Wolfram Alpha:

Radioactive Decay
-----------------

Unstable isotopes emit radiation as they decay into other
isotopes.

α Decay
-------

The reason alpha decay occurs is because the nucleus has
too many protons which cause excessive repulsion. In an
attempt to reduce the repulsion, a Helium-4 nucleus with
2 protons and 2 neutrons is emitted (42He2+). The way it
works is that the Helium nuclei are in constant collision
with the walls of the nucleus and because of their energy
and mass, there exists a nonzero probability of transmission.
That is, an alpha particle will tunnel out of the nucleus.

α decay results in an atom with  a mass number 4 less
and atomic number 2 less.

Because of their very large mass and its charge, α particles
have very short range.

β Decay
-------

Beta decay occurs when the neutron to proton ratio in the
nucleus is either too great or too small and causes instability.
There are 2 types of β decay:  β- and β+

In β- decay there are too many neutrons.  A neutron is turned
into a proton and an electron and electron antineutrino are
emitted.  Thus,
_
n -> p + e- + νe

In β+ decay there are too many protons.  A proton is turned
into a neutron and a positron and electron neutrino are emitted.
Thus,

p -> n + e+ + νe

A competing process for β+ decay is electron capture.  The
nucleus captures an electron which basically turns a proton
into a neutron and in the process an electron neutrino is
emitted.  Thus,

p + e+ -> n + νe

The high energy electrons/positrons have greater range of
penetration than alpha particles.

γ Decay
-------

Gamma decay occurs because the nucleus is at too high an
energy. The nucleus falls down to a lower energy state
and, in the process, emits a high energy photon known as
a gamma particle.

The number of protons (and neutrons) in the nucleus does
not change in this process, so the parent and daughter
atoms are the same chemical element.

γ rays are very high energy.  They are distinguished from
x-rays only by the fact that they come from the nucleus
rather than the result of electrons transitions.

Decay Formula
-------------

dN/dt = -λN

Where

N = Number of atoms present
λ = Decay constant

∴ dN/N = -λdt

Integrating we get:

lnN = -λt + c

∴ N = exp(c)exp(-λt)

At t = 0, N = exp(c) = N0

∴ N = N0exp(-λt)

N0/2 = N0exp(-λt1/2)

∴ 1/2 = exp(-λt1/2)

∴ 2 = exp(λt1/2)

∴ ln2 = λt1/2

∴ t1/2 = (ln2)/λ

∴ t1/2 = 0.693/λ

Define mean lifetime as τ = 1/λ

∴ t1/2 = 0.693τ

The unit of radioactive activity is the Becquerel (Bq).
One Bq is defined as one transformation (or decay or
disintegration) per second.

Velocities
----------

E = E0 + KE

Therefore,

KE = E - E0

= mc2 - m0c2

Now,

m = m0/√(1 - v2/c2) = γm0

Therefore,

KE = γm0c2 - m0c2

= (γ - 1)m0c2

We can use the binomial expansion to get:

KE = m0c2[1 + v2/2c2 + ...] - m0c2

For v << c we get:

KE = mv2/2

Typically the energy of an α particle ~ 5 MeV and a β particle
~ 1 MeV.  The velocities calculated using the above formulae
are ~ 2 x 107 m/s and 3 x 108 m/s respectively.

back to equilibrium forming a shock front that radiates energy.