electron)

For atoms that are way too big like uranium, the nucleus just spits out a chunk of itself - a particle consisting of 2 protons and 2 neutrons known as an alpha-particle.

In addition to spitting out these particles, these unstable nuclei usually emit a high frequency photon known as a gamma particle.

There are other more exotic modes of radioactive decay but these are by far the most common types encountered.
 
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In your example of H-3, more commonly known as tritium, this is a hydrogen nucleus that contains one proton and two neutrons. A normal hydrogen nucleus is just one proton so this nucleus contains too many neutrons and it decays by Beta-Minus decay, converting one of those neutrons into a proton and emitting a Beta-Minus particle as we talked about above.
 
Just as a little addendum to the explanations of geometer i would like to stress the fact that the emitted electron does NOT come "out" of the nucleus. Basically, following the rules of QFT it is created out of the so called fysical vacuum (out of nothing, if you wish). The energy needed for this creation come from the decaying mother-nucleus itself via the well known formula E=mc²...

marlon
 
Certainly this is very "smooth" pedantic explanation of stability and instability of isotopes, which offer geometer and marlon.
I assume, that sub-forum Theory Development should pursue other purpose – to offer an explanation for those phenomena which have no a satisfactory explanation in handbooks.
Let's look at stable and long-living isotopes of those elements, which are located before noble gases in the Periodic table:
1H (1p) – stable;
2H (1p+1n) – stable;
3H (1p+2n) – unstable: emanation of electron (beta–);

18F (9p+9n) – unstable: emanation of proton (beta+); electron capture;
19F (9p+10n) – stable;

35Cl (17p+18n) – stable;
36Cl (17p+19n) – unstable: beta+; beta–; electron capture;
37Cl (17p+20n) – stable;

77Br (35p+42n) – unstable: electron capture; beta+; gamma;
79Br (35p+44n) – stable;
81Br (35p+46n) – stable;
82Br (35p+47n) – unstable: beta+; gamma;

123I (53p+70n) – unstable: electron capture; gamma;
125I (53p+72n) – unstable: electron capture; gamma;
127I (53p+74n) – stable;
129I (53p+76n) – unstable: beta–; gamma;
131I (53p+78n) – unstable: beta–; gamma;

210At (85p+125n) – unstable: electron capture; alpha;
211At (85p+126n) – unstable: electron capture; alpha;

As you see, even this limited list of isotopes does not submit to the common rule. Especially it concerns to other isotopes (chlorine has 13 isotopes, bromine – 28, iodine – 37).
I think, there are big opportunities to explain stability and instability of isotopes due to features of a design of atoms (due to features of a geometry), but not amount of protons and neutrons in their center.