Re: Why is tritium radioactive?

Dear Jim,

Part 1: Why is tritium unstable?

An isolated particle can only decay into particles which are less
massive than the original particle.  The rest mass energy of the
original particle E=mc^2 is converted into the masses of the decay
products and also into their kinetic energies.  The decay must
conserve energy, linear momentum, angular momentum, electric charge,
lepton number, and baryon number.

The tritium nucleus, or triton, is unstable simply because it is more
massive than its decay products which are a helium-3 nucleus, an
electron, and an electron anti-neutrino.  This is called beta decay
because an electron ejected from the nucleus was named a beta
particle by early researchers.  Symbolically,

       3          3       0       _
      H   -->   He   +   e    +   v   +   kinetic energy
     1         2       -1

where the superscripts label the "atomic mass" A which is the number
of nucleons (proton and neutrons), while the subscripts label the
"atomic number" Z which is the charge in units such the proton charge
is +1.

Part 2: Why is helium-4 stable?

Now you might be wondering why any nucleus is stable.  Why doesn't the
helium-4 isotope, for example, dissociate into its constituent two
protons and 2 neutrons, or maybe into two deuterium nuclei?  The
answer is that 2 protons and 2 neutrons are MORE massive than a
helium-4 nucleus; and 2 deuterons are also MORE massive than a
helium-4 nucleus.  In fact, any way that you can imagine partitioning
the helium-4 nucleus will lead to an increase in mass: a triton and a
proton; helium-3 and a neutron.

The helium-4 nucleus is bound very tightly and the binding energy
decreases the mass of the bound nucleus.  Think of the binding energy
as negative potential energy in reactions (with zero potential energy
when the constituents are infinitely far apart).  Remember, Einstein
tells us that mass is just a very concentrated form of energy.

By the way, this negative binding energy is not unique to the nuclear
realm; it is important for atomic physics and chemistry as well.
Hydrogen (a bound state of a proton and an electron) is slightly less
massive than its constituent parts.  The binding energy is -13.6 eV
(electron volts) or -2.18 x 10^-18 joules when the electron is in
its lowest energy, or ground, state.

Part 3: Why is uranium-238 unstable?

As you say, "Physics and Chemistry textbooks speak of the tension
between the electromagnetic repulsive forces of the protons in the
nucleus and the short-range strong forces of all the nucleons."
This is a convenient model to keep in mind, but some subtle points
are missing.  For example, since neutrons are uncharged they do
not contribute to the Coulomb repulsion, but neutrons do respond
to the strong nuclear force which glues the nucleus together.  So
why can't I add enough neutrons to the 92 protons in uranium to
make a stable uranium nucleus?  For that matter, why can't I make
a nucleus of only neutrons and no protons?

Neutrons have an intrinsic (as opposed to orbital) angular momentum
of one half Planck's constant divided by 2*pi; in the jargon of
particle physics, neutrons have "spin one half".  The Spin-Statistics
Theorem says that particles of odd half-integer spin are fermions,
and no more than one fermion may occupy a given quantum state.
So as you add neutrons to the nucleus, you can put one neutron with 
spin "up" in the lowest energy state, and one neutron with spin "down" 
in the lowest energy state.  Then the lowest energy state is full.
Additional neutrons added to the nucleus must go in successively higher
energy states.

Protons are also spin one-half fermions.  By the time you add 92 protons
and 146 neutrons to make a U-238 nucleus, the most recently added 
particles are in such high-energy states that even the strong nuclear
force cannot bind them permanently.

When applied to electrons, which are also spin one-half fermions, the
Spin-Statistics Theorem yields the Pauli Exclusion Principle:
No two electrons in an atom can be at the same time in the same state 
or configuration.  This principle is at the heart of chemistry.
If electrons did not obey the Pauli Exclusion Principle, then they
would all collapse into the lowest energy electron state and all
species of atom would be virtually identical.


--Dr. Randall J. Scalise    http://www.phys.psu.edu/~scalise/